coordinated motion control of multiple manipulatorschemori/temp/francois/control... · roughly...

20Coordinated MotionControl of Multiple

Manipulators

Kazuhiro KosugeTohoku University

Yasuhisa HirataTohoku University

20.1 Introduction 20.2 Manipulation of an Object 20.3 Coordinated Motion Control of Multiple

Manipulators for Handling an Object Motion of Object and Control of Internal Force/Moment• Load Sharing Problem

20.4 Coordinated Motion Control Based on ImpedanceControl Law Design of Impedance for Handling an Object • Designof Impedance for RCC Dynamics

20.5 Decentralized Motion Control of Multiple MobileManipulators Multiple Mobile Manipulators • Leader-Follower Type ControlAlgorithm

20.1 Introduction

With the development of robot technologies, many control algorithms for multiple manipulators havebeen proposed so far to realize several tasks using manipulators in coordination as humans do using theirarms. Especially the control algorithms proposed by Nakano et al. [1] and by Kurono [2] are known asthe pioneering research or handling of a single object by multiple manipulators in coordination. By usingmultiple manipulators in a cooperative way, we could execute several tasks which could not be done by asingle manipulator in addition to the handling of a single object. Figure 20.1 shows typical examples oftasks executed by multiple manipulators.

Figure 20.1 (a) shows the handling of an object by multiple manipulators in coordination. With anappropriately designed control system, multiple manipulators in coordination could handle a large and/orheavy object, which could not be handled by a single manipulator because the load to each manipulator isreduced by distributing it among the manipulators. Figure 20.1 (b) shows an example of tasks using a toolby two manipulators in coordination. In this case, the use of the tool becomes possible with the improvedrigidity of the system. The kinematic closed loop is formed by the manipulators, and the tool is graspedby them in coordination. Figure 20.1 (c) shows an assembly task of two parts, as an example of dexteroustasks which could not be done by a single manipulator. Two parts could be assembled without using anyjig and additional devices by the dual manipulators.

Copyright © 2005 by CRC Press LLC

20-2 Robotics and Automation Handbook

(a)

(b)

(c)

FIGURE 20.1 Tasks of manipulators: (a) load capacity, (b) rigidity, and (c) dexterity.

In this chapter, we first consider problems of coordinated motion control of multiple manipulatorshaving physical interaction among the manipulators including an object. Then, we outline several types ofcontrol algorithms for multiple manipulators for handling an object in coordination. The impedance-basedmotion control algorithm is introduced for the manipulation of an object in coordination and the assemblyof two parts. Finally, we introduce decentralized control algorithms of multiple mobile manipulators incoordination.

20.2 Manipulation of an Object

Consider motion of an object supported by multiple manipulators. Suppose that n manipulators graspan object rigidly and each manipulator applies force/moment to the object as shown in Figure 20.2. Wedefine coordinate systems as shown in Figure 20.3 and several parameters as follows:

Ou − xu yuzu absolute coordinate systemO0 − x0 y0z0 object coordinate systemOhi − xhi yhi zhi i th arm coordinate systemr0 ∈ R3 position vector from the origin of the absolute coordinate system

to the origin of the object coordinate system


Coordinated Motion Control of Multiple Manipulators 20-3

Object Trajectory

Force/moment

FIGURE 20.2 Manipulation of an object.

v0 ∈ R3 linear velocity of the objectw0 ∈ R3 angular velocity of the objectm mass of the objectM ∈ R3 inertia matrix of the objectg ∈ R3 acceleration of gravityIN ∈ R N×N N × N identity matrix

As is well known, the motion equation of an object supported by multiple manipulators is expressed asfollows:

mr0 = F0 + mg (20.1)

Mw0 + w0 × (Mw0) = N0 (20.2)

where F0 and N0 are the resultant force and the resultant moment applied to the object by all manipulatorsand are expressed as follows:

F0 =n∑

i=1

Fi (20.3)

N0 =n∑

i=1

Ni (20.4)

xu

yu

Absolute Coordinate System

ith Arm Coordinate System

ith Arm

Object Coordinate System

zu

Ou

O0x0

y0

z0

xhi

yhi

zhi

ohi

r0

FIGURE 20.3 Coordinate systems.



