Determining 3-D Relative Transformations for Any Combination of Range and Bearing Measurements

In this paper, we address the problem of motion-induced 3-D robot-to-robot extrinsic calibration that is based on ego-motion estimates and combinations of interrobot measurements (i.e., distance and/or bearing observations from either or both of the two robots, recorded across multiple time steps). In particular, we focus on solving minimal problems, where the unknown 6-degree-of-freedom (DOF) transformation between the two robots is determined based on the minimum number of measurements necessary to find a finite set of solutions. In order to address the very large number of possible combinations of interrobot observations, we identify symmetries in the measurement sequence and use them to prove that any extrinsic robot-to-robot calibration problem can be solved based on the solutions of only 14 (base) minimal problems. Moreover, we provide algebraic (closed-form) and efficient symbolic-numerical (analytical) solution methods to these minimal problems. Finally, we evaluate the performance of our proposed solvers through extensive simulations and experiments.