Handling stability analysis and collaborative optimization design for an off-road vehicle with compound suspension

Abstract

Enhancing the handling and stability of off-road vehicles is critical for ensuring driving safety and performance in harsh terrain. This paper proposes a novel integrated analysis and collaborative optimization methodology tailored to address the handling stability limitations of off-road vehicles. A multi-body dynamics model of the full vehicle, including the front torsion bar spring and the rear leaf spring suspension, has been built and validated through the K&C test. On this basis, the vehicle handling stability has been generally analyzed and evaluated with a low score in steady-state steering characteristics. To improve the comprehensive handling stability, a co-simulation framework integrating modeFRONTIER, Adams/Car, and MATLAB has been established to perform the sensitivity analysis and develop an optimal design of a compound suspension system. A surrogate modeling approach combined with the multi-objective particle swarm optimization (MOPSO) algorithm has been employed to execute the collaborative optimization. The results show that the selected design factors prove to show important influence on the handling stability. The handling stability of the vehicle is significantly improved after optimization with the score of the constant-radius cornering test NW increasing by 19.92% and the overall evaluation score N_z increasing by 3.62%. The results validate the feasibility of the proposed methodology, which provides a scalable and efficient reference for enhancing the handling stability of off-road vehicles.

Introduction

Handling stability is of high importance for the off-road vehicle to ensure safety and efficient driving performance on various challenging terrains. The suspension system plays a crucial role in determining the handling characteristics of the off-road vehicle.^1,2 In recent years, compound suspension systems have gained significant attention due to their potential to combine the advantages of different types of suspension systems, effectively improving the handling stability and ride comfort of the vehicle.^3,4 These systems can range from combinations of different spring types, as in the present study, to more complex architectures such as hydraulically interconnected suspensions. ⁵

With the increasing research focus, a number of studies have investigated the handling stability performance of the off-road vehicle in the design, modeling, simulation, and control. Li et al., ⁶ developed a nonlinear 8-DOF model for ME-wheel off-road vehicles and proposed a predictive load transfer rate index, which is beneficial for early detection of rollover risk. Meanwhile, Yun et al., ⁷ established a 7-DOF model to investigate how suspension parameters affect vehicle response, thereby providing support for simulation-based testing. Gao et al., ⁸ constructed an evaluation indicator system for the vehicle’s steady-state and transient responses by implementing quadratic response surface fitting for the critical parameters. In addition, they introduced a generalized multi-dimensional adaptive learning particle swarm optimization algorithm to identify the global optimal solution for this indicator system. ⁹ Consequently, the vehicle’s handling and stability were enhanced by refining the vehicle model according to the optimization results. Jiang et al., ¹⁰ developed a nonlinear 14-DOF vehicle model and validated its accuracy through comparisons with ADAMS and CarSim models, and then important factors affecting the vehicle’s handling stability were analyzed. Wu presented a kinetic dynamic suspension system to achieve enhanced cooperative control of the roll and warp motion modes for on-road and off-road sports utility vehicles. ¹¹ In recent years, the introduction of intelligent algorithms and multidisciplinary optimization methods has further enhanced suspension system performance. For example, deep reinforcement learning-based active suspension control strategies can optimize damping force distribution in real-time,^12,13 and novel control strategies like fractional-order PID have been developed for semi-active suspensions, ¹⁴ while topology optimization techniques significantly reduce suspension mass while improving structural strength. ¹⁵ Additionally, in-wheel motor-driven electric off-road vehicles pose new challenges for suspension design, requiring consideration of increased unsprung mass effects on handling stability.^16,17

According to the previous studies, it can be concluded that the suspension and steering systems play a major role to achieve a good compromise between the conflicting performance goals, including vehicle handling, stability, safety, and ride comfort. ¹⁸ Thus, several efforts have been made to come up with the collaborative optimization of the suspension and steering systems. Zhang et al., ¹⁹ conducted research on optimizing the rear wheel steering mechanism. They effectively coordinated the motion relationship between the steering mechanism and the suspension system, and validated the optimization outcomes through simulations in ADAMS. Javanshir et al., ²⁰ optimized the geometric characteristics of suspension system using geometric equations governing the suspension system with the aim of minimizing camber angle changes by adding anti-roll bar and optimized torsion bar. In compound suspension systems, parameter matching directly determines handling stability and ride comfort, but existing studies suffer from insufficient comprehensiveness in parameter analysis and screening. For instance, most works only focus on individual parameters such as stiffness or damping, especially for commercial vehicle leaf spring compound suspensions.

