Automatica 43 (2007) 842–851www.elsevier.com/locate/automatica

Brief paper

Robust computationally efficient control of cooperative closed-chainmanipulators with uncertain dynamics�

Wail Gueaieba,∗, Salah Al-Sharhanb, Miodrag Bolica

aSchool of Information Technology and Engineering, University of Ottawa, Ottawa, Ont., Canada K1N 6N5bDepartment of Computer Science, Gulf University for Science and Technology, Hawally 32093, Kuwait

Received 23 September 2005; received in revised form 26 July 2006; accepted 27 October 2006Available online 26 March 2007

Abstract

This article presents a decentralized control scheme for the complex problem of simultaneous position and internal force control incooperative multiple manipulator systems. The proposed controller is composed of a sliding mode control term and a force robustifyingterm to simultaneously control the payload’s position/orientation as well as the internal forces induced in the system. This is accomplishedindependently of the manipulators dynamics. Unlike most controllers that do not require prior knowledge of the manipulators dynamics,the suggested controller does not use fuzzy logic inferencing and is computationally inexpensive. Using a Lyapunov stability approach, thecontroller is proven to be robust in the face of varying system’s dynamics. The payload’s position/orientation and the internal force errors arealso shown to asymptotically converge to zero under such conditions.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Coordinated robots; Uncertain systems; Hybrid position and force control; Sliding mode control; Robust control

1. Introduction

Advanced control of cooperative multiple manipulators hasbeen witnessing an increasing interest due to their versatility inthe tasks they can perform and due to their high productivityand potential of reducing the cost in many industrial applica-tions (Canadarm2TM; Special purpose dexterous manipulator).In particular, the bulk of the research has focused on the con-trol aspect of weakly coupled cooperative manipulators. On theother hand, relatively less work has been done in the area ofthe control of tightly coupled manipulators where two or morerobot arms cooperate to simultaneously move a certain objectalong a predefined path. When multiple manipulators grasp acommon object, they form a closed kinematic chain mecha-nism. This implicitly imposes a set of kinematic and dynamicconstraints on the position and velocity of the manipulators as

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor ToshiharuSugie under the direction of Editor Mituhiko Araki.

∗ Corresponding author.E-mail address: wgueaieb@site.uottawa.ca (W. Gueaieb).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2006.10.025

they have to remain in contact with the object while in motion.As a result, the degrees of freedom of the whole system de-crease leading to the generation of internal forces. Such forces,which have to be controlled appropriately, stem from the di-rect interaction between the end-effectors and the object. Asa result, controlling multiple manipulators interacting with anobject, or a given environment, is usually a much more com-plex problem than that of a single robot control. The motionof the manipulators has to be kinematically and dynamicallycoordinated. The kinematic coordination refers to the fact thatall the end-effectors involved have to move synchronously totrack a certain prespecified desired position and orientation ofthe manipulated object without losing contact of it. In the con-text of this paper, the dynamic coordination means that theend-effectors have to move in a certain manner as to controlthe internal forces induced between the payload and the end-effectors. Because of that and because of the high complexityof closed-chain robotic systems, in general, the efficient controlof such systems still stands as one of the challenging problemsin the field of robotics control.

Since the emergence of the first cooperative manipulatormodels, the control problem of such systems has drawn the

W. Gueaieb et al. / Automatica 43 (2007) 842–851 843

attention of several researchers (Fujii & Kurono, 1975). Tarn,Bejczy, and Yun (1987) proposed a linear transformationmethod that transforms the nonlinear dynamic equations of thesystem to a linear one with decoupled matrix-form equations.A nonlinear feedback controller was proposed in Yun (1989),and a controller based on a PD plus gravity compensationwas discussed in Wen and Kreutz (1989). Lately, a distributedimpedance control technique was proposed in Szewczyk,Plumet, and Bidaud (2002). Such types of controllers alongwith most non-adaptive control schemes for the coordinatedcontrol of multiple manipulator systems, usually assume afull knowledge of the system’s dynamics (Yukawa, Uchiyama,Nenchev, & Inooka, 1996). Although this is true for somecases, it might not be for many others since complex systems,in general, are usually subject to the ubiquitous presence ofuncertainties (Namvar & Aghili, 2005). These uncertaintiesmay have a dramatic effect on the controller’s performance andmay even induce instability. To deal with such uncertainties,several adaptive control schemes were proposed (Chiu, Lian,& Wu, 2004; Damaren, 2003; Hu & Goldenberg, 1993; Liu &Arimoto, 1998; Parra-Vega, Arimoto, Liu, Hirzinger, & Akella,2003; Sun & Mills, 2002; Tseng & Chen, 2003; Vukobratovic& Tuneski, 1999). These control algorithms approximate thesystem’s dynamics using a continuous online estimation ofa set of the plant’s physical parameters through well-definedadaptation laws. For it to provide a satisfactory performance,a typical adaptive control algorithm assumes that the dynamicmodel is perfectly known and free of significant external (un-modeled) disturbances. In other words, the controller is onlyrobust to parametric, or structured, uncertainties and to minorunstructured uncertainties. In addition, the unknown physicalparameters must have a constant or slowly varying nominalvalues. An explicit linear parameterization of the uncertaindynamics parameters has also to exist, and even if it does, itmight not be trivial especially with complex dynamic systems.Although the latter condition is guaranteed for every roboticdynamic equation, it might not be the case for many othersystems. It is worth mentioning that all the aforementionedconditions are necessary but may not be sufficient for a satis-factory performance and stability of a large number of adaptivecontrollers.

