Toward Simultaneous Coordinate Calibrations of AX=YB Problem by the LMI-SDP Optimization

Abstract

Accurate calibration of the robot hand-eye ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> ) and robot-world ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</b> ) relationships is extremely important for visually-guided robotic systems, and is usually symbolized by the AX <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=$</tex-math> </inline-formula> YB equation. The existing methodologies always calibrate the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> and <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</b> matrices using the separation of the rotational and translational components, causing the error propagation and accumulation. While the simultaneous calibration solves the derived linear matrix equation by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SVD</i> (Singular Value Decomposition) based approach, which produces unreliable results depending on the smallest singular value of the regression matrix. To this end, the work contained herein proposes a novel and generic calibration methodology for solving the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AX</b> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=$</tex-math> </inline-formula> <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">YB</b> problem using the LMI-SDP (Linear Matrix Inequality and Semi-definite Programming) optimization. In this approach, the linear form of the calibration equation is retrieved by means of the Kronecker product, and formulated as an optimization problem involving the unknown variable matrices <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</b> and <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</b> with convex constraints, in which the simultaneous solution is obtained via the LMI-SDP techniques. The results procured via the simulation analysis, accounting for the presence of noise levels and different data pairs, as well as the calibration experiments, are compared to those produced using the classical iterative method and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DQ</i> (Dual Quaternion)-based approach, thereby verifying the accuracy and efficacy of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The motivation behind this work stems from the simultaneous calibration issues pertaining to robot-eye and robot-workpiece coordinate relationships, that are present in vision-guided robotic systems. Considering the inaccuracy and robustness deficiencies of the existing methodologies, due to the separated calibration of the rotational and translational components, this paper proposes a generic and efficient calibration methodology to deal with the AX <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=$</tex-math> </inline-formula> YB problem, using the Kronecker product and the LMI-SDP optimization. Simulation analysis reveals that the proposed algorithms exhibit robustness under different noise levels and data pairs. Moreover, the practicability of the algorithm has also been verified via practical experiments. The average errors with 16 sets of calibration data can reach 0.0056rad in the rotational component, and 0.2529mm in the translational component. The proposed methodology can be extended to the practical applications of the coordinate calibration involving the multi-robot systems with visual sensors.