Analytical solution of the spacecraft magnetometer calibration problem using the method of least squares

In this paper, an analytical method is proposed for solving the problem of magnetometer calibration for the model considered in [1]. When solving the problem of determining the calibration parameters of the magnetometer unit, it is taken into account that for measurements with any spatial orientation of the magnetometer unit, the value of the measured magnetic induction vector is preserved and is a known model value. A penalty function of 12 variables equal to the sum of the squares of the residuals is introduced into consideration. The algorithm for solving the problem of calibrating the measuring axes of the magnetometer unit is reduced to searching, by the method of least squares, for such values of the variables of this function that, for a given set of magnetometer measurement vectors, provide it with a minimum. For this purpose, the specified function is examined for a conditional extremum in the presence of three equality constraints. The Lagrangian function is compiled and, based on the necessary condition for the extremum of this function, the system of 15 equations in the 15 variables is formed. It is proved that the system has four solutions. Formulas are derived that make it possible to obtain the components of each of these four solutions. As a solution to the magnetometer calibration problem, it is recommended to choose a solution to the specified system that provides a minimum of the Lagrangian function.