Data-Driven Physics Embedded Dynamics with Predictive Control and Reinforcement Learning for Quadrupeds

State of the art quadrupedal locomotion approaches integrate Model Predictive Control (MPC) with Reinforcement Learning (RL), enabling complex motion capabilities with planning and terrain adaptive behaviors. However, they often face compounding errors over long horizons and have limited interpretability due to the absence of physical inductive biases. We address these issues by integrating Lagrangian Neural Networks (LNNs) into an RL MPC framework, enabling physically consistent dynamics learning. At deployment, our inverse dynamics infinite horizon MPC scheme avoids costly matrix inversions, improving computational efficiency by up to 4x with minimal loss of task performance. We validate our framework through multiple ablations of the proposed LNN and its variants. We show improved sample efficiency, reduced long-horizon error, and faster real time planning compared to unstructured neural dynamics. Lastly, we also test our framework on the Unitree Go1 robot to show real world viability.

STRIDE: Structured Lagrangian and Stochastic Residual Dynamics via Flow Matching

Robotic systems operating in unstructured environments must operate under significant uncertainty arising from intermittent contacts, frictional variability, and unmodeled compliance. While recent model-free approaches have demonstrated impressive performance, many deployment settings still require predictive models that support planning, constraint handling, and online adaptation. Analytical rigid-body models provide strong physical structure but often fail to capture complex interaction effects, whereas purely data-driven models may violate physical consistency, exhibit data bias, and accumulate long-horizon drift. In this work, we propose STRIDE, a dynamics learning framework that explicitly separates conservative rigid-body mechanics from uncertain, effectively stochastic non-conservative interaction effects. The structured component is modeled using a Lagrangian Neural Network (LNN) to preserve energy-consistent inertial dynamics, while residual interaction forces are represented using Conditional Flow Matching (CFM) to capture multi-modal interaction phenomena. The two components are trained jointly end-to-end, enabling the model to retain physical structure while representing complex stochastic behavior. We evaluate STRIDE on systems of increasing complexity, including a pendulum, the Unitree Go1 quadruped, and the Unitree G1 humanoid. Results show 20% reduction in long-horizon prediction error and 30% reduction in contact force prediction error compared to deterministic residual baselines, supporting more reliable model-based control in uncertain robotic environments.

Real-Time Gait Adaptation for Quadrupeds using Model Predictive Control and Reinforcement Learning

Model-free reinforcement learning (RL) has enabled adaptable and agile quadruped locomotion; however, policies often converge to a single gait, leading to suboptimal performance. Traditionally, Model Predictive Control (MPC) has been extensively used to obtain task-specific optimal policies but lacks the ability to adapt to varying environments. To address these limitations, we propose an optimization framework for real-time gait adaptation in a continuous gait space, combining the Model Predictive Path Integral (MPPI) algorithm with a Dreamer module to produce adaptive and optimal policies for quadruped locomotion. At each time step, MPPI jointly optimizes the actions and gait variables using a learned Dreamer reward that promotes velocity tracking, energy efficiency, stability, and smooth transitions, while penalizing abrupt gait changes. A learned value function is incorporated as terminal reward, extending the formulation to an infinite-horizon planner. We evaluate our framework in simulation on the Unitree Go1, demonstrating an average reduction of up to 36.48\% in energy consumption across varying target speeds, while maintaining accurate tracking and adaptive, task-appropriate gaits.