Trajectory drawing method and system for mechanical arm based on aztnd model

Abstract

The present application relates to the technical field of mechanical arm control, and particularly relates to a mechanical arm trajectory drawing method and system based on an AZTND model; the method comprises the following steps: acquiring mechanical arm curve drawing task parameters and an AZTND solver; establishing a mechanical arm time-varying kinematics equation, introducing joint angular velocity constraints and acceleration constraints, and converting the curve drawing problem into a time-varying nonlinear equation; defining an error function as the residual of the time-varying nonlinear equation, constructing an AZTND dynamics equation, and determining the parameters of a segmented adaptive coefficient function according to the curve drawing stage; using an ODE numerical solution method to solve the AZTND dynamics equation in real time, obtaining a mechanical arm joint angular velocity control instruction, and driving a mechanical arm end effector to track an expected curve trajectory. The present application solves the problems of slow convergence speed, weak anti-interference ability and low geometric precision in traditional control in time-varying trajectory tracking, and realizes high-precision real-time drawing of complex artistic curves.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, specifically to a method and system for drawing robotic arm trajectories based on the AZTND model. Background Technology

[0002] In modern industrial automation, the trajectory tracking accuracy and real-time performance of robotic arms directly impact production quality and efficiency. Especially in high-end fields such as aerospace and precision instrument manufacturing, robotic arms need to achieve micron-level trajectory tracking accuracy in noisy, highly disruptive industrial environments. Traditional control methods based on fixed parameters often suffer from response lag and decreased accuracy when faced with time-varying trajectories, model uncertainties, and external disturbances.

[0003] As intelligent manufacturing evolves towards flexibility and intelligence, the tasks of robotic arms have expanded from simple repetitive movements to complex trajectory tracking requiring real-time adjustments. This dynamic characteristic demands that the control system possess online adaptive adjustment capabilities, a requirement that traditional control methods struggle to meet. While neural network control methods have addressed time-varying issues to some extent, they still have shortcomings in noise suppression and convergence speed.

[0004] Currently, the main technical solutions used for industrial robotic arm control are as follows:

[0005] Traditional PID control achieves control through a linear combination of proportional, integral, and derivative components. It has a simple structure but fixed parameters, making it difficult to adapt to time-varying systems and external disturbances.

[0006] Neural network control based on gradient descent: It uses neural networks to approximate nonlinear functions and adjusts weights through the gradient descent algorithm. However, it suffers from slow convergence speed and is prone to getting trapped in local optima in time-varying systems.

[0007] The basic model of the null neural network: by constructing an error function and designing an evolutionary rule to make it converge to zero, it can handle time-varying problems but has weak anti-interference ability.

[0008] Adaptive sliding mode control: It addresses system uncertainties by designing sliding surfaces and adaptive laws, but chattering occurs, affecting control accuracy.

[0009] The existing technology has the following main drawbacks:

[0010] Parameter fixation problem: Traditional control methods use fixed control parameters, which cannot be adjusted in real time according to the system state, resulting in performance degradation in time-varying environments.

[0011] Noise sensitivity: Existing neural network control is prone to error accumulation under sensor noise and external interference, which may even lead to system instability.

[0012] Insufficient convergence performance: Gradient-based methods converge slowly, making it difficult to meet the real-time requirements of highly dynamic environments.

[0013] Constraint handling is difficult: Most methods struggle to handle multiple constraints on a robotic arm, such as joint limits and speed constraints, simultaneously. Summary of the Invention

[0014] To address the aforementioned technical problems, this invention provides a method and system for drawing robotic arm trajectories based on the AZTND model, aiming to achieve high-precision real-time tracking and drawing of complex artistic curves (especially butterfly curves), significantly improving the convergence speed, anti-interference ability, and geometric accuracy of robotic arms in time-varying trajectory tracking.

