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This value is often rounded up to 36,000 km for approximate calculations. In practice, a precise geostationary orbit cannot be attained because of disturbance forces in space and the effects of the earth s equatorial bulge. The gravitational fields of the sun and the moon produce a shift of about 0.85 /year in inclination. Also, the earth s equatorial ellipticity causes the satellite to drift eastward along the orbit. In practice, stationkeeping maneuvers have to be performed periodically to correct for these shifts, as described in Sec. 7.4. An important point to grasp is that there is only one geostationary orbit because there is only one value of a that satisfies Eq. (2.3) for a periodic time of 23 h, 56 min, 4 s. Communications authorities throughout the world regard the geostationary orbit as a natural resource, and its use is carefully regulated through national and international agreements.

FIGURE 8.5 This program turns away if the wall is seen by either the infrared sensor or the bumper sensor.

3.2 Antenna Look Angles The look angles for the ground station antenna are the azimuth and elevation angles required at the antenna so that it points directly at the satellite. In Sec. 2.9.8 the look angles were determined in the general case of an elliptical orbit, and there the angles had to change in order to track the satellite. With the geostationary orbit, the situation is much simpler because the satellite is stationary with respect to the earth. Although in general no tracking should be necessary, with the large earth stations used for commercial communications, the antenna beamwidth is very narrow (see Chap. 6), and a tracking mechanism is required

By example (c) in Section 36.1, lim x ln x = 0. Hence,

the main program in Fig. 8.1. This variable acts as a switch. If it is 1 the robot will follow the wall to the left and if it is 1 the robot will follow the wall to the right. We will also have to add some code to allow for this switch. Figure 8.5 shows the algorithm that achieves all this. Notice that we now check to see if the front bumper is closed as well as checking for the infrared sensors. Also notice how we set the variable FN to be used in the rFeel() & FN statement. Remember if you are following the wall to the right then the right 45 and front infrared sensors need to be considered (i.e., 00110 6), and if you follow the wall to the left then the left 45 and front sensors are to be tested (i.e., 01100 12). Run the program and try changing TurnDir to 1 and see how the robot now follows the wall to the right instead of to the left. The program in Fig. 8.5 performs reasonably well. As mentioned earlier though, the robot still arcs far away from the wall when it rounds a sharp corner. The algorithm can still be improved further.

to compensate for the movement of the satellite about the nominal geostationary position. With the types of antennas used for home reception, the antenna beamwidth is quite broad, and no tracking is necessary. This allows the antenna to be fixed in position, as evidenced by the small antennas used for reception of satellite TV that can be seen fixed to the sides of homes. The three pieces of information that are needed to determine the look angles for the geostationary orbit are 1. The earth-station latitude, denoted here by lE 2. The earth-station longitude, denoted here by fE 3. The longitude of the subsatellite point, denoted here by fSS (often this is just referred to as the satellite longitude) As in Chap. 2, latitudes north will be taken as positive angles, and latitudes south, as negative angles. Longitudes east of the Greenwich meridian will be taken as positive angles, and longitudes west, as negative angles. For example, if a latitude of 40 S is specified, this will be taken as 40 , and if a longitude of 35 W is specified, this will be taken as 35 . In Chap. 2, when calculating the look angles for low-earth-orbit (LEO) satellites, it was necessary to take into account the variation in earth s radius. With the geostationary orbit, this variation has negligible effect on the look angles, and the average radius of the earth will be used. Denoting this by R: R 6371 km (3.5)