Nonholonomic Behavior in Robotic Systems
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In this chapter, we study the effect of nonholonomic constraints on the behavior of robotic systems. These constraints arise in systems such as multifingered robot hands and wheeled mobile robots, where rolling contact is involved, as well as in systems where angular momentum is conserved. We discuss the problem of determining when constraints on the velocities of the configuration variables of a robotic system are integrable, and illustrate the problem in a variety of different situations. The emphasis of this chapter is on the basic tools needed to analyze nonholonomic systems and the application of those tools to problems in robotic manipulation. These tools are drawn both from some basic theorems in differential geometry and from nonlinear control theory.
Chapter Summary
- Nonholonomic constraints are linear velocity constraints
of the form
which cannot be integrated to give constraints on the configuration variables
alone. By choosing 
to
be a basis for the null space of the linear velocity constraints, we
get the associated control system
The problem of nonholonomic motion planning consists of finding a trajectory
, given 
and
.
- The Lie bracket between two vector fields
and 
on
is a new vector field 
defined by
- A distribution
is a smooth assignment of a
subspace of the tangent space to each point 
. One
important way of generating it is as the span of a number of vector
fields:
The distribution
is said to be regular if the dimension
of 
does not vary with . The distribution is
said to be involutive if it is closed under the Lie bracket,
that is if for all 
, we have 
.
- A distribution
of dimension 
is said to be integrable if there exist 
independent functions whose
differentials annihilate the distribution. Frobenius' theorem
asserts that a regular distribution is integrable if and only if it is
involutive. A Pfaffian system or codistribution 
is completely nonholonomic if the involutive closure of the distribution
spans for all .
- Consider the system
The controllability Lie algebra is the Lie algebra generated by the vector fields
. It is the smallest Lie algebra
containing . Chow's theorem asserts that if
the controllability Lie algebra is full rank, we can steer this system
from any initial to any final point.
- Given a distribution , the filtration associated
with is defined by
and
where
The filtration is said to be regular if each of the
are regular. For a regular filtration, the smallest integer 
at which rank 
is equal to that of 
is called the degree of nonholonomy
of the distribution. The growth vector 
for a regular filtration is defined as 
. The relative growth vector 
is defined as 
with 
.
- Given
, a Lie
product is any nested set of Lie brackets of the generators 
. A
Lie algebra generated by is said to be nilpotent if
there exists an integer such that all Lie products of length
greater than are zero. A Philip Hall basis is an ordered
set of Lie products chosen by a set of rules so as to keep track of
the restrictions imposed by the properties of the Lie bracket, namely
skew-symmetry and the Jacobi identity.
