Adaptive Robust

8/12/2019 Adaptive Robust

1/13

International Journal of Intelligent Computing and CyberneticsEmerald Article: Adaptive robust tracking control for mobile manipulatorsn the task-space under uncertainties

Mohamad Boukattaya, Tarak Damak, Mohamed Jallouli

Article information:

To cite this document: Mohamad Boukattaya, Tarak Damak, Mohamed Jallouli, (2011),"Adaptive robust tracking control for mobile

manipulators in the task-space under uncertainties", International Journal of Intelligent Computing and Cybernetics, Vol. 4 Iss: pp. 81 - 92

Permanent link to this document:

ttp://dx.doi.org/10.1108/17563781111115804

Downloaded on: 06-08-2012

References: This document contains references to 13 other documents

To copy this document: [email protected]

Users who downloaded this Article also downloaded: *

Hui Chen, Miguel Baptista Nunes, Lihong Zhou, Guo Chao Peng, (2011),"Expanding the concept of requirements traceability: The rolef electronic records management in gathering evidence of crucial communications and negotiations", Aslib Proceedings, Vol. 63

ss: 2 pp. 168 - 187

ttp://dx.doi.org/10.1108/00012531111135646

Guo Cai-fen, Jing Ran-zhe, (2011),"Modeling and controlling work-in-progress in discrete manufacturing systems", Kybernetes, Vol.

0 Iss: 5 pp. 842 - 847

ttp://dx.doi.org/10.1108/03684921111142386

Franois Des Rosiers, Jean Dub, Marius Thriault, (2011),"Do peer effects shape property values?", Journal of Property

nvestment & Finance, Vol. 29 Iss: 4 pp. 510 - 528

ttp://dx.doi.org/10.1108/14635781111150376

Access to this document was granted through an Emerald subscription provided by AMIRKABIR UNIVERSITY OF TECHNOLOGY

For Authors:

f you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service.

nformation about how to choose which publication to write for and submission guidelines are available for all. Please visit

www.emeraldinsight.com/authors for more information.

About Emerald www.emeraldinsight.com

With over forty years' experience, Emerald Group Publishing is a leading independent publisher of global research with impact in

usiness, society, public policy and education. In total, Emerald publishes over 275 journals and more than 130 book series, as


2/13

Adaptive robust tracking controlfor mobile manipulators in thetask-space under uncertainties

Mohamad Boukattaya and Tarak DamakControl of Industrial Process Unit, Sfax Engineering School, Sfax, Tunisia, and

Mohamed JallouliIntelligent Control Design and Optimization of Complex System Unit,

Sfax Engineering School, Sfax, Tunisia

Abstract

Purpose The purpose of this paper is to address the trajectory tracking control in task space of anon-holonomic wheeled mobile manipulator with parameter uncertainties and disturbances. Theproposed algorithm is robust adaptive controlstrategy where parametric uncertaintiesare compensatedby adaptive update techniques and the disturbances are suppressed. The system stability and theconvergence of tracking errors to zero are rigorously proved using a Lyapunov theory.

Design/methodology/approach The proposed algorithm is derived based on the advantage ofthe robot regressor dynamics that express the highly non-linear robot dynamics in a linear form interms of the known and unknown robot parameters. The update law for the unknown dynamicparameters is obtained using Lyapunov theory.

Findings Simulation experiments show the effectiveness of the proposed robust adaptive basedcontroller in comparison with a classical passivity based controller.

Originality/value The proposed adaptive approach is interesting for the control of the mobilemanipulators in the task space coordinate even in the presence of dynamic uncertainties and external

disturbances.

KeywordsTracking, Control technology, Robots, Trajectories, Uncertainty management

Paper typeResearch paper

1. IntroductionMobile manipulators refer to robotic manipulators (or arms) mounted on mobile platforms(or vehicles). Such systems combine the advantages of mobile platforms and robotic armsand reduce their drawbacks. The mobile platform extends the arms workspace, whereasthe arm offers much operational functionality. Applications for such systems could befound in mining, construction, forestry, planetary exploration and military Pavlov andTimofeyev (1976), Chung and Velinsky (1998) and Watanabe et al. (2000). With the

assumption of known dynamics, much research has been carried out. In Yamamoto andYun (1996), non-linear feedback control for the mobile manipulator was developed tocompensate for the dynamic interaction between the mobile platform and the arm toachieve tracking performance. In Khatib (1999), coordination and control of mobilemanipulators were presented with two basic task-oriented controls: end-effector taskcontrol and platform self posture control. In Hootsmanns and Dubowsky (1991), a controlmethod based on an extended Jacobian transpose was proposed to compensate fordynamic interactions between the manipulator and platform. In Spong (2005), a computed

