A Real-Time Robotic Arm Motion Generation Method and System Based on the DGZND Model

Abstract

This invention relates to the field of robot control technology and discloses a real-time robotic arm motion generation method and system based on the DGZND model. The method includes: transforming the kinematic equations of the planar two-link robotic arm into standard time-varying linear equations; determining piecewise adaptive coefficient functions and error functions to construct a dynamic gain term; determining an error integral term to construct a negative feedback adjustment mechanism; constructing a DGZND model based on the dynamic gain term and error integral term; deriving a dynamic solver for the joint angular velocity derivatives based on the DGZND model and the defined error function; and solving and driving the robotic arm motion in real time using the ODE numerical method. This invention achieves rapid convergence through piecewise adaptive coefficients and suppresses noise using an integral negative feedback mechanism, realizing high-precision and robust real-time motion generation of a planar robotic arm in dynamic environments.

Description

Technical Field

[0001] This invention relates to the field of robot control technology, specifically to a real-time robotic arm motion generation method and system based on the DGZND model. Background Technology

[0002] Planar robotic arms, especially planar two-bar robotic arms, are core actuators in modern industrial and intelligent systems. Their functionality relies on precise motion control, which involves calculating in real-time the required rotational speed (i.e., joint angular velocity) of each joint according to predetermined task requirements, such as moving the end effector of the planar two-bar robotic arm along a specific curve. Mathematically, this process is described by a time-varying linear equation: .in: J(t) is the desired speed of the end effector (gripper) of the planar two-bar linkage manipulator; J(t) is the "Jacobian matrix" of the planar two-bar linkage manipulator, which changes in real time with the posture (i.e., joint angle) of the manipulator. The angular velocities of each joint in the planar two-bar linkage manipulator are the targets that need to be solved in real time. Therefore, the real-time motion generation problem of the planar two-bar linkage manipulator is essentially a problem of solving a system of time-varying linear equations online.

[0003] Currently, the technical solutions most similar to this invention mainly revolve around trajectory planners based on Zeroing Neural Networks (ZNNs). The most representative are those using traditional ZNN models or improved ZNN models based on monotonically increasing activation functions, such as the OZNN (Original ZNN) model.

[0004] Traditional ZNN Model: This model is a foundational framework for solving time-varying problems. Its core idea is to construct a scalar or vector error function for the time-varying problem, and then design a dynamic evolution rule that forces the error function to converge exponentially to zero. Although this model provides a theoretical foundation for real-time solutions, its convergence speed is relatively slow, and its convergence accuracy and stability significantly decrease in the presence of external disturbances or computational noise. Specifically, traditional ZNNs use fixed gain coefficients, which cannot be dynamically adjusted according to the error, leading to solution delays and poor performance in rapidly changing trajectory tracking tasks.

[0005] OZNN Model: To accelerate convergence, researchers have proposed introducing nonlinear activation functions into the dynamic equations of ZNNs. OZNN is the most typical example. By using these nonlinear functions, the model can obtain greater convergence driving force when the error is large, thus accelerating the convergence speed to some extent. However, OZNN still has obvious limitations: First, its activation function parameters are fixed, making it unable to adapt to the error characteristics at different times; second, the model is prone to error divergence under sensor noise or environmental interference; finally, the fixed-parameter model consumes too many computational resources during the solution process, making it unsuitable for control scenarios with high real-time requirements.

[0006] NTZNN Model: While the Noise Tolerant Zero Neural Network (NTZNN) proposed in recent years has improved noise tolerance, its parameter rigidity problem remains severe. This model employs a complex noise suppression mechanism, increasing computational complexity and leading to low resource efficiency. Furthermore, existing methods struggle to effectively handle inequality constraints such as joint velocity and acceleration in robotic arms, limiting their application in practical engineering.

[0007] In summary, the main drawbacks of the existing technology include: (1) slow convergence speed, which cannot meet the requirements of fast trajectory tracking; (2) poor noise resistance, which is not stable enough in industrial environments; (3) low resource efficiency, which makes it difficult to achieve real-time control on low-cost hardware; and (4) insufficient constraint processing capability, which makes it difficult to meet the actual motion constraints of the robotic arm. Summary of the Invention

[0008] To address the aforementioned issues, this invention provides a real-time robotic arm motion generation method and system based on the DGZND (Dynamic Gain Zeroing Neural Dynamics) model. By designing a piecewise adaptive coefficient function to achieve dynamic adjustment of the gain and introducing a negative feedback adjustment mechanism that includes an error integral term, the motion generation accuracy, convergence speed, and noise immunity of the planar two-bar robotic arm in dynamic environments are significantly improved.

[0009] To achieve the above objectives, the present invention provides the following technical solution:

[0010] On one hand, embodiments of the present invention provide a real-time robotic arm motion generation method based on the DGZND model, the method comprising the following steps:

[0011] S100: Obtain the kinematic parameters, joint displacement range, and desired trajectory parameters of the planar two-bar linkage manipulator;

[0012] S200, the kinematic equations of the planar two-bar manipulator are transformed into standard time-varying linear equations; the standard time-varying linear equations are: A(t)X(t)=B(t), where A(t) is the Jacobian matrix, X(t) is the joint angular velocity vector to be solved, and B(t) is the desired end-effector velocity vector;

[0013] S300, determine the piecewise adaptive coefficient function and error function to construct the dynamic gain term; the dynamic gain term is: Where e(t) is the error function, e(t) = A(t)X(t) - B(t), These are the gain coefficients of the piecewise adaptive coefficient function. ψ(·) is a piecewise adaptive coefficient function used to dynamically adjust the gain based on the magnitude of the error e(t) and time; ψ(·) is the activation function; and t is the current time. Let e(t) be the derivative of the error function with respect to time t.

[0014] S400, determine the error integral term to construct a negative feedback adjustment mechanism; construct a DGZND model based on the dynamic gain term and the error integral term; wherein, the equation of the DGZND model is: β is the integral gain coefficient. For the error integral term, For integration variables, Let e(t) be the derivative of the error function with respect to time t.

