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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 589267 16 pageshttpdxdoiorg1011552013589267
Research ArticleNonlinear Control for Trajectory Tracking ofa Nonholonomic RC-Hovercraft with Discrete Inputs
Dictino Chaos David Moreno-Salinas Rociacuteo Muntildeoz-Mansilla and Joaquiacuten Aranda
Departament of Computer Science and Automatic Control UNED Madrid Spain
Correspondence should be addressed to Dictino Chaos dchaosdiaunedes
Received 17 June 2013 Accepted 23 October 2013
Academic Editor Guanghui Sun
Copyright copy 2013 Dictino Chaos et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This work studies the problem of trajectory tracking for an underactuated RC-hovercraft the control of which must be done bymeans of discrete inputs Thus the aim is to control a vehicle with very simple propellers that produce only a discrete set of controlcommands and with minimal information about the dynamics of the actuators The control problem is approached as a cascadecontrol problem where the outer loop stabilizes the position error and the inner loop stabilizes the orientation of the vehicleStability of the controller is theoretically demonstrated and the robustness of the control law against disturbances and noise isestablished Simulation examples and experiments on a real setup validate the control law showing the real system to be robustagainst disturbances noise and outdated dynamics
1 Introduction
Autonomous vehicles (AV) in general and autonomoussurface vehicles (ASV) in particular are becoming ubiquitousin many fields of engineering research due in part to theflexibility and versatility that a number of them display inthe execution of individual and cooperative tasks Thesecharacteristics coupled with the fact that their use avoidsplacing human lives at risk makes them quite attractivein multiple missions Central to the operation of theseautonomous vehicles is the availability of good trackingcontrollers that allow the AVs to execute demanding tasksin different operational scenarios Moreover in the presenteconomical scenario in which many research projects sufferwithdrawal of funds it is of the utmost importance to offerreliable low cost systems
The three main problems that arise in control theory ofautonomous vehicles according to the classification of [1] arepath following [2] point stabilization [3 4] and trajectorytracking As part of this trend the tracking control problemfor underactuated vehicles represents a very challengingresearch topic being thus the problem studied in the presentwork
Underactuated vehicles are of special interest due to theirreduced cost and the relative easiness of their construction
furthermore a fully actuated vehicle is only needed forsome special operations making its usage unnecessary fora wide number of actions In addition the possibility ofusing the same vehicle for terrestrial and maritime opera-tions put underactuated hybrid vehicles in a very attractivesituation In this sense the most common hybrid vehicleis the hovercraft Hovercraft is usually used as test vehiclefor control algorithms of underactuated marine systems dueto its dynamic properties the possibility of being tested onground and its relative (and mission dependent) low cost asin the experimental testbed developed on [5] or [6]
In addition to the low cost and the possible hybrid natureof the vehicle at hand the nonholonomic condition of theunderactuated vehicles is by itself interesting enough andcomplex for a detailed study An usual solution for fullyactuated vehicles consists in using sliding mode controllerssuch as the solution described in [7] The main limitation ofthis control strategy is the impossibility of acting over all thestates of the vehicle at the same time when an underactuatedvehicle as an hovercraft is considered The control problemof underactuated vehicles is much more challenging than thefully actuated one because of the limitation on the controllaws that can be implemented This is a consequence of thefamous result stated by Brockett [8] that states that some
2 Mathematical Problems in Engineering
kind of underactuated vehicles (that includes the hovercraft)cannot be stabilized to a certain pose with a continuous andtime invariant control law The reader is referred to [9] foran interesting survey in control problems for nonholonomicvehicles
Representative examples of tracking controllers applied tounderactuated hovercrafts are stated in [10] where the track-ing control problem is solved using backstepping techniquesachieving practical stability results in [11] that solve the sameproblem exploiting the differential flatness properties of avessel with respect to its position and in [12] where theauthors extend previous results for the tracking and pathfollowing problems to more general marine vehicles takinginto account the parametric uncertainty on the models
Other remarkable examples are found in [13] where thetracking problem of general underactuated ships is studiedin [14] that explains how to select outputs when generalizedforces act on the vehicle in [15] where Barbalatrsquos lemma andbackstepping techniques are combined to achieve asymptotictracking in [16] that solves the tracking problemof awheeledrobot in [17] where the authors address the tracking problemfor a surface craft and in [18] in which the challengingproblem of solving both point stabilization and trackingproblem at the same time is studied
It must be remarked that the control strategies describedabove achieve a good control performance assuming thatforce and torque imparted by propellers can be perfectlycontrolled Nevertheless thrust allocation is in general avery complex task in surface crafts (see eg [19]) andmoreover expensive equipments are usually required Thusthe availability of low cost vehicles equipped with simpleactuators and with a limited knowledge of their dynamicsis of the utmost importance for the problem studied inthe present work and a very interesting alternative from apractical point of view
In the last few years the interest on the control of under-actuated vehicles where only a discrete set of control inputs isavailable has increased significantly See for example [20 21]for a description on the challenges that arise in quantizedcontrol of mobile vehicles and remote control of systemsover networks with state quantization respectively Anotherremarkable example is found in the works [22 23] where anonholonomic underactuated hovercraft is controlled by adiscrete set of inputs In these works the problem of pointstabilization is solved using receding horizon predictive con-trollers and approaching the continuous output by the closestallowed control action This strategy works adequately whenthe optimal control law is bang-bang but it is not suitablewhen the control signals take intermediate values betweentwo allowed ones something impossible to achieve withall-nothing-reverse signals Unfortunately this is the actualtracking problem scenario because forces and torques shouldbe close to a continuous reference far away from saturation
In order to solve the control problem of underactuatedvehicles some authors consider second order sliding-modecontrol laws See for example [24] where this technique isapplied to the tracking problem of a surface vessel and [11]where the trajectory tracking problem for an underactuatedhovercraft is solved These control laws tend to produce
control actions that are similar to bang-bang actions There-fore they are good candidates to be approximated with all-nothing-reverse thrusters The later are the kind of thrustersused in the vehicle (hovercraft) dealt in this work due to theirsimplicity of control and implementation and also their lowcost
This kind of problem has been partially studied in theground robotic field Some interesting results on the field ofwheeled robots show that in the case of Dubinrsquos vehicles it ispossible to design a tracking controller that makes the vehicleconverge to a reference trajectory using only a discrete setof control actions (turn right and left accelerate and brake)See for example [25] that uses a sliding-mode control and[16] where a controller has been developed using optimalsynthesis
An interesting practical problem is the extension of theabove results for an underactuated low cost vehicle as ahovercraft The main difference between a wheeledgroundunderactuated vehicle and a marine underactuated vehicle isthe nonholonomic restriction on the trajectories that can befollowed by each vehicle The nonholonomic restriction in avehicle is the relationship between the acceleration and theorientation of the vehicle The sliding condition for marinevehicles (second-order nonholonomic restriction) is morecomplex than the nonsliding condition (first-order nonholo-nomic restriction) for wheeled robots Thus the aim of thiswork is to control a vehicle with very simple propellers thatproduce only a discrete set of control commands and witha minimal information about the dynamics of the actuatorsThe availability simplicity and low cost of these kind ofsystems even the hybrid nature of the hovercraft makethem very attractive to be used in a number of operationalscenarios Furthermore a group of these vehicles can be usedfor cooperative andor coordinated tasks with relatively lowprice
The sole conditions that are imposed to the actuators usedon this work can be summarized as follows
(1) The available control actions can turn right and leftwhile the vehicle go forward and backwards
(2) The available force and torque are bigger than theforce and torque necessary to exactly track the ref-erence (forces and torques can be dominated by thecontrol actions)
Notice that the first condition is important to ensure localcontrollability (see [26]) while second condition is necessaryto ensure feasibility of the trajectory
Thekey contribution of the paper is two-fold (i) we reflectthe fact that the inputs are discrete in the controller designprocess and (ii) we take explicitly into account the existenceof unknown bounded input disturbances and measurementnoise Theoretical stability proofs are obtained and in addi-tion the control law is successfully tested on a real low costvehicle In the literature to the best knowledge of the authorsthere are no attempts to solve the trajectory tracking problemfor a second order nonholonomic vehicle as the hovercraftwith a discrete set of inputs Furthermore in the few caseswhere a discrete set of inputs is considered in a vehicle of this
Mathematical Problems in Engineering 3
120595
r
Fp
Fs
o
X
Y
l(x y)
U
u
V = [x t]T
Figure 1 Inertial reference frame and body fixed frame
kind the effect of this practical limitation and the effect ofdisturbances and noise have never been studied in detail
The paper is organized as follows Section 2 describes themodel of the vehicle and the set of trajectories that can betracked with a discrete set of inputs The control problem isformulated in Section 3 In Section 4 a nonlinear controlleris designed Practical implementation and the effect of dis-turbances and noise are discussed in Section 5 The stabilityand robustness of the controller are analyzed in Section 6Simulation and experimental tests are shown in Section 7Finally conclusions are commented on in Section 8
2 Vehicle Model
The model of the vehicle operating over a solid surface isdefined in two reference frames one inertial global referenceframe attached to the floor where the RC-hovercraft movesand a body-fixed reference frame shown in Figure 1 Accord-ing to the model (the details of which can be found in [27])the dynamics of the system can be described in the globalreference frame by
= V119909 (1)
119910 = V119910 (2)
V119909= 119865 cos (120595) minus 119889
119906V119909+ 119901V119909 (3)
V119910= 119865 sin (120595) minus 119889
119906V119910+ 119901V119910 (4)
120595 = 119903 (5)
119903 = 120591 minus 119889
119903119903 + 119901
119903 (6)
The state of the vehicle is given by x = [119909 119910 V119909 V119910 120595 119903]
119879where 119909 and 119910 are the position coordinates of the centre ofmass of the RC-hovercraft 120595 is the orientation V
119909and V
119910
are the linear velocities and 119903 is the yaw rate The parameters119889
119906and 119889
119903of the model are the normalized drag coefficients
where 119889
119906= 119863
119906119898 119889
119903= 119863
119903119869 119898 is the mass of the
hovercraft 119869 the moment of inertia and 119863
119906and 119863
119903are
the linear and rotational drag coefficients respectively Thecontrol actions are the normalized force 119865 = (119865
119901+119865
119904)119898 and
Table 1 Model parameters
Parameter Value119898 0995Kg119869 0014Kgm2
119897 0075m119906max 0545N119906min 0347N119889
11990603588 sminus1
119889
11990317357 sminus1
Table 2 Force selection table
120591 gt 0 120591 lt 0
119865 gt 0 119865
119901= 119906max 119865119904 = minus119906min 119865
119901= minus119906min 119865119904 = 119906max
119865 lt 0 119865
119901= 0 119865
119904= 119906min 119865
119901= 119906min 119865119904 = 0
torque 120591 = 119897(119865
119901minus 119865
119904)119869 where 119865
119904and 119865
119901are the starboard
and port forces produced by the fans and 119897 is the distancebetween the propellers and the symmetry axis as Figure 1shows The dynamics of the vehicle are also subject to theunknown disturbance vector p = [119901V119909 119901V119910 119901119903]
119879 and theoutput of the system is corrupted by measurement noise n =
[119899
119909 119899
119910 119899V119909 119899V119910 119899120595 119899119903]
119879 such that the measured state is y =
x + n Noise and disturbances are unknown but bounded asfollows
sup119905ge0
p (119905) le 119901
119898 sup
119905ge0
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119889p119889119905
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
sup119905ge0
n (119905) le 119899
119898
(7)
The value of the parameters for the experimental deviceare summarized in Table 1
21 Physical Constraints The experimental vehicle is undersome physical constraints as it is usual in any real deviceThecontrol inputs are forces produced by thrusters the valuesof which belong to a discrete set 119865
119904119901isin minus119906min 0 119906max
with 119906max gt 119906min gt 0 Under these considerations thecontrol actions 119865 and 120591 belong to the discrete set of ninecombinations of 119865
119904and 119865
119901shown in Figure 2 It is important
to notice that these control actions can move the hovercraftforward and backwards and right and left Moreover it willbe shown that it is possible to control the RC-hovercraft witha reduced set of control actions and without knowledge ofthe exact values of 119906min and 119906max The key point is that oncethe sign of the force and torque is known then there alwaysexists a proper selection of 119865
119904and 119865
119901such that the desired
values of sign(119865) and sign(120591) are obtained As Figure 2 showsthere exist more than one solution In Table 2 one possibleselection scheme is shown
22 Feasible Trajectories Dynamics of the RC-hovercraft areflat with respect to the position coordinates as demonstratedin [11] The reader is referred to [28] for further informationabout flat systems Thus when the reference spatial trajec-tories 119909
119903(119905) and 119910
119903(119905) are set then the whole state vector
4 Mathematical Problems in Engineering
[umax umax]
F =Fp minus Fs
m
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 2 Discrete control inputs on the laboratory setup
and the control actions that must be applied to follow thereference can be computed For this purpose considering (3)and (4) in the absence of disturbances and defining 119865
119903as the
force needed to track the reference the ideal orientation 120595
119903
needed to exactly track the reference is given by the followingnonholonomic restrictions
119865
119903cos (120595
119903) = V119909119903+ 119889
119906V119909119903
119865
119903sin (120595
119903) = V119910119903+ 119889
119906V119910119903
(8)
The orientation is well defined if 119865119903
= 0 It is important tonotice that actually there exist two families of solutionsdepending on how the trajectory is followed by the vehiclegoing forward (119899 odd) or backwards (119899 even) as follows
120595
119903= atan2 ( 119910
119903+ 119889
119906119910
119903
119903+ 119889
119906
119903) + 119899120587
119865
119903= (minus1)
119899radic
(
119903+ 119889
119906
119903)
2
+ ( 119910
