Super-exponential convergence zero neural network method for position tracking of redundant mechanical arm

Abstract

The invention discloses a super-exponential convergence zero neural network method for position tracking of a redundant mechanical arm, and belongs to the technical field of robot motion control and intelligent computing. A zero neural network fusing a time-varying dynamic attenuation coefficient and a power-sigmoid type nonlinear activation function is constructed, and by designing an error evolution kinetic equation based on a Jacobian matrix Moore-Penrose pseudo-inverse, a network state is driven to rapidly converge to a time-varying pseudo-inverse truth value in a super-exponential form. And the output of the network serves as a pseudo-inverse online estimated value, a joint speed instruction can be generated in real time in combination with tail end position tracking error feedback, and high-precision, rapid and stable tracking of the tail end of the redundant mechanical arm to an expected trajectory is achieved. The method only depends on a kinematic model, does not need complex dynamic modeling or offline training, has the advantages of being simple in structure, small in calculation amount and high in real-time performance, and is suitable for tracking control scenes with different structural parameters and various complex trajectories.

Description

Technical Field

[0001] This invention relates to the field of robot motion control and intelligent computing technology, specifically to a redundant robotic arm end-effector position tracking control method based on a nullable neural network, and more particularly to a super-exponential convergent nullable neural network control method that integrates time-varying dynamic decay coefficients and nonlinear activation functions. Background Technology

[0002] Redundant robotic arms, with their multi-degree-of-freedom structure, possess high flexibility and adaptability in complex tasks. However, due to the redundancy of degrees of freedom, their inverse kinematics solution is not unique, which can easily lead to joint angle drift, causing the end effector position to deviate from the expected trajectory, increasing the risk of self-collision and affecting control stability. Therefore, high-precision, fast-convergence end effector position tracking control has become a key challenge in the control of redundant robotic arms.

[0003] At the differential kinematics level, end-effector position tracking requires solving the Moore-Penrose pseudoinverse of the Jacobian matrix to map the desired end-effector velocity to joint velocity. The Jacobian matrix exhibits nonlinear time-varying characteristics, and its pseudoinverse needs to be solved online in real time. Traditional numerical methods (such as singular value decomposition) are computationally burdensome under time-varying parameters and struggle to meet real-time requirements; therefore, an efficient and stable online solution strategy is urgently needed.

[0004] Nullable neural networks are widely used for online solving of time-varying matrices due to their advantages such as strong parallel computing capabilities and simple structure. Existing methods mostly use constant decay coefficients and fixed activation functions, which make it difficult to balance convergence speed and steady-state accuracy in strongly nonlinear and high-frequency time-varying scenarios. Moreover, the parameters are sensitive to changes in operating conditions, which limits their practical application in redundant robotic arm control.

[0005] Therefore, it is necessary to synergistically improve the decay coefficient and activation function in the null neural network and design a control method that can adaptively adjust and has super-exponential convergence characteristics to achieve fast, stable, and high-precision online solution of the pseudo-inverse of the time-varying Jacobian matrix, thereby improving the real-time performance, convergence speed, and robustness of the position tracking of the redundant robotic arm end effector. Summary of the Invention

[0006] The purpose of this invention is to address the aforementioned problems by providing a super-exponential convergent nullable neural network method for redundant robotic arm position tracking. This method constructs a nullable neural network that integrates a time-varying dynamic decay coefficient and a power-sigmoid nonlinear activation function. Based on the Moore-Penrose pseudo-inverse of the Jacobian matrix, a time-varying error matrix is ​​built, and an error evolution dynamic equation is designed accordingly. This allows the system error to decay rapidly in a super-exponential manner during network evolution, thereby achieving efficient and stable online solution of the Moore-Penrose pseudo-inverse of the time-varying Jacobian matrix.

[0007] The technical solution of the present invention is as follows: A super-exponential convergent nullable neural network method for redundant robotic arm position tracking includes the following steps: Based on the geometry of the redundant robotic arm, a velocity mapping relationship between the change of its end-effector position and the joint motion is established, i.e., a differential kinematic model. Based on the desired end-point reference trajectory and the actual end-point position, calculate the position tracking error, and construct the desired end-point velocity command based on this error; Construct and run a super-exponential convergent nullable neural network for online estimation of the generalized inverse of the time-varying Jacobian matrix in a differential kinematics model; The output of the super-exponential convergent null neural network is used as an online estimate of the generalized inverse. Combined with the desired end-effector velocity command, the joint angular velocity control command is calculated. The robotic arm joints are driven to move according to the joint angular velocity control command, forming a closed-loop control, so that the end position tracks the reference trajectory.

[0008] Furthermore, the differential kinematic model is specifically expressed as follows: , in, End position vector rate of change, The joint angular velocity vector. Joint angle vector, This represents the time-varying Jacobian matrix determined by the structural parameters and joint angles of the robotic arm.

