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Abstract

U SED as a four-parameter representation of a rotation, a quaternion affords considerable computational advantages compared to a direction-cosine matrix, its nineparameter equivalent. To realize the quaternion's advantages, it is sometimes necessary to extract the quaternion from an existing matrix. As an example, for space shuttle steering the commanded attitude is readily computed as a direction-cosine matrix, whereas actual attitude is most compactly computed as a quaternion. Attitude error is computed as the quaternion product once the commanded-attitude quaternion is extracted from the corresponding matrix and conjugated. The virtues of an error quaternion are that the vector part coincides with the error axis and the length of the vector part is the sine of half the error angle. The half-angle property means length reaches maximum (unity) at 180° error, a very useful feature. The quaternion extraction algorithm is also used in several additional space shuttle flight-software applications.Singularities in flight software are at best highly undesirable. Two algorithms for extracting a quaternion from a direction-cosine matrix have been found in the literature. The algorithm by Grubin l is the simplest, but it degrades in precision for rotation angles in the vicinity of 180° because it involves quotients in which both the dividend and divisor approach zero by subtraction of nearly equal numbers. At 180°i t fails. The algorithm proposed by Rupp 2 and by Hendley 3 fails at 180° by yielding any one of eight solutions, six of which are incorrect. All algorithms must lose directional precision in the vicinity of null. Since the algorithm presented here is otherwise free from singularities, it has been adopted by the organizations involved in space shuttle guidance and control. The vector part of the algorithm can be used separately as a compact singularity-free method for finding the eigenvector of a direction-cosine matrix.A quaternion and a direction-cosine matrix are associated by their transformations,where V A and V B are either the same three-component vector expressed in frames A and B or are vectors of equal length within a single frame, M B A is the direction-cosine matrix, q B A is the quaternion, and (q B A ) * is its conjugate. A matrix product is implied by Eq.(l) and quaternion products by Eq. (2).Denoting the quaternion scalar element by q 0 and the vector elements by q h q 2 , q$, a direction-cosine matrix M is related to the quaternion q by Af=2(3)Two properties of use in extracting the quaternion from the matrix are: 1) a rotation quaternion is of unit length, and 2) a quaternion and its negative are interchangeable. A quaternion with positive scalar represents a rotation of 180° or less; its negative represents a rotation of opposite sense about the same axis through 360° >minus the same angle. Since these rotations are equivalent, the quaternions are interchangeable. Thus, although there exist two quaternions corresponding to a single direction-cosine matrix, one quaternion can arbitrarily be se...

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