Centering and Observability in Attitude-Independent Magnetometer-Bias Determination
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Centering and Observability in Attitude-Independent Magnetometer-Bias Determination Roberto Alonso! and Malcolm D. Shuster" Abstract The TWOSTEP algorithm is examined for the case where the centered portion of the negative-log-likelihood function provides incomplete observability of the magnetometerbias vector. In those cases where the full negative-log-likelihood function provides a complete estimate, the TWOSTEP algorithm can be modified to provide an estimate of all three components of the magnetometer bias vector. However, the procedure leads to a discrete degeneracy of the estimate which can be resolved only by explicit evaluation of the negativelog-likelihood.
Introduction The TWOSTEP algorithm [1] is a very efficient and robust algorithm for the estimation inflight of the magnetometer-bias vector without knowledge of the attitude. Numerous comparisons [2] have shown that TWOSTEP is superior to all other attitude-independent algorithms providing an estimate of the magnetometerbias vector in hundreds of simulations, in many of which the other algorithms failed completely. Only Acufia's algorithm [2,3] was seen to be superior to TWOSTEP in some cases, namely, those in which a reference magnetic field was not available, in which case TWOSTEP cannot be applied, or when the measurement noise levels had been severely mismodelled, in which case Acufia's algorithm was sometimes marginally better. Acufia's algorithm, however, requires that the spacecraft be made to spin rapidly about two different axes in succession, which limits its range of applicability. Thus, the TWOSTEP algorithm is the clear algorithm of choice for nearEarth missions, while Acufia's algorithm is unchallenged for interplanetary missions.
IJefe, Grupo de Control de Actitud, Comisi6n Nacional de Actividades Espaciales (CONAE), Avenida Paseo Col6n 751, (1063) Buenos Aires, Argentina. 2Director of Research, Acme Spacecraft Company, 13017 Wisteria Drive, Box 328, Germantown, Maryland, 20874. email: [email protected].
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TWOSTEP has been extended also to estimate linear parameters of the magnetometer calibration [4]. The TWOSTEP algorithm depends on the separation of the negative-loglikelihood function into two pieces. Therefore, there is always the possibility that neither piece will have complete information on the magnetometer-bias vector even if the estimation is possible with the full negative-log-likelihood function. These cases will occur mostly when the bias vector is barely observable, since the first step of TWOSTEP works quite well in general. It is these cases of poor observability which we treat in the present work. The TWOSTEP algorithm [1] assumes a measurement model of the form k = 1, ... ,N
(1)
where B, is the measurement of the magnetic field (more exactly, magnetic induction) by the magnetometer at time tk; H, is the corresponding value of the geomagnetic field with respect to an Earth-fixed coordinate system; A k is the attitude of the magnetometer with respect to the Earth-fixed coordinates; b is t
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