A flexible robot arm fractional order super-spiral sliding mode control method and system based on modal compensation

Abstract

This invention relates to a fractional-order superspiral sliding mode control method and system for a flexible robotic arm based on modal compensation. The method includes: constructing a rigid-flexible coupled dynamic model containing elastic deformation modes; defining the servo angular errors of the first and second-level linkage systems based on the rigid-flexible coupled dynamic model; and obtaining a fractional-order sliding surface based on modal compensation by combining the real-time modal compensation terms of the flexible linkages; calculating an equivalent control law using the fractional-order sliding surface; constructing a fractional-order superspiral sliding mode control model using the equivalent control law and a switching control law based on the fractional-order superspiral sliding mode algorithm; and outputting control commands using the fractional-order superspiral sliding mode control model to control the trajectory of the flexible robotic arm. This invention enables active compensation for elastic deformation, chatter suppression, and high-precision robust tracking control of the flexible robotic arm.

Description

Technical Field

[0001] This invention relates to the field of flexible robotic arm control technology, and in particular to a fractional-order superspiral sliding mode control method and system for flexible robotic arms based on modal compensation. Background Technology

[0002] With continuous breakthroughs in robotics application technology, robotic arms are widely used in aerospace, precision assembly, medical devices, and other fields. Flexible robotic arms, characterized by fast response speed, high flexibility, and low energy consumption, have attracted widespread attention and research from scholars and institutions both domestically and internationally. A flexible robotic arm consists of joint drive mechanisms and flexible links. Compared to the bulky size and heavy weight of traditional rigid robotic arms, flexible robotic arms are manufactured using lightweight, flexible materials, resulting in a smaller size and achieving higher operating speeds and larger working volumes within the same physical area. However, due to the flexible nature of the links, even slight changes in shape can cause displacement at the link ends, leading to elastic deformation during operation and affecting the control accuracy of the flexible robotic arm.

[0003] In recent years, flexible robotic arm controllers, such as PID control, robust control, fuzzy control, and neural network control, have been widely studied and applied due to their relatively simple structure, ability to suppress some disturbances, and ease of engineering implementation. However, due to the highly nonlinear and strongly coupled characteristics of robotic arm systems, existing controllers often rely on precise dynamic models or empirical parameter tuning in actual working conditions. This results in limited ability to cope with load changes and external disturbances, affecting the accuracy of trajectory tracking and the stability of the system, thus restricting their control performance under actual working conditions.

[0004] Furthermore, existing controllers typically ignore the elastic deformation of the links during design, thus failing to actively suppress the displacement caused by deformation. This displacement acts on the joint drive device through joint coupling, causing the actual output angle of the joint drive device to deviate from the expected value, resulting in deviations in the rotation angle of the robotic arm and making it impossible to accurately track the target trajectory.

[0005] Therefore, it is necessary to improve the existing controller and design a controller that can actively suppress the coupling interference of the elastic deformation of the flexible link on the joint drive device, so as to achieve high-precision tracking of the end effector trajectory of the robotic arm. Summary of the Invention

[0006] To address the problems existing in the prior art, the present invention aims to provide a fractional-order superspiral sliding mode control method and system for flexible robotic arms based on modal compensation. This method designs a fractional-order sliding surface module based on modal compensation to address the elastic deformation of the flexible robotic arm. The real-time modal compensation term of the flexible link is introduced into the fractional-order sliding surface to construct a fractional-order sliding surface capable of characterizing the historical motion state of the robotic arm system, thus achieving active compensation for the elastic deformation of the robotic arm. Based on this, a fractional-order superspiral sliding mode control module based on modal compensation is designed. This control method combines the superspiral algorithm with fractional-order calculus operators, smoothing the output signal through the superspiral structure to suppress physical chattering of the joint drive device, and utilizing the memory and adjustable order characteristics of fractional-order operators to enhance the control system's ability to suppress external disturbances. Finally, based on fractional-order Lyapunov theory, the Mittag-Leffler stability of the designed fractional-order superspiral sliding mode control system for flexible robotic arms based on modal compensation is analyzed.