Fi and Ni are the force and the moment applied to the object by the i th manipulator in the object coordinatesystem with respect to the absolute coordinate system. Note that the motion of the object is generatedbased on the force F0 and the moment N0, which are the resultant force and the resultant moment appliedto the object by the manipulators.

Putting Equation (20.1) and Equation (20.2) together, we have the following equation:

L =[

mI3 0

0 M

][r0l

w0

]+

[−mg

w0 × (Mw0)

](20.5)

where

L ≡[

F0

N0

](20.6)

F0 and N0 are also expressed as

[F0

N0

]= KF (20.7)

where

K = [I6 I6 · · · I6] ∈ R6×6n (20.8)

F = [F T

1 NT1 F T

2 NT2 · · · F T

n NTn

]T ∈ R6n (20.9)

When each manipulator applies the force Fi and the moment Ni to the object, the resultant force F0 andthe resultant moment N0 applied to the object by all of the manipulators are shown by Equation (20.3)and Equation (20.4). Thus, the motion of the object is defined uniquely by Equation (20.5) based onthe resultant force and the resultant moment applied to the object coordinate system with respect to theabsolute coordinate system.

On the other hand, when the motion of the object is generated by multiple manipulators in coordination,we cannot determine uniquely the force Fi and the moment Ni , which are applied by each manipulatorto the object from Equation (20.7), although we can derive the resultant force and the resultant momentapplied to the object by all of the manipulators. The force/moment in the null space of K represented byEquation (20.8) is referred to as the internal force/moment [3]. The internal force/moment is the forceand the moment that do not contribute to the motion of the object.

The solution of Equation (20.7) depends on how to distribute the load required for the manipulationof the object among the manipulators and how to apply the internal force/moment to the object. Thefollowing shows an example of the solutions of Equation (20.7) for the dual manipulators case (n = 2)proposed by Uchiyama [3].

Solving Equation (20.7) using the generalized inverse matrix, we have the following equation:

F = W−L + [I6 − I6]T ξ (20.10)

where ξ(∈ R12) is any vector and W− is the generalized inverse matrix defined as

W− =[

R

I6 − R

](20.11)

From W−1 L in Equation (20.10) and Equation (20.11), we can see that the load of the manipulator L isdistributed to two manipulators based on ratio of R : I6 − R.



When we distribute the load of the manipulator L equally to two manipulators, we can solve Equa-tion (20.7) with R = 1

2 I6 as

F = 1

2

[I6

I6

]L +

[I6

−I6

]ξ (20.12)

In the general case (n ≥ 3), we can solve Equation (20.7) using the pseudo inverse matrix of K , K + as

F = K +L + (I6n − K +K )ξ (20.13)

The second terms on the right hand sides of Equation (20.10), Equation (20.12), and Equation (20.13) donot influence the force/moment applied to the object; that is, these terms do not affect the motion of theobject. By using these terms, we can specify the internal force/moment applied to the object.

20.3 Coordinated Motion Control of Multiple Manipulatorsfor Handling an Object

When multiple robots manipulate a single object in coordination as shown in Figure 20.2, we have toconsider the following problems:

(1) How to grasp the object(2) How to generate the motion of the object(3) How to apply the internal force/moment to the object(4) How to distribute the load of the object among robots

In this section, we consider the problems of (2), (3), and (4), under the assumption that the manipulatorsgrasp the object firmly and can apply the force/moment to the object arbitrarily.

20.3.1 Motion of Object and Control of Internal Force/Moment

Many researchers have discussed the coordinated motion control problems of multiple manipulators.Roughly speaking, the coordinated motion control algorithms for manipulators proposed so far couldbe classified into five types: the master-slave type of control algorithms, the hybrid type of control algo-rithms, the compliance-based control algorithms, the object dynamics–based control algorithms, and theaugmented dynamics–based control algorithms. In this section, we briefly introduce the outlines of thesecontrol algorithms.

