J. Cent. South Univ. (2016) 23: 2001-2007

DOI: 10.1007/s11771-016-3257-z

Reliability-based design optimization for flexible mechanism with particle swarm optimization and advanced extremum response surface method

ZHANG Chun-yi(张春宜)¹, SONG Lu-kai(宋鲁凯)¹, FEI Cheng-wei(费成巍)^2,3,

HAO Guang-ping(郝广平)¹, LIU Ling-jun(刘令君)¹

1. School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, China;

2. School of Computer Science and Engineering, Beihang University, Beijing 100191, China;

3. Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: To improve the computational efficiency of the reliability-based design optimization (RBDO) of flexible mechanism, particle swarm optimization-advanced extremum response surface method (PSO-AERSM) was proposed by integrating particle swarm optimization (PSO) algorithm and advanced extremum response surface method (AERSM). Firstly, the AERSM was developed and its mathematical model was established based on artificial neural network, and the PSO algorithm was investigated. And then the RBDO model of flexible mechanism was presented based on AERSM and PSO. Finally, regarding cross-sectional area as design variable, the reliability optimization of flexible mechanism was implemented subject to reliability degree and uncertainties based on the proposed approach. The optimization results show that the cross-section sizes obviously reduce by 22.96 mm² while keeping reliability degree. Through the comparison of methods, it is demonstrated that the AERSM holds high computational efficiency while keeping computational precision for the RBDO of flexible mechanism, and PSO algorithm minimizes the response of the objective function. The efforts of this work provide a useful sight for the reliability optimization of flexible mechanism, and enrich and develop the reliability theory as well.

Key words: reliability-based design optimization; flexible robot manipulator; artificial neural network; particle swarm optimization; advanced extremum response surface method

1 Introduction

Flexible mechanisms are widely used in many fields such as aerospace, robot, and so forth. With the ever-increasing requirements of light weight, small size and suffering from many complex loads in operation, it is urgent to implement the reliability-based design optimization (RBDO) of flexible mechanism reasonably. The objective of RBDO is to minimize the masses (or sizes) of flexible mechanism with respect to the random variables of material parameters, stochastic loads and reliability degree, in order to balance the contradiction between reliability and mass [1].

Recently, a lot of researches focus on the development of the RBDO method of structures [2-4]. For instance, FEI et al [5-6] proposed four RBOD models for mechanical dynamic assembly structure based on distributed collaborative response surface method, and validated the four RBDO models by the reliability optimization of high pressure turbine blade-tip radial running clearance. EOM et al [7] developed the RBDO method on the foundation of standard response surface method, which was successfully applied to the RBDO of 3-D structure. MARANO [8] discussed the multi-objective RBDO of random vibration structures and studied on the RBDO of tuned mass damper with this method. Current RBDO methods basically satisfy the requirement of structural reliability design and optimization.

With these RBDO approaches, the reliability optimization of flexible mechanism gets a rapid development to some extent, and some achievements have emerged [9-11]. ZHANG and OUYANG [12] focused on the reliability optimization design for complex mechanism based on reliability topology optimization model and level set method; MENG et al [13] finished the reliability optimization of hydraulic transmission mechanism by subset simulation approximate to reliability degree and sequential optimization. However, because of the nonlinearity and strong coupling in the inertia force and dynamic equations of flexible mechanism under operation, it is very difficult to implement the RBDO of flexible mechanism with acceptable optimization efficiency and accuracy. Currently, the investigation of flexible mechanism mainly comprises of reliability analysis and optimization design [14-16]. However, few researches on the RBDO of flexible mechanism are found except that ZHANG et al [17] established the mean model of reliability optimization on flexible mechanism based on the extremum response surface method (ERSM) and HAN et al [18] proposed the decomposition coordinated method of reliability optimization for two-link flexible robot manipulator by combining mean-probability model and decomposition coordinated theory [18]. However, the computational efficiency and accuracy of flexible mechanism in those efforts are unacceptable yet for achieving the desired requirement because they almost take flexible mechanism reliability optimization model as a linear programming model.

In fact, the response surface model and optimization method are two important factors for the improvement of the efficiency and accuracy of flexible mechanism RBDO. Therefore, seeking an effective response surface model and optimization approach is a key way to address the above issue. BP artificial neural network (ANN) technique holds the high-nonlinear mapping ability which is promising to effectively deal with the nonlinear problem of flexible mechanism and improve the precision [19]. Furthermore, particle swarm optimization (PSO) algorithm has been proved to have high optimization speed and precision in complex nonlinear loop optimization [20]. The advanced approaches provide a heuristic insight to improve the RBDO efficiency and precision of flexible mechanism.

