CN109343345B - Mechanical arm polynomial interpolation track planning method based on QPSO algorithm - Google Patents
Mechanical arm polynomial interpolation track planning method based on QPSO algorithm Download PDFInfo
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Abstract
The invention provides a QPSO algorithm-based mechanical arm polynomial interpolation track planning method, which is used for carrying out time optimal track planning on a 3-5-3 segmented interpolation track of a multi-degree-of-freedom mechanical arm by utilizing the QPSO algorithm and is characterized by comprising the following steps of: step S1, constructing a target function required by the polynomial interpolation track planning of the mechanical arm; and step S2, performing iterative evolution on the particle population according to the individual optimal position and the global optimal position of the particles according to the QPSO algorithm, thereby selecting the positions of the particles with the global optimal fitness and the corresponding fitness values. The invention can ensure that the tail end track of the six-degree-of-freedom mechanical arm meets the principle that the operation process is as smooth as possible and the tail end position, the angular velocity and the angular acceleration have no sudden change; the precision of the simulation result is superior to that of a double-layer PSO optimization algorithm, the convergence rate is higher than that of the traditional PSO optimization algorithm, and the time consumed in the calculation process is less than that of the double-layer PSO algorithm and the PSO algorithm.
Description
Technical Field
The invention relates to the technical field of robot control, in particular to a mechanical arm polynomial interpolation track planning method based on a QPSO algorithm.
Background
A certain configuration requires a set of joint angles of the robot manipulator and all possible sets of joint angles to be referred to as a configuration space, the constraints encompassing the physical constraints of the robot.
The common sectional interpolation methods have 3-3-3-3-3 polynomial interpolation methods and 4-3-4 polynomial interpolation methods mentioned in Wanyan and the like, and for the former, the three-polynomial interpolation method is used, and the track pulsation is easy to be discontinuous due to the low order; compared with the two methods, the 3-5-3 polynomial interpolation method proposed by the xueyinget al has the characteristics of high order, no position, no speed and no acceleration mutation of the track, and the front and rear two sections are three-term interpolation, aiming at the complexity of a mechanical arm model, the formula conversion calculation amount can be greatly reduced, however, the track planning based on the polynomial difference has the characteristics of high order, no convex hull and the like, the traditional method is difficult to optimize, the particle swarm algorithm can just solve the problem, Lixiaet al proposes a PSO time optimal track planning under the speed constraint aiming at a six-degree-of-freedom mechanical arm, but the target value convergence time and the calculation time of the PSO (particle swarm algorithm) are both longer, in order to improve the convergence speed of the particle swarm optimization, a self-adaptive double-layer particle swarm optimization algorithm is provided by Zhongshiwary and the like, but when the self-adaptive double-layer particle swarm optimization algorithm is applied to time optimal trajectory planning of a six-degree-of-freedom mechanical arm, the precision is poor, and the self-adaptive double-layer particle swarm optimization algorithm is easy to precocious and cannot reach an optimal value.
Compared with the general PSO, QPSO (quantum particle swarm optimization) is more suitable for time-optimal trajectory planning of industrial mechanical arms.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a mechanical arm polynomial interpolation track planning method based on a QPSO algorithm, which solves the problems that the track planning of a polynomial difference has the characteristics of higher order, no convex hull and the like and is difficult to optimize by using a traditional method; the tail end track of the six-degree-of-freedom mechanical arm can be ensured to meet the principle that the operation process is as smooth as possible, and the tail end position, the angular velocity and the angular acceleration do not have sudden change; the precision of the simulation result is superior to that of a double-layer PSO optimization algorithm, the convergence rate is higher than that of the traditional PSO optimization algorithm, and the time consumed in the calculation process is less than that of the double-layer PSO algorithm and the PSO algorithm. The technical scheme adopted by the invention is as follows:
a QPSO algorithm-based mechanical arm polynomial interpolation track planning method is characterized by comprising the following steps of:
step S1, constructing a target function required by the polynomial interpolation track planning of the mechanical arm;
and step S2, performing iterative evolution on the particle population according to the individual optimal position and the global optimal position of the particles according to the QPSO algorithm, thereby selecting the positions of the particles with the global optimal fitness and the corresponding fitness values.
