10.4 Correlation Angles
We have seen that the attitude uncertainty depends on the measurement uncertainties, the measurement densities, and the correlation angle (or angle of intersection of the attitude loci). In this section, we give a formal definition of the correlation angle, expressions for the correlation angles among the arc-length and rotation angle measurements described above, and an example of the application of correlation angles and measurement densities to determine the accuracy of the Sun position in a two-axis Sun sensor.
To specify the-angle of intersection between two loci, several choices are available. Geometrically, it is convenient to define the correlation angle as the acute angle between the tangents to the loci as was done in Section 10.1. However, for computer work or algebraic manipulation, this involves continuous tests on the range of an angle and adjustments when it falls outside the range 0 to 90 deg. Thus, for algebraic use, it is more convenient to define a unique correlation angle covering the range 0 to 360 deg. Given two arbitrary loci, L, and Ljt we formally define the correlation angle between them, @t/j, as the rotation angle at the intersection of the loci from the positive gradient of L, counterclockwise (as viewed from infinity toward the spacecraft) to the positive gradient of Lj, as illustrated in Fig. 10-11. This is equivalent in its effect on attitude uncertainties to defining ©(/,y as the acute angle between the tangents to L, and Lj. Note that from the formal definition, we have
As an example of the correlation angle for two arc length measurements, consider the Sun angle/nadir angle correlation angle, ©«/,,, shown in Fig. 10-11.
equals the angle between the radii of the two small circles at their intersection; however, this is just the Sun-Earth rotation angle, i>, defined in Fig. 10-7. Thus,
When the correlation angle is 0 deg or 180 deg, the two small circles are tangent and the two measurements give essentially the same information about the attitude. Thus, when the correlation angle is small (or near 180 deg), the attitude uncertainty is largest, because the component of the attitude tangent to the two circles is essentially unknown. In contrast, when the correlation angle is near 90 deg or 270 deg, the two measurements are independent and the attitude uncertainty is smallest.
As an example of the correlation angle for arc length/rotation angle intersections, we consider the Sun angle and Sum Earth rotation angle measurements. Figures 10-8 through 10-10 are convenient for studying the general character of these loci intersections. The latitude lines on the underlying coordinate grid in these figures are curves along which the Sun angle is constant, because a fixed Sun angle, j8, implies that the attitude lies on a small circle centered on the Sun. The 0 = 60 deg locus is marked on Fig. 10-8. Thus, in Figs. 10-8 through 10-10, ©i/4> is simply the angle between the constant 4> curves and the constant P curves as indicated in Fig. 10-8 between the /? = 60 deg and $ = 30 deg loci. The value of the angle at any point on the celestial sphere is derived in Section 11.3 as
Qp/n, = 0 implies that the constant /? and constant 4> curves are tangent. As is most easily seen in Figs. 10-8 and 10-9 (c), this occurs when the attitude lies on the great circle containing the Earth and the null* (shown as a dashed line on the figures). Consequently, along this great circle, no information is available on the component of the attitude tangent to the constant p and 4> curves.
Because the Sun angle and the nadir angle are the same type of measurement, a similar relationship must hold for the nadir angle/rotation angle correlation angle, ©V4>:
Also by symmetry with /?/$, 0,/<d = O when the attitude lies on the Sun-null great circle.
The set of all possible correlation angles relating any set of attitude measurements satisfies an addition theorem. For example, if G^, Gv and are the directions of the gradients of the constant jS, tj, and $ curves, respectively, then we can see from Fig. 10-12 that
where n= 1 jf the vectors are in the order G^, G,,, G^ and n = 2 if they are in the order G^, G0, G,,. Equation (10-9) is particularly useful for the approximate evaluation of correlation angles, because frequently one or two of them are easy to estimate.
*This is true everywhere along the Earth-null great circle except at the Earth, Zenith, null, and antinull, where the tangent to the constant 4> curve is undefined.
Correlation angles will be used extensively in Chapter II to determine the attitude accuracy from various measurement types. To illustrate the versatility of the correlation angle and measurement density concepts, we analyze here the internal accuracy of the solid angle Sun sensors described in Section 6.1. Specifically, given a sensor with a circular field of view 128 deg in diameter, and a uniform reticle pattern with an 0.5-deg step size (or least significant bit) on both axes at the boresight, we wish to determine the maximum inaccuracy in the measured position of the Sun, assuming that there is no error in the sensor measurements. Our procedure will be first to determine the type of measurement made by the Sun sensor and then to determine the measurement densities and the correlation angle between the two sensor measurements.
