##

4.2.2 3D Rigid Bodies:

One might expect that defining for a 3D rigid body is an obvious
extension of the 2D case; however, 3D rotations are significantly more
complicated. The resulting C-space will be a six-dimensional
manifold,
. Three dimensions come from
translation and three more come from rotation.

The main quest in this section is to determine the topology of
. In Section 3.2.3, yaw, pitch, and roll were used
to generate rotation matrices. These angles are convenient for
visualization, performing transformations in software, and also for
deriving the DH parameters.
However, these were concerned with applying a single rotation, whereas
the current problem is to characterize the set of all rotations. It
is possible to use , , and to parameterize the
set of rotations, but it causes serious troubles. There are some
cases in which nonzero angles yield the identity rotation matrix,
which is equivalent to
. There are also
cases in which a continuum of values for yaw, pitch, and roll angles
yield the same rotation matrix. These problems destroy the topology,
which causes both theoretical and practical difficulties in motion
planning.

Consider applying the matrix group concepts from Section
4.2.1. The general linear group is
homeomorphic to
. The orthogonal group, , is determined
by imposing the constraint . There are
independent equations that require distinct columns to be orthogonal,
and three independent equations that force the magnitude of each
column to be . This means that has three dimensions, which
matches our intuition since there were three rotation parameters in
Section 3.2.3. To obtain , the last constraint,
, is added. Recall from Example 4.12 that
consists of two circles, and the constraint
selects one of them. In the case of , there are two
three-spheres,
, and
selects one of
them. However, there is one additional complication: Antipodal points
on these spheres generate the same rotation matrix. This will be seen
shortly when quaternions are used to parameterize .

**Subsections**
Steven M LaValle
2020-08-14