A 7-DOF Robotic Arm Analytical Solution Optimization Method Based on DMSPSO-CIL Algorithm

Abstract

The present invention provides a seven-degree-of-freedom manipulator analytical solution optimization method based on the DMSPSO-CIL algorithm, comprising the following steps: Step 1: Obtain the current manipulator joint angle information θ cur , the next moment (posture) position information P of the robot arm T ,R T ; Step 2: Input the current pose information into the inverse kinematics solver (ASIK-RMLO) to obtain the redundant circle and redundant angle #imgabs0# Step 3: Use #imgabs1# as the position of the particle in the DMSPSO-CIL algorithm and perform update iterations; Determine whether the termination condition is met. If so, jump to step 4, otherwise repeat step 3; Step 4: Give the optimal solution #imgabs2# and its corresponding robot arm joint angle value θ nest The present invention solves the inverse kinematics problem of a seven-freedom manipulator. Based on the system requirement of "minimizing the change in joint angles", a corresponding objective function is established, and the proposed DMSPSO-CIL algorithm is used to solve and optimize this practical problem.

Description

Technical Field

[0001] The present invention relates to the technical field of human-computer interaction, and in particular to a method for optimizing an analytical solution of a seven-degree-of-freedom robotic arm based on a DMSPSO-CIL algorithm. Background Art

[0002] Medical robots are robots used to perform medical procedures or assist in healthcare settings such as hospitals, clinics, and rehabilitation centers. They can assist doctors and expand their capabilities, possessing three key characteristics: medical usability, clinical adaptability, and excellent interactivity. In recent years, numerous research institutions and commercial companies have dedicated themselves to the research and implementation of medical robotic systems, conducting in-depth research on the application of teleoperated robots in the medical industry. In actual robotic operations, a 6-DOF robot can freely move or rotate around an axis in three-dimensional space, meeting the operational requirements of most scenarios. However, in complex, unstructured applications, a 7-DOF redundant robot offers an additional degree of freedom, allowing the robotic arm to avoid specific obstacles and facilitating the end effector's arrival at a desired location, offering greater flexibility. For example, in ultrasound diagnosis, the ultrasound probe requires flexible, multi-angle rotation for comprehensive lesion identification. However, this increased flexibility inevitably complicates the robotic arm structure, leading to a variety of inverse kinematic solutions. In three-dimensional space, for a given end effector position, redundant degrees of freedom yield an infinite number of solutions. The set of all infinite solutions is referred to as a closed-form solution. In the practical application of robotic arms, closed-form solutions play a key role in solving problems such as joint constraints, active obstacle avoidance, singular configurations, and robotic arm dynamics optimization.

[0003] Solving the inverse kinematics of a robotic arm is a difficult and challenging task. The most common method is to use the numerical iteration method of the Jacobian matrix to solve the inverse kinematics of the robotic arm. However, since this method requires calculating the inverse Jacobian matrix first, it is a very complicated process. Especially for redundant arms, the Jacobian matrix is ​​not a square matrix, which undoubtedly increases the difficulty of solving the problem. Based on this, researchers have also explored other numerical iteration methods that do not rely on the Jacobian matrix. However, these methods still cannot avoid the problems of slow solution speed and low convergence accuracy caused by complicated iterations. Therefore, finding another solution method (i.e., a closed-form analytical solution) has become a research hotspot for solving the inverse kinematics problem of robotic arms. Summary of the Invention

[0004] The purpose of the present invention is to solve the problem of complex inverse solution of a seven-degree-of-freedom manipulator and the problems of slow convergence speed and low search accuracy of traditional particle swarm.

[0005] A novel particle swarm optimization algorithm, DMSPSO-CIL, is proposed to address the problems of slow convergence and low search accuracy. It primarily incorporates two strategies: first, the dynamic multi-subswarm strategy (DMS). In DMS, the population size is gradually reduced to one according to established rules as the algorithm iterates, thereby balancing the global and local search capabilities of the swarm and enhancing search diversity. Second, the comprehensive interactive learning strategy (CIL). This strategy utilizes four interactive learning modes: intra-subswarm learning (INTRA-SL), inter-subswarm learning (INTER-SL), optimal learning (OL), and fast convergence (FC). These modes allow particles within each subswarm to learn from each other, from particles in other subswarms, or from the optimal subswarm within the entire subswarm. This further enhances the swarm's local and global search capabilities while accelerating convergence.