[0015] To achieve the above objectives, the present invention provides the following technical solution:

[0016] On one hand, embodiments of the present invention provide a method for drawing the trajectory of a robotic arm based on the AZTND model, the method comprising the following steps:

[0017] S100, acquire the robot arm curve drawing task parameters and AZTND solver, the AZTND solver includes a piecewise adaptive coefficient function, a negative feedback integral term and a noise term;

[0018] S200: Establish the time-varying kinematic equations of the robotic arm, introduce joint angular velocity constraints and acceleration constraints, and transform the curve drawing problem into a time-varying nonlinear equation;

[0019] S300, define the error function as the residual of the time-varying nonlinear equation, construct the AZTND dynamic equation, the AZTND dynamic equation includes a product term of the piecewise adaptive coefficient function and the activation function, a negative feedback integral term and a noise term, the negative feedback integral term eliminates steady-state deviation by integrating and accumulating historical errors;

[0020] S400, the parameters of the piecewise adaptive coefficient function are determined according to the curve drawing stage. The piecewise adaptive coefficient function uses a first gain coefficient in the initial stage to ensure the system is fast and stable, a second gain coefficient in the main drawing stage to ensure high-precision tracking of complex curvature changes, and a third gain coefficient in the completion stage to maintain accuracy and save computing resources.

[0021] S500 uses the ODE numerical solution method to solve the AZTND dynamic equation in real time, obtains the joint angular velocity control command of the robotic arm, and drives the end effector of the robotic arm to track the desired curve trajectory.

[0022] Optionally, in S200, the step of establishing the time-varying kinematic equations of the robotic arm, introducing joint angular velocity constraints and acceleration constraints, and transforming the curve drawing problem into a time-varying nonlinear equation includes:

[0023] Establish time-varying kinematic equations ,in, The rate of change of the desired position of the robotic arm's end effector. For the Jacobian matrix of the robotic arm, The joint angular velocity to be solved;

[0024] Introducing joint angular velocity constraints and acceleration constraints;

[0025] The constrained optimization problem is transformed into a complementary-condition form using KKT conditions, and then smoothed using the NCP function to obtain time-varying nonlinear equations. ,in, The time-varying coefficient matrix, For time-varying bias vectors, The state variables include the joint angular velocities to be determined. And Lagrange multipliers.

[0026] Optionally, in S300, the definition of the error function as the residual of the time-varying nonlinear equation, and the construction of the AZTND dynamic equation, includes:

[0027] Define error function ;

[0028] Constructing the dynamic equations of AZTND ,in, For piecewise adaptive coefficient functions, For activation function, This is the integral gain coefficient. For noise terms, These are the exponential coefficients, This is the noise term.

[0029] Optionally, in S300, the product term of the piecewise adaptive coefficient function and the activation function is defined piecewise according to the drawing time:

[0030] When the rendering time is in the initial stage, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm and superimposed with the first constant gain to ensure that the system stabilizes quickly.

[0031] When the rendering time is in the main rendering phase, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm and superimposed with the second constant gain to ensure high-precision tracking of complex curvature changes;

[0032] When the rendering time is in the completion stage, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm, which saves computing resources while maintaining accuracy;

[0033] The first constant gain is less than the second constant gain.

[0034] Optionally, in S500, the step of using the ODE numerical solution method to solve the AZTND dynamic equation in real time to obtain the joint angular velocity control command of the robotic arm and drive the end effector of the robotic arm to track the desired curve trajectory includes:

[0035] The AZTND dynamic equations are transformed into a system of ordinary differential equations concerning the state variables;

[0036] The ordinary differential equations are solved in discrete time using numerical integration to obtain the joint angular velocity values ​​at each time step.

[0037] The robot arm joint state is updated based on the obtained joint angular velocity to achieve closed-loop feedback control.

[0038] Optionally, the curve drawing task is a butterfly curve drawing task, and the desired curve trajectory is defined by a parametric equation, which is:

[0039] ;

[0040] ;

[0041] ;

[0042] Where A is the scaling factor, The scaling factor controls the curve size, and the angular frequency controls the drawing speed.

[0043] Optionally, the method further includes:

[0044] Set the parameters of the robotic arm, using the UR5 robotic arm as the controlled object, and configure the initial joint angle and joint angular velocity limits;

[0045] Set the butterfly curve parameters, and configure the scaling factor and angular frequency to control the curve size and drawing speed;

[0046] Noise terms were added to the simulation environment to simulate random disturbances during the actual drawing process, thus verifying the anti-interference capability of the AZTND model.

