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visibility |CQ| nor the boundary segment (Xj C), and so the obstacle (and the corresponding segment of the Bug2 path) will either curve inside the triangle, with |QT| left outside of it, or curve outside the triangle, with |QT| lying inside. The rest is identical to case 2. 3c. Point C is not on the obstacle boundary. Then, a curvilinear quadrangle is formed, Xj Ti CQ (Figure 3.19). Again, the obstacle will either curve inside the quadrangle, with |QT| outside of it, or curve outside the quadrangle, with |QT| lying inside. Since neither the lines of visibility |CTi | and |CQ| nor the boundary segment (Xj Ti ) can be crossed, the obstacle (and the corresponding segment of the Bug2 path) will eventually cross |Xj Q| before intersecting |QT| and form the leave point Lj . The rest of the argument is identical to case 2. Q.E.D. If the robot is currently located in the secondary semiplane, then it is indeed possible that a point that lies on the M-line and seems otherwise a good candidate for the next intermediate target Ti does not lie on the Bug2 path. This means that point should not even be considered as a candidate for Ti . Such an example is shown in Figure 3.15: While at location C, the robot will reject the seemingly attractive point Q (Step 2 of the algorithm) because it does not lie on the Bug2 path. We are now ready to establish convergence of algorithm VisBug-21.

S (a) (b)

9.2 Pipelined Architecture ShiftValSwapS0 <= iFP2[30:23]-iFP1[30:23] ((iFP1[30:23] == 0) & (iFP2[30:23] != 0));

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Theorem 3.6.1. Algorithm VisBug-21 is convergent. Proof: By the de nition of intermediate target point Ti , for any Ti that corresponds to the robot s location C, Ti is reachable from C. According to the algorithm, the robot s next step is always in the direction of Ti . This means that if the locus of points Ti is converging to T , so will the locus of points C. In turn, we know that if the locus of points Ti lies on the Bug2 path, then it indeed converges to T . The question now is whether all the points Ti generated by VisBug-21 lie on the Bug2 path. All steps of the procedure Compute Ti -21, except Step 4, explicitly test each candidate for the next Ti for being contiguous to the previous Ti and belonging to the Bug2 path. The only questionable points are points Ti on the M-line that are produced in Step 4 of the procedure: They are not required to be contiguous to the previous Ti . In such cases, points in question are chosen as Ti points only if the robot s location C lies in the main semiplane, in which case the conditions of Lemma 3.6.2 apply. This means that all the intermediate targets Ti generated by algorithm VisBug-21 path lie on the Bug2 path, and therefore converge to T. Q.E.D. Compared to a tactile sensing-based algorithm, the advantage of using vision is of course in the extra information due to the scene visibility. If the robot is thrown

Figure 3.20 Example of a walk (dashed line) in a maze under Algorithm VisBug-21 (compare with Figure 3.11). S, Start; T , Target.

RModeS0 <= iRMode; // Stage 1 {s1, e1, f1} {s2, e2, f2} RModeS1 IneAddS1 Man2ShiftedS1 Man1S1 // Stage 2 Sign1S2 Sign2S2 ExpS2 RModeS2 IneAddS2 ManZeroS2