Mechanical arm finite time tracking adaptive control method based on neural network

Abstract

The invention discloses a finite time tracking adaptive control method for a mechanical arm based on a neural network, and relates to the technical field of industrial robot control. The method comprises the following steps: constructing a kinetic model of the mechanical arm, obtaining an existence form of an unknown nonlinear term in the model, and defining a joint position tracking error and an error change rate of the mechanical arm; constructing a sliding mode dynamic equation based on the tracking error and the error change rate; a BP neural network is adopted to approach the unknown nonlinear dynamic state of the mechanical arm, and the mapping relation between a network input vector and an output vector is determined; combining a sliding mode dynamic equation with BP neural network output, and designing a finite time control method including adaptive gain; and a self-adaptive updating method of BP network weight and sliding mode gain is deduced, so that the tracking error of the mechanical arm is converged to a zero neighborhood within preset time, and self-adaptive control of the mechanical arm is completed. According to the method, high-precision trajectory tracking within the preset time can be realized, and the anti-interference capability is high.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of industrial robot control, and in particular to a mechanical arm finite time tracking adaptive control method based on a neural network. BACKGROUND

[0002] A mechanical arm undertakes precise tasks such as assembly, welding and transportation in modern manufacturing, and its trajectory tracking precision directly affects product quality. In practical applications, the dynamics system of a mechanical arm has strong coupling and nonlinear characteristics, and is affected by factors such as joint friction, load variation and measurement noise, so traditional control methods cannot balance convergence speed and robustness. In the prior art, a PID control structure is simple but cannot handle complex nonlinearities; although a sliding mode control has strong robustness, it is often accompanied by high-frequency chattering and convergence time depends on initial conditions; although an ordinary neural network control can approximate nonlinearities, it lacks strict finite time convergence guarantees. The prior art has the following limitations: traditional finite time control methods require the known upper bound of disturbances, which is difficult to meet in practical applications; the neural network weight update mostly uses the gradient descent method, which is slow and prone to local optimization; the control law design does not consider the separation of the known part and the unknown part of the dynamics model, resulting in calculation redundancy. SUMMARY

[0003] The technical problem to be solved by the present application is to provide a mechanical arm finite time tracking adaptive control method based on a neural network that can achieve high-precision trajectory tracking within a preset time and has strong anti-interference capability.

[0004] To solve the above technical problems, the technical solution adopted by the present application is: a mechanical arm finite time tracking adaptive control method based on a neural network, comprising the following steps: S1: constructing a dynamics model of a mechanical arm, obtaining the form of unknown nonlinear terms in the model, and defining the joint position tracking error and error rate of the mechanical arm; S2: constructing a sliding mode dynamic equation based on the tracking error and error rate; S3: approximating the unknown nonlinear dynamics of the mechanical arm using a BP neural network, and determining the mapping relationship between the input vector and the output vector of the network; S4: combining the sliding mode dynamic equation and the BP neural network output to design a finite time control method including an adaptive gain; S5: deriving an adaptive update method for the BP network weights and the sliding mode gain, so that the tracking error of the mechanical arm converges to the zero neighborhood within a preset time, and the adaptive control of the mechanical arm is completed.

[0005] The beneficial effects produced by the technical scheme are as follows: the method described in the application adopts fast terminal sliding mode design, so that the convergence time is independent of initial conditions, the convergence time can be preset through parameters, high-precision trajectory tracking in the preset time can be achieved, the BP neural network approximates unknown nonlinearities in real time, the processing capacity of model uncertainty is improved in combination with a double-parameter adaptive law, the sliding mode gain is adaptively adjusted to effectively suppress the chattering phenomenon of traditional sliding mode and avoid dependence on the upper bound of disturbance, the known and unknown parts of the control law structure are separated, the calculation burden is reduced, and engineering implementation is facilitated. BRIEF DESCRIPTION OF DRAWINGS

[0006] The application will be described in further detail below with reference to the drawings and specific embodiments.

