Fixed time optimal control method and system for single-connecting-rod mechanical arm

Abstract

The invention provides a fixed-time optimal control method and system of a single-connecting-rod mechanical arm, and relates to the technical field of optimal control of the single-connecting-rod mechanical arm. The fixed-time optimal control method comprises the steps that a kinetic model of the single-connecting-rod mechanical arm to be controlled is obtained, and the model is converted into a state model by introducing a state variable; according to the state model, an active-disturbance-rejection control strategy is adopted, and an expansion state observer is established and used for observing disturbance state information of the single-connecting-rod mechanical arm; based on the state model and the extended state observer, a backstepping method technology and a reinforcement learning algorithm are combined, an optimal controller for the single-connecting-rod robot mechanical arm is designed, and self-adaptive optimal backstepping control is conducted on the single-connecting-rod robot mechanical arm; the active disturbance rejection control technology and the fixed time optimal control are combined, a new control strategy and an optimization path are developed, and the control performance and the stability of the mechanical arm system are further improved.

Description

Technical Field

[0001] The present disclosure relates to the technical field of optimal control of a single-link robotic arm, and in particular to a fixed-time optimal control method and system for a single-link robotic arm. Background Art

[0002] In actual engineering practice, the rapid convergence of tracking errors and excellent transient performance are indispensable indicators for evaluating the performance of the controlled system. For this reason, fixed-time control technology has won great attention in the engineering community due to its unique advantages. The biggest feature of fixed-time control technology is that it can ensure that the convergence time of the robotic arm system is fixed, and this convergence time is not affected by the initial state of the system, which means that no matter what the initial state of the system is, the system can reach a stable state or achieve synchronization within a predetermined time; this feature is particularly important in engineering applications because it allows engineers to accurately estimate the response time of the system during the design phase to meet specific time requirements; therefore, the current research is to accelerate the convergence speed in actual engineering through fixed-time control technology.

[0003] In the framework of fixed-time optimal control, the system often faces disturbance challenges caused by unknown factors such as turbulence, wind changes, mechanical resistance, and internal dynamic uncertainty. In order to meet these challenges, a variety of compensation strategies have emerged, and the active disturbance rejection control technology has stood out among many methods due to its unique response mechanism and has become the preferred solution for compensating disturbance terms. The essence of the active disturbance rejection control technology lies in its two core components: the extended state observer and the tracking differentiator. The extended state observer can accurately capture and process disturbances from outside the system and internal nonlinear uncertainties, thereby ensuring the accuracy and stability of system control; and the tracking differentiator can cleverly process input signals that lack clear mathematical expressions, ensuring that the system can smoothly track and respond to these signals, avoiding system oscillations that may be caused by sudden changes in signals.

[0004] It is worth noting that, although ADRC technology has shown its great potential in many fields, it is itself highly complex, and its algorithm and implementation process are relatively cumbersome. In the fixed-time optimal control of a single-link manipulator, it is necessary to combine the ADRC technology with the dynamic characteristics and control objectives of the manipulator, which undoubtedly increases the difficulty of research. As a result, the research on the fixed-time optimal control of a single-link manipulator is still relatively small, resulting in insufficient performance and stability of the fixed-time control technology for the single-link manipulator system. Summary of the invention

[0005] In order to solve the above-mentioned problems, the present invention proposes a fixed-time optimal control method and system for a single-link robotic arm, which combines the anti-disturbance control technology with the fixed-time optimal control, opens up a new control strategy and optimization path for the single-link robotic arm, and further improves the control performance and stability of the robotic arm system.

[0006] According to some embodiments, the present disclosure adopts the following technical solutions:

[0007] A fixed-time optimal control method for a single-link robotic arm, comprising:

[0008] Obtain the dynamic model of the single-link robotic arm to be controlled, and transform the model into a state model by introducing state variables;

[0009] According to the state model, an extended state observer is established by adopting the anti-disturbance control strategy to observe the disturbance state information of the single-link manipulator.

