Mechanical arm trajectory tracking control method based on improved African vulture algorithm

Abstract

The invention discloses a mechanical arm trajectory tracking control method based on an improved African vulture algorithm. The method comprises the steps that original mechanical arm data are acquired; obtaining a trajectory tracking result of the mechanical arm based on a pre-trained mechanical arm trajectory tracking control model according to the original mechanical arm data; the mechanical arm trajectory tracking control model is trained by using an improved African vulture algorithm. The parameter adjustment problem of the controller in the mechanical arm trajectory tracking control model is solved, the improved African vulture algorithm has the advantages of being high in convergence speed, high in global search capability and high in solving precision, the global search capability and the local search capability are balanced, the performance effect of parameters obtained through optimization is excellent, and the method is suitable for large-scale popularization and application. And the parameter adjusting time and energy of scientific researchers can be effectively reduced.

Description

Technical Field

[0001] The invention relates to the technical field of robot control, and in particular to a robot arm trajectory tracking control method based on an improved African vulture algorithm. Background Art

[0002] With the rapid development of industrial automation and robotics, robotic arms are increasingly used in manufacturing, aerospace, medical and other fields. The trajectory tracking control of robotic arms is a key technology to ensure that the robotic arms can move accurately along the predetermined trajectory, and its performance directly affects production efficiency, quality and safety. However, in practical applications, robotic arm systems often have factors such as nonlinearity, time-varying, model uncertainty and external interference, which pose severe challenges to the accuracy and stability of trajectory tracking.

[0003] Traditional robot trajectory tracking control methods, such as PID control and sliding mode control, can achieve trajectory tracking to a certain extent, but it is difficult to achieve precise control when dealing with the above complex problems. As an advanced control method, Linear Active Disturbance Rejection Control (LADRC) estimates and compensates for the system's internal dynamics and external disturbances in real time through an extended state observer, effectively improving the system's anti-interference ability and control accuracy. The performance of the LADRC algorithm depends to a large extent on the tuning of its parameters. However, the LADRC algorithm has the problem of difficulty in adjusting parameters, and conventional parameter tuning methods often cannot obtain parameters with good control effects. Summary of the invention

[0004] The purpose of the present invention is to overcome the deficiencies in the prior art and to provide a robot arm trajectory tracking control method based on an improved African vulture algorithm, which can solve the parameter adjustment problem of the controller in the robot arm trajectory tracking control model.

[0005] To achieve the above object, the present invention is implemented by adopting the following technical solutions:

[0006] On the one hand, the present invention provides a robot arm trajectory tracking control method based on an improved African vulture algorithm, comprising:

[0007] Get the original robot arm data;

[0008] According to the original robot arm data, based on a pre-trained robot arm trajectory tracking control model, a trajectory tracking result of the robot arm is obtained;

[0009] The robot arm trajectory tracking control model is trained using the improved African vulture algorithm, including:

[0010] Access to African vulture populations;

[0011] Updating the individual position of each vulture in the African vulture population, and calculating the fitness value of the vulture using the untrained tracking control model of the robotic arm;

[0012] The global exploration phase and the local exploration phase were divided by using the satiety rate calculation method considering linear variables.

[0013] The positions of the vultures are updated in the global exploration stage and the local exploration stage respectively, and the individual positions of the optimal vultures are obtained according to the fitness values; and an adaptive coefficient is introduced in the global exploration stage;

[0014] The parameters represented by the individual positions of the optimal vultures are used as parameters of the robot arm trajectory tracking control model.

[0015] Optionally, before updating the individual position of each vulture in the African vulture population, the African vulture population is initialized using Kent chaotic mapping to generate the individual position of each vulture in the African vulture population, such as formula:

[0016] ;

[0017] In the formula, For the The individual position of the vulture at the iteration; For the The individual position of the vulture at the iteration; is the adjustment parameter of Kent chaotic mapping, ; is the current iteration number.

[0018] Optionally, the Cauchy reverse learning strategy is used to update the individual position of each vulture in the African vulture population, and the formula is:

[0019] ;

[0020] In the formula, is a random function; is the upper bound of each vulture individual in the African vulture population in the dim dimension; is the lower bound of each vulture individual in the African vulture population in the dim dimension; is the position of each vulture individual in the African vulture population in the dim dimension.

