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

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_iX=XB_i when the angles corresponding to rotational parts of A_i and B_i are near or equal to π 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.