Backstepping Control for Nonlinear Dual Smart Drive

Roemi Fernandez*, Joao Hespanha†, Teodor Akinfiev‡ and Manuel Armada

Abstract— Intended for being used in legged robots, aspecially designed nonlinear actuator, Dual Smart Drive, whichoffers continuously changing transmission ratio and dualproperties, and that is very suitable for situations where thesame drive is required to perform two different types ofstart-stop motions of the mobile link, is introduced. Then,the associated control problem to this nonlinear actuator isestablished and a backstepping design strategy is adopted todevelop a Lyapunov-based nonlinear controller that ensuresasymptotic tracking of the desired laws of motion, which havebeen properly selected using time optimal control. Finally,experimental results are presented to show the effectivenessand feasibility of the proposed nonlinear control method forthe Dual Smart Drive.

I. INTRODUCTION

Presently legged robots performance is characterised bothby very low speeds and high energy expense, resulting inlow efficiency for these machines. This feature, due mainlyto their actuation being realised with conventional drives,restricts notably their potential [1] and limits the time ofautonomous robot operation. In this respect the selection ofappropriated drive systems is one of the most determiningfactors to overcome the stated drawbacks [2], [3]. Severalauthors [4], [5], [6] have demonstrated that the employmentof quasi-resonance drives instead of conventional onesrepresents a good way for increasing the robot efficiency.They use variable geometry to accomplish variable reductionand the arrangement gives different transmission ratios atdifferent positions of the output joint.

The drive described here, the Dual Smart [7], is alsoa further development of the quasi-resonance drives andit provides a continuously changing transmission ratioaccording to the angular position of the mobile link.Its nonlinear transmission ratio varies smoothly from aminimum value at the middle position of the mobile link,ad infinitum value in the extreme positions of the mobilelink. Nevertheless, this drive has the additional advantageof presenting two possible magnitudes of the reduction ratiofor each position of the mobile link due to the particularconfiguration of the mechanism linkages. This dual property

*Supported by the Spanish Ministry of Education under Grant F.P.U.†Supported by the National Science Foundation under Grant No. ECS-

0242798.‡Supported by the Spanish Ministry of Science and Technology under

Grant Ramon y Cajal.R. Fernandez, T. Akinfiev and M. Armada are with the Industrial

Automation Institute, Spanish Council for Scientific Research, La Poveda28500 Arganda del Rey, Madrid, Spain roemi@iai.csic.es,teodor@iai.csic.es, armada@iai.csic.es

J. Hespanha is with the Center for Control Engineering and Com-putation, University of California, Santa Barbara, CA 93106-9560 USAhespanha@ece.ucsb.edu

permits to arrange the linkage mechanism within the limitsof one angle, when the load (or the external force) issmall (and to gain for high speeds of displacement), orwithin the limits of another angle, when the load is greater(having, correspondingly, smaller displacement velocities).Consequently, the utilisation of this drive allows to increasethe motion speed of walking machines by enabling highacceleration and deceleration at the beginning and at theend of the driving (high absolute magnitude of the reductionratio), and for maintaining high speeds in the intermediatepart of the driving trajectory both for legs and body (lowabsolute magnitude of the reduction ratio).

The use of such a kind of nonlinear actuator augmentscontrol system complexity. In order to solve that problem,a combined backstepping/time-optimal control strategy isproposed. The basic idea is to use the backstepping designtechnique to develop a Lyapunov-based nonlinear controllerfor the Dual Smart Drive that achieves asymptotic tracking ofthe reference trajectories, which have been suitable selectedusing time optimal control method [8].

The rest of the paper is organized as follows. SectionII describes the Dual Smart Drive and its nonlinearmathematical model. Section III is devoted to the timeoptimal control problem for the calculation of the referencetrajectories. Section IV explains the nonlinear controllerdesign using backstepping technique. The performance ofthe resulting controller is validated in section V throughexperimental testing. Finally section VI summarises majorconclusions.

