# Optimal Dimensional Synthesis of Ackermann Steering Mechanisms for Three-Axle, Six-Wheeled Vehicles
## 4 Optimization Synthesis of the Four-Bar Steering Mechanism of a Three-Axle, Six-Wheealed Vehicle
### The Objective Function and Constrictions  
The optimization objective function for the design of the four-bar steering mechanism in the front-axle steering mode is defined as the mean of the squared steering errors at all selected positions during the steering motion, combined with a penalty function for constraint violations, as follows:  
\[Min\ f(X)=\left\{\sum_{i=1}^{N}\frac{\left[\delta_{out\ f}(i)-\delta_{outA\ f}(i) \right]^{2}}{N}\right\}+Mh_{1} \tag{35}\]  
In the reverse-phase steering mode, the optimization objective function is defined as the mean of the squared steering errors at all selected positions during the steering motion for four-bar steering mechanisms of the front and rear axles, combined with a penalty function for constraint violations, as follows:  
\[Min\ f(X)=\sum_{i=1}^{N}\biggl{(}w_{1}\times\frac{\left[\delta_{out\ f}(i)- \delta_{outA\ f}(i)\right]^{2}}{N}+w_{3}\times\frac{\left[\delta_{out\ r}(i)- \delta_{outA\ r}(i)\right]^{2}}{N}\biggr{)}+Mh_{1}+Mh_{2} \tag{36}\]  
In Equations (35) and (36), \(X\) is the vector of design parameters, \(N\) is the number of selected steering angular positions of the inner front wheel, \(w_{1}\) and \(w_{3}\) are the weighting factor on the placement of the intermediate axle, which can be expressed as a function of the transmission ratio of front to rear axle, \(K\), as follows:  
\[w_{1}=\frac{L_{1}}{L}=\frac{1}{K+1},\ w_{3}=\frac{L_{3}}{L}=\frac{K}{K+1} \tag{37}\]  
\(\delta_{out\ f}(i)\) and \(\delta_{out\ r}(i)\) are the practically generated steering angles of the outer wheels of the synthesized front and rear four-bar steering mechanisms, evaluated by Equations (33) and (34) and \(\delta_{outA\ f}(i)\) and \(\delta_{outA\ r}(i)\) are the ideal Ackermann steering angles of the outer wheels of the front and rear axles, evaluated by Equations (18)-(20). And \(\delta_{out\ f}(i)-\delta_{outA\ f}(i)\), \(\delta_{out\ r}(i)-\delta_{outA\ r}(i)\) are the steering errors of the outer wheels of the front and rear axles when the steering mechanisms move to the _i-th_ steering angular  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Steering System**} \\ \cline{2-6}
**Front Steering** & \multicolumn{2}{c}{**Front-Axle**} & \multicolumn{2}{c}{**Reverse-Phase (\(K\) = 1)**} & \multicolumn{2}{c}{**Reverse-Phase (\(K\)\(\
eq\)1)**} \\
**Angle (\({}^{\circ}\))** & \multicolumn{2}{c}{**Steering Mode**} & \multicolumn{2}{c}{**Steering Mode**} \\ \cline{2-6}  & \(R_{\mathbf{c}}\) & \(R_{\mathbf{c1}}\) & \(R_{\mathbf{c3}}\) & \(R_{\mathbf{c1}}\) & \(R_{\mathbf{c3}}\) \\ \hline
40 & 5.046 & 3.568 & 1.896 & 5.199 & 2.947 \\ \hline
30 & 6.892 & 4.575 & 3.365 & 6.730 & 4.802 \\ \hline
20 & 9.955 & 6.939 & 5.564 & 9.841 & 8.114 \\ \hline
10 & 19.209 & 12.873 & 11.830 & 19.149 & 17.554 \\ \hline \(-10\) & 17.619 & 11.641 & 12.650 & 17.712 & 19.050 \\ \hline \(-20\) & 8.246 & 5.335 & 6.434 & 8.139 & 9.638 \\ \hline \(-30\) & 4.846 & 2.780 & 4.339 & 4.664 & 6.931 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Turning radii for two steering modes (unit: m).  
position. That is the structural error of the function generation mechanism. \(Mh_{1}\) and \(Mh_{2}\) are the penalty functions for violating the constraints, and the value of \(M\) is usually set to \(10^{6}\), when the constraints are not met, \(h\) is set to 1; otherwise, it is set to 0.  
The [PERSON] four-bar steering mechanisms of the front and rear axles must simultaneously satisfy the constraint of a triple rocker mechanism of [PERSON]'s criteria [41], the constraint inequalities of the link lengths are expressed as follows:  
\[h_{1}:\;2\Big{[}Max\Big{(}r_{1f},r_{2f},r_{3_{f}},r_{4f}\Big{)}+ Min\Big{(}r_{1f},r_{2f},r_{3f},r_{4f}\Big{)}\Big{]}>Sum\Big{(}r_{1f},r_{2f},r_{3f},r_{4f} \Big{)} \tag{38}\] \[h_{2}:\;2[Max(r_{1r},r_{2r},r_{3r},r_{4r})+Min(r_{1r},r_{2r},r_{3r},r_{4r} )]>Sum(r_{1r},r_{2r},r_{3r},r_{4r}) \tag{39}\]