280
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014
A New Performance Index for the Repetitive
Motion of Mobile Manipulators
Lin Xiao and Yunong Zhang,
Member, IEEE
Abstract—A
mobile manipulator is a robotic device composed
of a mobile platform and a stationary manipulator fixed to the
platform. To achieve the repetitive motion control of mobile
manipulators, the mobile platform and the manipulator have
to realize the repetitive motion simultaneously. To do so, a novel
quadratic performance index is, for the first time, designed and
presented in this paper, of which the effectiveness is analyzed
by following a neural dynamics method. Then, a repetitive
motion scheme is proposed by combining the criterion, physical
constraints, and integrated kinematical equations of mobile
manipulators, which is further reformulated as a quadratic
programming (QP) subject to equality and bound constraints.
In addition, two important Bridge theorems are established to
prove that such a QP can be converted equivalently into a linear
variational inequality, and then equivalently into a piecewise-
linear projection equation (PLPE). A real-time numerical algo-
rithm based on PLPE is thus developed and applied for the online
solution of the resultant QP. Two tracking-path tasks demonstrate
the effectiveness and accuracy of the repetitive motion scheme. In
addition, comparisons between the nonrepetitive and repetitive
motion further validate the superiority and novelty of the
proposed scheme.
Index Terms—Mobile
manipulators, neural dynamics,
quadratic programming (QP), real-time numerical algorithm,
repetitive motion scheme.
I. Introduction
A
ROBOT manipulator is said to be redundant when more
degrees of freedom (DOF) are available than the min-
imum number that is required to perform a specific end-
effector primary task. The potential application of a redundant
manipulator is determined, in some sense, by the DOF number,
as well as by the manipulator’s structure and its control
scheme. The end-effector’s motion will not be accurate or even
cannot be fulfilled if a manipulator does not have sufficient
DOF number. The use of redundant manipulators is expected
to increase dramatically in the future because of their ability
Manuscript received September 26, 2012; revised February 24, 2013;
accepted March 14, 2013. Date of publication May 7, 2013; date of current
version January 13, 2014. This work was supported in part by the National
Natural Science Foundation of China, under Grants 61075121 and 60935001,
and the Specialized Research Fund for the Doctoral Program of Institutions
of Higher Education of China with Project 20100171110045. This paper was
recommended by Associate Editor H. M. Schwartz.
The authors are with the School of Information Science and Tech-
nology, Sun Yat-Sen University, Guangzhou 510006, China (e-mail:
xiaolin860728@163.com; zhynong@mail.sysu.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCYB.2013.2253461
to avoid the internal singularity configurations and obstacles,
and to optimize dynamic performance.
Fixed-based redundant manipulators have long been studied
and widely applied in factory automation [1]–[6], such as
working in hazardous or cluttered environments, carrying
radioactive materials, doing repetitive and dull work within
a limited workspace. In addition, many techniques have been
developed for solving the inverse kinematics of fixed-based
redundant manipulators. The most popular method is to apply
the pseudoinverse formulation for obtaining a general solu-
tion at the joint-velocity level, which contains a minimum-
norm particular solution and a homogeneous solution [7], [8].
However, even though the joint velocities are instantaneously
minimized, there is no guarantee that the kinematic singu-
larities are avoided. Moreover, this method has the gener-
ally undesirable property that repetitive end-effector motions
do not necessarily yield repetitive joint motions. Thus, the
manipulator’s behavior is difficult to predict when the end-
effector traces a closed path in its workspace. In addition, it is
less efficient to readjust the manipulator’s configuration after
every cycle via self-motion such that the joint angles return
to their initial values. Klein and Huang [9] were the first to
observe this phenomenon for the case of the pseudoinverse
control to a planar three-link manipulator. A large number of
research on this topic have been produced in the last three
decades. For example, Brockett [10] and Shamir and Yomdin
[11] provided differential geometric interpretations along with
necessary and sufficient conditions for repeatability. In [12],
an extended Jacobian matrix method was for the first time
presented to achieve repetitive motion control of fixed-based
redundant manipulators. Roberts and Maciejewski [13] were
devoted to the optimal synthesis of repeatable extended Jaco-
bian inverse kinematics algorithms with prescribed properties.