1. Master-slave type of control algorithms [1,4]: Master-slave type of control algorithm has beenproposed by Nakano [1] as the pioneering research of multiple robots coordination. In this method,a manipulator referred to as a master controls the pose of the object based on the position controllaw, and the other manipulators referred to as slave control the force/moment applied to the objectbased on the force control law as shown in Figure 20.4. This method controls the position ofthe object and the force/moment applied to the object precisely. In this method, however, thedistribution of the load among robots could not be realized appropriately. The servo stiffness of

Force Control

Position Control

Master ArmSlave Arm

FIGURE 20.4 Master-slave type of control algorithm.



Compliance/Impedance Compliance/Impedance

FIGURE 20.5 Compliance based control algorithm.

the position-controlled master manipulator is very high, the servo stiffness of the force-controlledslave manipulators is almost zero, and the object is supported through the stiffness. Eventually,the position-controlled master manipulator has to support the entire load required for the objectmotion.

2. Hybrid type of control algorithms [5–7]: To control the motion of the object in 3D space, the robothas to have six degrees of freedom (6-DOF). In addition, to control the internal force/momentapplied to the object, the robot has to have 6-DOF. In the hybrid type of control algorithms,multiple robots control the 6-DOF with respect to the motion of the object and the 6-DOF withrespect to the internal force/moment applied to the object by using the 6n-DOF of n manipulators.

This type is similar to the hybrid position/force control algorithm and is regarded as a general-ization of the master-slave type of control algorithm. Takase [5] has proposed this type of controlalgorithm, which is derived based on how to constrain the motion of the object by manipulators. Inthe practical use of this method, however, we have to consider several problems such as position andorientation errors among the coordinate systems of manipulators or kinematic modeling errorsof each robot and the manipulated object. Unless we could control the manipulators exactly, themanipulators might apply excessive internal force/moment to the object.

3. Compliance-based control algorithms [2,8,9]: In the compliance-based control algorithm, theobject is compliantly grasped through manipulator compliances or impedances realized by thehardware or the software as shown in Figure 20.5. This type of control algorithm is robust againstthe kinematic modeling errors of the manipulators and the object. Even if the modeling errorsexist, the robots would not apply the excessive internal force/moment to the object. The effect ofthe modeling errors is reduced by the compliances or the impedances, through which the robotscontrol the position/orientation of the object and the internal force/moment applied to the object.

4. Object dynamics–based control algorithms [10]: In this algorithm, the motion of the object iscontrolled dynamically based on Equation (20.5), under the assumption that the robots could beregarded as actuators which generate the force/moment at grasping points of the object as shown inFigure 20.6. The control algorithm has been proposed by Nakamura et al. [10]. When the mass ofthe manipulated object is small, the precise manipulation of the object is not easy. To manipulate the

Object Trajectory

Force ActuatorForce Actuator

Resultant Force

FIGURE 20.6 Object dynamics–based control algorithm.



Object Trajectory

Force Actuator Force Actuator

Desired Motion

Joint Input

Joint Input

FIGURE 20.7 Augmented dynamics–based control algorithm.

object with small mass, the manipulators have to apply small force/moment to the object precisely.In the practical robots; however, it is difficult to apply the small force/moment precisely to themanipulated object at their grasping points.

5. Augmented dynamics–based control algorithm [11]: This control algorithm has been proposedby Khatib [11]. This is an extension of their hybrid position/force control algorithm based onthe manipulator dynamics at its endeffector. They derived a resultant dynamics including themanipulator dynamics and the object dynamics at the representative point of the object as shownin Figure 20.7. Then, they designed a hybrid position/force control algorithm so that both thepose of the object and the force/moment applied to the object are controlled. Note that they donot consider the internal force/moment applied to the object, but they do consider force/momentapplied among grasping points.

20.3.2 Load Sharing Problem

When two manipulators manipulate an object, the load sharing between them is realized based on Equa-tion (20.10). Several researches for the load sharing have been proposed so far. Zheng and Luh [12] haveproposed an optimal algorithm for load distribution with minimum exerted forces on the object. Thesealgorithms require less computational time, which makes them attractive for real-time applications. Theadaptive load-sharing controller developed by Pittelkau [13] optimizes the distribution of joint torquesover two manipulators to increase the efficiency and combined load-carrying capacity of the two ma-nipulators. Uchiyama and Yamashita [14] have considered the adaptive load sharing of two cooperativemanipulators holding a common object from the viewpoint of robustness of the holding.

