MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

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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:

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For a PPP arm, C I 1 I 1 I 1 , the unit cube. = For a PRP or RPP arm, C S 1 I 1 I 1 , a pipe. = For an RRP arm, C S 1 S 1 I 1 , a solid torus. =

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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 identi ed, or as a solid torus. The obstacle monotonicity property is preserved in con guration space. This is simply because the equivalent relation that de nes C and Cf from J and Jf has no effect on the third joint axis, l3 . Thus we have the following statement: Theorem 6.3.10. The con guration space obstacle OC possesses the monotonicity property along l3 axis. As with the subset Jf , Cp C can be de ned as the set {l3 = 0}; OC1 , OC2 , OC3 , OC3+ , OC3 , Pc , Pm , Cf , Cpf , and Cf can be de ned accordingly. Theorem 6.3.11. Let Q1 = OC3 Cf and Q2 = Cp Cf . Then, Bf = Q1 Q2 is a deformation retract of Cf .

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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 identi ed; and (b) as a solid torus.

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THREE-LINK XXP ARM MANIPULATORS

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The proof is analogous to that of Theorem 6.3.5. Let B denote the 2D space obtained by patching all the holes in Bf so that B Cp . It is obvious that = B Cp T is a deformation retract of C. We obtain the following statement = = parallel to Theorem 6.3.6. Theorem 6.3.12. Given two points js , jt Bf , if there exists a path pC Cf connecting js and jt , then there must exist a path pB Bf , such that pB pJ . A path between two given points js = (j1s , j2s , l3s ), jt = (j1t , j2t , l3t ) Cf can be obtained by nding the path between the two points js = (j1s , j2s , l3s ), js = (j1t , j2t , l3t ) Bf . Because of the monotonicity property (Theorem 6.3.10), js and jt always exist and can be respectively connected within Cf with js and jt via vertical line segments. Hence the following statement: Corollary 6.3.4. The problem of nding a path in Cf between points js , jt Cf can be reduced to that of nding a path in its subset Bf . 6.3.6 Connectivity Graph At this point we have reduced the problem of motion planning for an XXP arm in 3D space to the study of a 2D subspace B that is homeomorphic to a common torus. Consider the problem of moving a point on a torus whose surface is populated with obstacles, each bounded by simple closed curves. The torus can be represented as a square with its opposite sides identi ed in pairs (see Figure 6.17a). Note that the four corners are identi ed as a single point. Without loss of generality, let the start and target points S and T be respectively in the center and the corners of the square. This produces four straight lines connecting S and T , each connecting the center of the square with one of its corners. We call each line a generic path and denote it by gi , i = 1, 2, 3, 4. De ne a connectivity graph G on the torus by the obstacle-free portions of any three of the four generic paths and the obstacle boundary curves. We have the following statement: Theorem 6.3.13. On a torus, if there exists an obstacle-free path connecting two points S and T , then there must exist such a path in the connectivity graph G. Without loss of generality, let g1 , g2 , and g3 be the complete set of generic paths, as shown in Figure 6.17a, where the torus is represented as a unit square with its opposite sides identi ed. The generic paths g1 , g2 , and g3 cut the unit square into three triangular pieces. Rearrange the placements of the three pieces by identifying the opposite sides of the square in pairs along edges a and b, respectively (see Figures 6.17b and 6.17c). We thus obtain a six-gon (hexagon), with three pairs of S and T as its vertices and generic paths g1 , g2 , and g3 as its edges. The hexagon presentation is called the generic form of a torus.

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