When is a mechanical control system kinematic?

Abstract

The author deals with simple mechanical control systems (A.D. Lewis and R.M. Murray, 1997), possibly with constraints (A.D. Lewis, 1997). These systems are naturally characterised by having forces as inputs. However, in the current literature there is a tendency to replace force inputs with velocity inputs. This is sometimes a justifiable step, and serves to simplify somewhat certain applications. However, one must be careful as: (1) sometimes, very often in fact, this simplification is comprehensively not possible (as we demonstrate); and (2) the two different control problems which result, one with force inputs and one with velocity inputs, will never be equivalent with the same class of inputs (unless one makes the impractical restriction of smooth inputs). Nevertheless, if one properly considers the classes of inputs for both problems, it is possible to give a simple, computationally tractable condition which determines when a system with force inputs can be reduced to one with velocity inputs. One of the main points made is that there is no correspondence, be it complete or approximate, between mechanical control systems, and so-called nonholonomic (i.e., driftless) control systems. A precise connection is made between those mechanical systems which are dynamic (i.e., have forces as inputs) and those which are kinematic (i.e., have velocities as inputs).