## A

0.23 deg

0.23 deg

Fig. 5-19. Apparent Dally Motion of a Satellite In Geosynchronous Orbit.

In this coordinate frame, we can handle the motion relative to the fixed inertial background just the same as we do the apparent motion of the Moon or planets. Many introductory texts on celestial mechanics treat this issue. See, for example, Roy [1991], Green [1985], or Smart [1977], or Wertz [2001].

### 5.4 Development of Mapping and Pointing Budgets

Nearly all spacecraft missions involve sensing or interaction with the world around them, so a spacecraft needs to know or control its orientation. We may conveniently divide this problem of orientation into two .areas of pointing and mapping. Pointing means orienting the spacecraft, camera, sensor, or antenna to a target having a specific geographic position or inertial direction. Mapping is determining the geographic position of die look point of a camera, sensor, or antenna. Satellites used only for communications will generally require only pointing. Satellites having some type of viewing instrument, such as weather, ground surveillance, or Earth resources satellites, will ordinarily require both pointing ("point the instrument at New Yoik") and mapping ("determine the geographic location of the tall building in pixel 2073").

The goal of this section is to develop budgets for pointing and mapping. A budget lists all the sources of pointing and mapping errors and how much they contribute to the overall pointing and mapping accuracy. This accuracy budget frequently drives both the cost and performance of a space mission. If components in the budget are left out or incorrectly assessed, the satellite may not be able to meet its performance objectives. More commonly, people who define the system requirements make the budgets for pointing and mapping too stringent and, therefore, unnecessarily drive up the cost of the mission. As a result, we must understand from the start the components of mapping and pointing budgets and how they affect overall accuracy. In this section we will emphasize Earth-oriented missions, but the same basic rules apply to inertially-oriented missions.

The components of the pointing and mapping budgets are shown in Fig. 5-20 and defined in Table 5-5. Basic pointing and mapping errors are associated with spacecraft navigationâ€”that is, knowledge of its position and attitude in space. But even if the position and attitude are known precisely, a number of other errors will be present For example, an error in the observation time will result in an error in the computed location of the target, because the target frame of reference moves relative to the spacecraft A target fixed on the Earth's equator will rotate with the Earth at 464 m/s. A 10-sec error in the observation time would produce an error of 5 km in the computed geographic location of the target. Errors in the target altitude, discussed below, can be a key component of pointing and mapping budgets. The instrument-mounting error represents the misalignment between the pointed antenna or instrument and the sensor or sensors used to determine the attitude. This error is extremely difficult to remove. Because we cannot determine it from the attitude data alone, we must view it as a critical parameter and keep it small while integrating the spacecraft

Pointing errors differ from mapping errors in the way they include inaccuracies in attitude control and angular motion. Specifically, the pointing error must include the entire control error for the spacecraft. On the other hand, the only control-related component of the mapping error is the angular motion during the exposure or observation time. This short-term jitter results in a blurring of the look point of the instrument or antenna.

As discussed earlier in Sec. 42.2, we may achieve an accuracy goal for either pointing or mapping in many ways. We may, in one instance, know the position of the spacecraft precisely and the attitude only poorly. Or we may choose to allow a larger error in position and make the requirements for determining the attitude more stringent In an ideal world we would look at all components of the pointing and mapping budgets and adjust them until a small increment of accuracy costs the same for each component For example, assume that a given mission requires a pointing accuracy of 20 milliradians, and that we tentatively assign 10 milliradians to attitude determination and 5 milliradians to position determination. We also find more accurate attitude would cost \$ 100,000 per milliradian, whereas more accurate position would cost only \$50,000 per milliradian. In this case we should allow the attitude accuracy to degrade and improve the position-accuracy requirement until the cost per milliradian is the same for both. We will then have the lowest cost solution.

TABLE 5-5. Sources of Pointing and Mapping Errors.

SPACECRAFT POSITION ERRORS:

AI In- or along-track Displacement along the spacecraft's velocity vector AC Cross-track Displacement normal to the spacecraft's orbit plane

A/j Elevation Error in angle from nadir to sensing axis

Sensing axis orientation errors include errors in (1) attitude determination, (2) instrument mounting, and (3) stability for mapping or control for pointing.

OTHER ERRORS:

ARt Target altitude Uncertainty in the altitude of the observed object

A T Clock error Uncertainty in the real observation time (results in uncertainty in the rotational position of the Earth)

In practice we can seldom follow the above process. For example, we cannot improve accuracy continuously. Rather, we must often accept large steps in both performance and cost as we change methods or techniques. Similarly, we seldom know precisely how much money or what level of performance to budget In practice the mission designer strives to balance the components, often by relying on experience and intuition as much as analysis. But the overall goal remains correct We should try to balance the error budget so that incrementally improving any of the components results in approximately comparable cost.

A practical method of creating an error budget is as follows. We begin by writing down all of the components of the pointing and mapping budgets from Table 5-5. We assume that these components are unrelated to each other, being prepared to combine them later by taking the root sum square of the individual elements. (We will have to examine this assumption in light of the eventual mission design and adjust it to take into account how the error components truly combine.) The next step is to spread the budget equally among all components. Thus, if all seven error sources listed in Table 5-5 are relevant to the problem, we will initially assign an accuracy requirement for each equal to the total accuracy divided by Jl. This provides a starting point for allocating errors. Our next step is to look at normal spacecraft operations and divide the error sources into three categories:

(A) Those allowing very little adjustment

(B) Those easily meeting the error allocation established for them, and

(C) Those allowing increased accuracy at increased cost

Determining the spacecraft position using ground radar is a normal operation, and the ground station provides a fixed level of accuracy. We cannot adjust this error source without much higher cost, so we assign it to category (A) and accept its corresponding level of accuracy. A typical example of category (B) is the observation time for which an accuracy of tens of milliseconds is reasonable with modem spacecraft clocks. Therefore, we will assign an appropriately small number (say 10 ms) to the accuracy associated with the timing error. Attitude determination ordinarily falls into

TABLE 5-6. Mapping and Pointing Error Formulas, s Is the grazing angle and lat is the latitude of the targets the target azimuth relative to the ground track, A Is the Earth central angle from the target to the satellite, 0 is the distance from the satellite to the target, Rj is the distance from the Earth's center to the target (typically - RB the Earth's radius), and Rs Is the distance from the Earth's center to the satellite. See Fig. 5-20.

TABLE 5-6. Mapping and Pointing Error Formulas, s Is the grazing angle and lat is the latitude of the targets the target azimuth relative to the ground track, A Is the Earth central angle from the target to the satellite, 0 is the distance from the satellite to the target, Rj is the distance from the Earth's center to the target (typically - RB the Earth's radius), and Rs Is the distance from the Earth's center to the satellite. See Fig. 5-20.

 Error Source Error Magnitude (units) Magnitude of Mapping Error (km) Magnitude of Pointing Error (fad) Direction of Error Attitude Errors
0 0