Page 246 - Autonomous Mobile Robots

232 Autonomous Mobile Robots

physical insights, we present the adaptive laws to design the outputs of the
“rules” numerically using adaptive (NN) control techniques. It is shown that
the motion tracking error converges to zero, the force tracking error is uniformly
bounded, and the closed-loop stability is guaranteed without the requirement
of the PE condition.
The rest of the chapter is organized as follows. The dynamics of mobile
robot systems subject to nonholonomic constraints are briefly described
in Section 6.2. Multilayer NF systems as the key design tool are introduced
in Section 6.3. The main results of the adaptive NF control design are presented
in Section 6.4, and a simulation example is provided in Section 6.5. Concluding
remarks are given in Section 6.6.

6.2 DYNAMICS OF NONHOLONOMIC MOBILE ROBOTS

In general, a nonholonomic mobile robot system having an n-dimensional con-
T
figuration space with generalized coordinates q =[q 1 , ... , q n ] and subject to
(n − m) constraints can be described by [35]
(6.1)
M(q)¨q + C(q, ˙q)˙q + G(q) = B(q)τ + f + τ d
T
where M(q) ∈ R n×n is the inertia matrix and M(q) = M(q)> 0, C(q, ˙q) ∈
n
R n×n is the centripetal and coriolis matrix, G(q) ∈ R is the gravitation force
vector, B(q) ∈ R n×r is the full-rank input transformation matrix and is assumed
r
to be known, as it is a function of fixed geometry of the system, τ ∈ R is the
n
input vector of forces and torques, f ∈ R is the constrained force vector,
n
and τ d ∈ R denotes bounded unknown disturbances including unstructured
unmodeled dynamics. The dynamic system (6.1) has the following properties
[32,36]:
Property 6.1 MatricesM(q),G(q)are uniformly boundedand uniformly con-
tinuous if q is uniformly bounded and continuous, respectively. Matrix C(q, ˙q)
is uniformly bounded and uniformly continuous if ˙q is uniformly bounded and
continuous.

T ˙
˙
Property 6.2 Matrix M − 2C is skew-symmetric, that is, x (M − 2C)x = 0,
∀x = 0.
When the system is subjected to nonholonomic constraints, the (n − m)
nonintegrable and independent velocity constraints can be expressed as

J(q)˙q = 0 (6.2)

where J(q) ∈ R (n−m)×n is the matrix associated with the constraint.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 232 — #4