The performance of the suspension system is one of the most important factors in the vehicle design. For the double wishbone suspension system, the conventional deterministic optimization does not consider any deviations of design parameters, so design sensitivity analysis and robust optimization design are proposed. In this study, the design parameters of the robust optimization are the positions of the key points, and the random factors are the uncertainties in manufacturing. A simplified model of the double wishbone suspension is established by software ADAMS. The sensitivity analysis is utilized to determine main design variables. Then, the simulation experiment is arranged and the Latin hypercube design is adopted to find the initial points. The Kriging model is employed for fitting the mean and variance of the quality characteristics according to the simulation results. Further, a particle swarm optimization method based on simple PSO is applied and the tradeoff between the mean and deviation of performance is made to solve the robust optimization problem of the double wishbone suspension system.
Suspension used in an automobile is a system mediating the interface between the vehicle and the road, and the functions of it are related to a wide range of drivability such as handing ability, stability, and comfortability [
With reference to automobile suspension system, a number of researches have devoted considerable efforts to design optimization. Many important relationships have been highlighted among vehicle suspension parameters and suspension performance indices [
Robust optimization design is essentially multiple objectives:
This paper is organized as follows: in Section
Robust design based on particle swarm optimization.
The doublewishbone suspension system is a group of space RSSR (revolute joint—spherical join—spherical join—revolute joint) fourbar linkage mechanisms. Its kinematics relations are complicated, kinematics visualization analysis is difficult, and its performance is poor. Thus, rational settings of the position parameters of the guiding mechanism are crucial to assuring good performance of the independent doublewishbone suspension.
The vehicle’s right and left suspensions are symmetrical, so choose the left or the right part of the suspension system which is studied to simulate the entire mechanism, excluding the variation of wheel centre distance (WCD) which is advisable. The key design parameters are the coordinates of the key points (see Table
Key point.
Key point 




Lca front  −200  −400  150 
Lca back  200  −450  155 
Lca outer  0  −750  100 
Uca front  100  −450  525 
Uca back  250  −490  530 
Uca outer  40  −675  525 
Shaft inner  0  −200  225 
Spring lower  0  −600  150 
Subframe front  −400  −450  150 
Subframe back  400  −450  150 
Tie rod inner  200  −400  300 
Tie rod outer  150  −750  300 
Spring upper  40  −500  650 
The center of the wheel  0  −800  300 
Double wishbone suspension model.
Make the following assumptions on double wishbone suspension.
The composition members of the suspension are rigid body, and the elastic deformation is ignored.
Rigid connection between the various components is used and ignores the internal clearance and friction.
Only consider the ground roughness, without regarding to the dynamic factors.
Add an incentive on the test platform to simulate the unevenness of the ground; the tires are always in contact with the test bench.
Add an excitation source on the test platform,
Sideways displacement.
Toe angle.
As shown in Figure
In this section, the model of the robust design is built. A fullfactor test and sensitivity analysis are utilized to determine main design variables. The Latin hypercube design is adopted to find the initial point, and the database is created for fitting the kriging model of the robust design.
Robust design has become a powerful tool to aid designers in making judicious selection and control of variation. The fundamental principle of robust design is to improve the quality of a product by eliminating the variation of controllable factors (i.e., dimension, assemble gap, material properties, etc.) and uncontrollable factors (i.e., applied loadings, environment, aging, etc.). Consequently, compared with traditional optimization design, robust design can make the product maintain good performance [
A standard engineering optimization problem is normally formulated as follows:
There are 12 key points, and each one of them has 3 coordinate values. So, there are 36 coordinate parameters. If every one of the coordinate is selected as design variables, it needs much iteration. In order to reduce time of analysis and save resources, the fullfactor test is utilized to determine main design variable, and the impact of every dependent variable is in Table
Impact of each variable.
Coordinates of key points  Impact  

Toe angle  Sideways displacement  
Lca front 
1  1 
Lca front 
2  2 
Lca front 
3  3 
Lca back 
1  2 
Lca back 
1  3 
Lca back 
2  3 
Lca outer 
1  1 
Lca outer 
1  3 
Lca outer 
3  2 
Uca front 
1  1 
Uca front 
1  1 
Uca front 
3  3 
Uca back 
1  1 
Uca back 
1  2 
Uca back 
2  3 
Uca outer 
1  1 
Uca outer 
1  3 
Uca outer 
3  3 
Controllable factors.
Key point  Level 1  Level 2  Level 3 