119903+ 119889
119906119910
119903)
2
(9)
Once the family of solutions of interest is selected theorientation120595
119903and its derivatives 119903
119903and 119903
119903 can be computed
Straightforward computations yield the requested torque120591
119903= 119903
119903+119889
119903119903
119903to track the reference Now we consider a spatial
trajectory x119903= [119909
119903(119905) 119910
119903(119905)]
119879 four times differentiable with119865
119903= 0 Two kinds of trajectories of interest can be defined as
follows
(1) A feasible trajectory where 119865
119903and 120591
119903are such that
119865
119904119901isin [minus119906min 119906max]
(2) A D-feasible trajectory where there always exist fourcombinations of allowed forces and torques (119865
1 120591
1)
(119865
2 120591
2) (1198653 120591
3) and (119865
4 120591
4) such that
1003816
1003816
1003816
1003816
119865
119903(119905)
1003816
1003816
1003816
1003816
lt min (1198651 119865
2 minus119865
3 minus119865
4)
1003816
1003816
1003816
1003816
120591
119903(119905)
1003816
1003816
1003816
1003816
lt min (minus1205911 120591
2 minus120591
3 120591
4)
(10)
On one hand we consider that a trajectory is feasible when itcan be followed by the vehicle that is when the necessaryforces and torques to track the trajectory can be producedby values of 119865
119904and 119865
119901smaller than the maximum available
Notice that if a trajectory is not feasible the only way to trackit is to increase the size of the thrusters in order to incrementthe values of 119865
119904and 119865
119901 allowing the trajectory to be feasible
againThe control actions 119865119903(119905) and 120591
119903(119905) that define a feasible
trajectory lie on the dotted region of Figure 3On the other hand we consider that a trajectory is D-
feasible when the forces and torques selected fromTable 2 aregreater in absolute value than the forces and torques neededto track the trajectory Notice that all the trajectories thatare D-feasible are also feasible but the opposite is not trueThe control actions 119865
119903(119905) and 120591
119903(119905) that define a D-feasible
trajectory lie on the dashed region of Figure 3
Mathematical Problems in Engineering 5
F =Fp + Fs
m
[umax umax]
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
1
2
3
4
5
6
7
8
9
(Fr 120591r)
F0
minus1205910 1205910
minusF0
[0 0]
A
Figure 3 Feasible (dotted) and D-feasible (dashed) regions A trajectory such as the green one is feasible because (119865119903(119905) 120591
119903(119905)) lies on the
dotted region but is not D-feasible because it lies outside the dashed region
3 Problem Formulation
The key objective of the present work is to design and toimplement a feedback control law for 119865
119904and 119865
119901to solve the
robust trajectory tracking problem for a vehicle commandedwith a discrete set of inputs In other words the aim is thedevelopment of a controller that is able to track a desiredtrajectory in the presence of disturbances and noise withonly a limited knowledge of the model of the vehicle and theactuators
Themain practical restriction is that the vehicle is under-actuated and the control law must only employ a discrete setof inputs produced by the thrusters 119865
119904119901isin minus119906min 0 119906max
Notice that the controller is oriented to be used on smallautonomous vehicles equipped with simple thrusters becausethe addition of more thrusters would add complexity to thesystem and would increase the power consumption
Consider the dynamics of the hovercraft describedby (1)ndash(6) a D-feasible spatial reference trajectoryx119890 and the spatial tracking error defined by e
119901=
[119909 minus 119909
119903 119910 minus 119910
119903 V119909minus V119909119903 V119910minus V119910119903]
119879 Then the requirementsfor a robust trajectory tracking controller are as follows
(1) The error e119901
is bounded in the presence of thebounded noise and disturbances described by (7)
(2) The final bound of e119901 can be made arbitrary small
if the noise 119899
119898and the disturbances 119901
119898are small
enough (or alternatively if force and torque are largeenough)
4 Nonlinear Control Design
The control design is based on the cascade structure illus-trated in Figure 4 In the outer loop the force and the
orientation are used as virtual control inputs for the positioncontroller to stabilize the position tracking errorThepositioncontroller defines an orientation reference 120595
119888to be tracked
by the inner loop the orientation controller that controlsthe orientation with the torque 120591 Therefore the positioncontroller determines the sign of the force 119865 to be appliedon the RC-hovercraft whereas the orientation controllerdetermines the sign of the torque 120591 Once sign(119865) and sign(120591)are computed the values of 119865
119904and 119865
119901are selected from
Table 2
41 Position Control The objective of this subsection is thedesign of a controller for the position of the hovercraft Theerror variables are defined by
119890
119909= 119909 minus 119909
119903
119890
119910= 119910 minus 119910
119903
119890V119909 = V119909minus V119909119903
119890V119910 = V119910minus V119910119903
(11)
And their dynamics can be written as
119890
119909= 119890V119909
119890
119910= 119890V119910
119890V119909 = 119865 cos (120595) minus 119889
119906119890V119909 minus 119865
119909119903+ 119901V119909
119890V119910 = 119865 sin (120595) minus 119889
119906119890V119910 minus 119865
119910119903+ 119901V119910
(12)
where 119865119909119903
= 119865
119903cos(120595
119903) and 119865
119910119903= 119865
119903sin(120595119903) are the nominal
forces needed to track exactly the reference in absence of
6 Mathematical Problems in Engineering
References
x
Positioncontrol
120595 r
120595c rc
Orientationcontrol Sign(120591)
Selector
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J
25
2
15
1
05
0
minus05
minus1
minus15
F=(F
b+Fe)m
Fe
Fb
Hovercraft
x
Sign(F)
[umax umax]
[umax 0]
[umax minusumin]
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 4 Control architecture
disturbances At this point a positive control constant 1198961 and
the variables 119904119909and 119904119910are introduced in order to proceedwith
the error stabilization as follows
119904
119909= 119890
119909+ 119896
1119890V119909
119904
119910= 119890
119910+ 119896
1119890V119910
(13)
It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896
1if 119904119909
and 119904
119910are both equal to zero Furthermore the problem of
tracking error stabilization is reduced to the stabilization of119904
119909and 119904
119910 Computing their dynamics it is clear that 119865 and
the orientation 120595 are coupled Consider the following
119904
119909= 119896
1119865 cos (120595) + (1 minus 119889
119906119896
1) 119890V119909 minus 119896
1119865
119909119903+ 119896
1119901V119909
119904
119910= 119896
1119865 sin (120595) + (1 minus 119889
119906119896
1) 119890V119910 minus 119896
1119865
119910119903+ 119896
1119901V119910
(14)
Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates
119911
1= 119904
119909cos (120595) + 119904
119910sin (120595)
119911
2= minus119904
119909sin (120595) + 119904
119910cos (120595)
(15)
Then simple computations yield
1= 119896
1119865 minus 119896
1119865
119903cos (120595 minus 120595
119903) + 119911
2119903
+ (1 minus 119889
119906119896
1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896
1119901
1
(16)
2= 119896
1119865
119903sin (120595 minus 120595
119903) minus 119911
1119903
+ (1 minus 119889
119906119896
1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896
1119901
2
(17)
where 1199011and 119901
2are bounded disturbances given by
119901
1= 119901V119909 cos (120595) + 119901V119910 sin (120595)
119901
2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)
p =1003817
1003817
1003817
1003817
1003817
1003817
[119901
1 119901
2]
1198791003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
(18)
Analyzing the dynamics of 1199111and 119911
2 given by (16) and
(17) it must be noticed that (17) does not depend on 119865 so 119865
must be used to control 1199111 In order to solve this problem the
sign of 119865must be opposed to the sign of 1199111 thus we propose
sign (119865) =
minus sign (1199111) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
lt 120576
(19)
where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911
1|
is small Notice that this condition imposes a minimal timebetween switches because |119911
1|must change from minus120576 to +120576 (or
from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911
1| lt 120576
In order to stabilize 1199112 the sign of 119865
119903sin(120595 minus 120595
119903)must be
the opposite of the sign of 1199112 For this purpose the following
reference for the orientation is defined as follows
120595
119888= 120595
119903minus 119896
2tanh (119911
2) sign (119865
119903) (20)
119903
119888=
120595
119888=
120595
119903minus
2119896
2(1 minus tanh (119911
2)
2
) sign (119865119903) (21)
The saturation of 1199112given by the operation tanh in (20) is
introduced to guarantee that when120595 = 120595
119888and 0 lt 119896
2lt 1205872
then 119865
119903sin(120595119888minus 120595
119903) = minus|119865
119903| sin(119896
2tanh(119911
2)) that is opposed
to 1199112In general the orientation reference 120595
119888is not perfectly
tracked by 120595 Thus the orientation errors are defined by
119890
120595= 120595 minus 120595
119888 (22)
119890
119903= 119903 minus 119903
119888 (23)
and the dynamics of z finally become
1= 119896
1119865 minus 119896
1119865
119903cos (119890
120595+ 120595
119888minus 120595
119903) + 119911
2119903 + 119896
1119901
1
+ (1 minus 119889
119906119896
1) (119890V119909 cos (119890120595 + 120595
119888) + 119890V119910 sin (119890120595 + 120595
119888))
(24)
2= minus 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119911
1119903 + 119896
1119865
119903119908 + 119896
1119901
2
+ (1 minus 119889
119906119896
1) (119890V119910 cos (119890120595 + 120595
119888) minus 119890V119909 sin (119890120595 + 120595
119888))
(25)
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
kind of underactuated vehicles (that includes the hovercraft)cannot be stabilized to a certain pose with a continuous andtime invariant control law The reader is referred to [9] foran interesting survey in control problems for nonholonomicvehicles
Representative examples of tracking controllers applied tounderactuated hovercrafts are stated in [10] where the track-ing control problem is solved using backstepping techniquesachieving practical stability results in [11] that solve the sameproblem exploiting the differential flatness properties of avessel with respect to its position and in [12] where theauthors extend previous results for the tracking and pathfollowing problems to more general marine vehicles takinginto account the parametric uncertainty on the models
Other remarkable examples are found in [13] where thetracking problem of general underactuated ships is studiedin [14] that explains how to select outputs when generalizedforces act on the vehicle in [15] where Barbalatrsquos lemma andbackstepping techniques are combined to achieve asymptotictracking in [16] that solves the tracking problemof awheeledrobot in [17] where the authors address the tracking problemfor a surface craft and in [18] in which the challengingproblem of solving both point stabilization and trackingproblem at the same time is studied
It must be remarked that the control strategies describedabove achieve a good control performance assuming thatforce and torque imparted by propellers can be perfectlycontrolled Nevertheless thrust allocation is in general avery complex task in surface crafts (see eg [19]) andmoreover expensive equipments are usually required Thusthe availability of low cost vehicles equipped with simpleactuators and with a limited knowledge of their dynamicsis of the utmost importance for the problem studied inthe present work and a very interesting alternative from apractical point of view
In the last few years the interest on the control of under-actuated vehicles where only a discrete set of control inputs isavailable has increased significantly See for example [20 21]for a description on the challenges that arise in quantizedcontrol of mobile vehicles and remote control of systemsover networks with state quantization respectively Anotherremarkable example is found in the works [22 23] where anonholonomic underactuated hovercraft is controlled by adiscrete set of inputs In these works the problem of pointstabilization is solved using receding horizon predictive con-trollers and approaching the continuous output by the closestallowed control action This strategy works adequately whenthe optimal control law is bang-bang but it is not suitablewhen the control signals take intermediate values betweentwo allowed ones something impossible to achieve withall-nothing-reverse signals Unfortunately this is the actualtracking problem scenario because forces and torques shouldbe close to a continuous reference far away from saturation
In order to solve the control problem of underactuatedvehicles some authors consider second order sliding-modecontrol laws See for example [24] where this technique isapplied to the tracking problem of a surface vessel and [11]where the trajectory tracking problem for an underactuatedhovercraft is solved These control laws tend to produce
control actions that are similar to bang-bang actions There-fore they are good candidates to be approximated with all-nothing-reverse thrusters The later are the kind of thrustersused in the vehicle (hovercraft) dealt in this work due to theirsimplicity of control and implementation and also their lowcost
This kind of problem has been partially studied in theground robotic field Some interesting results on the field ofwheeled robots show that in the case of Dubinrsquos vehicles it ispossible to design a tracking controller that makes the vehicleconverge to a reference trajectory using only a discrete setof control actions (turn right and left accelerate and brake)See for example [25] that uses a sliding-mode control and[16] where a controller has been developed using optimalsynthesis
An interesting practical problem is the extension of theabove results for an underactuated low cost vehicle as ahovercraft The main difference between a wheeledgroundunderactuated vehicle and a marine underactuated vehicle isthe nonholonomic restriction on the trajectories that can befollowed by each vehicle The nonholonomic restriction in avehicle is the relationship between the acceleration and theorientation of the vehicle The sliding condition for marinevehicles (second-order nonholonomic restriction) is morecomplex than the nonsliding condition (first-order nonholo-nomic restriction) for wheeled robots Thus the aim of thiswork is to control a vehicle with very simple propellers thatproduce only a discrete set of control commands and witha minimal information about the dynamics of the actuatorsThe availability simplicity and low cost of these kind ofsystems even the hybrid nature of the hovercraft makethem very attractive to be used in a number of operationalscenarios Furthermore a group of these vehicles can be usedfor cooperative andor coordinated tasks with relatively lowprice
The sole conditions that are imposed to the actuators usedon this work can be summarized as follows
(1) The available control actions can turn right and leftwhile the vehicle go forward and backwards
(2) The available force and torque are bigger than theforce and torque necessary to exactly track the ref-erence (forces and torques can be dominated by thecontrol actions)
Notice that the first condition is important to ensure localcontrollability (see [26]) while second condition is necessaryto ensure feasibility of the trajectory
Thekey contribution of the paper is two-fold (i) we reflectthe fact that the inputs are discrete in the controller designprocess and (ii) we take explicitly into account the existenceof unknown bounded input disturbances and measurementnoise Theoretical stability proofs are obtained and in addi-tion the control law is successfully tested on a real low costvehicle In the literature to the best knowledge of the authorsthere are no attempts to solve the trajectory tracking problemfor a second order nonholonomic vehicle as the hovercraftwith a discrete set of inputs Furthermore in the few caseswhere a discrete set of inputs is considered in a vehicle of this
