A flexible robot arm preset time trajectory tracking control method based on an interference observer

Abstract

The application discloses a preset time trajectory tracking control method for a flexible mechanical arm based on an interference observer, and the method is as follows: step 1, for a type of connecting rod flexible mechanical arm, a dynamic model in the Lagrange form is established; step 2, a preset time time-varying function is designed to be smoothly approximated; step 3, for the lumped uncertainty existing in the system, a preset time sliding mode interference observer is designed to be observed and compensated; step 4, an instruction filter is introduced and an error compensator is designed; step 5, for the interference observer, the second-order instruction filter and the error compensator designed in steps 3 and 4, a preset time trajectory tracking control law based on the backstepping method is designed. The method adopts the Lagrange dynamic model of the flexible mechanical arm, considers the model parameter uncertainty and external environmental interference of the flexible mechanical arm, and realizes the preset time trajectory tracking control of the mechanical arm by designing the preset time sliding mode interference observer and further deducing the control law.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system control and relates to a robotic arm trajectory tracking control method, specifically a flexible robotic arm preset time trajectory tracking control method based on a disturbance observer. Background Technology

[0002] Flexible joint robotic arms are highly nonlinear and complex dynamic systems. In real-world working environments, robotic arms inevitably experience parameter uncertainties and external environmental disturbances, posing significant challenges to high-precision trajectory tracking control. Traditional preset-time control often generates infinite gain at the convergence end, easily causing control torque chattering. Furthermore, when designing traditional backstepping methods for high-order nonlinear systems, it is necessary to repeatedly differentiate the virtual control quantity, resulting in a severe "differential explosion" problem, which drastically increases the difficulty and computational burden of controller design. Summary of the Invention

[0003] This invention provides a preset time trajectory tracking control method for a flexible robotic arm based on an interference observer. The method adopts the Lagrangian dynamics model of the flexible robotic arm, considers the uncertainty of the model parameters of the flexible robotic arm and external environmental interference, and achieves preset time trajectory tracking control of the robotic arm by designing a preset time sliding mode interference observer and further deriving the control law.

[0004] The objective of this invention is achieved through the following technical solution:

[0005] A method for tracking and controlling a flexible robotic arm's preset time trajectory based on an interference observer includes the following steps:

[0006] Step 1, targeting a class A Lagrangian dynamic model of a linkage-based flexible robotic arm is established:

[0007]

[0008] in, For the angles of each link, The rotation angle of each joint motor, The inertia matrix, For Coriolis matrix, For gravity torque matrix, Here is the inertia matrix of the motor. For elastic torque, For external disturbance torque, Input torque; , , , For the nominal model, To address uncertainty, we separate the nominal model from uncertainty and define lumped uncertainty. By defining system state variables , =1,2,3,4, which can be transformed into the following strictly feedback state-space equation:

[0009]

[0010] in, For equivalent set total uncertainty, Here is the stiffness matrix of the motor;

[0011] Step 2: Design a time-varying function that smoothly approximates the preset time:

[0012] Design the following smooth approximation function and time-varying control gain function:

[0013]

[0014] in, For preset time, This is the lower limit of gain saturation. As a smooth transition factor, For constant parameters, For time, For intermediate functions, To smoothly approximate the function, The time-varying control gain function, It is a time-varying scaling function;

[0015] Step 3: To address the lumped uncertainty in the system, a preset time-sliding mode disturbance observer is designed for observation and compensation. The dynamics and adaptive law design of the preset time-sliding mode disturbance observer are as follows:

[0016]

[0017]

[0018] in, , This is an estimate of the state. This is an estimate of the disturbance. For state estimation error, For adjustable positive constant parameters, Preset a time-varying gain function for the observer. To enable online adjustment of integral gain using an adaptive law, It is an adjustable constant parameter;

[0019] Step 4: Introduce the command filter and design the error compensator:

[0020] Step 4-1: Introduce a second-order instruction filter to solve the differential explosion problem in the backstepping method. The second-order instruction filter... and ,in, For input to the virtual control law, and For the state variables of the filter, For natural frequency, For the damping ratio, =1,2,3, the filtered output signal is ;