Modeling imperfection of complex systems, such as closed-chain robotic manipulators, is inevitable in most cases. Thismakes the development of a robust control approach for theincreasingly complex cooperative manipulator systems a nec-essary step to keep up with the increasingly demanding designrequirements of such systems. Most efforts for compensatingfor modeling uncertainties have focused on the use of fuzzylogic-based controllers given their high credibility in control-ling ill-defined systems (Karray & de Silva, 2004). In spite ofthe significant advances made in this direction (Burn, Short,& Bicker, 2003; Chang & Chen, 2000, 2005; Gueaieb, Karray,& Al-Sharhan, 2001, 2003; Hsu & Fu, 2000; Kim, 2004; Yoo& Ham, 2000), the proposed fuzzy logic controllers still sufferfrom their relatively high computational complexities, whichmakes their integrability within an embedded control system achallenge in itself.

Even with the increasing number of control systems proposedin the literature for the control of closed-chain manipulatorsystems, overlooking the dynamic coordination of the manip-ulators and failing to study the controller’s stability criteria inthe existence of parametric and modeling uncertainties are stillamong the common deficiencies in designing these control sys-tems. On the other hand, most of the controllers that take intoaccount the manipulators dynamic coordination (hybrid posi-tion/force controllers), divide the robots working space into twoorthogonal sub-spaces in which position/orientation and forceare independently controlled as they happen to be in the nullcontrol space of each other. Although the sub-space orthogo-nality condition may be satisfied in a number of applications,it is not generally the case for strongly coupled multiple ma-nipulator systems, and hence such control techniques becomeinapplicable.

In this manuscript, we extend the pioneering work previouslypresented in Parra-Vega et al. (2003) to the control of closed-chain multi-manipulator systems with no a priori knowledge ofthe system’s dynamics. In here we propose a decentralized hy-brid position/force tracking controller whose objectives are totrack a predefined desired position and orientation of the pay-load, while controlling the internal forces of the closed-chainsystem and make them converge to their predefined desiredvalues. The proposed controller is based on a decentralizedadaptive control scheme forcing each robot to operate indepen-dently of the others saving the communication time overheadbetween the robots. This full autonomy of the controller is oneof the main reasons behind its high computational efficiency,which makes its embedding and real-time execution within anintegrated control system quite easy to achieve. In addition, theproposed controller does not require the orthogonality of theend-effectors position and the internal force control sub-spacessaving the need to decouple the internal force and the payload’sposition feedback signals. The controller is also independent ofthe manipulators dynamics, which makes it suitable for a largevariety of cooperative manipulator systems.1 The controller’sflexibility extends to giving the designer the luxury of notusing internal force feedback signals. In this case, mountingforce sensors at the manipulators’ wrists becomes unnecessary,which makes the realization of the controller in a real-worldcooperative manipulator system simpler and more economi-cal. The rest of the paper is organized as follows. Section 2formally defines the control problem in hand and outlinesthe dynamic and kinematic modeling aspects of coordinatedclosed-chain manipulator systems. The formulation of theproposed controller and its stability analysis are discussed inSection 3. Numerical simulations are carried out to assess theperformance of the proposed controller and to illustrate its ro-bustness and ability to meet the predefined goals independentlyof the manipulators dynamics. The results are illustrated andanalyzed in Section 4. Eventually, we provide a brief summary

1 It should be understood that although the proposed controller is inde-pendent of the manipulators dynamics, porting it to different cooperative ma-nipulators with different dynamics might need fine tuning of the controller’sparameters to maintain a satisfactory performance.

844 W. Gueaieb et al. / Automatica 43 (2007) 842–851

of the main contributions of this work and conclude with afew concluding remarks in Section 5.

2. Kinematics and dynamics of cooperative manipulators

2.1. Problem statement

Consider m cooperative manipulators handling a commonobject as shown in Fig. 1. The objectives of the coordinatedrobots are to simultaneously (i) move that object so that its cen-ter of mass tracks a predefined trajectory (position and orienta-tion), and (ii) control the internal forces between the object andthe end-effectors so that they converge to their predetermineddesired values. These objectives are to be achieved within adecentralized control scheme and with no prior knowledge ofthe robot and payload’s dynamics. That is, the control law �i

for each robot i, for i = 1, . . . , m, has to be independent ofthe other robot parameters. To facilitate the formulation of theclosed-chain system’s dynamics, the payload is supposed to berigidly grasped by the end-effectors. In other words, there is norelative motion between the end-effectors and the object in or-der not to increase the system’s degrees of freedom. Also, themanipulators forward kinematics are supposed to be known.The manipulators inverse kinematics, on the other hand, are notneeded as they are not used by the controller.

2.2. Kinematics

At any instant of time, the location of the manipulated ob-ject can be defined by the vector x =[xT

(p), xT(o)]T ∈ Rk0 , where

x(p) ∈ Rp and x(o) ∈ Ro are vectors defining the position andorientation of the object, respectively. The superscript integersp and o belong to the set {0, 1, 2, 3}, where R0 denotes theempty vector, and k0 (k0 =p+o) is the object’s total degrees offreedom. In what follows, all positions and orientations of theobject and the end-effectors coordinates are expressed relative

E2

Em

E1

Robot m

Robot 2

Robot 1

Object

Fig. 1. Multiple cooperative robots manipulating a common object.

to a common reference frame unless otherwise stated. Us-ing the manipulators forward kinematics equations, x can berewritten as

x = �1(q1) = �2(q2) = · · · = �m(qm), (1)

where qi ∈ Rki and ki are the respective joint coordinates andthe degrees of freedom of robot i, for i = 1, . . . , m. Differenti-ating Eq. (1) with respect to time yields

x = �i (qi) = J�i(qi)qi , i = 1, . . . , m, (2)

where J�i(qi) ∈ Rk0×ki is the Jacobian matrix from the object’s

center of mass to qi . Differentiating Eq. (2) with respect to timeresults in

x = J�i(qi)qi + J�i

(qi)qi , i = 1, . . . , m. (3)

Eqs. (1)–(3) represent the kinematics equations of cooperativerobotic systems.

