# Multi-Criteria Optimization of an Innovative Suspension System for Race Cars
## 4 Synthesis of the Proposed Solution  
The innovative solution proposed in this work, for both the front and rear suspension systems, corresponded to the guiding mechanism shown schematically in Figure 7, whose spatial variant (Figure 20) enabled the minimization (even canceling) of the camber and caster angles variations by using a guiding rocker (4) mounted between the upper arm (3) and the chassis (0). The guiding rocker was arranged so as to replace/substitute the two fictive kinematic restrictions (\(\Delta\gamma=0\), \(\Delta\beta=0\)) used in the kinematic optimization depicted in the previous section of the work. The location of the point/end M on the guiding rocker was previously determined for the initial position of the suspension mechanism (when the car was in rest), along with the trajectory of this point during the kinematic analysis (see Figures 14 and 19). The problem raised by the proposed solution was the fact that we needed to determine where the focal point (M\({}_{0}\)) of the guiding rocker was on the chassis, so that is would assure the cancelation of the camber and caster variations. To do this, in the following, we proposed a synthesis algorithm based on the method of least squares, through which, starting from the imposed movement of point M (the trajectory \(\Gamma\) described through a series of successive positions M\({}_{k}\)), the global coordinates of point M\({}_{0}\) on the chassis (and thus the spatial arrangement of the guiding rocker) was determined. Figure 19: The interest point trajectory during the kinematic analysis of the rear suspension. Figure 20: The spatial scheme of the innovative suspension system. The constraint of point M on the guiding rocker consisted in the fact that it had to be on a fixed surface or curve with point M\({}_{0}\) on the chassis. In the case of guiding point M on a sphere with the center in M\({}_{0}\) (Figure 21), the guiding (constraint) equation had the form:  
\[(X_{\text{M}}-X_{\text{M}0})^{2}+(Y_{\text{M}}-Y_{\text{M}0})^{2}+(Z_{\text{M} }-Z_{\text{M}0})^{2}-l_{4}^{2}=0 \tag{6}\]  
where \(l_{4}\) is the length of the guiding rocker. The point coordinates (X\({}_{\text{M}}\), Y\({}_{\text{M}}\), Z\({}_{\text{M}}\)) in the global reference frame (attached to chassis) were input data of the synthesis problem. The coordinates of the chassis M\({}_{0}\) point (X\({}_{\text{M}0}\), Y\({}_{\text{M}0}\), Z\({}_{\text{M}0}\)) and the length of the guiding rocker (\(l_{4}\)) were unknown and needed to be computed. Equation (6) can be rewritten under the following form:  
\[X_{\text{M}}^{2}+Y_{\text{M}}^{2}+Z_{\text{M}}^{2}+X*X_{\text{M}}+Y*Y_{\text{M} }+Z*Z_{\text{M}}+R=0 \tag{7}\]  
where: \(X=-2*X_{\text{M}0}\), Y = \(-2*Y_{\text{M}0}\), Z = \(-2*Z_{\text{M}0}\), R = X\({}_{\text{M}0}^{2}+Y_{\text{M}0}^{2}+Z_{\text{M}0}^{2}-l_{4}^{2}\). Rewriting Equation (7) for the \"m\" number of positions that could be imposed to the wheel and subtracting the first equation, \"m \(-\) 1\" equations with 3 unknown (X, Y, Z) were obtained:  
\[X\cdot[(X_{\text{M}})_{k+1}-(X_{\text{M}})_{1}]+Y\cdot[(Y_{\text{M}})_{k+1}-( Y_{\text{M}})_{1}]+Z\cdot[(Z_{\text{M}})_{k+1}-(Z_{\text{M}})_{1}]=(r)_{1}-(r)_{k+1} \tag{8}\]  
where \((r)_{k}=\left(X_{\text{M}}^{2}+Y_{\text{M}}^{2}+Z_{\text{M}}^{2}\right)_{k}\), which leads to a function of form:  
\[F_{k}\left(X,Y,Z\right)=L_{k},k=1...\text{ m }-1. \tag{9}\]  
For m = 4 imposed positions, the system (4) was a linear one, with 3 equations and 3 unknowns. However, as there had to be more than 3 imposed finite positions, we obtained an over-determined system that could be solved by means of the least squares approach.