Constraint in Attitude Estimation Part II: Unconstrained Estimation
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Constraint in Attitude Estimation Part II: Unconstrained Estimation 1 Malcolm D. Shuster
We compute for insight, not for numbers.
- Richard W. Hamming (1915-1998)
Abstract The consequences of ignoring the norm constraint in quatemion estimation or the orthogonality constraint in the estimation of the attitude matrix are examined within the framework of batch maximum-likelihood estimation. Unconstrained estimation of the attitude matrix is shown to be a useful first step to the constrained estimation, because it confers global convergence to the estimation process. Unconstrained estimation of the quatemion, however, is shown to be fraught with problems. Apart from the fact that the unconstrained quatemion estimate can introduce errors which cannot be eliminated later, it can also lead to a singular (hence, noninvertible) inverse covariance matrix when used with a realistic measurement model. Consequently, the unconstrained quatemion estimate cannot be constructed. This has the additional consequence that there can be no recovery from an arbitrary initial condition. Certain practices associated with unconstrained quatemion estimation in the Kalman filter, in particular, the partial reset, are shown to interfere with the process of the correct restoration of the quatemion norm. Thus, one is forced to conclude that unconstrained quatemion estimation should be avoided in practice.
Introduction As stated in the introduction to Part I [1] this work seeks to provide a more rigorous foundation for research on unconstrained attitude estimation [2-6] and to show many of the pitfalls of unconstrained quatemion estimation. The first part of this work [1] presented the foundations of batch least-squares attitude estimation within the framework of Maximum-Likelihood Estimation and a 'This and the preceding 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].
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detailed account of the attitude measurement sensitivity matrix. It was shown there that the unconstrained quaternion measurement sensitivity matrix was ambiguous, and, therefore, would lead to meaningless unconstrained estimates of the quaternion. In particular, it was shown that for one very physical case of quaternion estimation the estimation process broke down completely. In the present part we examine unconstrained attitude matrix and quaternion estimation in more detail. The present Part II concentrates on specific examples of unconstrained attitude estimation. Our studies are restricted to the attitude matrix and the quaternion, since these (apart from the Cayley-Klein parameters, which differ only trivially from the quaternion) are the only higher-dimensional attitude representations in common use. Our attention is restricted again to batch attitude estimation, since batch estimation is more transparent then the jumble of equations in the Kalman filter. Thus, we limit out
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