CN105046004B - Based on the permanent magnet spherical motor inverse kinematics method for improving particle cluster algorithm - Google Patents

Abstract

The present invention relates to a kind of based on the permanent magnet spherical motor inverse kinematics method for improving particle cluster algorithm, carry out in accordance with the following steps：The first step：Initial position co-ordinates and the Eulerian angles tried to achieve by rotor of output shaft axle, determine its postrotational coordinate position, and rotor of output shaft axle gives the distance of coordinate position required by coordinate position and reality and is used as fitness function after being rotated using rotor；Second step：Eulerian angles corresponding to PSO Algorithm permanent magnet spherical motor inverse kinematics are improved with based on simulated annealing.The present invention can effectively jump out locally optimal solution, have higher solving precision.

Description

Inverse kinematics solution method of permanent magnet spherical motor based on improved particle swarm optimization algorithm

technical field

The invention belongs to the technical field of solving the inverse kinematics of a permanent magnet spherical motor, and relates to a method for solving the inverse kinematics of a permanent magnet spherical motor based on an improved particle swarm algorithm.

Background technique

With the development of high-precision and complex control systems such as mechanical joints, the requirements for the accuracy and stability of the driving mechanism are increasing. Traditionally, the accumulation of errors in a control system composed of multiple single-degree-of-freedom motors and complex mechanical transmission mechanisms leads to a decrease in the accuracy of the entire control system, and even affects the overall stability of the system. The multi-degree-of-freedom motor with permanent magnets can greatly increase the magnetic energy product of the motor, effectively improve the operating efficiency of the motor, reduce the volume of the motor, and improve the controllability of the motor. The emergence of the above problems has promoted the research of multi-degree-of-freedom spherical motors and development. At the same time, with the continuous deepening of control theory and motor theory research and the continuous development of computer technology and power electronics technology, the development of multi-degree-of-freedom spherical motor control technology has attracted the attention of many scholars. The inverse kinematics of permanent magnet spherical motor, as the basis of dynamic control, motion analysis, off-line programming and trajectory planning, has become an urgent problem to be solved.

Chinese patent announcement number CN101520857A, the announcement date is September 2, 2009, and the name is "a method for solving the inverse kinematics of permanent magnet spherical motor based on neural network", which discloses the forward kinematics model of permanent magnet spherical motor, and proposes A method for solving inverse kinematics based on feed-forward neural network. Its disadvantages are complex solution process, low solution accuracy and poor robustness. Particle Swarm Optimization (PSO) is an evolutionary computing method based on swarm intelligence. PSO was proposed by Dr. Kennedy and Eberhart in 1995. The PSO algorithm is a kind of evolutionary algorithm. It starts from a random solution, finds the optimal solution through iteration, evaluates the quality of the solution through fitness, and finds the global optimum through the optimal value currently searched. The advantage of PSO is that it has the characteristics of parallel processing, good robustness, easy to implement, and high computational efficiency, and has been successfully applied to various complex optimization problems. However, as a general random global search algorithm, the standard particle swarm optimization algorithm cannot take into account the convergence speed, global and local fine search capabilities, and it also has the disadvantages of premature convergence and easy to fall into local optimum.

Contents of the invention

Purpose of the invention: In view of the shortcomings of the existing inverse kinematics solution method for permanent magnet spherical motors, the present invention provides a solution method for inverse kinematics of permanent magnet spherical motors based on the improved particle swarm optimization algorithm, which can effectively jump out of the local optimal solution and has a relatively High solution accuracy.

