A CONTROL THEORY FRAMEWORK

FOR PERFORMANCE EVALUATION OF

MOBILE MANIPULATORS

Katarzyna Zadarnowska and Krzysztof Tchon ∗

Abstract

We propose a new, control theoretic framework for defining perfor-

mance measures of mobile manipulators. As a guiding principle we

assume that the kinematics or the dynamics of a mobile manipulator

are represented by the end point map of a control system with outputs,

and that a controllable system yields nontrivial performance measures.

In the paper we focus on two categories of dynamic performance mea-

sures: the compliance measure and the admittance measure. In both

categories the following local and global performance characteristics

are introduced: the agility ellipsoid, the agility and mobility, the condi-

tion number and the distortion. The usefulness of new local measures

∗Institute of Computer Engineering, Control and Robotics, Wroc law Univer-sity of Technology, ul. Janiszewskiego 11/17, 50-372 Wroc law, Poland, e-mail:kz|tchon@ict.pwr.wroc.pl.

1

is demonstrated on example problems of determining optimal motion

patterns of a mobile robot.

1 Introduction

Differently to the stationary manipulators, that have been the subject of in-

tense research for over 30 years, the mobile manipulators appeared in the

robotic literature around the nineties of the XX century. Since then the mo-

bile manipulators have become one of the most challenging topics of robotics.

A mobile manipulator is defined as a complex robotic system consisting of

a mobile platform and a stationary manipulator mounted on the platform.

By design, the mobile manipulator has enhanced locomotion and manipu-

lation capabilities. This synergy of locomotion properties of the platform

and manipulation capacities of the arm causes that mobile manipulators are

commonly used in the industrial, service, and educational and entertainment

robotics, as well as in more dedicated fields like the Earth and space explo-

ration, demining, antiterrorist actions or rescue operations. The diversity of

applications of mobile manipulators calls for new modeling tools and control

algorithms.

The performance evaluation of stationary manipulators belongs to a well es-

tablished domain of robotics that has a vital significance for the design and

exploitation of manipulators. Usually, performance measures are defined as

some functionals of the manipulator Jacobian or the inertia matrix. The

2

measures describing the performance at a given configuration are called lo-

cal, local measures averaged over the configuration space are named global.

There exists a rich robotic literature concerned with construction and appli-

cations of performance measures for stationary manipulators [19], [26], [34],

[18], [12], [6], [15], [24].

The choice of a particular performance measure depends on the task that

should be executed by the manipulator. Performance measures serve a num-

ber of purposes. They may be used as additional criteria in inverse kinematics

algorithms [18] or as a means for performance evaluation and optimization

of the manipulator. In the latter case, local measures detect optimal (e.g.

far from singular, isotropic, avoiding obstacles and workspace boundaries)

configurations, whereas global performance measures support the manipula-

tor design [14]. Moreover, performance measures may be used in order to

determine optimal workpoints in the taskspace [18] or may help in keeping

the manipulator joints within their mechanical limits [19], [23]. Performance

measures are useful in selecting optimal velocities of the manipulator joints

[24] or in optimal planning of the end-effector trajectory [1], [2]. Performance

measures facilitate a collision-free control of a pair of cooperating manipula-

tors mounted on a common platform [32].

In this paper we are concerned with mobile manipulators composed of a non-

holonomic mobile platform and a holonomic on-board manipulator. In com-

parison to performance measures for stationary manipulators, the literature

3

dealing with the performance of mobile manipulators is rather modest. A

dominant approach consists in a direct transfer to mobile manipulators of the

methodology that proved to be successful for stationary manipulators. An

attempt at transferring the concepts of manipulability as well as kinematic

and dynamic manipulability ellipsoid from stationary to mobile manipula-

tors was undertaken by Yamamoto and Yun [31]. These authors used the

introduced measures to establish a coordination between manipulation and

locomotion capabilities of the mobile manipulator. Gardner and Velinsky

presented in [14] an application of the manipulability ellipsoid to the optimal

design of the mobile manipulator executing a specific task. Various concepts

of manipulability for mobile manipulators have been introduced by Foulon

at all [13], Bayle at all [4] and Yamamoto and Fukuda [33]. In [5] Bayle and

coauthors proposed the ellipsoid’s eccentricity as a measure of the shape of

the manipulability ellipsoid and recommended an application of this measure

to the optimal motion planning of the mobile manipulator.

Restricted to the kinematic performance, the approach assumed in the lit-

erature can be reconstructed in the following way. Suppose that in certain

coordinate systems the platform kinematics and the end-effector location of

the on-board manipulator are described as a control system with outputs, of

the form

q = G(q)u, y = k(q, x), (1)

where vectors q ∈ Rn, x ∈ Rp, y ∈ Rr, u ∈ Rm represent, respectively, the

4

platform coordinates, the manipulator joint position, the end-effector posi-

tion and orientation, and the platform controls. After the time differentiation

of the output function along a system trajectory we obtain

y =∂k

∂q(q, x)q +

∂k

∂x(q, x)x =

[

∂k∂q

(q, x)G(q) ∂k∂x

(q, x)

]

u

x

. (2)

The matrix

J(q, x) =

[

∂k∂q

(q, x)G(q) ∂k∂x

(q, x)

]

(3)

of dimension r × (m + p) may be treated as a Jacobian matrix of the mobile

manipulator. Using this matrix, we get a sort of manipulability matrix

M(q, x) = J(q, x)JT (q, x) =

= ∂k∂x

(q, x)(

∂k∂x

)T(q, x) + ∂k

∂q(q, x)G(q)GT (q)

(

∂k∂q

)T

(q, x).(4)

that may serve as a starting point for defining performance measures in

a way analogous as for stationary manipulators. While defining performance

measures for stationary or mobile manipulators, it is expected that the per-

formance deteriorates at the points (q, x) where matrix (4) becomes sin-

gular. However, for r > m + p this matrix is always singular, what im-

plies e.g. that performance measures of any nonholonomic mobile platform

(r = n > m, p = 0) would be void. Obviously, the last conclusion contradicts

to the fact that the nonholonomic platform is controllable, so its motion ca-

pabilities are not impaired. As another example, let us consider a simple

5

device that might be called a wheeled jack, consisting of a unicycle-type mo-

bile platform equipped with a 1 degree of freedom jack able to move up and

down in the vertical direction. The mathematical model of the kinematics of

the wheeled jack is the following (we have m = 2, p = 1, r = m + p = 3)

q1 = u1 cos q3

q2 = u1 sin q3

q3 = u2

y = k(q, x) = (q1, q2, x).

