A mechanical arm trajectory tracking control method under a preset time constraint

Abstract

This application relates to the field of programmable robotic arms, and particularly to a method for trajectory tracking control of a robotic arm under preset time constraints. The method includes: establishing a second-order nonlinear system based on a robotic arm dynamics model; defining joint position tracking error and joint velocity tracking error; scaling the joint position tracking error; designing a virtual control law and an actual control law; designing a preset time disturbance observer, which is used to observe and compensate for comprehensive disturbances in real time, ensuring that the error converges to zero within a preset time; and integrating these into a complete controller, which is then applied to the robotic arm system to achieve full-state tracking control under preset time constraints. This application, by introducing a preset time disturbance observer into the robotic arm trajectory tracking control, enables rapid and accurate observation and compensation of disturbances within a user-defined time period, effectively addressing the high-speed and high-precision requirements of full-state tracking for robotic arms under complex working conditions with time-varying disturbances.

Description

Technical Field

[0001] This application relates to the field of program-controlled robotic arms, specifically to a method for tracking and controlling the trajectory of a robotic arm under a preset time constraint. Background Technology

[0002] Tracking control of robotic arms is a core technology in industrial automation and intelligent manufacturing, and is widely used in precision assembly, material handling, and flexible manufacturing. Current typical robotic arm control methods can be mainly classified into the following categories:

[0003] 1. Model-based precision control method.

[0004] Represented by radial basis function (RBF) neural network control, this type of method adaptively approximates the nonlinear characteristics of a robotic arm (such as complex friction and parameter perturbations) through neural networks, without relying on a completely accurate dynamic model. However, the network training accuracy and convergence speed are limited by the network structure and learning rate, and control delays are prone to occur in highly dynamic scenarios. In addition, its generalization ability is highly dependent on the coverage of the training data; when the actual working conditions exceed the distribution of the training data, the control accuracy drops significantly. Therefore, the implicit dependence of this method on model accuracy has not been completely eliminated, and factors such as nonlinear friction, parameter perturbations, and changes in external loads in reality can still easily lead to deterioration of control performance.

[0005] 2. Robust control-based methods.

[0006] Sliding mode control, for example, is highly robust to matching uncertainties. By using discontinuous control laws to force the system state asymptotically to the sliding surface, it can suppress uncertainties to some extent. However, traditional sliding mode control suffers from chattering, which causes high-frequency oscillations in the robotic arm joints, reducing tracking accuracy and exacerbating actuator wear. Although there are improvement strategies, they often come at the cost of sacrificing robustness or increasing complexity, tending to be conservative. Furthermore, it is difficult to simultaneously achieve both tracking accuracy and response speed, making it hard to meet the demands of high-dynamic and high-precision operations.

[0007] 3. Control method based on disturbance observer.

[0008] This type of method improves control performance by observing and compensating for disturbances. However, the convergence characteristics of traditional disturbance observers are limited by system bandwidth or observer parameter design, making it difficult to achieve accurate observation and compensation of disturbances within a preset time. As a result, the requirements for speed and accuracy in tracking the robot arm's position, speed, and other states cannot be met simultaneously.

[0009] Existing robotic arm trajectory tracking control methods mainly suffer from the following two key problems:

[0010] First, in complex scenarios with time-varying disturbances, it is difficult for the robotic arm to achieve high-precision convergence in tracking its position, speed, and other states within a preset time.

[0011] Secondly, complex centralized disturbance compensation architectures or slow disturbance observation convergence speeds can lead to system response lag, making it unsuitable for high-speed, highly dynamic operations. For example, control strategies based on traditional linear disturbance observers rely on the observer gain for observation error convergence speed and are susceptible to high-frequency noise interference. This makes it difficult to ensure rapid and accurate tracking of the robotic arm across all states under strong disturbances, failing to meet the stringent requirements for dynamic performance and accuracy in high-end manufacturing.

[0012] Therefore, there is an urgent need for a new full-state tracking control scheme for robotic arms. Summary of the Invention

[0013] In view of this, this application discloses a robotic arm trajectory tracking control method under a preset time constraint to solve the problems in the prior art, including:

[0014] S1. Establish a second-order nonlinear system with integrated disturbances corresponding to the dynamic model of the robotic arm;

[0015] S2. Based on the joint motion trajectory of the robotic arm in the nonlinear system, define the joint position tracking error and the joint velocity tracking error;

[0016] S3. Establish a preset performance function and scale the joint position tracking error based on the preset performance function;

[0017] S4. Based on the joint velocity tracking error and the scaled joint position tracking error, design the virtual control law and the actual control law;

[0018] S5. Design a preset time disturbance observer; the preset time disturbance observer is used to observe and compensate for the comprehensive disturbance in real time, so that the error converges to zero within a preset time; the preset time is not affected by the control parameters.

