Analytical solution to a time-varying LIP model for quadrupedal walking on a vertically oscillating surface

This paper introduces an analytically tractable and computationally efficient model for legged robot dynamics during locomotion on a dynamic rigid surface (DRS), along with an approximate analytical solution and a real-time walking pattern generator synthesized based on the model and solution. By relaxing the static-surface assumption, we extend the classical, time-invariant linear inverted pendulum (LIP) model for legged locomotion on a static surface to dynamic-surface locomotion, resulting in a time-varying LIP model termed as DRS-LIP. Sufficient and necessary stability conditions of the time-varying DRS-LIP model are obtained based on the Floquet theory. This model is also transformed into Mathieu's equation to derive an approximate analytical solution that provides reasonable accuracy with a relatively low computational cost. Using the extended model and its solution, a walking pattern generator is developed to efficiently plan physically feasible trajectories for quadrupedal walking on a vertically oscillating surface. Finally, simulations and hardware experiments from a Laikago quadrupedal robot walking on a pitching treadmill (with a maximum vertical acceleration of 1 m/s2) confirm the accuracy and efficiency of the proposed analytical solution, as well as the efficiency, feasibility, and robustness of the pattern generator, under various surface motions and gait parameters.

Automated summary

This paper introduces a model for legged robot dynamics during locomotion on a dynamic rigid surface, along with an approximate analytical solution and a real-time walking pattern generator synthesized based on the model and solution. The model extends the classical LIP model for legged locomotion on a static surface to dynamic-surface locomotion, and stability conditions are obtained based on Floquet theory. The model is transformed into Mathieu's equation to derive an approximate analytical solution. A walking pattern generator is developed using the model and solution to efficiently plan physically feasible trajectories for quadrupedal walking. Simulations and hardware experiments confirm the accuracy and efficiency of the proposed solution, as well as the efficiency, feasibility, and robustness of the pattern generator, under various surface motions and gait parameters.