Swimming Under Constraints: A Safe Reinforcement Learning Framework for Quadrupedal Bio-Inspired Propulsion

Soft robots are valuable robophysical platforms for studying body-caudal undulatory locomotion, but their compliant bodies are difficult to control precisely under changing hydrodynamic loading. Conventional proportional-integral-derivative (PID) feedback stabilizes periodic undulation in static water, but can accumulate flow-dependent tracking delay and increasing inter-trial variability when environmental flow becomes non-trivial. Here, we evaluate whether augmenting PID control with a Linear Repetitive Learning Estimation Scheme (PID-LRLES) recovers tracking accuracy and repeatability under dynamic flow. The LRLES generalizes classical integral action from constant to periodic, non-constant references, while using a stable transfer-function realization whose poles have negative real parts to avoid the long-term instability issues of classical repetitive control. Closed-loop experiments were carried out in a recirculating flow tank at five bulk flow speeds spanning 0 to 32.6 cm s^-1, using an embedded soft capacitive bending sensor at a 1 kHz control-loop rate. With controller gains tuned once in static water and then held fixed across all conditions, PID-LRLES tracked the periodic bending-envelope reference more closely than the PID baseline and significantly reduced the inter-trial spread of the per-trial RMSE (paired Wilcoxon signed-rank test, p = 1.8 x 10^-4, n = 25). Embedded soft proprioception and cycle-to-cycle learning act as complementary contributors to robustness: the sensor exposes the periodic hydrodynamic bias in body deformation, while the learning term absorbs it over recent oscillation cycles. By reducing flow-dependent control-induced variability, the approach provides an enabling layer for future robophysical studies seeking to isolate the effects of morphology, sensing, and environmental flow on aquatic locomotion.

Reinforcement learning combined with imitation learning has significantly advanced biomimetic quadrupedal locomotion. However, scaling these frameworks to massive, multi-source datasets exposes fundamental bottlenecks. First, traditional GAN-based discriminators are prone to mode collapse, struggling to capture diverse motion distributions from uncurated datasets. Second, existing kinematic priors suffer from out-of-distribution (OOD) tracking conflicts, leading to severe unintended heading drifts during complex maneuvers. Furthermore, deploying unconstrained priors to physical hardware poses critical safety risks by disregarding actuator dynamics. To overcome these challenges, we propose Diff-CAST (Diffusion-guided Constraint-Aware Symmetric Tracking), a novel motion prior framework leveraging the multi-modal distribution modeling capabilities of diffusion models for stylistic rewards. Diff-CAST effectively replaces traditional GAN discriminators, unlocking robust data scaling on heterogeneous collections. To ensure high-fidelity intent execution and reliable real-world deployment, we introduce a comprehensive Sim2Real architecture integrating Symmetric Augmented Command Conditioning (SACC) for drift-free tracking, and Constrained RL for hardware safety. Experiments on a quadruped demonstrate that Diff-CAST mitigates mode collapse, enables seamless transitions between diverse skills, and ensures robust, hardware-compliant locomotion.

Legged locomotion in unstructured environments demands not only high-performance control policies but also formal guarantees to ensure robustness under perturbations. Control methods often require carefully designed reference trajectories, which are challenging to construct in high-dimensional, contact-rich systems such as quadruped robots. In contrast, Reinforcement Learning (RL) directly learns policies that implicitly generate motion, and uniquely benefits from access to privileged information, such as full state and dynamics during training, that is not available at deployment. We present ContractionPPO, a framework for certified robust planning and control of legged robots by augmenting Proximal Policy Optimization (PPO) RL with a state-dependent contraction metric layer. This approach enables the policy to maximize performance while simultaneously producing a contraction metric that certifies incremental exponential stability of the simulated closed-loop system. The metric is parameterized as a Lipschitz neural network and trained jointly with the policy, either in parallel or as an auxiliary head of the PPO backbone. While the contraction metric is not deployed during real-world execution, we derive upper bounds on the worst-case contraction rate and show that these bounds ensure the learned contraction metric generalizes from simulation to real-world deployment. Our hardware experiments on quadruped locomotion demonstrate that ContractionPPO enables robust, certifiably stable control even under strong external perturbations.