Lca front 
−405  −400  −395 
Lca front 
145  150  155 
Lca back 
195  200  205 
Lca back 
−455  −450  −445 
Lca back 
150  155  160 
Lca outer 
−755  −750  −745 
Lca outer 
95  100  105 
Uca front 
520  525  530 
Uca back 
−495  −490  −485 
Uca back 
525  530  535 
Uca outer 
−680  −675  −670 
Uca outer 
520  525  530 
Engineering optimization problems often need enormous computation time for several programs running at the same time. We cannot provide the evaluation of the objective function and constraints to execute such large scale of exact analysis. So the application of approximation is necessary. In this paper, the Kriging model is adopted to build the approximation. Kriging model, one of the response surface models (RSM), has such advantages as unbiased estimator at the training sample point, desirably strong nonlinear approximating ability, and flexible parameter selection of the model, and thus it is quite suitable for approximate models [
In this paper the Kriging model is introduced into the robust design. In the conventional Kriging model, the performance
From a Bayesian perspective, the prior knowledge of the performance
The conventional Kriging model assumes that the Gaussian process has a stationary covariance, with the covariance function defined as follows:
The variance
In order to innovate or improve and develop a new product and confirm a new technical parameter experiments usually need to be done repeatedly in the process of production and scientific research. It is very important to reasonably arrange experimental procedures to reduce the times of experiments and shorten the time of each experiment and avoid blindness. It requires two aspects of works to be done in order to solve the problem mentioned above. One is to design an experiment that can fully reflect the effect of all factors, which can reduce time of experiments and save resources. Another is to analyze the experimental results, in order to acquire reasonable conclusions and the error analysis.
DOE can analyze a design space and provide a rough estimate of an optimal design, which can be used as a starting point for numerical optimization. The Latin hypercube design could cover the design space more evenly than other DOE methods and generate more evenly distributed points. Therefore, in this paper, the Latin hypercube design is adopted to find the initial point and created the database for approximation model. For this problem, the inputs are the 12 main design variables, the outputs are the mean and their variance of the toe angle and sideways displacement, and 200 sample points from an LHS design are used to fit the kriging model. A set of 393 verification points, randomly selected across the domain, is used to evaluate the RMSE for each kriging model. Kriging model is established to fit the multiobjective robust design. The rsquare of the kriging model is 0.87, so it can fit the virtual model.
The particle swarm optimization (PSO) is one of the evolutionary computation techniques introduced by Kennedy and Eberhart in 1995 [
Particle swarm optimization is usually used as a traditional optimization method which is inspired from the social behaviour of flocks of birds. It is more competitive in various aspects, for example, due to its simplicity. Particle swarm optimization, genetic algorithms, and other evolutionary algorithms are all artificial life calculated. But particle swarm optimization is different from other evolutionary algorithms, using group iterative solution of cooperation mechanisms to generate the optimal solution instead of using group iterative solution of competing mechanisms. In PSO algorithm, each individual is called “particle,” which represents a potential solution. The algorithm achieves the best solution by the variability of some particles in the tracing space. The particles search in the solution space following the best particle by changing their positions and the fitness frequently; the flying direction and velocity are determined by the objective function.
The procedure of PSO is as follows:
initialize the original position and velocity of particle swarm;
calculate the fitness value of each particle;
for each particle, compare the fitness value with the fitness value of pbest; if current value is better, then renew the position with current position, and update the fitness value simultaneously;
determine the best particle of group with the best fitness value; if the fitness value is better than the fitness value of gbest, then update the gbest and its fitness value with the position;
check the finalizing criterion; if it has been satisfied, quit the iteration; otherwise, return to step
It can be shown as Figure
Particle swarm optimization.
Assuming
The first term of (
In experiment, the population of group particle is 40;
Particle swarm optimization was used to search the optimal solution. A particle swarm optimization is created, the maximum iterations are set to 50, the number of particles is set to 15, and the objectives are the values and their variations of the toe angle and sideways displacement. V. Pareto, the French economist, who studied the multiobjective optimization problem of economics first, proposed the concept of Pareto solution set. There are 27 Pareto solutions in the results of the optimization.
In multiobjective optimization, each optimization objective is often conflicting, which requires coordination between the optimal solutions of each target. Considering the importance of each target, choose one Pareto optimal solution, and the design values of it are shown in Table
Robust results.
Key point  Initial value  Robust results 

Lca front 
−400  −400 
Lca front 
150  150.33 
Lca back 
200  205 
Lca back 
−450  −450 
Lca back 
155  151.67 
Lca outer 
−750  −752.5 
Lca outer 
100  101.33 
Uca front 
525  519.67 
Uca back 
−490  −489.67 
Uca back 
530  532.17 
Uca outer 
−675  −679.83 
Uca outer 
525  523.33 
Sideways displacement.
Toe angle.
It can be seen that the maximum deviation in the toe angle for the optimal design has been reduced by 52 percent, compared with the base design. As the discussion of the results, the most concerned factor is the relationship between objective function and design parameter. By comparing the experimental results, the robust design based on particle swarm significantly improved the robust of the toe angle and the sideways displacement, ensuring the reasonable of the design performance.
In this study, a robust design based on bioinspired computation is presented and illustrated by the design of a double wishbone suspension system in order to reduce the effect of variations due to uncertainties in fabrication. As they are directly related to fabrication errors, the coordinates of key points were taken as design variables and at the same time are considered as random variables. So the robust design optimization problem had 13 design variables (joint positions) and 13 random constants (fabrication errors of joint positions). In this paper, the Latin hypercube design is adopted to make DOE design matrix of the 13 design variables. The Kriging model is built according to the result of DOE, and then the particle swarm is used to search optimal solution of the robust design. Particle swarm is implemented in a test case and the results show that the method can decrease the solution’s time. The robustness of solution is improved. The improvement in robustness became larger as the amount of fabrication errors increases.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under the Grant no. 51165007 and by the Natural Science Foundation of Jiangxi Province under the Grant no. 20132BAB206025. The authors would like to thank the reviewers for their valuable comments and suggestions.