A methodology to quantify the stochastic distribution of friction coefficient required for level walking

Abstract

The required friction coefficient is defined as the minimum friction needed at the shoe and floor interface to support human locomotion. The available friction is the maximum friction coefficient that can be supported without a slip at the shoe and floor interface. A statistical model was recently introduced to estimate the probability of slip and fall incidents by comparing the available friction with the required friction, assuming that both the available and required friction coefficients have stochastic distributions. This paper presents a methodology to investigate the stochastic distributions of the required friction coefficient for level walking. In this experiment, a walkway with a layout of three force plates was specially designed in order to capture a large number of successful strikes without causing fatigue in participants. The required coefficient of friction data of one participant, who repeatedly walked on this walkway under four different walking conditions, is presented as an example of the readiness of the methodology examined in this paper. The results of the Kolmogorov–Smirnov goodness-of-fit test indicated that the required friction coefficient generated from each foot and walking condition by this participant appears to fit the normal, log-normal or Weibull distributions with few exceptions. Among these three distributions, the normal distribution appears to fit all the data generated with this participant. The average of successful strikes for each walk achieved with three force plates in this experiment was 2.49, ranging from 2.14 to 2.95 for each walking condition. The methodology and layout of the experimental apparatus presented in this paper are suitable for being applied to a full-scale study.

Introduction

It is estimated that the annual direct cost of occupational injuries due to slips, trips and falls in the USA exceeds 6 billion US dollars (Courtney et al., 2001). Falls on the same level accounted for 65% of claim cases and 53% of claim costs in total direct workers’ compensation for occupational injuries due to slips and falls (Leamon and Murphy, 1995). Friction plays an important role in falls on the same level as summarized by Chang et al. (2001b).

Required friction is the minimum friction needed at the shoe and floor interface to support different types of human activities. When the required friction for an activity exceeds the available friction at the shoe and floor interface, a slip may occur (Redfern et al., 2001). The available friction represents the maximum frictional force that can be supported without a slip at the shoe and floor interface. The required coefficient of friction (RCOF) is typically measured on dry surfaces with a force plate and is one of the maximum values in the friction coefficient obtained by dividing the component of the measured ground reaction force (GRF) tangent to the floor surface by the normal component during a step (Redfern et al., 2001). A mechanical device, known as a slipmeter, is typically used to measure the available coefficient of friction (ACOF) at the contact interfaces (Chang et al., 2001a, Chang et al., 2001b). Hanson et al. (1999) developed a model to estimate the probability of a fall in which actual fall incidents in a laboratory environment were related to the differences between the means of the ACOF and RCOF with a logistic regression model.

The ACOF measured with slipmeters always has random variations even when identical samples are used (Chang and Matz, 2001). In addition, there are variations in friction with different samples of identical floor and shoe materials as shown by Chang and Matz (2001). Therefore, the ACOF for a particular interface is not a constant, but has a stochastic distribution. The concept of a stochastic distribution in the ACOF was discussed by Barnett (2002) and Marpet (2002).

During human locomotion, there is variation across steps. Hsiang and Chang (2002) investigated variations in the normal force at the shoe and floor interface during repeated walking. They reported the mean, standard deviation, skewness and kurtosis of the distributions in the normal force during level normal walking on a treadmill. Although results from treadmill walking might not be applicable to normal walking, most of the results on level walking reported in the literature included only the means and standard deviations (Perkins, 1978; Strandberg, 1983; Redfern and DiPasquale, 1997; Hanson et al., 1999; Cham and Redfern, 2002; Lockhart et al., 2005; Kim et al., 2005).

Recently, a statistical model was developed to compare the ACOF and RCOF in estimating the probability of a slip or fall incident (Chang, 2004). In this model, a stochastic distribution was assumed for both the ACOF (p_a) and RCOF (p_r) as shown in Fig. 1. The cumulative probability for the RCOF (μ_r) to exceed the ACOF (μ_a), according to Chang (2004), is

$$P({\mu }_{\mathrm{a}}⩽{\mu }_{\mathrm{r}})={\mathit{\int }}_{-\infty }^{+\infty }{p}_{\mathrm{a}}(x){\mathit{\int }}_x^{+\infty }{p}_{\mathrm{r}}(y)\mathrm{d}y\mathrm{d}x\text{,}$$

where p_a and p_r are the probability density functions for the ACOF and RCOF, respectively. Typically, the mean of the RCOF was compared with the mean of the ACOF and the situation was determined to be safe if the mean of the RCOF was less than the mean of the ACOF. However, in addition to differences in the mean values, the variations in the RCOF and ACOF could also contribute to the probability of a slip, as illustrated by Chang (2004). By using a stochastic distribution for the RCOF and ACOF, the estimate of the probability of a slip or fall incident may improve.

The RCOF of human locomotion has been widely investigated, as partially summarized by Redfern et al. (2001). Variations in the RCOF across and within participants were observed by Brough et al. (1979), Soames and Richardson (1985), Hoang et al. (1987) and Redfern and DiPasquale (1997). However, they did not attempt to quantify the variation statistically beyond calculating the means and standard deviations. Although the stochastic distributions of the normal force at the shoe and floor interface were quantified by Hsiang and Chang (2002), they did not investigate the friction coefficient at the same interface.

In this paper, a methodology to investigate the stochastic distributions of the RCOF for level walking at the shoe and floor interface is presented as the first step toward the estimate of slip probability based on the statistical model introduced by Chang (2004). To calculate the RCOF when a participant walks repeatedly on a leveled dry surface, a special walkway with multiple force plates was constructed for the measurement of the GRF at the shoe and floor interface. Due to the large number of successful strikes needed to investigate the stochastic distributions, the main objective of the experimental setup was to achieve this goal without fatiguing participants.