[0010] Based on the state model and extended state observer, the backstepping technique is combined with the reinforcement learning algorithm to design the optimal controller for the single-link robot arm, and the single-link robot arm is adaptively controlled with optimal backstepping.

[0011] According to some embodiments, the present disclosure adopts the following technical solutions:

[0012] A fixed-time optimal control system for a single-link robotic arm, comprising:

[0013] The model building module is configured to: obtain a dynamic model of the single-link robotic arm to be controlled, and transform the model into a state model by introducing state variables;

[0014] The observer establishment module is configured to: establish an extended state observer based on the state model and adopt an anti-disturbance control strategy to observe the disturbance state information of the single-link manipulator;

[0015] The controller design module is configured as follows: based on the state model and the extended state observer, the backstepping technique is combined with the reinforcement learning algorithm to design the optimal controller for the single-link robot arm and perform adaptive optimal backstepping control on the single-link robot arm.

[0016] According to some embodiments, the present disclosure adopts the following technical solutions:

[0017] A computer program product comprises a computer program, wherein when the computer program is executed by a processor, the fixed-time optimal control method of a single-link mechanical arm is implemented.

[0018] According to some embodiments, the present disclosure adopts the following technical solutions:

[0019] A non-transitory computer-readable storage medium is used to store computer instructions. When the computer instructions are executed by a processor, the fixed-time optimal control method of a single-link robotic arm is implemented.

[0020] According to some embodiments, the present disclosure adopts the following technical solutions:

[0021] An electronic device comprises: a processor, a memory and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory so that the electronic device executes the fixed-time optimal control method for a single-link robotic arm.

[0022] Compared with the prior art, the present invention has the following beneficial effects:

[0023] 1. The technical solution of the present invention adopts the fixed time technology to design the extended state observer ESO, which can not only accurately estimate the disturbance in the manipulator system with time delay (i.e. d(t) in the dynamic model and Δ(t) in the state model), but also observe the state of the system (i.e. x in the state model). 1 、x 2 、x 3 ), due to the combination of fixed-time technology, the observer's signal can converge within a fixed time interval, which can significantly improve the convergence speed; compared with the infinite-time control method, this method shows superior performance in the convergence speed of the tracking error in the system tracking task.

[0024] 2. The technical solution of the present invention proposes a fixed-time optimal backstepping tracking control strategy, which integrates the advantages of tracking control and optimal control. By constructing the optimal performance indicator function, it ensures that the single-link robotic arm completes the tracking control task and optimizes the system performance, such as energy consumption, response time, etc.; the core is to use the optimal control theory to improve the tracking control effect, so that the robotic arm can operate efficiently and accurately in a complex environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The accompanying drawings constituting a part of the present disclosure are used to provide a further understanding of the present disclosure. The illustrative embodiments of the present disclosure and their descriptions are used to explain the present disclosure and do not constitute an improper limitation on the present disclosure.

[0026] Figure 1 This is a method flow chart of a single-link robotic arm according to Example 1.

[0027] Figure 2 Schematic diagram of the single-link robotic arm of Example 1.

[0028] Figure 3This is the observation curve of the single-link robot arm x1 state based on ESO in Example 1.

[0029] Figure 4 This is the observation curve of the x2 state of the single-link robot arm based on ESO in Example 1.

[0030] Figure 5 This is the observation curve of the x3 state of the single-link robot arm based on ESO in Example 1.

[0031] Figure 6 This is a comparison chart of the tracking error curves of the single-link robotic arm system based on ESO under fixed time and infinite time in Example 1.

[0032] Figure 7 This is a control input curve diagram of the single-link robotic arm system based on ESO in Example 1.

[0033] Figure 8 This is a weight convergence curve diagram of the execution neural network of Example 1.

[0034] Fig. 9 This is a weight convergence curve diagram of the evaluation neural network of Example 1. DETAILED DESCRIPTION

[0035] The present disclosure is further described below in conjunction with the accompanying drawings and embodiments.

[0036] It should be noted that the following detailed descriptions are exemplary and are intended to provide further explanation of the present disclosure. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present disclosure belongs.

