Recall from Chapter 5 that many sampling-based planning algorithms rely on measuring distances or volumes in . If , as in the wheeled systems from Section 13.1.2, then the concepts of Section 5.1 apply directly. The equivalent is needed for a general state space , which may include phase variables in addition to the configuration variables. In most cases, the topology of the phase variables is trivial. For example, if , then each component is constrained to an interval of . In this case the velocity components are just an axis-aligned rectangular region in , if is the dimension of . It is straightforward in this case to extend a measure and metric defined on up to by forming the Cartesian product.
A metric can be defined using the Cartesian product method given by (5.4). The usual difficulty arises of arbitrarily weighting different components and combining them into a single scalar function. In the case of , this has involved combining translations and rotation. For , this additionally includes velocity components, which makes it more difficult to choose meaningful weights.