[0006] To address the complex inverse kinematics problem of a seven-degree-of-freedom (7DOF) manipulator, we employed a geometric analytical method based on ASIK-RMLO (Regression-Oriented Linear Logic) to analyze and solve the inverse kinematics problem for a Barrett WAM seven-DOF robot, unlike the traditional numerical iteration method for six-DOF manipulators. Furthermore, for redundant robots with infinite inverse solutions, we established a corresponding objective function based on the system requirement of "minimizing joint angle changes" and applied the proposed DMSPSO-CIL algorithm to solve and optimize this practical problem.

[0007] To achieve the above object, the present invention provides the following technical solution: a method for optimizing the analytical solution of a seven-degree-of-freedom manipulator based on the DMSPSO-CIL algorithm, comprising the following steps:

[0008] Step s1: Get the current robot arm joint angle information θ cur , the next moment position information P of the robot T ,R T ;

[0009] Step s2: Input the current pose information into the inverse kinematics solver to obtain the redundant circle and redundant angle

[0010] Step s3: As a novel particle swarm optimization algorithm, DMSPSO-CIL updates and iterates the positions of particles. DMSPSO-CIL includes two strategies: the dynamic multi-subswarm strategy (DMS) and the comprehensive interactive learning strategy (CIL). In DMS, the population size gradually decreases to one according to established rules as the algorithm iterates. CIL uses four interactive learning modes: intra-subswarm learning (INTRA-SL), inter-subswarm learning (INTER-SL), optimal learning (OL), and fast convergence (FC), allowing particles in each subswarm to learn from each other, from particles in other subswarms, or from the optimal subswarm within the entire subswarm.

[0011] Step s4: Give the optimal solution and its corresponding robotic arm joint angle value θ nest ;

[0012] The optimal solution is defined as the one with the smallest difference in joint angle between the solution set and the previous solution. The specific mathematical model is described as follows:

[0013]

[0014] in, is the redundant circle redundant angle parameter, θ nest ,θ cur are the joint angle values ​​of the robot arm at the current moment and the feasible solution of the joint angle at the next moment, P T and R T is the expected position of the robot arm at the next moment, That is, the formula for the closed-form solution method for the inverse kinematics of a redundant manipulator with link offset.

[0015] Further, DMSPSO-CIL algorithm:

[0016] s301: In DMSPSO-CIL, first randomly initialize the speed and position of all N particles and evaluate their corresponding fitness values. At the same time, gbest is also initialized. The N particles are divided into n subgroups, and the gbestj corresponding to each subgroup is also initialized. The maximum number of iterations is set to t max , divided into m stages, each stage includes iterations, m is defined as m=log2 n +1;

[0017] s302: Phase l, l∈[1,m], including t iterations, phase l tth iteration, The entire group includes subgroups, for subgroup j, have particles;

[0018] s303: For particle i of subgroup j, its velocity update is based on INTRA - SL, INTER - SL, and OL modes. If and p < r both hold, then each particle updates its velocity according to INTRA - SL. If and p ≥ r, then each particle's velocity update performs INTEA - SL. If then each particle updates its velocity according to the FC mode. Then, update the position of each particle, pbesti, and gbestj in subgroup j;

[0019] s304: If gbestj in subgroup j has not changed for ρ consecutive iterations and the number of subgroups in the whole group is greater than 1, that is then perform the OL mode for gbestj;

[0020] s305: For the t - th iteration of stage l, update gbest in the whole population;

[0021] s306: Determine whether the termination condition is met. If so, jump to step s4; otherwise, loop through steps s302 - s305.

[0022] Furthermore, the process of the DMS mechanism is as follows:

[0023] s3011: The whole population contains N particles {X1, X2, …, X N}; Divide the N particles into n subgroups {S1, S2, …, S n}, where each subgroup consists of N / n particles; For j ∈ [1, n], S j is denoted as S j = X (j-1)N / n+1 , X (j-1)N / n+2 , …, X N / n+N / n}; The whole iteration process includes t max iterations; Divide the t max iterations into m stages, where m = log2 n + 1; Each stage includes t<​​​​​​​​​​​​​​​​j , update all particles and gbest j ;

[0025] s3013: If the number of subgroups becomes 1, the algorithm terminates; otherwise, jump to s3012.

