MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

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A b* b2 B l2 B a2 a 2 A C b1 b 2 b* b1 b 2 b2 B

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Figure 5.39 An illustration for the proof of Theorem 5.2.1.

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arm endpoint (and not any other point of the arm body) is in contact with some obstacle, A. The dead end position may occur only if one or two other obstacles, B and C, appear as shown in Figure 5.39a. Clearly, it is always possible to move from P to a position distinct from P1 (here, P2 or P2 , respectively). Thus, P cannot be a dead end position. Case 3 (Figure 5.39c); a = a1 , b = b1 . In this case, the segments l2 in both positions P1 and P intersect each other. This may occur only if l2 is rolling around some obstacle, A. Here, P may be a dead end only if one or two other obstacles, B and C, appear as shown in Figure 5.39c. Observe that positions P2 and P2 , respectively, are good alternatives to P1 . Therefore, P is not a dead-end position. This exhausts all possible cases and completes the proof. Q.E.D.

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Proof of Lemma 5.2.2 (Section 5.2.1). Torus is a closed orientable manifold. The maximum number of closed curves needed to divide a given closed orientable manifold into two separate domains is determined by its connectivity numbers. The rst connectivity number is known to de ne the maximum number of closed cuts that can be made on the surface without dividing it into separate domains. On the torus, the rst connectivity number is equal to two [105]. The only arrangement for two closed cuts (two closed curves), a and b, that can be made on the torus without dividing it into separate domains is shown in Figure 5.40. According to Theorem 5.2.1, a virtual boundary consists of simple closed curves and thus cannot have self-intersections. Any other arrangement of two closed curves on the torus such that they do not touch or intersect each other produces at least two separate domains. Similarly, more than two simple closed curves produce more than two separate domains. Therefore, if some area on the torus is separated from the rest of it by simple closed curves then the boundary of this area consists of no more that two such curves. Q.E.D.

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APPENDIX

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Figure 5.40 Closed cuts a and b do not divide the surface of the torus into separate areas but leave it as one area. However, addition of any other closed curve would cut it into two separate areas.

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Separation Theorems on the Common Torus. (See Section 5.8.4). The purpose of this subsection is to prove Theorem 5.8.4. The proof for its planar counterpart can be found in Ref. 110, VI.16.2, which uses the concept of regular grating as the fundamental tool. We will use an analogous strategy to prove Theorem 5.8.4. Since the topology of a torus is different from that of a plane, we start with the modi ed de nitions of regular grating, k-chains, and kcycles in torus T 1 , and proceed with the corresponding operations and properties (Theorems 5.9.1 to 5.9.7). Several intermediate results are needed in order to prove Theorem 5.8.4. The proofs for some of these are the same as their planar counterparts, in which case we simply restate the statements and cite the source [110]. Proofs will be given to statements that are valid only for T 1 . Regular grating is a convenient tool for studying the connectivity of a subset of T 1 . We show in Theorem 5.9.8 that a 1-cycle (a simple closed curve) does not necessarily separate a torus into two halves as it would in a plane. This major fact makes the proof of Theorem 5.8.4 different from its planar counterpart. Finally, to prove Theorem 5.8.4 we need to show that if a region (a connected subset) D in T 1 is uniformly connected, then the connectivity of any of its boundary components is destroyed by the removal of two single points. This is done by drawing a cross-cut L connecting the same boundary component of D and showing that D-L has exactly two components (Theorem 5.9.12). The proof of Theorem 5.9.12 in turn requires the intermediate results of Theorems 5.9.9 to 5.9.11.

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