[0021] Step 4-2: Design the error compensator subsystem:

[0022]

[0023] in, Let the compensator state vector be... =1,2,3,4 Preset a time-varying function for the controller. For filtering error, =1,2,3 Adjustable normal parameters ;

[0024] Step 5: For the interference observer, second-order command filter, and error compensator designed in Steps 3 and 4, design a preset time trajectory tracking control law based on the backstepping method:

[0025]

[0026] in, For the first Virtual control variables in step-by-step and reverse-step design For the dynamic tracking error of the system, =1,2,3,4 For the desired location trajectory, This refers to the actual input control torque that ultimately acts on the robotic arm motor.

[0027] Compared with the prior art, the present invention has the following advantages:

[0028] This invention addresses the trajectory tracking problem of flexible joint robotic arms by designing a control method based on preset time and command filtering backstepping. By designing a smooth preset time time-varying function, the method effectively avoids the problems of infinite gain and torque chattering at the convergence end of traditional preset time control. By introducing a second-order command filter and its error compensator, the "differential explosion" problem in traditional backstepping design is completely solved, reducing the computational burden. The designed preset time sliding mode disturbance observer can estimate and compensate for the lumped uncertainty of the system online without requiring a known upper bound on the disturbance derivative, greatly improving the robustness and trajectory tracking accuracy of the flexible robotic arm system. Attached Figure Description

[0029] Figure 1 A block diagram of a time trajectory tracking control system for a flexible robotic arm based on an interference observer;

[0030] Figure 2 for , , , The preset time-varying function;

[0031] Figure 3 for , Tracking curves of each joint angle;

[0032] Figure 4 for , The tracking error curves for each joint angle;

[0033] Figure 5 for , The joint perturbation estimation curves;

[0034] Figure 6 for , Error curves for perturbation estimation of each joint;

[0035] Figure 7 for , The input control torque curves for each joint;

[0036] Figure 8 for , Tracking curves of each joint angle;

[0037] Figure 9 for , The tracking error curves for each joint angle;

[0038] Figure 10for , The joint perturbation estimation curves;

[0039] Figure 11 for , Error curves for perturbation estimation of each joint;

[0040] Figure 12 for , The input control torque curves for each joint;

[0041] Figure 13 for , Tracking curves of each joint angle;

[0042] Figure 14 for , The tracking error curves for each joint angle;

[0043] Figure 15 for , The joint perturbation estimation curves;

[0044] Figure 16 for , Error curves for perturbation estimation of each joint;

[0045] Figure 17 for , The input control torque curves for each joint. Detailed Implementation

[0046] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0047] This invention provides a method for tracking and controlling a flexible robotic arm's preset time trajectory based on an interference observer, such as... Figure 1 As shown, the method includes the following steps:

[0048] Step 1, targeting a class The dynamic model of the linkage-driven flexible robotic arm, in Lagrangian form, is shown below:

[0049]

[0050] in, For the angles of each link, The rotation angle of each joint motor, The inertia matrix, For Coriolis matrix, For gravity torque matrix, and These are the inertia matrix and stiffness matrix of the motor, respectively. For elastic torque, For external disturbance torque, This is for input torque. , , , For the nominal model, To address uncertainty, we separate the nominal model from uncertainty and define lumped uncertainty. By defining system state variables , =1,2,3,4, which can be transformed into the following strictly feedback state-space equation:

[0051]

[0052] in, This represents the uncertainty of the equivalent set.

[0053] Step 2: Design a smooth preset time-varying function.

[0054] To avoid the infinite gain phenomenon that occurs when the traditional preset-time convergence function approaches the preset time, a smooth approximation function is first designed:

[0055]

[0056] in, Convergence time is user-defined. This is the lower limit of gain saturation. As a smooth transition factor, For time, For intermediate functions, To achieve a smooth approximation function, the time-varying control gain function is designed as follows: ,in This is a constant parameter. It also defines a hidden time-varying scaling function. From this definition, we can know that the core property is .

