Research Article Attitude Stabilization Control of a Quadrotor UAV …downloads.hindawi.com/journals/mpe/2014/749803.pdf · 2019-07-31 · time and remain therea er. e sliding mode

Research ArticleAttitude Stabilization Control of a Quadrotor UAV byUsing Backstepping Approach

Xing Huo1 Mingyi Huo2 and Hamid Reza Karimi3

1 College of Engineering Bohai University Jinzhou 121000 China2Department of Control Science and Engineering Harbin Institute of Technology Harbin 150001 China3Department of Engineering Faculty of Engineering and Science The University of Agder N-4898 Grimstad Norway

Correspondence should be addressed to Xing Huo hmyi888163com

Received 29 December 2013 Accepted 13 January 2014 Published 19 February 2014

Academic Editor Xudong Zhao

Copyright copy 2014 Xing Huo et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The modeling and attitude stabilization control problems of a four-rotor vertical takeoff and landing unmanned air vehicle (UAV)known as the quadrotor are investigated The quadrotorrsquos attitude is represented by the unit quaternion rather than Euler angles toavoid singularity problem Taking dynamical behavior of motors into consideration and ignoring aerodynamic effect a nonlinearcontroller is developed to stabilize the attitude The control design is accomplished by using backstepping control technique Theproposed control law is based on the compensation for the Coriolis and gyroscope torques Applying Lyapunov stability analysisproves that the closed-loop attitude system is asymptotic stable Moreover the controller can guarantee that all the states of thesystem are uniformly ultimately bounded in the presence of external disturbance torque The effectiveness of the proposed controlapproach is analytically authenticated and also validated via simulation study

1 Introduction

In the past decade the small UAV market has grown rapidlySmall UAVs are applied in various areas such as surveil-lance reconnaissance and aerial photography In particularquadrotor helicopters are an emerging rotorcraft concept forUAV platforms The particular interest of the research com-munity in the quadrotor design can be linked to two mainadvantages over comparable vertical takeoff and landing(VTOL) UAVs such as helicopters First quadrotors do notrequire complex mechanical control linkages for rotor actua-tion relying instead on fixed pitch rotors and using variationin motor speed for vehicle control This simplifies both thedesign andmaintenance of the vehicle Second the use of fourrotors ensures that individual rotors are smaller in diameterthan the equivalent main rotor on a helicopter relative to theairframe size Consequently the last decade has seen manysuccessfully developed platforms of micro-UAVs especiallythe quadrotor UAV such as Australia National UniversityrsquosX-4 flyer aircraft prototype [1] the Stanford UniversityrsquosSTARMAC quadrotor [2] the Swiss Federal Institutersquos OS4

aircraft prototype [3] and the University of PennsylvaniarsquosGrasp micro-UAV test bed [4]

To guide a quadrotor to accomplish the planned missiona control system needs to be designed It consists of positioncontrol for translational motion and attitude control forrotation motion In this study the design of controller forattitude control is investigatedOne distinct feature of attitudedynamics of a quadrotor is that its configuration manifold isnot linear it evolves on a nonlinear manifold referred to asthe special orthogonal group This yields important andunique properties that cannot be observed fromdynamic sys-tems evolving on a linear space As a result the attitude con-trol problem of a rigid body has been investigated by severalresearchers and a wide class of controllers based on classic ormodern control theory has been proposed For instancebased on the conventional proportional-integer-derivative(PID) approach an model-independent PD controller byusing quaternion-based feedback was proposed to stabilizethe quadrotor attitude system [5] In [6] the design of a PIDcontrol algorithm for the quadrotor attitude system was pre-sented The model of the vehicle was modified to simplify

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 749803 9 pageshttpdxdoiorg1011552014749803

2 Mathematical Problems in Engineering

the controller design Considering faults occurring on themounted motors a control strategy by using gain schedulingincorporating PID was developed in [7] It successfullyachieved attitude and position control even in the presence ofactuator faults In another recent work [8] the problemof sta-bilization and disturbance rejection of attitude subsystem of aquadrotor was addressed The controller was designed in theframework of PID control scheme similar control methodsalso can be found in [9ndash13]

In addition to the above PID-based attitude controldesign many modern control approaches have also beenapplied to quadrotor attitude controller design In [14] anattitude control lawwas proposed for triple tilting rotormini-UAV It was robust with respect to dynamical couplings andadverse torquesThe work in [15] investigated the applicationof model reference adaptive control for quadrotor even in thepresence of actuator uncertainties An adaptive law and con-trol laws based on adaptive technique were developed andapplied for quadrotor UAV in case of actuator partial loss ofeffectiveness faults [16] In [17] the problem of fault tolerantcontrol for quadrotor was addressed by using backsteppingcontrol approach In [18] the output-feedback control systemdesign for a quadrotor was discussed The position and attit-ude control was designed by using model reference controland nonlinear control allocation techniques The positioncontrol of vertical take-off and landing UAV without linearvelocity measurements was also investigated in [19] In [20]an adaptive nonlinear attitude stabilization control schemewas synthesized for a quadrotor The problem of parametricuncertainties in the quadrotor model was investigated andthe controller was designed based on model reference adap-tive control technique In [21] an attitude free position con-trol design was proposed for a quadrotor by using dynamicinversion In another related study [22] an adaptive lawwas designed to asymptotically follow an attitude commandwithout the knowledge of the inertia matrix The proposedcontrol was verified by using a quadrotorUAVmodel In [23]a quaternion-based feedback was developed for the attitudestabilization of quadrotor The control design took intoaccount a priori input bound and is based on nested satura-tion approach It forced the closed-loop trajectories to enterin some a priori fixed neighborhood of the origin in a finitetime and remain thereafter

The sliding mode control (SMC) is a powerful theory forcontrolling uncertain systems [24] The main advantages arethat the SMC system has great robustness with respect touncertain parameters and external disturbances Henceapplying SMC to design attitude control for quadrotor hasbeen intensively carried out in recent years In [25] integralSMC and reinforcement learning control were presented astwo design techniques for accommodating the nonlinear dis-turbances of an outdoor quadrotor In [26] SMC was used tocontrol a quadrotor UAV in the presence of disturbance andactuator fault The proposed approach was able to achievedisturbance rejection in the fault-free condition and also ableto recover some of performances when a fault occurred Toregulate the attitude angle of a quadrotor to the desired sig-nals an SMC-based attitude controller was proposed in [27]In that approach an adaptive estimatorwas incorporated and

the adaptive technique was used to estimate the upper boundof the virtual command The work in [28] explained thedevelopments of the use of SMC for a fully actuated subsys-tem of a quadrotor to obtain attitude control stability In[29] an SMC attitude controller was developed for a quadro-tor It allowed for a continuous control robust to externaldisturbance andmodel uncertainties to be computed withoutthe use of high control gain In another work [30] an aug-mented SMC-based fault-tolerant control was designed the-oretically implemented practically and tested experimentallyin a quadrotor The problems of propeller damage and actu-ator fault conditions for tracking control were discussedMoreover many other nonlinear control techniques basedcontroller design were also proposed for quadrotor as sug-gested in [31ndash33]

Based on the results available in the literature this workwill investigate the attitude control design of a quadrotorUAVThe attitude orientation of the quadrotor is representedby using unit quaternion Backstepping technique is adoptedto develop the controller It is shown that the controller canasymptotically stabilize the attitude system Explicitly takingexternal disturbance torque acting on the quadrotor into con-sideration the controller is able to guarantee the uniformlyultimately bounded stability of the attitude system All thestates of the closed-loop attitude system are governed tobe uniformly ultimately bounded The remainder of thispaper is organized as follows In Section 2 mathematicalmodel of a quadrotorUAV attitude system and problem state-ment are summarized A backstepping-based attitude controlapproach is presented in Section 3 and also the stabilityof closed-loop system is provided In Section 4 simulationresults with the application of the designed control schemeto a quadrotor are presented Section 5 presents some con-cluding remarks and future work