Marcos
et al.
[14], [15] provided a technique that combines
the closed-loop pseudoinverse method with genetic algorithms,
leading to an optimization criterion for repeatable control
of redundant manipulators, and avoiding the joint-angle drift
problem. Zhang
et al.
[16]–[18] proposed a cyclic-motion
generation scheme, which can remedy effectively the joint-
angle drift phenomenon of fixed-base redundant manipulators
and make the joint variables kept within their limits.
With the evolution of the complex technological society and
the introduction of new notions and innovative theoretical tools
in the field of intelligent systems, mobile manipulators are
attracting significant interest in the industrial, military, and
public service communities [19]–[21], because of their large-
2168-2267 c 2013 IEEE
XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS
281
scale mobility and manipulation abilities, as compared to these
of the fixed-base manipulators. However, the integration of a
redundant manipulator and a mobile platform gives rise to
many new difficulties; for example, how to coordinate a given
task into fine motions to be carried out by the manipulator
and the gross motions to be achieved by the mobile platform;
and how to define the repeatability of an inverse-kinematic
scheme for a mobile manipulator. It is well known that,
for a stationary manipulator, an inverse-kinematic scheme is
called repetitive, if it maps closed paths in the task space
to closed trajectories in the configuration space. However,
for a mobile manipulator, if the mobile platform does not
return to the initial position, a repetitive inverse-kinematic
scheme for a stationary manipulator is no longer fit for a
mobile manipulator. Note that repetitive motion control of
mobile manipulators starts to play a more and more important
role in practical applications, which urgently requires an
effective scheme for solving the nonrepetitive problem of
mobile manipulators. In [22], Tcho´ was the first to introduce
n
the concept of repeatability of inverse kinematics algorithms
for mobile manipulators by exploiting the endogenous con-
figuration space approach; and further presented repeatable
inverse kinematics algorithms [23]–[25], which provides an
insight into the mechanism of repeatability. However, among
these inverse kinematics algorithms for repeatability of mobile
manipulators, the physical constraints (e.g., joint-angle limits
and joint-velocity limits) are usually not taken into account.
If these physical limits are not considered, a saturation may
occur in some cases. Thus, these schemes may be less effective
to control mobile manipulators for generating cyclic motion.
To achieve the repetitive motion control of mobile manipu-
lators (with physical constraints considered), a novel quadratic
performance index is for the first time designed and presented
in this paper, of which the effectiveness is analyzed by fol-
lowing Zhang
et al.’s
neural dynamics method. Then, a repet-
itive motion scheme is proposed by combining the criterion,
physical constraints, and integrated kinematical equations of
mobile manipulators, and further reformulated as a quadratic
programming (QP) subject to equality and bound constraints.
In addition, two important Bridge theorems are established,
which prove that such a QP can be converted equivalently into
a linear variational inequality (LVI) and then equivalently into
a piecewise-linear projection equation (PLPE). A real-time
numerical algorithm based on PLPE (or, say, discrete-time QP
solver) is thus developed and applied for the online solution
of the resultant QP. As an illustrative example, a wheeled
mobile manipulator is studied, which is composed of a mobile
platform driven by two independent wheels, and a six-DOF
spatial manipulator mounted on the platform. Tracking-path
tasks based on the wheeled mobile manipulator are performed,
which further verify the effectiveness and accuracy of the
proposed scheme, as well as the discrete-time QP solver.
The remainder of this paper is organized into five sections.
The kinematics model of the mobile manipulator is established
in Section II. In Section III, a repetitive motion scheme is
proposed and analyzed. A real-time numerical algorithm is
developed in Section IV for the solution of the resultant QP.
Section V illustrates tracking-path results based on a wheeled
Fig. 1.
