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Mathematical aspects of calibration by diffeomorphisms

Orozco Roberto, Ratajczak Joanna

International Journal of Applied Mathematics and Computer Science

2025

Vol. 35, no. 3

391--402

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.

Calibration by diffeomorphisms of robot manipulator kinematics: a novel approach

Orozco Roberto, Ratajczak Adam

Bulletin of the Polish Academy of Sciences. Technical Sciences

2025

Vol. 73, nr 2

art. no. e153230

Our paper presents a nonparametric data-driven technique that can enhance the accuracy of robot kinematics models by reducing geometric and nongeometric inaccuracies. We propose this approach based on the theory of singular maps and the Large Dense Diffeomorphic Metric Mapping (LDDMM) framework, which has been developed in the field of Computational Anatomy. This framework can be thought of as a method for identifying nonlinear static models that encode a priori knowledge as a nominal model that we deform using diffeomorphisms. To tackle the kinematic calibration problem, we implement Calibration by Diffeomorphisms and obtain a solution using an image registration formalism. We evaluate our approach via simulations on double pendulum robot models, which account for both geometric and nongeometric discrepancies. The simulations demonstrate an improvement in the precision of the kinematics results for both types of inaccuracies. Additionally, we discuss the potential application of physical experiments. Our approach provides a fresh perspective on robot kinematics calibration using Calibration by Diffeomorphisms, and it has the potential to address inaccuracies caused by unknown or difficult-to-model phenomena.