To achieve the above object, the present invention adopts the following technical solutions:

A method for solving the inverse kinematics of a permanent magnet spherical motor based on the improved particle swarm optimization algorithm is carried out according to the following steps:

Step 1: From the initial position coordinates (x _i , y _i , _zi ) of the rotor output shaft and the obtained Euler angles, determine its rotated coordinate position (x _e , y _e , z _e ), to After the rotor rotates, the distance between the given coordinate position of the rotor output shaft and the actual coordinate position is used as the fitness function;

The second step: use the improved particle swarm algorithm based on the simulated annealing algorithm to solve the Euler angle corresponding to the inverse kinematics of the permanent magnet spherical motor, including the following steps:

1) Initialization parameters

Set particle population size N, inertia weight ω, maximum and minimum values of particle velocity V, annealing start and end temperatures T and T ₀ ;

2) Determine the search space, randomly generate a population of N particles, that is, randomly generate N initial solutions X _i (l), i=1, 2,..., N and N initial velocities V _j (l), j =1,2,...,N, l is the number of iterations, the initial number of iterations is 0, Xi (l) is the position of the i- _{th particle after the l-th iteration, V j (} l) is the l-th iteration The velocity change rate of the jth particle after that;

3) Calculate the fitness value f(X _i (l)) of each particle, find the individual extremum P _best and the global extremum P _gbest , record the individual extremum position P _cbest and the global extremum position P _cgbest ;

4) Let the current temperature t=T, when t≥T ₀ , execute the following loop operation:

a) Update the speed and position of all particles according to the following formula to get the next generation of particles:

In the formula, d=1,2,...,D, D is the optimization space dimension; is the value of the d-th dimension of the speed change rate v _i of the i-th particle after the l-th iteration; c ₁ , c ₂ are the improved learning factors, r ₁ , r ₂ are evenly distributed in the (0,1) interval the random number; Indicates the value of the d-th dimension of the best position p _i that the i-th particle has searched so far after the l-th iteration, Indicates the value of the d-th dimension of the best position p _g that all particles have searched so far after the l-th iteration; is the value of the d-th dimension of the position x _i of the i-th particle after the l-th iteration;

The improved learning factor is:

In the formula, c _1s and c _2s represent the initial values of iterations of c ₁ and c ₂ , c _1e and c _2e represent the final values of iterations of c ₁ and c ₂ , l is the current number of iterations, and l _max is the maximum number of iterations;

b) For the individual extremum, calculate the fitness value f( _xi (l+1)) of the updated particle, and calculate the increment of f( _xi (l+1)) ΔE=f(X _i (l+1) ))-f(X _i (l));

c) If ΔE≤0, accept the new point as the initial point of the next simulation, if ΔE>0, calculate the new acceptance probability: if exp(-ΔE/kt)>ε, k is the Boltzmann constant in the Metropolis criterion, ε It is a [0,1] random number, and accepts the new value, otherwise it rejects and maintains the value of the previous point;

d) Update the individual extremum and the position of the individual extremum;

e) find out and record the new global extremum and global extremum position;

f) cooling down, that is, the current temperature t=αt, where α is a constant less than 1, increase the number of iterations, and judge whether t has reached T ₀ , if yes, then terminate the algorithm, otherwise return to step (a) to continue execution;

5) Output the fitness function value and the corresponding Euler angle after the iteration is completed.

The beneficial effects of the present invention are:

1. The present invention proposes to use the improved particle swarm algorithm based on the simulated annealing algorithm to solve the inverse kinematics problem of the permanent magnet spherical motor, adopts the idea of heuristic algorithm, introduces a special solution close to the optimal solution, and guides the particles to approach the optimal solution. In the iterative cycle, any constant temperature can reach thermal equilibrium, and when it is cooled to a low temperature, it will reach the minimum state of internal energy at this low temperature, which has high solution accuracy.

2. Introduce the learning factor arccosine strategy, so that the algorithm retains a certain learning factor weight in the later stage, maintains the diversity of the population and better local search performance, and the overall performance is better.

3. The improved particle swarm algorithm based on the idea of simulated annealing can accept bad values according to the probability, so it is not easy to fall into the local optimum, speed up the global search ability, and has strong robustness and global convergence.

4. Customize the given Euler angle in the Cartesian space, use the improved particle swarm algorithm to discretize the continuous change of the Euler angle, avoid complex analytical algorithms, and ensure the continuity of the rotor motion.

