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268 Autonomous Mobile Robots

to overcome the obstruction to stabilizability due to Brockett’s theorem, and
a smooth time-invariant feedback is used to stabilize the transformed system.
In the original coordinates, the resulting feedback control is discontinuous.
Various time varying controllers have been proposed in the literature [4,9]. The
kinematic nonholonomic control systems can be asymptotically stabilized to
an equilibrium point by smooth time-periodic static state feedback. However,
the convergence rate for this method is comparatively slow. Hybrid controllers
combine continuous time features with either discrete event features or discrete
time features [6,10].
Among the many control strategies that have been proposed for various non-
holonomic systems, research results can generally be classified into two classes.
The first class is kinematic control, which provides the solutions only at the pure
kinematic level, where the systems are represented by their kinematic models
and velocity acts as the control input. Based on exact system kinematics, dif-
ferent control strategies have been proposed [4,5,8]. Recently, a few research
works have been carried out to design controllers against possible existence
of modeling uncertainties and external disturbances [11–13]. Robust exponen-
tial regulation is proposed in Reference 11 by assuming known bounds of the
nonlinear drifts. It is also required that the x 0 -subsystem is Lipschitz. To relax
this condition, adaptive state feedback control is proposed in Reference 12 for
systems with strong nonlinear drifts.
It is noted that one commonly used approach for control system design
of nonholonomic systems is to convert, with appropriate state and input trans-
formations, the original systems into some canonical forms for which controller
design can be carried out more easily [14–17]. The chained form [14] and the
power form [15] are two of the most important canonical forms of nonholo-
nomic control systems. The class of nonholonomic systems in chained form was
first introduced by Murray and Sastry [14] and has been studied as a benchmark
example in the literatures. It is well known that many mechanical systems with
nonholonomic constraints can be locally, or globally, converted to the chained
form under coordinate change and state feedback [5,14]. The typical examples
include tricycle-type mobile robots and cars towing several trailers. A new
canonical form, called extended nonholonomic integrators (ENI) was presen-
ted in Reference 17, and it was shown that nonholonomic systems in ENI form,
chained and power forms are equivalent, and can thus be dealt with in a unified
framework. Using the special algebraic structures of the canonical forms, vari-
ous feedback strategies have been proposed to stabilize nonholonomic systems
in the literature [16–21].
The second class is dynamic control, taking inertia and forces into account,
where the torque and force are taken as the control inputs. Different researchers
have investigated this problem. Sliding mode control is applied to guarantee
the uniform ultimate boundedness of tracking error in Reference 24. In Ref-
erence 23, stable adaptive control is investigated for dynamic nonholonomic
chained systems with uncertain constant parameters. In Reference 24, adaptive

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