2.3. Dynamics

In a closed-chain robotic system, the dynamics of the ithmanipulator in the joint space is given by

Mi(qi)qi + Qi(qi, qi)qi + Wi(qi) − J Tei(qi)Fei

= �i , (4)

where �i ∈ Rki denotes the joint torque/force applied by theactuators on the ith manipulator, Mi(qi) ∈ Rki×ki is the pos-itive definite inertial matrix, Qi(qi, qi) ∈ Rki×ki is a matrixrepresenting the Coriolis and centrifugal forces and joint fric-tion coefficients, Wi(qi) ∈ Rki represents the vector of grav-itational forces, Jei

(qi) ∈ Rk0×ki is the manipulator Jacobianmatrix from the end-effector Ei to qi , and Fei

∈ Rk0 is theforce exerted by the object on the end-effector Ei . The dynam-ics equation of the object in the task space is given by

M0(x)x + Q0(x, x)x + W0(x) = F0, (5)

where M0(x) ∈ Rk0×k0 is the object’s symmetric positive def-inite inertial matrix, Q0(x, x) ∈ Rk0×k0 denotes the object’sCoriolis and centrifugal terms and the coefficient terms due tothe friction between the object and the environment. The vectorW0(x) ∈ Rk0 represents the gravitational force acting on theobject, and F0 ∈ Rk0 is the resulting force of the m manipula-tors acting on the object’s center of mass. The force F0 can beexpressed as F0 = −∑m

i=1Fcei, where Fcei

∈ Rk0 is the force“indirectly” applied by the object’s center of mass on the ithmanipulator, and is known as the interaction force. The forcesFcei

and Feiare related by

Fcei= J T

cei(x)Fei

, (6)

where Jcei(x) ∈ Rk0×k0 is the Jacobian matrix from the

end-effector Ei to the object’s center of mass. Noticethat the Jacobian matrices J�i

(qi), Jei(qi), and Jcei

(x),are related by Jei

(qi) = Jcei(x)J�i

(qi), i = 1, . . . , m. Theforce Fcei

can be regarded as the sum of an internalforce fi and an external force �i .

Fcei= fi + �i . (7)

W. Gueaieb et al. / Automatica 43 (2007) 842–851 845

From the property of internal forces it is known that

m∑i=1

fi = 0. (8)

The internal forces cancel each other and only external forcescontribute to the motion of the object. Hence, Eq. (5) can nowbe rewritten as

M0(x)x + Q0(x, x)x + W0(x) = −m∑

i=1

�i . (9)

The external force �i can then be formulated as

�i = −ci(t)(M0(x)x + Q0(x, x)x + W0(x)), (10)

where t �0 denotes the time variable, and ci(t) ∈ Rk0×k0 isa positive definite diagonal matrix representing the load dis-tribution of the object onto the ith manipulator. An importantphysical property of load distribution matrices is that they sumup to the identity matrix Ik0 ∈ Rk0×k0 . That is,

m∑i=1

ci(t) = Ik0 for t �0. (11)

Substituting Eq. (9) into (10) results in

�i = ci(t)

m∑j=1

�j . (12)

Substituting Eq. (12) into (7) yields fi = Fcei− ci(t)

∑mj=1�j .

To support the following derivation of the manipulators’dynamics, we will consider ci(t) to be in the form of ci(t) =�i (t)Ik0 , where �i (t) is a positive real scalar. Hence, usingEqs. (4), (6), (7), (12), and the kinematics equations (1), (2),and (3) the dynamics equation of the ith manipulator can berewritten as

�i = Di(qi)qi + Ci(qi, qi )qi + Gi(qi) − J T�i

(qi)fi , (13)

where

Di(qi) = Mi(qi) + ci(t)M0(qi),

Gi(qi) = Wi(qi) + ci(t)W0(qi),

Ci(qi, qi ) = Qi(qi, qi) + ci(t)(M0(qi) + Q0(qi, qi )),

M0(qi) = J T�i

(qi)M0(x)J�i(qi), W0(qi) = J T

�i(qi)W0(x),

Q0(qi, qi ) = J T�i

(qi)Q0(x, x)J�i(qi).

The matrix {2Ci(qi, qi )− (Di(qi)− ci (t)Q0(qi, qi ))} is a skewsymmetric matrix. In other words, ∀r ∈ Rki ,

rT{2Ci(qi, qi ) − (Di(qi) − ci (t)Q0(qi, qi ))}r = 0. (14)

Assumption 1. The matrix (ci(t)Q0(qi, qi )) is uniformly con-tinuous and bounded. Thus, there exists a positive constant �i

such that

12‖ci (t)Q0(qi, qi )‖��i ∀t �0. (15)

Since the matrices Di(qi), Ci(qi, qi ), and Gi(qi), are linearin terms of the manipulators physical parameters, the first termsof the dynamics equation (13) can be rewritten as

Di(qi)qri + Ci(qi, qi )qri + Gi(qi)

= Yi(qi, qi , qri , qri , ci(t))�i (16)

for i = 1, . . . , m, where �i ∈ Rk�i is a vector of the phys-ical parameters (mass, moments of inertia, friction coeffi-cients, etc.) of the ith manipulator and the payload, k�i

isa positive integer denoting the number of such parameters,Yi(qi, qi , qri , qri , ci(t)) ∈ Rki×k�i is the regression matrix,which is independent of the physical parameters in �i , andqri ∈ Rki is a design nominal reference signal to be deter-mined later. Substituting (16) into (13) yields the followingerror dynamics equation:

Di(qi)sri + Ci(qi, qi )sri

= �i − Yi(qi, qi , qri , qri ci(t))�i + J T�i

(qi)fi (17)

with sridef= q − qri (∈ Rki ) being a residual error signal corre-

sponding to robot i.

