3D rigid bodies

For the case of a 3D rigid body to which any transformation in $ SE(3)$ may be applied, the same general principles apply. The quaternion parameterization once again becomes the right way to represent $ SO(3)$ because using (4.20) avoids all trigonometric functions in the same way that (4.18) avoided them for $ SO(2)$. Unfortunately, (4.20) is not linear in the configuration variables, as it was for (4.18), but it is at least polynomial. This enables semi-algebraic models to be formed for $ {\cal C}_{obs}$. Type FV, VF, and EE contacts arise for the $ SE(3)$ case. From all of the contact conditions, polynomials that correspond to each patch of $ {\cal C}_{obs}$ can be made. These patches are polynomials in seven variables: $ x_t$, $ y_t$, $ z_t$, $ a$, $ b$, $ c$, and $ d$. Once again, a special primitive must be intersected with all others; here, it enforces the constraint that unit quaternions are used. This reduces the dimension from $ 7$ back down to $ 6$. Also, constraints should be added to throw away half of $ {\mathbb{S}}^3$, which is redundant because of the identification of antipodal points on $ {\mathbb{S}}^3$.

Steven M LaValle 2012-04-20