[0026] Further, comprehensive interactive learning strategies are as follows:

[0027] s3021: In INTRA-SL mode, particles in a subswarm can update their velocities by learning their personal best positions and the best positions in the subswarm. Therefore, the algorithm focuses more on local search at this stage. Its velocity update formula is as follows:

[0028] V i =ωV i +c1R1*(pbest i -X i )+c2R2*(gbest j -X i ), r≤p

[0029] Where V i and pbest i They represent the speed and personal best position of the i-th particle in the j-th subgroup respectively; gbest j represents the best position in the jth subgroup;

[0030] s3022: In INTER-SL mode, particles in a subswarm can update their velocities by learning their personal best positions and ebests. Therefore, the algorithm focuses more on global search at this time. Its velocity update formula is as follows:

[0031] V i =ωV i +c1R1*(pbest i -X i )+c2R2*(ebest j -X i ), r>p

[0032] Among them, ebest j It represents the average of the best individual positions of particles selected from all subpopulations, as detailed below:

[0033]

[0034] in, Indicates that from the jth subgroup The best personal position of a randomly selected particle in represents the total number of subgroups in the entire population; X jrepresents the subscript index of a particle randomly selected from the jth subgroup;

[0035] s3023: In OL mode, if the best position of a subgroup is not updated during the iteration, it is called particle stagnation, and the number of stagnations is recorded as ρ. If ρ of a subgroup exceeds the set value, the best positions of two different subgroups are randomly selected from all subgroups to update the best position of the stagnant subgroup:

[0036]

[0037] Among them gbest j is the best particle of the jth subgroup, R is a random number in the interval [0,1], and From two different subgroups S a and S b The best particle selected from S a and S b The selection of is also random. If the optimal particle in a subgroup does not improve within ρ consecutive iterations, it can be considered that the subgroup is trapped in a local optimum. By learning the optimal particles from other subgroups, the OL mode can help the subgroup avoid premature convergence and jump out of the local optimum.

[0038] S3024: In FC mode, every particle in the entire population learns from gbest to speed up its convergence and improve solution accuracy, as shown below:

[0039] V i =ωV i +c2R2*(gbest-X i )

[0040] More specifically, if the number of subgroups is greater than 1, i.e. Then the CIL solution will execute INTRA-SL, INTER-SL and OL modes; otherwise The CIL solution will only execute in FC mode.

[0041] Compared with the prior art, the present invention has the following beneficial effects:

[0042] In the application of optimizing the analytical solution of inverse kinematics of a seven-degree-of-freedom manipulator, the proposed DMSPSO-CIL algorithm has excellent real-time performance, accuracy and stability. In particular, compared with the traditional traversal search (SWSA) method, the DMSPSO-CIL solution speed is increased by nearly 100%. At the same time, in terms of accuracy and stability, the proposed DMSPSO-CIL algorithm also has the best performance among all compared algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 Schematic diagram of DMS strategy.

[0044] Figure 2 It is a comprehensive interactive learning strategy (CIL).

[0045] Figure 3 These are the implementation steps of the DMSPSO-CIL algorithm.

[0046] Figure 4 This is the principle block diagram of ASIK-RMLO based on DMSPSO-CIL algorithm.

[0047] Figure 5 It is an analytical solution optimization method based on the DMSPSO-CIL algorithm. DETAILED DESCRIPTION

[0048] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. The specific embodiments described herein are only used to explain the technical solution of the present invention and are not intended to limit the present invention.

[0049] The present invention provides a technical solution: a seven-degree-of-freedom manipulator analytical solution optimization method based on the DMSPSO-CIL algorithm, the process of which is as follows: Figure 5 As shown, the following steps are included:

[0050] Step s1: Get the current robot arm joint angle information θ cur , the next moment (posture) position information P of the robot arm T ,R T ;

[0051] Given the pose information of the end effector (palm or tool) of the robotic arm: position vector P T =[0.25, -0.7, 0.1], rotation matrix TR Z =[0, 0, π / 2].

[0052] Step s2: Input the current pose information into the inverse kinematics solver to obtain the redundant circle and redundant angle (like Figure 4 );

[0053] S201: ASIK-RMLO: Redundant Arm Redundant Circle

[0054] Assume that the desired pose of the end of the manipulator (palm / tool) is given, and the position vector P T =[0.25, -0.7, 0.1] and the rotation matrix R T =[TR x , TR y , TR z], we can get the wrist position P of the robot arm W :

[0055] P W =P T -TR Z ×L3

[0056] Where L3 is the length of the robot arm palm (or tool length). The distance from wrist to shoulder is obtained as follows:

[0057]

[0058] Where “·” is the vector dot product.