2 Mathematical Model andControl Problem Statement

21 Attitude Dynamic Model of a Quadrotor The quadrotorUAV under consideration consists of a rigid cross frameequipped with four rotors as shown in Figure 1 Those fourrotors are divided into front (119872

2) back (119872

4) left (119872

1) and

right (1198723) motors Motors 119872

2and 119872

4rotate in counter-

clockwise direction while the other two in clockwise direc-tion All of themovements can be controlled by the changes ofeach rotor speed If a yawmotion is desired one has to reducethe thrust of one set of rotors and increase the thrust ofthe other set while maintaining the same total thrust to avoidan up-down motion Hence the yaw motion is then realizedin the direction of the induced reactive torque To accomplishroll motion it should decrease (increase) the speed of motor1198723and increase (decrease) the speed of motor 119872

1 On the

other hand pitch motion can be maneuvered by decreasing(increasing) the speed of motor 119872

1while increasing

(decreasing) the speed of motor1198724

LetF119894(119883119868 119884119868 119885119868) fixed with the Earth denote an inertial

frame and letF119887(119883119861 119884119861 119885119861) denote a frame rigidly attached

to the quadrotor body as shown in Figure 1 The vector

Mathematical Problems in Engineering 3

ZI

OIXI

YI

f1

f2f3

f4

M1

M2M3

M4

XB

ZB

YB

OB

Figure 1 The structure of quadrotor UAV and its coordinatereference frames

Θ = [120601 120579 120595]119879 is the orientation of body frame F

119887with

respect to F119894called Euler angles These angles are bounded

as follows

minus120587

2lt 120601 lt

120587

2 minus

120587

2lt 120579 lt

120587

2 minus

120587

2lt 120595 lt

120587

2 (1)

When using Euler angles to represent quadrotor attitudedirection cosinematrix will be used to describe the kinematicmotion of attitudeHowever one of the drawbackswith appli-cation of the direction cosinematrix is the inherent geometricsingularity It is known that the four-parameter descriptionof the orientation called the quaternion representation canbe applied to avoid that drawback Consequently the unitquaternion will be used in this work for attitude represen-tation of quadrotor UAV Ignoring aerodynamic effect themathematical model of quadrotor attitude system can bedescribed as follows [5]

1199020= minus

1

2q119879120596 (2)

q =1

2[1199020E3+ S (q)]120596 (3)

I = minusS (120596) I120596 minus G119886+ u (4)

119868119898119894119894= 120591119894minus 119891119894 119894 = 1 2 3 4 (5)

whereQ = [1199020q119879]119879

isin R4 with q isin R3 is the unit quaterniondenoting the attitude of quadrotor with respect to the inertialframeF

119894 It is subject to the equation 119902

2

0+ q119879q = 1 120596 isin R3

denotes the angular velocity of the quadrotor with respect toF119894and expressed in F

119887 I isin R3times3 is the positive-definite

symmetric inertia matrix of the UAV 119868119898

isin R and 120603119894isin R are

the moment of inertia and the speed of the rotor119872119894 respec-

tively G119886

isin R3 denotes the gyroscope torques due to thecombination of the rotation of the quadrotor and the fourmotors it is given by

G119886=

4

sum

119894=1

119868mi (120596 times e119868) (minus1)

119894+1120603119894

(6)

and e119868= [0 0 1]

119879 denotes the unit vector inF119894 The matrix

S(x) is a skew-symmetric matrix such that S(x)y = x times y forany vector for x y isin R3 where ldquotimesrdquo denotes the vector cross-product E

3denotes the 3 times 3 identity matrix

In (4) u = [1199061 1199062

1199063]119879

isin R3 is the total torque actingon the UAV and it is generated by four motors

Consider the following

u =

[[[[

[

119889119887 (1206032

3minus 1206032

1)

119889119887 (1206032

2minus 1206032

4)

120581 (1206032

2+ 1206032

4minus 1206032

1minus 1206032

3)

]]]]

]

(7)

where 119889 isin R is the distance from the rotors to the center ofmass of the quadrotor and 119887 isin R and 120581 isin R are two positiveparameters depending on the density of air the radius of thepropeller the number of blades and the geometry lift anddrag coefficients of the blades [34]The reaction torque119891

119894isin R

in (5) generated in free air by themotor119872119894due tomotor drag

is given by

119891119894= 1205811206032

119894 (8)

Moreover the four-control input of the attitude system is 120591119894

119894 = 1 2 3 4 which denotes the torques produced by therotors

22 Problem Statement Given any initial attitude Q andangular velocity120596 the control objective to be achieved can bestated as consider the quadrotor UAV attitude dynamicsdescribed by (2) and (5) design an control law 120591

119894for themotor

119872119894 119894 = 1 2 3 4 to stabilize the resulting closed-loop attitude

system that is

lim119905rarrinfin

Q (119905) = [1 0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(9)

3 Attitude Stabilization Control Design

Define new state variables as x1= [1 minus |119902

0| q119879]

119879

x2= 120596 and

x3= [1206032

11206032

21206032

3]119879 Then the attitude system equations (2)ndash

(5) can be rewritten as follows

x1= [minus sgn (119902

0) 1199020q119879]119879 (10)

Ix2= minusS (x

2) Ix2minus G119886+ u (11)

I119898x3= X (120591 minus f) (12)

where I119898

= diag(1198681198981

1198681198982

1198681198983

) X = diag(21206031 21206032 21206033) 120591 =

[1205911 1205912

1205913]119879 f = [1198911 119891

21198913]119879 and sgn(119910) is defined as the

nonzero sigum function as follows

sgn (119910) = minus1 119910 lt 0

1 119910 ge 0(13)

For the transformed system equations (10)ndash(12) back-stepping control technique will be applied in this section to


rrdReferencepath

trajectory

Positioncontrol

The desiredattitude

Desired thrust T

Attitudecontrol

Motordynamics

The attitude and angular velocity

dynamicsRigid body

Feedback of the actual position

Figure 2 The control structure of the quadrotor UAV

design attitude controller and the following change of coor-dinates will be firstly introduced

z1= x1 (14)

z2= x2minus 1205721 (15)

z3= x3minus 1205722 (16)

where 1205721isin R3 and 120572

2isin R119873 are virtual control inputs to be

designed laterBased on the preceding coordinates changes the follow-

ing procedures can be followed to design a controller tostabilize the attitude of the considered quadrotor UAV

Step 1 Westartwith (14) by considering x2as the control vari-

able It is obtained from (2) (3) and (10) that

z1=

1

2[sgn(119902

0)q119879120596[119902

0E3+ S (q)]119879120596119879]

119879

=1

2[sgn (119902

0) qT

1199020E3+ S (q)

]120596

= P (Q)120596

(17)

where P(Q) = (12) [sgn(1199020)q1198791199020E3+S(q)

]The task in this step is to design a virtual control law 120572

1

to guarantee lim119905rarrinfin

z1(119905) = 0 Choose a Lyapunov candidate

function as 1198811= (12)z119879

1z1 and design the virtual control 120572

1

as 1205721= minus1198971P119879(Q)z

1 where 119897

1isin R is a positive scalar Then

applying (15) and (17) yields

1= z1198791P (Q)120596

= z1198791P (Q) (z

2minus 1198971P119879 (Q) 119911

1)

