Kurfess T.R. Robotics and Automation Handbook

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-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 manipulators

having physical interaction among the manipulators including an object. Then, we outline several types of

controlalgorithmsfor multiple manipulators for handling an object in coordination.The impedance-based

motion control algorithm is introduced for the manipulation of an object in coordination and the assembly

of two parts. Finally, we introduce decentralized control algorithms of multiple mobile manipulators in

coordination.

20.2 Manipulation of an Object

Consider motion of an object supported by multiple manipulators. Suppose that n manipulators grasp

an object rigidly and each manipulator applies force/moment to the object as shown in Figure 20.2. We

deﬁne coordinate systems as shown in Figure 20.3 and several parameters as follows:

− x

absolute coordinate system

− x

object coordinate system

− x

ith arm coordinate system

∈ R

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.

∈ R

linear velocity of the object

∈ R

angular velocity of the object

m mass of the object

M ∈ R

inertia matrix of the object

g ∈ R

acceleration of gravity

∈ R

N×N

N × N identity matrix

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

follows:

= F

+ mg

(20.1)

+ w

× (Mw

) = N

(20.2)

where F

and N

are the resultant force and the resultant moment applied to the object by all manipulators

and are expressed as follows:



i=1

(20.3)



i=1

(20.4)

Absolute Coordinate System

ith Arm Coordinate System

ith Arm

Object Coordinate System

FIGURE 20.3 Coordinate systems.

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and N

are the force and the moment applied to the object by the ith manipulator in the object coordinate

system with respect to the absolute coordinate system. Note that the motion of the object is generated

based on the force F

and the moment N

, which are the resultant force and the resultant moment applied

to the object by the manipulators.

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

L =



0 M







−mg

× (Mw

)



(20.5)

where

L ≡





(20.6)

and N

are also expressed as





= KF (20.7)

where

K = [I

···I

] ∈ R

6×6n

(20.8)

F =



···F



∈ R

(20.9)

When each manipulator applies the force F

and the moment N

to the object, the resultant force F

and

the resultant moment N

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 on

the resultant force and the resultant moment applied to the object coordinate system with respect to the

absolute coordinate system.

On the other hand, when the motion of the object is generated by multiple manipulatorsin coordination,

we cannot determine uniquely the force F

and the moment N

, which are applied by each manipulator

to the object from Equation (20.7), although we can derive the resultant force and the resultant moment

applied to the object by all of the manipulators. The force/moment in the null space of K represented by

Equation (20.8) is referred to as the internal force/moment [3]. The internal force/moment is the force

and 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 manipulation

of the object among the manipulators and how to apply the internal force/moment to the object. The

following 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 + [I

− I

]

ξ (20.10)

where ξ(∈R

)isanyvectorandW

−

is the generalized inverse matrix deﬁned as

−



− 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 is

distributed to two manipulators based on ratio of R : I

− R.

Coordinated Motion Control of Multiple Manipulators 20

-5

When we distribute the load of the manipulator L equally to two manipulators, we can solve Equa-

tion (20.7) with R =

F =





L +



−I



ξ (20.12)

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

F = K

L + (I

− K

K )ξ (20.13)

The second terms on the right hand sides of Equation (20.10), Equation (20.12), and Equation (20.13) do

not inﬂuence the force/moment applied to the object; that is, these terms do not affect the motion of the

object. By using these terms, we can specify the internal force/moment applied to the object.

20.3 Coordinated Motion Control of Multiple Manipulators

for Handling an Object

When multiple robots manipulate a single object in coordination as shown in Figure 20.2, we have to

consider 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 manipulators

grasp 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 could

be 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 the

augmented dynamics–based control algorithms. In this section, we brieﬂy introduce the outlines of these

control algorithms.

1. Master-slave type of control algorithms [1,4]: Master-slave type of control algorithm has been

proposed 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 control

law, and the other manipulators referred to as slave control the force/moment applied to the object

based on the force control law as shown in Figure 20.4. This method controls the position of

the object and the force/moment applied to the object precisely. In this method, however, the

distribution of the load among robots could not be realized appropriately. The servo stiffness of

Force Control

Position Control

Master Arm

Slave Arm

FIGURE 20.4 Master-slave type of control algorithm.