3. Controller design

Consider the following joint reference signal of robot i:

qridef= J+

�i(qi)(xd − �i x) + sdi

− �ii

with

i = sgn(sqi). (18)

The following residual errors corresponding to the ith robot arethen defined as

sidef= J+

�i(qi)( ˙x + �i x), (19)

sdi

def= si(t0)e−i (t−t0), (20)

sqi

def= si − sdi= sri − �ii , (21)

where J+�i

(qi)def= J T

�i(qi)(J�i

(qi)JT�i

(qi))−1 is the pseudo-

inverse of the Jacobian matrix J�i(qi), i is a positive con-

stant, �i and �i are positive definite matrices, xd is the desiredposition of the object’s center of mass, and x is the corre-

sponding position error, i.e., xdef= x − xd . Note that J+

�i(qi)

exits only if robot i is not at a singularity position, i.e., only ifrank[J�i

(qi)] = k0.

Assumption 2. At every instant of time,

rank[J�i(qi)] = k0, i = 1, . . . , m. (22)

Consider the following control law corresponding to the ithmanipulator:

�i = −Kdisri − J T

�i(qi)(fdi

− Kfifi) (23)

846 W. Gueaieb et al. / Automatica 43 (2007) 842–851

for i=1, . . . , m, where Kdiand Kfi

are positive definite square

matrices of appropriate dimensions and fidef= fi − fdi

is theinternal force error at manipulator i.

Lemma 1. If, for i = 1, . . . , m, sri , qi , J�i(qi), and

Yi(qi, qi , qri , qri ci(t))�i , are bounded, then fi is bounded.

Proof. Premultiplying the residual error signal sri by J�i(qi)

results in

J�i(qi)sri = J�i

(qi)si + J�i(qi)[−sdi

+ �ii]= ( ˙x + �i x) + J�i

(qi)[−sdi+ �ii].

Taking the time-derivative of both sides leads to

J�i(qi)sri + J�i

(qi)sri = ( ¨x + �i˙x) + J�i

(qi)[−sdi+ �ii]

+ J�i(qi)[−sdi

+ �i i].Thus,

J�i(qi)sri = ( ¨x + �i

˙x) + J�i(qi)(−si)

+ J�i(qi)[−sdi

+ �i sgn(sqi)].

Now, premultiplying the error dynamics (17) by J�i(qi)

D−1i (qi) and substituting control law (23), yields

J�i(qi)D

−1i (qi)J

T�i

(qi)(Kfi+ 1)fi

= J�i(qi)sri + J�i

(qi)D−1i (qi)(Kdi

+ Ci(qi, qi ))sri

+ J�i(qi)D

−1i (qi)Yi(qi, qi , qri , qri ci(t))�i .

Letting D′i (qi) = J�i

(qi)D−1i (qi)J

T�i

(qi)(Kfi+ 1) leads to

D′i (qi)fi = ( ¨x + �i

˙x) + bisi + ui , (24)

where

bi = J�i(qi)D

−1i (qi)(Kdi

+ Ci(qi, qi )) − J�i(qi),

ui = J�i(qi)D

−1i (qi)[(Kdi

+ Ci(qi, qi ))(−sdi+ �ii )

+ Yi(qi, qi , qri , qri ci(t))�i]+ J�i

(qi)[−sdi+ �i sgn(sqi

)].

By computing [D′1(q1)f1 − D′

i (qi)fi] for i = 2, . . . , m, andfrom the internal force property (8), we obtain R(q)f = Q(t),

where f T = [f T1 , . . . , f T

m], qT def=[qT1 , . . . , qT

m], and R(q) andQ(t) are defined as follows:

R(q) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

D′1(q1) −D′

2(q2) 0 · · · 0

D′1(q1) 0 −D′

3(q3) · · · 0

......

......

...

D′1(q1) 0 0 · · · −D′

m(qm)

I I I · · · I

⎤⎥⎥⎥⎥⎥⎥⎥⎦

,

Q(t) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

(b1s1 − b2s2) + (u1 − u2)

(b1s1 − b3s3) + (u2 − u3)

...

(b1s1 − bmsm) + (u1 − um)

0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

From the structure of matrix R(q) and since Di(qi),i = 1, . . . , m, is positive definite, it can be deduced thatR(q) has full rank. The matrices J�i

(qi) and D−1i (qi) are

bounded. Thus, R(q) and R−1(q) are also bounded. The ma-trix Ci(qi, qi ) is bounded for a bounded joint velocity qi .Since J�i

(qi) is bounded, the term bi must also be bounded.sdi

�si(t0), so sdiis bounded. sri is bounded which implies

that si and i are bounded as well. sdi� − i si(t0), so sdi

isbounded. Since Yi(qi, qi , qri , qri ci(t))�i is bounded, the termui must also be bounded. Therefore, Q(t) is bounded, whichimplies that f = R−1(q)Q(t) is bounded. �

Theorem 2. Consider the robot dynamics (13) and control law(23). If, for i=1, . . . , m, Kdi

and the eigenvalues of �i are set tobe large enough (more details in the proof), then the payload’sposition/orientation error x exponentially converges to zero.

Proof. Consider the Lyapunov function candidate V =∑mi=1Vi ,

where Vi = 12 sT

riDi(qi)sri . Since Di(qi) is a positive definite

matrix, it becomes clear that Vi , and hence V, are non-negativescalars:

Vi = sTriDi(qi)sri + 1

2 sTriDi(qi)sri .

From error dynamics (17),

Vi = sTri[−Ci(qi, qi )sri + �i − Yi(qi, qi , qri , qri ci(t))�i

+ J T�i

(qi)fi] + 12 sT

riDi(qi)sri .