For off-road vehicles, handling stability control turns into a key technology for safe operation in rough environments due to dynamic load distribution and terrain irregularities.^21–24 For extreme terrains (e.g. mud, sand, rock surfaces), compound suspensions must dynamically adjust parameters using terrain recognition algorithms. ²⁵ For example, vision- or LiDAR-based preview systems can proactively adjust suspension stiffness, ²⁶ while distributed electric-drive off-road vehicles improve stability in extreme conditions through coordinated optimization of torque vectoring and suspension systems.^27–30 Although numerous studies have been performed on handling stability analysis and optimization, two core research gaps remain unaddressed. First, research on leaf spring-type compound suspensions for commercial vehicles is still scarce. Existing studies primarily focus on multi-link or hybrid compound suspensions for passenger vehicles and off-road utility vehicles, while neglecting leaf spring compound suspensions that are critical for commercial vehicle load-bearing and handling stability. Second, existing studies lack comprehensive analysis and screening of parameters affecting suspension performance, and fail to establish a systematic analysis and optimization process. It is also noteworthy that while advanced control strategies like steer-by-wire offer promising avenues for stability enhancement, ³¹ the foundational optimization of the passive suspension geometry and parameters remains a critical and underexplored aspect for commercial vehicle design, which is the focus of this work.

To fill these gaps, this study focuses on the handling stability of commercial vehicle leaf spring compound suspensions. In this paper, we propose an integrated analysis and collaborative optimization method for improving the handling stability of an off-road vehicle with compound suspension. Additionally, we have performed a sensitivity analysis to identify the key parameters that have the most significant impact on handling stability and propose optimized parameter settings. The findings of this study are expected to provide valuable insights and practical guidance for the design and improvement of composite suspension systems in off-road vehicles.

Modeling and simulation

Full vehicle modeling

To investigate the handling stability of the off-road vehicle with compound suspension, we developed a high-fidelity multi-body dynamics (MBD) model of the full vehicle in Adams/Car, as shown in Figure 1. The model integrates a front torsion bar spring suspension and a rear leaf spring suspension, accurately capturing their dynamic interactions under various handling conditions. Key geometric and load parameters of the vehicle, including centroid coordinates and axle loads, are detailed in Tables 1 and 2. Prior to full-vehicle simulation, critical sub-models (e.g. suspension and tire subsystems) were experimentally validated through K&C tests and tire mechanical characteristic bench tests, ensuring the reliability of the MBD model for subsequent analysis.

Figure 1.

The suspension subsystem MBD model. (a) Front torsion bar spring suspension. (b) Rear leaf spring suspension.

Table 1.

Geometry parameters of the vehicle.

Parameters	Value (mm)
Vehicle length	5365
Vehicle width	2120
Vehicle height	2130
Centroid coordinates	(2072, −0.5, 456)
Wheelbase	3351

Table 2.

Load parameters of the vehicle.

Parameters	Value (kg)
Full quality	5010
Front axle shaft load	2990
Rear axle shaft load	2020
Total sprung mass	4920
Front axle sprung mass	2933
Rear axle sprung mass	1987

The front suspension subsystem, featuring a torsion bar spring, is modeled as a flexible body in Adams/Car to precisely replicate its elastic characteristics, as shown in Figure 1(a). One end of the torsion bar spring is connected to the vehicle frame through a fixed joint, while the other end is linked to the control arm. The rear suspension subsystem, equipped with a variable-stiffness leaf spring, is constructed using the equivalent mid-section principle and the flexible beam principle, as shown in Figure 1(b). Each leaf of the leaf spring is defined as a flexible beam, and its material properties are set accordingly. The stiffness and damping characteristics of springs and shock absorbers are defined based on their actual physical properties, ensuring that the model can faithfully reflect the dynamic response of the suspension under various operating conditions.

