and consequent attitude ambiguities. Ambiguity is a frequent problem when dealing with inverse trigonometric functions and must be carefully considered in mission analysis. Although from Fig. 12-6, <PS and all the rotation angles in Eq. (12-30) are in the first quadrant by. inspection, the generalization of Eq. (12-35) for arbitrary angles is not apparent. From the form of Eq. (12-35), it would appear that there is a fourfold ambiguity in however, some of these ambiguities may be resolved by applying the rules for quadrant specification given in Appendix A. There is, however, a true ambiguity in the sign of A which may be seen by redrawing Fig. 12-6 for <Psas—70 deg and noting that, in this case,
The ambiguity between Eqs. (12-30) and (12-38) is real if only pitch or roll measurements are available and must be differentiated from apparent ambiguities which may be resolved by proper use of the spherical triangle relations. However, if both Q£ and i) measurements are available, the ambiguity may be resolved by the sign of 0£ because 0£ is positive for Eq. (12-30) and negative for Eq. (12-38).
The algebraic method is based on the rotation matrix representation of the attitude. Any two vectors, n and v, define an orthogonal coordinate system with the basis vectors, q, r, and s given by q=u (12-39a)
provided that u and r are not parallel, i.e.,
At a given time, two measured vectors in the spacecraft body coordinates (denoted by the subscript B) itB and vB, determine the body matrix, MB:
For example, the measured vectors may be the Sun position from two-axis Sun sensor data, an identified star position from a star tracker, the nadir vector from an infrared horizon scanner, or the Earth's magnetic field vector from a magnetometer. These vectors may also be obtained in an appropriate reference frame (denoted by the subscript R) from an ephemeris, a star catalog, and a magnetic field model. The reference matrix, MR, is constructed from uA and \R. by
As defined in Section 12.1, the attitude matrix, or direction cosine matrix, A, is given by the coordinate transformation,
because it carries the column vectors of MR into the column vectors of MB. This equation may be solved for A to give
Because MR is orthogonal, MRl= Mj and, hence (see Appendix C),
Nothing in the development thus far has limited the choice of the reference frame or the form of the attitude matrix. The only requirement is that MR possess an inverse, which follows because the vectors q, r, and s are linearly independent provided that Eq. (12-40) holds. The simplicity of Eq. (12-45) makes it particularly attractive for onboard processing. Note that inverse trigonometric functions are not required; a unique, unambiguous attitude is obtained; and computational requirements are minimal.
The preferential treatment of the vector u over v in Eq. (12-39) suggests that u should be the more accurate measurement;* this ensures that the attitude matrix transforms u from the reference frame to the body frame exactly and v is used only to determine the phase angle about u. The four measured angles that are required to specify the two basis vectors are used to compute the attitude matrix which is parameterized by only three independent angles. Thus, some information is implicitly discarded by the algebraic method. The discarded quantity is the measured component of v parallel to u, i.e., ua vfl. This measurement is coordinate independent, equals the known scalar and is therefore useful for data validation as described in Section 9.3. All of the error in is assigned to the less accurate measurement \B, which accounts for the lost information.
Three reference coordinate systems are commonly used: celestial, ecliptic, and orbital (see Section 3.2). The celestial reference system, Mc, is particularly convenient because it is obtained directly from standard ephemeris and magnetic field model subroutines such as EPHEMX and MAGFLD in .Section 20.3. An ecliptic reference system, ME, defined by the Earth-to-Sun vector, S, and the ecliptic north pole, P£, is obtained by the transformation
where S and PE are in celestial coordinates,
and €»23.44 deg is the obliquity of the ecliptic.
An orbital reference system, M0, is defined by the nadir vector, E, and the negative orbit normal, — ft, in celestial coordinates,
•If both measurements are of comparable accuracy, basis vectors constructed from u + v and 6—i would provide the advantage of symmetry.
Any convenient representation may be used to parameterize the attitude matrix. Quaternions and various Euler angle sequences are commonly used as described in the previous section.
The construction of vector measurements from.sensor data is generally straightforward, particularly for magnetometers (Section 7.5), Sun sensors (Section 7.1), and star sensors (Section 7.6). For Earth-oriented spacecraft using horizon scanners, the nadir vector may be derived from the measured quantities by reference to the orbital coordinate system defined in Fig. 12-7. The ZQ axis is along the nadir vector and the Y0 axis is along the negative orbit normal. The scanner measures both (1) the pitch angle, Q£, about the scanner axis (the spacecraft Y axis, Ya) from the spacecraft Z axis, ZB, to the YaZ0 plane, and (2) ftE, the angle from the scanner axis to the nadir minus 90 deg.*
Solving the quadrantal spherical triangles, XB YBZQ and YBZBZQ, gives
Hence, the nadir vector in body coordinates is
EB=(sinfl£cos/3£, -sin/?£,cosfl£cos/?£)T ('2-51)
A major disadvantage of the attitude determination methods described thus far is that they are basically ad hoc. That is, the measurements are combined to provide an attitude estimate but the combination is not optimal in any statistical
•The angles Qc and PE are analogous to pitch and roll, respectively, as they are defined in Chapter 2. Because standard definitions of pitch, roll, and yaw do not exist, the sign of the quantities here may differ from that used on some spacecraft (See Section 2.2.)
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