[0019] Furthermore, the preset time perturbation observer is defined by the following formula:

[0020]

[0021] in, This represents the estimated value of the joint angular velocity vector. This represents the update law for the estimated joint angular velocity vector. This represents the estimated value of the combined disturbance. This represents the rate of change of the estimated combined disturbance over time. and Represents the nonlinear correction term. This represents the error scaling factor. This represents the control gain constant. Preset the convergence time for the system; Represents the time-varying adjustment function rate of change over time Indicates the actual control input;

[0022] S6. Integrate the virtual control law, the actual control law, and the preset time disturbance observer into a complete controller; realize the full state tracking control of the robotic arm system under the preset time based on the controller.

[0023] The beneficial effects of this application include:

[0024] This application addresses the inherent oscillation and asymptotic convergence problems of traditional control methods (such as RBF neural network adaptive control, sliding mode control, etc.) by providing a preset time disturbance observer. Based on the function design of this preset time disturbance observer, the system can accurately estimate the comprehensive external disturbance of the system within a preset time, thereby effectively enhancing the transient response capability and robustness of the system.

[0025] The biggest difference from the prior art is that the preset time in this application does not depend on the initial state of the system and is independent of the control design parameters. The convergence time of the system disturbance estimation error is directly preset by the user, avoiding the oscillation and noise amplification problems caused by high-gain feedback. By combining the preset performance function to constrain the tracking error of the robotic arm, it can be ensured that the system state is always within the preset performance boundary, which significantly improves the accuracy and dynamic performance of trajectory tracking. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the robotic arm trajectory tracking control method under preset time constraints in the embodiments of this application;

[0027] Figure 2 This is a schematic diagram of the dynamic system model parameters of a two-degree-of-freedom robotic arm in an embodiment of this application;

[0028] Figure 3 This is a schematic diagram of the nominal model parameters of the two-degree-of-freedom robotic arm system in the embodiments of this application.

[0029] Figure 4 This is a schematic diagram of the initial state parameters in an embodiment of this application;

[0030] Figure 5 This is a schematic diagram of the relevant control parameters in the embodiments of this application;

[0031] Figure 6 This is a schematic diagram of the position tracking trajectories of joint 1 and joint 2 in an embodiment of this application;

[0032] Figure 7This is a schematic diagram of the velocity tracking trajectories of joint 1 and joint 2 in an embodiment of this application;

[0033] Figure 8 This is a schematic diagram illustrating the position tracking errors of joints 1 and 2 in an embodiment of this application.

[0034] Figure 9 This is a schematic diagram of the interference estimation curve and its error variation characteristics in the embodiments of this application;

[0035] Figure 10 This is a schematic diagram of the output characteristics of the system control torque in an embodiment of this application. Detailed Implementation

[0036] To make the objectives, technical solutions, features, and advantages of this application clearer and to facilitate a better understanding of the technical solutions of this application by those skilled in the art, the following detailed description of this application is provided in conjunction with the accompanying drawings and embodiments.

[0037] Example 1:

[0038] This embodiment includes a robotic arm trajectory tracking control method under a preset time constraint, such as... Figure 1 As shown, it includes:

[0039] S1. Establish a second-order nonlinear system with integrated disturbances corresponding to the dynamic model of the robotic arm.

[0040] The dynamic model of the robotic arm is defined by the following formula:

[0041]

[0042] in, This represents the positive definite inertia matrix of the robotic arm. The matrix representing the centrifugal force and Coriolis force of the robotic arm. This represents the gravitational torque vector of the robotic arm. This indicates the control torque applied to the joint; This represents unknown external disturbances and frictional forces under time-varying conditions. These represent the joint position, velocity, and acceleration vectors, respectively. This indicates the number of joints in the robotic arm.

[0043] The formula for the second-order nonlinear system containing the combined disturbance is:

[0044]

[0045] in, This represents the known system control gain. The combined disturbance representing the time-varying uncertainty term. Indicates system output, This represents the joint position vector of the robotic arm. This represents the joint velocity vector of the robotic arm.

[0046] S2. Based on the joint motion trajectory of the robotic arm in the nonlinear system, define the joint position tracking error and the joint velocity tracking error, with the following formulas:

[0047]

[0048] in, It is the joint position tracking error vector. This represents the joint velocity tracking error vector. Indicates the time of the robotic arm joint. The actual position vector, Indicates the time of the robotic arm joint. The expected position vector, Indicates the time of the robotic arm joint. The actual velocity vector; Indicates the time of the robotic arm joint. The virtual control law.