The tire subsystem is also a key part of the MBD model. In this work, a PAC2002 tire model is adopted to describe the complex relationship between tire forces and tire performance parameters including the slip angles, camber angles, and vertical loads. Compared with the typical Magic Formula model and FTire model, the adopted PAC2002 model matches better for the moderate working conditions of commercial vehicle in this work and shows higher computational efficiency. Further, Characteristic parameters such as tire stiffness, damping, and friction coefficients can be obtained from bench test and directly used for MBD modeling, seeing in Table 3.

Table 3.

Characteristic parameters of PAC2002 tire model.

Parameters	Value
Vertical stiffness (kN/m)	1430
Vertical damping (kN s/m)	20
Unloaded radius (m)	1.0
Tire width (m)	0.75
Aspect ratio	0.65
Slip stiffness (kN)	800
Cornering stiffness (kN m/°)	20
Minimum friction	0.8
Maximum friction	0.9

The other subsystems of the vehicle, including steering and car body are also constructed after proper simplification. The subsystem models are assembled together through input and output communicators, and them the assembly MBD model of the full vehicle is established, as shown in Figure 2.

Figure 2.

The full vehicle MBD model.

Model verification

To validate the accuracy of the suspension MBD model, the suspension performance parameters obtained from the simulation are compared with those from the test. The tests were carried out on a K&C characteristic test bench, mainly focusing on the parallel wheel travel test for the front suspension.^32–34 During the parallel wheel travel analysis, a parallel excitation with a stroke of −90 to 80 mm was applied simultaneously at the left and right wheels, causing the suspension to move. The suspension alignment parameters were obtained. Figure 3 depicts the curves of toe angle and camber angle versus left wheel travel, respectively. It can be seen that the simulation results accord well with the test results, verifying the accuracy of the suspension model. Additionally, it can be seen from Figure 3 that, during wheel travel, the overall change of the toe angle and camber angle is <1°/50 mm, which is beneficial to the understeer characteristics of the vehicle.

Figure 3.

Test and simulation results for the parallel wheel travel analysis: (a) Toe angle variation versus wheel travel and (b) Camber angle variation versus wheel travel.

The validation of the tire MBD model was conducted on a multifunctional tire mechanical characteristic test bench. The tests mainly focus on the tire cornering and longitudinal slip performance under different vertical forces. In the cornering performance test, various lateral forces were applied to measure the tire’s slip angles as shown in Figure 4(a). In the longitudinal slip test, the tire’s slip ratios under different longitudinal forces were obtained as shown in Figure 4(b). By comparing the simulation results of the PAC2002 model with the test results, a good achievement can be observed in Figure 4 and the reliability of the tire model can be verified.

Figure 4.

Test and simulation results for the tire mechanical analysisl: (a) Lateral force versus slip angle and (b) Longitudinal force versus slip ratio.

Steering stability analysis

Constant-radius cornering test

In the constant-radius cornering test simulation, the vehicle is assumed to travel at a constant speed v along a circular path with a fixed radius R. The steering wheel angle is adjusted to maintain the vehicle’s circular motion. The simulation parameters include a vehicle speed increasing from 2.5 to 25 m/s. The vehicle model incorporates the nonlinear tire force characteristics and considers the influence of road adhesion coefficient μ, which is set to 0.8 for a dry asphalt road condition. The time history of the simulation results of the yaw velocity and the lateral acceleration are shown in Figure 5(a) and (b), respectively. The simulation results show that as the vehicle speed increases, the lateral acceleration increases monotonically and the yaw velocity of the vehicle’s center of gravity increases initially and then tends to stabilize. The ratio of cornering radius R_k/R₀ and the difference between the front and rear axle side slip angles (δ₁–δ₂) are also monitored and depicted in Figure 5(c) and (d), respectively, showing an obvious understeer characteristics.