Mathematical Problems in Engineering 3
120595
r
Fp
Fs
o
X
Y
l(x y)
U
u
V = [x t]T
Figure 1 Inertial reference frame and body fixed frame
kind the effect of this practical limitation and the effect ofdisturbances and noise have never been studied in detail
The paper is organized as follows Section 2 describes themodel of the vehicle and the set of trajectories that can betracked with a discrete set of inputs The control problem isformulated in Section 3 In Section 4 a nonlinear controlleris designed Practical implementation and the effect of dis-turbances and noise are discussed in Section 5 The stabilityand robustness of the controller are analyzed in Section 6Simulation and experimental tests are shown in Section 7Finally conclusions are commented on in Section 8
2 Vehicle Model
The model of the vehicle operating over a solid surface isdefined in two reference frames one inertial global referenceframe attached to the floor where the RC-hovercraft movesand a body-fixed reference frame shown in Figure 1 Accord-ing to the model (the details of which can be found in [27])the dynamics of the system can be described in the globalreference frame by
= V119909 (1)
119910 = V119910 (2)
V119909= 119865 cos (120595) minus 119889
119906V119909+ 119901V119909 (3)
V119910= 119865 sin (120595) minus 119889
119906V119910+ 119901V119910 (4)
120595 = 119903 (5)
119903 = 120591 minus 119889
119903119903 + 119901
119903 (6)
The state of the vehicle is given by x = [119909 119910 V119909 V119910 120595 119903]
119879where 119909 and 119910 are the position coordinates of the centre ofmass of the RC-hovercraft 120595 is the orientation V
119909and V
119910
are the linear velocities and 119903 is the yaw rate The parameters119889
119906and 119889
119903of the model are the normalized drag coefficients
where 119889
119906= 119863
119906119898 119889
119903= 119863
119903119869 119898 is the mass of the
hovercraft 119869 the moment of inertia and 119863
119906and 119863
119903are
the linear and rotational drag coefficients respectively Thecontrol actions are the normalized force 119865 = (119865
119901+119865
119904)119898 and
Table 1 Model parameters
Parameter Value119898 0995Kg119869 0014Kgm2
119897 0075m119906max 0545N119906min 0347N119889
11990603588 sminus1
119889
11990317357 sminus1
Table 2 Force selection table
120591 gt 0 120591 lt 0
119865 gt 0 119865
119901= 119906max 119865119904 = minus119906min 119865
119901= minus119906min 119865119904 = 119906max
119865 lt 0 119865
119901= 0 119865
119904= 119906min 119865
119901= 119906min 119865119904 = 0
torque 120591 = 119897(119865
119901minus 119865
119904)119869 where 119865
119904and 119865
119901are the starboard
and port forces produced by the fans and 119897 is the distancebetween the propellers and the symmetry axis as Figure 1shows The dynamics of the vehicle are also subject to theunknown disturbance vector p = [119901V119909 119901V119910 119901119903]
119879 and theoutput of the system is corrupted by measurement noise n =
[119899
119909 119899
119910 119899V119909 119899V119910 119899120595 119899119903]
119879 such that the measured state is y =
x + n Noise and disturbances are unknown but bounded asfollows
sup119905ge0
p (119905) le 119901
119898 sup
119905ge0
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119889p119889119905
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
sup119905ge0
n (119905) le 119899
119898
(7)
The value of the parameters for the experimental deviceare summarized in Table 1
21 Physical Constraints The experimental vehicle is undersome physical constraints as it is usual in any real deviceThecontrol inputs are forces produced by thrusters the valuesof which belong to a discrete set 119865
119904119901isin minus119906min 0 119906max
with 119906max gt 119906min gt 0 Under these considerations thecontrol actions 119865 and 120591 belong to the discrete set of ninecombinations of 119865
119904and 119865
119901shown in Figure 2 It is important
to notice that these control actions can move the hovercraftforward and backwards and right and left Moreover it willbe shown that it is possible to control the RC-hovercraft witha reduced set of control actions and without knowledge ofthe exact values of 119906min and 119906max The key point is that oncethe sign of the force and torque is known then there alwaysexists a proper selection of 119865
119904and 119865
119901such that the desired
values of sign(119865) and sign(120591) are obtained As Figure 2 showsthere exist more than one solution In Table 2 one possibleselection scheme is shown
22 Feasible Trajectories Dynamics of the RC-hovercraft areflat with respect to the position coordinates as demonstratedin [11] The reader is referred to [28] for further informationabout flat systems Thus when the reference spatial trajec-tories 119909
119903(119905) and 119910
119903(119905) are set then the whole state vector
4 Mathematical Problems in Engineering
[umax umax]
F =Fp minus Fs
m
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 2 Discrete control inputs on the laboratory setup
and the control actions that must be applied to follow thereference can be computed For this purpose considering (3)and (4) in the absence of disturbances and defining 119865
119903as the
force needed to track the reference the ideal orientation 120595
119903
needed to exactly track the reference is given by the followingnonholonomic restrictions
119865
119903cos (120595
119903) = V119909119903+ 119889
119906V119909119903
119865
119903sin (120595
119903) = V119910119903+ 119889
119906V119910119903
(8)
The orientation is well defined if 119865119903
= 0 It is important tonotice that actually there exist two families of solutionsdepending on how the trajectory is followed by the vehiclegoing forward (119899 odd) or backwards (119899 even) as follows
120595
119903= atan2 ( 119910
119903+ 119889
119906119910
119903
119903+ 119889
119906
119903) + 119899120587
119865
119903= (minus1)
119899radic
(
119903+ 119889
119906
119903)
2
+ ( 119910
119903+ 119889
119906119910
119903)
2
(9)
Once the family of solutions of interest is selected theorientation120595
119903and its derivatives 119903
119903and 119903
119903 can be computed
Straightforward computations yield the requested torque120591
119903= 119903
119903+119889
119903119903
119903to track the reference Now we consider a spatial
trajectory x119903= [119909
119903(119905) 119910
119903(119905)]
119879 four times differentiable with119865
119903= 0 Two kinds of trajectories of interest can be defined as
follows
(1) A feasible trajectory where 119865
119903and 120591
119903are such that
119865
119904119901isin [minus119906min 119906max]
(2) A D-feasible trajectory where there always exist fourcombinations of allowed forces and torques (119865
1 120591
1)
(119865
2 120591
2) (1198653 120591
3) and (119865
4 120591
4) such that
1003816
1003816
1003816
1003816
119865
119903(119905)
1003816
1003816
1003816
1003816
lt min (1198651 119865
2 minus119865
3 minus119865
4)
1003816
1003816
1003816
1003816
120591
119903(119905)
1003816
1003816
1003816
1003816
lt min (minus1205911 120591
2 minus120591
3 120591
4)
(10)
On one hand we consider that a trajectory is feasible when itcan be followed by the vehicle that is when the necessaryforces and torques to track the trajectory can be producedby values of 119865
119904and 119865
119901smaller than the maximum available
Notice that if a trajectory is not feasible the only way to trackit is to increase the size of the thrusters in order to incrementthe values of 119865
119904and 119865
119901 allowing the trajectory to be feasible
againThe control actions 119865119903(119905) and 120591
119903(119905) that define a feasible
trajectory lie on the dotted region of Figure 3On the other hand we consider that a trajectory is D-
feasible when the forces and torques selected fromTable 2 aregreater in absolute value than the forces and torques neededto track the trajectory Notice that all the trajectories thatare D-feasible are also feasible but the opposite is not trueThe control actions 119865
119903(119905) and 120591
119903(119905) that define a D-feasible
trajectory lie on the dashed region of Figure 3
Mathematical Problems in Engineering 5
F =Fp + Fs
m
[umax umax]
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
1
2
3
4
5
6
7
8
9
(Fr 120591r)
F0
minus1205910 1205910
minusF0
[0 0]
A
Figure 3 Feasible (dotted) and D-feasible (dashed) regions A trajectory such as the green one is feasible because (119865119903(119905) 120591
119903(119905)) lies on the
dotted region but is not D-feasible because it lies outside the dashed region
3 Problem Formulation
The key objective of the present work is to design and toimplement a feedback control law for 119865
119904and 119865
119901to solve the
robust trajectory tracking problem for a vehicle commandedwith a discrete set of inputs In other words the aim is thedevelopment of a controller that is able to track a desiredtrajectory in the presence of disturbances and noise withonly a limited knowledge of the model of the vehicle and theactuators
Themain practical restriction is that the vehicle is under-actuated and the control law must only employ a discrete setof inputs produced by the thrusters 119865
119904119901isin minus119906min 0 119906max
Notice that the controller is oriented to be used on smallautonomous vehicles equipped with simple thrusters becausethe addition of more thrusters would add complexity to thesystem and would increase the power consumption
Consider the dynamics of the hovercraft describedby (1)ndash(6) a D-feasible spatial reference trajectoryx119890 and the spatial tracking error defined by e
119901=
[119909 minus 119909
119903 119910 minus 119910
119903 V119909minus V119909119903 V119910minus V119910119903]
119879 Then the requirementsfor a robust trajectory tracking controller are as follows
(1) The error e119901
is bounded in the presence of thebounded noise and disturbances described by (7)
(2) The final bound of e119901 can be made arbitrary small
if the noise 119899
119898and the disturbances 119901
119898are small
enough (or alternatively if force and torque are largeenough)
4 Nonlinear Control Design
The control design is based on the cascade structure illus-trated in Figure 4 In the outer loop the force and the
orientation are used as virtual control inputs for the positioncontroller to stabilize the position tracking errorThepositioncontroller defines an orientation reference 120595
119888to be tracked
by the inner loop the orientation controller that controlsthe orientation with the torque 120591 Therefore the positioncontroller determines the sign of the force 119865 to be appliedon the RC-hovercraft whereas the orientation controllerdetermines the sign of the torque 120591 Once sign(119865) and sign(120591)are computed the values of 119865
119904and 119865
119901are selected from
Table 2
41 Position Control The objective of this subsection is thedesign of a controller for the position of the hovercraft Theerror variables are defined by
119890
119909= 119909 minus 119909
119903
119890
119910= 119910 minus 119910
119903
119890V119909 = V119909minus V119909119903
119890V119910 = V119910minus V119910119903
(11)
And their dynamics can be written as
119890
119909= 119890V119909
119890
119910= 119890V119910
119890V119909 = 119865 cos (120595) minus 119889
119906119890V119909 minus 119865
119909119903+ 119901V119909
119890V119910 = 119865 sin (120595) minus 119889
119906119890V119910 minus 119865
119910119903+ 119901V119910
(12)
where 119865119909119903
= 119865
119903cos(120595
119903) and 119865
119910119903= 119865
119903sin(120595119903) are the nominal
forces needed to track exactly the reference in absence of
6 Mathematical Problems in Engineering
References
x
Positioncontrol
120595 r
120595c rc
Orientationcontrol Sign(120591)
Selector
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J
25
2
15
1
05
0
minus05
minus1
minus15
F=(F
b+Fe)m
Fe
Fb
Hovercraft
x
Sign(F)
[umax umax]
[umax 0]
[umax minusumin]
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 4 Control architecture
disturbances At this point a positive control constant 1198961 and
the variables 119904119909and 119904119910are introduced in order to proceedwith
the error stabilization as follows
119904
119909= 119890
119909+ 119896
1119890V119909
119904
119910= 119890
119910+ 119896
1119890V119910
(13)
It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896
1if 119904119909
and 119904
119910are both equal to zero Furthermore the problem of
tracking error stabilization is reduced to the stabilization of119904
119909and 119904
119910 Computing their dynamics it is clear that 119865 and
the orientation 120595 are coupled Consider the following
119904
119909= 119896
1119865 cos (120595) + (1 minus 119889
119906119896
1) 119890V119909 minus 119896
1119865
119909119903+ 119896
1119901V119909
119904
119910= 119896
1119865 sin (120595) + (1 minus 119889
119906119896
1) 119890V119910 minus 119896
1119865
119910119903+ 119896
1119901V119910
(14)
Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates
119911
1= 119904
119909cos (120595) + 119904
119910sin (120595)
119911
2= minus119904
119909sin (120595) + 119904
119910cos (120595)
(15)
Then simple computations yield
1= 119896
1119865 minus 119896
1119865
119903cos (120595 minus 120595
119903) + 119911
2119903
+ (1 minus 119889
119906119896
1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896
1119901
1
(16)
2= 119896
1119865
119903sin (120595 minus 120595
119903) minus 119911
1119903
+ (1 minus 119889
119906119896
1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896
1119901
2
(17)
where 1199011and 119901
2are bounded disturbances given by
119901
1= 119901V119909 cos (120595) + 119901V119910 sin (120595)
119901
2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)
p =1003817
1003817
1003817
1003817
1003817
1003817
[119901
1 119901
2]
1198791003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
(18)
Analyzing the dynamics of 1199111and 119911
2 given by (16) and
(17) it must be noticed that (17) does not depend on 119865 so 119865
must be used to control 1199111 In order to solve this problem the
sign of 119865must be opposed to the sign of 1199111 thus we propose
sign (119865) =
minus sign (1199111) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
lt 120576
(19)
where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911
1|
is small Notice that this condition imposes a minimal timebetween switches because |119911
1|must change from minus120576 to +120576 (or
from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911
1| lt 120576
In order to stabilize 1199112 the sign of 119865
119903sin(120595 minus 120595
119903)must be
the opposite of the sign of 1199112 For this purpose the following
reference for the orientation is defined as follows
120595
119888= 120595
119903minus 119896
2tanh (119911
2) sign (119865
119903) (20)
119903
119888=
120595
119888=
120595
119903minus
2119896
2(1 minus tanh (119911
2)
2
) sign (119865119903) (21)
The saturation of 1199112given by the operation tanh in (20) is
introduced to guarantee that when120595 = 120595
119888and 0 lt 119896
2lt 1205872
then 119865
119903sin(120595119888minus 120595
119903) = minus|119865
119903| sin(119896
2tanh(119911
2)) that is opposed
to 1199112In general the orientation reference 120595
119888is not perfectly
tracked by 120595 Thus the orientation errors are defined by
119890
120595= 120595 minus 120595
119888 (22)
119890
119903= 119903 minus 119903
119888 (23)
and the dynamics of z finally become
1= 119896
1119865 minus 119896
1119865
119903cos (119890
120595+ 120595
119888minus 120595
119903) + 119911
2119903 + 119896
1119901
1
+ (1 minus 119889
119906119896
1) (119890V119909 cos (119890120595 + 120595
119888) + 119890V119910 sin (119890120595 + 120595
119888))
(24)
2= minus 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119911
1119903 + 119896
1119865
119903119908 + 119896
1119901
2
+ (1 minus 119889
119906119896
1) (119890V119910 cos (119890120595 + 120595
119888) minus 119890V119909 sin (119890120595 + 120595
119888))
(25)
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
120595
r
Fp
Fs
o
X
Y
l(x y)
U
u
V = [x t]T
Figure 1 Inertial reference frame and body fixed frame
kind the effect of this practical limitation and the effect ofdisturbances and noise have never been studied in detail
