## 2 Human-Vehicle Model and Road Profile for Stochastic Analysis
### Approximate Model Based on Generalized Polynomial Chaos  
Suppose the uncertain parameter of the stochastic model is expressed in a set of random variables, \(\vec{\xi}=(\xi_{1},\xi_{2},\ldots,\xi_{n})\), the model response can also be expressed using the same set of variables [44]. Therefore, the output \(\widetilde{y}(t,\xi)\) of the stochastic model can be approximated using truncated generalized polynomial chaos expansion (gPC) expressed as the set of \(\xi\), as given by [29; 45]  
\[\widetilde{y}(t,\vec{\xi})=\sum_{j=0}^{S}a_{j}(t)\phi_{j}(\xi) \tag{8}\]  
where \(\phi_{j}(\xi)\) are the generalized Askey-Wiener polynomial chaos basis functions in terms of multi-dimensional random variables \(\vec{\xi}=(\xi_{1},\xi_{2},\ldots,\xi_{n})\). Additionally, \(a_{j}\) are the deter  
Figure 6: The effects of road roughness and coupling on the reaction forces and vertical displacement of the seat floor. (**a**) Reaction force in A- and E-class roads. (**b**) Vertical displacement for the case of full and partial coupling analysis on an E-class road. ministic coefficients to be estimated. Further, the total number of polynomial terms, \(S\), can be obtained as follows:  
\[S=\frac{(p+n)!}{p!n!}-1 \tag{9}\]  
here, \(p\) indicates the order of the polynomial chaos, and \(n\) indicates the number of uncertain parameters. The orthogonal polynomial functions according to the Askey-Wiener polynomial basis are listed in Table 2.  
In this study, we assumed that the uncertain parameter has a Gaussian distribution. For the random variable with a Gaussian distribution, generalized polynomial chaos expansion employs the Hermite polynomials. The general expression of Hermite polynomials of order \(p\) is expressed as follows:  
\[\phi_{p}\big{(}\xi_{1},\xi_{2},\ldots,\xi_{p}\big{)}=(-1)^{p}e^{\frac{1}{2} \xi^{T}\xi}.\frac{\partial^{p}}{\partial\xi_{1}\cdot\partial\xi_{2}\cdots \partial\xi_{p}}\cdot e^{-\frac{1}{2}\xi^{T}\xi} \tag{10}\]  
where superscript \({}^{T}\) indicates the transpose of the random variable vector. The unknown coefficients of polynomial chaos expansion are commonly obtained through the use of the Galerkin method. However, it is not easy to apply the Galerkin method when the stochastic model equation is complicated and difficult to tractable. Therefore, in this study, the unknown coefficient was estimated using the collocation method. The method is accomplished by computing the output of the stochastic model at a set of collocation points. The collocation points of the random variable are the roots of the next higher-order polynomial [36; 44]. For example, in a one-dimensional random field, the collocation points of the second-order polynomial chaos are calculated from the root of the third-order Hermite polynomial \(\left(-\sqrt{3},0,\sqrt{3}\right)\). In the collocation method, the output of the stochastic model and approximation model based on gPC are the same at the collocation point, as follows:  
\[\mathbf{y}(t,\mathbf{\xi})=\widetilde{\mathbf{y}}(t,\mathbf{\xi})=\mathbf{T}(\mathbf{\xi})\mathbf{a}(t) \tag{11}\]  
where \(\mathbf{y}(t,\mathbf{\xi})\) represent the output vectors of the stochastic model at the collocation points, \(\mathbf{\xi}=(\xi_{1},\xi_{2},\cdots,\xi_{M})\). The subscript of \(M\) denotes the number of collocation points. The transform matrix \(\mathbf{T}\) and unknown coefficients \(\mathbf{a}(t)\) are as follows:  
\[\mathbf{T}(\mathbf{\xi})= \begin{pmatrix}\phi_{0}(\xi_{1})&\phi_{1}(\xi_{1})&\cdots &\phi_{S}(\xi_{1})\\ \phi_{0}(\xi_{2})&\phi_{1}(\xi_{2})&\cdots&\phi_{S}(\xi_{2})\\ \vdots&\vdots&\ddots&\vdots\\ \phi_{0}(\xi_{M})&\phi_{1}(\xi_{M})&\cdots&\phi_{S}(\xi_{M})\end{pmatrix} ,\ \mathbf{a}(t)= \begin{pmatrix}a_{1}(t)\\ a_{2}(t)\\ \vdots\\ a_{M}(t)\end{pmatrix} \]  
Applying the least square method to Equation (10), the vectors of unknown coefficient can be easily obtained, as follows:  
\[\mathbf{a}(t)=\Big{\{}\mathbf{T}(\mathbf{\xi})^{T}\mathbf{T}(\mathbf{\xi})\Big{\}}^{-1} \mathbf{T}(\mathbf{\xi})^{T}\mathbf{y}(t,\mathbf{\xi}) \tag{12}\]