[0021] Optionally, the calculating the fitness value of the vulture includes:

[0022] Call the untrained robot arm trajectory tracking control model and use the ITAE index as the objective function to calculate the fitness value of the vulture. The formula is:

[0023] ;

[0024] Where t is time, and e is the input and output error signal of the robot.

[0025] Optionally, the formula for calculating the satiety rate considering linear variables is:

[0026] ;

[0027] ;

[0028] ;

[0029] In the formula, is the satiety rate of the vulture; is a random number between [-1, 1]; is a random number between [-1, 1]; is the current iteration number; is the maximum number of iterations; is the satiety rate coefficient; is a random number between [-2, 2]; is a linear variable.

[0030] Optional, when When , the vulture is in the global exploration stage, and the position of the vulture is updated in the global exploration stage, as shown in the formula:

[0031] ;

[0032] ;

[0033] ;

[0034] In the formula, is the vulture position vector in the next iteration of the global exploration phase; is the adaptive coefficient; For one of the best vultures; is the first variable; Exploring parameters globally; is a random number between [0, 1]; is a random number between [0, 1]; Find the upper bound for the global exploration phase; Find the lower bound for the global exploration phase; is a random number between [0, 1]; is a random number between [0, 2]; is the vulture position vector in the current iteration.

[0035] Optional, when When , the vulture is in the first local exploration stage. In the first local exploration stage, the position of the vulture is updated, as shown in the formula:

[0036] ;

[0037] ;

[0038] In the formula, is the vulture position vector in the next iteration of the first local exploration phase; is the first variable; is a random number between [0, 1]; is the second variable; One of the best vultures; is the vulture position vector in the current iteration; is the first local exploration parameter; is a random number between [0, 1]; is the third variable; is a random number between [0, 1]; is the fourth variable.

[0039] Optional, when When , the vulture is in the second local exploration stage. In the second local exploration stage, the position of the vulture is updated, as shown in the formula:

[0040] ;

[0041] ;

[0042] In the formula, is the vulture position vector in the next iteration of the second local exploration phase; is the fifth variable; is the sixth variable; For the best vulture in the first group; is the vulture position vector in the current iteration; is the second local exploration parameter; is a random number between [0, 1]; for the best vulture in the second group; For one of the best vultures; is the second variable; For levy flight operations.

[0043] In a second aspect, the present invention provides a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, the steps of the robot arm trajectory tracking control method based on the improved African vulture algorithm described in the first aspect are implemented.

[0044] In a third aspect, the present invention provides a computer system, comprising:

[0045] Memory, for storing computer instructions;

[0046] A processor is used to execute the computer instructions to implement the steps of the robot arm trajectory tracking control method based on the improved African vulture algorithm described in the first aspect.

[0047] Beneficial Effects

[0048] Compared with the prior art, the beneficial effects achieved by this aspect are:

[0049] The present invention solves the problem of parameter adjustment of the controller in the robot arm trajectory tracking control model. The improved African vulture algorithm has the advantages of fast convergence speed, strong global search capability, and high solution accuracy. It achieves a balance between global search and local search capabilities, and has excellent performance on the parameters obtained by optimization, which can effectively reduce the parameter adjustment time and energy of scientific researchers. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 Shown is a schematic diagram of the working principle of the improved African vulture algorithm of the present invention in one embodiment;

[0051] Figure 2 Shown is a schematic diagram of the flow of an improved African vulture algorithm of the present invention in one embodiment;

[0052] Figure 3 The figure shows a schematic diagram of the structure of a robot arm trajectory tracking control model in one embodiment of the present invention;

[0053] Figure 4 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F1 in one embodiment;

[0054] Figure 5 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F2 in one embodiment;

[0055] Figure 6 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F3 in one embodiment;

[0056] Figure 7 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F4 in one embodiment;

[0057] Figure 8 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F5 in one embodiment;

[0058] Fig. 9 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F6 in one embodiment;

[0059] Fig.10 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F7 in one embodiment;

[0060] Fig.11 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F8 in one embodiment;

[0061] Fig.12 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F9 in one embodiment;

[0062] Fig.13 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F10 in one embodiment;

[0063] Fig.14 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F11 in one embodiment;

[0064] Fig.15 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F12 in one embodiment;

[0065] Fig.16 The figure shows a comparison of the convergence curves of the improved African vulture algorithm of the present invention and the existing algorithm for the test function F13 in one embodiment;

[0066] Fig.17 Shown is a comparison diagram of the robot arm trajectory tracking control angle error of the improved African vulture algorithm of the present invention and the African vulture algorithm in one embodiment. DETAILED DESCRIPTION

[0067] The technical solution of the present invention is described in detail below through the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations on the technical solution of the present invention. The embodiments of the present invention and the technical features in the embodiments may be combined with each other unless there is a conflict.