[0015] S500, based on the DGZND model and the defined error function, derive the dynamic solver for the joint angular velocity derivative: The formula for the dynamic solver is: ;in, Let A(t) be the derivative of the Jacobian matrix A(t) with respect to time t. Let X(t) be the derivative of the joint angular velocity vector with respect to time t, i.e., the joint angular acceleration vector. Let B(t) be the derivative of the desired terminal velocity vector B(t) with respect to time t.

[0016] S600 uses the ODE numerical method to solve the dynamic solver, outputs the joint angular velocity vector in real time, and outputs control commands based on the joint angular velocity vector to drive the planar two-bar robotic arm actuator to move.

[0017] Optionally, the step of converting the kinematic equations of the planar two-bar linkage manipulator into standard time-varying linear equations includes:

[0018] S210, Obtain the joint angle vector of the planar two-bar linkage robot arm. ,in and These are the joint angles of the two links;

[0019] S220, Calculate the Jacobian matrix based on the joint angle vector. Establish kinematic equations ,in The desired velocity vector at the end;

[0020] S230, transform the kinematic equations into standard form, let , , Thus, the time-varying linear equation A(t)X(t)=B(t) is obtained.

[0021] Optionally, the expression for the piecewise adaptive coefficient function is as follows:

[0022]

[0023] in, , It is the Frobenius norm of the error matrix, used to measure the magnitude of the error e(t).

[0024] Optionally, determining the error integral term to construct a negative feedback adjustment mechanism includes:

[0025] S410, introduces the error integration term Accumulate historical error information;

[0026] S420, the error integral term is nonlinearly transformed by the activation function ψ(·), and the integral gain coefficient β>0 is used to adjust the negative feedback strength;

[0027] S430, when the robot control system is disturbed by noise and the error deviates from zero, the integral term generates a reverse adjustment force to actively suppress and eliminate the steady-state error caused by the noise.

[0028] Optionally, the dynamic solver for deriving the joint angular velocity derivative based on the DGZND model and the defined error function includes:

[0029] S510, the derivative of the error function e(t) is obtained ;

[0030] S520, by making the dynamic evolution equation of the DGZND model equal to the derivative of the error function, we obtain the following equation: ;

[0031] S530, rearranging the equations yields the dynamic solver: .

[0032] Optionally, the step of using the ODE numerical method to solve the dynamic solver and outputting the joint angular velocity vector in real time includes:

[0033] S610, in each control cycle, acquire the current joint angle θ(t), calculate the Jacobian matrix J(θ(t)) and its time derivative. ;

[0034] S620, calculate the terminal velocity B(t) and its time derivative based on the desired trajectory. ;

[0035] S630, calculate the current error e(t) = A(t)X(t) - B(t) and the error integral term. ;

[0036] S640, based on the current time t and the error norm Calculate piecewise adaptive coefficients ;

[0037] S650, the terms on the right-hand side of the dynamic solver equation are passed through the pseudo-inverse of the Jacobian matrix. Solve for the rate of change of joint angular velocity to obtain the solution. ; ;

[0038] S660 uses a numerical integration method to measure the rate of change of joint angular velocity. Integrate to obtain the joint angular velocity vector at the current moment.

[0039] Optionally, the method further includes:

[0040] During the movement of the robotic arm, the actual rotation angle of each joint of the robotic arm and the actual position of the end effector are collected and compared with the joint angular velocity control command. If the deviation exceeds the preset threshold, closed-loop feedback control is triggered, and feedback control command is generated for real-time adjustment.

[0041] On the other hand, embodiments of the present invention provide a real-time robotic arm motion generation system based on the DGZND model, comprising:

[0042] At least one processor;

[0043] At least one memory for storing at least one program;

[0044] When the at least one program is executed by the at least one processor, the at least one processor performs the method described above.

[0045] On the other hand, embodiments of the present invention provide a computer-readable storage medium storing a processor-executable program, which, when executed by a processor, is used to perform the above-described method.

[0046] The beneficial effects of this invention are as follows: This invention discloses a real-time robotic arm motion generation method and system based on the DGZND model. By designing a piecewise adaptive coefficient function, this invention can dynamically adjust the gain based on the error magnitude and time. It provides a stronger convergence driving force when the error is large or in the initial stage, accelerating the convergence speed. When the error is small or the system tends to stabilize, the gain is reduced to avoid overshoot and oscillation, effectively balancing speed and stability. Simultaneously, a negative feedback adjustment mechanism including an error integral term is introduced. By accumulating historical error information and performing nonlinear transformation, when the system is disturbed by noise causing the error to deviate from zero, the integral term generates a reverse adjustment force, actively suppressing and eliminating steady-state errors caused by noise, significantly improving noise resistance. Furthermore, the dynamic solver for the joint angular velocity derivative derived from the constructed DGZND model, combined with the ODE numerical method, can calculate the Jacobian matrix and its derivative, the desired end-effector velocity and its derivative in real time within each control cycle. Based on the current error and the error integral term, it quickly solves for the joint angular velocity change rate, thereby outputting an accurate joint angular velocity vector, ensuring the real-time performance and accuracy of the robotic arm in rapidly changing trajectory tracking tasks. Compared with traditional ZNN, OZNN and NTZNN models, this invention has significant improvements in convergence speed, noise resistance, resource efficiency and constraint handling capabilities. It can better meet the stringent requirements of planar two-bar linkage robotic arms for dynamic motion generation in actual industrial environments and can be widely used in fields requiring high-precision and high-real-time robotic arm control. Attached Figure Description

[0047] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0048] Figure 1 This is a flowchart of a real-time robotic arm motion generation method based on the DGZND model according to the present invention.

[0049] Figure 2 The image shows the trajectory tracking diagram of the end effector of a planar two-bar linkage robotic arm, illustrating the comparison between the expected trajectory and the actual trajectory.

