On the inverse problem of determining Mars lander coordinates using Phobos eclipse observations

This paper deals with an inverse problem concerning the determination of the MetNet Precursor Mission landing site coordinates on Mars using Phobos eclipses detection. The main difficulties are that this problem corresponds to the inversion of a multievaluated and nonlinear function which models the latitudinal and longitudinal Phobos shadow motion across the Mars surface and the use of experimental data involving measurement errors. We consider two different algorithms for the inverse problem: the Levenberg–Marquardt algorithm and the block relaxation method, and we perform numerical tests in order to compare the results obtained. We show the efficiency of an algorithm which employs a recursive least squares estimator based on a coordinate relaxation method. The precision in coordinate determination has been analyzed for different uncertainty values of the observed contact times and for different number of observed eclipses. The numerical results for the planned MetNet Precursor Mission show how, within a landing site band of δ φ = ± 5 ° around the equator, a maximum difference of 40×60 m between the true position and the solution is attainable with 1 s uncertainties in eclipse time observations.