A mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization

Abstract

The application discloses a mechanical arm fuzzy PID control method based on quantum pigeon group optimization, first motion trajectory planning is carried out, and a mechanical arm is controlled to move according to the planned motion trajectory through a fuzzy PID controller; a QAPIO algorithm is used to optimize fuzzy PID control parameters, and each pigeon can be used as a PID control parameter of each joint of the mechanical arm. The QAPIO algorithm of the application modifies a compass operator update equation, a probability density function in quantum representation of real number coding and a quantum rotation gate in the original algorithm, so that the algorithm has a continuous updating capability, can continuously update an optimal value, improves global optimization capability, avoids premature local optimization, and solves the problem that the original algorithm is prone to falling into a [0, 0,..., 0, 0]T point, causing the optimization algorithm to fail. The application effectively realizes optimization of fuzzy PID controller parameters of the mechanical arm, and has good practicability.

Description

Technical Field

[0001] This invention belongs to the technical field of robotic arm control, specifically relating to a fuzzy PID control method for robotic arms based on quantum pigeon swarm optimization. Background Technology

[0002] With the development of new-generation information technology, fuzzy PID controllers have gained significant importance in the field of engineering control due to their simplicity and convenience. They are widely used in electronics, automation control, aerospace, and other fields, especially in three-axis robotic arm control systems. The purpose of a three-axis robotic arm PID controller is to find a set of Δ... kp Δ who Δ kd This allows the system's performance to reach its optimal level. However, the most critical aspect of PID control is the tuning of the PID control parameters. Conventional parameter optimization methods are complex and difficult to implement, often resulting in oscillations.

[0003] During the movement of the robotic arm, the system stiffness changes constantly, necessitating simultaneous parameter optimization for all controllers to ensure control accuracy and efficiency. A single fuzzy PID controller requires optimization of 7 parameters, while the three-axis robotic arm fuzzy PID controller requires optimization of 21 parameters, significantly more than the 3 parameters required for a typical single PID controller. Because the Quantum Pigeon Swarm Optimization (QPIO) algorithm expresses the most population characteristics with the fewest individuals, it can solve large-scale optimization problems with a small population size.

[0004] However, existing quantum pigeon swarm optimization (QPIO) algorithms have the following problems: In the correction of real-number encoded quantum representations, when solving for the probability density function of the original algorithm, the probability density function value approaches 0 infinitely. Using this probability density function value to solve for the new optimal solution observation will cause the coordinate to approach zero infinitely. In quantum rotation gates, during the actual operation of the algorithm, the value of a certain dimension is always updated in each iteration, which will cause... α i The value decreases from the beginning of the algorithm to the end, causing the algorithm to prematurely fall into a local optimum. In the basic pigeon flocking algorithm, only the flight vector direction is affected by the original flight direction and the optimal pigeon direction when the compass operator is updated.

[0005] Therefore, this invention is based on QPIO and improves upon it to obtain the Quantum Modified Pigeon Swarm Optimization Algorithm (QAPIO), which is used for parameter self-tuning of fuzzy PID. It has continuous update capability, can continuously update the optimal value, improve the global optimization capability, avoid prematurely obtaining local optima, and solve the problem that the original algorithm is prone to getting stuck at the [0,0,...,0,0]T point too early, causing the optimization algorithm to fail. Summary of the Invention

[0006] The purpose of this invention is to provide a fuzzy PID control method for robotic arms based on quantum pigeon flock optimization, which aims to solve the problem of large-scale optimization of fuzzy PID control parameters for robotic arms.

[0007] This invention is mainly achieved through the following technical solutions:

[0008] A fuzzy PID control method for a robotic arm based on quantum pigeon swarm optimization is proposed. First, motion trajectory planning is performed, and a fuzzy PID controller is used to control the robotic arm to move according to the planned trajectory. Then, the quantum pigeon swarm algorithm is employed to optimize the fuzzy PID control parameters. Each pigeon can serve as a PID control parameter for each joint of the robotic arm. Particles in the quantum pigeon swarm algorithm are selected... x Update the PID control parameters for each joint. x This includes the following steps:

[0009] Step S100: Initialize parameters t , N p0 , α 0 Δ i , R With location information,

[0010] in, t This represents the current iteration number. N p0 The total number of pigeons α 0 Let Δ be the initial pseudostate probability. i For the quantum rotating door to change angle, R For compass counting;

[0011] Step S200: Set random speed and position information for each pigeon, compare the fitness of each pigeon, and find the current optimal solution;

[0012] Step S300: Solve for the coordinates of the central landmark and operate the quantum rotating gate to update the pseudo-state probability;

[0013] Step S400: Observe the optimal solution Solve the problem and obtain the optimal solution observations. Update;

[0014] Step S500: Operate the compass operator to update the pigeon's position and speed information, then compare the fitness of all pigeons to find a new optimal solution;

[0015] Step S600: If the number of iterations reaches the upper limit of the compass operator, stop the current iteration and switch to operating the landmark operator, proceeding to step S700; otherwise, jump to step S300.

[0016] Step S700: Sort the pigeons according to their fitness values, operate the landmark operator, and store the best position and the optimal function value;

[0017] Step S800: Determine whether the number of iterations exceeds the upper limit of the landmark operator. If it does, output the result; otherwise, jump to step S700.

[0018] To better implement this invention, further, step S300, which involves solving for the coordinates of the central landmark and operating the quantum rotation gate to update the pseudo-state probability, includes:

[0019] Calculate the distance between the central landmark and the pigeon with the best fitness, and solve for the L2 norm of the coordinate difference:

[0020] (16)

[0021] in, distance The distance between the central landmark and the pigeon with the best adaptability;

[0022] x c ( t +1) is the first t +1 generation central landmark coordinates;

[0023] x gb The location coordinates of the pigeon with the best fitness, i.e., the optimal solution;

[0024] The position of the central landmark changes with the positions of all pigeons. When the distance between the optimal pigeon and the central landmark decreases, the probability amplitude will decrease, and it will be processed according to formulas (12) and (13). Conversely, the probability amplitude will be reset.