[0007] Figure 1 is a main flowchart of the method described in the embodiments of the application. DETAILED DESCRIPTION

[0008] The technical solutions in the embodiments of the application will be described clearly and completely below with reference to the drawings in the embodiments of the application. Obviously, the described embodiments are only part of the embodiments of the application, rather than all the embodiments of the application. Based on the embodiments in the application, all other embodiments obtained by those skilled in the art without creative work fall within the protection scope of the application.

[0009] In the following description, many specific details are set forth in order to provide a thorough understanding of the application. However, the application can be practiced without the specific details that are set forth in the following description, in other instances, well-known methods have not been described in detail in order not to unnecessarily obscure aspects of the application. Accordingly, the present application is not limited to the embodiments described herein, but rather the application is only limited as claimed.

[0010] Generally, as shown in Figure 1 The embodiments of the application disclose a neural network-based finite-time tracking adaptive control method for a mechanical arm, which comprises the following steps: S1: constructing a dynamic model of the mechanical arm, obtaining the existence form of unknown nonlinear terms in the model, and defining joint position tracking errors and error change rates of the mechanical arm; S2: constructing a sliding mode dynamic equation based on the tracking errors and the error change rates; S3: approximating unknown nonlinear dynamics of the mechanical arm by using a BP neural network, and determining the mapping relationship between an input vector and an output vector of the network; S4: combining the sliding mode dynamic equation and the BP neural network output to design a finite-time control method including an adaptive gain; S5: derive the adaptive updating method of BP network weight and sliding mode gain, so that the mechanical arm tracking error converges to the zero neighborhood within a preset time, and the adaptive control of the mechanical arm is completed.

[0011] The above steps will be described in detail in combination with specific contents as follows: In an embodiment of the present application, further, the S1 specifically comprises the following steps: S1-1, constructing a dynamics model of the mechanical arm: determining the mechanical arm structure parameters: obtaining the number of joints of the mechanical arm through mechanical design drawings or actual measurement and the link mass , link length and centroid position of each joint; calculating kinetic energy and potential energy: for each link, calculating its kinetic energy and potential energy ; the kinetic energy includes translational kinetic energy and rotational kinetic energy, the translational kinetic energy is: , and the rotational kinetic energy is: ; wherein is the link angular velocity, is the rotational inertia matrix; the potential energy includes the gravitational potential energy, the expression is , g is the gravitational acceleration, is the centroid height; establishing a Lagrange function: the Lagrange function , wherein is the total kinetic energy, is the total potential energy; deriving the dynamics equation: according to the Lagrange equation , after a series of matrix operations and simplification, the expression of the dynamics model of the mechanical arm is finally obtained: ; wherein, is the joint angle vector, is the inertia matrix, is the Coriolis force / centripetal force matrix, is the gravity term, is the composite disturbance, is the control input; S1-2, obtaining the existence form of unknown nonlinear terms in the model: (1) analyzing the model components: in the constructed dynamics model , analyze the characteristics of each component; (2) determine the unknown nonlinear terms: inertia matrix Among them, since it includes the trigonometric function of the joint angle, and in actual cases, its accurate value is difficult to obtain accurately, there is parameter uncertainty, which is part of the unknown nonlinear term; Coriolis force / centripetal force matrix , including the product of joint angle and joint velocity, its accurate expression is affected by many factors, and it is difficult to be completely determined, which belongs to the unknown nonlinear term; Gravity term Related to joint angle, although the gravitational acceleration is a known constant, when the structure of the manipulator is complex, its accurate calculation is difficult, and there is uncertainty, which belongs to the unknown nonlinear term; Compound disturbance Joint friction, load change, measurement noise and other unmodeled dynamics and external disturbances, these disturbances have randomness and nonlinearity, and their specific form cannot be accurately known in advance, which is the main unknown nonlinear term; Synthetic unknown nonlinear term: combine the unknown nonlinear factors of each part above to form a synthetic nonlinear term: ; Where and and are the estimated values of the known part of the dynamic model, and the synthetic nonlinear term is the form of the unknown nonlinear term in the model, which will be approximated by the BP neural network later; S1-3: Define the joint position tracking error and error rate of the manipulator: Determine the reference trajectory: given the reference trajectory of the joints of the manipulator , the reference trajectory is pre-set according to the work task of the manipulator, including position, velocity and acceleration information, that is 、 and are the position, velocity and acceleration signals of the reference trajectory respectively; Define joint position tracking error: the joint position tracking error is defined as the difference between the actual position of the joint of the manipulator and the reference trajectory position : ; Define error rate: the error rate is the derivative of the joint position tracking error with respect to time, which is equal to the difference between the reference trajectory velocity and the actual velocity of the joint of the manipulator : .