Description of drawings:

Figure 1 is a comparison effect diagram of the two algorithms;

Figure 2(a) is the comparison between the given Euler angle α and the actual Euler angle in the nutating motion;

Figure 2(b) is the comparison between the given Euler angle β and the actual Euler angle in the nutating motion;

Figure 2(c) is the comparison between the given Euler angle γ and the actual Euler angle in the nutating motion;

Figure 3 is the comparison between the given trajectory of the rotor output shaft and the actual trajectory in the nutating motion;

Figure 4(a) is the comparison between the given Euler angle α and the actual Euler angle in complex trajectory motion;

Figure 4(b) is the comparison between the given Euler angle β and the actual Euler angle in complex trajectory motion;

Figure 4(c) is the comparison between the given Euler angle γ and the actual Euler angle in complex trajectory motion;

Figure 5 is a comparison between the given trajectory of the rotor output shaft and the actual trajectory in the complex trajectory movement;

detailed description:

The Particle Swarm Hybrid Algorithm (SA-PSO) based on the idea of simulated annealing accepts bad values according to the probability, so that it is not easy to fall into the local optimum, and has high solution accuracy. The improved particle swarm algorithm is used to discretize the continuous change of Euler angle Solve and simulate to verify the effectiveness of this method.

The present invention will be further described below in combination with two embodiments and accompanying drawings.

Example 1

The permanent magnet spherical motor can greatly increase the magnetic energy product of the motor, effectively improve the operating efficiency of the motor, reduce the volume of the motor, and improve the controllability of the motor. It has a wide range of applications in fields such as robots and intelligent flexible manufacturing systems that require three-dimensional space. . The inverse kinematics of permanent magnet spherical motor, as the basis of dynamic control, motion analysis, off-line programming and trajectory planning, has become an urgent problem to be solved.

The solution method of inverse kinematics is divided into analytical method and intelligent algorithm. The former is a set of very complex nonlinear equations about generalized Euler angles, the calculation is more complicated and the solution is more difficult. In the intelligent algorithm, in view of the complex, difficult, poor robustness of the neural network and the insufficient precision of the ant colony algorithm, a particle swarm hybrid algorithm based on the idea of simulated annealing is proposed to solve the inverse kinematics problem of the permanent magnet spherical motor. Compared with other intelligent algorithms , the particle swarm optimization algorithm needs to adjust few parameters, the structure is simple, and it is easy to implement.

For permanent magnet spherical motors, the rotor position can be defined by a set of Euler angles α, β, and γ. The rotation transformation matrix A is expressed as follows:

In the formula, the angle cosine cos is abbreviated as c, and the angle sine sin is abbreviated as s. This rotation matrix satisfies the following relationship:

(x _e ,y _e ,z _e ) ^T ＝A(x _i ,y _i ,z _i ) ^T

In the formula, (x _i , y _i , _zi ) are the initial position coordinates of the rotor output shaft, and (x _e , y _e , z _e ) are the coordinates of the position point after the rotor rotates.

At a certain time t, the position vector X(t)=(x _e (t), y _e (t), z _e (t)) ^T of a certain point on the rotor and the Euler angle vector θ(t)=(α( t), β(t), γ(t)) The relationship between ^T can be expressed as follows:

X=F(θ)

The inverse kinematics problem is a solution problem of a three-dimensional function F(θ), that is, the values of the three Euler angles α, β and γ are obtained through an optimization algorithm. From the initial position coordinates (x _i , y _i , _zi ) of the rotor output shaft and the obtained Euler angles, the positive kinematics equation of the permanent magnet spherical motor is used to determine the coordinate position after rotation (x _e , y _e , z _e ). Let the objective function be

f＝((x _d -x _e ) ² +(y _d -y _e ) ² +(z _d -z _e ) ² ) ^1/2

In the formula, (x _d , y _d , z _d ) are the coordinates of the given rotor output shaft position. The smaller the value of the objective function, the closer the (x _e , y _e , z _e ) obtained by the inverse kinematics solution is to the given (x _d , y _d , z _d ), that is, the higher the solution accuracy of the solution.