[0047] On the other hand, embodiments of the present invention provide a robotic arm trajectory drawing system based on the AZTND model, comprising:

[0048] At least one processor;

[0049] At least one memory for storing at least one program;

[0050] When the at least one program is executed by the at least one processor, the at least one processor performs the method described above.

[0051] 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.

[0052] The embodiments of the present invention have the following beneficial effects:

[0053] The AZTND model proposed in this invention innovatively integrates a piecewise adaptive coefficient function and a negative feedback integral mechanism, which can dynamically adjust control parameters without relying on offline trajectory planning. It effectively solves the parameter solidification problem in traditional control in time-varying trajectory tracking, and its convergence speed and robustness are significantly better than traditional OZNN (Original Zeroing Neural Network) and GNN (Gradient Neural Network) methods.

[0054] The piecewise adaptive coefficient function designed in this invention intelligently adjusts the gain coefficient according to different stages of curve drawing (start, main drawing, and completion). In the start stage, it quickly stabilizes the system to avoid jitter; in the main drawing stage, it enhances the gain to ensure high-precision tracking of complex curvature changes; and in the completion stage, it saves computing resources, thus achieving an optimal balance between control accuracy and computational efficiency.

[0055] The negative feedback integral term introduced in this invention can accumulate and eliminate historical errors, which is crucial for suppressing continuous low-frequency interference during the drawing process, effectively preventing the overall shape of the curve from being distorted, ensuring geometric symmetry, and significantly improving the quality of artistic drawing.

[0056] This invention achieves real-time, high-precision tracking of complex artistic curves through an end-to-end adaptive learning mechanism without requiring offline trajectory planning. This simplifies system deployment, enhances the robotic arm's real-time perception and control capabilities in noisy environments, and achieves an error accuracy of [missing information]. Meter level. Attached Figure Description

[0057] 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.

[0058] Figure 1This is a flowchart illustrating the robotic arm trajectory drawing method based on the AZTND model in an embodiment of the present invention.

[0059] Figure 2 This is an analysis diagram of the joint motion angles of a six-degree-of-freedom robotic arm in an embodiment of the present invention;

[0060] Figure 3 This is an analysis diagram of the joint angular velocity of a six-degree-of-freedom robotic arm in an embodiment of the present invention;

[0061] Figure 4 This is a three-axis position trajectory diagram of the end effector of the robotic arm in an embodiment of the present invention;

[0062] Figure 5 This is an analysis diagram of the end effector speed of the robotic arm in an embodiment of the present invention;

[0063] Figure 6 This is a three-axis acceleration analysis diagram of the end effector of the robotic arm in an embodiment of the present invention;

[0064] Figure 7 This is a diagram showing the three-axis position error analysis of the end effector of the robotic arm in this embodiment of the invention;

[0065] Figure 8 This is a diagram showing the three-axis speed error analysis of the end effector of the robotic arm in this embodiment of the invention;

[0066] Figure 9 This is a diagram showing the triaxial acceleration error analysis of the robotic arm end effector in an embodiment of the present invention.

[0067] Figure 10 This is a comparison diagram of the expected trajectory and the actual trajectory of the robotic arm in an embodiment of the present invention;

[0068] Figure 11 This is a simulation diagram of the three-dimensional spatial motion trajectory of the end effector of the robotic arm in an embodiment of the present invention;

[0069] Figure 12 This is a diagram illustrating the effect of a robotic arm drawing a butterfly trajectory in an embodiment of the present invention. Detailed Implementation

[0070] 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.

[0071] 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.

[0072] 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.”

[0073] 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.

[0074] 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.

[0075] refer to Figure 1 ,like Figure 1 The figure shows a method for drawing the trajectory of a robotic arm based on the AZTND model provided by an embodiment of the present invention. The method includes the following steps:

[0076] S100, acquire the robot arm curve drawing task parameters and AZTND solver, the AZTND solver includes a piecewise adaptive coefficient function, a negative feedback integral term and a noise term;

[0077] S200: Establish the time-varying kinematic equations of the robotic arm, introduce joint angular velocity constraints and acceleration constraints, and transform the curve drawing problem into a time-varying nonlinear equation;

[0078] Specifically, the time-varying kinematic equations of the robotic arm are established. ,in The rate of change of the desired position of the robotic arm's end effector. For the Jacobian matrix of the robotic arm, The joint angular velocity is to be solved. To ensure a safe and smooth drawing process while maintaining artistic quality, joint angular velocity constraints are added. And acceleration constraints. The constrained optimization problem is transformed into a complementary-condition form using KKT conditions, and then smoothed using the NCP function to obtain the time-varying nonlinear equations. ,in Includes the joint angular velocity to be determined And Lagrange multipliers.

