Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 9815469, 9 pages

https://doi.org/10.1155/2018/9815469

## A Hybrid Strategy of Differential Evolution and Modified Particle Swarm Optimization for Numerical Solution of a Parallel Manipulator

^{1}State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China^{2}Lappeenranta University of Technology, 53810 Lappeenranta, Finland

Correspondence should be addressed to Bingyan Mao

Received 19 July 2017; Revised 8 January 2018; Accepted 22 January 2018; Published 22 February 2018

Academic Editor: Giuseppe Fedele

Copyright © 2018 Bingyan Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a hybrid strategy combined with a differential evolution (DE) algorithm and a modified particle swarm optimization (PSO), denominated as DEMPSO, to solve the nonlinear model of the forward kinematics. The proposed DEMPSO takes the best advantage of the convergence rate of MPSO and the global optimization of DE. A comparison study between the DEMPSO and the other optimization algorithms such as the DE algorithm, PSO algorithm, and MPSO algorithm is performed to obtain the numerical solution of the forward kinematics of a 3-RPS parallel manipulator. The forward kinematic model of the 3-RPS parallel manipulator has been developed and it is essentially a nonlinear algebraic equation which is dependent on the structure of the mechanism. A constraint equation based on the assembly relationship is utilized to express the position and orientation of the manipulator. Five configurations with different positions and orientations are used as an example to illustrate the effectiveness of the proposed DEMPSO for solving the kinematic problem of parallel manipulators. And the comparison study results of DEMPSO and the other optimization algorithms also show that DEMPSO can provide a better performance regarding the convergence rate and global searching properties.

#### 1. Introduction

Differential evolution (DE) is a heuristic and straightforward strategy with the prominent features of global optimization and only a few control parameters [1, 2]. Particle swarm optimization (PSO) is a group theory based algorithm that was inspired by fish schools, bird flocks, and others. The disadvantage of PSO is that the individuals are easily influenced by the best particle and best position so it may only get the local optimum [3, 4]. The control parameters of PSO can be modified according to the specific optimization problems and applications [5–7]. This paper presents a new method integrated with differential evolution (DE) and modified particle swarm optimization (PSO). In particular, this strategy aims to combine the advantages of DE and the modified PSO together and then apply the hybrid algorithm to the numerical solution of a parallel manipulator.

Parallel robots have been a hot research topic for many years due to their superior performance such as high response speed, high stiffness, high payload/weight ratio, low inertia, and good dynamic performance [8–10]. This paper focuses on a spatial parallel manipulator with 3 DOFs which was developed by Lee and Shah [11]. The 3-RPS parallel manipulator has three legs and each branch is a serial kinematic chain [12]. To obtain the position and orientation of the moving platform, it is necessary to define the forward kinematics of the parallel manipulator based on the length of the branches [13]. It should be noted that forward kinematics of parallel manipulators is more complicated than that of serial ones, and vice versa [14–16].

Analysis of kinematics can be divided into two approaches: analytical methods and numerical methods [17, 18]. In numerical methods, the forward kinematic solution is found by solving a nonlinear global optimization problem to find the numerical solution [19]. Many numerical calculation methods for solving nonlinear equations have been utilized in kinematic problems. Newton’s iteration method, together with its improvements, is a common method that is very efficient as regards convergence speed. However, Newton’s iteration method is an arduous procedure which is sensitive to initial values and involves a large number of calculation steps [8]. Compared to PSO, the differential evolution (DE) algorithm has received much more attention due to its capability of global optimization [20, 21].

The nonlinearity and multiple solution properties of the forward kinematics of a parallel manipulator make the analytical method more difficult and sophisticated than the numerical one. The numerical solution of the forward kinematics can be obtained using optimization methods such as DE and PSO or other hybrid optimization methods [22, 23]. In this paper, some modern intelligent optimization methods will be utilized and compared with each other. A time-saving hybrid strategy combined by differential evolution and modified particle swarm optimization is developed for the numerical solution of the forward kinematics of a 3-DOF parallel manipulator.

#### 2. A Hybrid Strategy of Differential Evolution and Particle Swarm Optimization

The differential evolution (DE) optimization algorithm is a simulation of the biological evolution process. It is capable of handling nondifferentiable, nonlinear, and multimodal objective functions. To start an optimization process, an initial population must be randomly generated within a predefined bound, and then a new population in the next generation is generated through mutation, crossover, and selection operations.

