Trajectory planning and control method and system for underwater manipulator based on pnvrgznd model

Abstract

This invention relates to the field of underwater robotic arm control technology, specifically to a method and system for trajectory planning and control of an underwater robotic arm based on the PNVRGZND model. The method includes: obtaining the joint angle vectors, desired velocity vectors, and Jacobian matrix of a six-DOF cascaded underwater robotic arm; transforming the kinematic equations into time-varying linear equations and defining an error function; constructing a PNVRGZND model containing a novel double-integral structure, novel time-varying parameters that are monotonically increasing and bounded, and a novel nonlinear activation function containing exponential, linear, and power terms; substituting the derivative of the error function into the PNVRGZND model to obtain the dynamic evolution equation of the joint angular velocity vector; determining known quantities based on the real-time state of the robotic arm and the desired trajectory; numerically solving the dynamic evolution equation to obtain the joint angular velocity vector and outputting it to the joint actuator. This invention significantly improves the trajectory tracking accuracy and robustness of underwater robotic arms in environments with strong interference.

Description

Technical Field

[0001] This invention relates to the field of underwater robot motion control technology, and in particular to an underwater robotic arm trajectory planning and control method and system based on the PNVRGZND model. Background Technology

[0002] Underwater six-degree-of-freedom tandem robotic arms, with their flexible joint configuration and pressure and corrosion resistance, have become core actuators in fields such as marine pipeline inspection and repair, marine structure installation, marine rescue and salvage, and marine observation network maintenance. Their fundamental task is to precisely coordinate the movement of each rotary joint to achieve real-time and accurate control of the end effector's position and attitude in marine environments characterized by strong currents, low visibility, and strong electromagnetic interference, enabling dynamic tasks such as flange alignment, pipeline weld tracking, and cable retrieval in the marine environment.

[0003] However, underwater robotic arms exhibit highly nonlinear and strongly coupled dynamic characteristics, and their operating environment is often accompanied by time-varying ocean current disturbances and complex secondary noise interference from thrusters and acoustic equipment. This makes it difficult for traditional methods based on analytical inverse kinematics or numerical iteration to simultaneously guarantee solution accuracy, real-time performance, and robustness. For example, the Jacobian matrix pseudo-inverse method is prone to failure near singular configurations; numerical optimization methods have slow convergence speeds and may get trapped in local maxima. In recent years, intelligent methods based on neurodynamics have attracted attention due to their advantages in parallel computing and adaptive capabilities. However, traditional gradient neurodynamic (GND) models have insufficient convergence accuracy, and zeroing neurodynamic (ZND) models lack stability in noisy environments. Therefore, there is an urgent need to develop a novel neurodynamic method that can balance strong steady-state performance, fast convergence, and strong resistance to underwater secondary noise to achieve high-precision motion control of underwater robotic arms in complex time-varying environments.

[0004] Currently, the relevant technical solutions mainly revolve around GND and ZND models, with the most representative being single-integral neurodynamic models, time-varying parameter neurodynamic models, nonlinear function activation neurodynamic models, and double-integral neurodynamic models.

[0005] Single-integral neurodynamic models are typically constructed by adding an integral term to the error function, based on the traditional ZND model or the GND model with a velocity compensation term. While these models offer improvements in noise resistance and convergence accuracy compared to traditional methods, they struggle to effectively suppress secondary noise common in underwater environments, leading to significant tracking errors under strong noise interference.

[0006] In neurodynamic models with time-varying parameters, a common approach is to use monotonically increasing and unbounded time-varying parameters (such as exponential functions). While this accelerates initial convergence, the infinite growth of the parameters over time prevents the system from achieving steady-state convergence and significantly increases computational complexity and runtime. Another approach designs monotonically decreasing time-varying parameters with lower bounds. While this design ensures steady-state accuracy, its decreasing characteristic weakens error decay. Therefore, existing time-varying parameter models still face an inherent contradiction between steady-state performance and convergence accuracy, making it difficult to achieve both simultaneously.

[0007] The neurodynamic models activated by nonlinear functions are mainly divided into two categories: finite-time convergence and predefined-time convergence. Although the former can achieve fast convergence, its convergence time depends on the initial state of the system and cannot be predetermined. In practical applications, it is difficult to meet the scenarios with clear time constraints on the convergence process. Although the latter can set the upper bound of the convergence time by adjusting the parameters of the activation function, its convergence speed is usually relatively slow.