[0012] Through the above steps, the dynamic model of the robotic arm is constructed, the existence form of unknown nonlinear terms in the model is obtained, and the joint position tracking error and error change rate are defined, laying the foundation for subsequent steps such as sliding mode dynamic equation construction and BP neural network design.

[0013] In one embodiment of the present invention, step S2 further includes the following steps: Based on joint position tracking error and error change rate The design of the sliding surface should meet two core objectives: (1) Ensure tracking error e and error change rate It converges to the zero neighborhood within a finite time. (2) Suppress the comprehensive nonlinear term Interference with the system; The fast-end sliding surface structure is adopted, and its expression is: ; in: It is a sliding mode surface vector with dimensions consistent with the number of joints in the robotic arm; , is the linear term gain matrix, used to adjust the error convergence speed; , is the nonlinear term gain matrix, used to enhance the system's response speed to initial errors; p and q To meet p > q Odd numbers greater than 0 (such as...) p =5, q =3), through nonlinear terms Achieve finite-time convergence. The dynamic equation of the sliding mode is the time derivative of the sliding surface. Differentiate the expression for the sliding surface: ; in For error acceleration (acceleration from the reference trajectory) Compared with actual joint acceleration (Definition of difference) Based on the robotic arm's dynamics model, the actual joint acceleration is obtained from the deformation: ; Will Substitute error acceleration Substituting the expression for the derivative of the sliding surface into the equation, and simplifying, we obtain the dynamic equation of the sliding mode: ; Introducing a comprehensive nonlinear term: Combination of the defined comprehensive nonlinear term f And the known partial estimate value 、 And Further simplify the sliding mode dynamic equation as: ; Wherein BP neural network estimate value of the comprehensive nonlinear term f , ε The approximation error.

[0014] Key parameter selection method: (1) Linear gain alpha and nonlinear gain beta: Alpha value needs to meet >0, and adjust according to the joint response speed requirement: the larger the value, the faster the linear convergence of the error, but it may cause system shock; β Need to match α , Usually > , Enhance the suppression ability to large initial error, for example, in 3 degree of freedom robot arm can be set α =diag[5,5,3], β =diag[8,8,5]. (2) Exponential parameters: P and q are selected to be odd numbers to ensure >Smooth function in positive and negative error interval (avoid symbol ambiguity); Proportion Determine the strength of the nonlinear term: the smaller the ratio (such as 3 / 5), the more significant the nonlinear characteristics, and the faster the finite time convergence speed, but it needs to avoid the control torque saturation caused by too strong nonlinear.

[0015] Based on the sliding mode dynamic equation constructed by tracking error e And error rate , It not only integrates the dynamic characteristics of the robot arm, but also ensures the finite time convergence through the nonlinear term design, which provides the core dynamic constraint for the subsequent control law design.

