Vibration control method for movable flexible arm with non-uniform tension

Abstract

The invention relates to the technical field of mobile flexible arm system control, in particular to a vibration control method for a mobile flexible arm with non-uniform tension, which comprises the following steps of: establishing a mobile flexible arm system model with non-uniform tension, and aiming at the mobile flexible arm system model with non-uniform tension, establishing a vibration control method for the mobile flexible arm with non-uniform tension; a boundary vibration control algorithm based on the obstacle Lyapunov function is constructed, a control-dedicated Lyapunov function J (t) is constructed based on the boundary vibration control algorithm so as to determine the stability of the mobile flexible arm system model, if the Lyapunov function J (t) is positive definite and # imgabs0 # is negative definite, it is judged that the system model is stable, numerical simulation verification is conducted on the algorithm, a numerical simulation graph is obtained, and the numerical simulation graph is calculated. And verifying the availability of the control algorithm. According to the method, the model and the algorithm are constructed, the availability of the control algorithm is determined through a mathematical analysis calculation method and a simulation method, the tension influence of the flexible arm, especially the connecting point of the flexible arm and the tail end load is reduced, and the control performance and stability of the movable flexible arm are improved.

Description

Technical Field

[0001] The invention relates to the technical field of mobile flexible arm system control, and in particular to a vibration control method for a mobile flexible arm with non-uniform tension. Background Art

[0002] In recent years, with the development of computer technology and artificial intelligence technology, automation technology has also made great strides. With the advent of the intelligent era, flexible arms have replaced rigid arms with their advantages of low mass, low energy consumption, high flexibility and the ability to work under various difficult and even extreme conditions. They play an important role in various fields such as automated production, space exploration, medical surgery, disaster relief and exploration. The mobile flexible arm system built by mounting the flexible arm on a mobile platform such as a car breaks through the limitations of the limited operating space of the traditional fixed robotic arm, and its application field is also more extensive.

[0003] China Patent Publication No.: CN113400348A, discloses a mobile and rotating multi-flexible arm vibration control device and method, including: environmental perception, obstacle representation and autonomous motion; the environmental perception part includes: image acquisition module, calibration module and multi-sensor fusion mechanism: obstacle representation part includes: first classify static and dynamic obstacles, then perform posture recognition, establish obstacle model, and simplify the model boundary representation, predict the trajectory and range of dynamic obstacles, and extract the characteristic contour of obstacles; autonomous motion includes obstacle avoidance decision and motion control. If there are dynamic obstacles, collision detection and trajectory replanning algorithms are used to adjust the moving route in real time in the local environment, and then output motion control to the robot based on the planned route to realize the functions of obstacle avoidance, path selection and motion. Through the cooperation of various modules and mechanisms, a reliable, fast, highly autonomous and high-precision real-time obstacle avoidance system is realized.

[0004] Chinese patent publication number: CN108827571A, discloses a two-degree-of-freedom flexible arm vibration detection control device and method, including a first flexible arm, a second flexible arm, a third flexible arm, a fourth flexible arm, a first rotating mechanism, a second rotating mechanism, a linear moving mechanism and a control component; the first end of the second flexible arm is fixedly connected to the first end of the third flexible arm, and they are arranged together on the linear moving mechanism, the first end of the first flexible arm is rotationally connected to the second end of the second flexible arm through the first rotating mechanism, the first end of the fourth flexible arm is rotationally connected to the second end of the third flexible arm through the second rotating mechanism, the first flexible arm, the second flexible arm, the third flexible arm and the fourth flexible arm are all provided with piezoelectric sensors and acceleration sensors, and the second flexible arm and the third flexible arm are provided with piezoelectric drivers. The present invention effectively eliminates the influence of other uncontrollable factors on the vibration control effect, and is conducive to improving the detection accuracy.

[0005] However, there are still the following problems in the prior art:

[0006] In reality, in various control studies on flexible arms, the tension of the arm is always ignored. In actual production, if the tension of the arm, especially the tension at the connection point with the end load, is not taken into account, it will cause the flexible arm to vibrate during operation, and even affect the control performance and stability of the system during the control process, thereby reducing work efficiency and increasing danger. Summary of the invention

[0007] To this end, the present invention provides a vibration control method for a mobile flexible arm with non-uniform tension, so as to solve the problem that in actual situations, in various control studies on flexible arms, the tension of the arm is always ignored. In actual production, if the tension of the arm, especially the tension at the connection point with the end load, is not taken into account, it will cause the flexible arm to vibrate during operation, and even affect the control performance and stability of the system during the control process, thereby reducing work efficiency and increasing danger.