[0057] Step 3: Implement the preset time sliding mode interference observer.

[0058] Addressing lumped uncertainties in the system's state-space equations Let its dynamic subsystem be denoted as The corresponding state observer and disturbance observer are designed as follows:

[0059]

[0060] in, and These are the estimated values ​​for the state and the disturbance, respectively. For state estimation error, A time function is preset for the first stage. To overcome the problem of unknown boundaries of the derivative of the true perturbation, an adaptive law is designed to adjust the integral gain online. The initial value satisfies Introducing intermediate auxiliary variables and At this point, the estimation error dynamics can be written as: .

[0061] Step 4: Introduce an instruction filter and design an error compensator.

[0062] To overcome the differential explosion caused by differentiation, a second-order instruction filter is introduced. and ,in, For input to the virtual control law, , =1,2,3. The filtered output signal is To eliminate filtering errors To mitigate the impact, the following compensator subsystem is constructed:

[0063]

[0064] The controller's preset time scaling function is: .

[0065] Step 5: Design a trajectory tracking control law based on the backstepping method for the interference observer, second-order command filter and error compensator designed in Steps 3 and 4.

[0066] Define the tracking errors of the system as follows: , , , ,in The desired position trajectory is given. The control law is designed as follows:

[0067]

[0068] Example: To verify the effectiveness of the above method, for a class of The dynamic model of the linkage-driven flexible robotic arm, in Lagrangian form, is shown below:

[0069]

[0070] in, For the angles of each link, The rotation angle of each joint motor, The inertia matrix, For Coriolis matrix, For gravity torque matrix, and These are the inertia matrix and stiffness matrix of the motor, respectively. For elastic torque, For external disturbance torque, This is for input torque. , , Separating the nominal model from uncertainty With uncertainty And define lumped uncertainty By defining system state variables , =1,2,3,4, which can be transformed into the following strictly feedback state-space equation:

[0071]

[0072] in, This represents the equivalent settling uncertainty. In order to... To compensate, a preset time-varying disturbance observer is introduced. First, a preset time-varying function is designed:

[0073]

[0074] in, Convergence time is user-defined. This is the lower limit of gain saturation. As a smooth transition factor, It is a constant. From this definition, the core property is... .

[0075] Design a pre-defined time-sliding mode disturbance observer. Address the lumped uncertainties in the system's state-space equations. Let its dynamic subsystem be denoted as .set up , , , for aggregate uncertainty The observer can be designed according to the following expression:

[0076]

[0077] in, and These are the estimated values ​​for the state and the disturbance, respectively. To estimate the error, A preset time function is provided for the observer, where all adjustable parameters satisfy... To overcome the problem of unknown boundaries of the derivative of the true perturbation, an adaptive law is designed to adjust the integral gain online. The initial value satisfies Introducing intermediate auxiliary variables and At this point, the estimation error dynamics can be written as: .

[0078] To solve the differential explosion problem and obtain the first derivative of the virtual control signal as accurately as possible, an instruction filter is introduced:

[0079]

[0080] in, For the first Virtual control law input in step-back step design; and For the state variables of the filter; For natural frequency, Let be the damping ratio. Define the filter's output signal as... Its first derivative is By selecting appropriate parameters and This can guarantee the error of the filtered signal. It converges within a very small error range, and the amplitude and derivative of the filtered signal are both bounded.

[0081] Based on filter (5), the following system tracking error is defined:

[0082]

[0083] in, This represents the desired position trajectory curve.

[0084] To achieve the preset time convergence, a time-varying function of equation (3) is introduced. All adjustable parameters meet the requirements. The tracking error (6) can then be corrected to the following expression:

[0085]

[0086] in, This is the output of the error compensator in the instruction filtering method; the specific design will be given later.