The objective of this work attempts to propose the PSO-advanced extremum response surface method (PSO-AERSM) based on BP-ANN technique and PSO algorithm for the RBDO of flexible mechanism. The RBDO of two-link flexible robot manipulator (TFRM) is taken as an example to validate the superiority of the proposed methods with the comparison of methods.

2 Advanced extremum response surface method

2.1 Basic thought

AERSM is a dynamic probabilistic analysis method for complex mechanism. Based on BP-ANN model in Fig. 1, where x_j=[x₁, x₂, …, x_l], and l is the number of variables. The basic thought of AERSM [21] is shown Fig. 2, where x_j is the ith input sample; x is the set of input samples; Y_j(t, x_j) is the output response corresponding to x_j; Y_j,max(x_j) is the maximum value of Y_j(t, x_j) in time domain[0, T]; Y(x) is the extremumresponse surface model (BP-ANN model). Firstly, a BP-ANN model is structured with the relationship between input variables (n is the total number of samples) and system extremum output response Y(x) during time domain [0, T]. The mapping relationship between input variables and output response is obtained by searching for the optimal initial weights of ANN model based on PSO and the network training by Bayesian regularization algorithm. And then the reliability analysis is completed by the established ANN model instead of complex nonlinear dynamic differential equations. Due to the random process of output response transformed into random variables for the dynamic reliability analysis of complex system, the computational efficiency is promising to be improved based on the AERSM.

Fig. 1 BP artificial neural network model

Fig. 2 Basic principle of AERSM

2.2 Precision improvement for advanced ERSM

The computational accuracy of reliability is decided by the model of AERSM, which is mainly influenced by the generalization ability of ANN model. Thus, it has a great significance to improve the generalization of ANN. But the generalization of ANN is seriously affected by the defects of general BP network, such as slow velocity for training convergence and local extremum. Therefore, to solve this issue, this work improves the generalization ability by searching the initial weights, threshold value and network training algorithm with strong generalization ability.

2.2.1 Initial optimal weight and threshold value for ANN

PSO algorithm is able to improve the speed of network training and avoid local extremum value by searching for initial optimal weights and threshold values. Firstly, the PSO algorithm initializes a group of particles in space by taking the weights and threshold values of network as particle dimension. Each of particles is a potential solution [19]. All the particles track the current optimal particle to search in the solution space, and then their individual extremum are updated by tracking the individual and population extremums. After finite iterations, the optimal particle is searched, which is initial optimal weights and threshold values of ANN. The renewal formula of particle position and velocity are

     (1)

where w is the inertia weight; d is the search space dimension; i is the the i-th particle; k is the current iteration number; X_id is the current particle position; V_id is the current particle velocity; P_id is the current individual extremum; P_gd is the current population extremum; c₁ and c₂ are the non-negative acceleration factors; r₁ and r₂ are the random numbers during time domain [0, 1].

Inertia weight w reflects the degree of the current velocity inheriting the previous velocity. Large inertia weight is beneficial to global search, while small inertia weight makes for local search. In order to better balance the global and local search ability, this work adopts the adaptive inertia weight which is changed with iteration number. The adaptive inertia weight is shown as

                       (2)

where w₁ is the initial inertia weight; w₂ is the inertia weight at the largest number of iteration; t is the current iteration number; T is the largest iteration number.

2.2.2 Bayesian regularization algorithm

Gradient decent method is general BP network training algorithm, which does not have a good application in complex nonlinear function approximation due to low approximation accuracy and weak generalization ability. For the problem of the algorithm, Bayesian regularization algorithm is chosen to train ANN model in this work. The algorithm is able to effectively improve the generalization of ANN through solving the over-fitting problem by continuously reducing the weights and threshold values in training process. Its performance function is

                            (3)

where

          (4)

where k₁ and k₂ are the proportional coefficients; w_j the weight of ANN; ε the expected error function of output response; W the vector of weight and threshold value for network layers; K the iteration number; Z the Jacobian matrix of ε; λ the iteration variable.

2.3 Reliability sensitivity analysis

The reliability sensitivity reflects the influence level of the variation of random variables on failure probability. It is an important part of reliability optimization design. Its value determines the search direction of the optimal solution of reliability optimization design.

