Singularity Analysis for the Existing Closed-Form Solutions of the Hand-Eye Calibration

Abstract

In this paper, a class of singular phenomenon is first found for the existing closed-form solutions of the hand–eye calibration problem with the form ${{{A}}_{i}{{X}}={{X}}{{ B}}_{i}}$ when the angles corresponding to rotational parts of ${{A}}_{i}$ and ${{B}}_{i}$ are near or equal to $\pi $ radian. A universal observability index is put forward to detect when this singularity would undoubtedly occur. For avoiding this singularity, a novel analytical solution based on a new cost function is proposed to estimate the hand–eye matrix in the presence of measurement errors. Simulation and experimental results can illustrate the feasibility and benefits of the proposed observability index and the singular-free closed-form solution. In addition, the other kind of singular phenomenon is also discovered for the existing closed-form solution, where the orientation of the unknown hand–eye matrix is parameterized by modified Rodrigues parameters. Therefore, in order to obtain the non-singular analytical solution based on modified Rodrigues parameters, a novel additional rotation theory is introduced and verified by the hand–eye calibration of a novel surgical robot.