(5)

A straightforward computation based on (5) shows that matrix (4) is always

singular, so again the corresponding performance measures become void. On

the other hand, it is easy to show that with a suitable control (u1, u2, x)

the wheeled jack can reach any point in its taskspace, hence its performance

measures should be non-trivial.

Both the example of the mobile platform as well as that of the wheeled

jack unveil an immanent flaw of the approach toward defining performance

measures for mobile manipulators that dominates in the literature. As an al-

ternative, in this paper we propose to use the endogenous configuration space

approach set forth in [28], [29]. This approach will provide performance mea-

sures not only for all mobile manipulators to which the existing approach may

be applicable, but also for mobile robots and mobile manipulators that can-

not be managed in a satisfactory way by the existing approach. This paper

focuses on defining dynamic performance measures for mobile manipulators.

6

The reader interested in kinematic measures may consult [30], [36], [37]. As

a point of departure we assume a control system representation of the mobile

manipulator and postulate that performance of a controllable system is non-

trivial. Depending on the form of the output function of the control system

representation of the mobile manipulator we introduce two types of dynamic

performance measures: the compliance and the admittance. Other dynamic

measures can be introduced according to the same definitional scheme, af-

ter a suitable modification of the output function. The main results of the

paper refer to five performance characteristics of compliance or admittance

type: the agility ellipsoid, the agility, the mobility, the condition number, and

the distortion. These measures may be regarded as local or global. As an

illustration we present a number of computational results that provide mo-

tion patterns in the taskspace of MK mobile robot designed in our robotics

laboratory. The core idea of the control theoretic framework is presented

below.

1.1 The main idea

Our approach relies on the following analogy between the kinematics of a

stationary manipulator and the end point map of a control system. For sim-

plicity, in the presentation we shall use coordinate systems in the jointspace

and in the taskspace. Suppose that

y = k(x)

7

denotes a coordinate representation of kinematics of a redundant manipulator

with x ∈ Rp, y ∈ Rr. If 4x is a tangent vector to the jointspace (an

infinitesimal motion in the jointspace) at x then the corresponding tangent

vector 4y to the taskspace (an infinitesimal motion in the taskspace) at k(x)

satisfies the Jacobian equation

4y = J(x)4x =d

dα|α=0 k(x + α4x),

where J(x) = ∂k∂x

(x) is the manipulator Jacobian. The Jacobian equation

defines a transmission of infinitesimal motions from the joints to the end-

effector, that takes place in the configuration x. Formally, this transmission

can be characterized by the manipulability ellipsoid

E(x) = J(x)(4x ∈ Rp |‖4x‖= 1).

Using the Jacobian pseudoinverse J#(x) = JT (x)M−1(x) we solve the Jaco-

bian equation and obtain

E(x) = η ∈ Rr | ηT M−1(x)η = 1,

where M(x) = J(x)JT (x) denotes the manipulability matrix at configuration

x. Various functions of eigenvalues of M(x) may be taken as local perfor-

mance measures of the manipulator.

8

Now let

z = f(z) + g(z)u

y = h(z)(6)

be a smooth affine control system with outputs, where z ∈ Rn, u ∈ Rm,

y ∈ Rr, and admissible control functions are Lebesgue square integrable over

a time interval [0, T ], u(·) ∈ L2m[0, T ]. For a fixed initial state z0 we compute

the state trajectory z(t) = ϕz0,t(u(·)) and define the end point map

y(T ) = Hz0,T (u(·)) = h(ϕz0,T (u(·)))

that computes the output at T of system (6) steered by u(·). Suppose that

4u(·) (a variation of control) is a tangent vector to the control space at u(·).

Then, the resulting tangent vector to the output space (a variation of system

output) at Hz0,T (u(·)) is equal to

4y(T ) = DHz0,T (u(·))4u(·) =d

dα|α=0 h(ϕz0,T (u(·) + α4u(·))). (7)

The derivative DHz0,T (u(·)) can be expressed by means of the linear approx-

imation of the control system along (u(t), z(t)), called the variational system

ζ = A(t)ζ + B(t)4u(t)

η = C(t)ζ,(8)

9

where A(t) = ∂∂z

(f(z(t)) + g(z(t))u(t)), B(t) = g(z(t)), C(t) = ∂∂z

h(z(t)).

Denoting by Φ(t, s) the transition matrix, ∂Φ(t,s)∂t

= A(t)Φ(t, s), Φ(s, s) = In,

we obtain

DHz0,T (u(·))4u(·) = C(T )

∫ T

0

Φ(T, t)B(t)4u(t)dt.

It is easily observed that DHz0,T (u(·)) plays the role analogous to the manip-

ulator Jacobian, so we set DHz0,T (u(·)) = Jz0,T (u(·)). The Jacobian equation

4y(T ) = Jz0,T (u(·))4u(·)

describes how an infinitesimal change in the control function is transmitted

into a change of the system output at T . The image

Ez0,T (u(·)) = Jz0,T (u(·))(4u(·) ∈ L2m[0, T ] |‖4u(·)‖= 1),

of the unit sphere in L2m[0, T ], is an ellipsoid in R

r. Using the Jacobian

pseudoinverse (J#z0,T (u(·))η)(t) = BT (t)ΦT (T, t)CT (T )G−1

z0,T (u(·))η = 4u(t)

that solves the equation (7) we show that

Ez0,T (u(·)) = η ∈ Rr | ηT G−1

z0,T (u(·))η = 1,

where Gz0,T (u(·)) = C(T )∫ T

0Φ(T, t)B(t)BT (t)ΦT (T, t)dtCT (T ) denotes the

output controllability Gramian [25] for the linearized control system (8). Uti-

10

lizing an analogy between the Gram matrix and the manipulability matrix,

various functions of eigenvalues of the Gram matrix can be adapted as perfor-

mance measures of the control system at a given control u(·). Depending on

the physical meaning of state, input and output variables and on the form of

functions defining control system (6), a system of this kind represents either

the kinematics (input=velocity) or the dynamics (input=force) of a mobile

manipulator. More specifically, we may speak of kinematic, compliance, ad-

mittance and other performance measures.

The remaining part of this paper is composed in the following way. Section 2

summarizes basic concepts of the endogenous configuration space approach.

Section 3 defines compliance and admittance performance measures of mobile

manipulators. Section 4 presents computation of performance measures for

an example mobile robot. The paper is concluded with section 5.