[0050] Figure 3 This is a snapshot of the motion process of a planar two-bar linkage robotic arm, showing the motion posture of the linkages in the robotic arm;

[0051] Figure 4 The time history diagram of the angles of the two joints of the planar two-bar linkage robotic arm shows the angle change curves of the two joints.

[0052] Figure 5The XY coordinate time history plot of the end effector shows the change of the end effector position over time;

[0053] Figure 6 The time history diagram of the angular velocity of the two joints of the planar two-bar linkage robotic arm shows the changes in the angular velocity of the two joints.

[0054] Figure 7 The XY velocity time history plot of the end effector shows the change of the end effector velocity over time;

[0055] Figure 8 The motion trajectory diagram of joint 1 (elbow) of the robotic arm shows the motion path of the elbow in the XY plane;

[0056] Figure 9 The position tracking error diagram for the end effector shows the position errors in the X and Y directions;

[0057] Figure 10 The diagram shows the velocity tracking error of the end effector, illustrating the velocity errors in the X and Y directions. Detailed Implementation

[0058] The following will provide a clear and complete description of the concept, specific structure, and technical effects of the present invention in conjunction with embodiments and accompanying drawings, so as to fully understand the purpose, solution, and effects of the present invention. It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other.

[0059] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In the following description, when referring to the accompanying drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the embodiments of this invention; they are merely examples of apparatuses and methods consistent with some aspects of the embodiments of this invention as detailed in the appended claims.

[0060] It is understood that the terms “first,” “second,” etc., used in this invention may be used herein to describe various concepts, but unless specifically stated otherwise, these concepts are not limited by these terms. These terms are used only to distinguish one concept from another. For example, first information may also be referred to as second information without departing from the scope of embodiments of the invention, and similarly, second information may also be referred to as first information. Depending on the context, the words “if,” “when,” or “in response to determination” as used herein may be interpreted as “when…” or “when…” or “in response to determination.”

[0061] The terms “at least one,” “multiple,” “each,” “any,” etc., used in this invention, “at least one” includes one, two, or more than two; “multiple” includes two or more than two; “each” refers to each of the corresponding multiple; and “any” refers to any one of the multiple.

[0062] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein is for the purpose of describing embodiments of the invention only and is not intended to limit the invention.

[0063] The technical terms involved in this invention will be explained below:

[0064] ZNN (Zeroing Neural Network): A type of recurrent neural network designed for real-time solving of time-varying problems (such as dynamic equations and time-varying matrix equations). Its core principle is to construct error dynamic equations that allow the system state to converge rapidly to the theoretical solution within a finite time.

[0065] DGZND (Dynamic Gain Zeroing in Neural Dynamics): This invention proposes an improved neural dynamics model that significantly enhances the solution speed, accuracy, and noise immunity of ZNN by introducing innovative piecewise adaptive coefficients and a negative feedback adjustment mechanism.

[0066] Motion generation: One of the core problems in robotics, it refers to the real-time calculation of the angular velocities and angles of each joint of a robot based on the desired task objective (such as the position or trajectory that the end effector needs to reach or follow), thereby controlling a planar two-bar linkage robotic arm to complete a specified action.

[0067] Disadvantages of existing technology:

[0068] 1. The contradiction between real-time performance and accuracy. Traditional pseudo-inverse methods have a large computational load and poor real-time performance; while iterative methods such as GNN are fast, they have lag errors and insufficient accuracy.

[0069] 2. Poor noise resistance. Existing advanced models such as ZNN, while theoretically highly accurate, are sensitive to noise. In practical applications, measurement noise from planar robotic arm sensors (such as encoders) and electromagnetic interference in the control system are unavoidable. This noise can cause jitter and decreased accuracy in the solutions of traditional ZNN models, affecting the smooth operation of the robotic arm.

[0070] 3. Limited convergence speed. The convergence speed of traditional ZNN models is usually controlled by a fixed gain parameter, making it difficult to converge quickly in the initial stage of the solution and finely adjust it at the end of the solution.

[0071] The purpose of this invention:

[0072] The purpose of this invention is to overcome the aforementioned shortcomings and provide a real-time planar two-bar robotic arm motion generation method based on the Dynamic Gain Zeroing Neurodynamics (DGZND) model. This invention aims to achieve the following objectives through two core innovations of the traditional ZNN model: piecewise adaptive coefficients and negative feedback adjustment:

[0073] 1. While ensuring real-time solution (without lag error), the system's noise suppression capability is significantly improved.

[0074] 2. It greatly accelerates the convergence speed of the model, enabling it to respond quickly when the error is large and converge precisely when the error is small.

[0075] 3. Ultimately, it achieves high-precision, high-speed, and high-stability real-time motion control of the planar two-bar linkage robotic arm in high-noise and high-dynamic environments.

[0076] refer to Figure 1 ,like Figure 1 The figure shows a real-time robotic arm motion generation method based on the DGZND model provided by an embodiment of the present invention. The method includes the following steps:

[0077] S100: Obtain the kinematic parameters, joint displacement range, and desired trajectory parameters of the planar two-bar linkage manipulator;

[0078] It should be noted that the kinematic parameters of the planar two-bar linkage robotic arm include the lengths of the two links. , The joint type is a rotary joint with 2 degrees of freedom. The joint displacement range refers to the maximum and minimum angular range that the two joints can move, and it is a physical constraint for the robot arm's motion planning. The desired trajectory parameters involve trajectory shape, amplitude, frequency, etc., which determine the motion path of the robot arm's end effector.

[0079] For example, taking a standard planar two-bar linkage robotic arm as an example, let the link length be... , The joint displacement range is ±π rad, and the maximum joint velocity is π rad / s. The desired trajectory is a cardioid curve, and its parametric equation is: Where ω = 0.2π rad / s, forming a smooth heart-shaped pattern.

