## 2 The Optimization Design of Parameters and Parallel Wheel Travel Simulation
Given the fixed length of the wheel radius \(\parallel\overrightarrow{AI}\parallel\), it is possible to determine the coordinates of the wheel contact point I. A local rigid coordinate system \(o_{3}xyz\) is established at the center of the wheel, with the unit vector \(\widehat{z}_{3}^{\text{s}}\) in the \(z\)-direction oriented perpendicular to the wheel plane pointing outward. The \(y\)-axis is directed upward, and the \(x\)-axis direction is determined by the right-hand rule. Then:  
\[\widehat{z}_{3}=-\overrightarrow{AB}/\parallel\overrightarrow{AB}\parallel =\left[ \begin{array}{cc}\widehat{z}_{3}^{y}&\widehat{z}_{3}^{z}&\widehat{z }_{3}^{z}\end{array} \right] \tag{16}\]  
Based on this, the transformation matrix \(E_{o_{3}o}\) expressing the relative transformation of the body coordinate system \(o_{3}\) concerning the global coordinate system \(o\) can be determined:  
\[E_{o_{3}o}= \begin{bmatrix}C_{\alpha}&S_{\beta}&0\\ -C_{\alpha}S_{\beta}&C_{\alpha}C_{\beta}&S_{\alpha}\\ S_{\alpha}S_{\beta}&-S_{\alpha}C_{\beta}&C_{\alpha}\end{bmatrix}  \tag{17}\]  
in which:  
\[\alpha=\arccos(\widehat{z}_{3}^{\text{s}}),\,\beta=\arctan(-\widehat{z}_{3}^{ \text{s}}/\widehat{z}_{3}^{\text{s}})\]  
The coordinate matrix in the global coordinate system \(o\) after the wheel oscillation can be expressed as follows:  
\[\overrightarrow{oI}=E_{o_{3}o}\times(\overrightarrow{AI}^{0}+\overrightarrow {oA})\times E_{o_{3}o}^{-1} \tag{18}\]  
By substituting the obtained coordinates of the hard points during the wheel travel process into Equations (1)-(3), the lateral slip \(\Delta S_{\text{lat}}\), camber angle \(\psi\), toe angle \(\theta\), and other suspension motion characteristics at the center of the wheel can be solved based on the aforementioned formulas. According to the current technical specifications, the preliminary design coordinates of the hard points are presented in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hard Point** & **X** & **Y** & **Z** \\ \hline hpl\_tiered\_inner & \(-1300\) & \(-1846\) & \(0\) \\ hpl\_tiered\_outer & \(-1236\) & \(-3800\) & \(0\) \\ hpl\_uca\_outer & \(20\) & \(-3300\) & \(347.66\) \\ hpl\_uca\_rear & \(500\) & \(-1846\) & \(400\) \\ hpl\_uca\_front & \(-965\) & \(-1846\) & \(400\) \\ hpl\_uca\_outer & \(-50\) & \(-3400\) & \(-566.99\) \\ hpl\_uca\_rear & \(200\) & \(-1646.4\) & \(-500\) \\ hpl\_uca\_front & \(-950\) & \(-1646.4\) & \(-500\) \\ hpl\_wheel\_center & \(0\) & \(-3628.41\) & \(-77\) \\ hpl\_top\_mount & \(0\) & \(-2116.94\) & \(2055.09\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial Hardpoint Coordinate. Based on the preliminary design coordinates of the hard points, a physical model of the double-wishbone front suspension was created in SolidWorks 2023, as depicted in Figure 1(a). Subsequently, this model was imported into ADAMS/Car 2020, as illustrated in Figure 1(b). Performing a parallel-wheel travel simulation on the suspension model simulates the actual conditions of suspension motion when the wheel encounters obstacles or other situations that cause the tire to bounce during its operation. Based on the simulation results, a comprehensive analysis of the suspension characteristics under these conditions is conducted. Setting the simulation wheel bounce from \(-50\) mm to \(50\) mm, the variation in major parameters with wheel travel range is depicted in Figure 3 after the simulation. Utilizing the double-wishbone suspension kinematic model developed in this paper for motion analysis, the comparison in the figure shows a close match between the simulation results and the model's predictions, thus validating the correctness of the virtual prototype model of the double-wishbone suspension presented in this study. 1. In simulated experiments conducted in ADAMS, the curve of the wheel's lateral slip with wheel bounce is illustrated in Figure 2(a).