2 Basic concepts

In this paper we shall concentrate on performance measures for mobile ma-

nipulators composed of a nonholonomic platform and a holonomic on-board

manipulator. In the case when either the on-board manipulator or the plat-

form is absent the mobile manipulator specifies, respectively, to the nonholo-

nomic mobile platform (a mobile robot) or to the stationary manipulator.

This implies that performance measures for mobile manipulators should gen-

11

eralize the existing measures for holonomic manipulators and provide new

measures for nonholonomic mobile platforms. For the later use we shall need

the following basic concepts originated from the endogenous configuration

space approach.

2.1 Dynamics of mobile manipulator

The kinematic performance measures for mobile manipulators have been pro-

vided in [28], [30]. In this subsection we shall adapt the approach described

in subsection 1.1 to mobile manipulators whose control system representa-

tion includes not only the kinematics, but also the dynamics of the platform

and the on-board manipulator. To this aim, let us consider a mobile manip-

ulator with n generalized coordinates q = (q1, . . . , qn) ∈ Rn of the platform,

and p joint coordinates x = (x1 . . . , xp) ∈ Rp of the on-board manipulator.

The control system representation of the platform kinematics is defined by a

driftless control system

q = G(q)µ =

m∑

i=1

gi(q)µi, (9)

12

where µ ∈ Rm. Using the Lagrange-d’Alembert principle, we can write down

the dynamics of the mobile manipulator in the form

M(q, x)

q

x

+C(q, x, q, x)

q

x

+D(q, x) = B(q, x)

u

v

+

F (q)

0

,

(10)

where M(q, x) denotes the inertia matrix, C(q, x, q, x)

q

x

is a vector of

Coriolis, centripetal and frictional forces, D(q, x) is the gravity vector, and

B(q, x) stands for the control matrix. The term F (q) denotes forces (torques)

exerted by the nonholonomic constraints. The control functions of the plat-

form u(·) ∈ L2m[0, T ] and of the on-board manipulator v(·) ∈ L2

p[0, T ] have

the meaning of forces or torques, and therefore pairs u(·) = (u(·), v(·)) ∈

X = L2m[0, T ]×L2

p[0, T ] will be called dynamic endogenous configurations of

the mobile manipulator.

After a nonholonomic reduction of the dynamics (10) and an addition of an

output function characterizing the task of the mobile manipulator, we obtain

an affine control system representation of the kinematics and dynamics of the

13

mobile manipulator [36]

q = G(q)µ

x = π

µ

π

= P1(q, x, µ, π) + P2(q, x)u

y = k(q, x),

(11)

where

P1(q, x, µ, π) =

GT 0

0 Ip

M

G 0

0 Ip

−1

−

GT 0

0 Ip

M

G 0

0 0

+ C

G 0

0 Ip

µ

π

−

GT 0

0 Ip

D

,

P2(q, x) =

GT 0

0 Ip

M

G 0

0 Ip

−1

GT 0

0 Ip

B

and

y =

R(q, x) d(q, x)

0 1

∈ SE(3)

describes position and orientation of the end-effector. Elements of the special

Euclidean group SE(3) will be identified with matrices

R d

0 1

, where R ∈

SO(3), the special orthogonal group, and d ∈ R3. We assume analyticity of all

14

vector fields and functions appearing in (11). The group SE(3) endowed with

a Riemannian metric <Y1, Y2 > (y) = Y T1 Q(y)Y2, where Y1, Y2 ∈ R6 denote

a pair of twists and Q(y) = QT (y) > 0 is a symmetric, positive definite

matrix, will be referred to as the taskspace of the mobile manipulator.

The dynamic endogenous configuration space X may also be equipped with

a Riemannian metric

〈4u1(·),4u2(·)〉(u(·)) =

∫ T

0

4uT1 (t)R(u(·), t)4u2(t)dt, (12)

where R(u(·), t) = RT (u(·), t) > 0 denotes a weight matrix of size (m + p) ×

(m + p). The norm of a tangent vector 4u(·) at u(·), resulting from (12), is

equal to

‖4u(·)‖2 (u(·)) =

∫ T

0

4uT (t)R(u(·), t)4u(t)dt. (13)

A proper choice of the Riemannian metrics in the configuration space and

in the taskspace will produce physically consistent and gauge-invariant per-

formance measures [10], [11], and will provide to a natural definition of the

weighted pseudoinverse Jacobian [22], [11] for mobile manipulators.

Let z = (q, x, µ, π) ∈ Rs, s = n+ m+ 2p, and suppose that for a given initial

state z0 and every control u(·) there exists a state trajectory

z(t) = ϕz0,t(u(·)) = (q(t), x(t), µ(t), π(t)),

15

and an output trajectory y(t) = k(z(t)) of system (11). The end point map

of this system

y(T ) = Cz0,T (u(·)) = k(z(T )) (14)

describes the end-effector position and orientation at time T . Using the result

of Sontag [27] we can prove that (14) is Frechet continuously differentiable.

The derivative of Cz0,T that represents a transmission of the force variation

4u(·) into the position and orientation variation 4y at T will be called the

compliance map of the mobile manipulator. In order to compute this map

for a given configuration u(·) ∈ X , we need the variational system

ξ = A(t)ξ + B(t)4u(t), (15)

along (z(t), u(t)), associated with (11), where

A(t) =

∂(G(q(t))µ(t))∂q

0 G(q(t)) 0

0 0 0 Ip

∂(P1(q,x,µ,π)+P2(q,x)u(t))∂q

∂(P1(q,x,µ,π)+P2(q,x)u(t))∂x

∂P1(q,x,µ,π)∂µ

∂P1(q,x,µ,π)∂π

,

B(t) =

0

0

P2(q(t), x(t))

.

The compliance map is a linear transformation of the tangent vectors to the

dynamic endogenous configuration space at u(·) into the tangent space to the

16

taskspace at Cz0,T (u(·)) (identified with the Lie algebra se(3) ∼= R6 of the

special Euclidean group), defined through the differentiation

JC

z0,T (u(·))4u(·) =d

dα|α=0 Cz0,T (u(·) + α4u(·)). (16)

A variation 4u(·) of u(·) produces an infinitesimal motion 4y of the end-

effector that will be characterized by a linear and an angular component

4y =

vT

ωT

= JC

z0,T (u(·))4u(·), (17)

where

vT = ddα

|α=0 d(ϕz0,T (u(·) + α4u(·))),

ωT =]( ddα

|α=0 R(ϕz0,T (u(·) + α4u(·))))RT (ϕz0,T (u(·)))[,

where for a skew symmetric 3 × 3 matrix Ω the symbol ]Ω[ denotes the

isomorphism of the Lie algebra so(3) and R3.

