Formulation 14.1 assumed that feedback is not necessary. If the initial state is given, then the solution takes the form of an action trajectory, which upon integration yields a time-parametrized path through . This extended the Piano Mover's Problem of Section 4.3.1 to include phase spaces and differential constraints. Now suppose that feedback is required. The reasons may be that the initial state is not given or the plan execution might not be predictable due to disturbances or errors in the system model. Recall the motivation from Section 8.1.
With little effort, the feedback motion planning framework from Chapter 8 can be extended to handle differential constraints. Compare Formulations 8.2 and 14.1. Feedback motion planning under differential constraints is obtained by making the following adjustments to Formulation 8.2:
The most important difference with respect to Chapter 8 is that is replaced with , which allows complicated differential models of Chapter 13 to be used. The vector field that results from must satisfy the differential constraints imposed by . In Section 8.4.4, simple constraints on the allowable vector fields were imposed, such as velocity bounds or smoothness; however, these constraints were not as severe as the models in Chapter 13. For example, the Dubins car does not allow motions in the reverse direction, whereas the constraints in Section 8.4.4 permit motions in any direction.
Steven M LaValle 2012-04-20