[0008] To achieve the above object, the present invention provides a vibration control method for a mobile flexible arm with non-uniform tension, which comprises:

[0009] Determine the kinetic energy of the mobile flexible arm system with non-uniform tension, the potential energy of the mobile flexible arm system with non-uniform tension, and the virtual work of the mobile flexible arm system with non-uniform tension, and establish a mobile flexible arm system model with non-uniform tension in combination with the Hamiltonian principle;

[0010] For the mobile flexible arm system model with non-uniform tension, a boundary vibration control algorithm based on the obstacle Lyapunov function is constructed to ensure that the boundary tension constraint condition of the flexible arm is satisfied, wherein the boundary vibration control algorithm includes the control force exerted by the flexible arm hub, the control force exerted by the trolley, and the control force exerted by the end load;

[0011] Based on the boundary vibration control algorithm, a control-specific Lyapunov function J(t) is constructed to determine the stability of the mobile flexible arm system model. If the Lyapunov function J(t) is positive definite and If negative, the system model is judged to be stable.

[0012] The algorithm is numerically simulated and verified to obtain a numerical simulation diagram, which is used to determine the stability of the trolley based on the rotation angle of the flexible arm and the motion displacement of the trolley in the numerical simulation diagram, and to verify the feasibility of the control algorithm.

[0013] Furthermore, the kinetic energy of the mobile flexible arm system with non-uniform tension is expressed by formula (1):

[0014]

[0015] In formula (1), F m (t) represents the kinetic energy of the mobile flexible arm system with non-uniform tension at time t, t represents time, ρ represents the uniform mass per unit length of the flexible arm, L represents the length of the flexible arm, w represents the spatial position, K(w, t) represents the absolute displacement function of the flexible arm with respect to the spatial position w and time t, express The transpose of m s represents the mass of the end load, J represents the moment of inertia of the flexible arm, β(t) represents the rotation angle of the flexible arm at time t, m v represents the mass of the car, and s(t) represents the displacement of the car at time t.

[0016] Furthermore, the potential energy of the mobile flexible arm system with non-uniform tension is expressed by formula (2):

[0017]

[0018] In formula (2), F n (t) represents the potential energy of the mobile flexible arm system with non-uniform tension at time t, EI represents the bending stiffness of the flexible arm, q(w, t) represents the elastic deformation function of the flexible arm with respect to spatial position w and time t, and g(w, t) represents the non-uniform tension function of the flexible arm that changes with time and space with respect to spatial position w and time t.

[0019] Furthermore, the virtual work of the mobile flexible arm system with non-uniform tension is expressed by formula (3):

[0020] F v (t) = [d x (t)+u x (t)]β(t)+[d y (t)+u y (t)]s(t)+[d z (t)+u z (t)]q(L,t) (3)

[0021] In formula (3), F v (t) represents the virtual work of the mobile flexible arm system with non-uniform tension at time t, d x (t) represents the time-varying external disturbance to the flexible arm hub, u x (t) represents the control force exerted by the flexible arm hub, d y (t) represents the external disturbance to the car that changes with time, u y (t) represents the control force exerted by the car, d z(t) represents the external disturbance to the end load that changes with time, u z (t) represents the control force exerted by the end load, and q(L, t) represents the elastic deformation function at the end of the robot arm with respect to the length L of the flexible arm and time t.

[0022] Furthermore, the mobile flexible arm system model with non-uniform tension includes formula (4) to formula (8),

[0023] q(0,t)=q'(0,t)=q″(L,t)=0 (4)

[0024]

[0025]

[0026] In formulas (4) to (8), q represents elastic deformation, s represents the displacement of the trolley, r(L) represents the initial tension of the flexible arm with a length of L, r represents the initial tension, and h(L) represents the nonlinear elastic modulus of the flexible arm with a length of L.