II. SYSTEM DESCRIPTION

The design of the actuator was presented in [9]. It consistsof a crank connected to the reducing gear of a DC motor, amobile link that rotates around a fixed point, and a slider thatslips along the mobile link in a radial direction thanks to themovement of the crank to which it is connected. An essentialcharacteristic of this actuator is that the length of the crankis smaller than the distance between the mobile link rotationaxis and the crank rotation axis (see Fig. 1). The kinematicsis determined by the following parameters: �ML - distancebetween the mobile link rotation axis and the crank rotationaxis, �C - length of the crank, ϕ - angular position of the rotormeasured clockwise from the ox axis, α - angular positionof the crank measured clockwise from the ox axis, and β -angular position of the mobile link measured clockwise fromthe ox axis. Taking into account that �C < �ML, the mobilelink will move between two extreme positions (see Fig. 1):

−β0 < β < β0 where β0 = arcsin

(�C

�ML

). (1)

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TB1.2

0-7803-9354-6/05/$20.00 ©2005 IEEE 773

A displacement of the mobile link from one end positionto the other one can be carried out in two different ways -by displacement of the crank within the limits of the angleΓ1 (so-called first regime) or within the limits of the angleΓ2 (so-called second regime). As the relationship betweenthe angular position of the rotor and the angular position ofthe crank is ϕ = KGα , (KG - constant transmission of thereduction gear), the variation of the angular position of themobile link β as function of the angular position of the rotorϕ is given by:

β = arctg

(Sin

(ϕ

/KG

)(�ML/�C)+Cos

(ϕ

/KG

))

. (2)

The nonlinear transmission ratio KD between the reducing

EncoderDC Motor

Crank

Mobile Link

Gearhead

Slider

b0 a0

G1G2

Fig. 1. Elements and kinematical schema of the Dual Smart Drive.

gear and the mobile link can be calculated using KD = α/β ,where α is the angular velocity of the crank and β is theangular velocity of the mobile link. As a result,

KD =1+(�ML/�C)2 +2(�ML/�C)Cosα

1+(�ML/�C)Cosα. (3)

This last equation has a 2π periodic character consistingin two different parts (see Fig. 3): one with negative values

BODY OF ROBOT

MOTOR

SLIDER

CRANK

REDUCTION GEAR

MOBILE LINK

90º REDUCTION GEAR

PULLEYBELT

SCREWNUT

LEG

Fig. 2. Example of Dual Smart Drive connection on a legged robot.

(part Γ2), and another one with positive values (part Γ1).The negative magnitude of the reduction ratio means that thecrank and the mobile link are rotating in opposite directions.It is interesting to notice that the reduction ratio tendsto infinity at the end points, −β0, β0, where the crankis perpendicular to the mobile link. In these points, thedeviation of the mobile link from its medium position ismaximal. That is why, the movement of the mobile linkfrom one end position to the other one ensures the mostfavourable change of the reduction ratio for maintaining highaccelerations at the beginning and at the end of driving(high absolute magnitude of the reduction ratio) and formaintaining high speeds in the intermediate part of thetrajectory (low absolute magnitude of the reduction ratio).Moreover, when the motion of the mobile link is carriedout by displacement of the crank within the limits of theangle Γ1, the absolute average magnitude of the reductionratio will be greater than when displacement is within thelimits of the angle Γ2 (see Fig. 4). This enables to use themovement of the crank within the limits of the angle Γ2 whenthe load is small (i.e. for the motion of the robot’s leg) andto gain high speeds of displacement, or within the limits ofthe angle Γ1 when the load is great (i.e. for the motion of therobot’s body), having correspondingly, smaller velocities ofdisplacement. Therefore, the drive allows easily to performa displacement of the mobile link from one end position toanother and to have two different laws for changing drive’sreduction ratio [9]. It is important to point out that at thesingular points, where the drive changes its working regime,the velocities are null. This essential feature gives us thepossibility to model the Dual Drive dynamics independentlyfor each working regime and to combine both of them fora global representation. Additionally, a horizontal operationof the drive is assumed in order to achieve a gravitationaldecoupling. Bearing these facts in mind, the mobile linkequation is given by:

Jiβi = KDKGMi −biβi −MFRisign(

βi

), (4)

where Ji is the equivalent inertia of the mobile link in eachworking regime i = 1, 2, βi is the angular acceleration of the

774

1 2 3 4 5 6 7 8-300

-200

-100

0

100

200

300

Angular position of the crank [rad]

Nonlin

ear

transm

issi

on r

atio

Γ1

Γ2

Fig. 3. Nonlinear transmission ratio as function of the angular position ofthe crank.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

0

1

2

3

4

5

6

7

8

9

10

Angular position of the mobile link [rad]

