A Physics-Guided Neural Network Framework for Prediction and Control of Spring-Mass Running.

The spring-mass template acts as a fundamental bridge between animal locomotion and legged robotic platforms. However, controlling spring-mass dynamics involves a persistent trade-off: numerical integration offers accuracy but high computational cost, while approximate analytical solutions provide efficiency but suffer from linearization errors. To resolve this, we propose a physics-guided hybrid neural network framework for the spring-mass running template. Unlike closed-box predictors trained end-to-end, our architecture enforces flight phase physics by embedding exact analytical solutions for the tractable ballistic dynamics directly into the network structure, and learning only the non-integrable stance dynamics. This ``analytically augmented" design enhances interpretability without sacrificing accuracy. For control, we introduce a mixed-objective training strategy that combines supervised imitation with goal-conditioned learning, enabling precise trajectory tracking without the computational burden of per-step optimization. We evaluated the framework's prediction accuracy, control precision, and runtime efficiency through extensive simulations. Furthermore, we validate the robustness of the learned policies through a comprehensive Basin of Attraction (BoA) analysis, demonstrating that the proposed networks stabilize contiguous manifolds of highly perturbed initial conditions compared to existing methods. Finally, we assessed the framework's real-world adaptability by validating our predictor against an archival experimental dataset from a physical one-legged hopper. Our results demonstrate that this hybrid approach outperforms existing analytical and data-driven methods in accuracy and robustness while maintaining the low inference times required for real-time embedded control.