Figure 10-13 shows the locus of Sun positions corresponding to given output angles T and A. (Compare Fig. 10-13 with Fig. 7-9.) Clearly, the loci of Sun positions corresponding to a given T or A output signal are great circles at a constant rotation angle about the X and Y axes, respectively, from the sensor boresight which defines the center of the field of view of the sensor.* (These rotation angle loci are different from those of Figs. 10-8 through 10-10 because the rotation angle is being measured about the sensor axis rather than about the position of the Sun.)
To determine the measurement density on the celestial sphere, we note that the separation between the sensor input slit and the reticle pattern on the back of the sensor is a constant. Therefore, equal steps along the reticle pattern correspond to equal steps in the tangent of the angle from the boresight to the Sun along the two axes, i.e., tanT and tan A. Therefore, the density of the step boundaries on the celestial sphere is the derivative of the tangent of the measurement angles. (Compare with Eq. (10-4).) Thus, dT = l/cos2r, ds = 1 /cos2A, and the measurement step
* The locus of Sun positions for constant T or A is a great circle only if the index of refraction, n, of the material inside the sensor is 1. If n ^ 1, then the loci will deviate slightly from great circles and A and T will not be independent In this example, we will assume n= I, as is commonly true for high resolution sensors.
size on the celestial sphere is 0.5"/d. At the boresight the measurement density is 1 and the Sun angle is being measured in 0.5-deg steps; along an axis at the edge of the sensor (for example, at T = 0, A = 64 deg) the density is 1 /cos264° = 5.20, and the steps are 0.5°/5.20 = 0.096 deg. Thus, ignoring problems of diffraction, reduced intensity, and manufacturing imperfections, all of which tend to be worse at the edge, the resolution at the edge of the sensor is approximately five time better than at the center.
To evaluate the uncertainty in the position of the Sun, we still need to determine the correlation angle, 0A/r, between the two loci. This can be obtained by inspecting Fig. 10-13. Specifically, 0A/r equals the rotation angle about the Sun from the sensor + Y axis to the - X axis. The angular separation between the X and y axes is 90 deg; therefore, the correlation angle at any point in the sensor field of view may be evaluated using Fig. 10-10 with the Sun, Earth, and null replaced by the sensor + Y axis, the — X axis, and the boresight, respectively, as shown in Fig. 10-14 with the center of the view shifted to the boresight axis.
Figure 10-14 shows that 0A/r is near 90 deg in the vicinity of the boresight and along the X and Y axes. For ©A/r=90 deg, the uncertainty in the position of the Sun, Us, is smallest and is equal to half the length of the diagonal of a rectangle whose sides are the angular step size. At the boresight, t/s = 0.5x0.5°xV5" =0.354 deg. At the sensor boundary along one of the axes. i/s=0.5x(0.52 + 0.0962)2 = 0.255 deg. Along a line midway between the X and Y axes, steps in A and T are of equal size, but the measurement loci do not intersect at right angles. Along this line we may use Napier's Rules (Appendix A) to obtain tan A = tan T=cos 45° tan/} (JO-lOa)
where /} is the angle from the boresight to the Sun. At the sensor boundary, /? = 64° and. therefore, 0A/r=47.34°, A = r=55.40°, dK = dT=3.10, and the step size is
COMPOUND MEASUREMENTS—SUN-TO-EARTH HORIZON CROSSING ROTATION ANGLE
SENSOR «V MUS
COMPOUND MEASUREMENTS—SUN-TO-EARTH HORIZON CROSSING ROTATION ANGLE
SENSOR «V MUS
0.5°/3.10 = 0.161°. This may be further evaluated using the upper form of Eq. (11-14) to give l/s = 0.201 deg. (If the loci at the boundary midway between the two axes formed a rectangle on the celestial sphere rather than a parallelogram, the attitude uncertainty there would be 0.114 deg.) Thus, although the single-axis resolution varies by a factor of 5 over the full range, the attitude uncertainty fluctuates by only about 50%. A similar analysis for a Sun sensor with a 32-deg "square" field of view (Fig. 10-14) and other properties as above gives Us = 0.354 deg at the boresight, 0.308 deg at the center of each edge, and 0.300 deg at the corners.
