## 2 The Optimization Design of Parameters and Parallel Wheel Travel Simulation
\begin{split}& A=a_{1}^{2}\cdot a_{2}^{2}+a_{2}^{2}\cdot c_{1}^{2}+a_{ 1}^{2}\cdot c_{2}^{2}\\ & B=-2\cdot\left(a_{2}^{2}\cdot b_{1}\cdot c_{1}+a_{1}^{2}\cdot b _{2}\cdot c_{2}\right)\\ & C=a_{2}^{2}\cdot b_{1}^{2}+a_{1}^{2}\cdot b_{2}^{2}-R_{N}^{2} \cdot a_{1}^{2}\cdot a_{2}^{2}\\ & a_{1}=k_{NC}^{y}\cdot k_{NF}^{z}-k_{NF}^{y}\cdot k_{NC}^{z}\\ & a_{2}=k_{NF}^{y}\cdot k_{NC}^{z}-k_{NC}^{y}\cdot k_{NF}^{z}\\ & b_{1}=P\cdot k_{NF}^{z}-Q\cdot k_{NC}^{z}\\ & b_{2}=P\cdot k_{NF}^{y}-Q\cdot k_{NC}^{y}\\ & c_{1}=k_{NC}^{x}\cdot k_{NF}^{z}-k_{NF}^{x}\cdot k_{NC}^{z}\\ & c_{2}=k_{NC}^{x}\cdot k_{NF}^{y}-k_{NF}^{x}\cdot k_{NC}^{y}\\ & P=\left(R_{C}^{2}-k_{NC}^{x}\;2-k_{NC}^{y}\;2-k_{NC}^{y}\;2-R_{ N}^{2}\right)/2\\ & Q=\left(R_{F}^{2}-k_{NF}^{x}\;2-k_{NF}^{y}\;2-k_{NF}^{y}\;2-R_{ N}^{2}\right)/2\\ \end{split}  \tag{12}\]  
In the process of vertical wheel travel, point B serves as the semi-axis power input point. However, its distance from the longitudinal arm endpoint \(M\), as well as the connecting points \(C\) and \(J\) of the wheel support, remains constant. Let the distance BM be denoted as Rm and the distances \(BC\) and \(BJ\) be denoted as \(R_{BC}\) and \(R_{BJ}\). Similarly, applying the method used to solve for point M as described above, the coordinates of point B can be determined:  
\[\Sigma(X_{B}-X_{i})^{2}=R_{i}^{2} \tag{13}\]  
in which: \(X=x,y,z;i=N,BC,BJ\). As the sought point represents an intersection point in the spatial sphere, employing different computational methods may yield solutions that satisfy the constraint equations but do not correspond to the desired point. Therefore, the coordinates of points B \((x_{B},y_{B},z_{B})\) are determined using the method described in the reference. Set:  
\[ \begin{split}\overrightarrow{JC}&=\left(\begin{array} []{cc}k_{JC}^{x}\cdot\widehat{x}&k_{JC}^{y}\widehat{y}&k_{JC}^{z}\cdot \widehat{z}\end{array}\right)\\ \overrightarrow{MC}&=\left(\begin{array}{cc}k_{MC}^{x}\cdot\widehat{x}&k_{ MC}^{y}\widehat{y}&k_{MC}^{z}\cdot\widehat{z}\end{array}\right)\end{split} \]  
Then:  
\[ \begin{split} z&=\frac{-S+\sqrt{S^{2}-4\cdot R\cdot\overline{T}}}{2 \cdot R}\\ x&=U_{1}+V_{1}\cdot z\\ y&=U_{2}+V_{2}\cdot z\end{split}  \tag{14}\]  
in which:  
\[ \begin{split} R&=V_{1}^{2}+V_{2}^{2}+1\\ S&=2\left[\left(U_{1}-x_{I}\right)\cdot V_{1}+\left(U_{2}-y_{I} \right)\cdot V_{2}\right]\\ T&=\left(U_{1}-x_{J}\right)^{2}+\left(U_{2}-y_{I}\right)^{2}+ \left(z_{J}^{2}-R_{BJ}^{2}\right)\\ U_{1}&=\frac{k_{1}k_{JC}^{y}-k_{2}k_{MC}^{y}}{2\left(k_{ JC}^{y}\cdot k_{MC}^{y}-k_{MC}^{z}\cdot k_{JC}^{y}\right)}\\ V_{1}&=\frac{k_{JC}^{y}+k_{MC}^{y}-k_{MC}^{z}\cdot k_{NC}^{y}}{k_{ MC}^{y}-k_{MC}^{z}\cdot k_{MC}^{y}}\\ U_{2}&=\frac{k_{1}k_{JC}^{z}-k_{2}k_{MC}^{z}}{2\left(k_{ MC}^{y}\cdot k_{NC}^{y}-k_{CC}^{z}\cdot k_{MC}^{y}\right)}\\ V_{2}&=\frac{k_{1}k_{JC}^{z}-k_{MC}^{z}\cdot k_{NC}^{z}}{k_{ MC}^{z}\cdot k_{CC}^{y}-k_{CC}^{y}\cdot k_{NC}^{z}}\\ k_{1}&=\left(x_{C}^{2}-x_{M}^{2}\right)+\left(y_{C}^{2}-y_{M}^{2} \right)+\left(z_{C}^{2}-z_{M}^{2}\right)\\ k_{2}&=\left(x_{C}^{2}-x_{I}^{2}\right)+\left(y_{C}^{2}-y_{I}^{2} \right)+\left(z_{C}^{2}-z_{I}^{2}\right)\end{split} \]In the process of vertical wheel travel, the center point of the wheel undergoes vertical movement in response to the angular displacement of the upper and lower pivoting arms, causing changes around the kingpin _CF_. Meanwhile, point \(A\) maintains a constant distance from the longitudinal arm endpoint M and the external endpoints C and F of the wheel support. Let the distance _AM_ be denoted as _RA_, and the distances _AC_ and _AF_ be denoted as _RAC_ and _RAF_. Similarly, following the method used to solve for point B as described above, the coordinates of point A can be determined:  
\[\Sigma(X_{A}-X_{i})^{2}=R_{i}^{2} \tag{15}\]  
in which: \(X=x,y,z;i=A,AC,AF\). During the suspension motion, vector \(\overrightarrow{AB}\) consistently remains perpendicular to the wheel plane xo3y, even as vector \(\overrightarrow{AI}\) maintains orthogonality with vector \(\overrightarrow{AB}\).