[0027] Furthermore, the control force exerted by the flexible arm hub is expressed by formula (9):

[0028]

[0029] In formula (9), k1 represents the first auxiliary variable control parameter, φ a (t) represents the first auxiliary variable, ξ1 represents the control parameter of the flexible arm rotation angle, λ represents the control parameter, β d represents the predetermined position of the flexible arm rotation angle, sgn(·) represents the sign function, d xm represents the external disturbance d that varies with time x The upper bound of (t).

[0030] Furthermore, the control force exerted by the car is expressed by formula (10):

[0031]

[0032] In formula (10), k2 represents the second auxiliary variable control parameter, φ b (t) represents the second auxiliary variable, ξ2 represents the displacement control parameter of the car motion, s d represents the predetermined position of the vehicle's motion displacement, d ym represents the external disturbance d that varies with time y The upper bound of (t).

[0033] Furthermore, the control force exerted by the end load is expressed by formula (11):

[0034]

[0035] In formula (11), k3 represents the elastic deformation change speed control parameter of the flexible arm end, ξ3 represents the elastic deformation control parameter of the flexible arm end, In(·) represents the logarithmic function, M represents the constraint value, d zm represents the external disturbance d that varies with time Z The upper bound of (t).

[0036] Furthermore, the control-specific Lyapunov function J(t) is expressed by formula (12):

[0037] J(t)=J a (t)+J b (t)+J c (t) (12)

[0038] In formula (12), J(t) represents the control-specific Lyapunov function, J a (t) represents the Lyapunov function of the energy term, J b (t) represents the Lyapunov function of the cross term, J c (t) represents the Lyapunov function with respect to the auxiliary term.

[0039] Furthermore, the process of verifying the availability of the control algorithm includes:

[0040] Obtaining a three-dimensional graph of the vibration of the flexible arm when control is applied to determine the amplitude of the flexible arm;

[0041] acquiring the rotation angle of the flexible arm when the control is applied to determine the angular stop position;

[0042] Obtain a two-dimensional map of the movement displacement of the trolley when the control is applied to determine the stopping position of the trolley;

[0043] If the amplitude of the flexible arm is less than or equal to the preset amplitude, the angle stop position is at the preset angle position and the trolley stop position is at the preset trolley position, it is determined that the control algorithm is feasible.

[0044] Compared with the prior art, the present invention establishes a mobile flexible arm system model with non-uniform tension, and for the mobile flexible arm system model with non-uniform tension, constructs a boundary vibration control algorithm based on the obstacle Lyapunov function, and constructs a control-specific Lyapunov function J(t) based on the boundary vibration control algorithm to determine the stability of the mobile flexible arm system model, wherein, if the Lyapunov function J(t) is positive definite and

[0045] If the value is negative, the system model is judged to be stable, the algorithm is numerically simulated and verified, a numerical simulation diagram is obtained, and the availability of the control algorithm is verified. The present invention constructs a model and an algorithm, determines the availability of the control algorithm by mathematical analysis and calculation and simulation, reduces the influence of the tension of the flexible arm, especially at the connection point with the end load, and improves the control performance and stability of the mobile flexible arm.

[0046] In particular, the present invention constructs a boundary vibration control algorithm for a mobile flexible arm system model with non-uniform tension, so as to control the tension of the mobile flexible arm to keep it stable. In actual situations, when controlling the mobile flexible arm, the tension is seldom taken into consideration, but the tension of the arm, especially the tension at the connection point with the end load, if not controlled, will cause the mobile flexible arm to vibrate, affecting work efficiency. Based on this, it is considered to construct a boundary vibration control algorithm, and the tension of the flexible arm is controlled from three aspects: the control force exerted by the flexible arm hub, the control force exerted by the trolley, and the control force exerted by the end load, thereby reducing the influence of the tension on the flexible arm, especially at the connection point with the end load, and improving the control performance and stability of the mobile flexible arm.

