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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.

S (a) (b)

Clock gating is a direct means for reducing dynamic power dissipation but creates dif culties in implementation and timing analysis.

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.

But I knew what I wanted I wanted to do what these people did Now, I remembered my trade Seeing all these people in a state of violent anxiety over their trades made me feel part of the group A member of a brotherhood of traders who worried about whether their trade was profitable Just like me Hey, Mack, you lookin for somebody a voice called out to me I could not tell where it had come from, so I just answered while I was looking around the room, Anderson Looking for Anderson Over in front of you, keep walking, to the conference room You look like you stepped in something If I had been able to identify who was talking to me, I would have asked about the current price on the British Pound Copies of the Wall Street Journal were everywhere, but most of them were unopened.

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.

Before we proceed to the implementation details, it is important to note how important careful clock planning is in FPGA design. The system clock is central

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.

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 320 and more on this topic in Section 36) 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.