Robot trajectory control method fusing delay performance and finite-time command filter

Abstract

The application belongs to the technical field of robot control, and particularly relates to a robot trajectory control method fusing delay performance and finite time command filter, comprising: establishing a dynamic model of the robot; introducing a preset performance boundary with a delay function to constrain a position tracking error; in the control law design, combining a backstepping method with a second-order finite time command filter to obtain a virtual control signal and a derivative thereof; processing the filter error by constructing a dynamic compensation signal at a position layer, and directly feeding forward the derivative signal of the filter when designing a control torque at a speed layer, so that the differential error at the speed layer is eliminated; meanwhile, an external time-varying disturbance is estimated on line by using a RBF neural network; and finally, the actual finite time stability of a closed loop system is proved based on Lyapunov theory. The application can realize high-precision trajectory tracking of the robot, and effectively overcome the influence of command filter compensation coupling and external disturbance on the system.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, specifically relating to a robot trajectory control method that integrates delay performance and finite-time command filter. Background Technology

[0002] High-precision trajectory tracking control for robots is a core technology in industrial automation and intelligent manufacturing, widely used in precision assembly, material handling, and laser processing. In actual operation, robot systems are often affected by complex time-varying disturbances such as model parameter perturbations, external load changes, and nonlinear friction. Current typical nonlinear robot control methods can be categorized as follows: Control methods based on traditional preset performance, while allowing for the artificial setting and constraint of the transient evolution and steady-state convergence accuracy of the tracking error, place extremely stringent requirements on the initial state of the system; that is, the initial tracking error must strictly remain within the preset performance boundaries. In complex real-world conditions, the robot's initial pose often fails to precisely meet this constraint, easily leading to computational singularities in the control algorithm or even system instability. Backstepping, as a classic architecture for nonlinear system control, requires repeated analytical differentiation operations when designing the virtual control law. As the system order increases, this repetitive differentiation leads to a severe "differential expansion" (differential explosion) problem. This not only increases the computational complexity of the controller but also easily causes high-frequency chattering in the control output signal, exacerbating actuator wear. Control methods based on a combination of command filtering and neural networks can effectively avoid analytical differentiation by introducing a command filter to obtain the derivative of the virtual control law. However, existing filtering control strategies typically employ a two-layer error compensation mechanism for both the position and velocity layers. In Eulerian-Lagrange physical systems, further dynamic analysis of the velocity layer compensation signal leads to a composite coupling term with the unknown nonlinear inertia matrix. This coupling term not only increases the complexity of the control law but also severely pollutes the adaptive neural network's approximation channel for purely external time-varying disturbances, resulting in oscillations in network weight updates, decreased disturbance observation accuracy, and weakened system immunity.

[0003] Existing methods face two major challenges: First, under arbitrary initial poses and high dynamic operation requirements, it is difficult for robot trajectory tracking to simultaneously overcome the performance bottlenecks of strictly limited initial error and differential expansion. Second, when introducing filters to compensate for system errors, it is inevitable to cause coupling between the compensation signal and the unknown inertial matrix, which makes it impossible for neural networks to achieve high-precision approximation of external time-varying disturbances and adapt to the operational requirements under complex and highly disturbed conditions. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a robot trajectory control method that integrates delay performance and a finite-time command filter, comprising:

[0005] S1: Establish the Lagrangian dynamics model of the robot, transform the robot dynamics model into a second-order nonlinear system, and obtain the joint position and velocity state variables of the system;

[0006] S2: Calculate the joint position tracking error based on the desired trajectory of the robot joint, and introduce a delay preset performance function to transform the original position error;

[0007] S3: Design a virtual control law for the closed-loop control system based on the transformed position error, and construct a second-order finite-time command filter to filter the virtual control law in order to avoid the computational complexity caused by direct differentiation.

[0008] S4: Establish a position layer error compensation mechanism to compensate for the inherent filtering error of the finite-time command filter, and define the compensated position tracking error;

[0009] S5: Construct the velocity tracking error and extract the derivative signal inside the filter as the torque feedforward signal;

[0010] S6: Introduce an RBF neural network to approximate the external time-varying disturbances in the system, and design an adaptive law for perturbation to update the network weights online;

[0011] S7: Combining the derivative feedforward signal from step S5 with the RBF neural network compensation signal from S6, design the actual joint control torque, and prove the global uniform boundedness of all states of the closed-loop system in a finite time based on Lyapunov stability theory.

