**1. Introduction**

The smoothness of a vehicle during driving is one of its important evaluation indicators. When a vehicle is driven in irregular road conditions, uncomfortable vibrations may be transmitted to the driver. As unpaved roads make up the majority of an agricultural mobile robot's working environment, considerable of vibration can be transmitted to the operating equipment carried by an agricultural mobile robot, reducing its working precision and shortening its lifespan. As an important system in a vehicle that has a vibration mitigation function, a properly designed suspension system can absorb some of the vibrations and reduce the impact on the driver or working equipment, and has therefore long been the subject of research and optimization by academics. There have been many studies on conventional vehicle suspensions. The optimal design of an agricultural robot suspension system is essentially a constrained optimization problem. Lagrange multiplier methods, evolutionary algorithms and machine learning are commonly applied to solve constrained optimization problems. These methods are utilized in many areas, such as path planning [1], lesion diagnosis [2–4], defect detection [5,6], structural design [7] and resource allocation [8,9]. There are very few studies on the optimal design of agricultural robot suspensions. Therefore, it is helpful and instructive to carry out research on the optimal

**Citation:** Qu, Z.; Zhang, P.; Hu, Y.; Yang, H.; Guo, T.; Zhang, K.; Zhang, J. Optimal Design of Agricultural Mobile Robot Suspension System Based on NSGA-III and TOPSIS. *Agriculture* **2023**, *13*, 207. https:// doi.org/10.3390/agriculture13010207

Academic Editors: Vadim Bolshev, Vladimir Panchenko and Alexey Sibirev

Received: 21 December 2022 Revised: 11 January 2023 Accepted: 12 January 2023 Published: 14 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

design of the suspension of agricultural robots by investigating the existing research on the suspension optimization design of passenger cars.

Ciro Moreno Ramírez et al. [10] investigated the effect of two different structures of suspension systems on vehicle dynamics, and investigated the stability of the systems using root trajectory analysis methods. Li et al. [11] used the response surface method to establish an approximate model of the MacPherson suspension systems and verified the reliability of it, and then used the NSGA-II algorithm to optimize the key hard-point coordinates of the suspension. To optimize the design of the pertinent parameters, Qian and Jin [12] examined the performance of an SUV double wishbone suspension and applied the D-optimal method based on the response surface model. Guang Li et al. [13] used the NSGA-II algorithm to optimize the suspension parameters of high-speed locomotives. The locomotive dynamics model established by SIMPACK software was used to optimize the design of six parameters by the NSGA-II algorithm, and the three evaluation indexes designed were minimum indicators. None of the above studies established an accurate non-linear mathematical model, which may lead to problems such as inaccurate solution results or reduced solution efficiency in the solution process.

Chen et al. [14] established a mathematical model of the double wishbone suspension matching an in-wheel motor with a cosine matrix method. Zhang and Li [15] established a nonlinear mathematical model of a double wishbone independent suspension using a kinematic analysis method, and then used simulation analysis in Adams software and bench testing to demonstrate that the developed mathematical model accurately expressed the dynamic properties of the suspension. The nonlinear motion equation for flexible double wishbone suspension was developed by Abdelrahman et al. [16] using the concept of imaginary displacement, but the main focus of this study was on how flexible structures respond to various road unevenness, vehicle speed, and material damping coefficients. By utilising laser scanning to create 3D models of the suspension parts, and Adams to simulate their dynamics, Prastiyo and Fiebig [17] compared the benefits of linear and progressive double wishbone suspensions and found no discernible differences between them, which showed that it is more practical to use a traditional linear double wishbone suspension. A mathematical model of the suspension system was created by the aforementioned analysis, but further optimization of the current suspension system has not been done.

