7.4.1 Adaptation of Motion Planning Algorithms

All of the components from the general motion planning problem of Formulation 4.1 are included: $ {\cal W}$, $ {\cal O}$, $ {\cal A}_1$, $ \ldots $, $ {\cal A}_m$, $ {\cal C}$, $ {q_{I}}$, and $ {q_{G}}$. It is assumed that the robot is a collection of $ r$ links that are possibly attached in loops.

It is assumed in this section that $ {\cal C}= {\mathbb{R}}^n$. If this is not satisfactory, there are two ways to overcome the assumption. The first is to represent $ SO(2)$ and $ SO(3)$ as $ {\mathbb{S}}^1$ and $ {\mathbb{S}}^3$, respectively, and include the circle or sphere equation as part of the constraints considered here. This avoids the topology problems. The other option is to abandon the restriction of using $ {\mathbb{R}}^n$ and instead use a parameterization of $ {\cal C}$ that is of the appropriate dimension. To perform calculus on such manifolds, a smooth structure is required, which is introduced in Section 8.3.2. In the presentation here, however, vector calculus on $ {\mathbb{R}}^n$ is sufficient, which intentionally avoids these extra technicalities.

Steven M LaValle 2012-04-20