Robust adaptive control of multiple robots in cooperative motion using ...

Proceedings of the 1991 IEEE Intemational Conference on Robotics and Automation Sacramento, California - April 1991

Robust Adaptive Control of Multiple Robots in Cooperative Motion Using o Modification Mohamed Zribi, Shaheen Ahmad School of Electrical Engineering Purdue University West Lafayette, IN 47907-0501

Abstract

control for rigid robots. Slotine and Li [7] exploited the structural properties of the rigid manipulator dynamics to design an adaptive controller capable of general trajectory tracking control. Bayard and Wen [2] introduced a class of asymptotically stable adaptive control laws for rigidly jointed robotic manipulators. Seraji [6] proposed a decentralieed adaptive control algorithm for robot manipulators. His controller is based on independent joint control. A PID controller and position velocity acceleration feedforward controller both with adjustable gains are used to control each joint independently.

In this paper, we address the problem of multiple robots manipulating a rigid object cooperatively. An adaptive controller which is designed based on Lyapunov’s second method is proposed. This controller takes into account the dynamics of both the object and the manipulators. The proposed adaptive controller does not require feedback of joint accelerations and inversion of inertia or Jacobian matrices. Global stability in the sense of Lyapunov of the overall system is shown. The effects of bounded disturbances on the multi-robot system are also considered. An adaptive control law which guarantees the convergence of the tracking error to a bounded set is proposed. The siee of this set is shown to depend on the bounds of the disturbances. 1. Introduction

In this paper we address the problem of adaptive control of multiple robots during cooperative manipulation of a rigid object. We first start by developing the dynamic models of the manipulators and the load. Properties of this dynamic system which we exploit are also clarified. An adaptive controller which is designed based on the Lyapunov’s second method is then proposed. This adaptive control law guarantees the global stability of the system in the sense of Lyapunov. Finally the issue of robustness in the presence of bounded disturbances is addressed. An adaptive law which guarantee the convergence of the tracking errors to a bounded set is proposed. The siee of the set depends upon the bounds on the disturbances. 1. C o o p e r a t i v e M u l t i - R o b o t S y s t e m Model 1.1 D y n a m i c s model

Recently considerable amount of research has focused on the problem of cooperative control and coordination of multiple robots. Interest in multirobot systems has arisen because several tasks require the use of two or more robots. However the multirobot systems are more difficult to control than single robots. Additional problems arise as the parameters of the robots and the manipulated load may not be known exactly. Few control schemes for cooperative robots have been proposed. Yun, Tarn and Bejcey [9] used exact linearization and output decoupling to design a controller for the two robot manipulators. Yoshikawa and Zheng [12] proposed a cooperative dynamic hybrid control method; this method takes into consideration both the manipulator dynamics and the object dynamics. Guo and Ahmad [l]looked a t the control problems when the robots have compliant joints.

The general dynamic model for n cooperative multi-robot system has been investigated thoroughly in the literature, and is also described in the below for completeness. The dynamic equation of the ith manipulator in cooperative manipulation is given as: H ~+ < cipi ~ + G ;= ri+

..

n

(1)

where, qi(t) E R”’ is the vector of joint displacements, and n; is the number of joints of the ith robot. The inertia matrix of the ith robot is H;(q;(t)) E Rnixni, this is a positive definite and symmetric matrix. The

Hsia [3], and Ortega and Spong [5] presented an extensive summary of recent papers on adaptive

CH2969-4/91/0000/2160$01.OO 0 1991 IEEE

~ 7 ‘ ~i = l ,

2160

matrix

and ii;(rj) is a skew symmetric matrix; r ; = I r;,,rjylrizIT represents the vector from the center of mass of the object to the contact point of the object and the ith manipulator.

of

centrifugal and Coriolis forces is the vector of gravity forces is G,(q,(t))E R"', and the manipulator Jacobian is J , ( q , ( t ) )E R"""'. The control input torques for the ith robot is r , E R"'. The forces/moments applied by the ith manipulator on the object at the point of contact is F, and it is assumed to be exactly measurable. The contact forces/moments F , E RIn can be written in terms of the contact forces f , ER'"' and contact moments 7 , E Rnr2, (where, m = m i + m2, and m is the dimension of the Cartesian work space), such that C,(q,(t),q,(t))E R"""';