[0025] (12)

[0026] (13)

[0027] in, x c Coordinates of the central landmark;

[0028] α i ( t ) is the first i Only one pigeon in the 1st t The probability of pseudostates in the generation;

[0029] α i ( t +1) is the first i+1 pigeon in the 1st t +1 generation pseudo-state probability;

[0030] Δ i For the quantum rotating door to change angle;

[0031] ε is an infinitesimal value.

[0032] To better implement the present invention, further, in step S400, the observation value of the optimal solution is... Update:

[0033] The probability density function of the quantum representation of real-number encoding is as follows:

[0034] (14)

[0035] As can be seen from formula (14), the wave function will only be affected by the standard deviation, and the wave function value will be limited to 4. s Within the range; Equation for generating the optimal solution observations:

[0036] (15)

[0037] As can be seen from formula (15), the optimal solution observations are generated around the central landmark according to the 4σ principle, and the generation range gradually shrinks as the number of iterations increases;

[0038] in, The observations are the optimal solution.

[0039] x c Coordinates of the central landmark;

[0040] t This represents the current iteration number;

[0041] T This represents the maximum number of iterations.

[0042] s i Standard deviation,

[0043] It is the probability density function;

[0044] This is a normalization factor used to ensure that the integral of the probability density function is 1;

[0045] rand It is a random number;

[0046] i Number the current pigeon;

[0047] n Number the largest pigeon.

[0048] To better implement the present invention, further, in step S500, the position information and speed information of the pigeon are updated:

[0049] No. i Only one pigeon in the 1st t The location information update strategy is as follows:

[0050] (2)

[0051] No. i Only one pigeon in the 1st t The speed information update strategy is as follows:

[0052] (17)

[0053] in, x i ( t ) is the first i Only one pigeon in the 1st t Location information of the generation;

[0054] x i ( t +1) is the first i Only one pigeon in the 1st t +1 generation location information;

[0055] v i ( t ) is the first i Only one pigeon in the 1st t Speed ​​information of the generation;

[0056] v i ( t +1) is the first i Only one pigeon in the 1st t Speed ​​information for +1 generation;

[0057] e -Rt It is an exponential decay factor;

[0058] rand It is a random number;

[0059] x gb The location coordinates of the pigeon with the best fitness, i.e., the optimal solution;

[0060] These are the observations for the optimal solution.

[0061] The compass operator vector is influenced by the original flight direction, the direction of the optimal solution observation, and the direction of the optimal pigeon.

[0062] To better realize the present invention, further, in step S700, the landmark operator is operated according to formulas (3), (4), and (5).

[0063] (3)

[0064] (4)

[0065] (5)

[0066] in, N p For the first t Number of individuals in the population in the next iteration

[0067] fitness ( x The fitness function () is used to evaluate the optimization effect. The smaller the fitness function value, the better the optimization effect achieved by adjusting the parameters.

[0068] x c It serves as the central landmark.

[0069] To better realize the present invention, further, the fitness function fitness ( x )as follows:

[0070]

[0071] in, t This represents the current iteration number.

[0072] T The maximum number of iterations,

[0073] e ( t This represents the joint motion error of the robotic arm.

[0074] oh For weights.

[0075] To better realize the present invention, further, since the maximum absolute deviation is much smaller than the integral of its product with time over time, it is set to... oh =5000.

[0076] To better realize the present invention, the PID control parameters of each joint further include the PID parameters of each joint. k p0 , k i0 ,k d0 and the rate of change of angular displacement error of each joint ec and the PID adjustment parameters Δ for each joint k p Δ k i Δ k d scaling factor k ec , α , β , c The three-axis robotic arm has 21 control parameters. The 21 parameters of the fuzzy PID controller are optimized to find the optimal value, and this optimal value is then used for the motion control of the robotic arm. The process includes the following steps:

[0077] Step A1: Initialize the algorithm;

[0078] Set the total number of pigeons N p0 Maximum number of iterations for the compass operator T m Maximum number of iterations for the landmark operator T n Number of iterations t Initial pseudo-state probability α 0 The quantum rotating gate changes angle Δ i Compass Calculator R And set optimization dimensions dim ;

[0079] Randomly generate pigeon spawn coordinates and initial speed;

[0080] Step A2: Input the corresponding parameters of the pigeon into the fuzzy PID controller built in Simulink, and perform motion simulation of the robotic arm in conjunction with the Simulink model to obtain the motion response curves of each joint of the robotic arm;

[0081] The fitness of the motion response curves of each joint of the robotic arm is calculated and compared with the current fitness of the pigeon. The solution with the minimum fitness is taken to update the current pigeon solution.

[0082] The fitness is compared with the fitness of the optimal solution, and the optimal parameters are updated accordingly.

[0083] Step A3: Solve for the coordinates of the central landmark and operate the quantum rotating gate to update the pseudo-state probability α;

[0084] Step A4: Observations of the optimal solution Solve the problem and obtain the optimal solution observations. Update;

[0085] Step A5: Use the compass operator to update the pigeon's position, updating both the pigeon's position and speed information;

[0086] Solve for fitness and find the optimal pigeon;

[0087] Step A6: Determine if the maximum number of iterations for the compass operator has been reached. T m If not, proceed to step A3; otherwise, proceed to step A7.

[0088] Step A7: Sort the pigeons according to their fitness, pause the update of the coordinates of the second half of the pigeons, and use the landmark operator to update the positions of the pigeons; solve for the fitness and find the optimal pigeon;

[0089] Step A8: Determine whether the maximum number of iterations of the landmark operator has been reached or the minimum limit has been met. If not, proceed to step A7; otherwise, output the result.

[0090] To better realize the present invention, further, in step A1, the total number of pigeons is set. N p0 The maximum number of iterations for the compass operator is 100. T m The maximum number of iterations for the landmark operator is 80. T n The initial number of iterations is 20. t= 0, initial pseudo-state probability α 0 = The quantum rotating gate changes angle Δ i =14°, compass operator R =0.2, set optimization dimension dim =21.

[0091] To better implement the present invention, further, in step A1, parameters are set. k p0 , k i0 , k d0 The solution is randomly generated within the range [0, 30] according to a uniform distribution, and the parameters are set. k ec , α , β , c The solutions are randomly generated in the range [0,3] according to the average distribution, and the fitness of all solutions is initialized to infinite.

[0092] Quantum pigeon swarm optimization algorithm:

[0093] The pigeon flocking algorithm consists of two parts: the compass operator and the landmark operator.

