Using a simplest passive walking model to analyze the friction force and the slip boundary
Date Issued
2012
Date
2012
Author(s)
Lin, Hsin-Ying
Abstract
In recent years, the population of elders is increasing day by day and the fall of elders has become a common and serious problem. Nearly one third of elders fall each year and half of these elders fall more than once. Fall caused lots of problems. The most important is that falls cause a huge economic burden. In the concept of prevention is better than cure, it is highly desired to prevent and reduce the probability of elders falling so that the huge economic burden which is caused by falls can be reduced. In this thesis, it is attempted to simulate the slip situation and to find the slip boundary. According to literatures, the risk of walking downward a hill is higher than downward a stairs. Thus, in this paper, the forces when the elders walk downward a hill are investigated.
First, we built a simplest passive walking model on an inclined slope. The passive walking model has two point feet and two knee-less legs each having a point mass. It also has a third-point mass at the hip joint which represents the mass of upper body. The walking model has two degrees of freedom. One is at the sole of stance leg from horizontal to the stance leg, and the other is at the hip joint from horizontal to the swing leg. The walking model has four state variables which are the two angles and the differential of the angles. Euler-Lagrange equation is used to find the equations of motion of the passive walking model. Then, the initial conditions of the four state variables are given in Matlab so that the value of state variables can be derived at each moment with regular internal by interative integration. When the sole of the swing leg contacts with the inclined slope, impact occurs and the angular velocities become discontinuous. We assume that when impact occurs, the duration of leg change is infinitesimally small. Thus, the angular momentum of the whole system to the contact point and that of the rear leg to the hip joint are conserved. Because there are too many parameters in the equations of motion, we defined three dimensionless parameters and assumed that the masses of the stance leg and legs are equal. We can obtain different walking models by varying the dimensionless parameters which are the mass ratio and the length ratio, respectively. Finally, we can analyze the slip situation by finding the friction force and normal force during walking. If the value of friction force is smaller than the value of normal force multiplying the coefficient of static friction, we can conclude that the walking model is under a non-slip situation.
In this thesis, we use limit cycle and Poincare’s map to determine whether there are periodic solutions or not. Different slopes and mass ratios are given to investigate how the limit cycle changes. The total energy of the passive walking model is conserved in a walking cycle and is dissipated when the inelastic impact occurs. During a walking cycle, the kinetic energy and the potential energy convert to each other. Finally, we use the force balance to find the normal force and friction force during a step and use the ratio of the friction force and normal force to find the needed coefficient of static friction.
In order to simplify the walking model, we can use an inverted pendulum whose mass is the total mass of the walking model. The force acting on the inverted pendulum is the resultant force of the walking model. Because the whole system of the walking model is not a rigid body, the moment of the inverted pendulum is different from that of the walking model. Thus, the slip boundaries between these two models should be different. If we assume the walking model moves very slowly, then the discrepancy of the slip boundaries would be tolerable.
Subjects
passive walking mode
limit cycle
friction force
inverted pendulum
Type
thesis
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