10.5 Compound Measurements—Sun-to-Earth Horizon Crossing Rotation Angle
Sections 10.1 through 10.4 described measurements involving one or two reference vectors whose orientation in inertial space is known. The technique that we have used is to examine the locus of possible attitudes for any given measurement to classify that measurement. However, some common attitude measurements do not fall into the basic categories that we have established thus far. One example of such compound measurements is the rotation angle about the attitude from the Sun to the Earth's horizon, The horizon sensor which produces this measurement is assumed to have a field of view which is a point on the celestial sphere, which thus sweeps out a small circle as the spacecraft rotates and provides an output pulse upon crossing the Earth's horizon.
The Sun-to-Earth horizon crossing rotation angle differs from other rotation angle measurements in that the location of the horizon crossing on the celestial sphere is unknown. We know only that the horizon crossing is a given arc-length distance from the nadir vector. Thus, is neither a rotation angle measurement nor an arc-length measurement and the attitude loci corresponding to constant values of do not have the same form as the loci corresponding to /? or <t> measurements.
Let ip, y, and p be the Sun-nadir separation, the sensor mounting angle (relative to the attitude), and the angular radius of the Earth, respectively. Figures 10-15 and 10-16 show the shape of several constant 4>w curves for ip>y + p and \p< y + p, respectively. The solid curves are the attitude loci for a constant Sun to Earth-in horizon crossing angle and the dashed curves are the attitude loci for a constant Sun to Earth-out horizon crossing angle. Earth-in, or in-triggering, denotes
i s a sensor crossing from space onto the disk of the Earth and Earth-out, or out-triggering, denotes a crossing from the disk of the Earth to space.
An examination of the figures shows several characteristics of the loci. The loci are entirely contained between small circles of radii y + p and y—p, because these are the only conditions under which the sensor will cross the Earth. In the limit p->0, the set of all <bH loci lie on a small, circle of radius y about the Earth; that is, ij = y. The constant loci are neither small circles nor constant rotation angle curves, but a third, distinct measurement type. Note that in Fig. 10-16 the 45-deg locus consists of two, discrete, nonintersecting, closed curves. In this case, it is possible to have four discrete, ambiguous attitude solutions when combining <PH with the nadir angle measurement. Unfortunately, the formulae for the various correlation angles involving and other attitude measurements are inconvenient to use. However, the location of several of the correlation angle singularities can be identified from the figures. The <PH measurement may be combined with either the Sun angle or the nadir angle measurement to determine the attitude. Attitude singularities will occur whenever the 4»w loci are tangent to small circles centered on the Sun or the Earth (L^), respectively. Reference to Figs. 10-15 and 10-16 shows that for the horizon angle/nadir angle method, the constant loci are tangent to small circles centered on the Earth at the transitions from solid to dashed lines. Thus, an i}/<I>w singularity occurs when there is a transition from Earth-in crossing to Earth-out crossing. Equivalently, 0l/4>j| = O whenever the sensor field-of-view small circle is tangent to the Earth. Similarly, =0
whenever the constant 4»w loci are tangent to small circles centered on the Sun. Again, these Lp curves are the latitude lines of the underlying grid. Representative points where 0|s/4>JI = O have been marked by the letter,! on Figs. 10-15 and 10-16. This occurs when the Sun vector, the nadir vector, and the horizon crossing vector are coplanar.
Finally, we may determine the attitude by using two horizon crossing measurements, an Earth-in crossing, and an Earth-out crossing. (For example, an Earth-in rotation of 45 deg and an Earth-out rotation angle of 75 deg implies that the attitude must be at one of the two points marked B on Fig. 10-16.) For a spherical Earth, this gives us the same information as an Earth-width measurement plus a Sun-to-nadir vector rotation angle measurement. As discussed in Section 10.3, a singularity occurs in the Earth-width/rotation angle method whenever the attitude lies on the Sun-null great circle shown as a dotted line in Fig. 10-16. At any point along this line, the <PH loci passing through that point are mutually tangent. Although this cannot be clearly established from the figure, it is at least consistent with the shape of the attitude loci along the Sun-null great circle.
