Analysis on the number of local minima for 3D SLAM problem

Three dimensional Simultaneous Localization and Mapping (SLAM) is one of the fundamental task for autonomous robots, to operate successfully in unknown environment. Recently convexity analysis for 2D mobile-robot SLAM system has been analyzed whereas for highly nonlinear problems i.e. 3D SLAM for aerial robotic, the understanding of convex structure of the system is much of interest to robotics community. In particular, the study of number of local minima can help to provide a guaranteed global minimum solution to highly non-convex problems. In this work, we have shown that, given one of the orientation angle is assumed to be known, the two-pose 5DOF SLAM is equivalent to solving a problem with two unknown variables. We also have provided an upper bound for the number of local minima in a 5DOF SLAM. For two-pose 3D SLAM problem, the analysis reveals that there exist at most four local minima.