A Mathematical Model of Human Dynamic Locomotion: Theoretical Bases of the Model

Abstract

The most current models of dynamic locomotion involve the use of a simple or damped spring-mass system (McMahon and Green, 1979 and Blickhan, 1989). Each of these models uses rather simple approximations (point-like mass, and massless spring) of the complex human anatomy. They use the dynamic variables but neglect the control process completely. These models do not describe a realistic behavior of the system at some instant in time. For example, previous models have kept the system stiffness k, as a constant during the support phase. In reality, however, a complicated process depending on anatomy, posture, and muscle control gives rise to a wide variation in system stiffness as the takeoff leg moves over the support foot. Therefore, the problem faced in developing an analytical approach for coaching is to develop a mathematical model that accurately describes support phase mechanisms. The purpose of this study is to create a mathematical model that reflects all features that determine jump distance. In order to create a more realistic model, it has been necessary to derive equations of motion in a spring-mass system with stiffness k, as a function of time and posture. System stiffness k(t) was calculated from jump data collected using a Bertec force plate. Jump data was also used to test the accuracy of the model by comparing calculations to measurements of a 3D Motion Analysis System. The input parameters used for our model were the touchdown angle, the velocity at touchdown, the mass of the subject, the leg and foot length, and the system stiffness kW. We found the actual jump distance and the calculated distance in agreement. Also the calculated coordinates and velocities as functions of time match the measured data. The very first tests suggest a relative deviation of less than 5%. This refined model is more accurate than previous models of dynamic locomotion. It contains all the features necessary to accurately predict flight distance as a function of initial value parameters and support phase parameters. This model now becomes a tool for coaches to design individual performance in a heuristic manner.