A Fixed-Time Control Method for a Flexible Double-Link Robotic Arm

Abstract

This invention provides a fixed-time control method for a flexible double-link manipulator, applied in the field of robotics intelligence. The method includes: establishing a dynamic model of a flexible system with dead-zone input; deriving the initial control torque dependent on the dynamic model of the flexible system with dead-zone input using backstepping; designing a first radial basis function neural network update rate based on the unknown dynamic information in the initial control torque approximated by a radial basis function neural network; designing a second radial basis function neural network update rate based on the approximation of the unknown dead-zone function by the radial basis function neural network; and obtaining the fixed-time control torque of the novel flexible double-link manipulator based on the two radial basis function neural network update rates and the initial control torque. The fixed-time control strategy proposed in this invention ensures the rapid convergence of the flexible double-link manipulator, and can simultaneously improve trajectory tracking accuracy and vibration suppression performance even under operating conditions involving system constraints.

Description

Technical Field

[0001] This invention relates to the field of robot intelligence, and more particularly to a fixed-time control method for a flexible double-link robotic arm. Background Technology

[0002] The control research of flexible robotic arms first requires the establishment of its dynamic model, which is usually described by partial differential equations (PDEs), increasing the complexity of control design. To simplify the complexity of the model, researchers often convert PDEs into ordinary differential equations (ODEs) and perform dimensionality reduction through methods such as the hypothetical mode method.

[0003] In practical applications of flexible robotic arms, problems such as input dead zone, output constraints, and system model uncertainties are common. Ignoring these issues can not only affect tracking performance in flexible two-link robotic arm (FTLM) systems but may also lead to system instability. Neural network control methods are widely used to address system uncertainties and nonlinearities.

[0004] In existing vibration control of flexible robotic arms, the elastic deformation that occurs during movement makes it difficult to achieve ideal control effects, and the flexibility of the robotic arm itself makes modeling more challenging. Secondly, nonlinear inputs also affect the stability and effectiveness of the control system. An efficient controller is needed to adjust the control system during the control process, adaptively adjusting the control torque to achieve better vibration suppression and transient performance. Furthermore, previous inventions have not considered the impact of input dead zones in the vibration suppression control of flexible systems. Since the nonlinear function of the actuator is unknown, ignoring dead zone nonlinearity will affect the control of the flexible robotic arm system, leading to system unreliability. Moreover, the lack of constraints on the system's state errors may result in poor transient performance. Summary of the Invention

[0005] To address the aforementioned problems, this invention provides a fixed-time control method for a flexible double-link robotic arm, improving the vibration suppression capability of the flexible robotic arm system during trajectory tracking. Compared to traditional neural network control frameworks, this invention optimizes the neural network update rate to adapt to a fixed-time control framework, enabling the controller to have better self-learning and resistance to dead-zone nonlinear inputs. Specifically, it includes:

[0006] A fixed-time control method for a flexible double-link robotic arm, the method comprising:

[0007] S1. Based on Hamilton's principle, Euler-Bernoulli beam theory, and the rotational inertia, angular velocity, and elastic vibration of the flexible double-link manipulator system, a dynamic model of the flexible system with dead zone input is established.

[0008] S2. Based on the dynamic model of a flexible system with dead zone input, the initial control torque that depends on the dynamic model of the flexible system with dead zone input is derived by combining the backstepping method.

[0009] S3. Based on the unknown dynamic information in the approximate initial control torque using a radial basis function neural network, design the update rate of the first radial basis function neural network;

[0010] Based on the approximation of the unknown dead zone function by a radial basis function neural network, the update rate of a second radial basis function neural network is designed.

[0011] S4. Based on the update rate of the first radial basis function neural network, the update rate of the second radial basis function neural network, and the initial control torque, the fixed-time control torque of the novel flexible double-link robotic arm is obtained.

[0012] Optionally, the formula for the dynamic model of the flexible system in S1 is formula (1):

[0013] (1)

[0014] in, , where is the stiffness matrix;

[0015] Torque The dead zone nonlinear function;

[0016] , For time-varying generalized coordinates, The rotation angle of the joint link in the flexible double-link robotic arm;

[0017] Matrix J(Q) is the inertia matrix;

[0018] matrix For the Coriolis matrix and centripetal effect;

[0019] The first derivative of the state variables of the FTLM system;

[0020] It is the second derivative of the state variables of the FTLM system.

