# Multi-Criteria Optimization of an Innovative Suspension System for Race Cars
## 3 The Kinematic Optimization of the Suspension Mechanism
For each of the three design objectives, the monitored value was the root mean square (RMS) during simulation:  
\[RMS=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+\cdots v_{n}^{2}}{n}} \tag{1}\]  
where \(v_{1}\), \(v_{2}\),..., \(v_{n}\) represent the successive values of the parameter of interest, depending on the number of steps (\(n\)) that the simulation takes to reach the end. Before running the optimization algorithm, the file that contained the MBS model designed in ADAMS/View was transferred in ADAMS/Insight. The design variables (factors) and design objectives (responses) were set up after opening the saved model in ADAMS/Insight. To run the optimization process, the responses and the factors were promoted from the candidates in the inclusion list (\"Promote to Inclusions\" in ADAMS/Insight). Each design objective was analyzed based on a regression function that approximated the model in such way that the error between the predicted and the measured values were minimal. For example, if we only considered two design variables, the regression functions would have the following forms:  
* Linear model:  
\[R=a_{1}*F_{1}+a_{3}*F_{3}+e \tag{2}\]  
* Linear model with factors interaction:  
\[R=a_{1}+a_{2}*F_{1}+a_{3}*F_{2}+a_{4}*F_{1}*F_{2}+e \tag{3}\]  
* Quadratic model:  
\[R=a_{1}+a_{2}*F_{1}+a_{3}*F_{2}+a_{4}*F_{1}*F_{2}+a_{5}*F_{1}^{2}+a_{6}*F_{2}^ {2}+e \tag{4}\]  
* Cubic model:  
\[R=a_{1}+a_{2}*F_{1}+a_{3}*F_{2}+a_{4}*F_{1}*F_{2}+a_{5}*F_{1}^{2}+a_{6}*F_{2}^ {2}+\cdots \tag{5}\] \[\ldots+a_{7}*F_{1}*F_{2}^{2}+a_{8}*F_{1}^{2}*F_{2}+a_{9}*F_{1}^{3 }+a_{10}*F_{2}^{3}+e\]  
where \(F_{1}\) and \(F_{2}\) were the design variables (factors), \(a_{i}\)--the terms coefficients, \(R\)--the system response, and \(e\)--the error (that would be minimized). Depending on the selected DOE investigation strategy, we obtained the design space and the workspace of the experiment. The design space was a matrix that presented, in a normalized form, the combinations between the factors. Thus, the \"\(-1\)\" value corresponded to the minimum value of the factor, and \"\(1\)\" to the maximum value [41], according to the previously defined variation ranges. The effective values of the factors were seen in the workspace, along the corresponding responses. The program ran a simulation in ADAMS/View for each trial specified in this matrix to determine the responses values. After the simulations were run, the results appeared automatically in the workspace, by filling in the response columns with numerical values. Based on these results, ADAMS/Insight automatically generated the corresponding regression function, according to the selected type of regression model. The response analysis was done by using the evaluation methods presented in reference [43]. To begin, the kinematic optimization algorithm was implemented for the front wheels' suspension mechanism (the selected tuning strategy would then be used for the rear wheels' suspension optimization as well). The MBS model developed in ADAMS/View for the front axle suspension system is shown in Figure 10. The lower control arm (1) was linked to the chassis in points A and B through spherical joints, the rocker (5) was linked to the chassis in point C by a revolute joint, and point D was the location of the spherical joint that linked the toe link/steering rod (6) to the steering rack. In the kinematic model of the suspension system, the steering rack was fixed, so it was rigidly connected to chassis (in other words, the steering wheel is not actuated). The global coordinates of the four points were defined as design variables for the kinematic optimization of the front suspension system, as presented in Figure 9:0  
The initial values of the design variables are (in [mm]): DV_1 = \(-\)270.37, DV_2 = 38.5, DV_3 = 108.37, DV_4 = \(-\)193.5, DV_5 = 38.5, DV_6 = \(-\)279.43, DV_7 = \(-\)297.64, DV_8 = 214.89, DV_9 = 38.21, DV_10 = \(-\)288.5, DV_11 = 101.5, DV_12 = 135.01. For each variable, the variation domain was \(\pm\)20 mm relative to the initial value.