A combined control method, system and electronic equipment for a flexible robotic arm

Abstract

This invention relates to a combined control method, system, and electronic device for a flexible robotic arm. The method includes: Step S1: Constructing a dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method; Step S2: Decoupling the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model; Step S3: Designing a trajectory tracking controller based on a novel reaching law for the slow-varying subsystem model; and simultaneously designing a vibration suppression controller for the fast-varying subsystem model; Step S4: Superimposing the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the combined control output, and applying the combined control output to the flexible robotic arm. This invention can effectively improve the control accuracy of the flexible robotic arm and suppress vibration.

Description

Technical Field

[0001] This invention relates to the field of flexible robotic arm control technology, and in particular to a combined control method, system and electronic device for a flexible robotic arm. Background Technology

[0002] Flexible robotic arms, with their advantages of being lightweight and efficient, are widely used in modern industry and aerospace. However, their modeling is complex and highly coupled. This strong coupling manifests as elastic vibrations occurring during movement. These elastic vibrations possess infinite-dimensional modes, and due to model uncertainties and the influence of unknown external disturbances, accurately establishing a coupled system model is difficult. Sliding mode control, with its strong robustness to external disturbances and uncertain parameters, can address the uncertainties and susceptibility to external disturbances in flexible robotic arm models to some extent. However, due to the switching characteristics of sliding mode control, the system state trajectory traverses both sides of the sliding surface, easily generating chattering. Therefore, effectively suppressing chattering is crucial for improving the control quality of sliding mode control, thereby enhancing the control accuracy of the flexible robotic arm.

[0003] During the movement of a flexible robotic arm, it is necessary to both track the trajectory and suppress vibration to ensure high precision in its control. Existing flexible arm control methods can be mainly divided into two types: one is to design a controller based on the coupled dynamics equations of the flexible robotic arm; the other method is to decouple the flexible arm model, considering trajectory tracking and vibration control separately, rather than relying on the rigid-flexible coupling characteristics of the flexible robotic arm and suppressing vibration through the coupling effect during movement. However, the existing methods involve complex controller design processes based on the coupled equations of the flexible robotic arm, and the vibration suppression effect is not ideal. Summary of the Invention

[0004] Therefore, the technical problem to be solved by the present invention is to overcome the problems of complex process and poor vibration suppression effect in the design of controllers for flexible robotic arms using coupling equations in the prior art.

[0005] To address the aforementioned technical problems, this invention provides a combined control method for a flexible robotic arm, comprising:

[0006] Step S1: Construct a dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method;

[0007] Step S2: Decouple the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model;

[0008] Step S3: Design a trajectory tracking controller based on a novel reaching law for the slow-changing subsystem model; simultaneously design a vibration suppression controller for the fast-changing subsystem model;

[0009] Step S4: Superimpose the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the output of the combined control, and apply the output of the combined control to the flexible robotic arm.

[0010] In one embodiment of the present invention, in step S1, a dynamic model of the flexible robotic arm is constructed based on the assumed modal method and the Lagrange method, and the formula is:

[0011]

[0012] in, It is a positive definite mass matrix. It is a cross-coupling matrix and , Here is the damping matrix. Here is the stiffness matrix. The coefficient matrix, For the flexible robotic arm to turn corners, and They are respectively The first and second derivatives, To describe the modal coordinates of the elastic vibration of the flexible robotic arm, and They are respectively The first and second derivatives, For system control input.

[0013] In one embodiment of the present invention, step S2 employs singular perturbation theory to decouple the dynamic model, obtaining a slowly varying subsystem model. The method includes:

[0014] Multiply both sides of the equation of the dynamic model by the left. Get about The transformation expression is expressed as:

[0015]

[0016] in, ;

[0017] Then regarding the above The transformation expansion yields:

[0018]

[0019]

[0020] in, for Matrix elements;

[0021] Based on singular perturbation theory, perturbation parameters are introduced into the dynamic model. , , Stiffness matrix The elements in the table introduce new state variables. , and bring in about formula and formula get:

[0022] Transformation Formula 1: ;

[0023] Transformation Formula 2: ;

[0024] Let the perturbation parameters From the transformation formula 2, we get:

[0025]

[0026] Among them, with subscript The variables are the components of the slowly varying subsystem model, and then the formula is... Substituting into the transformation formula 1, we obtain the slow-varying subsystem model, which is expressed as:

[0027]

[0028] in, , for matrix elements, for The estimated value, Input for the slow-varying subsystem model.