By using MC simulation [20], the failure probability is achieved as

                            (5)

where μ_g and D_g are the mean and variance matrixes of the limit state function, respectively; Φ(·) the standard normal distribution function.

The sensitivity of the mean matrix of random variables is represented by

                     (6)

in which

                             (7)

where E(·) is the function of mean values; μ_ij the j-th datum in the i-th input variable; μ_i0 the mean value of the i-th input variable; the variance of the i-th input variable; y_j the j-th output response; [y] the allowable deformation.

The larger sensitivity values are selected as the objective of reliability optimization, which can provide the gradient information for reliability optimization and greatly improve the optimization efficiency.

3 PSO-AERSM

The RBDO model is denoted by

                            (8)

where f(x) is the objective function; R_s is the system reliability; is the allowable system reliability; g(x) is the constraint conditions for design variables; a_i and b_i are the lower and upper boundary of the design variable, respectively.

The nonlinear optimization model is solved by using PSO. The particle would be adjusted when it does not meet the constraint conditions. The updating for particle is that the j-th digit is adjusted when the particle X_i does not satisfy the constraints

                     (9)

where N_ar is the arbitrary random number in [0,1].

In PSO model, the reliability degree of system needs to be calculated many times in optimization process. The AERSM model is potential to greatly reduce computation task and improve computational efficiency. The procedure of the RBDO of flexible mechanism with PSO-AERSM is shown in Fig. 3.

Fig. 3 RBDO procedure of flexible mechanism with PSO- AERSM

4 Example

The simplified model of two-link flexible robot manipulator (TFRM) is shown in Fig. 4. The members of the manipulator are assumed as a homogeneous Euler beam. The loads at the point and arm-end are postulated as concentrated masses with ignoring rotational inertia and damping of the rotor motor. The x₁-y₁ local coordinate is built for member-1, the x₂-y₂ local coordinate is established for member-2, where y₁ and y₂ represent the elastic deformation for member-1 and member-2, respectively and t is the movement time. The movements of two moving local coordinates are described by using the azimuths θ₁(t) and θ₂(t).

Fig. 4 Two-link flexible robot manipulator

The shape functions of the components are

                          (10)

The elastic deformation for component changes with time. The elastic deformation of member-1 and member-2 on direction y are y₁(t, x₁) and y₂(t, x₂), respectively, which are expressed in Eq. (11). The generalized coordinate q(t) is shown in Eq. (12).

                    (11)

   (12)

where g_i(t) is the i-th order elastic coordinate of member-1 and u_i(t) is the i-th order elastic coordinate of member-2.

According to Lagrange equations, the dynamics equation of TFRM is

       (13)

where U_g is the gravitational potential energy of the system; M is the mass matrix; K is the stiffness matrix; Q_k(t) is the total force corresponding to the moment of rotation calculated by the virtual work method.

The mass M, length L and driving torque τ are the basic parameters of TFRM, as shown in Table 1. The density ρ, elastic modulus E and section size h and b are the random parameters of TFRM, as shown in Table 2. In the analysis, the random variables include material density, elastic modulus and section sizes, which are assumed to follow the normal distributions and be independent on each other. Comparing with the mean of length, the variance of length is very small. So the length is considered a constant.

Table 1 Basic parameters of component 1 and component 2

Table 2 Random parameters of component 1 and component 2

Within the variance range of random variables, 100 groups of input data are extracted by MC method. The output response (maximum deformation) of member-1 and member-2 are calculated by limit state function. The normalized data are taken as training sample of ANN. Through the comparison of the network training error, the hidden layer neuron are chosen as “3”, which is shown in Table 3. And the 4-3-1 three layer network structure is chosen as BP-ANN model where the transfer function from input layer to hidden layer is ‘tansig’, the transfer function from the hidden layer to the output layer is ‘purelin’, and the training function is ‘trainbr’. Then, the initial optimal weights and threshold levels are searched by using PSO. The particle dimension v is the total number of weights and threshold values in which v=19 and the number of particle N=40 are selected in this work.

Table 3 Network training error with different hidden neuron numbers

The initial optimal weights and threshold values are inputted into ANN model. The learning rate and training error are 0.1 and 10^-5, respectively. After network training with Bayesian regularization algorithm, the advanced extremum response surface function is obtained where the weight and threshold levels of two members are given as follows.

