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The International Journal of Robotics Research
DOI: 10.1177/02783649030227007
2003; 22; 565 The International Journal of Robotics Research
B. Bayle, J. -Y. Fourquet and M. Renaud
Manipulability of Wheeled Mobile Manipulators: Application to Motion Generation
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B. Bayle
LSIIT
Strasbourg, France
bernard@eavr.u-strasbg.fr
J.-Y. Fourquet
ENIT
Tarbes, France
fourquet@enit.fr
M. Renaud
LAAS-CNRS
Toulouse, France
renaud@laas.fr
Manipulability of
Wheeled Mobile
Manipulators:
Application to
Motion Generation
Abstract
We propose a systematic modeling of the nonholonomic mobile ma-
nipulators built from a robotic arm mounted on a wheeled mobile
platform. We use the models derived to generalize the standard def-
inition of manipulability to the case of mobile manipulators. The
effects of mounting a robotic arm on a nonholonomic platform are
shown through the analysis of the manipulability thus defined. Sev-
eral applications are evoked, particularly applications to control.
The optimization of criteria inherited from manipulability consider-
ations are given to generate the controls of our system when its end
effector motion is imposed.
KEY WORDS—mobile manipulation, manipulability, non-
holonomy, motion planning
1. Introduction
Nowadays, mobile manipulator is a widespread term which
refers to robots built from a robotic arm mounted on a mo-
bile platform. Such systems allow the most usual missions of
roboticsystemswhichrequirebothlocomotion and manipula-
tionabilities(Arai1996;Khatib1997).Actually,suchsystems
combine the advantages of mobile platforms and roboticarms
and reducetheir drawbacks.Forinstance, the mobile platform
extendsthe workspace,while the arm offersmany operational
functionalities (simply opening a door for the robot). So, they
seem particularly suited for field and service robotics. Al-
thoughquiteoldin the robotics history (Pavlovand Timofeyev
The International Journal of Robotics Research
Vol. 22, No. 7–8, July–August 2003, pp. 565-581,
©2003 Sage Publications
1976), this concept has mainly been studied for less than 15
years (Dubowsky and Tanner 1988; Liu and Lewis 1990; Pin
and Culioli 1990).
The publications devotedto these systems are numerous—
see, for example, Bayle, Fourquet, and Renaud (2001a) for a
survey—and cover a wide range of research areas. The clas-
sification of the various contributions is made difficult by the
poor description of the original problems related to mobile
manipulation. Among these contributions, we can classify
some major concerns:
• control (Cameron, MacKenzie, and Ward 1993; Chong
et al. 1997; Chung, Velinsky, and Hess 1998; Dong,
Xu, and Wang 2000; Hootsmans and Dubowsky 1991;
Huang, Sugano, and Tanie 2000; Liu and Lewis 1990;
Pissard-Gibolet 1993), optimal control (Desai et al.
1996; Lee, Kim, and Cho 1996), elastic band (Brock
and Khatib 1997; Quinlan 1994);
• path optimization problems (Carriker, Khosla, and
Krogh 1991; Chen and Zalzala 1997; Pin, Culioli, and
Reister 1994; Pin et al. 1997; Zhao, Ansari, and Hou
1994);
• operational motion planning problem (Seraji 1993,
1998), generalized motion planning problem (Foulon,
Fourquet,and Renaud 1998), operational path planning
problem (Foulon, Fourquet, and Renaud 1998, 1999)
or generalized path planning problem (Bayle, Four-
quet,and Renaud 2000a;Foulon,Fourquet,andRenaud
1999; Fourquet and Renaud 1996);
• dynamic modeling (Dubowsky and Tanner 1988;
Fukudaet al. 1992;Inoue, Miyamoto,and Okawa1996;
565
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566 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July–August 2003
Yamamoto and Yun 1996, 1997), stability, effects of
mechanical disturbances (Huang, Sugano, and Tanie
1998, 2000; Rey and Papadopoulos 1997);
• multi-robot cooperation (Desai and Kumar 1996;
Hashimoto, Oba,and Zenitani1995; Khatib et al. 1996;
Sugar and Kumar 1999), etc.
In spite of the great number of studies, quite few efforts
have been made on the systematic modeling ofthe mobile ma-
nipulators. Among the works on the modeling of a particular
mobile manipulator, we can cite Yamamoto and Yun (1994,
1997) and Tcho
´
n and Muszy
´
nski (2000). There is a simple
explanation for this. Indeed, if the systematic modeling of
robotic arms is well known, this is not necessarily the case for
mobile platforms.