Computer-aided design model of the mobile manipulator.
mobile manipulator. Section VI concludes this paper with final
remarks. Before ending this section, the main contributions of
this paper are listed as follows.
1) This paper proposes and analyzes a new performance
index for the repetitive motion of mobile manipulators
by defining three different error functions and exploiting
an exponent-type design formula. To the best of the
authors’ knowledge, it is the first time for researchers to
design such a performance criterion, which may give new
insights into the repetitive motion of mobile manipulators.
2) A repetitive motion scheme is proposed and presented
in this paper, which can coordinate simultaneously the
mobile platform and the manipulator to complete the end-
effector task, and achieve the repetitive motion control.
3) Two important Bridge theorems are established to connect
QP, LVI, and PLPE. Thus, a real-time numerical algo-
rithm based on PLPE is finally developed and applied for
the online solution of the resultant QP. In addition, its
convergence analysis is provided for the completeness of
this paper.
4) Two tracking-path tasks are conducted to demonstrate
the effectiveness and accuracy of the repetitive motion
scheme. In addition, this paper also makes a comparison
between the nonrepetitive and repetitive motions of the
wheeled mobile manipulator.
II. Kinematics Modeling of Mobile Manipulators
In this section, a wheeled mobile manipulator is developed
to demonstrate the effectiveness of the repetitive motion
scheme [26]–[29]. The computer-aided design model of the
mobile manipulator is shown in Fig. 1, which is composed of
a wheeled mobile platform and a six-DOF spatial manipulator.
The mobile platform includes two independent drive wheels
and two omni-directional passive supporting wheels. In this
paper, only the end-effector position is considered, and thus the
manipulator becomes a functionally redundant manipulator.
In the following subsections, the kinematics equation of the
mobile platform is first analyzed and presented with respect
to the world coordinate system. Then, the forward kinematics
equation of the manipulator is given directly with respect to the
base coordinate system (termed
X
C
Y
C
Z
C
coordinate system).
Finally, the velocity-level integrated kinematics model of the
282
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014
TABLE I
D-H Parameters of the Six-DOF Manipulator
#
1
2
3
4
5
6
θ
i
(rad)
θ
1
θ
2
θ
3
θ
4
θ
5
θ
6
d
i
(m)
0.698
0.065
0
0.190
0.130
0.082
a
i
(m)
0
0
0.555
0
0
0
α
i
(rad)
0
π/2
0
π/2
π/2
−π/2
Fig. 2. Kinematical model of the mobile platform.
wheeled mobile manipulator is obtained by combining the
equations of the manipulator and the mobile platform.
A. Kinematics of Mobile Platform
In this subsection, the mathematical model of the mobile
platform is first established under the nonholonomic constraint
condition. For a better understanding, the simplified model of
the mobile platform is shown in Fig. 2 with the following
notations used in the process of modeling.
1)
P
0
: the middle point of the two-drive wheel axis; its
coordinate in the world coordinate system is (x
0
, y
0
, z
0
);
2)
P
C
: the point connecting the manipulator and the mobile
platform; its coordinate in the world coordinate system
is (x
C
, y
C
, z
C
) with
z
C
= 0;
3)
d:
the distance between point
P
0
and point
P
C
;
4)
b:
the distance between the drive wheels and point
P
0
;
5)
r:
the radius of the drive wheels;
6)
φ:
the heading angle of the mobile platform; its time
˙
derivative is the heading velocity
φ;
7)
Q:
a hypothetical point around which the mobile plat-
form rotates;
8)
R:
the distance between point
Q
and the left drive wheel;
9)
ω:
the rotational velocity with which the mobile platform
˙
rotates around point
Q;
numerically
ω
=
φ;
10)
ϕ
l
,
ϕ
r
: the rotational velocities of the left and right
˙ ˙
wheels.