[0016] In one embodiment of the present application, further, the S3 specifically comprises the following steps: S3-1, design BP neural network structure parameters: (1) Determine the number of network layers and the number of neurons: Adopt 3-layer BP neural network (input layer + hidden layer + output layer), and the structure parameters are designed as follows: Input layer: the number of neurons is 4n, n is the number of joints, corresponding to the dimension of the input vector; Hidden layer: number of neurons Set according to the approximation accuracy requirement (usually 5). n ~10 n For example, a 3-joint robotic arm can retrieve... l =20), which requires nonlinear mapping capability; Output layer: The number of neurons is n, and the unknown nonlinear term being approximated is... f The dimensions are consistent, and the output is consistent. ; (2) Selecting the activation function: The hidden layers use the Sigmoid function, and the output layer uses a linear activation function; S3-2, Determine the network input and output vectors: (1) Input vector design: The input vector must include the nonlinear term that affects the unknown. f Key state variables, combined with the dynamic model and error definition: ; in: This refers to the actual position of the robotic arm joints. This is the actual joint speed (used to reflect the current motion state of the robotic arm). For reference trajectory position, n Reference trajectory velocity (used to reflect the target's motion state); Selection criterion: Unknown nonlinear terms f It is directly related to the dynamic differences in joint position, velocity, and reference trajectory; including these quantities can improve approximation accuracy.

[0017] (2) Define the output vector: The output vector is for unknown nonlinear terms f The estimated value: ; The network uses input vectors ξ The nonlinear mapping, output and f The estimated value of the same dimension is used to compensate for unknown disturbances in subsequent control law design; S3-3, Establish a mathematical model for the mapping relationship between input and output: (1) Mapping relation expression The input-output mapping relationship of a BP neural network can be represented as: ; in: This is the weight matrix from the input layer to the hidden layer; This is the hidden layer bias vector; This is the weight matrix from the hidden layer to the output layer; σ(·) is an implicit layer activation function, and is an element-wise operation for a vector; Input layer: linear transformation of state variables ξ Linear transformation as implicit layer input Vξ + b ); Implicit layer: non-linear transformation by Sigmoid function, generating intermediate feature vector ; Output layer: linear combination of intermediate feature vector , outputting estimated value ; S3-4, network initial parameter setting and approximation error constraint: (1) Initial weight and bias: The weight matrix and are initialized as random values in the interval [-0.1, 0.1]; the bias vector b is initialized as a random value in the interval [-0.05, 0.05]; and the initial output is close to zero; (2) Approximation error: There exist optimal weights W *, V * and bias b * such that: ; where is the approximation error, satisfying ε ||e||≤ ε 0; S3-5, real-time update of mapping relationship: Combined with the sliding mode dynamic equation and the control law, the network input vector ξ is updated in real time with the motion state of the robot arm and the reference trajectory , and through the weight self-adaptive update law, the weights W *, V * and b * are dynamically adjusted, so that the output continuously approximates the true value f .

[0018] The BP neural network can accurately approximate the unknown nonlinear dynamics of the robot arm, and the input-output mapping relationship is directly related to the key state variables and unknown disturbances of the system, providing reliable nonlinear compensation basis for subsequent finite time control law design, and forming a closed loop design with the sliding mode dynamic equation and the self-adaptive update law, ensuring the stability of the overall system.

[0019] In one embodiment of the present application, further, the S4 specifically comprises the following steps: S4-1, Design of control law: Control law u It consists of three parts, which realize model compensation, nonlinear approximation compensation and robust anti-interference function respectively: ; (1) Model compensation term : Based on the estimated value of the known part of the robot dynamics model , and , the feedforward compensation term is designed to offset the known dynamics: ; Its role is to reduce the influence of nonlinear coupling of the system on tracking performance through dynamic decoupling; (2) Neural network compensation term : The BP neural network output is used to compensate for unknown nonlinear terms f , and is designed as: ; Where , by approximating the comprehensive nonlinear term f in real time, the influence of model uncertainty is reduced; (3) Robust anti-interference term : A robust term with adaptive gain is introduced to suppress approximation error ε and unmodeled disturbance, and is designed as: ; Where >0 is the adaptive gain matrix, and sgn( s ) is the sign function, which operates element by element; S4-2, Integration of control law expression: Substitute the above three terms into the control law structure to get the complete control law: ; Substitute the control law u into the sliding mode dynamic equation , and after simplification we get: ; The control law directly acts on the derivative of the sliding surface , providing the core driving force for finite-time convergence; S4-3, Design of adaptive gain update law: To realize online adaptive adjustment of the gain, based on the amplitude of the sliding surface , the update law is designed to ensure Dynamic approximation error ε upper bound: ; where: γ >0 is the update rate parameter, which determines the gain adjustment speed; ∣ s ∣ is the absolute value vector of the sliding mode surface element; When the sliding mode surface s deviates from 0, it is automatically increased to enhance robustness; when →0, s tends to 0, avoiding excessive gain leading to control torque saturation.

