Since visualization is still possible with one more dimension, suppose
there are three links,
. The C-space can be
visualized as a 3D cube with opposite faces identified. Each
coordinate ranges from 0 to , for which
. Suppose that each link has length to obtain
. A point
is transformed as
To obtain polynomials, let
, which results in
for which the constraints
be satisfied. This preserves the torus topology of , but
now it is embedded in
. Multiplying the matrices yields the
, defined as
for the and coordinates, respectively.
Again, consider imposing a single constraint,
which constrains the point
to traverse the -axis.
The resulting variety is an interesting manifold, depicted in Figure
4.24 (remember that the sides of the cube are
Increasing the required value for the constraint on the final
point causes the variety to shrink. Snapshots for and
are shown in Figure 4.25. At , the
variety is not a manifold, but it then changes to
this sphere is reduced to a point at , and then for
the variety is empty.
The variety for the three-link chain
with is a 2D manifold.
If , then the variety shrinks.
If , the variety is a sphere. At it is a point,
and for it completely vanishes.
If two constraints, and , are imposed, then the varieties are intersected to obtain a
1D set of solutions. The example is equivalent to a
well-studied four-bar mechanism.
Instead of the constraint , we could instead constrain the
coordinate of to obtain . This yields another 2D
variety. If both constraints are enforced simultaneously, then the
result is the intersection of the two original varieties. For
example, suppose and . This is equivalent to a
kind of four-bar mechanism , in which the
, is fixed along the -axis from 0 to . The
is depicted in Figure 4.26. Using the
coordinates, the solution may be easily
parameterized as a collection of line segments. For all
, there exist solution points at
. Note that once again the variety is not a manifold.
A family of interesting varieties can be generated for the four-bar
mechanism by selecting different lengths for the links. The
topologies of these mechanisms have been determined for 2D and a
3D extension that uses spherical joints (see ).
Steven M LaValle