Designing a complex mechatronic product involves multiple design variables, objectives, constraints, and evaluation criteria as well as their nonlinearly coupled relationships. The design space can be very big consisting of many functional design parameters, structural design parameters, and behavioral design (or running performances) parameters. Given a big design space and inexplicit relations among them, how to design a product optimally in an optimization design process is a challenging research problem. In this paper, we propose a systematic optimization design method based on design space reduction and surrogate modelling techniques. This method firstly identifies key design parameters from a very big design space to reduce the design space, secondly uses the identified key design parameters to establish a system surrogate model based on datadriven modelling principles for optimization design, and thirdly utilizes the multiobjective optimization techniques to achieve an optimal design of a product in the reduced design space. This method has been tested with a highspeed train design. With comparison to others, the research results show that this method is practical and useful for optimally designing complex mechatronic products.
It is very difficult to optimally design a complex mechatronic product for many reasons. First, there are a big number of design parameters, usually greater than 100. Second, there are many key performance indicators as either goals or constraints. Third, these parameters are multipledisciplines related and their determination needs multidisciplinary collaborative efforts. Furthermore, the coupled relations among these parameters and performance indicators are highly nonlinear and vague. In a word, optimally designing a complex mechatronic product is very challenging in a big design space. Therefore, the optimization efficiency is low and it is difficult to obtain a satisfactory solution. In addition, complex mechatronic products are usually composed of many subsystems having different parameters, and their performances are correlated. It is difficult to have an effective system model to describe the relationships between parameters, subsystem performances, and the whole system performances. Thus, DataDriven Modelling techniques such as artificial neural networks (ANNs) based surrogate modelling provide alternative solutions to this problem.
Designing a complex mechatronic product optimally requires considering numerous design parameters and ways of identifying a set of best design variables and obtaining a best design solution effectively under major constraints such as safety/security and stability. The challenges are threefold:
This paper presents a systematic optimization design method for designing complex mechatronic products. At the beginning, it uses each subsystem dynamics model to conduct design parameter sensitivity analysis and identify key design parameters for design space reduction. Then, a neural networkbased surrogate model is established between the identified key design parameters and key performance indicators to describe the whole system performance. Upon the neural network surrogate model, an optimization design model is developed, and finally, an optimal design computing is realized with an improved optimization algorithm for better quality and efficiency. In the current design practice, designing a typical complex mechatronic product such as a highspeed train is mainly by the trial and error method. It lacks a systematic design optimization method for its design. Therefore, we take the optimal design of a highspeed train as a case study to verify the effectiveness of the proposed method.
The paper is structured as follows. Section
Multiobjective and multidisciplinary optimization in engineering is closely related to our research problems. Many scholars or engineers have conducted a lot of research on optimization frameworks and algorithms. Fabio et al. [
The above methods usually need to establish a large number of equations and formulas for derivation. Thus, their efficiency is low and the solution is not guaranteed. Some scholars proposed using surrogate models to solve optimization problems. Wang and Shan [
However, the optimization design of complex mechatronic products is a systematic problem, which involves parameter identification, design space reduction, and optimization strategies. Some scholars studied some of the related problems. Wang [
In summary, the surrogate model technology is useful for the optimization of complex electromechanical systems, and it has a successful application in the optimization of some products. There are a body of work on surrogate modelling and multiobjective optimization. However, for optimally designing complex mechatronic products, it still lacks a systematic approach to guide and guarantee the optimization design of such complex systems. Therefore, this paper proposes a systematic optimization method based on integral design space reduction and system surrogate modelling techniques for designing complex mechatronic products optimally.
In order to improve the design efficiency and reduce the difficulty of performance evaluation (simulation) calculation, this paper puts forward a new systematic optimization design method based on the surrogate model and intelligent multiobjective optimization techniques for designing complex mechatronic products. It includes five stages. In Stage 1, according to the topology structure, design parameters, and boundary conditions of a complex system, the design parameters of the complex system are extracted. In Stage 2, the design parameter space is reduced by expert knowledge or by the parameter sensitivity analysis with each subsystem evaluation (simulation) model. In Stage 3, the key design parameters affecting the whole system design objectives (or running performances) are obtained, forming a reduced design space. In Stage 4, a surrogate model describing the relationship between the whole system performances and the key design parameters is established, and then a corresponding optimization model is developed based on the surrogate model. In Stage 5, intelligent optimization algorithms are used for the optimization design of a complex mechatronic product.
The system optimization design flow is shown in Figure
Optimization design flow.
Figure
Optimal design problem space of a complex mechatronic product.
