To make the optimal design of the multilink transmission mechanism applied in mechanical press, the intelligent optimization techniques are explored in this paper. A preference polyhedron model and new domination relationships evaluation methodology are proposed for the purpose of reaching balance among kinematic performance, dynamic performance, and other performances of the multilink transmission mechanism during the conceptual design phase. Based on the traditional evaluation index of single target of multicriteria design optimization, the robust metrics of the mechanism system and preference metrics of decisionmaker are taken into consideration in this preference polyhedron model and reflected by geometrical characteristic of the model. At last, two optimized multilink transmission mechanisms are designed based on the proposed preference polyhedron model with different evolutionary algorithms, and the result verifies the validity of the proposed optimization method.
To improve work efficiency of mechanical press and acquire specific kinematic and dynamic output of slider, multilink transmission mechanisms are applied to replace the traditional cranklink mechanisms. The determination of structure parameters is the most important link in the design process of the mechanism, which determines the ultimate characteristics and performances of the product to a great extent, especially in the conceptual design phase. The conceptual development of the product accomplishes almost 70 percent of the performances and characteristics while occupying only 1 percent of the whole lifecycle cost as shown in Figure
Lifecycle cost committed and performances and characteristics of product versus incurred lifecycle phase.
The design optimization of multilink transmission mechanism is competitive not only for kinematic and dynamic performance, but also for manufacturability, serviceability, and overall lifecycle cost. Considering the conflicting objectives, as well as the highly complex search space and constraints, a rigorous and quantitative multidisciplinary design methodology and evaluation standard of design scheme are needed for solving such multiobjective optimization problem (MOOP). The evolutionary algorithms (EAs) could provide efficient solutions to the above problems [
The feasibility of applying evolutionary algorithms to the solutions of multiobjective engineering optimization problems has been explored by many previous researches.
Marler and Arora [
Saravanan et al. [
Castillo et al. [
However, the researches mentioned above are based on traditional ranking and domination method. This research aims to extend the evaluation process of Pareto front solutions with the incorporation of robust design and concept of multiobjective risk decision. A preference polyhedron model is proposed to provide further dominant relationship information [
This section will introduce the main concepts and previous research works related to this project, covering topics like basic evolutionary algorithm theory, multiobjective optimization, existing Pareto optimal front, and design optimization of multilink transmission mechanism. Moreover, the basic conception of preference polyhedron model will also be introduced briefly in this section.
Evolutionary algorithms are random exploring optimization algorithm based on the idea of the biological evolutionary. Different from traditional optimization algorithms, evolutionary algorithms have no strict requirements for the problem. After a repetitive loop or a series of generations, it can find the fittest individual or individuals to solve the particular problem. Thus, evolutionary algorithms are widely used to solve complex optimization problems [
Genetic algorithm is a typical example of evolutionary algorithms, which regards each solution to the particular optimization problem as an individual in the evolution process of large population. The fitness of each individual is determined by a given fitness function which evaluates the level of aptitude that particular individual solves the given optimization problem, while each generation in the evolution process will create a new set of individuals through genetic operators: crossover and mutation operation. New child individuals produced by the above operations will then be selected by a selection method and finally reinserted into the population by a replacement method. Such process will be repeated generationally at a user defined number. While solving real engineering design problems, for example, to observe the optimal structure parameters of a complex mechanical system, the individual characteristics always can be coded into a finite set of such design parameters. These parameters make up the chromosome with genetic information that represents the realworld structure of the individual, which in this case is a solution to the particular optimization problem. The process can be concluded as shown in Figure
Standard process of evolutionary algorithms.
In a real complex engineering optimization problem, lots of various performance factors or characteristic factors should be taken into consideration while such factors are always difficult to be quantified and competing with each other [
However, the above method always overlooks the stability of the optimization result in the optimum iterative procedure. Figure
Deterministic optimal design point and robust optimal design point.
Considering the interaction of different objectives in the multiobjective optimization problems, it is necessary to evaluate the importance of multiple objectives and determine the degree to which objectives need to be modified [
Preference polyhedron model.
The geometric size and shape parameters of the polygon models in Figure
Moreover, the above polygon model can offer more geometric size and shape information which can be used to build extra evaluation function to improve the Pareto front of optimal results when the evaluation function
The design optimization problem of multilink transmission mechanism is a typical multiobjective optimization problem, where various factors such as the kinematic performance, the dynamic performance, the lifecycle cost, manufacturability, and serviceability should be considered. This means that the evolutionary should optimize possible structure parameters based on at least such following criteria: kinematic performance, dynamic performance, manufacturing costs, and structural stability. In addition, the design parameters of multilink transmission mechanism include the length of each link and the angle parameters of component placement, which will ultimately influence the performance of the mechanism. The design parameters are usually defined as a design vector
The objective of the design of multilink transmission mechanism can be represented as a vector
It is the most important thing to determine an optimal parameter set
Multiobjective optimization design (MOOD) framework.
This section briefly introduces the input and output parameters in the conceptual design process of multilink transmission mechanism. Moreover, the preference polyhedron model and related dominant relationship will also be introduced in detail in this section. By the iteration of the proposed optimization model, new multiobjective optimal solution based on risk assessment and robust design will ultimately be adopted. Figure
The traditional cranklink and multilink transmission mechanism.
This simplified model of multilink transmission mechanism was derived from a real multilink mechanical press, so the design variables depend upon the original value of real design parameters. The design vector
The basic kinematic and dynamic performances and other performances concerned in this optimization are all defined by the elements of the vector
So, the optimization model is simplified to determine the optimal set of vectors
As a result of the error of the manufacturing and assembling, the design parameters always deviate from the ideal optimal design value and bring about the fluctuation of the output performance. The evaluation function
The above multiobjective optimization problems were ultimately expressed by the design vector
Preference polyhedron model.
The practical engineering problems always have many upper and lower limits, and the inside and outside red lines of the polyhedron in Figure
When the value of
As shown in Figure
In this definition,
Out of limit.
When
The confidence function can be ultimately revised as the following form:
In this way, the individual with high confidence level will be easier to be adopted to compare with the lower ones. Based on the definitions above, two individuals in the Pareto front (as shown in Figure
Individuals in Pareto front.
It is assumed that
Apparently, it can be easily proved that
Different metal materials show different malleability under different working conditions including working speed and working temperature. In order to obtain high forming quality products with less energy consumption, the end effector of the transmission mechanism must satisfy some specific kinematic and dynamic performance requirements. More specifically, the slider of press should maintain constant working velocity and acceleration during the working process. In addition, the transmission mechanism should also have a good effect on the reinforcement to consume less energy. Based on the original structure design parameters and performance parameters, this section will discuss the establishment of the multiobjective optimization model.
The main design objectives in this research are demonstrated in Table
Main design objectives.
Design objectives  Description  Value 