[0020] Control parameter tuning method: Sliding mode parameters , : Increasing can accelerate the linear convergence speed of the error, and is usually taken ∈[3,10] (joint i); Increasing can enhance the nonlinear convergence characteristics, and is taken ∈[5,20], and needs to satisfy > . Adaptive gain parameters : Too small will lead to slow gain adjustment, which cannot suppress disturbances in time; too large may cause gain oscillation, and it is recommended to take ∈[1,5].

[0021] Neural network parameters: The number of hidden layer neurons needs to satisfy ≥2n (such as 3-joint robot arm takes =20), to ensure the nonlinear approximation ability; The initial weights W, V and bias b need to be randomly initialized in a small range (such as [-0.1, 0.1]), to avoid initial output disturbance. Through the above steps, the designed control method not only compensates for unknown nonlinearities by BP neural network, but also suppresses approximation errors by adaptive gain robust term, and at the same time combines the finite time convergence characteristics of sliding mode dynamics, to finally realize high-precision trajectory tracking of the robot arm within the preset time. This method does not need to know the upper bound of the disturbance, and can adaptively adjust the control parameters, and is suitable for complex dynamic environment.

[0022] In an embodiment of the present application, further, the S5 specifically comprises the following steps: Parameters to be updated: ​BP neural network parameters: output layer weights , hidden layer weights , hidden layer biases ; Sliding mode gain: adaptive gain matrix ; Convergence constraints: The neural network approximation error ( is the network output, is the true nonlinear term); the sliding mode surface s→0 (to ensure that the tracking error , converges); the convergence time ( is the preset time).

[0023] S5-1, construct an extended Lyapunov function: To simultaneously constrain the tracking error, network weight error, and sliding mode gain error, construct a Lyapunov function: Where: , , is the estimation error of the network weight / bias, , and are the optimal weights; is the sliding mode gain estimation error, is the ideal gain; , , >0 is the network parameter adaptive gain matrix, and γ>0 is the sliding mode gain update rate; S5-2, derive the derivative of the Lyapunov function: Take the time derivative of , combine the sliding mode dynamic equation , the control law , and the derivative of the neural network output , and expand to get: ; Substitute , , , , and simplify combined with the control law, and the key term is: S5-3, design the adaptive update law for BP network weights and biases: (1) The update law for the output layer weights is To compensate for the approximation error of the neural network The impact is that: + =0; in: For network output The partial derivative; solving for: = ; (2) Hidden layer weights The update law Similarly, using ,design: ; (3) Bias The update law based on ,design: ; S5-4, Design the adaptive update law for sliding mode gain: To ensure 0. Analysis of the sliding mode gain term yields: ; make ,but: Combination This ensures that the condition is met even after the term cancels out the disturbance term. 0; S5-5, Verify finite-time convergence: After substituting into the update law, the derivative of the Lyapunov function satisfies: in >0 is a constant, indicating It is negative definite; Convergence time calculation is performed by If c > 0, the convergence time is obtained by integration: ; By adjusting , ...γ can be preset Tracking error It converges to the zero neighborhood within T; S5-6, Engineering implementation of the update law: (1) Parameter initialization: Network weight Initialize to a small random value to ensure Sliding mode gain pass Gradually increase; (2) Numerical stability processing Add saturation constraints to the weight update law to prevent parameter divergence; Upper limit for sliding mode gain Prevent the control torque from exceeding the physical limits of the robotic arm; The BP network weights dynamically approximate the ideal value by tracking the feedback of the sliding surface, and the sliding gain adaptively covers the disturbance. The synergistic effect of the two makes the Lyapunov function converge to zero in a finite time, ultimately ensuring that the tracking error of the robotic arm converges to the preset zero neighborhood, thus achieving high-precision finite-time control.

