*3.2. Simulation Experiments*

Further comparative validation of the above optimization results was carried out. A parametric quarter suspension model was built in Adams software and the dynamics of the optimized parameter combinations were analysed to compare results.

The Adams model used in the dynamic simulation is shown in Figure 3, and its relevant parameters are shown in Table 7. In the Adams model, we used RigidBody: Link to represent frame and horizontal arm in the suspension system; RigidBody: Cylinder to represent damper, and RigidBody: Box to represent the body, under-spring mass (e.g., wheel) and the ground. Artificial masses of all entities were used to ensure that the model parameters in the Adams simulation were the same as the PDE model parameters, involving the parameterized frame, horizontal arm and damper corresponding RigidBody: Link density calculated by linear density shown in Table 1. Table 8 compares the parameters of the globally optimal solution (obtained from NSGA-III) in the PDE model with those in the Adams model. Since the parameters of the Adams model were calculated based on the parameters of the PDE model, the quality parameters of the two models were exactly the same. In Adams View, add fixed, revolute and translational joints to the model at the corresponding position.

**Table 7.** Main parameters of the Adams parametric model for dynamics simulation.


\* These values relate to the optimization results.

**Figure 3.** Adams parametric model for dynamics simulation (1. Body, 2. Frame, 3. Damper, 4. Horizontal arm, 5. Under-spring mass, 6. Ground).

**Table 8.** Parameters of the globally optimal solution (obtained by NSGA-III) in the partial differential equation model compared with those in the Adams model.


A step signal was applied at the wheel-ground contact position, the amplitude of which was the sum of the suspension gravity and the design load of the individual wheels. Three parameters: suspension centre of mass acceleration, suspension centre of mass height and damper spring deflection coefficient, were evaluated for comparison over a period of 1.5 s. The parameter response curves over time are shown in Figures 4–6.

Analysis of Figure 4 shows that the optimal combination of suspension parameters derived from NSGA-III can quickly reduce the body acceleration while keeping the acceleration within a small fluctuation range, which has a positive impact on maintaining smoothness in the work of agricultural mobile robots. By analysing Figure 5, it can be seen that the NSGA-III optimal combination of suspension parameters allows the suspension to reach a steady state in the shortest possible stabilisation time, while keeping the suspension's centre of mass variation range to a minimum, which is beneficial to the mobile robot's ability to effectively maintain and quickly recover a steady state when encountering bumps. Analysis of Figure 6 shows that the optimal combination of suspension parameters

obtained by NSGA-III enables the spring deflection coefficient of the suspension to be kept at a low value. The deflection coefficient was chosen as a measure of spring deflection rather than a direct measure of deflection because an increase in the length of the cross-arm causes an increase in spring deflection when other parameters are held constant. The use of a deflection factor avoids this problem affecting the optimization process. The results of the combined dynamics simulations show that the NSGA-III combined with the TOP-SIS method is indeed the best solution in the Pareto solution set of the multi-objective evolutionary algorithm for the multi-objective suspension optimization problem.

**Figure 4.** The quarter suspension modal barycenter's acceleration time response curve.

**Figure 5.** The quarter suspension modal barycenter's height displacement time response curve.

**Figure 6.** The quarter suspension modal spring deformation factor time response curve.

Combining Tables 4 and 5 and Figures 4–6, it is obvious that no solution is optimal for all evaluation criteria. For example, when we wish to reduce the deformation factor (DF) of the damper, we can do so by increasing the length of the horizontal arm (l2). However, at the same time, increasing the horizontal length (l2) leads to an increase in the total mass of the suspension (Msum). The conflict between the objectives makes it difficult to select an optimal solution from the non-dominated solution set, which is why we introduced the TOPSIS method for optimal solution selection. The TOPSIS method measured the advantages and disadvantages of different non-dominated solutions in a scientific and rational way and scored them. Table 4 shows that one of the solutions obtained by NSGA-III after analysis using the TOPSIS method received the highest score. The four objective values corresponding to the top five scoring solutions are shown in Table 5. We identified that both of the four objectives of the highest scoring NSGA-III solution were not the smallest, i.e., optimal. The smallest total mass (Msum) and the lowest centre-of-mass height (Ycm) appeared in the solution obtained by SMPSO and the smallest degree of centre-ofmass fluctuation(σ) and the smallest degree of damper deformation (DF) appeared in the solution obtained by GDE3. Meanwhile, the time response curves for the NSGA-III derived solution in Figures 4–6, respectively, were not optimal. In the centre-of-mass acceleration time response curves and the damper deformation factor time response curves, the GDE3 solution exhibited better attenuation than NSGA-III, while both the peak acceleration and deformation factor were smaller than the NSGA-III solution. In the centre-of-mass height time response curve, another solution derived by NSGA-III (ranked 4th) was able to obtain the lowest centre-of-mass height, which meant that it has the best stability.

The above results do not seem to indicate that the solution found by NSGA-III is optimal, but when the four objectives were considered collectively, the solution found by NSGA-III showed superiority over the other solutions. We calculated the difference between each objective and their minimum of the sample set according to the TOPSIS method. This meant that any solution will only receive a higher score if all four objectives are as close as possible to their minimum of the sample set. The SMPSO derived solutions had the largest degree of centre-of-mass fluctuation (σ) and damper deformation factor (DF), the GDE3 derived solutions had the second largest total mass (Msum) and the largest height of mass (Ycm); therefore, their scores were weakened. Under the comprehensive scoring of the TOPSIS method, the solution obtained by NSGA-III had the highest score, i.e., the optimal solution.

### **4. Conclusions**

We proposed a multi-objective optimization design method for the suspension of agricultural robots that can balance different performance indicators and obtain the optimal parameter combination of the suspension. We utilized the Lagrangian equation to establish the partial differential equation(PDE) model of the agricultural robot double wishbone suspension, including structural parameters and performance parameters, and utilized the multi-objective evolutionary algorithm NSGA-III and TOPSIS method to carry out the agricultural robot double wishbone optimum design of the suspension. We established four evaluation indicators to evaluate the performance of the suspension, including the total mass of the suspension system, the barycenter's height of the suspension in a stable state, the fluctuation degree of the suspension under the step response, and the deformation degree of the damper. We selected eight typical multi-objective evolutionary algorithms to solve the multi-objective suspension optimization design problem, took advantage of the TOPSIS method to score the non-dominated solution set, and selected the solution with the highest score as the optimal solution. The results show that the optimal solution obtained by NSGA-III and TOPSIS method has advantages in a comprehensive performance. Therefore, we come to the conclusion that NSGA-III combined with the TOPSIS method can effectively obtain a high-quality design of an agricultural mobile robot's suspension system.

**Author Contributions:** Conceptualization, methodology, writing—original draft preparation, writing—review and editing Z.Q.; methodology and visualization, P.Z.; review & editing, supervision, funding acquisition and project administration, Y.H.; validation and formal analysis, H.Y. and T.G.; software, K.Z.; review & editing, supervision, funding acquisition and project administration, J.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Talent start-up Project of Zhejiang A&F University Scientific Research Development Foundation (2021LFR066) and the National Natural Science Foundation of China (32171894(C0043619), 31971787(C0043628)).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
