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Figure 2.16 (a) When robot orientation is added to a 2D task, the (b) resulting 3D C-space of parameters (x, y, ) is nonlinear, even if the original robot and obstacles are polygons. Here the robot A is a line segment of length l, and the obstacle is a horizontal table line.
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of approximation can cause a dramatic change in the number and positions of nodes of the approximated surfaces and eventually in the generated paths. Measuring the computational burden in terms of complexity of the connectivity graph may create peculiar situations where the derived computational complexity of a given task contradicts our intuitive notion of problem complexity. Consider, for example, a circular obstacle A in Figure 2.17. Assume that the motion planning algorithm that we plan to use requires polygonal obstacles. Then, obstacle A is rst approximated say, by one of the polygons B or C in Figure 2.17. Now, according to Piano Mover s algorithms, planning a path around obstacle C is computationally more dif cult than planning a path around the obstacle B, because of the greater number of nodes in C. Moreover, in the limit, increasing the accuracy of polygon approximation takes the computational burden to in nity. But, from the human and from the robotics control viewpoints, walking around the circle A is actually easier than walking around obstacles B or C, because the latter require special decisions at the corners of the obstacle. Also, from the dynamics standpoint, there is an undesirable sharp change in the velocity vector at the corners of obstacles B and C. One can, of course, solve this speci c example by including circular objects in the list of those allowed by the algorithm, but this will only shift this discussion to some other shapes. For more detail on the Piano Mover s model, the reader is referred to the literature. The model s computational complexity for cases of rigid or hinged bodies has been studied extensively. The problem was shown to be computationally prohibitive [15 17]. A 2D case has been studied in Refs. 18 20. Cases where objects to be moved are polygons (polyhedra) or discs (spheres) moving amidst polygonal (polyhedral) obstacles are considered in [15, 18, 19, 21 23]. The rst attempt to study the case of moving an object with a number of free-hinged links was initiated in 1968 by Pieper [24], in the context of motion planning for robot arm manipulators. Exact algorithms for this problem have been described [15, 16, 20], as have various heuristics (e.g., see Refs. 25 and 26). The computational complexity of the problem was rst reported by Reif [15], who showed that the general Piano Mover s problem is PSPACE-hard. He also
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sketched a possible solution for moving a solid object in polynomial time, by direct computation of the forbidden volumes in spaces of higher dimensions.4 Reif also demonstrated that even the preliminary operation of approximating the robot workspace with a speci ed accuracy carries a high computational cost. Schwartz and Sharir [18] presented a polynomial-time algorithm for a 2D Piano Mover s problem with convex polygonal obstacles. It has been shown in a number of works (e.g., Lozano-Perez and Wesley [27]) that the process of moving in the task s con guration space carries additional computational costs. In general, even if the original obstacles are polyhedral, obstacles in the con guration space have nonplanar walls. In order to keep the problem manageable, various constraints are typically imposed. Moravec [28] considered a path planning algorithm for a mobile robot moving in two dimensions, with the robot presented as a circle. In his treatment of a 2D path planning problem with a convex polygonal robot and convex polygonal obstacles, Brooks and Binford [29, 30] used a generalized cylinder presentation to reduce the planning problem to a graph search. A generalized cylinder is formed by a volume swept by a cross section (in general, of varying shape and size) moving along the cylinder axis, which in turn can be some spine curve. The version of the Piano Movers problem where the robot can consist of a number of free-hinged links is more dif cult. On the heuristic level this version was started by Pieper [24] and further investigated by Paul [31]. Both were attracted to the problem s obvious relation to control of robot arms with multiple degrees of freedom. Later, new approaches for this version have been considered in Refs. 16 and 20. The most general (although very expensive computationally) algorithm for moving a free-hinged body was given by Schwartz and Sharir [16]. The technique is based on the general method of cell decomposition; the robot and obstacles are assumed to be limited by algebraic surfaces. A more economical (but still prohibitive for many practical tasks) algorithm for the general case was reported by Canny [32]. A variety of special cases shown to lead to simpler algorithms were described by Hopcroft et al. [20]. 2.9 MOTION PLANNING WITH INCOMPLETE INFORMATION By the mid-1980s it became clear that the inherent uncertainty of a realistic robot environment and the subsequent need for real-time sensing called for a paradigm of motion planning that would fundamentally differ from the Piano Mover s paradigm. It was further realized that uncertainty and sensing were not some small irritating engineering details but major factors in the theoretical foundation of motion planning algorithms. As it turned out, uncertainty and sensing became the very center around which the new paradigm would be built. The result was the theory and practice of robot motion planning with incomplete information, or the SIM (Sensing Intelligence Motion) paradigm.
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Higher dimensions d appear when one takes into account the moving rigid object s orientation along its way; d = 3 for the 2D case, and d = 6 for the 3D case.
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