Although
nonholonomic mechanics has a long history, dating back at least to
the work of Hertz and H¨older towards
the end of the 19th century, it is still
today a very active domain of research, both for its theoretical interest and its
applications, e.g. in wheeled vehicles, robotics, and motion generation. In the
past decade or so, a flurry of activity has concerned the study of
nonholonomicsystems as nonlinear dynamic systems to which control theory methods could
be profitably applied. As a result, the control of classical nonholonomic mechanical
systems such as cars, trucks with trailers, rolling 3D objects, underactuated
mechanisms, satellites, etc., has made a definite progress, and often met a satisfactory
level.
Systems considered in classical nonholonomic mechanics are smooth, continuous
time systems, i.e., they can be described by ODEs on a smooth manifold of
configurations, on which smooth (most often, analytic) constraints apply. However,
nonholonomic-like
behaviours can be recognized in more general systems,
some of great practical relevance, which may present for instance discontinuities
of the dynamics, discreteness of the time axis, and discreteness (e.g., quantization)
of the input space. For these systems, some very basic control problems
such as the analysis of reachability and the synthesis of steering control sequences
still pose quite challenging problems.
This paper attempts at providing a general conceptual framework capable of
capturing the notion of nonholonomy for a broad class of systems, allowing for
discrete and hybrid (mixed continuous and discrete) configurations and transitions.
Upon the analysis of few simple but significant examples, a unique definition
encompassing all “intuitively nonholonomic” behaviours in hybrid systems,
does not appear to be feasible, or practical. Hence we propose the definition
of two different types of nonholonomic behaviours, which we call internal and
external, respectively. These two types are not obviously reducible to a single
one, and indeed we show examples of simple mechanical systems exhibiting only
internal, only external, or both internal and external nonholonomy, respectively.
Although our definitions are not always directly computable, we provide equivalent,
or sufficient conditions for some specific classes of systems, which allow for
practical tests to be applied.