The mathematical description of the standard particle swarm optimization algorithm is as follows: suppose the search space is D-dimensional, the number of particles is n, and the position of the i-th particle is represented by Xi = (x _i1 , x _i2 ,..., x _iD ); the i-th _particle The velocity change rate of the particle is represented by v _i =(v _i1 ,v _i2 ,...,v _iD ); the best position searched by the i-th particle so far is p _i =(p _i1 ,p _i2 ,.. .,p _iD ), the best position searched by all particles so far is p _g =(p _g1 ,p _g2 ,...,p _gD ), then the position of the particle at time t+1 is updated by the following formula:

v _id (t+1)＝ωv _id (t)+c ₁ rand()·[p _id (t)-x _id (t)]+c ₂ rand()·[p _gd (t)-x _id ( t)]

x _id (t+1)＝x _id (t)+v _id (t+1) 1≤i≤n，1≤d≤D

In the formula, t represents the number of iterations; ω is called the inertia factor; c ₁ and c ₂ are called the learning factors; v _id (t) is the d-th dimension of the velocity change rate v _i of the i-th particle after the t-th iteration Numerical value; rand() is a random number between [0,1]; p _id (t) is the d-th dimension value of the best position p _i searched by the i-th particle so far after the t-th iteration; x _id (t) is the value of the d-th dimension of the position X _i of the i-th particle after the t-th iteration; p _gd (t) is the d-th dimension of the best position p _g that all particles have searched so far after the t-th iteration Dimension value; the change range of the position and velocity of the d-th dimension is [-x _dmax , x _dmax ] and [-v _dmax , v _dmax ], if the x _id and v _id iterated in a certain dimension exceed the boundary value The range is valued according to the boundary value.

Particle swarm algorithm based on the idea of simulated annealing algorithm: PSO algorithm is simple and easy to implement, does not need to adjust too many parameters and does not require gradient information, and the convergence speed is fast in the early stage, but it will be affected by random oscillation in the later stage, making it at the global optimal value Nearby requires a long search time, the convergence speed is slow, and it is easy to fall into a local minimum, which reduces the accuracy and is easy to diverge. The technology of adding simulated annealing can greatly improve system performance, increase information throughput and increase computing speed. To this end, a particle swarm algorithm model based on simulated annealing particles is established, and the idea of simulated annealing is introduced into the particle swarm algorithm, so that the simulated annealing mechanism is added to the update process of the speed and position of each particle, and the fitness value of the particle swarm after evolution is based on the Metropolis criterion While accepting the optimized solution, the deteriorated solution is accepted according to the probability. The algorithm jumps out of the local extremum region and adjusts the annealing temperature adaptively. As the temperature drops, the particles gradually form a low-energy ground state and converge to the global optimal solution.

Improvement of the learning factor: In the particle swarm optimization algorithm, the learning factors c ₁ and c ₂ determine the impact of the particle’s own experience and the experience of the group on the particle’s trajectory, reflecting the information exchange between particles, setting a larger or smaller c ₁ and c ₂ are not conducive to the collection of particles. Ideally, at the beginning of the search, the particles should explore the entire space as much as possible. At the end of the search, the particles avoid falling into the local extremum. The nonlinear strategy adjusts the learning factor to make the learning factor change nonlinearly to control the local and global search of the algorithm. The basic idea is to speed up the change speed of c ₁ and c ₂ in the early stage, so that the algorithm can quickly enter the local search, and in the later stage, the larger c ₂ is used to make the algorithm pay more attention to the group information and maintain the diversity of particles. The characteristic of the arccosine strategy is that ideal values of c ₁ and c ₂ are set in the later stage of the algorithm to keep the particles at a certain search speed and avoid premature convergence. The construction method of the arccosine acceleration factor is as follows:

In the formula, c _1s and c _2s represent the initial iteration values of c ₁ and c ₂ , c _1e and c _2e represent the final iteration values of c ₁ and c ₂ , t is the current iteration number, and t _max is the maximum iteration number.