A mathematical model of an independent steering-suspension guidance mechanism was developed by Chen et al. [18] based on the theory of spatial mechanics, and the structural hard-point coordinates of the suspension were optimized using sensitivity analysis. Sancibrian et al. [19] developed a model containing a large number of structural parameters, mainly the lengths of the individual links, for the structure of a double wishbone suspension. Seven functional parameters were applied to evaluate the design parameters. This study was based on the gradient descent method to find the optimal solution. Shi et al. [20] investigated the issue of optimizing the MacPherson suspension's hard-point coordinates, established the relationship between the hard-point coordinates of the suspension and the evaluation index based on a dynamics model and support vector regression (SVR) established in the Adams/Car software, and designed a double-loop multi-objective particle swarm algorithm to optimize the hard-point coordinates of the suspension. The results showed that the improved algorithm outperformed the traditional multi-objective particle swarm algorithm and the genetic algorithm. Totu and Alexandru [21] presented a comprehensive design method based on the least squares approach for the optimal design of an innovative racing car suspension system. This study used Adams/View and Adams/Insight to build a regression model and carry out the optimization design. Zhu et al. [22] provided a sliding mode control method for the equivalent two-degrees-of-freedom model and calibrate the unknown parameters in the equivalent two-degrees-of-freedom model by parameter identification for a quarter double wishbone suspension model. In this study, the equivalent two-degreesof-freedom suspension model was constructed by parameter identification, allowing for the analysis of the suspension response without the need to create a dynamic model. The

above studies did not consider the influence of suspension performance parameters (such as damping and stiffness) on suspension performance during the optimization process.

Issa and Samn [23] used the Harris Hawk Optimization (HHO) algorithm to optimize the damping and stiffness coefficients of passive suspension, which they simplified to a two-degrees-of-freedom model. The result showed that the optimized passive suspension performance was improved. Huang et al. [24] simplified the double wishbone suspension to a single-wheel two-degrees-of-freedom model and employed evolutionary algorithms to find its parameters. Based on the suspension equivalent two-degrees-of-freedom model and the seat-passenger equivalent eight-degrees-of-freedom model, Papaioannou and Koulocheris [25] improved the solution speed and quality of evolutionary algorithms by dividing the optimization targets into primary optimization objectives and auxiliary ones. Drehmer et al. [26] modeled the motion of the whole vehicle by introducing the influence on the driver's seat to establish an 8-degrees-of-freedom motion model. Particle swarm optimization algorithms and sequential quadratic programming algorithms were used to optimize the damping and stiffness of the four independent suspensions of the complete vehicle under different road conditions. Gobbi et al. [27] used a 2-degrees-of-freedom linear model to analytically describe the dynamic behaviour of a vehicle travelling on a randomly contoured road, and respectively optimized the damping and stiffness of the passive suspension and the damping, stiffness and controller gain of the active suspension based on multi-objective planning theory and robust design theory. Kwon et al. [28] developed a mathematical model of a hydro-pneumatic suspension system and a whole vehicle model for the design of hydro-pneumatic suspension in heavy vehicles. This study developed an agent model to reduce the computational effort in the optimal design process, and used it for multi-objective optimization solutions. Zheng et al. [29] designed an active, tuned inertial damper (TID) suspension systems based on a combination of active actuator and inertializer, and proposed a parameter optimization method based on an analytical solution. Stability algebraic analysis was carried out using Hurwitz's criterion and it was verified that the parameters obtained with this method could guarantee the stability of the suspension. Yang et al. [30] used a decomposition-based multidisciplinary optimization approach for the optimal design of passive suspension systems based on an equivalent two-degrees-offreedom model. In terms of the selection of design parameters, this study focused on the refinement of the parameters affecting damping and stiffness in the suspension system, such as the number of coils of the spring, the spring diameter and the piston diameter, among other structural parameters. Li et al. [31] proposed a dimensionless hybrid index based on safety probability to evaluate suspension performance for the optimal design of energy-harvesting suspension systems, transformed the multi-objective optimization problem into a single-objective optimization problem, and then used a genetic algorithm to solve the multi-parameter optimization problem. Truong and Dao [32] proposed a hybrid HNSGA-III & MOPSO algorithm based on the MOPSO algorithm and NSGA-III for the optimal design of the stiffness and damping of a powertrain suspension model. HNSGA-III & MOPSO algorithms showed better efficiency and solution quality than MOPSO and NSGA-III in this study. Grotti et al. [33] proposed a multi-objective archive-based Quantum Particle Swarm Optimizer (MOQPSO) algorithm for the optimization of suspension systems. The results were compared with NSGA-II algorithm and COGA-II algorithm. It was shown that the MOQPSO algorithm proposed in this study could obtain better non-dominated set solutions than the NSGA-II and COGA-II algorithms. Bingul and Yildiz [34] carried out a multi-objective optimized design of a non-linear suspension system for an electric vehicle based on the NSGA-II algorithm. Three suspension systems, including passive suspension, active suspension based on PD control, and active suspension based on FL control, were optimized. This study designed seven minimum optimization objectives to evaluate the degree of influence of the active suspension on the driver under road vibration conditions. Prasad et al. [35] optimized and improved the stiffness and PID control parameters of a semi-active suspension system in a quarter suspension model. They proposed a fast convergence optimization algorithm for saving computational costs. Compared with NSGA-II algorithm, the geometry-inspired genetic algorithm proposed in this study converges quickly under small population conditions. The above studies did not consider the influence of suspension structural parameters on suspension performance during the optimization process.

