## 4 Kinematic Analysis of the Wheel Guidance Mechanisms  
The current spatial position and orientation of the wheel, relative to the global reference frame attached to the car body (OXYZ), is completely determined by three non-collinear characteristic points (see Figure 4): the center G of the wheel (\(\rm G_{s/d}\) for the left/right wheel), and the projections \(\rm G^{\prime}\), \(\rm G^{\prime\prime}\) of the lower guidance point \(\rm M_{1}\) on the transverse and vertical axes of the wheel carrier reference frame (\(\rm PX_{\rm P}Y_{\rm P}Z_{\rm P}\)). The axes of the global reference frame OXYZ are directed in parallel to the car's axles (with the positive X-axis towards the front of the car, the positive Y-axis towards the left, and the positive Z-axis upwards), where the origin O is located at the intersection of the median longitudinal-vertical and vertical-transversal plans. The wheel carrier reference frame \(\rm PX_{\rm P}Y_{\rm P}Z_{\rm P}\) has the origin P in the center of the spindle axis, along which the local transverse axis \(\rm Y_{\rm P}\) is directed. The system of position functions for determining the global coordinates of the three points defining the local frame includes the condition that the distances between these points are constant (given that the wheel carrier is a rigid body) and the constraints imposed on the points \(\rm M_{i}\) by which the wheel carrier is guided, as follows:  
\[ \begin{array}{rcl}\rm F_{1}&=&\left(X_{G^{\prime\prime}}-X_{G^{\prime}}\right)^{ 2}+\left(Y_{G^{\prime\prime}}-Y_{G^{\prime}}\right)^{2}+\left(Z_{G^{\prime \prime}}-Z_{G^{\prime}}\right)^{2}-G^{\prime\prime}{G^{\prime}}^{2}=0,\\ \rm F_{2}&=&\left(X_{G^{\prime\prime}}-X_{G}\right)^{2}+\left(Y_{G^{\prime \prime}}-Y_{G}\right)^{2}+\left(Z_{G^{\prime\prime}}-Z_{G}\right)^{2}-G^{ \prime\prime}{G^{2}}=0,\\ \rm F_{3}&=&\left(X_{G^{\prime}}-X_{G}\right)^{2}+\left([PERSON]_{G^{\prime}}-Y_{G} \right)^{2}+\left([PERSON]_{G^{\prime}}-[PERSON]_{G}\right)^{2}-G^{\prime}{G^{2}}=0,\\ \rm F_{i}&=&\left(X_{Mi}-X_{Mi0i}\right)^{2}+\left(Y_{Mi}-Y_{Mi0i}\right)^{2} +\left(Z_{Mi}-Z_{Mi0i}\right)^{2}-\rm I_{i}^{2}=0.\end{array}  \tag{19}\]  
The guidance on the circle of the point \(\rm M_{i}\) (e.g., the guidance arm 1 in Figure 4) introduces two equations of the form \(\rm F_{i}\) in Equation (19), namely the pairs \(\rm M_{i}\,-\,M^{\prime}_{0i}\) and \(\rm M_{i}\,-\,M^{\prime\prime}_{0i}\). The global coordinates of the guidance points \(\rm M\) are determined according to the positions of the points \(\rm G\), \(\rm G^{\prime}\), and \(\rm G^{\prime\prime}\), as follows:  
\[ \begin{array}{c}\begin{bmatrix}X_{Mi}\\ Y_{Mi}\\ Z_{Mi}\end{bmatrix}=\begin{bmatrix}X_{P}\\ Y_{P}\\ Z_{P}\end{bmatrix}+\begin{bmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{bmatrix}\cdot\begin{bmatrix}X_{Mi}\\ Y_{Mi}\\ Z_{Mi}\end{bmatrix}_{P}\end{array}  \tag{20}\]  
The column matrix \(\rm[r_{P}]\) represents the position vector of the origin \(\rm P\) of the wheel carrier reference frame, expressed in the global reference frame. The global coordinates of \(\rm P\) are determined from the intersection of the spindle axis \(\rm GG^{\prime}\) with the plane perpendicular to \(\rm GG^{\prime}\) passing through \(\rm G^{\prime\prime}\):  
\[ \begin{array}{c}\left(X_{P}-X_{G^{\prime\prime}}\right)\,\cdot\,\left(X_{G^ {\prime}}-X_{G}\right)+\left(Y_{P}-Y_{G^{\prime\prime}}\right)\,\cdot\,\left( Y_{G^{\prime}}-Y_{G}\right)\,+\left(Z_{P}-Z_{G^{\prime\prime}}\right)\,\cdot\, \left(Z_{G^{\prime}}-Z_{G}\right)=0,\\ \frac{X_{P}-X_{G}}{X_{G^{\prime}}-X_{G}}=\frac{Y_{P}-Y_{G}}{Y_{G^{\prime}}-Y_{G }}=\frac{Z_{P}-Z_{G}}{Z_{G^{\prime}}-Z_{G}}.\end{array}