[0008] The dual-integral neurodynamic model is derived by introducing a dual-integral structure into the traditional ZND model. Compared to the traditional ZND model and the single-integral ZND model, this type of model exhibits higher steady-state accuracy and stronger robustness under constant, linear, and even quadratic noise environments, making it particularly suitable for high-precision real-time control scenarios sensitive to noise. However, existing dual-integral models are all built on the ZND framework, and there is no dual-integral model based on gradient neurodynamics (GND), which may limit its applicability in problems that utilize gradient information for optimization. Summary of the Invention

[0009] To address the aforementioned technical problems, this invention provides a method and system for trajectory planning and control of an underwater robotic arm based on the PNVRGZND model. The aim is to significantly improve the trajectory tracking accuracy and robustness of the underwater robotic arm under strong secondary noise and time-varying ocean current interference through the synergistic design of a novel double-integral structure, bounded time-varying parameters, and composite nonlinear activation functions.

[0010] To achieve the above objectives, the present invention provides the following technical solution: On one hand, embodiments of the present invention provide a method for trajectory planning and control of an underwater robotic arm based on the PNVRGZND model, the method comprising the following steps: Obtain the joint angle vectors of the underwater six-degree-of-freedom serial manipulator, the desired velocity vector of the end effector, and the Jacobian matrix of the manipulator; transform the kinematic equations of the manipulator into time-varying linear equations; and define an error function based on the time-varying linear equations. A PNVRGZND model is constructed, which is a novel variable-parameter robust gradient basis nulling neurodynamic model with predefined time. The PNVRGZND model includes a novel double-integral structure, novel time-varying parameters, and novel nonlinear activation function. The novel double-integral structure is constructed by introducing a velocity compensation term and constructing auxiliary variables on the basis of the traditional gradient neurodynamic model. The novel time-varying parameters are monotonically increasing and bounded. The novel nonlinear activation function includes exponential, linear, and power terms. Substituting the derivative of the error function into the PNVRGZND model, we obtain the dynamic evolution equation for the joint angular velocity vector. The known quantities in the time-varying linear equation are determined based on the real-time state of the robotic arm and the desired trajectory. The joint angular velocity vector is obtained by numerically solving the dynamic evolution equation. Control commands are generated based on the joint angular velocity vector to drive the end effector to track the desired trajectory.

[0011] Optionally, the step of converting the kinematic equations of the robotic arm into time-varying linear equations and defining an error function based on the time-varying linear equations includes: The kinematic equation of the robotic arm is expressed as the desired velocity vector being equal to the product of the Jacobian matrix and the joint angular velocity vector. The Jacobian matrix is ​​used as the coefficient matrix of the time-varying linear equation, the joint angular velocity vector is used as the state vector to be solved in the time-varying linear equation, and the desired velocity vector is used as the known vector of the time-varying linear equation. The error function is obtained by subtracting the known vector from the product of the coefficient matrix and the state vector to be solved.

[0012] Optionally, constructing the PNVRGZND model includes: The derivative of the error function is expressed as the product of the negative gain coefficient and the coefficient matrix and its transpose, multiplied by the error function, minus the first integral term, the second integral term, and the third integral term; The first integral term is the integral of the time variable from zero to the current time, and the integrand is the product of the new time-varying parameter and the auxiliary variable; The second integral term is the integral of the time variable from zero to the current time, and the integrand is the product of the new time-varying parameter and the result of the new nonlinear activation function applied to the auxiliary variable; The third integral term is a double integral of the time variable from zero to the current time. The inner integral variable is an auxiliary variable, the outer integral variable is a time variable, and the integrand is the product of the square of the new time-varying parameter and the result of the new nonlinear activation function applied to the auxiliary variable. The auxiliary variable is the sum of the product of the derivative of the error function, the gain coefficient, the coefficient matrix and its transpose, and the error function.

[0013] Optionally, the formula for the novel time-varying parameter is: ; in, For a novel time-varying parameter, a>0, d>0, where a and d are constants, e is the natural base, and t is the time variable. .

[0014] Optionally, the formula for the novel nonlinear activation function is: ; in, It is a novel nonlinear activation function. , , , All are constants greater than zero. A constant between 0 and 1 For any input vector, This represents the natural exponential function. This indicates taking the absolute value of the vector elements. This is a sign function that outputs 1 when the input is greater than 0, 0 when the input is equal to 0, and -1 when the input is less than 0.

[0015] Optionally, substituting the derivative of the error function into the PNVRGZND model to obtain the dynamic evolution equation for the joint angular velocity vector includes: Taking the derivative of the error function, we find that the derivative of the error function is equal to the product of the derivative of the coefficient matrix and the state vector to be solved, plus the product of the derivative of the coefficient matrix and the state vector to be solved, minus the derivative of the known vector. Substituting the derivative of the error function into the PNVRGZND model and rearranging, we obtain that the product of the derivative of the coefficient matrix and the state vector to be solved is equal to the derivative of the known vector minus the product of the derivative of the coefficient matrix and the state vector to be solved, minus the product of the gain coefficient and the coefficient matrix and its transpose multiplied by the error function, and then minus the first integral term, the second integral term, and the third integral term. By left-multiplying both sides of the equation with the pseudo-inverse of the coefficient matrix, we obtain an expression for the derivative of the state vector to be solved.