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2

(18)

Hence if z2

= 0 then 1

= minus1198971z1198791P(Q)P119879(Q)z

1=

minus1198971P119879(Q)z

12

lt 0 for all P119879(Q)z1

= 0 By using Lyapunovstability theory [35] it can prove that

lim119905rarrinfin

10038171003817100381710038171003817P119879 (Q) z

1

10038171003817100381710038171003817= 0 (19)

On the other hand it can be obtained from the definitionof P(Q) that

P119879 (Q) z1=

1

2[sgn (119902

0) q119879

1199020E3+ S (q)

]

119879

[1 minus10038161003816100381610038161199020

1003816100381610038161003816 q119879]119879

=1

2sgn (119902

0) q

(20)

Hence it leaves (19)-(20) as lim119905rarrinfin

sgn(1199020)q = 0 Then

using (13) leads to lim119905rarrinfin

q = 0

The quadrotor UAV actually is an underactuated systembecause it has six degrees of freedom while it has only fourinputs The collective input (or throttle input) is the sum ofthe thrusts of each motor Hence the control structure of anrigid quadrotor is illustrated in Figure 2 Position control andattitude control are included Assumed that the desired totalthrust supplied by the position control is 119879 which is gener-ated by the four rotors and given by

119879 = 119887

4

sum

119894=1

1206032

119894 (21)

Accordingly the desired speed of the four motors can beobtained from (7) and (21) that is 120578 = Nx

3with 120578 =

[1199061 1199062

1199063

119879]119879 and

N =

[[[

[

0 119889119887 0 minus119889119887

119889119887 0 minus119889119887 0

120581 minus120581 120581 minus120581

119887 119887 119887 119887

]]]

]

(22)

where N is nonsingular as long as 119889119887120581 = 0

Step 2 From (22) one has u = ΞNx3 where

Ξ =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

(23)

We now differentiate the second error z2 using (15) to give

Iz2= Ix2minus I1= minusS (x

2) Ix2minus G119886+ ΞNx

3minus J1 (24)

Choose another Lyapunov candidate function 1198812

= 1198811+

(12)z1198792Iz2 and design the virtual control law 120572

2as

1205722= (ΞN)

dagger[S (x2) Ix2minus P119879 (Q) z

1minus 1198972z2+ G119886+ I1]

(25)


where 1198972isin R is a positive constant and (sdot)

dagger denotes the pseudoinverse of a full-row rank matrix

Differentiating both sides of 1198812and inserting (24) and

(25) yield

2= 1+ z1198792Iz2

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2

+ z1198792[minusS (x

2) Ix2minus G119886+ ΞN (z

3+ 1205722) minus I

1]

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3

(26)

Again if z3= 0 one has

2= minus1198971P119879(Q)z

12

minus 1198972z22 and

thus both z1and z2will converge to zero asymptotically

According to the analysis in Steps 1 and 2 if z3can be

driven to zero by designing an appropriate controller then z1

and z2will also be governed to zero and thus the quadrotor

attitude will be stabilized In the following theorem we sum-marize our control solution to drive z

3to zero by incorporat-

ing backstepping-based control action

Theorem 1 Consider the quadrotor attitude system describedby (2)ndash(5) design a nonlinear backstepping controller as fol-lows

120591 = Xminus1 [minus(ΞN)119879z2minus 1198973z3+ I1198982] + f (27)

where 1198973isin R is an positive control gain Then the closed-loop

attitude system is asymptotically stable that is lim119905rarrinfin

q(119905) =[0 0 0]

119879 lim119905rarrinfin120596(119905) = [0 0 0]

119879

Proof It can be obtained from (12) and (16) that

I119898z3= I119898x3minus I1198982= X (120591 minus f) minus I

1198982 (28)

Consider another candidate Lyapunov function1198813as follows

1198813= 1198812+

1

2z1198793I119898z3 (29)

Differentiating (29) and inserting (26) and (28) result in

3= 2+ z1198793I119898z3

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2

+ z1198792ΞNz3+ z1198793[X (120591 minus f) minus I

1198982]

(30)

With the developed controller equation (27) (30) can besimplified into the following form

3= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2minus 1198973z1198793z3le 0 (31)

Investigating of 1198813in (29) shows clearly that 119881

3gt 0 for all z

119894

119894 = 1 2 3 and also that 1198813rarr infin when z

119894rarr infin Therefore

in accordancewith LaSalle-Yoshizawarsquos theorem [35] 3from

(31) fulfills

3le minus119882(z

1 z2 z3) le 0 (32)

where 119882(z1 z2 z3) is a continuous function Then all solu-

tion z119894(119905) are uniformly globally bounded and

lim119905rarrinfin

119882(z1 z2 z3) = 0 (33)

Hence the controller is uniformly globally asymptoticallystable since additionally119882(z

1 z2 z3) gt 0 Moreover it can be

concluded from (31)ndash(33) that

lim119905rarrinfin

z119894(119905) = 0 (34)

To this end it can be obtained from (14)ndash(16) the defini-tion of 120572

1and 120572

2 and (34) that

lim119905rarrinfin

q (119905) = [0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(35)

Summarizing the above analysis it can come to the conclu-sion that the closed-loop attitude system of the consideredquadrotor UAV is asymptotically stable Thereby the proof iscompleted here

The result obtained fromTheorem 1 can only be guaran-teed when the quadrotor UAV is free of external disturbanceHowever this case will never be true in practical flyingespecially when the quadrotor flies outdoor Assume that theexternal disturbance torque acting on the quadrotor is u

119889isin

R3 then the dynamics of the quadrotor equation (4) or (11)will be changed as follows

Ix2= minusS (x

2) Ix2minus G119886+ u + u

119889 (36)

Although the disturbance torque u119889is inevitable it is

always bounded in practice Hence it is reasonable to assumethat there exists a positive scalar 119889max isin R such that u

119889 le

119889max

Theorem 2 Consider the quadrotor attitude system describedby (2)ndash(5) in the presence of external disturbance torque u

119889

with application of the controller equation (27) suppose thatthe control gain 119897

2is chosen to satisfy

1198972minus 120576 gt 0 (37)

where 120576 isin R is a positive scalar specified by the designer Thenthe closed-loop attitude system is uniformly ultimately boundedstable

Proof Using the almost the same analysis as in the proof ofTheorem 1 one has

2= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3+ z1198792u119889

(38)

Then it leaves 3in (30) as

3= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2minus 1198973z1198793z3+ z1198792u119889 (39)

Applying the Youngrsquos inequality the following inequality canbe established

z1198792u119889le 120576

1003817100381710038171003817z21003817100381710038171003817

2

+1

4120576

1003817100381710038171003817u1198891003817100381710038171003817

2

le 1205761003817100381710038171003817z2

1003817100381710038171003817

2

+1

41205761198892

max (40)


Table 1 Quadrotor UAV model parameters

Parameters Description Value Units119892 Gravity 981 ms2

119898 Mass 0468 kg119889 Distance 0225 m119868119898119894 119894 = 1 2 3 Rotor inertia 34 times 10

minus5 kgm2

119868119909

Roll inertia 49 times 10minus3 kgm2

119868119910

Pitch inertia 49 times 10minus3 kgm2

119868119911

Yaw inertia 88 times 10minus3 kgm2

119877119886

Motor resistance 067 Ω

119896119898

Motor constant 43 times 10minus3 NmA

119896119892

Gear ratio 56119887 Proportionality constant 29 times 10

minus5

120581 Proportionality constant 11 times 10minus6

It is thus obtained from (40) that 3is bounded by

3= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2minus 1198973z1198793z3+ z1198792u119889

le minus1198971z1198791P (Q)P119879 (Q) z

1minus (1198972minus 120576) z1198792z2minus 1198973z1198793z3+

1

41205761198892

max

(41)