A further computation shows that

vT =∂d

∂z(q(T ), x(T ))Dϕz0,T (u(·))4u(·) = J1

z0,T (q(T ), x(T ))Dϕz0,T (u(·))4u(·)

17

and

ωT =

[

] ∂R∂z1

(q(T ), x(T ))RT (q(T ), x(T ))[, . . . , ] ∂R∂zs

(q(T ), x(T ))RT (q(T ), x(T ))[

]

×

Dϕz0,T (u(·))4u(·) = J2z0,T (q(T ), x(T ))Dϕz0,T (u(·))4u(·).

These computations result in the compliance map

JC

z0,T (u(·))4u(·) =

J1z0,T (q(T ), x(T ))

J2z0,T (q(T ), x(T ))

∫ T

0

Φ(T, t)B(t)4u(t)dt, (18)

where Φ(t, s) denotes the transition matrix of system (15), satisfying the evo-

lution equation ∂Φ(t,s)∂t

= A(t)Φ(t, s), Φ(s, s) = Is . By definition the com-

pliance map is a kind of Jacobian map that transforms increments of force

into position and orientation increments of the end-effector.

Consequently, a dynamic endogenous configuration u(·) ∈ X is called com-

pliance regular, if the compliance map JC

z0,T (u(·)) is surjective. At regular

configurations Cz0,T is an open map what makes the system (11) locally out-

put controllable. Now, suppose that u(·) is regular and consider a Jacobian

type equation

JC

z0,T (u(·))4u(·) = η. (19)

The equation can be solved by the least squares method that consists in

minimizing the norm

min4u(·)

1

2‖4u(·)‖2 (u(·)) (20)

18

defined by (13) under the equality constraint (19). Having performed neces-

sary computations we obtain the compliance pseudoinverse JC#z0,T (u(·)) : R6 →

X defined as

(JC#z0,T (u(·))η)(t) =

= R−1(u(·), t)BT (t)ΦT (T, t)

J1z0,T (q(T ), x(T ))

J2z0,T (q(T ), x(T ))

T

(DC

z0,T )−1(u(·))η,(21)

where

DC

z0,T (u(·)) =

J1z0,T (q(T ), x(T ))

J2z0,T (q(T ), x(T ))

MC

z0,T (u(·))

J1z0,T (q(T ), x(T ))

J2z0,T (q(T ), x(T ))

T

.

(22)

The matrix (22) will be called the compliance dexterity matrix of the mo-

bile manipulator. The compliance dexterity matrix includes the compliance

mobility matrix of the mobile platform

MC

z0,T (u(·)) =

∫ T

0

Φ(T, t)B(t)R−1(u(·), t)BT (t)ΦT (T, t)dt. (23)

We notice that the compliance dexterity matrix constitutes an equivalent to

the output controllability Gramian of the linear approximation of (15) (see

subsection 1.1), while the compliance mobility matrix corresponds to the

respective controllability Gramian in the state space.

It is easily shown that compliance regularity of the dynamic endogenous con-

19

figuration u(·) ∈ X means that detDC

z0,T (u(·)) 6= 0, otherwise the configura-

tion u(·) ∈ X is singular. Performance measures related to the compliance

dexterity matrix will be called compliance performance measures.

Next, we can make a prolongation of the output function in (11) to get the

following representation of the dynamics of a mobile manipulator

q = G(q)µ

x = π

µ

π

= P1(q, x, µ, π) + P2(q, x)u

y =

v

ω

= k(q, x, µ, π),

(24)

where

y = k(q, x, µ, π) =

J1z0,T (q(t), x(t))

J2z0,T (q(t), x(t))

z (25)

denotes the time derivative of the original output function. Observe, that

since k does not depend on µ and π, the prolongation k is independent from

the control u(t).

With notations defined above, the end point map associated with system

20

(24) assumes the form

y(T ) =

v(T )

ω(T )

= Az0,T (u(·)) = k(ϕz0,T (u(·))).

A corresponding admittance map of the mobile manipulator provides the

variation 4y of the velocity resulting from the force variation 4u(·)

4y = JA

z0,T (u(·))4u(·) (26)

and is defined as

JA

z0,T (u(·))4u(·) =∂k

∂z(z(T ))

∫ T

0

Φ(T, t)B(t)4u(t)dt. (27)

By analogy to the compliance map we obtain the admittance dexterity matrix

of the mobile manipulator

DA

z0,T (u(·)) =

∂k∂z

(z(T ))∫ T

0Φ(T, t)B(t)R−1(u(·), t)BT (t)ΦT (T, t)dt

(

∂k∂z

(z(T )))T

.(28)

A dynamic endogenous configuration u(·) ∈ X will be referred to as ad-

mittance regular, when the admittance map JA

z0,T (u(·)) is surjective (i.e.

detDA

z0,T (u(·)) 6= 0), otherwise u(·) is singular. Performance measures as-

sociated with the admittance map will be called admittance performance

measures.

21

3 Performance measures of mobile manipu-

lators

As we have already stated, our methodology of defining performance mea-

sures relies on analogy between the manipulability matrix of a stationary

manipulator and the controllability Gramian of a control system associated

with the kinematics or dynamics of a mobile manipulator. In this section we

shall pursue this analogy in order to derive local and global dynamic mea-

sures for mobile manipulators. Kinematic performance measures have been

provided in [30].

3.1 Compliance performance measures

The compliance performance measures refer to the transmission of variations

of input forces into variations of position and orientation in the taskspace.

Let us begin with the compliance agility ellipsoid

Ez0,T (u(·)) = JC

z0,T (u(·))Sz0,T (u(·)) = ζ ∈ R6 | ζT (DC

z0,T )−1(u(·))ζ = 1

(29)

that describes the image by the mobile manipulator compliance map (18) of

a unit sphere

Sz0,T (u(·)) = 4u(·) ∈ X |‖4u(·)‖ (u(·)) = 1 (30)

22

of motion directions in the configuration space at u(·) ∈ X into the motion

directions in the taskspace at Cz0,T (u(·)).

The ellipsoid is determined by the compliance dexterity matrix DC

z0,T (u(·))

defined by (22). It is straightforward to observe that minimal and max-

imal half axes of this ellipsoid are equal, respectively, to the square root√

λDC

z0,T(u(·)) of the minimal and

√

λDC

z0,T(u(·)) of the maximal eigenvalue of

the compliance dexterity matrix. In the case when all eigenvalues of the com-

pliance dexterity matrix DC

z0,T (u(·)) are equal, i.e. λDC

z0,T(u(·)) = λDC

z0,T(u(·)), we

call u(·) compliance isotropic configuration of the mobile manipulator. Oth-

erwise, the configuration is called compliance anisotropic. At the isotropic

configuration there are no privileged directions of motion in the taskspace,

so the compliance agility ellipsoid becomes a sphere. At a compliance singu-

lar endogenous configuration the minimal eigenvalue equals 0, therefore the

compliance agility ellipsoid collapses.

