# Hybrid Improved PSO Algorithm for Soil Property Parameter Estimation
## 2 Off-Road Wheel-Soil Interaction Force Model  
Before estimating soil parameters, it is necessary to explore the mechanical relationship between the wheel and the soil. Figure 1 depicts [PERSON]'s semi-empirical model [4], with the parameters defined in Table 1. Empirical models are commonly used in research because they balance model precision with reduced research difficulty. Based on Dr. [PERSON]'s findings [3], the static sinkage \(z_{0}\) can be expressed by Equation (1):  
\[z_{0}=\left[\frac{3W}{b(3-n)\left(\frac{k_{c}}{b}+k_{\varphi}\right)\sqrt{D}} \right]^{\lfloor\frac{2}{(2n+1)}\rfloor} \tag{1}\]  
In the equation, \(W\) represents the vertical wheel load (in newtons, N), \(b\) denotes the wheel width (in meters, m), and \(k_{c}\), \(k_{\varphi}\), and \(n\) refer to the soil cohesion modulus (kPa\(\cdot\)m\({}^{-n}\)), the soil friction angle modulus (kPa\(\cdot\)m\({}^{-n}\)), and the soil sinkage coefficient (dimensionless), respectively. The term \(D\) represents the wheel diameter (in meters, m), defined as twice the radius (\(D=2r\)), where \(r\) is the wheel radius. As indicated by the equation, \(z_{0}\) reflects only the static sinkage resulting from wheel-soil interaction, without accounting for the dynamic sinkage caused by wheel slip. Researcher [PERSON] [5] further highlighted the significance of dynamic sinkage in his study, emphasizing its dependence on slip ratio and transient soil behavior. To capture the dynamic sinkage component, [PERSON] [2] proposed a modified sinkage exponent \(n\) that incorporates the dynamic effect:  
\[n=n_{0}+n_{1}s \tag{2}\]  
where \(n_{0}\) is the constant value obtained from direct soil testing, \(n_{1}\) is the fitting constant, and \(s\) is the slip ratio. Regarding the normal stress at the contact surface between a rigid wheel and the soil, [PERSON] expressed it using the following equation:  
\[\sigma=\bigg{(}\frac{k_{c}}{b}+k_{\varphi}\bigg{)}z^{n} \tag{3}\]  
Since the theoretical determination of the maximum stress point is not well established, [PERSON] [4] used an empirical equation to determine this point:  
\[ \begin{cases}\theta_{m}=\theta_{1}(C_{1}+C_{2}s)\\ \theta_{2}=C_{3}\theta_{1}\end{cases}  \tag{4}\]  
where \(C_{1}\), \(C_{2}\), and \(C_{3}\) are coefficients obtained from fitting experimental data. These coefficients can be adjusted iteratively to improve the curve fitting. However, to reduce computational complexity and enhance efficiency, some researchers have conducted studies on these three values. [PERSON] [15] found that the \(\theta_{2}\) obtained through visual methods may be larger than the calculated \(\theta_{2}\). Since \(\theta_{2}\) is much smaller than \(\theta_{1}\), the stress in the region from \(\theta_{2}\) to 0 degrees is minimal, leading some researchers to consider \(\theta_{2}\) as approximately 0. [PERSON] [16] studied the three coefficients in scenarios where experiments or simulations are not feasible, providing suitable values. He found that the predicted results align well with the fitted model when \(C_{1}=0.5\) and \(C_{2}=C_{3}=0\). According to the geometric relationship, the equations relating \(z_{0}\) and \(\theta_{1}\) in Figure 1 are given as (5) and (6):  
\[z_{0}=r(1-cos\theta_{1}) \tag{5}\]  
\[\theta_{1}=arccos\Big{(}1-\frac{z_{0}}{r}\Big{)} \tag{6}\]  
Figure 1: [PERSON]’s semi-empirical model of wheel–soil interaction. By combining Equations (1), (3) and (5), and for the sake of computational convenience, \(C_{3}\) is set to 0, leading to the following equation:  
\[ \begin{cases}\sigma_{1}(\theta)=\Big{(}\frac{k_{\epsilon}}{b}+k_{\varphi}\Big{)} r^{n}(cos\theta-cos\theta_{1})^{n}&(\theta_{m}\leq\theta\leq\theta_{1})\\ \sigma_{2}(\theta)=\Big{(}\frac{k_{\epsilon}}{b}+k_{\varphi}\Big{)}r^{n} \Big{[}cos\Big{(}\theta_{1}-\theta\Big{(}\frac{1}{C_{1}+C_{2}s}-1\Big{)} \Big{)}-cos\theta_{1}\Big{]}^{n}&(0\leq\theta\leq\theta_{m})\end{cases}  \tag{7}\]  
The magnitude of shear stress between the wheel and the soil depends on the shear deformation of the soil.