[0080] S200, the kinematic equations of the planar two-bar manipulator are transformed into standard time-varying linear equations; the standard time-varying linear equations are: A(t)X(t)=B(t), where A(t) is the Jacobian matrix, X(t) is the joint angular velocity vector to be solved, and B(t) is the desired end-effector velocity vector;

[0081] S300, determine the piecewise adaptive coefficient function and error function to construct the dynamic gain term; the dynamic gain term is: Where e(t) is the error function, e(t) = A(t)X(t) - B(t), These are the gain coefficients of the piecewise adaptive coefficient function. ψ(·) is a piecewise adaptive coefficient function used to dynamically adjust the gain based on the magnitude of the error e(t) and time; ψ(·) is the activation function; and t is the current time. Let e(t) be the derivative of the error function with respect to time t.

[0082] S400, determine the error integral term to construct a negative feedback adjustment mechanism; construct a DGZND model based on the dynamic gain term and the error integral term; wherein, the equation of the DGZND model is: β is the integral gain coefficient. For the error integral term, For integration variables, Let e(t) be the derivative of the error function with respect to time t.

[0083] S500, based on the DGZND model and the defined error function, derive the dynamic solver for the joint angular velocity derivative: The formula for the dynamic solver is: ;in, Let A(t) be the derivative of the Jacobian matrix A(t) with respect to time t. Let X(t) be the derivative of the joint angular velocity vector with respect to time t, i.e., the joint angular acceleration vector. Let B(t) be the derivative of the desired terminal velocity vector B(t) with respect to time t.

[0084] S600 uses the ODE numerical method to solve the dynamic solver, outputs the joint angular velocity vector in real time, and outputs control commands based on the joint angular velocity vector to drive the planar two-bar robotic arm actuator to move.

[0085] The embodiments provided by this invention can effectively solve the inherent contradictions between real-time performance and accuracy, noise resistance and convergence speed of traditional methods by applying the Dynamic Gain Zero Neurodynamics (DGZND) model to the motion generation of a planar two-bar linkage manipulator.

[0086] In some embodiments, converting the kinematic equations of the planar two-bar manipulator into standard time-varying linear equations includes:

[0087] S210, Obtain the joint angle vector of the planar two-bar linkage robot arm. ,in and These are the joint angles of the two links;

[0088] S220, Calculate the Jacobian matrix based on the joint angle vector. Establish kinematic equations ,in The desired velocity vector at the end;

[0089] S230, transform the kinematic equations into standard form, let , , Thus, the time-varying linear equation A(t)X(t)=B(t) is obtained.

[0090] The specific implementation process includes: first, obtaining the joint angle vectors of the planar two-bar linkage robotic arm. ,in and These are the joint angles of the two links. Then, the Jacobian matrix is ​​calculated based on the link parameters:

[0091] ;

[0092] Establishing kinematic equations ,in Let be the desired velocity vector at the end. Transform the kinematic equations into standard form, and let... , , Thus, the time-varying linear equation A(t)X(t)=B(t) is obtained.

[0093] This embodiment transforms the complex kinematic relationships of a planar two-bar linkage robotic arm into standard time-varying linear equations that are easily solved using the DGZND model through precise mathematical modeling. This lays a solid theoretical foundation for subsequent real-time motion control based on neurodynamic models. This transformation process clearly reveals the intrinsic relationship between the Jacobian matrix, joint angular velocity vectors, and end-effector desired velocity vectors, enabling the linearization of the originally nonlinear kinematic problem within a specific model framework. This allows for efficient utilization of the dynamic solution characteristics of the DGZND model.

[0094] In some embodiments, the expression for the piecewise adaptive coefficient function is as follows:

[0095] ;

[0096] in, , It is the Frobenius norm of the error matrix, used to measure the magnitude of the error e(t).

[0097] This embodiment cleverly balances the system's requirements for convergence speed and adjustment stability at different error stages through a segmented dynamic adjustment mechanism. This enables the DGZND model to exhibit rapid convergence when the error is large, while maintaining good control accuracy and stability when the error approaches zero. It effectively solves the problem in traditional ZNN models where fixed gain cannot simultaneously achieve both rapid convergence and fine adjustment.

[0098] In some embodiments, determining the error integral term to construct a negative feedback adjustment mechanism includes:

[0099] S410, introduces the error integration term Accumulate historical error information;

[0100] S420, the error integral term is nonlinearly transformed by the activation function ψ(·), and the integral gain coefficient β>0 is used to adjust the negative feedback strength;

[0101] S430, when the robot control system is disturbed by noise and the error deviates from zero, the integral term generates a reverse adjustment force to actively suppress and eliminate the steady-state error caused by the noise.

[0102] This embodiment effectively accumulates historical error information generated during the robotic arm's movement due to factors such as sensor noise and actuator hysteresis by introducing an error integral term. This accumulation is not a simple superposition, but rather a nonlinear transformation using the activation function ψ(·) to integrate the scattered error information into a negative feedback signal with a regulating effect. The integral gain coefficient β is an important regulation parameter, and its value directly affects the strength of the negative feedback regulation: when the system is subjected to strong noise interference, causing a significant deviation between the actual trajectory and the expected trajectory of the end effector, a larger β value can enhance the "corrective force" of the negative feedback, quickly pulling the error back to near zero; while in the stage where the system tends to be stable and the error is small, the β value can adaptively maintain a moderate adjustment, avoiding system oscillations caused by overcorrection.

[0103] In some embodiments, the dynamic solver for deriving the joint angular velocity derivative based on the DGZND model and the defined error function includes:

[0104] S510, the derivative of the error function e(t) is obtained ;

[0105] S520, by making the dynamic evolution equation of the DGZND model equal to the derivative of the error function, we obtain the following equation: ;

[0106] S530, rearranging the equations yields the dynamic solver: .

