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Else if Target is unreachable the procedure stops. Else if C = Ti go to step S1. Procedure Compute Ti -21 includes the following steps:
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Step 1: If T is visible, de ne Ti = T ; procedure stops. Else if Ti is on an obstacle boundary go to Step 3. Else go to Step 2. Step 2: De ne point Q as the endpoint of the maximum length contiguous segment |Ti Q| of M-line that extends from point Ti in the direction of point T . If an obstacle has been identi ed crossing the M-line at point Q, then de ne a hit point, H = Q; assign X = Q, de ne Ti = Q; go to Step 3. Else de ne Ti = Q; go to Step 4. Step 3: De ne point Q as the endpoint of the maximum length contiguous segment of obstacle boundary, (Ti Q), extending from Ti in the local direction. If an obstacle has been identi ed crossing M-line at a point P (Ti Q), |PT| < |HT|, assign X = P ; if, in addition, |PT| does not cross the obstacle at P , de ne a leave point, L = P , de ne Ti = P , and go to Step 2. If the lastly de ned hit point, H , is identi ed again and H (Ti Q), then Target is not reachable; procedure stops. Else de ne Ti = Q; go to Step 4. Step 4: If Ti is on the M-line de ne Q = Ti , otherwise de ne Q = X. If some points {P } on the M-line are identi ed such that |S T | < |QT|, S {P }, and C is in the main semiplane, then nd the point S {P } that produces the shortest distance |S T |; de ne Ti = S ; go to Step 2. Else procedure stops.
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In brief, procedure Compute Ti -21 operates as follows. Step 1 is executed at the last stage, when target T becomes visible (as at point A, Figure 3.15). A special case in which points of the M-line noncontiguous to the previously considered sets of points are tested as candidates for the next intermediate Target Ti is handled in Step 4. All the remaining situations relate to choosing the next point Ti among the Bug2 path points contiguous to the previously de ned point Ti ; these are treated in Steps 2 and 3. Speci cally, in Step 2 candidate points along the M-line are processed, and hit points are de ned. In Step 3, candidate points along obstacle boundaries are processed, and leave points are de ned. The test for target reachability is also performed in Step 3. It is conceivable that at some locations C of the robot the procedure will execute, perhaps even more than once, some combinations of Steps 2, 3, and 4. While doing this, contiguous and noncontiguous segments of the Bug2 path along the M-line and along obstacle boundaries may be considered before the next intermediate target Ti is de ned. Then the robot makes a physical step toward Ti .
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Figure 3.14 Scene 1: Path generated by algorithms VisBug-21 or VisBug-22. The radius of vision is rv .
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Analysis of VisBug-21. Examples shown in Figures 3.14 and 3.15 demonstrate the effect of radius of vision rv on performance of algorithm VisBug-21. (Compare this with the Bug2 performance in the same environment, Figure 3.13). In the analysis that follows, we rst look at the algorithm performance and then address the issue of convergence. Since the path generated by VisBug-21 can diverge signi cantly from the path that would be produced under the same conditions by algorithm Bug2, it is to be shown that the path-length performance of VisBug-21 is never worse than that of Bug2. One would expect this to be true, and it is ensured by the following lemma.
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Lemma 3.6.1. For a given scene and a given set of Start and Target points, the path produced by algorithm VisBug-21 is never longer than that produced by algorithm Bug2. Proof: Assume the scene and start S and target T points are xed. Consider the robot s position, Ci , and its corresponding intermediate target, Ti , at step i of the path, i = 0, 1, . . . . We wish to show that the lemma holds not only for the whole path from S to T , but also for an arbitrary step i of the path. This amounts to showing that the inequality {SCi } + |Ci Ti | [STi ] (3.15)
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