1.2 Kinematic Model Occasionally, we might be interested in controlling the manipulators in some predefined Cartesian task space such that: q ( t ) = Ki(qi(t)) i = 1,...n where Ki(.) : R ~ ' ~ + Ris ~ the ~ ~ transformation from the joint angle space of qi(t) to the task space containing zi(t), and zi(t) E R"' is the position and orientation of the point of contact of the ith manipulator, with the load, expressed in the world coordinate frame. Notice ] ~ that &(t) = vi(t)' ~ i ( t ) E~Rm. If we define the manipulator Jacobian Ji(qi(t)) to be the differential map of qj(t) space to zi(t) space, then we can write zi(t) = J;(qi(t))qi(t) i = 1, ...n We will also define T i : Rm-rRm to be a transformation between the center of mass of the load and the contact point with the ith end-effector, expressed in the world coordinate frame, then we can write: ~

~ , = [ f ? ~ I , ' ] ' , I = I ,n .

The equations of motion of the object (load) are obtained from the Newton-Euler equations.

I;

+ WX(IW)

[

.

I="

=

C(oi+ rixfi)

i= 1

(3)

Where the position of the center of mass of the object expressed in world coordinate frame is x(t) E R"" . The rotational velocity of the center of mass of the object in world coordinate frame is W(t)ER"'L, and the gravity vector for the object, expressed in world coordinate reference frame is g(x(t)) E R"". The mass matrix M E RmlXml and the inertia matrix I E Rwxm2 are diagonal matrices whose diagonal elements are respectively the mass and inertia of the load. The position of the end-effector of the ith manipulator with respect to the object center of mass, expressed in the world coordinate frame is

where the angular velocity of the point of contact of the ith manipulator, with the load, expressed in world coordinate frame is vi(t) E Rml. The rotational velocity of the point of contact of the ith manipulator, with the load, expressed in world coordinate frame is wi(t) E Rnr2. The angular velocity of the center of mass of the object expressed in world coordinate frame is v(t) E Rml, and z(t) E Rm is the position and orientation vector of the center of mass of the object expressed in world coordinate frame.

r;(t) E R"".

The motion of the load expressed by equations (2) and (3) can be written as i=n

Sa

+ N2 = XBiF; i= 1

We require each robot manipulator t o follow a predefined trajectory z;d(t) and qid(t) = K-'(aid(t)), then the trajectory tracking error of the ith robot, ei(t) E R"', is:

(4)

Where i(t) = [V(t)T ~ ( t ) ~ ]s ~€Itmxm , is the augmented matrix whose diagonal elements are the mass matrix and the inertia matrix of the load. S , Bi and nj(ri) are defined as follows

ej(t)

~

2161

- qid(t)

i = 1,...n

(6)

where the vector of desired joint displacements for the ith manipulator is qjd(t) E R"'. Similarly, the velocity tracking error of the object is defined as: where the e,(t) = ( v(t) - vd(t) ) ( w(t) - Wd(t) ) desired angular velocity of the center of mass of the object expressed in world coordinate frame is vd(t) E RI"' , and the desired rotational velocity of the ~ ~ of mass of the object in world coordinate center frame is wd(t) E R " ~ . The load's tracking velocity error is given as,

[

and RYE Rsx3 is the rotation matrix that converts any vector expressed in i-th base coordinate frame into a vector in the world reference frame, and R ~ E isR the rotation matrix that converts any vector expressed in i-th base coordinate frame into a vector in the center of mass of the object reference frame,

= qi(t)

eters will be estimated by the proposed adaptive scheme. The matrix Y,(qi,qi,q;)E Rniir represents the structure of the ith robot dynamics, hence its elements are combinations of the elements of the inertia and centrifugal/Coriolis matrices, and the gravity vector. Similar parameterieation for load dynamics is also possible, see later.

where vid(t) E R"" and wid(t) E R"" are the desired angular and rotational velocities of the robot manipulator. From the previous equation, we can see that there is a transformation between joint velocity tracking error and the load's Cartesian velocity tracking error; the above an equation can be written as

e&)

=

1.4 Definition of variables used for the C o n t r o l Law

T ; ' J ~ ~+~(T;'J; - T $ J ~ , ~ ) ~ ~ ~( 7 )

The error vector of the parameter estimates , a; is defined as the difference the between the estimated parameters , ii , and their exact values , a; , (hence &(t) = ii(t) - a;(t) ). Similarly, we can write the parameter estimation error vector for the load as

It should be noted that in general the Jacobian matrices are not square matrices. If the ith robot is redundant (i.e.m < ni) and if the robot has insufficient degrees of freedom to carry out the required cartesian motion ( n ; < m ) then the corresponding Jacobian will not be a square matrix. In both cases the control problem is further complicated. It should be noted that our controller does not require Jacobian inverses, and problems related to inverses of nonsquare systems are not addressed here.

