Constraint in Attitude Estimation Part I: Constrained Estimation
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Constraint in Attitude Estimation Part I: Constrained Estimation 1 Malcolm D. Shuster
Felix qui potuit cognoscere de rerum causas,' - Publius Vergilius Maro (70-19 B.C.E.)
Abstract A complete and careful foundation is presented for maximum-likelihood attitude estimation and the calculation of measurement sensitivity matrices with the intent of revealing heretofore undisclosed pitfalls associated with unconstrained quaternion estimation. Efficient formulas are developed for computing the measurement sensitivity matrix for any attitude representation for which an efficient formula for the inverse kinematic equation is known. In particular, it is shown that the measurement sensitivity matrix for the quaternion is ambiguous and may take on a wide range of values. Hence, estimates of a quaternion which do not take correct account of the norm constraint will be physically meaningless. It is shown also that within Maximum Likelihood Estimation the form of the Wahba cost function for attitude estimation is incorrect when the attitude constraint is relaxed. A simple physical example is presented for quaternion estimation from noise-free vector measurements which fails when the norm constraint on the quaternion is relaxed. Part I of this work provides the basis for more detailed investigations of unconstrained attitude estimation in Part II [1].
Introduction This is the first of two articles whose purpose is to dissuade practitioners of spacecraft attitude estimation from estimating a quatemion or quatemion correction which does not satisfy the appropriate norm constraint, at least to first order in the estimation error. This was also the intent of an earlier conference report of this work [2], where the intent was stated less bluntly. IThis and the succeeding article [1] are an expansion of an earlier conference report [2], presented in August 1993. 2Director of Research, Acme Spacecraft Company, 13017 Wisteria Drive, Box 328, Germantown, Maryland 20874. email: [email protected]. "Iranslation: Happy [is he] who has been able to know the causes of things.
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By "appropriate" we mean unit-norm for a quatemion which would be a quaternion of rotation if the norm constraint were enforced, and for a quaternion correction that the corrected quatemion have unit norm to this order. We say first order in the estimation error, because the linearization process of batch estimators or the extended Kalman filter discards terms of second order. In general, by quaternion of rotation [3] we mean an element of the quatemion group (i.e., homomorphic to SO(3)), whose domain is S3, the unit sphere in four dimensions. By a quaternion in general, we mean an element of the quatemion skew field (algebra, division ring), whose domain is R 4 • We will sometimes discard the tag "of rotation" when it is obvious from the context. The distinguishing feature of quaternions, of course, is the nature of the multiplication operation [3]. It has long been the feeling of this writer that unconstrained quatemion estimation" has been lacki
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