The paper is organized as follows Section 2 describes themodel of the vehicle and the set of trajectories that can betracked with a discrete set of inputs The control problem isformulated in Section 3 In Section 4 a nonlinear controlleris designed Practical implementation and the effect of dis-turbances and noise are discussed in Section 5 The stabilityand robustness of the controller are analyzed in Section 6Simulation and experimental tests are shown in Section 7Finally conclusions are commented on in Section 8
2 Vehicle Model
The model of the vehicle operating over a solid surface isdefined in two reference frames one inertial global referenceframe attached to the floor where the RC-hovercraft movesand a body-fixed reference frame shown in Figure 1 Accord-ing to the model (the details of which can be found in [27])the dynamics of the system can be described in the globalreference frame by
= V119909 (1)
119910 = V119910 (2)
V119909= 119865 cos (120595) minus 119889
119906V119909+ 119901V119909 (3)
V119910= 119865 sin (120595) minus 119889
119906V119910+ 119901V119910 (4)
120595 = 119903 (5)
119903 = 120591 minus 119889
119903119903 + 119901
119903 (6)
The state of the vehicle is given by x = [119909 119910 V119909 V119910 120595 119903]
119879where 119909 and 119910 are the position coordinates of the centre ofmass of the RC-hovercraft 120595 is the orientation V
119909and V
119910
are the linear velocities and 119903 is the yaw rate The parameters119889
119906and 119889
119903of the model are the normalized drag coefficients
where 119889
119906= 119863
119906119898 119889
119903= 119863
119903119869 119898 is the mass of the
hovercraft 119869 the moment of inertia and 119863
119906and 119863
119903are
the linear and rotational drag coefficients respectively Thecontrol actions are the normalized force 119865 = (119865
119901+119865
119904)119898 and
Table 1 Model parameters
Parameter Value119898 0995Kg119869 0014Kgm2
119897 0075m119906max 0545N119906min 0347N119889
11990603588 sminus1
119889
11990317357 sminus1
Table 2 Force selection table
120591 gt 0 120591 lt 0
119865 gt 0 119865
119901= 119906max 119865119904 = minus119906min 119865
119901= minus119906min 119865119904 = 119906max
119865 lt 0 119865
119901= 0 119865
119904= 119906min 119865
119901= 119906min 119865119904 = 0
torque 120591 = 119897(119865
119901minus 119865
119904)119869 where 119865
119904and 119865
119901are the starboard
and port forces produced by the fans and 119897 is the distancebetween the propellers and the symmetry axis as Figure 1shows The dynamics of the vehicle are also subject to theunknown disturbance vector p = [119901V119909 119901V119910 119901119903]
119879 and theoutput of the system is corrupted by measurement noise n =
[119899
119909 119899
119910 119899V119909 119899V119910 119899120595 119899119903]
119879 such that the measured state is y =
x + n Noise and disturbances are unknown but bounded asfollows
sup119905ge0
p (119905) le 119901
119898 sup
119905ge0
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119889p119889119905
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
sup119905ge0
n (119905) le 119899
119898
(7)
The value of the parameters for the experimental deviceare summarized in Table 1
21 Physical Constraints The experimental vehicle is undersome physical constraints as it is usual in any real deviceThecontrol inputs are forces produced by thrusters the valuesof which belong to a discrete set 119865
119904119901isin minus119906min 0 119906max
with 119906max gt 119906min gt 0 Under these considerations thecontrol actions 119865 and 120591 belong to the discrete set of ninecombinations of 119865
119904and 119865
119901shown in Figure 2 It is important
to notice that these control actions can move the hovercraftforward and backwards and right and left Moreover it willbe shown that it is possible to control the RC-hovercraft witha reduced set of control actions and without knowledge ofthe exact values of 119906min and 119906max The key point is that oncethe sign of the force and torque is known then there alwaysexists a proper selection of 119865
119904and 119865
119901such that the desired
values of sign(119865) and sign(120591) are obtained As Figure 2 showsthere exist more than one solution In Table 2 one possibleselection scheme is shown
22 Feasible Trajectories Dynamics of the RC-hovercraft areflat with respect to the position coordinates as demonstratedin [11] The reader is referred to [28] for further informationabout flat systems Thus when the reference spatial trajec-tories 119909
119903(119905) and 119910
119903(119905) are set then the whole state vector
4 Mathematical Problems in Engineering
[umax umax]
F =Fp minus Fs
m
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 2 Discrete control inputs on the laboratory setup
and the control actions that must be applied to follow thereference can be computed For this purpose considering (3)and (4) in the absence of disturbances and defining 119865
119903as the
force needed to track the reference the ideal orientation 120595
119903
needed to exactly track the reference is given by the followingnonholonomic restrictions
119865
119903cos (120595
119903) = V119909119903+ 119889
119906V119909119903
119865
119903sin (120595
119903) = V119910119903+ 119889
119906V119910119903
(8)
The orientation is well defined if 119865119903
= 0 It is important tonotice that actually there exist two families of solutionsdepending on how the trajectory is followed by the vehiclegoing forward (119899 odd) or backwards (119899 even) as follows
120595
119903= atan2 ( 119910
119903+ 119889
119906119910
119903
119903+ 119889
119906
119903) + 119899120587
119865
119903= (minus1)
119899radic
(
119903+ 119889
119906
119903)
2
+ ( 119910
119903+ 119889
119906119910
119903)
2
(9)
Once the family of solutions of interest is selected theorientation120595
119903and its derivatives 119903
119903and 119903
119903 can be computed
Straightforward computations yield the requested torque120591
119903= 119903
119903+119889
119903119903
119903to track the reference Now we consider a spatial
trajectory x119903= [119909
119903(119905) 119910
119903(119905)]
119879 four times differentiable with119865
119903= 0 Two kinds of trajectories of interest can be defined as
follows
(1) A feasible trajectory where 119865
119903and 120591
119903are such that
119865
119904119901isin [minus119906min 119906max]
(2) A D-feasible trajectory where there always exist fourcombinations of allowed forces and torques (119865
1 120591
1)
(119865
2 120591
2) (1198653 120591
3) and (119865
4 120591
4) such that
1003816
1003816
1003816
1003816
119865
119903(119905)
1003816
1003816
1003816
1003816
lt min (1198651 119865
2 minus119865
3 minus119865
4)
1003816
1003816
1003816
1003816
120591
119903(119905)
1003816
1003816
1003816
1003816
lt min (minus1205911 120591
2 minus120591
3 120591
4)
(10)
On one hand we consider that a trajectory is feasible when itcan be followed by the vehicle that is when the necessaryforces and torques to track the trajectory can be producedby values of 119865
119904and 119865
119901smaller than the maximum available
Notice that if a trajectory is not feasible the only way to trackit is to increase the size of the thrusters in order to incrementthe values of 119865
119904and 119865
119901 allowing the trajectory to be feasible
againThe control actions 119865119903(119905) and 120591
119903(119905) that define a feasible
trajectory lie on the dotted region of Figure 3On the other hand we consider that a trajectory is D-
feasible when the forces and torques selected fromTable 2 aregreater in absolute value than the forces and torques neededto track the trajectory Notice that all the trajectories thatare D-feasible are also feasible but the opposite is not trueThe control actions 119865
119903(119905) and 120591
119903(119905) that define a D-feasible
trajectory lie on the dashed region of Figure 3
Mathematical Problems in Engineering 5
F =Fp + Fs
m
[umax umax]
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
1
2
3
4
5
6
7
8
9
(Fr 120591r)
F0
minus1205910 1205910
minusF0
[0 0]
A
Figure 3 Feasible (dotted) and D-feasible (dashed) regions A trajectory such as the green one is feasible because (119865119903(119905) 120591
119903(119905)) lies on the
dotted region but is not D-feasible because it lies outside the dashed region
3 Problem Formulation
The key objective of the present work is to design and toimplement a feedback control law for 119865
119904and 119865
119901to solve the
robust trajectory tracking problem for a vehicle commandedwith a discrete set of inputs In other words the aim is thedevelopment of a controller that is able to track a desiredtrajectory in the presence of disturbances and noise withonly a limited knowledge of the model of the vehicle and theactuators
Themain practical restriction is that the vehicle is under-actuated and the control law must only employ a discrete setof inputs produced by the thrusters 119865
119904119901isin minus119906min 0 119906max
Notice that the controller is oriented to be used on smallautonomous vehicles equipped with simple thrusters becausethe addition of more thrusters would add complexity to thesystem and would increase the power consumption
Consider the dynamics of the hovercraft describedby (1)ndash(6) a D-feasible spatial reference trajectoryx119890 and the spatial tracking error defined by e
119901=
[119909 minus 119909
119903 119910 minus 119910
119903 V119909minus V119909119903 V119910minus V119910119903]
119879 Then the requirementsfor a robust trajectory tracking controller are as follows
(1) The error e119901
is bounded in the presence of thebounded noise and disturbances described by (7)
(2) The final bound of e119901 can be made arbitrary small
if the noise 119899
119898and the disturbances 119901
119898are small
enough (or alternatively if force and torque are largeenough)
4 Nonlinear Control Design
The control design is based on the cascade structure illus-trated in Figure 4 In the outer loop the force and the
orientation are used as virtual control inputs for the positioncontroller to stabilize the position tracking errorThepositioncontroller defines an orientation reference 120595
119888to be tracked
by the inner loop the orientation controller that controlsthe orientation with the torque 120591 Therefore the positioncontroller determines the sign of the force 119865 to be appliedon the RC-hovercraft whereas the orientation controllerdetermines the sign of the torque 120591 Once sign(119865) and sign(120591)are computed the values of 119865
119904and 119865
119901are selected from
Table 2
41 Position Control The objective of this subsection is thedesign of a controller for the position of the hovercraft Theerror variables are defined by
119890
119909= 119909 minus 119909
119903
119890
119910= 119910 minus 119910
119903
119890V119909 = V119909minus V119909119903
119890V119910 = V119910minus V119910119903
(11)
And their dynamics can be written as
119890
119909= 119890V119909
119890
119910= 119890V119910
119890V119909 = 119865 cos (120595) minus 119889
119906119890V119909 minus 119865
119909119903+ 119901V119909
119890V119910 = 119865 sin (120595) minus 119889
119906119890V119910 minus 119865
119910119903+ 119901V119910
(12)
where 119865119909119903
= 119865
119903cos(120595
119903) and 119865
119910119903= 119865
119903sin(120595119903) are the nominal
forces needed to track exactly the reference in absence of
6 Mathematical Problems in Engineering
References
x
Positioncontrol
120595 r
120595c rc
Orientationcontrol Sign(120591)
Selector
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J
25
2
15
1
05
0
minus05
minus1
minus15
F=(F
b+Fe)m
Fe
Fb
Hovercraft
x
Sign(F)
[umax umax]
[umax 0]
[umax minusumin]
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 4 Control architecture
disturbances At this point a positive control constant 1198961 and
the variables 119904119909and 119904119910are introduced in order to proceedwith
the error stabilization as follows
119904
119909= 119890
119909+ 119896
1119890V119909
119904
119910= 119890
119910+ 119896
1119890V119910
(13)
It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896
1if 119904119909
and 119904
119910are both equal to zero Furthermore the problem of
tracking error stabilization is reduced to the stabilization of119904
119909and 119904
119910 Computing their dynamics it is clear that 119865 and
the orientation 120595 are coupled Consider the following
119904
119909= 119896
1119865 cos (120595) + (1 minus 119889
119906119896
1) 119890V119909 minus 119896
1119865
119909119903+ 119896
1119901V119909
119904
119910= 119896
1119865 sin (120595) + (1 minus 119889
119906119896
1) 119890V119910 minus 119896
1119865
119910119903+ 119896
1119901V119910
(14)
Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates
119911
1= 119904
119909cos (120595) + 119904
119910sin (120595)
119911
2= minus119904
119909sin (120595) + 119904
119910cos (120595)
(15)
Then simple computations yield
1= 119896
1119865 minus 119896
1119865
119903cos (120595 minus 120595
119903) + 119911
2119903
+ (1 minus 119889
119906119896
1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896
1119901
1
(16)
2= 119896
1119865
119903sin (120595 minus 120595
119903) minus 119911
1119903
+ (1 minus 119889
119906119896
1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896
1119901
2
(17)
where 1199011and 119901
2are bounded disturbances given by
119901
1= 119901V119909 cos (120595) + 119901V119910 sin (120595)
119901
2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)
p =1003817
1003817
1003817
1003817
1003817
1003817
[119901
1 119901
2]
1198791003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
(18)
Analyzing the dynamics of 1199111and 119911
2 given by (16) and
(17) it must be noticed that (17) does not depend on 119865 so 119865
must be used to control 1199111 In order to solve this problem the
sign of 119865must be opposed to the sign of 1199111 thus we propose
sign (119865) =
minus sign (1199111) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
lt 120576
(19)
where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911
1|
is small Notice that this condition imposes a minimal timebetween switches because |119911
1|must change from minus120576 to +120576 (or
from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911
1| lt 120576
In order to stabilize 1199112 the sign of 119865
119903sin(120595 minus 120595
119903)must be
the opposite of the sign of 1199112 For this purpose the following
reference for the orientation is defined as follows
120595
119888= 120595
119903minus 119896
2tanh (119911
2) sign (119865
119903) (20)
119903
119888=
120595
119888=
120595
119903minus
2119896
2(1 minus tanh (119911
2)
2
) sign (119865119903) (21)
The saturation of 1199112given by the operation tanh in (20) is
introduced to guarantee that when120595 = 120595
119888and 0 lt 119896
2lt 1205872
then 119865
119903sin(120595119888minus 120595
119903) = minus|119865
119903| sin(119896
2tanh(119911
2)) that is opposed
to 1199112In general the orientation reference 120595
119888is not perfectly
tracked by 120595 Thus the orientation errors are defined by
119890
120595= 120595 minus 120595
119888 (22)
119890
119903= 119903 minus 119903
119888 (23)
and the dynamics of z finally become
1= 119896
1119865 minus 119896
1119865
119903cos (119890
120595+ 120595
119888minus 120595
119903) + 119911
2119903 + 119896
1119901
1
+ (1 minus 119889
119906119896
1) (119890V119909 cos (119890120595 + 120595
119888) + 119890V119910 sin (119890120595 + 120595
119888))
(24)
2= minus 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119911
1119903 + 119896
1119865
119903119908 + 119896
1119901
2
+ (1 minus 119889
119906119896
1) (119890V119910 cos (119890120595 + 120595
119888) minus 119890V119909 sin (119890120595 + 120595
119888))
(25)
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
[umax umax]
F =Fp minus Fs
m
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 2 Discrete control inputs on the laboratory setup
and the control actions that must be applied to follow thereference can be computed For this purpose considering (3)and (4) in the absence of disturbances and defining 119865
119903as the
force needed to track the reference the ideal orientation 120595
119903
needed to exactly track the reference is given by the followingnonholonomic restrictions
119865
119903cos (120595
119903) = V119909119903+ 119889
119906V119909119903
119865
119903sin (120595
119903) = V119910119903+ 119889
119906V119910119903
(8)
The orientation is well defined if 119865119903
