Robotica (2006) volume 24, pp. 333–335. © 2005 Cambridge University Press
doi:10.1017/S0263574705002225 Printed in the United Kingdom
Trajectory optimization of flexible mobile manipulators
H. Ghariblu
∗
and M. H. Korayem†
(Received in Final Form: August 7, 2005, first published online 31 October 2005)
SUMMARY
A computational algorithm is developed to find a dynamic
motion trajectory of a mobile manipulator with flexible links
and joints that will allow the robot to carry a maximum load
between two specified end positions. A compact form of
the linearized state space dynamic equations is organized as
well as constraint equations. Then, the problem of finding
a maximum load carrying capacity on flexible mobile
manipulators is formulated as a trajectory optimization
problem.
KEYWORDS: Trajectory optimization; Mobile manipula-
tors; Linearized trajectory.
I. INTRODUCTION
Finding the full load motion for a specified point-to-point
task can increase the productivity and economic usage of
mobile manipulators. In a classical fixed base manipulator,
the dynamic load carrying capacity (DLCC) is defined as a
maximum load, which a manipulator can carry repeatedly
in its fully extended configuration, while the dynamics of
the load and manipulator itself must be taken into account.
1
In a mobile manipulator with lighter weight and higher
motion speed, the existence of the flexible structure on joints
and links degrades the motion accuracy. Accordingly, an
additional constraint must be imposed, when the DLCC is
determined for flexible mobile manipulators. The constraint
should limit the end effector tracking error due to the
deflection or oscillation of the mobile manipulator.
Wang and Ravani
1,2
presented an algorithm to determine
the load carrying capacity on fixed base robots. The limitation
of the rigid joint and link assumption in the formulation and
analysis of flexible manipulators were investigated by the
other researchers.
3−7
This paper, presents a new method to determine the load
carrying capacity for mobile manipulators with flexible links
and joints during the motion between two end positions.
For an analysis of flexibility, the Lagrangian assumed mode
is employed. Then, the problem of finding the maximum
load carrying capacity for flexible mobile manipulators
is converted into a trajectory optimization problem. An
objective function is defined and dynamic equations are
* Corresponding author. Inteligent Systems Laboratory, Faculty of
Mechanical Engineering, Zanjan University, Zanjan (Iran). E-mail:
ghariblu@mail.znu.ac.ir
† Mechanical Engineering Department, Iran University of Science
and Technology, Tehran (Iran).
linearized, as well as other constraints on its state space form
to solve with Iterative linear Programming (ILP) method.
II. LINEARIZED STATE SPACE REPRESENTATION
OF DYNAMIC EQUATIONS
The Lagrangian assumed mode formulation is used to model
the dynamics of the link flexibility on a mobile manipulator
8
(Figure 1). Also, the elastic mechanical coupling between
the i th joint and link is modeled as a linear torsional spring
with a constant stiffness coefficient k
ti
(Figure 2).
Then, the compact form of extended equation of motion,
9
can be represented as:
[M]
e
¨
q
e
+ C( q
e
,
˙
q
e
) + G( q
e
) + [K
t
]
e
[ q ]
e
= τ
e
(1)
If the load is treated as a variable, the dynamic Equation (1)
is rearranged as:
¨
q
e
= [M( q
e
,m
L
)]
−1
e
( τ
e
− C( q
e
,
˙
q
e
,m
L
) − G( q
e
,m
L
)
− [K
t
]
e
q
e
) =
f ( q
e
,
˙
q
e
, τ
e
,m
L
) (2)
Defining the state vector
X = [x
1
x
2
]
T
, where x
1
= q
e
and
x
2
=
˙
q
e
, the state space representation of Equation (2) can
be rewritten in the following form:
˙
X =
˙
x
1
˙
x
2
=
x
2
f (
X(j ), τ
e
(j ),m
L
)
=
F (
X(j ), τ
e
(j ),m
L
) (3)
Discretized form of the equation (3) is:
X(j + 1) −
X(j )
h
=
F (
X(j ), τ
e
(j ),m
L
), j = 1, 2,...,p
(4)
where, h =
T
p
and T is the overall motion time and p is the
total number of time intervals. The nonlinear function f at
the (k + 1)th trajectory is expanded in Taylor series about the
kth trajectory. After neglecting the higher order (non linear)
terms the following equation is obtained in a matrix form:
X(j + 1) = [G
j
]
X(j ) + [H
j
] τ
e
(j ) +
B
j
m
L
+
D
j
(5)
The elements of matrices [G
j
], [H
j
] and
B
j
,
D
j
are found in
Reference [10].
X(j + 1) can be written as a linear