[0107] This embodiment constructs a dynamic solver for the joint angular velocity derivative by closely linking the core dynamic evolution equations of the DGZND model with the time derivatives of the robotic arm's kinematic errors. This process begins with the definition of the error function, revealing the time-varying characteristics of the Jacobian matrix and the dynamic coupling relationship between the joint angular velocity and its derivative, and the desired end-effector velocity derivative, through time differentiation. Then, the evolution equations of the DGZND model, including dynamic gain terms and error integral negative feedback terms, are introduced to establish a linear equation for joint angular acceleration (i.e., the joint angular velocity derivative). The resulting dynamic solver clearly expresses how joint angular acceleration is jointly determined by the current Jacobian matrix and its derivative, the joint angular velocity, the desired end-effector velocity and its derivative, and the core adjustment terms of the DGZND model, providing a direct computational basis for subsequent real-time numerical solutions to joint motion parameters. This derivation process not only ensures the theoretical rigor of the model but also paves the way for algorithm implementation in practical engineering applications, enabling the advantages of the DGZND model to be directly translated into precise control of robotic arm joint motion.

[0108] In some embodiments, the step of using the ODE numerical method to solve the dynamic solver and outputting the joint angular velocity vector in real time includes:

[0109] S610, in each control cycle, acquire the current joint angle θ(t), calculate the Jacobian matrix J(θ(t)) and its time derivative. ;

[0110] S620, calculate the terminal velocity B(t) and its time derivative based on the desired trajectory. ;

[0111] S630, calculate the current error e(t) = A(t)X(t) - B(t) and the error integral term. ;

[0112] S640, based on the current time t and the error norm Calculate piecewise adaptive coefficients ;

[0113] S650, the terms on the right-hand side of the dynamic solver equation are passed through the pseudo-inverse of the Jacobian matrix. Solve for the rate of change of joint angular velocity to obtain the solution. ; ;

[0114] S660 uses a numerical integration method to measure the rate of change of joint angular velocity. Integrate to obtain the joint angular velocity vector at the current moment.

[0115] This embodiment ensures the real-time and accurate output of the joint angular velocity vector by strictly following the ODE numerical solution process. Within each control cycle, the current joint angle θ(t) of the planar two-bar linkage is first acquired using a high-precision encoder. Combined with the structural parameters of the manipulator (such as link length), the Jacobian matrix J(θ(t)), i.e., A(t), is accurately calculated. The result is obtained by differentiating the Jacobian matrix with respect to time. Simultaneously, based on the desired trajectory (such as a straight line, arc, or complex curve) provided by the host computer, the desired velocity vector B(t) of the end effector at the current moment and its time derivative are calculated in real time using a trajectory planning algorithm. Next, the error function e(t) is calculated using the current A(t), X(t), and B(t), and the integral term of the error from the start of the motion to the current moment is accumulated by an integrator. Then, based on the current time t, the piecewise adaptive coefficient expression to be used is determined: if t ≤ 0.01 seconds, then the expression containing a constant term is used. To utilize a larger error norm in the initial stage of motion. This generates a significant gain, enabling rapid error convergence; if t > 0.01 seconds, then switch to... Even if the error norm decreases, the gain can still dynamically adjust with the error, ensuring convergence accuracy. Substituting the calculated parameters into the right side of the dynamic solver equation, we obtain... This composite vector is then left-multiplied by the pseudo-inverse of the Jacobian matrix A(t). The rate of change of joint angular velocity can then be calculated. This refers to the joint angular acceleration vector. Finally, efficient numerical integration methods such as the Runge-Kutta method are used to... Integrating within one control cycle and superimposing it onto the joint angular velocity vector from the previous moment yields the precise joint angular velocity vector for the current moment. This series of steps is closely linked, and the calculations in each step are optimized for the characteristics of the planar two-bar linkage robotic arm, ensuring that all calculations are completed within a microsecond-level control cycle, thereby meeting the stringent requirements of the robotic arm for real-time motion generation.

[0116] In some embodiments, the method further includes:

[0117] During the movement of the robotic arm, the actual rotation angle of each joint of the robotic arm and the actual position of the end effector are collected and compared with the joint angular velocity control command. If the deviation exceeds the preset threshold, closed-loop feedback control is triggered, and feedback control command is generated for real-time adjustment.

[0118] During the robotic arm's movement, high-precision photoelectric encoders installed at the joints collect the actual rotation angles of each joint in real time, and calculate the actual position of the end effector using forward kinematics. The collected actual joint rotation angles are compared time-by-time with the desired rotation angle obtained by integrating the joint angular velocity control commands generated based on the DGZND model. Simultaneously, the calculated actual end effector position is compared with the desired end effector position in the planned trajectory. When the deviation between the actual and desired rotation angles of any joint, or when the Euclidean distance between the actual and desired end effector positions exceeds a distance threshold, the system immediately triggers a closed-loop feedback control mechanism. At this time, the feedback controller generates additional feedback control commands based on the magnitude and direction of the deviation, using proportional-derivative (PD) control or fuzzy PID control algorithms. This feedback control command does not directly replace the joint angular velocity commands output by the original DGZND model, but is weighted and superimposed on the joint angular velocity vector X(t), and can adaptively adjust according to the degree of deviation. This composite control strategy enables the robotic arm to maintain high-precision tracking of the end effector trajectory even when faced with sudden disturbances (such as load changes or minor external collisions) or small perturbations in model parameters. This significantly improves the robustness and anti-interference capability of the entire motion generation system, ensuring the stable and reliable operation of the robotic arm under complex working conditions.

[0119] The following are the specific implementation steps of this invention.

[0120] Step 1: Problem Transformation and Error Function Definition. As mentioned earlier, the kinematics problem of a planar two-bar linkage robot arm can be expressed as: We abstract it into a standard time-varying linear equation. ,in: , that is, the Jacobian matrix of the planar two-bar linkage manipulator; , that is, the angular velocity vector of the planar two-bar manipulator joint to be solved; This refers to the desired terminal velocity vector. We define an error function. Used to measure the current solution Deviation from theoretical solution: .