Metropolis criterion: Assuming that a new state j is generated from the current state i, if the internal energy of the new state is less than the internal energy of the state i (that is, E _j < E _i ), then accept the new state j as the new current state; otherwise, the probability Accepting states, where k is the Boltzmann constant. The fitness value after particle swarm evolution accepts the optimized solution according to the Metropolis criterion and accepts the deteriorated solution according to the probability. The algorithm jumps out of the local extremum region and adjusts the annealing temperature adaptively. As the temperature gradually decreases, the particles gradually form a low-energy state. Converge to the global optimal solution.

The following is a simulation study of the improved particle swarm optimization algorithm. Let the radius of the rotor sphere be R. In order not to lose generality, the initial coordinates of the position of the rotor output shaft are ( _xi ,y _i , _zi )=(0.64R,0.48R ,0.6R), the coordinates of this point after the rotor rotates are (x _e ,y _e ,z _e )=(0.3570R,0.7028R,0.6153R), and the Euler angle variation corresponding to the rotor in the above rotation is calculated To solve, take R=1 in the simulation process.

The detailed steps of the improved particle swarm algorithm based on the idea of simulated annealing algorithm are as follows:

1) Initialization parameters

Set the particle population size N to 50, the maximum value of the particle velocity V to be 1 and the minimum value to -1, the learning factor adopts the arc cosine strategy, the maximum value of the inertia weight is 0.9, and the minimum value is 0.4, and according to the following formula way to decrement:

Among them, t is the iteration variable, w _start is the initial value of the inertia weight, and w _end is the final value of the inertia weight; the function of the inertia weight is to improve the convergence performance of the particle swarm optimization algorithm and avoid the algorithm from falling into local optimum, so that the particle swarm optimization algorithm In the initial iteration process, it tends to global optimization search, and then gradually turns to local optimal search, so as to adjust the solution in the local area. The value of inertia weight used in this algorithm is decreasing.

The annealing start and stop temperature T is 13000 and T ₀ is 0.01. In order to speed up the convergence speed, the Boltzmann constant k in the Metropolis criterion is 2;

2) Determine the search space, randomly generate a population of N particles, that is, randomly generate N initial solutions X _i (l) (i=1,2,...,N) and N initial velocities V _j (l)( j=1,2,...,N);

3) Calculate the fitness value f(X _i (l))(i=1,2,...,N) of each particle, find the individual extremum P _best and the global extremum P _gbest , and record the position of the individual extremum P _cbest and global extremum position P _cgbest ;

4) Let the current temperature t=T, when t≥T ₀ , execute the following loop operation:

a) Update the speed and position of all particles according to the following formula to get the next generation of particles:

In the formula, d=1,2,...,D, D is the optimization space dimension; i=1,2,...,N; l is the number of iterations, c ₁ , c ₂ are the improved learning factors , r ₁ , r ₂ are random numbers uniformly distributed in the (0,1) interval; p _i represents the individual extremum, p _g represents the global extremum, x represents the position of the particle, v represents the velocity of the particle; for the velocity and position The update change of , when the update value exceeds its boundary range, take its boundary value.

b) For the individual extremum, calculate the fitness value f(x _i (l+1))(i=1,2,...,N) of the updated particle, and get ΔE=f(X _i (l+1 ))-f(X _i (l));

c) If ΔE≤0, accept the new point as the initial point of the next simulation, if ΔE>0, calculate the new acceptance probability: if exp(-ΔE/kt)>ε, ε is a random number in [0,1] , also accept the new value, otherwise reject and maintain the value of the previous point;

d) Update the individual extremum and the position of the individual extremum;

e) find out and record the new global extremum and global extremum position;

f) cooling t=αt, increasing the number of iterations, judging whether t has reached T ₀ , if yes, then terminate the algorithm, otherwise return to step (a) to continue execution;

Among them, α is a constant, and its value determines the cooling process. A small amount of attenuation may lead to an increase in the number of iterations of the algorithm process, so that the algorithm process accepts more changes, visits more fields, searches a larger range of solution space, and returns a better final solution. In the present invention, α is taken as 0.99 , combined with the annealing start and end temperatures T and T ₀ , it can be calculated that the maximum number of iterations of this algorithm can reach 1400, which improves the accuracy of the solution.