[0049] S3. Establish a preset performance function and scale the joint position tracking error based on the preset performance function.

[0050] To ensure that the system's position tracking error remains within preset performance boundaries, and to constrain its transient and steady-state performance, preset performance constraints are established as follows:

[0051]

[0052] in, This indicates the number of joints in the robotic arm.

[0053] Establish an exponential preset performance function The formula is:

[0054]

[0055] in, Represents the initial performance boundary. Represents the steady-state performance boundary. Indicates the convergence rate. This represents an exponential function with the natural constant e as its base.

[0056] Exponential preset performance function The first and second derivatives are expressed as follows:

[0057]

[0058] Furthermore, the original error is normalized to obtain the scaled position tracking error:

[0059]

[0060] in, This represents the scaled position tracking error. This indicates the original position tracking error. This indicates the preset performance function.

[0061] S4. Based on the joint velocity tracking error and the scaled joint position tracking error, design the virtual control law and the actual control law.

[0062] In this embodiment, the virtual control law is designed based on the backstepping method, and the formula is:

[0063]

[0064] in, Represents a virtual control law. This represents the virtual control adjustment coefficient, used to adjust the virtual control feedback gain. This represents the desired joint angular velocity vector. This indicates the preset performance function. This represents the first derivative of the preset performance function with respect to time. This represents the position tracking error after scaling.

[0065] The actual control law is designed using Lyapunov theory to determine the actual control torque of the system. The formula is:

[0066]

[0067] in, Indicates the actual control input. This represents the known system control gain. This represents the positive definite inertia matrix of the robotic arm. Indicates joint position. This represents the actual control adjustment coefficient, used to adjust the actual control gain. This represents the joint velocity tracking error vector. Representing virtual control law The derivative of It is an estimate of the combined disturbance.

[0068] Furthermore, It can be represented as:

[0069]

[0070] in, This represents the desired joint angle acceleration vector; This represents the derivative of the scaling tracking error.

[0071] S5. Design a preset time disturbance observer; the preset time disturbance observer is used to observe and compensate for the comprehensive disturbance in real time, so that the error converges to zero within a preset time; the preset time is not affected by the control parameters.

[0072] To address the non-vanishing and time-varying uncertainties in robotic arm systems, this application designs a pre-set time disturbance observer, with the following formula:

[0073]

[0074] in, This represents the estimated value of the joint angular velocity vector. This represents the update law for the joint angular velocity vector estimate, i.e., the derivative of the angular velocity vector estimate with respect to time t. This represents the estimated value of the combined disturbance. This represents the rate of change of the estimated combined disturbance over time. and Represents the nonlinear correction term. This represents the error scaling factor. This represents the control gain constant. Preset convergence time for the system. It is set manually and is independent of system control parameters; Represents the time-varying adjustment function Over time The rate of change, and the formula for the time-varying adjustment function are:

[0075]

[0076] in, Indicates the initial gain. Indicates the current time; hour, Joint angular velocity estimation error Overall disturbance estimation error: The error converges to zero within a preset time.

[0077] Nonlinear correction term in the formula: , This is designed to improve the convergence speed of observation errors within a preset time and enhance robustness to initial errors and disturbance changes. It acts on the joint velocity state error and the comprehensive disturbance estimation error, respectively, and is specifically expressed as follows:

[0078]

[0079] in, Represents the double tangent function. Represents the standard Euclidean norm. This represents the scaling error of the composite estimate from the joint velocity observer. ; This represents the scaling error of the composite estimate from the integrated perturbation observer. .

[0080] Composite estimation error of joint velocity observer The combined estimation error of the integrated disturbance observer and the combined disturbance observer They are defined as follows:

[0081]

[0082] in, This represents the actual overall disturbance. This represents the joint velocity vector of the robotic arm. The composite estimation error of the joint velocity observer. The combined estimation error of the integrated disturbance observer and the combined disturbance observer The derivative form is expressed as:

[0083]

[0084] in, This represents the time derivative of the actual composite disturbance. Represents the identity matrix.

[0085] Compared to the uncertain convergence time of asymptotic convergence schemes and the convergence time of traditional fixed-time schemes coupled with design parameters, the method in this application compensates for external disturbances in a coordinated manner with a preset time disturbance observer and adaptive control. It can accurately estimate the comprehensive external disturbances of the system within a preset time, and this time does not depend on the initial state of the system and is independent of the control design parameters.