[0029] In one embodiment of the present invention, step S2 employs singular perturbation theory to decouple the dynamic model, obtaining a fast-changing subsystem model. The method includes:

[0030] because and If different time characteristics are present, then fast timescales are introduced. In the fast time marker Next, the slow variable and Treat it as a constant:

[0031] make Substituting into the transformation formula 2, we obtain the fast-changing subsystem model, which is expressed as:

[0032] .

[0033] In one embodiment of the present invention, the method for designing a trajectory tracking controller based on a novel reaching law for the slowly varying subsystem model in step S3 includes:

[0034] A new type of reaching law is constructed, with the following formula:

[0035]

[0036] in, For sliding surface, are system state variables and , , , , ;

[0037] Define the trajectory tracking error of the flexible robotic arm as:

[0038]

[0039] in, For the desired angular displacement, This represents the actual angular displacement.

[0040] make , We can obtain:

[0041]

[0042] The sliding surface in sliding mode control is defined as follows:

[0043]

[0044] in, ;

[0045] The trajectory tracking controller, constructed by combining the aforementioned slow-varying subsystem model, novel reaching law, and sliding surface, is expressed as follows:

[0046] .

[0047] In one embodiment of the present invention, the method for designing a vibration suppression controller for the rapidly changing subsystem model in step S3 includes:

[0048] According to the aforementioned slowly varying subsystem model, let , , ;

[0049] The state-space expression of the fast-changing subsystem model is then:

[0050]

[0051] Solving for the control quantity Make the objective function The smallest is represented as:

[0052]

[0053] in, , They are respectively and The weighted matrix, and , ;

[0054] According to the principle of extrema, make the objective function The smallest vibration suppression controller is:

[0055]

[0056] in, For optimal feedback gain, And satisfy .

[0057] In one embodiment of the present invention, in step S4, the outputs of the trajectory tracking controller and the vibration suppression controller are superimposed, as shown in the following formula:

[0058]

[0059] in, This is a trajectory tracking controller used to control the rigid motion of a flexible robotic arm. This is a vibration suppression controller used to control the elastic vibration of a flexible robotic arm.

[0060] To address the aforementioned technical problems, this invention provides a combined control system for a flexible robotic arm, comprising:

[0061] Module: Used to construct the dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method;

[0062] Decoupling module: used to decouple the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model;

[0063] Design module: used to design a trajectory tracking controller based on a novel reaching law for the slow-varying subsystem model; and also used to design a vibration suppression controller for the fast-varying subsystem model;

[0064] Superposition module: used to superimpose the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the output of the combined control, and apply the output of the combined control to the flexible robotic arm.

[0065] To address the aforementioned technical problems, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the combined control method for the flexible robotic arm described above.

[0066] To address the aforementioned technical problems, the present invention provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the combined control method for the flexible robotic arm described above.

[0067] The technical solution of the present invention has the following advantages compared with the prior art:

[0068] The novel approach law constructed in this invention can not only improve the system response speed, enabling the system to track the desired trajectory more quickly, but also effectively suppress chattering. This novel approach law improves the control quality of sliding mode control and enhances the system control accuracy.

[0069] For a slowly varying subsystem model, this invention designs a trajectory tracking controller based on a novel reaching law to achieve trajectory tracking; for a rapidly varying subsystem model, it designs a vibration suppression controller to achieve vibration suppression.

[0070] Compared with the pure sliding mode control method, the combined control method for flexible robotic arms of this invention can suppress the elastic vibration of the flexible arm more quickly.

[0071] This invention is highly practical, can effectively improve the control accuracy of flexible arms, and can be widely applied in practice. Attached Figure Description

[0072] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0073] Figure 1 This is a flowchart of the method of the present invention;

[0074] Figure 2 This is a schematic diagram comparing the hyperbolic tangent function image and the sign function image in an embodiment of the present invention;

[0075] Figure 3 This is a control block diagram of the flexible robotic arm assembly in an embodiment of the present invention;

[0076] Figure 4 This is a comparison diagram of the rotation angle curves of the flexible robotic arm in the embodiments of the present invention;

[0077] Figure 5 This is a control input curve diagram of the combined control in an embodiment of the present invention;

[0078] Figure 6 This is a control input curve diagram of pure sliding mode control in an embodiment of the present invention;

[0079] Figure 7 This is a diagram showing the end-effector vibration curve of the flexible robotic arm under combined control in an embodiment of the present invention.