[0009] Furthermore, the calculation of the position tracking error specifically includes: Set the end-position tracking error to ,in For reference trajectory, It is a position vector; The desired end-velocity command, constructed based on the error, is as follows: ,in, The rate of change of the reference trajectory, This is the error feedback gain.

[0010] Furthermore, the construction and operation of a super-exponential convergent nullification neural network specifically includes: For online solving of time-varying Jacobian matrices The right Moore-Penrose pseudoinverse, the time-varying error matrix of the network is set as follows: ,in, To nullify the state matrix of the neural network, Time-varying Jacobian matrix The right Moore-Penrose pseudo-inverse; The evolutionary dynamic equations for designing convergent driving matrices are as follows: ,in, It is the time-varying dynamic decay coefficient. It is a power-sigmoid type nonlinear activation function that acts on the error matrix; Substituting the time derivative of the error matrix into the evolutionary dynamics equation, we obtain the state update law of the null neural network: .

[0011] Furthermore, the time-varying dynamic attenuation coefficient Designed to grow exponentially over time: ,in, This is the initial amplitude of the attenuation coefficient, used to adjust the convergence strength of the error system in the initial stage; This is the exponential growth rate parameter, used to control the rate at which the decay coefficient increases over time.

[0012] Furthermore, the power-sigmoid type nonlinear activation function The formula for applying an independent nonlinear mapping to each element of the matrix is ​​as follows: , Among them, activation function For the error matrix Each element acts one by one; It is an odd-order power exponent and This is used to enhance the nonlinear suppression capability in regions with large errors; This is the gain coefficient for the hyperbolic sine term, used to improve the convergence driving force in the large error region; This is an adjustment parameter for the hyperbolic tangent function, used to improve smoothness and steady-state accuracy when the error approaches zero.

[0013] Furthermore, the calculated joint angular velocity control command specifically includes: The real-time state matrix of the null neural network As an estimate of the pseudo-inverse of the Jacobian matrix, the joint angular velocity control command is calculated using the following formula: .

[0014] Furthermore, joint angular velocity control commands are obtained. Subsequently, to ensure that the joint velocities meet the physical execution constraints of the robotic arm, the joint velocities can be solved... Based on the minimum norm solution, it is saturated to satisfy: ,in, and These represent the minimum and maximum values ​​of the joint velocity, respectively.

[0015] Furthermore, the initial amplitude of the attenuation coefficient exponential growth rate upper limit of saturation value .

[0016] Furthermore, the parameters of the power-sigmoid type nonlinear activation function are preferably set as follows: polynomial order Hyperbolic sine term gain Hyperbolic tangent function adjustment parameters .

[0017] Compared with existing technologies, the advantages of this invention are: 1. This invention constructs a nullified neural network that integrates time-varying dynamic decay coefficients and power-sigmoid type nonlinear activation functions, and combines the Moore-Penrose pseudo-inverse of the Jacobian matrix to design the error evolution dynamic equation, so that the system error can decay rapidly in a super-exponential form during the network evolution process, thereby achieving efficient and stable online solution of the time-varying Jacobian matrix Moore-Penrose pseudo-inverse.

[0018] 2. Based on the constructed super-exponential convergent null neural network, this invention proposes a redundant robotic arm end-effector position tracking control method, which can achieve fast and stable tracking of the desired reference trajectory by the end-effector while ensuring system stability, and significantly improves the convergence speed and tracking accuracy of redundant robotic arm position tracking control.

[0019] 3. The redundant robotic arm end-effector position tracking control method described in this invention only relies on the kinematic model and real-time position information of the robotic arm. It does not require complex dynamic modeling of the system or offline training of neural network parameters, thereby reducing the dependence on system modeling accuracy and computing resources. It has the advantages of simple implementation structure, low computational load, and good real-time performance, and is easy to deploy and apply in actual engineering systems.

[0020] 4. The present invention has good adaptability to the linkage structure parameters, initial joint state and reference trajectory form of redundant robotic arms. It can achieve stable end position tracking control under different structural parameter configurations and various trajectory conditions, thereby improving the versatility and engineering application value of the control method. Attached Figure Description

[0021] Figure 1 This is a flowchart of a super-exponential convergent null neural network control method for position tracking of a redundant robotic arm end effector.

[0022] Figure 2 Time-varying dynamic decay coefficient under different parameter conditions The curve showing the change.

[0023] Figure 3 Power-sigmoid type nonlinear activation functions under different parameter conditions function curve Figure 4 This is a redundant robotic arm end-effector position tracking control structure based on a super-exponential convergent nullified neural network.

[0024] Figure 5 This diagram illustrates the evolution of the actual trajectory and reference trajectory of the end effector of a redundant robotic arm based on a super-exponential convergent null neural network.

[0025] Figure 6 A comparison of the overall tracking performance between the actual trajectory and the reference trajectory of the redundant robotic arm end effector.

[0026] Figure 7 The curve showing the change in position tracking error of the redundant robotic arm end effector over time.