[0094] 1. Compass Operator

[0095] The map and compass analogue mimics the role of the sun and Earth's magnetic field as navigational tools for pigeons. Pigeons sense magnetic fields through magnetoception, thus creating a mental map and using the sun as a compass to adjust their direction. As the flock approaches its destination, it gradually reduces its reliance on the sun and magnetic particles.

[0096] At this stage, the pigeon swarm algorithm is somewhat similar to the particle swarm algorithm (PSO), where each pigeon is represented by its position and velocity information. The i-th pigeon is in the... t The update strategy for speed and location information is as follows:

[0097] (1)

[0098] (2)

[0099] in R This is the compass operator, with a value of 0.2.

[0100] 2. Landmark Operator

[0101] This phase primarily focused on improving and optimizing the algorithm's exploration capabilities. The landmark operator mimics the influence of navigational landmarks on pigeons. As the flock approaches its destination, pigeons rely on nearby landmarks for navigation. If a pigeon is familiar with a landmark, it will fly directly to the destination; conversely, if it is unfamiliar with a landmark and far from its destination, it will follow other pigeons familiar with the landmark to reach its destination.

[0102] First, the pigeons are sorted by their fitness values. Then, in each iteration, the population size is halved. It is assumed that the central pigeon is familiar with the terrain and can fly directly to the destination, while the other pigeons follow the central pigeon towards the destination. The position update strategy is as follows:

[0103] (3)

[0104] (4)

[0105] (5)

[0106] in, Np ( t ) is the first t The number of individuals in the population in the next iteration;

[0107] Np ( t -1) is the firstt-1 The number of individuals in the population in the next iteration;

[0108] fitness (·) represents the fitness function, which is used to evaluate the optimization effect. The smaller the fitness function value, the better the optimization effect achieved by the set of parameters.

[0109] x c Coordinates of the central landmark;

[0110] x i ( t ) is the first i Only one pigeon in the 1st t Location information of the generation;

[0111] x i ( t +1) is the first i Only one pigeon in the 1st t +1 generation location information.

[0112] Basic pigeon flocking algorithm process (simplified):

[0113] Start—

[0114] Step 1: Initialize parameters and location information;

[0115] Step 2: Set random speed and position information for each pigeon, compare the fitness of each pigeon, and find the current optimal solution;

[0116] Step 3: Operate the compass operator. Update the pigeon's position and speed information according to formulas (1) and (2), then compare the fitness of all pigeons to find the new optimal solution;

[0117] Step 4: If the number of iterations reaches the iteration limit of the map and compass operators, stop the current iteration and switch to operating the landmark operator; otherwise, go to Step 3.

[0118] Step 5: Sort the pigeons according to their fitness values, operate the landmark operator according to formulas (3)(4)(5), and store the best position and the optimal function value;

[0119] Step 6: Determine if the number of iterations exceeds the iteration limit. If it does, output the result; otherwise, jump to Step 5.

[0120] Output: Optimal solution;

[0121] —end.

[0122] Real-number encoded quantum representation (RCQ):

[0123] Quantum quanta are represented by "0" and "1" states, namely normal and pseudo-states. The state of a qubit is represented by the following equation:

[0124] |ψ>=α|0>+β|1>

[0125] in α and β Let each represent the probability of one of the two states, satisfying the equation

[0126]

[0127] Here, the optimal solution is considered as a linear superposition of the probabilities of two states, namely the "0" state and the "1" state. The quantum representation of the optimal solution can be updated as follows:

[0128]

[0129] The convergence direction of each pigeon can then be redefined as:

[0130] (6)

[0131] in, For observations with candidate solutions, a function is introduced. oh ( x ), will | oh ( x )| 2 The observations are used as the probability density function to calculate the optimal candidate solution.

[0132] (7)

[0133] in, m i and s i Let represent the expected value and standard deviation, respectively. Using the current optimal solution as the expected value, the standard deviation can be expressed by the following formula:

[0134] (8)

[0135] (9)

[0136] in, r u A random number between 0 and 1;

[0137] The final observed value of the current optimal solution is:

[0138] (10)

[0139] No. jOnly one pigeon in the 1st t The velocity update formula for the next iteration is changed to:

[0140] (11).

[0141] Quantum Rotating Gate (QRG):

[0142] In quantum genetic algorithms, because chromosomes under quantum encoding are no longer in a single state, traditional selection, crossover, and mutation operations cannot be used. Instead, quantum rotation gates are used to act on the ground states of quantum chromosomes, causing them to interfere with each other and undergo phase changes, thereby altering the genetic structure. α i The distribution range.

[0143] Here, QRG is also used to update the probability magnitude of the optimal solution. By increasing the rotation angle, the probability is improved. α i The probability magnitude is increased, thereby improving the convergence speed of individuals towards the global optimum. At the start of the algorithm, α i and β i The corresponding probability magnitudes are all [values ​​to be filled in]. If the global optimal solution changes after the iteration ends, it is changed through a quantum rotation gate. α i Otherwise, reset all probability magnitudes to their initial values ​​to prevent the algorithm from getting trapped in local optima. The specific update strategy for QRG is as follows:

[0144] (12)

[0145] (13)

[0146] in, x c Coordinates of the central landmark;

[0147] α i ( t ) is the first i Only one pigeon in the 1st t The probability of pseudostates in the generation;

[0148] α i ( t +1) is the first i +1 pigeon in the 1st t +1 generation pseudo-state probability;

[0149] Δ i For the quantum rotating door to change angle;

[0150] ε is an infinitesimal value;

[0151] t This represents the current iteration number.

[0152] The beneficial effects of this invention are as follows:

[0153] The QAPIO algorithm of this invention is an improved version of QPIO. The QAPIO algorithm of this invention modifies the compass operator update equation, the probability density function in the real-number encoded quantum representation, and the quantum rotation gate in the original algorithm. This gives the algorithm continuous update capability, allowing it to continuously update the optimal value, improving global optimization ability, avoiding premature attainment of local optima, and solving the problem that the original algorithm easily gets trapped in the [0,0,...,0,0]T point too early, causing optimization algorithm failure. This invention effectively optimizes the parameters of the fuzzy PID controller for robotic arms and has good practicality. Attached Figure Description

[0154] Figure 1 This is a schematic diagram of a three-axis robotic arm.

