## 5 Simulation Test of the Grouser Effect
### Establishment of the Regression Equation of the Grouser Effect
\begin{table}
\begin{tabular}{c c c c c} \hline
**Definitions** & **Estimated Value** & **Standard Error** & \(\mathbf{T}\) & \(\mathbf{p}\) & **Sig.** \\ \hline Intercept & 85.568 & 2.764663 & 20.95 & \(<\)0.0001 & \(\ast\) \\ \(h\) & 56.29 & 2.54312 & 22.13 & 0.00000 & \(\ast\) \\ \(l\) & \(-\)12.827 & 2.54312 & \(-\)5.04 & 0.00050 & \(\ast\) \\ \(h\times h\) & \(-\)3.545 & 2.843294 & \(-\)1.25 & 0.01563 & \(\ast\) \\ \(h\times l\) & \(-\)4.095 & 4.849539 & \(-\)0.84 & 0.01716 & \(\ast\) \\ \(\lambda\times h\) & 0.535 & 4.849539 & \(-\)0.11 & 0.04026 & \(\ast\) \\ \(\lambda\) & \(-\)5.595 & 2.54312 & \(-\)2.20 & 0.05244 \(\char 130\) & \(\ast\) \\ \(\lambda\times l\) & 14.1 & 4.849539 & 2.91 & 0.24089 & \\ \(l\times l\) & \(-\)8.1125 & 2.843294 & \(-\)2.85 & 0.41818 & \\ \(\lambda\times\lambda\) & \(-\)6.6975 & 2.843294 & \(-\)2.36 & 0.91434 & \\ \(S\) = 8.0421 & \(R^{2}\) = 0.982056 & Adjustment \(R^{2}\) = 0.965906 & \(R\)-\(sq.\) (predict) = 80.28\% \\ \hline \end{tabular}
\end{table} 
Table 7: Parameter estimates and ANOVA table for the model. \begin{table}
\begin{tabular}{c c c c c} \hline
**Definitions** & **Estimated Value** & **Standard Error** & \(\mathbf{T}\) & \(\mathbf{p}\) \\ \hline Intercept & 84.856 & 2.475419 & 34.28 & \(<\)0.0001 \\ \(h\) & 56.29 & 2.475419 & 22.74 & 0.00000 \\ \(l\) & \(-\)12.827 & 2.475419 & \(-\)5.18 & 0.00018 \\ \(h\times h\) & 11.964 & 3.500771 & 3.42 & 0.00459 \\ \(h\times l\) & \(-\)8.1125 & 2.767602 & \(-\)2.93 & 0.01169 \\ \(\lambda\times h\) & \(-\)6.6975 & 2.767602 & \(-\)2.42 & 0.03091 \\ \(\lambda\) & 84.856 & 2.475419 & \(-\)2.26 & 0.04161 \\ \(S\) = 7.828 & \(R^{2}\) = 0.97898 & Adjustment \(R^{2}\) = 0.967697 & \(R\)-\(sq.\) (predict) = 90.84\% \\ \hline \end{tabular}
\end{table} 
Table 8: Parameter estimates and ANOVA table for the reduced model. According to the different status of the data residuals after simplification of the model, it was determined whether the constructed model was normal. In Figure 14a, the residuals' scatter points are irregularly distributed, and there is no special funnel shape or trumpet shape, so it can be judged that the fitting model of the response is normal. By analyzing the normal distribution of the residual, the \(p\)-value was 0.161, which is greater than 0.05, the residuals conform to a normal distribution, as shown in Figure 14b [25]. By observing whether the scatterplot of the residual error and each response was curved, we could judge whether the model was abnormal. In Figure 15, the scatter diagram of the residual error and the response values of the grouser's height \(h\), the track shoe's pitch \(l\), and the ratio \(\lambda\) of the grouser's thickness to the track shoe's length shows that a normal graph, and thus the model was determined to be normal. The regression equation constructed for the measured traction force data is  
\[ \begin{split} FX_{1}&=85.568-5.595\lambda_{1}+56.2 9h_{1}-12.827l_{1}\\ &-6.6975\lambda_{1}h_{1}-3.545\lambda_{1}l_{1}-h_{1}l_{1}\\ &+0.531\lambda_{1}^{2}+14.1h_{1}^{2}-4.095l_{1}^{2}\end{split}  \tag{15}\]  
Figure 14: Analysis of residual error. (**a**) Scatterplot of the predicted values versus the residuals; (**b**) normal probability plot. Figure 13: Normal comparison chart of the measured values and the predicted values of traction. where  
\[ \begin{array}{l}\lambda_{1}=\frac{\lambda-0.2}{0.1},\ \lambda\in[0.1,\ 0.3]\\ h_{1}=\frac{h-15.3}{10.3},\ h\in[5,\ 25.6]\\ l_{1}=\frac{-16}{4},\ l\in[12,\ 20]\end{array} \]  
After optimization of each response, the regression equation constructed from the predicted traction data is  
\[ \begin{array}{l}FX_{2}=84.856-5.595\lambda_{1}+56.29h_{1}-12.827l_{1}\\ -6.6975\lambda_{1}h_{1}-8.1125h_{1}l_{1}+11.964\lambda_{1}^{2}\end{array}  \tag{16}\]