[0087] The first step, in In this case, equation (7) can be expressed as Taking the first derivative with respect to time, we get:

[0088]

[0089] Virtual input and error compensator It can be designed as:

[0090]

[0091]

[0092] Bundle and Substitution We can obtain:

[0093]

[0094] The second step, in In this case, equation (7) can be expressed as Taking the first derivative with respect to time, we get:

[0095]

[0096] Virtual input and error compensator It can be designed as:

[0097]

[0098]

[0099] in These are the perturbation observations obtained from observer (4). and Substitution We can obtain:

[0100]

[0101] in This represents the disturbance error.

[0102] The third step, In this case, equation (7) can be expressed as Taking the first derivative with respect to time, we get:

[0103]

[0104] Virtual input and error compensator It can be designed as:

[0105]

[0106]

[0107] Bundle and Substitution We can obtain:

[0108]

[0109] Fourth step, in In this case, equation (7) can be expressed as Taking the first derivative with respect to time, we get:

[0110]

[0111] Actual input and error compensator It can be designed as:

[0112]

[0113]

[0114] Bundle and Substitution We can obtain:

[0115]

[0116] In summary, the error compensation subsystem and control law can be designed as follows:

[0117]

[0118]

[0119] Before proving stability, we introduce the following theorem:

[0120] Theorem 1: If there exists a Lyapunov function satisfy Given that the external disturbance is bounded Then at the preset time Physical error Strictly converges to the residual set .

[0121] Proof of observer stability: Considering the Lyapunov function for the estimation error dynamics Differentiation yields Because the energy of a physical system is finite, the first-to-last value of the derivative of the lumped perturbation is limited. It must exist. Because the adaptive gain is monotonically constant, it must exist at a finite time point. Make In the initial approach phase Integrating over time and scaling the integral yields the following result. This indicates that this stage is strictly bounded. During the sliding hold stage... ,analyze The dynamic evolution direction was found, if Then its time derivative satisfies At this point, it must be monotonically decreasing; if ,but At this point, it must be monotonically increasing. Therefore, in the sliding mode, the absolute value will not exceed the maximum value during the approach phase. In the global time domain, we have... According to Theorem 1, the disturbance error can finally be proved. Can be set at a preset time It strictly converges to a small neighborhood near the origin. Inside.

[0122] Proof of controller stability: A weighted Lyapunov analysis method is introduced to avoid gain explosion caused by canceling cross terms. An upper bound constant for the maximum gain of the stiffness coupling term is defined. , Representation matrix The minimum eigenvalue lower bound. Let the constant weight coefficients... Consider the Lyapunov function of the weighted error compensator. By combining the dynamics of the compensator and using Young's inequality for scaling, we can obtain... ,in A state vector containing weights. These are the minimum gain parameters for each term. Let be the bounded equivalent generalized filtering error vector. Define the scaling weighted compensation error. and constructor , According to Theorem 1, its value at the preset time can be obtained. Strict convergence upper bound is Similarly, define system tracking state variables that include weights. Its upper bound for convergence is Finally, the physical tracking error vector is defined. From the triangle inequality, we can obtain Since the weighting coefficients are typically large, the boundedness of the weighted error vector sufficiently guarantees that the actual physical tracking error will necessarily be within a specified preset time. It converges strictly to a bounded set of residuals.

[0123] A two-degree-of-freedom flexible joint manipulator was selected for simulation verification. The mathematical model is shown in model (1), and its nominal coefficient matrix is... , , The specific expanded form is as follows:

[0124]

[0125] The expressions for each parameter in the coefficient matrix are as follows:

[0126]

[0127] Considering the existence of problems in the actual operation of the system The uncertainty of model parameters, i.e., the quality during actual operation. With the moment of inertia of the connecting rod All are nominal parameters The nominal parameters of the robotic arm's various physical quantities are as follows: Link mass: , Link length: Centroid length: , Moment of inertia of the connecting rod: , Motor moment of inertia: Joint stiffness matrix: Acceleration due to gravity: .

[0128] Set the desired trajectory:

[0129]

[0130] The external time-varying disturbance torque of the system is set as follows:

[0131]

[0132] Set the observer gain and smoothing parameters to: preset time. , , , , , Adaptive learning rate Set the controller and command filter parameters to: control preset time. , , , , , , , , Command filter natural frequency Damping ratio .

