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General closed-form solutions of the position self-calibration problem

The work in [36] investigates the anchors and sources position self-calibration problem in the 3D space based on range measurements and without any prior restriction on the network configuration. Using a well known low-rank property of Euclidean distance matrices, we first reduce the problem to finding 12 unknowns ascribed in a 3×3 transformation matrix and a 3×1 translation vector. In order to estimate them, we then introduce a polynomial parametrization with 9 unknowns that are estimated by solving a linear system. Afterwards, we identify an intrinsic matrix polynomial system that encodes the solution set of the problem and provide a direct method for solving it. The resulting procedure is simple and straightforward to implement using standard numerical tools. We also show that closed-form solutions can always be obtained when the reference frame is fixed. This is illustrated by adopting reference frames from the literature and by introducing a triangular reference frame whose constraints are imposed only on one position set (anchor or source). Experimental results on synthetic and real sound data show that the proposed closed-form solutions efficiently solve the position self-calibration problem.