[0007] To achieve the above objectives, the present invention provides the following solution: A fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation, comprising: A rigid-flexible coupling dynamic model containing elastic deformation modes is constructed. Based on the rigid-flexible coupling dynamic model, the servo angle error of the first-level and second-level linkage system is defined. Combined with the real-time modal compensation term of the flexible linkage, a fractional-order sliding surface based on modal compensation is obtained. The equivalent control law is calculated using the fractional-order sliding surface, and a fractional-order superspiral sliding mode control model is constructed using the equivalent control law and the switching control law based on the fractional-order superspiral sliding mode algorithm. The fractional-order superspiral sliding mode control model is used to output control commands to control the trajectory of the flexible robotic arm.

[0008] Optionally, constructing the rigid-flexible coupling dynamic model includes: ; ; ; ; Where t represents the current time, , These are the link servo angular accelerations for the first and second stage linkage systems, respectively. , These are the link deflection angular accelerations of the first and second stage linkage systems, respectively. , These refer to the link servo angular velocities of the first-stage and second-stage linkage systems, respectively. , These refer to the link deflection angular velocities of the first and second stage linkage systems, respectively. , These are the input currents for the first and second stage linkage systems of the flexible robotic arm, respectively. This refers to the self-damping coefficient on the joint drive device side of the first-stage linkage system. This represents the self-damping coefficient on the link side of the first-stage linkage system. This refers to the self-damping coefficient on the joint drive device side of the two-stage linkage system. This refers to the self-damping coefficient on the link side of the two-stage linkage system. It is the current gain term on the joint drive side of the first-stage linkage system. It is the link-side current gain term of the first-stage linkage system. It is the current gain term on the joint drive device side of the two-stage linkage system. It is the link-side current gain term of the two-stage linkage system. This refers to the joint load disturbance of the first-stage linkage system. This refers to the joint elastic-resistance coupling disturbance of the first-stage linkage system. For joint load disturbances in a two-stage linkage system. This refers to the joint elastic-resistance coupling disturbance of the two-stage linkage system.

[0009] Optionally, the servo angle error of the first-level and second-level linkage systems is defined as follows: ; ; in, , These are the servo angle errors of the first-stage and second-stage linkage systems, respectively. Given a reference trajectory angle, and These are the link servo angles of the flexible robotic arm.

[0010] Optionally, obtaining the fractional-order sliding surface includes: ; ; in, and These are the sliding surfaces of the primary and secondary linkage systems of the flexible robotic arm. and These are sliding mode fractional operators, where... and These are the fractional orders of the sliding mode, and These are the proportional gains of the first-stage and second-stage linkage system errors, respectively. and These are the modal compensation terms for the first and second stage linkage systems, respectively. and These represent the modal compensation gains of the first-stage and second-stage linkage systems, respectively. and These are the link deflection angles of the flexible robotic arm.

[0011] Optionally, calculating the equivalent control law using the fractional-order sliding surface includes: ; ; in, and These are the equivalent control laws for the first and second stage linkage systems of the flexible robotic arm, respectively. This refers to the self-damping coefficient on the joint drive device side of the first-stage linkage system. This refers to the self-damping coefficient on the joint drive device side of the two-stage linkage system. This refers to the current gain term on the joint drive side of the first-stage linkage system. This refers to the current gain term on the joint drive device side of the two-stage linkage system. Indicates the angular acceleration of the reference trajectory. and These are the servo angular velocities of the first and second stage linkages, respectively. and It is a flexible modal fractional-order sliding surface control operator. and It is the order of the fractional-order sliding surface control operator for flexible modes.