[0012] The beneficial effects of this invention are:

[0013] This invention introduces a finite-time command filter to directly extract the smoothed derivative signal of the virtual control law for torque feedforward, avoiding the computational complexity caused by repeated analytical differentiation in traditional backstepping methods. Simultaneously, an error compensation mechanism effectively eliminates inherent filtering errors, ensuring high-precision trajectory tracking performance while reducing the controller's computational load. Furthermore, addressing the limitation of traditional preset performance control requiring initial errors to be strictly within performance boundaries, this invention incorporates a delayed preset performance control strategy. By introducing a shift-delay attenuation function, it allows the system's initial tracking error to remain outside the preset boundary within a limited time. This mechanism fundamentally avoids the surge in initial control gain and actuator torque saturation caused by excessive initial errors, thereby improving the robot's dynamic stability and practical engineering feasibility during startup. Attached Figure Description

[0014] Figure 1 This is a flowchart of the robot trajectory control method that integrates delay performance and finite-time command filter in this invention;

[0015] Figure 2 This is a schematic diagram of the position tracking of joint 1 in this invention;

[0016] Figure 3 This is a schematic diagram of the position tracking of joint 2 in this invention;

[0017] Figure 4 This is a schematic diagram showing the positional error of joints 1 and 2 and the preset performance boundary in this invention;

[0018] Figure 5 This is a schematic diagram of the velocity tracking of joint 1 in this invention;

[0019] Figure 6 This is a schematic diagram of the velocity tracking of joint 2 in this invention;

[0020] Figure 7 This is a schematic diagram of the control torque output of joints 1 and 2 in this invention;

[0021] Figure 8 This is a schematic diagram of the approximation curves of the RBF neural network for the time-varying perturbations of joints 1 and 2 in this invention;

[0022] Figure 9 This is a residual error diagram of the RBF neural network fitting in this invention. Detailed Implementation

[0023] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0024] This embodiment uses a two-degree-of-freedom rotating robot as the controlled object. The robot trajectory tracking control method of this invention, which integrates the preset delay performance and the finite-time command filter, is employed to achieve trajectory tracking control. Simulation verification is performed using MATLAB 2018b. The specific implementation steps are as follows:

[0025] In this embodiment, step S1 constructs a robot dynamics model, transforming it into a second-order nonlinear system. The robot dynamics model is constructed as follows:

[0026]

[0027] in, This represents the positive definite inertia matrix of the robot. The matrix representing the centrifugal force and Coriolis force of the robot; This represents the robot's gravitational torque vector. It is the control torque applied to the joint. This represents unknown time-varying interference caused by external loads; , These represent the joint angle vector, joint angular velocity vector, and joint angular acceleration vector, respectively.

[0028] To perform control design and stability analysis, the following lemmas and assumptions are given:

[0029] Lemma 1: For any real number and fractional powers There must exist a positive constant. This makes the following inequality hold globally:

[0030]

[0031] Lemma 2: For the Lyapunov function estimator defined in a compact set There exists a constant The following inequality always holds:

[0032]

[0033] This lemma is used to deal with the order mismatch problem of adaptive weight terms.

[0034] Lemma 3: Consider a continuous positive definite function defined in the neighborhood of the origin. If there exists a real number fractional constants and bounded constants Make it satisfy the following Lyapunov differential inequality:

[0035]

[0036] This indicates that the nonlinear closed-loop system is stable in finite time, and the system's state trajectory will be stable in finite time. Within, it rapidly converges to a compact set near the origin. It is internal and remains bounded within the set.

[0037] Among them, the residual compact set with steady-state convergence The specific boundary is defined as follows:

[0038]

[0039] Upper limit of convergence time for the system state to reach the compact boundary satisfy:

[0040]

[0041] In the formula, The Lyapunov function value at the initial moment of the system. To meet Any adjustable parameter.

[0042] In this embodiment, step S2 constructs a preset performance transformation mechanism containing a delay function based on the desired trajectory.