[0080] Figure 8 This is a diagram showing the end-effector vibration curve of the flexible robotic arm under pure sliding mode control in an embodiment of the present invention.

[0081] Figure 9 This is a position tracking curve of the flexible robotic arm in an embodiment of the present invention;

[0082] Figure 10 This is a position tracking error curve of a flexible robotic arm in an embodiment of the present invention;

[0083] Figure 11 This is a control input curve diagram of combined control under applied torque in an embodiment of the present invention;

[0084] Figure 12 This is a control input curve diagram of pure sliding mode control under applied torque in an embodiment of the present invention;

[0085] Figure 13 This is a diagram showing the end-effector vibration curve of the flexible robotic arm under combined control in an embodiment of the present invention.

[0086] Figure 14 This is a diagram showing the end-effector vibration curve of the flexible robotic arm under sliding mode control in an embodiment of the present invention. Detailed Implementation

[0087] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0088] Example 1

[0089] Reference Figure 1 As shown, the present invention relates to a combined control method for a flexible robotic arm, comprising:

[0090] Step S1: Construct a dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method;

[0091] Step S2: Decouple the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model;

[0092] Step S3: Design a trajectory tracking controller based on a novel reaching law for the slow-changing subsystem model; simultaneously design a vibration suppression controller for the fast-changing subsystem model;

[0093] Step S4: Superimpose the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the output of the combined control, and apply the output of the combined control to the flexible robotic arm.

[0094] The following is a detailed description of this embodiment:

[0095] Step S1 includes:

[0096] The dynamic model of the flexible robotic arm is constructed based on the hypothetical modal method and the Lagrange method, and the formula is as follows:

[0097] (1)

[0098] in, It is a positive definite mass matrix. It is a cross-coupling matrix and , Here is the damping matrix. Here is the stiffness matrix. The coefficient matrix, For the flexible robotic arm to turn corners, and They are respectively Regarding time The first and second derivatives, To describe the modal coordinates of the elastic vibration of the flexible robotic arm, and They are respectively Regarding time The first and second derivatives, For system control input.

[0099] Step S2 includes:

[0100] because Zhengding, record Multiplying both sides of equation (1) by the left side simultaneously, we get:

[0101] (2)

[0102] Expanding equation (2) yields:

[0103] (3)

[0104] (4)

[0105] in, for Matrix elements.

[0106] Based on singular perturbation theory, perturbation parameters are introduced into the model. , , Stiffness matrix The elements in the table introduce new state variables. , Substituting these equations into equations (3) and (4), we get:

[0107] (5)

[0108] (6)

[0109] Let the perturbation parameters From equation (6), we get:

[0110] (7)

[0111] Among them, including The variables are the components of the slow-varying subsystem model. Substituting equation (7) into equation (5) yields the slow-varying subsystem model, which is expressed as:

[0112] (8)

[0113] in, , For matrix elements, for The estimated value, Input for the slow-varying subsystem model.

[0114] because and If different time characteristics are present, then fast timescales are introduced. In the fast time marker Next, treating the slow variable as a constant, we obtain...

[0115] (9) Order Substituting into the transformation formula 2, we obtain the fast-changing subsystem model, which is expressed as:

[0116] (10)

[0117] According to Tikhonov's theory, we have:

[0118] (11)

[0119] (12)

[0120] in, for The higher-order infinitesimals. The decoupled subsystem model is simplified compared to the original system model, making controller design easier.

[0121] The flexible arm system is decoupled using the singular perturbation method, resulting in two subsystems, each with its own controller. For trajectory tracking, a sliding mode control based on a novel reaching law is used to design the trajectory tracking controller. For elastic vibration, a vibration suppression controller is designed based on optimal control theory. The combined control input for the flexible arm system is obtained by superimposing the control signals from these two controllers.

[0122] Step S3 includes:

[0123] The novel reaching law in this embodiment is based on the power reaching law, and the formula for the power reaching law is:

[0124]

[0125] The novel reaching law, based on the power-law reaching law, adds a variable-speed reaching term, introduces a system state variable, and dynamically improves the reaching effect. It also adds an exponential term to increase the reaching speed. To further eliminate chattering, it employs... replace The novel approach law design is as follows:

[0126] (13)

[0127] in, For sliding surface, are system state variables and , , , , .

