## 3 Road Friction Estimation Method  
In order to continuously measure the road surface friction characteristics, it is necessary to continuously measure the characteristics with a fixed slip ratio. Furthermore, it is necessary to confirm the method for estimating the overall \(\mu\)-\(s\) characteristics from some combinations of the \(\mu\) and \(s\) set. In this study, the magic formula (MF) proposed by Prof. [PERSON] in Delft University of Technology [5] is used to estimate the road friction characteristics. This equation is a function introduced from empirical research, and it is shown as a transcendental function. The original MF has been complicated and improved so that the detailed characteristics of the tire can be expressed, but this basic concept is used in this paper. The basis of this equation is a combination of the sin function and the arctangent function of Equation (2), which is expressed here using three parameters as follows:  
\[\mu=a\sin\Bigl{\{}b\ \tan^{-1}(c\ s)\Bigr{\}} \tag{2}\]  
where \(a\), \(b\), and \(c\) are parameters for determining the characteristic shape, and \(s\) is the slip ratio. Since it is necessary to identify the entire property with this MF, we need to identify the parameters \(a\), \(b\), \(c\) using at least 3 sets of \(\mu\)-\(s\) values. To clarify the relationship between these characteristics and each parameter, the MF is differentiated by the slip ratio \(s\) to obtain the following Equation (3):  
\[\frac{d\mu}{ds}=\frac{a\ b\ c\cos\bigl{\{}b\ \tan^{-1}(c\ s)\bigr{\}}}{c^{2}\ s ^{2}+1} \tag{3}\]  
Here, since the differentiation of MF at \(s=0\) represents \(K_{B}\), the following Equation (4) is introduced:  
\[K_{B}=\frac{\partial\mu(s)}{\partial s}\Big{|}_{s=0}=a\ b\ c \tag{4}\]  
Next, at the point where Equation (3) becomes 0, the value at which this value peaks is peak \(\mu\) (\(\mu\)-max), so the slip ratio at which peak \(\mu\) occurs using this is shown by Equation (5):  
\[s_{P}=\frac{\tan\bigl{(}\frac{\pi}{2b}\bigr{)}}{c} \tag{5}\]  
Using Equation (5), the peak \(\mu\) (\(\mu\)-max) is expressed by the following Equation (6):  
\[\mu_{\max}=a\sin\Bigl{[}b\ \tan^{-1}\Bigl{\{}\tan\Bigl{(}\frac{\pi}{2b} \Bigr{)}\Bigr{\}}\Bigr{]} \tag{6}\]  
Further, lock \(\mu\) is obtained by substituting \(s=1\) for Equation (2), and is shown in the following Equation (7):  
\[\mu_{L}=a\sin\Bigl{(}b\ \tan^{-1}c\Bigr{)} \tag{7}\]  
In order to express each characteristic shown here, it is necessary to identify the parameters \(a\), \(b\), and \(c\) in the MF shown in Equation (2), using the experimental data. We use the experimental results of three sets of \(\mu\) and \(s\). This analysis is carried out by iteration using the steepest descent method programed by MATLAB. As an example, Figure 10 shows a comparison between the result identified using MF and the experimental result. From this result, it can be seen that although the value of lock \(\mu\) is slightly different, the \(\mu\)-\(s\) characteristics can be generally identified by using MF. The parameters of this identification result are as follows:  
\(a=0.8232\), \(b=1.6450\), \(c=21.1680\). Therefore, the results of determining these three parameters \(a\), \(b\), \(c\) based on the obtained experimental data are shown in Figure 11. Since the parameter \(a\) represents the value of the peak \(\mu\), it is compared with this value. Here, since the identification is performed using the steepest descent method, the relationship with other parameters is also examined, but from the relationship in Figure 11a, the correlation coefficient between\(\mu\)-max and the parameter \(a\) is about 0.994. It shows a very high correlation. The relationship between the remaining parameters \(b\) and \(c\) is shown in Figure 11b, and although there are large variations due to differences in shape, etc., the range that these parameters can take is clarified.