Step S1 specifically includes:
s1.1, constructing a polynomial interpolation function; the general interpolation formula of the 3-5-3 polynomial of the multi-degree-of-freedom mechanical arm joint i is as follows:
wherein S isi1(t)、Si2(t)、Si3(t) joints i of the multi-degree-of-freedom mechanical arm are at ti1、ti2、ti3Difference of these three segmentsAngle change locus corresponding to time, aij、bij、cijJ is the jth coefficient of the three sections of polynomial interpolation functions, and j is 0,1, 2, 3, 4 and 5;
step S1.2, conversion of a known condition and an objective function;
the angle change track of the joint i is divided into three sections by a 3-5-3 polynomial, in order to ensure that the joint of the three sections has no sudden change of position, angular velocity and angular acceleration, the joint angle, the angular velocity and the angular acceleration of each section of the end point are equal to those of the starting point of the next section, and the angular velocity and the angular acceleration of the starting point of the first section and the end point of the third section are both 0, so that the following formulas (2), (3) and (4) exist;
wherein theta isi0、θi1、θi2And thetai3Known as ti1、ti2、ti3Respectively representing the time from the starting point to the end point of each three-section angle track;
firstly, two sections of trinomial conversion are carried out, and then, five-item conversion is carried out; firstly, substituting formula (2) into formula (1), and obtaining S through conversion and reductioni1(t) about ai3Expression of (1), ti1In respect of ai3The expression of (1); similarly, formula (4) is substituted into formula (1) to respectively obtain Si3(t) about ci3Expression of (1), ti3In respect of ci3The expression of (1); then, according to the conditions obtained at present, the formula (3) is substituted into the formula (1), and S is obtained through conversion and reductioni2(t) with respect to ti2、ai3、ci3And bi5The expression of (1); finally, as requiredIf the optimization target is time optimal trajectory planning, the fitness value is the sum of three sections of trajectory time, and the objective function is obtained as follows:
fitness＝f(ai3,bi5,ci3)＝ti1+ti2+ti3 (5)
s1.3, setting constraint conditions;
setting the maximum limits of angular velocity and angular acceleration of each joint, and ai3,bi5,ci3The value range of (a).
Step S2 specifically includes:
step S2.1, the basic idea of the QPSO algorithm includes:
in a target search space of D dimension, a certain range (low)d,highd) D1, 2,.. times, D, let k 1,2,.. times, M, randomly taking M particles to form a particle group X (h) X ═ X1(h),X2(h),...,XM(h) H iteration, where the particle k at any position is denoted Xk(h)＝{Xk,1(h),Xk,2(h),...Xk,d(h),...Xk,D(h) In the current iteration, the individual optimal position of each particle in the population of particles is denoted as Pk(h)＝{Pk,1(h),Pk,2(h),...Pk,d(h),...Pk,D(h) }; the globally optimal particle position for all particles in this population is denoted Gk(h)＝{Gk,1(h),Gk,2(h),...Gk,d(h),...Gk,D(h) And g (h) ═ Pg(h) Wherein g is a particle subscript at a global optimal position, and g belongs to {1, 2.. multidot.M };
at time t, the individual optimal positions of particle k are:
at time t, the global optimal position of particle k is:
gdparticle index, G, for global optimum position in d-dimensiond(h) The h-th iteration global optimal particle position of the d-th dimension is obtained;
comparing the objective function value corresponding to the current optimal position of the particle with the objective function value corresponding to the global optimal position each time the particle is updated, if the former is more optimal, the former replaces the latter to be the global optimal position, and the updating formula of the particle position is as follows:
wherein alpha is a contraction-expansion coefficient;
u (0,1) is a random variable;
s2.2, realizing time optimal trajectory planning based on a QPSO algorithm;
realizing a QPSO algorithm-based time optimal trajectory planning for obtaining the target function (5); equation (5) is considered to be a three-dimensional fitness function, i.e., D is 3, and let Xk(h)＝{ai3,k,bi5,k,ci3,k}；
Taking M particles from each dimension for initialization, and setting the optimal position of each particle as the current particle position;
substituting each particle into the objective function, calculating the fitness value of each particle, and taking the particle position corresponding to the optimal fitness value as the initial global optimal position;
the iteration times are accumulated to 1, the positions of the particles are updated by using a formula (11), and the fitness value corresponding to each particle is solved;
respectively updating the individual optimal position and the global optimal position of the particle according to the particle fitness value of the current layer by using the formulas (9) and (10);
when the global optimal fitness value is smaller than a preset threshold value or the iteration times reach the maximum preset times, jumping out of the loop; obtaining the optimal position of the particles as G, namely the global optimal position, and the global optimal fitness value is f (G);
finally, repeating the whole process for a plurality of times, and taking respective average values of the global optimal position and the global optimal fitness value to obtain
And
the invention has the advantages that: compared with the PSO and the double-layer particle swarm algorithm, the QPSO optimization algorithm of the invention compares the effect of solving the three-section track time optimal value of each joint of the six-degree-of-freedom mechanical arm, and the simulation effect shows that the convergence speed of the QPSO is better than that of the common PSO, the precision of the QPSO is better than that of the double-layer PSO and the common PSO, and the calculation time consumption is the least, so the comprehensive effect of the QPSO is better than that of the common PSO when the complex optimization problem is solved; and the precision and the time consumption of the industrial mechanical arm play a crucial role in the operation efficiency of the industrial mechanical arm, and compared with the general PSO, the QPSO is more suitable for time optimal trajectory planning of the industrial mechanical arm.