A volume measure of the compliance agility ellipsoid will define the local

compliance agility of the endogenous configuration u(·)

aC

z0,T (u(·)) =√

detDC

z0,T (u(·)) =

√

√

√

√

r∏

i=1

λiDC

z0,T(u(·)). (31)

By definition, the compliance agility of compliance singular configurations

of the mobile manipulator is zero. From the control theory point of view,

nonzero local compliance agility implies output controllability of the vari-

ational system (15) and local output controllability of system (11). The

23

maximization of local compliance agility defines a motion pattern of the mo-

bile manipulator (i.e. a trajectory of the platform and of the end-effector

generated by the optimal u(·)) that ensures the highest sensitivity of the

position and orientation of the end-effector assumed at T to variations of

the platform and the on-board manipulator control forces. This interpreta-

tion corresponds to the manipulability of stationary manipulators. It is easy

to show, that a 2R type manipulator (arm and forearm of a human hand)

achieves its maximal manipulability when the angle between arms equals π2.

This configuration pattern is assumed by the hand of a writing man, because

it provides the highest sensitivity of the hand keeping a pen to small motions

in its joints.

The average value of the local compliance agility (31)

AC

z0,T (Cz0,T ) =1

vol(X )

∫

X

aC

z0,T (u(·))du(·). (32)

over some region X ⊂ X defines the global compliance agility of the mobile

manipulator. Maximization of this measure with regard to the end point

map Cz0,T , enables one to design a mobile manipulator whose average agility

in executing various tasks would be maximal.

In the case when there is no on-board manipulator, the compliance dexterity

matrix (22) coincides with the compliance mobility matrix (23), which allows

24

us to define the local compliance mobility of the platform

mC

z0,T (u(·)) =√

detMC

z0,T (u(·)). (33)

The global compliance mobility arises as a result of averaging (33) over X ⊂

X

MC

z0,T (Cz0,T ) =1

vol(X )

∫

X

mC

z0,T (u(·))du(·). (34)

To quantitatively deal with the compliance isotropy and the anisotropy of

endogenous configurations of mobile manipulators, we introduce the local

compliance condition number of the endogenous configuration

κC

z0,T (u(·)) =

√

√

√

√

λDC

z0,T(u(·))

λDC

z0,T(u(·))

, (35)

that provides information about the shape of the compliance agility ellipsoid.

The global compliance condition number can be defined as

KC

z0,T (Cz0,T ) =1

vol(X )

∫

X

κC

z0,T (u(·))du(·), (36)

where X ⊂ X .

Eventually, by analogy to the performance measures of stationary manipula-

tors, we define the compliance distortion of a mobile manipulator. The dis-

tortion is an imitation of the left Cauchy-Green tensor occurring in the theory

of elasticity [21]. Consider the end point map Cz0,T : X → SE(3) as a con-

25

figuration map. Then, the mobile manipulator compliance map JC

z0,T (u(·)) :

X → R6 may be regarded as the deformation gradient of the end point map.

The adjoint compliance map becomes a map (JC

z0,T )?(u(·)) : R6 → X defined

as

((JC

z0,T )?(u(·))p)(t) =

= R−1(u(·), t)BT (t)ΦT (T, t)

J1z0,T (q(T ), x(T ))

J2z0,T (q(T ), x(T ))

T

p.

Using (18) and (22) we compute the left Cauchy-Green tensor

JC

z0,T (u(·))(JC

z0,T )?(u(·))p = DC

z0,T (u(·))p.

The local compliance distortion

dC

z0,T (u(·)) = TrDC

z0,T (u(·)). (37)

of the end point map Cz0,T at the configuration u(·) is equal to the squared

Frobenius norm of the so called left stretch tensor [21], and can be interpreted

as a local measure of elasticity of this end point map. By averaging of (37)

over an X ⊂ X , we obtain the global compliance distortion

DC

z0,T (Cz0,T ) =1

vol(X )

∫

X

dC

z0,T (u(·))du(·). (38)

The concept of distortion for stationary manipulators has been introduced

in [24].

26

3.2 Admittance performance measures

By analogy to the compliance performance measures for mobile manipulators,

we can introduce admittance performance measures. To this aim it is enough

to notice, that admittance performance measures are based on the admittance

dexterity matrix (28). Thus, substituting (28) into (29), (31), (32), (35), (36),

(37) and (38) we obtain, respectively, admittance agility ellipsoid, local and

global admittance agility, local and global admittance condition number and

local and global admittance distortion. In general the admittance performance

measures refer to the transmission of increments of input forces into incre-

ments of taskspace velocities.

4 Optimal motion patterns

Local and global dynamic dexterity measures introduced in the previous

section can be exploited in order to determine the optimal configurations

(motion patterns), geometries or dynamics of mobile manipulators. Below we

shall exemplify only the local compliance and admittance dynamic measures.

A mobile platform (no on-board manipulator) will serve as a testbed. A

compliance–like global dynamic measure was examined in [36], [37].

Since the endogenous configuration space is infinite-dimensional, in order to

carry out effective computations we use a finite parameterization of platform

controls u(·) by truncated orthogonal expansions. Such a representation will

27

be called band-limited [8]. In this way we obtain a band-limited endogenous

configuration space X and band-limited performance measures.

4.1 Motion patterns of MK mobile robot

In order to illustrate the compliance and admittance performance measures

we shall consider MK mobile robot, recently designed in our Robotics Labora-

tory. The robot is shown in Figure 1, whose left part defines all parameters of

Figure 1: MK mobile robot

the robot. MK is a differential drive type mobile robot actuated with respect

to its swinging cylindric body. The coordinate vector q = (x, y, ϕ1, ϕ2, α)

describes position of the middle point on the robot axle in the inertial frame,

the revolution angles of the wheels, and the swing angle of the robot body.

