## 5 Empirical Equation for Slip Sinkage of a Track System  
During off-road tracked vehicle driving, normalized slip sinkage was similar regardless of changes in surcharge mass and relative density of the soil; the normalized slip sinkage increased with slip displacement and gradually converged. Accordingly, normalized slip sinkage in this study is shown as a function of slip displacement, allowing slip sinkage to be predicted based on the static sinkage calculated according to driving conditions (Figure 11). Figure 10: Normalized slip sinkage and slip displacement relationships for (**a**) \(D_{r}\) = 40%, (**b**) \(D_{r}\) = 60%, and (**c**) \(D_{r}\) = 80%. Considering that the normalized slip sinkage-slip displacement relationship shows a sigmoid function form, a regression analysis was conducted with various sigmoid functions (Boltzmann, Logistic, and Weibull). The Weibull function showed the best regression results, as evidenced by the highest \(R^{2}\) (Equation (5)), which was then used to evaluate the normalized slip sinkage-slip displacement relationship equation. \[y=A_{2}\Big{(}1-e^{(-(k(x-A_{1}))^{j})}\Big{)} \tag{5}\]  
Here, \(A_{1}\) and \(A_{2}\) represent the minimum and maximum values of the graph, respectively, while \(k\) and \(p\) are constants determined by the shape of the curve. As shown in Figure 11, the constants of Equation (5) were determined after assessing the best-fit curve, that is, the curve that best predicts the normalized slip sinkage-slip displacement relationship. The determined constants were substituted into Equation (5). The following equation to evaluate the slip displacement of the off-road tracked vehicle was proposed. \[z_{j}=z_{0}\Big{(}2.78-2.78e^{(-0.009j.177)}\Big{)} \tag{6}\]  
Here, the unit of \(j\) is mm. As mentioned earlier, the extent of slip displacement (\(j\)) occurring in one continuous-track system is proportional to the longitudinal distance traveled by the track system (\(x\)). As confirmed by the model experiment results, slip sinkage increases with slip displacement; it increases with the longitudinal distance traveled by the track system. By substituting Equation (1) into Equation (6), Equation (7) can be used to estimate the slip sinkage according to the longitudinal position (\(x\)) and slip ratio (\(i\)) of the track system. \[z_{j}=z_{0}\Big{(}2.78-2.78e^{(-0.009(ix)^{1.77})}\Big{)} \tag{7}\]  
Here, the unit of \(x\) is mm. That is, as shown in Figure 12, slip sinkage does not occur at the frontmost part (\(x=0\)) of the soil-track interface; it occurs predominantly at the end part (\(x=1\)) of the soil-track interface. On the other hand, static sinkage due to vehicle weight (\(z_{0}\)) occurs evenly throughout the entire soil-track interface. Therefore, the total ground sinkage (\(z_{t}\)) increases by the amount of slip sinkage (\(z_{j}\)) induced when the track system moves longitudinally. To verify the applicability of the slip sinkage assessment results and proposed equation, additional model experiments were carried out using two model track systems with material properties different from those used thus far in this research. The properties of model track systems for verification are shown in Table 4. Soil samples with relative densities similar to those used thus far (40% and 80%) were used for the model ground. The slip sinkage-slip displacement curve was assessed from the  
Figure 11: Curve fitting of the normalized slip sinkage and slip displacement relationships. verification experiment results obtained using the same procedure as described above. Figure 13 shows the comparison between experimental values and those predicted using the proposed model (Equation (6)). Figure 12: Ground sinkage developed on the soil–track interface. Figure 13: Comparison between experimental values and the values predicted by the proposed model: (**a**) Model (**a**); and (**b**) Model (**b**).