[0047] In particular, the present invention verifies the usability of the boundary vibration control algorithm from two aspects: mathematical analysis and numerical simulation. In terms of mathematical analysis and calculation, the present invention constructs a control-specific Lyapunov function J(t) based on the boundary vibration control algorithm, determines the Lyapunov function J(t) and If the Lyapunov function J(t) is positive definite and If the value of the boundary vibration control algorithm is negative, the system model is judged to be stable. In terms of numerical simulation, the present invention verifies the stability of the mobile flexible arm system model with non-uniform tension by numerical simulation through MATLAB. If both aspects indicate that the system model is stable, it means that the boundary vibration control algorithm is available. By using the boundary vibration control algorithm, the influence of the tension on the flexible arm, especially at the connection point with the end load, is reduced, thereby improving the control performance and stability of the mobile flexible arm. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 A schematic diagram of the method steps of a method for controlling vibration of a mobile flexible arm with non-uniform tension according to an embodiment of the invention;

[0049] Figure 2 is a schematic diagram of the operation of the mobile flexible arm;

[0050] Figure 3 This is a vibration diagram of the flexible arm without control applied in the present invention;

[0051] Figure 4 is an angle position diagram of the flexible arm without control of the present invention;

[0052] Figure 5 This is a motion position diagram of the trolley without control in the present invention;

[0053] Figure 6 Vibration diagram of the flexible arm to which the present invention applies control;

[0054] Figure 7 A diagram of the angle position of the flexible arm to which the present invention applies control;

[0055] Figure 8 The motion position diagram of the trolley to which the present invention applies control. DETAILED DESCRIPTION

[0056] In order to make the objects and advantages of the present invention more clearly understood, the present invention is further described below in conjunction with embodiments; it should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0057] The preferred embodiments of the present invention are described below with reference to the accompanying drawings. It should be understood by those skilled in the art that these embodiments are only used to explain the technical principles of the present invention and are not intended to limit the protection scope of the present invention.

[0058] To facilitate reading of this plan, some of the terms involved in this plan are explained below:

[0059] Tension: It is a physical concept that describes the internal resistance of an object due to tension, and has different meanings in different fields.

[0060] Non-uniform tension: The phenomenon in which tension is unevenly distributed in a material or structure.

[0061] Mobile flexible arm: Based on the traditional robotic arm, by combining a lightweight high-degree-of-freedom robotic arm with an autonomous mobile platform, the working range and flexibility of the robotic arm are expanded. It has a high degree of freedom and can adapt to changing work tasks. It has important research and application value in the fields of industry, flexible manufacturing systems, medical health, home services, and human-machine collaborative operations.

[0062] To facilitate understanding, the meaning of some contents in the article is explained.

[0063] (·) represents the derivative with respect to time t, and (·)' represents the derivative with respect to spatial position w. For example,

[0064]

[0065] Formula (13) to Formula (19) can be applied to the entire text and will not be repeated here.

[0066] See also Figures 1 to 8 As shown, Figure 1 This is a schematic diagram of the method steps of a method for controlling vibration of a mobile flexible arm with non-uniform tension according to an embodiment of the invention. Figure 2 The schematic diagram of the operation of the mobile flexible arm is shown in FIG. Figure 3 This is the vibration diagram of the flexible arm without control of the present invention. Figure 4 is the angle position diagram of the flexible arm without control in the present invention, Figure 5 This is the motion position diagram of the trolley without control in the present invention. Figure 6 Vibration diagram of the flexible arm to which the present invention applies control, Figure 7 The angle position diagram of the flexible arm to which the present invention applies control, Figure 8 The present invention provides a method for controlling vibration of a mobile flexible arm with non-uniform tension, which includes:

[0067] Step S1, determining the kinetic energy of the mobile flexible arm system with non-uniform tension, the potential energy of the mobile flexible arm system with non-uniform tension, and the virtual work of the mobile flexible arm system with non-uniform tension, and establishing a mobile flexible arm system model with non-uniform tension in combination with the Hamiltonian principle;

[0068] Step S2, for the mobile flexible arm system model with non-uniform tension, construct a boundary vibration control algorithm based on the obstacle Lyapunov function to ensure that the boundary tension constraint condition of the flexible arm is satisfied, wherein the boundary vibration control algorithm includes the control force exerted by the flexible arm hub, the control force exerted by the trolley, and the control force exerted by the end load;

[0069] Step S3, constructing a control-specific Lyapunov function J(t) based on the boundary vibration control algorithm to determine the stability of the mobile flexible arm system model, wherein if the Lyapunov function J(t) is positive definite and If negative, the system model is judged to be stable.