= 0 It is important tonotice that actually there exist two families of solutionsdepending on how the trajectory is followed by the vehiclegoing forward (119899 odd) or backwards (119899 even) as follows
120595
119903= atan2 ( 119910
119903+ 119889
119906119910
119903
119903+ 119889
119906
119903) + 119899120587
119865
119903= (minus1)
119899radic
(
119903+ 119889
119906
119903)
2
+ ( 119910
119903+ 119889
119906119910
119903)
2
(9)
Once the family of solutions of interest is selected theorientation120595
119903and its derivatives 119903
119903and 119903
119903 can be computed
Straightforward computations yield the requested torque120591
119903= 119903
119903+119889
119903119903
119903to track the reference Now we consider a spatial
trajectory x119903= [119909
119903(119905) 119910
119903(119905)]
119879 four times differentiable with119865
119903= 0 Two kinds of trajectories of interest can be defined as
follows
(1) A feasible trajectory where 119865
119903and 120591
119903are such that
119865
119904119901isin [minus119906min 119906max]
(2) A D-feasible trajectory where there always exist fourcombinations of allowed forces and torques (119865
1 120591
1)
(119865
2 120591
2) (1198653 120591
3) and (119865
4 120591
4) such that
1003816
1003816
1003816
1003816
119865
119903(119905)
1003816
1003816
1003816
1003816
lt min (1198651 119865
2 minus119865
3 minus119865
4)
1003816
1003816
1003816
1003816
120591
119903(119905)
1003816
1003816
1003816
1003816
lt min (minus1205911 120591
2 minus120591
3 120591
4)
(10)
On one hand we consider that a trajectory is feasible when itcan be followed by the vehicle that is when the necessaryforces and torques to track the trajectory can be producedby values of 119865
119904and 119865
119901smaller than the maximum available
Notice that if a trajectory is not feasible the only way to trackit is to increase the size of the thrusters in order to incrementthe values of 119865
119904and 119865
119901 allowing the trajectory to be feasible
againThe control actions 119865119903(119905) and 120591
119903(119905) that define a feasible
trajectory lie on the dotted region of Figure 3On the other hand we consider that a trajectory is D-
feasible when the forces and torques selected fromTable 2 aregreater in absolute value than the forces and torques neededto track the trajectory Notice that all the trajectories thatare D-feasible are also feasible but the opposite is not trueThe control actions 119865
119903(119905) and 120591
119903(119905) that define a D-feasible
trajectory lie on the dashed region of Figure 3
Mathematical Problems in Engineering 5
F =Fp + Fs
m
[umax umax]
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
1
2
3
4
5
6
7
8
9
(Fr 120591r)
F0
minus1205910 1205910
minusF0
[0 0]
A
Figure 3 Feasible (dotted) and D-feasible (dashed) regions A trajectory such as the green one is feasible because (119865119903(119905) 120591
119903(119905)) lies on the
dotted region but is not D-feasible because it lies outside the dashed region
3 Problem Formulation
The key objective of the present work is to design and toimplement a feedback control law for 119865
119904and 119865
119901to solve the
robust trajectory tracking problem for a vehicle commandedwith a discrete set of inputs In other words the aim is thedevelopment of a controller that is able to track a desiredtrajectory in the presence of disturbances and noise withonly a limited knowledge of the model of the vehicle and theactuators
Themain practical restriction is that the vehicle is under-actuated and the control law must only employ a discrete setof inputs produced by the thrusters 119865
119904119901isin minus119906min 0 119906max
Notice that the controller is oriented to be used on smallautonomous vehicles equipped with simple thrusters becausethe addition of more thrusters would add complexity to thesystem and would increase the power consumption
Consider the dynamics of the hovercraft describedby (1)ndash(6) a D-feasible spatial reference trajectoryx119890 and the spatial tracking error defined by e
119901=
[119909 minus 119909
119903 119910 minus 119910
119903 V119909minus V119909119903 V119910minus V119910119903]
119879 Then the requirementsfor a robust trajectory tracking controller are as follows
(1) The error e119901
is bounded in the presence of thebounded noise and disturbances described by (7)
(2) The final bound of e119901 can be made arbitrary small
if the noise 119899
119898and the disturbances 119901
119898are small
enough (or alternatively if force and torque are largeenough)
4 Nonlinear Control Design
The control design is based on the cascade structure illus-trated in Figure 4 In the outer loop the force and the
orientation are used as virtual control inputs for the positioncontroller to stabilize the position tracking errorThepositioncontroller defines an orientation reference 120595
119888to be tracked
by the inner loop the orientation controller that controlsthe orientation with the torque 120591 Therefore the positioncontroller determines the sign of the force 119865 to be appliedon the RC-hovercraft whereas the orientation controllerdetermines the sign of the torque 120591 Once sign(119865) and sign(120591)are computed the values of 119865
119904and 119865
119901are selected from
Table 2
41 Position Control The objective of this subsection is thedesign of a controller for the position of the hovercraft Theerror variables are defined by
119890
119909= 119909 minus 119909
119903
119890
119910= 119910 minus 119910
119903
119890V119909 = V119909minus V119909119903
119890V119910 = V119910minus V119910119903
(11)
And their dynamics can be written as
119890
119909= 119890V119909
119890
119910= 119890V119910
119890V119909 = 119865 cos (120595) minus 119889
119906119890V119909 minus 119865
119909119903+ 119901V119909
119890V119910 = 119865 sin (120595) minus 119889
119906119890V119910 minus 119865
119910119903+ 119901V119910
(12)
where 119865119909119903
= 119865
119903cos(120595
119903) and 119865
119910119903= 119865
119903sin(120595119903) are the nominal
forces needed to track exactly the reference in absence of
6 Mathematical Problems in Engineering
References
x
Positioncontrol
120595 r
120595c rc
Orientationcontrol Sign(120591)
Selector
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J
25
2
15
1
05
0
minus05
minus1
minus15
F=(F
b+Fe)m
Fe
Fb
Hovercraft
x
Sign(F)
[umax umax]
[umax 0]
[umax minusumin]
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 4 Control architecture
disturbances At this point a positive control constant 1198961 and
the variables 119904119909and 119904119910are introduced in order to proceedwith
the error stabilization as follows
119904
119909= 119890
119909+ 119896
1119890V119909
119904
119910= 119890
119910+ 119896
1119890V119910
(13)
It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896
1if 119904119909
and 119904
119910are both equal to zero Furthermore the problem of
tracking error stabilization is reduced to the stabilization of119904
119909and 119904
119910 Computing their dynamics it is clear that 119865 and
the orientation 120595 are coupled Consider the following
119904
119909= 119896
1119865 cos (120595) + (1 minus 119889
119906119896
1) 119890V119909 minus 119896
1119865
119909119903+ 119896
1119901V119909
119904
119910= 119896
1119865 sin (120595) + (1 minus 119889
119906119896
1) 119890V119910 minus 119896
1119865
119910119903+ 119896
1119901V119910
(14)
Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates
119911
1= 119904
119909cos (120595) + 119904
119910sin (120595)
119911
2= minus119904
119909sin (120595) + 119904
119910cos (120595)
(15)
Then simple computations yield
1= 119896
1119865 minus 119896
1119865
119903cos (120595 minus 120595
119903) + 119911
2119903
+ (1 minus 119889
119906119896
1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896
1119901
1
(16)
2= 119896
1119865
119903sin (120595 minus 120595
119903) minus 119911
1119903
+ (1 minus 119889
119906119896
1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896
1119901
2
(17)
where 1199011and 119901
2are bounded disturbances given by
119901
1= 119901V119909 cos (120595) + 119901V119910 sin (120595)
119901
2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)
p =1003817
1003817
1003817
1003817
1003817
1003817
[119901
1 119901
2]
1198791003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
(18)
Analyzing the dynamics of 1199111and 119911
2 given by (16) and
(17) it must be noticed that (17) does not depend on 119865 so 119865
must be used to control 1199111 In order to solve this problem the
sign of 119865must be opposed to the sign of 1199111 thus we propose
sign (119865) =
minus sign (1199111) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
lt 120576
(19)
where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911
1|
is small Notice that this condition imposes a minimal timebetween switches because |119911
1|must change from minus120576 to +120576 (or
from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911
1| lt 120576
In order to stabilize 1199112 the sign of 119865
119903sin(120595 minus 120595
119903)must be
the opposite of the sign of 1199112 For this purpose the following
reference for the orientation is defined as follows
120595
119888= 120595
119903minus 119896
2tanh (119911
2) sign (119865
119903) (20)
119903
119888=
120595
119888=
120595
119903minus
2119896
2(1 minus tanh (119911
2)
2
) sign (119865119903) (21)
The saturation of 1199112given by the operation tanh in (20) is
introduced to guarantee that when120595 = 120595
119888and 0 lt 119896
2lt 1205872
then 119865
119903sin(120595119888minus 120595
119903) = minus|119865
119903| sin(119896
2tanh(119911
2)) that is opposed
to 1199112In general the orientation reference 120595
119888is not perfectly
tracked by 120595 Thus the orientation errors are defined by
119890
120595= 120595 minus 120595
119888 (22)
119890
119903= 119903 minus 119903
119888 (23)
and the dynamics of z finally become
1= 119896
1119865 minus 119896
1119865
119903cos (119890
120595+ 120595
119888minus 120595
119903) + 119911
2119903 + 119896
1119901
1
+ (1 minus 119889
119906119896
1) (119890V119909 cos (119890120595 + 120595
119888) + 119890V119910 sin (119890120595 + 120595
119888))
(24)
2= minus 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119911
1119903 + 119896
1119865
119903119908 + 119896
1119901
2
+ (1 minus 119889
119906119896
1) (119890V119910 cos (119890120595 + 120595
119888) minus 119890V119909 sin (119890120595 + 120595
119888))
(25)
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
F =Fp + Fs
m
[umax umax]
[umax 0]
[umax minusumin]
120591 =l(Fp minus Fs)
J
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
1
2
3
4
5
6
7
8
9
(Fr 120591r)
F0
minus1205910 1205910
minusF0
[0 0]
A
Figure 3 Feasible (dotted) and D-feasible (dashed) regions A trajectory such as the green one is feasible because (119865119903(119905) 120591
119903(119905)) lies on the
dotted region but is not D-feasible because it lies outside the dashed region
3 Problem Formulation
The key objective of the present work is to design and toimplement a feedback control law for 119865
119904and 119865
119901to solve the
robust trajectory tracking problem for a vehicle commandedwith a discrete set of inputs In other words the aim is thedevelopment of a controller that is able to track a desiredtrajectory in the presence of disturbances and noise withonly a limited knowledge of the model of the vehicle and theactuators
Themain practical restriction is that the vehicle is under-actuated and the control law must only employ a discrete setof inputs produced by the thrusters 119865
119904119901isin minus119906min 0 119906max
Notice that the controller is oriented to be used on smallautonomous vehicles equipped with simple thrusters becausethe addition of more thrusters would add complexity to thesystem and would increase the power consumption
Consider the dynamics of the hovercraft describedby (1)ndash(6) a D-feasible spatial reference trajectoryx119890 and the spatial tracking error defined by e
119901=
[119909 minus 119909
119903 119910 minus 119910
119903 V119909minus V119909119903 V119910minus V119910119903]
119879 Then the requirementsfor a robust trajectory tracking controller are as follows
(1) The error e119901
is bounded in the presence of thebounded noise and disturbances described by (7)
(2) The final bound of e119901 can be made arbitrary small
if the noise 119899
119898and the disturbances 119901
119898are small
enough (or alternatively if force and torque are largeenough)
4 Nonlinear Control Design
The control design is based on the cascade structure illus-trated in Figure 4 In the outer loop the force and the
orientation are used as virtual control inputs for the positioncontroller to stabilize the position tracking errorThepositioncontroller defines an orientation reference 120595
119888to be tracked
by the inner loop the orientation controller that controlsthe orientation with the torque 120591 Therefore the positioncontroller determines the sign of the force 119865 to be appliedon the RC-hovercraft whereas the orientation controllerdetermines the sign of the torque 120591 Once sign(119865) and sign(120591)are computed the values of 119865
119904and 119865
119901are selected from
Table 2
41 Position Control The objective of this subsection is thedesign of a controller for the position of the hovercraft Theerror variables are defined by
119890
119909= 119909 minus 119909
119903
119890
119910= 119910 minus 119910
119903
119890V119909 = V119909minus V119909119903
119890V119910 = V119910minus V119910119903
(11)
And their dynamics can be written as
119890
119909= 119890V119909
119890
119910= 119890V119910
119890V119909 = 119865 cos (120595) minus 119889
119906119890V119909 minus 119865
119909119903+ 119901V119909
119890V119910 = 119865 sin (120595) minus 119889
119906119890V119910 minus 119865
119910119903+ 119901V119910
(12)
where 119865119909119903
= 119865
119903cos(120595
119903) and 119865
119910119903= 119865
119903sin(120595119903) are the nominal
forces needed to track exactly the reference in absence of
6 Mathematical Problems in Engineering
References
x
Positioncontrol
120595 r
120595c rc
Orientationcontrol Sign(120591)
Selector
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J
25
2
15
1
05
0
minus05
minus1
minus15
F=(F
b+Fe)m
Fe
Fb
Hovercraft
x
Sign(F)
[umax umax]
[umax 0]
[umax minusumin]
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 4 Control architecture
disturbances At this point a positive control constant 1198961 and
the variables 119904119909and 119904119910are introduced in order to proceedwith
the error stabilization as follows
119904
119909= 119890
119909+ 119896
1119890V119909
119904
119910= 119890
119910+ 119896
1119890V119910
(13)
It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896
1if 119904119909
and 119904
119910are both equal to zero Furthermore the problem of
tracking error stabilization is reduced to the stabilization of119904
119909and 119904
119910 Computing their dynamics it is clear that 119865 and
the orientation 120595 are coupled Consider the following
119904
119909= 119896
1119865 cos (120595) + (1 minus 119889
119906119896
1) 119890V119909 minus 119896
1119865
119909119903+ 119896
1119901V119909
119904
119910= 119896
1119865 sin (120595) + (1 minus 119889
119906119896
1) 119890V119910 minus 119896
1119865
119910119903+ 119896
1119901V119910
(14)
Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates
119911
1= 119904
119909cos (120595) + 119904
119910sin (120595)
119911
2= minus119904
119909sin (120595) + 119904
119910cos (120595)
(15)
Then simple computations yield
1= 119896
1119865 minus 119896
1119865
119903cos (120595 minus 120595
119903) + 119911
2119903
+ (1 minus 119889
119906119896
1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896
1119901
1
(16)
2= 119896
1119865
119903sin (120595 minus 120595
119903) minus 119911
1119903
+ (1 minus 119889
119906119896
1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896
1119901
2
(17)
where 1199011and 119901
2are bounded disturbances given by
119901
1= 119901V119909 cos (120595) + 119901V119910 sin (120595)
119901
2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)
p =1003817
1003817
1003817
1003817
1003817
1003817
[119901
1 119901
2]
1198791003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
(18)
Analyzing the dynamics of 1199111and 119911
2 given by (16) and
(17) it must be noticed that (17) does not depend on 119865 so 119865
must be used to control 1199111 In order to solve this problem the
sign of 119865must be opposed to the sign of 1199111 thus we propose