[0012] Optionally, the switching control law includes: ; ; in, and These are the switching control laws for the first and second stage linkage systems of the flexible robotic arm. , These are the integral compensation state variables for the first and second stage flexible linkage systems, respectively. It is the control gain of the nonlinear proportional term of the sliding mode surface in the modal compensation of a single-stage linkage system. It is the control gain of the nonlinear proportional term of the sliding surface in the modal compensation of the two-stage linkage system. It is the control gain of the sliding surface for modal compensation in a first-stage linkage system. It is the control gain of the sliding surface for modal compensation in a two-stage linkage system. This is a symbolic function used to determine the real-time direction of action of the joint drive device.

[0013] Optionally, constructing the fractional-order superspiral sliding mode control model includes: ; ; in, , These are control commands for the first and second stage linkage systems of the arm, respectively.

[0014] Optionally, the method further includes: The Mittag-Leffler stability of the fractional-order superspiral sliding mode control model is analyzed based on fractional-order Lyapunov theory. According to fractional-order Lyapunov theorem, it is proved that the fractional-order superspiral sliding mode control model satisfies Mittag-Leffler stability and the system error asymptotically converges to the equilibrium point.

[0015] To achieve the above objectives, the present invention also provides a fractional-order superspiral sliding mode control system for a flexible robotic arm based on modal compensation, comprising: The dynamic model building module is used to build rigid-flexible coupled dynamic models containing elastic deformation modes; The fractional-order sliding surface design module is used to define the servo angle error of the first-level and second-level linkage system according to the rigid-flexible coupling dynamic model, and to obtain the fractional-order sliding surface based on modal compensation by combining the real-time modal compensation term of the flexible linkage. The control model construction module is used to calculate the equivalent control law using the fractional-order sliding surface, and to construct a fractional-order superspiral sliding mode control model using the equivalent control law and the switching control law based on the fractional-order superspiral sliding mode algorithm. The control model output module is used to output control commands using the fractional-order superspiral sliding mode control model to control the trajectory of the flexible robotic arm.

[0016] The beneficial effects of this invention are as follows: 1. This invention introduces the elastic modes of the flexible link into the fractional-order sliding surface, realizing active compensation for elastic deformation, effectively suppressing the residual vibration excited by the flexible structure, reducing the convergence neighborhood of the link servo angle error, and significantly improving the transient and steady-state accuracy of the flexible robotic arm trajectory tracking.

[0017] 2. This invention reconstructs the discontinuous terms in sliding mode control through a superspiral algorithm structure and utilizes the memory effect of fractional calculus to make the current state of the system depend on all past states rather than a single instantaneous state. This design smooths the controller's output signal, reduces the input current oscillation of the joint drive device, suppresses high-frequency chattering, and improves the convergence speed and anti-interference capability of the control system under external disturbances.

[0018] 3. The control system designed in this invention features active deformation suppression through modal compensation and smooth output through fractional-order superhelical propagation. This control system improves the positioning accuracy of the linkage, suppresses chattering through smoothed signals, and extends the lifespan of the joint drive device. The control system of this invention exhibits superior performance in flexible robotic arms, including high stability and strong anti-interference capabilities, and has significant engineering practical value. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the 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.

[0020] Figure 1 This is a structural block diagram of a two-degree-of-freedom flexible robotic arm according to an embodiment of the present invention; Figure 2 This is a flowchart of a fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation, according to an embodiment of the present invention. Figure 3 This is a structural block diagram of a fractional-order superspiral sliding mode control system for a flexible robotic arm based on modal compensation, according to an embodiment of the present invention. Figure 4 This is a trajectory tracking diagram of the flexible robotic arm according to an embodiment of the present invention; Figure 5 This is a trajectory tracking diagram of a flexible robotic arm after external interference is added, according to an embodiment of the present invention. Detailed Implementation

[0021] 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.

[0022] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0023] This embodiment discloses a fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation, including the following steps: Step 1: Construct a dynamic model containing elastic deformation modes. Based on the output torque of the joint drive device and the structural dynamic equations of the flexible link, the Euler-Lagrange modeling method is used to establish a rigid-flexible coupled dynamic model of a two-degree-of-freedom flexible manipulator containing elastic deformation modes, in order to characterize the coupling relationship between joint rotation and elastic deformation of the flexible link.

