Underactuated and nonholonomic systems

Many interesting systems cannot be expressed in the form ${\ddot q}= h(q,{\dot q},u)$ with $n$ independent action variables because of underactuation or other constraints. For example, the models in Section 13.1.2 are underactuated and nonholonomic. In this case, it is not straightforward to convert the equations into a vector of double integrators because the dimension of $U(q,{\dot q})$ is less than $n$, the dimension of ${\cal C}$. This makes it impossible to use grid-based sampling of $U(q,{\dot q})$. Nevertheless, it is still possible in many cases to discretize the system in a clever way to obtain a lattice. If this can be obtained, then a straightforward resolution-complete approach based on classical search algorithms can be developed. If $X$ is bounded (or a bounded region is obtained after applying the phase constraints), then the search is performed on a finite graph. If failure occurs, then the resolution can be improved in the usual way to eventually obtain resolution completeness. As stated in Section 14.2.2, obtaining such a lattice is possible for a large family of nonholonomic systems [762]. Next, a method is presented for handling reachability graphs that are not lattices.