Substituting (23) and using (14) leads to

Vi = sTri[−Kdi

sri − J T�i

(qi)(fdi− Kfi

fi)

− Yi(qi, qi , qri , qri ci(t))�i + J T�i

(qi)fi]+ 1

2 sTrici (t)Q0(qi, qi )sri

= sTri[(−Kdi

+ 12 ci (t)Q0(qi, qi ))sri

− Yi(qi, qi , qri , qri ci(t))�i]+ sT

riJ T�i

(qi)(1 + Kfi)fi .

From (21), we get

Vi = sTri[(−Kdi

+ 12 ci (t)Q0(qi, qi ))sri

− Yi(qi, qi , qri , qri ci(t))�i]+ (J�i

(qi)si)T(1 + Kfi

)fi

+ [J�i(qi)(�ii − sdi

)]T(1 + Kfi)fi

= Vi1 + Vi2 + Vi3,

W. Gueaieb et al. / Automatica 43 (2007) 842–851 847

where

Vi1 = sTri[(−Kdi

+ 12 ci (t)Q0(qi, qi ))sri

− Yi(qi, qi , qri , qri ci(t))�i],Vi2 = (J�i

(qi)si)T(1 + Kfi

)fi ,

Vi3 = [J�i(qi)(�ii − sdi

)]T(1 + Kfi)fi .

Eq. (19) yields Vi2=( ˙x+�i x)T(1+Kfi)fi . Let Kf be an upper

bound of Kfifor i = 1, . . . , m, i.e., Kf

def= maxi=1,...,m Kfi.

Thus,

m∑i=1

Vi2 = ( ˙x + �i x)T

(m∑

i=1

(1 + Kfi)fi

)

�( ˙x + �i x)T(1 + Kf )

(m∑

i=1

fi

).

Taking into account property (8) of the system’s internal forcesleads to

∑mi=1 fi = 0, and so

∑mi=1 Vi2 �0.

In addition, �ii =�i

∫ t

t0sgn(sqi

(�)) d���i (t−t0). Similarly,(20) implies that sdi

�si(t0). Therefore, there exists a state-independent constant vector � ∈ Rki such that J�i

(qi)(�ii −sdi

)� �. Hence,

m∑i=1

Vi3 � �Tm∑

i=1

((1 + Kfi)fi)

�(1 + Kf )�Tm∑

i=1

(fi) = 0.

It can be easily concluded following the results in Parra-Vegaet al. (2003) that for a bounded payload’s desired trajectory, xd ,Yi(qi, qi , qri , qri ci(t))�i � i (t), where i (t) ∈ Rki is a time-variant state-dependent vector.

Let Kidef= Kdi

− �iIki, where �i is defined in (15). If Kdi

ischosen in such a way that Kdi

> �iIki, then Ki is a positive

definite matrix. Hence, there exists a matrix Ki1 such that Ki =KT

i1Ki1. Thus, Vi1 may be rewritten as

Vi1 = − sTriKisri − sT

riYi(qi, qi , qri , qri ci(t))�i

� − ‖Ki1sri ‖2 + ‖sri ‖ i (t).

Hence, setting Ki1 > ‖ i (t)‖ guarantees Vi1, and so V , to benon-positive. Therefore, Vi1 and V are bounded functions. Thisimplies that sri is bounded. Using (21) and (22), it follows thati , ˙x, x, ˙qi , and qi , are bounded. Lemma 1 then implies thatfi is bounded for i = 1, . . . , m. Therefore, it can be concludedfrom Eq. (17) that sri is bounded, i.e., there exists �i ∈ Rki

such that sri � �i , for i = 1, . . . , m.So far, we have proved that the system’s control signals are

bounded. The rest of the proof is dedicated to proving that asliding mode is induced on the surface sqi

=0 with equilibriumat qi = ˙qi = 0, for i = 1, . . . , m.

Consider the energy function corresponding to robot i, Vqi=

12 sT

qisqi

. From (18) and (21) we get sqi= sri − �i sgn(sqi

). Let

�im be the minimum eigenvalue of �i and �iN

def= ‖�i‖. Hence,

Vqi= sT

qisqi

= sTqi

(sri − �i sgn(sqi))

�‖sqi‖‖sri ‖ − �im‖sqi

‖= (�iN − �im)‖sqi

‖ = −�i‖sqi‖,

where �idef= �im − �iN . Choosing �i large enough so that �i > 0,

enforces a sliding mode with sliding mode condition sTqi

sqi� −

�i‖sqi‖ (Parra-Vega et al., 2003). The sliding mode is estab-

lished at time tsi = sqi(t0)/�i =0, since sqi

(t0)=0. This impliesthat the payload’s trajectory/orientation tracking errors are al-ways constrained within a manifold that represents an expo-nential solution towards trajectory/orientation, xd . Therefore,limt→∞ x = 0 and limt→∞ ˙x = 0. �

Lemma 1 and Theorem 2 show that using control law (23) forthe cooperative robotic system defined by (13) leads to asymp-totic convergence of the payload’s position and orientation totheir respective predefined desired values. In the following, wewill show that it also forces the internal force errors to asymp-totically decay to zero.

Theorem 3. If a common gain �i = � for all m robots, thencontroller (23) also leads to the asymptotic converge of theinternal force error, fi , to zero, for i = 1, . . . , m.

Proof. Using (24) and since D′i (qi) is non-singular, the internal

force error, fi , can be expressed as

fi = (D′i (qi))

−1( ¨x + �i˙x)

+ (D′i (qi))

−1bisi + (D′i (qi))

−1ui . (25)

Using the internal force property (8), we get

limt→∞

m∑i=1

[(D′i (qi))

−1( ¨x + �i˙x) + (D′

i (qi))−1bisi

+ (D′i (qi))

−1ui] = limt→∞

m∑i=1

fi = 0.

This implies that

limt→∞

m∑i=1

(D′i (qi))

−1( ¨x + � ˙x) + limt→∞

m∑i=1

(D′i (qi))

−1bisi

+ limt→∞

m∑i=1

(D′i (qi))

−1ui = 0.

