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Adaptive Neural-Fuzzy Control of Mobile Robots 233

The constraint (6.2) is referred to as the classical nonholonomic constraint
when it is not integrable. In the chapter, constraint (6.2) is assumed to be
completely nonholonomic and exactly known. The effect of the constraints can
be viewed as restricting the dynamics on the manifold nh as

nh ={(q, ˙q)|J(q)˙q = 0}

It is noted that since the nonholonomic constraint (6.2) is nonintegrable, there
is no explicit restriction on the values of the configuration variables.
Based on the nonholonomic constraint (6.2), the generalized constraint
forces in the mechanical system (6.1) can be given by

T
f = J (q)λ (6.3)

where λ ∈ R n−m is known as friction force on the contact point between the
rigid body and environmental surfaces.
Since J(q) ∈ R (n−m)×n , it is always possible to find an m rank matrix
R(q) ∈ R n×m formed by a set of smooth and linearly independent vector fields
spanning the null space of J(q), that is,

T T
R (q)J (q) = 0 (6.4)

Denote R(q) =[r 1 (q), ... , r m (q)] and define an auxiliary time function ˙z(t) =
m
T
[˙z 1 (t), ... , ˙z m (t)] ∈ R such that
˙ q = R(q)˙z(t) = r 1 (q)˙z 1 (t) + ··· + r m (q)˙z m (t) (6.5)

Equation (6.5) is the so-called kinematic model of nonholonomic systems in the
literature. Usually, ˙z(t) has physical meaning, consisting of the linear velocity
T
v and the angular velocity ω, that is, ˙z(t) =[v ω] . Equation (6.5) describes
the kinematic relationship between the motion vector q(t) and the velocity
vector ˙z(t).
Differentiating (6.5) yields

˙
¨ q = R(q)˙z + R(q)¨z (6.6)

From (6.5), ˙z can be obtained from q and ˙q as

T −1 T
˙ z =[R (q)R(q)] R (q)˙q (6.7)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 233 — #5