In the case of wheeled mobile platforms, which include a
large part of the land mobile manipulators, except mainly hu-
manoids and all-terrain systems, the rolling without slipping
(r.w.s.) of the wheels on the ground introduces specific dif-
ficulties in the modeling. The platform, which cannot move
instantly in any arbitrary direction, is then said to be non-
holonomic. If we restrict our study to that large category of
wheeled mobile platforms, we must evoke the excellent con-
tribution of Campion, Bastin, and D’Andréa-Novel (1996),
which offers good tools for the generic modeling of robotic
systems built from wheeled mobile platforms. We propose
this modeling in this paper after we introduce the kinematic
modeling of the subsystems: platform and robotic arm. This
is the purpose of Section 2. This reveals, in particular, the
existence of the control of mobility of the mobile manipu-
lator, which represents the control producing instantaneous
velocities of the end effector (EE) of the mobile manipulator.
Also we obtain the instantaneous kinematic location model
(IKLM) of the mobile manipulator, which sets the derivative
of the EE location as a function of the control of mobility.
The previous modeling is then used in Section 3 to extend
the theory of manipulability of Yoshikawa (1985), proposed
for the robotic arms, to the case of wheeled mobile manipula-
tors. This proves a very appropriate tool to characterize their
instantaneous kinematics.
Section 4 points out some possible applications of this
instantaneous kinematic analysis, particularly from a con-
trol point of view. Notably we address the operational mo-
tion planning problem (Bayle, Fourquet, and Renaud 2000b).
Since our system is redundant, the inversion of the IKLM of
the mobilemanipulator allows us to solve thisproblem and, in
the meantime, to take into account an additional criteria based
on a measure of the mobile manipulator manipulability.
This study aims at understanding the use ofmanipulability,
usually defined for robotic arms, in the case of nonholonomic
mobile manipulators.
2. Modeling
Inthissection,weillustratethegenericmodelingofthemobile
manipulator using the example of a planar mobile manipula-
tor, represented in Figure 1. We first model independently its
platform in Section 2.1 and its robotic arm in Section 2.2. The
modelingof the mobilemanipulator willbe finallyestablished
in Section 2.3.
As we will see later, the ellipsoids of manipulability are
six-dimensional in the most general case. For obvious rea-
sons of visualization and thus of pedagogy, we only provide
examples of planar systems in this paper; in this case, the el-
lipsoids of manipulability are simple ellipses. Nevertheless,
the techniques of modeling and manipulability analysis de-
tailed here are absolutely general and thus apply to systems
with more complex structures. For instance, they were suc-
cessfullyappliedto the mobilemanipulator of theLAAS. This
mobile manipulator is built from a two independently driven
wheel mobile platform and a six revolute joint robotic arm
(Bayle, Fourquet, and Renaud 2002).
2.1. Wheeled Mobile Platform Case
Wheeled mobile platforms are properly described and mod-
eled by Campion, Bastin, and D’Andréa-Novel (1996). In this
section, we introduce only the elements which are necessary
in our study. Only minor changes have been done purposely
to be consistent with the modeling of mobile manipulators.
2.1.1. Description
We assume that the mobile platform moves on a planar hori-
zontal surface. LetR = (O, x, y, z) be anyfixed frame with
z vertical and R
= (O
, x
, y
, z
) a mobile frame linked to
the platform. The origin of R
is usually chosen as a remark-
able point of the platform (e.g., the midpoint of the rear axle).
The location
1
(Fourquet and Renaud 1999) of the platform
is given by a vector ξ
ξ
ξ
p
of m
p
= 3 coordinates which define
the position and the orientation of the platform in R. They are
called the operational coordinates of the platform. We write
ξ
ξ
ξ
p
=[xyϑ]
T
, where x and y are, respectively, the abscissa
and the ordinate of O
in R and ϑ is the angle (x, x
). The
set of all the locations constitutes the operational space of the
platform, denoted by M
p
.
The mobile platform wheels can be classified into four
categories:
• the fixed wheels for whichthe axle has a fixed direction;
• the steering wheels, for which the orientation axis
passes through the center of the wheel;
1. It is called as posture in Campion, Bastin, and D’Andréa-Novel (1996) but
the choice of location allows us to use the same name for the robotic arms,
the platforms and later for the mobile manipulators.
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Bayle, Fourquet, and Renaud / Manipulability of Wheeled Mobile Manipulators 567
Fig. 1. Planar mobile manipulator with a car-like platform.