[0070] Step S4, performing numerical simulation verification on the algorithm, obtaining a numerical simulation diagram, for determining the stability of the trolley based on the rotation angle of the flexible arm and the motion displacement of the trolley in the numerical simulation diagram, and verifying the availability of the control algorithm.

[0071] Specifically, the Hamiltonian principle is a basic principle in physics, which is used to describe the motion laws of physical systems. It indicates that among all possible motion paths, the actual motion path of the physical system is the path that makes the action take an extreme value (usually a minimum value). This is a prior art and will not be elaborated on.

[0072] Specifically, the barrier Lyapunov function is a scalar function designed to ensure that the state of the system always remains within a certain safety set, thereby preventing the system state from reaching undesirable boundaries or violating constraints. This is a prior art and will not be described in detail.

[0073] Specifically, there is no limitation on the method used for numerical simulation verification. In this embodiment, the boundary vibration control algorithm is numerically simulated and verified by MATLAB. Of course, Cadence Incisive digital logic design simulator, Synopsys VCS simulation tool and OpenFOAM software package can also be used. Those skilled in the art can make a choice according to the actual situation, which will not be elaborated here.

[0074] Specifically, in the implementation, the constraint problem of the boundary tension constraint condition g(L, t) of the flexible arm is equivalent to the constraint problem of q'(L, t), and M>0 is set as the constraint value of q'(L, t), that is, |q'(L, t)|<M. Formula (20) shows that g(L, t) is bounded.

[0075] g(L,t)=r(L)+h(L)[q'(L,t)] 2 (20)

[0076] In formula (20), r(L) represents the initial tension of the flexible arm with a length of L, h(L) represents the nonlinear elastic modulus of the flexible arm with a length of L, and q(L, t) represents the elastic deformation function at the end of the robotic arm with respect to the flexible arm length L and time t.

[0077] Specifically, the kinetic energy of the mobile flexible arm system with non-uniform tension is expressed by formula (1),

[0078]

[0079] In formula (1), F m (t) represents the kinetic energy of the mobile flexible arm system with non-uniform tension at time t, t represents time, ρ represents the uniform mass per unit length of the flexible arm, L represents the length of the flexible arm, w represents the spatial position, K(w, t) represents the absolute displacement function of the flexible arm with respect to the spatial position w and time t, express The transpose of m s represents the mass of the end load, J represents the moment of inertia of the flexible arm, β(t) represents the rotation angle of the flexible arm at time t, m v represents the mass of the car, and s(t) represents the displacement of the car at time t.

[0080] Specifically, at time t∈[0,+∞), spatial position w∈[0,L], the absolute displacement function K(w,t) of the flexible arm is expressed by formula (1-1):

[0081]

[0082] In formula (1-1), q(w, t) represents the elastic deformation function of the flexible arm with respect to spatial position w and time t.

[0083] Specifically, the potential energy of the mobile flexible arm system with non-uniform tension is expressed by formula (2):

[0084]

[0085] In formula (2), F n (t) represents the potential energy of the mobile flexible arm system with non-uniform tension at time t, EI represents the bending stiffness of the flexible arm, q(w, t) represents the elastic deformation function of the flexible arm with respect to spatial position w and time t, and g(w, t) represents the non-uniform tension function of the flexible arm that changes with time and space with respect to spatial position w and time t.

[0086] Specifically, the non-uniform tension function g(w, t) of the flexible arm that changes with time and space is expressed by formula (2-1):

[0087] g(w,t)=r(w)+h(w)[q'(w,t)] 2 (2-1)

[0088] In formula (2-1), r(w) represents the initial tension of the flexible arm at the spatial position w, and h(w) represents the nonlinear elastic modulus at the spatial position w.

[0089] Specifically, r(w)>0, h(w)≥0,

[0090] Specifically, the virtual work of the mobile flexible arm system with non-uniform tension is expressed by formula (3):

[0091] F v (t) = [d x (t)+u x (t)]β(t)+[d y (t)+u y (t)]s(t)+[d z (t)+u z (t)]q(L,t) (3)

[0092] In formula (3), F v (t) represents the virtual work of the mobile flexible arm system with non-uniform tension at time t, d x(t) represents the time-varying external disturbance to the flexible arm hub, u x (t) represents the control force exerted by the flexible arm hub, d y (t) represents the external disturbance to the car that changes with time, u y (t) represents the control force exerted by the car, d z (t) represents the external disturbance to the end load that changes with time, u z (t) represents the control force exerted by the end load, and q(L, t) represents the elastic deformation function at the end of the robot arm with respect to the length L of the flexible arm and time t.