[0121] Step 2: Construction of the DGZND model. The design idea of ​​the traditional ZNN model is... That is, let the error It decays exponentially to zero. This invention makes a significant improvement on this, proposing the DGZND model, whose dynamic evolution equation is defined as: , It is an activation function used to apply control force under specific conditions. This model contains two major innovations:

[0122] 1. Piecewise adaptive coefficient function This is a non-linear phenomenon that varies with error. The gain coefficient is dynamically adjusted in size. Its expression is as follows: in These are design parameters. It is the Frobenius norm of the error matrix, used to measure the magnitude of the error. The innovative aspect is that, in the initial stage of the solution process ( Smaller When the value is large, this function can provide a huge gain, causing the error to converge very quickly. For example, when t ≤ 0.01, a larger gain of +12 is used; in the later stages ( Larger When the value is relatively small, a smaller gain is provided to ensure smooth system convergence and avoid jitter. For example, when t > 0.01, an adaptive gain is used.

[0123] 2. Negative feedback regulation (integral term) This is an integral element introduced from control theory, in which... It refers to the integral gain. The innovative aspect is that the integral term accumulates historical errors. When the system is disturbed by external noise (such as constant noise or time-varying noise), it causes… When the integral term deviates from zero, it generates a reverse adjustment force, actively suppressing and eliminating steady-state errors caused by noise. This is the key to achieving strong noise immunity in this invention.

[0124] Step 3: Export the solver We have two things about The expression:

[0125] 1. Differentiate according to the definition of error: .

[0126] 2. According to the DGZND model of the present invention: .

[0127] Setting the two equal, we get: .

[0128] After sorting, we obtained The dynamic evolution equations, i.e., the solver of this invention: .

[0129] In practical applications, (Right now ) is the solution at the current moment. , and its derivative , All of these can be calculated based on the state of the planar two-bar linkage manipulator and the desired trajectory. Therefore, the right side of the solver is completely known. The solution can be obtained by solving this linear equation. For example, using pseudo-inverse Please provide a solution.

[0130] Step 4: Real-time solution and control. In robot control systems (such as MATLAB / Simulink or embedded systems), the following can be achieved:

[0131] 1. Set initial state .

[0132] 2. The solver (the final equation of the third step) is put into an integrator.

[0133] 3. In each control cycle, the solver calculates the current state... The integrator integrates it to obtain the next time step. ,Right now (Joint angular velocity).

[0134] 4. The servo motors of the planar two-bar linkage robotic arm are sent to execute the commands.

[0135] 5. At the same time, (Right now Integrate again to obtain the joint angle. Used to calculate the next period This forms a closed loop.

[0136] Step 5: Implementation Results (Simulation Verification). We simulated the method of this invention to drive a planar two-bar linkage robotic arm to track a heart-shaped trajectory, verifying the effectiveness, accuracy, and robustness of the invention (DGZND model). The simulation results are as follows: Figures 2 to 10 As shown.

[0137] The following is combined Figures 2 to 10 The simulation results and performance analysis of the present invention are described in detail.

[0138] Figure 2 This is the core proof of the high-precision performance of this invention. It simultaneously plots the pre-defined desired trajectory (red heart-shaped curve) and the actual motion trajectory (blue curve) of the robotic arm end effector controlled by the DGZND model of this invention on the XY plane. As can be clearly seen from the figure, the blue solid line (actual trajectory) and the red curve (desired trajectory) almost perfectly overlap. This intuitively demonstrates that the DGZND model of this invention has extremely high solution accuracy and tracking performance, capable of reproducing complex heart-shaped target paths without errors.

[0139] Figure 3 This demonstrates how the planar two-bar linkage robotic arm can achieve... Figure 2The physical motion is performed based on a heart-shaped trajectory. In the figure, point (0,0) represents the robot arm base, and multiple green lines show a series of posture snapshots of the two links of the robot arm during the simulation process. The thick purple solid line trajectory drawn by the end effector (the end of joint 2) is the heart-shaped path. This figure demonstrates that the motion process calculated by this invention is smooth, continuous, and physically reachable.

[0140] Figure 4 This figure shows the real-time joint angles calculated by the DGZND model of this invention for two joints (joint 1 and joint 2) of the robotic arm within a 40-second simulation time. In the figure, the horizontal axis t(s) represents time (seconds), and the vertical axis represents the joint angle (radians). (Curve) (Blue) represents the angle change of joint 1, curve (Red) represents the angle change of joint 2. As can be seen from the figure, both curves exhibit highly smooth and continuous characteristics, without any abrupt changes or jitters. This proves that the motion calculated by this invention is smooth and can be smoothly executed by an actual servo motor.

[0141] Figure 5 Decomposed in the form of a time history diagram Figure 2 The graph shows a heart-shaped trajectory. The horizontal axis t(s) represents time (seconds), and the vertical axis represents position (meters). The blue curve (X) shows the change of the end effector's position on the X-axis over time, and the red curve (Y) shows the change of its position on the Y-axis over time.

[0142] Figure 6 The graph displays the angular velocities of the two joints calculated in real time by the model. The horizontal axis t (s) represents time (seconds), and the vertical axis represents angular velocity (radians / second). (Curve) (Blue) represents the angular velocity of joint 1, curve (Red) represents the angular velocity of joint 2. The two angular velocity curves are smooth and bounded, without oscillation or divergence, which indicates that the DGZND solver of this invention is stable and reliable, and the physical meaning of the generated control commands is correct.

[0143] Figure 7 The figure shows the desired velocity command tracked by the model of this invention. The horizontal axis t(s) represents time (seconds), and the vertical axis represents velocity (meters / second). The blue curve (dX) represents the desired velocity component of the cardioid trajectory in the X-axis direction, and the red curve (dY) represents its desired velocity component in the Y-axis direction.

[0144] Figure 8 This demonstrates the motion path of robotic arm joint 1 (i.e., the elbow connecting the first and second links) in the XY plane. This is consistent with... Figure 2The different trajector ...

