Towards a closed-form solution of constraint networks for maximum likelihood mapping

Several recent algorithms address simultaneous localization and mapping as a maximum likelihood problem. According to such formulation map estimation is achieved by finding the configuration that minimizes the error function associated to the constraint network representing the map. Almost all the proposed algorithms exploit iterative fixed-point techniques. Less attention has been paid to the evaluation of the structure of the problem. In this paper, we derive the closed form solution for a constraint network of planar poses when the error function is given in a particular form. The derivation of the exact solution allows a better knowledge of the problem and a more detailed investigation of the performance of existing numerical techniques. The system of equations, that are satisfied by the solution, are computed and a general algorithm to solve such equations is provided.