[0079] S300, define the error function as the residual of the time-varying nonlinear equation, construct the AZTND dynamic equation, the AZTND dynamic equation includes a product term of the piecewise adaptive coefficient function and the activation function, a negative feedback integral term and a noise term, the negative feedback integral term eliminates steady-state deviation by integrating and accumulating historical errors;

[0080] Specifically, define the error function. Constructing the AZTND dynamic equations:

[0081] ,in For piecewise adaptive coefficient functions, For activation function, This is the integral gain coefficient. The noise term is represented by the equation. The equation comprises three core components: a piecewise adaptive coefficient function that dynamically adjusts the convergence strength based on the error magnitude and the plotting stage; a negative feedback integral term that accumulates historical errors to eliminate steady-state bias; and a noise term that simulates random disturbances in the real-world environment.

[0082] S400, the parameters of the piecewise adaptive coefficient function are determined according to the curve drawing stage. The piecewise adaptive coefficient function uses a first gain coefficient in the initial stage to ensure the system is fast and stable, a second gain coefficient in the main drawing stage to ensure high-precision tracking of complex curvature changes, and a third gain coefficient in the completion stage to maintain accuracy and save computing resources.

[0083] Specifically, the piecewise adaptive coefficient function is defined in segments according to the plotting time:

[0084] When the rendering time is in the initial stage, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm and superimposed with the first constant gain to ensure that the system stabilizes quickly.

[0085] When the rendering time is in the main rendering phase, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm and superimposed with the second constant gain to ensure high-precision tracking of complex curvature changes;

[0086] When the rendering time is in the completion stage, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm, which saves computing resources while maintaining accuracy.

[0087] The first constant gain is less than the second constant gain.

[0088] S500 uses the ODE numerical solution method to solve the AZTND dynamic equation in real time, obtains the joint angular velocity control command of the robotic arm, and drives the end effector of the robotic arm to track the desired curve trajectory.

[0089] Specifically, the AZTND dynamic equations are transformed into equations concerning state variables. The system of ordinary differential equations is solved in discrete time using numerical integration. The obtained joint angular velocities are used as control commands to drive the robotic arm's movement, achieving closed-loop feedback control.

[0090] This invention provides a method and system for drawing robotic arm trajectories based on the AZTND model. By dynamically adjusting control parameters according to the drawing stage using a piecewise adaptive coefficient function, it effectively solves the parameter fixation problem in time-varying trajectory tracking in traditional control, avoiding response lag or excessive oscillation caused by fixed gain. The innovative introduction of a historical error accumulation mechanism in the negative feedback integral term eliminates steady-state errors through integral action, significantly suppressing the impact of low-frequency interference on drawing accuracy. This invention effectively addresses the core challenges of time-varying characteristics, constraints, and environmental interference in robotic arm control. Through the synergistic effect of adaptive parameter adjustment and integral compensation, it achieves high-precision real-time drawing of complex artistic curves, providing more reliable technical support for precision trajectory tracking tasks in industrial automation.

[0091] The core of this embodiment lies in constructing a real-time controller based on the AZTND (Activated Zeroing-Type Neural Dynamics) model. The technical solution process is as follows:

[0092] Step 1: Task modeling and problem transformation for drawing the butterfly curve of the robotic arm;

[0093] Establish time-varying kinematic equations: ,in It is the Jacobian matrix of the robotic arm. It is the joint angular velocity to be solved.

[0094] Introducing artistic drawing constraints: To ensure a safe and smooth drawing process while maintaining artistic quality, joint angular velocity constraints need to be added. And acceleration constraints.

