# Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage
## 2 Optimization Design Problem Formulation  
The multi-objective optimization problem of a six-bar steering linkage has been proposed in previous work [17]. The objective functions are derived from the side take-off model (STO) leading type, as shown in Figure 4. The combination of the drawing and mathematical techniques is used to define a connected point B of two circles with radius (steering arm length \(L_{a}\), and tie rod length \(L_{t}\)) and known length (rack length \(L_{r}\) and is the off-set distance \(H\)). The changing position C caused by rack movements (\(b\)) can be formulated mathematically due to the shape changing of the linkage, which has been detailed in Sleesongsom and Bureerat [17].  
In cases of the car turning right, the two circles traced by the steering arms and tie rods are intersected at two points \(B\) and \(B^{\prime}\):  
\[x^{2}+y^{2}=L_{a}^{2} \tag{7}\]  
and  
\[(x-C_{x2})^{2}+(y-C_{y2})^{2}=L_{t}^{2} \tag{8}\]  
where \(b\) is the displacement of a rack to the positive right-hand side, the center \(C\) (\(C_{x2}\), \(C_{y2}\)) = (\(W_{t}\)/2 \(L_{b}\), \(H\)) and \(L_{b}\) = \(L_{r}\)/2 \(-b\). This \(L_{b}\) value will be used for computing \(\theta_{2}\) and \(\theta_{3}\). In cases of turning left, \(\theta_{5}\) and \(\theta_{6}\) are computed by setting \(L_{b}\) = \(L_{r}\)/2 + \(b\).  
Rearranging (7) into \(x=(L_{a}^{2}-y^{2})^{1/2}\) and substituting into (8) forms a quadratic equation  
\[a_{2}y^{2}+a_{1}y+a_{0}=0 \tag{9}\]  
where \(a_{2}=1-B^{2}\), \(a_{1}=-2 CB\), \(a_{0}=C^{2}-{L_{a}}^{2}\), \(C=\frac{(L_{a}^{2}-L_{t}^{2}+C_{x2}^{2}+C_{y2}^{2})}{2C_{x2}}\), and \(B=\frac{C_{y2}}{C_{x2}}\).  
The solutions of Equation (9) are  
\[y_{1}=\left(-a_{1}+\sqrt{a_{1}^{2}-4a_{2}a_{0}}\right)/2a_{2} \tag{10}\]  
\[y_{2}=\left(-a_{1}-\sqrt{a_{1}^{2}-4a_{2}a_{0}}\right)/2a_{2} \tag{11}\]  
From Equation (7), if \(y_{1}\) and \(y_{2}\) are known, \(x_{1}\) and \(x_{2}\) can be computed.  
Figure 4: Six-bar steering linkage configuration (reproduced from [17]).  
The highest value between \(y_{1}\) and \(y_{2}\) are selected to be the solution. An initial angle on the left arm \(\theta_{2}=\theta_{20}\) is similar to the right arm \(\theta_{6}=\theta_{60}\) only in cases of the car moving in a straight motion. However, in Figure 4, the angles \(\theta_{2}\), \(\theta_{3}\), \(\theta_{5}\), and \(\theta_{6}\) have the trigonometric relations as follows:  
\[\theta_{2}\;\;or\;\;\theta_{6}\;=\tan^{-1}\left(\frac{y_{1}}{x_{1}}\right) \tag{12}\]  
and  
\[\theta_{3}\;\;or\;\theta_{5}\;=\tan^{-1}\left(\frac{y_{1}-H}{W_{t}/2-L_{r}-x_{ 1}}\right) \tag{13}\]  
where \(L_{b}=L_{r}/2+b\) for \(\theta_{5}\) and \(\theta_{6}\), and \(L_{b}=L_{r}/2-b\) for computing \(\theta_{2}\) and \(\theta_{3}\).  
The outer wheel angle and inner front wheel angle are \(\theta_{O}=\theta_{20}-\theta_{2}\), and \(\theta_{I}=\theta_{6}-\theta_{60}=\theta_{6}-\theta_{20}\), respectively. Then, STE in Equation (6) is computed while the limiting position of steering linkage is our consideration to protect our design result from useless mechanisms [17; 33; 34].