[0086] S6. Integrate the virtual control law, the actual control law, and the preset time disturbance observer into a complete controller; realize the full state tracking control of the robotic arm system under the preset time based on the controller.

[0087] By achieving full-state tracking error convergence within a preset time and bounded closed-loop signal through the controller, the overall controller can be represented as follows:

[0088]

[0089] Based on the control law of the above formula, a controller is constructed, which ultimately realizes the full-state tracking control of the robotic arm system under a preset time.

[0090] Example 2:

[0091] This embodiment includes a robotic arm trajectory tracking control method under preset time constraints. The difference from Embodiment 1 is that this embodiment will combine Matlab2018b simulation experiments to illustrate the effectiveness of the robotic arm trajectory tracking control method. Specifically, through numerical simulation experiments of a two-degree-of-freedom robotic arm, the full-state tracking control method of the robotic arm based on a preset time disturbance observer is systematically verified.

[0092] The parameters of the two-degree-of-freedom robotic arm dynamic system model are as follows: Figure 2 As shown, the total system simulation time is set to 10, that is... Sampling period , Figure 3 The nominal model parameters of a two-degree-of-freedom robotic arm system are listed. Figure 4 The initial state parameters are listed, and other relevant control parameters are as follows: Figure 5 As shown.

[0093] exist Figure 2 and Figure 3 middle, This represents the combined parameter vector of the dual-arm dynamics model. These are the masses of link 1 and link 2, respectively. These are the moments of inertia of link 1 and link 2, respectively. These are the distances from link 1 and link 2 to their centers of mass, respectively. The lengths of link 1 and link 2 are given respectively; this simulation only provides the algebraic relationships between the physical parameters of the dual-arm system.

[0094] Figure 4 The parameters in the figure represent the initial state conditions for the simulation of the dual-joint robotic arm, where the initial joint angles are... The spatial orientation of the two joints and the initial joint angular velocity are defined at the start of the simulation. The motion rates of the two joints at the start of the simulation were defined.

[0095] Figure 5 middle, Used to adjust the response characteristics of joint position tracking. Used to adjust the control characteristics for joint velocity tracking These are the scaling parameters for the composite observer; This indicates the preset convergence time of the preset time perturbation observer. This represents the initial gain of the preset time-perturbation observer. It is a parameter for controlling performance gain; As the initial preset performance boundary, To ultimately preset performance boundaries, This is the convergence rate of the preset performance function.

[0096] Furthermore, the dynamic model matrix form of the dual-arm system is expressed as follows:

[0097]

[0098]

[0099]

[0100]

[0101] in, Here is the positive definite inertia matrix of the robotic arm. Let be the matrix of the centrifugal force and the Coriolis force of the robotic arm; Let be the gravitational torque vector of the robotic arm. It is the control torque applied to the joint. This represents unknown external disturbances and frictional forces under time-varying conditions. These represent the joint angle vector, joint angular velocity vector, and joint angular acceleration vector, respectively. The simulation is set with a preset convergence time. It takes 1 second.

[0102] Furthermore, the desired trajectory is set as follows: The simulation was performed, and the simulation results are as follows: Figures 6-10 As shown.

[0103] Figure 6 The position tracking trajectories of joints 1 and 2 are shown, demonstrating that the actual position of the robotic arm can quickly and accurately track the desired trajectory.

[0104] Figure 7 The velocity tracking trajectories of joint 1 and joint 2 are given, showing that the actual velocity can also achieve accurate tracking of the desired velocity.

[0105] Figure 8 The position tracking errors of joints 1 and 2 are shown. It can be seen that the position tracking errors are always within the preset performance boundary and eventually converge to the preset performance boundary. Internally, this indicates that the proposed control method improves the system's response speed.

[0106] Figure 9 The interference estimation curve and its error variation characteristics are presented. It can be clearly seen that the interference estimation error converges rapidly within the preset time, providing accurate interference information support for the implementation of subsequent anti-interference control strategies.

[0107] Figure 10 The output characteristics of the system's control torque were demonstrated. The results showed that the torque peak was lower and the fluctuation was smoother. This characteristic helps to reduce the drive loss of the robotic arm joints and improve the system's energy efficiency and operational stability.

[0108] In summary, this study proposes a robotic arm trajectory tracking control method under preset time constraints by integrating a preset time and a disturbance observation mechanism. Addressing the uncertainties of time-varying loads, the collaborative design of a preset time disturbance observer and adaptive control effectively compensates for external disturbances, significantly improving the system's robustness in unstructured operating environments. This method can effectively solve the problem of accurate tracking across all states in complex scenarios such as dynamic control of high-precision manufacturing equipment, providing a technical solution with both real-time convergence and robust anti-interference capabilities for engineering fields such as industrial automated flexible production lines and precision control of defense equipment, demonstrating significant practical engineering application value.