[0155] Figure 2 The schematic diagram for optimizing the parameters of a fuzzy PID controller.

[0156] Figure 3 The flowchart shows the quantum-corrected pigeon flock optimization algorithm.

[0157] Figure 4 This is a flowchart of Example 4. Detailed Implementation

[0158] Example 1:

[0159] A fuzzy PID control method for a robotic arm based on quantum pigeon flock optimization, such as... Figure 3 As shown, motion trajectory planning is first performed, and the robotic arm is controlled to move according to the planned trajectory using a fuzzy PID controller. The quantum pigeon swarm algorithm is then used to optimize the fuzzy PID control parameters. Each pigeon can serve as a PID control parameter for each joint of the robotic arm. Particles in the quantum pigeon swarm algorithm are selected... x Update the PID control parameters for each joint. x This includes the following steps:

[0160] Step S100: Initialize parameters t , N p0 , α 0 Δ i , R With location information,

[0161] in, t This represents the current iteration number.N p0 The total number of pigeons α 0 Let Δ be the initial pseudostate probability. i For the quantum rotating door to change angle, R For compass counting;

[0162] Step S200: Set random speed and position information for each pigeon, compare the fitness of each pigeon, and find the current optimal solution;

[0163] Step S300: Solve for the coordinates of the central landmark and operate the quantum rotating gate to update the pseudo-state probability;

[0164] Step S400: Observe the optimal solution Solve the problem and obtain the optimal solution observations. Update;

[0165] Step S500: Operate the compass operator to update the pigeon's position and speed information, then compare the fitness of all pigeons to find a new optimal solution;

[0166] Step S600: If the number of iterations reaches the upper limit of the compass operator, stop the current iteration and switch to operating the landmark operator, proceeding to step S700; otherwise, jump to step S300.

[0167] Step S700: Sort the pigeons according to their fitness values, operate the landmark operator, and store the best position and the optimal function value;

[0168] Step S800: Determine whether the number of iterations exceeds the upper limit of the landmark operator. If it does, output the result; otherwise, jump to step S700.

[0169] Preferably, step S300, which involves solving for the coordinates of the central landmark and operating the quantum rotating gate to update the pseudo-state probability, includes:

[0170] Calculate the distance between the central landmark and the pigeon with the best fitness, and solve for the L2 norm of the coordinate difference:

[0171] (16)

[0172] in, distance The distance between the central landmark and the pigeon with the best adaptability;

[0173] x c ( t +1) is the first t +1 generation central landmark coordinates;

[0174] x gb The location coordinates of the pigeon with the best fitness, i.e., the optimal solution;

[0175] The position of the central landmark changes with the positions of all pigeons. When the distance between the optimal pigeon and the central landmark decreases, the probability amplitude will decrease, and it will be processed according to formulas (12) and (13). Conversely, the probability amplitude will be reset.

[0176] (12)

[0177] (13)

[0178] in, x c Coordinates of the central landmark;

[0179] α i ( t ) is the first i Only one pigeon in the 1st t The probability of pseudostates in the generation;

[0180] α i ( t +1) is the first i +1 pigeon in the 1st t +1 generation pseudo-state probability;

[0181] Δ i For the quantum rotating door to change angle;

[0182] ε is an infinitesimal value, which can be less than or equal to 10. -7 Replace the value with .

[0183] Preferably, in step S400, the observation values ​​of the optimal solution are... Update:

[0184] The probability density function of the quantum representation of real-number encoding is as follows:

[0185] (14)

[0186] As can be seen from formula (14), the wave function will only be affected by the standard deviation, and the wave function value will be limited to 4. s Within the range; Equation for generating the optimal solution observations:

[0187] (15)

[0188] As can be seen from formula (15), the optimal solution observations are generated around the central landmark according to the 4σ principle, and the generation range gradually shrinks as the number of iterations increases;

[0189] in, The observations are the optimal solution.

[0190] x c Coordinates of the central landmark;

[0191] t This represents the current iteration number;

[0192] T This represents the maximum number of iterations.

[0193] s i Standard deviation,

[0194] It is the probability density function;

[0195] This is a normalization factor used to ensure that the integral of the probability density function is 1;

[0196] rand It is a random number;

[0197] i Number the current pigeon;

[0198] n Number the largest pigeon.

[0199] Preferably, in step S500, the pigeon's position and speed information are updated:

[0200] No. i Only one pigeon in the 1st t The location information update strategy is as follows:

[0201] (2)

[0202] No. i Only one pigeon in the 1st t The speed information update strategy is as follows:

[0203] (17)

[0204] in, x i ( t ) is the first i Only one pigeon in the 1st t Location information of the generation;

[0205] x i ( t +1) is the first i Only one pigeon in the 1stt +1 generation location information;

[0206] v i ( t ) is the first i Only one pigeon in the 1st t Speed ​​information of the generation;

[0207] v i ( t +1) is the first i Only one pigeon in the 1st t Speed ​​information for +1 generation;

[0208] e -Rt It is an exponential decay factor;

[0209] rand It is a random number;

[0210] x gb The location coordinates of the pigeon with the best fitness, i.e., the optimal solution;

[0211] These are the observations for the optimal solution.

[0212] Preferably, in step S700, the landmark operator is operated according to formulas (3), (4), and (5).

[0213] (3)

[0214] (4)

[0215] (5)

[0216] in, N p For the first t Number of individuals in the population in the next iteration

[0217] fitness ( x The fitness function () is used to evaluate the optimization effect. The smaller the fitness function value, the better the optimization effect achieved by adjusting the parameters.

[0218] x c The coordinates of the central landmark.

[0219] Preferably, the fitness function fitness ( x )as follows:

[0220]

[0221] in, t This represents the current iteration number.

[0222] T The maximum number of iterations,

[0223] e ( t This represents the joint motion error of the robotic arm.

[0224] oh For weights.

[0225] Preferably, since the maximum absolute deviation is much smaller than the integral of its product with time over time, it is set to... oh =5000.