,

-

a,

It should be noted that we can write k i ( q i R i + ei(qitii)4i

where

q;(t) - qjr(t)

i=1, ...n

+ 6i(qi)

yi(qi%ii,ii)ii

is a vector of r robot parameters Where are the estimates of the inertia, Coriolis and centrifugal matrices; 6 ; is the estimate of the gravity vector. Also it should be noted that Hi(qi)qi+ Ei(qi,qi)qi+ d i ( q i ) = yi(qi,qi,qi)ii, where Gi , ti are the errors in the inertia, Coriolis and centrifugal matrices; di is the error in the gravity vector.

(8)

1.8 P r o p e r t i e s of M a n i p u l a t o r D y n a m i c s W h i c h Will be E x p l o i t e d in t h e C o n t r o l Design

a; E Rr

Now define

Few important properties of the inertia matrix, the centrifugal/Coriolis matrix and the gravity vector need to be mentioned. P 1 I R e l a t i o n s h i p b e t w e e n H;(q;) and Ci(qi,&)

YiGi

4 fii(qi)iir + bi(qir4i)qir + Gi(qi) - H.(". 91d - Ai(ii-4id)) + Ci(qid - Ai(qi3id)) + Gi (12)

= Yidqi, qi,qids qid>iiid)ii

f T ( l ~ H ;- c i ) f

=

o v < E R"'

i=1, ...n

(9)

where a; E R ' is a vector of r robot parameters It is important to note that the regressor matrix YiD is independent of the acceleration of the manipulator. 2. Adaptive C o n t r o l l e r in the Absence of Disturbances Theorem 1 The adaptive controller given below guarantees the global stability in the sense of Lyapunov of the multi-robot system in cooperative motion.

The above property implies that %Hi - c; can be chosen to be skew-symmetric. It should be noted that the choice of c ; is not unique. P 2 I Parameterisation of robot d y n a m i c s Another property which will be used in the subsequent development is the linearity of Hi(qi), Ci(q;,qi), and Gi(qi) with respect to the manipulator dynamic parameters ai. H.( i 91 .)". qi + Ci(qinii)ii

+ Gi(qi) = Yi(qipqi>ii)ai

(11)

ki , ti

We define the augmented trajectory tracking error for the ith robot as, =

a, - a,

The matrices ri E RniXni ,ToE R""'"' , hi E ~ n i X n i and K OP E Rni'(~iii + K HI .I ~. )c;l'(~i t J ~ -Fcipi ~ - G ;- H ~ { ciii ~ ~- ci&)

t

=

If we choose P i= z

(19)

V(t)

=

'I'

,

T.. + %k,Se, + es,,,e,,

e:l

-Szd

- N2

+ CBjF; + lhSeo + K,e,] i= I

t=n

= -

edY,a,,

--+

+ C~,,B~F~ i=

(20)

1

here we have set Yoao

=

Y&,, - io)

sad

Obviously Y, is a function of

+ N~ - %Si, Z,z,Sd,id,'ad

- K,e,

(21)

.

Thus v(t) becomes

I="

1 + e:~,,a, + c~:(B~F~ - -Y,Q

.T

+ a,r,a,

(22)

i= 1

Because of the linearity of H, C, and G with respect to the manipulators parameters, we can write

Substituting the controller given in equation (13), V(t) becomes: i=n I= 1

i=n

-

Y&i

write

i=n

- xeTh3iDAje; i= 1

.

