# 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
#### 4.3.3 Optimal Design of the Reverse-Phase \((K\
eq 1)\) Four-Bar Steering Mechanism  
The design parameters of this steering mode are \(\theta_{20~{}f}\), \(r_{2f}\), \(\theta_{20~{}r}\), \(r_{2r}\) and \(K\). Table 8 presents the statistical results of two hundred optimization experiments for the reverse-steering \((K\
eq 1)\) four-bar mechanism of a three-axle, six-wheeled vehicle. The DE-gr method achieves the best performance, with the lowest mean value (\(1.41\times 10^{-4}\)), standard deviation (\(1.61\times 10^{-5}\)), and worst error value (\(1.58\times 10^{-4}\)). Its minimum error (\(1.256\times 10^{-4}\)) is nearly identical to HPSO and L-EPSDE's (\(1.254\times 10^{-4}\)). The HPSO method shows slightly better mean and standard deviation values than the L-EPSDE method, but its worst value is slightly inferior. PSO method performs the poorest on best error (\(1.298\times 10^{-4}\)) and standard deviation (\(2.15\times 10^{-3}\)). Figure 22 shows the convergence speed of the four optimization methods, where the DE-gr method converges fastest, followed by L-EPSDE and PSO, and HPSO converges more slowly. Overall, the DE-gr method excels in accuracy, converge speed, and stability, with L-EPSDE and HPSO performing comparable performance and PSO slightly lagging. Figure 21: Turning radii of the gravity center of the vehicle’s rear part (Reverse-phase, \(K=1\)). Figure 20: Transmission angle of the optimized rear four-bar steering mechanisms (Reverse-phase, \(K=1\)). Table 9 shows the objective function values, optimal dimensions, and maximum steering errors for the three-axle, six-wheeled reverse-phase four-bar steering mechanisms synthesized using four optimization methods. The steering angle range of the front steering wheel is a little different from the literature [33]. The result revealed that the front and rear-optimized four-bar mechanisms have distinct dimensions. The red marks in Table 9 highlight the maximum steering errors (\(\Delta_{f}\),\(\Delta_{r}\)) for the front and rear axles. For the front axle, PSO and DE-gr methods performed best, with maximum errors of 2.0286\({}^{\circ}\) and 2.0646\({}^{\circ}\), respectively, while the HPSO method was slightly worse at 2.1978\({}^{\circ}\). For the rear axle, DE-gr and L-EPSDE methods achieved the best results, with maximum errors of 3.3272\({}^{\circ}\) and 3.3301\({}^{\circ}\), respectively, whereas HPSO and PSO methods performed slightly poor at 3.8740\({}^{\circ}\) and 3.8985\({}^{\circ}\). Overall, the maximum steering errors of front and rear axles in this study are improved compared to the literature [33]. Comparing the convergence trends in Figure 13 (\(K=1\)) and Figure 22 (\(K\
eq 1\)), it is evident that all four methods converge more rapidly to the optimal solution in the \(K\
eq 1\) case. For example, the DE-gr method reaches the optimal solution after 50 iterations for \(K=1\), but only 25 iterations for \(K\
eq 1\). Additionally, referring to Table 6 (\(K=1\)) and Table 8 (\(K\
eq 1\)), the optimal objective function values for all four methods (e.g., 1.254 \(\times\) 10\({}^{-4}\) for L-EPSDE) are lower in the \(K\
eq 1\) case compared to \(K=1\) (9.997 \(\times\) 10\({}^{-3}\) for L-EPSDE). In the optimal synthesis of the reverse-phase steering mechanism for a three-axle, six-wheeled vehicle, considering \(K\) as a design parameter not only enhances design flexibility for intermediate axle placement but also improves convergence efficiency and solution accuracy.