# Hybrid Improved PSO Algorithm for Soil Property Parameter Estimation
## 2 Off-Road Wheel-Soil Interaction Force Model
The relationship between shear stress and soil deformation can be expressed using [PERSON]'s simplified formula [6]:  
\[\tau=(c+\sigma tan\varphi)\Big{(}1-e^{-\frac{l}{k}}\Big{)} \tag{8}\]  
The shear displacement of the soil, \(j\)[4] is as follows:  
\[j(\theta)=r[(\theta_{1}-\theta)-(1-s)(sin\theta_{1}-sin\theta)] \tag{9}\]  
By combining Equations (7)-(9), in the active region from \(\theta_{m}\) to \(\theta_{1}\):  
\[\tau_{1}=(c+\sigma_{1}(\theta)tan\varphi)\Big{(}1-e^{-\frac{r[(\theta_{1}- \theta)-(1-s)(sin\theta_{1}-sin\theta)]}{k}}\Big{)} \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Definition** & **Unit** \\ \hline \multirow{4}{*}{\(\theta\)} & The angle formed between the line connecting any & \\  & point on the contact surface to the wheel center and & \\  & the vertical direction, serving as the independent & \\  & variable in theoretical integration calculations & \\ \hline \multirow{4}{*}{\(\theta_{1}\)} & The angle between the line connecting the front & \\  & contact point of the tire–soil interface to the wheel & \\  & center and the vertical direction & \\ \hline \multirow{4}{*}{\(\theta_{2}\)} & The angle formed between the line connecting the & \\  & rear contact point of the tire–soil interface to the & \\  & wheel center and the vertical direction & \\ \hline \multirow{2}{*}{\(\theta_{m}\)} & The angular position on the contact surface where & \\  & the normal stress reaches its peak value & \\ \hline \multirow{2}{*}{\(z\)} & The vertical deformation of soil at the front contact & \\  & point of the tire–soil interface & \\ \hline \multirow{2}{*}{\(z_{2}\)} & The vertical deformation of soil at the rear contact & \\  & point of the tire–soil interface & \\ \hline \multirow{2}{*}{\(r\)} & Wheel radius & \\ \hline \multirow{2}{*}{\(T\)} & Wheel torque & N–m \\ \hline \multirow{2}{*}{\(F_{DP}\)} & Traction force & N \\ \hline \multirow{2}{*}{\(\tau_{1}\)} & Normal stress at \(\theta_{1}\) & kPa \\ \hline \multirow{2}{*}{\(\tau_{2}\)} & Normal stress at \(\theta_{2}\) & kPa \\ \hline \multirow{2}{*}{\(\sigma_{1}\)} & Shear stress at \(\theta_{1}\) & kPa \\ \hline \multirow{2}{*}{\(\sigma_{2}\)} & Shear stress at \(\theta_{2}\) & kPa \\ \hline \end{tabular}
\end{table} 
Table 1: [PERSON]’s semi-empirical model of wheel–soil interaction. In the active region from 0 to \(\theta_{m}\):  
\[\tau_{2}=(c+\sigma_{2}(\theta)tan\varphi)\times\left(1-e^{-\frac{r[(\theta_{1}- \theta)-(1-s)(sin\theta_{1}-sin\theta)]}{2}}\right) \tag{11}\]  
From the mechanical equilibrium of the soil and the analysis of soil stress, it can be concluded that:  
\[W=rb\bigg{[}\int_{0}^{\theta_{m}}\sigma_{2}(\theta)cos\theta d\theta+\int_{ \theta_{m}}^{\theta_{1}}\sigma_{1}(\theta)cos\theta d\theta+\int_{0}^{\theta _{m}}\tau_{2}(\theta)sin\theta d\theta+\int_{\theta_{m}}^{\theta_{1}}\tau_{1}( \theta)sin\theta d\theta\bigg{]} \tag{12}\]  
\[F_{DP}=rb\bigg{[}\int_{0}^{\theta_{m}}\tau_{2}(\theta)cos\theta d\theta+\int_{ \theta_{m}}^{\theta_{1}}\tau_{1}(\theta)cos\theta d\theta-\int_{0}^{\theta_{m }}\sigma_{2}(\theta)sin\theta d\theta-\int_{\theta_{m}}^{\theta_{1}}\sigma_{ 1}(\theta)sin\theta d\theta\bigg{]} \tag{13}\]  
\[T=r^{2}b\bigg{[}\int_{0}^{\theta_{m}}\tau_{2}(\theta)d\theta+\int_{\theta_{m} }^{\theta_{1}}\tau_{1}(\theta)d\theta\bigg{]} \tag{14}\]  
Performing calculations for Equations (12)-(14) using computational tools is highly complex and not suitable for real-time applications. To address this issue, the three equations were simplified. Simpson's Rule, represented by Equation (15), is a numerical integration method that approximates the definite integral by using quadratic polynomials to fit the integrand. Its accuracy is relatively high because it takes into account the behavior of the integrand over the interval, not just the values at the endpoints. \[\int_{a}^{b}f(x)dx=\frac{b-a}{6}\bigg{[}f(a)+4f\bigg{(}\frac{a+b}{2}\bigg{)}+f (b)\bigg{]} \tag{15}\]  
By substituting Equation (15) into Equations (12)-(14), the following mathematical model can be established:  
\[\left\{