Estimation of leg stiffness using an approximation to the planar spring–mass system in high-speed running

Abstract

Leg stiffness plays a critical role in legged robots’ speed regulation. However, the analytic solutions to the differential equations of the stance phase do not exist, of course not for the exact analytical solution of stiffness. In view of the challenge in dealing with every circumstance by numerical methods, which have been adopted to tabulate approximate answers, the “harmonic motion model” was used as approximation of the stance phase. However, the wide range leg sweep angles and small fluctuations of the “center of mass” in fast movement were overlooked. In this article, we raise a “triangle motion model” with uniform forward speed, symmetric movement, and straight-line center of mass trajectory. The characters are then shifted to a quadratic equation by Taylor expansion and obtain an approximate analytical solution. Both the numerical simulation and ADAMS-Matlab co-simulation of the control system show the accuracy of the triangle motion model method in predicting leg stiffness even in the ultra-high-speed case, and it is also adaptable to low-speed cases. The study illuminates the relationship between leg stiffness and speed, and the approximation model of the planar spring–mass system may serve as an analytical tool for leg stiffness estimation in high-speed locomotion.

The influence of leg length and individual mass in the spring-mass running model

“…Their research provides different insights into the mechanics of running on different terrains. Guo W, Cai C et al [12] a new "triangular motion model" is proposed to estimate leg stiffness during high-speed running. This provides another perspective for the study of biomechanics of high speed motion.…”

Controlling a One-Legged Robot to Clear Obstacles by Combining the SLIP Model with Air Trajectory Planning

“…This also explains why it is difficult to obtain good performance when using fixed SLIP model spring stiffness to describe the dynamic characteristics of robot legs. To address this issue, methods for qualitatively analyzing the change in spring stiffness with the increase in running speed have been proposed [ 42 , 43 ].…”