Singleaxis Attitude Determination Methods

11.1 Methods (or Spinning Spacecraft

General Requirements, Specific Solution Methods, Nonin-tersecting Loci

112 Solution Averaging

11J Single-Axis Attitude Determination Accuracy

Attitude Accuracy for Measurements With Unconvicted Uncertainties, Attitude Accuracy for Measurements With Correlated Uncertainties, Measurement Densities and Correlation Angles

11.4 Geometrical Limitations on Single-Axis Attitude Accuracy

Limitations on the Attitude Direction Due to Attitude Accuracy Requirements, Limitations on Reference Vector Direction, Applications

11.5 Attitude Uncertainty Due to Systematic Errors Behavior of Single-Frame Solutions, Identification of Singularities, State Vector Formulation

This chapter describes standard procedures for determining the orientation in space of any single spacecraft axis. For illustration we will normally assume that this is the spin axis of a spin-stabilized spacecraft. However, this axis could equally well be that of an attitude sensor, such as the rotation axis of a scanning horizon sensor, or any axis in a three-axis stabilized spacecraft.

The methods presented here are all deterministic in that they use the same number of observations as variables (normally the two parameters required to specify the orientation of a single axis). The models presented have all been used for the operational support of a variety of spacecraft. The directions to the Sun and to the center of the Earth or to a point on the Earth's horizon are used as reference directions for illustration; however, the techniques presented may equally well be applied to any known reference vectors. All of the models given involve different observations made at the same time. However, if the attitude is assumed constant or if a dynamic model for attitude motion is available, these methods may be applied to observations made at different times.

Section 11.1 describes the basic, deterministic single-axis methods and the problem of nonintersecting loci. Section 11.2 describes the resolution of solution ambiguities, data weighting, and solution averaging. Sections 11.3 and 11.4 then provide analytic expressions for single-axis uncertainties, limitations on solution accuracy due to the relative geometry of reference vectors, and application of this information to mission analysis. Finally, Section 11.5 describes the behavior of single-axis solutions in the presence of systematic biases, identifies the specific singularity conditions for each of the models in Section 11.1, and introduces the need for state estimation procedures to resolve the biases characteristic of real spacecraft data.

11.1 Methods for Spinning Spacecraft

Peter M. Smith

Determining the attitude of a spin-stabilized spacecraft in the absence of nutation is equivalent to fixing the orientation of the unit spin vector axis with respect to some inertial coordinate system. The most common system used is that of the celestial coordinates right ascension and declination, described in Section 2.2. In general, ambiguous solutions for the attitude are obtained, due to multiple intersections of the attitude loci, and must be resolved either by comparison with an a priori attitude or by using the method of block averaging described in Section 11.2

11.1.1 General Requirements

For a deterministic, two-component attitude solution, we require two reference vectors with their origin at the spacecraft and either (a) an arc-length measurement from the spin vector to each reference vector, or (b) one arc-length measurement and a rotation angle measurement about the spin axis between the reference vectors.

As shown in Fig. 11-1, each arc-length measurement for case (a) defines a cone* about each reference vector; the intersections of these cones are possible attitude solutions. For concreteness, let us assume that the two known reference

Fig. 11-1. Single-Axis Attitude Solution Using Two Arc-Length Measurements, Case (a)

• Recall .from Chapter 2 that we may define single-axis attitude either by three components of a unit vector, A, or by the coordinates (a,S) of the point at which that vector intersects the unit celestial sphere. In the former case, we think of an arc-length measurement as determining a cone about the reference vector; in the latter case, as determining a small circle (the intersection of the cone with the celestial sphere) on the celestial sphere about the reference point Because the two representations are equivalent, we will use them interchangeably as convenient. See Section 2.2.1 for a discussion of the relative merits of the spherical and rectangular coordinate systems.

Fig. 11-1. Single-Axis Attitude Solution Using Two Arc-Length Measurements, Case (a)

• Recall .from Chapter 2 that we may define single-axis attitude either by three components of a unit vector, A, or by the coordinates (a,S) of the point at which that vector intersects the unit celestial sphere. In the former case, we think of an arc-length measurement as determining a cone about the reference vector; in the latter case, as determining a small circle (the intersection of the cone with the celestial sphere) on the celestial sphere about the reference point Because the two representations are equivalent, we will use them interchangeably as convenient. See Section 2.2.1 for a discussion of the relative merits of the spherical and rectangular coordinate systems.

vectors are the Sun and nadir vectors, S and E. The cone about S has a half angle, j8, equal to the angular separation of this vector and the unknown attitude vector, A; similarly, the cone about E has a half angle, 17, equal to the angular separation between E and A. The possible solutions for the attitude are A, and A2.

Analytically, this geometrical problem is specified by three simultaneous equations in three unknowns, A,-, AJt Ak :

These three equations may be solved using the following technique due to Grubin [1977]. Let cos/3-EScostj jr=-7-t-2— (ll-3a)

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