Page 318 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THE CASE OF THE PPP (CARTESIAN) ARM 293

l 3

O 1
T
O 2
d
H
b
c
a
o l 2
T′
L
S
d′
L′
H′
a′
b′ c′
V-plane S′

l 1
Figure 6.10 C-space in the case when both front and rear parts of link l 3 interact with
obstacles, producing a single obstacle that is a combination of a Type III + and Type III −
obstacles.

Simultaneous Interaction of Combinations of Links with Obstacles.
Since Type I obstacles are trivial from the standpoint of motion planning—they
can be simply treated as walls parallel to the sides of the C-space cubicle—we
focus now on the combinations of Type II and Type III obstacles. When both
links l 2 and l 3 are simultaneously in contact with obstacles, the C-point is at
the intersection curve between Type II and Type III obstacles, which presents a
simple closed curve. (Refer, for example, to the intersection of obstacles O 2 and
O 3 , Figure 6.11.) Observe that the Type III monotonicity property is preserved
in the union of Type II and Type III obstacles. Hence,

Corollary 6.2.3. If there is a path around the union of a Type II and a Type III
obstacles, then there must be a path around them along their intersection curve.

As in the case of intersection between the Type III obstacle and the V-plane
(see above), one of the two possible local directions is clearly preferable to the