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Adaptive Neural-Fuzzy Control of Mobile Robots 243

it is worth mentioning that different control objectives may also be pursued,
such as stabilization to manifolds of equilibrium points (as opposed to a single
equilibrium position) or to trajectories.
m
By appropriate selection, a set of vector ˙z(t) ∈ R , the control objective can
be specified as: given a desired z d (t), ˙z d (t), and desired constraint λ d , determine
a control law such that for any (q(0), ˙q(0)) ∈ , z(t) and ˙q asymptotically
converge to a manifold nhd specified as

nhd ={(q, ˙q)|z(t) = z d , ˙q = R(q)˙z d (t)} (6.21)
while the constraint force error (λ − λ d ) is bounded in a certain region. The
variable z(t) can be thought as m “output equations” of the nonholonomic
system.

Assumption 6.4 The desired reference trajectory z d (t) is assumed to be
bounded and uniformly continuous, and has bounded and uniformly continuous
derivatives up to the second order. The desired λ d (t) is bounded and uniformly
continuous.

Let us define the following notations as

e z = z − z d (6.22)
e λ = λ − λ d (6.23)
˙ z r =˙z d − ρ 1 e z (6.24)
s =˙e z + ρ 1 e z (6.25)

where ˙z r is the reference trajectory described in internal state space.
Apparently, we have

˙ z =˙z r + s (6.26)
For force control, define µ as

J λ
˙ µ =−ρ 2 µ − ρ −1 T (6.27)
3
n
where µ ∈ R . For the convenience of controller design, combining s and µ to
form the following new hybrid variables
σ = Rs + µ (6.28)

ν = R˙z r − µ (6.29)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 243 — #15