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234 Autonomous Mobile Robots

The dynamic equation (6.1), which satisfies the nonholonomic constraint (6.2),
can be rewritten in terms of the internal state variable ˙z as

˙ T
M(q)R(q)¨z +[M(q)R(q) + C(q, ˙q)R(q)]˙z + G(q) = B(q)τ + J (q)λ + τ d
(6.8)

T
Substituting (6.5) and (6.6) into (6.1), and then premultiplying (6.1) by R (q),
T
the constraint matrix J (q)λ can be eliminated by virtue of (6.4). As a
consequence, we have the transformed nonholonomic system

˙ q = R(q)˙z = r 1 (q)˙z 1 + ··· + r m (q)˙z m (6.9)
M 1 (q)¨z + C 1 (q, ˙q)˙z + G 1 (q) = B 1 (q)τ + τ d1 (6.10)

where

T
M 1 (q) = R M(q)R
T
˙
C 1 (q, ˙q) = R [M(q)R + C(q, ˙q)R]
T
G 1 (q) = R G(q)
T
B 1 (q) = R B(q)
T
τ d1 = R τ d
which is more appropriate for the controller design as the constraint λ has been
eliminated from the dynamic equation.
Exploiting the structure of the dynamic equation (6.10), some properties
are listed as follows.

Property 6.3 Matrix D 1 (q) is symmetric and positive-definite.

˙
Property 6.4 Matrix D 1 (q) − 2C 1 (q, ˙q) is skew-symmetric.
Property 6.5 D(q),G(q),J(q), and R(q) are bounded and continuous if
˙
z is bounded and uniformly continuous. C(q, ˙q) and R(q) are bounded if
˙
˙ z is bounded. C(q, ˙q) and R(q) are uniformly continuous if ˙z is uniformly
continuous [37].
In the following, the kinematic nonholonomic subsystem (6.5) is converted
into the chained canonical form. The nonholonomic chained system considered

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 234 — #6