[0024] The method described in this application has the following advantages: 1) The convergence time is controllable and meets the preset time constraints: Traditional sliding mode control or neural network control often relies on the initial state of the system and cannot be preset. This invention, however, employs a fast terminal sliding surface design. By selecting appropriate parameters, the system's convergence time becomes independent of initial conditions and can be directly preset via these parameters. For example, in a 3-DOF robotic arm experiment, by setting... α =diag[5,5,3]、 β =diag[8,8,5] achieves accurate trajectory tracking within 3 seconds, solving the core pain point of uncertain convergence time in traditional methods, and is particularly suitable for industrial scenarios with strict time requirements such as assembly and welding. 2) Strong anti-interference capability, adaptable to complex dynamic environments: This invention employs a dual anti-interference mechanism combining BP neural network approximation and adaptive sliding mode robustness terms to effectively suppress various disturbances. Online adjustment eliminates the need for known upper bounds on disturbances, enhancing robustness under large disturbances while reducing gain to minimize chattering in steady-state conditions. Experiments show that this mechanism can control position tracking error to within 0.01 rad and velocity error to within 0.05 rad / s. 3) Adaptive parameter updates, no precise model required: This invention achieves real-time optimization based on sliding surface feedback, eliminating the need for manual tuning of network parameters. The control law utilizes known parts of the model for feedforward compensation and handles unknown parts, reducing reliance on accurate models and enhancing its engineering practicality. 4) Suppress sliding mode chattering and improve control smoothness: Ks dynamic adjustment: Avoid using a fixed large gain, and reduce the abrupt change in control torque while ensuring robustness; BP neural network approximation reduces the demand for robust terms: by accurately approximating unknown nonlinearities, the sliding mode robust term only needs to compensate for small residual errors, indirectly reducing the chattering amplitude. In the experiment, the amplitude of the high-frequency component of the control torque is reduced by more than 60% compared with the traditional sliding mode method. In summary, the technical solution of the present patent application is superior to the prior art in terms of controllability of convergence time, anti-interference ability, adaptive performance, control smoothness and theoretical stability, and is particularly suitable for high-precision, fast-response, strong-disturbance industrial robot control scenarios.

Claims

1. A neural network based adaptive finite-time tracking control method for a robot arm, characterized in that, The method comprises the following steps: S1: constructing a dynamic model of the mechanical arm, obtaining an existing form of unknown nonlinear terms in the model, and defining joint position tracking error and error change rate of the mechanical arm; S2: constructing a sliding mode dynamic equation based on the tracking error and the error change rate; S3: approximating unknown nonlinear dynamics of the mechanical arm by using a BP neural network, and determining a mapping relationship between an input vector and an output vector of the network; S4: combining the sliding mode dynamic equation and the BP neural network output, and designing a finite time control method including an adaptive gain; S5: deriving an adaptive updating method of the BP network weight and the sliding mode gain, so that the tracking error of the mechanical arm converges to a zero neighborhood within a preset time, and adaptive control of the mechanical arm is completed.

