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that after having de ned a point H j , MA will never de ne this point again as an H or an L point. Therefore, on each of the subsequent local cycles (if any), point H j will be passed not along the M-line but along the obstacle boundary. After having left point H j , MA can expect one of the following to occur: MA will never return again to H j ; this happens, for example, if it leaves the current obstacle altogether or reaches the Target T . MA will de ne at least the rst pair of points (Lj , H j +1 ), . . . , and will then return to point H j , to start a new local cycle. MA will come back to point H j without having de ned a point Lj on the previous cycle. This means that MA could nd no other intersection point Q of the line (H j , T ) with the current obstacle such that Q would be closer to the point T than H j , and the line (Q, T ) would not cross the current obstacle at Q. This can happen only if either MA or point T are trapped inside the current obstacle (see Figure 3.10). The condition is both necessary and suf cient, which can be shown similar to the proof in the target reachability test for algorithm Bug1 (Section 3.3.1). Based on this observation, we now formulate the test for target reachability for algorithm Bug2.
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Figure 3.10 Examples where no path between points S and T is possible (traps), algorithm Bug2. The path is the dashed line. After having de ned the hit point H2 , the robot returns to it before it de nes any new leave point. Therefore, the target is not reachable.
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Test for Target Reachability. If, on the pth local cycle, p = 0, 1, . . . , after having de ned a hit point H j , MA returns to this point before it de nes at least the rst two out of the possible set of points Lj , H j +1 , . . . , H k , this means that MA has been trapped and hence the target is not reachable. We have learned that in in-position situations algorithm Bug2 may become inef cient and create local cycles, visiting some areas of its path more than once. How can we characterize those situations Does starting or ending inside the obstacle that is, having an in-position situation necessarily lead to such inef ciency This is clearly not so, as one can see from the following example of Bug2 operating in a maze (labyrinth). Consider a version of the labyrinth problem where the robot, starting at one point inside the labyrinth, must reach some other point inside the labyrinth. The well-known mice-in-the-labyrinth problem is sometimes formulated this way. Consider an example4 shown in Figure 3.11.
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Figure 3.11 Example of a walk (dashed line) in a maze under algorithm Bug2. S, Start; T , Target.
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To t the common convention of maze search literature, we present a discrete version of the continuous path planning problem: The maze is a rectangular cell structure, with each cell being a little square; any cell crossed by the M-line (straight line (S, T )) is considered to be lying on the line. This same discussion can be carried out using an arbitrary curvilinear maze.
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Given the fact that no bird s-eye view of the maze is available to MA (at each moment it can see only the small cell that it is passing), the MA s path looks remarkably ef cient and purposeful. (It would look even better if MA s sensing was something better than simple tactile sensing; see Figure 3.20 and more on this topic in Section 3.6.) One reason for this is, of course, that no local cycles are produced here. In spite of its seeming complexity, this maze is actually an easy scene for the Bug2 algorithm. Let s return to our question, How can we classify in-position situations, so as to recognize which one would cause troubles to the algorithm Bug2 This question is not clear at the present time. The answer, likely tied to the topological properties of the combination (scene, Start, Target), is still awaiting a probing researcher. 3.4 COMBINING GOOD FEATURES OF BASIC ALGORITHMS Each of the algorithms Bug1 and Bug2 has a clear and simple, and quite distinct, underlying idea: Bug1 sticks to every obstacle it meets until it explores it fully; Bug2 sticks to the M-line (line (Start, Target)). Each has its pluses and minuses. Algorithm Bug1 never creates local cycles; its worse-case performance looks remarkably good, but it tends to be overcautious and will never cover less than the full perimeter of an obstacle on its way. Algorithm Bug2, on the other hand, is more human in that it can take a risk. It takes advantage of simpler situations; it can do quite well even in complex scenes in spite of its frighteningly high worst-case performance but it may become quite inef cient, much more so than Bug1, in some unlucky cases. The dif culties that algorithm Bug2 may face are tied to local cycles situations when the robot must make circles, visiting the same points of the obstacle boundaries more than once. The source of these dif culties lies in what we called in-position situations (see the Bug2 analysis above). The problem is of topological nature. As the above estimates of Bug2 average behavior show, its performance in out-positions situations may be remarkably good; these are situations that mobile robots will likely encounter in real-life scenes. On the other hand, xing the procedure so as to handle in-position situations well would be an important improvement. One simple idea for doing this is to attempt a procedure that combines the better features of both basic algorithms. (As always, when attempting to combine very distinct ideas, the punishment will be the loss of simplicity and elegance of both algorithms.) We will call this procedure BugM1 (for modi ed ) [59]. The procedure combines the ef ciency of algorithm Bug2 in simpler scenes (where MA will pass only portions, instead of full perimeters, of obstacles, as in Figure 3.5) with the more conservative, but in the limit the more economical, strategy of algorithm Bug1 (see the bound (3.7)). The idea is simple: Since Bug2 is quite good except in cases with local cycles, let us try to switch to Bug1 whenever MA concludes that it is in a local cycle. As a result, for a given point on a BugM1 path, the number of local cycles
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