[0059] When the wrist posture is determined, the center and radius of the redundant circle can be calculated using the above geometric structure. The upper arm length of the robotic arm is L1, the forearm length is L2, and the distance between the shoulder joint and the wrist joint is d. According to the triangle cosine theorem, the angles α1 and α2 between the upper arm L1, forearm L2, and the horizontal line can be obtained:

[0060]

[0061]

[0062] Further obtain d c and R C :

[0063] d c =L1×cos(α1)

[0064] R c =L1×sin(α1)

[0065] Due to the existence of the connecting rod offset, the distances between the shoulder joint, wrist joint, and elbow joint will change, denoted as NL1 and NL2. According to the Pythagorean theorem of triangles, it is easy to derive:

[0066]

[0067]

[0068] Therefore, the angles between NL1, NL2 and the horizontal line l are:

[0069]

[0070]

[0071] The distance d from the center C of the generated redundant circle to the shoulder joint S c , the radius of the circle R c (i.e. the distance from the center C to the elbow joint) can be calculated by the following formula:

[0072] d c =NL1×cos(α1)

[0073] R c =NL1×sin(α1)

[0074] Therefore, we get the redundant circle, denoted as Circle:

[0075]

[0076] The middle interval is 1°. According to the wrist normalized rotation matrix RW Z Rotate the redundant circle, and the final redundant circle is:

[0077] C R = = Circle × RW z

[0078] S202: ASIK-RMLO: Calculating joint angles:

[0079] The position vector P of the end effector of the known robot arm T =[0.25, -0.7, 0.1], posture description matrix R T =[TR x , TR y , TR z ], solve the joint angles when reaching this state. The seven degrees of freedom correspond to seven joint angles, denoted as θ = [θ1, θ2, θ3, θ4, θ5, θ6, θ7]. Among them, the first four joint angles can determine the position of the wrist, and the last three joint angles give the posture of the end effector.

[0080] s2021: Calculate wrist position

[0081] The center and radius of the redundant circle have been obtained. The joint angles θ1 and θ2 are the azimuth and elevation angles of the upper arm relative to the shoulder joint (basic position), respectively. The position P of the node E1 can be obtained by first E1 Solve:

[0082] EC R =C R / ||C R ||2

[0083] EN cw =C R ×P w / ||C R ×P W ||2

[0084]

[0085] Therefore, the azimuth angle (θ1) and elevation angle (θ2) can be obtained:

[0086]

[0087]

[0088] The joint angle θ2 is the torsional degree of freedom of the upper arm and can be calculated by normalizing the vector:

[0089] N E1Z =(P E1 ×EZ)×P E1

[0090]

[0091] The joint angle θ4 corresponds to the base height of the forearm, also known as the elbow angle. Due to the flexibility of the link offset, theoretically, if the redundant circle is described as the elbow-out posture, there is also a corresponding elbow-in posture. This also means that a second redundant circle can be generated, but it is not considered because it does not conform to the characteristics of the human skeletal structure (the human arm cannot reach this posture). Therefore, θ4 for the elbow-out posture is:

[0092]

[0093] s2022: Calculating the pose of the end effector (hand)

[0094] The joint angles θ5 and θ6 correspond to the azimuth and elevation angles of the end effector, respectively. The first four joint angles [θ1, θ2, θ3, θ4] have been calculated, so the homogeneous transformation matrix of the link relative to the base coordinate system can be obtained and the rotation matrix as follows:

[0095]

[0096] Then, we take the wrist position P W Construct the end effector (hand / tool) coordinate system for the base coordinate system:

[0097]

[0098] Where, is the position of the end effector in the base coordinate system with the wrist as the origin, and further we can get:

[0099]

[0100]

[0101] The joint angle θ7 corresponds to the rotation angle of the hand. We have previously obtained the first six joint angles [θ1, θ2, θ3, θ4, θ5, θ6], so the wrist rotation matrix can be calculated. The hand rotation joint angle can be obtained from the following formula:

[0102] θ7 = arccos(WR x ·TR y )

[0103] Step s3: Take as the position of the particles in the DMSPSO-CIL algorithm and perform update iterations (as Figure 3 ).