[0021] Optionally, the formula for the initial control torque in S2 is formula (2):

[0022] (2)

[0023] k b For the state-constrained parameters of the Lyapunov function; for transpose; For the state error of the FTLM system; Add virtual control variables to the FTLM system to introduce state errors; The first derivative of the virtual control quantity; This is a virtual control variable; These are the control parameters for the FTLM system; for transpose; These are the control parameters for the FTLM system.

[0024] Optionally, the update rate of the first radial basis function neural network in S3 is given by formula (3):

[0025] (3)

[0026] The first radial basis function is the neural network update rate; The scaling factor for the update rate of the first radial basis function neural network; The Gaussian function is the first radial basis function of the neural network. for The i-th component in; These are the estimated values ​​of the neural network weights for the first radial basis function; for transpose; It is a positive positive number.

[0027] Optionally, the update rate of the second radial basis function neural network in S3 is given by formula (4):

[0028] (4)

[0029] The second radial basis function is the neural network update rate; This is the scaling factor for the update rate of the second radial basis function neural network. The Gaussian function is the second radial basis function of the neural network. Add virtual control variables to the FTLM system to introduce state errors; These are the estimated values ​​of the neural network weights for the second radial basis function; for transpose; It is a positive positive number.

[0030] Optionally, the fixed-time control torque of the novel flexible double-link robotic arm in S4 is given by formula (5):

[0031] (5)

[0032] For the fixed-time control torque of a novel flexible double-link robotic arm;

[0033] The Gaussian function is the first radial basis function of the neural network. The Gaussian function is the second radial basis function of the neural network.

[0034] The Lyapunov function of the fixed-time control torque based on the novel flexible double-link robotic arm is given by formula (6):

[0035] (6)

[0036] By continuously differentiating and transforming formula (6), we obtain According to the theory of fixed-time control, there exists a constant. and satisfy , obtain a fixed time For formula (7):

[0037] (7)

[0038] Among them, T max The maximum time for system convergence. These are control parameters for system stability. These are control parameters for system stability.

[0039] The above technical solution has at least the following advantages compared with the existing technology:

[0040] This invention considers a system modeling method for flexible double-link manipulators under nonlinear disturbances with dead-zone input, enabling a more accurate description of the dynamic behavior of flexible linkage systems with dead-zone inputs. Compared to flexible single-link manipulators, this invention has been experimentally verified on flexible double-link manipulators with strong nonlinear coupling, achieving good trajectory tracking and vibration suppression effects.

[0041] To address the interference of dead-time input nonlinearity encountered by FTLM (Flexible Transmission Model), this invention designs a novel neural network update rate that satisfies the fixed-time control framework. This eliminates the impact of actuator dead time on the system, improving its robustness and tracking performance.

[0042] This invention considers the uncertainties of FTLM systems and designs a fixed-time control method based on RBFNN to compensate for system uncertainties. Furthermore, to handle the output constraints of FTLM, a logarithmic barrier Lyapunov function is employed. This ensures that the system output remains strictly within the specified range, preventing control instability caused by constraint violations and improving the system's dynamic response characteristics. Extensive experiments demonstrate that the proposed control method can effectively suppress the elastic vibration of the flexible linkage system and accurately track the desired trajectory. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0044] Figure 1 A flowchart of the present invention provided for embodiments of the present invention;

[0045] Figure 2 A schematic diagram of the dynamic model of the flexible double-link manipulator with input dead zone provided in an embodiment of the present invention;

[0046] Figure 3 This is a block diagram of the fixed-time control algorithm for a flexible double-link robotic arm provided in an embodiment of the present invention;

[0047] Figure 4 A schematic diagram of the FTLM experimental equipment provided in an embodiment of the present invention;

[0048] Figure 5 This is a schematic diagram comparing the tracking trajectory performance of two control methods provided in an embodiment of the present invention;

[0049] Figure 6 This is a schematic diagram comparing the tracking errors of two control methods provided in an embodiment of the present invention;

[0050] Figure 7 A schematic diagram comparing the elastic vibration of two control methods provided in an embodiment of the present invention;

[0051] Figure 8 This is a schematic diagram comparing the control inputs of two control methods provided in an embodiment of the present invention. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0053] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms “first,” “second,” and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Similarly, the terms “an,” “a,” or “the,” and similar terms do not indicate a quantity limitation, but rather indicate the presence of at least one. The terms “comprising,” “including,” or “including,” and similar terms mean that the element or object preceding the word encompasses the element or object listed following the word and its equivalents, without excluding other elements or objects. The terms “connected,” “linked,” or “connected,” and similar terms are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect.

