# Optimal Dimensional Synthesis of Ackermann Steering Mechanisms for Three-Axle, Six-Wheeled Vehicles
###### Abstract
[32] conducted the dimensional synthesis of steering mechanisms for a three-axle semi-trailer truck with front and rear steerable axles. The steering system, consisting of four-bar linkages, was optimized using the interior-point algorithm, which required an initial guess solution, to minimize the root-mean-square (RMS) error between the steering angles calculated from the Ackermann condition and the Freudenstein equation. The research offers design insights and practical solutions to enhance the steering performance and maneuverability of three-axle semi-trailers. [PERSON] et al. [33] examined the optimal dimensional synthesis of a four-bar steering mechanism for a three-axle, six-wheeled steering system with front and rear steerable axles. The system comprised two four-bar mechanisms, one for each steerable axle, coordinated by a constant-pitch gear system. The optimization objective function consisted of two sets of quadratic polynomials involving five design parameters of the front and rear steering mechanisms, along with a weighting factor for the wheelbase. However, the paper does not specify the optimization method employed. Additionally, by simplifying the nonlinear relationship between the output and input steering angles into a linear one. For the optimally synthesized steering mechanism, the maximum steering error of the front four-bar mechanism was 6.8 degrees, while that of the rear four-bar mechanism was 15 degrees. [PERSON] et al. [34] developed an independent steering system for a three-axle, six-wheeled electric vehicle powered by six in-wheel motors, aiming to enhance the flexibility of the vehicle on complex off-road terrains. The system integrates both mechanical and electronic control steering systems, each offering multiple steering modes. The study introduces these modes, builds a detailed CAD model using SolidWorks, and establishes a kinematic model for analysis. A segmented steering trapezoid mechanism was implemented to enhance the performance of the reverse-phase steering mode. Results show the steering system significantly improves vehicle flexibility, enabling stable and efficient maneuvering on complex terrains. Traditional optimization methods, such as the Newton-Raphson, SQP, and Hooke-Jeeves methods, etc., require gradient calculations and an initial estimated solution, complicating the design process for complex nonlinear objective functions. Modern, well-developed, and well-popular metaheuristic optimizations, such as swarm-based algorithms: PSO, HPSO, MKH, CS, APT-FPSO, BAS, etc. [35], and evolutionary-based algorithms: GA, GA-KK, GA-CSP, DE, GA-FL, GA-DE, MUMSA, NSAGII, IOA*-at, ICA-SA, DE-SRT, TS-MBFOA, SAP-TLBO, ICA-GA, TLBO, CMDE, CPF-DE, HLIDE, ADELI, DEC, ImHS, GSA, ATLBO-DA, GSEF-IAA, ODSRA+CP, REA, HCDJ, etc. [35], overcome these challenges by eliminating the need for deriving the gradient of the objective function or providing the initial guess solution while efficiently and accurately finding the optimal results. [PERSON] et al. [36] proposed the Particle Swarm Optimization (PSO) algorithm in 1995, inspired by the predatory behavior of bird flocks. In 1997, [PERSON] and [PERSON] [37] proposed the Differential Evolution (DE) algorithm to address the Chebyshev polynomial fitting problem. [PERSON] et al. [35] developed the Hybrid Particle Swarm Optimization (HPSO) algorithm to enhance the convergence and diversity of improved PSO. HPSO incorporates the constriction factor (C) and inertia weight (w) from improved PSO while selecting the top 30% of particles with the best fitness values for the next generation. The remaining 70% undergo crossover and selection by Genetic Algorithms (GA). [PERSON] et al. [35] incorporated the golden ratio values (0.618:0.382) into the DE algorithm as crossover and mutation rates. Mutation is performed first with a rate of 0.382, followed by crossover with a rate of 0.618, leading to the Differential Evolution with golden ratio (DE-gr).