With the choice of control gains in (37) it leads to

3le minus2119897119881

3+

1198892

max4120576

(42)

where 119897 = min1198971120582min(P(Q)P119879(Q)) (119897

2minus 120576)120582max(I)

1198973120582max(I119898) 120582min(sdot) and 120582max(sdot) denote the minimum and

themaximum eigenvalue ofmatrix respectively UsingTheo-rem 418 (page 172) in [35] it can be concluded from (42) that1198813is uniformly ultimately bounded togetherwith the states z

119894

119894 = 1 2 3 More precisely there exists a finite-time 1199050isin R+

such that z119894(119905) le 120576

lowast

1 z119886(119905) le 120576

lowast

1for any 120576

lowast

1gt 119889max2

radic119897120576

and 119905 ge 1199050 119894 = 1 2 3 In other words the closed-loop attitude

system is uniformly ultimately bounded stable Hence theproof of Theorem 2 is completed

4 Numerical Example

In this section the properties of the proposed attitude stabil-ization approach is evaluated and simulated numerically bythe quadrotor model presented in [5] The physical param-eters of this model are listed in Table 1 At time 119905 = 0 theinitial attitude of the quadrotor isΘ(0) = [35 minus45 18]

119879degcorresponding to the unit quaternion values q(0) =

[03315 minus03170 00242]119879 the initial angular velocity is of

120596(0) = [0 0 0]119879 radsecThe control gains for the controller

equation (27) are chosen as 1198971= 002 119897

2= 0075 and 119897

3=

0008 Moreover attitude and angular velocity sensor noises(1205902119901) 00001(1120590) aremodeled as zero-meanGaussian random

variables with variance 1205902119901

In the ideal case that the quadrotor is free of external dis-turbance torques with application of the controller equation

02

04

Time (s)

The q

uate

rnio

nq

q1q2q3

minus02

minus040

0

2 4 6

Figure 3 The response of the unit quaternion q in the absence ofexternal disturbances

0

20

40

Time (s)0 2 4 6

minus20

minus40

120601

120579120595

The E

uler

angl

es (d

eg)

Figure 4 The response of Euler attitude angles in the absence ofexternal disturbances

(27) it leads to the control performance shown in Figures 3ndash5It is observed from the response of the unit quaternion andthe Euler attitude angles shown in Figures 3 and 4 respec-tively that the attitude is successfully accomplished while theresulted angular velocity is illustrated in Figure 5

For a quadrotor flying outdoor it is always under theeffect of external disturbances such as induced by windTherefore simulation is further carried out in the presence ofexternal disturbances Here disturbances are introduced on


0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596

Figure 5 The angular velocity 120596 in the absence of externaldisturbances

02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3

Figure 6 The response of the unit quaternion q in the presence ofexternal disturbances

the pitch roll and yaw at the 3th 65th and the 8th secondsrespectivelyThe attitude control performance obtained fromthe developed controller equation (27) is shown in Figures 6ndash8 From Figures 6-7 it is known that the disturbance torquehas little effect on the performance of the attitude and that isbecause the controller equation (27) is able to guarantee theuniformly ultimately bounded stability of the closed-loopsystem in the presence of disturbance It is seen in Figure 8that although there exists some overshoot for the angular

Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12

Figure 7 The response of Euler attitude angles in the presence ofexternal disturbances

Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596

Figure 8 The angular velocity 120596 in the presence of externaldisturbances

velocity when disturbance is introduced to the quadrotor theangular velocity can be stabilized soon

5 Conclusions

In this work the design of a quadrotor UAV attitude stabi-lization controller using quaternion feedback and integratorbackstepping was presented With appropriate choice ofintegrator backstepping variables both equilibrium points in


the closed-loop system are proved to be asymptotically stableFurthermore taking external disturbance torque into consid-eration the attitude and the angular velocity of the quadro-tor were governed to be uniformly ultimately bounded bythe proposed controller Simulations of a quadrotor UAVwere also presented to illustrate the performance of the con-troller and that the attitude of the UAV was regulated to theclosest equilibrium point Future work will emphasize theposition or path following control of the quadrotor even inthe presence of external disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This present work was supported partially by National Natu-ral Science Foundation of China (Project no 61304149) andNatural Science Foundation of Liaoning China (Project no2013020044) The authors highly appreciate the above finan-cial supports

References

[1] C Shen Z J Rong andM Dunlop ldquoDetermining the full mag-netic field gradient from two spacecraft measurements underspecial constraintsrdquo Journal of Geophysical Research A vol 117no 10 2012

[2] W C Tu S R Elkington X L Li W L Liu and J BonnellldquoQuantifying radial diffusion coefficients of radiation belt elec-trons based on globalMHDsimulation and spacecraftmeasure-mentsrdquo Journal of Geophysical Research A vol 117 no 10 2012

[3] Y D Song and W C Cai ldquoNew intermediate quaternion basedcontrol of spacecraft part Imdashalmost global attitude trackingrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 10 pp 7307ndash7319 2012

[4] GM Li and LD Liu ldquoCoordinatedmultiple spacecraft attitudecontrol with communication time delays and uncertaintiesrdquoChinese Journal of Aeronautics vol 25 no 5 pp 698ndash708 2012

[5] F Pantellini S Belheouane N Meyer-Vernet and A ZaslavskyldquoNano dust impacts on spacecraft and boom antenna chargingrdquoAstrophysics and Space Science vol 341 no 2 pp 309ndash314 2012

[6] X Y Gao K L Teo andG RDuan ldquoRobust119867infincontrol of spa-

cecraft rendezvous on elliptical orbitrdquo Journal of the FranklinInstitute vol 349 no 8 pp 2515ndash2529 2012

[7] H R Williams R M Ambrosi N P Bannister P Samara-Ratna and J Sykes ldquoA conceptual spacecraft radioisotope ther-moelectric and heating unit (RTHU)rdquo International Journal ofEnergy Research vol 36 no 12 pp 1192ndash1200 2012

[8] S Laborde and A Calvi ldquoSpacecraft base-sine vibration testdata uncertainties investigation based on stochastic scatterapproachrdquoMechanical Systems and Signal Processing vol 32 pp69ndash78 2012

[9] S Yin S X Ding A H A Sari andH Hao ldquoData-drivenmon-itoring for stochastic systems and its application on batch pro-cessrdquo International Journal of Systems Science vol 44 no 7 pp1366ndash1376 2013

[10] S Yin S X Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[11] S Yin H Luo and S-X Ding ldquoReal-time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2013

[12] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronics2013

[13] S Yin X Yang and H R Karimi ldquoData-driven adaptive obse-rver for fault diagnosisrdquoMathematical Problems in Engineeringvol 2012 Article ID 832836 21 pages 2012

[14] E V Morozov and A V Lopatin ldquoDesign and analysis of thecomposite lattice frame of a spacecraft solar arrayrdquo CompositeStructures vol 94 no 10 pp 3106ndash3114 2012

[15] F Nemec O Santolik M Parrot and J S Pickett ldquoMagneto-spheric line radiation event observed simultaneously on boardCluster 1 Cluster 2 and DEMETER spacecraftrdquo GeophysicalResearch Letters vol 39 no 18 2012

[16] L Palin C Jacquey J A Sauvaud et al ldquoStatistical analyzis ofdipolarizations using spacecraft closely separated along Z in thenear-Earth magnetotailrdquo Journal of Geophysical Research A vol117 no 9 2012