[0226] Example 2:

[0227] A fuzzy PID control method for a robotic arm based on quantum pigeon flock optimization, such as... Figure 2 As shown, the motion control of a three-axis robotic arm first requires motion trajectory planning, and the robotic arm moves according to the above trajectory. A fuzzy PID controller is used as the motion controller for the robotic arm, and the control parameters of the fuzzy PID controller are optimized using the quantum modified pigeon flocking optimization algorithm (QAPIO).

[0228] like Figure 1 As shown, B1 is the base, B2 is the boom, and B3 is the end effector. J 1 represents the moment of inertia of the base. J 2 represents the moment of inertia of the boom. J 3 represents the moment of inertia of the end stage. mr 1 represents the mass of the base motor shaft. mr 2 represents the mass of the boom motor shaft. mr 3 represents the mass of the end-stage worktable motor shaft. my 1 represents the mass of the base. my 2 represents the weight of the boom. my 3 represents the mass of the end worktable. I 1 represents the base moment of inertia. I 2 represents the moment of inertia of the boom. I 3 represents the moment of inertia of the end stage. i 1 represents the base rotation angle. i 2 represents the boom rotation angle. i 2 represents the rotation angle of the end worktable. l 1 represents the length of member B1. l 2 represents the length of member B2. d 1. d 2 represents the distance between the centroids.

[0229] like Figure 2 As shown, Input=[q 1, q 2, q 3] represents the input angular displacement, Output=[ q 1', q 2', q 3'] represents the output angular displacement. e =[ e 1, e 2, e [3] represents the angular displacement error of each component. ec =[ ec 1, ec 2, ec [3] represents the rate of change of angular displacement error for each joint, Δ kp =[Δ kp 1, Δ kp 2, Δ kp 3]、Δ who =[Δ who 1, Δ who 2, Δ who 3]、Δ kd =[Δ kd 1, Δ kd 2, Δ kd [3] These are the PID adjustment parameters for each joint output by the fuzzy controller, k ec =[k ec 1,k ec 2,k ec 3]、 α =[ α 1, α 2, α 3]、 β =[ β 1, β 2, β 3]、 c =[ c 1, c 2, c 3]. These are respectively ec Δ kp Δ who Δ kd The scaling factor. kp 0=[ kp 01, kp 02, kp 03]、 who 0=[ who 01, who 02, who 03]、 kd 0=[ kd 01, kd 02, kd 03] represents the three parameters of PID.U =[ U 1, U 2, U 3] The controller controls the output torque of the three motors.

[0230] The fuzzy controller will input and output parameters e , ec Δ Kp Δ Ki Δ Kd Divided into 7 fuzzy partitions: Negative Large (NB), Negative Medium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), and Positive Large (PB). Input / Output Parameters e , ec Δ Kp Δ Ki Δ Kd The fuzzy subset is {NB, NM, NS, Z, PS, PM, PB}. A triangular function is chosen as the membership function, which is derived from... a , b , c Three parameters determine the shape, trimf( x ,[ a b c The equation is expressed as follows:

[0231]

[0232] The stiffness of the robotic arm system changes constantly during its movement, necessitating simultaneous parameter optimization for all controllers to ensure control accuracy and efficiency. A single fuzzy PID controller requires optimization of 7 parameters, while the 3-axis robotic arm fuzzy PID controller in this paper requires optimization of 21 parameters, significantly more than the 3 parameters required for a typical single PID controller. Since the Quantum Pigeon Swarm Optimization (QPIO) algorithm expresses the most population characteristics with the fewest individuals, it can solve large-scale optimization problems with a small population size. This invention selects QPIO and improves it to obtain the Quantum Modified Pigeon Swarm Optimization (QAPIO) algorithm for parameter self-tuning of fuzzy PID controllers. QAPIO is an improved algorithm based on QPIO. QAPIO modifies the compass operator update equation, the probability density function in the real-number encoded quantum representation, and the quantum rotation gate in the original algorithm, enabling continuous updating of the optimal value, improving global optimization capabilities, avoiding premature attainment of local optima, and solving the problem of the original algorithm easily getting trapped in [0,0,...,0,0] too early. T This issue causes the optimization algorithm to fail.

[0233] The quantum pigeon swarm algorithm process is as follows:

[0234] Start —

[0235] Step 1: Initialize parameters t , Np 0、 α 0, Δ i , R With location information;

[0236] Step 2: Set random speed and position information for each pigeon, compare the fitness of each pigeon, and find the current optimal solution;

[0237] Step 3: Operate the quantum rotating gate; update the pseudo-state probability α according to formulas (12) and (13) in sequence;

[0238] Step 4: Solve for the optimal solution observations; solve for the optimal solution observations sequentially according to formulas (9)(8)(7)(10)(6). Update;

[0239] Step 5: Operate the compass operator; update the pigeon's position and speed information according to formulas (11) and (2), then compare the fitness of all pigeons to find a new optimal solution;

[0240] Step 6: If the number of iterations reaches the upper limit of the compass operator, stop the current iteration and switch to operating the landmark operator; otherwise, go to Step 3.

[0241] Step 7: Sort the pigeons according to their fitness values, operate the landmark operator according to formulas (3)(4)(5), and store the best position and the optimal function value;

[0242] Step 8: Determine if the number of iterations exceeds the iteration limit. If it does, output the result; otherwise, jump to Step 7.

[0243] Output: Optimal solution;

[0244] — End.

[0245] Preferably, the quantum representation of the real number encoding is modified:

[0246] When solving the probability density function of the original algorithm, when the... i When a pigeon is far from the current optimal candidate solution, the formula (7) contains ( x i - u i ) 2The term will be very large. Since formula (7) belongs to the normal distribution function, this will cause the probability density function value to be infinitely close to 0. When using the probability density function value to solve for the new optimal solution observation value using formula (10), it will cause the coordinate to be infinitely close to zero. Therefore, assuming that the optimal solution observation value is infinitely close to zero, and combining formulas (6), (11), and (2), we will get the following equation:

[0247]

[0248] in, e -Rt The term will approach 0 infinitely with the number of iterations, so the probability density function used in the original algorithm is very likely to trap the algorithm at a zero point.

[0249] The probability density function of the quantum representation encoded by real numbers is modified to obtain the following formula:

[0250] (14)

[0251] in, It is the probability density function;

[0252] This is a normalization factor used to ensure that the integral of the probability density function is 1;

[0253] rand It is a random number;

[0254] i Number the current pigeon;

[0255] n Number the largest pigeon;

[0256] As can be seen from this formula, the corrected wavefunction will only be affected by the standard deviation, and the wavefunction value will be limited to 4. s Within the range.