In order to solve the optimization design problem for a complex mechatronic product, it is necessary to define what the design parameters (variables) are and what the objective indicators are. The objectives of the optimization are defined by considering the comprehensive requirements of the performances. Usually setting up a set of the design objectives is very important, while the design goal (comprehensive objective) function is usually formed by summing up all of the weighted objectives. Weights for each individual objective are determined based on their relative importance and previous design knowledge. So the system optimization is transformed into solving the maximum or minimum value of the comprehensive objective function.
Within a huge design space, there are many parameters and the coupled relationships between these design parameters and objectives are very complex and usually nonlinear. If
To solve this problem, a method for design space reduction is proposed with two rounds. In round 1, important parameters are chosen by experts with related domain knowledge. In round 2, the important parameters will be used to establish surrogate models against each performance indicator, and then sensitivity analysis is conducted based on the established surrogate models. Finally, according to the sensitivity analysis results, the key design parameters are determined based on a predefined sensitivity threshold. The specific process is shown in Figure
The flow of design space reduction.
For a complex mechatronic product, usually, there is no practical model for describing whole system performances, and instead, there are several subsystem performance (simulation) models within multidisciplinary fields. The whole system performances are nonlinearly coupled with the subsystem performances. Thus, the direct use of subsystem performance simulation models in the optimal design process is difficult because it requires a coupled system simulation method with spatiotemporal synchronization process control over subsystem simulation computing [
The surrogate model is a mathematical model for fitting discrete data using an approximation approach, which can determine key parameters and their value ranges with less training samples and advanced trial design methods. [
The framework of the BPN.
The design performances of a complex product are considered at the same time with different performance indexes. When performing an optimal design, it is necessary to meet all these indexes at the same time and thus synthesize these indexes into a comprehensive goal function. Lastly, a multiobjective optimization model can be established based on the surrogate model.
Another aspect is to establish a radial basis function network for the parameters. The establishment of the backpropagation neural network (BPN) parameters includes the accuracy of the model and diffusion factor. The mean square error of the same model can also affect the adjustment of the BPN surrogate model.
For having a surrogate model with high accuracy, RRMSE (Relative Root Mean Square Error) error criterion is used in training. RRMSE of the model is defined as follows:
Among them,
The optimization model can be described with the surrogate model as follows:
Design variables (key design parameters):
Optimization goal function with subperformance functions (indicators):
When constructing the optimization goal function
The optimization of complex mechatronic products is a very complex problem, involving many parameters, indicators, and boundary conditions. Meanwhile, it also requires considering the efficiency and accuracy of optimization. Intelligent optimization algorithm is a good method for speeding up the optimization. Intelligent optimization algorithms include genetic algorithm, differential evolution algorithm, and particle swarm algorithm. Cai and Aref [
Each algorithm has its own characteristics. DE (differential evolution) is a parallel optimization algorithm evolved from GA (genetic algorithm), and it has excellent characteristics for global optimization. We select DE in our application because DE reportedly has a more effective evolutionary strategy [
The novelty of this systematic optimization design method has twofold. Firstly, it can effectively couple design space reduction and the system surrogate modelling techniques into the system optimization modelling. In the existing literatures, these two techniques are discussed separately; thus, when applying the surrogate modelling technique to develop a surrogate model for describing complex system relationships, there is a general difficulty in determining what parameters in both inputs and outputs should be chosen to establish an effective surrogate model for use in the optimization. Secondly, it demonstrates that the identified key variables from the space reduction are well qualified as the input variables for establishing the corresponding system surrogate model and the optimization model, thus, providing a general form of the system optimization modelling and solving techniques for complex mechatronic product optimal design.
Here, we demonstrate the proposed optimal design method with a highspeed train design. We validate the system optimization from the following three aspects. The first is to approve that the proposed systematic method is necessary and solutionguaranteed by the comparison of a direct surrogate model without design space reduction and the one with this operation. The second is to compare the convergence speed and accuracy of the surrogate model under different design parameters. This indicates that, without a design space reduction operation, the resultant surrogate model will perform quite differently. The last is to prove the effectiveness of the system method in terms of its efficiency and accuracy. We overall demonstrate the usefulness of the method by comparing three different tests.
Highspeed train is a typical complex mechatronic product. As a means of rapid transportation, it has been widely appreciated and vigorously developed. However, design and development of a highspeed train need to evaluate its dynamics behaviors and performances under various running environments to meet its safety and running performance requirements. It has several subsystems (shown in Figure
Highspeed train and its running environment.
Therefore, here, we only take dynamics performancerelated design as an example because it is the top level of design for highspeed trains. The running performances of a highspeed train include 7 performance indicators as our design objectives; the number of initial design parameters possibly affecting the performance indicators is more than 100 (shown in Table
The all design parameters of highspeed train.
Parameter  Name  Unit 