Under assumption of (lateral and longitudinal) non-slipping motion of the

robots wheels, the mathematical model of its kinematics and dynamics takes

28

the following form [16]

x = η123 cos θ, y = η123 sin θ, ϕ1 = 2Rη1, ϕ2 = 2

Rη2, α = 1

Rη3,

M(α)η + C(α, η) = Bu,

y = k(x, y, α, ϕ1, ϕ2) =

= (x + (` + rc) sin α cos θ, y + (` + rc) sin α sin θ, R − (` + rc) cos α) ,

(39)

where θ = R2d

(ϕ2−ϕ1) and η123 = η1 +η2 +η3. Variables η1, η2 are determined

by linear velocities of the wheels at their contact points with the ground, η3

is a component of the robot linear velocity resulting from swinging motion

of the robot body. The output function describes the Cartesian position of

a chosen point at the robot body. The inertia matrix M(α), the vectors of

gravity, Coriolis, centripetal and frictional forces C(α, η), and control matrix

B are the following:

M (α) =

(

mc + 43mo + 8mk

)

d2 + mkR2 + mc

(

`2 sin2 α + 13`2

c + 12r2

c

)

(

mc + 23mo

)

d2 − mkR2 − mc

(

`2 sin2 α + 13`2

c + 12r2

c

)

(

mc + mo + 4mk + mc`R

cos α)

d2

(

mc + 23mo

)

d2 − mkR2 − mc

(

`2 sin2 α + 13`2

c + 12r2

c

)

(

mc + 43mo + 8mk

)

d2 + mkR2 + mc

(

`2 sin2 α + 13`2

c + 12r2

c

)

(

mc + mo + 4mk + mc`R

cos α)

d2

(

mo + mc + 4mk + mc`R

cos α)

d2

(

mc + mo + 4mk + mc`R

cos α)

d2

(

mc + mo + 4mk + 2mc`R

cos α + mc

R2

(

`2 + 12r2

c

))

d2

,

(40)

29

C (α, η) =

−mc` sin α (2η2 + η3) (η2 − η1)−

2mc`2

Rsin α cos αη3 (η2 − η1) − mcd2` sin α

R2 η23 + 4d2k1

R2 η1,

mc` sin α (2η1 + η3) (η2 − η1) +

2mc`2

Rsin α cos αη3 (η2 − η1) − mcd2` sinα

R2 η23 + 4d2k2

R2 η2,

−mc` sin αR

(η2 − η1)2 (R + ` cos α) − mc`d2

R2 sin αη23+

mcg`d2

Rsin α

, B = 2d2

R

I2

0

.

(41)

In the representation (39) the controls have been chosen in the form of trun-

cated Fourier series

ui(t) =

k∑

j=0

ci2j−1 sin jωt + ci2j cos jωt, i = 1, 2, (42)

with either k = 0 (constant controls) or k = 1 (first order harmonics). The

corresponding band-limited endogenous configuration spaces are X = R2 and

X = R6.

Specific calculations have been accomplished for real values of geometric

parameters R = 0.254[m], d = 0.225[m], ` = 0.12[m], rc = 0.035[m],

lc = 0.165[m], mass parameters mo = 0.5[kg], mk = 2[kg], mc = 7[kg],

fixed initial posture q0 = 0 of the platform, initial velocities η0 = 0, and with

constraints imposed on admissible configurations | cij |≤ 0.5 for j = 0 and

|cij |≤ 0.25 for j = 1, 2. Weight matrices R(u(·), t) and Q(y) are assumed to

be the unity matrices.

30

4.1.1 The compliance mobility of MK mobile robot

The problem to be solved consists in the determination of control functions

in the form (42) providing the maximal value of the local compliance mobility

(33). Numerical values of optimal configurations (i.e. control parameters)

obtained for different values of time horizon T are collected in Tables 1 and 2.

For short they will be referred to as compliance mobile configurations. Ap-

plied to the mobile robot, these configurations generate characteristic motion

patterns in the taskspace, shown in Figures 2 and 3. The results demonstrate

that after crossing a certain time horizon (T = 3), compliance optimal mo-

tion patterns change from straight line into curves. Looking at the shape of

mobility ellipsoids we notice that anisotropy of compliance mobile configu-

rations of the mobile robot increases along with the time horizon. It also

insignificantly increases after adding harmonics to the control functions.

time horizon mobile configurations

T = 3(c10, c20) = (±0.5,±0.5)

(mC

q0,η0,T )(c) = 0.0112

T = 7(c10, c20) = (±0.4432,±0.5), (±0.5,±0.4432)

(mC

q0,η0,T )(c) = 40.7099

T = 12(c10, c20) = (±0.4824,±0.5), (±0.5,±0.4824)

(mC

q0,η0,T )(c) = 6.6540 ∗ 104

T = 22(c10, c20) = (±0.4973,±0.5), (±0.5,±0.4973)

(mC

q0,η0,T )(c) = 1.8854 ∗ 108

Table 1: Compliance mobile configurations of MK mobile robot (X = R2)

31

time horizon mobile configurations

T = 3(c10, c11, c12, c20, c21, c22) =

= (±0.5,±0.25,±0.25,±0.5,±0.25,±0.25)(mC

q0,η0,T )(c) = 0.0306

T = 7

(c10, c11, c12, c20, c21, c22) == (±0.4589,±0.1802,±0.25,±0.5,±0.25,±0.25),

(±0.5,±0.25,±0.25,±0.4589,±0.1802,±0.25)(mC

q0,η0,T )(c) = 168.5453

Table 2: Compliance mobile configurations of MK mobile robot (X = R6)

4.1.2 The admittance mobility of MK mobile robot

The aim of this subsection consist in applying the concept of local admittance

mobility in order to determine admittance optimal motion patterns of MK

mobile robot. Here, instead of k(x, y, α, ϕ1, ϕ2) we take

k(x, y, α, ϕ1, ϕ2, η1, η2, η3).

time horizon mobile configurations

T = 3(c10, c20) = (±0.5,±0.5)

(mA

q0,η0,T )(c) = 0.7098

T = 7(c10, c20) = (±0.5,±0.5)

(mA

q0,η0,T )(c) = 47.5042

T = 12(c10, c20) = (±0.4822,±0.5), (±0.5,±0.4822)

(mA

q0,η0,T )(c) = 103.7116

T = 22(c10, c20) = (±0.4355,±0.5), (±0.5,±0.4355)

(mA

q0,η0,T )(c) = 2.8158 ∗ 108

Table 3: Admittance mobile configurations of MK mobile robot (X = R2)

Tables 3 and 4 collect the results for different values of time horizon T .