It is worth pointing out that the mobile platform and each
link of the manipulator are rigid, and the motion of the mobile
platform is only in the XOY-plane. Under the assumption that
no-slipping and no-sideways sliding occur to the platform (i.e.,
rolling without slipping), the velocities of left and right wheels
are both strictly perpendicular to the drive wheel axle. Thus,
according to the geometrical relationship of Fig. 2, we have
˙
˙
˙
˙
˙
rϕ
l
=
x
0
cos
φ
+
y
0
sin
φ
=
rϕ
r
φ
=
R
R
+
b
R
+ 2b
(1)
mobile platform, which is perpendicular to the drive wheel
axle (i.e., the X
C
-axis). In the direction along the drive wheel
axle (i.e., the Y
C
-axis), the velocity of the mobile platform
has to equal zero in view of the assumption of rolling without
slipping. So, the velocity of the mobile platform in Y
C
-axis
can be given as
˙
x
C
sin
φ
−
y
C
cos
φ
+
d φ
= 0.
˙
˙
(2)
Combining (1) and (2), one can obtain the following important
equations:
r
˙
(
ϕ
r
−
ϕ
l
)
˙
˙
φ
=
2b
x
C
=
˙
y
C
=
˙
dr
r
cos
φ
+
sin
φ ϕ
l
+
˙
2
2b
r
dr
sin
φ
−
cos
φ ϕ
l
+
˙
2
2b
dr
r
cos
φ
−
sin
φ ϕ
r
˙
2
2b
r
dr
sin
φ
+
cos
φ ϕ
r
˙
2
2b
where
x
C
and
y
C
denote the time derivatives of
x
C
and
y
C
,
˙
˙
respectively. Therefore, the mathematical models of the mobile
platform are presented as follows by rewriting the above
equations in a compact matrix form:
Aϕ
=
φ
˙ ˙
Bϕ
=
p
C
˙ ˙
(3)
(4)
where
ϕ
= [
ϕ
l
, ϕ
r
]
T
, and
p
C
= [˙
C
, y
C
]
T
. In addition, matrices
˙
˙ ˙
˙
x
˙
A
and
B
are defined as follows:
r
−1
A
=
2b 1
T
,B
=
r
cos
φ
2 sin
φ
−
sin
φ
cos
φ
1
−d/b
1
.
d/b
Note that
r
= 0.1025 m,
b
= 0.32 m, and
d
= 0.1 m for the
mobile platform shown in this paper.
B. Kinematics of Six-DOF Manipulator
It is well known that, for a stationary manipulator, its
forward kinematical equation (i.e., the transformation of joint-
space vector
θ
∈
R
n
to end-effector position-and-orientation
vector
r
b
∈
R
m
) can be established directly with respect to the
base coordinate system (i.e.,
X
C
Y
C
Z
C
coordinate system)
f
(θ) =
r
b
where
f
(·) is a differentiable nonlinear function with a known
structure and parameters for a given manipulator. For example,
where
x
0
cos
φ
+
y
0
sin
φ
=
x
C
cos
φ
+
y
C
sin
φ
in view of
˙
˙
˙
˙
the velocity projection theorem that the projections of the
velocities of
P
C
and
P
0
in the X
C
-axis are equivalent to each
other. Note that (1) describes the velocity constraint of the
XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS
283
the specific kinematics equation of the manipulator investi-
gated in this paper is (i.e.,
n
= 6 and
m
= 3 with D-H
parameters shown in Table I [30], [31])
r
b
=
f
(θ)
∈
R
m
⎤
⎡
l
65
(c
5
s
32
c
1
−
s
5
c
4
c
32
c
1
+
s
5
s
4
s
1
) +
l
43
s
32
c
1
+
l
2
s
2
c
1
=
⎣
l
65
(c
5
s
32
s
1
−
s
5
c
4
c
32
s
1
−
s
5
s
4
c
1
) +
l
43
s
32
s
1
+
l
2
s
2
s
1
⎦
(5)
l
65
(s
5
c
4
s
32
+
c
5
c
32
) +
l
43
c
32
+
l
2
c
2
+
l
1
+
l
0
where
l
i
denotes the length of the
ith
link,
c
i
:= cos
θ
i
and
s
i
:= sin
θ
i
with
i
= 1, 2,
· · ·
,
6. In addition,
l
0
denotes the
height of the first joint from the bottom of the mobile platform,
l
65
:=
l
6
+
l
5
,
l
43
:=
l
4
+
l
3
,
s
32
:= sin(θ
3
+
θ
2
) and
c
32
:=
cos(θ
3
+
θ
2
). In this paper, parameters
l
0
= 0.698 m,
l
1
= 0.065
m,
l
2
= 0.555 m,
l
3
= 0.190 m,
l
4
= 0.130 m,
l
5
= 0.082 m,
and
l
6
= 0.133 m. Note that such a manipulator is redundant
(i.e.,
m < n),
so (5) is under-determined and generally admits
an infinite number of solutions in terms of inverse kinematics.