Parameter 1  Nominal wheel radius 

Parameter 2  Distance between backs of wheel flanges 

Parameter 3  Wheelset roll moment of inertia  Kg⋅ 
Parameter 4  Wheelset yaw moment of inertia  Kg⋅ 
Parameter 5  Longitudinal stiffness of primary suspension per axle side 

Parameter 6  Vertical damping of primary suspension per axle side  KN⋅s/m 
Parameter 7  Longitudinal stiffness of axle box tumbler joint per axle side 

Parameter 8  Yaw damper lateral span 

Parameter 9  Lateral stiffness of secondary suspension per bogie side 

Parameter 10  Vertical stiffness of secondary suspension per bogie side 

Parameter 11  Secondary vertical damper  KN⋅s/m 
Parameter 12  Secondary lateral damper  KN⋅s/m 
Parameter 13  Wheelset mass 

Parameter 14  Lateral stiffness of axle box tumbler joint per axle side 

Parameter 15  Longitudinal stiffness of secondary suspension per bogie side 

Parameter 16  Longitudinal stiffness of Yaw damper joint per bogie side 

Parameter 17  Carbody roll moment of inertia  Kg⋅ 
Parameter 18  Lateral distance between the secondary suspension of the two sides of the bogie 

Parameter 19  Carbody mass 

Parameter 20  Lateral stiffness of primary suspension per axle side 

Parameter 21  Longitudinal distance between bogie centers 

Parameter 18  Carbody yaw moment of inertia  Kg⋅ 
Parameter 19  Vertical distance from the rail surface to the center of gravity 

Parameter 20  Wheelbase 

Parameter 21  Vertical stiffness of primary suspension per axle side 

Parameter 22  Carbody pitch moment of inertia  Kg⋅ 
Parameter 23  Framework mass 

Parameter 24  Wheelset pitch moment of inertia  Kg⋅ 
Parameter 25  Vertical damping joint stiffness per axle side 

Parameter 26  Nominal wheel radius 

Parameter 27  Distance between backs of wheel flanges 

Parameter 28  Wheelset roll moment of inertia  Kg⋅ 
Parameter 29  Air spring vertical stiffness (per spring) 




Parameter 100  Lateral damper joint stiffness of secondary suspension per bogie side 

