A Koopman-Based Digital Twin Approach for Fault Detection in Cable Slab Dynamics of Wafer Scanners

This paper presents a higher-dimensional state-space model that employs Koopman operators to predict and monitor the nonlinear dynamics of a cable slab in wafer scanners or other precision motion systems. Developed as a digital twin, our approach not only approximates the complex behavior of the cable slab but also facilitates early fault detection by comparing predicted dynamics with real-time sensor data. To address the absence of a definitive analytical model for cable slab dynamics, we systematically evaluate various Koopman observable functions to minimize tracking errors. The resulting model achieves an error margin of approximately ±1% over the specified motion range and demonstrates robust performance in predicting untrained, acyclic randomized cable slab motion and reaction forces. The system was also compared against the state-of-the-art neural network model for the cable slab, showing a reduction in reaction force error prediction of 75.4%. A sensor noise fault with a signal-to-noise of 20 was detected in 0.35 s using only the reaction force measurement. A position-based noise fault (signal-to-noise of 30) was detected in 0.04 s using position-based fault detection. A hardware malfunction fault test is also performed by removing some tubing material inside the cable slab and is detected in 0.38 s through position-based fault detection with the Koopman model. Experimental results confirm that the Koopman-based framework effectively approximates the nonlinear dynamics and detects faults in the cable slab, thereby enhancing the reliability of precision motion systems.