Solving the AXB=YCZ Problem for a Dual-Robot System With Geometric Calculus

Multi-manipulator systems are becoming increasingly indispensable in modern manufacturing scenes. To achieve the highest accuracy in a dual-robot manufacturing system, it is essential to calibrate the transformation relationships between several coordinate frames, namely hand-to-eye, base-to-base, and tool-to-flange transformations. This process is classified as the problem of
$\mathbf{AXB}=\mathbf{YCZ}$
. In this article, we propose a simultaneous calibration method based on geometric algebra and geometric calculus as a solution to this problem. The proposed method consists of three steps. In the first step, an initial value for the rotation part is found in closed form. Starting from this reasonable initial value, a nonlinear constrained optimization problem is defined and solved using a stochastic gradient descent algorithm. The iterative part is formulated entirely in geometric algebra elements instead of matrices, and the gradient is calculated using geometric calculus instead of vector calculus. These substitutions enable the proposed method to find the optimal result with less than 1/3.5 of the computational footprint and to achieve approximately 4.5 times the runtime speed boost compared to previous methods. The translational component is derived last by solving a least square problem. Simulations and experiments are conducted to test the performance of the proposed method, and the results verify the proposed method’s capability to find optimal solutions and clear superiority in computational time.
Note to Practitioners
— A robotic manufacturing system requires calibration before deployment to achieve the best processing accuracy. In a dual-robot scenario, determining the hand-eye, base-base, and tool-flange transformation relationships is necessary. In this article, we propose a novel method that can perform this calibration efficiently and effectively. The proposed method employs a new mathematical tool called geometric algebra to encode rotations, which enables this method to find the optimal result with significantly less time and memory. Further experiments show that the proposed method achieves identical accuracy performance but runs approximately 4.5 times faster than previous approaches. Such outstanding efficiency makes this proposed method a great approach for systems that require frequent calibration. The final measurement experiments also showed that the system calibrated by the proposed method can achieve an accuracy of 0.3mm.