[0016] Optionally, obtaining the joint angular velocity vector by numerically solving the dynamic evolution equation includes: Set the initial joint angle, gain coefficient, positive coefficients in the novel time-varying parameters, and positive coefficients and power exponents in the novel nonlinear activation function; Set the desired trajectory of the end effector in underwater three-dimensional space; The Jacobian matrix is ​​calculated based on the real-time joint angles of the robotic arm, the desired velocity vector is calculated based on the desired trajectory, and the derivatives of the coefficient matrix and the known vector are determined based on the real-time joint angles and the desired trajectory. Substituting the known quantities into the dynamic evolution equation, the joint angular velocity vector is obtained by numerical integration.

[0017] On the other hand, embodiments of the present invention provide an underwater robotic arm trajectory planning and control system based on the PNVRGZND model, including: 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 described above.

[0018] 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.

[0019] The embodiments of the present invention have the following beneficial effects: The PNVRGZND model proposed in this invention, by constructing a novel double-integral structure, can theoretically completely eliminate secondary noise interference, significantly improving the system's robustness in complex marine environments. The end effector's errors in position, velocity, and acceleration are stably maintained at [values ​​to be filled in]. rice, meters per second and meters per second 2 The magnitude of the data verified the model's advantage in achieving ultra-high precision tracking and control even in a simulated marine environment with strong secondary noise, fully meeting the precision requirements of tasks such as underwater pipeline precision inspection.

[0020] The novel time-varying parameters introduced in this invention exhibit monotonically increasing characteristics with an upper bound, overcoming the inherent contradiction between steady-state convergence and control accuracy in existing time-varying parameters. The angle and acceleration curves of the six joints all show regular, smooth periodic changes, with amplitudes strictly controlled within a safe range, without abrupt changes or divergence. This fully demonstrates that the time-varying parameters can maintain high-precision control performance throughout the entire motion cycle while ensuring the system resists time-varying ocean current interference and achieves steady-state convergence.

[0021] The composite nonlinear activation function designed in this invention achieves predefined time-based convergence characteristics while significantly improving the convergence speed. Position, velocity, and acceleration errors all converge rapidly to near zero within an extremely short time and remain stable throughout the entire motion cycle without divergence or cumulative drift. This verifies that while achieving rapid convergence, the model can effectively suppress dynamic shocks and vibrations in underwater operations, ensuring the stability and real-time response capability of the robotic arm in high-speed, high-dynamic tasks such as emergency ascent and obstacle avoidance. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the embodiments will be briefly described below.

[0023] Figure 1 This is a flowchart illustrating the steps of an underwater robotic arm trajectory planning and control method based on the PNVRGZND model provided in an embodiment of the present invention. Figure 2 This is a three-dimensional effect diagram of the trajectory tracking of the underwater robotic arm end effector provided in an embodiment of the present invention; Figure 3 This is an instantaneous state diagram of underwater robotic arm trajectory tracking provided in an embodiment of the present invention; Figure 4 This is a position error diagram of the end effector of the underwater robotic arm provided in an embodiment of the present invention; Figure 5 This is a speed error diagram of the end effector of the underwater robotic arm provided in an embodiment of the present invention; Figure 6 This is an acceleration error diagram of the end effector of the underwater robotic arm provided in an embodiment of the present invention; Figure 7 This is a diagram showing the angular velocity trajectory of the joints of an underwater robotic arm provided in an embodiment of the present invention. Figure 8 This is a diagram of the joint acceleration trajectory of an underwater robotic arm provided in an embodiment of the present invention; Figure 9 This is a velocity trajectory diagram of the end effector of the underwater robotic arm provided in an embodiment of the present invention; Figure 10 This is an acceleration trajectory diagram of the end effector of the underwater robotic arm provided in an embodiment of the present invention. Detailed Implementation

[0024] To make the objectives, technical solutions, and advantages of this invention clearer, the following will provide a clear and complete description of the disclosed concepts and technical effects of this invention in conjunction with embodiments and accompanying drawings, so as to fully understand the objectives, solutions, and effects of this invention. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other.

[0025] 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.

[0026] Unless otherwise defined, all technical and scientific terms used in this invention have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used in this invention is for descriptive purposes only and is not intended to limit the invention.

