Asymptotic Solution of a Boundary Value Problem for a Spring\u2013Mass Model of Legged Locomotion

Hanna Okrasińska-Płociniczak,Łukasz Płociniczak

doi:10.1007/s00332-020-09641-w

Abstract

Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.

Highlights

One of the fundamental achievements of evolution is the ability of various animals to move efficiently throughout the natural terrain
Building a complete model of running can be prohibitively difficult and, to proceed one may just focus on some particular features or simplify matters with a use of low-dimensional conceptual models (Dickinson et al 2000)
In our previous work (Płociniczak and Wróblewska 2020), we have proved that it has a unique solution for sufficiently small angles of attack

Summary

Introduction

One of the fundamental achievements of evolution is the ability of various animals to move efficiently throughout the natural terrain. As was noted in Blickhan and Full (1993), despite of tremendous diversity of various locomotion modes, anatomical differences, and sizes of particular animals, the spring–mass model describes the fundamental mechanical principles of bouncing gaits very accurately. This simple model is very robust and has been investigated in many studies from different vantage points: experimental (Farley et al 1991, 1993; Dickinson et al 2000) and biomechanical modelling (He et al 1991; Geyer et al 2005). We consider the mathematically interesting singular case of large velocity in which we have to use a two-parameter perturbation These results give very simple expressions that reveal some biomechanical properties of the running or hoping leg. Throughout the paper, we include some numerical examples that illustrate the theory

Model and Problem Statement

Asymptotic Solution

Small Angle

Large Velocity

Conclusion