-+

(23 1

Y;uaj = Y i G j - Y;&

V(t) = ,xcT[YiDii - KiDfj+Tireg

- cei Ki&

(27)

(28)

Thus v(t) is a negative semidefinite function. Now using the same arguments as Spong et al. [8] and Slotine and Li [7] we can conclude that ej -+ o a8 t -+ 00 , and ei -+ o as t 00 . Also from equation (7) we can conclude that e, -+ o a8 t -+ 00 Finally because of the rigid grasp, we can conclude that e, --c o as t -+ 00 . ( As the steady state joint position errors tend to zero, the steady state load position error tends to zero ). Thus the proposed control yields a system which is globally stable in the sense of Lyapunov. Remarks 1. Equation (14) has an infinite number of solutions. Thus to get a unique solution, we have to minimize a specific criterion (this is a planning problem). 2. If the parameters are known exactly then the system is globally asymptotically stable. This property can be seen from the fact that e, is dependent on e;, and if e; -+ o => e, 0. This can be proved by a Taylor series expansion of B about !&d. 3. The internal forces which are forces that produce no motion of the object can be controlled through the design of rjreg. This will be addressed in a later paper. 4. It should be noted that Ti and B; are dual transformations (i.e. T ; ~= B~ ). 8. Adaptive Control in the Presence of

,=n

=

A ~ ~ we D , can

i=1

Also it should be noted that k,Sd,

i= 1

I=n

JTF~- Yiua;)

I="

+ i=xe';'P;e; 1

' + Y i ~ i ;+] i=xiiTria; 1

2163

Bounded Disturbances

The previous section considered the control problem of the multi-robot system in the absence of disturbances. The proposed controller does not guarantee the stability of the system and the convergence of the errors to zero when disturbances are present. Peterson and Narendra [lo], and Reed and Ioannou [ l l ] showed that when disturbances are present, a controller can be designed such that errors remain bounded.

The Lyapunov function candidate used in the previous section will be used again

In this section, the effects of bounded disturbances on the multi-robot system will be considered. We will assume that the disturbances enter the multi-robot system through the dynamics of the manipulators. It should be noted that disturbances added to the load can be transformed into disturbances which act on the robots.

i= 1

+ %e,Se, T . + %e;fK,pe, + %azroao

Now if we differentiate V with respect to time, we get i= 1

+ Gi - dj = ~

i +J

P;

=

i=1

i=l

a. - - r:lyTE.I

1 -

where

ci

2 a ; -2

/ ~ [ l b i l ? + A ~ , , I k ; l5~ ] E - d i

{ e ; , e ; for i = 1, ...n

I

I

,+.r;lii I

Bi

fi =

i

O

fiz

< a;, aiz < IIijII< II i j II > 2a;,

if

I="

i= 1

i=1

i=n

+ riiil+ Ce;rPii; i= I

+ i:[YTeo + r,&] + x c T d i i= 1

(34)

It should be noted that bii = & for i = 1, ...n and io = io; this is the case because a; = 0, and io = o as the parameters are assumed to be time invariant. Now using the adaptation laws given by equations (30) and (31) we obtain,

If we choose P i = 2 i z T K i ~ ,we can write,

}

i=l,,,,

Now, it should be mentioned that because of the choice of Ti and fo, we have ciiTii 2 0 , for i = l . . n , and foiTi0 2 o , i.e.

if II

if

(33)

,=U

is defined such that,

/I ai II fiz(---l) aiz

i=n

v(t) = - x c ~ +~ XaTlyTt; ~ ~ c ~

The idea of improving the controller by using a 0 modification scheme was first introduced by Ioannou and Kokotovic [4]. Theorem 2 The control law given by Theorem 1 equations (13) and (14), and with the adaptation laws given by (30) and (31) guarantees boundedness of the states errors to a bounded set D D

.T

using the controller given in theorem 1 , we get

where d; E Rni represents the bounded input disturbance vector. It is assumed that bounds on the disi=l, ...n . turbances, di, are known, I t i l l c di 8.1 Improvement of the controller using the n modification scheme

i=n

i= 1

T .

+ e o S e o + HeoSeo + eTKopeU+ i,T,a,

(29)

I=n

i= 1

T .