It is shown in Theorem 2 that limt→∞ x =0 and limt→∞ ˙x =0.Then, limt→∞ ¨x = 0 and limt→∞ si = 0. Hence, the aboveequation implies that

limt→∞

m∑i=1

(D′i (qi))

−1ui = 0. (26)

848 W. Gueaieb et al. / Automatica 43 (2007) 842–851

The matrix D′i (qi), and so (D′

i (qi))−1, is linearly indepen-

dent of ui . Hence, the joint velocity qi , and hence the ma-trix (D′

i (qi))−1, can be continuously varying in time and

so may assume an infinite number of possible values fori = 1, . . . , m. Since Eq. (26) has to hold for all those val-ues and since, both, (D′

i (qi))−1 and ui are bounded, the

only solution to (26) is limt→∞ ui = 0. Going back to(25) leads to limt→∞ fi = limt→∞(D′

i (qi))−1( ¨x + �i

˙x) +limt→∞(D′

i (qi))−1bisi + limt→∞(D′

i (qi))−1ui = 0. �

It is important to mention that the proposed control schemedoes not explicitly compensate for external disturbances.However, it does compensate for the viscous friction in themanipulators’ joints as it can be implicitly incorporated withinmatrix Qi(qi, qi) in (4). Although this may also apply to otheradaptive algorithms (Chiu et al., 2004; Damaren, 2003; Hu &Goldenberg, 1993; Liu & Arimoto, 1998; Parra-Vega et al.,2003; Sun & Mills, 2002; Tseng & Chen, 2003; Vukobratovic& Tuneski, 1999), the main advantage of the proposed tech-nique lies in its independence of a precise explicit mathemat-ical model of the system’s dynamics without the need of acomputationally demanding soft computing engine.

It is also worth pointing out that it is possible to chooseKfi

= 0, for i = 1, . . . , m, in which case controller (23) be-comes independent of fi , and hence it would be unnecessaryto measure the internal force fi at manipulator i. This makesthe use of noisy force sensors unnecessary provided there isa certain mechanism to ensure that the manipulators remainin contact with the object. Although this is possible, the rateof convergence of fi to zero is proportional to (Kfi

+ 1).Hence, a larger Kfi

would lead to a faster convergence of fi

to zero.

4. Numerical results and discussion

4.1. Experimental setup

To demonstrate the performance of the proposed controller(23), numerical simulations are carried out on two 3-DOF iden-tical Adept manipulators. Both manipulators are to cooperateto handle an object whose mass m=11.2 kg, length d =0.24 m,moment of inertia Ic = 1

12md2, and stiffness kc = 300 kN/m. Itis important to bring to the reader’s attention the fact that theobject’s stiffness is only assumed to be known to compute the

Table 1Values of the manipulators physical parameters

Manipulator’s parameter Value

Ij , j = 1, 2, 3—rotational inertia of link j I1 = 1.1 kg m2, I2 = 0.3 kg m2, I3 = 0.01 kg m2

m∗j , j = 1, 2, 3—mass of link j m∗

1 = 30 kg, m∗2 = 15 kg, m∗

3 = 5 kgIcol.—rotational inertia of the inner transmission column for joint 2 Icol. = 0.57 kg m2

l1 and l2—lengths of links 1 and 2 l1 = 0.425 m, l2 = 0.375 mr2—distance of the center of mass of link 2 from the axis of joint 2 r2 = 0.165 mVVj , j = 1, 2, 3—viscous friction coefficients VVj = 1.2VSj , j = 1, 2, 3—Coulomb friction coefficients VSj = 0.01

contact and the internal forces during the simulation. In areal robotic system, and whenever needed, such forces aredetermined through means of force sensors mounted at themanipulators wrists. The manipulators physical parametersare summarized in Table 1. The bases of the two ma-nipulators are located at (X1(base), Y1(base)) = (0, 0) and(X2(base), Y2(base)) = (1, 0), respectively. The manipulated ob-ject is set to move in a straight line in the horizontal planefollowing the desired trajectory:⎛⎜⎝

xd

yd

�d

⎞⎟⎠=

⎛⎜⎝

0.4 − 0.2e(−t/6.6) + 0.5e(−t/6.8)

0.4√3

− 0.2 − 0.2√3

e(−t/6.6) + 0.5√3

e(−t/6.8)

�/3

⎞⎟⎠ ,

where �d denotes the object’s desired orientation in rad,(xd, yd)T is the desired position of its center of mass in meters,and t denotes the time index in seconds. The object initiallystarts at (0.6, (0.6/

√3) − 0.2, �/3 − 0.1)T, that is (0.1 m,

0.1 m, 0.1 rad)T off its desired initial position.The manipulators dynamics are defined as follows:

Mi(qi) =⎡⎢⎣

a b�i 0

b�i c I3

0 I3 I3

⎤⎥⎦ , Wi(qi) =

⎛⎜⎝

VS1 sign(qi1)

VS2 sign(qi2)

VS3 sign(qi3)

⎞⎟⎠ ,

Qi(qi, qi) =⎡⎢⎣

VV1 −b�i qi2 0

b�i qi1 VV2 0

0 0 VV3

⎤⎥⎦ ,

where i =1, 2, �i =cos(qi2 −qi1), �i =sin(qi2 −qi1), a=I1 +m∗

2l21 +m∗

3l21 , b=m∗

2l1r2 +m∗3l1l2, c= Icol. + I2 + I3 +m∗

2r22 +

m∗3l

22 . The position and the orientation of end-effectors E1 and

E2 are defined by the following forward kinematics equation:⎛⎜⎝

xie

yie

�ie

⎞⎟⎠=

⎛⎜⎝

l1 cos qi1 + l2 cos qi2 + Xi(base)

l1 sin qi1 + l2 sin qi2 + Yi(base)

qi2 + qi3

⎞⎟⎠ .