Member-1,

      (14)

and Member-2,

  (15)

Through the reliability sensitivity analysis, the results indicate that the stiffness of member-1 is not a failure and the input variables sensitivity is 0; the reliability sensitivities of member-2 are shown in Eq. (16) and Fig. 5. From Eq.(16) and Fig. 5, it is seen that the section size hold the greatest influence on member-2 due to its sensitivity is more than 99 %. Thus, it is necessary to optimize the section size of TFRM.

 (16)

The reliability optimization mathematical model is established by taking the mean value of section size as the objective function and system stiffness reliability R_s as the constraints, as shown in Eq. (17).

                     (17)

Fig. 5 Sensitivity of component 2

where R_s1 and R_s2 are the reliabilities of member-1 and member-2, respectively;is the system allowable reliability.

The design requirement of TFRM is to achieve the minimum of section size on TFRM when the allowable deflection is 18 mm, the variance is 0.36 mm, and the system reliability is 0.953.

The mathematical model is solved by taking the optimal variable number v=4, which is particle dimension, and the population number N=40. After 100 iterations, the fitness value of optimal individual curve is shown in Fig. 6. After extracting 1000 samples, the results of TFRM reliability analysis with different methods are listed in Table 4. And the comparison of different optimization methods is summarized in Table 5.

As revealed in Table 4, the computational time of ERSM and AERSM are far less than those of the MC method, while the AERSM has the highest efficiency; the calculation accuracy of AERSM is significantly consistent with MC method, which is 0.8 % higher than ERSM. When the reliability degree is calculated by using AERSM, BP neural network with “trainbr” training function is selected to fit the limit state function that can promote the network to converge to the global optimal solution effectively. Thus the accuracy of AERSM is high.

Fig. 6 Change curve of optimal adaptive value

Table 4 Results of TFRM reliability analyses with different methods

Table 5 Comparison of different optimization methods

As demonstrated in Table 5, the optimized section size by PSO-AERSM is much smaller than the original data, which reduces by 29.83 mm² compared to the first-order second-moment method and by 22.96 mm² compared to the decomposed coordinated method. While the reliability requirement is achieved with PSO- AERSM, the section size of member-1 is reduced for large reliability margin. While the section size of member-2 is increased due to small reliability margin. It is demonstrated that the section size of member-1 is reduced by 59 mm² and the section size of member-2 is reduced by 26.42 mm² under the reduction of total section size. The total section area of TFRM is reduced. It is obvious that PSO algorithm can search the optimal solution, and has the advantage of high precision and fast convergence speed.

In conclusion, the proposed AERSM is promising to improve the efficiency and precision for the RBDO of flexible mechanism, which sufficiently validates the effectiveness of PSO algorithm.

5 Conclusions

The objective of this effort is to develop the particle swarm optimization-advanced extremum response surface method (PSO-AERSM) with high computational efficiency and precision for the reliability-based design optimization (RBDO) of flexible mechanism. Through the RBDO of two-link flexible robot manipulator(TFRM) to minimize the section size of TFRM subject to reliability and random variables, some conclusions are summarized as follows:

1) The reliability sensitivity analysis indicates that the safety margin of member-1 is excessive and the sensitivity of failure probability to section size of member-2 is over 99 %.

2) Through the RBDO of TFRM with the PSO-AERSM, it is demonstrated that the section size of member-1 is reduced by 59 mm² and the section size of member-1 is reduced by 26.42 mm² under the reduction of total section size.

3) As revealed from the comparison of methods, the computational accuracy of RBDO with AERSM is significantly consistent with MC method, and better than ERSM. Moreover, AERSM has the highest computational efficiency. Comparing with the first-order second-moment method and decomposed coordinated method, the developed PSO-AERSM has the optimal result due to its high-efficiency and high-precision in the RBDO of flexible mechanism.

4) The efforts of this work provide a promising approach for the RBDO of flexible mechanism, and enrich and develop the theory and method of mechanical reliability optimization design.

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(Edited by FANG Jing-hua)

Foundation item: Projects(51275138, 51475025) supported by the National Natural Science Foundation of China; Project(12531109) supported by the Science Foundation of Heilongjiang Provincial Department of Education, China; Projects(XJ2015002, G-YZ90) supported by Hong Kong Scholars Program, China; Project(2015M580037) supported by Postdoctoral Science Foundation of China

Received date: 2015-06-01; Accepted date: 2015-11-15

Corresponding author: FEI Cheng-wei, PhD; E-mail: Feicw544@163.com