Optimal motion patterns of MK mobile robot corresponding to the maximal

admittance mobility along with the associated admittance mobility ellipsoids

32

a)

-1.5 -1 -0.5 0.5 1 1.5x

-1.5

-1

-0.5

0.5

1

1.5y

−4

−2

0

2

4 −4

−3

−2

−1

0

1

2

3

4

−0.05

0

0.05

y

x

b)

-4 -3 -2 -1 1 2 3 4x

-4

-3

-2

-1

1

2

3

4

y

−60

−40

−20

0

20

40

60 −60

−40

−20

0

20

40

60

−0.05

0

0.05

y

x

Figure 2: Motion pattern on x − y plane corresponding to the compliancemobile configurations of MK mobile robot along with the compliance mobilityellipsoid X = R2: a) T = 3; b) T = 7

are shown in Figures 4 and 5. It is worth observing that also in the case of

admittance mobility the anisotropy of configurations of the mobile robot in-

creases with the time horizon as well as after adding harmonics to the control

functions. It turns out that both in case of compliance and admittance mo-

bility, after crossing a certain value of the time horizon T , the optimal motion

patterns of the mobile robot change from the straight lines into curves. A

difference between compliance and admittance motion patterns is that in the

later case the mobile robot obeys the straight line pattern for longer times

33

a)

-6 -4 -2 2 4 6x

-6

-4

-2

2

4

6

y

−4

−2

0

2

4 −4

−3

−2

−1

0

1

2

3

4

−0.05

0

0.05

y

x

b)

-4 -3 -2 -1 1 2 3 4x

-4

-3

-2

-1

1

2

3

4

y

−60

−40

−20

0

20

40

60 −60

−40

−20

0

20

40

60

−0.05

0

0.05

y

x

Figure 3: Motion pattern on x − y plane corresponding to the compliancemobile configurations of MK mobile robot along with the compliance mobilityellipsoid X = R6: a) T = 3; b) T = 7

(T = 7).

5 Conclusion

The main objective of this paper has been to propose a new control theoretic

framework for defining performance measures of mobile manipulators. A cor-

nerstone of our approach is the end point map of a control system represen-

tation of mobile manipulator kinematics and dynamics. In this framework,

34

time horizon mobile configurations

T = 3(c10, c11, c12, c20, c21, c22) =

= (±0.5,∓0.25,±0.25,±0.5,∓0.25,±0.25)(mA

q0,η0,T )(c) = 1.8897

T = 7(c10, c11, c12, c20, c21, c22) =

= (∓0.5,±0.25,∓0.25,∓0.5,±0.25,∓0.25)(mA

q0,η0,T )(c) = 137.3991

T = 12

(c10, c11, c12, c20, c21, c22) == (±0.5,∓0.25,∓0.25,∓0.5,±0.2496,±0.25),

(∓0.5,±0.2496,±0.25,±0.5,∓0.25,∓0.25)(mA

q0,η0,T )(c) = 2.0094 ∗ 108

Table 4: Admittance mobile configurations of MK mobile robot (X = R6)

we can treat in a uniform way all types of mobile manipulators consisting

of holonomic or nonholonomic mobile platforms and holonomic or nonholo-

nomic on-board manipulators, as well as separate stationary manipulators

and mobile platforms. Kinematic performance measures were discussed in

[30]. In this paper we have concentrated on two types of dynamic measures:

the compliance measures and the admittance measures. Computer simu-

lations show optimal motion patterns of a mobile robot provided by these

measures. Other dynamic measures can be introduced according to the same

definitional scheme, utilizing higher order prolongations of the output func-

tion. The approach set forth in this paper on the one hand provides new

design tools for mobile manipulators, on the other opens new research per-

spectives and states new questions. One such a question is concerned with

the construction of a kind of canonical performance measures, that would

be independent both of the task and the configuration space coordinate sys-

35

a)

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

y

−5

0

5 −5

0

5

−0.5

0

0.5

y

x

b)

-10 -7.5 -5 -2.5 2.5 5 7.5 10x

-10

-7.5

-5

-2.5

2.5

5

7.5

10

y

−150

−100

−50

0

50

100

150 −150

−100

−50

0

50

100

150

−0.2

0

0.2

y

x

Figure 4: Motion pattern on x − y plane corresponding to the admittancemobile configurations of MK mobile robot along with admittance mobilityellipsoid X = R2: a) T = 3; b) T = 12

tems, and of the representation of control signals driving the robot. We

believe that this question may be approached along the lines suggested for

stationary manipulators in [10]. Potentially, the introduction of Rieman-

nian structures to the configuration space and to the taskspace of mobile

manipulators will serve this purpose (it enables defining weighted Jacobian

pseudoinverses [10], [22]), however this problem is beyond the scope of the

present paper. Another important question concerns applicability of the

control theoretic framework to the performance evaluation of systems whose

36

a)

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1x

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

y

−5

0

5 −5

0

5

−1

0

1

y

x

b)

-0.06 -0.04 -0.02 0.02 0.04 0.06x

-0.06

-0.04

-0.02

0.02

0.04

0.06

y

−1000

−500

0

500

1000 −1000

−500

0

500

1000−20

0

20

y

x

Figure 5: Motion pattern on x − y plane corresponding to the admittancemobile configurations of MK mobile robot along with admittance mobilityellipsoid X = R6: a) T = 3; b) T = 12

motion violates nonholonomic constraints. An auspicious result would pro-

vide performance measures for motor vehicles [17] and other systems moving

with permitted slip of the wheels [9], [7]. Preliminary results in this direc-

tion have been reported in [38]. Eventually, a construction of performance

measures independent of the time horizon and the initial state of the mobile

platform would be of considerable interest and might open a way toward

defining task-oriented performance measures. A first step toward this aim

has been made in [37].

37

6 Acknowledgment

This research has been supported by the Foundation for Polish Science.

References

[1] J. Angeles, The robust design of parallel manipulators, Proc.1st Int.

Coll. of Collaborative Research Centre 562, Braunschweig, Germany,

2002, pp. 9–30.

[2] N. A. Aspragathos and S. Foussias, Optimal location of a robot path

when considering velocity performance, Robotica, 20, 2002, pp. 139–147.

[3] G. L. Baker, J. P. Gollub, Chaotic Dynamics: An Introduction, Cam-

bridge, 1990.

[4] B. Bayle, I. Y. Fourquet, M. Renaud, Manipulability analysis for mo-

bile manipulators, Proc. 2001 IEEE Int. Conf. Robot. Automat., Seoul,

Korea, 2001, pp. 1251 - 1256.

[5] B. Bayle, J.-Y. Fourquet and M. Renaud, Manipulability of wheeled

mobile manipulators: application to motion generation, Int. J. Robot.

Res., 22(7–8), 2003, pp. 565–581.

