Page 282 - Autonomous Mobile Robots

Adaptive Control of Mobile Robots 269

robust stabilization is considered for dynamic nonholonomic chained systems
with external disturbances. Using geometric phase as a basis, control of
Caplygin dynamical systems was studied in Reference 2, and the closed-loop
system was proved to achieve the desired local asymptotic stabilization of
a single equilibrium solution. The principal limitation associated with these
schemes is that controllers are designed at the velocity input level or torque
input level and the actuator dynamics are excluded.
AsdemonstratedinReference25, actuatordynamicsconstituteanimportant
component of the complete robot dynamics, especially in the case of high-
velocity movement and highly varying loads. Many control methods have
therefore been developed to take into account the effects of actuator dynamics
(see, for instance, References 26–29). However, the literature is sparse on the
control of the nonholonomic systems including the actuator dynamics.
In this chapter, the stabilization problem is considered for general nonholo-
nomic mobile robots at the actuator level, taking into account the uncertainties
in dynamics and the actuators. The controller design consists of two stages. In
the first stage, to facilitate control system design, the nonholonomic kinematic
subsystem is transformed into a skew-symmetric form and the properties of the
overall systems are discussed. Then, a virtual adaptive controller is presented
to compensate for the parametric uncertainties of the kinematic and dynamic
subsystems. In the second stage, an adaptive controller is designed at the actu-
ator level and the controller guarantees that the configuration state of the system
converges to the origin.
This chapter is organized as follows: the model and model transformation
of the system including actuator dynamics are presented in Section 7.2. The
adaptive control law and stability analysis are presented in Section 7.3. Simu-
lation studies are presented in Section 7.4 to show that the proposed method is
effective. The conclusions are given in Section 7.5.

7.2 DYNAMIC MODELING AND PROPERTIES
In general, a nonholonomic system including actuator dynamics, having
an n-dimensional configuration space with generalized coordinates q =
T
[q 1 , ... , q n ] and subject to n − m constraints can be described by [30]

J(q)˙q = 0 (7.1)
T
M(q)¨q + C(q, ˙q)˙q + G(q) = B(q)K N I + J (q)λ (7.2)
dI
L + RI + K a ω = ν (7.3)
dt
where M(q) ∈ R n×n is the inertia matrix which is symmetric positive def-
inite, C(q, ˙q) ∈ R n×n is the centripetal and coriolis matrix, G(q) ∈ R n
is the gravitation force vector, B(q) ∈ R n×r is the input transformation

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 269 — #3