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.