Drawings
FIG. 1 is a flow chart of the algorithm of the present invention.
FIG. 2 is a schematic diagram of a simulated joint angle change trajectory according to the present invention.
Fig. 3 is a schematic diagram of the simulated joint angular velocity change trajectory.
FIG. 4 is a schematic diagram of a simulated joint angular acceleration change trajectory according to the present invention.
FIG. 5 is a comparison graph of the iterative evolutionary effects of the three algorithms of the present invention.
Detailed Description
The invention is further illustrated by the following specific figures and examples.
The invention provides a QPSO algorithm-based mechanical arm polynomial interpolation track planning method, which is used for carrying out time optimal track planning on a 3-5-3 segmented interpolation track of a six-degree-of-freedom mechanical arm by utilizing the QPSO algorithm and solves the problems that the track planning of polynomial difference values has the characteristics of higher order, no convex hull and the like and is difficult to optimize by utilizing a traditional method.
Step S1, constructing an objective function;
the QPSO algorithm needs to determine the fitness of each particle according to an objective function, and then can select the particle position with the optimal fitness according to the fitness of each particle in each iteration;
s1.1, constructing a polynomial interpolation function;
the trajectory planning of the six-degree-of-freedom mechanical arm can plan the angular trajectories of 6 joints into 1 joint, namely, the joint i is considered, i is 1,2, n, n is 6, and the spatial coordinates of two intermediate points are introduced under the condition that the spatial coordinates of an initial point and an end point are determined according to a 3-5-3 polynomial interpolation method; based on the mechanical arm model, the joint angle S corresponding to the initial point, the intermediate point 1, the intermediate point 2 and the terminal point of the joint i can be obtained by utilizing inverse kinematics solutioni1(ti0)、Si1(ti1)、Si2(ti2) And Si3(ti3) (ii) a The 3-5-3 polynomial interpolation formula of the joint i is as follows:
wherein S isi1(t)、Si2(t)、Si3(t) joints i of the six-DOF robot arm at ti1、ti2、ti3The angle change trace corresponding to the three differential time segments, aij、bij、cijJ is the jth coefficient of the three sections of polynomial interpolation functions, and j is 0,1, 2, 3, 4 and 5;
step S1.2, conversion of a known condition and an objective function;
the angle change track of the joint i is divided into three sections by a 3-5-3 polynomial, because the space coordinates of an initial point, a middle point 1, a middle point 2 and an end point are known, the angles of the starting point and the end point of the three sections of tracks are also known, and in order to ensure that the joint position, the angular velocity and the angular acceleration of the joint at the joint of the three sections of tracks have no sudden change of the position, the angular velocity and the angular acceleration, the joint angle and the angular velocity of the end point of each section and the starting point of the next section are differentSince the angular velocities and angular accelerations at the start point of the first stage and the end point of the third stage are both 0, the following expressions (2), (3), and (4) exist; in the formula Si1(t)、Si2(t)、Si3(t) one point above represents derivation (i.e. angular velocity), and two points represent derivation after derivation (i.e. angular acceleration);
wherein theta isi0、θi1、θi2And thetai3Known as ti1、ti2、ti3Respectively representing the time from the starting point to the end point of each three-section angle track;
in order to simplify the calculation process, two sections of trinomial conversion are firstly carried out, and then quinomial conversion is carried out; firstly, substituting formula (2) into formula (1), and obtaining S through conversion and reductioni1(t) about ai3Expression of (1), ti1In respect of ai3The expression of (1); similarly, formula (4) is substituted into formula (1) to respectively obtain Si3(t) about ci3Expression of (1), ti3In respect of ci3The expression of (1); then, according to the conditions obtained at present, the formula (3) is substituted into the formula (1), and S is obtained through conversion and reductioni2(t) with respect to ti2、ai3、ci3And bi5The expression of (1); and finally, if the optimization target is required to be achieved as the time optimal trajectory planning, the fitness value is the sum of three sections of trajectory time, and the objective function is obtained as follows:
