# 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
#### 4.3.2 Optimal Design of the Reverse-Phase (\(K=1\)) Four-Bar Steering Mechanism  
In reverse-phase (\(K=1\)) steering mode, with the front and rear steerable axles, and a non-steerable middle axle. The design parameters of this steering mode are \(\theta_{2o\ f}\), \(r_{2f}\) and \(\theta_{2o\ r}\), \(r_{2p}\). Table 6 summarizes the statistical results of two hundred optimization experiments for the reverse-phase (\(K=1\)) four-bar steering mechanism using four optimization methods. L-EPSDE achieves the smallest standard deviation (2.53 \(\times\) 10\({}^{-4}\)), worst error value (1.51 \(\times\) 10\({}^{-2}\)), mean value (1.022 \(\times\) 10\({}^{-2}\)), and minimum error (9.9997 \(\times\) 10\({}^{-3}\)). DE-gr and PSO also achieve the same minimum error, with DE-gr ranking second in mean value and standard deviation, and PSO has a maximum mean value (1.062 \(\times\) 10\({}^{-2}\)) and standard deviation (3.85 \(\times\) 10\({}^{-3}\)). Figure 13 illustrates the convergence speed of the four optimization methods. The DE-gr method is the fastest to converge to the optimal solution, followed by L-EPSDE, while HPSO and PSO converge slightly slowly but still achieve the optimal solution within 200 iterations. Overall, DE-gr and L-EPSDE exhibit the fastest convergence, highest accuracy, and precision, while HPSO and PSO perform slightly worse.  
Table 7 shows the optimal objective function values, dimensions, and maximum steering errors for the reverse-phase (\(K=1\)) four-bar steering mechanisms synthesized from two hundred optimization experiments using four optimization methods. Among them, \(\Delta_{f}\) and \(\Delta_{r}\) mean maximum steering errors of the front and rear steering mechanisms. It is observed that the dimensions of the optimized four-bar steering mechanisms are different between the front and rear sets, the maximum steering error of the front axle is the smallest (6.3850\({}^{\circ}\)) by the DE-gr, L-EPSDE, and PSO methods, and the largest (6.4167\({}^{\circ}\)) by the HPSO method. The maximum steering error of the rear axle is also minimized by the DE-gr, L-EPSDE, and PSO methods (17.4121\({}^{\circ}\)) and maximized by the HPSO method (17.4145\({}^{\circ}\)).  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Method** & **Best Error** & **Worst Error** & **Median** & **Mean** & **SD** \\ \hline PSO & \(9.997\times 10^{-3}\) & \(5.66\times 10^{-2}\) & \(1.007\times 10^{-2}\) & \(1.062\times 10^{-2}\) & \(3.85\times 10^{-3}\) \\ HPSO & \(1.000\times 10^{-2}\) & \(8.22\times 10^{-2}\) & \(1.052\times 10^{-2}\) & \(1.035\times 10^{-2}\) & \(3.09\times 10^{-4}\) \\ DE-gr & \(9.997\times 10^{-3}\) & \(1.52\times 10^{-2}\) & \(1.002\times 10^{-2}\) & \(1.025\times 10^{-2}\) & \(2.57\times 10^{-4}\) \\ L-EPSDE & \(9.997\times 10^{-3}\) & \(1.51\times 10^{-2}\) & \(1.026\times 10^{-2}\) & \(1.022\times 10^{-2}\) & \(2.53\times 10^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 6: Statistical results of the four optimization methods for the reverse-phase steering mode (\(K=1\)) (unit: rad.\({}^{2}\)).  
Figure 13: Convergence speed of the four optimization methods (Reverse-phase, \(K=1\)).  
Figure 16: Transmission angle of the optimized front four-bar steering mechanisms (Reverse-phase, \(K=1\)).  
Figure 17: Turning radii of the gravity center of the vehicle’s front part (Reverse-phase, \(K=1\)).  
Figure 15: Steering error of the optimized front four-bar steering mechanisms (Reverse-phase, \(K=1\)).