Figure 5.

Simulation results of the constant-radius cornering test: (a) Time history of the yaw velocity, (b) Time history of the lateral acceleration, (c) R_k/R₀ versus lateral acceleration and (d) (δ₁–δ₂) versus lateral acceleration.

Step cornering test

For the step cornering test, a sudden step change of 32.5° in the steering wheel angle is applied to the vehicle model. The initial vehicle speed is set as 60 km/h, and the step input magnitude is 0.1 s. The simulation duration is 5 s, capturing the vehicle’s transient response immediately after the steering input. The time history of the simulation results of the yaw velocity and the lateral acceleration are shown in Figure 6(a) and (b), respectively. According to Figure 6, under the step cornering test, the simulation results reveal that the vehicle’s yaw velocity and lateral acceleration exhibit an immediate response to the steering wheel angle step input, reaching its peak value within 0.5–1 s and then gradually stabilizing after about 1 s.

Figure 6.

Simulation results of the step cornering test. (a) Time history of the yaw velocity. (b) Time history of the lateral acceleration.

The other typical test including pylon course slalom test and pulse input test have also been conducted and not been elaborated here. On basis of the comprehensive analysis of the results from all the above-mentioned vehicle handling stability tests, the evaluation scores for each handling stability test are summarized in Table 4. From Table 4, the overall evaluation score N_Z can be rated as 92.43, which is calculated as the average score of all test. Among all tests, it can be concluded that the analyzed off-road vehicle possesses a high body roll angle K_Φ under constant-radius cornering test, showing that the stable steering characteristic of the vehicle needs improvement.

Table 4.

Evaluation of the vehicle handling stability.

Test	Indice	Value	Score	Weighted score	Evaluation
Constant-radius cornering test	Lateral acceleration, an (m/s2)	6.02	80.2	N W = 80.74	Average
	Body roll angle, KΦ (°/(m/s2))	1.02	74.62		Fair
	Understeer rate, U (°/(m/s2))	0.24	87.41		Good
Step cornering test	Response time, T (s)	0.21	91.54	N J = 91.54	Excellent
Pylon course slalom test	Averaged peak steering wheel angle, δ (°)	27.71	100	N S = 100	Excellent
	Averaged peak yaw velocity, r (°/s)	5.86	100		Excellent
Pulse input test	Resonant frequency, fp (Hz)	0.75	92.70	N M = 94.85	Excellent
	Resonant level, D (dB)	0.47	100		Excellent
	Phase lag angle, α (°)	73.45	91.03		Excellent

Sensitivity analysis of the compound suspension

Design of experiments

Suspension structural parameters, including coordinates of suspension hard points, spring stiffness, shock absorber damping, torsion bar spring, and so on, are of paramount importance to the handling stability of off-road vehicles. When conducting multi-objective optimization design, simultaneous analysis of all parameters would entail enormous computational burdens, compromise optimization efficiency, and diminish the targeted focus of the design process. This paper therefore focuses on investigating how the structural parameters of front and rear suspension systems affect vehicle performance. Specifically, we have conducted a sensitivity analysis of these parameters using the design of experiments (DOE) method, with handling stability indices adopted as key evaluation metrics. Furthermore, the TOPSIS-entropy weight method was employed to compute the comprehensive contribution coefficients. First, a parameter performance index matrix was constructed and standardized, and then the objective weights were calculated on basis of index variability. Then the proximity of each parameter to positive/negative ideal solutions were quantified. The coefficients were derived from coupling these weights and proximities. Ultimately, this process enabled the identification and selection of system parameters exerting the most significant impact on suspension performance.