sign (119865) =
minus sign (1199111) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
lt 120576
(19)
where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911
1|
is small Notice that this condition imposes a minimal timebetween switches because |119911
1|must change from minus120576 to +120576 (or
from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911
1| lt 120576
In order to stabilize 1199112 the sign of 119865
119903sin(120595 minus 120595
119903)must be
the opposite of the sign of 1199112 For this purpose the following
reference for the orientation is defined as follows
120595
119888= 120595
119903minus 119896
2tanh (119911
2) sign (119865
119903) (20)
119903
119888=
120595
119888=
120595
119903minus
2119896
2(1 minus tanh (119911
2)
2
) sign (119865119903) (21)
The saturation of 1199112given by the operation tanh in (20) is
introduced to guarantee that when120595 = 120595
119888and 0 lt 119896
2lt 1205872
then 119865
119903sin(120595119888minus 120595
119903) = minus|119865
119903| sin(119896
2tanh(119911
2)) that is opposed
to 1199112In general the orientation reference 120595
119888is not perfectly
tracked by 120595 Thus the orientation errors are defined by
119890
120595= 120595 minus 120595
119888 (22)
119890
119903= 119903 minus 119903
119888 (23)
and the dynamics of z finally become
1= 119896
1119865 minus 119896
1119865
119903cos (119890
120595+ 120595
119888minus 120595
119903) + 119911
2119903 + 119896
1119901
1
+ (1 minus 119889
119906119896
1) (119890V119909 cos (119890120595 + 120595
119888) + 119890V119910 sin (119890120595 + 120595
119888))
(24)
2= minus 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119911
1119903 + 119896
1119865
119903119908 + 119896
1119901
2
+ (1 minus 119889
119906119896
1) (119890V119910 cos (119890120595 + 120595
119888) minus 119890V119909 sin (119890120595 + 120595
119888))
(25)
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
References
x
Positioncontrol
120595 r
120595c rc
Orientationcontrol Sign(120591)
Selector
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25120591 = (Fb minus Fe)J
25
2
15
1
05
0
minus05
minus1
minus15
F=(F
b+Fe)m
Fe
Fb
Hovercraft
x
Sign(F)
[umax umax]
[umax 0]
[umax minusumin]
[0 minusumin]
[minusumin minusumin]
[minusumin 0]
[minusumin umax]
[0 umax]
[0 0]
Figure 4 Control architecture
disturbances At this point a positive control constant 1198961 and
the variables 119904119909and 119904119910are introduced in order to proceedwith
the error stabilization as follows
119904
119909= 119890
119909+ 119896
1119890V119909
119904
119910= 119890
119910+ 119896
1119890V119910
(13)
It is easy to check from (13) that the tracking errorvariables converge exponentially to zero with rate 1119896
1if 119904119909
and 119904
119910are both equal to zero Furthermore the problem of
tracking error stabilization is reduced to the stabilization of119904
119909and 119904
119910 Computing their dynamics it is clear that 119865 and
the orientation 120595 are coupled Consider the following
119904
119909= 119896
1119865 cos (120595) + (1 minus 119889
119906119896
1) 119890V119909 minus 119896
1119865
119909119903+ 119896
1119901V119909
119904
119910= 119896
1119865 sin (120595) + (1 minus 119889
119906119896
1) 119890V119910 minus 119896
1119865
119910119903+ 119896
1119901V119910
(14)
Therefore the stabilization of both variables at the sametime is an interesting and difficult problem To overcomeit both control actions are decoupled by rotating the errorvector an angle defined by the orientation angle 120595 in thebody fixed frame of the vehicle providing the following newcoordinates
119911
1= 119904
119909cos (120595) + 119904
119910sin (120595)
119911
2= minus119904
119909sin (120595) + 119904
119910cos (120595)
(15)
Then simple computations yield
1= 119896
1119865 minus 119896
1119865
119903cos (120595 minus 120595
119903) + 119911
2119903
+ (1 minus 119889
119906119896
1) (119890V119909 cos (120595) + 119890V119910 sin (120595)) + 119896
1119901
1
(16)
2= 119896
1119865
119903sin (120595 minus 120595
119903) minus 119911
1119903
+ (1 minus 119889
119906119896
1) (minus119890V119909 sin (120595) + 119890V119910 cos (120595)) + 119896
1119901
2
(17)
where 1199011and 119901
2are bounded disturbances given by
119901
1= 119901V119909 cos (120595) + 119901V119910 sin (120595)
119901
2= minus119901V119909 sin (120595) + 119901V119910 cos (120595)
p =1003817
1003817
1003817
1003817
1003817
1003817
[119901
1 119901
2]
1198791003817
1003817
1003817
1003817
1003817
1003817
le 119901
119898
(18)
Analyzing the dynamics of 1199111and 119911
2 given by (16) and
(17) it must be noticed that (17) does not depend on 119865 so 119865
must be used to control 1199111 In order to solve this problem the
sign of 119865must be opposed to the sign of 1199111 thus we propose
sign (119865) =
minus sign (1199111) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
lt 120576
(19)
where 120576 is a small constant to avoid chattering by holding thesign of the force to its previous value sign(119865minus) when |119911
1|
is small Notice that this condition imposes a minimal timebetween switches because |119911
1|must change from minus120576 to +120576 (or
from +120576 to minus120576) before the force119865 changes its sign In additionthis control law tries to make |119911
1| lt 120576
In order to stabilize 1199112 the sign of 119865
119903sin(120595 minus 120595
119903)must be
the opposite of the sign of 1199112 For this purpose the following
reference for the orientation is defined as follows
120595
119888= 120595
119903minus 119896
2tanh (119911
2) sign (119865
119903) (20)
119903
119888=
120595
119888=
120595
119903minus
2119896
2(1 minus tanh (119911
2)
2
) sign (119865119903) (21)
The saturation of 1199112given by the operation tanh in (20) is
introduced to guarantee that when120595 = 120595
119888and 0 lt 119896
2lt 1205872
then 119865
119903sin(120595119888minus 120595
119903) = minus|119865
119903| sin(119896
2tanh(119911
2)) that is opposed
to 1199112In general the orientation reference 120595
119888is not perfectly
tracked by 120595 Thus the orientation errors are defined by
119890
120595= 120595 minus 120595
119888 (22)
119890
119903= 119903 minus 119903
119888 (23)
and the dynamics of z finally become
1= 119896
1119865 minus 119896
1119865
119903cos (119890
120595+ 120595
119888minus 120595
119903) + 119911
2119903 + 119896
1119901
1
+ (1 minus 119889
119906119896
1) (119890V119909 cos (119890120595 + 120595
119888) + 119890V119910 sin (119890120595 + 120595
119888))
(24)
2= minus 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119911
1119903 + 119896
1119865
119903119908 + 119896
1119901
2
+ (1 minus 119889
119906119896
1) (119890V119910 cos (119890120595 + 120595
119888) minus 119890V119909 sin (119890120595 + 120595
119888))
(25)
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
where 119908 is the disturbance produced by the orientationtracking error 119908 = sin(120595 minus 120595
119903) minus sin(120595
119888minus 120595
119903) bounded by
|119908| le
1003816
1003816
1003816
1003816
120595 minus 120595
119888
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
119890
120595
1003816
1003816
1003816
1003816
1003816
(26)
Theorem 2 will show that for small 119890120595 (24)-(25) is a stable
system with disturbances 119890120595 1199011 and 119901
2
42 Orientation Control The objective of this subsection isthe design of an orientation controller so that 120595 can be ableto track the control reference 120595
119888 The error dynamics yields
119890
120595=
120595 minus
120595
119888= 119903 minus
120595
119888= 119890
119903
119890
119903= 120591 minus 120591
119888minus 119889
119903119890
119903+ 119901
119903
(27)
where 120591119888= 119889
119903
120595
119888minus
120595
119888is the control torque necessary to track
the orientation reference 120595119888 At this point a positive control
gain 119896
3is introduced in order to define the variable 119904
120595as
follows
119904
120595= 119890
120595+ 119896
3119890
119903 (28)
It is clear that the orientation error converges exponentiallyto zero with rate 1119896
3if 119904120595is equal to zero The computation
of its dynamics yields
119904
120595= 119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903) 119890
119903 (29)
Thus according to (29) in order to stabilize 119904120595 the sign of 120591
must be the opposite of the sign of 119904120595 This fact motivates the
following control law
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(30)
where chattering is avoided in a similar way as in (19) holdingthe sign of the torque to its previous value sign(120591minus) when |119904
120595|
is small Notice that this control law tries to make |119904120595| lt 120576
5 Practical Implementation and Noise
The positions and velocities of the RC-hovercraft are mea-sured to compute the control law However due to thepresence of bounded noise n only the estimates of the abovevariables (119909 119910 120595 V
119909 V119910 and 119903) are available
Using the estimates as the actual values of the state theerror variables are computed with (11) then 119904
119909and 119904
119910are
computed using (13) and rotated with (15) to obtain theestimates
1= 119911
1+ 119899
1and
2= 119911
2+ 119899
2 where
1003816
1003816
1003816
1003816
119899
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119899
2
1003816
1003816
1003816
1003816
le (z + radic
2 (1 + 119896
1)) 119899
119898 (31)
The term radic2(1 + 119896
1) in (31) comes from the definition of 119904
119909
and 119904119910 (13) while the term z appears from (15) that involves
the estimate 120595
The computation of 2is carried out with (17) assuming
that the unknown disturbance 1199012is equal to 0 The estimated
orientation reference 120595
119888is computed using (20) and its
derivative
120595
119888is computed with (21) These values are finally
used in (28) to obtain the estimate 119904120595= 119904
120595+ 119891
3 where
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 119896
3119896
2119896
1119901
119898+ 119899
119898(1 + 119896
3+ 119899
119898+ 119896
1
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ |119903|)
+ 119896
2119899
119898(1 + 119896
3
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
) (z + radic
2 (1 + 119896
1))
+ 2
radic
2119899
119898
1003816
1003816
1003816
1003816
1 minus 119889
119906119896
1
1003816
1003816
1003816
1003816
radic119890
2
V119909 + 119890
2
V119910
(32)
Thus the control law (19) and (30) become
sign (119865) =
minus sign (1) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
ge 120576
sign (119865minus) if 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
lt 120576
(33)
sign (120591) =
minus sign (119904120595) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
ge 120576
sign (120591minus) if 1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
lt 120576
(34)
The control equations (33) and (34) define only the sign of119865 and 120591 In order to determine the actual control actions tobe applied to the vehicle 119865
119890and 119865
119901are selected according to
Table 2
6 Stability Analysis
In this section theoretical stability results and its practicalapplications are explained Detailed proofs of the theoremsare given in the Appendix
Notation In what follows we consider that a function 120572(119909) isa K class function if 120572(0) = 0 and 120572 is strictly increasing Afunction 120573(119909 119905) is classKL if it isK class with respect to 119909
and 120573 rarr 0 when 119905 rarr infin [29]
Lemma 1 The velocities of the vehicle are globally ultimatelybounded Proof in Section A1
The first result states that the velocities are boundedbecause the control actions and the disturbances that canincrease the kinetic energy are bounded and the terms thattend to reduce the kinetic energy are unbounded
61 Position Stability
Theorem 2 Consider the system (24)-(25) with control action(33) in the presence of bounded disturbances and noise (7)If disturbances and noise are small enough then for any D-feasible trajectory x
119903 there exist three positive constants 119896
1 1198962
and 119908
119898 and two functions 120573 and 120574 of class KL and K
respectively such that if |119908| le 119908
119898 then
z (119905) le 120573 (z (0) 119905) + 120574 (119908
119898+ 119899
119898+ 119901
119898+ 120576) (35)
Proof on Section A2
Corollary 3 Under the conditions of Theorem 2 there existtwo functions 120573
2and 1205742 of classKL andK respectively such
that for |119908| le 119908
119898
1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
2(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
2(119908
119898+ 119899
119898+ 119901
119898+ 120576) (36)
Proof in Section A3
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Equation (35) means that for a small orientation error(and thus a small 119908) if noise and disturbances are not toolarge compared with the control action 119865 then z is boundedFurthermore as (36) states the steady state tracking error(when 120573
2vanishes) is a growing function depending on
the orientation error disturbances and noise Thus whilethe noise disturbances and orientation error increase thetracking performance decreases
62 Orientation Stability
Theorem 4 Consider the dynamics described by (29) withcontrol action (34) in the presence of bounded disturbances andnoise (7) If disturbances and noise are small enough then thereexist five positive constants 119911
119898 1198963 119886 119887 and 119888 and a K class
function 120574
3such that for any D-feasible trajectory x
119903and any
positive gain 119896
2 if z le 119911
119898 then
1003816
1003816
1003816
1003816
1003816
119904
120595(119905)
1003816
1003816
1003816
1003816
1003816
le max (1003816100381610038161003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
minus 119886119905 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898120576)
(37)
Proof in Section A4
Corollary 5 Under conditions of Theorem 4 there existsanother constant 119889 such that for z(119905) le 119911
119898
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le 119889 sdot 119890
minus1199051198963+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576 (38)
Proof in Section A5
Equation (37) means that if the tracking error is boundedby 119911119898 and disturbances and noise are not too large compared
with the control input 120591 then the variable 119904120595is stabilized in
finite timeThus according to (38) the orientation error con-verges exponentially Notice that the performance decreaseswhile noise and disturbance levels increase as it occurs withthe position tracking
63 Overall Convergence Previous results have demon-strated that the orientation error 119890
120595 converges exponentially
to a neighborhood of the origin while z is bounded by 119911119898
Thus (26) implies that119908 converges to a neighborhood of theorigin also Furthermore when119908 is smaller than an arbitrarymaximum value 119908
119898 then z converges to a neighborhood of
the origin tooThe following result states that control laws (33) and (34)
can stabilize the tracking errorwhen both are used at the sametime
Theorem 6 Consider the dynamics of the hovercraft (1)ndash(6)in the presence of bounded disturbances and noise (7) andsubject to control actions (33)-(34) For any positive constant119872 if the initial condition of the hovercraft is such that z(0) lt119872 and the reference trajectory x
119903is D-feasible then it is
possible to choose the control constants 1198961 1198962 1198963 and 120576 such
that1003817
1003817
1003817
1003817
1003817
e119901(119905)
1003817
1003817
1003817
1003817
1003817
le 120573
3(
1003817
1003817
1003817
1003817
1003817
e119901(0)
1003817
1003817
1003817
1003817
1003817
119905) + 120574
4(119899
119898+ 119901
119898+ 120576) (39)
where 1205744is a classK function and 120573
3is a classKL function
Proof in Section A6
This result states that given any bound for the initialcondition 119872 and a trajectory that is D-Feasible then theconstants of the tracking controller can be set in order tostabilize the tracking error
There exists a compromise between the tracking perfor-mance and the control effort imposed by 120576 If 120576 is too largethen the tracking performance is poor as shown in (39) If 120576is too small then (33) and (34) will produce fast switches inthe control law
Remark 7 It must be noticed that the control gains andthus the tracking performance depend on the bounds ofthe initial conditions (see (A24) of the Appendix for moredetails) This result is conservative in the sense that if wewant to increase the region of attraction (increasing119872) thenthe gain 119896