[0024] Step 2: Design a fractional-order sliding surface module based on modal compensation. The real-time modal compensation term of the flexible link is introduced into the fractional-order sliding surface to construct a fractional-order sliding mode that can characterize the historical motion state of the robotic arm system, thereby achieving active compensation for the elastic deformation of the robotic arm. Define the servo angle errors of the first and second level linkages of the system as follows: and ,in Given a reference trajectory angle, and It is the link servo angle of the flexible robotic arm.

[0025] ; ; in, and It is the sliding surface of the primary and secondary linkage systems of the flexible robotic arm. and For sliding mode fractional operators, where, and For sliding mode fractional order, and The proportional gain for the error of the first and second stage linkage system. and For the modal compensation terms of the first and second stage linkage systems, among which, and This represents the modal compensation gain of the first and second stage linkage systems. and It is the deflection angle of the link in the flexible robotic arm.

[0026] Step 3: Design a fractional-order superspiral sliding mode control module based on modal compensation. Using the modal compensation sliding mode generated in Step 2 as input, a control law is constructed by combining the superspiral algorithm and fractional-order operators. The superspiral structure is used to smooth the output to suppress chattering of the joint drive device. The adjustable order and memory characteristics of the fractional-order operators are used to compensate for external disturbances, and control commands for the joint drive device are output, thus completing the construction of the control system.

[0027] The equivalent control law derived from the equivalent control principle is as follows: ; ; in, and These are the equivalent control laws for the first and second stage linkage systems of the flexible robotic arm, respectively. This refers to the self-damping coefficient on the joint drive device side of the first-stage linkage system. Let be the self-damping coefficient on the joint drive device side of the secondary linkage system; It is the current gain term on the joint drive side of the first-stage linkage system. It is the current gain term on the joint drive device side of the two-stage linkage system. Indicates the angular acceleration of the reference trajectory. and These are the servo angular velocities of the first and second stage linkages, respectively. and It is a flexible modal fractional-order sliding surface control operator, in which, and It is the order of the flexible modal fractional sliding surface control operator, and its value range is... .

[0028] The switching control law based on the fractional-order superspiral sliding mode algorithm is: ; ; in, and These are the switching control laws for the first and second stage linkage systems of the flexible robotic arm. , For the integral compensation state variables of the first and second stage flexible linkage systems It is the control gain of the nonlinear proportional term of the sliding mode surface in the modal compensation of a single-stage linkage system. It is the control gain of the nonlinear proportional term of the sliding surface in the modal compensation of the two-stage linkage system. It is the control gain of the sliding surface for modal compensation in a first-stage linkage system. It is the control gain of the sliding surface for modal compensation in a two-stage linkage system. This is a symbolic function, which in this invention is a low-level logic reversing switch used to determine the real-time operating direction of the joint drive device.

[0029] The total control output of the two-degree-of-freedom flexible robotic arm system is: ; ; Step 4: Based on fractional-order Lyapunov theory, the Mittag-Leffler stability of the designed modal compensation-based flexible manipulator fractional-order super-spiral sliding mode control system is analyzed.

[0030] Based on the fractional Lyapunov theorem, it is proved that the fractional super-helical sliding mode control system of the flexible robotic arm based on modal compensation satisfies Mittag-Leffler stability, and the system error asymptotically converges to the equilibrium point.

[0031] Step 5: Conduct anti-interference and trajectory tracking experiments on the fractional-order super-spiral sliding mode control system, output control commands, and verify the system's stability and tracking accuracy.

[0032] In one possible implementation, this embodiment discloses a fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation, such as... Figure 2 As shown, it includes: Step 1: Construct a dynamic model containing elastic deformation modes.