C. Integrated Kinematics of Mobile Manipulator
Based on the kinematics equations of the manipulator and
the mobile platform, the integrated mathematical model of the
mobile manipulator is developed in this subsection.
First, for a mobile manipulator, the motion of the end-
effector is relative to the world coordinate system instead of
the base coordinate system. Thus, by adopting a transformation
matrix from the base coordinate system to the world coordinate
system, a new integrated kinematical equation with respect to
the world coordinate system can be obtained for the mobile
manipulator
⎤
⎡ ⎤
x
C
x
C
⎦
f
(θ) :=
⎣
y
C
⎦
+
g(θ, φ).
y
C
1
0
0
(6)
Note that the above equation is the kinematics model of
the mobile manipulator with respect to the world coordinate
system at the position level.
By differentiating (6) with respect to time
t,
the integrated
kinematics of the mobile manipulator at the velocity level is
obtained in the following form:
˙
φ
p
˙
r
w
=
C
+
J(θ, φ)
˙
˙
(7)
0
θ
−
sin
φ
cos
φ
0
0
0
1
where Jacobian matrix
J(θ, φ)
is defined as
J(θ, φ)
=
∂g(θ, φ)/∂w
∈
R
m×(n+1)
with
w
= [φ;
θ]
∈
R
n+1
. To further
˙
simplify (7), one can eliminate the intermediate variable
φ
in
view of (3) and (4). Thus, the above equation can be rewritten
as
r
w
=
˙
Aϕ
˙
Bϕ
˙
+
J(θ, φ)
˙
.
0
θ
(8)
⎡
cos
φ
⎣
sin
φ
r
w
=
0
where
I
denotes the identity matrix and the augmented matrix
K
is defined as follows:
B
0
A
0
K
=
+
J(θ, φ)
∈
R
m×(n+2)
.
0 0
0
I
The velocity-level integrated kinematics model (9) of the
wheeled mobile manipulator is thus obtained.
III. Design of Repetitive Motion Scheme
As discussed previously, for mobile manipulators, we have
to consider the effect of the mobile platform on repeatability
of an inverse kinematics algorithm. Thus, the aforementioned
repetitive motion scheme for stationary manipulators may
not be effective for the repetitive motion control of mobile
manipulators. In addition, the physical constraints of mobile
manipulators have to be considered. Motivated by these points,
we design and propose a novel repetitive motion scheme for
mobile manipulators. Such a scheme can not only solve the
joint-angle drift problem, but also make the mobile platform
return to the initial position. That is to say, the final state of
the mobile manipulator is repetitive with its initial state after
completing a closed-path task.