[0027] refer to Figure 1 ,like Figure 1 The figure shows an underwater robotic arm trajectory planning and control method based on the PNVRGZND model provided by an embodiment of the present invention. The method includes the following steps: S100: Obtain the joint angle vector of the underwater six-degree-of-freedom serial manipulator, the desired velocity vector of the end effector, and the Jacobian matrix of the manipulator; transform the kinematic equation of the manipulator into a time-varying linear equation; and define an error function based on the time-varying linear equation. S200, Construct the PNVRGZND model. The PNVRGZND model is a novel variable-parameter robust gradient basis nulling neurodynamic model with predefined time. The PNVRGZND model includes a novel double-integral structure, novel time-varying parameters, and novel nonlinear activation function. The novel double-integral structure is constructed by introducing a velocity compensation term and constructing auxiliary variables on the basis of the traditional gradient neurodynamic model. The novel time-varying parameters are monotonically increasing and bounded. The novel nonlinear activation function includes exponential, linear, and power terms. S300, Substitute the derivative of the error function into the PNVRGZND model to obtain the dynamic evolution equation for the joint angular velocity vector; S400: Based on the real-time state of the robotic arm and the desired trajectory, the known quantities in the time-varying linear equation are determined, the joint angular velocity vector is obtained by numerically solving the dynamic evolution equation, and control commands are generated based on the joint angular velocity vector to drive the end effector to track the desired trajectory.

[0028] This invention provides a method and system for trajectory planning and control of an underwater robotic arm based on the PNVRGZND model. It constructs a novel double-integral structure that integrates the characteristics of traditional gradient neurodynamics and traditional nullification neurodynamics, introduces a novel time-varying parameter that is monotonically increasing and bounded, and a novel nonlinear activation function containing exponential, linear, and power terms, thus constructing the PNVRGZND model. Theoretically, this model can completely eliminate secondary noise through double integration, balance steady-state convergence and high-precision control through bounded time-varying parameters, and achieve rapid convergence within a predefined time using a composite nonlinear activation function. This invention effectively addresses strong underwater current disturbances and strong secondary noise interference generated by thrusters and acoustic equipment, significantly improving the trajectory tracking accuracy, robustness, and real-time response capability of underwater robotic arms in complex dynamic environments. It provides reliable technical support for tasks such as precision inspection of underwater pipelines, structural installation, and rescue and salvage operations in marine environments.

[0029] In some embodiments, S100, the step of converting the kinematic equations of the robotic arm into time-varying linear equations and defining an error function based on the time-varying linear equations includes: S110, the kinematic equation of the robotic arm is expressed as the desired velocity vector being equal to the product of the Jacobian matrix and the joint angular velocity vector, the Jacobian matrix is ​​used as the coefficient matrix of the time-varying linear equation, the joint angular velocity vector is used as the state vector to be solved in the time-varying linear equation, and the desired velocity vector is used as the known vector of the time-varying linear equation. S120, the product of the coefficient matrix and the state vector to be solved, minus the known vector, is used as the error function.

[0030] Specifically, for an underwater six-DOF robotic arm, its joint angle vector can be expressed as: , Let represent the rotation angle of the nth joint at time t, where n is 6 for the six-DOF robotic arm in this embodiment. Let the desired speed of the end effector be... Then the motion tracking problem of the robotic arm can be transformed into solving the joint angular velocities. This makes the kinematic equations of the robotic arm Established, Let be the Jacobian matrix of the robotic arm at time t, obtained by differentiating the forward kinematics of the robotic arm with respect to the joint angles. This equation can be abstracted into a general time-varying linear equation. ,in Let Jacobian matrix be the value of the robotic arm. Let be the joint angular velocity vector to be solved. Let be the desired terminal velocity vector. To measure the deviation between the current solution and the theoretical solution, an error function is defined. ,when hour, This is the joint angular velocity solution that satisfies the desired velocity.

[0031] In some embodiments, S200, constructing the PNVRGZND model includes: The derivative of the error function is expressed as the product of the negative gain coefficient and the coefficient matrix and its transpose, multiplied by the error function, minus the first integral term, the second integral term, and the third integral term; The first integral term is the integral of the time variable from zero to the current time, and the integrand is the product of the new time-varying parameter and the auxiliary variable; The second integral term is the integral of the time variable from zero to the current time, and the integrand is the product of the new time-varying parameter and the result of the new nonlinear activation function applied to the auxiliary variable; The third integral term is a double integral of the time variable from zero to the current time. The inner integral variable is an auxiliary variable, the outer integral variable is a time variable, and the integrand is the product of the square of the new time-varying parameter and the result of the new nonlinear activation function applied to the auxiliary variable. The auxiliary variable is the sum of the product of the derivative of the error function, the gain coefficient, the coefficient matrix and its transpose, and the error function.

[0032] The design concept of the traditional GND model is ,in This is a positive constant, called the convergence factor, used to adjust the rate of error decay. The design philosophy of the traditional ZND model is... ,in For positive integers, The derivative of the error function, i.e., let the error... It decays exponentially to zero. This invention makes a significant change, combining the characteristics of traditional GND and ZND models while introducing novel time-varying parameters and nonlinear activation functions, ultimately constructing the PNVRGZND model, whose dynamic evolution model is defined as: ; in, The gain coefficient is greater than zero. This is a novel time-varying parameter that satisfies the properties of monotonically increasing and having an upper bound. For inner-layer integration variables, For outer integral variables, The novel nonlinear activation function designed for this invention Auxiliary variables introduced into the model.

