利用最簡走路模型分析摩擦力與滑倒之界線

Abstract

近年來，老年人的比例日益增加，而跌倒成為老年人一個很普遍且嚴重的問題。每一年，有將近三分之一的老年人曾經發生跌倒的情況，而這些跌倒的老年人中又有二分之一甚至跌倒超過一次。跌倒會造成許多後續的問題，而其中最重要的是，跌倒造成了龐大的社會成本。在預防勝於治療的理念下，本論文希望能藉由模擬長者走路時磨擦力與正向力之關係，了解產生可能產生滑倒之情況，並探討產生滑倒之臨界狀況，以期能預防並減少長者滑倒的機率，以減少因跌倒而產生之巨大的社會成本。由文獻我們發現，老年人走路時，下坡比下樓梯更容易造成危險，因此，本論文將探討老年人走在斜坡上之受力情況。 首先，我們建立一最簡被動走路模型，此模型的腳為點狀，並具有兩支無膝蓋之腳，每支腳具有一點質量，而第三個點質量為上半身之質量集中在臀部。此模型具有兩個自由度，分別在站立腳之腳底與斜坡之接觸點，其角度為從水平至站立腳之角度；另一自由度則在臀部之關節，其角度為從水平至搖擺腳之角度。此外，此模型具有四個狀態變數，分別為兩個角度以及其微分值。我們利用尤拉拉格朗日法找出此最簡被動走路模型之動態方程式，並利用Matlab將狀態變數之初始值代入，利用不斷的積分以及迭代，我們即可得到在每一間隔時的狀態變數之值。當搖擺腳之腳底與地面產生接觸時，衝擊會產生並造成角速度之不連續。我們假設衝擊產生時，站立腳與搖擺腳互換之時間非常小，並由於沒有受到其他外力，在換腳的過程中，整體系統對接觸點之角動量和後腳對臀部關節的腳動量皆會守恆。由於此動態方程式之參數很多，為了將參數減少，我們定義了三個無因次的參數，並假設站力腳與搖擺腳之質量相等。因此藉由調整質量比和長度比，我們可得到不同的走路模型。最後，我們利用此走路模型對摩擦力進行分析，若磨擦力之值小於正向力乘上最大靜摩擦係數之值，則我們可推論此走路模型不會產生滑倒之現象。 我們利用極限圈和Poincare圖進行走路模型之週期解的探討，並利用不同的無因次化參數去觀察極限圈改變之情形;而在一走路週期中，由於沒有受到其他外力，因此總能量會守恆，但在產生衝擊時會產生消散，此消散之能量則用位能去彌補;最後利用力平衡之概念找出行走一步時磨擦力與正向力之值，由其比值找出所需之地面的摩擦力係數。 為了簡化走路模型，我們可將此走路模型簡化成倒單擺模型，此倒單擺之重量和走路模型之重量相等，而其受力則為走路模型的合力。由於在走路過程中，此走路模型不是剛體，因此造成力矩之不相等，若我們假設走路之過程非常緩慢，則此誤差則可忽略。

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.