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In the second algorithm, the robot will constantly move forward, but try to make its way back to the line after it has lost it due to a sharp curve You might think that the second strategy is better After all, it should allow the robot to reach the end of the line more quickly if we can implement it properly However, consider for a moment that the robot in question could be a car driving down a road and not just following a line on the screen The second algorithm would indeed let the car take a shorter path to the end of the road, but it does so by letting the car take short-cuts by driving off the road when the road makes a sharp turn and then getting back on the road a little further on.

(2.44)

(a) ax > 0, by de nition (Section 35.1). (b) Dx ex = ex > 0, and a function with positive derivative is increasing. More generally, the de nition of ax , together with the fact that ln y is increasing, implies that ax is increasing if a > 1. (c) We know (Problem 34.13(a)) that ln u < u. Hence, x = ln ex < ex . (d) This is a direct consequence of part (c). (e) Let u = x; then, by Property (V), 1 ex = e u = u e As x , u + , and the denominator on the right becomes arbitrarily large [by part (d)]. Hence, the fraction becomes arbitrarily small.

(2.45)

It is important to realize that neither of these strategies is necessarily better than the other Each one has advantages and disadvantages depending on the situation and the environment One of the advantages of using a simulator is that you can test and improve a variety of algorithms very quickly and test them in various environments just as easily We will develop two algorithms to implement both strategies discussed above 732 A FIRST STRATEGY Figure 76 shows a subroutine that implements the rst strategy discussed above If you run the program, you will see the robot behaving exactly as predicted For simplicity the code does not check if the end of the line has been reached Compare Figs 74 and 76, the subroutine in Fig 7.

The coordinates l and z given in Eqs. (2.40) and (2.42) are known in terms of the earth station height and latitude, and hence the range vector is known in terms of these quantities and the LST. As a point of interest, for zero height, the angle yE is related to lE by tan

CHAP. 35]

4 lost the line in a fast turn because the robot only checks once (with an if-statement) to see if it needs to turn, and only turns once (if needed) before proceeding forward This means that the robot can lose the line if the line turns faster than the robot can turn..

E(H50) 5

(2.46)

FollowLine: while true rForward 1 while rSense() & 1 rTurn 1 wend while rSense() & 4 rturn -1 wend wend Return

35.8 Sketch the graph of y = ex , and show that it is the re ection in the diagonal line y = x of the graph of y = ln x.

Here, eE is the earth s eccentricity, equal to 0.08182. The difference between the geodetic and geocentric latitudes reaches a maximum at a geocentric latitude of 45 , when the geodetic latitude is 45.192 . The magnitude of the range is 2

FollowLine: while true if rSense() & 1 rTurn 1 LastTurn = 1 //remember which direction we WERE turning endif if rSense() & 4 rTurn -1 LastTurn = -1 // remember which direction we WERE turning endif rForward 2 // since we don't care if we lose the line, // move forward twice if rSense()=0 rTurn 3*LastTurn // if we lose the line make a BIG endif // turn back towards it wend Return

(2.47)