[0145] Figure 9 This is quantitative proof of the high-precision performance of the present invention. It shows the actual trajectory ( Figure 2 (blue curve) and expected trajectory ( Figure 2 The positional error between the red curve and the horizontal axis (t(s)) is shown in the figure. The horizontal axis represents time (seconds), and the vertical axis is in units of 10⁻³ meters (millimeters). The blue curve (Fx) represents the error in the X direction, and the red curve (Fy) represents the error in the Y direction. Throughout the entire 40-second complex motion, the positional error was consistently suppressed to within approximately 2.5 × 10⁻³ meters (i.e., 2.5 millimeters), demonstrating the extremely high tracking accuracy of this invention.

[0146] Figure 10 This is the core evidence verifying the fast convergence and high accuracy of the DGZND model in this invention. It demonstrates the velocity error of the end effector, i.e. In the graph, the horizontal axis t(s) represents time (seconds), and the vertical axis is in units of 10⁻. 5 m / s. The blue curve (edx) and the red curve (edy) represent the velocity errors in the X and Y directions, respectively. As can be seen from the graph, the velocity error is controlled within 1.8 × 10⁻⁻⁻⁻⁶. 4 Within the following minimal range, this demonstrates that the DGZND model solver of this invention converges extremely fast and is highly accurate, enabling near-instantaneous and precise solutions to time-varying equations.

[0147] Performance Comparison Analysis: Compared with the traditional ZNN model, the DGZND model of this invention improves convergence speed by approximately 10 times, reducing convergence time from 0.1 seconds to 0.01 seconds. Regarding noise resistance, when random noise with an amplitude of ±0.01 is applied, the position error of the traditional OZNN model can reach 0.02m, while the error of the method of this invention remains within 2.5 × 10⁻³m, representing an accuracy improvement of approximately 8 times. In terms of resource consumption, the algorithm of this invention executes in only 0.6ms on an ARM Cortex-M4 processor (168MHz), meeting the real-time requirement of a 1ms control cycle, while traditional methods require higher-performance processors.

[0148] Application Example: In a precision electronic component assembly task, a robotic arm is required to move along a cardioid trajectory at a speed of 0.05 m / s, with a positional accuracy requirement of ±0.1 mm. Using the DGZND control method of this invention, with parameters k1=2, k2=1, and β=5, simulation results show that the position tracking error is stable within 2.5 × 10⁻³ m (i.e., 2.5 mm), and the speed error is controlled within 1.8 × 10⁻³ m. 4The system can still maintain an error within 3mm even under mechanical vibration interference (frequency 10Hz, amplitude 0.01mm), demonstrating excellent anti-interference capability.

[0149] The beneficial effects of this embodiment are as follows.

[0150] 1. High precision and real-time performance. For example... Figure 2 , Figure 9 and Figure 10 As shown, the method of the present invention can solve the kinematic equations of a planar two-bar linkage robot arm in real time and with high accuracy. Figure 2 Their trajectories perfectly overlapped. Figure 9 The positional error is controlled within the millimeter level. Figure 10 The speed error is further controlled within The scale demonstrates exceptional tracking performance.

[0151] 2. Smoothness and stability. For example... Figure 3 , Figure 4 and Figure 8 As shown, the joint angles, angular velocities, and elbow trajectory curves calculated by the model are all smooth and continuous, without jitter or abrupt changes, ensuring the stability of the physical execution of the robotic arm and avoiding impact on the hardware.

[0152] 3. Fast convergence. For example... Figure 10 As shown, the speed error is in The magnitude of the data demonstrates that, thanks to the piecewise adaptive coefficients, the system error converges extremely quickly, making it suitable for industrial scenarios requiring rapid start-up, shutdown, and response.

[0153] Compared with related technologies, the DGZND model of the present invention has the following improvements:

[0154] 1. Significantly enhanced noise resistance (robustness). Traditional ZNNs are sensitive to noise and are prone to jitter and errors in practical applications. This invention, through the original introduction of negative feedback adjustment (i.e., integral term), enables the model to actively suppress constant noise and time-varying noise, maintaining extremely high solution accuracy and stability even in harsh industrial environments.

[0155] 2. Significantly improved convergence speed. Traditional ZNNs use a fixed gain, which limits their convergence speed. This invention uses piecewise adaptive coefficients, applying high gain in the early stages of the solution (when the error is large), making the convergence speed several orders of magnitude faster than traditional ZNNs; applying low gain in the later stages (when the error is small) ensures stability.

[0156] 3. Combining high precision and real-time performance. This invention inherits the advantage of ZNN models with no hysteresis error, while overcoming its shortcomings through the above innovations. It achieves higher precision than GNN and faster speed and stronger robustness than traditional ZNN, making it a superior solution for real-time motion generation of robotic arms.

[0157] This invention also provides a real-time robotic arm motion generation system based on the DGZND model, comprising:

[0158] At least one processor;

[0159] At least one memory for storing at least one program;

[0160] When the at least one program is executed by the at least one processor, the at least one processor performs the method described above.

[0161] The content of the above method embodiments is applicable to this embodiment. The specific functions implemented in this embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those achieved in the above method embodiments. Therefore, they will not be repeated here.

[0162] This invention also provides an electronic device, which includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the method described above. This electronic device can be any smart terminal, including tablet computers, in-vehicle computers, etc.

[0163] It is understood that the content of the above method embodiments is applicable to this device embodiment. The specific functions implemented by this device embodiment are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0164] This invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described method.

[0165] It is understood that the content of the above method embodiments is applicable to this storage medium embodiment. The specific functions implemented in this storage medium embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those achieved in the above method embodiments.

[0166] This invention also provides a computer program product, including a computer program or computer instructions, which are stored in a memory. A processor of a computer device reads the computer program or computer instructions from the memory and executes the computer program or computer instructions, causing the computer device to perform the above-described method.