Ab

solu

te m

ag

nitu

de

of

the

no

nlin

ea

r

tr

an

smis

sio

n r

atio

Γ2

Γ1

Fig. 4. Absolute magnitudes of the nonlinear transmission ratio as functionof the angular position of the mobile link.

mobile link, Mi is the moment that actuates over the mobilelink, bi is the equivalent viscosity friction coefficient andMFRi is the moment of dry friction during the movement ineach regime. The rotor equation is given by:

JMϕi =km

RM[ui − kE ϕi]−Mi −bMϕi, (5)

where JM is the rotor inertia, ϕi is the angular accelerationof the rotor, km is the torque constant, ui is the input controlmotor terminal voltage, RM is the motor resistance, kE isthe back - EMF constant, and bM is the viscosity frictioncoefficient on the motor shaft.

Combining all these equations the dynamic model of thesystem with a nonlinear transmission ratio results in:

x1i = x2i,x2i = f (x1i,x2i)+K (x1i)ui.

(6)

where x1i denotes the angular position of the rotor ϕ , x2i thecorresponding angular velocity ϕ , and,

f (x1i,x2i) = JDi(x1i)

[(JGiKD(x1,x2)

K3D(x1)

−KMa−

− bGi

K2D(x1)

)x2i − MFRGi

KD(x1)sign

(x2i

KD(x1)KG

)],

K (x1i) = KMbJDi(x1i),

JGi =Ji

K2G

, KMa = bM +kEkm

RM,

bGi =bi

K2G

, MFRGi =MFRi

KG,

KMb =km

RM, JDi (x1i) =

K2D(x1)

JMK2D(x1) + JGi

.

III. TIME OPTIMAL CONTROL

In the previous section it has been argued that themovement from one end position to the other one ensuresa favorable change of the reduction ratio for each workingregime. Then, the desired control objective is to make thisdisplacement in a minimum time using all the capabilitiesthat the electromotor and the transmission have available.For this reason, a time optimal control [8] will be used forthe calculation of the reference trajectories.

For nonlinear systems, optimal control theory onlyprovides necessary conditions for optimality. Hence, only aset of candidate controls can be deduced using the generaltheory. So, once the equations of motion have been derived,the Pontryagin’s Minimum Principle is applied to obtainthe necessary conditions for optimality. Then, the equationsfor the state and co-state vector that satisfy the necessaryconditions are determined and subsequently, the controlsequences that can be candidates for time optimal controlare obtained.

The control problem is to minimize the cost functional

Ψ(ui) =T∫

t0

dt = T − t0, T is free, (7)

subject to a magnitude input constraint of the form |ui(t)| ≤umax, ∀t ∈ [t0,T ]. The Hamiltonian function for the system(6) and the cost functional (7) is given by:

Hi(x, p,u) = 1+ x2i p1i + f (x1i,x2i) p2i++K (x1i)ui p2i.

(8)

Since the Hamiltonian is linear on ui, the optimal controlis of the form:

u∗i (t) = umaxsign[K(x∗1i(t))p∗2i(t)] (9)

almost everywhere on [t0,T ∗], where T ∗ is the minimumtime, and x∗1i(t) and p∗2i(t) are the state and co-statetrajectories under the optimal control law. Thus the timeoptimal control is bang-bang. This means that one canpartition the state space into two regions, one on whichui = umax and another one on which ui = −umax. Theboundary between them is called the switching curve. Forsecond-order systems such as this one, one can determinethe switching curve by plotting system trajectories in the

775

-150 -100 -50 0 50 100 150

-150

-100

-50

0

50

100

150

x2i

x1i

c1

- c1

γ+

γ−

Fig. 5. State plane trajectories for the system given by (6). For illustrationpurposes we used umax = 1.