[0104] S301: As shown in Figure 1 , in DMSPSO-CIL, first randomly initialize the velocities and positions of all N particles, evaluate their corresponding fitness values, and initialize gbest at the same time. The N particles are divided into n subgroups. The gbest corresponding to each subgroup j will also be initialized. Set the maximum number of iterations to t max , divided into m stages, each stage includes iterations, and m is defined as m = log2 n +1.

[0105] S302: In stage l, l ∈ [1, m], it includes t iterations. In the t-th iteration of stage l, the entire population includes subgroups. For subgroup j, there are particles.

[0106] s303: As shown in Figure 2 , for particle i in subgroup j, its velocity is updated according to the INTRA-SL, INTER-SL, and OL modes. Specifically, if and p < r both hold, then each particle updates its velocity according to INTRA-SL. If and p ≥ r are satisfied, then each particle updates its velocity by executing INTEA-SL; if then each particle updates its velocity according to the FC mode. Then, update the position of each particle, pbest i and gbest in subgroup j j .

[0107] s304: If gbest in subgroup j j has not changed in ρ consecutive iterations and the number of subgroups in the entire population is greater than 1 (i.e., ), then for gbestj Execute OL mode.

[0108] s305: For the tth iteration of stage l, Will update gbest in the entire group.

[0109] s306: Determine whether the termination condition is met (the number of algorithm iterations reaches the maximum or the obtained solution meets the accuracy requirement). If so, jump to step s4; otherwise, repeat steps s302 to s305;

[0110] If the position information of the end effector (palm or tool) of the robot arm is given: the position vector P T =[0.25,-0.7,0.1], rotation matrix TR Z =[0,0,π / 2]. Send the pose information to the inverse kinematics solver to obtain multiple sets of redundant circle parameters under the same pose. Corresponding different inverse solutions. The corresponding inverse solutions are listed. The joint angles of the WAM robot arm have a limited range, so all the solutions obtained meet the restriction conditions.

[0111] Step s4: Give the optimal solution and its corresponding robotic arm joint angle value θ nest .

[0112] The optimal solution is defined as follows:

[0113]

[0114] in, is the redundant circle redundant angle parameter, θ nest ,θ cur are the joint angle values ​​of the manipulator at the current moment (posture) and the feasible solution of the joint angle at the next moment (posture), P T and R T is the expected position of the hand (tool) of the robot arm at the next moment (posture), This is the formula for the closed-form solution of the inverse kinematics of a redundant manipulator with link offset that we analyzed. There exists a set of inverse solutions θ nest , so the process of finding the optimal solution is transformed into: find the solution that minimizes the objective equation under certain conditions (termination conditions). And return its corresponding inverse solution θ nest process.

[0115] The above description merely represents the preferred embodiments of the present invention, and while the description is relatively specific and detailed, it should not be construed as limiting the scope of the present invention. It should be noted that a person skilled in the art would be able to make various modifications, improvements, and substitutions without departing from the spirit of the present invention, all of which fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention shall be determined by the appended claims.