[0027] Figure 8 The curves show the changes in the joint angles of the redundant robotic arm over time.

[0028] Figure 9 The curves showing the change of angular velocity of each joint of the redundant robotic arm over time.

[0029] Figure 10 This is a schematic diagram illustrating the evolution of the actual trajectory and reference trajectory of the end effector of a redundant robotic arm based on a super-exponential convergent null neural network.

[0030] Figure 11 This is a comparison of the overall tracking performance between the actual trajectory and the reference trajectory of the redundant robotic arm end effector in Example 2.

[0031] Figure 12 The curve showing the change in position tracking error of the redundant robotic arm end effector over time.

[0032] Figure 13 The curves show the changes in the joint angles of the redundant robotic arm over time.

[0033] Figure 14 The curves showing the change of angular velocity of each joint of the redundant robotic arm over time. Detailed Implementation

[0034] It should be noted that relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0035] The features and performance of the present invention will be further described in detail below with reference to embodiments.

[0036] Please see Figure 1-14 A super-exponential convergent nullable neural network method for position tracking of redundant robotic arms, such as... Figure 1 As shown, it includes the following steps: S1. Construct a redundant robotic arm motion model: S11. Set the structural parameters and motion state variables of the redundant robotic arm. Set the redundant robotic arm as a planar three-degree-of-freedom robotic arm, with all links having the same length and being constant. Set the time variable to The position of the end effector in the Cartesian coordinate system is set as follows: The joint angle vector is set to The joint angular velocity vector is set to .in, and These represent the position components of the end effector in a Cartesian coordinate system. and They represent the first The rotation angle and angular velocity of each joint.

[0037] S12. Establish the forward kinematics model of the redundant robotic arm: Based on the set redundant robotic arm structural parameters and motion state variables, a forward kinematics model of the redundant robotic arm is established, and the mapping relationship between the end effector position and the joint angle is obtained as follows: , in, This represents the nonlinear positive kinematic mapping function determined by the geometry of the robotic arm and the parameters of the links.

[0038] Establish a differential kinematic model for the redundant robotic arm: The mapping formula with respect to time Taking the derivative, we obtain the differential kinematic model of the redundant robotic arm as follows: , in, This represents the time-varying Jacobian matrix determined by the structural parameters and joint angles of the robotic arm.

[0039] S2. Establish a redundant robotic arm end-effector position tracking and control model: S21. Set the end effector reference trajectory and position tracking error: The reference trajectory of the end effector in a Cartesian coordinate system is ,in and The reference position component of the end effector in this coordinate system, with respect to time. derivative Exists. The end effector position tracking error is set as follows: , To ensure position tracking error Asymptotic convergence is achieved by setting the following desired first-order stable error dynamic equation: , in, This is the position tracking error feedback gain.

[0040] S22. Establish the minimum norm solutions for the desired velocity of the end effector and the joint velocity: The desired terminal velocity is obtained as follows: , Due to the redundant robotic arm Jacobian matrix For non-square matrices, the minimum norm solution for joint velocities can be obtained using the right Moore-Penrose pseudo-inverse: , in, Jacobian matrix The right Moore-Penrose pseudo-inverse.

[0041] To ensure that the joint velocities meet the physical execution constraints of the robotic arm, the joint velocities can be solved... Based on the minimum norm solution, it is saturated to satisfy: , in, and These represent the minimum and maximum values ​​of the joint velocity, respectively.

[0042] S23. Construct the inverse kinematics model of the velocity layer of the redundant robotic arm: The inverse kinematics expression of the velocity layer of the redundant robotic arm based on the differential kinematics model is as follows: , Joint angular velocity The calculation depends on the right Moore–Penrose pseudoinverse of the Jacobian matrix. Online solution; through the Online solution can be performed to calculate joint angular velocities in real time. This drives the position tracking error. Gradually converge to zero to realize the end effector position vector For reference trajectory Stable and accurate tracking.

[0043] S3. Configure a super-exponential convergent null neural network: To address the need for online solution of the Moore-Penrose pseudoinverse of the time-varying Jacobian matrix in the end-effector position tracking control of redundant robotic arms, a nullable neural network with super-exponential convergence is designed. By constructing the time-varying error matrix and its evolution dynamic equation, and combining the dynamic decay coefficient with the power-sigmoid nonlinear activation function, stable and fast online calculation of the Moore-Penrose pseudoinverse is achieved, providing joint velocity solutions for the end-effector position tracking control of redundant robotic arms.

[0044] S31. Construct the time-varying error matrix of the right pseudo-inverse of the Jacobian matrix of the redundant robotic arm: For the Jacobian matrix of redundant robotic arms The right Moore-Penrose pseudoinverse, denoted by its theoretical objective solution, is... And assume that the corresponding nullified neural network state matrix is The time-varying error matrix of the nullification neural network is set as follows: , When the time-varying error matrix satisfies: At that time, the nullified neural network state matrix The right Moore-Penrose pseudoinverse of the Jacobian matrix can be asymptotically approximated. , and serve as its online approximation. Among them, This represents the Frobenius norm.