From the above handling stability analysis, it can be adopted that the stable steering characteristic of the vehicle needs improvement. Meanwhile, to balance the “sensitivity” and “hysteresis” characteristics of the off-road vehicle in the neutral position range, the on-center handling test is also introduced. The considered performance indicators include understeer rate U, body roll angle K_Φ, yaw velocity gain G_ω, and corresponding response lag time T_D. In accordance with the constraints of engineering design, 12 initial design parameters are selected, which are as follows: rear suspension leaf spring stiffness, positions of the front and rear points of the leaf spring, damping coefficients of the front and rear shock absorbers, diameters of the front and rear torsion bar springs, lengths of the front and rear buffers, stiffness of the front and rear buffers, and stiffness of the front torsion bar spring. The initial design parameters and the allowable variation ranges are listed in Table 5.

Table 5.

Parameters and values of the suspension model.

Parameter	Symbol	Value	Variation range
Stiffness coefficient of rear suspension leaf spring	SR	1	0.6–1.3
Front point location of leaf spring (mm)	FR	66	30–96
Rear point location of leaf spring (mm)	RR	109.5	70.5–139.5
Diameters of front torsion bar spring (mm)	DFS	24	16–30
Diameters of rear torsion bar spring (mm)	DRS	22	16–30
Length coefficient of front buffer	LFB	1	0.7–1.4
Length coefficient of rear buffer	LRB	1	0.7–1.4
Stiffness coefficient of front buffer	SFB	1	0.7–1.3
Stiffness coefficient of rear buffer	SRB	1	0.7–1.3
Damping coefficient of front shock absorber	DF	1	0.5–1.5
Damping coefficient of rear shock absorber	DR	1	0.5–1.5
Stiffness of front torsion bar spring (N m/rad)	ST	95,000	66,500–123,500

There are two primary types of design of experiments (DOE) techniques: classic designs and space-filling designs. The most widely utilized classic designs encompass central composite design (CCD), fractional factorial design (FFD), and Box–Behnken designs. Prominent space-filling designs include maximum entropy designs, mini-max and maxi-min designs, Latin hypercube sampling (LHS) designs, and orthogonal arrays18. In the present study, the CCD method was adopted for sensitivity analysis due to its advantages in terms of sampling space and nonlinear fitting. A CCD orthogonal array (L₂₀₀2¹²) involving twelve factors with two levels was designed.

Integrated simulation platform

In previous stage of sensitivity analysis, the DOE factors table has been established. However, the substantial number of sample points in this table renders individual simulation analyses insufficient to meet the study requirements. To address this challenge, it is imperative to develop a co-simulation platform that can automate the processes of model establishment, simulation execution, and response output based on the parameters of the sample points. Thus, we built an integrated simulation platform combining modeFRONTIER, Adams/Car, and MATLAB, as shown in Figure 7. This multi-tool co-simulation approach is consistent with the current state-of-the-art for tackling complex off-road suspension optimization problems. ³⁵

Figure 7.

Integrated simulation platform for sensitivity analysis.

The specific steps for running the integrated simulation platform are as follows: First, Adams/Car is invoked via a CMD file to execute the simulation program, which includes commands for calling the vehicle assembly model, road surface model, control file, and extracting calculation results. After completing the vehicle simulation, result files for the performance indicators are generated. MATLAB and custom post-processing scripts are then used to read the simulation results, processing the data into the target format and finally transferring the results to modeFRONTIER to generate a data table. This integrated platform will not only streamline the workflow but also enhance the efficiency and accuracy of the sensitivity analysis, thereby ensuring comprehensive and systematic evaluation of the suspension parameters.

Sensitivity analysis

The sensitivity analysis of the handling stability of the vehicle has been executed on basis of the established simulation platform. The importance of each objective is comprehensively weighed and the sensitivity analysis results are depicted in Figure 8. This figure shows the influence of design parameters on the performance indicators within a heat map. Among them, red indicates a positive influence, and blue indicates a negative influence. A sensitive correlation exists between two variables only when the absolute value of the influence coefficient reaches at least 0.3, and thus the sensitive parameters can be screened out.

