Nonholonomic planning

The term nonholonomic planning was introduced by Laumond [593] to describe the problem of motion planning for wheeled mobile robots (see [595,633] for overviews). It was informally explained in Section 13.1 that nonholonomic refers to differential constraints that cannot be completely integrated. This means they cannot be converted into constraints that involve no derivatives. A more formal definition of nonholonomic will be given in Section 15.4. Most planning research has focused on velocity constraints on ${\cal C}$, as opposed to a phase space $X$. This includes most of the models given in Section 13.1, which are specified as nonintegrable velocity constraints on the C-space ${\cal C}$. These are often called kinematic constraints, to distinguish them from constraints that arise due to dynamics.

In mechanics and control, the term nonholonomic also applies to nonintegrable velocity constraints on a phase space [112,113]. Therefore, it is perfectly reasonable for the term nonholonomic planning to refer to problems that also involve dynamics. However, in most applications to date, the term nonholonomic planning is applied to problems that have kinematic constraints only. This is motivated primarily by the early consideration of planning for wheeled mobile robots. In this book, it will be assumed that nonholonomic planning refers to planning under nonintegrable velocity constraints on ${\cal C}$ or any phase space $X$.

For the purposes of sampling-based planning, complete integrability is actually not important. In many cases, even if it can be theoretically established that constraints are integrable, it does not mean that performing the integration is practical. Furthermore, even if integration can be performed, each constraint may be implicit and therefore not easily parameterizable. Suppose, for example, that constraints arise from closed kinematic chains. Usually, a parameterization is not available. By differentiating the closure constraint, a velocity constraint is obtained on ${\cal C}$. This can be treated in a sampling-based planner as if it were a nonholonomic constraint, even though it can easily be integrated.