[0095] Convert to equation form: Through the KKT conditions and the NCP function, the above butterfly curve drawing problem is finally converted into a problem of solving a time-varying nonlinear equation: , where is a time-varying coefficient matrix, which is composed of the Jacobian matrix, the gradient of the constraint conditions, and the Jacobian matrix of the NCP function; is a time-varying bias vector, which includes the expected end velocity and the constant term of the constraint conditions; is the state variable, which includes the joint angular velocity to be solved and the Lagrange multiplier.

[0096] Step 2: Design an AZTND solver for artistic curve drawing;

[0097] Based on the equation obtained in the first step, define the error function . Construct an AZTND model dedicated to artistic curve drawing:

[0098] ;

[0099] Define the expected butterfly curve: Use the parametric equation of the butterfly curve as the drawing path:

[0100] ;

[0101] ;

[0102] ;

[0103] where A is the scaling factor (controlling the size of the curve), is the angular frequency, which controls the drawing speed.

[0104] Elaborate on the key components of the above model:

[0105] Piecewise adaptive coefficient function : This is the core of realizing "intelligent artistic drawing". This function dynamically adjusts the convergence strength according to different stages of the butterfly curve drawing and the current error.

[0106] ;

[0107] 0 < t < 5: Use . This stage starts drawing from the starting end of the butterfly curve, and it is necessary to quickly stabilize the system to avoid the starting jitter affecting the formation of subsequent complex patterns.

[0108] (main drawing stage): Use This stage is crucial for mapping the complex wing structure of a butterfly curve. High gain is required to ensure high-precision tracking of the complex curvature changes of the curve, especially maintaining a fast response at the turning points of the wing edges.

[0109] t>10 (Drawing completion stage): Using The butterfly curve is still being plotted, the system remains stable, and a gain proportional to the error is used to maintain accuracy while conserving computational resources.

[0110] Negative feedback integral term This feature can accumulate and eliminate historical errors, which is crucial for suppressing continuous, low-frequency interference during the drawing process. It can effectively prevent the overall shape of the butterfly curve from being distorted, maintain geometric symmetry, and ensure that the left and right wings are perfectly symmetrical.

[0111] Noise item The AZTND model designed in this invention is highly robust to simulate various random disturbances during the drawing process, ensuring that clear and symmetrical butterfly patterns can still be drawn even in noisy environments.

[0112] Step 3: Setting up and simulating the butterfly curve;

[0113] Curve parameters: Use the butterfly curve as the desired drawing path.

[0114] Robotic arm parameters: Using the UR5 robotic arm as the controlled object, set its initial joint angle, joint angular velocity limit, etc.

[0115] AZTND parameters: settings , , , .

[0116] Step 4: Simulation and comparison of butterfly curve drawing effects;

[0117] We demonstrate the superior rendering effect of this invention through multiple sets of simulation diagrams:

[0118] like Figure 2 As shown, this graph records the angle changes of each joint of a six-DOF robotic arm during a 20-second simulation. Six curves of different colors illustrate the angle changes of each joint during the 20-second motion simulation. Overall, joints q1 and q2 have the largest swing amplitude and are responsible for the main body's positional movement; q3, q4, and q5 perform smaller but more complex coordinated movements, mainly responsible for fine-tuning the end effector's posture; while the angle of q6 remains almost constant, its role clearly being to maintain the stability of the end effector's direction. All curves are continuous and smooth, reflecting good motion planning. This graph is an important tool for analyzing and verifying the motion coordination and performance of the robotic arm's joints.

[0119] like Figure 3 As shown in the figure, this graph illustrates the variation of the angular velocities of each joint (dq1 to dq6) of a six-DOF robotic arm over time during a 20-second motion simulation. Corresponding to the angle graph, the angular velocity visually reflects the speed and direction of each joint's movement. The angular velocities of all joints in the graph exhibit continuous, smooth sinusoidal fluctuations, indicating that the robotic arm is performing a periodic reciprocating motion, and the motion is smooth and shock-free. By comparison, the angular velocity amplitudes of joints dq1 and dq2 are significantly larger than those of other joints, which is consistent with their undertaking of large-amplitude main motion tasks; while the angular velocity of dq6 remains slightly fluctuating near zero, further confirming its role in maintaining end-effector stability. This graph is a crucial basis for analyzing joint motor performance and evaluating motion smoothness and vibration.

