Complete Linear Attitude-Independent Magnetometer Calibration
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Complete Linear AttitudeIndependent Magnetometer Calibration Roberto Alonso' and Malcolm D. Shuster Abstract The TWOSTEP algorithm, which has been applied successfully to the attitude independent estimation of magnetometer biases is extended to estimate scale factors and nonorthogonality corrections. For the case of a spinning spacecraft, the algorithm performs well. This goes against the common wisdom that not much more than biases can be determined without knowledge of the attitude.
Introduction The TWOSTEP algorithm [1] was developed to determine magnetometer biases inflight without knowledge of the attitude under any possible conditions. Exhaustive studies [2] have shown it to be more robust and more efficient than other existing methods. In fact, the authors have been unable up to now to create a non-trivial scenario in which the algorithm will not perform well. In the past, only the determination of magnetometer biases was attempted from such data, it being widely believed that additional parameters were not observable. The present work seeks to reverse that opinion. The TWOSTEP algorithm relies on a centering procedure for its first step. Such a procedure was first applied to magnetometer bias determination by Gambhir [3, 4]. To understand the need for such a procedure we examine the model for the magnetometer measurement, which we write as (1)
where B, is the measurement of the magnetic field (more exactly, magnetic induction) by the magnetometer at time tk; H, is the correspondingvalue of the geomagnetic field with respect to an Earth-fixed coordinate system; A k is the attitude of the 1Jefe, Grupo de Control de Actitud, Comision Nacional de Actividades Espaciales (CONAE), Avenida Paseo Colon 751, (1063) Buenos Aires, Argenina. 2Director of Research, Acme Spacecraft Company, 13017 Wisteria Drive, Box 328, Germantown, Maryland 20874. email: [email protected].
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magnetometer with respect to the Earth-fixed coordinates; b is the magnetometer bias; and Ek is the measurement noise. The measurement noise, which includes both sensor errors and geomagnetic field model uncertainties, is generally assumed to be white and Gaussian. If the attitude matrix A k is not known, we can work instead with derived scalar measurements and scalar measurement noise defined by
/B
Zk =
/H
2 k1 -
k/
2
(2a)
Vk = 2(Bk - b) · Ek -
IEkl
2
(2b)
Then we can write Zk = 2Bk
•
b - /b/2
+ Vk,
k = 1, ... , N
(3)
If we assume that Ek is white and Gaussian (4)
u~
= E{v~} -
ILk IL~
= E{Vk} =
= 4(B k -
(5a)
-tr(~k)
b)T~k(Bk - b)
+ 2(tr ~~)
(5b)
This effective measurement model is presented in greater detail in [1]. The estimation of b according to the criterion of maximum likelihood leads us to find the value which minimizes the negative-log-likelihood function [5] J(b) = -1 2
L N
k=l
[
1 2(Zk - 2Bk · b a;
+ /b/2
-
ILk)2
+ log u~ + log 27T ]
(6)
of which only the first term under the summation depends on the magnetometer bias. The minimization of equation (6) is complicated by
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