[0257] Modify the equation for generating the optimal solution observations:

[0258] (15)

[0259] in, The observations are the optimal solution.

[0260] x c Coordinates of the central landmark;

[0261] t This represents the current iteration number;

[0262] T This represents the maximum number of iterations.

[0263] s i Standard deviation,

[0264] It is the probability density function;

[0265] As can be seen from formula (15), the optimal solution observations are generated around the central landmark according to the 4σ principle, and the generation range gradually shrinks as the number of iterations increases.

[0266] Preferably, improvements are made to the quantum rotating gate:

[0267] The original algorithm for the quantum rotating gate will change after the global optimal solution is altered. α i Otherwise, reset all probability magnitudes to their initial values ​​to prevent the algorithm from getting trapped in local optima. (Δ in the quantum rotating gate) i The value is generally set at around 10° to allow α i The value follows gradient descent, indirectly making the wavefunction value smaller and smaller. However, in the actual operation of the algorithm, the value of a certain dimension will always be updated in each iteration, which will cause... α i The value decreases from the beginning to the end of the algorithm, causing it to prematurely fall into a local optimum. An alternative method for resetting the probability magnitude is provided here. α i The method.

[0268] The algorithm adds the distance calculation between the central landmark and the pigeon with the best fitness, i.e., solving for the L2 norm of the coordinate difference:

[0269] (16)

[0270] in, distance The distance between the central landmark and the pigeon with the best adaptability;

[0271] x c ( t +1) is the first t +1 generation central landmark coordinates;

[0272] x gb The location coordinates of the pigeon with the best fitness, i.e., the optimal solution;

[0273] The position of the central landmark changes with the position of all pigeons. When the distance between the optimal pigeon and the central landmark decreases, the probability amplitude will decrease, i.e., it will be processed according to formulas (12) and (13). Otherwise, the probability amplitude will be reset.

[0274] Preferably, the update equation for the compass operator is improved as follows:

[0275] The basic pigeon flocking algorithm compass operator update formula only has and The flight vector direction is influenced by both the original flight direction and the optimal pigeon direction, and the quantum pigeon flocking algorithm update formula has... and The flight vector direction is affected by both the original flight direction and the direction of the optimal solution observation.

[0276] QAPIO combines the above update formulas to obtain the following formula:

[0277] (17)

[0278] in, x i ( t ) is the first i Only one pigeon in the 1st t Location information of the generation;

[0279] x i ( t +1) is the first i Only one pigeon in the 1st t +1 generation location information;

[0280] v i ( t ) is the first i Only one pigeon in the 1st t Speed ​​information of the generation;

[0281] v i ( t +1) is the first i Only one pigeon in the 1st t Speed ​​information for +1 generation;

[0282] e -Rt It is an exponential decay factor;

[0283] rand It is a random number;

[0284] x gb The location coordinates of the pigeon with the best fitness, i.e., the optimal solution;

[0285] The observations are the optimal solution.

[0286] As shown in the formula above, the compass operator vector is influenced by the original flight direction, the direction of the optimal solution observation, and the optimal pigeon direction. This will greatly enhance the ability of pigeon flocks to navigate using geomagnetism.

[0287] Example 3:

[0288] A fuzzy PID control method for a robotic arm based on quantum pigeon flocking optimization is proposed. First, motion trajectory planning is performed, and the robotic arm moves according to this trajectory. A fuzzy PID controller is used as the motion controller for the robotic arm, and the control parameters of the fuzzy PID controller are optimized using the quantum modified pigeon flocking optimization algorithm (QAPIO). Figure 3 As shown, it includes the following steps:

[0289] Step 1: Initialize parameters t , N p0 , α 0 Δ i , R With location information;

[0290] Step 2: Set random speed and position information for each pigeon, calculate and compare the fitness of each pigeon, and find the current optimal solution;

[0291] Step 3: Solve for the coordinates of the central landmark according to formula (4), and operate the quantum rotating door to update the pseudo-state probability α according to formulas (16)(12)(13) in sequence;

[0292] Step 4: Solve for the optimal solution observations; solve for the optimal solution observations in sequence according to formulas (9), (8), (14), and (15). Update;

[0293] Step 5: Operate the compass operator; update the pigeon's position and speed information according to formulas (17) and (2), then compare the fitness of all pigeons to find a new optimal solution;

[0294] Step 6: If the number of iterations reaches the upper limit of the compass operator, stop the current iteration and switch to operating the landmark operator; otherwise, go to Step 3.

[0295] Step 7: Sort the pigeons according to their fitness values, operate the landmark operator according to formulas (3), (4), and (5), and store the best position and the optimal function value;

[0296] Step 8: Determine if the number of iterations exceeds the iteration limit. If it does, output the result; otherwise, jump to Step 7.

[0297] Step 9: Output: Optimal solution.

[0298] Preferably, in Step 3, the coordinates of the central landmark are calculated:

[0299] The location update strategy is as follows:

[0300] (4)

[0301] in, x c Coordinates of the central landmark;

[0302] N p ( t ) is the first t Number of individuals in the population in the next iteration;

[0303] x i ( t ) is the first i Only one pigeon in the 1st t Location information of the generation;

[0304] fitness ( x The fitness function is used to evaluate the optimization effect. The smaller the fitness function value, the better the optimization effect achieved by changing the parameters.

[0305] Calculate the distance between the central landmark and the pigeon with the best fitness, i.e., solve for the L2 norm of the coordinate difference:

[0306] (16)

[0307] The position of the central landmark changes with the positions of all pigeons. When the distance between the optimal pigeon and the central landmark decreases, the probability amplitude will decrease, i.e., it will be processed according to formulas (12) and (13). Conversely, the probability amplitude will be adjusted accordingly. α Reset:

[0308] (12)

[0309] (13)

[0310] in, x c Coordinates of the central landmark;

[0311] α i ( t ) is the first i Only one pigeon in the 1st t The probability of pseudostates in the generation;

[0312] α i ( t +1) is the first i +1 pigeon in the 1st t+1 generation pseudo-state probability;

[0313] Δ i For the quantum rotating door to change angle;

[0314] ε is an infinitesimal value.

