Page 353 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

328 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

side, the RRR linkage is only one out of the five 3D linkages shown in Figure 6.1,
Section 6.1, which together comprise the overwhelming majority of robot arm
manipulators that one finds in practice. Still, RRR is a popular arm, and knowing
how to do sensor-based motion planning for it would be of much interest. Judging
by our analysis of the RR arm in Chapter 5, it is also likely the most difficult
arm for sensor-based motion planning.
Conceptually, the difficulty with the RRR arm, and to some extent with other
XXR arms, is of the same kind that we discussed in the Section 6.1, when describ-
ing a fly moving around an object in three-dimensional space. The fly has an
infinite number of ways to go around the object. Theoretically, it may need to
try all those ways in order to guarantee getting “on the other side” of the object.
We have shown in this chapter that, thanks to special properties of mono-
tonicity of the corresponding configuration space, no infinite motion will ever
be necessary for any XXP arm manipulator in order to guarantee convergence.
No matter how complex are the 3D objects in the arm’s workspace, the motion
planning algorithm guarantees a finite (and usually quick) path solution. No such
properties have been found so far for the XXR arm manipulators. On the other
hand, similar to XXP arms considered in this chapter, motion planning for XXR
arms seems to be reducible to motion along curves that are similar to curves we
have used for XXP algorithms (such as an intersection curve between an obstacle
and an M-plane or V-plane, etc.). Even in the worst case, this would require a
search of a relatively small graph.
Provided that the arm has the right whole-body sensing, in practice one can
handle an XXR arm by using the motion planning schemes developed in this
chapter for XXP arms, perhaps with some heuristic modifications. Some such
attempts, including physical experiments with industrial RRR arm manipulators,
are in described Chapter 8 (see also Ref. 115).