[17] YMiyake H Usui H Kojima andHNakashima ldquoPlasma par-ticle simulations on stray photoelectron current flows arounda spacecraftrdquo Journal of Geophysical Research A vol 117 no 92012

[18] M J Aschwanden J P Wuelser N V Nitta J R Lemen M LDeRosa andAMalanushenko ldquoFirst three-dimensional recon-structions of coronal loopswith the STEREO119860+119861 spacecraft IVMagnetic modeling with twisted force-free fieldsrdquo The Astro-physical Journal vol 756 no 2 article 124 2012

[19] R E Denton B U O Sonnerup J Birn et al ldquoTest of Shi et almethod to infer the magnetic reconnection geometry fromspacecraft data MHD simulation with guide field and antipar-allel kinetic simulationrdquo Journal of Geophysical Research A vol117 no 9 2012

[20] K D Bilimoria and E R Mueller ldquoHandling qualities of acapsule spacecraft during atmospheric entryrdquo Journal of Space-craft and Rockets vol 49 no 5 pp 935ndash943 2012

[21] H Lu Y-H Li and C-L Zhu ldquoRobust synthesized control ofelectromechanical actuator for thrust vector system in space-craftrdquo Computers amp Mathematics with Applications vol 64 no5 pp 699ndash708 2012

[22] S-J Dong Y-Z Li J Wang and J Wang ldquoFuzzy incrementalcontrol algorithm of loop heat pipe cooling system for space-craft applicationsrdquoComputers ampMathematics with Applicationsvol 64 no 5 pp 877ndash886 2012

[23] A T Basilevsky ldquoAnalysis of suspicious objects in TVpanoramataken byVenera-9 spacecraftrdquo Solar SystemResearch vol 46 no5 pp 374ndash378 2012

[24] C Edwards and S Spurgeon Sliding Mode Control Theory andApplications Taylor amp Francis Boca Raton Fla USA 1998

[25] K B McCoy I Derecho TWong et al ldquoInsights into the extre-motolerance of Acinetobacter radioresistens 50v1 a gram-neg-ative bacterium isolated from the Mars Odyssey spacecraftrdquoAstrobiology vol 12 no 9 pp 854ndash862 2012


[26] K Yong S Jo and H Bang ldquoA modified rodrigues parameter-based nonlinear observer design for spacecraft gyroscope para-meters estimationrdquo Transactions of the Japan Society For Aero-nautical and Space Sciences vol 55 no 5 pp 313ndash320 2012

[27] J X Fan and D Zhou ldquoNonlinear attitude control of flexiblespacecraft with scissored pairs of control moment gyrosrdquo Jour-nal of Dynamic Systems Measurement and Control vol 134 no5 Article ID 054502 5 pages 2012

[28] Y C Xu andQ Chen ldquoMinimization ofmass for heat exchangernetworks in spacecrafts based on the entransy dissipation the-oryrdquo International Journal of Heat andMass Transfer vol 55 no19-20 pp 5148ndash5156 2012

[29] H-T Cui and X-J Cheng ldquoAnti-unwinding attitude maneuvercontrol of spacecraft considering bounded disturbance andinput saturationrdquo Science China Technological Sciences vol 55no 9 pp 2518ndash2529 2012

[30] A-M Zou K D Kumar and Z-G Hou ldquoAttitude coordinationcontrol for a group of spacecraft without velocity measure-mentsrdquo IEEE Transactions on Control Systems Technology vol20 no 5 pp 1160ndash1174 2012

[31] R P Praveen M H Ravichandran V T S Achari V P J RajG Madhu and G R Bindu ldquoA novel slotless Halbach-arraypermanent-magnet brushless dc motor for spacecraft applica-tionsrdquo IEEE Transactions on Industrial Electronics vol 59 no 9pp 3553ndash3560 2012

[32] K G Klein G G Howes J M TenBarge S D Bale C H KChen and C S Salem ldquoUsing synthetic spacecraft data to inter-pret compressible fluctuations in solar wind turbulencerdquo TheAstrophysical Journal vol 755 no 2 article 159 2012

[33] G M Chen J Y Xu W B Wang J H Lei and A G BurnsldquoA comparison of the effects of CIR- and CME-induced geo-magnetic activity on thermospheric densities and spacecraftorbits case studiesrdquo Journal of Geophysical Research A vol 117no 8 2012

[34] Z Y Tan L Dong and F L Tang ldquoMonte Carlo calculations ofcharacteristic quantities of low-energy electron irradiation tospacecraft dielectricsrdquo Nuclear Instruments amp Methods in Phys-ics Research Section B vol 285 pp 86ndash93 2012

[35] H K Khalil Nonlinear Systems Prentice Hall Upper SaddleRiver NJ USA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the controller design Considering faults occurring on themounted motors a control strategy by using gain schedulingincorporating PID was developed in [7] It successfullyachieved attitude and position control even in the presence ofactuator faults In another recent work [8] the problemof sta-bilization and disturbance rejection of attitude subsystem of aquadrotor was addressed The controller was designed in theframework of PID control scheme similar control methodsalso can be found in [9ndash13]

In addition to the above PID-based attitude controldesign many modern control approaches have also beenapplied to quadrotor attitude controller design In [14] anattitude control lawwas proposed for triple tilting rotormini-UAV It was robust with respect to dynamical couplings andadverse torquesThe work in [15] investigated the applicationof model reference adaptive control for quadrotor even in thepresence of actuator uncertainties An adaptive law and con-trol laws based on adaptive technique were developed andapplied for quadrotor UAV in case of actuator partial loss ofeffectiveness faults [16] In [17] the problem of fault tolerantcontrol for quadrotor was addressed by using backsteppingcontrol approach In [18] the output-feedback control systemdesign for a quadrotor was discussed The position and attit-ude control was designed by using model reference controland nonlinear control allocation techniques The positioncontrol of vertical take-off and landing UAV without linearvelocity measurements was also investigated in [19] In [20]an adaptive nonlinear attitude stabilization control schemewas synthesized for a quadrotor The problem of parametricuncertainties in the quadrotor model was investigated andthe controller was designed based on model reference adap-tive control technique In [21] an attitude free position con-trol design was proposed for a quadrotor by using dynamicinversion In another related study [22] an adaptive lawwas designed to asymptotically follow an attitude commandwithout the knowledge of the inertia matrix The proposedcontrol was verified by using a quadrotorUAVmodel In [23]a quaternion-based feedback was developed for the attitudestabilization of quadrotor The control design took intoaccount a priori input bound and is based on nested satura-tion approach It forced the closed-loop trajectories to enterin some a priori fixed neighborhood of the origin in a finitetime and remain thereafter

The sliding mode control (SMC) is a powerful theory forcontrolling uncertain systems [24] The main advantages arethat the SMC system has great robustness with respect touncertain parameters and external disturbances Henceapplying SMC to design attitude control for quadrotor hasbeen intensively carried out in recent years In [25] integralSMC and reinforcement learning control were presented astwo design techniques for accommodating the nonlinear dis-turbances of an outdoor quadrotor In [26] SMC was used tocontrol a quadrotor UAV in the presence of disturbance andactuator fault The proposed approach was able to achievedisturbance rejection in the fault-free condition and also ableto recover some of performances when a fault occurred Toregulate the attitude angle of a quadrotor to the desired sig-nals an SMC-based attitude controller was proposed in [27]In that approach an adaptive estimatorwas incorporated and

the adaptive technique was used to estimate the upper boundof the virtual command The work in [28] explained thedevelopments of the use of SMC for a fully actuated subsys-tem of a quadrotor to obtain attitude control stability In[29] an SMC attitude controller was developed for a quadro-tor It allowed for a continuous control robust to externaldisturbance andmodel uncertainties to be computed withoutthe use of high control gain In another work [30] an aug-mented SMC-based fault-tolerant control was designed the-oretically implemented practically and tested experimentallyin a quadrotor The problems of propeller damage and actu-ator fault conditions for tracking control were discussedMoreover many other nonlinear control techniques basedcontroller design were also proposed for quadrotor as sug-gested in [31ndash33]