20.4 Coordinated Motion Control Based on ImpedanceControl Law

In this section, we introduce the control algorithm based on the impedance control to realize the handlingof a single object and the assembly of two parts.

20.4.1 Design of Impedance for Handling an Object

Kosuge et al. have proposed a coordinated motion control algorithm of manipulators handling an objectbased on impedance control of each manipulator [9,15]. In the algorithm based on the impedance controllaw, the object is supported by multiple manipulators that are controlled so as to have impedance dynamics



FIGURE 20.8 Compliant support of object.

around desired trajectories of their endeffectors as shown in Figure 20.8. The compliant support of theobject using mechanical impedance is used for decreasing the modeling errors of the robot system.

In this section, we are going to reconsider the algorithm and introduce an alternative coordinatedmotion control algorithm of manipulators handling an object in coordination, so that we can control amechanical impedance of a manipulated object as well as the external force/moment sharing of the objectamong the manipulators.

First, let us consider a control problem of an apparent mechanical impedance of the manipulated objectby multiple manipulators. We consider a problem to realize a mechanical impedance of the object aroundits desired compliance center as shown in Figure 20.9. We assume that each manipulator supports theobject rigidity with its mechanical impedance around offset force/moment required for the manipulationof the object which consists of the desired internal force/moment and the shared load.

Let the mechanical impedance of the object be expressed by the following equation:

M�x + D�x + K �x = Fext (20.14)

where �x is the deviation of the manipulated object from the desired trajectory of the compliance centerof the object, Fext is the external force/moment applied to the object around its desired compliance center,and M, D, K are 6 × 6 positive definite matrices.

We assume that each manipulator grasps the object firmly and that no relative motion between the objectand each manipulator exists. To realize the impedance of the manipulated object, we consider the case inwhich each manipulator is controlled, based on the impedance control law around the desired compliancecenter of the object as shown in Figure 20.9. Let the mechanical impedance of the i th manipulator aroundthe desired compliance center of the object be expressed by

Mi �xi + Di �xi + Ki �xi = Fei (20.15)

x3

x2

x1

FIGURE 20.9 Desired object impedance.



where Fei is the external force/moment applied to the i th manipulator around the desired compliance centerof the object and �xi is the deviation of the compliance center of the i th manipulator. From the assumptionthat there is no relative motion between each manipulator and the object, we have the following relation:

�x = �xi (i = 1, 2, . . . , n) (20.16)

The external force/moment Fext is shared by each manipulator which is as follows:

Fei = δi Fext (i = 1, 2, . . . , n) (20.17)

whereδi > 0. Concerned with the external force/moment applied to the object, the following relation holds:

n∑i=1

Fei = Fext (20.18)

and we haven∑

i=1

δi = 1 (δi > 0) (20.19)

From these equations, we obtain the mechanical impedance of each manipulator, which is necessary to real-ize the desired mechanical impedance of the manipulated object expressed by Equation (20.14), as follows:

Mi = δi M (i = 1, 2, . . . , n) (20.20)

Di = δi D (i = 1, 2, . . . , n) (20.21)

Ki = δi K (i = 1, 2, . . . , n) (20.22)

20.4.2 Design of Impedance for RCC Dynamics

As shown in Figure 20.10, the remote compliance center (RCC) is a well-known device for realizing theassembly tasks of two parts. We are going to consider realizing a dynamic motion of the RCC betweentwo parts by using dual manipulators. Let the dynamics of the RCC device be expressed by the followingequation:

MRCC �x + DRCC �x + K RCC �x = F (20.23)

where MRCC , DRCC , K RCC are 6 × 6 positive definite matrices.

Peg

Hole

Compliance Center

FIGURE 20.10 Remote compliance center.



Arm 1 Arm 2

F1

− F2

FIGURE 20.11 Assembly task of two parts by two arms.

