Real-time robust nonlinear model predictive control with monotonically increasing weight for quadruped locomotion

Nonlinear model predictive control (NMPC) has demonstrated significant success for enhancing the dynamics and robustness of legged robots traversing complex terrains. However, when implementing closed-loop control, the performance and stability of NMPC are not guaranteed in each control cycle. Additionally, incorporating terminal constraint or constraint conditions to ensure closed-loop stability introduces a substantial computational burden. Addressing this problem, a monotonically increasing weight (MIW) NMPC approach with reduced computational burden used in legged locomotion is proposed and proved its closed-loop stability. The equality constraint of MIW-NMPC is formulated as a nonlinear system based on the single rigid body dynamic (SBRD) model, with its reachability validated using geometric control theory. Then the approach utilizes time-varying monotonically increasing stage cost penalties, which ensures stability without any terminal constraint. Furthermore, to further reduce the computational burden and improve real-time performance of MIW-NMPC, a fast sequential quadratic programming (SQP) solver supplemented by a warm-start strategy and full Newton steps is proposed. For this purpose, four simulation experiments used the Unitree A1 quadruped robot as the hardware have been meticulously designed. To achieve a comprehensive comparison, six distinct NMPC approaches were tested separately under the same experimental conditions. The stability, trajectory tracking capability, real-time performance and robustness of MIW-NMPC were demonstrated in comparison of the results. It is crucial to highlight that proposed MIW-NMPC with fast SQP approach avoids the stability issues and the computational burden that can be induced by the use of terminal constraint. Finally, the paper develops robust controllers for quadruped robots and provides insights into the stability considerations when designing NMPC.