Page 283 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

258 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

Proof: CS of any 2-DOF robot can be considered as a closed subset of a torus,
1 −1 1
written as C ⊆ T .Let C be the complement of C in T and O C ⊂ C be CSO.
−1 1

Then, the set O = O C ∪ C is open and locally connected in T ; thus, accord-
C
1
ing to Theorems 5.8.3 and 5.8.4, O is bounded by simple closed curves in T .

C
Q.E.D.
Now consider kinematic configurations of 2-DOF robots other than RR
arms—that is, arms (b) to (e) in Figure 5.1. By Lemma 5.8.3, CS of each of
these robot arms is homeomorphic to either the surface of a cylinder (RP or PR
arms) or a disk (PP arm). In each of these cases, CS can be thought of as a
closed subset of the torus. This also applies to 2-DOF arms with two revolute
joints, one or both of which are constrained. The physical constraints on the joint
range can be due to either the robot design or the obstacles in WS. This indicates
that the constraints on joint limits can be treated as obstacles.
One might argue that, since the information about joint limits is known before-
hand, there is no reason to treat them as unknown obstacles. This is true,
especially if incorporating those limits is easy. Note, however, that the joint
limits are not necessarily mutually independent. There are commercial robots in
which the limit values of one joint depend on the values of other joints. This
dependence is a function of the robot design and may be quite complex. Treating
joint limits as obstacles is an elegant way to combine simplicity with universality.
To conclude, we have shown that if the spatial relationship between the
robot arm and obstacles satisfies some reasonable and nonrestrictive for prac-
tice conditions, as defined by Conditions 5.8.1 to 5.8.4, then the corresponding
configuration space obstacle (CSO) is uniformly locally connected. In particular,
for the case of 2-DOF robot arms, this property guarantees that the free config-
uration space obstacle (FCS) is bounded by simple closed curves, which is an
important feature upon which various sensor-based motion strategies developed
above are based.
Both the simplicity and closedness of the boundary curves are important to
these algorithms: It is these features that allow the algorithms to solve the motion
planning problem with very little input information (local sensing only) about
the robot environment. This is true for the simpler Bug family algorithms pre-
sented in Chapter 3, as well as for the more sophisticated algorithms developed
in this chapter. The robot can correctly conclude that the target position is not
reachable—a global property—by circumnavigating only parts of the obstacles
involved.

5.9 APPENDIX
Proof of Theorem 5.2.1 (Section 5.2.1). Suppose the statement of the theorem
does not hold, and the virtual boundary of some obstacle is formed in C-space
by one or more simple (which is guaranteed by Lemma 5.2.1) but not closed
curves. Take one such simple open curve and consider one of its two endpoints