The load distribution matrices are set to c1(t) = c2(t) =diag(0.5, 0.5, 0.5). The desired internal forces are given thevalues of fd1 =−fd2 =−10 N acting on the line connecting thetwo end-effectors. The manipulators are also set to have a jointfriction term in the form of Fr(qi)= diag(0.15, 0.06, 0.015)qi ,for i = 1, 2. The controller is programmed to operate at 80 Hz.

W. Gueaieb et al. / Automatica 43 (2007) 842–851 849

0 5 10 15 200.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Time (s)

x (

m)

Desired

Actual

0 5 10 15 200.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Time (s)

y (

m)

Desired

Actual

0 5 10 15 2010

20

30

40

50

60

70

80

90

Time (s)

ψ (

deg)

Desired

Actual

0 5 10 15 20−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

Time (s)

Inte

rnal F

orc

e f

1 (

N)

Desired

Actual

Fig. 2. Tracking errors of the payload’s x–y position, orientation, and internal force.

4.2. Experimental results

The performance of the proposed controller is depicted inFigs. 2 and 3. It can be seen from Fig. 2 that the controller is ableto deem down the errors of the payload’s position/orientationand those of the internal forces to zero within about 2 s. To betterillustrate the robustness of the controller, an unknown externaldisturbance is instantaneously applied to the system halfwaythrough the payload’s trajectory, i.e.,at t = 10 s. At this time,the payload’s mass is also dropped to half its original value tobe 5.6 kg, which results in an abrupt change of the payload’sinertia as well. Once again, and as Fig. 2 shows, it took about2 s for the controller to bring all control signals back to theirdesired values.

5. Conclusion

In this article, a decentralized hybrid position/force controlscheme is proposed for the control of multiple tightly coupled

manipulators handling a common object. The controller isproven to track, both, the payload’s position/orientation and thesystem’s internal forces to their respective desired values evenwhen they share the same lines of action. This is accomplishedwithout a prior knowledge of the system’s dynamics, whichmakes the controller highly modular and portable to varioustypes of cooperative manipulators with different dynamics.Thanks to its decentralized architecture and its independenceof any fuzzy logic inferencing mechanism, unlike many othercontrollers suggested in the literature for similar control objec-tives, the proposed controller is computationally inexpensive.This drastically eases its embedding and enhance its real-timeexecution performance within an integrated control system.In addition, it offers the flexibility of not using the internalforce feedback signals, which makes its real-world realizationsimpler and more economical, since it saves the need for forcesensors in this case. Finally, numerical simulations are carriedout to illustrate the validity of the proposed controller and toconfirm its proven anonymous position/force tracking ability

850 W. Gueaieb et al. / Automatica 43 (2007) 842–851

Manipulator 1 Manipulator 2

Jo

int

1

0 5 10 15 2080

60

40

20

0

20

40

60

80

Time (s)

Torq

ue a

t Join

t 1 (

N.m

)

0 5 10 15 20−60

−40

−20

0

20

40

60

80

100

120

140

Time (s)

Torq

ue a

t Join

t 1 (

N.m

)

Join

t 2

0 5 10 15 20−60

−40

−20

0

20

40

60

80

Time (s)

Torq

ue a

t Join

t 2 (

N.m

)

0 5 10 15 20−80

−60

−40

−20

0

20

40

60

Time (s)

Torq

ue a

t Join

t 2 (

N.m

)

Join

t 3

0 5 10 15 20

−60

−40

−20

0

20

40

60

80

100

120

Time (s)

Torq

ue a

t Join

t 3 (

N.m

)

0 5 10 15 20−100

−80

−60

−40

−20

0

20

40

Time (s)

Torq

ue a

t Join

t 3 (

N.m

)

Fig. 3. Controller’s output torque.

W. Gueaieb et al. / Automatica 43 (2007) 842–851 851

under the aforementioned operating conditions. A potentialfuture research avenue to extend this work is to consider multi-fingered manipulators where fingertips just contact the objectinstead of rigidly grasping it.

References

Burn, K., Short, M., & Bicker, R. (2003). Adaptive and nonlinear fuzzy forcecontrol techniques applied to robots operating in uncertain environments.Journal of Robotic Systems, 20, 391–400.

Canadarm2TM. URL: http://www.mdrobotics.ca/ssrms_frame.htmlChang, Y. C., & Chen, B. S. (2000). Robust tracking designs for both

holonomic and nonholonomic constrained mechanical systems: Adaptivefuzzy approach. IEEE Transactions on Fuzzy Systems, 8(1), 46–66.

Chang, Y. C., & Chen, B. S. (2005). Intelligent robust tracking controlsfor holonomic and nonholonomic mechanical systems using only positionmeasurements. IEEE Transactions on Fuzzy Systems, 13(4), 491–507.

Chiu, C.-S., Lian, K.-Y., & Wu, T.-C. (2004). Robust adaptive motion/forcetracking control design for uncertain constrained robot manipulators.Automatica, 40(12), 2111–2119.

Damaren, C. J. (2003). An adaptive controller for two cooperating flexiblemanipulators. Journal of Robotic Systems, 20(1), 15–21.

Fujii, S., & Kurono, S. (1975). Coordinated computer control of a pairof manipulators. Proceedings of the fourth World congress of theinternational federation for theory of machines and mechanisms (IFToMM)(pp. 411–417).

Gueaieb, W., Karray, F., & Al-Sharhan, S. (2001). An adaptive fuzzy controlapproach for cooperative manipulators. In Proceedings of the internationalsymposium on intelligent control (ISIC’01) (pp. 167–172). Mexico City,Mexico.

Gueaieb, W., Karray, F., & Al-Sharhan, S. (2003). A robust adaptivefuzzy position/force control scheme for cooperative manipulators. IEEETransactions on Control Systems Technology, 11(4), 516–528.

