For the case of a 3D rigid body to which any transformation in
may be applied, the same general principles apply. The quaternion
parameterization once again becomes the right way to represent
because using (4.20) avoids all trigonometric functions
in the same way that (4.18) avoided them for
.
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
. Type FV, VF, and EE contacts arise for the
case.
From all of the contact conditions, polynomials that correspond to
each patch of
can be made. These patches are polynomials in
seven variables:
,
,
,
,
,
, and
. 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
back down to
. Also, constraints
should be added to throw away half of
, which is redundant
because of the identification of antipodal points on
.
Steven M LaValle 2020-08-14