[0315] Preferably, in Step 4, the observation value of the optimal solution is... Update:

[0316] in, m i and s i Let represent the expected value and standard deviation, respectively. Using the current optimal solution as the expected value, the standard deviation can be expressed by the following formula:

[0317] (8)

[0318] (9)

[0319] in, r u A random number between 0 and 1;

[0320] No. i Only one pigeon in the 1st t The location information update strategy is as follows:

[0321] (2)

[0322] The convergence direction of each pigeon can be redefined as:

[0323] (6)

[0324] No. j Only one pigeon in the 1st t The velocity update formula for the next iteration is changed to:

[0325] (11)

[0326] Assuming that the observed value of the optimal solution approaches zero infinitely, combining equations (6), (11), and (2) will yield the following equation:

[0327]

[0328] Among them, the term will approach 0 infinitely as the number of iterations increases, so the probability density function used in the original algorithm is very likely to trap the algorithm at the zero point.

[0329] The probability density function of the quantum representation encoded by real numbers is modified to obtain the following formula:

[0330] (14)

[0331] As can be seen from this formula, the corrected wave function will only be affected by the standard deviation, and the wave function value will be limited to the range of 4σ.

[0332] Modify the equation for generating the optimal solution observations:

[0333] (15)

[0334] From the above formula, we know that the optimal solution observations revolve around the central landmark according to 4 s The principle is generated, and the range of generation gradually shrinks as the number of iterations increases.

[0335] Example 4:

[0336] A fuzzy PID control method for robotic arms based on quantum pigeon flock optimization is proposed. The method optimizes 21 parameters of the fuzzy PID controller and finds the optimal value, and then uses the obtained optimal value for the motion control of the robotic arm.

[0337] like Figure 4 As shown, the algorithm optimizes the fuzzy PID controller based on the following principles:

[0338] 1) Initialize and run the QAPIO optimization algorithm;

[0339] 2) Settings: Total number of pigeons N p0 The number of iterations for the compass operator is 100. T m The landmark operator iteration count is 80. T n Set to 20; Initialization: number of iterations t= 0, initial pseudo-state probability α 0 = Δ i =14°, R =0.2 Set optimization dimension dim =21;

[0340] 3) Randomly generate pigeon spawn coordinates and initial speed, where kp 0、 who 0、 kd The solution for 0 is randomly generated in the range [0, 30] according to a uniform distribution. kec , α , β , c The solutions are randomly generated in the range [0,3] according to the average distribution, and the fitness of all solutions is initialized to infinite;

[0341] 4) Input the corresponding parameters of the pigeon into the fuzzy PID controller built in Simulink, and perform motion simulation of the robotic arm in conjunction with the Simulink model to obtain the motion response curves of each joint of the robotic arm.

[0342] 5) Calculate the fitness of the obtained robotic arm motion response curve and compare it with the current fitness of the pigeon. Take the solution with the minimum fitness to update the current pigeon solution. This value can be used to comprehensively evaluate the dynamic performance of the system.

[0343] 6) Compare the fitness with the fitness of the optimal solution and update the optimal parameters accordingly;

[0344] 7) Solve for the coordinates of the central landmark according to formula (4), and operate the quantum rotating gate to update the pseudo-state probability α according to formulas (16)(12)(13) in sequence;

[0345] 8) Solve for the optimal solution observations. Update the optimal solution observations sequentially according to formulas (9), (8), (14), and (15);

[0346] 9) Operate the compass operator. Update the pigeon's position and speed information according to formulas (17) and (2), and return to step 4;

[0347] 10) Determine if the termination condition is met, i.e., the number of iterations has reached the maximum number of iterations for the compass operator. T m If the termination condition is met, proceed to step 11; otherwise, return to step 4.

[0348] 11) Input the corresponding parameters of the pigeon into the fuzzy PID controller built in Simulink, and perform motion simulation of the robotic arm in conjunction with the Simulink model to obtain the motion response curves of each joint of the robotic arm.

[0349] 12) Calculate the fitness of the obtained robotic arm motion response curve and compare it with the current fitness of the pigeon. Update the current pigeon solution with the solution corresponding to the minimum fitness. This value can be used to comprehensively evaluate the dynamic performance of the system.

[0350] 13) Compare the fitness with the fitness of the optimal solution and update the optimal parameters accordingly;

[0351] 14) Sort the pigeons according to their fitness and pause the update of the coordinates of the second half of the pigeons. Update the pigeons' positions by operating the landmark operator according to formulas (3)(4)(5).

[0352] 15) Determine whether the termination condition is met, i.e., the number of iterations reaches the maximum number of iterations of the landmark operator or the fitness threshold. If the termination condition is met, stop the iteration and output the optimal solution; otherwise, return to step 10.

[0353] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any simple modifications or equivalent changes made to the above embodiments based on the technical essence of the present invention shall fall within the protection scope of the present invention.

Claims

1. A mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization, characterized in that, Firstly, the motion trajectory is planned, and the mechanical arm is controlled to move according to the planned motion trajectory through a fuzzy PID controller; the quantum pigeon swarm algorithm is used to optimize the fuzzy PID control parameters, each pigeon can be used as the PID control parameter of each joint of the mechanical arm, the particle x for joint PID control parameters in the quantum pigeon swarm algorithm is selected, and the set x is updated, including the following steps: Step S100: initialize parameters t , N p0 , α 0 , Δ θ , R with location information, wherein, t is the current iteration number, N p0 is the total number of pigeons, α 0 is the initial probability of false state, Δ θ is the quantum rotation gate change angle, R is the compass operator; Step S200: setting random speed and position information of each pigeon, comparing fitness of each pigeon, finding out current optimal solution; Step S300: solving central landmark coordinate, operating quantum rotation gate, updating pseudo-state probability; Step S400: updating the optimal solution observation value solving, updating the optimal solution observation value solving, updating the optimal solution observation value Step S500: operating compass operator, updating position information and speed information of pigeons, then comparing fitness of all pigeons, finding new optimal solution; Step S600: if iteration number reaches upper limit of compass operator, stopping current iteration, operating landmark operator, entering step S700; otherwise, jumping to step S300; Step S700: sorting pigeons according to pigeon fitness value, operating landmark operator, storing best position and optimal function value; Step S800: judging whether iteration number exceeds upper limit of landmark operator, if yes, outputting result, otherwise, jumping to step S700; In the step S300, the central landmark coordinate is solved, the quantum rotation gate is operated, and the pseudo-state probability is updated, including: calculating the distance between the central landmark and the pigeon with the best fitness, and solving the two-norm of the coordinate difference: (16) wherein dist is the distance of the central landmark from the best fitness pigeon; x c ( t +1) is the coordinate of the central landmark of the first t +1 generation; x gb The position coordinates of the best pigeon, i.e. the optimal solution; The position of the central landmark changes with the positions of all pigeons, and when the distance between the optimal pigeon and the central landmark decreases, the probability amplitude will decrease, which is processed according to formula (12) and formula (13); otherwise, the probability amplitude is reset: (12) (13) wherein x c is the central landmark coordinate; α i ( t ) for the first i generation of pigeons; and t the probability of pseudo for the first generation of pigeons; α i ( t +1) is the number of the first i +1 generation; and t +1 generation; and Δ θ for quantum rotation gate change angle; ε is an infinitesimal.

2. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 1, characterized in that, The step S400, the optimal solution observation value is updated: The probability density function of the quantum representation of real coding is as follows: (14) From equation (14), it is known that the wave function will only be affected by the standard deviation and will limit the wave function value within the range of 4 σ The optimal solution observation value generation equation: (15) According to formula (15), the optimal solution observation value is generated around the central landmark according to the 4σ principle, and the generation range gradually decreases with the increase of the iteration number; wherein is the optimal solution observation; x c is the central landmark coordinate; t n is the current iteration number; T is the maximum number of iterations; σ i for standard deviation, is the probability density function; is a normalization factor to ensure that the integral of the probability density function is 1; rand is a random number; i CurrentPigeonNumber; n MaxPigeonNumber is the maximum pigeon number.

3. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 2, characterized in that, In the step S500, the position information and speed information of the pigeons are updated: The first i Only pigeons in the first t The position information update strategy of the first generation is as follows: （2） The first i Only pigeons in the first t Speed information update strategy of the following generation: (17) wherein x i ( t ) is the first i only pigeon in the position information of the first t generation; x i ( t +1) is the position information of the first i only pigeon in the first t +1 generation; v i ( t ) for the first i generation of pigeons; and t speed information for the first v i ( t +1) is the first i Only pigeons in the first t +1 generation of speed information; e -Rt is an exponential decay factor; rand is a random number; x gb The position coordinates of the best-adapted pigeon, i.e. the optimal solution; is the optimal solution observation.

4. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 1, characterized in that, In the step S700, the landmark operator is operated according to formula (3), formula (4) and formula (5), （3） （4） （5） wherein x c is the central landmark coordinate; N p ( t ) is the first t Number of individuals in the population in the next iteration x i ( t ) for the first i Pigeon in the first t Generation of location information; fitness x ) is a fitness function for evaluating the optimization effect, and the smaller the fitness function value is, the better the optimization effect obtained by the set of parameters is.​ 5. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 4, characterized in that, Fitness function fitness ( x ) is as follows: wherein t is the current iteration number, T The maximum number of iterations, e t ) is a mechanical arm joint motion error,​ ω is the weight.

6. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 5, characterized in that, Setting ω = 5000.

7. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to any one of claims 1-6, characterized in that, The joint PID control parameters include PID parameters of each joint k p0 、 k i0 、 k d0 and the change rate of the joint angular displacement error ec and the joint PID adjustment parameters Δ k p 、Δ k i 、Δ k d scaling factor k ec 、 α 、 β 、 γ The three-axis mechanical arm has 21 control parameters, the 21 parameters of the fuzzy PID controller are optimized to find the optimal value, and the optimal value obtained is used for the motion control of the mechanical arm; including the following steps: Step A1: initializing the algorithm; Set total number of pigeons N p0 , maximum number of iterations for compass operator T m , maximum number of iterations for landmark operator T n , number of iterations t , initial pseudo state probability α 0 , quantum rotation gate change angle delta θ , compass operator R , and set optimization dimension dim ; Randomly generating pigeon generation coordinates and initial speed; Step A2: inputting the pigeon dimension parameters into the fuzzy PID controller established by simulink, jointly simulating the motion of the mechanical arm with the simulink model, and obtaining the motion response curve of each joint of the mechanical arm; Calculating the fitness of the obtained motion response curve of each joint of the mechanical arm, and comparing it with the current fitness of the pigeon, and updating the current pigeon solution corresponding to the minimum fitness; Comparing the fitness with the optimal solution fitness, and updating the optimal parameters; Step A3: solving the central landmark coordinate, and operating the quantum rotation gate to update the pseudo-state probability α; Step A4: Update the optimal solution observation Solve for the optimal solution observation Update; Step A5: operating the compass operator to update the pigeon position, and updating the position information and speed information of the pigeon; Solving the fitness and finding the optimal pigeon; Step A6: Determine if the maximum number of iterations of the compass operator has been reached T m If not, then go to Step A3, otherwise go to Step A7; Step A7: sorting the pigeons according to the pigeon fitness, pausing the update of the coordinates of the second half of the pigeons, operating the landmark operator to update the position of the pigeons; solving the fitness and finding the optimal pigeon; Step A8: judge whether the maximum iteration number of the landmark operator is reached or the minimum limit is met, if not, go to step A7, otherwise, output the result.

8. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 7, characterized in that, The total number of pigeons is set in step A1 N p0 = 100, the maximum number of iterations of the compass operator T m = 80, the maximum number of iterations of the landmark operator T n = 20, the initial number of iterations t = 0, the initial probability of pseudo state α 0 = = Δ, the angle of change of the quantum rotation gate θ = 14°, the compass operator R = 0.2, the set optimization dimension dim = 21.

9. The mechanical arm fuzzy PID control method based on quantum pigeon swarm optimization according to claim 7, characterized in that, In the step A1, the parameter is set k p0 , k i0 , k d0 The solution in the range of [0, 30] is randomly generated according to the average distribution, the parameter is set k ec , α , β , γ The solution in the range of [0, 3] is randomly generated according to the average distribution, and the fitness of all solutions is initialized to infinity.

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