[0045] S32. Constructing the error evolution dynamics and state evolution equations of a null neural network: To achieve the defined time-varying error matrix To achieve asymptotic convergence to zero, this application constructs a nullification neural network error evolution dynamic equation with super-exponential convergence characteristics, which takes the form: , in, It is the time-varying dynamic decay coefficient. It is a power-sigmoid type nonlinear activation function that acts on the error matrix, with each element acting one by one while keeping the matrix dimension unchanged.

[0046] The error matrix formula and its relation to time derivative Substituting into the above equation, we obtain the state evolution equation of the super-exponential convergent nullable neural network as follows: , in, is the state matrix of the nullified neural network, used to approximate the right Moore-Penrose pseudoinverse of the Jacobian matrix online.

[0047] S33. Set the time-varying dynamic decay coefficient and the power-sigmoid nonlinear activation function: To obtain the super-exponential convergence characteristics of the error evolution dynamics equation and the null neural network state evolution equation, this application synergistically designs the time-varying dynamic decay coefficient and the nonlinear activation function.

[0048] The optimal design for the time-varying dynamic attenuation coefficient is as follows: , in, This is the initial amplitude of the attenuation coefficient, used to adjust the convergence strength of the error system in the initial stage; The exponential growth rate parameter is used to control the rate at which the decay coefficient increases over time, thereby gradually accelerating the convergence process of the error over time.

[0049] Simultaneously, a power-sigmoid type nonlinear activation function is set, the expression of which is: , Among them, activation function For the error matrix Each element acts one by one; It is an odd-order power exponent and Used to enhance the nonlinear suppression capability in regions with large errors; This is the gain coefficient for the hyperbolic sine term, used to improve the convergence driving force in the large error region; This is an adjustment parameter for the hyperbolic tangent function, used to improve smoothness and steady-state accuracy when the error approaches zero.

[0050] Through the synergistic effect of the time-varying dynamic decay coefficient and the power-sigmoid nonlinear activation function, the constructed null neural network has a strong convergence driving force in the stage of large error and maintains good smoothness and stability in the stage of error approaching zero, thus providing a basis for the theoretical analysis of subsequent super-exponential convergence characteristics.

[0051] Furthermore, to facilitate understanding of the hyperbolic function form involved in the power-sigmoid type nonlinear activation function, examples of its commonly used mathematical definitions are shown in Table 1, where... Indicates about time A continuous real-valued function.

[0052] Table 1. Examples of definitions of hyperbolic functions involved in power-sigmoid type nonlinear activation functions

[0053] Different parameter values ​​will affect the time-varying dynamic decay coefficient and power-sigmoid type nonlinear activation function The changing characteristics of [the parameter] are affected, and their changing trends under different parameter conditions are illustrated as follows: Figure 2 and Figure 3 As shown.

[0054] Fusion time-varying dynamic decay coefficient With power-sigmoid type nonlinear activation function The overall structure of the super-exponential convergent null neural network is shown in the figure below. Figure 4 As shown.

[0055] S4. Analyze the global asymptotic stability and super-exponential convergence characteristics of super-exponential convergent nullable neural networks: This step aims to verify the global asymptotic stability and super-exponential convergence characteristics of the constructed super-exponential convergent nullification neural network under different initial conditions through theoretical analysis. Theoretically, this ensures the stability, speed, and reliability of the nullification neural network in the online solution of the Moore-Penrose pseudo-inverse of the variable Jacobian matrix, providing a theoretical basis for the engineering application of the redundant robotic arm end-effector position tracking control method.

[0056] S41. Analyze the global asymptotic stability of a dynamic system with nulled neural network errors: To verify the stability of the constructed null neural network, this application conducts a theoretical analysis of its corresponding error dynamic system, showing that the error system has global asymptotic stability under different initial conditions, thus providing a basis for the subsequent analysis of super-exponential convergence characteristics.

[0057] Construct the following Lyapunov function: , in, This represents the trace operation of a matrix. Clearly, the Lyapunov function satisfies... , and only if It takes a value of zero.

[0058] For Lyapunov functions Trajectory of the error dynamic system expression relative to time Taking the derivative and combining it with the definition of the error matrix and the defined power-sigmoid type nonlinear activation function, we can obtain: , in, and Let each represent the error matrix. and activation function output matrix The , Each element.

[0059] Since the designed power-sigmoid nonlinear activation function is a monotonically increasing odd function, for any All of the following are available: , If and only if The value is zero at time. Furthermore, due to the time-varying dynamic decay coefficient... For any This holds true for all, therefore we can conclude that: , And the equal sign only applies to It was established at that time.

[0060] Therefore, the constructed error dynamic system exhibits asymptotic stability in a global sense, and the time-varying error matrix... It gradually converges to the zero matrix over time, that is: , And because The above result is equivalent to the Frobenius norm of the error matrix satisfying the convergence condition, thus nulling the neural network state matrix. The right Moore-Penrose pseudoinverse that can asymptotically approximate the Jacobian matrix of a redundant robotic arm This enables online and stable solutions.

