Special Euclidean group

Now that the complicated part of representing $ SO(3)$ has been handled, the representation of $ SE(3)$ is straightforward. The general form of a matrix in $ SE(3)$ is given by (4.16), in which $ R \in SO(3)$ and $ v \in {\mathbb{R}}^3$. Since $ SO(3) { \;\cong\; }{\mathbb{RP}}^3$, and translations can be chosen independently, the resulting C-space for a rigid body that rotates and translates in $ {\mathbb{R}}^3$ is

$\displaystyle {\cal C}= {\mathbb{R}}^3 \times {\mathbb{RP}}^3 ,$ (4.31)

which is a six-dimensional manifold. As expected, the dimension of $ {\cal C}$ is exactly the number of degrees of freedom of a free-floating body in space.

Steven M LaValle 2012-04-20