fitness＝f(ai3,bi5,ci3)＝ti1+ti2+ti3 (5)
s1.3, setting constraint conditions;
this step sets the maximum limits of angular velocity and angular acceleration for each joint, and ai3,bi5,ci3The value range of (a);
considering kinematic constraint, the angular velocity and the angular acceleration of each joint of the six-freedom-degree mechanical arm are subjected to the maximum velocity vmaxWith maximum acceleration amaxThe limitation of (2) should make each joint angular velocity and angular acceleration all be less than the maximum value for guaranteeing the normal operating of arm operation, so, have:
according to equation (7) in consideration of t1、t2、t3Are all greater than 0, can respectively obtain ai3,bi5,ci3Within a desirable range of
Step S2, path planning of QPSO algorithm;
performing iterative evolution on the particle population according to the individual optimal position and the global optimal position of the particles by the QPSO algorithm, thereby selecting the positions of the particles with global optimal fitness and the corresponding fitness values; the QPSO algorithm has strong global search capability at the beginning and relatively weak local search capability, and the local search capability is enhanced as the global search capability is weakened along with iteration;
step S2.1, the basic idea of the QPSO algorithm includes:
in a target search space of D dimension, a certain range (low)d,highd) D1, 2,.. times, D, let k 1,2,.. times, M, randomly taking M particles to form a particle group X (h) X ═ X1(h),X2(h),...,XM(h) H iteration, where the particle k at any position is denoted Xk(h)＝{Xk,1(h),Xk,2(h),...Xk,d(h),...Xk,D(h)}In the current iteration, the individual optimal position of each particle in the population of particles is denoted as Pk(h)＝{Pk,1(h),Pk,2(h),...Pk,d(h),...Pk,D(h) }; the globally optimal particle position for all particles in this population is denoted Gk(h)＝{Gk,1(h),Gk,2(h),...Gk,d(h),...Gk,D(h) And g (h) ═ Pg(h) Wherein g is a subscript of the particle at the global optimal position, g belongs to {1, 2.. multidot.m }, and if the particle is optimal, an objective function of fitness ═ f (X) needs to be substituted into the subscriptk,1,Xk,2,...,XK,D) The smaller the value, the better;
at time t, the individual optimal positions of particle k are:
at time t, the global optimal position of particle k is:
gdparticle index, G, for global optimum position in d-dimensiond(h) The h-th iteration global optimal particle position of the d-th dimension is obtained;
comparing the objective function value corresponding to the current optimal position of the particle with the objective function value corresponding to the global optimal position each time the particle is updated, if the former is more optimal, the former replaces the latter to be the global optimal position, and the updating formula of the particle position is as follows:
wherein alpha is a contraction-expansion coefficient;
u (0,1) is a random variable;
s2.2, realizing time optimal trajectory planning based on a QPSO algorithm;
in order to obtain the QPSO algorithm-based time-optimal trajectory planning implementation of the objective function (5), taking the joint i in the six-degree-of-freedom manipulator as an example, equation (5) can be regarded as a three-dimensional fitness function, i.e., D is 3, and let X bek(h)＝{ai3,k,bi5,k,ci3,kV is set according to the actual condition of the mechanical armmax＝2πrad/s,amax＝4πrad/s2And set ai3,bi5,ci3The value range of (a); as shown in fig. 1, h in fig. 1 is the iteration number, and L is the repetition number;
taking M particles from each dimension for initialization, and setting the optimal position of each particle as the current particle position; m ═ 20;
substituting each particle into the objective function, calculating the fitness value of each particle, and taking the particle position corresponding to the optimal fitness value as the initial global optimal position;
the iteration times are accumulated to 1, the positions of the particles are updated by using a formula (11), and the fitness value corresponding to each particle is solved;
respectively updating the individual optimal position and the global optimal position of the particle according to the particle fitness value of the current layer by using the formulas (9) and (10);