[0061] Therefore, it can be seen that the constructed super-exponential convergent nullification neural network has asymptotic stability in the global sense, which can provide stable and reliable theoretical support for the online solution of the pseudo-inverse of the Jacobian matrix in the position tracking control of redundant robotic arms.

[0062] S42. Analyze the super-exponential convergence characteristics of a dynamic system with zero-error neural networks. Based on the global asymptotic stability analysis, the convergence rate of the constructed zero-error neural network dynamic system is further analyzed to verify its super-exponential convergence characteristics.

[0063] (1) Analysis of the higher-order growth characteristics of nonlinear activation functions Because the power-sigmoid type nonlinear activation function contains both odd-power polynomial terms and hyperbolic sine-form nonlinear terms, its output exhibits a significant nonlinear growth characteristic as the error magnitude increases. Specifically, for any error matrix element... Under the condition that the parameters are reasonably selected, there exists a constant. and integer order This makes the following inequality hold: , in, Indicates activation function In the , Output at each element.

[0064] Therefore, when the error amplitude is large, the output of the activation function exhibits a high-order nonlinear growth trend relative to the error, thus providing a stronger error suppression effect in the error dynamic system and helping to accelerate the convergence process of the system.

[0065] (2) Super-exponential decay characteristics of Lyapunov function in error system Combining the obtained expression for the time derivative of the Lyapunov function along the trajectory of the error dynamic system, and substituting the higher-order growth characteristics of the nonlinear activation function into it, we get: , Furthermore, by utilizing the equivalence relationship between matrix element norms and Lyapunov functions, and combining... It can be seen that there exists a constant. and index This makes the time derivative of the Lyapunov function satisfy the following inequality: , Time-varying dynamic attenuation coefficient Under the influence of this, the decay rate of the Lyapunov function continuously increases with time, and its decay speed is faster than that of the conventional exponential form, providing a theoretical basis for the super-exponential convergence of the error system.

[0066] (3) The promoting effect of time-varying dynamic decay coefficient on super-exponential convergence In this application, the time-varying dynamic decay coefficient is designed to grow exponentially. Because The decay weights, which monotonically increase over time, are introduced into the Lyapunov differential inequality, causing the decay rate of the error system to continuously increase during network evolution. Therefore, under the synergistic effect of the time-varying dynamic decay coefficient and the power-sigmoid nonlinear activation function, the convergence process of the error system is significantly faster than that of the conventional exponential form, thus exhibiting super-exponential convergence characteristics.

[0067] In summary, the proposed super-exponential convergent nullification neural network can significantly accelerate the error convergence process while ensuring system stability in the online solution of the Jacobian matrix Moore-Penrose pseudo-inverse of the redundant robotic arm, thus providing an efficient and reliable online computational foundation for subsequent position tracking control of the redundant robotic arm.

[0068] S5. Construct a redundant robotic arm end-effector position tracking control method based on a super-exponential convergent nullable neural network: This step presents a scheme for implementing position tracking control of the redundant robotic arm end effector based on the nulled neural network. This scheme is used to solve the Jacobian matrix Moore-Penrose pseudo-inverse online and construct the velocity-layer inverse kinematics control law. The control implementation process is as follows: Figure 4 As shown.

[0069] S51. Construct an online mechanism for solving the right pseudo-inverse of the Jacobian matrix based on a nullified neural network: According to the definition of a super-exponential convergent nullable neural network, assume that the state matrix of the nullable neural network is... Its state evolution is updated by the corresponding state equation. Combining the stability and convergence analysis results, it can be seen that the designed time-varying dynamic decay coefficient... and power-sigmoid type nonlinear activation function Under the influence of the zeroing neural network state matrix The right Moore-Penrose pseudoinverse of the Jacobian matrix of the redundant robotic arm can asymptotically approximate it in a global sense. It also exhibits super-exponential convergence properties.

[0070] Therefore, during the movement of the redundant robotic arm, the state matrix of the null neural network is reduced. As the right Moore–Penrose pseudoinverse of the Jacobian matrix The online approximation is used to update the joint velocity solution in real time, thereby reducing the computational complexity caused by direct analytical inversion and avoiding numerical instability problems.

[0071] S52. Constructing the inverse kinematics control law for the velocity layer based on a nullable neural network: Based on the established definition of the end effector position tracking error and the expression for the desired end effector velocity, and combined with the determined differential kinematics model of the redundant manipulator, the nulled neural network state matrix obtained during the control implementation process will be used. Right pseudoinverse of the Jacobian matrix The inverse kinematic control law for the velocity layer based on the null neural network is constructed as follows: , in, For position error feedback gain, This serves as the reference trajectory for the end effector.

