## Threeaxis Attitude Determination Methods

12.1 Parameterization of the Attitude

12.2 Three-Axis Attitude Determination Geometric Method, Algebraic Method, q Method

### 12 J Covariance Analysis

Chapter 11 described deterministic procedures for computing the orientation of a single spacecraft axis and estimating the accuracy of this computation. The methods described there may be used either to determine single-axis attitude or the orientation of any single axis on a three-axis stabilized spacecraft. However, when the three-axis attitude of a spacecraft is being computed, some additional formalism is appropriate. The attitude of a single axis can be parameterized either as a three-component unit vector or as a point on the unit celestial sphere, but three-axis attitude is most conveniently thought of as a coordinate transformation which transforms a set of reference axes in inertial space to a set in the spacecraft. The alternative parameterizations for this transformation are described in Section 12.1. Section 12.2 then describes three-axis attitude determination methods, and Section 12.3 introduces the covariance analysis needed to estimate the uncertainty in three-axis attitude.

12.1 Parameterization of the Attitude

### F.L. Markley

Let us consider a rigid body in space, either a rigid spacecraft or a single rigid component of a spacecraft with multiple components moving relative to each other. We assume that there exists an orthogonal, right-handed triad u, v, w of unit vectors fixed in the body, such that uXv=w (12-1)

The basic problem is to specify the orientation of this triad, and hence of the rigid body, relative to some reference coordinate frame, as illustrated in Fig. 12-1.

It is clear that specifying the components of ft, v, and w along the three axes of the coordinate frame will fix the orientation completely. This requires nine parameters, which can be regarded as the elements of a 3x3 matrix, A, called the attitude matrix:

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