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272 Autonomous Mobile Robots

Until now we have brought the kinematics (7.1), dynamics (7.2), and actuator
dynamics (7.3) of the considered nonholonomic system from the generalized
n m
coordinatesystemq ∈
tofeasibleindependentgeneralizedvelocitiesv ∈
without violating the nonholonomic constraint (7.1).
For ease of controller design in this chapter, the existing results for the con-
trol of nonholonomic canonical forms in the literature are exploited. In the fol-
lowing, the kinematic nonholonomic subsystem (7.8) is first converted into the
chainedcanonicalform, andthentotheskew-symmetricchainedformforwhich
a very nice controller structure [18] exists in the literature and can be utilized.
This will be detailed later. The nonholonomic chained subsystem considered
in this chapter is m-input, (m − 1)-chain, single-generator chained form given
by [9,24]

˙ x 1 = u 1
˙ x j,i = u 1 x j,i+1 (2 ≤ i ≤ n j − 1)(1 ≤ j ≤ m − 1) (7.13)

˙ x j,n j = u j+1
T
n
Notethat, inEquation(7.13), X =[x 1 , X 2 , ... , X m ] ∈ R withX j =[x j−1,2 , ... ,
T
] (2 ≤ j ≤ m) are the states and u =[u 1 , u 2 , ... , u m ] are the inputs of
x j−1,n j−1
the kinematic subsystem.
The chained form is one of the most important canonical forms of nonholo-
nomic control systems. It has been shown in References 5 and 14 and references
therein that many nonlinear mechanical systems with nonholonomic constraints
on velocities can be transformed, either locally or globally, to the chained form
system via coordinates and state feedback transformation. The necessary and
sufficient conditions for transforming system (7.8) into the chained form are
given in Reference 33. The following assumption is made in this chapter.

Assumption 7.1 The kinematic model of a nonholonomic system given by
Equation (7.8) can be converted into chained form (7.13) by some diffeomorphic
coordinate transformation X = T 1 (q) and state feedback v = T 2 (q)u where u
is a new control input.

The existence and construction of the transformation for these systems have
been established in the literature [9,34]. It is given here for completeness of the
presentation. For detailed explanations of the notations on differential geometry
used below, readers are referred to Reference 35.

Proposition 7.1 Consider the drift-free nonholonomic system

˙ q = s 1 (q)v 1 + ··· + s m (q)v m

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 272 — #6