[0072] Using the aforementioned control law, the joint speed commands of the redundant robotic arm can be obtained in real time without explicitly calculating the pseudo-inverse of the Jacobian matrix, thus enabling online tracking control of the end effector position relative to the reference trajectory.

[0073] S53. Achieve closed-loop tracking control of the position of the redundant robotic arm end effector: During the movement of the redundant robotic arm, based on the constructed forward kinematics model and utilizing real-time acquired joint angle information... Calculate the current end effector position and reference trajectory By comparison, the position tracking error of the end effector can be obtained. Based on this, and according to the constructed inverse kinematics control law of the velocity layer based on the nullable neural network, the state matrix of the nullable neural network is used... As an online approximation of the right Moore-Penrose pseudoinverse of the Jacobian matrix, the joint angular velocities of the redundant robotic arm are calculated in real time. The system updates the joint angle state based on the joint angular velocity to drive the redundant robotic arm. Simultaneously, the null neural network continuously updates its state matrix according to the state evolution equation. This enables synchronous online approximation of the right Moore-Penrose pseudoinverse of the Jacobian matrix of a redundant robotic arm.

[0074] In the aforementioned closed-loop control process, the end effector position tracking error Under the constraints of the error dynamics equation, the position of the redundant robotic arm end effector gradually converges to zero, thereby achieving closed-loop and stable tracking control of the reference trajectory.

[0075] Since the asymptotic stability and super-exponential convergence characteristics of the constructed zero-negation neural network error dynamic system have been theoretically proven in the global sense, the online calculation error of the right Moore-Penrose pseudo-inverse of the Jacobian matrix can be rapidly decayed and remain stable during the position tracking control of the constructed redundant robotic arm. This significantly improves the response speed and tracking accuracy of the end-position tracking control while ensuring system stability, and enhances its real-time performance and robustness under high-speed motion and complex trajectory tracking conditions.

[0076] In another specific embodiment, a method for end-effector position tracking control of a three-degree-of-freedom redundant robotic arm based on a super-exponentially convergent nullable neural network includes: This embodiment uses a three-degree-of-freedom planar redundant robotic arm as an example to illustrate the specific application process of the redundant robotic arm end-effector position tracking control method based on super-exponential convergent nullable neural network proposed in this invention in a real system, so as to verify the feasibility and effectiveness of the method.

[0077] 1) Robotic arm model and parameter settings The redundant robotic arm is a three-link planar robotic arm, with each link having a length of l = 1m; the joint angle vector of the robotic arm is defined as follows: The joint angular velocity vector is defined as ,in and They represent the first The rotation angle and angular velocity of each joint. Based on the established forward kinematics model, the position vector of the end effector in the Cartesian coordinate system is represented as: ,in and These represent the position components of the end effector in a Cartesian coordinate system. The differential kinematic model of the redundant robotic arm satisfies the following relationship: , in, This represents the time-varying Jacobian matrix determined by the structural parameters and joint angles of the robotic arm.

[0078] (2) Reference trajectory and initial condition settings Assuming the reference trajectory of the end effector is set as a planar trefoil rose curve, its expression is: , The corresponding reference trajectory vector is ,in and The reference position component of the end effector in this coordinate system, with respect to time. derivative It exists continuously.

[0079] The initial joint angle of the redundant robotic arm is set to The end effector position tracking error is defined as: , And it satisfies the following expected first-order stable error dynamic equation: , Wherein, the position tracking error feedback gain is taken .

[0080] 3) Implementation of inverse kinematics control based on velocity layer of null neural network During the movement of the robotic arm, based on the established end-effector position tracking control model, the desired end-effector velocity is set as follows: , To avoid the computational complexity and numerical instability issues caused by directly calculating the Moore-Penrose pseudoinverse of the Jacobian matrix, this embodiment uses the designed super-exponential convergent nullification neural network to solve the right Moore-Penrose pseudoinverse of the Jacobian matrix online. The parameters of the nullification neural network are shown in Table 2.

[0081] Table 2. Parameter settings for the super-exponential convergent nullable neural network used for tracking planar three-leaf rose curves.

[0082] During the control process, the state matrix of the null neural network is reduced. As the right pseudo-inverse of the Jacobian matrix The online approximation is obtained, and the joint angular velocity is calculated in real time based on the constructed control law: , In this process, the null neural network synchronously updates its state matrix under the influence of a time-varying dynamic decay coefficient and a nonlinear activation function. To achieve the right Moore-Penrose pseudoinverse of the Jacobian matrix. The continuous online approach.

[0083] (4) Closed-loop control process and operation mode: During the closed-loop control process, the system cyclically executes the following steps according to a preset control cycle: ① Collect the current joint angle And calculate the current position of the end effector based on the forward kinematics model. ; ②Based on the reference trajectory Calculate end-position tracking error ; ③ The state matrix is ​​updated online by a nullified neural network. And calculate the joint angular velocity accordingly. ; ④ Update the joint angle state according to the joint angular velocity, thereby driving the robotic arm to move.