[0128] The novel reaching law in this embodiment includes power-law reaching terms, variable-speed reaching terms, and exponential reaching terms. At this time, the variable-speed approach term and the exponential approach term play a dominant role. The last two terms of the reaching law are equivalent to the exponential reaching law, with a higher reaching rate; when At this time, the power-law approaching term and the variable-speed approaching term play a dominant role. ,Right now exist The surrounding area converges to The addition of the variable-speed approaching term not only increases the approaching rate, causing the system state to converge to the origin more quickly, but also, when approaching the sliding surface, the approaching rate increases with the magnitude of the system state variable. Reducing the approach speed can avoid chattering caused by excessively fast approach velocity. The new approach law employs... replace The graphs of the two functions are as follows: Figure 2 As shown in the figure. It can be seen from the figure that... The changes were gradual, and Compared to the absence of mutations, therefore use It can suppress chattering to a certain extent, resulting in better controller performance and a more stable system state.

[0129] Stability analysis

[0130] Constructing Lyapunov functions:

[0131] (14)

[0132] Based on the fundamental principles of sliding mode control, and verifying the reachability condition, the derivative of the Lyapunov function is obtained as follows:

[0133] (15)

[0134] If and only if hour, Therefore, the novel reaching law satisfies the sliding mode reachability condition, and the system is stable.

[0135] The trajectory tracking controller (sliding mode controller) is designed as follows:

[0136] Define the trajectory tracking error of the flexible robotic arm as:

[0137] (16)

[0138] in, For the desired angular displacement, This represents the actual angular displacement.

[0139] make , We can obtain:

[0140] (17)

[0141] The sliding surface in sliding mode control is defined as follows:

[0142] (18)

[0143] in, .

[0144] Combining equations (8), (13), and (18), we obtain the trajectory tracking controller, which is expressed as:

[0145] (19)

[0146] The vibration suppression controller (optimal controller) is designed as follows:

[0147] According to the aforementioned slowly varying subsystem model, let , , ;

[0148] Then the state-space expression of equation (10) is:

[0149] (20)

[0150] because The controllable, rapidly changing subsystem can be considered a linear time-invariant system, therefore, the LQR (Low-Quickness Reduction) method can be used to design a vibration suppression controller. The LQR solution has a unified analytical expression, and optimal closed-loop control of the system can be achieved through linear state feedback.

[0151] Solving for the control quantity Make the objective function The smallest is represented as:

[0152] (twenty one)

[0153] in, , They are respectively and The weighted matrix, and , .

[0154] According to the principle of extrema, make the objective function The smallest vibration suppression controller is:

[0155] (twenty two)

[0156] in, For optimal feedback gain, And it satisfies the Ricatti equation:

[0157] (twenty three)

[0158] therefore, The design is reduced to the Ricatti equation. Solve for it.

[0159] Step S4 includes:

[0160] For the decoupled subsystem model, a trajectory tracking controller and a vibration suppression controller are designed separately. The control output obtained by combining the two controllers is:

[0161] (twenty four)

[0162] in, This is a trajectory tracking controller used to control the rigid motion of a flexible robotic arm. This is a vibration suppression controller used to control the elastic vibration of a flexible robotic arm. Figure 3 This is a block diagram of the combined control of a flexible robotic arm.

[0163] According to singular perturbation theory and Tikhonov's theorem, the state variables of the original system can be used for the controller design of the subsystems obtained after decoupling, and the combined control of the two subsystems can make the closed-loop stability of the original system stable.

[0164] To verify the performance of this invention, simulations were performed in MATLAB. The specific experimental analysis is as follows:

[0165] The parameters for the combined control of the novel reaching law are set as follows: , , , , , , , The parameters for pure sliding mode control with power-sum approach are set as follows: , , Taking the first vibration mode is sufficient to meet the accuracy requirements.

[0166] Figure 4 The graph shows the rotation angle curve of the flexible robotic arm. As can be seen from the graph, the combined control based on the novel approach law (this invention) enables the flexible robotic arm to reach the target position faster than pure sliding mode control, and its response speed is also faster than pure sliding mode control. Figure 5 and Figure 6 The figures show the control input curves for combined control (the present invention) and pure sliding mode control, respectively. As can be seen from the figures, the combined control input based on the novel reaching law can effectively suppress chattering compared to pure sliding mode control. Figure 7 and Figure 8 The figures show the end-effector vibration curves of the flexible robotic arm under combined control (the present invention) and pure sliding mode control, respectively. As can be seen from the figures, the proposed combined control based on the novel reaching law can effectively suppress its elastic vibration.