[0033] like Figure 1 As shown, the two-degree-of-freedom flexible robotic arm in this embodiment is a dual-input, four-output system, comprising two joint drive devices and two flexible links. Under input, the joint drive devices within the two flexible link systems generate torque. , This allows the two flexible linkage systems to output linkage servo angles respectively. , This causes the two flexible connecting rods to deflect at different angles. , . For a global coordinate system, This is a local coordinate system centered on the second flexible linkage system. and These are local rotating reference frames centered on the first flexible linkage system and the second flexible linkage system, respectively.

[0034] Based on the dynamics of the joint drive mechanism and the Euler-Lagrange equations of the connecting rod, a rigid-flexible coupling dynamic model of the system is established: ; ; ; ; Where t represents the current time. , For the linkage servo angular acceleration of the first and second stage linkage systems, , It refers to the angular acceleration of the link deflection in the first and second stage linkage system. , It refers to the link servo angular velocity of the primary and secondary linkage systems. , It refers to the linkage servo angle of the primary and secondary linkage systems. , It refers to the angular velocity of the connecting rods in the first and second stage linkage system. , The moment of inertia of the joint drive mechanism. , The coefficient of rotational friction of the joint drive mechanism. , The torque constant of the joint drive device. , For the rotational inertia of the flexible connecting rod, , The elastic coefficient of the flexible link is... , The coefficient of friction at the end of the flexible connecting rod. and It is the input current of the first and second linkage systems of the flexible robotic arm.

[0035] To facilitate controller design, the model is simplified to the following form: ; ; ; ; in, This refers to the self-damping coefficient on the joint drive device side of the first-stage linkage system. This represents the self-damping coefficient on the link side of the first-stage linkage system. This refers to the self-damping coefficient on the joint drive device side of the two-stage linkage system. This refers to the self-damping coefficient on the link side of the two-stage linkage system. It is the current gain term on the joint drive side of the first-stage linkage system. It is the link-side current gain term of the first-stage linkage system. It is the current gain term on the joint drive device side of the two-stage linkage system. It is the link-side current gain term of the two-stage linkage system. This refers to the joint load disturbance of the first-stage linkage system. This refers to the joint elastic-resistance coupling disturbance of the first-stage linkage system. For joint load disturbances in a two-stage linkage system. This refers to the joint elastic-resistance coupling disturbance of the two-stage linkage system.

[0036] Step 2: Design a fractional-order sliding surface module based on modal compensation.

[0037] Define the servo angle errors of the first and second level linkages of the system as follows: and ,in The reference trajectory angle is given.

[0038] Using the Caputo definition of fractional calculus and introducing flexible modal variables as real-time compensation, the sliding surface of the first-stage linkage system of a flexible robotic arm is designed. Sliding surface of the second-stage linkage system : ; ; in, , Construct fractional-order operators for flexible modal sliding modes, where, and For the construction order of flexible modes, and The proportional gain for the error of the first and second stage linkage system. and This represents the modal compensation gain of the first and second stage linkage systems. and This refers to the modal compensation terms for the first and second stage linkage systems.

[0039] Step 3: Design a modal compensation fractional-order superspiral sliding mode controller and build a control system for the flexible robotic arm.

[0040] The control system of this invention is constructed from a dynamic model, a fractional-order sliding surface module based on modal compensation, and a fractional-order superspiral sliding mode control module. The output of the sliding mode controller is equivalently controlled by a flexible robotic arm system. Fractional-order superspiral sliding mode switching control composition: .

[0041] The equivalent control law derived from the equivalent control principle is as follows: ; ; in, and These are the equivalent control laws for the first and second stage linkage systems of the flexible robotic arm, respectively. Indicates the angular acceleration of the reference trajectory. and It is a flexible modal fractional-order sliding surface control operator, in which, and It is the order of the flexible modal fractional sliding surface control operator, and its value range is... .

