基于运动规划方法的车辆有限时间编队跟踪控制
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Finite-time Formation Tracking Control for Vehicles Based on Motion
Planning Approach*
Yongfang Liu and Zhiyong Geng
Abstract— This paper addresses the finite-time formation
tracking problem for vehicles with dynamics model on SE(3)
(the specific Euclidean group of rigid body motions), under
the condition that the tracking time is given according to
task requirements in advance. By using Pontryagin’s maximum
principle (PMP) on Lie groups, a finite-time optimal tracking
control law is designed for vehicles to track a desired trajectory
at the given time. Simultaneously, the corresponding cost
function is guaranteed to be optimal. To demonstrate the
effectiveness of the proposed control law, an illustrative example
is provided.
I. INTRODUCTION
In the past few years, control and coordination of multiple
autonomous vehicles are of great interest to the various
scientific communities, due to its broad applications in such
fields as robot manipulators, motion control of underwater
and aerospace autonomous vehicles. One of the distinct
feature for these systems is that their configuration mani-
folds are not linear. Considering only rotation of the above
systems, it evolves on the nonlinear manifold SO(3), where
the special orthogonal group SO(3) is the matrix Lie group
of 3 × 3 orthogonal matrices with determinant of one.
For the case when translation and rotation are considered
simultaneously, the configuration manifold is the matrix Lie
group SE(3). These nonlinear manifolds yield important and
unique properties that cannot be observed from dynamics
systems evolving on linear spaces [1].
A common problem in controlling above nonlinear control
systems on Lie groups is to make the rigid bodies follow
a desired trajectory, which is referred to as the trajectory
tracking problem [2]. In the past few years, the tracking
problem of rigid bodies has received much attention in the
literatures. However, most of the prior work is based on
parametrization, where three-parameter representation of at-
titude, such as Euler angles, Modified Rodriguez parameters,
or Unit quaternions (four-parameter) are used to describe the
configuration manifold. Parameterization methods convert
the configuration space from nonlinear space to normal
Euclidean space by identifying the different velocity spaces
as the same Euclidean space essentially. For the stability
problems or tracking problems in a neighborhood, these
methods are good approximations. However, for the systems
* This work is supported by National Nature Science Foundation of China
under Grants 61374033, 11072002.
Y. Liu, and Z. Geng (Corresponding author) are with the State Key
Laboratory for Turbulence and Complex Systems of Peking University,
Department of Mechanics and Engineering Science, College of Engi-
neering, Peking University, Beijing 100871, P. R. China (e-mail: zy-
geng@pku.edu.cn)
with rigid bodies which can not be considered locally in a
neighborhood, these methods present difficulties for different
rigid bodies to keep rigid formation when the non-linear
trajectory tracking problem is considered. Besides, these
methods cause singularities or ambiguities, as there is no
three-parameter representation for Lie group SO(3), that is
both global and without singularities [3]. Quaternions do not
have singularities, but they have ambiguities in representing
an attitude, as the three-sphere S
3
double covers SO(3)
[1]. Furthermore, considering that computations for rigid
bodies performed with different choices of coordinates will
produce different results, it is desirable to take into account
the geometry structure of the nonlinear manifold, i.e. the
configuration space of rigid bodies and work with it directly.
On the nonlinear manifold, like the group of rigid body
motions SE(3), the tracking problem of rigid bodies has
also been studied extensively in the past. Some pioneering
work can be found in [1], [2], [4], [5], [6], [7], where
the asymptotical tracking control laws are designed and the
obtained results are coordinate-free. The difference is that
the trajectory tracking results are derived from the general
Riemannian framework to Lie groups in [2], [4]. Note that
this approach may fail to fully exploit the additional structure
which is available in Lie group. Thus, many researchers stud-
ied the tracking problem of rigid bodies on the corresponding
group in [1], [5], [6], [7]. Specially, in [6], [7], the group
structure is used to transform the trajectory tracking problem
into the better understood problem of stabilizing the identity
element. This is impossible in a general Riemanning setting.
In [8], [9], the optimal control method is used to study
the attitude tracking problem of rigid bodies. For attitude
control of rigid bodies, the optimal control law is described
by solving differential equations in [8]. Similarly, the attitude
control of a spin-stabilized spacecraft is considered and the
control law is derived by solving a unconstrained numerical
parameter optimization in [9]. For the case that translation
and rotation of rigid bodies are considered simultaneously,
the finite-time optimal formation problem for kinematic
systems on SE(3) has been investigated in [10].
Motivated by the aforementioned results, we focus on
the finite-time formation tracking problem for vehicles with
dynamic models on Lie group SE(3) in this paper. Like-
wise to [10], the translational and rotational are considered
simultaneously for rigid bodies motions, which is more
complex than only focusing on attitude. Using the group
structure of SE(3), the tracking problem is converted into
the problem of finite-time motion planning. Then, a finite-
time optimal tracking control law is designed based on
2014 11th IEEE International
Conference on Control & Automation (ICCA)
June 18-20, 2014. Taichung, Taiwan
978-1-4799-2837-8/14/$31.00 ©2014 IEEE 326
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