[0120] like Figure 4 As shown in the figure, this graph records the trajectory of the spatial position (X, Y, Z coordinates) of the robotic arm's end effector over 20 seconds. The three curves represent the displacement of the end effector in the X, Y, and Z directions, respectively, all exhibiting continuous, smooth, and regular periodic fluctuations. This motion pattern indicates that the robotic arm is performing a repetitive task. The smoothness of the trajectory reflects the good performance of the motion control algorithm, while the range and synchronization of position changes along each axis directly determine the accuracy and stability of the end effector's final pose, serving as a key basis for evaluating the robotic arm's performance.

[0121] like Figure 5 As shown in the figure, this graph illustrates the velocity changes of the robotic arm's end effector along the X, Y, and Z axes during a 20-second motion. The dX (blue) and dY (red) curves exhibit complete periodic positive and negative fluctuations, indicating that the end effector performs regular reciprocating motion along the X and Y axes; while the dZ (yellow) curve remains close to the zero velocity line, indicating almost no displacement along the Z-axis (vertical direction). The overall curves are smooth, demonstrating the stability of the end effector's motion and good trajectory planning.

[0122] like Figure 6 As shown in the figure, the graph illustrates the changes in acceleration (denoted as ddX, ddY, and ddZ) along the X, Y, and Z spatial coordinate axes of the robotic arm's end effector over time during a 20-second movement. All three acceleration curves exhibit regular but complex periodic oscillations, indicating that the robotic arm underwent frequent acceleration, deceleration, and direction changes during task execution, demonstrating significant dynamic characteristics. This graph serves as a crucial basis for analyzing the robotic arm's motion smoothness, dynamic load, and control system response performance.

[0123] like Figure 7As shown in the figure, this graph records the changes in the deviation (i.e., position error) between the actual position and the target position of the robotic arm's end effector in the X, Y, and Z directions over time. As the figure shows, the error curves (eX, eY, eZ) of the X, Y, and Z axes fluctuate within a very small range, closely surrounding the zero error line. This indicates that the robotic arm's closed-loop control system responds quickly and adjusts effectively, rapidly correcting deviations and achieving high-precision end-point control. Such graphs are crucial for evaluating and verifying the trajectory tracking accuracy and control algorithm performance of robotic arms.

[0124] like Figure 8 As shown in the graph, this figure reflects the speed control accuracy of the robotic arm's end effector in the X, Y, and Z directions. As the figure shows, the curves edX, edY, and edZ, representing the three-axis speed errors, remain perfectly aligned with the horizontal axis representing zero error throughout the entire 20-second motion. This indicates that the robotic arm's actual speed can track the target speed accurately in real time, without producing any observable steady-state errors in any of the three dimensions. This result clearly verifies that the robotic arm's speed control system possesses extremely high accuracy and stability.

[0125] like Figure 9 As shown in the graph, this chart illustrates the variation of acceleration control error of the robotic arm's end effector in the X, Y, and Z directions over time. The three error curves (eddx, eddy, eddz) exhibit regular periodic fluctuations within the positive and negative ranges, indicating high overall control accuracy. The fluctuation amplitude along the Z-axis (eddz) is slightly larger than that along the X and Y axes. This suggests that the system can track the target acceleration well, but some periodic deviations still exist during dynamic processes.

[0126] like Figure 10 As shown in the figure, the motion trajectory of the robotic arm's end effector in a two-dimensional plane (XY plane) is clearly illustrated. In the figure, the blue desired trajectory and the red actual trajectory almost completely overlap, together forming a precise and smooth butterfly curve. This high degree of overlap indicates that the robotic arm's motion control system possesses excellent precision and tracking performance; its actual movement perfectly replicates the pre-set desired path, demonstrating outstanding control performance.

[0127] like Figure 11As shown in the figure, this diagram illustrates the simulation results of the motion trajectory of a robotic arm's end effector in three-dimensional space. In the figure, a red three-dimensional cone represents the base or fixed reference point of the robotic arm, and the fan-shaped blue line segments emanating from its side visually simulate the motion direction of the robotic arm joints. These trajectory lines ultimately converge on the left side to form a three-dimensional spatial curve composed of a dense cloud of colored points. This curve clearly outlines the complete planned motion path of the robotic arm's end effector within a specific working area. This simulation diagram intuitively presents the complete kinematic relationship from the base to the end effector, serving as an effective visualization tool for trajectory planning, reachability analysis, and control algorithm verification.

