# Optimal Dimensional Synthesis of Ackermann Steering Mechanisms for Three-Axle, Six-Wheeled Vehicles
###### Abstract
This approach significantly improves the accuracy and convergence efficiency of traditional DE, making DE-gr suitable for low- and medium-dimension optimization problems [35; 38]. EPSDE, proposed by [PERSON] et al. [39] in 2011, is an efficient metaheuristic optimization method known for its adaptive parameter adjustment, minimizing the need for extensive manual tuning. However, when applied to high-dimension engineering optimization problems, it may experience incomplete or slow convergence, requiring an increased population size or iteration count to achieve satisfactory results. To improve this, [PERSON] et al. [35] introduced the concept of linear population size reduction (LPSR) into the EPSDE method, resulting in the L-EPSDE method. It eliminates poor individuals during iterations, linearly reduces population size, and enhances search efficiency and convergence speed. In [35], the performance of five metaheuristic optimization methods (PSO, HPSO, DE-gr, EPSDE, and L-EPSDE) was compared in solving the optimal dimensional synthesis problems of five representative four-bar path generators. The synthesis problems of four-bar path generators essentially are medium- or high-dimension problems. Results demonstrated that four metaheuristic methods (HPSO, DE-gr, EPSDE, and L-EPSDE) efficiently converged to the global optimal solution with minimal iterations. L-EPSDE was verified as particularly effective for high-dimension optimization problems. Due to the limited literature addressing the optimal dimensional synthesis of steering mechanisms for three-axle, six-wheeled vehicles, this study focuses on optimizing Ackermann steering mechanisms for two steering modes in a three-axle, six-wheeled vehicle: front-axle steering (front axle steerable) and reverse-phase steering (central axle non-steerable). A comparison of turning radii between vehicles with one and two steerable axles reveals that two steerable axles significantly reduce the turning radius and enhance maneuverability. Due to space limitations, the steering mode involving the first two steerable axles will be addressed in a separate study. Four metaheuristic optimization methods known for their convergence efficiency and ease of implementation, involving two swarm-based algorithms: Particle Swarm Optimization (PSO) and Hybrid Particle Swarm Optimization (HPSO), along with two evolutionary-based algorithms: Differential Evolution with golden ratio (DE-gr) and Linearly Ensemble of Parameters and Mutation Strategies in Differential Evolution (L-EPSDE), are employed to compare their effectiveness and efficiency in solving the optimal dimensional synthesis of Ackermann trapezoidal steering mechanisms for such vehicles. The details of the algorithms and flowcharts of the employed four metaheuristic optimization methods can be referred to in the literature [35]. Two hundred optimization numerical experiments were conducted using the four metaheuristic optimization methods. Statistical analysis reveals that evolutionary-based algorithms (DE-gr and L-EPSDE) outperform swarm-based ones (PSO and HPSO) in accuracy and efficiency. Additionally, intermediate-axle placement was considered a weighted design parameter in the optimal synthesis of the reverse-phase steering mechanism, allowing for more flexible design outcomes. The results of this study validate that the evolutionary-based optimization methods are effective and efficient approaches for synthesizing Ackermann steering mechanisms in three-axle, six-wheeled vehicles.