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6.3.9 Discussion As demonstrated in this section, the kinematic constraints of any XXP arm major linkage result in a certain property called monotonicity of the arm joint space and con guration space (C-space or Cf ). The essence of the monotonicity property is that for any point on the surface of a C-space obstacle, there exists at least one direction in C-space that corresponds to one of the joint axes, such that no other points in space along this direction can be reached by the arm. The monotonicity property allows the arm to infer some global information about obstacles based on local sensory data. It thus becomes an important component in sensor-based motion planning algorithms. We concluded that motion planning for a three-dimensional XXP arm can be done on a 2D compact surface, Bf , which presents a deformation retract of the free con guration space Cf . We have further shown that any convergent 2D motion planning algorithm for moving a point on a compact surface (torus, in particular) can be lifted into 3D for motion planning for three-joint XXP robot arms. The strategy is based on the monotonicity properties of C-space. Given the arm s start and target points js , jt Cf and the notions above and below as de ned in this section, the general motion planning strategy for an XXP arm can be summarized as consisting of these three steps: 1. Move from js to js , where js Bf is directly above or below js ; 2. nd a path between js and jt within Bf , where jt Bf is directly above or below jt ; and 3. move from jt to jt . Because of the monotonicity property, motion in Steps 1 and 3 can be achieved via straight line segments. In reality, Step 2 does not have to be limited to the plane: It can be lifted into 3D by modifying the 2D algorithm respectively, thus resulting in local optimization and shorter paths. With the presented theory, and with various speci c algorithms presented in this and previous chapters, one should have no dif culty constructing one s own sensor-based motion planning algorithms for speci c XXP arm manipulators. 6.4 OTHER XXX ARMS One question about motion planning for 3D arm manipulators that still remains unanswered in this chapter is, How can one carry out sensor-based motion planning for XXR arm manipulators that is, arms whose third joint is of revolute type At this time, no algorithms with a solid theoretical foundation and with guaranteed convergence can be offered for this group. This exciting area of research, of much theoretical as well as practical importance, still awaits for its courageous explorers. In engineering terms, one kinematic linkage from the XXR group, namely RRR, is of much importance among industrial robot manipulators. On a better
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side, the RRR linkage is only one out of the ve 3D linkages shown in Figure 6.1, Section 6.1, which together comprise the overwhelming majority of robot arm manipulators that one nds 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 5, it is also likely the most dif cult arm for sensor-based motion planning. Conceptually, the dif culty 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 describing a y moving around an object in three-dimensional space. The y has an in nite 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 monotonicity of the corresponding con guration space, no in nite 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 nite (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 modi cations. Some such attempts, including physical experiments with industrial RRR arm manipulators, are in described 8 (see also Ref. 115).
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