• the castor wheels, for which the orientation axis does
not pass through the center of the wheel;
• the Swedish wheels, which are similar to the fixed
wheels, with the exception of an additional parameter
that describes the direction, with respect to the wheel
plane, of the zero component of the velocity at the con-
tact point.
It is assumed that the wheels always keep their shape, which
is sensible in indoor robotics. We will consider that they all
have the same radius, denoted by r.
Let N = N
f
+ N
s
+ N
c
+ N
sw
be the total number of
wheels. N
f
, N
s
, N
c
, and N
sw
represent, respectively, the num-
ber of fixed, steering, castor and Swedish wheels. From now
on, these indices correspond to the different types of wheels.
Let ϕ
ϕ
ϕ
f
, ϕ
ϕ
ϕ
s
, ϕ
ϕ
ϕ
c
, and ϕ
ϕ
ϕ
sw
be the vectors giving the rotation an-
gles of the wheels about their horizontal axle; they all are
variables. Let β
β
β
f
, β
β
β
s
, β
β
β
c
, and β
β
β
sw
be the vectors giving the
orientation of the wheels; β
β
β
f
and β
β
β
sw
are constants whereas
β
β
β
s
and β
β
β
c
are variables. The N-dimensional rotation vector
is ϕ
ϕ
ϕ =[ϕ
ϕ
ϕ
T
f
ϕ
ϕ
ϕ
T
s
ϕ
ϕ
ϕ
T
c
ϕ
ϕ
ϕ
T
sw
]
T
. Then, the platform configuration is
given by the n
p
-dimensional vector, q
q
q
p
=[ϕ
ϕ
ϕ
T
ξ
ξ
ξ
T
p
β
β
β
T
s
β
β
β
T
c
]
T
,
with n
p
= N + m
p
+ N
s
+ N
c
3.
2.1.2. Models
We assume that the conditions of r.w.s. of the wheels on the
ground are always satisfied. Campion, Bastin, and D’Andréa-
Novel (1996) show that there exists a systematic writing of
the r.w.s. constraints.
By writing the r.w.s. constraints, we find out that there
exist a vector, denoted by η
η
η
p
, that we call control of mobility
of the platform, and a matrix (β
β
β
s
), which allow us to set the
derivative of ξ
ξ
ξ
p
˙
ξ
ξ
ξ
p
= R(ϑ)(β
β
β
s
)η
η
η
p
, (1)
where
R(ϑ) =
C
ϑ
−S
ϑ
0
S
ϑ
C
ϑ
0
001
,
with C
ϑ
= cos ϑ and S
ϑ
= sin ϑ, is the rotation matrix ex-
pressing the orientation of R
with respect to R. The model
(1) relates the derivative of the platform location, in a given
configuration, to its control of mobility. So it is called the
IKLM of the mobile platform. The dimension δ
m
p
of η
η
η
p
is
called the degree of mobility of the platform.
Additionally there exists an N
s
-dimensional vector, de-
noted byζ
ζ
ζ
p
, that wecall controlof steerabilityof the platform,
which represents the kinematic control of the steering wheels
orientation:
˙
β
β
β
s
= ζ
ζ
ζ
p
. (2)
The dimension δ
s
p
= N
s
of ζ
ζ
ζ
p
is called the degree of steer-
ability of the platform.
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568 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July–August 2003
The control (of maneuvrability) of the mobile platform is
then u
u
u
p
=[η
η
η
T
p
ζ
ζ
ζ
T
p
]
T
. Its dimension, δ
M
p
= δ
m
p
+ δ
s
p
, is called
the degree of maneuvrability of the platform.
If we define z
z
z
p
=[ξ
ξ
ξ
T
p
β
β
β
T
s
]
T
, then
˙
z
z
z
p
= B
p
(ϑ, β
β
β
s
)u
u
u
p
,
with
B
p
(ϑ, β
β
β
s
) =
R(ϑ)(β
β
β
s
) 0
0 I
N
s
.
where I
N
s
is the N
s
order identity matrix.
The r.w.s. constraints also allow us to set the derivative
of q
q
q
p
˙
q
q
q
p
= S
p
(ϑ, β
β
β
s
,β
β
β
c
)u
u
u
p
, (3)
which relates the derivative of the platform configuration,
in a given configuration, to its control (of maneuvrability).
The corresponding model is called the instantaneous kine-
matic configuration model (IKCM)ofthe platform(Campion,
Bastin, and D’Andréa-Novel 1996).
From now on, we assume that the mobile platform al-
ways has a number of actuators equal to its degree of
maneuvrability.
