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Map joining is an efficient strategy for solving feature based SLAM problems. This paper demonstrates that joining of two 2D local maps, formulated as a nonlinear least squares problem has at most two local minima, when the associated uncertainties can be described using spherical covariance matrices. Necessary and sufficient condition for the existence of two minima is derived and it is shown that more than one minimum exists only when the quality of the local maps used for map joining is extremely poor. The analysis explains to some extent why a number of optimization based SLAM algorithms proposed in the recent literature that rely on local search strategies are successful in converging to the globally optimal solution from poor initial conditions, particularly when covariance matrices are spherical. It also demonstrates that the map joining problem has special properties that may be exploited to reliably obtain globally optimal solutions to the SLAM problem.
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