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/1756-378X.htm

Adaptivetracking control

81

Received 25 January 2010Revised 10 October 2010

Accepted 16 October 2010

International Journal of Intelligent

Computing and Cybernetics

Vol. 4 No. 1, 2011

pp. 81-92

q Emerald Group Publishing Limited

1756-378X

DOI 10.1108/17563781111115804


3/13

torque control was developed using as examples a car-like mobile manipulator and adifferentially driven mobile manipulator. Most previous approaches require a preciseknowledge of the dynamics of the system and ignore external disturbance. These issuesmake the proposed schemes inappropriate for realistic applications. To handle these

difficulties, adaptive and intelligent schemes were investigated to deal with mobilemanipulators with unknown parameters and disturbances. In Xu et al. (2008), a robustcontrol scheme using neural network combined withslidingmodecontroller is proposed tosolve trajectory tracking problem for mobile manipulators. This control scheme not onlyovercomes the unstructured uncertainties, but also has the capability of disturbancerejection in the presence of unknown bounded disturbances. In Gao et al. (2008), anintelligent controller based on radial basic function neural network is introduced for thecoordinating control of mobile manipulators when external disturbances acting on thesystem. In Tan et al. (2008), an adaptive hybrid control scheme combining the kinematicsof the mobile platform and the unified dynamic model of the mobile manipulator wasproposed. This controller consists of two parts: one is responsible for the tracking controlof the mobile platform in kinematics. The other part is for the robot arm in dynamics. Forfurther consideration of unmodeled dynamics and external disturbances, a radial basisfunction neural-network (RBFNN) is adopted in the adaptive controller. An unified modeland robust neural-network controller was also investigated in Tan et al.(2007), where acomputed torque method and RBFNN were used to approximate the unstructured orstructured uncertainties of the mobile manipulator. In Liet al.(2010), an adaptive robustcontrol strategy extended to the actuator level for multiplemobile manipulators carrying acommon object was presented. The proposed controls are robust not only to parametricuncertainties including mass variation and electrical parameters, but also to externaldisturbances. In Tsai et al. (2007), a robust tracking control scheme incorporating theadvantages of the sliding mode control together with the NN approach, for achievingtrajectory trackingcontrol of a non-holonomic wheeled mobile platform withdual onboard

multi-link arms was proposed. The significant features of the proposed controller hinge onno prior knowledge of the mobile manipulators dynamics.

Most of the previous research work designed the controller in joint space and used acomplicated and a computationally expensive learning algorithm. In order to overcomethese difficulties, an adaptive control in task space will be proposed based on theadvantage of the robot regressor dynamics that express the highly nonlinear robotdynamics in a linear form in terms of the known and unknown robot parameters.The proposed adaptive algorithm does not rely on precise prior knowledge of dynamicsparameters, and it can suppress disturbances and modeling errors caused by parametersuncertainties.

This paper is organized as follows. Section 2 is devoted to kinematic and dynamicmodeling of the mobile manipulator with non-holonomic constraints. Section 3 presents

the design of the robust adaptive controller. Section 4 presents computer simulationresults to illustrate the effectiveness of the proposed theory. Conclusions are formulatedin Section 5.

2. Modeling of a mobile manipulator2.1. Kinematic modelingConsider the mobile manipulator system shown in Figure 1.

IJICC4,1

82


4/13

The mobile platform moves by driving two independent wheels. Assuming that thetwo driven wheels do not slip, the following constraint equation may be derived:

_xFsinw2 _yFcos w _wd 0: 1

where xFand yF are the coordinates of the point F, base point of the manipulator inworld coordinate system, w is the heading angle of the d is the distance between thewheel axis and F.