when the global optimal fitness value is smaller than a preset threshold value or the iteration times reach the maximum preset times, jumping out of the loop; obtaining the optimal position of the particles as G, namely the global optimal position, and the global optimal fitness value is f (G);
and finally, repeating the whole process for 10 times, namely L is 10, and taking the average value of the global optimal position and the global optimal fitness value to obtain the global optimal position and the global optimal fitness value
And
modeling and simulating a six-degree-of-freedom mechanical arm;
the selected object is a typical industrial mechanical arm PUMA560 manipulator; kinematic modeling of PUMA560 using a D-H coordinate system; establishing a mechanical arm model by using MATLAB Toolbox, and carrying out related simulation in MATLAB;
the angle change tracks of six joints of the industrial mechanical arm PUMA560 are shown in FIG. 2, and the angle tracks are smooth in the running process of the six joints;
angular velocity change trajectories and angular acceleration change trajectories of six joints can be obtained by performing derivation and secondary derivation on the formula (1), as shown in fig. 3 and 4, at each interpolation point, the angular velocity and the angular acceleration change continuously and have certain smoothness, so that the time optimal trajectory planning based on the QPSO algorithm meets the principle that the tail end position, the angular velocity and the angular acceleration of the mechanical arm do not have sudden change.
In the embodiment, the optimal value solution is performed on the sum of three sections of track time of each joint of the six-degree-of-freedom mechanical arm by using a QPSO optimization algorithm, namely, the formula (5), and simultaneously the same time optimal track planning is performed on the same object by using a double-layer PSO algorithm and a PSO algorithm respectively; performing 100 iterations on 20 particles, circulating the whole process for 10 times in order to observe the accuracy of each algorithm result, and finally taking the average value of 10 results; the three algorithms compare the iterative evolution effects of 6 joints at the optimal time, and the comparison result of the joint 1 is shown in FIG. 5;
according to the invention, the QPSO algorithm is utilized to carry out time optimal trajectory planning on the 3-5-3 segmented interpolation trajectory of the six-degree-of-freedom mechanical arm, the tail end running process is smooth, the tail end position, the angular velocity and the angular acceleration have no mutation, and the trajectory effect is good in the optimal time; in the process of solving the three-section track time optimal value of each joint of the six-degree-of-freedom mechanical arm by utilizing the QPSO optimization algorithm, the optimal value solving effect is compared with the double-layer PSO algorithm and the PSO algorithm, and the simulation effect shows that when the time optimal track planning is carried out on the 3-5-3 segmented interpolation track of the six-degree-of-freedom mechanical arm, the convergence speed of the QPSO is superior to that of the common PSO, the precision of the QPSO is better than that of the double-layer PSO and the common PSO, and the calculation time consumption is the least; therefore, the QPSO has better comprehensive effect than the general PSO when solving the complex optimization problem; the QPSO algorithm can better converge on the global optimal point, is not easy to sink into the local optimal point, has stronger global searching capability and local searching capability, has strong global searching capability at the beginning, has weaker local searching capability, has weakened global searching capability along with iteration, has strengthened local searching capability, plays a vital role in the operation efficiency of the industrial mechanical arm due to the precision and the time consumption of the industrial mechanical arm, and is more suitable for time optimal trajectory planning of the industrial mechanical arm than the general PSO.
Finally, it should be noted that the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting, and although the present invention has been described in detail with reference to examples, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, which should be covered by the claims of the present invention.
