# 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
### Optimal Results of the Four-Bar Steering Mechanism
#### 4.3.1 Optimal Design of the Front-Axle, Four-Bar Steering Mechanism  
In this steering mode, design parameters are \(r_{2f}\), \(\theta_{2g}\), \(\theta_{2g}\), assume \(L_{1}=L_{3}\), \(a_{1}=a_{3}=L_{1}/2\). Table 4 summarizes the statistical results of two hundred optimization experiments for the front-axle (or called front two-wheel) four-bar steering mechanism using four optimization methods. The red-marked values in each row represent the minimum values among the statistical results of the four methods. The DE-gr method achieves the smallest standard deviation (SD) (1.98 \(\times\) 10\({}^{-6}\)), mini-worst value (1.70 \(\times\) 10\({}^{-4}\)), and the second-best error value (1.51 \(\times\) 10\({}^{-4}\)). PSO method also achieves the smallest SD but slightly large mean value (1.58 \(\times\) 10\({}^{-4}\)) and max-best value (1.56 \(\times\) 10\({}^{-4}\)). The HPSO method has a slightly large SD (4.29 \(\times\) 10\({}^{-6}\)) and is superior to the PSO method in terms of mean value, and best value. While the L-EPSDE method delivers the smallest best value (1.47 \(\times\) 10\({}^{-4}\)) and mean value (1.49 \(\times\) 10\({}^{-4}\)) but a slightly large SD (3.61 \(\times\) 10\({}^{-6}\)). Table 5 presents the optimal objective function values, dimensions, and maximum steering errors for the front-axle four-bar steering mechanisms synthesized from two hundred optimization experiments using four optimization methods. The L-EPSDE method achieves the smallest optimal objective function value (1.472 \(\times\) 10\({}^{-4}\)) and maximum steering error (8.9699\({}^{\circ}\)), and the HPSO method has the second smallest maximum steering error (8.9871\({}^{\circ}\)), while even the DE-gr method has the minimum SD (1.98 \(\times\) 10\({}^{-6}\)) and the second \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimization** & **PSO** & **HPSO** & **DE-gr** & **L-EPSDE** \\
**Method** & & & & \\ \hline Size of population & & & & \\ (_NP_) & 100 & 100 & 100 & 100 \\ Minimum NP & & & & \\ (_NPmin_) & & – & – & 5 \\ Crossover method & – & Single-point/ uniform & Multiple-point & Multiple-point \\ Crossover rate & – & – & 0.618 & 0.1–0.9 \\ (_CR_) & – & – & DE/best/2 & DE/rand/1 \\ Mutation method & – & – & DE/best/1 & DE/current-to-rand/1 \\ Mutate rate & – & – & 0.382 & 0.4–0.9 \\ Selection method & – & – & Competition & Competition \\ Learning & & & rule & rule \\ coefficients & 2.05 & 2.05 & – & – \\ (\(C_{1}\), \(C_{2}\)) & & & & \\ Inertia weight & 0–1 & 0–1 & – & – \\ (\(\omega\)) & & & 1000 & \\ Iteration number & & & 200 & \\ Experimental times & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameter setting of the four optimization methods. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & \(r_{2f}\) **(mm)** & \(\theta_{2 df}\) (\({}^{\circ}\)) & \(r_{2r}\) **(mm)** & \(\theta_{2 qr}\) (\({}^{\circ}\)) & **K** \\ \hline Range of & & & & \\ \(K\) = 1 & 200–300 & 60–120 & 200–300 & 60–120 & 1 \\ Range of & & & & \\ \(K\) \(\
eq\) 1 & 200–300 & 60–90 & 200–300 & 60–150 & 0.3–0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design range of four-bar steering mechanisms. best optimal objective function value (1.511 \(\times\)\(10^{-4}\)), the maximum steering error is the largest (9.1156\({}^{\circ}\)). Figure 9 illustrates the convergence speed of the four optimization methods, with L-EPSDE showing the fastest convergence, followed by DE-gr. From Table 4 and Figure 9, overall, L-EPSDE and DE-gr methods outperform PSO and HPSO methods in accuracy, precision, and convergence speed for the front-axle steering mode. The steering angle relation, the turning radius, and the steering error of the optimized front-axle four-bar steering mechanism are shown in Figure 10, Figure 11, and Figure 12, respectively. For comparison, the part data of the turning radius (Figure 11) at the center of gravity of  
Figure 11: Turning radius of optimized four-bar steering mechanisms (Front-axle steering mode).