A Tutorial for the Analysis of the Piecewise-Smooth Dynamics of a Constrained Multibody Model of Vertical Hopping

Roland Zana,Ambrus Zelei,László Bencsik,Bálint Bodor

doi:10.3390/mca23040074

Abstract

Contradictory demands are present in the dynamic modeling and analysis of legged locomotion: on the one hand, the high degrees-of-freedom (DoF) descriptive models are geometrically accurate, but the analysis of self-stability and motion pattern generation is extremely challenging; on the other hand, low DoF models of locomotion are thoroughly analyzed in the literature; however, these models do not describe the geometry accurately. We contribute by narrowing the gap between the two modeling approaches. Our goal is to develop a dynamic analysis methodology for the study of self-stable controlled multibody models of legged locomotion. An efficient way of modeling multibody systems is to use geometric constraints among the rigid bodies. It is especially effective when closed kinematic loops are present, such as in the case of walking models, when both legs are in contact with the ground. The mathematical representation of such constrained systems is the differential algebraic equation (DAE). We focus on the mathematical analysis methods of piecewise-smooth dynamic systems and we present their application for constrained multibody models of self-stable locomotion represented by DAE. Our numerical approach is demonstrated on a linear model of hopping and compared with analytically obtained reference results.

Highlights

We further developed a minimally complex model of hopping resulting in a constrained model, on which our dynamic analysis methodology was demonstrated
The constrained models are described by a non-minimum set of coordinates and the equations of motion are augmented with geometric constraint equations
This modeling approach, which is typical in multibody dynamics, allows us to build high DoF models effectively

Summary

Introduction

Many descriptive models of legged locomotion—such as walking, running, and hopping—can be found in the literature. References [8,9] introduce models for the investigation of foot impact intensity at forefoot, mid-foot and rear-foot ground collision. The SLIP model in [11] has been developed for the understanding of the essential dynamics of running and hopping locomotion. The further detailed variation of the SLIP model is presented in [12]: the segmented leg provides a geometry very similar to the human leg; the inertia of leg segments is neglected; the ground-foot collision and the impact-induced energy absorption are not investigated. The control of foot placement and human balancing problems are discussed in [14], where a predictive model is applied

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