Low-order controllability and kinematic reduction

Abstract

The discussions of controllability in Chapter 7 are of a quite general nature. However, as we saw in Exercise E7.14 (see also Example 7.22 for a control-affine example), even these very general results can fail to provide complete characterizations of the controllability of a system, even in rather simple examples. In this chapter we provide low-order controllability results that are quite sharp. These are extensions of results initially due of Hirschorn and Lewis [2001], and considered in the context of motion planning by Bullo and Lewis [2003b]. One of the interesting features of these results is that their hypotheses are “feedback-invariant,” a notion that we do not discuss here; see [Lewis 2000a] for an introduction in the setting of affine connection control systems. Interestingly, these controllability results are related to a notion called “kinematic controllability” by Bullo and Lynch [2001], and considered here in Section 8.3. While the controllability results of Section 8.2 have more restrictive hypotheses than those of Chapter 7, it turns out that the restricted class of systems are those for which it is possible to develop some simplified design methodologies for motion planning. These methodologies are presented by way of some examples in Chapter 13. The emphasis in this chapter is on understanding the sometimes subtle relationships between various concepts; the design issues and serious consideration of examples are postponed to Chapter 13.