2must be small and thus the effect of noise and
disturbances is larger The result is also conservative in thesense that it restricts its applicability toD-feasible trajectoriesbut in practice the system is stable from any initial conditionwith a good selection of the control gains
7 Results
In this section simulations and experiments on a real setuphave been developed in order to test and illustrate thepotential of the control design of Section 4
71 Simulation Results The forthcoming simulation exam-ples consist in tracking a circular reference with and withoutnoise and disturbances Then a more general trajectorytracking problem is studied to show that the control designcan overcome some of the limitations that are assumed onthe stability demonstrations The control parameters used inthe simulations are 119896
1= 6 s 119896
2= 1 s 119896
3= 1 s and 120576 = 001
711 Circular Trajectory without Noise and Disturbances Thereference trajectory is a circumference with its center at theorigin of the inertial coordinate frame and with a radius119877 of 2 meters The trajectory is tracked with a speed of119881 = 05ms The force and torque needed to track perfectlythis trajectory (computed as described in Section 22) areconstants (119865
119903= 02179ms2 and 120591
119903= 04339 rads2) so the
trajectory isD-feasible because they do not saturate actuatorsThe trajectory of the hovercraft is depicted in Figure 5The vehicle starts from the origin without disturbances andnoise This figure shows how the trajectory followed by thehovercraft converges and tracks almost perfectly the referencetrajectory
The evolution of the position is shown in Figure 6 andvelocities are shown in Figure 7 It must be noticed that theactual trajectory converges to the reference trajectory in atime close to 20 s
The orientation 120595 and its control reference 120595119888are shown
in Figure 8The orientation converges faster than the position(less than 5 seconds) The reason is intuitively clear becausethe position only converges when the orientation error 119890
120595 is
small enoughOnce 119890120595is smaller than119908
119898 z starts to converge
to zero and the control references 120595119888and 119903
119888converge to the
references 120595119903and 119903
119903 too
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
minus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
minus2
minus15
minus1
minus05
0
05
1
15
2
x(m
)
y (m)
Figure 5 Simulated trajectory (solid) and circular reference(dashed)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
x(m
)
t (s)
(a)
0 10 20 30 40 50 60minus3
minus2
minus1
0123
y(m
)
t (s)
(b)
Figure 6 Simulated position (solid) versus references (dashed)
712 Effect of Noise and Disturbances Noise and distur-bances are introduced in the simulations to study theireffects on the control performance In order to make thesimulation more realistic the controller is simulated with asampling period of 117 s that corresponds to the frame rateof the camera in the real setup In addition a disturbancep = [0ms2 002ms2 002ms2] and random noise with anamplitude of 005 (meters for 119909 and 119910 ms for V
119909and V119910 rad
for 120595 and rads for 119903) are introduced in the simulationThe effect of noise and disturbances on the performance
depends strongly on the reference trajectory that must betracked To illustrate this effect Figure 9 shows the trajecto-ries followed by the system when the reference speed is fixedand the radius varies from 1 to 4 meters This figure shows
0 10 20 30 40 50 60
minus04minus02
0020406
x(m
s)
t (s)
(a)
0 10 20 30 40 50 60
minus04minus02
0020406
t (s)
y(m
s)
(b)
Figure 7 Simulated velocities (solid) versus references (dashed)
0 10 20 30 40 50 60minus4
minus2
0
2
4
6120595
(rad
)
t (s)
(a)
0 10 20 30 40 50 60minus08minus06minus04minus02
0020406
r(r
ads
)
t (s)
(b)
Figure 8 Simulated orientation and yaw rate (solid) vs references(dashed)
how the performance decreases for small radii for two rea-sons The first one is that the reduction of the radius reducesthe signalnoise ratio The second one deals with centripetalforces While the radius decreases centripetal forces increasewith a factor 1198812119877 This implies that forces increase and thetrajectories are pushed to the unfeasible region In this casethe impact of noise and disturbance becomes more evidentbecause the control authority is degraded
Figure 10 confirms the behavior predicted byTheorem 6First the trajectories experiment a transient that dependson the initial conditions This transient is of approximately25 s during which the tracking error decreases to the steady
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus4
minus3
minus2
minus1
0
1
2
3
4
x(m
)
y (m)
Figure 9 Simulated trajectories with noise and disturbances fordifferent circle radius 119877 = 1 2 3 4 meters in red violet green andblue respectively
0 10 20 30 40 50 600
05
1
15
2
25
3
35
4
e p(m
)
t (s)
Figure 10 Tracking error for different circle radius 119877 = 1 2 3 4
meters in red violet green and blue respectively
state value After this time only a residual tracking errorremains It shows how the tracking error increases for smallradii because the trajectory is more difficult to track (close tobe unfeasible)
713 General Trajectory For this third scenario a new refer-ence trajectory is set 119909
119903(119905) = 2 sin(38119905) 119910
119903(119905) = 3 cos(28119905)
in order to test the control design The necessary forces andtorques to track the trajectory are depicted in Figure 11 It canbe noticed that the forces lie outside the D-feasible regionhowever the trajectory is feasible This example is used toillustrate that the D-feasible condition (10) is conservativeFurthermore Figure 12 shows that the same control designused for the circular trajectory converges to this non-D-feasible reference too
minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5minus08
minus06
minus04
minus02
0
02
04
06
08
1
12
120591 (rads2)
F(m
s2)
Figure 11 Force map for a general trajectory that is feasible but notD-feasible
minus3 minus2 minus1 0 1 2minus25
minus2
minus15
minus1
minus05
0
05
1
15
2x
(m)
y (m)
Figure 12 General trajectory followed by the vehicle (blue) andreference (red)
This behavior can be explained from an intuitive pointof view D-feasible conditions (10) indicate that the referenceforce and torque must be smaller in absolute value that anyforce and torque than can be selected by the control law fromTable 2However this is a sufficient condition but not always anecessary condition For example suppose that the trajectoryis such that119865
119903is positive for all the time (the current scenario
as Figure 11 shows) then it is possible to select a positiveforce that is larger than 119865
119903 with a larger torque in absolute
value than 120591
119903 This is enough to stabilize the trajectory
because during the trajectory tracking a negative force isnever needed
72 Experimental Results
721 Experimental Setup The experimental setup consistsin an underactuated radio controlled RC-hovercraft a CCD
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Figure 13 Experimental setup
minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1
minus06
minus04
minus02
0
02
04
06
08
x(m
)
y (m)
Figure 14 Actual tracking Trajectory (solid) and references(dashed)
camera to capture images from a zenithal point of viewand a computer to send the control commands via a radiocontrol interfaceThewhole system is shown in Figure 13Thelaboratory has similar capacities to the experimental testbeddeveloped on [5] or [6]
The CCD camera is equipped with wide angle optics thatcover a region of approximately 3m times 4m when the camerais placed at 4 meters over the floor The frame rate of thecomputer-camera system is 17 fps (frames per second)
The position of the vehicle is tracked by the camera bydetecting three LEDs that are attached to the RC-hovercraftThese 3 LEDs compose an isosceles triangle to determine thevehicle orientation The triangle size is 25 cm of length in itsmajor edge and 16 cm in its minor edge
The control signals are sent to the vehicle through acommunication interface that connects the parallel port ofthe computer with the radio controller The control lawis implemented in LabVIEW programming language Thecontrol algorithm works as follows
(1) Once a frame is captured the image is binarized andthe centroid of each individual LED is computed
0 5 10 15 20 250
01
02
03
04
05
06
07
08
09
1
Dist
ance
(m)
t
Figure 15 Tracking error
(2) The barrel distortion introduced by the lenses is cor-rected and the undistorted positions of the LEDs areobtained by solving the perspective transformation
(3) The centre of the mass of the vehicle and its ori-entation are computed from the positions of thetriangle vertexThe velocities are estimated by a finitedifference
(4) Finally the control law computes the forces 119865119904and
119865
119901 and the corresponding commands are sent to the
radio controller
For amore detailed description of the experimental setupsee [30]
722 Real Tracking Test The control design is now testedon the experimental setup described above To check therobustness of the controller a worst case analysis has beenrealized A trajectory with a very small radius 119877 = 06mis tracked with a constant speed of 119881 = 05ms Notice thatthis trajectory is very hard to track because of its small radiusas above mentioned in Section 712 Figure 14 shows thatthe vehicle is able to track this very challenging trajectorywith an acceptable tracking error As far as the authors knowthe performance achieved is comparable to the best resultsobtained with a laboratory hovercraft (see [10]) Notice theadded hard limitation imposed by the all-nothing actuatorswhose dynamics are unknown Figure 16 shows the evolutionof the position of the system and demonstrates that thecontrol law is able to track the trajectory Figure 17 shows thevelocities Notice that the measured velocity is noisy due tothe all-nothing-reverse action of the propellersThis problemcan be reduced by filtering the position measurements butthe aim of this example is to show that this controller candeal with these problems and the vehicle is able to trackthe velocity reference with great accuracy Figure 15 showsthat the final bound of the tracking error is about 04m andthus compatible with simulation results shown on Figure 10Finally Figure 18 shows the input thrust applied to the vehicle
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
0 5 10 15 20 25minus06minus04minus02
0020406
x(m
)
t (s)
(a)
0 5 10 15 20 25minus06minus04minus02
002040608
y(m
)
t (s)
(b)
Figure 16 Actual positions (solid) versus references (dashed) in areal setup
t (s)0 5 10 15 20 25
minus1
minus05
0
05
1
x(m
s)
(a)
0 5 10 15 20 25minus1
minus05
0
05
1
y(m
s)
t (s)
(b)
Figure 17 Actual velocities (solid) versus references (dashed) in areal setup
8 Conclusions
In this paper a tracking control law for an underactuatednonholonomic vehicle with a discrete set of inputs has beendesigned and testedThe control lawhas been designed takinginto account the fact that the actuators of the vehicle can onlyproduce thrust in an all-nothing reverse fashion
Despite this hard limitation on the real setup this papershows how this control design can overcome these difficul-ties Furthermore this work demonstrates that a low cost
0 5 10 15 20 25minus04
minus02
0020406
Fs
(N)
t (s)
(a)
0 5 10 15 20 25minus04
minus02
0020406
t (s)
F p(N
)
(b)
Figure 18 Control actions 119865119904and 119865
119901applied in a real setup
underactuated vehicle with very simple actuators (capableof moving the vehicle forward and backwards and simul-taneously turn right and left) can be controlled in order totrack anyD-feasible trajectory under the sole assumption thatcontrol actions are large enough compared with noise anddisturbances
Theoretical proofs of stability and robustness againstnoise and disturbances have been obtained showing thatthe control law makes the position tracking error to besemiglobal finitely bounded if the vehicle starts with abounded tracking error and the trajectory is D-feasible
Simulations and experimental results state that the con-trol strategy is robust enough to be applied to a real environ-ment with disturbances noise and unmodeled dynamics Inaddition this control law is very suitable to be applied to lowcost and simple vehicleswhere precisemodels of the actuatorsand dynamics are not available
Appendix
A Proofs
This section contains the proofs of the theoretical results
A1 Proof of Lemma 1 Consider the functions 1198711= V21199092
119871
2= V21199102 and 119871
3= 119903
2
2 and compute the derivative alongthe trajectories of the vehicle
119871
1= minusV119909(119889
119906V119909minus119865 cos(120595)+119901V119909)
Defining 119865max and 120591max as the maximum values of |119865| and |120591|
that can be selected from Table 2 then
119871
1can be bounded by
119871
1lt minus119889
119906V2119909(1 minus
119865max + 119901
119898
119889
119906
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
) (A1)
Equation (A1) shows that
119871
1is strictly negative when
119889
119906|V119909| gt (119865max + 119901
119898) This implies that V
119909is ultimately
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
bounded by (119865max + 119901
119898)119889
119906 The same analysis can be done
with 119871
2and 119871
3showing that the set
Ω = x isin R6
1003816
1003816
1003816
1003816
V119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
V119910
1003816
1003816
1003816
1003816
1003816
le
119865max + 119901
119898
119889
119906
|119903| le
120591max + 119901
119898
119889
119903
(A2)
is a positively invariant set of the system (1)ndash(6)
A2 Proof of Theorem 2 Without loose of generality theanalysis is restricted to the region defined by Ω If noise anddisturbances are small enough then there exists a positiveconstant 120575 such that |119865| minus |119865
119903| minus 119901
119898ge 120575 by definition of
a D-feasible trajectory Then for any positive 1205761 it is always
possible to find a constant 1198961such that
radic
2
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(1 minus 119889
119906119896
1)
119889
119906119896
1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(2119865max + 119901
119898) le 120575120576
1 (A3)
Moreover if disturbances are small enough then it is alsopossible to select 119908
119898so that
radic
2 (119901
119898+ 120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119908
119898) lt min (120575 100381610038161003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962)) (A4)
Consider the following Lyapunov candidate function 119881 =
(119911
2
1+ 119911
2
2)2 The computation of its derivative yields
119881 = 119896
1119911
1(119865 minus 119865
119903cos (120595 minus 120595
119903) + 119901
1)
minus 119896
1119911
2(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (119911
2)) minus 119865
119903119908 minus 119901
2)
+ (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
(A5)
For an appropriate stability analysis two cases must be takeninto account
Case A (|1199111| gt 119899
1+ 120576) In this analysis our aim is to bound all
the terms in (A5) The term |119865
119903cos(120595 minus 120595
119903)| can be bounded
by |119865119903| and |119911
1| can be bounded by z Then we define the
following function
119892 (119909) = min( 120575
radic2
119865
119903
radic2
sin(1198962tanh( 119909
radic2
))) (A6)
Condition (A3) in combination with (A6) implies thatminus120575|119911
1| minus |119865
119903119911
2| sin(119896
2tanh (|119911
2|)) le minusz119892(z) so
119881 le minus119896
1z (119892 (z) minus (120575120576
1+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
|119908| + 119901
119898)) (A7)
Equation (A7) is negative when 119892(z) gt 120576
1+ |119865
119903|119908
119898+ 119901
119898
This condition holds when z gt 119911max 1 where 119911max 1 isdefined as
119911max 1 = radic
2 tanhminus1 ( 1
119896
2
sinminus1 (radic2(
119901max + 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119908
119898)))
(A8)
Note that condition (A4) implies that 119911max 1 is finite and 119881
decreases when z gt 119911max 1
Case B (|1199111| le 119899
1+ 120576) In this case we have to establish the
bound of z On one hand we suppose that |1199111| gt |119911
2| and
with (31) we find
z le radic
2
1003816
1003816
1003816
1003816
119911
1
1003816
1003816
1003816
1003816
le
radic
2119899