[0109] Finally, it should be noted that the above description only depicts some embodiments of this application. For those skilled in the art, various changes, modifications, substitutions, and variations can be conceived of these embodiments without departing from the principles and spirit of this application. The scope of protection of this application is defined by the appended claims and their equivalents, and all the above-mentioned behaviors should be covered within the scope of protection of this application.

Claims

1. A method for tracking and controlling the trajectory of a robotic arm under a preset time constraint, characterized in that, include: S1. Establish a second-order nonlinear system with integrated disturbances corresponding to the dynamic model of the robotic arm; S2. Based on the joint motion trajectory of the robotic arm in the nonlinear system, define the joint position tracking error and the joint velocity tracking error; S3. Establish a preset performance function and scale the joint position tracking error based on the preset performance function; S4. Based on the joint velocity tracking error and the scaled joint position tracking error, design the virtual control law and the actual control law; S5. Design a preset time disturbance observer; The preset time disturbance observer is used to observe and compensate for the comprehensive disturbance in real time, so that the error converges to zero within a preset time; the preset time is not affected by the control parameters. S6. Integrate the virtual control law, the actual control law, and the preset time disturbance observer into a complete controller; realize the full state tracking control of the robotic arm system under the preset time based on the controller.

2. The robotic arm trajectory tracking control method under a preset time constraint according to claim 1, characterized in that, The preset time perturbation observer is defined by the following formula: ； in, This represents the update law for the estimated joint angular velocity vector. This represents the estimated value of the joint angular velocity vector. This represents the estimated value of the combined disturbance. Represents the control gain constant. This indicates the system's preset convergence time. Set by humans, Represents the time-varying adjustment function Over time rate of change, Indicates the actual control input; This represents the rate of change of the estimated combined disturbance over time. and Represents the nonlinear correction term. This represents the error scaling factor.

3. The robotic arm trajectory tracking control method under preset time constraints according to claim 2, characterized in that, The time-varying adjustment function is formulated as follows: ； in, Indicates the initial gain. Indicates the current time.

4. The robotic arm trajectory tracking control method under preset time constraints according to claim 2, characterized in that, The nonlinear correction term, , These factors, acting on the joint velocity state error and the comprehensive disturbance estimation error respectively, are expressed by the following formula: ； in, Represents the double tangent function. Represents the standard Euclidean norm. This represents the scaling error of the composite estimate from the joint velocity observer. , This represents the composite estimation error of the joint velocity observer; This represents the scaling error of the composite estimate from the integrated perturbation observer. , This represents the composite estimation error of the integrated disturbance observer.

5. The robotic arm trajectory tracking control method under a preset time constraint according to claim 4, characterized in that, Composite estimation error of joint velocity observer The combined estimation error of the integrated disturbance observer and the combined disturbance observer The formula is: ； in, This represents the joint velocity vector of the robotic arm. This represents the actual overall disturbance.

6. The robotic arm trajectory tracking control method under a preset time constraint according to claim 1, characterized in that, The establishment of a preset performance function includes establishing an exponential preset performance function. The formula is: ； in, Indicates the current time. Indicates the initial performance boundary. Represents the steady-state performance boundary. Indicates the convergence rate. This represents an exponential function with the natural constant e as its base.

7. The robotic arm trajectory tracking control method under a preset time constraint according to claim 1, characterized in that, The scaling of the joint position tracking error based on a preset performance function is calculated using the following formula: ； in, This represents the scaled position tracking error. This indicates the original position tracking error. This indicates the preset performance function.

8. The robotic arm trajectory tracking control method under a preset time constraint according to claim 1, characterized in that, The virtual control law, designed based on the backstep method, is as follows: ； in, Represents a virtual control law. This represents the desired joint angular velocity vector. This represents the virtual control adjustment coefficient, used to adjust the virtual control feedback gain. This indicates the preset performance function. This represents the first derivative of the preset performance function with respect to time. This represents the scaled position tracking error.

9. The robotic arm trajectory tracking control method under a preset time constraint according to claim 1, characterized in that, The actual control law is obtained by designing the actual control torque of the system using Lyapunov theory, and the formula is: ； in, Indicates the actual control input. This represents the known system control gain. This represents the positive definite inertia matrix of the robotic arm. Indicates joint position. This represents the actual control adjustment coefficient, used to adjust the actual control gain. This represents the joint velocity tracking error vector. Representing virtual control law The derivative, It is an estimate of the combined disturbance.

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