Figure 8.

Heat map of the influence of design parameters on the performance indicators.

It can be seen from Figure 8 that the rear suspension leaf spring stiffness S_R only has a relatively significant impact on the body roll angle K_Φ, with an influence coefficient of −0.626, indicating that the greater the stiffness, the smaller the roll angle. Meanwhile, the rear shock absorber damping D_R also affects the yaw velocity response lag time G_ω. The diameters of both the front and rear torsion bar spring have a certain impact on K_Φ, with influence coefficients of −0.425 and −0.316, respectively. Increasing the diameters of the front and rear torsion bar spring help to reduce K_Φ. It can be deduced that the increase of rear suspension leaf spring stiffness S_R and the torsion bar spring diameter enhances the roll stiffness and thus reduces the body roll angle K_Φ, which is consistent with the theory that higher stiffness improves anti-roll capability. The stiffness of the front suspension torsion bar spring S_T has varying degrees of impact on other optimization objectives except for the understeer rate U. On basis of the dynamic coupling theory, the front suspension torsion bar spring stiffness S_T affects yaw dynamics by decreasing yaw velocity gain G_ω and response lag T_D. Reducing S_T can decrease the yaw velocity gain G_ω and the corresponding response lag time T_D, while increasing the body roll angle K_Φ. The other design parameters have little impact on each performance indicator. The sensitivity analysis identifies the most influential parameters for the system, which enables us to focus on key variables in the subsequent collaborative optimization, improving the efficiency and accuracy of the optimization process.

Collaborative optimization design

Mathematical modeling

In general, it is not feasible to carry out optimization design solely based on sensitivity analysis results, as some parameters have been shown to exert contradictory effects on performance indicators.^36,37 Therefore, to achieve the optimal design of suspension parameters for the handling stability scenarios, a meta-model-based collaborative optimization process is implemented in this section. Meta-modeling refers to a technique that constructs simplified surrogate models to replace complex, computationally expensive simulation models, aiming to improve analysis efficiency while retaining key system characteristics. Computationally, direct MBD simulation is inefficient for multi-parameter optimization because it requires solving complex multi-body coupling and nonlinear contact equations, taking minutes to hours per run. While meta-model method avoids this by first training on a small set of MBD simulation data and subsequent performance predictions take only seconds. Thus in this paper, meta-modeling is used to quickly fit the relationship between the suspension design parameters and the handling stability indices. The main flowchart of the proposed process is structured as follows: mathematical modeling for the collaborative optimization problem, meta-modeling for the objectives, and solving via the multi-objective optimization algorithm.

On basis of the above sensitivity analysis results, the body roll angle K_Φ under the constant-radius cornering test and the yaw velocity response lag time T_D under the on-center handling test are taken as the main optimization objectives. The corresponding value ranges of the design parameters, the yaw velocity gain G_ω under the on-center handling test and the total weighted root-mean-square acceleration a_w under the ride comfort test are used as constraints. This formulation reflects the common need in suspension design to make trade-offs between conflicting performance objectives, such as handling versus ride comfort, a challenge also addressed in recent research on advanced suspension systems. ³⁸ Thus, the collaborative optimization problem can be defined as a constraint satisfaction problem in which the target is to find a series of solutions minimizing K_Φ and T_D. The mathematical model of this optimization problem can be described as follows:

$$\{\begin{array}{c}\min (K_{\varphi },T_D)\hfill \\ \multicolumn{1}{c}{\begin{array}{c}s.t{x}_i^{\mathrm{lower}\mathrm{bound}}\le x_i\le x_i^{\mathrm{upper}\mathrm{bound}},i=1,\cdots ,5\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\begin{array}{c}0.6\le S_R\le 1.3\hfill \\ \multicolumn{1}{c}{66500\le S_T\le 123500}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}{D}_F\ge 0.5,D_R\le 1.5\hfill \\ \multicolumn{1}{c}{D_{\mathrm{FS}}\ge 16,D_{\mathrm{RS}}\le 30}\end{array}}\\ \multicolumn{1}{c}{\begin{array}{c}0.18\le G_{\omega }\le 0.35\hfill \\ \multicolumn{1}{c}{a_w\le 0.315}\end{array}}\end{array}}\end{array}}\end{array}$$

(1)

where $x_i$ can take continuous values in the given variation range.