1+ 120576 le
radic
2 (z + radic
2 (1 + 119896
1)) 119899
119898+ 120576
(A9)
So for any positive 1205762 if 119899119898is small enough it yields
z leradic2 (
radic2 (1 + 119896
1) 119899
119898+ 120576)
(1 minusradic2119899
119898)
= 120576
2 (A10)
On the other hand if |1199111| le |119911
2| then z le radic
2|119911
2| so we
find in both cases that z leradic2|119911
2| + 120576
2and |119911
1| le 2(|119911
2| +
1 minus 119896
1)119899
119898+ 120576 Thus the derivative of 119881 can be bounded as
119881 le 119896
1(2 (
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 1 minus 119896
1) 119899
119898+ 120576) (
1003816
1003816
1003816
1003816
119865max1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 119901
119898)
minus 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
)) minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898)
+ 2 (
radic
2
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
+ 120576
2) 119889
119906119896
1120575120576
1
(A11)
Grouping all terms that can be made arbitrary small byreducing noise and disturbances in 120576
3and 120576
4 (A11) can be
rewritten as
119881 le minus 119896
1(
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
sin (1198962tanh (100381610038161003816
1003816
119911
2
1003816
1003816
1003816
1003816
))
minus
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
minus 119901
119898minus 120576
3) + 120576
4)
(A12)
Now we define
119911max 2 = tanhminus1 ( 1
119896
2
sinminus1 (119901
119898
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
+ 120576
3+ 119908
119898+ 120575)) (A13)
Combining (A13) with
119881 and grouping all the small termsin 1205765 then
119881 le 119896
1(minus|119911max 2|120575+1205765) that is negative if 1205765 is small
enough compared with 120575
Conclusion Defining 119911max = max(119911max 1 119911max 2) 119881 decreasesoutside the bounded region z le 119911max for small enough 119899
119898
Since 119881 is definite positive and radially unbounded then zis globally finitely bounded by 119911max 1 Furthermore as 119911max 1and 119911max 2 are class K functions of 119908
119898 119899119898 119901119898 and 120576 then
(35) holds
A3 Proof of Corollary 3 The bound of z implies that sis also bounded because (15) shows that s = z Then bydefinition of 119904
119909and 119904
119910
119896
1119890
119909= 119904
119909minus 119890
119909
119896
1119890
119910= 119904
119910minus 119890
119910
(A14)
This system is linear and stable thus it is ISS with respect to sthat is globally ultimately bounded since s = zThereforee119901is ISS with respect to 119899
119898 119901119898 119908119898 and 120576 then (36) holds
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
A4 Proof ofTheorem4 Without loose of generality considerthe initial state of the system (1)ndash(6) lying inside the setΩ From (23) |119890
119903| le |119903| + |119903
119903| le |120591
119898|119889
119903+ |119903
119903| The
velocities are bounded in Ω so (24)-(25) imply that thefirst derivatives of z can also be bounded and differentiating(25) it is straightforward to check that there exist positiveconstants 119888
1ndash1198886such that
z le 1003816
1003816
1003816
1003816
1
1003816
1003816
1003816
1003816
+
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
1+ 119888
2119911
119898+ 119888
3119901
119898
1003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
le 119888
4+ 119888
5119911
119898+ 119888
6119901
119898
(A15)
By the definition of 120591119888 it yields
120591
119888= 120591
119903minus 119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) (119889
119903
2+
2)
+ 2119896
2sign (119865
119903) (1 minus tanh2 (119911
2)) tanh (119911
2)
2
2
(A16)
Substituting (A15) into (A16) and (32) the effect of distur-bances and noise on the computation of 119904
120595and 120591
119888can be
bounded with a positive constant 1198887 and two polynomial
bounded functions 1205744and 120574
5 of classK as follows
1003816
1003816
1003816
1003816
120591
119888
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
120591
119903
1003816
1003816
1003816
1003816
+ 119896
2120574
4(119911
119898+ 119901
119898) (A17)
1003816
1003816
1003816
1003816
119891
3
1003816
1003816
1003816
1003816
le 120574
5(119888
7+ 119911
119898+ 119901
119898) 119899
119898+ 119896
3119896
2119896
1119901
119898 (A18)
Then |120591119903| lt |120591| because the reference trajectory is D-feasible
and thus there exists a positive constant 120575 such that |120591|minus|120591119903| le
120575 Now we define 119911119898as
119911
119898= minus
1
2
120574
4
minus1
(
120575
4119896
2
) (A19)
When z lt 119911
119898 the disturbances are small enough such that
119896
2120574
4(2119901
119898)+119901
119898lt 1205754 and with (A17) then |120591
119894|minus|120591
119888|minus|119901
119898| ge
120575 minus 1205754 minus 1205754 = 1205752 Now we select 1198963such that
1003816
1003816
1003816
1003816
1 minus 119896
3119889
119903
1003816
1003816
1003816
1003816
119896
3119889
119903
(
1003816
1003816
1003816
1003816
120591
119898
1003816
1003816
1003816
1003816
+ 119889
119903
1003816
1003816
1003816
1003816
119903
119903
1003816
1003816
1003816
1003816
) le
120575
4
(A20)
And we define the Lyapunov function119881
2= 119904
2
1205952 whose time
derivative is
119881
2= 119904
120595(119896
3(120591 minus 120591
119888+ 119901
119903) + (1 minus 119896
3119889
119903)119890
119903) Consider
first that |119904120595| ge 119891
3+120576 then the control law implies that the sign
of 120591 is the opposite of the sign of 119904120595 Condider the following
119881
2= 119904
120595119904
120595le minus120575119896
3(
1
2
minus
1
4
)
1003816
1003816
1003816
1003816
1003816
119904
120595
1003816
1003816
1003816
1003816
1003816
997888rarr 119904
120595le minus
120575119896
3
4
sign (119904120595)
(A21)
Solving (A21) for the equality and applying the comparisonlemma (see chapter 3 of [29]) we find that |119904
120595(119905)| le |119904
120595(0)| minus
120575119896
34119905 If 119904
120595lies inside the region |119904
120595| lt 119891
3+120576 (A21) does not
hold Nevertheless1198812decreases on the frontier of this region
preventing 119904120595from becoming larger than 119891
3+ 120576 Then in any
case |119904120595(119905)| le max(|119904
120595(0)| minus 120575119896
34119905 119891
3+ 120576) and substituting
the bound (A18) then (37) is obtained
A5 Proof of Corollary 5 By the definition of 119904120595in (28) the
orientation error dynamics become 1198963119890
120595= 119904
120595minus 119890
120595 whose
solution is
119890
120595(119905) = 119890
120595(0) 119890
minus1199051198963+
1
119896
3
int
119905
0
119890
minus(119905minus120591)1198963119904
120595(120591) 119889120591 (A22)
If the terms inside the integral are bounded by (37) then
1003816
1003816
1003816
1003816
1003816
119890
120595(119905)
1003816
1003816
1003816
1003816
1003816
le
1003816
1003816
1003816
1003816
1003816
119890
120595(0)
1003816
1003816
1003816
1003816
1003816
119890
minus1199051198963+
1003816
1003816
1003816
1003816
1003816
119904
120595(0)
1003816
1003816
1003816
1003816
1003816
(119890
|119904120595(0)|1198861198963minus 1) 119890
minus1199051198963
+ 120574
3(119887 + 119911
119898+ 119901
119898) 119899
119898+ 119888119901
119898+ 120576
(A23)
That takes the form of (38) with 119889 = supΩ|119890
120595| + |119904
120595|119890
|119904120595|1198861198963
A6 Proof of Theorem 6 The key point of the proof is that ifthe vehicle starts from a bounded initial condition z(0) lt
119872 where 119872 is an arbitrary constant it is always possible tochoose the control gain 119896
2such that during the time that 119908
needs to converge to119908 lt 119908
119898 z never reaches 119911
119898Therefore
the conditions of Theorem 4 always hold and the trackingerror is finitely bounded
Select 1198962so that (A19) of Theorem 4 is satisfied for 119911
119898=
2119872 as follows
119896
2=
120575
4120574
4(4119872)
(A24)
As z(0) lt 119872 it will take sometime for 119879 to go fromz = 119872 to z = 2119872 This time 119879 can be lower boundedby considering the maximum derivative of 119881
119881 le 119896
1
1003816
1003816
1003816
1003816
119911
2
1003816
1003816
1003816
1003816
(
1003816
1003816
1003816
1003816
119865
119903119908
1003816
1003816
1003816
1003816
+ 119901
2) + (1 minus 119889
119906119896
1) (119904
1119890V119909 + 119904
2119890V119910)
le 119887
1|z| + 119887
2
(A25)
for some constants 1198871and 119887
2 Select 119872 gt 119887
2119887
1then z gt
119887
2119887
1and thus
119881 = zz le 2119887
1z rarr z le 2119887
1rarr z le
z(0) + 2119887
1119905 Hence z can only grow with a finite rate of
2119887
1and therefore 119879 gt 1198722119887
2
Consider noise and disturbances small enough such that120574
3(119887 + 119872 + 119901
119898)119899
119898+ 119888119901
119898+ 120576 le 119889119890
minus1198791198963 From (38) it yieldsthat |119890
120595(119879)| le 2119889 sdot 119890
minus1198791198963 In addition for small enoughdisturbances compared with the control action consider thefollowing
max(119901
119898+ 120576
5
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
119901
119898+ 120575120576
1003816
1003816
1003816
1003816
119865
119903
1003816
1003816
1003816
1003816
) lt 119889 sdot 119890
minus1198791198963 (A26)
Substituting (A26) into (A8) and (A13) it yields
119911max (119879) = radic
2 tanhminus1 (4120574
4(4119872)
120575
sinminus1 (3radic2119889 sdot 119890
minus119872211988821198963))
(A27)
Function 120574
4is a polynomial order function in z bounded
by119872 Therefore when119872 rarr infin then 119911max rarr 0 and for alarge enough119872 then 119911max(119879) le 2119872 = 119911
119898
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
The result is now proved by contradiction Suppose thatthere exists some time 119905
1for which z(119905
1) gt 2119872 then there
must exist another time 119905
2for which z(119905
2) = 2119872 and 119881
grows in 119905
2 But 119881 only grows if z lt 119911max so according
to (26) 119911max(|119908(1199052)|) le 119911max(|119890120595(1199052)|) = 119872 However|119890
120595(119905
2)| le 2119889119890
1198791198963 and therefore z(1199052) lt 119911max(|119908(1199052)|) le
119911max(|119890120595(1199052)|) = 119872 because z lt 119911
119898for all 119879 le 119905
1le 119905
2 This
drives to a contradiction because z(1199052) = 2119872
Therefore z never reaches 119911119898and conditions of The-
orem 4 always hold This means that before some time 119905
2
conditions ofTheorem 2 holds thus we can conclude by (39)that e
119901 is finitely bounded
Acknowledgments
The authors would like to acknowledge the Spanish Min-istry of Economy and Competitiveness for its financialsupport under Grant DPI 2009-14552-C02-02 and theUNED University for the support under Research Project2012VPUNED0003
References
[1] P Encarnacao and A Pascoal ldquoCombined trajectory trackingand path following an application to the coordinated controlof autonomous marine craftrdquo in Proceedings of the 40th IEEEConference on Decision and Control (CDC rsquo01) vol 1 pp 964ndash969 December 2001
[2] K D Do and J Pan ldquoRobust and adaptive path following forunderactuated autonomous underwater vehiclesrdquo in Proceed-ings of the American Control Conference (ACC rsquo03) vol 3 pp1994ndash1999 June 2003
[3] K Y Pettersen and T I Fossen ldquoUnderactuated dynamicpositioning of a shipmdashexperimental resultsrdquo IEEE Transactionson Control Systems Technology vol 8 no 5 pp 856ndash863 2000
[4] W Oelen H Berghuis H Nijmeijer and C C de Wit ldquoHybridstabilizing control on a real mobile robotrdquo IEEE Robotics andAutomation Magazine vol 2 no 2 pp 16ndash23 1995
[5] R M Murray J Hickey E Klavins and S Low ldquoThe CaltechMulti-Vehicle Wireless Testbed (MVWT)rdquo Tech Rep Cali-fornia Institute of Technology Pasadena Calif USA 2003httpwwwcdscaltechedusimmurrayprojectsdurip01-mvwt
[6] V Vladimerou A Stubbs J Rubel A Fulford J Strickand G Dullerud ldquoA hovercraft testbed for decentralized andcooperative controlrdquo in Proceedings of the American ControlConference (AAC rsquo04) vol 6 pp 5332ndash5337 July 2004
[7] T Chatchanayuenyong and M Parnichkun ldquoNeural networkbased-time optimal sliding mode control for an autonomousunderwater robotrdquo Mechatronics vol 16 no 8 pp 471ndash4782006
[8] RWBrockett ldquoAsymptotic stability and feedback stabilizationrdquoin Differential Geometric Control Theory vol 27 pp 181ndash191Birkhaauser Boston Mass USA 1983
[9] I Kolmanovsky and N H McClamroch ldquoDevelopments innonholonomic control problemsrdquo IEEE Control Systems Mag-azine vol 15 no 6 pp 20ndash36 1995
[10] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft controller designand experimental resultsrdquo in Proceedings of the 42nd IEEE
Conference on Decision and Control (CDC rsquo03) vol 4 pp 3858ndash3863 December 2003
[11] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[12] A P Aguiar and J P Hespanha ldquoTrajectory-tracking and path-following of underactuated autonomous vehicles with para-metric modeling uncertaintyrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1362ndash1379 2007
[13] K Y Pettersen and H Nijmeijer ldquoUnderactuated ship trackingcontrol theory and experimentsrdquo International Journal of Con-trol vol 74 no 14 pp 1435ndash1446 2001
[14] G Toussaint T Basar and F Bullo ldquoTracking for nonlin-ear underactuated surface vessels with generalized forcesrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo00) pp 355ndash3360 2000
[15] W Dong and Y Guo ldquoNonlinear tracking control of underac-tuated surface vesselrdquo in Proceedings of the American ControlConference (ACC rsquo05) vol 6 pp 4351ndash4356 June 2005
[16] A Balluchi A Bicchi B Piccoli and P Soueres ldquoStability androbustness of optimal synthesis for route tracking by Dubinsrsquovehiclesrdquo in Proceedings of the 39th IEEE Confernce on Decisionand Control (CDC rsquo00) vol 1 pp 581ndash586 December 2000
[17] E Lefeber K Y Pettersen and H Nijmeijer ldquoTracking controlof an underactuated shiprdquo IEEETransactions on Control SystemsTechnology vol 11 no 1 pp 52ndash61 2003
[18] K D Do Z P Jiang and J Pan ldquoUniversal controllers forstabilization and tracking of underactuated shipsrdquo Systems ampControl Letters vol 47 no 4 pp 299ndash317 2002
[19] T I Fossen andM Blanke ldquoNonlinear output feedback controlof underwater vehicle propellers using feedback form estimatedaxial flow velocityrdquo IEEE Journal of Oceanic Engineering vol 25no 2 pp 241ndash255 2000
[20] J Liu and N Elia ldquoQuantized control with applications tomobile vehiclesrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control (CDC rsquo02) vol 3 pp 2391ndash2396December 2002
[21] H Ishii and T Basar ldquoRemote control of LTI systems overnetworks with state quantizationrdquo Systems amp Control Lettersvol 54 no 1 pp 15ndash31 2005
[22] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an RC hovercraftrdquo in Proceedings of the IEEEInternational Conference on Control Applications (ACC rsquo02) vol2 pp 1076ndash1081 September 2002
[23] H Seguchi and T Ohtsuka ldquoNonlinear receding horizoncontrol of an underactuated hovercraftrdquo International Journalof Robust and Nonlinear Control vol 13 no 3-4 pp 381ndash3982003
[24] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008
[25] D Chwa ldquoSliding-mode tracking control of nonholonomicwheeled mobile robots in polar coordinatesrdquo IEEE Transactionson Control Systems Technology vol 12 no 4 pp 637ndash644 2004
[26] A Megretski ldquoLocal controllabilityrdquo Dynamics of NonlinearSystems 2003 httpocwmiteducourseselectrical-engineer-ing-and-computer-science6-243j-dynamics-of-nonlinear-sys-tems-fall-2003lecture-noteslec12 6243 2003pdf
[27] L Cremean W B Dunbar D van Gogh et al ldquoThe Caltechmulti-vehicle wireless testbedrdquo in Proceedings of the 41st IEEE
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
Conference on Decision and Control (CDC rsquo02) vol 1 pp 86ndash88 December 2002
[28] M Fliess J Levine P Martin and P Rouchon ldquoFlatnessand defect of non-linear systems introductory theory andexamplesrdquo International Journal of Control vol 61 no 6 pp1327ndash1361 1995
[29] H K Khalil Nonlinear Systems Prentice Hall New York NYUSA 3rd edition 2002
[30] D Chaos Control no lineal de vehculos subactuados marinosno-holonmicos [PhD thesis] Department of Informatica yAutomatica of UNED University Madrid Spain 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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