## 2 Human-Vehicle Model and Road Profile for Stochastic Analysis
### Model Description and Derivation of Equation of Motion  
To calculate the responses of the seated human, the nonlinear five-degree-of-freedom human model, which was used in the previous works [38], was considered in this study. The schematic of the proposed human model is illustrated in Figure 1. The lumped parameter of the human model is composed of a head, a trunk, and a thigh with mass and moment of inertia. Each segment is connected using a torsional spring and a damper, and the trunk and thigh contact the seat through a spring and dampers, which are effective parameters combined with seat cushion and human skin. Regarding the generalized coordinates of the hip joint, \(x_{h}\) and \(z_{h}\) indicate the horizontal and vertical displacement of the hip joint, respectively, and \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) denote the angular displacement of the thigh, trunk, and head, respectively. Only a normal deflection of the springs and dampers is considered for the calculation of the spring and damping forces, respectively (see Figure 2). In fact, the effective stiffness, \(k_{i}\), connected to the thigh has different values depending on the contact location according to the experimental results [39]. However, due to the fact that the effect of the uncertainty of the design parameters on human-vehicle ride comfort is the main interest of our study, we adopted an identical value for the simplicity of analysis. Figure 1: The schematic diagram of the five-degree-of-freedom human model. The nonlinear equation of the five-degree-of-freedom human model is derived using [PERSON]'s equation. Kinetic energy (\(T\)), potential energy (\(V\)), and [PERSON]'s dissipation (\(D\)) of each segment are derived based on the displacement of the center of gravity, as follows:  
\[ \begin{array}{c}T=\frac{1}{2}\sum\limits_{i=1}^{3}{m_{i}\left({{\dot{x}}_{i}^{2 }+{z}_{i}^{2}}\right)}+\frac{1}{2}\sum\limits_{i=1}^{3}{{J_{i}}{\dot{\theta}_{ i}^{2}}}\\ V=\frac{1}{2}\sum\limits_{i=1}^{4}{{k_{i}}{\delta_{i}^{2}}}+\frac{1}{2}\sum \limits_{i=1}^{2}{{k_{i}^{T}}{\left({{\theta_{i}^{T}}}\right)^{2}}}+\sum \limits_{i=1}^{3}{{m_{i}}{g_{i}}{\Delta_{i}}}\\ D=\frac{1}{2}\sum\limits_{i=1}^{4}{{c_{i}}{\dot{\omega}_{i}^{2}}}+\frac{1}{2} \sum\limits_{i=1}^{2}{{c_{i}^{T}}{\left({{\dot{\theta}_{i}^{T}}}\right)^{2}}} \end{array}  \tag{1}\]  
where \(m_{i}\) and \(J_{i}\) denote the mass and mass moment of inertia of each segment, respectively. Additionally, \(k_{i}\) and \(c_{i}\) are translational, and \(k_{i}^{T}\) and \(c_{i}^{T}\) are torsional stiffness and damping coefficients, respectively. The translational and angular displacements of the springs are represented by \(\delta_{i}\) and \(\theta_{i}^{T}\), respectively. In addition, \(g\) is the gravitational acceleration and \(\Delta_{i}\) is the vertical displacement of each segment. The positions of the center of gravity at each segment are as follows:  
\[ \begin{array}{c}x_{1}=x_{h}+\frac{1}{2}L_{th}{\cos}{\theta_{1}},\\ x_{2}=x_{h}+\frac{1}{2}L_{tr}{\cos}{\theta_{2}},\\ x_{3}=x_{h}+L_{tr}{\cos}{\theta_{2}}+\frac{1}{2}L_{th}{\cos}{\theta_{3}},\\ z_{1}=z_{h}+\frac{1}{2}L_{th}{\sin}{\theta_{1}},\\ z_{2}=z_{h}+\frac{1}{2}L_{tr}{\sin}{\theta_{2}},\\ z_{3}=z_{h}+L_{tr}{\sin}{\theta_{2}}+\frac{1}{2}L_{h}{\sin}{\theta_{3}}\end{array} \]  
where \(x_{h}\) and \(z_{h}\) are the horizontal and vertical displacement of the hip joint. The length of the thigh, trunk, and head are denoted as \(L_{th}\), \(L_{tr}\), and \(L_{th}\), respectively. Accordingly, the translational and angular displacements of each spring are given below. \[\begin{array}{c}\delta_{1}=z_{h}-\frac{1}{2}T_{th}{\cos}{\theta_{1}}+L_{1}{ \sin}{\theta_{1}}-z_{b}-\delta_{1}^{*}\\ \delta_{2}=z_{h}-\frac{1}{2}T_{th}{\cos}{\theta_{1}}+L_{2}{\sin}{\theta_{1}}- z_{b}-\delta_{2}^{*}\\ \delta_{3}=x_{h}{\sin}{\theta_{2}^{*}}-z_{h}{\cos}{\theta_{2}  
where the thickness of the thigh and trunk are denoted as \(T_{th}\) and \(T_{tr}\), respectively.