[0042] The switching control law derived from the fractional-order superspiral sliding mode algorithm is as follows: ; ; in, and These are the switching control laws for the first and second stage linkage systems of the flexible robotic arm. , For the integral compensation state variables of the first and second stage flexible linkage systems It is the control gain of the nonlinear proportional term of the modal compensation sliding surface. It is the control gain of the modal compensation sliding surface. This is a symbolic function, which in this invention is a low-level logic reversing switch used to determine the real-time operating direction of the joint drive device.

[0043] The total control output of the two-degree-of-freedom flexible robotic arm system is: ; ; Step 4: Based on fractional-order Lyapunov theory, the Mittag-Leffler stability of the designed modal compensation-based flexible manipulator fractional-order super-spiral sliding mode control system is analyzed.

[0044] Taking the first-stage linkage system of a flexible robotic arm as an example, the fractional-order dynamic equation for the closed-loop sliding surface arrival stage is: ; in, External disturbances to the control system and L is the upper bound of the external disturbance.

[0045] The above formula can be simplified as: ; Where the auxiliary state vector Where T is the vector transpose. It is a nonlinear function of the sliding mode variable. These are auxiliary integral state variables, and the matrix of the flexible modal synthesis system. Decomposed into flexible modal nominal system matrix and flexible modal perturbation matrix ,in: ; Construct positive definite quadratic fractional Lyapunov functions : ; in, The matrix is ​​a positive definite decision matrix for flexible modes and: ; Find the fractional derivative of the first-stage linkage system of the flexible robotic arm under the condition of bounded external disturbance: ; The Lyapunov derivative term of the nominal matrix of a single-stage linkage system was calculated. Through matrix algebra calculations, this term exhibits a symmetric form: ; By scaling the system disturbance cross terms, the disturbance term matrix can be calculated, as shown below: ; According to the norm scaling principle, the quadratic form in the above equation is subject to the following constraints: ; Define the flexible modal stability criterion matrix The Lyapunov function satisfies: ; in, .

[0046] To ensure the asymptotic convergence of the closed-loop system state, the matrix... It must be strictly positive definite. The control gain of a single-stage linkage system. The following conditions must be met: ; Similarly, the control gain of a two-stage linkage system The following conditions must be met: ; Based on the fractional Lyapunov theorem, it is proved that the fractional super-helical sliding mode control system of the flexible robotic arm based on modal compensation satisfies Mittag-Leffler stability, and the system error asymptotically converges to the equilibrium point.

[0047] Step 5: Based on the hardware experimental platform of the two-degree-of-freedom flexible robotic arm system, conduct anti-interference and trajectory tracking experiments on the fractional-order super-spiral sliding mode control system of the flexible robotic arm based on modal compensation, output control commands, and verify the stability and tracking accuracy of the system.

[0048] like Figure 3 As shown, this embodiment discloses a fractional-order superspiral sliding mode control system for a flexible robotic arm based on modal compensation, comprising: The dynamic model building module is used to build rigid-flexible coupled dynamic models containing elastic deformation modes; The fractional-order sliding surface design module is used to define the servo angle error of the first and second-order linkage system based on the rigid-flexible coupling dynamic model, and to obtain the fractional-order sliding surface based on modal compensation by combining the real-time modal compensation term of the flexible linkage. The control model construction module is used to calculate the equivalent control law using the fractional-order sliding surface, and to construct the fractional-order superspiral sliding mode control model through the equivalent control law and the switching control law based on the fractional-order superspiral sliding mode algorithm. The control model output module is used to output control commands using a fractional-order superspiral sliding mode control model to control the trajectory of the flexible robotic arm.