[0054] To address the problem of traditional control methods' over-reliance on precise dynamic models, model-free control strategies have gradually emerged. Their core advantage lies in eliminating the need for precise dynamic models, achieving good control performance using only the input and output data of the control system, such as PID control, fuzzy control, and neural network control. However, existing methods fail to simultaneously guarantee transient performance (such as fast convergence and small overshoot) and vibration suppression. To a certain extent, fast convergence can lead to increased vibration amplitude. Therefore, in the presence of input nonlinearity, it is difficult to simultaneously achieve excellent error convergence and vibration suppression. Fixed-time control methods offer a feasible solution for achieving high precision and fast convergence. Compared to traditional finite-time control methods, fixed-time control algorithms have a significant advantage in convergence time, as it is independent of the robot's initial state and can be determined by the controller gain. The purpose of this invention using fixed-time control is to ensure better vibration suppression while maintaining good transient performance.

[0055] Flexible robotic arms face significant challenges in modeling and vibration suppression. To reduce controller complexity, the assumed modal method is used to construct the ordinary differential dynamic equations of the FTLM with dead-zone input. This invention aims to provide a fixed-time control algorithm for a state-constrained flexible double-link robotic arm that resists dead-zone input. This invention uses a dead-zone nonlinear function to constrain the actuator and the output state of the closed-loop system. Therefore, a novel neural network controller is designed to eliminate the influence of the dead-zone nonlinear function on the actuator. This invention designs a fixed-time control method with online learning capabilities, improving the vibration suppression capability of the flexible robotic arm system during trajectory tracking. Compared with traditional neural network control frameworks, this invention optimizes the neural network update rate to adapt to the fixed-time control framework, giving the controller better self-learning and resistance to dead-zone nonlinear input. The specific scheme is as follows:

[0056] like Figure 1 As shown, a fixed-time control method for a flexible double-link robotic arm includes:

[0057] S1. Based on Hamilton's principle, Euler-Bernoulli beam theory, and the rotational inertia, angular velocity, and elastic vibration of the flexible double-link manipulator system, a dynamic model of the flexible system with dead zone input is established.

[0058] S2. Based on the dynamic model of a flexible system with dead zone input, the initial control torque that depends on the dynamic model of the flexible system with dead zone input is derived by combining the backstepping method.

[0059] S3. Based on the unknown dynamic information in the approximate initial control torque using a radial basis function neural network, design the update rate of the first radial basis function neural network;

[0060] Based on the approximation of the unknown dead zone function by a radial basis function neural network, the update rate of a second radial basis function neural network is designed.

[0061] S4. Based on the update rate of the first radial basis function neural network, the update rate of the second radial basis function neural network, and the initial control torque, the fixed-time control torque of the novel flexible double-link robotic arm is obtained.

[0062] In one specific implementation, S1, based on Hamilton's principle, Euler-Bernoulli beam theory, and the rotational inertia, angular velocity, and elastic vibration of the flexible double-link robotic arm system, a dynamic model of the flexible system with dead-zone input is established:

[0063] like Figure 2As shown, based on Hamilton's principle and Euler-Bernoulli beam theory, and considering factors such as the moment of inertia, angular velocity, and elastic vibration of the flexible two-bar linkage (FTLM) system, a dynamic model of this flexible system with dead-zone input is established. XOYZ is the inertial coordinate system of the flexible manipulator, used to determine its spatial position. xoy is the rotational coordinate system of the flexible manipulator, used to observe elastic vibration. The flexible robotic arm is represented by the first The torque of each drive unit, Let represent the elastic vibration of the i-th link of the flexible robotic arm in the xoy coordinate system. This indicates the position of the i-th link of the flexible robotic arm. This represents the moment of inertia of the i-th link of the flexible robotic arm. This represents the equivalent viscous damping coefficient of the flexible robotic arm. It is the torsional stiffness constant of the flexible robotic arm.