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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-controlled

slave 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 object

motion.

2. Hybrid type of control algorithms [5–7]: To control the motion of the object in 3D space, the robot

has to have six degrees of freedom (6-DOF). In addition, to control the internal force/moment

applied 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 with

respect 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 control

algorithm, which is derived based on how to constrain the motion of the object by manipulators. In

the practical use of this method, however, we have to consider several problems such as position and

orientation errors among the coordinate systems of manipulators or kinematic modeling errors

of each robot and the manipulated object. Unless we could control the manipulators exactly, the

manipulators might apply excessive internal force/moment to the object.

3. Compliance-based control algorithms [2,8,9]: In the compliance-based control algorithm, the

object is compliantly grasped through manipulator compliances or impedances realized by the

hardware or the software as shown in Figure 20.5. This type of control algorithm is robust against

the kinematic modeling errors of the manipulators and the object. Even if the modeling errors

exist, the robots would not apply the excessive internal force/moment to the object. The effect of

the modeling errors is reduced by the compliances or the impedances, through which the robots

control 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 is

controlled dynamically based on Equation (20.5), under the assumption that the robots could be

regarded as actuators which generate the force/moment at grasping points of the object as shown in

Figure 20.6. The control algorithm has been proposed by Nakamura et al. [10]. When the mass of

the 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.

Coordinated Motion Control of Multiple Manipulators 20

-7

Object Trajectory

Force Actuator

Desired Motion

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 the

manipulated object at their grasping points.

5. Augmented dynamics–based control algorithm [11]: This control algorithm has been proposed

by Khatib [11]. This is an extension of their hybrid position/force control algorithm based on

the manipulator dynamics at its endeffector. They derived a resultant dynamics including the

manipulator dynamics and the object dynamics at the representative point of the object as shown

in Figure 20.7. Then, they designed a hybrid position/force control algorithm so that both the

pose of the object and the force/moment applied to the object are controlled. Note that they do

not consider the internal force/moment applied to the object, but they do consider force/moment

applied 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] have

proposed an optimal algorithm for load distribution with minimum exerted forces on the object. These

algorithms require less computational time, which makes them attractive for real-time applications. The

adaptive load-sharing controller developed by Pittelkau [13] optimizes the distribution of joint torques

over 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 cooperative

manipulators holding a common object from the viewpoint of robustness of the holding.

20.4 Coordinated Motion Control Based on Impedance

Control Law

In this section, we introduce the control algorithm based on the impedance control to realize the handling

of 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 object

based on impedance control of each manipulator [9,15]. In the algorithm based on the impedance control

law, the object is supported by multiple manipulators that are controlled so as to have impedance dynamics

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FIGURE 20.8 Compliant support of object.

around desired trajectories of their endeffectors as shown in Figure 20.8. The compliant support of the

object 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 coordinated

motion control algorithm of manipulators handling an object in coordination, so that we can control a

mechanical impedance of a manipulated object as well as the external force/moment sharing of the object

among the manipulators.

First, let us consider a control problem of an apparent mechanical impedance of the manipulated object

by multiple manipulators. We consider a problem to realize a mechanical impedance of the object around

its desired compliance center as shown in Figure 20.9. We assume that each manipulator supports the

object rigidity with its mechanical impedance around offset force/moment required for the manipulation

of 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 = F

ext

(20.14)

where x is the deviation of the manipulated object from the desired trajectory of the compliance center

of the object, F

ext

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 graspsthe object ﬁrmly and that no relative motion between the object

and each manipulator exists. To realize the impedance of the manipulated object, we consider the case in

which each manipulator is controlled, based on the impedance control law around the desired compliance

center of the object as shown in Figure 20.9. Let the mechanical impedance of the ith manipulator around

the desired compliance center of the object be expressed by



+ D



+ K

 x

= F

(20.15)

FIGURE 20.9 Desired object impedance.