It can be shown, that equation (1) is a non-holonomic constraint, i.e. a constraintwhich cannot be integrated analytically. For the mobile platform, the direct kinematicsequation relating linear velocity ofFto the wheel velocities is:

_xF

_yF

" #

r2b

bC0 dS0 r

2bbC0 2 dS0

r2b

bS0 2 dC0 r

2bbS0 dC0

24

35 _uR

_uL

" #: 2

where _uRand _uL, are the angular velocities of the right and the left wheels, respectively,

C0 cosw and S0 sinw. Displaying the rotation matrix explicitly, we can writeequation (2) as:

_xF

_yF

" #

C0 2S0

S0 C0

" # r2

r2

2dr2b

dr2b

24

35 _uR

_uL

" #: 3

The linear velocity of the end-effector is found using the fact that its base velocity isknown and given by equation (3). Therefore, the end-effector velocity is written as:

Figure 1.Mobile manipulator

system on adifferentially-driven

platform

qL

qR

j

q1

q2

L1

(m2 , I2)

(m1 , I1)

(mw, Iw)

(mc, Ic)

L2

E

Fb

b

y

xO

r

dO


83


5/13

_xE

_yE

" #

_xF

_yF

" #

C0 2S0

S0 C0

" # J11 J12

J21 J22

" # _u1 _w

_u2

" #: 4

where Jii i;

j 1;

2 terms are the elements of the fixed-base Jacobian of themanipulator employed, given by:

J11 2L1S1 2L2S12; J12 2L2S12; J21 L1C1 L2C12; J22 L2C12

whereL1,L2 are the lengths of the first, and the second arm, respectively, andu1, u2, arethe joint variables of the manipulator. With the notation: Cicosui, Sisinui,Cij cosui uj and Sij sinui uj, ;i;j 1; 2.

Combining equations (3) and (4), the forward differential kinematics of the mobilemanipulator is obtained as:

_xE

_yE

_xF

_yF

26666643777775

C0 2S0 0 0

S0 C0 0 0

0 0 C0 2S0

0 0 S0 C0

26666643777775

r

22

r

2bJ11

r

2

r

2bJ11 J11 J12

2 r

2bdJ21

r

2bdJ21 J21 J22

r

2

r

2 0 0

2

dr

2b

dr

2b 0 0

2666666666664

3777777777775

_uR

_uL

_u1

_u2

2666664

3777775

: 5

which can be expressed in the following form:

_x RJ_z Jz _z: 6

And its time derivative as:

x _Jz _zJz z: 7

2.2. Dynamic modeling

The dynamics of a mobile manipulator subject to non-holomonic constraints can beobtained using the Lagrangian approach in the following form:

Hq qVq; _q A Tql Eqt: 8

where q q1. . .qnT[ R

n1 is the generalized coordinates, Hq [ Rnn is asymmetric, positive definite inertia matrix, Vq; _q [ Rnn represents the vector ofcentripetal and Coriolis forces terms,Aq [ Rmn is the constraint matrix,l [ Rm1

is the Lagrange multiplier which denotes the vector of constraint forces, Eq [

IJICC4,1

84


6/13


7/13

3. Adaptive controller designSuppose that the desired trajectories in task space are described by xdand _xd. In order totrack the desired trajectories, we have to design a passivity based controller given bySpong (2005):

tx Hx _nVxnGx 2Kr: 15

wherenandrare given as:

e x 2xd

v _xd2 Le

r _x 2 v

: 16

With L andKare a diagonal matrices of positive gains.Substituting equations (15) and (16) into equation (14) yields:

Hx _rVxr Kr0: 17

The closed loop system equation (17) is non-linear and coupled. Consider the Lyapunovfunction:

V 1

2rTHxr e

TLKe: 18

The time derivative ofVcan be computed as:

_V 2_e TK_e 2 e TLKLe # 0: 19

It is obvious that errors will asymptotically converge to zero if the dynamics of thesystem are known exactly. On the other hand, the existence of disturbances anduncertainties influences the performance of the passivity based controller and make theclosed loop system unstable. To handle imperfect knowledge of the mobile manipulatorparameters and external disturbances, the passivity based controller equation (15) ismodified as:

tx ^Hx _v Vxv Gx 2Kr: 20

where ^Hx, Vx andGx are estimate of unknown parameters ^H, Vand

G.According to property equation (4), we can rewrite equation (20) as:

tx Yq;

_q;

v;

_v^u2Kr

:

21

Where uis estimates of unknown constant parametric vector u. Substituting equation(21) into (14), we may have the closed-loop error dynamics:

Hx _rVxrKrYq; _q; v;_v ~u: 22

Where ~u u2 u. If an appropriate adaptive update law for ucan be selected, we mayeasily to prove the convergence of the tracking errors to zero and the system stability.