[0084] (5) Explanation of implementation results: The total simulation time was set to 15s, the numerical integration step size was 0.01s, and the control law was updated numerically using the Euler method.

[0085] Figure 5 A schematic diagram illustrating the evolution process between the actual motion trajectory and the reference trajectory of the redundant robotic arm's end effector under the aforementioned control method is presented. It can be seen that even with a large initial positional error, the end effector can still quickly converge to the reference trajectory and maintain stable consistency with it during subsequent motion.

[0086] Furthermore, Figure 6 The overall comparison results of the actual tracking trajectory of the redundant robotic arm end effector and the reference trajectory are presented throughout the entire simulation time. Figure 6 As can be seen, the actual trajectory and the reference trajectory highly overlap in the entire time domain, indicating that the control method can achieve high-precision and stable tracking of the desired trajectory while ensuring system stability.

[0087] Under the constraint of the error dynamics equation, the end-point position tracking error gradually decreases over time and approaches zero, achieving stable closed-loop tracking of the reference trajectory. Its error convergence process is as follows: Figure 7 As shown.

[0088] Meanwhile, to verify the impact of the control method on the motion stability of the redundant robotic arm, Figure 8 The changes in the joint angles of the redundant robotic arm over time are presented. Figure 9 The corresponding changes in joint angular velocity are given. Figure 8 and Figure 9 It can be seen that during the entire control process, the joint angles of the robotic arm change continuously and smoothly, and the joint angular velocity remains within a bounded range without numerical divergence or high-frequency oscillation.

[0089] In summary, the redundant robotic arm end-effector position tracking control method based on super-exponential convergent nullable neural network can achieve fast and stable end-effector position tracking while ensuring system stability. It has good dynamic performance, numerical stability and engineering feasibility.

[0090] In another specific embodiment, a further specific implementation of the three-degree-of-freedom redundant robotic arm end-effector position tracking control method based on a super-exponentially convergent nullable neural network includes: Based on the previous embodiment, this embodiment further demonstrates that the redundant robotic arm end-effector position tracking control method proposed in this application, based on a super-exponential convergent nullable neural network, can also be effectively implemented under different robotic arm structural parameters, initial joint states, and reference trajectory conditions, thereby verifying the versatility and adaptability of the method.

[0091] In this embodiment, a three-degree-of-freedom planar redundant robotic arm is still used as the controlled object. Its structural form is the same as in the previous embodiment, but the length of each link is uniformly set to [value missing]. The joint angle vectors, end effector position vectors, and corresponding kinematic relationships of the robotic arm are consistent with those in the previous embodiment, and will not be described again here.

[0092] The reference trajectory of the end effector is set as a six-leaf clover-shaped planar curve with a value of 2m, and its mathematical expression is: , Corresponding reference trajectory vector And regarding time The first derivative of is continuous.

[0093] The initial joint angle of the redundant robotic arm is set to During the control process, the online solution method for the right Moore-Penrose pseudo-inverse of the Jacobian matrix based on a super-exponential convergent nullable neural network, as described in the previous embodiment, is still used. Combined with the same form of velocity layer inverse kinematics control law, closed-loop tracking control of the redundant robotic arm end effector on the reference trajectory is achieved. The structure and state evolution of the super-exponential convergent nullable neural network are the same as in the previous embodiment and will not be repeated here; its parameters are set according to Table 3.

[0094] Table 3. Parameter settings of the super-exponential convergent nullable neural network used for six-leaf clover-shaped plane trajectory tracking in Example 2

[0095] In this embodiment, the total simulation time is set to 8 seconds, and the numerical integration step size and control law numerical update method remain consistent with the previous embodiment. The redundant robotic arm end-effector position tracking control method based on a super-exponential convergent nullified neural network is verified under different link parameters, initial joint angle conditions, and a hexagonal plane reference trajectory.

[0096] Figure 10 The evolution of the actual motion trajectory of the redundant robotic arm end effector relative to the reference trajectory is shown under the conditions of this embodiment. Figure 11 The overall comparison results between the actual tracking trajectory and the reference trajectory over the entire simulation time range are presented. Figure 10 and Figure 11It can be seen that despite changes in the shape and initial state of the reference trajectory, the end effector can still quickly converge to the reference trajectory and maintain stable and accurate tracking performance throughout the entire time domain.

[0097] Furthermore, the variation of the end-position tracking error over time is as follows: Figure 12 As shown, the error gradually decreases and approaches zero under the constraint of error dynamics, indicating that the control method still has good convergence characteristics and stability under different trajectory conditions.

[0098] at the same time, Figure 13 and Figure 14 The changes in the joint angles and angular velocities of the redundant robotic arm over time are presented separately. Figure 13 and Figure 14 It can be seen that during the entire tracking process, the joint movement of the robotic arm was continuous and smooth, and the joint angular velocity remained within a bounded range without any numerical instability or high-frequency oscillation.