Coordinated Motion Control of Multiple Manipulators 20

-9

where F

istheexternalforce/momentapplied tothe ithmanipulatoraroundthedesiredcompliancecenter

of the object and x

is the deviation of the compliance centerof the ith manipulator. Fromthe assumption

that there is no relative motion between each manipulator and the object, we have the following relation:

 x = x

(i = 1, 2, ..., n) (20.16)

The external force/moment F

ext

is shared by each manipulator which is as follows:

= δ

ext

(i = 1, 2, ..., n) (20.17)

whereδ

> 0. Concerned withtheexternalforce/momentappliedtothe object, thefollowing relationholds:



i=1

= F

ext

(20.18)

and we have



i=1

= 1(δ

> 0) (20.19)

From these equations, we obtain the mechanical impedance of each manipulator, whichis necessary to real-

ize the desired mechanical impedance of the manipulated object expressed by Equation (20.14), as follows:

= δ

M (i = 1, 2, ..., n)

(20.20)

= δ

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

= δ

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 the

assembly tasks of two parts. We are going to consider realizing a dynamic motion of the RCC between

two parts by using dual manipulators. Let the dynamics of the RCC device be expressed by the following

equation:

RCC



x + D

RCC



x + K

RCC

 x = F (20.23)

where M

RCC

, D

RCC

, K

RCC

are 6 ×6 positive deﬁnite matrices.

Peg

Hole

Compliance Center

FIGURE 20.10 Remote compliance center.

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Arm 1

Arm 2

−

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 let

the impedance of each manipulator be expressed by the following equations:



+ D



+ K

 x

= F

(20.24)



+ D



+ K

 x

= F

(20.25)

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

=−F

(= F )

(20.26)

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

 x =x

− x

(20.27)

= M

= 2M

RCC

(20.28)

= D

= 2D

RCC

(20.29)

= K

= 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 tasks

successfully.

20.5 Decentralized Motion Control of Multiple

Mobile Manipulators

In this section, we consider the case of multiple mobile manipulators handling an object in coordination. As

explained below, the coordination of multiple mobile manipulators has several control problems different

from the coordination of multiple manipulators. We brieﬂy introduce the concept of a leader-follower

type of control algorithm designed for multiple mobile manipulators for handling a single object in

coordination 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 important

function to cover the working space in the ordinary environment. In the ordinary environment, multiple

small robots are more appropriate than a large and heavy robot because the small robot has less kinetic

energy than the large and heavy one. When they are moving with the same speed, the small one is less

harmful 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

Coordinated Motion Control of Multiple Manipulators 20

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is, a single controller is supposed to control all of the robots in a centralized way based on the global

information. The centralized control system may be effective in case of the coordinated motion control of

manipulators 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 single

controller could not control a large number of robots because of the real-time communication problem

around the robots and the computational burden of the single controller. A centralized controller is no

more 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 and

we could not position the mobile manipulator precisely. Therefore, we could not apply the same control

principle of manipulators for controlling the multiple mobile manipulators, and the control system of the

mobile manipulators has to be redesigned robust against the inevitable positioning error of each mobile

manipulator.

20.5.2 Leader-Follower Type Control Algorithm

Many control algorithms have been proposed for the handling of a single object by multiple robots in

coordination. Most of these control algorithms proposed so far have been designed under the assumption

that the geometric relations among the robots are known precisely. However, it is not easy to know

the geometric relations among them precisely, especially when the robots handle an unknown object in

coordination in a real environment.

Errors in position/orientation of each mobile robot detected by a dead reckoning system are inevitable

because of a slippage between a mobile mechanism and the ground. In addition to that, even if we knew

geometric relations among the robots, the geometric relations could not be kept precisely any more because

of the errors included in position/orientation information of each robot. To overcome these problems, we

have 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 desired

trajectory of the object is given to one of the mobile manipulators, referred to as a leader, and the other

robots, 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.