## 3 Stochastic Analysis of Ride Comfort Index
### Sensitivity Analysis of Ride Comfort  
To improve ride comfort, it is important to understand which segment of the vehicle or human body is closely related or sensitive to ride comfort. In the presence of uncertainties of a parameter, we performed human-vehicle simulation and obtained the probability distributions of ride index as illustrated in Figure 12. The results for translational parameters (\(k\) and \(c\)) are displayed in the left column, and those for torsional parameters (\(k^{T}\) and \(c^{T}\)) are in the right column. The simulations were performed when the standard deviation of each uncertain parameter has 5% of its mean value. As it may be expected, the shape of distributions is quite different depending on the uncertain parameters. Surprisingly, the range of the ride index for the translational parameters is at least 20 times wider than those for the torsional parameters. This implies that the translational parameters have much stronger effects on the uncertainty of the ride index than the torsional parameters. It can also be interpreted as the ride index is dominated by the fundamental frequency, which is mainly influenced by the translation parameter of human model [38]. In addition, it can  
Figure 11: Ride indices with respect to the road roughness for a different uncertain parameter, (**a**) \(k_{1}\) (**b**) \(k_{3}\) and (**c**) \(k_{1}^{T}\), where the error bar represents 3\(\sigma\). be seen that the distribution for \(k_{2}^{T}\) and \(c_{2}^{T}\) have a much shaper shape than \(k_{1}^{T}\) and \(c_{1}^{T}\), as shown in Figure 12b,d and Table 3. In other words, the hip joint is more sensitive than the neck part of human body. In a physical sense, it is easy to verify because the acceleration of the hip joint and trunk is more dominant than others in evaluating ride comfort [46]. In conclusion, it was found that the variations in \(k_{2}^{T}\) and \(c_{2}^{T}\), which connects the trunk and the head, has little effect on the movement of the hip joint and the trunk. It is also interesting to investigate the differences between the linear and nonlinear human models concerning human-vehicle simulation in terms of stiffness uncertainties. In this study, linearization was performed using Taylor expansion under a small motion assumption. The mean and standard deviation of the ride index are presented in Figure 13 at the top and bottom for the nonlinear and linear models, respectively. The ride indices were plotted as colormaps to simultaneously look over the effect of the uncertain parameters. For comparison, each parameter varied from 20% to 180% of the reference value and was expressed as a dimensionless value denoted by a superscript *. For example, \(k_{1}^{*}=1\) means an initial value, and \(k_{1}^{*}=0.2\) means the 20% of the initial value. The standard deviation of each uncertain parameter was also set to 5% of the mean value, and the human-vehicle simulation was performed on a C-class road. In conclusion, the mean values of the ride index for the linear and nonlinear models show similar trends. The higher the stiffness, the worse the ride comfort. However, the mean of the ride index of the nonlinear model shows a higher value than that of the linear model, and it is probably because the nonlinear model allows for large movements. For example, when \(k_{1}^{*}\) and \(k_{3}^{*}\) increases by 40%, the ride index increases by 25.4% from 0.451 to 0.604 in the nonlinear model (Figure 13A), while it increases by 20.5% from 0.421 to 0.529 in the linear model (Figure 13B). On the contrary, for damping, the higher the damping, the better the ride comfort. When \(c_{1}^{*}\) and \(c_{3}^{*}\) decreases by 40%, the ride index increases by 21.4% from 0.451 to 0.574 in the nonlinear model (Figure 14A), while it increases by 18.6% from 0.421 to 0.517 in the linear model (Figure 14B).