A movement criterion for running

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Abstract

The adjustment of the leg during running was addressed using a spring-mass model with a fixed landing angle of attack. The objective was to obtain periodic movement patterns. Spring-like running was monitored by a one-dimensional stride-to-stride mapping of the apex height to identify mechanically stable fixed points.

We found that for certain angles of attack, the system becomes self-stabilized if the leg stiffness was properly adjusted and a minimum running speed was exceeded. At a given speed, running techniques fulfilling a stable movement pattern are characterized by an almost constant maximum leg force. With increasing speed, the leg adjustment becomes less critical. The techniques predicted for stable running are in agreement with experimental studies.

Mechanically self-stabilized running requires a spring-like leg operation, a minimum running speed and a proper adjustment of leg stiffness and angle of attack. These conditions can be considered as a movement criterion for running.

Introduction

An almost sinusoidal pattern of the ground reaction force is observed in many types of fast animal and human locomotion (Cavagna et al (1964), Cavagna et al (1977); Alexander et al., 1986; Full et al., 1991; Farley et al., 1993). Although this includes both energy production (muscle fibers) and absorption (soft tissue, ligaments, muscles) the leg stiffness is surprisingly constant during the stance time.

Blickhan (1989) and McMahon and Cheng (1990) introduced a simple spring-mass model to approximate this generally observed force pattern. This representation of the leg by a linear spring was successfully applied by biologists (Blickhan and Full, 1993; Farley et al., 1993; Farley and Gonzalez, 1996), sport scientists (Arampatzis et al., 1999; Seyfarth et al., 1999), and bioengineers (Herr, 1998) to describe and predict animal and human locomotion. However, there is only little known about the advantages of this type of leg operation.

The spring-like loading of a segmented leg can be achieved by elastic operation of the leg joints. To guarantee a homogeneous loading of the leg joints, nonlinear torque-angular displacement characteristics (MΔϕν) with exponents ν>1.5 are necessary (Seyfarth et al., 2000). For forcibly loaded legs, this characteristic may be supported by passive properties of the tendons connecting the muscles to the skeleton. Experimentally observed exponents in tendon stress–strain relationships (Ker, 1981) are well suited to result in almost linear spring-like behavior of the leg (Seyfarth et al., 2001). Therefore, the linear leg spring is a concept to solve the kinematic redundancy problem of a segmented leg.

If a movement objective (e.g. performance criteria) is given, the spring-like leg behavior may guide to understand technical aspects of locomotion as shown for the long jump (Seyfarth et al., 1999). Here, the maximum jumping distance specifies the required adjustment of leg stiffness and angle of attack. Unfortunately, such an objective is not known for running yet.

An approach to predict the leg spring adjustment in running was made by Blickhan (1989) and McMahon and Cheng (1990), which showed that for given parameters (running speed, leg stiffness, angle of attack) the spring-mass model might produce symmetric trajectories of the center of mass. Nevertheless, they did not prove whether the predicted solutions are stable with respect to deviations in landing conditions or leg stiffness. More recently, Schwind (1998) showed that for a running spring-mass system only symmetric stance phases with respect to the vertical axis might result in cyclic movement trajectories. As there is no analytical solution of the planar spring-mass system known, he investigated the system by using nonlinear spring characteristics and adapted controllers. The stability of the system with a simple linear spring was not investigated.

The aim of this study is to investigate the stability of spring-like leg operation during running at a constant speed. Therefore, a stride-to-stride analysis of a conservative spring-mass model is used. At given initial conditions we identify appropriate leg adjustments (stiffness, angle of attack) resulting in a periodic running pattern. The number of successful strides serves as a measure for periodicity. The variation of the stride number for different leg adjustments provides a measure of running robustness. The predicted leg operation for periodic running movements is compared to an experimental study on human running.

Section snippets

Experimental data acquisition and analysis

In order to prove the predictions of the running model, the dynamic and kinematic parameters during bare-foot running were recorded in an experimental study with 12 students (body weight m=69.5±9.8 kg, height 1.77±0.08 m). The subjects were instructed to run across a force plate (initial leg length ℓ0=0.94±0.06 m) at moderate speed (vX=4.6±0.5 m/s). In total 67 contact phases were analyzed in terms of ground reaction forces and kinematic landmarks of the stance leg (hip, knee, ankle and ball of the

Results

The analysis of the spring-mass system revealed that there exist leg adjustments (leg stiffness, angle of attack), which lead into periodic limit cycles in the movement pattern. These solutions proved to be robust with respect to adjustment errors and variations in kinematic parameters (speed, initial apex height). After giving some representative examples we will analyze the mechanisms of self-stabilizing running by using a stride-to-stride analysis of the spring-mass model.

Discussion

Our investigation was based on the spring-mass model with a fixed adjustment of the landing leg. Due to the simplicity of this approach, it was possible to explore the configuration space of the system. It allowed us to identify periodic limit cycles using a stride number analysis and a stride-to-stride analysis in terms of a one-dimensional return map of the apex height.

Spring-like leg operation within the proper conditions facilitates control during periodic running exercises. The system

Acknowledgements

This work was supported by the German Science Foundation (DFG) within the Innovationskolleg ‘Bewegungssysteme’ (INKA22 Project C1).

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