14.3.1.1 Distance and volume in $X$

Recall from Chapter 5 that many sampling-based planning algorithms rely on measuring distances or volumes in ${\cal C}$. If $X = {\cal C}$, 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 $X$, 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 $x = (q,{\dot q})$, then each ${\dot q}_i$ component is constrained to an interval of ${\mathbb{R}}$. In this case the velocity components are just an axis-aligned rectangular region in ${\mathbb{R}}^{n/2}$, if $n$ is the dimension of $X$. It is straightforward in this case to extend a measure and metric defined on ${\cal C}$ up to $X$ 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 ${\cal C}$, this has involved combining translations and rotation. For $X$, this additionally includes velocity components, which makes it more difficult to choose meaningful weights.