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Figure 5.29 PR arm. (a) In W -space obstacle A interferes with the motion of link l1 . (b) In C-space this creates a band-like virtual obstacle. The arm can make a full circle around the obstacle, passing through points (1, 2, 3, 4, 5, 6, 1).
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Local Cycles. As another similarity with the RR arm, the path planning procedure for the PR arm can create local cycles. Recall that a local cycle is a situation where the arm passes through some segment of its path more than once. (For de nitions and further detail on local cycles, hit and leave points, and so on, see Section 5.2.1). In such a case, when the PR arm returns to the previously de ned hit point, its counter C2 will contain a value different from |n| 2 , n = 0, 1, . . . . An example with a local cycle is shown in Figure 5.31, where the W -space scene contains four obstacles. To see how a local cycle appears here, it is easier to follow the path rst in C-space (Figure 5.31b) and then see what the path looks like in W -space. Assume that the chosen local direction is left. The arm starts at point S and follows the M-line until it hits an obstacle, which happens to be obstacle D, and
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Figure 5.30 PR arm. Because of obstacles A and B, two band-like virtual obstacles appear in C-space and connect with each other in two spots. This creates two free areas of space disconnected from each other. As a result, point T cannot be reached from point S.
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de nes on it the rst hit point, point 1 (Figure 5.31b). It then turns left (in the gure this corresponds to going up) and starts passing around obstacle D. On this path segment it passes point 2, and it meets the M-line again at point 3. At point 3 the arm de nes its rst leave point; then it follows M-line until it hits obstacle B at point 4, which becomes the second hit point. Looking at Figure 5.31b, note that the correct way to proceed is so clear: One should turn right, go around obstacle C, meet there the M-line and follow it to T . So easy. Unfortunately, the arm has no information it would need to do this: It does not have the bene t of seeing the bird s-eye view of Figure 5.31 that we have. The only thing it knows is what comes to it from its sensors. Therefore, as required by the chosen local direction left, the arm will turn left and attempt to pass around obstacle C. On this path segment, while passing through point 5, it will
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Figure 5.31 PR arm. In this example with four obstacles, as the arm attempts to move from S to T under the planning algorithm, a local cycle is created.
encounter obstacle B and and will try to pass it around. This will take the arm to point 6, where it will encounter obstacle A. As the arm starts following obstacle A, at point 7 it will meet obstacle D. While following the latter, it will arrive to point 1, the rst hit point. This moment will signal that a local cycle has been produced. (This should not be confused with completing a closed curve of a virtual boundary. No closed curve of a virtual boundary has been explored so far in this example). As the algorithm will prescribe the arm to look for a leave point that is closer to T than the lastly de ned hit point, point 4, the arm will pass point 1 without even noticing it since the algorithm will not require remembering it. Nor will the counter C2 contain anything of interest at this point: Since C2 was last turned on at the second hit point, point 4, at the time of completion of the local cycle (point 1), its content will be some arbitrary number different from the prescribed |n| 2 , n = 1, 2, . . . .