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Stabilization of Nonholonomic Systems 193

In open loop strategies (see e.g., [5–7]) the control signal is calculated
off-line starting from the knowledge of the initial and the final configurations
of the system. By their own nature, these strategies are not able to compensate
for disturbances and model errors, therefore, in practise, the reached config-
uration may differ significantly from the desired final one. Nevertheless, it is
possible to include open loop strategies in an iterative design method, which
possesses some robustness properties. This approach is known as iterative state
steering [8].
In closed loop strategies (see e.g., [9–16] for time-varying feedbacks, [17–
21] for discontinuous ones, [12,22,23] for middle strategies [discontinuous and
time varying], [24,25] for hybrid control laws, and [26–28] for multi-rate meth-
ods) the control signal is computed online, based on the knowledge of the actual
configuration of the system and of the final one. They can potentially com-
pensate for model errors and disturbances. However, the result of Reference 29
states that there does not exist a continuous homogeneous controller that
robustly stabilizes nonholonomic systems against modeling uncertainties. This
has motivated further research in this direction. Many researchers have been
trying to solve this problem using discontinuous feedback (see [8,30–33]), or to
find special instances in which a continuous feedback can yield robust stability
(see e.g., [34]).
From the very brief discussion above it is apparent that several tools are
available for the control of nonholonomic systems. However, to date, it is not
possible to single-out a control strategy (or a set of tools) that performs better
than the other ones. This is mainly due to the following facts. A good control
law should have two basic features. First, it should drive the system from its
initial state to the final one in a simple way, second it should be robust against
model mismatches, noisy measurements, and the approximate knowledge of
initial conditions. Open loop strategies are generally able to grant the first item,
but nothing can be said on their robustness, although they can be exploited in
robust iterative designs. On the other hand, closed loop strategies are potentially
more robust, but the dynamics of the closed loop system may not be natural.In
particular the closed loop system may show oscillatory response, which is not
at all necessary or required to reach the desired final configuration. Note finally
that closed loop strategies are potentially more robust than open loop ones.
However, we will show that the robust stabilization problem for nonholonomic
systems has very special properties, and it is intrinsically hard.

5.2 PRELIMINARIES AND DEFINITIONS
In this chapter, we discuss the problem of designing stabilizing control laws for
nonholonomic systems described by equations of the form

˙ x = g(x)u (5.1)

© 2006 by Taylor & Francis Group, LLC

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