phase plane for the two extreme control values. Fig. 5shows trajectories of the system (6) for ui = umax (solidcurves) and ui =−umax (dashed curves). The arrows indicatedirection of motion of the state. One can see that all thetrajectories due to ui = +umax tend to the line x2i = c1 andall the trajectories due to ui = −umax tend to the line x2i =−c1. The trajectories which pass through the origin arelabelled as γ+ and γ− [8]. The γ+ curve is the locusof all points (x1i,x2i) which can be forced to (0,0) bythe control ui = +umax and the γ− curve is the locus ofall points (x1i,x2i) which can be forced to (0,0) by thecontrol ui =−umax. The γ curve, called the switching curve,is the union of the γ+ and γ− curves and it divides the stateplane into two regions R+ and R−. R+ consists of the pointsto the left of the switching curve γ and R− consists of thepoints to the right of the switching curve γ (cf. Fig. 6). Asthe bang-bang control has a finite number of switches onevery bounded time interval, it shall be demonstrated that theextremal controls for the system (6) can switch at most onceand that only the four control sequences {+umax}, {−umax},{+umax,−umax}, {−umax,+umax} can be candidates for timeoptimal control. The arguments are illustrated in Fig. 7. If theinitial state Ξ = (ξ1,ξ2) belongs to the γ+, by definition, thecontrol sequence {+umax} results in the trajectory Ξ0, whichreaches the origin. The control sequence {−umax} resultsin the trajectory ΞA′, which never reaches the origin.The control sequence {+umax,−umax} results in trajectoriesof the type ΞB′C′, which never reach the origin. Thecontrol sequence {−umax,+umax} results in trajectories ofthe type ΞD′E ′, which never reach the origin. Therefore,if the initial state is on the γ+ curve, from all the controlsequences which are candidates for minimum-time control,only the sequence {+umax} can force the state Ξ to0. Thus by elimination, it must be time optimal. Usinganalogous arguments, we can show that if the initial statebelongs to the γ− curve, then the time optimal controlis ui = −umax. Thus, the time optimal control law forinitial states on the γ curve has been derived. Let us nowconsider an initial state X which belongs to the region R+.

-20 -15 -10 -5 0 5 10 15 20

-150

-100

-50

0

50

100

150

0

R -

The switch curve

ui*

= - umax

ui*

= + umax

R+

γ

γ−

γ+

x2i

x1i

Fig. 6. Switching curve for the second order nonlinear system (6). Forillustration purposes we used umax = 1.

-20 -15 -10 -5 0 5 10 15 20

-150

-100

-50

0

50

100

150

0

A'

B' D'

E'

X

F'

G'

H'

I' J'

X'

L'

M'

N'

P' Q'

C'

Ξ

γ−

γ+

x2i

x1i

Fig. 7. Various trajectories generated by the four possible controlsequences. For illustration purposes umax = 1 was used.

If we use the control sequence {+umax}, the resultingtrajectory is XF ′, shown in Fig. 7, which never reaches theorigin. If the sequence {−umax} is applied, the resultingtrajectory XG′ never reaches the origin. If we apply thesequence {−umax,+umax}, the resulting trajectory is of thetype XH ′I′, which does not reach the origin. However, if weuse the sequence {+umax,−umax}, then one can reach theorigin along the trajectory XJ′0, provided that the transitionfrom the control ui = +umax to ui = −umax occurs at thepoint J′, that is, at the precise moment that the trajectorycrosses the γ switching curve. This is true for every statein R+. Thus, by the process of elimination, we have arrivedat the conclusion that the sequence {+umax,−umax} is timeoptimal for every state in R+, provided that the controlswitches from ui = +umax to ui = −umax at the γ switchingcurve. Using identical arguments one concludes that whenthe initial state belongs to R− the sequence {−umax,+umax}is time optimal with the transition from −umax to umax overγ . The time optimal control u∗i can therefore be written as afunction of the state as follows:

u∗i (x1i,x2i) = +umax ∀(x1i,x2i) ∈ γ+ ∪R+,u∗i (x1i,x2i) = −umax ∀(x1i,x2i) ∈ γ−∪R−.

(10)

776

Bang-bang control is useful for establishing a theoreticalbound on the best possible controlled system performance,but its practical is quite difficult [10], [11]. Generally, itsperformance degrades severely with modelling inaccuracies,unpredicted external disturbances or measurement noise.To add stability, a combination of time optimal controland backstepping is proposed, by using the time optimaltrajectories as reference values for a controller designedusing integrator backstepping. Thus, the approaches will bequasi time optimal rather than exactly time optimal.

IV. BACKSTEPPING

In order to solve the tracking problem, a nonlineartrajectory tracking controller is proposed following theintegrator backstepping technique [12], [13], [14]. Firstly, acoordinate transformation is introduced for the system (6):

e1i = e2i + xd2i − xd1i,

e2i = JDi [(GDi −KMa)(e2i + xd2i)−MZi]+

+KMbJDiui − xd2i.