Hsu, F. Y., & Fu, L. C. (2000). Intelligent robot deburring using adaptivefuzzy hybrid position/force control. IEEE Transactions on Robotics andAutomation, 16(4), 325–335.

Hu, Y. R., & Goldenberg, A. A. (1993). An adaptive approach to motion andforce control of multiple coordinated robots. Transactions of the ASME,Journal of Dynamic Systems Measurement and Control, 115(1), 60–69.

Karray, F., & de Silva, C. W. (2004). Soft computing and intelligent systemsdesign, theory, tools and applications. Essex, England: Addison-Wesley,Pearson Education Limited. URL: http://pami.uwaterloo.ca/soft_comp/textbook.html

Kim, E. (2004). Output feedback tracking control of robot manipulators withmodel uncertainty via adaptive fuzzy logic. IEEE Transactions on FuzzySystems, 12, 368–378.

Liu, Y., & Arimoto, S. (1998). Decentralized adaptive and nonadaptiveposition/force controllers for redundant manipulators in cooperations.International Journal of Robotics Research, 17(3), 232–247.

Namvar, M., & Aghili, F. (2005). Adaptive force–motion control ofcoordinated robots interacting with geometrically unknown environments.IEEE Transactions on Robotics, 21(4), 678–694.

Parra-Vega, V., Arimoto, S., Liu, Y.-H., Hirzinger, G., & Akella, P. (2003).Dynamic sliding pid control for tracking of robot manipulators: Theoryand experiments. IEEE Transactions on Robotics and Automation, 19(6),967–976.

Special purpose dexterous manipulator. URL: http://www.mdrobotics.ca/spdm_frame.html

Sun, D., & Mills, J. K. (2002). Adaptive synchronized control for coordinationof multirobot assembly tasks. IEEE Transactions on Robotics andAutomation, 18(4), 498–510.

Szewczyk, J., Plumet, F., & Bidaud, P. (2002). Planning and controllingcooperating robots through distributed impedance. Journal of RoboticSystems, 19(6), 283–297.

Tarn, T. J., Bejczy, A. K., & Yun, X. (1987). Design of dynamic control oftwo cooperating robot arms: Closed chain formulation. Proceedings of theIEEE international conference on robotics and automation (pp. 7–13).

Tseng, C.-S., & Chen, B.-S. (2003). A mixed H2/H∞ adaptive trackingcontrol for constrained non-holonomic systems. Automatica, 39(6),1011–1018.

Vukobratovic, M., & Tuneski, A. (1999). Contribution to the adaptive controlof multiple compliant manipulation of dynamic environments. Robotica,17, 97–109.

Wen, T. J., & Kreutz, K. (1989). Motion and force control for multiplecooperative manipulators. In Proceedings of IEEE international conferenceon robotics and automation (pp. 1246–1251). Phoenix, AZ, USA.

Yoo, B. K., & Ham, W. C. (2000). Adaptive control of robot manipulatorusing fuzzy compensator. IEEE Transactions on Fuzzy Systems, 8(2),186–199.

Yukawa, T., Uchiyama, M., Nenchev, D. N., & Inooka, H. (1996). Stabilityof control system in handling of a flexible object by rigid arm robots. InProceedings of IEEE international conference on robotics and automation(Vol. 3, pp. 2332–2339).

Yun, X. (1989). Nonlinear feedback control of two manipulators in presence ofenvironment constraints. In Proceedings of IEEE international conferenceon robotics and automation (pp. 1252–1257). Phoenix, AZ, USA.

Wail Gueaieb received the Bachelor and Mas-ters degrees in Computer Engineering and Infor-mation Science from Bilkent University, Turkey,in 1995 and 1997, respectively, and the Ph.D.degree in Systems Design Engineering from theUniversity of Waterloo, Canada, in 2001. He iscurrently an Assistant Professor in the School ofInformation Technology and Engineering (SITE)at the University of Ottawa, Canada. He is alsothe founder and director of the Machine Intel-ligence, Robotics, and Mechatronics (MIRaM)Laboratory at SITE, and the Associate Director

of the Ottawa–Carleton Institute of Electrical and Computer Engineering.Gueaieb’s research interests span the fields of intelligent mechatronics,robotics, and computational intelligence. He has been with the industry from2001 to 2004, where he contributed in the design and implementation of anew generation of smart automotive safety systems. He is also the author/co-author of three patents, and more than 40 articles in highly reputed interna-tional journals and conference proceedings.

Salah Al-Sharhan received his B.Sc. in Com-puter Science from Kuwait University, State ofKuwait, the M.A.Sc. and the Ph.D. degree inPattern Analysis and Machine Intelligence fromthe Department of Systems Design Engineeringat the University of Waterloo, Waterloo, Canada.He is currently the Head of Computer ScienceDepartment at the Gulf University for Scienceand Technology (GUST) and the director of theE-learning Centre of Excellence at GUST. Hisresearch interests include e-learning methodolo-gies, distributed and web-based learning, intelli-

gent control, and applying computational intelligence techniques to datamining and high-speed computer networks (routing, resource management,topology design and optimization).

Miodrag Bolic received the B.S. and M.S. de-grees in Electrical Engineering from the Uni-versity of Belgrade, Serbia, in 1996 and 2001,respectively, and his Ph.D. degree in ElectricalEngineering from Stony Brook University, NY,USA, in 2004. He is currently with the Schoolof Information Technology and Engineering atthe University of Ottawa, Canada. From 1996to 2000 he was Research Associate with the In-stitute of Nuclear Science Vinca, Serbia. From2001 to 2004 he worked part-time at SymbolTechnologies Inc., NY, USA on development of

RFID and Bluetooth solutions. His research is related to computer architec-tures, digital signal processing and RFID systems. He is a Chair of the jointchapter of signal processing, oceanic engineering, and geoscience and remotesensing for IEEE Ottawa section.