[0037] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present disclosure. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0038] Example 1

[0039] In one embodiment of the present disclosure, a fixed-time optimal control method for a single-link robotic arm is provided, comprising:

[0040] Step 1. Obtain the dynamic model of the single-link robotic arm to be controlled, and transform the model into a state model by introducing state variables;

[0041] Step 2. According to the state model, an extended state observer is established by adopting the anti-disturbance control strategy to observe the disturbance state information of the single-link manipulator;

[0042] Step 3. Based on the state model and the extended state observer, the backstepping technique is combined with the reinforcement learning algorithm to design the optimal controller for the single-link robot arm and perform adaptive optimal backstepping control on the single-link robot arm.

[0043] As an embodiment, the present invention discloses a fixed-time optimal control method for a single-link robotic arm, which combines the anti-disturbance control technology with the fixed-time optimal control, opens up a new control strategy and optimization path, and further improves the control performance and stability of the robotic arm system. Figure 1 The method shown is Figure 2 The single-link robot arm shown in the figure performs backstepping control, and the specific implementation process is as follows:

[0044] Step 1: Establish the dynamic model of the single-link robotic arm:

[0045]

[0046] Among them, q and are the joint angular position and acceleration of the robot arm respectively, y(t) is the angular position at time t, M is the mechanical inertia, m is the mass of the connecting rod, l is the length of the connecting rod, g is the gravitational acceleration, u(t) is the input torque on the joint of the robot arm, and d(t) is the unknown external disturbance.

[0047] Introduce state variable x 1 ,x 2 ,x 3 , let x 1 =q, The dynamic model of the single-link manipulator is converted into a state space equation, namely, a state model:

[0048]

[0049] Among them, x 1 ,x 2 ,x 3 Indicates the state of a single-link robot that is not measurable.

[0050] Step 2: Establish an extended state observer ESO to observe the disturbance state information of the single-link manipulator, that is, to compensate for external interference, which can be expressed as:

[0051]

[0052] in, They are state x 1 ,x 2 ,x3 The estimate of b 1 ,b 2 ,b 3 is the gain of the observer, and the estimation error is defined as where x=(x 1 ,x 2 ,x 3 ) T ,

[0053] Step 3: Combine fixed-time techniques and reinforcement learning algorithms to design an optimal controller:

[0054] The design of the controller depends on the state model of the robot system. The control strategy is obtained by solving the optimal control problem (such as minimizing the functional function of a certain performance indicator). For example, formulas (4) and (5) are designed through the state model, and these formulas are closely related to the controller. The same is true for the observer. The designed controller is designed based on the formula in the observer. According to the conversion between each variable, the required variables are brought into the controller.

[0055] Specifically, according to the state model of the given single-link robotic arm system, the optimal virtual controller and the optimal actual controller are designed respectively. First, the virtual controller and the actual controller are briefly described:

[0056] The virtual controller is usually simulated in a computer environment and does not directly control the actual robotic arm, while the actual controller is a hardware controller directly applied to the robotic arm. In the process of the robotic arm control algorithm, the virtual controller plays a vital role. The algorithm is simulated and tested in a virtual environment to ensure that its performance meets the requirements before it is deployed on the actual controller. In this way, the risk of failure of the algorithm in actual applications can be greatly reduced, and the reinforcement learning algorithm is used to train the robotic arm in a virtual environment to obtain a set of optimal control strategies. This set of strategies is then deployed on the actual controller to allow the robotic arm to perform tasks in the actual environment.

[0057] 1. Optimal Virtual Controller

[0058] The following optimal performance index function is established by 1 ) to minimize the cost value, thereby obtaining the optimal virtual controller

[0059]

[0060] Among them, Ω 1 is a compact set containing the origin, Ψ(Ω 1 ) is the admissible control set, α 1 It is a virtual controller. is the optimal virtual controller, is the cost function.

[0061] By taking the derivative of both sides of formula (4), we can obtain the Hamilton Jacobi Bellman (HJB) equation as follows:

[0062]

[0063] By solving The optimal virtual controller is obtained as follows:

[0064]

[0065] Will This breaks down into:

[0066]

[0067] in, K 11 ,K 12 is a positive design parameter.