(11)

where e1i = x1i−xd1i, e2i = x2i−xd2i denote the position andvelocity tracking errors, xd1i = x∗1i and xd2i = x∗2i, (for i = 1,2)the time optimal trajectories determined in Section 3 and

GDi =JGiKD(e1i+xd1i,z2i+xd1i−k1ie1i)

K3D(e1i+xd1i)

− bGi

K2D(e1i+xd1i)

, (12)

MZi =MFRGi

KD(e1i+xd1i)sign

(z2i + xd1i − k1ie1i

KD(e1i+xd1i)KG

). (13)

Now, a smooth positive definite Lyapunov-like function isdefined:

V1i =12

e21i. (14)

Its derivative is given by:

V1i = e1i (e2i + xd2i − xd1i) . (15)

Next, e2i is regarded as a virtual control law to make V1i

negative. This is achieved by setting e2i equal to −xd2i +xd1i − k1ie1i, for some positive constant k1i. To accomplishthis, an error variable z2i that we would like to set to zero isintroduced:

z2i = e2i + xd2i − xd1i + k1ie1i. (16)

Then V1i becomes:

V1i = z2ie1i − k1ie21i. (17)

To backstep, the system (11) is transformed into the form:

e1i = −k1ie1i + z2i,

z2i = JDi [(GDi −KMa)(z2i + xd1i − k1ie1i)−MZi]+

+KMbJDiui − xd1i + k1ie1i.

(18)

Now, a new control Lyapunov function V2i is built byaugmenting the control Lyapunov function V1i obtained in

the previous step using a stabilization function. This functionpenalizes the error between the virtual control and its desiredvalue. So, taking,

V2i = V1i +12

κiz22i, (19)

as a composite Lyapunov function, we obtain:

V2i = −k1ie21i +κiz2i

[JDi (GDi −KMa)(z2i + xd1i−

−k1ie1i)− JDiMZi+KMbJDiui +e1i

κi− xd1i + k1ie1i

].

(20)

The choice of control,

ui =

(JM

KMb+

JGi

KMbK2D(e1i+xd1i)

)[−e1i

κi+ xd1i−

−k1ie1i − JDi (GDi −KMa)(z2i + xd1i − k1ie1i)−

−JDiMZi − k2iz2i

],

(21)

yields,V2i = −k1ie

21i −κik2iz

22i, (22)

where k1i,k2i,κi> 0. This implies asymptotical stabilityaccording to Lyapunov stability theorem.

V. EXPERIMENTS

To evaluate the performance of the tracking controlalgorithm, different experiments were carried out using aprototype of the Dual Smart Drive. Pulse width modulationtechnique (PWM) was used to control the voltage deliveredto the motor. A 2000-pulse-per-revolution optical encoderwas attached to the motor drive to feedback angular positionto the controller. The control algorithms were implementeddirectly in a 486 Processor running real time operatingsystem QNX. The initial conditions for the Dual Smart Drivewere (x1i,x2i) = (−145.4 rad,0 rad/s). The time optimalreference trajectories were obtained using the bang-bangcontrol laws:

u∗1 = 10.8V for 0s < t ≤ 1.165su∗1 = −10.8V for 1.165s < t ≤ 1.183su∗2 = −10.8V for 0s < t ≤ 0.5su∗2 = 10.8V for 0.5s < t ≤ 0.518s

(23)

Fig. 8-11 show the results obtained using control law (21)with k11 = 50, k21 = 80, κ1 = 1 for the first working regime ofthe Dual Smart Drive. Red dotted lines represent the desiredvalues and blue solid lines represent the actual values. Inall the experiments a very good tracking performance wasobtained with a reasonable control effort. It was possibleto appreciate that the first state variable (position) comesvery close to the desired level almost immediately, whilethe second state variable (velocity) experiences lengthiertransients. It is also interesting to point out that despiteof the tracking errors of the motor, the mobile link tracks

777

its reference trajectory almost perfectly, due to the intrinsicproperties of the Dual Smart Drive.

0 0.2 0.4 0.6 0.8 1 1.2−150

−100

−50

0

50

100

150

Time [s]

Mot

or −

Ang

ular

Pos

ition

[rad

]

Fig. 8. Time evolution of the angular position of the rotor.

