The hybrid propulsion system of a robotic fish that combines propellers with bio-inspired oscillating fins promises both long-range sprint and low-disturbance maneuvering, yet the strongly coupled vortex fields make thrust prediction notoriously difficult. To tackle this modeling gap, we first perform high-fidelity CFD simulations of a multi-joint hybrid-driven robotic fish over a wide operating condition. The computations reveal that the interaction between the propeller jet and the fin-generated vortex sheet leads to pronounced nonlinear thrust modulation. A parsimonious parametric model derived from these data still leaves significant errors in some operating conditions. Therefore, we introduced Gaussian Process Regression to predict and compensate for the residual dynamics, effectively improving the accuracy and generalization ability of the model. The resulting hybrid model keeps physical interpretability while learning the remaining coupling dynamics. Validation results demonstrate that the enhanced hybrid model maintains excellent performance even under untrained operating conditions, achieving a goodness of fit exceeding 0.99 for the predicted thrust coefficient, lift coefficient, and moment coefficient of the tail, and above 0.95 for the propeller thrust coefficient. This study provides an effective solution to the actuator-coupling modeling problem of complex underwater robots and lays a solid foundation for the design of high-performance motion controllers.
Bai et al. (2026) studied this question.