Thus far we have described procedures for using two independent measurements to determine the orientation of a single spacecraft axis. For single-axis attitude, this is all of the information that is desired. However, to completely determine the orientation of a rigid spacecraft, three parameters must be determined and, therefore, an additional measurement is required. For three-axis-stabilized spacecraft, these three parameters are frequently chosen to be three angles, known as Euler angles, which define how the spacecraft-fixed coordinates are related to inertia! coordinates. This procedure is described in Section 12.1.
An alternative procedure frequently used for spinning spacecraft is to define the orientation in space of a single spacecraft axis (such as the spin axis) and then to define the rotational orientation of the spacecraft about this axis. This rotation angle, also called the azimuth or phase angle, may be specified as the azimuth of some arbitrary point in the spacecraft relative to some reference direction in inertial space, as illustrated in Fig. 10-17. In this figure, the underlying coordinate grid is fixed in inertial space.
To determine the three-axis attitude specified by the spin axis direction and azimuth, we first determine the orientation in inertial space of the spacecraft spin axis using any of the methods described in Sections 10.1 through 10.5. The one remaining attitude component is then measured by measuring the rotation angle about the attitude between some fixed direction in inertial space and an arbitrarily defined reference direction fixed in the spacecraft. For example, we might record the time at which a slit Sun sensor parallel to the spin axis sees, the Sun and assume that the spacecraft is rotating uniformly to determine its relative azimuth at any other time. Alternatively, if we are using a wheel-mounted horizon scanner (Section 6.2), we could measure the relative azimuth between the center of the disk of the Earth (midway between the telescope Earth-in and -out crossings for a spherical Earth) and some reference mark fixed in the body of the spacecraft, as has been done for the AE series of spacecraft. This pitch angle may be used directly for Earth-oriented satellites or, with ephemeris data, transformed into an inertial azimuthal measurement The reference point for the inertial azimuth is arbitrary. However, the perpendicular projection of the vernal equinox onto the spin plane is commonly used.
Another alternative procedure, frequently used on three-axis stabilized spacecraft, is to determine the attitude by measuring the orientation in spacecraft coordinates of two reference vectors fixed in inertial space. For example, three orthogonal magnetometers may be used to measure the orientation of the Earth's magnetic field in spacecraft coordinates. Similarly, a two-axis Sun sensor can provide the coordinates of the Sun vector in spacecraft coordinates. The specification of these two vectors in spacecraft coordinates fixes the orientation of the spacecraft in inertial space.
When using two reference vectors, the attitude problem is overdetermined because we have measured four parameters (two orientation parameters for each reference vector) but have only three independent variables. This is clearly shown by the Sun sensor/magnetometer example. The Sun sensor output defines a spacecraft axis which is pointing toward the Sun. It remains only to determine an azimuth about this axis. However, specifying the direction of the magnetic field vector in spacecraft coordinates determines both the azimuth of the spacecraft axis parallel to the magnetic field and also the angular separation of the magnetic field vector and the Sun vector. (See Fig. 10-18.) The latter quantity is not an independent parameter because it is fixed by knowing the direction in inertia) space of both the Sun vector and the magnetic field vector. (See Sections 9.3 and 12.2 for a discussion of using this fourth parameter as a test for invalid data.)
If we are determining three-axis attitude by determining the orientation in spacecraft coordinates of two reference vectors, then all the analysis of Sections 10.1 through 10.5 can be applied directly to determining the orientation of each reference vector. For example, the theory of correlation angles was applied to the output of a two-axis Sun sensor at the end of Section 10.4. Similarly, a single magnetometer measurement in a known magnetic field is an arc-length measurement specifying the angle between the external magnetic field and the magnetometer axis.* The same analytic procedures can be applied to other types of sensors as well.
'Ordinarily, magnetometer measurements are obtained from three mutually perpendicular magnetometers. The sum of the squares of the readings determines the overall field strength. Any two of the measurements may then be taken as the remaining independent numbers. These are two arc-length measurements which together determine the orientation of the magnetic field in spacecraft coordinates to within a discrete ambiguity which may be resolved by the sign of the third measurement z z
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