MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS in .NET

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Hence the arm will pass point 1 and continue following the obstacle boundary. It will pass point 2 and then point 3. At point 8 the arm will encounter again obstacle A (albeit at this new point), switch to following it, encounter at point 9 obstacle B, follow it to point 10 where it meets obstacle D, and follow that obstacle to point 11, where it nally! meets the M-line again. Since point 11 is closer to T than the last hit point (point 4), point 11 is de ned as the second leave point. Notice that by the time the arm reaches point 1 for the second time, on its way to point 11 it will have encountered all four obstacles A, B, C, and D. Since during this motion the arm is passing from one obstacle to the other in an uninterrupted obstacle following, with no path segments in free space, the arm will perceive all four as one obstacle. [This is very clear from the C-space picture (Figure 5.31b).] Thereafter, starting at point 11, the arm will switch to following the M-line and eventually will arrive happily at the target point T . The torturous exercise that we have just gone through is given only for those courageous souls who want to understand how a local cycle is formed and why it causes no dif culty to the algorithm. The arm will have no idea that it went through a local cycle. The theory developed here and in Section 5.2.1 guarantees that the cycle will not become an in nite loop. The different cases that we have considered thus far are necessary only to establish such a guarantee in the algorithm. We emphasize again that the described path has been produced by the arm equipped with simple tactile sensing. As described in Section 5.2.5, a more advanced sensing will in general improve the performance that is, shorten the path quite markedly.
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The Algorithm. We are now ready to formulate the sensor-based motion-planning algorithm for the prismatic-revolute (PR) arm. The following notation and parameters will be used in the procedure:
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Parameter F is used to handle an open curve of a Type II obstacle. It is set according to this rule: F = +1 when the arm, while following a virtual boundary, reaches the upper limit of joint value l1 . F = 1 when, while following a virtual boundary, the arm reaches the lower limit of joint value l1 . F = 0 at the rst hit point of the virtual boundary.
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A ag is used to distinguish between the rst and second open curves of the virtual boundary of a Type II obstacle. Counter C2 is used to handle closed curves of virtual boundaries. Complementary M-lines, M1 and M2 , are de ned as before. Hit, Hj , and leave, Lj , points are de ned as before (Section 5.2.1); Lo = S.
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TOPOLOGY OF ARM S FREE CONFIGURATION SPACE
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Distances are Euclidean distances along M-line in the plane (l1 , 2 ). For speci city only, the local direction for passing around an obstacle is taken as left when the rst hit point is de ned.
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The PR-Arm Algorithm consists of the following steps: 1. M1 -line is designated as the M-line. Set the ag down. Set j = 1. Go to Step 2. 2. Set counter C2 to zero. Set F = 0. From point Lj 1 , the arm moves along the M-line until one of the following occurs: (a) Target T is reached. The procedure stops. (b) An obstacle is encountered and a hit point, Hj , is de ned. Choose the local direction left . Turn on counter C2 . Go to Step 3. 3. The arm follows the virtual boundary until one of the following occurs: (a) Target is reached. The procedure stops. (b) M-line is met at a distance d from T such that d < d(Hj , T ); point Lj is de ned. Set the ag down. Increment j. Go to Step 2. (c) The arm reaches one of the limits of link l1 (this corresponds to an endpoint of one open curve of a virtual boundary) without ever meeting the M-line. Go to Step 4. (d) The arm returns to Hj (i.e., a closed curve along the virtual boundary has been completed) without ever encountering the M-line. The target cannot be reached. The procedure stops. 4. Depending on the value F , the ag condition, and the current arm position, one of the following occurs: (a) F = 0. Set F to +1 or 1 according to the rule above. Return to the last hit point. Choose the local direction right. Go to Step 3. (b) Value F corresponds to the current curve endpoint (i.e., +1 for the upper limit and 1 for the lower limit of l1 ); this means that the whole obstacle has been explored. The target cannot be reached. The procedure stops. (c) Value F does not correspond to the current curve endpoint, and the ag is down; this means the rst open curve of a Type II obstacle has been explored. Set j = 1. Set the ag up. Designate M2 -line as the M-line. Return to S. Go to Step 2. (d) Value F does not correspond to the current curve endpoint, and the ag is up; this means that the second open curve of a Type II obstacle has been investigated. The target cannot be reached. The procedure stops. 5.8 TOPOLOGY OF ARM S FREE CONFIGURATION SPACE In the previous sections of this chapter we have considered an exhaustive list of ve kinematic con gurations of two-link arm manipulators (see Figure 5.1).
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