Based on the results available in the literature this workwill investigate the attitude control design of a quadrotorUAVThe attitude orientation of the quadrotor is representedby using unit quaternion Backstepping technique is adoptedto develop the controller It is shown that the controller canasymptotically stabilize the attitude system Explicitly takingexternal disturbance torque acting on the quadrotor into con-sideration the controller is able to guarantee the uniformlyultimately bounded stability of the attitude system All thestates of the closed-loop attitude system are governed tobe uniformly ultimately bounded The remainder of thispaper is organized as follows In Section 2 mathematicalmodel of a quadrotorUAV attitude system and problem state-ment are summarized A backstepping-based attitude controlapproach is presented in Section 3 and also the stabilityof closed-loop system is provided In Section 4 simulationresults with the application of the designed control schemeto a quadrotor are presented Section 5 presents some con-cluding remarks and future work

2 Mathematical Model andControl Problem Statement

21 Attitude Dynamic Model of a Quadrotor The quadrotorUAV under consideration consists of a rigid cross frameequipped with four rotors as shown in Figure 1 Those fourrotors are divided into front (119872

2) back (119872

4) left (119872

1) and

right (1198723) motors Motors 119872

2and 119872

4rotate in counter-

clockwise direction while the other two in clockwise direc-tion All of themovements can be controlled by the changes ofeach rotor speed If a yawmotion is desired one has to reducethe thrust of one set of rotors and increase the thrust ofthe other set while maintaining the same total thrust to avoidan up-down motion Hence the yaw motion is then realizedin the direction of the induced reactive torque To accomplishroll motion it should decrease (increase) the speed of motor1198723and increase (decrease) the speed of motor 119872

1 On the

other hand pitch motion can be maneuvered by decreasing(increasing) the speed of motor 119872

1while increasing

(decreasing) the speed of motor1198724

LetF119894(119883119868 119884119868 119885119868) fixed with the Earth denote an inertial

frame and letF119887(119883119861 119884119861 119885119861) denote a frame rigidly attached

to the quadrotor body as shown in Figure 1 The vector


ZI

OIXI

YI

f1

f2f3

f4

M1

M2M3

M4

XB

ZB

YB

OB



119887with


as follows

minus120587

2lt 120601 lt

120587

2 minus

120587

2lt 120579 lt

120587

2 minus

120587

2lt 120595 lt

120587

2 (1)


1199020= minus

1

2q119879120596 (2)

q =1

2[1199020E3+ S (q)]120596 (3)


119868119898119894119894= 120591119894minus 119891119894 119894 = 1 2 3 4 (5)

whereQ = [1199020q119879]119879



2

0+ q119879q = 1 120596 isin R3






tively G119886


G119886=

4

sum

119894=1

119868mi (120596 times e119868) (minus1)

119894+1120603119894

(6)

and e119868= [0 0 1]




In (4) u = [1199061 1199062

1199063]119879



u =

[[[[

[

119889119887 (1206032

3minus 1206032

1)

119889119887 (1206032

2minus 1206032

4)

120581 (1206032

2+ 1206032

4minus 1206032

1minus 1206032

3)

]]]]

]

(7)


119894isin R


is given by

119891119894= 1205811206032

119894 (8)




119894for themotor


system that is

lim119905rarrinfin

Q (119905) = [1 0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(9)



0| q119879]

119879

x2= 120596 and

x3= [1206032

11206032

21206032




0) 1199020q119879]119879 (10)

Ix2= minusS (x

2) Ix2minus G119886+ u (11)

I119898x3= X (120591 minus f) (12)

where I119898

= diag(1198681198981

1198681198982

1198681198983

) X = diag(21206031 21206032 21206033) 120591 =

[1205911 1205912

1205913]119879 f = [1198911 119891



sgn (119910) = minus1 119910 lt 0

1 119910 ge 0(13)



rrdReferencepath

trajectory

Positioncontrol

The desiredattitude

Desired thrust T

Attitudecontrol

Motordynamics


dynamicsRigid body




z1= x1 (14)

z2= x2minus 1205721 (15)

z3= x3minus 1205722 (16)







z1=

1

2[sgn(119902

0)q119879120596[119902

0E3+ S (q)]119879120596119879]

119879

=1

2[sgn (119902

0) qT

1199020E3+ S (q)

]120596

= P (Q)120596

(17)

where P(Q) = (12) [sgn(1199020)q1198791199020E3+S(q)


1



function as 1198811= (12)z119879


1

as 1205721= minus1198971P119879(Q)z

1 where 119897



1= z1198791P (Q)120596

= z1198791P (Q) (z

2minus 1198971P119879 (Q) 119911

1)

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2

(18)

Hence if z2

= 0 then 1

= minus1198971z1198791P(Q)P119879(Q)z

1=

minus1198971P119879(Q)z

12



lim119905rarrinfin

10038171003817100381710038171003817P119879 (Q) z

1

10038171003817100381710038171003817= 0 (19)


P119879 (Q) z1=

1

2[sgn (119902

0) q119879

1199020E3+ S (q)

]

119879

[1 minus10038161003816100381610038161199020

1003816100381610038161003816 q119879]119879

=1

2sgn (119902

0) q

(20)


sgn(1199020)q = 0 Then


q = 0


119879 = 119887

4

sum

119894=1

1206032

119894 (21)


3with 120578 =

[1199061 1199062

1199063

119879]119879 and

N =

[[[

[

0 119889119887 0 minus119889119887

119889119887 0 minus119889119887 0

120581 minus120581 120581 minus120581

119887 119887 119887 119887

]]]

]

(22)



Ξ =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

(23)




3minus J1 (24)


= 1198811+


2as

1205722= (ΞN)


1minus 1198972z2+ G119886+ I1]

(25)





(25) yield

2= 1+ z1198792Iz2

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2



3+ 1205722) minus I

1]

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3

(26)


2= minus1198971P119879(Q)z

12













q(119905) =[0 0 0]

119879 lim119905rarrinfin120596(119905) = [0 0 0]

119879



1198982 (28)


1198813= 1198812+

1

2z1198793I119898z3 (29)


3= 2+ z1198793I119898z3

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2


1198982]

(30)


3= minus1198971z1198791P (Q)P119879 (Q) z



3gt 0 for all z

119894




(31) fulfills

3le minus119882(z

1 z2 z3) le 0 (32)



lim119905rarrinfin

119882(z1 z2 z3) = 0 (33)




lim119905rarrinfin

z119894(119905) = 0 (34)


1and 120572

2 and (34) that

lim119905rarrinfin

q (119905) = [0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(35)



119889isin


Ix2= minusS (x

2) Ix2minus G119886+ u + u

119889 (36)



119889 le

119889max


119889



1198972minus 120576 gt 0 (37)



2= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3+ z1198792u119889

(38)


3= minus1198971z1198791P (Q)P119879 (Q) z



z1198792u119889le 120576

1003817100381710038171003817z21003817100381710038171003817

2

+1

4120576

1003817100381710038171003817u1198891003817100381710038171003817

2

le 1205761003817100381710038171003817z2

1003817100381710038171003817

2

+1

41205761198892

max (40)





minus5 kgm2

119868119909


119868119910


119868119911


119877119886


119896119898


119896119892


minus5



3= minus1198971z1198791P (Q)P119879 (Q) z


le minus1198971z1198791P (Q)P119879 (Q) z


1

41205761198892

max

(41)