Parameter 101  Lateral stop clearance of secondary suspension 

Parameter 102  Vertical damping transverse span of secondary suspension 

Parameter 103  Yaw damper lateral span 

Parameter 104  Traction drawbar mass 

Parameter 105  Swing stiffness of traction joint of secondary suspension 

Due to the complexity of mechatronic products, the optimal design of a mechatronic product is a very complex problem. Some papers suggest that the surrogate modelling technique can be applied to this problem [
The number of initial design parameters of highspeed train is more than 100. The relationship between design parameters is very complicated and highly nonlinearly coupled in the wheel/rail contact model. The construction of a surrogate model of a complex electromechanical product requires big enough samples to train the model. Therefore, in order to get enough good samples, a Railway System Dynamics simulation Package SIMPACK Rail is utilized to generate a set of corresponding data between a set of inputs of design variables and a set of performance indicators. When we prepare the design variable values for simulation, there is a difficulty in knowing the right value range of each design parameter. Thus, we use experts’ guessed values as our references.
Figure
Topological structure and dynamics model in SIMPACK software of highspeed train.
Next, we take a unified approach to determine the range of each parameter. Based on the initial range value of the parameter, the upper and lower 10% are used as the initial range value of each parameter. We use this range to carry out the experimental design to get hundreds of test data sets. We use the experimental data as input and use the SIMPACK software to simulate performances of each sample design. Due to the mismatch of parameters, only 58 sets of 100 can produce simulation results. For the remaining 42 sets, we cannot get simulation results; thus, as a result, there are no enough samples to train the surrogate model with only 58% of the calculation results.
Therefore, it is not feasible to use the full parameters in the original design space to establish a surrogate model. In order to solve this problem, we need a system method to have a guaranteed solution. Thus, in this paper we propose a method with design space reduction as a key step.
The proposed design space reduction method has two rounds. In round 1, important parameters are selected by experts with related domain knowledge. In round 2, the important parameters will be used to establish surrogate models against each performance indicator, and then sensitivity analysis is conducted based on established surrogate models.
The details of the design space reduction technique have been reported in [
The design parameters sorted as the importance.
Parameter  Name  Unit  Section 


Nominal wheel radius 

395–430 

Distance between backs of wheel flanges 

1,351–1,355 

Wheelset roll moment of inertia  Kg⋅ 
500–750 

Wheelset yaw moment of inertia  Kg⋅ 
500–800 

Longitudinal stiffness of primary suspension per axle side 

800–1,150 

Vertical damping of primary suspension per axle side  KN⋅s/m  10–30 

Longitudinal stiffness of axle box tumbler joint per axle side 

5–10 

Yaw damper lateral span 

2,400–2,800 

Lateral stiffness of secondary suspension per bogie side 

100–200 

Vertical stiffness of secondary suspension per bogie side 

120–300 

Secondary vertical damper  KN⋅s/m  20–60 

Secondary lateral damper  KN⋅s/m  30–50 

Wheelset mass 

1,800–2,200 

Lateral stiffness of axle box tumbler joint per axle side 

4–10 

Longitudinal stiffness of secondary suspension per bogie side 

100–200 

Longitudinal stiffness of Yaw damper joint per bogie side 

5–13 

Carbody roll moment of inertia  Kg⋅ 
70,000–120,000 

Lateral distance between the secondary suspension of the two sides of the bogie 

2,400–2,500 

Carbody mass 

28,000–40,000 

Lateral stiffness of primary suspension per axle side 

800–1,200 

Longitudinal distance between bogie centers 

17,000–18,000 

Carbody yaw moment of inertia  Kg⋅ 
1,100,000–1,500,000 

Vertical distance from the rail surface to the center of gravity 

1,400–1,600 

Wheelbase 

2,400–2,600 

Vertical stiffness of primary suspension per axle side 

1,000–1,500 

Carbody pitch moment of inertia  Kg⋅ 
1,200,000–1,700,000 

Framework mass 

2,100–3,100 

Wheelset pitch moment of inertia  Kg⋅ 
65–100 

Vertical damping joint stiffness per axle side 

3–6 
Based on the results obtained from the design space reduction, the system surrogate model with the 16 key design parameters and performance indicators was established. In order to prove the system approach and justify the reason for choosing the 16 key design parameters in the system surrogate model (named as BPN2), we construct other 5 surrogate models for comparison. The result comparisons with the 6 surrogate models are detailed in the next section.
We use the same structure (see Figure
Six surrogate models.
BPN1  BPN2  BPN3  BPN4  BPN5  BPN6  