[0064] The formula for the dynamic model of a flexible system with dead zone nonlinearity is formula (1):

[0065] (1)

[0066] in, , where is the stiffness matrix;

[0067] Torque The dead zone nonlinear function;

[0068] , For time-varying generalized coordinates, The rotation angle of the joint link in the flexible double-link robotic arm;

[0069] Matrix J(Q) is the inertia matrix;

[0070] matrix The Coriolis matrix and centripetal effect are both of dimension 1. ;

[0071] The first derivative of the state variables of the FTLM system;

[0072] It is the second derivative of the state variables of the FTLM system.

[0073] Steps S2-S4 of this invention involve designing a fixed-time adaptive vibration controller with dead-zone input and state constraints, possessing RBFNN adaptive learning capability and a logarithmic barrier Lyapunov (BLF) constraint mechanism. The principle is primarily as follows: fixed-time control combined with backstepping design. The fixed-time control framework ensures system convergence within a preset maximum time. Backstepping, by defining virtual control quantities and error signals hierarchically, provides a control design method that balances fast convergence and singularity-free operation. Fixed-time control, through a nonlinear control law, ensures that the tracking error convergence time depends only on the controller gain, remaining independent of the initial state, gradually optimizing the convergence rate from the system's dynamic response to generate control commands that meet time constraints. Backstepping, by designing virtual control quantities to hierarchically process position and velocity errors, dynamically adjusts the control law based on the error signals, ensuring stable convergence of errors at each level and providing feedback to optimize overall control performance. This method effectively handles the strong coupling and underactuated characteristics of flexible robotic arms. To address the impact of dead-zone nonlinear input on the control strategy, an RBFNN is incorporated into the fixed-time control framework. A novel neural network update rate that satisfies the fixed-time framework is used to offset the impact of dead-zone input on vibration suppression. Simultaneously, the RBFNN approximates the system uncertainty, and the magnitude of the state error is constrained by the logarithmic barrier Lyapunov function. Ultimately, this achieves synergistic optimization of fast convergence, vibration suppression, and input dead-zone handling.

[0074] like Figure 3 As shown, the fixed-time frame control structure based on RBFNN of the present invention is illustrated: the system error is obtained from the reference trajectory and the FTLM input state, and is incorporated into the fixed-time control frame combined with the backstepping method. The fixed-time control ensures that the error converges with the gain within a preset time, and the backstepping method is strongly coupled through hierarchical processing. Within the frame, RBFNN uses a novel update rate to offset the input dead zone and approximate the system uncertainty. Combined with logarithmic BLF to constrain the error, and incorporating the sign function smoothing approximation to avoid chattering, the control command is finally generated, enabling the flexible robotic arm to achieve rapid convergence and vibration suppression according to the preset trajectory.

[0075] For continuous functions A radial basis function neural network can be represented as: ,in These are radial basis function neural network weights. This is the Gaussian function of the RBFNN. RBFNNs possess the ability to estimate unknown nonlinear functions and exhibit self-learning properties. In compact sets... Within this framework, RBFNN can estimate any continuous function to any required accuracy.

[0076] To maintain system stability, a specific virtual error term is incorporated into the controller. The RBFNN adjusts the flexible system's control strategy in real time based on system vibration error, tracking error, and the control torque from the previous moment, optimizing the system's vibration suppression and tracking performance. However, in actual control systems, various nonlinear inputs may occur, among which dead-zone nonlinearity can inhibit actuator operation and affect control performance. This invention constructs an RBFNN to counteract the impact of actuator dead-zone inputs on the control system. Thus, even with unknown dead-zone nonlinear inputs, the fixed-time controller can generate efficient control inputs, thereby enhancing the system's control performance.

[0077] In one specific implementation, S2, based on the dynamic model of a flexible system with dead zone input, the initial control torque dependent on the dynamic model of the flexible system with dead zone input is derived by combining the backstepping method.

[0078] The formula for the initial control torque is formula (2):

[0079] (2)

[0080] k b For the state-constrained parameters of the Lyapunov function; for transpose; For the state error of the FTLM system; Add virtual control variables to the FTLM system to introduce state errors; The first derivative of the virtual control quantity; This is a virtual control variable; Here is the stiffness matrix of the FTLM system; These are the control parameters for the FTLM system; for transpose; These are the control parameters for the FTLM system.