IJICC4,1

86


8/13

Let us define a Lyapunov function candidate as:

V 1

2rTHxre

TL

TKe1

2~uTG ~u: 23

Where G is a positive constant design matrix. The time derivative of V can becomputed as:

_V rTHx _r1

2_Hxr 2e

TL

TKe _~uTG ~u: 24

Substituting the closed-loop error dynamics equation (22) in the above equation, we get:

_V rTYq; _q; v;_v ~u2Kr2r _Hx 2 2Vx 2eTL

TKe _~uTG ~u: 25

Since _Hx 2 2Vx is a skrew symmetric, the above equation become:

_V 2_e TK_e 2 e TLTKLe ~urTYq; _q; v;_v _~uTG: 26

With selected update law as:

_u 2G

2TYTq; _q; v;_v r: 27

The time derivative of the Lyapunov function is negative _V# 0 and the system isasymptotically stable.

The proposed adaptive control system is shown in Figure 2.

4. Simulation results

To verify the effectiveness of the proposed adaptive robust control, let us consider themobile manipulator system shown in Figure 1. Simulation parameters are shown inTable I.

Figure 2.Structure of the

proposed controllerForwardkinematics

Forward

kinematics

Mobile

manipulator


87


9/13

The initial conditions are given as follow:

q0 0; 0;p2

; 0; 0; p4

; 2p2

h iT

and _z0 0; 0; 0; 0

h iT

Let the desired trajectory be:

xdt xdE; y

dE; x

dF; y

dF

h iT 0:2t0:3; 0:50:25 sin0:2pt; 0:2t; 0h iT

It consists of a sinusoidal path for the end-effector and a straight line for the mobileplatform.

We assume the disturbances as: tdt 0:5sint; 0:5 cost; 1:5sint;1:5costT.

The control gains used in proposed adaptive tracking controller were selected as:G 2I88 , L 2I44 and K100I44.

In order to validate the better performance of the proposed control, in the sameconditions, we compare:

. the passive based controller; and

. the proposed adaptive control.

The tracking performances of each control scheme are shown in Figures 3-8,respectively. From the comparison of both controls, it can be seen that the trackingresults of the passive based controller are not satisfactory and the tracking errorsfluctuate greatly in comparison with the adaptive robust control schemes which attaingood control performance, and the tracking error is much small because of adaptivemechanism. The simulation result thus verifies the effectiveness of the proposedcontrol in the presence of external disturbances.

Parameters (Units) Platform Manipulator

Dimension (m) b 0.182r 0.0508d 0.1

L1 0.514L2 0.362

Mass (kg) mc 17.25mw 0.159

m1 2.56m2 1.07

Inertia (kg m2) Ic 0.297Iw 0.0002

I1 0.148I2 0.0228

Table I.Parameters of the mobilemanipulator

Figure 3.Trajectory tracking withpassivity based controller

0.50.5

0

0.5

1

0.5 1.5 2.5 3.50 1 2 3 4 4.5

IJICC4,1

88


10/13

Figure 5.Trajectory tracking errors

for the platform withpassivity based controller

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

Time (sec)

X-axisY-axis

Trajectorytrakingerrors

fortheplatfrom(

m)

12 14 16 18 20

Figure 4.Trajectory tracking errors

for the end-effector withpassivity based controller

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

Time (sec)

X-axisY-axis

Trajectorytrakingerrorsfortheendef

fector(m)

12 14 16 18 20

Figure 6.Trajectory tracking with

adaptive based controller0.5

0.5

0

0.5

1

0.5 1.5 2.5 3.50 1 2 3 4 4.5


89


11/13

5. ConclusionIn this paper,a robust adaptive controller formobile manipulator system in the presence of

parametric uncertainties and external disturbances is proposed. The proposed controlstrategy was designed to drive simultaneously in task space desired end-effector andplatform trajectories without violating the non-holonomic constraints. The unknownparameters and the external disturbances are estimated by using update law in adaptivecontrol scheme. The effectiveness of the proposed controller is verified both analyticallyand in simulation. Future work will concentrate on the implementation of the proposedcontroller on a real mobile manipulator system and an adaptive controller whichintegrates the motion and the force should also be investigated.

Figure 7.Trajectory tracking errorsfor the end-effector with

adaptive based controller

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

Time (sec)

X-axisY-axis

Trajectorytrakingerrorsfortheende

ffector(m)

12 14 16 18 20

Figure 8.Trajectory tracking errorsfor the platform withadaptive based controller

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

Time (sec)

X-axisY-axis

Trajectorytrakin

gerrorsfortheplatfrom(

m)

12 14 16 18 20

IJICC4,1

90


12/13

References

Chung, J.H. and Velinsky, A. (1998), Modeling and control of a mobile manipulator, Robotica,Vol. 16 No. 6, pp. 607-13.

