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.

Steven M LaValle 2020-08-14