## 4 Kinematic Analysis of the Wheel Guidance Mechanisms
 \tag{21}\]  
By successive transformations, the linear system below is obtained:  
\[\rm n_{11}\,\cdot\,X_{P}+n_{12}\,\cdot\,Y_{P}+n_{13}\,\cdot\,Z_{P}=d_{1},\,n_{ 21}\,\cdot\,X_{P}+n_{22}\,\cdot\,Y_{P}+n_{23}\,\cdot\,Z_{P}=d_{2},\,n_{31}\, \cdot\,X_{P}+n_{32}\,\cdot\,Y_{P}+n_{33}\,\cdot\,Z_{P}=d_{3}, \tag{22}\]  
which is solved by the Cramer method [37], resulting in:  
\[X_{P}=\frac{\delta_{x}}{\delta},Y_{P}=\frac{\delta_{y}}{\delta},Z_{P}=\frac{ \delta_{z}}{\delta}, \tag{23}\]  
where:  
Figure 4: The global (car body) and local (wheel carrier) reference frames. \[\delta_{\rm x}= \begin{bmatrix}\rm d_{1}&n_{12}&n_{13}\\ \rm d_{2}&n_{22}&n_{23}\\ \rm d_{3}&n_{32}&n_{33}\end{bmatrix} ,\delta_{\rm y}= \begin{bmatrix}n_{11}&\rm d _{1}&n_{13}\\ n_{21}&\rm d_{2}&n_{23}\\ n_{31}&\rm d_{3}&n_{33}\end{bmatrix} ,\ \delta_{\rm z}= \begin{bmatrix}n_{11}&n_{12}&\rm d _{1}\\ n_{21}&n_{22}&\rm d_{2}\\ n_{31}&n_{32}&\rm d_{3}\end{bmatrix} ,\delta= \begin{bmatrix}n_{11}&n_{12}&\rm n _{13}\\ n_{21}&n_{22}&\rm n_{23}\\ n_{31}&n_{32}&\rm n_{33}\end{bmatrix} . \tag{24}\]  
The coefficients \(\rm a_{ij}\) (i,j = 1,2,3) of the connection matrix from the wheel carrier reference frame to the global reference frame are the cosines of the angles between the axes of the two reference frames, e.g., \(\rm a_{11}=\cos\angle(OX,\,YXp)\), having the following expressions:  
\[ \begin{array}{ll}\rm a_{11}=\frac{X_{G}-Xp}{\rm V_{G}},\ \rm a_{12}=\frac{X_{Gd}-Xp}{ \rm P_{Gd}},\ \rm a_{13}=\frac{(X_{G}-Yp)\cdot(Z_{Gd}-Zp)-(Y_{Gd}-Yp)\cdot(Z_{G}-Zp)}{R_{FG }-Xq},\\ \rm a_{21}=\frac{Y_{G}-Yp}{\rm V_{G}},\ \rm a_{22}=\frac{Y_{Gd}-Yp}{\rm P_{Gd}},\ \rm a_{23}=\frac{(X_{Gd}-Xp)\cdot(Z_{G}-Zp)-(Y_{G}-Xp)\cdot(Z_{Gd}-Zp)}{R_{FG }-Xq},\\ \rm a_{31}=\frac{Z_{G}-Zp}{\rm P_{G}},\ \rm a_{32}=\frac{Z_{Gd}-Zp}{\rm P_{Gd}},\ \rm a_{33}=\frac{(X_{G}-Xp)\cdot(Y_{Gd}-Yp)-(X_{Gd}-Xp)\cdot(Y_{G}-Yp)}{R_{FG }-Xq}\,.\end{array}  \tag{25}\]  
The column matrix \([\rm r_{Mi}]_{P}\) represents the position vector of the guidance point \(\rm M_{i}\) on the wheel carrier, expressed in the local reference frame. The local coordinates (\(X_{Mi}\), \(Y_{Mi}\), \(Z_{Mi}\))\({}_{P}\) are input data for analysis (as geometrical parameters defining the mechanism). The solution of the nonlinear system (19) is found using the Newton-Kantorovich method [38], starting from the neutral position of the mechanism (the car at rest), a position for which the initial solution of the system is established exactly, as follows:  
\[\begin{array}{ll}\rm X_{G}\rm=X_{P}^{0}\rm+X_{G(P)},\ Y_{G}\rm=Y_{P}^{0}\rm +Y_{G(P)},\ Z_{G}\rm=Z_{P}^{0}\rm+Z_{G(P)},\\ \rm X_{G^{\prime}}\rm=X_{P}^{0}\rm+X_{G^{\prime}(P)},\ Y_{G^{\prime}}\rm=Y_{P}^{0} \rm+Y_{G^{\prime}(P)},\ Z_{G^{\prime}}\rm=Z_{P}^{0}\rm+Z_{G^{\prime}(P)},\\ \rm X_{G^{\prime\prime}}\rm=X_{P}^{0}\rm+X_{G^{\prime\prime}(P)},\ Y_{G^{\prime \
* Testing the obtained error: \[ \begin{bmatrix}\text{X}_{\text{G}}\\ \text{Y}_{\text{G}}\\ \text{X}_{\text{G}^{\prime}}\\ \text{Y}_{\text{G}^{\prime}}\\ \text{Z}_{\text{G}^{\prime}}\\ \text{X}_{\text{G}^{\prime\prime}}\\ \text{Y}_{\text{G}^{\prime\prime}}\\ \text{Y}_{\text{G}^{\prime\prime}}\\ \text{Z}_{\text{G}^{\prime\prime}}\end{bmatrix} _{1}- \begin{bmatrix}\text{X}_{ \text{G}}\\ \text{Y}_{\text{G}}\\ \text{X}_{\text{G}^{\prime}}\\ \text{Y}_{\text{G}^{\prime}}\\ \text{Z}_{\text{G}^{\prime}}\\ \text{X}_{\text{G}^{\prime\prime}}\\ \text{Y}_{\text{G}^{\prime\prime}}\\ \text{Y}_{\text{G}^{\prime\prime}}\\ \text{Z}_{\text{G}^{\prime\prime}}\end{bmatrix} _{0}\leq\ \varepsilon.\] (29)  
If the expression is satisfied, then '1' is retained as the solution of the system; otherwise, the iterative process from point 'b' is resumed, considering the values of the unknowns from the previous iteration as the initial solution of the system in the new iteration. The iterative process 'a\(\rightarrow\)d' ends when the difference between the values of the unknowns in two consecutive iterations'm \(-\) 1' and'm' reach the required accuracy:  
\[