## 2 Human-Vehicle Model and Road Profile for Stochastic Analysis
### Random Road Profile and Human-Vehicle Model  
In order to analyze the effect of road surface on human-vehicle vibrations, we generated random road profiles according to ISO 8608, which classifies road roughness using power spectral density (PSD) [40]. An approximate form of the road displacement PSD can be written as follows:  
\[\Phi(\Omega)=\Phi(\Omega_{0})\bigg{(}\frac{\Omega}{\Omega_{0}}\bigg{)}^{-w} \tag{3}\]  
where \(\Omega\) is the angular spatial frequency and \(w\) is the waviness. The reference spatial frequency of \(\Omega_{0}\) is given by 1 rad/m. In addition, the PSD function can be modified as follows [41; 42]. \[\Phi(\Omega)=\frac{2\alpha\sigma^{2}}{\sigma^{2}+\alpha^{2}} \tag{4}\]  
where \(\sigma^{2}\) is the variance in the road roughness, and \(\alpha\) is the coefficient with respect to the type of road surface. The road roughness variances and coefficients are listed in Table 1. Additionally, the road profile can be approximated using the superposition of sinusoidal functions with respect to longitudinal distance, as follows:  
\[z(x)=\sum_{i=1}^{N}A_{i}\text{sin}(\Omega_{i}x-\phi_{i}) \tag{5}\]  
where the random phase angle is uniformly distributed in the range of \(0\leq\phi_{i}<2\pi\), and \(A_{i}\) denotes the amplitudes of the road profiles, defined as follows:  
\[A_{i}=\sqrt{\Phi(\Omega_{N})\frac{\Delta\Omega}{\pi}} \tag{6}\]  
where  
\[\Delta\Omega=\frac{\Omega_{N}-\Omega_{1}}{N-1}\]  
In this study, \(\Omega_{N6841. We generated the power spectral densities of five road profiles and compared them with eight road roughness references, as shown in Figure 3. Note that the higher the power spectral density, the rougher the road, and the resulting road profile falls into the ISO 8608 classification category. In addition, Figure 4 shows the road profiles generated from the power spectral densities for five different road roughness. As shown in Figure 5, to simulate the human-vehicle model, the mathematical human model is fully coupled with the vehicle model using CarSim 9.0.2 [43] in MATLAB/Simulink. The vibrations caused by the vehicle movement are transmitted to the  
Figure 4: Road profiles with respect to longitudinal distance for five road roughness. Figure 3: Power spectral densities of the generated road profiles according to ISO 8608 (gray dotted line: reference values of the road classification). human body model as a base excitation caused by the displacement of the vehicle seat floor. Additionally, the reaction forces caused by the human body are transmitted to the seat floor of the vehicle through the springs and dampers. The spring and damping forces are calculated considering the displacements between the thigh and the seat pad of the human model, as follows:  
\[F_{\mathrm{R}}=\sum_{i=1}^{4}k_{i}\delta_{i}+\sum_{i=1}^{4}c_{i}\dot{\delta}_{i} \tag{7}\]  
where \(\delta_{i}\) and \(\dot{\delta}_{i}\) indicate the translational displacements and their time rate of changes of each spring, respectively. Figure 6 illustrates the effects of road roughness and coupling on the reaction forces and the vertical displacement of the seat floor. Figure 6a shows the vertical forces generated between the human model and the vehicle model in the case of full coupling analysis. The average reaction force corresponding to the weight of the human model is about 500 N, regardless of the road class, either A or E. However, the reaction force in the E class, which is classified as a rough road, shows greater fluctuation than when the A class is used, which is classified as a smooth road. Its magnitude is more than 20% greater than average. In addition, Figure 6b shows the vertical displacement of the driver's seat when the human-vehicle model is connected with full coupling and partial coupling on an ISO E-class road. Full coupling means that the human-vehicle model transmits reaction forces in the mutual direction, and partial coupling means that only the vehicle transmits reaction forces to the human model.