0 0.2 0.4 0.6 0.8 1 1.2−50

0

50

100

150

200

250

300

Time [s]

Mot

or −

Ang

ular

Vel

ocity

[rad

/s]

Fig. 9. Time evolution of the angular velocity of the rotor.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Mobile Link − Angular Position [rad]

Mob

ile L

ink

− A

ngul

ar V

eloc

ity [r

ad/s

]

Fig. 10. Reference and actual mobile link trajectories in the phase plane.

VI. CONCLUSIONS

A nonlinear tracking controller which reflects the idealizeddynamics of time optimal control has been presented forthe Dual Smart Drive. This controller has been constructedusing a backstepping design procedure. Asymptotic stabilityand the desired tracking performance have been achieved.The limitations of bang-bang control due to modellinginaccuracies and unpredicted disturbances, have beenalleviated using backstepping. As modelling and timingaccuracies approach perfection, the controllers presented herecan approach true time optimal control. Experimental resultsshow the controller effectiveness and demonstrate that thecontrol objectives were accomplished.

0 0.2 0.4 0.6 0.8 1 1.2−15

−10

−5

0

5

10

15

Time [s]

Con

trol

Sig

nal −

Vol

tage

[V]

Fig. 11. Behaviour of the control signal.

VII. ACKNOWLEDGMENTS

R. Fernandez would like to acknowledge Spanish Ministryof Education and Science for funding her stay at the Centerfor Control Engineering and Computation, University ofCalifornia, Santa Barbara. R. Fernandez also would like tothank Professor Petar Kokotovic and all the other membersof the CCEC lab for a worthwhile and motivating stay.

REFERENCES

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[2] F. Pfeiffer, K. Loffler and M. Gienger, “Design aspects of walkingmachines,” Proceedings of the 3rd International Conference onClimbing and Walking Robots, Professional Engineering PublishingLimited London, UK, 2000, pp. 17-38.

[3] P. Sardin, M. Rostami and G. Besonet, “An anthropomorphic bipedrobot: dynamic concepts and technological design,” IEEE Transactionson Systems, Man and Cybernetics, Part A, Vol. 28, 1998.

[4] T. Akinfiev, and M. Armada, “Resonance and quasi-resonance drivefor start-stop regime,” Pergamon, Proc. 6th International ConferenceMECHATRONICS’ 98, Skovde, Sweden, 1998, pp. 91-96.

[5] O. Bruneau, J. P. Louboutin and J. G. Fontaine, “Optimal designof a leg-wheel hybrid robot actuated by a quasi-resonant system,”Proceedings of the 3rd International Conference on Climbingand Walking Robots, Professional Engineering Publishing LimitedLondon, UK, 2000, pp. 551-558.

[6] J. Roca, J. Palacin, J. Bradineras and J. M. Iglesias, “Lightweightleg design for a static biped walking robot,” Proceedings of the5th International Conference on Climbing and Walking Robots,Professional Engineering Publishing Limited London, UK, 2002, pp.383-390.

[7] T. Akinfiev, M. Armada and R. Fernandez, “Drive for workingelement, especially for a walking robot, and the control method,”Spanish Patent Office, 2002, Publication Number: 2195792.

[8] M. Athans, and P. L. Falb, Optimal Control, McGraw-Hill BookCompany, New York, 1966.

[9] R. Fernandez, T. Akinfiev and M. Armada, “Dual smart drive:analytical solution, simulation and experimental results,” Proceedingsof the 6th International Conference on Climbing and Walking Robotsand the Support Technologies for Mobile Machines, ProfessionalEngineering Publishing Limited London, UK, 2003, pp. 309-318.

[10] F. Song, and S. M. Smith, “Design of sliding mode fuzzy controllersfor an autonomous underwater vehicle without system model,”Oceans’2000 MTS/IEEE, pp. 835-840, 2000.

[11] P. H. Meckl, and W. Seering, “Active damping in a three-axis roboticmanipulator,” A.S.M.E. Journal of Vibration, Acoustic, Stress, andReliability in Design, 107, pp.38-46, 1985.

[12] P. V. Kokotovic, “The Joy of Feedback: Nonlinear and Adaptive,”IEEE Contr. Sys. Mag., vol. 12, pp. 7-17, 1992.

[13] M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, Nonlinear andAdaptive Control Design, New York: Wiley, 1995.

[14] H. K. Khalil, Nonlinear Systems, New York: Prentice Hall, 2002.

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