[0081] In one specific implementation, S3, based on the unknown dynamic information in the approximate initial control torque of the radial basis function neural network, the update rate of the first radial basis function neural network is designed;

[0082] Based on the approximation of the unknown dead zone function by a radial basis function neural network, the update rate of the second radial basis function neural network is designed as follows:

[0083] Since the dynamic information of the flexible robot is unknown, in order to solve this problem, this invention uses a radial basis function neural network to approximate the unknown dynamic information, and designs the first radial basis function neural network update rate, which is given by formula (3):

[0084] (3)

[0085] The first radial basis function is the neural network update rate; The scaling factor for the update rate of the first radial basis function neural network; The Gaussian function is the first radial basis function of the neural network. for The i-th component in; These are the estimated values ​​of the neural network weights for the first radial basis function; for transpose; It is a positive positive number.

[0086] To eliminate the influence of the dead zone function, the update rate of the second radial basis function neural network is designed, and its formula is Equation (4):

[0087] (4)

[0088] The second radial basis function is the neural network update rate; This is the scaling factor for the update rate of the second radial basis function neural network. The Gaussian function is the second radial basis function of the neural network. Add virtual control variables to the FTLM system to introduce state errors; These are the estimated values ​​of the neural network weights for the second radial basis function; for transpose; It is a positive positive number.

[0089] In one specific implementation, S4, based on the update rate of the first radial basis function neural network, the update rate of the second radial basis function neural network, and the initial control torque, the fixed-time control torque of the novel flexible double-link robotic arm is obtained:

[0090] Based on the ability of RBFNN to estimate unknown nonlinear functions and the update rate of the novel radial basis function neural network designed above, this invention uses RBFNN to approximate the unknown dynamic information of the system and eliminate the influence of dead zone input on the control system. Therefore, the fixed-time control torque of the novel flexible double-link manipulator is given by formula (5):

[0091] (5)

[0092] For the fixed-time control torque of a novel flexible double-link robotic arm;

[0093] The Gaussian function is the first radial basis function of the neural network. The Gaussian function is the second radial basis function of the neural network.

[0094] In addition, this invention constructs a new Lyapunov function as follows:

[0095] ;

[0096] By performing continuous differentiation and transformation on the function, this invention can obtain... According to the theory of fixed-time control, there exists a constant. and satisfy A fixed time can be obtained. As shown below:

[0097] .

[0098] Among them, T max The maximum time for system convergence. These are control parameters for system stability. These are control parameters for system stability.

[0099] Experimental verification was conducted on the fixed-time controller proposed in this invention:

[0100] To evaluate the algorithm's performance and stability, this invention was experimentally verified on a flexible dual-link robotic arm system. This platform can integrate testing of actual physical components, improving the reliability and design quality of the control system. The FTLM system, for example... Figure 4 As shown, the first and second stage linkages are made of flexible material, with widths of 3 inches and 1.5 inches respectively. The FTLM system is driven by two high-precision brushed DC motors. Angular position measurement uses a high-resolution orthogonal optical encoder with an accuracy of 1024 lines per revolution, sufficient to meet the stringent requirements of position detection. To achieve efficient integration of hardware and control algorithms, the experimental platform adopts a collaborative architecture of QUARC and Simulink software. During the experiment, the data acquisition equipment is responsible for collecting raw data from the sensors and encoders, the power amplifier conditions the system's drive signals, and the industrial computer serves as the carrier for the control algorithm, achieving precise control of the FTLM system through the designed control strategy. Detailed dynamic parameters of the FTLM system will be provided below, as shown in Table 1, which lists the parameters of the FTLM experimental platform.

[0101] Table 1

[0102]

[0103] To verify the control effect of a fixed-time controller based on RBFNN on a flexible double-link manipulator system with input dead zone and state constraints, this invention conducts experimental verification on trajectory tracking and vibration suppression for an FTLM system with unknown dynamics, input dead zone, and output constraints. Using the experimental results of PSF control as a benchmark, the advantages of fixed-time control are highlighted by analyzing the tracking performance and vibration suppression effect of the manipulator. To ensure a fair comparison, both control methods use the same desired signal, initial conditions, and dead zone function. To further emphasize the vibration suppression effect of the manipulator, the reference trajectories of both links are set as square wave signals with amplitudes of 15° and 10°, respectively. The parameters of the dead zone function are defined as follows: and Next, we will introduce the parameter configuration of the fixed-time control strategy in detail.

[0104] The control parameters of the fixed-time controller are , , , , , , , , , and Furthermore, the constraint parameters of the flexible double-link robotic arm are set as follows: and .