3le minus2119897119881

3+

1198892

max4120576

(42)


2minus 120576)120582max(I)



119894


such that z119894(119905) le 120576

lowast

1 z119886(119905) le 120576

lowast

1for any 120576

lowast

1gt 119889max2

radic119897120576



4 Numerical Example






2= 0075 and 119897

3=




02

04

Time (s)

The q

uate

rnio

nq

q1q2q3

minus02

minus040

0

2 4 6


0

20

40

Time (s)0 2 4 6

minus20

minus40

120601

120579120595

The E

uler

angl

es (d

eg)





0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596


02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3



Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12


Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596



5 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ZI

OIXI

YI

f1

f2f3

f4

M1

M2M3

M4

XB

ZB

YB

OB



119887with


as follows

minus120587

2lt 120601 lt

120587

2 minus

120587

2lt 120579 lt

120587

2 minus

120587

2lt 120595 lt

120587

2 (1)


1199020= minus

1

2q119879120596 (2)

q =1

2[1199020E3+ S (q)]120596 (3)


119868119898119894119894= 120591119894minus 119891119894 119894 = 1 2 3 4 (5)

whereQ = [1199020q119879]119879



2

0+ q119879q = 1 120596 isin R3






tively G119886


G119886=

4

sum

119894=1

119868mi (120596 times e119868) (minus1)

119894+1120603119894

(6)

and e119868= [0 0 1]




In (4) u = [1199061 1199062

1199063]119879



u =

[[[[

[

119889119887 (1206032

3minus 1206032

1)

119889119887 (1206032

2minus 1206032

4)

120581 (1206032

2+ 1206032

4minus 1206032

1minus 1206032

3)

]]]]

]

(7)


119894isin R


is given by

119891119894= 1205811206032

119894 (8)




119894for themotor


system that is

lim119905rarrinfin

Q (119905) = [1 0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(9)



0| q119879]

119879

x2= 120596 and

x3= [1206032

11206032

21206032




0) 1199020q119879]119879 (10)

Ix2= minusS (x

2) Ix2minus G119886+ u (11)

I119898x3= X (120591 minus f) (12)

where I119898

= diag(1198681198981

1198681198982

1198681198983

) X = diag(21206031 21206032 21206033) 120591 =

[1205911 1205912

1205913]119879 f = [1198911 119891



sgn (119910) = minus1 119910 lt 0

1 119910 ge 0(13)



rrdReferencepath

trajectory

Positioncontrol

The desiredattitude

Desired thrust T

Attitudecontrol

Motordynamics


dynamicsRigid body




z1= x1 (14)

z2= x2minus 1205721 (15)

z3= x3minus 1205722 (16)







z1=

1

2[sgn(119902

0)q119879120596[119902

0E3+ S (q)]119879120596119879]

119879

=1

2[sgn (119902

0) qT

1199020E3+ S (q)

]120596

= P (Q)120596

(17)

where P(Q) = (12) [sgn(1199020)q1198791199020E3+S(q)


1



function as 1198811= (12)z119879


1

as 1205721= minus1198971P119879(Q)z

1 where 119897



1= z1198791P (Q)120596

= z1198791P (Q) (z

2minus 1198971P119879 (Q) 119911

1)

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2

(18)

Hence if z2

= 0 then 1

= minus1198971z1198791P(Q)P119879(Q)z

1=

minus1198971P119879(Q)z

12



lim119905rarrinfin

10038171003817100381710038171003817P119879 (Q) z

1

10038171003817100381710038171003817= 0 (19)


P119879 (Q) z1=

1

2[sgn (119902

0) q119879

1199020E3+ S (q)

]

119879

[1 minus10038161003816100381610038161199020

1003816100381610038161003816 q119879]119879

=1

2sgn (119902

0) q

(20)


sgn(1199020)q = 0 Then


q = 0


119879 = 119887

4

sum

119894=1

1206032

119894 (21)


3with 120578 =

[1199061 1199062

1199063

119879]119879 and

N =

[[[

[

0 119889119887 0 minus119889119887

119889119887 0 minus119889119887 0

120581 minus120581 120581 minus120581

119887 119887 119887 119887

]]]

]

(22)



Ξ =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

(23)




3minus J1 (24)


= 1198811+


2as

1205722= (ΞN)


1minus 1198972z2+ G119886+ I1]

(25)





(25) yield

2= 1+ z1198792Iz2

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2



3+ 1205722) minus I

1]

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3

(26)


2= minus1198971P119879(Q)z

12













q(119905) =[0 0 0]

119879 lim119905rarrinfin120596(119905) = [0 0 0]

119879



1198982 (28)


1198813= 1198812+

1

2z1198793I119898z3 (29)


3= 2+ z1198793I119898z3

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2


1198982]

(30)


3= minus1198971z1198791P (Q)P119879 (Q) z



3gt 0 for all z

119894




(31) fulfills

3le minus119882(z

1 z2 z3) le 0 (32)



lim119905rarrinfin

119882(z1 z2 z3) = 0 (33)




lim119905rarrinfin

z119894(119905) = 0 (34)


1and 120572

2 and (34) that

lim119905rarrinfin

q (119905) = [0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(35)



119889isin


Ix2= minusS (x

2) Ix2minus G119886+ u + u

119889 (36)



119889 le

119889max


119889



1198972minus 120576 gt 0 (37)



2= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3+ z1198792u119889

(38)


3= minus1198971z1198791P (Q)P119879 (Q) z



z1198792u119889le 120576

1003817100381710038171003817z21003817100381710038171003817

2

+1

4120576

1003817100381710038171003817u1198891003817100381710038171003817

2

le 1205761003817100381710038171003817z2

1003817100381710038171003817

2

+1

41205761198892

max (40)





minus5 kgm2

119868119909


119868119910


119868119911


119877119886


119896119898


119896119892


minus5



3= minus1198971z1198791P (Q)P119879 (Q) z


le minus1198971z1198791P (Q)P119879 (Q) z


1

41205761198892

max

(41)


3le minus2119897119881

3+

1198892

max4120576

(42)


2minus 120576)120582max(I)



119894


such that z119894(119905) le 120576

lowast

1 z119886(119905) le 120576

lowast

1for any 120576

lowast

1gt 119889max2

radic119897120576



4 Numerical Example






2= 0075 and 119897

3=




02

04

Time (s)

The q

uate

rnio

nq

q1q2q3

minus02

minus040

0

2 4 6


0

20

40

Time (s)0 2 4 6

minus20

minus40

120601

120579120595

The E

uler

angl

es (d

eg)





0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596


02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3



Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12


Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596



5 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










rrdReferencepath

trajectory

Positioncontrol

The desiredattitude

Desired thrust T

Attitudecontrol

Motordynamics


dynamicsRigid body




z1= x1 (14)

z2= x2minus 1205721 (15)

z3= x3minus 1205722 (16)







z1=

1

2[sgn(119902

0)q119879120596[119902

0E3+ S (q)]119879120596119879]

119879

=1

2[sgn (119902

0) qT

1199020E3+ S (q)

]120596

= P (Q)120596

(17)

where P(Q) = (12) [sgn(1199020)q1198791199020E3+S(q)


1



function as 1198811= (12)z119879


1

as 1205721= minus1198971P119879(Q)z

1 where 119897



1= z1198791P (Q)120596

= z1198791P (Q) (z

2minus 1198971P119879 (Q) 119911

1)

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2

(18)

