Decomposed Algorithm for Spacecraft Attitude Estimation in Stellar Correction Mode

Abstract

The paper discusses the problem of a spacecraft attitude estimation and its solution by integrating the measurement data from an information-redundant gyro unit and a stellar navigation system. The traditional solution methods are based on the use of Kalman filter of n + 3 order, where n > 3 is the number of measurement channels (gyroscopes or angular rate sensors). The numerical implementation of the corresponding algorithm on an onboard computer requires significant computational burden. To considerably reduce the computational complexity of the algorithm without losing the accuracy, it is proposed to use a method of decomposing the filter of n + 3 order into three filters of the second order and n − 3 filters of the first order.

APPENDIX

Below is a method for constructing a pseudo-inverse matrix and the basis of the left null space of matrix G [6].

The pseudo-inverse matrix and the basis of the left null space of matrix G can be obtained by applying its QR-decomposition [13]:

$$G = QR = \left( {\begin{array}{*{20}{c}} {{{Q}_{1}}}&{{{Q}_{2}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{R}_{1}}} \\ {{{0}_{{}}}} \end{array}} \right) = {{Q}_{1}}{{R}_{1}}.$$

Since the matrix Q is orthogonal, $Q_{1}^{T}{{Q}_{2}} = 0,~~~Q_{2}^{T}{{Q}_{2}} = I.$

We define N = Q₂. Then N^TG = 0, ${{N}^{T}}N = I,~~{{G}^{ + }} = R_{1}^{{ - 1}}{{Q}_{1}},~~~~~{{G}^{ + }}G = I.$