[0105] This experiment used a square wave signal with a phase difference of 2.5 degrees as the input signal, which was converted and amplified for use in the control of the FTLM system. The SRV02 drive system controlled two drive motors respectively, while sensors collected and fed back the rotation angle and the elastic vibration of the robotic arm in real time. Based on these data, the trajectory tracking error was calculated. To verify the effectiveness of the fixed-time control scheme, in Figure 3 Comparative experiments were conducted on the experimental control platform shown to verify the performance of PSF control and fixed-time control algorithms. The experimental results are as follows: Figure 5-8 As shown.

[0106] This invention verifies both PSF control and fixed-time control algorithms. Figures 5 to 8 Experimental results of FTLM using two different control methods are described. Further evaluation will then be conducted based on these results.

[0107] The experimental results of trajectory tracking and tracking error of the FTLM system are as follows: Figure 5 and Figure 6 As shown. From Figure 6As can be seen, under PSF control, the maximum tracking error of the linkage is 12.05° and 1.51°; under fixed-time control, this value decreases to 11.56° and 0.71°. Furthermore, the maximum tracking error of both control strategies does not exceed the preset constraint range. This meets the system error requirements. From Figure 5 As can be seen, under PSF control, the steady-state tracking error of the linkage is 0.42° and 0.39°; under fixed-time control, this error is reduced to 0.16° and 0.12°. In addition, fixed-time control exhibits smaller overshoot and better transient performance.

[0108] The elastic vibration and control input of the FTLM system are respectively as follows: Figure 7 and Figure 8 As shown. For Figure 7 Analysis shows that under PSF control, the maximum elastic vibration amplitude of the link is 1.72° and 1.78°; under fixed-time control, these values ​​are 1.83° and 1.52°. Furthermore, the vibration eventually converges to within 0.3°, while the elastic vibration of the link under PSF control converges to within 1.0° and 0.8° respectively, with a slower decay rate. Figure 8 Observations show that the control inputs of fixed-time control and PSF control are basically the same, and the former improves performance without significantly increasing control energy consumption.

[0109] This invention proposes a fixed-time control method for a flexible dual-link manipulator with dead-zone input. Utilizing the approximation performance of RBFNN, a novel neural network update rate is designed to satisfy the fixed-time control framework, eliminating the influence of dead-zone nonlinear functions on the manipulator's trajectory tracking and approximating the unknown dynamic information of the flexible dual-link manipulator system. Then, a logarithmic barrier Lyapunov function (BLF) is used to ensure that the system output is strictly maintained within a specified range, preventing control instability caused by constraint violations. Ultimately, this enables the flexible manipulator to achieve trajectory tracking while suppressing elastic vibration.

[0110] The effectiveness of the proposed fixed-time controller based on RBFNN was verified through experiments on a flexible robotic arm experimental platform. Simultaneously, the stability of the closed-loop system was analyzed using logarithmic barrier Lyapunov stability, and it was proven that the closed-loop system satisfies a fixed-time frame. Physical experimental results show that:

[0111] In vibration suppression of flexible double-link robotic arms, fixed-time control based on RBFNN exhibits lower elastic vibration and better transient performance compared to PSF control.

[0112] In trajectory tracking of a flexible double-link robotic arm, fixed-time control based on RBFNN has smaller tracking error and faster settling time than PSF control, and the error signals are all within the constraint range.

[0113] In summary, the fixed-time control used in this invention offers faster convergence speed and higher trajectory tracking accuracy. It not only keeps the error of the FTLM system within a set range, improving the system's response efficiency, but also enables elastic vibrations to converge with smaller amplitudes and faster speeds. Furthermore, it effectively reduces system overshoot and optimizes transient performance. In contrast, PSF control exhibits larger steady-state tracking errors and poorer elastic vibration suppression. Therefore, the fixed-time control strategy proposed in this invention significantly improves the trajectory tracking performance and vibration suppression efficiency of the FTLM system, fully validating its effectiveness.

[0114] For the FTLM dynamic model with dead-zone input constructed using the assumed modal method, the fixed-time control strategy proposed in this invention not only ensures the rapid convergence of the flexible double-link manipulator but also simultaneously improves trajectory tracking accuracy and vibration suppression performance even under operating conditions involving system constraints. To enhance the applicability and robustness of the control scheme, multiple practical challenges, such as input dead zone, output constraints, and system uncertainty, are comprehensively addressed. A novel adaptive law is integrated within the fixed-time control framework, effectively compensating for model uncertainties and input nonlinearities, thereby significantly improving the system's adaptability in complex and uncertain environments. To handle the output constraints of the FTLM, a logarithmic barrier Lyapunov function (BLF) is employed. This ensures that the system output is strictly maintained within the specified range, preventing control instability caused by constraint violations. Furthermore, this method helps improve transient response characteristics and enhances overall control reliability.