[0068] Substituting formula (7) into formula (6) yields:

[0069]

[0070] because is unknown, and a neural network is used to approximate it. Then, can be rewritten as:

[0071]

[0072] Among them, Φ J1 (x 1 ,z 1 ) represents the basis function vector, and δ J1 (x 1 ,z 1 ) represents the approximation error.

[0073] Substituting formula (9) into formula (7) and (8) yields:

[0074]

[0075] Due to unknown The existence of Unable to use. To achieve the control objectives, reinforcement learning (RL) is used, which is implemented through two neural networks: the critic network and the actor network. The critic network is used to evaluate the control performance, while the actor network is responsible for executing the control behavior.

[0076] In order to obtain the optimized controller, a critic NN for evaluating the control performance and an actor NN responsible for executing the control behavior are designed:

[0077]

[0078] in, yes The estimated value of yes The estimated value of the virtual optimal controller is the neural network Approach Unknown dynamics in is an ideal estimate of the weights of a neural network; Approach Unknown dynamics in is an estimate of the ideal execution neural network weights, Φ J1 is the fuzzy basis function.

[0079] Given the fixed-time technique, the following is the fixed-time update rule for the evaluation neural network (critic) and the execution neural network (actor) weight design:

[0080]

[0081] Among them, γ c1 is the design parameter for judging the neural network, γ a1 and are the design parameters for executing the neural network.

[0082] Optimal Practical Controller

[0083] Construct the optimal performance index function by 2 ) to minimize the cost value, thereby obtaining the optimal actual controller u * :

[0084]

[0085] Among them, u is the actual controller, u * is the optimal actual controller, is the cost function.

[0086] The corresponding Hamilton Jacobi Bellman (HJB) equation is as follows:

[0087]

[0088] Similar to the optimal virtual controller, the approximate optimal actual controller is obtained by using the reinforcement learning algorithm

[0089]

[0090] in, is u * The estimated value of the neural network For approximation Unknown dynamics in represents the estimated value of the ideal weight of the executed neural network, Φ J2 is the fuzzy basis function.

[0091] Step 4: Based on the gradient descent method and the stability judgment method of the Lyapunov function, design the adaptive update rate of the execution-judgment neural network weights:

[0092]

[0093] Among them, γ c1 and γ c2 is the design parameter for judging the neural network, γ a1 and γ a2 are the design parameters for executing the neural network.

[0094] Neural network weights represent the connection strengths between different neurons in a neural network, and these connection strengths are continuously optimized through the training process to approximate the optimal control strategy. In a single-link robotic arm system, the neural network can be used to predict the motion state of the robotic arm or output control signals, and the optimization of the weights determines the accuracy of this prediction or control. In the optimal framework, the neural network weights, virtual controllers, and optimal actual controllers work together to achieve high-performance control tasks. Neural networks approximate optimal control strategies by learning and optimizing weights; virtual controllers provide a safe testing environment to verify the effectiveness of these strategies; and ultimately, optimal actual controllers apply these strategies to actual systems to achieve precise and stable control. In the optimal framework of a single-link robotic arm system, the neural network weights, virtual controllers, and optimal actual controllers together constitute an important component of achieving high-performance control tasks, and approximate the optimal control strategy through continuous optimization and iteration.

[0095] This embodiment uses the Lyapunov function to evaluate the stability of the single-link robotic arm system. Within the framework of the Lyapunov function, the convergence of the system is determined by analyzing the time derivative of the function. If the time derivative of the function tends to zero within a fixed time, it means that the system has reached a stable state within this period of time. Therefore, by designing the corresponding controller and neural network weights, the time derivative of the Lyapunov function can converge within a fixed time. That is to say, the fixed time technology can be embodied in this solution through the design of the controller and the neural network weights.

[0096] In order to prove the feasibility, effectiveness and correctness of this embodiment, this embodiment conducts the following simulation experiments:

[0097] In this simulation experiment, a fixed-time optimal controller based on ESO and reinforcement learning is designed for a single-link robotic arm system with external disturbances to achieve optimal tracking control of the single-link robotic arm.