[0068] The term "and / or" is only a description of the association relationship between related objects, indicating that there can be three relationships. For example, A and / or B can mean: A exists alone, A and B exist at the same time, and B exists alone. In addition, the character " / " generally indicates that the related objects are in an "or" relationship.

[0069] Example 1

[0070] like Figure 1 As shown, this embodiment introduces a robot arm trajectory tracking control method based on an improved African vulture algorithm, which specifically includes the following steps:

[0071] Get the raw robot data.

[0072] According to the original robot data, based on the pre-trained robot trajectory tracking control model, the robot trajectory tracking result is obtained. The structure of the robot trajectory tracking control model is shown as Figure 3 shown.

[0073] like Figure 2 As shown, the robot arm trajectory tracking control model is trained using the improved African vulture algorithm, including:

[0074] Get the African vulture population.

[0075] Set the African vulture population N and the maximum number of iterations , using Kent chaotic mapping to initialize the African vulture population, and generate the individual positions of each vulture in the African vulture population. Specifically:

[0076] Randomly generate a d-dimensional vulture individuals.

[0077] The generated vulture individuals are iterated to generate other individual chaotic sequences as shown in the formula:

[0078] ;

[0079] In the formula, For the The individual position of the vulture at the iteration; For the The individual position of the vulture at the iteration; is the adjustment parameter of Kent chaotic mapping, ; is the current iteration number.

[0080] After the iteration is completed according to the set vulture population number, it is mapped to the search space according to the formula. The specific formula is as follows:

[0081] ;

[0082] In the formula, and Respectively represent the lower and upper limits of the independent variable value; Represents the chaos factor generated by Kent chaos mapping.

[0083] The Cauchy reverse learning strategy is used to update the individual positions of each vulture in the African vulture population, and the fitness value of the vulture is calculated using the untrained robot tracking control model. The structure of the robot trajectory tracking control model is as follows: Figure 3 shown.

[0084] The optimization model formula of the Cauchy reverse learning strategy is:

[0085] ;

[0086] In the formula, is a random function; is the upper bound of each vulture individual in the African vulture population in the dim dimension; is the lower bound of each vulture individual in the African vulture population in the dim dimension; is the position of each vulture individual in the African vulture population in the dim dimension. The specific optimization steps are:

[0087] Define the Cauchy reverse jump probability Jr, set to 0.95, rand is A random number between

[0088] When rand≤Jr, generate the Cauchy reverse population of the current population, merge the current vulture population and the Cauchy reverse vulture population, and calculate the individual fitness values ​​of all vultures.

[0089] The untrained robot arm trajectory tracking control model built in Simulink is called, and the fitness value y of the vulture is calculated using the ITAE index as the objective function. The specific calculation formula is:

[0090] ;

[0091] Where t is time, and e is the input and output error signal of the robot.

[0092] The satiety rate calculation method considering linear variables is used to divide the exploration phase into global exploration phase and local exploration phase.

[0093] Fixed parameters Changed to a linearly decreasing variable, the specific calculation formula is:

[0094] ;

[0095] ;

[0096] ;

[0097] In the formula, is the satiety rate of the vulture; is a random number between [-1, 1]; is a random number between [-1, 1]; is the current iteration number; is the maximum number of iterations; is the satiety rate coefficient; is a random number between [-2, 2]; is a linear variable.

[0098] In this embodiment, when |F|≥1, the vulture is in the global exploration stage; when |F|<1, the vulture is in the local exploration stage.

[0099] The positions of the vultures are updated in the global exploration stage and the local exploration stage respectively, and the individual positions of the optimal vultures are obtained according to the fitness value; and an adaptive coefficient is introduced in the global exploration stage. The global search capability of the algorithm is enhanced by optimizing the satiety rate calculation method and introducing an adaptive coefficient in the global search stage, so as to avoid finding the local optimal solution too early in the global exploration stage and balance the global exploration and local exploration capabilities of the algorithm.

