An Investigation of 2nd-Order Fixed Point SLIP Behavior

Abstract

This paper introduces alternative behaviors described by the SLIP model when it is subject to a range of initial conditions. A non-dimensional SLIP model and a numerical return map search scheme are used to determine fixed points as a function of non-dimensional leg stiffness and vertical displacement under friction constraints. A SLIP model behavior analysis is performed, using an analytical stance phase approximation, by diverging from the fixed points, i.e. by increasing/decreasing initial horizontal velocity, and/or touchdown angle. The results show that beyond the regular fixed points, the SLIP model performs an alternative, stable behavior that repeats itself every two cycles of motion. We call these 2nd-order fixed points and the regular ones 1st-order fixed points. A numerical simulation scheme was developed to investigate 2nd-order fixed points for a wide range of horizontal velocities and touchdown angles. Results show that 2nd-order fixed points respecting the friction cone constraints exist that can lead to a number of different behaviors such as high jumps, obstacle avoidance of different heights, or backward motion.