[0098] In the optimal controller design process, the single-link robot system model parameters are set as follows:

[0099] Judging Neural Networks and and execute neural network and They all contain 6 neurons, the center points of the neural network are evenly distributed on [-3,3], and the width is: The initial weights of the neural network are:

[0100] The relevant parameters of the designed extended state observer ESO are: 1 =10, b 2 =160, b 3 =800;

[0101]

[0102] In order to achieve precise control of the dynamic behavior of the single-link robotic arm system, the parameters related to the optimal virtual controller and the optimal actual controller are designed as follows: K 11 =1,K 12 =20,K 21 =1,K 12 =10, thereby optimizing the performance of the controller and improving the response speed of the single-link robotic arm system, thus obtaining the optimal controller design.

[0103] Combined with the attached drawings, the effectiveness of this example simulation is further illustrated:

[0104] The simulation results are as follows Figure 3-Figure 9 , Figure 3-Figure 5 The observation performance of ESO on the external disturbance and state of the single-link manipulator system is demonstrated, from which it can be seen that the designed ESO has a good observation effect on the unmeasured state. Figure 6 The trajectories of the internal dynamics estimation errors under the fixed-time technique and the infinite-time technique are depicted. It is obvious that the fixed-time strategy converges faster than the infinite-time strategy, which proves the effectiveness of the strategy proposed in this invention. Figure 7 The control input curve graph of the single-link manipulator system is shown; Figure 8 and Fig. 9The weight convergence curves of the executive neural network and the judging neural network are presented respectively. It can be seen from the figure that each weight in the executive neural network and the judging neural network eventually tends to be stable and converged. Therefore, the designed reinforcement learning neural network control scheme can effectively achieve accurate approximation of the designed optimal controller and has a good convergence effect; therefore, the simulation proves the effectiveness of the proposed control scheme.

[0105] In summary, all signals in the single-link robotic arm system are consistently and ultimately bounded, and the simulation results demonstrate the effectiveness of the proposed optimal control scheme.

[0106] This embodiment uses backstepping and reinforcement learning technology as the design framework, and designs an extended state observer for the fixed-time optimal tracking control problem of a single-link manipulator system with unknown disturbances. The observer can not only estimate the disturbances existing in the system, but also estimate the state information of the system, thereby realizing comprehensive observation of the system state and disturbances.

[0107] Subsequently, a reinforcement learning method with an Actor-Critic framework is adopted to construct an optimal fixed-time controller based on the fixed-time technique, in which the evaluator and actor algorithms can converge within a fixed time interval, and the fixed-time technique also ensures that the system control signal converges to the vicinity of the origin within a fixed time interval.

[0108] Finally, by combining optimal control techniques, it is ensured that all signals achieve fixed-time convergence, thus ensuring that the system output effectively tracks the desired trajectory.

[0109] Example 2

[0110] In one embodiment of the present disclosure, a fixed-time optimal control system for a single-link robotic arm is provided, comprising:

[0111] The model building module is configured to: obtain a dynamic model of the single-link robotic arm to be controlled, and transform the model into a state model by introducing state variables;

[0112] The observer establishment module is configured to: establish an extended state observer based on the state model and adopt an anti-disturbance control strategy to observe the disturbance state information of the single-link manipulator;

[0113] The controller design module is configured as follows: based on the state model and the extended state observer, the backstepping technique is combined with the reinforcement learning algorithm to design the optimal controller for the single-link robot arm and perform adaptive optimal backstepping control on the single-link robot arm.

[0114] Example 3

[0115] In one embodiment of the present disclosure, a computer program product is provided, including a computer program, and when the computer program is executed by a processor, the fixed-time optimal control method of a single-link robotic arm is implemented.

[0116] Example 4

[0117] In one embodiment of the present disclosure, a non-transitory computer-readable storage medium is provided, wherein the non-transitory computer-readable storage medium is used to store computer instructions. When the computer instructions are executed by a processor, the fixed-time optimal control method of a single-link robotic arm is implemented.