Meta-model construction

As stated in equation (1), the optimization process is based on specific values of the objective functions and constraints. However, obtaining these values through finite element (FE) simulations is highly time-consuming. To address this issue, we employed the meta-model technique to efficiently evaluate the objective and constraint functions. Meta-models, or surrogate models, are a cornerstone of modern machine learning-based optimization frameworks, enabling rapid exploration of complex design spaces. ³⁹ Typically, in meta-model-based optimization problems, the primary challenge lies in constructing accurate meta-models for the objective and constraint functions, especially when these functions are highly complex and non-linear. Here three meta-models namely radial basis function (RBF), Kriging, and artificial neural networks (ANN) have been adopted to model the objective functions K_Φ and T_D, and the constraint function G_ω and a_w on basis of the train set which is established in sensitivity analysis section. The fitting indices for each meta-model have been calculated on basis of the validation set, including the coefficient of determination R², the root mean squared error (RMSE) and the, as listed in Table 6.

Table 6.

The coefficient of determination R² for the meta-models.

Functions	RBF			Kriging			ANN
	R 2	RMSE	MAE	R 2	RMSE	MAE	R 2	RMSE	MAE
K Φ	0.961	0.125	2.328	0.981	0.089	0.998	0.977	0.116	1.221
TD	0.881	0.231	3.174	0.832	0.281	4.291	0.951	0.101	2.109
G ω	0.882	0.352	2.551	0.901	0.182	1.289	0.832	0.572	3.981
aw	0.901	0.211	0.605	0.863	0.267	1.881	0.981	0.191	0.221

As can be seen from Table 6, the coefficient of determination R² values for different responses vary significantly depending on the objectives and constraints. The Kriging model shows the best adaptability for the responses of body roll angle K_Φ and yaw velocity gain G_ω to design variables. It can also be concluded that the neural network (ANN) model can effectively capture the relationship between design variables and the yaw velocity response lag time T_D. Meanwhile, the neural network (ANN) model has the highest credibility in responding to the total weighted root-mean-square acceleration a_w. Referring the definitions of the accuracy measurement metrics discussed earlier, it is clear that the above constructed meta-models exhibit a strong alignment with the high-fidelity simulation models. Consequently, these meta-models are deemed appropriate in the subsequent optimization process.

Multi-objective optimization

The collaborative optimization problem addressed in this work, which involves multiple objectives and constraints, is essentially the search for a set of non-dominated solutions—referred to as the Pareto front—that satisfy the constraints while optimizing the objective functions. Numerous optimization algorithms for multi-objective problems have been proposed in previous studies, including evolutionary algorithms, ant colony optimization, genetic algorithms, and others. Among these, the multi-objective particle swarm optimization (MOPSO) algorithm, inspired by the foraging behavior of birds, stands out as a prominent swarm intelligence optimization method. ⁴⁰ MOPSO is highly regarded for its well-distributed Pareto front and strong convergence capabilities, which stem from the collaborative behavior and information sharing among individuals within a swarm to achieve optimal solutions. In this study, a typical MOPSO algorithm has been employed to address the constrained multi-objective optimization problem. Table 7 shows the corresponding coefficients pre-specified in the MOPSO algorithm.

Table 7.

Coefficients of the MOPSO algorithm.