[0043] Define joint position tracking error:

[0044]

[0045] in For the desired position trajectory of the joint

[0046] To reduce position tracking error satisfy: Design an exponential preset performance boundary function:

[0047]

[0048] And introduce a delay decay function:

[0049]

[0050] Set initial values:

[0051]

[0052] Define conversion error So that it satisfies at the initial moment This removes the constraint of the system's initial position on the preset boundary.

[0053] In this embodiment, steps S3 and S4 use backstepping and finite-time command filters to design virtual control laws and construct a position layer compensation mechanism.

[0054] Design the virtual control law for the position subsystem:

[0055]

[0056] Will As the input to a second-order finite-time command filter, the state equation is:

[0057]

[0058]

[0059] Define the first-layer filter approximation error To reduce the impact of filtering errors on the system, a position layer error compensation signal is designed. Its dynamic equation is:

[0060]

[0061] Define the compensated position tracking error:

[0062]

[0063] In this embodiment, steps S5, S6, and S7 construct a velocity feedback mechanism and a neural network adaptive law, design the actual control torque, and perform stability analysis.

[0064] Define speed tracking error Extract the first derivative signal of the filter. The actual control torque is designed as follows:

[0065]

[0066] in, To approximate time-varying disturbances The output of the RBF neural network; the adaptive weight update law with correction is designed as follows:

[0067] Stability verification:

[0068] Construct a composite Lyapunov function of the master system that includes compensation error, and incorporate the actual error of the position layer after compensation. (in Velocity layer tracking error and the weight estimation error of the RBF neural network Combination:

[0069]

[0070] in, For the robot's inertia matrix, This is the adaptive learning rate matrix.

[0071] right Find the time derivative and the dynamic equation for the compensation error under the preset delay performance. Substitute. Benefiting from the compensated signal. The design of the first-layer command filtering error When precisely canceled out, we can obtain Simultaneously incorporating velocity layer dynamics and control torque and utilize Simplifying by the oblique symmetry, we get:

[0072]

[0073] In this formula, the nonlinear feedback term and the approximation residual of the neural network are... This constitutes the dominant error dynamic of the system.

[0074] Extract the minimum eigenvalue of the gain matrix and perform lower bound scaling, then use Young's inequality to adjust the cross-coupling terms (such as...). and approximating residuals ( Decoupling is performed; simultaneously, algebraic decomposition and fractional transformation are applied to the weight error term generated by the adaptive update law. This separates it into a negative definite fractional term and a bounded set of residuals.

[0075]

[0076] in, All of these are constant coefficients that are strictly greater than zero after eigenvalue scaling, and their specific mathematical definitions are as follows:

[0077]

[0078]

[0079]

[0080] In the formula, and Let represent the minimum and maximum eigenvalues ​​of the matrix, respectively. This represents the leakage correction coefficient in the adaptive update law of the neural network. To ensure system stability, the controller parameters must satisfy the following during tuning: and . For fractional powers, the system composite constant residual boundary value Includes external disturbances approaching the upper bound of the residual. Ideal network weights And the fractional algebraic scaling constant introduced to handle the mismatch in the order of adaptive terms, the specific mathematical derivation of which is as follows:

[0081]

[0082] By utilizing the compact set property to absorb the linear quadratic positive terms, and combining the above negative definite terms with the same power, the standard fractional Lyapunov differential inequality can be derived:

[0083]

[0084] in, To provide a comprehensive convergence coefficient, The extracted system set is the total bounded constant. The time interval is a fractional power. According to Lemma 3, all states of the entire closed-loop system are globally uniformly bounded in finite time.

[0085] In this embodiment, the specific internal analytical expressions of each matrix in the actual dynamic model of the two-degree-of-freedom robot are as follows:

[0086] The positive definite inertial matrix of the robot Expanded to:

[0087]

[0088] Coriolis force and centrifugal force matrix Expanded to:

[0089]

[0090] gravitational moment vector Expanded to:

[0091]

[0092] External time-varying disturbance vectors that the system is subjected to Set as:

[0093]

[0094] in, These represent the position angles of joint 1 and joint 2, respectively. These represent their angular velocities. The specific physical dimensions and corresponding values ​​of the above matrix parameters are substituted into the basic physical parameters set in Table 1 above for calculation.

[0095] In this simulation, the desired tracking trajectory of the system is set as follows: The external time-varying disturbance applied to the system is set to... .