Now we will add the disturbance vector d; to the dynamics of the ith manipulator. H;qi + Cjq;

(32)

i=1, ...n

2ai,

fiz is a positive design parameter, and For the load

hill < aiz

io = - r- Iy;fk,- for;lio

Similarly ,+,i:i, 2 0 . Thus we obtain,

where To is defined such

2164

The authors would like to express their sincere gratitude to Dr. John Chiasson for the support provided for Mohamed Zribi.

let pi be the minimum eigenvalue of KiD, be the minimum eigenvalue of Ai, and hi,,, be the maximum eigenvalue of A ~ ,and di be the bounded disturbance vector acting on the ith manipulators such that ~k.~~ll< 2;. Thus we can write,

I="

5

E[-pi Ih; I? - pix?Anlk;IF + lh; I1a; + x,i

Ik,II

Refer en ces S. Ahmad, and H. Guo "Dynamic Coordination of DualArm Robotic Systems With Joint Flexibility," Proc. IEEE. Int. Conf. on Robotics and Automation, Philadelphia, Pennsylvania, pp. 332 - 337, 1988.

ai I

121

i= 1

131

D. S. Bayard, and J. T. Wen, "New Class of Control Laws for Robotic Manipulators: P a r t 2. Adaptive case," Int. J . Control, Vol. 47, No. 5, pp. 1387-1406, 1985. T. C. Hsia "Adaptive Control of Robot Manipulators - A Review," Proceedings of the 1986 IEEE International Conference on Robotics and Automation, San Francisco, CA, pp. 183-189, 1986. P . A. Ioannou and P. V. Kokotovic, Adaptive Syefems un'fh Reduced Modela, New York: Springer-Verlag, 1983. R. Ortega and M. W. Spong, "Adaptive Motion Control of Rigid Robots: a Tutorial" Automatica, Vol. 25, No 6, pp. 877-888, 1989.

where,

01;

1 = --[I 2Pi

H. Seraji, "Decentralized Adaptive Control of manipulators: Theory, Simulation, and Experimentation," IEEE Trans. on Robotics and Automation, Vol. 5, No. 2, pp. 183-201, 1989.

x: n m

+ -I

I

Thus, V(t) < 0 , Vei, ei E D', where D" is the complement of D. This assures the global boundness of ei , e; . 4. Conclusion

J-J. E. Slotine, and W.Li, "Adaptive Manipulator Control: A Case Study," IEEE Trans. Automatic Control, Vol. AC33, NO. 11, pp. 995-1003, 1988. M. W. Spong, R. Ortega and R. Kelly, "Comments on Adaptive Manipulator Control: A Case Study," IEEE Trans. Automatic Control, Vol. AC-35, No. 6, pp. 761-762, 1990.

In this paper, we proposed an adaptive control scheme for the multi-robot system during cooperative motion. The proposed controller takes into account the dynamics of the object and dynamics of the manipulators. Few special properties such as the the linearity of the dynamics with respect to the parameters, were exploited during the derivation of the controller. The proposed controller guarantees global stability in the sense of Lyapunov. The advantage of this controller is that it does not require measurements of joint accelerations and inversion of inertia or Jacobian matrices. Next, we modified the control law to take into account bounded input disturbances. Note that the internal forces can be controlled through the selection of qreg "the load distribution torque" and will be detailed in the future.

X. Yun, T . J. Tarn, and A. K. Bejcay, "Dynamic Coordinated Control of Two Robot Manipulators," Proceedings of 28th Conference on Decision and Control, Tampa, FL., pp. 2476-2481, 1989. B. B. Peterson, and K. S. Narendra, " Bounded Error Adaptive Control," IEEE Trans. Automatic Control, Vol. AC-27, NO. 6, pp. 1161-1168, 1982. J. S. Reed and P . Ioannou, "Instability Analyeis and Robust Adaptive Control of Robotic Manipulators", IEEE. J. of Robotics and Automation, Vol. RA-5, No. 3, pp. 381-386, 1989. T . Yoshikawa and Xinahi Zheng, "Coordinated Dynamic Hybrid Position/Force Control for Multiple Robot Manipulators Handling One Constraint Object," Proceedings of the 1990 IEEE International Conference on Robotics and Automation, Cincinnati, Ohio, 1178-1183, 1990.

We modified the adaptation law by introducing a decay term. The decay term depends on the bounds of the unknown parameters. This adaptation scheme guarantees the convergence of the tracking errors t o a bounded set. In the case of no disturbances, this adaptation law guarantees global stability in the sense of Lyapunov of the system. Acknowledgement

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