The closure constraints introduced in Section 4.4 can be
summarized as follows. There is a set, , of polynomials
, , that belong to
and express
the constraints for particular points on the links of the robot. The
determination of these is detailed in Section 4.4.3. As
mentioned previously, polynomials that force points to lie on a circle
or sphere in the case of rotations may also be included in .
Let denote the dimension of . The closure space is
defined as

(7.21) 
which is an dimensional subspace of that corresponds to all
configurations that satisfy the closure constants. The obstacle set
must also be taken into account. Once again,
and
are
defined using (4.34). The feasible
space is defined as
, which are the configurations that satisfy
closure constraints and avoid collisions.
The motion planning problem is to find a path
such that
and
. The new
challenge is that there is no explicit parameterization of
,
which is further complicated by the fact that (recall that
is the dimension of
).
Steven M LaValle
20120420