[0033] This model contains three major innovations: First, a novel dual-integral structure. The dual-integral structure of this model... Based on the traditional GND model with a velocity compensation term, this model is constructed by integrating two auxiliary variables while ensuring that the auxiliary variables satisfy the integral form of the traditional ZND model. Theoretically, this dual-integral structure can completely eliminate secondary noise through integration, which is crucial for suppressing strong secondary noise interference from underwater thrusters, sonar, and other equipment, significantly improving the model's robustness in complex underwater electromagnetic and vibration environments.

[0034] Second, a new time-varying parameter. Inspired by the second important limit, this model introduces a novel time-varying parameter that is monotonically increasing and bounded above. Its specific form can be designed as Where a>0, d>0, and a and d are constant parameters, and Here, e is a natural constant. It can be seen that this time-varying parameter monotonically increases from its initial value and eventually converges stably to a constant e^d, without increasing indefinitely over time. Simultaneously, it maintains an increasing trend greater than the initial value throughout the convergence process, perfectly balancing rapid convergence in the initial stage and control accuracy in the steady-state stage. This resolves the inherent contradiction of traditional fixed or unbounded time-varying parameters, which either have large steady-state errors or excessive control gains that easily lead to system oscillations. This time-varying parameter overcomes the shortcomings of existing monotonically increasing and unbounded time-varying parameters, which cannot achieve steady-state convergence, and monotonically decreasing time-varying parameters with a lower bound, which have low convergence accuracy. It can achieve steady-state convergence and maintain high-precision control performance while ensuring the system's resistance to time-varying ocean current interference.

[0035] Third, a new nonlinear activation function. In this model... It is a nonlinear activation function that achieves predefined time convergence. Furthermore, through a composite design of exponential and power terms, it exhibits a faster convergence speed compared to traditional predefined time activation functions. This characteristic allows the underwater robotic arm to recover its stable motion within a preset safety time when encountering sudden ocean current impacts or target deviations, facilitating further operator intervention.

[0036] In some embodiments, the formula for the novel time-varying parameter is: ; in, For a novel time-varying parameter, a>0, d>0, where a and d are constants, e is the natural base, and t is the time variable. .

[0037] The novel time-varying parameter is equal to the result of a power operation on the sum of a fraction, where the numerator of the fraction is a positive constant and the denominator is the sum of another positive constant and the time variable plus one. The exponent of the power operation is equal to the sum of the other positive constant and the time variable plus one. The novel time-varying parameter monotonically increases with time and converges to a finite upper bound.

[0038] In some embodiments, the formula for the novel nonlinear activation function is: ; in, It is a novel nonlinear activation function. , , , All are constants greater than zero. A constant between 0 and 1 For any input vector, This represents the natural exponential function. This indicates taking the absolute value of the vector elements. This is a sign function that outputs 1 when the input is greater than 0, 0 when the input is equal to 0, and -1 when the input is less than 0.

[0039] Specifically, Represents the input vector The absolute value of the corresponding element. For a sign function, when The activation function assigns a value of 1 when the element is greater than 0, -1 when it is less than 0, and 0 when it is equal to 0. This activation function combines exponential, linear, and sign terms of different powers, which not only retains the predefined time convergence characteristics but also further accelerates the error convergence speed. It maintains a faster error decay rate than traditional activation functions under different error magnitudes, effectively improving the real-time response capability of underwater robotic arm trajectory tracking.

[0040] In some embodiments, in S300, substituting the derivative of the error function into the PNVRGZND model to obtain the dynamic evolution equation for the joint angular velocity vector includes: S310, Take the derivative of the error function and obtain that the derivative of the error function is equal to the product of the derivative of the coefficient matrix and the state vector to be solved, plus the product of the derivative of the coefficient matrix and the state vector to be solved, minus the derivative of the known vector. S320, Substitute the derivative of the error function into the PNVRGZND model, and after simplification, the product of the derivative of the coefficient matrix and the state vector to be solved is equal to the derivative of the known vector minus the product of the derivative of the coefficient matrix and the state vector to be solved, minus the product of the gain coefficient and the coefficient matrix and its transpose multiplied by the error function, and then minus the first integral term, the second integral term and the third integral term; S330, by left-multiplying both sides of the equation with the pseudo-inverse of the coefficient matrix, we obtain the expression for the derivative of the state vector to be solved.

[0041] Specifically, first, we take the derivative of the error function: These are the error function, the coefficient matrix, and the first derivative of the known vector with respect to time, respectively. Substituting the above results into the PNVRGZND model proposed in this invention: ; in, .