Hence if z2

= 0 then 1

= minus1198971z1198791P(Q)P119879(Q)z

1=

minus1198971P119879(Q)z

12



lim119905rarrinfin

10038171003817100381710038171003817P119879 (Q) z

1

10038171003817100381710038171003817= 0 (19)


P119879 (Q) z1=

1

2[sgn (119902

0) q119879

1199020E3+ S (q)

]

119879

[1 minus10038161003816100381610038161199020

1003816100381610038161003816 q119879]119879

=1

2sgn (119902

0) q

(20)


sgn(1199020)q = 0 Then


q = 0


119879 = 119887

4

sum

119894=1

1206032

119894 (21)


3with 120578 =

[1199061 1199062

1199063

119879]119879 and

N =

[[[

[

0 119889119887 0 minus119889119887

119889119887 0 minus119889119887 0

120581 minus120581 120581 minus120581

119887 119887 119887 119887

]]]

]

(22)



Ξ =[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

]]]

]

(23)




3minus J1 (24)


= 1198811+


2as

1205722= (ΞN)


1minus 1198972z2+ G119886+ I1]

(25)





(25) yield

2= 1+ z1198792Iz2

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2



3+ 1205722) minus I

1]

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3

(26)


2= minus1198971P119879(Q)z

12













q(119905) =[0 0 0]

119879 lim119905rarrinfin120596(119905) = [0 0 0]

119879



1198982 (28)


1198813= 1198812+

1

2z1198793I119898z3 (29)


3= 2+ z1198793I119898z3

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2


1198982]

(30)


3= minus1198971z1198791P (Q)P119879 (Q) z



3gt 0 for all z

119894




(31) fulfills

3le minus119882(z

1 z2 z3) le 0 (32)



lim119905rarrinfin

119882(z1 z2 z3) = 0 (33)




lim119905rarrinfin

z119894(119905) = 0 (34)


1and 120572

2 and (34) that

lim119905rarrinfin

q (119905) = [0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(35)



119889isin


Ix2= minusS (x

2) Ix2minus G119886+ u + u

119889 (36)



119889 le

119889max


119889



1198972minus 120576 gt 0 (37)



2= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3+ z1198792u119889

(38)


3= minus1198971z1198791P (Q)P119879 (Q) z



z1198792u119889le 120576

1003817100381710038171003817z21003817100381710038171003817

2

+1

4120576

1003817100381710038171003817u1198891003817100381710038171003817

2

le 1205761003817100381710038171003817z2

1003817100381710038171003817

2

+1

41205761198892

max (40)





minus5 kgm2

119868119909


119868119910


119868119911


119877119886


119896119898


119896119892


minus5



3= minus1198971z1198791P (Q)P119879 (Q) z


le minus1198971z1198791P (Q)P119879 (Q) z


1

41205761198892

max

(41)


3le minus2119897119881

3+

1198892

max4120576

(42)


2minus 120576)120582max(I)



119894


such that z119894(119905) le 120576

lowast

1 z119886(119905) le 120576

lowast

1for any 120576

lowast

1gt 119889max2

radic119897120576



4 Numerical Example






2= 0075 and 119897

3=




02

04

Time (s)

The q

uate

rnio

nq

q1q2q3

minus02

minus040

0

2 4 6


0

20

40

Time (s)0 2 4 6

minus20

minus40

120601

120579120595

The E

uler

angl

es (d

eg)





0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596


02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3



Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12


Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596



5 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(25) yield

2= 1+ z1198792Iz2

= minus1198971z1198791P (Q)P119879 (Q) z

1+ z1198791P (Q) z

2



3+ 1205722) minus I

1]

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3

(26)


2= minus1198971P119879(Q)z

12













q(119905) =[0 0 0]

119879 lim119905rarrinfin120596(119905) = [0 0 0]

119879



1198982 (28)


1198813= 1198812+

1

2z1198793I119898z3 (29)


3= 2+ z1198793I119898z3

= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2


1198982]

(30)


3= minus1198971z1198791P (Q)P119879 (Q) z



3gt 0 for all z

119894




(31) fulfills

3le minus119882(z

1 z2 z3) le 0 (32)



lim119905rarrinfin

119882(z1 z2 z3) = 0 (33)




lim119905rarrinfin

z119894(119905) = 0 (34)


1and 120572

2 and (34) that

lim119905rarrinfin

q (119905) = [0 0 0]119879

lim119905rarrinfin

120596 (119905) = [0 0 0]119879

(35)



119889isin


Ix2= minusS (x

2) Ix2minus G119886+ u + u

119889 (36)



119889 le

119889max


119889



1198972minus 120576 gt 0 (37)



2= minus1198971z1198791P (Q)P119879 (Q) z

1minus 1198972z1198792z2+ z1198792ΞNz3+ z1198792u119889

(38)


3= minus1198971z1198791P (Q)P119879 (Q) z



z1198792u119889le 120576

1003817100381710038171003817z21003817100381710038171003817

2

+1

4120576

1003817100381710038171003817u1198891003817100381710038171003817

2

le 1205761003817100381710038171003817z2

1003817100381710038171003817

2

+1

41205761198892

max (40)





minus5 kgm2

119868119909


119868119910


119868119911


119877119886


119896119898


119896119892


minus5



3= minus1198971z1198791P (Q)P119879 (Q) z


le minus1198971z1198791P (Q)P119879 (Q) z


1

41205761198892

max

(41)


3le minus2119897119881

3+

1198892

max4120576

(42)


2minus 120576)120582max(I)



119894


such that z119894(119905) le 120576

lowast

1 z119886(119905) le 120576

lowast

1for any 120576

lowast

1gt 119889max2

radic119897120576



4 Numerical Example






2= 0075 and 119897

3=




02

04

Time (s)

The q

uate

rnio

nq

q1q2q3

minus02

minus040

0

2 4 6


0

20

40

Time (s)0 2 4 6

minus20

minus40

120601

120579120595

The E

uler

angl

es (d

eg)





0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596


02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3



Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12


Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596



5 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













minus5 kgm2

119868119909


119868119910


119868119911


119877119886


119896119898


119896119892


minus5



3= minus1198971z1198791P (Q)P119879 (Q) z


le minus1198971z1198791P (Q)P119879 (Q) z


1

41205761198892

max

(41)


3le minus2119897119881

3+

1198892

max4120576

(42)


2minus 120576)120582max(I)



119894


such that z119894(119905) le 120576

lowast

1 z119886(119905) le 120576

lowast

1for any 120576

lowast

1gt 119889max2

radic119897120576



4 Numerical Example






2= 0075 and 119897

3=




02

04

Time (s)

The q

uate

rnio

nq

q1q2q3

minus02

minus040

0

2 4 6


0

20

40

Time (s)0 2 4 6

minus20

minus40

120601

120579120595

The E

uler

angl

es (d

eg)





0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596


02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3



Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12


Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596



5 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

04

08

Time (s)

minus04

minus080 2 4 6

Ang

ular

vel

ocity

(rad

s)

120596

1120596

2120596

3120596


02

04

Time (s)

minus02

minus040

0

2 4 6 8 10 12

The q

uate

rnio

nq

q1q2q3



Time (s)

120601

120579120595

The E

uler

angl

es (d

eg)

50

25

0

minus25

minus500 3 6 9 12


Time (s)

12

06

0

minus06

minus120 3 6 9 12

1120596

2120596

3120596

Ang

ular

velo

city

(deg

s)

120596



5 Conclusions






Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Attitude Stabilization Control of a Quadrotor UAV …downloads.hindawi.com/journals/mpe/2014/749803.pdf · 2019-07-31 · time and remain therea er. e sliding mode