[0128] like Figure 12 As shown, this image displays a gray and blue robotic arm whose end effector leaves a continuous red trail on a light-colored grid background, forming a butterfly pattern. The robotic arm's posture and trajectory are closely integrated, presenting a precise motion path and highlighting its technological capabilities in automated drawing or high-precision path planning. The overall image is simple yet technologically advanced, demonstrating the innovative application of mechanical systems in the fields of art or precision operation.

[0129] Compared with related technologies, the present invention has the following improvements:

[0130] 1. Application of the AZTND model: The activation-zero type neurodynamic model is adapted to the high-precision artistic curve drawing control of robotic arms. In particular, for the drawing needs of complex geometric patterns such as butterfly curves, a real-time solution framework from mathematical curves to joint control quantities is constructed.

[0131] 2. Piecewise adaptive coefficient function design: Using piecewise functions, the model parameters are intelligently adjusted according to different stages of butterfly curve drawing (start, petal drawing, completion), with particular optimization of response performance at petal turning points, and convergence speed optimized based on error norm and time.

[0132] 3. Negative feedback integral mechanism: By combining negative feedback integral adjustment, historical errors are accumulated and eliminated, effectively suppressing low-frequency and constant interference, solving the problems of geometric distortion and symmetry destruction of butterfly curves, and ensuring that the drawn butterfly curves always maintain perfect geometric symmetry.

[0133] 4. Integrated Robotic Arm Artistic Drawing: The method is specifically designed for dynamic trajectories and is suitable for high-precision artistic drawing tasks, such as butterfly curve drawing. It provides a complete implementation scheme for robotic arm artistic curve drawing system based on AZTND.

[0134] 5. Simulation initialization process: including setting butterfly curve parameters, configuring model parameters, and simulation verification methods under noisy environments to ensure experimental repeatability.

[0135] The AZTND model of the present invention has the following advantages:

[0136] Fast convergence speed: The piecewise adaptive coefficient function enables the error to converge quickly, in just 0.01 seconds, which is about 10 times faster than the traditional OZNN.

[0137] Strong noise resistance: Under constant and time-varying noise, the error is stable at the level of 10^{-4}, which is about 50 times better than the traditional model. The strong noise resistance and integral term design ensure that the drawn butterfly curve always maintains perfect geometric symmetry.

[0138] High resource efficiency: Dynamic parameter adjustment reduces computational consumption, making it suitable for low-cost hardware, with an execution time of only 0.8ms on the ARM Cortex-M4 processor.

[0139] Flexible applications: The model can handle robotic arm constraints, adapt to complex artistic drawing paths such as butterfly curves, and accurately reproduce the mathematical characteristics of butterfly curves, with symmetrical petals and accurate proportions.

[0140] High accuracy: tracking error reaches The level meets the needs of high-precision artistic drawing, and can draw smooth curves without jitter or burrs even in noisy environments, maintaining the artistic beauty of the butterfly curve.

[0141] Strong symmetry is maintained: The negative feedback integral mechanism effectively prevents the distortion of the butterfly's curved geometry, ensuring perfect symmetry between the left and right wings.

[0142] This invention also provides a robotic arm trajectory drawing and control system based on the AZTND model, comprising:

[0143] At least one processor;

[0144] At least one memory for storing at least one program;

[0145] When the at least one program is executed by the at least one processor, the at least one processor performs the method described above.

[0146] 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.

[0147] 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.

[0148] 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.

[0149] This invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described method.

[0150] 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.

[0151] 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.

[0152] 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.

[0153] 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.

[0154] 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.