[0042] After organizing, we obtained information about The dynamic evolution equations, i.e., the solver of this invention: ; In practical applications, the variable to be solved is the state vector. (i.e., the angular velocity of the robotic arm joints) ), and matrix ,vector and its derivative , All of these can be uniquely determined based on the real-time state of the underwater robotic arm and the desired trajectory. Therefore, all input quantities and model parameters in the above time-varying equations are known quantities and can be obtained through numerical solutions. For example, a pseudo-inverse can be used. Solve the problem, and the result is: .

[0043] In some embodiments, S400, obtaining the joint angular velocity vector by numerically solving the dynamic evolution equation includes: S410, set the initial joint angle, gain coefficient, positive coefficients in the new time-varying parameters, and positive coefficients and power exponents in the new nonlinear activation function; S420, Set the desired trajectory of the end effector in underwater three-dimensional space; S430, calculate the Jacobian matrix based on the real-time joint angles of the robotic arm, calculate the desired velocity vector based on the desired trajectory, and determine the derivatives of the coefficient matrix and the known vector based on the real-time joint angles and the desired trajectory; S440, Substitute the known quantities into the dynamic evolution equation and obtain the joint angular velocity vector by numerical integration.

[0044] Specifically, parameter settings and environment initialization include: Set the initial joint angle (simulating the idle posture of the underwater robotic arm): .

[0045] The model parameters are set as follows (optimized to suit simulated underwater dynamic environments with strong disturbances): Gain coefficient: h=800; Coefficients of time-varying parameters: d=10; a=50; Parameters of the activation function: , , , , p=0.1.

[0046] Expected trajectory: The end effector is set to track a pentagonal star-shaped trajectory in underwater three-dimensional space. This trajectory simulates the continuous scanning path of an underwater pipeline circumferential weld or the "star-shaped" alignment path of a group of bolt holes in an underwater structure, including complex movements such as straight lines and acute-angle turns, to fully verify the algorithm's tracking capability under complex underwater tasks.

[0047] The PNVRGZND model proposed in this invention is applied to an underwater six-DOF robotic arm using the MATLAB simulation platform to simulate its trajectory tracking in a precise "pentagonal star" scanning task of underwater pipeline welds. Strong secondary noise from the simulated thruster and ocean current interference is incorporated into the simulation. Simulation results are as follows: Figures 2 to 10 As shown.

[0048] Figure 2 This image shows a 3D simulation of the PNVRGZND model proposed in this invention applied to an underwater robotic arm tracking a pentagonal scanning path. As can be seen from the image, the actual trajectory (solid red line) and the desired trajectory (dashed blue line) almost completely overlap throughout the entire motion, with no significant deviation or lag between them. This result fully demonstrates the ultra-high tracking accuracy of this invention under complex time-varying underwater trajectories, providing a guarantee for precision operations such as pipeline weld inspection.

[0049] Figure 3 This image displays snapshots of the motion postures of each joint of an underwater robotic arm during a pentagram trajectory scanning task. Different colors distinguish the motion paths of each joint, clearly showing the complete dynamic evolution of the robotic arm from its initial standby position to the completion of the task. It can be seen that the trajectories of each joint are smooth and continuous without abrupt changes, and the transitions between joints are natural and stable. This visually demonstrates that the robotic arm's movement is smooth and continuous throughout the entire underwater operation cycle, effectively avoiding accidental collisions with surrounding underwater equipment or cables.

[0050] Figure 4The curves showing the position error of the end effector along the X, Y, and Z directions as a function of time are presented. The position errors in all three dimensions converge rapidly to near zero within a very short time and remain stable throughout the process. The accuracy is on the order of meters, far exceeding the millimeter-level precision requirements typically needed for underwater pipeline inspection. The error curve is smooth throughout the entire process, without significant fluctuations or abrupt changes, fully demonstrating the ultra-high control accuracy and excellent steady-state performance of the proposed PNVRGZND model in a noisy underwater environment.

[0051] Figure 5 The curves showing the velocity error of the end effector along the X, Y, and Z directions as a function of time are presented. The fluctuation amplitudes of the three error curves are all strictly limited to... On the order of meters per second. Combined with... Figure 4 The positional error shown in the figure further verifies that the invention can also achieve extremely high stability in terms of speed control, ensuring that the underwater robotic arm can make gentle contact with precision instruments or pipes when approaching them, avoiding damage caused by speed impact.

[0052] Figure 6 The curves showing the acceleration error of the end effector along the X, Y, and Z directions as a function of time are presented. Throughout the entire 20-second observation period, the acceleration errors along the X and Z axes are extremely small, and the acceleration error along the Y axis is also strictly limited to within a certain range. meters per second 2 The magnitude of the error curves is significant. All three error curves are continuous and smooth, without any jumps or abrupt changes. This result fully demonstrates that the controller can effectively ensure that the underwater robotic arm can still achieve high-precision control across all dimensions, from position and velocity to acceleration, even when facing ocean current interference.