[0118] Example 5

[0119] In one embodiment of the present disclosure, an electronic device is provided, including: a processor, a memory, and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory, so that the electronic device executes the fixed-time optimal control method for a single-link robotic arm.

[0120] The present disclosure 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 disclosure. 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.

[0121] 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.

[0122] Although the above describes the specific implementation methods of the present disclosure in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present disclosure. Technical personnel in the relevant field should understand that on the basis of the technical solution of the present disclosure, various modifications or variations that can be made by those skilled in the art without creative work are still within the scope of protection of the present disclosure.

Claims

1. A fixed-time optimal control method for a single-link robotic arm, characterized in that: include: Obtain the dynamic model of the single-link robotic arm to be controlled, and transform the model into a state model by introducing state variables; According to the state model, an extended state observer is established by adopting the anti-disturbance control strategy to observe the disturbance state information of the single-link manipulator. Based on the state model and extended state observer, the backstepping technique is combined with the reinforcement learning algorithm to design the optimal controller for the single-link robot arm, and the single-link robot arm is adaptively controlled with optimal backstepping.

2. A fixed time optimal control method for a single-link robotic arm as claimed in claim 1, characterized in that: The kinetic model is expressed as follows: Among them, q and are the angular position and acceleration respectively, M is the mechanical inertia, m is the mass of the connecting rod, l is the length of the connecting rod, g is the gravitational acceleration, u(t) is the input torque, and d(t) is the unknown external disturbance.

3. The fixed time optimal control method of a single-link robot arm according to claim 1, characterized in that: The state model, by introducing the state variable x1=q, The dynamic model of the single-link manipulator is transformed into a state space equation, and the space equation is rewritten as:

4. The fixed time optimal control method of a single-link robot arm according to claim 1, characterized in that: The extended state observer is expressed as follows: in, are the estimates of states x1, x2, and x3 respectively, b1, b2, and b3 are the gains of the observer, and the estimation error is defined as Where x=(x1,x2,x3) T , 5. The fixed time optimal control method of a single-link robot arm according to claim 1, characterized in that: The design is for an optimal controller of a single-link robot manipulator, including a virtual optimal controller and an actual optimal controller; The virtual optimal controller and the actual optimal controller are designed based on the gradient descent method and the stability judgment method of the Lyapunov function stability theory to design an adaptive update law for the execution-judgment neural network weights; based on the radial basis neural network, an adaptive law is designed for unknown nonlinear terms to form an optimal controller.

6. A fixed time optimal control method for a single-link robot arm as claimed in claim 5, characterized in that: The virtual optimal controller The formula is: The actual optimal controller u * , expressed as: Among them, K 11 ,K 12 ,K 21 ,K 22 is a positive design parameter, To perform neural network weight estimation, Φ J1 ,Φ J2 is the basis function vector.

7. A fixed-time optimal control system for a single-link robotic arm, characterized in that: include: The model building module is configured to: obtain a dynamic model of the single-link robotic arm to be controlled, and transform the model into a state model by introducing state variables; The observer establishment module is configured to: establish an extended state observer based on the state model and adopt an anti-disturbance control strategy to observe the disturbance state information of the single-link manipulator; The controller design module is configured as follows: based on the state model and the extended state observer, the backstepping technique is combined with the reinforcement learning algorithm to design the optimal controller for the single-link robot arm and perform adaptive optimal backstepping control on the single-link robot arm.

8. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the fixed-time optimal control method for a single-link robotic arm described in any one of claims 1 to 6 is implemented.

9. A non-transitory computer-readable storage medium, characterized in that: The non-transitory computer-readable storage medium is used to store computer instructions, and when the computer instructions are executed by the processor, a fixed-time optimal control method for a single-link robotic arm as described in any one of claims 1-6 is implemented.

10. An electronic device, characterized in that: include: A processor, a memory and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory so that the electronic device executes a fixed-time optimal control method for a single-link robotic arm as described in any one of claims 1-6.

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