[0093] Specifically, the model of the mobile flexible arm system with non-uniform tension includes equations (4) to (8),

[0094] q(0,t)=q'(0,t)=q″(L,t)=0 (4)

[0095]

[0096] In formulas (4) to (8), q represents elastic deformation, s represents the displacement of the trolley, r(L) represents the initial tension of the flexible arm with a length of L, r represents the initial tension, and h(L) represents the nonlinear elastic modulus of the flexible arm with a length of L.

[0097] Specifically, the control force exerted by the flexible arm hub is expressed by formula (9):

[0098]

[0099] In formula (9), k1 represents the first auxiliary variable control parameter, φ a (t) represents the first auxiliary variable, ξ1 represents the control parameter of the flexible arm rotation angle, λ represents the control parameter, β d represents the predetermined position of the flexible arm rotation angle, sgn(·) represents the sign function, d xm represents the external disturbance d that varies with time x The upper bound of (t).

[0100] Specifically, the auxiliary variable φ a (t) is expressed by formula (9-1),

[0101]

[0102] The control force exerted by the car is expressed by formula (10):

[0103]

[0104] In formula (10), k2 represents the second auxiliary variable control parameter, φ b (t) represents the second auxiliary variable, ξ2 represents the displacement control parameter of the car motion, s d represents the predetermined position of the vehicle's motion displacement, d ym represents the external disturbance d that varies with time y The upper bound of (t).

[0105] Specifically, the auxiliary variable φ b (t) is expressed by formula (10-1),

[0106]

[0107] The control force exerted by the end load is expressed by formula (11):

[0108]

[0109] In formula (11), k3 represents the control parameter of the elastic deformation change speed of the flexible arm end, ξ3 represents the control parameter of the elastic deformation of the end, In(·) represents the logarithmic function, M represents the constraint value, d zm represents the external disturbance d that varies with time Z The upper bound of (t).

[0110] Specifically, the present invention constructs a boundary vibration control algorithm for a mobile flexible arm system model with non-uniform tension, so as to control the tension of the mobile flexible arm to keep it stable. In actual situations, the tension is seldom taken into consideration when controlling the mobile flexible arm. However, if the tension of the arm rod, especially the tension at the connection point with the end load, is not controlled, it will cause the mobile flexible arm to vibrate, affecting the work efficiency. Based on this, it is considered to construct a boundary vibration control algorithm, and the tension of the flexible arm is controlled from three aspects: the control force exerted by the flexible arm hub, the control force exerted by the trolley and the control force exerted by the end load, thereby reducing the influence of the tension on the flexible arm, especially at the connection point with the end load, and improving the control performance and stability of the mobile flexible arm.

[0111] Specifically, the control-specific Lyapunov function J(t) is expressed by formula (12):

[0112] J(t)=J a (t)+J b (t)+J c (t) (12)

[0113] In formula (12), J(t) represents the control-specific Lyapunov function, J a (t) represents the Lyapunov function of the energy term, J b(t) represents the Lyapunov function of the cross term, J c (t) represents the Lyapunov function with respect to the auxiliary term

[0114] Specifically, J a (t) is expressed by formula (12-1),

[0115]

[0116] J b (t) is expressed by formula (12-2),

[0117]

[0118] J c (t) is expressed by formula (12-3),

[0119]

[0120] According to formula (12-4), J b (t) to scale up,

[0121]

[0122] In formula (12-4), δ1 represents the first control parameter, δ2 represents the second control parameter, σ1 represents the third control parameter, and σ2 represents the fourth control parameter.

[0123] Specifically, formula (12-5) represents the range of J(t) determined by combining the definition of the Lyapunov function,

[0124]

[0125] Formula (12-6) to Formula (12-9) are simultaneously valid to determine the values ​​of k1, k2, δ1, δ2, σ1, σ2, ξ1, ξ2, and λ to prove that the Lyapunov function J(t) is positive definite.

