Variational-Based Geometric Nonlinear Model Predictive Control for Robust Locomotion of Quadruped Robots

This paper proposes a novel nonlinear model predictive control (NMPC) method based on geometric variational calculus for high-dynamic and complex motion control of quadruped robots. By approximating system trajectory tracking error dynamics on the Special Euclidean group (SE(3)), the method avoids the singularities of Euler angles and the challenges of quaternion representation while capturing the coupling between rotational and translational dynamics for a more comprehensive motion description. Leveraging variational calculus, the resulting Geometric Nonlinear Model Predictive Controller (GNMPC) enables high-frequency updates while preserving essential nonlinear system characteristics. Experimental results across various scenarios validate the effectiveness and advantages of the proposed controller. Note to Practitioners—The primary motivation of this paper is to investigate the application of geometric methods in Model Predictive Control (MPC) and to validate their effectiveness in the context of quadruped robots, which exhibit nonlinear dynamics. In this work, the authors model the robot’s motion on a nonlinear manifold and linearize the system using variational methods. Sequential Quadratic Programming (SQP) is then applied to approximate the globally optimal solution. Experimental results demonstrate that this approach significantly improves the performance of quadruped robots, particularly in handling highly dynamic and robust motions.