Repetitive Motion Scheme:
For mobile manipulators with
physical constraints considered, the repetitive motion scheme
is designed as
minimize
subject to
(D˙ +
z)
T
(D˙ +
z)/2
q
q
K q
=
r
w
˙ ˙
−
q
q q
+
q
−
q q
+
˙
˙
˙
(10)
(11)
(12)
(13)
where
q
−
and
q
+
denote, respectively, the lower and upper
limits of the combined angle vector,
q
−
and
q
+
denote, re-
˙
˙
spectively, the lower and upper limits of the combined velocity
vector, and, matrix
D
and vector
z
are defined as
D
=
C
0
0
a
∈
R
(n+m)×(n+2)
, z
=
∈
R
n+m
I
c
with
a
= [λ
3
(p
C
−
p
C
(0)) ;
λ
2
(sin
φ
−
sin
φ(0))]
∈
R
m
,
C
=
[B;
A
cos
φ]
∈
R
m×2
and
c
=
λ
1
[θ
−
θ(0)]
∈
R
n
. Note that the
physical limits (i.e.,
q
−
,
q
+
,
q
−
and
q
+
) used in this wheeled
˙
˙
mobile manipulator can be seen in Table II.
A. Theoretical Analysis
In order to achieve the repetitive motion control of mobile
manipulators, let us consider three factors of mobile manipu-
lators, i.e., the joint angle of the manipulator
θ,
the rotational
angle of the platform
φ,
and the location of the manipulator
on the mobile platform
p
C
= [x
C
, y
C
]
T
. Evidently, the mobile
manipulator can return to the original state if and only if these
three variables return to their initial positions, when the end-
effector of mobile manipulators performs a closed trajectory.
That is to say, the repetitive motion of mobile manipulators is
equivalent to the repetitive motion of variables
θ, φ,
and
p
C
.
Thus, by following Zhang
et al.’s
design method [32], [33],
the repetitive motion of
θ, φ,
and
p
C
can be achieved through
the following design steps.
Let
q
= [ϕ;
θ]
∈
R
n+2
denote the combined angle vector, and
q
= [
ϕ; θ]
denote the time derivative of
q
(i.e., the combined
˙
˙ ˙
velocity vector). Equation (8) can be further rewritten in a
compact matrix form as
r
w
=
˙
B
0
0
0
ϕ
˙
A
0
˙
+
J(θ, φ)
0
I
θ
ϕ
˙
˙
˙
:=
Kq
θ
(9)
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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014
TABLE II
Physical Limits Used in the Wheeled Mobile Manipulator
−
q
i
(rad)
i
1
2
3
4
5
6
7
8
+
q
i
(rad)
q
i
(rad/s)
˙
+
+30.0000
+30.0000
+3.0000
+3.0000
+3.0000
+3.0000
+3.0000
+3.0000
q
i
(rad/s)
˙
−
−30.0000
−30.0000
−3.0000
−3.0000
−3.0000
−3.0000
−3.0000
−3.0000
−∞
−∞
−0.5236
−0.1745
−6.2832
−6.2832
−6.2832
−6.2832
+∞
+∞
+0.5236
+2.3562
+6.2832
+6.2832
+6.2832
+6.2832
where
λ
1
>
0,
λ
2
>
0, and
λ
3
>
0 denote scalar-valued
positive design parameters used for achieving the repetitive
motion. In addition, one can prove theoretically that, in (18)–
(20), as
t
→ ∞,
θ,
sin
φ
and
p
C
can converge to their initial
states globally and exponentially.
In addition, the variables of the integrated kinematics at
the velocity level for the mobile manipulator only include
ϕ
˙
˙
so (18)–(20) have to be converted into a [
ϕ; θ]-based
˙
and
θ,
˙
matrix–vector equation for meeting the needs of the simulation
modeling. In view of (3)
Aϕ
=
φ
and (4)
Bϕ
=
p
c
, (19) and
˙ ˙
˙
˙
(20) can be merged into an equation in the form of
B
λ
3
(p
C
−
p
C
(0))
ϕ
+
˙
= 0.
A
cos
φ
λ
2
(sin
φ
−
sin
φ(0))
(21)
Then, let
a
= [λ
3
(p
C
−
p
C
(0)),
λ
2
(sin
φ
−
sin
φ(0))]
T
,
C
=
[B,
A
cos
φ]
T
, and
c
=
λ
1
[θ
−
θ(0)],
(18) and (21) can be
combined into a unified matrix–vector equation as follows:
D˙
+
z
= 0
q
(22)
Step 1.
