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containing this point will never be larger than two; in other words, MA will never pass the same point of the obstacle boundary more than three times, producing the upper bound P D+3
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Algorithm BugM1 is executed at every point of the continuous path. Instead of using the xed M-line (straight line (S, T )), as in Bug2, BugM1 uses a straightj j j line segment (Li , T ) with a changing point Li ; here, Li indicates the j th leave point on obstacle i. The procedure uses three registers, R1 , R2 , and R3 , to store j intermediate information. All three are reset to zero when a new hit point Hi is de ned:
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Register R1 stores coordinates of the current point, Qm , of minimum distance between the obstacle boundary and the Target. j R2 integrates the length of the obstacle boundary starting at Hi . R3 integrates the length of the obstacle boundary starting at Qm . (In case of many choices for Qm , any one of them can be taken.)
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The test for target reachability that appears in Step 2d of the procedure is explained lower in this section. Initially, i = 1, j = 1; Lo = Start. The BugM1 o procedure includes these steps: 1. From point Li 1 , move along the line (Lo , Target) toward Target until one of these occurs: (a) Target is reached. The procedure stops. j (b) An ith obstacle is encountered and a hit point, Hi , is de ned. Go to Step 2. 2. Using the accepted local direction, follow the obstacle boundary until one of these occurs: (a) Target is reached. The procedure stops. j 1 j 1 (b) Line (Lo , Target) is met inside the interval (Lo , Target), at a point j ), and the line (Q, Target) does not Q such that distance d(Q) < d(H j cross the current obstacle at point Q. De ne the leave point Li = Q. Set j = j + 1. Go to Step 1. j 1 j 1 (c) Line (Lo , Target) is met outside the interval (Lo , Target). Go to Step 3. j (d) The robot returns to Hi and thus completes a closed curve (of the obstacle boundary) without having de ned the next hit point. The target cannot be reached. The procedure stops.
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MOTION PLANNING FOR A MOBILE ROBOT
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3. Continue following the obstacle boundary. If the target is reached, stop. Otherwise, after having traversed the whole boundary and having returned j j to point Hi , de ne a new leave point Li = Qm . Go to Step 4. 4. Using the contents of registers R2 and R3 , determine the shorter way along j j the obstacle boundary to point Li , and use it to get to Li . Apply the test for Target reachability (see below). If the target is not reachable, the j procedure stops. Otherwise, designate Lo = Li , set i = i + 1, j = 1, and i go to Step 1. As mentioned above, the procedure itself BugM1 is obviously longer and messier compared to the elegantly simple procedures Bug1 and Bug2. That is the price for combining two algorithms governed by very different principles. Note also that since at times BugM1 may leave an obstacle before it fully explores it, according to our classi cation above it falls into the Class 2. What is the mechanism of algorithm BugM1 convergence Depending on the scene, the algorithm s ow ts one of the following two cases. Case 1. If the condition in Step 2c of the procedure is never satis ed, then the algorithm ow follows that of Bug2 for which convergence has been j already established. In this case, the straight lines (Li , Target) always coincide with the M-line (straight line (Start, Target)), and no local cycles appear. Case 2. If, on the other hand, the scene presents an in-position case, then the condition in Step 2c is satis ed at least once; that is, MA crosses the j 1 j 1 straight line (Lo , Target) outside the interval (Lo , Target). This indicates that there is a danger of multiple local cycles, and so MA switches to a more conservative procedure Bug1, instead of risking an uncertain number of local cycles it might now expect from the procedure Bug2 (see Lemma 3.3.4). MA does this by executing Steps 3 and 4 of BugM1, which are identical to Steps 2 and 3 of Bug1. After one execution of Steps 3 and 4 of the BugM1 procedure, the last leave point j on the ith obstacle is de ned, Li , which is guaranteed to be closer to point T j than the corresponding hit point, Hi [see inequality (3.7), Lemma 3.3.1]. Then MA leaves the ith obstacle, never to return to it again (Lemma 3.3.1). From now on, the algorithm (in its Steps 1 and 2) will be using the straight line (Lo , i Target) as the leading thread. [Note that, in general, the line (Lo , Target) does i not coincide with the straight lines (Lo , T ) or (S, T )]. One execution of the i 1 sequence of Steps 3 and 4 of BugM1 is equivalent to one execution of Steps 2 and 3 of Bug1, which guarantees the reduction by one of the number of obstacles that MA will meet on its way. Therefore, as in Bug1, the convergence of this case is guaranteed by Lemma 3.3.1, Lemma 3.3.2, and Corollary 3.3.2. Since Case 1 and Case 2 above are independent and together exhaust all possible cases, the procedure BugM1 converges.
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