# Optimal Dimensional Synthesis of Ackermann Steering Mechanisms for Three-Axle, Six-Wheeled Vehicles
###### Abstract  
This study employs four metaheuristic optimization methods to optimize the dimensional synthesis of Ackermann steering mechanisms for three-axle, six-wheeled vehicles with front-axle steering mode and reverse-phase steering mode. The employed optimization methods include Particle Swarm Optimization (PSO), Hybrid Particle Swarm Optimization (HPSO), Differential Evolution with golden ratio (DE-gr), and Linearly Ensemble of Parameters and Mutation Strategies in Differential Evolution (L-EPSDE). With a front-wheel steering angle range of 70 degrees, two hundred optimization experiments were conducted for each method, and statistical analyses revealed that DE-gr and L-EPSDE methods outperformed PSO and HPSO methods in terms of standard deviation, mean value, and minimum error. These two methods exhibited superior convergence stability, faster convergence, and higher accuracy compared to PSO and HPSO. Reverse-phase (\(K\) = 1) steering mode outperformed front-axle steering mode, delivering reduced steering errors and turning radii. Considering the transmission ratio of front to rear axle (\(K\)) as a design variable in reverse-phase steering mode increased design flexibility and significantly lowered steering errors for the front and rear axle steering mechanisms. However, this comes with a slight increase in the turning radius of the vehicle's front part compared to when \(K\) = 1. The optimized mechanism, designed using the DE-gr method, was validated through kinematic simulations and steering analyses using MSC-ADAMS v2015 software, further confirming the effectiveness and reliability of the proposed design. three-axle six-wheeled vehicle; Ackermann steering mechanism; PSO; HPSO; DE-gr; L-EPSDE Article  
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tire wear. For example, a four-axle, eight-wheeled vehicle may be unsuitable for narrow paths due to its extended length. Conversely, a three-axle, six-wheeled vehicle, with a more compact design, is better equipped to navigate rugged terrain and narrow spaces, providing superior mobility and off-road capability. The problem of dimensional synthesis of linkages can be divided into three types: function generation, motion generation/rigid body guidance, and path generation [1]. In a vehicle, the steering system is one of the important subsystems, and the steering system has a great influence on the maneuverability and flexibility of the vehicle. The Ackermann trapezoidal steering mechanism for front-wheel drive in conventional four-wheeled vehicles essentially is a four-bar function generator designed to align the extensions of the front and rear wheels' axles at a common point during steering, minimizing tire wear and preventing sideslip instability. Generally, the steering mechanism of a vehicle is designed so that the axles of steerable wheels and rear wheels axle intercept at one point, which is called Ackermann's condition. The vehicle steering mechanism essentially is a function generator; its output and input steering angles should satisfy the Ackermann steering geometry to prevent tire abrasion and vehicle side-slip instability. This article investigates the optimal dimensional synthesis of the Ackermann steering mechanism for a three-axle, six-wheeled vehicle as a function generation problem. In general, there is no steering mechanism that can fully satisfy the Ackermann principle at every steering angle in the whole turning process, so it can only be designed to optimize the steering mechanism so that the overall steering errors are as small as possible. Many researchers have paid attention to finding the optimum solution for minimizing the steering error between the ideal Ackermann condition and the practical generated angle of the front outer wheel within the steering motion range.