[0126]

[0127] Formula (12-10) represents the derivation of the Lyapunov function J(t),

[0128]

[0129] Formula (12-11) represents the a (t) is derived,

[0130]

[0131] Formula (12-12) represents the b (t) Derivative,

[0132]

[0133] Formula (12-13) represents the c (t) Derivative,

[0134]

[0135] Formula (12-14) is combined with formula (12-10) to formula (12-13), and substituted into boundary control algorithm formula (9) to formula (11) to calculate, to prove The negative definiteness of

[0136]

[0137] The process of verifying the usability of the control algorithm includes,

[0138] Obtaining a three-dimensional graph of the vibration of the flexible arm when control is applied to determine the amplitude of the flexible arm;

[0139] acquiring the rotation angle of the flexible arm when the control is applied to determine the angular stop position;

[0140] Obtain a two-dimensional map of the movement displacement of the trolley when the control is applied to determine the stopping position of the trolley;

[0141] If the amplitude of the flexible arm is less than or equal to the preset amplitude, the angle stop position is at the preset angle position and the trolley stop position is at the preset trolley position, it is determined that the control algorithm is feasible.

[0142] It can be understood that the preset amplitude, preset angle position and preset trolley position are the amplitude, angle position and trolley position corresponding to the ideal operating conditions, and those skilled in the art can set them by themselves, which will not be elaborated here.

[0143] Specifically, the present invention verifies the usability of the boundary vibration control algorithm from two aspects: mathematical analysis and numerical simulation. In terms of mathematical analysis and calculation, the present invention constructs a control-specific Lyapunov function J(t) based on the boundary vibration control algorithm, determines the Lyapunov function J(t) and If the Lyapunov function J(t) is positive definite and If the value of the boundary vibration control algorithm is negative, the system model is judged to be stable. In terms of numerical simulation, the present invention verifies the stability of the mobile flexible arm system model with non-uniform tension by numerical simulation through MATLAB. If both aspects indicate that the system model is stable, it means that the boundary vibration control algorithm is available. By using the boundary vibration control algorithm, the influence of the tension on the flexible arm, especially at the connection point with the end load, is reduced, thereby improving the control performance and stability of the mobile flexible arm.

[0144] So far, the technical solutions of the present invention have been described in conjunction with the preferred embodiments shown in the accompanying drawings. However, it is easy for those skilled in the art to understand that the protection scope of the present invention is obviously not limited to these specific embodiments. Without departing from the principle of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will fall within the protection scope of the present invention.

Claims

1. A method for controlling vibration of a mobile flexible arm with non-uniform tension, characterized in that: include: Determine the kinetic energy of the mobile flexible arm system with non-uniform tension, the potential energy of the mobile flexible arm system with non-uniform tension, and the virtual work of the mobile flexible arm system with non-uniform tension, and establish a mobile flexible arm system model with non-uniform tension in combination with the Hamiltonian principle; For the mobile flexible arm system model with non-uniform tension, a boundary vibration control algorithm based on the obstacle Lyapunov function is constructed to ensure that the boundary tension constraint condition of the flexible arm is satisfied, wherein the boundary vibration control algorithm includes the control force exerted by the flexible arm hub, the control force exerted by the trolley, and the control force exerted by the end load; Based on the boundary vibration control algorithm, a control-specific Lyapunov function J(t) is constructed to determine the stability of the mobile flexible arm system model. If the Lyapunov function J(t) is positive definite and If negative, the system model is judged to be stable; The algorithm is numerically simulated and verified to obtain a numerical simulation diagram, which is used to determine the stability of the trolley based on the rotation angle of the flexible arm and the motion displacement of the trolley in the numerical simulation diagram, and to verify the feasibility of the control algorithm.

2. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 1, characterized in that: The kinetic energy of the mobile flexible arm system with non-uniform tension is expressed by formula (1): In formula (1), F m (t) represents the kinetic energy of the mobile flexible arm system with non-uniform tension at time t, t represents time, ρ represents the uniform mass per unit length of the flexible arm, L represents the length of the flexible arm, w represents the spatial position, K(w, t) represents the absolute displacement function of the flexible arm with respect to the spatial position w and time t, express The transpose of m s represents the mass of the end load, J represents the moment of inertia of the flexible arm, β(t) represents the rotation angle of the flexible arm at time t, m v represents the mass of the car, and s(t) represents the displacement of the car at time t.

3. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 2, characterized in that: The potential energy of the mobile flexible arm system with non-uniform tension is expressed by formula (2): In formula (2), F n (t) represents the potential energy of the mobile flexible arm system with non-uniform tension at time t, EI represents the bending stiffness of the flexible arm, q(w, t) represents the elastic deformation function of the flexible arm with respect to spatial position w and time t, and g(w, t) represents the non-uniform tension function of the flexible arm that changes with time and space with respect to spatial position w and time t.

4. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 3, characterized in that: The virtual work of the mobile flexible arm system with non-uniform tension is expressed by formula (3): F v (t)=[d x (t)+u x (t)]β(t)+[d y (t)+u y (t)]s(t)+[d z (t)+u z (t)]q(L,t) (3) In formula (3), F v (t) represents the virtual work of the mobile flexible arm system with non-uniform tension at time t, d x (t) represents the time-varying external disturbance to the flexible arm hub, u x (t) represents the control force exerted by the flexible arm hub, d y (t) represents the external disturbance to the car that changes with time, u y (t) represents the control force exerted by the car, d z (t) represents the external disturbance to the end load that changes with time, u z (t) represents the control force exerted by the end load, and q(L, t) represents the elastic deformation function at the end of the robot arm with respect to the length L of the flexible arm and time t.

5. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 4, characterized in that: The mobile flexible arm system model with non-uniform tension includes formula (4) to formula (8), q(0,t)=q'(0,t)=q″(L,t)=0 (4) In formulas (4) to (8), q represents elastic deformation, s represents the displacement of the trolley, r(L) represents the initial tension of the flexible arm with a length of L, r represents the initial tension, and h(L) represents the nonlinear elastic modulus of the flexible arm with a length of L.

6. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 5, characterized in that: The control force exerted by the flexible arm hub is expressed by formula (9): In formula (9), k1 represents the first auxiliary variable control parameter, φ a (t) represents the first auxiliary variable, ξ1 represents the control parameter of the flexible arm rotation angle, λ represents the control parameter, β d represents the predetermined position of the flexible arm rotation angle, sgn(·) represents the sign function, d xm represents the external disturbance d that varies with time x The upper bound of (t).

7. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 6, characterized in that: The control force exerted by the car is expressed by formula (10): In formula (10), k2 represents the second auxiliary variable control parameter, φ b (t) represents the second auxiliary variable, ξ2 represents the displacement control parameter of the car motion, s d represents the predetermined position of the vehicle's motion displacement, d ym represents the external disturbance d that varies with time y The upper bound of (t).

8. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 7, characterized in that: The control force exerted by the end load is expressed by formula (11): In formula (11), k3 represents the elastic deformation change speed control parameter of the flexible arm end, ξ3 represents the elastic deformation control parameter of the flexible arm end, In(·) represents the logarithmic function, M represents the constraint value, d zm represents the external disturbance d that varies with time Z The upper bound of (t).

9. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 8, characterized in that: The control-specific Lyapunov function J(t) is expressed by formula (12): J(t)=J a (t)+J b (t)+J c (t) (12) In formula (12), J(t) represents the control-specific Lyapunov function, J a (t) represents the Lyapunov function of the energy term, J b (t) represents the Lyapunov function of the cross term, J c (t) represents the Lyapunov function with respect to the auxiliary term.

10. The vibration control method of a mobile flexible arm with non-uniform tension according to claim 1, characterized in that: The process of verifying the availability of the control algorithm includes: Obtaining a three-dimensional graph of the vibration of the flexible arm when control is applied to determine the amplitude of the flexible arm; acquiring the rotation angle of the flexible arm when the control is applied to determine the angular stop position; Obtain a two-dimensional map of the movement displacement of the trolley when the control is applied to determine the stopping position of the trolley; If the amplitude of the flexible arm is less than or equal to the preset amplitude, the angle stop position is at the preset angle position and the trolley stop position is at the preset trolley position, it is determined that the control algorithm is feasible.

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