Suppose that each manipulator has the same impedance structure as the one of the RCC device and letthe impedance of each manipulator be expressed by the following equations:

M1�x1 + D1�x1 + K1�x1 = F1 (20.24)

M2�x2 + D2�x2 + K2�x2 = F2 (20.25)

The force/moment applied to each manipulator has the following relation as shown in Figure 20.11.

F1 = −F2(= F ) (20.26)

From Equation (20.23), Equation (20.24), and Equation (20.25), we have

�x = �x1 − �x2 (20.27)

M1 = M2 = 2MRCC (20.28)

D1 = D2 = 2DRCC (20.29)

K1 = K2 = 2K RCC (20.30)

By specifying the impedance parameters expressed in Equation (20.28), Equation (20.29), and Equa-tion (20.30), the impedance dynamics expressed in Equation (20.23) is realized between two parts sup-ported by two manipulators. Under the assumption that we could design the impedance dynamics appro-priately for realizing the assembly tasks of two parts, two manipulators could realize the assembly taskssuccessfully.

20.5 Decentralized Motion Control of MultipleMobile Manipulators

In this section, we consider the case of multiple mobile manipulators handling an object in coordination. Asexplained below, the coordination of multiple mobile manipulators has several control problems differentfrom the coordination of multiple manipulators. We briefly introduce the concept of a leader-followertype of control algorithm designed for multiple mobile manipulators for handling a single object incoordination as an example of the solution.

20.5.1 Multiple Mobile Manipulators

The mobile manipulators are expected to do tasks in an ordinary environment. Mobility is an importantfunction to cover the working space in the ordinary environment. In the ordinary environment, multiplesmall robots are more appropriate than a large and heavy robot because the small robot has less kineticenergy than the large and heavy one. When they are moving with the same speed, the small one is lessharmful to a human if its collisions occur.

As mentioned above, much research has been done for the motion control of multiple robots. However,most of the control algorithms proposed so far are designed based on the centralized control system; that



is, a single controller is supposed to control all of the robots in a centralized way based on the globalinformation. The centralized control system may be effective in case of the coordinated motion control ofmanipulators since the number of the manipulators in coordination is usually limited to two or three.

The use of mobile manipulators in the real world is one of the challenging topics in recent years [16–18].Considering the case where such mobile manipulators transport a single object in coordination, a singlecontroller could not control a large number of robots because of the real-time communication problemaround the robots and the computational burden of the single controller. A centralized controller is nomore realistic to control a large number of mobile manipulators.

In addition, deferent from the case of the manipulator, the dead reckoning system of a mobile manipula-tor is not so reliable, because the mobile manipulator has a slippage between its wheels and the ground andwe could not position the mobile manipulator precisely. Therefore, we could not apply the same controlprinciple of manipulators for controlling the multiple mobile manipulators, and the control system of themobile manipulators has to be redesigned robust against the inevitable positioning error of each mobilemanipulator.

20.5.2 Leader-Follower Type Control Algorithm

Many control algorithms have been proposed for the handling of a single object by multiple robots incoordination. Most of these control algorithms proposed so far have been designed under the assumptionthat the geometric relations among the robots are known precisely. However, it is not easy to knowthe geometric relations among them precisely, especially when the robots handle an unknown object incoordination in a real environment.

Errors in position/orientation of each mobile robot detected by a dead reckoning system are inevitablebecause of a slippage between a mobile mechanism and the ground. In addition to that, even if we knewgeometric relations among the robots, the geometric relations could not be kept precisely any more becauseof the errors included in position/orientation information of each robot. To overcome these problems, wehave to design a coordinated motion control algorithm robust against the inevitable positioning errors.

Kume et al. [18] have proposed the leader-follower type control algorithm of multiple mobile manipu-lators for handling a single object in coordination as shown in Figure 20.12. In this algorithm, the desiredtrajectory of the object is given to one of the mobile manipulators, referred to as a leader, and the otherrobots, referred to as followers, are controlled so as to have a virtual caster-like dynamics. The followers

FIGURE 20.12 Coordination of multiple mobile manipulators.



estimate the desired trajectory of the leader along the heading direction of the virtual caster to handle theobject in coordination with the leader. By using the virtual caster-like dynamics, which generates motion ofeach follower passively based on the force/moment applied to it similar to the real caster, multiple mobilemanipulators could handle a single object in coordination without using the geometric relations amongrobots. That is, this type of control algorithm is robust against the inevitable positioning errors of mobilemanipulators.