Claims

1. A method for optimizing the analytical solution of a seven-degree-of-freedom manipulator based on the DMSPSO-CIL algorithm, characterized by: The following steps are involved: Step s1: Get the current robot arm joint angle information θ cur , the expected pose information P of the robot at the next moment T ,R T ; Step s2: Input the current pose information into the inverse kinematics solver to obtain the redundant circle and redundant angle Step s3: As a novel particle swarm optimization algorithm, DMSPSO-CIL updates and iterates the positions of particles. DMSPSO-CIL includes two strategies: the dynamic multi-subswarm strategy (DMS) and the comprehensive interactive learning strategy (CIL). In DMS, the population size gradually decreases to one according to established rules as the algorithm iterates. CIL uses four interactive learning modes: intra-subswarm learning (INTRA-SL), inter-subswarm learning (INTER-SL), optimal learning (OL), and fast convergence (FC), allowing particles in each subswarm to learn from each other, from particles in other subswarms, or from the optimal subswarm within the entire subswarm. Step s4: Give the optimal solution and its corresponding robotic arm joint angle value θ nest ; The optimal solution is defined as the one with the smallest difference in joint angle between the solution set and the previous solution. The specific mathematical model is described as follows: in, is the redundant circle redundant angle parameter, θ nest ,θ cur are the joint angle values ​​of the robot arm at the current moment and the feasible solution of the joint angle at the next moment, P T and R T is the expected pose information of the robot arm at the next moment, That is, the formula for the closed-form solution of the inverse kinematics of a redundant manipulator with link offset; DMSPSO-CIL algorithm: s301: In DMSPSO-CIL, firstly, the speed and position of all N particles are randomly initialized and their corresponding fitness values ​​are evaluated. At the same time, gbest is also initialized. N particles are divided into n subgroups, and the gbest corresponding to each subgroup is j It will also be initialized, setting the maximum number of iterations to t max , divided into m stages, each stage includes iterations, m is defined as m=log2 n +1; s302: Phase l, l∈[1,m], including t iterations, phase l tth iteration, The entire group includes subgroups, for subgroup j, have particles; s303: For the particles of subgroup j Its velocity update is based on INTRA-SL, INTER-SL, and OL modes. If and both p < r hold, then each particle updates its velocity according to INTRA-SL. If it satisfies and p ≥ r, then each particle's velocity update executes INTEA-SL. If then each particle updates its velocity according to the FC mode. Then, update the position of each particle, pbest i and gbest in subgroup j j ; s304: If gbest in subgroup j j There is no change in consecutive ρ iterations, and the number of subgroups in the entire group is greater than 1, that is, gbest j Execute OL mode; s305: For the tth iteration of stage l, Will update gbest in the entire group; s306: Determine whether the termination condition is met. If so, jump to step s4; otherwise, loop through steps s302 to s305. The process of the DMS mechanism is as follows: s3011: The entire population contains N particles {X1, X2, ..., X N }; Divide N particles into n subgroups {S1, S2, ..., S n }, each subgroup consists of N / n particles; for j∈[1,n], S j Indicated as S j ={X (j-1)N / n+1 ,X (j-1)N / n+2 ,…,X N / n+N / n }; The entire iterative process includes t max Iterations; t max The iteration is divided into m stages, where m = log2 n +1; each stage includes t max / m iterations; the best solution of the j-th subgroup is denoted as gbest j ; Update all particles and gbest j ; s3012: In the lth stage l∈[1,m], N particles are divided into subgroups, for The jth subgroup contains particles, defined as The best solution of the jth subgroup is gbest j , update all particles and gbest j ; s3013: If the number of subgroups becomes 1, the algorithm terminates; otherwise, jump to s3012; The comprehensive interactive learning strategy is as follows: s3021: In INTRA-SL mode, particles in a subswarm update their velocities by learning their personal best positions and the best positions in the subswarm. Therefore, the algorithm focuses more on local search at this stage. Its velocity update formula is as follows: V i =ωV i +c1R1*(pbest i -X i )+c2R2*(gbest j -X i ),r≤p Where V i and pbest i They represent the speed and personal best position of the i-th particle in the j-th subgroup respectively; gbest j represents the best position in the jth subgroup; In INTER-SL mode, particles in a subgroup update their velocities by learning their personal best positions and ebests. Therefore, the algorithm focuses more on global search at this time. Its velocity update formula is as follows: V i =ωV i +c1R1*(pbest i -X i )+c2R2*(ebest j -X i ),r>p Among them, ebest j It represents the average of the best individual positions of particles selected from all subpopulations, as detailed below: in, represents the personal best position of a particle randomly selected from the jth subgroup, represents the total number of subgroups in the entire population; X j represents the subscript index of a particle randomly selected from the jth subgroup; s3023: In OL mode, if the best position of a subgroup is not updated during the iteration, it is called particle stagnation, and the number of stagnations is recorded as ρ. If ρ of a subgroup exceeds the set value, the best positions of two different subgroups are randomly selected from all subgroups to update the best position of the stagnant subgroup: Among them gbest j is the best particle of the jth subgroup, R is a random number in the interval [0,1], and From two different subgroups S a and S b The best position selected in S a and S b The selection of is also random. If the optimal particle in a subgroup does not improve within ρ consecutive iterations, the subgroup is considered to be trapped in a local optimum. By learning the optimal particles from other subgroups, the OL mode helps the subgroup avoid premature convergence and jump out of the local optimum. S3024: In FC mode, every particle in the entire population learns from gbest to speed up its convergence and improve solution accuracy, as shown below: V i =ωV i +c2R2*(gbest-X i ) gbest represents the best position in the entire population; if the number of subgroups is greater than 1, that is, Then the CIL solution will execute INTRA-SL, INTER-SL and OL modes; otherwise The CIL solution will only execute in FC mode.

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