## Geometrical Basis Of Attitude Determination

Lily C. Chen James R. Weriz

10.1 Single-Axis Attitude

10.2 Arc-Length Measurements

10.3 Rotation Angle Measurements

10.4 Correlation Angles

10.5 Compound Measurements—Sun to Earth Horizon Crossing Rotation Angle

### 10.6 Three-Axis Attitude

In Part II we described both the hardware and the process by which attitude data are gathered, transmitted to the attitude software system, and assembled in a manner appropriate for attitude determination. In Part HI we describe the procedures by whichthese data are processed to determine the spacecraft attitude. Chapter 10 introduces the types of attitude measurements and the geometrical meaning of these measurements. Chapters 11 and 12 describe methods for combining as many measurements as there are observables (usually two or three) to produce a single, possibly multivalued determination of the attitude. Chapters 13 and 14 then describe filtering methods to provide optimum estimates of the attitude, given many data points.

As discussed in Chapter 1, there are two types of attitude. Single-axis attitude is the specification of the orientation of a single spacecraft axis in inertial space. Ordinarily, this single axis is the spin axis of a spin-stabilized spacecraft. However, it could be any axis in either a spinning or a three-axis stabilized spacecraft. Single-axis attitude requires two independent numbers for its specification, such as the right ascension and declination of the spin axis. The attitude of a single axis may be expressed either as a unit vector in inertia! space or as a geometrical point on the unit celestial sphere centered on the spacecraft. (See Section 2.1.) Generally, we will use the vector representation of the attitude for numerical or computer calculations, and the geometrical representation for analytical work and physical arguments. However, because of the direct correspondence between the two representations, we will often move back and forth between them as convenient for the particular problem.

If the orientation of a single axis is specified, the complete spacecraft orientation is not fixed because the rotation of the spacecraft about the specified axis is still undetermined. A third independent attitude component, such as the azimuth about the spin axis of a point on the spacecraft relative to some object in inertial space, completely fixes the inertial orientation of a rigid spacecraft. Such a three-component attitude is commonly called three-axis attitude because it fixes the orientation of the three orthogonal spacecraft axes in inertial space.

Throughout Part III we will frequently ignore the distinction between single-and three-axis attitude. If we refer to the attitude as the orientation of a single axis, this may be taken as either single-axis attitude or one axis of a three-axis system. Specifically, in Sections 10.1 through 10.5 we will discuss the types of single-axis

measurements. Measurements concerned specifically with determining the third component in three-axis systems will then be discussed in Section 10.6.

Specifying the orientation of a single axis in space requires two independent attitude measurements. Therefore, if only one of these measurements is known, an infinite set of possible single-axis attitude orientations exists which maps out a curve, or locus, on the celestial sphere. This is illustrated in Fig. 10-1 for the Sun angle measurement, /?, which is the arc-length separation between the attitude and the Sun. Any two attitude measurements are equivalent if and only if they correspond to the same locus of possible altitudes on the celestial sphere.

Fig. 10-1. Locus of Attitudes Corresponding to Measured Sun Angle, ¡i (arbitrary inertial coordinates)

Given both independent attitude measurements, each having a distinct locus of possible values, the attitude must lie at their intersection. In general, there may be multiple intersections resulting in ambiguous attitude solutions. Because no measurement is exact, the possible attitudes corresponding to any real measurement lie in a band on the celestial sphere about the corresponding locus with the width of the band determined by the uncertainty in the measurement, as illustrated in Fig. 10-2. The Sun angle measurement, /?, with uncertainty Up, implies that the attitude must lie somewhere in a band centered on Lp of width AL^. Similarly, the nadir angle measurement, i) (i.e., the arc-length separation between the attitude and the center of the Earth), with uncertainty Uv, implies that the attitude lies in the band defined by L, and AL,. Clearly, the attitude must lie in one of the two parallelograms formed by the intersection of the two bands. We will assume that the region of intersection is sufficiently small that we may use plane geometry to describe these parallelograms. The correct parallelogram may be chosen and the attitude ambiguity resolved either from an a priori estimate of the attitude, or, if the attitude is constant in time, by processing many measurements from different times and selecting that solution which remains approximately constant. (See Section 11.2.)

10.1 Single-Axis Attitude

Fig. 10-1. Locus of Attitudes Corresponding to Measured Sun Angle, ¡i (arbitrary inertial coordinates)

Figure 10-3 shows an expanded view of the parallelogram of intersection formed by th^ two bands on the celestial sphere. The center of the parallelogram, where the two measurement loci intersect, is the measured value or estimate of the attitude. The size of the parallelogram is the uncertainty in the attitude result. For any measurement, m (either ft or ij in Figs. 10-2 and 10-3), the width, ALm, of the attitude uncertainty band on the celestial sphere is determined by the measurement uncertainty, Um, and the measurement density, dm, which is the change in measurement per unit arc-length change between adjacent loci, measured perpendicular to the loci. Thus, dm=UJALm

To obtain a more formal definition, let m, and m2 be two values of the measurement m (e.g., /), and fij), and let be the arc-length separation between Lmt and Lmi measured

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