Calibration of a spacecraft magnetometer taking into account the nature of the temperature dependence of the sensitivity matrix and the offset vector

In this paper, an analytical method is proposed for solving the problem of magnetometer calibration for a model that takes into account the vector of temperature dependence of zero offsets for each of the measuring axes of the magnetometer unit and the matrix of linear temperature dependence of each of the members of the sensitivity matrix, scaling the signal based on the actual sensitivity of each axis and including linear off-axis effects. 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 magnitude of the measured magnetic field strength vector is preserved and is a known model value. A penalty function of 24 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, with a given set of vectors of magnetometer measurements, provide it with a minimum. For this purpose, the specified function is examined for an extremum. Based on the necessary condition for the extremum of the penalty function, a system of 24 equations in the 24 variables is formed, which, for convenience, is divided into three systems (each of them is a system of 8 linear algebraic equations in the 8 variables). It is proved that the main matrix of each of these three systems is an invertible, from which it follows that each of them has a solution, and only one. The components of the solutions of these systems (the coordinates of the stationary point of the penalty function) are found using Cramer's rule. It is proved that the second differential of the penalty function at the found stationary point is positive, from which it follows that this point really provides the minimum of the specified function.