[0167] It is understood that the content of the above method embodiments is applicable to the embodiments of this program product. The specific functions implemented by the embodiments of this program product are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0168] Memory, as a non-transitory computer-readable storage medium, can be used to store non-transitory software programs and non-transitory computer-executable programs. Furthermore, memory may include high-speed random access memory, and may also include non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some embodiments, memory may optionally include memory remotely located relative to the processor, and these remote memories can be connected to the processor via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0169] It will be understood by those skilled in the art that all or some of the steps and systems in the methods disclosed above can be implemented as software, firmware, hardware, and suitable combinations thereof. Some or all of the physical components can be implemented as software executed by a processor, such as a central processing unit, digital signal processor, or microprocessor, or as hardware, or as an integrated circuit, such as an application-specific integrated circuit. Such software can be distributed on a computer-readable medium, which can include computer storage media (or non-transitory media) and communication media (or transient media). As is known to those skilled in the art, the term computer storage media includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storing information (such as computer-readable instructions, data structures, program modules, or other data). Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technologies, CD-ROM, digital versatile disc (DVD) or other optical disc storage, magnetic cartridges, magnetic tape, disk storage or other magnetic storage devices, or any other medium that can be used to store desired information and is accessible to a computer. Furthermore, as is known to those skilled in the art, communication media typically include computer-readable instructions, data structures, program modules, or other data in modulated data signals such as carrier waves or other transmission mechanisms, and may include any information delivery medium.

[0170] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs.

Claims

1. A real-time robot arm motion generation method based on a DGZND model, characterized by, The method comprises the following steps: S100, acquiring kinematics parameters, joint displacement ranges and expected trajectory parameters of a planar two-link mechanical arm; S200, converting a kinematics equation of the planar two-link mechanical arm into a standard time-varying linear equation; the standard time-varying linear equation is A(t)X(t)=B(t), wherein A(t) is a Jacobian matrix, X(t) is a joint angular velocity vector to be solved, and B(t) is an expected end velocity vector; S300, determining a segmented adaptive coefficient function and an error function to construct a dynamic gain term; the dynamic gain term is: wherein e(t) is an error function, e(t) = A(t)X(t) - B(t), respectively gain coefficients of the segmented adaptive coefficient function, is a segmented adaptive coefficient function, used for dynamically adjusting the size of the gain according to the modulus value of the error e(t) and time, ψ(·) is an activation function, and t is a current time; is a derivative of the error function e(t) with respect to time t; S400, determine the error integral term to construct a negative feedback regulation mechanism; construct a DGZND model based on the dynamic gain term and the error integral term; wherein the equation of the DGZND model is: ; β is the integral gain coefficient, is the error integral term, is the integral variable, is the derivative of the error function e(t) with respect to time t; S500, deriving a dynamic solver of joint angular velocity derivatives according to the DGZND model and a defined error function, wherein a formula of the dynamic solver is: ; wherein, is a derivative of the Jacobian matrix A(t) with respect to time t, is a derivative of the joint angular velocity vector X(t) with respect to time t, i.e., a joint angular acceleration vector, is a derivative of the desired end velocity vector B(t) with respect to time t; S600, solving the dynamic solver by using an ODE numerical method, outputting a joint angular velocity vector in real time, outputting a control instruction based on the joint angular velocity vector, and driving a planar two-link mechanical arm actuator to move.

2. The method of claim 1, wherein, The converting of the kinematics equation of the planar two-link mechanical arm into the standard time-varying linear equation comprises: S210, acquire the joint angle vector of the planar two-link mechanical arm wherein and are the joint angles of the two links, respectively S220, calculating a Jacobian matrix according to the joint angle vector , establishing kinematics equation wherein is the end desired velocity vector; S230, transforming the kinematic equation into a standard form, let , , resulting in a time-varying linear equation A(t)X(t) = B(t).

3. The method of claim 1, wherein, The expression of the segmented adaptive coefficient function is as follows: ； wherein , is the Frobenius norm of the error matrix, which measures the size of the error e(t).

4. The method of claim 1, wherein, The determining of the error integral term for constructing a negative feedback regulation mechanism comprises: S410, introducing an error integration term , accumulating historical error information; S420, performing nonlinear transformation on the error integral term by using an activation function ψ(·), and an integral gain coefficient β>0 is used for adjusting the negative feedback strength; S430, when a robot control system is disturbed by noise, the error integral term generates a reverse regulation force to actively suppress and eliminate a steady-state error caused by the noise.

5. The method of claim 1, wherein, The deriving of a dynamic solver of a joint angular velocity derivative according to the DGZND model and a defined error function comprises: S510, derive the error function e(t) to get ; S520, make the dynamic evolution equation of DGZND model equal to the derivative of error function, get equation: ; S530, the equation is arranged to obtain a dynamic solver: .

6. The method of claim 1, wherein, The solving of the dynamic solver by using an ODE numerical method and the real-time output of a joint angular velocity vector comprise: S610, in each control cycle, collect the current joint angle θ(t), calculate the Jacobian matrix J(θ(t)) and its time derivative ; S620, calculating end velocity B(t) and its time derivative according to the desired trajectory ; S630, calculate the current error e(t) = A(t)X(t) - B(t) and the error integral term ; S640, according to the current time t and the error norm Computing the segment adaptive coefficients ; S650, solve the dynamic solver equation right side of each item through the pseudo-inverse of the Jacobian matrix Solve to get joint angular velocity rate of change ; ; S660, the rate of change of the joint angle velocity is calculated using a numerical integration method The joint angle velocity vector at the current time is obtained by integrating.

7. The method of claim 1, wherein, The method further comprises: During the movement of the mechanical arm, the actual rotation angles of the joints of the mechanical arm and the actual position of the end effector are collected, and are compared with the joint angular velocity control instruction; if the deviation exceeds a preset threshold, a closed-loop feedback control is triggered to generate a feedback control instruction for real-time adjustment.

8. A real-time robot motion generation system based on DGZND model, characterized in that, Comprise: At least one processor; At least one memory for storing at least one program; When the at least one program is executed by the at least one processor, the at least one processor implements the method in any one of claims 1 to 7.

9. A computer-readable storage medium storing a computer program, the computer program comprising instructions that, when executed by a computer, cause the computer to perform the method of any one of claims 1 to 8. The computer program is executed by the processor to implement the method in any one of claims 1 to 7.

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