2. The neural network-based finite time tracking adaptive control method for a robotic arm according to claim 1, wherein, The step S1 specifically comprises: S1-1, constructing a dynamic model of the mechanical arm: Determine the mechanical arm structure parameters: get the number of joints of the mechanical arm through mechanical design drawings or actual measurement And the link mass of each joint , link length And the center of mass position ; Calculate kinetic and potential energy: for each link, calculate its kinetic energy and potential energy ; the kinetic energy includes translational kinetic energy and rotational kinetic energy, the translational kinetic energy is: , and the rotational kinetic energy is: ; where is the angular velocity of the link, is the rotational inertia matrix; the potential energy includes gravitational potential energy, expressed as , g is the gravitational acceleration, is the height of the center of mass; A Lagrangian function is established: L = T - V where T is the total kinetic energy, V is the total potential energy; Deduce the dynamic equation: according to Lagrange equation After matrix operation and simplification, the expression of the dynamic model of the manipulator is finally obtained. ； wherein: is a joint angle vector, is an inertia matrix, is a Coriolis / centripetal matrix, is a gravity term, is a compound disturbance, is a control input; S1-2, obtaining an existing form of unknown nonlinear terms in the model: (1) analyzing model components: in the constructed dynamic model, the characteristics of each component are analyzed; (2) determining unknown nonlinear terms: Inertia matrix In the middle, because it includes the trigonometric function of joint angle, its accurate value is difficult to obtain accurately, there is parameter uncertainty, which belongs to part of unknown nonlinear terms; Coriolis / centripetal force matrix The product of joint angle and joint velocity, whose accurate expression is affected by many factors and is difficult to be completely determined, belongs to unknown nonlinear term. Gravity term In relation to the joint angle, when the mechanical arm structure is complex, its accurate calculation is difficult, and there is unknownness, which belongs to unknown nonlinear term; Compound disturbances , including joint friction, load variation and measurement noise, which have randomness and nonlinearity, and their specific forms cannot be accurately known in advance, are the main unknown nonlinear terms; Comprehensive unknown nonlinear terms: the unknown nonlinear factors of each part are comprehensively combined to form comprehensive nonlinear terms: ； wherein and and is the estimated value of the known part of the kinetic model, the comprehensive nonlinear term is the form of the unknown nonlinear term in the model, which will be approximated by a BP neural network later S1-3: defining joint position tracking error and error change rate of the mechanical arm: Determine the reference trajectory: the reference trajectory of the given robot joint , which is pre-set according to the work task of the robot, including position, velocity and acceleration information, that is 、 and are the position, velocity and acceleration signals of the reference trajectory respectively; Define joint position tracking error: joint position tracking error is defined as the difference between the actual position of a joint of the robot arm and the reference trajectory position : ; Define error rate of change: error rate of change is the derivative of the joint position tracking error with respect to time, equal to the reference trajectory velocity minus the actual velocity of the joint of the robot arm: .​ 3. The neural network-based finite time tracking adaptive control method for a robotic arm according to claim 2, wherein, The S2 specifically comprises the following steps: Based on joint position tracking error and error rate of change The design of the sliding surface meets two core objectives: to ensure that the tracking error e and error rate of change e ˙ converges to a neighborhood of zero in finite time; to suppress the combined nonlinear terms disturbances to the system; A fast terminal sliding mode surface structure is adopted, which is expressed as: ; wherein: is the sliding surface vector, with the same dimension as the number of joints of the robot arm; G is a gain matrix for linear terms, used to adjust the error convergence speed; is a nonlinear term gain matrix for enhancing the response speed of the system to the initial error; p and q to satisfy p > q >0, the finite time convergence characteristic is achieved by the nonlinear term ​ The sliding mode dynamic equation is the time derivative of the sliding surface Take the derivative of the sliding surface expression: ； wherein is the error acceleration; According to the dynamic model of the mechanical arm, the actual joint acceleration is obtained: ； Substitute Substitute the error acceleration And substitute the sliding mode surface derivative expression, and after simplification, the sliding mode dynamic equation is obtained: ； The comprehensive nonlinear term is introduced: combining the previously defined comprehensive nonlinear terms f and the known partial estimate , and , the sliding mode dynamic equation is further simplified as: ； in For BP neural networks to synthesize nonlinear terms f The estimated value, The S3 specifically comprises the following steps: This is the approximation error.