[0049] Specifically, this embodiment discloses a fractional-order superspiral sliding mode control system for a flexible robotic arm based on modal compensation, comprising: Construct an integer-order rigid-flexible coupling dynamic model of a flexible manipulator with elastic deformation to describe the coupling characteristics between the elastic deformation of the flexible link and the joint rotation. An error model is constructed based on the dynamic model, and a fractional-order sliding surface based on modal compensation is constructed based on the error model, the modal compensation module, and the sliding mode. A fractional-order superspiral sliding mode control module is designed to combine the superspiral algorithm with fractional-order calculus operators to build a controller. The superspiral structure smooths the output signal to suppress the physical chattering of the servo motor, and the adjustable order and memory characteristics of the fractional-order operator are used to enhance the system's anti-interference performance against external disturbances. By combining a fractional-order sliding surface with a fractional-order superspiral sliding mode control module, a fractional-order superspiral sliding mode control system based on modal compensation is built. The trajectory of the flexible robotic arm is controlled by the output signal of the fractional-order superspiral sliding mode controller. A stability analysis module was designed to verify the Mittag-Leffler stability of the closed-loop system based on fractional Lyapunov theory.

[0050] To verify the actual effect of the present invention, the controller provided by the present invention was experimentally verified on the experimental platform of a two-degree-of-freedom flexible manipulator in this embodiment. The parameters are shown in Table 1.

[0051] Table 1 Controller parameters of the present invention In the experiment, the controller provided by this invention was compared with a reference trajectory, and the results were as follows: Figure 4 and Figure 5 The response curve data is shown below. Figure 5 This is an experiment simulating external interference, showing the response curve of a step signal of 0.1 rad / s added between 12s and 14s.

[0052] analyze Figure 4 and Figure 5 As shown in the response curves, the output curves of the controller in the first and second servo systems of this invention highly coincide with the reference trajectory, exhibiting fast dynamic response with no overshoot and a steady-state error approaching zero. This verifies the controller's accurate tracking performance under undisturbed conditions. When a 0.1 rad / s step disturbance is applied between 12 and 14 seconds, the controller can still strictly control the position output of the flexible robotic arm within the reference value. Within this range, no cumulative position drift, oscillation, or recovery overshoot was observed during the entire duration of the disturbance or after its removal.

[0053] from Figure 4 and Figure 5 It can be seen that the controller designed in this invention has strong anti-interference ability and high tracking accuracy, making it suitable for applications with high interference intensity.

[0054] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation, characterized in that, include: A rigid-flexible coupling dynamic model containing elastic deformation modes is constructed. Based on the rigid-flexible coupling dynamic model, the servo angle error of the first-level and second-level linkage system is defined. Combined with the real-time modal compensation term of the flexible linkage, a fractional-order sliding surface based on modal compensation is obtained. The equivalent control law is calculated using the fractional-order sliding surface, and a fractional-order superspiral sliding mode control model is constructed using the equivalent control law and the switching control law based on the fractional-order superspiral sliding mode algorithm. The fractional-order superspiral sliding mode control model is used to output control commands to control the trajectory of the flexible robotic arm.

2. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 1, characterized in that, Constructing the rigid-flexible coupled dynamic model includes: ； ； ； ； Where t represents the current time, , These are the link servo angular accelerations for the first and second stage linkage systems, respectively. , These are the link deflection angular accelerations of the first and second stage linkage systems, respectively. , These refer to the link servo angular velocities of the first and second stage linkage systems, respectively. , These refer to the link deflection angular velocities of the first and second stage linkage systems, respectively. , These are the input currents for the first and second stage linkage systems of the flexible robotic arm, respectively. This refers to the self-damping coefficient on the joint drive device side of the first-stage linkage system. This represents the self-damping coefficient on the link side of the first-stage linkage system. This refers to the self-damping coefficient on the joint drive device side of the two-stage linkage system. This refers to the self-damping coefficient on the link side of the two-stage linkage system. It is the current gain term on the joint drive side of the first-stage linkage system. It is the link-side current gain term of the first-stage linkage system. It is the current gain term on the joint drive device side of the two-stage linkage system. It is the link-side current gain term of the two-stage linkage system. This refers to the joint load disturbance of the first-stage linkage system. This refers to the joint elastic-resistance coupling disturbance of the first-stage linkage system. This refers to the joint load disturbance of the two-stage linkage system. This refers to the joint elastic-resistance coupling disturbance of the two-stage linkage system.

3. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 2, characterized in that, The servo angle error of the first and second stage linkage system is defined as follows: ； ； in, , These are the servo angle errors of the first-stage and second-stage linkage systems, respectively. Given a reference trajectory angle, and These are the link servo angles of the flexible robotic arm.

4. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 1, characterized in that, Obtaining the fractional-order sliding surface includes: ； ； in, and These are the sliding surfaces of the primary and secondary linkage systems of the flexible robotic arm. and These are sliding mode fractional operators, where... and These are the fractional orders of the sliding mode, and These are the proportional gains of the first-stage and second-stage linkage system errors, respectively. and These are the modal compensation terms for the first and second stage linkage systems, respectively. and These represent the modal compensation gains of the first-stage and second-stage linkage systems, respectively. and These are the link deflection angles of the flexible robotic arm.

5. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 1, characterized in that, Calculating the equivalent control law using the fractional-order sliding surface includes: ； ； in, and These are the equivalent control laws for the first and second stage linkage systems of the flexible robotic arm, respectively. This refers to the self-damping coefficient on the joint drive device side of the first-stage linkage system. This refers to the self-damping coefficient on the joint drive device side of the two-stage linkage system. This refers to the current gain term on the joint drive side of the first-stage linkage system. This refers to the current gain term on the joint drive device side of the two-stage linkage system. Indicates the angular acceleration of the reference trajectory. and These are the servo angular velocities of the first and second stage linkages, respectively. and It is a flexible modal fractional-order sliding surface control operator. and It is the order of the fractional-order sliding surface control operator for flexible modes.

6. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 5, characterized in that, The switching control law includes: ； ； in, and These are the switching control laws for the first and second stage linkage systems of the flexible robotic arm. , These are the integral compensation state variables for the first and second stage flexible linkage systems, respectively. It is the control gain of the nonlinear proportional term of the sliding mode surface in the modal compensation of a single-stage linkage system. It is the control gain of the nonlinear proportional term of the sliding surface in the modal compensation of the two-stage linkage system. It is the control gain of the sliding surface for modal compensation in a first-stage linkage system. It is the control gain of the sliding surface for modal compensation in a two-stage linkage system. This is a symbolic function used to determine the real-time direction of action of the joint drive device.

7. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 6, characterized in that, The fractional-order superspiral sliding mode control model is constructed as follows: ； ； in, , These are control commands for the first and second stage linkage systems of the arm, respectively.

8. The fractional-order superspiral sliding mode control method for a flexible robotic arm based on modal compensation according to claim 1, characterized in that, The method also includes: The Mittag-Leffler stability of the fractional-order superspiral sliding mode control model is analyzed based on fractional-order Lyapunov theory. According to fractional-order Lyapunov theorem, it is proved that the fractional-order superspiral sliding mode control model satisfies Mittag-Leffler stability and the system error asymptotically converges to the equilibrium point.

9. A fractional-order superspiral sliding mode control system for a flexible robotic arm based on modal compensation, used to implement the method described in any one of claims 1-8, characterized in that, include: The dynamic model building module is used to build rigid-flexible coupled dynamic models containing elastic deformation modes; The fractional-order sliding surface design module is used to define the servo angle error of the first-level and second-level linkage system according to the rigid-flexible coupling dynamic model, and to obtain the fractional-order sliding surface based on modal compensation by combining the real-time modal compensation term of the flexible linkage. The control model construction module is used to calculate the equivalent control law using the fractional-order sliding surface, and to construct a fractional-order superspiral sliding mode control model using the equivalent control law and the switching control law based on the fractional-order superspiral sliding mode algorithm. The control model output module is used to output control commands using the fractional-order superspiral sliding mode control model to control the trajectory of the flexible robotic arm.

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