[0053] Figure 7 The curves showing the angular velocities of the six joints over time are presented. The angular velocities of each joint exhibit regular, smooth, periodic fluctuations with stable and controllable amplitudes, not exceeding the motion limits of the underwater robotic arm. This result verifies that the controller can effectively coordinate the movements of each joint, enabling the end effector to complete complex spatial trajectories while ensuring the continuity and safety of underwater joint movements.

[0054] Figure 8 The curves showing the acceleration of six joints over time are displayed. Each joint acceleration curve falls within the range of [-0.6, 1.0] radians per second. 2 The changes within the range exhibit regular periodicity, and the changes are smooth, continuous, and without abrupt changes. This result verifies that the controller can effectively suppress the dynamic impact and vibration at the joint level of the underwater robotic arm caused by sudden changes in ocean currents, protect the underwater dynamic sealing structure, and extend the equipment's lifespan.

[0055] Figure 9The curves showing the velocity variation of the end effector along the X, Y, and Z directions over time are presented. Throughout the entire 20-second motion cycle, the velocities in all three directions exhibit regular periodic fluctuations within ±0.06 m / s, and the velocity curves in each direction maintain a stable phase relationship. This coordinated velocity variation pattern indicates that the end effector is achieving smooth motion strictly following the expected trajectory, further validating the practicality of the control method.

[0056] Figure 10 The curves showing the acceleration of the end effector along the X, Y, and Z axes as a function of time are displayed. Throughout the 20-second observation period, the acceleration curves along all three axes fall within the range of [-0.1, 0.08] m / s². 2 The device exhibits regular oscillations within the range, with smooth and continuous changes in acceleration along each axis and coordinated phase, indicating that the end effector is in a stable periodic motion state and can effectively resist simulated underwater ocean current disturbances.

[0057] Compared with related technologies, the beneficial effects of this embodiment are: First, it exhibits strong resistance to underwater noise and ultra-high tracking accuracy. The PNVRGZND model proposed in this invention, through the construction of a novel double-integral structure, can theoretically completely eliminate secondary noise interference, significantly improving the system's robustness in complex marine environments. For example... Figure 2 As shown, the actual trajectory (red solid line) and the expected trajectory (blue dashed line) almost completely overlap during the entire motion process; Figures 4 to 6 Furthermore, it is shown that the errors of the end effector in the three dimensions of position, velocity, and acceleration are stably maintained at [values ​​to be filled in]. rice, meters per second and meters per second 2 The magnitude of the data verified the model's advantage in achieving ultra-high precision tracking and control even under simulated strong secondary noise underwater environments, fully meeting the precision requirements of tasks such as precision inspection of underwater pipelines.

[0058] Second, it achieves a balance between steady-state convergence and high-precision control. This invention introduces a novel time-varying parameter. It exhibits monotonically increasing characteristics with an upper bound, overcoming the inherent contradiction between steady-state convergence and control accuracy in existing time-varying parameters. For example... Figure 7 and Figure 8 As shown, the angle and acceleration curves of the six joints all exhibit regular and smooth periodic changes, with the amplitude strictly controlled within a safe range, without any sudden changes or divergence. Figure 9 and Figure 10 The study demonstrates the smooth and coordinated changes at the terminal under ocean current disturbances, fully proving that the time-varying parameter can maintain high-precision control performance throughout the entire motion cycle while ensuring the system resists time-varying ocean current disturbances and achieves steady-state convergence.

[0059] Third, it achieves fast convergence within a predefined time and excellent dynamic performance. The composite nonlinear activation function designed in this invention achieves predefined time convergence characteristics while significantly improving the convergence speed. For example... Figures 4 to 6 As shown, the position, velocity, and acceleration errors all converged rapidly to near zero within a very short time and remained stable throughout the entire 20-second motion cycle, without divergence or cumulative drift. Combined with... Figure 3 The snapshot of the robotic arm's motion posture and the smoothness and continuity of the trajectory of each joint shown verify that while achieving rapid convergence, the model can effectively suppress dynamic impacts and vibrations in underwater operations, ensuring the stability and real-time response capability of the robotic arm in high-speed and high-dynamic tasks such as emergency ascent and obstacle avoidance.

[0060] This invention also provides an underwater robotic arm trajectory planning and control system based on the PNVRGZND model, comprising: 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 described above.

[0061] 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.

[0062] 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.

[0063] 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.

[0064] This invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described method.

[0065] 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.

[0066] 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.

[0067] 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.

[0068] 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.

[0069] 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.

