# Optimal Dimensional Synthesis of Ackermann Steering Mechanisms for Three-Axle, Six-Wheeled Vehicles
## 5 Conclusions  
This paper investigates the optimal dimensional synthesis of Ackerman trapezoidal steering mechanisms for a three-axle, six-wheeled vehicle in front-axle and reverse-phase steering modes. The optimization objective minimizes the mean squared steering errors within the steering range with a penalty for constraint violations of [PERSON]'s criteria. In reverse-phase (\(K\
eq 1\)) steering mode, the transmission ratio of the front to rear axle (\(K\)) is considered as a design parameter. Four metaheuristic optimization algorithms--PSO, HPSO, DE-gr, and L-EPSDE--were employed through two hundred optimization numerical experiments. For front-axle steering mode, the L-EPSDE method achieved the smallest maximum steering error (8.96986\({}^{\circ}\)). In reverse-phase (\(K=1\)) steering mode, DE-gr, L-EPSDE, and PSO methods minimized front axle steering errors (6.3850\({}^{\circ}\)) and rear-axle steering errors (17.4121\({}^{\circ}\)), respectively, while the HPSO method had slightly higher errors, 6.4167\({}^{\circ}\) and 17.4145\({}^{\circ}\), in the front and rear axles, respectively. Additionally, the reverse-phase (\(K=1\)) steering mode yielded a significantly smaller turning radius compared to the front-axle steering mode. In reverse-phase (\(K\
eq 1\)) steering mode, the DE-gr method performed best overall, reducing the steering errors of front and rear axles to 2.0646\({}^{\circ}\) and 3.3272\({}^{\circ}\), respectively. And the L-EPSDE method has comparable performance with steering error of front and rear axles of 2.0648\({}^{\circ}\) and 3.3301\({}^{\circ}\), respectively. However, for reverse-phase  
Figure 34: Analyses error between ADAMS and MATLAB for the front steering mechanism (\(K\
eq 1\)). Figure 35: Analyses error between ADAMS and MATLAB for the rear steering mechanism (\(K\
eq 1\)). (\(K\
eq 1\)) steering mode, the turning radii are slightly larger than those of the case \(K=1\). Statistical analysis of the two hundred optimization numerical experiments revealed that DE-gr and L-EPSDE methods had the fastest convergence, highest accuracy, and precision, with PSO and HPSO methods slightly behind but still better than previous studies [33]. For the optimal synthesis of the reverse-phase Ackermann four-bar steering mechanism of a three-axle, six-wheeled vehicle, considering \(K\) as a design parameter not only enhances design flexibility for intermediate axle placement but also significantly reduces the steering error and improves the accuracy and convergence efficiency. Overall, the study concludes that the Ackermann four-bar steering mechanism in front-axle and reverse-phase steering modes is effectively modeled and efficiently solved as a low-dimension optimization problem with four and five design parameters. Among the four utilized metaheuristic algorithms, DE-gr and L-EPSDE performed the best. Furthermore, the DE-gr-optimized reverse-phase Ackermann four-bar steering mechanisms were validated using multi-dynamic analysis software MSC-ADAMS v2015 and the MATLAB program, showing the analysis errors of the output steering angle of 0.222\({}^{\circ}\) (front axle) and 0.234\({}^{\circ}\) (rear axle), respectively, confirming the accuracy and feasibility of the proposed dimensional synthesis approach of the Ackermann trapezoidal steering mechanism for a three-axle, six-wheeled vehicle. The result of this study indicates that two metaheuristic optimization methods, DE-gr and L-EPSDE, with their high efficiency, accuracy, and fast convergence speed, are well-suited for further investigation on the optimal dimensional synthesis of a front-wheel drive two-axle four-wheeled vehicle's six-bar and eight-bar steering mechanisms and a three-axle, six-wheeled vehicle's four-bar and six-bar steering mechanisms (with two front steerable axles and a rear non-steerable axle) and a six-bar steering mechanism in reverse-phase steering mode. Conceptualization, Y.-H.K. ; Data curation, Y.-C.Z. ; Formal analysis, Y.-H.K. and Y.-C.Z. ; Investigation, Y.-H.K. and D.-C.P. ; Methodology, Y.-H.K. and D.-C.P.