Particle swarm optimization (PSO) is a computational algorithm that optimizes a problem by iteratively improving a candidate solution with regard to a given measure requirement. The movements of the particles mimic the movement of organisms in a bird flock or fish school and are guided by their own best known position in the search space as well as the entire swarm’s best known position.

Modified particle swarm optimization (MPSO) has the user-defined constant parameters , , and which have a great impact on the optimization performance. The parameter is utilized to adjust the velocity, the parameter is utilized to adjust , which is the best position achieved so far by every individual, and the parameter is utilized to adjust , which is the best value obtained so far by any particle in the population. In order to improve the performance of PSO, time-varying acceleration coefficients and time-varying inertia weight can be utilized. The inertia weight can provide a balance between the local search and global search during the optimization process:where and are the initial and the final inertia weight, respectively, Iter is the current iteration number, and is the maximum number of iterations.

By increasing the number of iterations, the weights of and can have different convergence rates:where and are the maximum and minimum acceleration coefficient of and , respectively.

The hybrid strategy of DE and MPSO (DEMPSO) is the new strategy. A key merit of DE algorithm is the efficient global optimization. Furthermore, the diversity of the entire population can be always maintained during the whole evolution process, which can prevent the individuals from falling into a local optimum. PSO, on the other hand, has the advantage of fast convergence speed. The individual with the best history and the best individual among the entire iteration are saved to obtain the lowest fitness values.

Combining the advantages of DE and PSO, a new hybrid DEMPSO strategy is proposed which aims to achieve both fast convergence speed and efficient global optimization. Since the population of PSO easily falls into a local optimum, the proposed DEMPSO method uses the DE algorithm to reduce the search space first, and then the obtained populations are taken over by the MPSO as an initial population to get a fast convergence rate to the final global optimum. The hybrid algorithm can obtain the global minimum value based on the fitness function, which is built for the numerical solution for forward kinematics of a parallel manipulator. The procedure of the DEMPSO algorithm is illustrated as follows.

*(**1) Population Initialization*. The individual , with the population number NP, is randomly generated to form an initial population in a -dimensional space. All the individuals should be generated within the bounds of the solution space. The initial individuals are generated randomly in the range of the search space. And the associated velocities of all particles in the population are generated randomly in the -dimension space. Therefore, the initial individuals and the initial velocity can be expressed as follows:where , , is the range of the search space, and is a random number chosen between 0 and 1.

*(**2) Iteration Loop of DE*. Let individual denote the mutation operation at time . By randomly choosing three individuals from the previous population, the mutant individual can be generated as the following equation:where is a differential weight between 0 and 1. It is a zoom factor to control the amplification of the mutation operation. , , and are random integers that have been selected from 1 to NP and , , , and are not the same as each other.

The crossover operation aims to construct a new population which is chosen from the current individuals and mutant individuals in order to increase the diversity of the generated individuals:where is a random number chosen from 0 to 1, is an integer chosen from 1 to randomly, and Cr is a crossover parameter that is randomly chosen from 0 to 1.

In the selection operation, the crossover vector is compared to the target vector by evaluating the fitness function value based on a greedy criterion, and the vector with a smaller fitness value is selected as the next generation vector:

Update the global best part with the minimum fitness value and the personal-best part . Let the value be and perform a comparison with the stopping tolerance value . If is less than , the iteration of DE has finished. All the population and positions will continue to the next loop of MPSO.

*(**3) Iteration Loop of MPSO*. Set the time-varying parameters , , and as the MPSO defined. Renew the individual velocity as follows:where the constant parameters , , and are defined and is a number randomly chosen between 0 and 1.

The new individuals are generated as follows:

Let the best value of MPSO be and perform a comparison with fitness value . If is less than , the iteration of MPSO has finished.

The initial population is generated by the DE algorithm, and the stopping criterion of DE is set as the fitness value less than a user-defined stopping tolerance value which is dependent on the specific kinematics of a parallel manipulator. When the fitness value is less than , the DE population will be taken over by the MPSO algorithm. The new velocity and new location of the population are updated in every generation until the fitness value becomes less than the bound condition satisfied.

In order to depict clearly the population moving process, the position and difference vector distribution of the population for the DE-based algorithm and MPSO-based algorithm are shown in Figures 1–4.