[0100] when When , the vulture is in the global exploration stage, and the position of the vulture is updated in the global exploration stage, as shown in the formula:

[0101] ;

[0102] ;

[0103] ;

[0104] In the formula, is the vulture position vector in the next iteration of the global exploration phase; is the adaptive coefficient; For one of the best vultures; is the first variable; Exploring parameters globally; is a random number between [0, 1]; is a random number between [0, 1]; Find the upper bound for the global exploration phase; Find the lower bound for the global exploration phase; is a random number between [0, 1]; is a random number between [0, 2]; is the vulture position vector in the current iteration.

[0105] when When , the vulture is in the first local exploration stage. In the first local exploration stage, the position of the vulture is updated, as shown in the formula:

[0106] ;

[0107] ;

[0108] In the formula, is the vulture position vector in the next iteration of the first local exploration phase; is the first variable; is a random number between [0, 1]; is the second variable; For one of the best vultures; is the vulture position vector in the current iteration; is the first local exploration parameter; is a random number between [0, 1]; is the third variable; is a random number between [0, 1]; is the fourth variable.

[0109] when When , the vulture is in the second local exploration stage. In the second local exploration stage, the position of the vulture is updated, as shown in the formula:

[0110] ;

[0111] ;

[0112] In the formula, is the vulture position vector in the next iteration of the second local exploration phase; is the fifth variable; is the sixth variable; For the best vulture in the first group; is the vulture position vector in the current iteration; is the second local exploration parameter; is a random number between [0, 1]; for the best vulture in the second group; For one of the best vultures; is the second variable; For levy flight operations.

[0113] The parameters w0, wc and b represented by the individual positions of the optimal vultures are used as the parameters w0, wc and b of the robot arm trajectory tracking control model.

[0114] Example 2

[0115] like Figure 4-16 As shown, this embodiment is a performance test simulation performed on the improved African vulture algorithm of the present invention.

[0116] As shown in Table 1, 13 standard algorithm test functions were selected, and the original African vulture algorithm (AVOA) and the improved African vulture algorithm (EAVOA) of the present invention were optimized and tested by MATLAB, and compared with the WOA algorithm and the PSO algorithm, where F1~F7 are unimodal test functions and F8~F13 are multimodal test functions. The unimodal test function has only one global optimal value and no local optimal value, which can be used to test the search ability and convergence speed of the optimization algorithm; while the local optimal point of the multimodal test function increases with the dimensionality exponentially, which can be used to test the global search ability of the algorithm.

[0117] Table 1 Test functions

[0118]

[0119] The parameter conditions of the four optimization algorithms were set to the same (population size 30, maximum number of iterations 300), and each test function was independently run 30 times by MATLAB and the results were processed, as shown in Table 2, which records the optimal value, average value, and standard deviation of the results obtained by the algorithm for solving 13 test functions; Figure 4-16 As shown, they are respectively the convergence curve comparison diagrams of the algorithm for the global optimal values ​​of 13 test functions. Through the comparison, it is verified that the improved African vulture algorithm of the present invention has stronger global search ability, faster convergence speed and higher accuracy.

[0120] Table 2 Experimental results data

[0121]

[0122] Example 3

[0123] This embodiment uses the robot arm trajectory tracking control method based on the improved African vulture algorithm described in Example 1, and sets the parameters w0, wc, and b representing the individual positions of the optimal vultures finally obtained to the controller in the robot arm trajectory tracking control model. The structure of the robot arm trajectory tracking control model is as follows: Figure 3 As shown, the output control angle error result is as follows Fig.17 As shown, the parameters obtained by improving the African vulture algorithm using the present invention have better effects on trajectory tracking, the input and output errors of the robot arm joint angles are smaller, and the control accuracy is higher.

[0124] Example 4

[0125] This embodiment introduces a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, the steps of the robot arm trajectory tracking control method based on the improved African vulture algorithm described in Example 1 or 3 are implemented.

[0126] Example 5

[0127] This embodiment introduces a computer system, including:

[0128] Memory, for storing computer instructions;

[0129] A processor is used to execute the computer instructions to implement the steps of the robot arm trajectory tracking control method based on the improved African vulture algorithm described in Example 1 or 3.

[0130] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application may adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.

[0131] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0132] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0133] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0134] The embodiments of the present invention are described above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned specific implementation methods. The above-mentioned specific implementation methods are merely illustrative and not restrictive. Under the enlightenment of the present invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the purpose of the present invention and the claims, which all fall within the protection of the present invention.