Gialleonardo et al. [36] used the NSGA-II algorithm for optimal design of control strategy parameters for active suspensions. They designed six minimum targets to evaluate the parameters of three different active control strategies and to compare the performance of them. Xu et al. [37] used a multi-objective particle swarm algorithm to optimize the design of key parameters of a straw back-throwing device. This study showed that the multi-objective evolutionary algorithm is effective and feasible for the optimal design of mechanical structures. Jiang et al. [38] used the Kriging model and NSGA-II algorithm for the multi-objective lightweight design of two parts for a control arm and torsion beam, which are widely used in passenger car suspensions. Dang et al. [39] used NSGA-III to optimize the parameters of the rotational speed and opening length of a fertiliser spreader with the objectives of accuracy, uniformity, adjustment time, and crushing rate. They chose the single-objective evolutionary algorithm GA and the multi-objective evolutionary algorithm MOEA-D-DE for comparison. The evaluation metrics showed that NSGA-III has significant advantages in solving this kind of problem. Li et al. [40] proposed a multi-objective optimal control method for an active suspension system aimed at solving the negative vibration problem generated by the in-wheel motor of an electric vehicle. An integrated model considering the electromechanical coupling between the electromagnetic excitation of the motor and the transient dynamics of the vehicle was established and developed. The Pareto solution set for the optimal parameters of the active suspension system was solved using the particle swarm optimization algorithm. Chen et al. [41] conducted a study on the multi-objective optimization problem of high-speed train suspension systems using the NSGA-II algorithm to optimize four suspension parameters with three minimum objectives. The research above did not offer a clear strategy on how to select the optimal solution from the Pareto solution set. Some studies [13,31–34,36,37,41] lacked comparisons with other multi-objective optimization algorithms.

To address the above problems, our study proposes an optimal design method for the suspension of the agricultural mobile robot based on NSGA-III and TOPSIS. First, a dynamics model of the suspension system of the agricultural mobile robot was established based on the first-class Lagrange equation, which includes structural and performance parameters of the suspension system. Four minimum objectives were established for evaluating the performance of the suspension system. Second, the dynamics model was multi-objective optimized using eight multi-objective optimization algorithms including NSGA-III. The optimal solution in the Pareto solution set obtained by the multi-objective optimization algorithm was selected using the TOPSIS method. Finally, the optimal solutions set obtained by the multiple evolutionary algorithms was scored and ranked using the TOPSIS method. The results show that the NSGA-III algorithm obtained the optimal parameter combinations in this study. The top five parameter combinations were simulated by Adams software to verify the feasibility and effectiveness of the method.

This work aimed to find the optimum design of an agricultural robot passive suspension system using the NSGA-III algorithm and TOPSIS method. The main contributions and innovations of this study are as follows:


composed was again scored and ranked using the TOPSIS method, and the scoring was more reasonable.
