Iterative relative error allocation and compensation in dual-robot collaboration based on dynamic vector valued Nash games

In dual-robot collaborative applications requiring high-precision trajectory synchronization, relative errors between robots often remain significant despite prior individual kinematic calibration using laser trackers. This paper identifies a critical issue wherein the robot pose transformation with respect to the tracker introduces projection discrepancies that are not reflected in the tracker-measured absolute errors. Consequently, although each robot exhibits sub-millimeter accuracy individually, contact-based measurements (via dial indicators) reveal relative errors of more than 2 mm when both robots execute identical circular Cartesian trajectories. It is discovered that the different servo performance cause asynchronous tracking, leading to movement direction-dependent relative deviation. To address this, we propose a novel game-theoretic iterative compensation strategy based on direct contact feedback. The relative errors are decomposed into motion corrections allocated to each robot according to a dynamic game model that reflects their error contribution and dynamic actuation capabilities. By applying iterative small-step trajectory adjustments, the system converges towards minimal relative error. Experimental results demonstrate the feasibility and superiority of the proposed method in achieving sub-millimeter relative accuracy. This study introduces a new paradigm that combines feedback-driven error decomposition with iterative learning and dynamic games, offering practical insights into high-precision dual-robot collaboration.