2.1.3. Example
We consider the case of the car-like platform shown in Fig-
ure 2. The car is reduced to an equivalent vehicle (car-like
platform) with only one steering wheel in the front, on the
longitudinal axis of the vehicle,
2
and two fixed wheels on the
same axle, in the back. The wheels are numbered in Figure 2.
We denote by β
3
= β
s
1
the orientation of the steering wheel,
by ϕ
1
= ϕ
f
1
, ϕ
2
= ϕ
f
2
, and ϕ
3
= ϕ
s
1
the rotation angles of the
right fixed wheel, of the left fixed wheel and of the steering
wheel, respectively (we hide indices .
f
and .
s
for simplicity).
So we obtain ϕ
ϕ
ϕ =[ϕ
1
ϕ
2
ϕ
3
]
T
and q
q
q
p
=[ϕ
1
ϕ
2
ϕ
3
xyϑβ
3
]
T
.
If we write the r.w.s. constraints, we obtain (Bayle 2001)
R
T
(ϑ)
˙
ξ
ξ
ξ
p
=
−DS
β
3
0
C
β
3
η
p
,
with C
β
3
= cos β
3
and S
β
3
= sin β
3
. The control of mobility
η
p
is a scalar and then δ
m
p
= 1. The platform IKLM is thus
˙
ξ
ξ
ξ
p
=
−DC
ϑ
S
β
3
−DS
ϑ
S
β
3
C
β
3
η
p
. (4)
As
˙
β
3
= ζ
p
(then δ
m
p
= 1), in this case, u
u
u
p
=[η
p
ζ
p
]
T
and
then δ
M
p
= 2.
2. This model is convenient to describe the system equipped with two front
wheels sharing the same instantaneous center of rotation.
The platform IKCM computation is detailed in Bayle
(2001) (see Figure 2):
˙
q
q
q
p
=
˙
ϕ
ϕ
ϕ
˙
ξ
ξ
ξ
p
˙
β
3
= S
p
(ϑ, β
3
)u
u
u
p
, (5)
with
S
p
(ϑ, β
3
) =
1
r
(DS
β
3
− LC
β
3
) 0
−
1
r
(DS
β
3
+ LC
β
3
) 0
−
D
r
0
−DC
ϑ
S
β
3
0
−DS
ϑ
S
β
3
0
C
β
3
0
01
.
2.2. Robotic Arm Case
2.2.1. Description
We consider a robotic arm built from n
a
mobile bodies (sup-
posed perfectly rigid) articulated by n
a
revolute and/or pris-
matic joints. The most usual way to model a robotic arm con-
sists of using the Denavit–Hartenberg modified parameters
(Craig 1989; Khalil and Kleinfinger 1986). We associate the
frame R
i
= (O
i
, x
i
, y
i
, z
i
), with i = 0, 1, ... , n
a
,tothe
ith body of the robotic arm. So, the frame R
0
is linked to
the base. The center of the EE (tool or grip) is denoted by
O
n
a
+1
. So, both points O
n
a
and O
n
a
+1
are linked to the EE.
The Denavit–Hartenberg modified parameters define the lo-
cation(positionandorientation) of all the bodies of the robotic
arm, i.e., its whole geometry.
The robotic arm configuration is known when the posi-
tion of all its points in R
0
are known (Neimark and Fufaev
1972). It is defined by a vector q
q
q
a
of n
a
independent coor-
dinates. These coordinates, called generalized coordinates of
the robotic arm, characterize the values of the different joints:
rotation angles for the revolute ones, translations for the pris-
matic joints. The configuration q
q
q
a
=[q
a
1
q
a
2
... q
a
n
a
]
T
is an
elementof theconfigurationspace of theroboticarm, denoted
by N
a
.
The location of the robotic arm EE is given by an m
a
-
dimensional vector, denoted by ξ
ξ
ξ
a
=[ξ
a
1
ξ
a
2
... ξ
a
m
a
]
T
. Its
m
a
coordinates are the operational coordinates of the robotic
arm. They define the position and the orientation of the
EE in R
0
. The set of the locations constitutes the opera-
tional space of the robotic arm, denoted by M
a
. We define:
(i) the position of the EE in R
0
by the Cartesian coordi-
nates of O
n
a
+1
, which is the most usual choice, [ξ
a
1
ξ
a
2
ξ
a
3
]
T
;
(ii) the orientation of the EE with the Euler classical angles
(Paul 1981), [ξ
a
4
ξ
a
5
ξ
a
6
]
T
=[ψθϕ]
T
. This choice is particu-
larly relevant in this study (see Section 2.3).
R
EMARK. The location of the robotic arm EE can be defined
in different ways according to the task to achieve. So, for a
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