## 3 Stochastic Analysis of Ride Comfort Index
### gPC Results for Human-Vehicle Model with Uncertain Parameters  
A fully coupled human-vehicle simulation was performed by combining CarSim and MATLAB/Simulink in the presence of the parametric uncertainty of the human body model. In this stochastic analysis, it is assumed that there are three uncertain parameters, such as two translational stiffness (\(k_{1}\), \(k_{3}\)) and one torsional stiffness (\(k_{1}^{T}\)). It also assumes that the translational springs connected to the same human body are identical to reduce computational costs, that is, \(k_{1}=k_{2}\) and \(k_{3}=k_{4}\) (see Figure 1). A single uncertain parameter was selected for the dynamic simulations in turn, and the accelerations of the human model were extracted with respect to the corresponding parameter. The second order of the polynomial chaos was considered to obtain the stochastic responses of the human model. In other words, single-dimensional second-order polynomial chaos expansion was used in the approximate model based on gPC. In addition, those with uncertain stiffness were assumed to be the Gaussian distributed variable. They can be represented as follows:  
\[ \begin{array}{l}k_{1}(\xi_{1})=\mu_{k_{1}}+\sigma_{k_{1}}\xi_{1}\\ k_{3}(\xi_{2})=\mu_{k_{3}}+\sigma_{k_{3}}\xi_{2}\\ k_{1}^{T}(\xi_{3})=\mu_{k_{1}^{T}}+\sigma_{k_{1}^{T}}\xi_{3}\end{array}  \tag{14}\]  
where \(\mu_{k}\) and \(\sigma_{k}\) represent the mean and standard deviation values of the corresponding stiffness, respectively. In this study, the standard deviation of the uncertain stiffness was assumed to be 5% of the corresponding mean value. The conditions of the human-vehicle simulations are as follows. The simulated vehicle model used in CarSim was an E-class sedan. In each segment, the accelerations of the human model were calculated when the vehicle was traveling at a constant speed of 60 km/h in the longitudinal direction. The steering angle of the vehicle was controlled to maintain straight-line driving. The total running time and sampling frequency were set to 50 s and 1000 Hz, respectively. However, in the dynamic simulations, only times between 5 and 50 s are analyzed to eliminate transient effects due to accelerations fluctuating to the target speed. Since MCS requires a huge number of simulations to achieve satisfactory convergence, we also developed an automatic process for human-vehicle simulations using the COM interface in CarSim. Automation is implemented by generating a set of samples for uncertain parameters and controlling the COM interface with a MATLAB script. A detailed description of the automatic process follows. First, generate the samples corresponding to the probability distribution in the MATLAB script. The number of samples at this stage depends on the number of MCS and gPC runs. Second, launch MATLAB/Simulink in CarSim after executing the COM interface of CarSim in the MATLAB script. Third, provide the generated set of samples as input to the function corresponding to the human model in MATLAB/Simulink. Fourth, once the simulation is performed, save the response of generalized coordinates of the human model for each sample in the workspace of MATLAB. Then, terminate the CarSim and MATLAB/Simulink via MATLAB script. Finally, repeat steps 2 to 4 as many times as the number of simulations of MCS and gPC. For the three uncertain parameters \(k_{1}\), \(k_{3}\), and \(k_{1}^{T}\), the variations of the vertical acceleration at the hip joint of the human model were calculated using gPC and MCS, as shown in Figure 8. Vertical accelerations close to zero are displayed in light gray to indicate the distribution of probability densities (see the color bar on the right). Several results are observed. First, it is clear that the accelerations calculated by the approximation model based on gPC show good agreement with those obtained through the use of MCS for all three types of uncertain parameters. To obtain a satisfactory convergent result, gPC requires only three times simulations for three collocation points, while MCS does 500 times simulations.