## 4 Discussion
With the introduction of wet or soapy substrate conditions, the COF dramatically decreased as the adhesion dropped, and the COF values oscillated as hysteresis increased. There was a slight decreasing trend for the COF values at the speed of 50 mm/s and a slight increase at the speed of 300 mm/s. The regression analysis, which considered all performed measurements (dry, wet, and soapy, at all speeds, and on smooth and anti-slip tiles) at the same time, resulted in Equation (1). The coefficients of the noted equation indicated the following: as the coefficient of hysteresis of the rubber was 0.686 with a negative sign, it showed that with an increase in hysteresis, the COF decreased (considering all measurements, dry, wet, and soapy, at all speeds, and on smooth and anti-slip tiles). The rubber hardness had a coefficient of 0.000012, indicating that an increase in hardness slightly increased the COF. The same effect was observed with the roughness of the substrate, with a coefficient of 0.018 but with a negative sign (the COF decreased). By the regression analysis results, speed had a small coefficient of influence on the COF (0.0000318), and the last factor was the coefficient of the substrate conditions (dry, wet, and soapy). Again, by the regression analysis results, the substrate conditions had a significant impact on the COF, with a coefficient of \(-0.459\). The regression analysis results did not align well with the experimental data, showing that regression was inadequate for analyzing these experimental results. This discrepancy arose because regression assumes a linear relationship between input and output parameters, which does not apply to this specific case of rubber friction. However, the regression analysis clearly indicated that rubber hysteresis significantly affects the COF of rubber. Due to the inadequacy of regression analysis, an additional statistical method, the Taguchi method, was included in the analysis, providing results that were more consistent with the experimental data. According to the Taguchi method, considering all parameters except rubber hardness (Figure 5), due to its high degree of correlation with other parameters, the COF was clearly strongly influenced by rubber hysteresis. The Taguchi method showed that the surface condition had a predominant influence, which was logical and in accordance with experimental results. Also, the surface conditions impacted the significance of other parameters, as seen from Figures 5 and 6, as the parameters showed different trends for the dry surface on one side and for wet and soapy surfaces on the other. As already noted, under dry conditions, the primary friction mechanism is linked to adhesion, which can explain the difference. The introduction of lubrication (water, soap) meant that adhesion decreased and other parameters gained significance and changed their overall influence. Nevertheless, the increase in hysteresis to a certain point (0.35) led to a decrease in the COF. With a further hysteresis increase up to 0.39, the COF increased and then decreased with a further hysteresis increase. This effect was also observed in the experimental results in most cases, except under dry conditions, where the experiments showed that there was a COF increase for the anti-slip tile at all hysteresis values above 0.35. One can only speculate the reason behind it, and it is necessary to conduct further research into how and why the specific values of hysteresis (between 0.35 and 0.39) lead to COF change. From the practical side, it is beneficial to know that the best anti-slip properties would be achieved with the noted hysteresis values in dangerous (slip prone) situations (wet, soapy). As the results of the Taguchi method showed, an increase in roughness led to an increase in the COF under wet and soapy conditions (Figure 6) due to a decrease in adhesion. Under dry conditions, the roughness increase acted negatively on the COF, which was also clear from the Taguchi method results shown in Figure 5, where all surface conditions are represented.