Claims

1. A robot arm trajectory tracking control method based on an improved African vulture algorithm, characterized in that: include: Get the original robot arm data; According to the original robot arm data, based on a pre-trained robot arm trajectory tracking control model, a trajectory tracking result of the robot arm is obtained; The robot arm trajectory tracking control model is trained using the improved African vulture algorithm, including: Access to African vulture populations; Updating the individual position of each vulture in the African vulture population, and calculating the fitness value of the vulture using the untrained tracking control model of the robotic arm; The global exploration phase and the local exploration phase were divided by using the satiety rate calculation method considering linear variables. The positions of the vultures are updated in the global exploration stage and the local exploration stage respectively, and the individual positions of the optimal vultures are obtained according to the fitness values; and an adaptive coefficient is introduced in the global exploration stage; The parameters represented by the individual positions of the optimal vultures are used as parameters of the robot arm trajectory tracking control model.

2. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 1 is characterized in that: Before updating the individual position of each vulture in the African vulture population, the African vulture population is initialized using Kent chaotic mapping to generate the individual position of each vulture in the African vulture population, as shown in the formula: ； In the formula, For the The individual positions of the vultures at the iteration; For the The individual positions of the vultures at the iteration; is the adjustment parameter of Kent chaotic mapping, ; is the current iteration number.

3. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 1 is characterized in that: The Cauchy reverse learning strategy is used to update the individual positions of each vulture in the African vulture population, and the formula is: ； In the formula, is a random function; is the upper bound of each vulture individual in the African vulture population in the dim dimension; is the lower bound of each vulture individual in the African vulture population in the dim dimension; is the position of each vulture individual in the African vulture population in the dim dimension.

4. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 1 is characterized in that: The step of calculating the fitness value of the vulture comprises: Call the untrained robot arm trajectory tracking control model and use the ITAE index as the objective function to calculate the fitness value of the vulture. The formula is: ； Where t is time, and e is the input and output error signal of the robot.

5. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 1 is characterized in that: The formula for calculating the satiety rate considering linear variables is: ； ； ； In the formula, is the satiety rate of the vulture; is a random number between [-1, 1]; is a random number between [-1, 1]; is the current iteration number; is the maximum number of iterations; is the satiety rate coefficient; is a random number between [-2, 2]; is a linear variable.

6. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 5 is characterized in that: when When , the vulture is in the global exploration stage, and the position of the vulture is updated in the global exploration stage, as shown in the formula: ； ； ； In the formula, is the vulture position vector in the next iteration of the global exploration phase; is the adaptive coefficient; For one of the best vultures; is the first variable; Exploring parameters globally; is a random number between [0, 1]; is a random number between [0, 1]; Find the upper bound for the global exploration phase; Find the lower bound for the global exploration phase; is a random number between [0, 1]; is a random number between [0, 2]; is the vulture position vector in the current iteration.

7. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 5 is characterized in that: when When , the vulture is in the first local exploration stage. In the first local exploration stage, the position of the vulture is updated, as shown in the formula: ； ； In the formula, is the vulture position vector in the next iteration of the first local exploration phase; is the first variable; is a random number between [0, 1]; is the second variable; For one of the best vultures; is the vulture position vector in the current iteration; is the first local exploration parameter; is a random number between [0, 1]; is the third variable; is a random number between [0, 1]; is the fourth variable.

8. The robot arm trajectory tracking control method based on the improved African vulture algorithm according to claim 5 is characterized in that: when When , the vulture is in the second local exploration stage. In the second local exploration stage, the position of the vulture is updated, as shown in the formula: ； ； In the formula, is the vulture position vector in the next iteration of the second local exploration phase; is the fifth variable; is the sixth variable; For the best vulture in the first group; is the vulture position vector in the current iteration; is the second local exploration parameter; is a random number between [0, 1]; for the best vulture in the second group; For one of the best vultures; is the second variable; For levy flight operations.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the robot arm trajectory tracking control method based on the improved African vulture algorithm described in any one of claims 1 to 8 are implemented.

10. A computer system, characterized in that: include: Memory, for storing computer instructions; A processor, used to execute the computer instructions to implement the steps of the robot arm trajectory tracking control method based on the improved African vulture algorithm according to any one of claims 1 to 8.

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