Nominal press (KN)  2500 

Slider stroke per minute (times/minute)  10 

Slider stroke (mm)  350~500 

Nominal stroke (mm)  8 

Stroke speed radio  ≥1.4 

Ideal working speed of slider (mm/s)  70 
The initial values of input and output parameters for the multilink transmission mechanisms in Figure
Initial structural design parameters for multilink mechanism.








490 mm  420 mm  360 mm  115 mm  190 mm  1160 mm  45° 
Output performance parameters.
Output parameter  Description  Value 


Peak torque of the motion (N·mm) 


Average torque of the motion (N·mm)  8.16 × 10^{7} 

Stroke speed radio  1.45 

Average working velocity of slider (mm/s)  125.20 

Maximum deviation of working speed  36.19 

Average working acceleration of slider (mm^{2}/s)  40.47 
In order to confirm the value range of design parameters, correlation analysis is implemented between input and output parameters. Figure
Correlation graph.
From Figure
The preference of design parameters.
Parameter  Value  

Lower  Base  Upper  

450  490  550 

350  420  450 

330  360  390 

100  115  130 

175  190  250 

1000  1160  1500 

40  45  50 
It is supposed that the error of design parameters in the processing and installation is ±5 mm and ±2° and follows the standard distribution. The effect caused by such errors is shown in Table
Output performance variation.
Performance  Value  

Min  Max  

122.47  139.98 

35.55  43.68 



The basic process of this optimization is just as shown in Figure
The optimization flow of the model.
The optimization process based on the evolutionary algorithms can be deterministic as you want by either changing the number of iterations or other termination criteria. This experiment adopts NSGAII multiobjective algorithms as the basic framework and takes the number of iterations as the terminal condition. Relative evolutionary parameters are set as Table
NSGAII evolutionary parameters.
Option  Value 

Population size  40 
Numbers of generations  200 
Crossover probability  0.8 
Crossover distribution index  10 
Mutation distribution index  20 
It must be said that the initial values of output parameters
The information about new evaluation parameters
Comparison of output performance.
Output parameters  Initial value  Optimized value  Performance improvement 


1  0.76~0.85  15%~24% 

1  0.43~0.64  36%~57% 

1  0.53~0.57  43%~47% 

1  0.54~0.58  42%~46% 
The history data of
The history data of
The history data of
The history data of
This paper explored the use of intelligent optimization techniques to obtain optimum design of a multilink transmission mechanism. In contrast with traditional evolutionary algorithms, this paper adopted a polygon model into the iterative optimization process to describe the domination relationships of the individuals on the Pareto front. In addition, robust design optimization was used in the optimization process. From the experiment results, the output performance of the multilink transmission mechanism was improved apparently. Interesting future work will include a more detailed study of the polygon model on the description of the individuals on the Pareto front.
The authors declare that they have no competing interests.
This work was supported by the National Natural Science Foundation of China (no. 51505218) and the Fundamental Research Funds for the Central Universities (no. NS2015050). The supports are gratefully acknowledged.