Gao, C., Zhang, M. and Liu, R. (2008), Research on the method and experiment of the coordinated

control for wheeled mobile manipulators, Proceedings of the IEEE InternationalConference on Automation and Logistics, Qingdao, China, September.

Hootsmanns, N. and Dubowsky, S. (1991), The motion control of manipulators on mobilevehicles, Proceedings of IEEE International Conference on Robotics and Automation,Vol. 3, pp. 2336-41.

Khatib, O. (1999), Mobile manipulation: the robotic assistant, Robotics and AutonomousSystems, Vol. 26 Nos 2/3, pp. 157-83.

Li, Z., Li, J. and Kang, Y. (2010), Adaptive robust coordinated control of multiplemanipulators interacting with rigid environments, Automatica Journal, Vol. 46 No. 12,pp. 2028-34.

Pavlov, V. and Timofeyev, A. (1976), Construction and stabilization of programmed movements

of a mobile robot-manipulator, Engineering Cybernatics, Vol. 14 No. 6, pp. 70-9.Spong, W. (2005),Motion Control of Robot Manipulators, The Coordinated Science Laboratory,

University of Illinois, Chicago, IL.

Tan, X., Zhao, D., Yi, J. and Xu, D. (2007), Unified model and robust neural-network control ofomnidirectional mobile manipulators, Proceedings of the 6th IEEE InternationalConference on Cognitive Informatics (ICCI07).

Tan, X., Zhao, D., Yi, J. and Xu, D. (2008), Adaptive hybrid control for omnidirectional mobilemanipulators using neural-network, paper presented at American Control ConferenceWestin Seattle Hotel, Seattle, Washington, DC, USA, June 11-13.

Tsai, C., Cheng, M. and Li, S. (2007), Robust tracking control for a wheeled mobile manipulatorwith dual arms using hybrid sliding mode neural network,Asian Journal of Control,Vol.9No. 4, pp. 377-89.

Watanabe, K., Sato, K., Izumi, K. and Kunitake, Y. (2000), Analysis and control for anomnidirectional mobile manipulator,Journal of Intelligent and Robotic Systems, Vol. 27No. 1, pp. 3-20.

Xu, D., Zhao, D., Yi, J., Tan, X. and Chen, Z. (2008), Trajectory tracking control ofomnidirectional wheeled mobile manipulators: robust neural network based sliding modeapproach, paper presented at IEEE International Conference on Robotics andAutomation, Pasadena, USA, May 19-23.

Yamamoto, Y. and Yun, X. (1996), Effect of the dynamic interaction on coordinated control ofmobile manipulators, IEEE Transaction on Robotics and Automation , Vol. 12 No. 5,pp. 816-24.

About the authors

Mohamad Boukattaya received an Electromechanical Engineering Diploma fromthe Sfax Engineering School, Tunisia, in 2002. From the same school in 2006 hereceived a Masters degree in Automatic and Industrial Informatics. He is currentlya Researcher at the Control of Industrial Process Unit at the same school. Hisresearch interests include robot control and simulation, especially in mobilemanipulators, mobile robotics, and redundant systems.Mohamad Boukattaya is thecorresponding author and can be contacted at: [email protected]


91


13/13

Tarak Damak received an Electrical Engineering Diploma from the SfaxEngineering School, Tunisia, in 1989, a DEA degree in Automatic control fromthe Institut National des Sciences Appliquees de Toulouse-France in 1990, and aPhD from the UniversitePaul Sabatier de Toulouse-France in 1994. In 2006, he

obtained the University Habilitation from the Sfax Engineering School. Hiscurrent research interests are in the fields of distributed parameter systems,sliding mode control and observers, and adaptive non-linear control.

Mohamed Jallouli received a DEA degree in Automatics from University ofValenciennes, France in 1986 and a PhD in Robotics Engineering fromUniversity Paris XII, France, in 1991. He is currently an Assistant Professor ofElectric and Computer Engineering at Sfax Engineering School. His currentinterests include the implementation of intelligent methods in robotic and visionsystem as well as in multisensory data fusion mobile bases.

IJICC4,1

92

To purchase reprints of this article please e-mail: [email protected] visit our web site for further details: www.emeraldinsight.com/reprints

Adaptive Robust

Documents

Transcript of Adaptive Robust