5) Output the fitness function value and the corresponding Euler angle after the iteration is completed;

Compare the change of the minimum value of the objective function with the number of iterations when using the standard particle swarm optimization algorithm and the improved particle swarm optimization algorithm. The comparison results are shown in Figure 1. The simulation results show that the optimization result of the improved particle swarm optimization algorithm is better than that of the standard particle swarm optimization algorithm, with faster convergence speed and finer local search ability, which shows that the idea of combining other algorithms to improve the particle swarm optimization algorithm is correct and has further research significance.

The situation of solving the inverse kinematics of the permanent magnet spherical motor by the improved particle swarm optimization algorithm is investigated when the initial value of the Euler angle is not zero. Nutating motion is one of the working conditions that can best examine the torque controllability of spherical motors. In the simulation, the initial coordinates of the position of the rotor output shaft are ( _xi , y _i , z _i )=(0.6,0.8,0) , the given Euler angles are defined in Cartesian space as:

α(n+1)=sin[π/2(t _n +0.02)]

β(n+1)=cos[π/2(t _n +0.02)] t ₀ =0, t ₁₅₀ =3

γ(n+1)=0.25(t _n +0.02)

A set of Euler angles θ corresponds to a set of coordinate values (x _e (t), y _e (t), z _e (t)) ^T from the positive kinematic equation of the permanent magnet spherical motor. When the angle changes continuously, the discretization solution is carried out. Comparing the given Euler angle change trajectory with the Euler angle change obtained by the improved particle swarm optimization algorithm, the comparison results are shown in Figure 2, where Figure 2 (a), (b), and (c) correspond to The comparison of three Euler angles α, β, γ.

It can be seen from Figure 2 that when the initial value of the Euler angle is not zero, the improved particle swarm optimization algorithm can perform discretization and solution according to the given Euler angle trajectory defined in the Cartesian space, and has high solution accuracy.

Using the obtained Euler angle change trajectory, combined with the positive kinematics model of the permanent magnet spherical motor, the actual motion trajectory of the rotor output shaft is obtained, and compared with the given nutating motion, the comparison result is shown in Figure 3, where The red dotted line is the motion track of the rotor output shaft, and the green dotted line is the given track of the rotor output shaft.

It can be seen from Figure 3 that the motion trajectory of the rotor output shaft obtained basically coincides with the given trajectory, which further illustrates the solution accuracy of the improved particle swarm optimization algorithm.

Example 2

In order not to lose generality, the case where the rotor output shaft moves on a complex trajectory when the initial value of the Euler angle is not zero is investigated. In the simulation, the initial coordinates of the position point of the rotor output shaft are (x _i , y _i , z _i )=( 0,0.6,0.8), the given Euler angle is defined in Cartesian space as:

α(n+1)=2*sin(1.5*(t _n +0.02))

β(n+1)=sin(pi/2*(t _n +0.02)-pi/6)+cos(1.5*pi*(t _n+ 0.02)) t ₀ =0, t ₁₅₀ =3

γ(n+1)=2*cos(2*(t _n +0.02))*sin(0.5*(t _n +0.02))

The improved particle swarm optimization algorithm is used to discretize the solution to the continuous change of Euler angles. Comparing the given Euler angle change trajectory with the Euler angle change obtained by the improved particle swarm optimization algorithm, the comparison results are shown in Figure 4, where Figure 4 (a), (b), and (c) correspond to The comparison of three Euler angles α, β, γ. Using the obtained Euler angle change trajectory, combined with the positive kinematics model of the permanent magnet spherical motor, the actual motion trajectory of the rotor output shaft is obtained, and compared with the given nutating motion, the comparison result is shown in Figure 5, where The red dotted line is the motion track of the rotor output shaft, and the green dotted line is the given track of the rotor output shaft.

