11.4 Continuous State Spaces

This section takes many of the concepts that have been developed in Sections 11.1 and 11.2 and generalizes them to continuous state spaces. This represents an important generalization because the configuration space concepts, on which motion planning was defined in Part II, are all based on continuous state spaces. In this section, the state space might be a configuration space, $X = {\cal C}$, as defined in Chapter 4 or any other continuous state space. Since it may be a configuration space, many interesting problems can be drawn from robotics.

During the presentation of the concepts of this section, it will be helpful to recall analogous concepts that were already developed for discrete state spaces. In many cases, the formulations appear identical. In others, the continuous case is more complicated, but it usually maintains some of the properties from the discrete case. It will be seen after introducing continuous sensing models in Section 11.5.1 that some problems formulated in continuous spaces are even more elegant and easy to understand than their discrete counterparts.