[0096] Simulation results are as follows Figures 2 to 9 As shown. Figure 2 and Figure 3 The position tracking performance of joint 1 and joint 2 is shown. As can be observed from the figure, the actual position curve of the robot joint can track the expected sine and cosine trajectory in a short time, indicating that the proposed control method has good steady-state and transient tracking accuracy. Figure 4 The evolution relationship between position tracking error and the preset performance boundary of delay is demonstrated. Due to the introduction of the delay function, the initial error of the system can be outside the initial performance boundary, and the error curve is strictly constrained within the preset boundary range after a certain period of time, effectively removing the limitation of the initial state imposed by traditional methods.

[0097] Figure 5 and Figure 6This reflects the system's tracking response characteristics at the velocity level. The actual velocity trajectory can track the desired trajectory within a finite time and converges to the neighborhood of the equilibrium point. Figure 7 The actual control torque output characteristics of the system are demonstrated. Because this method directly extracts the derivative signal of the filter for torque feedforward, it avoids the analytical coupling caused by velocity layer compensation errors and the differentiation of the system inertia matrix. The torque output curve is smooth and free of high-frequency pulse jitter, verifying the effectiveness of this feedforward strategy in reducing actuator losses and improving system executability.

[0098] The observation and compensation effects on external uncertain disturbances Figure 8 and Figure 9 This is reflected in the text. Figure 8 This indicates that the RBF neural network can effectively approximate time-varying disturbances, and its output curve highly coincides with the actual disturbance trajectory. Figure 9 The estimation errors of the RBF network weights for joints 1 and 2 are presented. The results show that the errors converge rapidly in the early stages of system operation and fluctuate slightly around zero. This verifies that the introduced error compensation mechanism effectively eliminates the inherent filtering error of the command filter, reduces the online approximation load on the neural network, and thus ensures the accuracy and robustness of the parameter adaptation process.

[0099] In summary, the control method proposed in this invention can effectively solve the stability analysis and loop proof problem in the command filtering backstepping method, as well as the error convergence problem under complex disturbances and nonlinear coupling, providing an effective and stable technical solution for high-precision trajectory tracking control of robots.

[0100] The simulation results and analysis above verify the technical effects of the present invention. It should be understood that the above simulation scenarios are merely typical application examples of the present invention and are not intended to limit the invention. Any modifications, equivalent substitutions, optimizations, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0101] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A robot trajectory control method integrating delay performance and finite-time command filter, characterized in that, include: S1: Establish the Lagrangian dynamics model of the robot, transform the robot dynamics model into a second-order nonlinear system, and obtain the joint position and velocity state variables of the system; S2: Calculate the joint position tracking error based on the desired trajectory of the robot joint, and introduce a delay preset performance function to transform the original position error; S3: Design a virtual control law for the closed-loop control system based on the transformed position error, and construct a second-order finite-time command filter to filter the virtual control law in order to avoid the computational complexity caused by direct differentiation. S4: Establish a position layer error compensation mechanism to compensate for the inherent filtering error of the finite-time command filter, and define the compensated position tracking error; S5: Construct the velocity tracking error and extract the derivative signal inside the filter as the torque feedforward signal; S6: Introduce an RBF neural network to approximate the external time-varying disturbances in the system, and design an adaptive law for perturbation to update the network weights online; S7: Combining the derivative feedforward signal from step S5 with the RBF neural network compensation signal from S6, design the actual joint control torque, and prove the global uniform boundedness of all states of the closed-loop system in a finite time based on Lyapunov stability theory.

2. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that, include: Establish a Lagrangian dynamics model for the robot, transforming the robot's dynamics model into a second-order nonlinear system, including: in, Represents the symmetric positive definite inertia matrix of the robot; These represent the actual position, velocity, and acceleration vectors of the robot's joints, respectively. The matrix representing the Coriolis force and centrifugal force of the robot; This represents the robot's gravitational torque vector. This represents the actual control torque vector of the joint; Unknown time-varying interference caused by external loads Indicates the number of joints in the robot; Transform the robot dynamics model into a second-order nonlinear system: in, These represent the position and velocity vectors of the robot joints, respectively.

3. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that... Based on the desired trajectory of the robot joints, the joint position tracking error is calculated. A delay preset performance function is introduced to transform the original position error, including: Define joint position tracking error for: in, This is the position tracking error vector; This is the actual joint position vector; The desired joint position trajectory vector; Design an exponential preset performance boundary function: in, Preset performance boundary functions; The initial value vector for the boundary; The boundary steady-state value vector; The constant is the exponential convergence rate constant; It is a time variable; Introduce a delay function: in, It is a delay decay function; This is the initial value vector for the delay function; The decay rate constant; and These represent the values ​​of the position tracking error and the preset performance boundary function at the initial time, respectively. Joint position tracking error Normalization processing And the error is transformed through a delay function: in, This is the normalized error vector; The result after the delay function transformation is the position error vector; This is the position tracking error vector; This is a preset performance boundary function.

4. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that... A virtual control law for a closed-loop control system is designed based on the transformed position error. A second-order finite-time command filter is constructed to filter the virtual control law to avoid the computational complexity caused by direct differentiation. This includes: Delay conversion error Regarding time The derivative represents the actual joint velocity. To treat the position tracking subsystem as a virtual control input and derive and design a virtual control law to ensure the system satisfies the finite-time stability condition, the virtual control law is considered as such. for: in, The virtual control law vector; The desired joint velocity vector; This is the position tracking error vector; Preset performance boundary functions; and These are the first-order time derivative vectors of the boundary function and the delay function, respectively; It is a positive definite diagonal gain matrix; It is a finite-time nonlinear function; It is a fractional continuous sign function; Standard symbolic functions; It is a constant power; The virtual control law As the input to a second-order finite-time command filter, its state equation is: in, This is the filtered virtual control law; for The time derivative; The derivative vector of the virtual control law state; for The time derivative; and These are the first gain constant and the second gain constant, respectively.

5. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that... A position layer error compensation mechanism is established to compensate for the inherent filtering error of the finite-time command filter, and the compensated position tracking error is defined, including: The dynamic equation for the location layer error compensation signal is as follows: in, This is the location layer error compensation signal vector; for The first time derivative; It is a positive definite diagonal gain matrix; It is a finite-time nonlinear function; It is a fractional continuous sign function; This is the normalized error vector; The result after the delay function transformation is the position error vector; This is the position tracking error vector; Preset performance boundary functions; Define the compensated position tracking error: in, This is the compensated position tracking error vector.

6. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that, Constructing velocity tracking error includes: in, This is the speed tracking error vector; This is the actual joint velocity vector; This is the filtered virtual control law.

7. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that, An RBF neural network is introduced to approximate external time-varying disturbances in the system, and an adaptive law for perturbation is designed to update the network weights online, including: The form of approximating the external perturbation using an RBF neural network is as follows: in, Unknown time-varying interference caused by external loads; The ideal network weight matrix; To approximate the residual vector; Let Gausky function vector be the vector. Let T be the number of nodes in the RBF neural network, and T be the matrix transpose. The adaptive update law with correction is designed as follows: in, To estimate the weight matrix; for The first time derivative; It is a positive definite diagonal learning rate matrix; Input feature vectors into the network; This is a correction coefficient constant.

8. The robot trajectory control method according to claim 1, which integrates delay performance and finite-time command filter, is characterized in that... By combining derivative-feedforward signals with RBF neural network compensation signals, practical joint control torques are designed. Based on Lyapunov stability theory, the globally uniform boundedness of all states of the closed-loop system within a finite time interval is proved, including: The actual control torque law is designed as follows: in, This is the actual control torque vector of the joint; Represents the symmetric positive definite inertia matrix of the robot; The matrix of Coriolis force and centrifugal force for the robot; Represents the robot's gravitational torque vector; The velocity feedback gain matrix; Constructing the main system Lyapunov functions: in, Scalars of Lyapunov functions for the main system; This is the weight estimation error matrix; The ideal network weight matrix; To estimate the weight matrix; Represents the trace operation of a matrix; Simplify its derivative by utilizing the oblique symmetry property of the dynamics, and combine it with the standard form of Young's inequality. Proof status and Boundedness; Constructing the Lyapunov function for the compensation subsystem: in, For compensating the Lyapunov function scalar of the subsystem; Here is the location layer error compensation signal vector; T is the matrix transpose. Substitution Relationship Combining inequality scaling, in Prove the compensation signal under bounded conditions Boundedness, and therefore based on Analyze the actual finite-time stability of the system state signals.

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