[0155] 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 method for drawing the trajectory of a robotic arm based on the AZTND model, characterized in that, The method includes the following steps: S100, acquire the robot arm curve drawing task parameters and AZTND solver, the AZTND solver includes a piecewise adaptive coefficient function, a negative feedback integral term and a noise term; S200: Establish the time-varying kinematic equations of the robotic arm, introduce joint angular velocity constraints and acceleration constraints, and transform the curve drawing problem into a time-varying nonlinear equation; S300, define the error function as the residual of the time-varying nonlinear equation, construct the AZTND dynamic equation, the AZTND dynamic equation includes a product term of the piecewise adaptive coefficient function and the activation function, a negative feedback integral term and a noise term, the negative feedback integral term eliminates steady-state deviation by integrating and accumulating historical errors; S400, the parameters of the piecewise adaptive coefficient function are determined according to the curve drawing stage. The piecewise adaptive coefficient function uses a first gain coefficient in the initial stage to ensure the system is fast and stable, a second gain coefficient in the main drawing stage to ensure high-precision tracking of complex curvature changes, and a third gain coefficient in the completion stage to maintain accuracy and save computing resources. S500 uses the ODE numerical solution method to solve the AZTND dynamic equation in real time, obtains the joint angular velocity control command of the robotic arm, and drives the end effector of the robotic arm to track the desired curve trajectory.

2. The method according to claim 1, characterized in that, In S200, the establishment of the time-varying kinematic equations of the robotic arm, introducing joint angular velocity constraints and acceleration constraints, transforms the curve plotting problem into a time-varying nonlinear equation, including: Establish time-varying kinematic equations ,in, The rate of change of the desired position of the robotic arm's end effector. For the Jacobian matrix of the robotic arm, The joint angular velocity to be solved; Introducing joint angular velocity constraints and acceleration constraints; The constrained optimization problem is transformed into a complementary-condition form using KKT conditions, and then smoothed using the NCP function to obtain time-varying nonlinear equations. ,in, The time-varying coefficient matrix, For time-varying bias vectors, The state variables include the joint angular velocities to be determined. And Lagrange multipliers.

3. The method according to claim 1, characterized in that, In S300, the error function is defined as the residual of the time-varying nonlinear equation, and the AZTND dynamic equation is constructed, including: Define error function ; Constructing the dynamic equations of AZTND ,in, For piecewise adaptive coefficient functions, For activation function, This is the integral gain coefficient. For noise terms, These are the exponential coefficients, This is the noise term.

4. The method according to claim 1, characterized in that, In S300, the product term of the piecewise adaptive coefficient function and the activation function is defined piecewise according to the drawing time: When the rendering time is in the initial stage, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm and superimposed with the first constant gain to ensure that the system stabilizes quickly. When the rendering time is in the main rendering phase, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm and superimposed with the second constant gain to ensure high-precision tracking of complex curvature changes; When the rendering time is in the completion stage, the piecewise adaptive coefficient function uses coefficients that are proportional to the error norm, which saves computing resources while maintaining accuracy; The first constant gain is less than the second constant gain.

5. The method according to claim 1, characterized in that, In S500, the step of using the ODE numerical solution method to solve the AZTND dynamic equation in real time to obtain the joint angular velocity control command of the robotic arm and drive the end effector of the robotic arm to track the desired curve trajectory includes: The AZTND dynamic equations are transformed into a system of ordinary differential equations concerning the state variables; The ordinary differential equations are solved in discrete time using numerical integration to obtain the joint angular velocity values ​​at each time step. The robot arm joint state is updated based on the obtained joint angular velocity to achieve closed-loop feedback control.

6. The method according to claim 1, characterized in that, The curve drawing task is a butterfly curve drawing task, and the desired curve trajectory is defined by parametric equations, which are: ； ； ； Where A is the scaling factor, The scaling factor controls the curve size, and the angular frequency controls the drawing speed.

7. The method according to claim 6, characterized in that, The method further includes: Set the parameters of the robotic arm, using the UR5 robotic arm as the controlled object, and configure the initial joint angle and joint angular velocity limits; Set the butterfly curve parameters, and configure the scaling factor and angular frequency to control the curve size and drawing speed; Noise terms were added to the simulation environment to simulate random disturbances during the actual drawing process, thus verifying the anti-interference capability of the AZTND model.

8. A robotic arm trajectory drawing system based on the AZTND model, characterized in that, include: 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 performs the method as described in any one of claims 1 to 7.

9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the method of any one of claims 1 to 7.

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