Coefficients	Value
Particle swarm size, $n$	100
Particle dimensions	4
$k_{\max }$	1000
${\omega }_{\max }$	0.8
${\omega }_{\min }$	0.3
$c_{1,\max }$	2
$c_{1,\min }$	0.5
$c_{2,\max }$	2
$c_{2,\min }$	0.5

The detailed optimization process was implemented using the integrated simulation platform. It took a total of 88 iterations, spanning ∼42 min, to achieve the desired results. The resulting Pareto front comprising 455 solutions is illustrated in Figure 9 with red marks. It is evident that the MOPSO algorithm effectively addressed the problem, yielding a well-distributed set of optimal solutions. As shown in Figure 9, the objective function K_Φ has opposite trend comparing with that of objective function T_D as selecting a higher K_Φ from optimal solutions would sacrifice the values of T_D. It should be pointed out that our current work focuses on the deterministic optimization of the compound suspension system under baseline operating conditions, and we acknowledge that we have not fully addressed the sensitivity of vehicle dynamics to parameter variations (e.g. tire properties, payload, terrain friction coefficient μ, and damping tolerances) as noted. This limitation may affect whether the optimized design can maintain satisfactory handling stability within the typical variation range of uncertain parameters, and shall be addressed in our future work.

Figure 9.

Pareto front solution set for collaborative optimization.

The obtained Pareto front provides designers with a range of optimal design options to select from based on specific practical requirements. From an engineering application perspective, to obtain a trade-off design compromising the two objectives, a satisfactory function has been developed. The trade-off optimal design has been identified as the one with the maximum satisfactory value, as presented in Table 8. Additionally, the trade-off optimal design has been assessed by performing handling stability analysis, and the results are also presented in Table 8. It can be seen from Table 8 that the performance indicators of the trade-off optimal design are all within the acceptable range. Compared with the initial design, a comprehensive promotion is achieved as K_Φ increasing by 22.5% and T_D decreasing by 3.4% simultaneously.

Table 8.

The trade-off optimal design for collaborative optimization.

Design version	Design parameter						Performance indicator
	SR	DF	DR	DFS	DRS	ST	K Φ	TD
Optimal design	1	1	1	22	24	95,000	0.791	0.169
Initial design	1.3	0.8	0.9	30	29	100,000	1.02	0.175

A comprehensive handling stability analysis has been performed for the optimal design, and the evaluation scores for each handling stability test are summarized in Table 9. From Table 9, it can be concluded that the handling stability of the vehicle is significantly improved after optimization with the score of the constant-radius cornering test increasing by 19.92% and the overall evaluation score increasing by 3.62%.

Table 9.

Evaluation of the vehicle handling stability after optimization.

Test	Weighted score	Initial design	Optimal design	Improvement
Constant-radius cornering test	N W	80.74	96.82	19.92%
Step cornering test	N J	91.54	93.61	2.26%
Pylon course slalom test	N S	100	100	0.00%
Pulse input test	N M	94.85	96.4	1.63%
Overall evaluation score	N Z	92.43	95.78	3.62%

Conclusions

In this paper, an integrated analysis and collaborative optimization method has been proposed to improve the handling and stability performance of an off-road vehicle with compound suspension. A multi-body dynamics model of the full vehicle, including the front torsion bar spring and the rear leaf spring suspension, has been built and validated through the K&C test. On this basis, the vehicle handling stability has been generally analyzed and improved. The results are summarized as follows:

1. The typical handling stability test have been conducted to evaluated the handling stability of the off-road vehicle. The results show that the analyzed off-road vehicle possesses a high body roll angle K_Φ under constant-radius cornering test, showing that the stable steering characteristic of the vehicle needs improvement.

2. By performing the sensitivity analysis of the handling stability, the influence of the suspension parameters on the handling stability performance have been comprehensively weighed and the sensitive parameters have screened out for the subsequent optimization process.

3. We have performed the collaborative optimization by using an surrogate modeling method and a multi-objective particle swarm optimization (MOPSO) algorithm. The results show that the selected design factors prove to show important influence on the handling stability. The handling stability of the vehicle is significantly improved after optimization with the score of the constant-radius cornering test N_W increasing by 19.92% and the overall evaluation score N_z increasing by 3.62%. The proposed integrated method proves to provide an extensive reference for the handling stability optimization for the off-road vehicle.