References

[1] Nakano, E., Ozaki, S., Ishida, T., and Kato, I., Cooperational control of the anthropomorphousmanipulator “MELARM,” Proc. 4th Int. Symp. Industrial Robots, pp. 251–260, 1974.

[2] Kurono, S., Coordinated computer control of a pair of artificial arms, Biomechanism, Vol. 3, pp. 182–193, Tokyo University Press, Tokyo, 1975.

[3] Uchiyama, M., A unified approach to load sharing, motion decomposing, and force sensing of dualarm robots, robotic research, 5th Int. Symp., pp. 225–232, MIT Press, Cambridge, MA, 1990.

[4] Luh, J.Y.S. and Zheng, Y.F., Constrained relations between two coordinated industrial robots formotion control, Int. J. Robotics Res., Vol. 6, No. 3, pp. 60–70, 1987.

[5] Takase, K., Representation of constrained motion and dynamic control of manipulators underconstrained, Trans. SICE, Vol. 21, No. 5, pp. 508–513, 1985.

[6] Hayati, S., Hybrid position/force control of multi-arm cooperating robots, Proc. IEEE Int. Conf.Robotics and Automation, pp. 82–89, 1986.

[7] Tarn, T.J., Bejczy, A.K., and Yun, X., Design of dynamic control of two coordinated robots in motion,Proc. 24th IEEE Conf. Decision and Control, pp. 1761–1765, 1985.

[8] Hanafusa, H. and Asada, H., A robot hand with elastic fingers and its application to assembly process,Proc. IFAC Symp. Inf. Control Probl. Manuf. Technol., pp. 127–138, 1977.

[9] Koga, M., Kosuge, K., Furuta, K., and Nosaki, K., Coordinated motion control of robot arms basedon the virtual internal model, IEEE Trans. Robotics and Automation, Vol. 8, No. 1, pp. 77–85, 1992.

[10] Nakamura, Y., Nagai, K., and Yoshikawa, T., Mechanics of coordinative manipulation by multiplerobotic mechanisms, Proc. IEEE Int. Conf. Robotics and Automation, pp. 991–998, 1987.

[11] Khatib, O., Object manipulation in a multi-effector robot system, Int. Symp. Robotics Res., SantaCruz, CA, August 1987.

[12] Zheng, Y.F. and Luh, J.Y.S., Optimal load distribution for two industrial robots handling a singleobject, Proc. 1988 IEEE Int. Conf. Robotics and Automation, pp. 344–349, 1988.

[13] Pittelkau, M.E., Adaptive load sharing force control for two-arm manipulators, Proc. Int. Conf.Robotics and Automation, pp. 498–503, 1988.

[14] Uchiyama, M. and Yamashita, T., Adaptive load sharing for hybrid control two cooperative manip-ulators, Proc. IEEE Int. Conf. Robotics and Automation, pp. 986–991, 1991.

[15] Kosuge, K., Ishikawa, J., Furuta, K., and Sakai, M., Control of single-master multi-slave manipulatorsystem using VIM, Proc. 1990 IEEE Int. Conf. Robotics and Automation, pp. 1172–1177, 1991.

[16] Khatib, O., Yokoi, K., Chang, K., Ruspini, D., Holmberg, R., and Casal, A., Vehicle/arm coordinationand multiple mobile manipulator decentralized cooperation, Proc. IEEE/RSJ Int. Conf. IntelligentRobots Syst., pp. 546–553, 1996.

[17] Yamamoto, Y. and Xiaoping, Y., Effect of the dynamic interaction on coordinated control of mobilemanipulators, IEEE Trans. Robotics and Automation, Vol. 12, No. 5, pp. 816–824, 1996.

[18] Kume, Y., Hirata, Y., Wang, Z., and Kosuge, K., Decentralized control of multiple mobile manip-ulators handling a single object in coordination, Proc. IEEE/RSJ Int. Conf. Intelligent Robots Syst.,pp. 2758–2763, 2002.


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