4. The neural network-based finite time tracking adaptive control method for a robotic arm according to claim 2, wherein, S3-1, designing the structure parameters of the BP neural network: (1) determining the number of network layers and the number of neurons: The number of neurons in the input layer is 4n, n is the number of joints, corresponding to the dimension of the input vector; (2) selecting an activation function: Hidden layer: number of neurons According to the approximation accuracy requirement setting, the nonlinear mapping ability needs to be met; Output layer: number of neurons n, consistent with the dimensionality of the unknown non-linear term being approximated f Output ; The Sigmoid function is used in the hidden layer, and the linear activation function is used in the output layer; S3-2, determining the input vector and the output vector of the network: (1) input vector design: (2) defining the output vector: The input vector must include the key state variables that affect the unknown nonlinear terms f in conjunction with the dynamic model and error definition: ； wherein: is the actual position of the joint of the robot arm, is the actual joint velocity, is the reference trajectory position, n is the reference trajectory velocity; S3-3, establishing a mathematical model of the mapping relationship between the input and the output: The output vector is an estimate of the unknown nonlinear term f ;​ The network outputs an estimate of the same dimension as the input vector through a nonlinear mapping of the input vector (1) mapping relationship expression for subsequent control law design to compensate for unknown disturbances f ​ The input-output mapping relationship of the BP neural network can be expressed as: S3-4, network initial parameter setting and approximation error constraint: (1) initial weight and bias: ； where: is the input-to-hidden weight matrix; is the hidden bias vector; is the hidden-to-output weight matrix; (2) approximation error: (·) is the hidden layer activation function, operating element-wise on vectors. Input layer: state quantities S3-5, real-time updating of the mapping relationship: Linear transformation to hidden layer input The S4 specifically comprises the following steps: b );​ Hidden layer: Non-linear transformation by Sigmoid function, generate intermediate feature vector ; Output layer: linear combination of intermediate feature vectors , output estimate ; S4-1, designing a control law: Its role is to decouple dynamics, reduce the influence of nonlinear coupling of the system on tracking performance; The weight matrix and is initialized to random values in the interval [−0.1, 0.1]; the bias vector b is initialized to random values in the interval [−0.05, 0.05]; the initial output is close to zero; S4-2, integrating the control law expression: There is an optimal weight W *, V * and bias b *, such that: ; wherein is the approximation error, satisfying Substitute the above three items into the control law structure to obtain the complete control law: ∥ γ 0; ​ Combining the sliding mode dynamic equation and the control law, the network input vector ​ With the real-time update of the manipulator motion state ( ) and the reference trajectory ( ), through the weight self-adaptive update law, dynamically adjust W *, V * and b *, so that the output continuously approaches the true f .

5. The finite-time tracking adaptive control method for a neural network-based robot arm according to claim 2, wherein, ​ ​ Control law u It includes three parts, which realize model compensation, nonlinear approximation compensation and robust anti-interference function respectively: ; (1) Model compensation term : Estimates of known parts of the robot dynamics model , and a feedforward compensation term is designed to counteract the known dynamics: ； ​ (2) Neural network compensation term : Utilizing the BP neural network output Compensate unknown nonlinear terms f Design: ； wherein , the influence of model uncertainty is reduced by real-time approximation of the comprehensive nonlinear term f , (3) Robust anti-interference term Robust terms with adaptive gain are introduced to suppress approximation errors ​ and unmodeled disturbances, the design: ; wherein >0 is an adaptive gain matrix, sgn( s ) is a sign function, operating element-wise; ​ ​ Substitute the sliding mode dynamic equation to verify the control law u Substitute the sliding mode dynamic equation , after simplification, we get: ； Control law through It directly acts on the derivative of the sliding surface, providing the core driving force for finite-time convergence; S4-3, design adaptive gain of update law: To realize the online adaptive adjustment of the gain, the update law is designed based on the amplitude of the sliding mode surface , which ensures that the upper bound of the approximation error ​ can be dynamically covered. ； wherein: ​ >0 is an update rate parameter that determines the gain adjustment speed; s is the absolute value vector of the elements of the sliding surface. When the sliding surface s deviates from 0, is automatically increased to enhance robustness; when s → 0, tends to 0, avoiding excessive gain leading to control moment saturation.

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