[0099] In summary, the simulation results of this embodiment further verify that the redundant robotic arm end-effector position tracking control method based on super-exponential convergent nullable neural network has good stability, robustness and engineering feasibility under different structural parameters, initial conditions and reference trajectories.

[0100] The embodiments described above merely illustrate specific implementation methods of this application, and while the descriptions are detailed and specific, they should not be construed as limiting the scope of protection of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the technical solution of this application, and these modifications and improvements all fall within the scope of protection of this application.

Claims

1. A super-exponential convergent nullable neural network method for position tracking of a redundant robotic arm, characterized in that, Includes the following steps: Based on the geometry of the redundant robotic arm, a velocity mapping relationship between the change of its end-effector position and the joint motion is established, i.e., a differential kinematic model. Based on the desired end-point reference trajectory and the actual end-point position, calculate the position tracking error, and construct the desired end-point velocity command based on this error; Construct and run a super-exponential convergent nullable neural network for online estimation of the generalized inverse of the time-varying Jacobian matrix in a differential kinematics model; The output of the super-exponential convergent null neural network is used as an online estimate of the generalized inverse. Combined with the desired end-effector velocity command, the joint angular velocity control command is calculated. The robotic arm joints are driven to move according to the joint angular velocity control command, forming a closed-loop control, so that the end position tracks the reference trajectory.

2. The super-exponential convergent nullable neural network method for redundant robotic arm position tracking according to claim 1, characterized in that, The differential kinematic model is specifically represented as follows: ， in, End position vector rate of change, The joint angular velocity vector. Joint angle vector, This represents the time-varying Jacobian matrix determined by the structural parameters and joint angles of the robotic arm.

3. The super-exponential convergent nullable neural network method for redundant robotic arm position tracking according to claim 1, characterized in that, The calculation of position tracking error specifically includes: Set the end-position tracking error to ,in For reference trajectory, It is a position vector; The desired end-velocity command, constructed based on the error, is as follows: ,in, The rate of change of the reference trajectory, This is the error feedback gain.

4. The super-exponential convergent nullable neural network method for redundant robotic arm position tracking according to claim 1, characterized in that, The construction and operation of a super-exponential convergent nullification neural network specifically includes: For online solving of time-varying Jacobian matrices The right Moore-Penrose pseudoinverse, the time-varying error matrix of the network is set as follows: ,in, To nullify the state matrix of the neural network, Time-varying Jacobian matrix The right Moore-Penrose pseudo-inverse; The evolutionary dynamic equations for designing convergent driving matrices are as follows: ,in, It is the time-varying dynamic decay coefficient. It is a power-sigmoid type nonlinear activation function that acts on the error matrix; Substituting the time derivative of the error matrix into the evolutionary dynamics equation, we obtain the state update law of the null neural network: .

5. The super-exponential convergence nullable neural network method for redundant robotic arm position tracking according to claim 4, characterized in that, The time-varying dynamic attenuation coefficient Designed to grow exponentially over time: ,in, This is the initial amplitude of the attenuation coefficient, used to adjust the convergence strength of the error system in the initial stage; This is the exponential growth rate parameter, used to control the rate at which the decay coefficient increases over time.

6. The super-exponential convergence nullable neural network method for redundant robotic arm position tracking according to claim 4, characterized in that, The power-sigmoid type nonlinear activation function The formula for applying an independent nonlinear mapping to each element of the matrix is ​​as follows: ， Among them, activation function For the error matrix Each element acts one by one; It is an odd-order power exponent and This is used to enhance the nonlinear suppression capability in regions with large errors; This is the gain coefficient for the hyperbolic sine term, used to improve the convergence driving force in the large error region; This is an adjustment parameter for the hyperbolic tangent function, used to improve smoothness and steady-state accuracy when the error approaches zero.

7. The super-exponential convergence nullable neural network method for redundant robotic arm position tracking according to claim 4, characterized in that, The calculated joint angular velocity control command specifically includes: The real-time state matrix of the null neural network As an estimate of the pseudo-inverse of the Jacobian matrix, the joint angular velocity control command is calculated using the following formula: .

8. The super-exponential convergent nullable neural network method for redundant robotic arm position tracking according to claim 7, characterized in that, Receive joint angular velocity control command Subsequently, to ensure that the joint velocities meet the physical execution constraints of the robotic arm, the joint velocities can be solved... Based on the minimum norm solution, it is saturated to satisfy: ,in, and These represent the minimum and maximum values ​​of the joint velocity, respectively.

9. The super-exponential convergence nullable neural network method for redundant robotic arm position tracking according to claim 5, characterized in that, The initial amplitude of the attenuation coefficient exponential growth rate upper limit of saturation value .

10. The super-exponential convergence nullable neural network method for redundant robotic arm position tracking according to claim 6, characterized in that, The parameters of the power-sigmoid type nonlinear activation function are preferably set as follows: polynomial order Hyperbolic sine term gain Hyperbolic tangent function adjustment parameters .

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