[6] A. Bowling, O. Khatib, The Dynamic Capability Equations: A New Tool

for Analyzing Robotic Manipulator Performance, IEEE Trans. Robot.,

21(1), 2005, pp. 115 - 123.

38

[7] L. Caracciolo, A. De Luca and S. Iannitti, Trajectory tracking control

of a four-wheel differentially driven mobile robot, Proc. 1999 IEEE Int.

Conf. Robot. Automat., Detroit, MI, 1999, pp. 2632–2638.

[8] G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Non-

comutative Harmonic Analysis, CRC Press, Boca Raton, 2001.

[9] B. D‘Andrea-Novel, G. Campion and G. Bastin, Control of wheeled

mobile robots not satisfying ideal velocity constraints: a singular per-

turbation approach, Int. J. Robust Nonlin. Contr., 5, 1995, pp. 243–267.

[10] K. L. Doty, C. Melchiorri, E.M. Schwartz, C. Bonivento, Robot manip-

ulability, IEEE Trans. Robot. Automat., 11(2), 1995, pp. 462 - 468.

[11] K. L. Doty, C. Melchiorri, C. Bonivento,, A Theory of Generalized In-

verses Applied to Robotics, Int. J. Robot. Res., 12(1), 1993, pp. 1 -

19.

[12] K. Van Den Doel, D. K. Pai, Performance measures for robot manip-

ulators: A unified approach, Int. J. Robot. Res., 15(1), 1998, pp. 92 -

111.

[13] G. Foulon, I. Y. Fourquet, M. Renaud, On coordinated tasks for non-

holonomic mobile manipulators, Proc. 5th IFAC Symp. Robot Control,

Nantes, France, 1997, pp. 491 - 498.

[14] J. F. Gardner, S. A. Velinsky, Kinematics of mobile manipulators and

implications for design, J. Robot. Syst., 17(6), 2000, pp. 309 - 320.

39

[15] C. Gosselin, J. Angeles, A global performance index for the kinematic

optimization of robotic manipulators, Trans. ASME, vol. 113, 1991, pp.

220 - 226.

[16] M. Kaba la, R. Muszynski, M. Wnuk, Singularity robust, dynamic lin-

earization control algorithm for MK mobile robot, In: Robot Control

2003, ed. by: I. Duleba i J. Z. Sasiadek, Elsevier, Oxford, 2003, pp. 557

- 562.

[17] U. Kiencke and L. Nielsen, Automotive Control Systems, Springer Ver-

lag, 2000, Berlin.

[18] C. A. Klein, B. E. Blaho, Dexterity measures for the design and control

of kinematically redundant manipulators, Int. J. Robot. Res., 6(2), 1987,

pp. 72-82.

[19] A. Liegeois, Automatic supervisory control for the configuration and

behavior of multibody mechanisms, IEEE Trans. Syst. Man Cybernet.,

7(12), 1977, pp. 842 - 868.

[20] O. Ma, J. Angeles, Optimum design of manipulators under dynamic

isotropy conditions, Proc. 1993 IEEE Int. Conf. Robot. Automat., At-

lanta, GA, pp. 470 - 475.

[21] J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elastic-

ity, Prentice-Hall, Englewood Cliffs, N.J., 1983.

40

[22] Y. Nakamura, Advanced Robotics: Redundancy and Optimization, Ad-

dison Wesley, 1991, Reading, MA.

[23] J. A. Pamanes-Garcia, A criterion for optimal placement of robotic ma-

nipulators, IFAC Proc. 6th Int. Symp. on Information Control Problems

in Manufacturing Technology, Madrid, Spain, 1989, pp. 183–187.

[24] F.C. Park and R.W. Brockett Kinematic Dexterity of Robotic Mecha-

nisms, Int. J. Robot. Res., 13(1), 1994, pp. 1 - 15.

[25] W. J. Rugh, Linear System Theory, Prentice-Hall, 1997.

[26] J. K. Salisbury and J. J. Craig, Articulated hands: Force control and

kinematic issues, Int. J. Robot. Res., 1(1), 1982, pp. 4–17.

[27] E.D. Sontag, Mathematical Control Theory, Springer-Verlag, New York,

1990.

[28] K. Tchon, R. Muszynski, Instantaneous kinematics and dexterity of mo-

bile manipulators, Proc. 2000 IEEE Int. Conf. Robot. Automat., San

Francisco, CA, 2000, pp. 2493-2498.

[29] K. Tchon and J. Jakubiak, Endogenous configuration space approach

to mobile manipulators: A derivation and performance assessment of

Jacobian inverse kinematics algorithms, Int. J. Control, 76(14), 2003,

pp. 1387–1419.

41

[30] K. Tchon, K. Zadarnowska, Kinematic dexterity of mobile manipulators:

an endogenous configuration space approach, Robotica, 21, 2003, pp. 521

- 530.

[31] Y. Yamamoto, X. Yun, Unified analysis of mobility and manipulability

of mobile manipulators, Proc. 1999 IEEE Int. Conf. Robot. Automat.,

Detroit, MI, 1999, pp. 1200-1206.

[32] Y. Yamamoto, X. Yun, Coordinating locomotion and manipulation of

a mobile manipulator, IEEE Trans. Automat. Contr., 39(6), 1994, pp.

1326 - 1332.

[33] Y. Yamamoto, S. Fukuda, Trajectory planning of multiple mobile ma-

nipulators with collision avoidance capability., Proc. 2002 IEEE Int.

Conf. Robot. Automat., Washington, DC, 2002, pp. 3565-3570.

[34] T. Yoshikawa, Manipulability of robotic mechanisms, Int. J. Robot. Res.,

4(2), 1985, pp. 3 - 9.

[35] T. Yoshikawa, Dynamic manipulability of robot manipulators, J.

Robotic Systems., 1(1), 1985, pp. 113 - 124.

[36] K. Zadarnowska, Dynamic dexterity and isotropy of mobile robots: an

endogenous configuration space approach, In: On Advances in Robot

Kinematics, ed. by: J. Lenarcic i C. Galletti, Kluwer Academic Publish-

ers, Dordrecht, The Netherlands, 2004, pp. 477 - 484.

42

[37] K. Zadarnowska, Performance measures of mobile manipulators, Doc-

toral Dissertation, Inst. Eng. Cybernetics, Wroc law Univ. of Technology,

Poland, 2005 (in Polish).

[38] K. Zadarnowska, A Control Theory Framework for Performance Eval-

uation of Wheeled Mobile Robots Subjected to Slipping: compliance

performance measures, Report of Inst. of Computer Engineering, Con-

trol and Robotics, Wroc law Univ. of Technology, Poland, 2006.

43