Page 345 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

320 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

We have thus reduced the motion planning problem in the arm workspace to
the one of moving a point from start to target position in C-space.
The following characteristics of the C-space topology of XXP arms are direct
results of Theorem 6.3.7:

1
1
1
∼
• For a PPP arm, C = I × I × I , the unit cube.
1
1
1
• For a PRP or RPP arm, C = S × I × I , a pipe.
∼
1
1
1
∼
• For an RRP arm, C = S × S × I , a solid torus.
Figure 6.16 shows the C-space of an RRP arm, which can be viewed either
as a cube with its front and back, left and right sides pairwise identified, or as a
solid torus.
The obstacle monotonicity property is preserved in configuration space. This
is simply because the equivalent relation that defines C and C f from J and J f
has no effect on the third joint axis, l 3 . Thus we have the following statement:
Theorem 6.3.10. The configuration space obstacle O C possesses the monotonic-
ity property along l 3 axis.
As with the subset J f , C p ⊂ C can be defined as the set {l 3 = 0}; O C1 , O C2 ,

, P c , P m , C f , C pf ,and C can be defined accordingly.
O C3 , O C3 + , O C3 − f

∩ C f and Q 2 = C p ∩ C f . Then,
Theorem 6.3.11. Let Q 1 = ∂O C3 −

B f = Q 1 ∪ Q 2
is a deformation retract of C f .

(a) (b)
Figure 6.16 Two views of C-space of an RRP arm manipulator: (a) As a unit cube with
its front and back, left and right sides pairwise identified; and (b) as a solid torus.