We can define an indefinite vector/sclar-valued error
function
E(t)
according to the corresponding problem solving.
˙
Step 2.
We can choose its time derivative
E(t)
via the follow-
ing exponent-type design formula so that
E(t)
exponentially
converges to zero (with
λ
denoting a scalar-valued positive
design parameter):
˙
E(t)
=
−λE(t).
(14)
Step 3.
We can expand the above design formula in terms
of the specific error function, and thus obtain the desired
dynamics equations.
Specifically, in this paper, we define three different error
functions as follows:
E(t)
:=
θ
−
θ(0)
E(t)
:= sin
φ
−
sin
φ(0)
E(t)
:=
p
C
−
p
C
(0)
(15)
(16)
(17)
where
θ(0), φ(0),
and
p
C
(0) denote the initial states of
θ, φ,
and
p
C
, respectively. It is worth pointing out that, when the
mobile platform moves circularly (i.e., the heading angle
φ
makes 360-degree turns), the heading angle can return to the
initial position. In this situation, the value of the heading angle
is
φ
=
φ(0)
+ 2kπ (k =
±1, ±2, · · ·
). If
φ
−
φ(0)
is used
[instead of sin
φ
−
sin
φ(0)],
the restrictions become harsh and
the resultant repetitive motion scheme is difficult to realize.
Thus, the sine function is exploited, and the simulation and
modeling of the scheme become easy. Then, by combining
the design formula (14) and the above three error functions
respectively, three corresponding differential equations can be
obtained
˙
θ
+
λ
1
[θ
−
θ(0)]
= 0
˙
φ
cos
φ
+
λ
2
[sin
φ
−
sin
φ(0)]
= 0
p
C
+
λ
3
[p
C
−
p
C
(0)] = 0
˙
(18)
(19)
(20)
where the augmented matrix
D
and vector
z
are just what we
have defined before.
Therefore, from the above derivation, one knows that
solving (18)–(20) is equivalent to solving (22). In addition,
when (22) is solved, the resultant solutions can achieve the
repetitive motion control of mobile manipulators. However,
because the end-effector motion-trajectory requirement has to
be considered and physical constraints always exist in mobile
manipulators, it is better to minimize
D˙
+
z
2
/2
(with
q
2
·
2
denoting the two-norm of a vector), rather than using
D˙
+
z
= 0 directly. Thus, we can obtain the novel quadratic
q
performance criterion (10) for achieving the repetitive motion
planning and control of mobile manipulators.
Based on the above analysis, performance criterion (10)
is effective for repetitive motion control of mobile manipu-
lators. Furthermore, the integrated kinematics equation and
the physical limits of mobile manipulators are incorporated
as constraints to guarantee that the primary task will be
completed successfully with all variables being kept within
their limits. Therefore, we can conclude that the repetitive
motion scheme (10)–(13) is effective.
B. Scheme Reformulation
As the repetitive motion scheme (10)–(13) is resolved at the
˙
combined-vector velocity level [i.e., in terms of
q
], the limited
−
+
combined-angle vector range [q
, q
] has to be converted into
a
q
-based expression. The following conversion technique is
˙
adopted:
μ(q
−
−
q)
q
˙
μ(q
+
−
q)
(23)
where
μ >
0 is used to scale the feasible region of
q
. Note that
˙
large values of
μ
may cause sudden deceleration for joints and
wheels when the mobile manipulator approaches their physical
limits, and that, in mathematically,
μ
2max
1
i n+2
{˙
i
/(q
i
−
q
+ +
−
−
q
i
),
−˙
i
/(q
i
−
q
i
)}. Therefore, the following new combined
q
−
+
bound constraints can be used to replace (12)–(13):
ξ
−
q
˙
ξ
+
(24)
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