The simulation results show that when the initial value of the Euler angle is not zero and the motion trajectory of the rotor output shaft is relatively complex, the particle swarm hybrid algorithm discrete solution based on the simulated annealing idea still has high precision and good robustness.

Claims (2)

1. A permanent magnet spherical motor inverse kinematics solving method based on an improved particle swarm algorithm is carried out according to the following steps:

the first step is as follows: from the initial position co-ordinate (x) of the rotor output shaft _i ,y _i ,z _i ) And the obtained Euler angle, and determining the rotated coordinate position (x) _e ,y _e ,z _e ) Taking the distance between the given coordinate position of the rotor output shaft and the actually calculated coordinate position after the rotor rotates as a fitness function;

the second step is that: the method for solving the Euler angle corresponding to the inverse kinematics of the permanent magnet spherical motor by using the improved particle swarm optimization algorithm based on the simulated annealing algorithm comprises the following steps of:

1) Initialization parameters

Setting the particle population size N, the inertia weight omega, the maximum value and the minimum value of the particle speed V, the annealing start and stop temperatures T and T ₀ ；

2) Determining a search space, randomly generating a population of N particles, i.e. randomly generating N initial solutions X _i (l) I =1, 2.., N and N initial speeds V _j (l) J =1, 2.. And N, l is an iterationThe number of times, the number of initial iterations is 0, X _i (l) Is the position of the ith particle after the l iteration, V _j (l) The velocity change rate of the jth particle after the ith iteration;

3) Calculating a fitness value f (X) for each particle _i (l) Find individual extremum P) _best And a global extremum P _gbest Recording the position P of the individual extremum _cbest And a global extremum position P _cgbest ；

4) Let the current temperature T = T, when T ≧ T ₀ Then, the following loop operation is performed:

a) And updating the speed and the position of all the particles according to the following formula to obtain the next generation of particles:

wherein D =1, 2.,. D, D is the optimization space dimension;is the rate of change v of the velocity of the ith particle after the l-th iteration _i The value of the d-th dimension of (1); c. C ₁ ，c ₂ For improved learning factor, r ₁ ，r ₂ Random numbers uniformly distributed in the interval (0, 1);represents the best position p searched for by the ith particle so far after the l-th iteration _i The value of the d-th dimension of (c),represents the best position p searched so far for all particles after the l-th iteration _g The value of the d-th dimension of (1);is the position x of the ith particle after the ith iteration _i The value of the d-th dimension of (1);

the improved learning factors are:

in the formula, c _1s ，c _2s Is shown by c ₁ ，c ₂ Initial value of iteration of c _1e ，c _2e Denotes c ₁ ，c ₂ Is the current iteration number, l _max Is the maximum number of iterations;

b) Calculating the fitness value f (x) of the updated particles for the individual extreme value _i (l + 1)), f (x) is calculated _i (l + 1)) increment Δ E = f (X) _i (l+1))-f(X _i (l))；

c) If delta E is less than or equal to 0, a new point is accepted as an initial point of the next simulation, and if delta E is greater than 0, a new acceptance probability is calculated: if exp (-. DELTA.E/kt) > epsilon, k is a Boltzmann constant in the Metropolis criterion, epsilon is a [0,1] random number, a new value is also accepted, otherwise, the value of the previous point is rejected and maintained;

d) Updating the individual extreme value and the position of the individual extreme value;

e) Finding and recording a new global extreme value and a global extreme value position;

f) Cooling, namely enabling the current temperature T = alpha T and alpha to be a constant less than 1, increasing the iteration times, and judging whether T reaches T or not ₀ If yes, terminating the algorithm, otherwise, returning to the step (a) to continue execution;

5) And outputting the fitness function value and the corresponding Euler angle after the iteration is finished.

2. The method according to claim 1, wherein in the first step the fitness function is made as:

f＝((x _d -x _e ) ² +(y _d -y _e ) ² +(z _d -z _e ) ² ) ^1/2 wherein (x) _d ,y _d ,z _d ) Is given asRotor output shaft position coordinates.