Data set  U  V  W  X  Y  Z 
The number of parameters  12  16  18  20  25  29 
The number of iterations  68  93  126  195  256  291 
The surrogate model structure of highspeed train based on different parameter groups.
Figure
The convergence rates of these surrogate models are shown in Figure
The convergence rate of BPN surrogate models.
BPN1
BPN2
BPN3
BPN4
BPN5
BPN6
Design variables are corresponding to the 6 BPN models, and they are a subset of
According to the design requirements and specifications of highspeed train in China, the range of the 7 indexes can be obtained. In this way, we regard the performance requirements as the boundary conditions of the optimization design. Constraints are
The function is
In order to compare the optimization results, we use each of the 6 surrogate models as an optimization analyzer to get the corresponding results between the input variables and 7 key indicator outcomes; in this way, we establish 6 optimization models corresponding to the BPN1, BPN2, BPN3, BPN4, BPN5, and BPN6 surrogate models. We use the same optimization algorithm (differential evolution algorithm) to solve the 6optimization models. The optimization algorithm is implemented within MATLAB.
The optimization times for each case are shown in Figure
The time histogram required for obtaining optimal results based on six surrogate models.
The optimization results are shown in Table
Comparison table of optimization results.
Lateral stability  Vertical stability  Derailment 
The ratio of wheel load 
Lateral wheelset force/10 (KN)  Overturning 
Critical speed/200 (km/h)  

Initial performances  2.3804  2.0245  1.497  1.978  1.2232  1.98  2.98 
BPN1  2.2162  2.0235  1.568  2.104  1.12513  2.054  3.03 
BPN2  2.1935  2.0021  1.272  1.828  1.02541  1.939  3.43 
BPN3  2.1925  2.001  1.253  1.82  1.02465  1.915  3.40 
BPN4  2.192  2.0003  1.248  1.806  1.01683  1.867  3.44 
BPN5  2.178  1.9986  1.232  1.79  1.01485  1.819  3.35 
BPN6  2.1715  1.9927  1.211  1.781  1.01312  1.804  3.39 
The table of performance improvement percentage (%).
Lateral stability  Vertical stability  Derailment coefficient  The ratio of wheel load reduction  Lateral wheel force  Overturning coefficient  Critical speed  

BPN1  6.90%  0.05% 


8.02% 

1.68% 
BPN2  7.85%  1.11%  15.03%  7.58%  16.17%  2.07%  15.22% 
BPN3  7.89%  1.11%  16.30%  7.58%  16.23%  3.28%  14.28% 
BPN4  7.91%  1.19%  16.63%  8.70%  16.87%  5.70%  15.69% 
BPN5  8.50%  1.28%  17.70%  9.50%  16.87%  8.13%  12.62% 
BPN6  8.78%  1.36%  19.11%  9.96%  17.17%  8.89%  14.30% 
In order to show the optimization results more intuitively, the optimized performance results in Table
Comparison chart of optimization results.
The chart of performance improvement percentage.
Figures
In summary, using just the key variables can give a system surrogate model with a better balance between the quality of optimization results and computational costs. This is a balanced system solution. In our case study, the surrogate model BPN 2 with 16 key variables is identified as the system surrogate model and supports the system optimization very well with a good balance between the optimization quality and time. It can be concluded that the system optimization method based on the system surrogate model with only key design parameters is more effective in terms of optimization accuracy and computational cost. The importance of the design space reduction and the effectiveness of the system optimization method for complex mechatronic products are proved.
Due to the complexity of complicated mechatronic products, the multiplicity of parameters, and the intricate relationship between design parameters and performance indicators, the optimal design of such a product is very sophisticated with hard problems. Therefore, it is almost impossible to ensure that all parameters are optimal. Because of the complex relationship among subsystems, the whole system simulation model is difficult to build, and the coupled simulation time is too long and the cost is huge, it is necessary to provide a datadriven modelling and optimization design method for complex mechatronic product design from a systemic perspective to balance the system executing time and effectiveness.
This paper has proposed a systematic design optimization method for complex mechatronic products from the identification of initial design parameters and objectives, to design space reduction for key variable identification, setting up a system surrogate model with just the key variables, establishing a system optimization model, and optimization computing.
The implementation of this method has been demonstrated through a case study of China highspeed train design with 6 different surrogate models. From the comparison study, it can be seen that the appropriate design space reducing is very important, which leads to not only the identification of key variables but also the establishment of the system surrogate and optimization models. The system optimization method based on the system surrogate model established with just key design parameters is more effective in terms of optimization accuracy and computational cost. The ability to produce a better design solution with the proposed method has been validated and demonstrated.
The authors declare that they have no conflicts of interest.
This research is supported by the National Natural Science Fund of Intelligent Mapping Research for Requirement Driven Fine Design Process of HighSpeed Railway Vehicle (no. 51575461), China.