Claims (1)
1. A QPSO algorithm-based mechanical arm polynomial interpolation track planning method is characterized by comprising the following steps of:
step S1, constructing a target function required by the polynomial interpolation track planning of the mechanical arm;
step S2, according to the QPSO algorithm, carrying out iterative evolution on the particle population according to the individual optimal position and the global optimal position of the particles, thereby selecting the positions of the particles with global optimal fitness and the corresponding fitness values;
step S1 specifically includes:
s1.1, constructing a polynomial interpolation function; the general interpolation formula of the 3-5-3 polynomial of the multi-degree-of-freedom mechanical arm joint i is as follows:
wherein S isi1(t)、Si2(t)、Si3(t) joints i of the multi-degree-of-freedom mechanical arm are at ti1、ti2、ti3The angle change trace corresponding to the three differential time segments, aij、bij、cijJ is the jth coefficient of the three sections of polynomial interpolation functions, and j is 0,1, 2, 3, 4 and 5;
step S1.2, conversion of a known condition and an objective function;
the angle change track of the joint i is divided into three sections by a 3-5-3 polynomial, in order to ensure that the joint of the three sections has no sudden change of position, angular velocity and angular acceleration, the joint angle, the angular velocity and the angular acceleration of each section of the end point are equal to those of the starting point of the next section, and the angular velocity and the angular acceleration of the starting point of the first section and the end point of the third section are both 0, so that the following formulas (2), (3) and (4) exist;
wherein theta isi0、θi1、θi2And thetai3Known as ti1、ti2、ti3Respectively representing the time from the starting point to the end point of each three-section angle track;
firstly, two sections of trinomial conversion are carried out, and then, five-item conversion is carried out; firstly, substituting formula (2) into formula (1), and obtaining S through conversion and reductioni1(t) about ai3Expression of (1), ti1In respect of ai3The expression of (1); similarly, formula (4) is substituted into formula (1) to respectively obtain Si3(t) about ci3Expression of (1), ti3In respect of ci3The expression of (1); then, according to the conditions obtained at present, the formula (3) is substituted into the formula (1), and S is obtained through conversion and reductioni2(t) with respect to ti2、ai3、ci3And bi5The expression of (1); and finally, if the optimization target is required to be achieved as the time optimal trajectory planning, the fitness value is the sum of three sections of trajectory time, and the objective function is obtained as follows:
fitness＝f(ai3,bi5,ci3)＝ti1+ti2+ti3 (5)
s1.3, setting constraint conditions;
setting the maximum limits of angular velocity and angular acceleration of each joint, and ai3,bi5,ci3The value range of (a);
step S2 specifically includes:
step S2.1, the basic idea of the QPSO algorithm includes:
in a target search space of D dimension, a certain range (low)d,highd) D1, 2,.. times, D, let k 1,2,.. times, M, randomly taking M particles to form a particle group X (h) X ═ X1(h),X2(h),...,XM(h) H iteration, where the particle k at any position is denoted Xk(h)＝{Xk,1(h),Xk,2(h),...Xk,d(h),...Xk,D(h) In the current iteration, the individual optimal position of each particle in the population of particles is denoted as Pk(h)＝{Pk,1(h),Pk,2(h),...Pk,d(h),...Pk,D(h) }; the globally optimal particle position for all particles in this population is denoted Gk(h)＝{Gk,1(h),Gk,2(h),...Gk,d(h),...Gk,D(h) And g (h) ═ Pg(h) Wherein g is a particle subscript at a global optimal position, and g belongs to {1, 2.. multidot.M };
at time t, the individual optimal positions of particle k are:
at time t, the global optimal position of particle k is:
gdparticle index, G, for global optimum position in d-dimensiond(h) The h-th iteration global optimal particle position of the d-th dimension is obtained;
comparing the objective function value corresponding to the current optimal position of the particle with the objective function value corresponding to the global optimal position each time the particle is updated, if the former is more optimal, the former replaces the latter to be the global optimal position, and the updating formula of the particle position is as follows:
wherein alpha is a contraction-expansion coefficient;
u (0,1) is a random variable;
s2.2, realizing time optimal trajectory planning based on a QPSO algorithm;
realizing a QPSO algorithm-based time optimal trajectory planning for obtaining the target function (5); equation (5) is considered to be a three-dimensional fitness function, i.e., D is 3, and let Xk(h)＝{ai3,k,bi5,k,ci3,k}；
Taking M particles from each dimension for initialization, and setting the optimal position of each particle as the current particle position;
substituting each particle into the objective function, calculating the fitness value of each particle, and taking the particle position corresponding to the optimal fitness value as the initial global optimal position;
the iteration times are accumulated to 1, the positions of the particles are updated by using a formula (11), and the fitness value corresponding to each particle is solved;
respectively updating the individual optimal position and the global optimal position of the particle according to the particle fitness value of the current layer by using the formulas (9) and (10);
when the global optimal fitness value is smaller than a preset threshold value or the iteration times reach the maximum preset times, jumping out of the loop; obtaining the optimal position of the particles as G, namely the global optimal position, and the global optimal fitness value is f (G);
finally, repeating the whole process for a plurality of times, and taking respective average values of the global optimal position and the global optimal fitness value to obtain
And
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