[0115] The following points need to be explained:

[0116] (1) The accompanying drawings of the embodiments of the present invention only involve the structures involved in the embodiments of the present invention. Other structures can refer to the general design.

[0117] (2) For clarity, the thickness of layers or regions is enlarged or reduced in the drawings used to describe embodiments of the invention, i.e., these drawings are not drawn to scale. It is understood that when an element such as a layer, film, region or substrate is referred to as being “above” or “below” another element, the element may be “directly” located “above” or “below” the other element or there may be intermediate elements.

[0118] (3) Where there is no conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other to obtain new embodiments.

[0119] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. The scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A fixed-time control method for a flexible double-link robotic arm, characterized in that, The method includes: S1. Based on Hamilton's principle, Euler-Bernoulli beam theory, and the rotational inertia, angular velocity, and elastic vibration of the flexible double-link manipulator system, a dynamic model of the flexible system with dead zone input is established. S2. Based on the dynamic model of a flexible system with dead zone input, the initial control torque that depends on the dynamic model of the flexible system with dead zone input is derived by combining the backstepping method. S3. Based on the unknown dynamic information in the approximate initial control torque using a radial basis function neural network, design the update rate of the first radial basis function neural network; Based on the approximation of the unknown dead zone function by a radial basis function neural network, the update rate of a second radial basis function neural network is designed. S4. Based on the update rate of the first radial basis function neural network, the update rate of the second radial basis function neural network, and the initial control torque, the fixed-time control torque of the novel flexible double-link manipulator is obtained. The formula for the dynamic model of the flexible system in S1 is formula (1): ；（1） in, , where is the stiffness matrix; Torque The dead zone nonlinear function; , For time-varying generalized coordinates, The rotation angle of the joint link in the flexible double-link robotic arm; Matrix J(Q) is the inertia matrix; matrix For the Coriolis matrix and centripetal effect; The first derivative of the state variables of the FTLM system; For the second derivative of the state variables of the FTLM system; The formula for the initial control torque in S2 is formula (2): ；（2） k b For the state-constrained parameters of the Lyapunov function; for Transpose of; For the state error of the FTLM system; Add virtual control variables to the FTLM system to introduce state errors; The first derivative of the virtual control quantity; This is a virtual control variable; These are the control parameters for the FTLM system; for Transpose of; These are the control parameters for the FTLM system; The update rate of the first radial basis function neural network in S3 is given by formula (3): ；（3） The first radial basis function is the neural network update rate; The scaling factor for the update rate of the first radial basis function neural network; The Gaussian function is the first radial basis function of the neural network. for The i-th component in; These are the estimated values ​​of the neural network weights for the first radial basis function; for transpose; It is a positive positive number; The update rate of the second radial basis function neural network in S3 is given by formula (4): ；（4） The second radial basis function is the neural network update rate; This is the scaling factor for the update rate of the second radial basis function neural network. The Gaussian function is the second radial basis function of the neural network. Add virtual control variables to the FTLM system to introduce state errors; These are the estimated values ​​of the neural network weights for the second radial basis function; for Transpose of; It is a positive positive number.

2. The fixed-time control method for the flexible double-link robotic arm according to claim 1, characterized in that, The fixed-time control torque of the novel flexible double-link robotic arm in S4 is given by formula (5): ；（5） For the fixed-time control torque of a novel flexible double-link robotic arm; The Gaussian function is the first radial basis function of the neural network. The Gaussian function is the second radial basis function of the neural network.

3. The fixed-time control method for the flexible double-link robotic arm according to claim 2, characterized in that, The Lyapunov function of the fixed-time control torque based on the novel flexible double-link robotic arm is given by formula (6): ；（6） By continuously differentiating and transforming formula (6), we obtain According to the theory of fixed-time control, there exists a constant. and satisfy , obtain a fixed time For formula (7): （7） Among them, T max The maximum time for system convergence. These are control parameters for system stability. These are control parameters for system stability.

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