Mathematical Aspects of Calibration by Diffeomorphisms

In this paper, we present a novel approach to calibrating robotic manipulators—calibration by diffeomorphisms. The method is carried out in detail, placing special emphasis on the mathematical basis of the algorithm. The main idea is based on the synergy of the theory of singular mappings and the large dense diffeomorphic metric mapping framework, a method previously unused in robotic applications, together with reproducing kernel Hilbert spaces. The proposed solution allows the determination of appropriate diffeomorphisms, which, as it were, adjust the arbitrarily chosen kinematics to match a real one, thus taking into account inaccuracies arising from inaccurately determined parameters or a previously unmodelled phenomenon, for example, due to high complexity or nonlinearities. The effectiveness of the calibration by diffeomorphisms is illustrated using a numerical experiment for a manipulator with two degrees of freedom.