[0070] 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 underwater manipulator trajectory planning and control based on PNVRGZND model, characterized in that, The method includes the following steps: Obtain the joint angle vectors of the underwater six-degree-of-freedom serial manipulator, the desired velocity vector of the end effector, and the Jacobian matrix of the manipulator; transform the kinematic equations of the manipulator into time-varying linear equations; and define an error function based on the time-varying linear equations. A PNVRGZND model is constructed, which is a novel variable-parameter robust gradient basis nulling neurodynamic model with predefined time. The PNVRGZND model includes a novel double-integral structure, novel time-varying parameters, and novel nonlinear activation function. The novel double-integral structure is constructed by introducing a velocity compensation term and constructing auxiliary variables on the basis of the traditional gradient neurodynamic model. The novel time-varying parameters are monotonically increasing and bounded. The novel nonlinear activation function includes exponential, linear, and power terms. Substituting the derivative of the error function into the PNVRGZND model, we obtain the dynamic evolution equation for the joint angular velocity vector. The known quantities in the time-varying linear equation are determined based on the real-time state of the robotic arm and the desired trajectory. The joint angular velocity vector is obtained by numerically solving the dynamic evolution equation. Control commands are generated based on the joint angular velocity vector to drive the end effector to track the desired trajectory.

2. The method of claim 1, wherein, The step of converting the kinematic equations of the robotic arm into time-varying linear equations and defining an error function based on the time-varying linear equations includes: The kinematic equation of the robotic arm is expressed as the desired velocity vector being equal to the product of the Jacobian matrix and the joint angular velocity vector. The Jacobian matrix is ​​used as the coefficient matrix of the time-varying linear equation, the joint angular velocity vector is used as the state vector to be solved in the time-varying linear equation, and the desired velocity vector is used as the known vector of the time-varying linear equation. The error function is obtained by subtracting the known vector from the product of the coefficient matrix and the state vector to be solved.

3. The method of claim 1, wherein, The construction of the PNVRGZND model includes: The derivative of the error function is expressed as the product of the negative gain coefficient and the coefficient matrix and its transpose, multiplied by the error function, minus the first integral term, the second integral term, and the third integral term; The first integral term is the integral of the time variable from zero to the current time, and the integrand is the product of the new time-varying parameter and the auxiliary variable; The second integral term is the integral of the time variable from zero to the current time, and the integrand is the product of the new time-varying parameter and the result of the new nonlinear activation function applied to the auxiliary variable; The third integral term is a double integral of the time variable from zero to the current time. The inner integral variable is an auxiliary variable, the outer integral variable is a time variable, and the integrand is the product of the square of the new time-varying parameter and the result of the new nonlinear activation function applied to the auxiliary variable. The auxiliary variable is the sum of the product of the derivative of the error function, the gain coefficient, the coefficient matrix and its transpose, and the error function.

4. The method of claim 3, wherein, The formula for the novel time-varying parameter is: ； in, For a novel time-varying parameter, a>0, d>0, where a and d are constants, e is the natural base, and t is the time variable. .

5. The method according to claim 3, characterized in that, The formula for the novel nonlinear activation function is as follows: ； in, It is a novel nonlinear activation function. , , , All are constants greater than zero. A constant between 0 and 1 For any input vector, This represents the natural exponential function. This indicates taking the absolute value of the vector elements. This is a sign function that outputs 1 when the input is greater than 0, 0 when the input is equal to 0, and -1 when the input is less than 0.

6. The method according to claim 1, characterized in that, The step of substituting the derivative of the error function into the PNVRGZND model to obtain the dynamic evolution equation for the joint angular velocity vector includes: Taking the derivative of the error function, we find that the derivative of the error function is equal to the product of the derivative of the coefficient matrix and the state vector to be solved, plus the product of the derivative of the coefficient matrix and the state vector to be solved, minus the derivative of the known vector. Substituting the derivative of the error function into the PNVRGZND model and rearranging, we obtain that the product of the derivative of the coefficient matrix and the state vector to be solved is equal to the derivative of the known vector minus the product of the derivative of the coefficient matrix and the state vector to be solved, minus the product of the gain coefficient and the coefficient matrix and its transpose multiplied by the error function, and then minus the first integral term, the second integral term, and the third integral term. By left-multiplying both sides of the equation with the pseudo-inverse of the coefficient matrix, we obtain an expression for the derivative of the state vector to be solved.

7. The method according to claim 1, characterized in that, The process of obtaining the joint angular velocity vector by numerically solving the dynamic evolution equation includes: Set the initial joint angle, gain coefficient, positive coefficients in the novel time-varying parameters, and positive coefficients and power exponents in the novel nonlinear activation function; Set the desired trajectory of the end effector in underwater three-dimensional space; The Jacobian matrix is ​​calculated based on the real-time joint angles of the robotic arm, the desired velocity vector is calculated based on the desired trajectory, and the derivatives of the coefficient matrix and the known vector are determined based on the real-time joint angles and the desired trajectory. Substituting the known quantities into the dynamic evolution equation, the joint angular velocity vector is obtained by numerical integration.

8. A trajectory planning and control system for an underwater robotic arm based on the PNVRGZND 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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