[0133] Setting the system's initial state: Initial joint angle Motor initial angle The initial velocities are all Adaptive perturbation upper bound initial estimate .

[0134] Figures 3-17 Simulation results show that even when the model exists Under conditions of severe parameter uncertainty and time-varying external environmental disturbances, the designed preset-time sliding mode disturbance observer can still observe for the specified preset time. , The unknown lumped disturbance is accurately estimated within the control system. The internal drive joint accurately tracks the desired trajectory, and the tracking error strictly converges to a minimal residual set near zero. Furthermore, the command filter effectively eliminates the differential explosion problem caused by the traditional backstepping method, resulting in smooth control torque throughout the process without significant chattering. This effectively verifies the strong robustness and transient / steady-state performance of the present invention.

Claims

1. A method for tracking and controlling a flexible robotic arm's preset time trajectory based on an interference observer, characterized in that... The method includes the following steps: Step 1, targeting a class A Lagrangian dynamic model of a linkage-based flexible robotic arm is established: in, For the angles of each link, The rotation angle of each joint motor, The inertia matrix, For Coriolis matrix, For gravity torque matrix, Here is the inertia matrix of the motor. For elastic torque, For external disturbance torque, Input torque; , , , For the nominal model, To address uncertainty, we separate the nominal model from uncertainty and define lumped uncertainty. By defining system state variables , =1,2,3,4, which can be transformed into the following strictly feedback state-space equation: in, For equivalent set total uncertainty, Here is the stiffness matrix of the motor; Step 2: Design a time-varying function that smoothly approximates the preset time: in, For preset time, This is the lower limit of gain saturation. As a smooth transition factor, For constant parameters, For time, For intermediate functions, To smoothly approximate the function, The time-varying control gain function, It is a time-varying scaling function; Step 3: To address the lumped uncertainty in the system, design a preset time sliding mode disturbance observer for observation and compensation; Step 4: Introduce the command filter and design the error compensator: Step 4-1: Introduce a second-order instruction filter to solve the differential explosion problem in the backstepping method. The second-order instruction filter... and ,in, For input to the virtual control law, and For the state variables of the filter, For natural frequency, For the damping ratio, =1,2,3, the filtered output signal is ; Step 4-2: Design the error compensator subsystem: in, Let the compensator state vector be... =1,2,3,4 Preset a time-varying function for the controller. For filtering error, =1,2,3 Adjustable normal parameters ; Step 5: Design a preset time trajectory tracking control law based on the backstepping method for the interference observer, second-order command filter and error compensator designed in Steps 3 and 4.

2. The method for tracking and controlling a flexible robotic arm's preset time trajectory based on an interference observer according to claim 1, characterized in that... In step 2, ,in For constant parameters; .

3. The method for tracking and controlling a flexible robotic arm's preset time trajectory based on an interference observer according to claim 1, characterized in that... In step 3, the dynamics and adaptive law design of the preset time sliding mode disturbance observer are as follows: in, , This is an estimate of the state. This is an estimate of the disturbance. For state estimation error, For adjustable positive constant parameters, Preset a time-varying gain function for the observer. To enable online adjustment of integral gain using an adaptive law, It is an adjustable constant parameter.

4. The method for tracking and controlling a flexible robotic arm with a preset time trajectory based on an interference observer according to claim 3, characterized in that... The initial value of the integral gain satisfies Introducing intermediate auxiliary variables and At this point, the estimation error dynamics can be written as: .

5. The method for tracking and controlling a flexible robotic arm's preset time trajectory based on an interference observer according to claim 1, characterized in that... In step 4 .

6. The method for tracking and controlling a flexible robotic arm with a preset time trajectory based on an interference observer according to claim 1, characterized in that... In step 5, the preset time trajectory tracking control law based on the backstepping method is: in, For the first Virtual control variables in step-by-step and reverse-step design For the dynamic tracking error of the system, =1,2,3,4 For the desired location trajectory, This refers to the actual input control torque that ultimately acts on the robotic arm motor.

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