[0167] In simulation experiment 2, an application was made to the flexible robotic arm. The torque, the initial state of the flexible robotic arm is , Simulation results are as follows Figures 9 to 14 As shown.

[0168] Figure 9 and Figure 10The figures show the position tracking curve and tracking error curve of the flexible robotic arm. As can be seen from the figures, the combined control (this invention) can track the desired trajectory faster and has no fluctuations compared with pure sliding mode control. Figure 11 and Figure 12 The figures show the control input curves of the combined control and pure sliding mode control systems, respectively. As can be seen from the figures, the combined control based on the novel reaching law (this invention) can effectively suppress chattering compared to pure sliding mode control. Figure 13 and Figure 14 The figures show the end-effector vibration curves of the flexible robotic arm under combined control and pure sliding mode control, respectively. As can be seen from the figures, the combined control method based on the novel reaching law (this invention) can quickly suppress its elastic vibration.

[0169] Compared to power-law approach laws, the novel approach law of this invention not only improves the system response speed, enabling the system to track the desired trajectory more quickly, but also effectively suppresses chattering. This novel approach law improves the control quality of sliding mode control and enhances the system control accuracy. Furthermore, compared to pure sliding mode control, the combined control method of this invention can suppress the elastic vibration of the flexible arm more quickly.

[0170] Example 2

[0171] This embodiment provides a combined control system for a flexible robotic arm, including:

[0172] Module: Used to construct the dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method;

[0173] Decoupling module: used to decouple the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model;

[0174] Design module: used to design a trajectory tracking controller based on a novel reaching law for the slow-varying subsystem model; and also used to design a vibration suppression controller for the fast-varying subsystem model;

[0175] Superposition module: used to superimpose the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the output of the combined control, and apply the output of the combined control to the flexible robotic arm.

[0176] Example 3

[0177] This embodiment provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the combined control method for the flexible robotic arm described in Embodiment 1.

[0178] Example 4

[0179] This embodiment provides a computer-readable storage medium storing a computer program thereon. When the computer program is executed by a processor, it implements the steps of the combined control method for the flexible robotic arm described in Embodiment 1.

[0180] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code. The solutions in the embodiments of this application can be implemented in various computer languages, such as the object-oriented programming language Java and the interpreted scripting language JavaScript.

[0181] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0182] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0183] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0184] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0185] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A combined control method for a flexible robotic arm, characterized in that: include: Step S1: Construct a dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method; Step S2: Decouple the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model; Step S3: Design a trajectory tracking controller based on a novel reaching law for the slow-changing subsystem model; simultaneously design a vibration suppression controller for the fast-changing subsystem model; The novel reaching law formula is as follows: Where S is the sliding surface, and x is the system state variable and k1>0, 0<α<1, k2>0, k3>0; Step S4: Superimpose the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the output of the combined control, and apply the output of the combined control to the flexible robotic arm.

2. The combined control method for the flexible robotic arm according to claim 1, characterized in that: In step S1, a dynamic model of the flexible robotic arm is constructed based on the assumed modal method and the Lagrange method, and the formula is: Where M is the positive definite mass matrix, and H is the cross-coupling matrix and C is the damping matrix, K is the stiffness matrix, F is the coefficient matrix, and θ is the rotation angle of the flexible robotic arm. and Take the first and second derivatives of θ, respectively, and q = [q1, q2, ..., q]. n ] T To describe the modal coordinates of the elastic vibration of the flexible robotic arm, and These are the first and second derivatives of q, respectively, and u is the system control input.

3. The combined control method for the flexible robotic arm according to claim 2, characterized in that: In step S2, singular perturbation theory is used to decouple the dynamic model to obtain a slowly varying subsystem model. The method includes: Multiplying both sides of the dynamic model's formula by N on the left yields a transformation expression for N, which is expressed as: Where N = M -1 ; Expanding the transformation expression for N, we get: Where, N 11 N 12 N 21 N 22 Let N be the matrix elements; Based on singular perturbation theory, perturbation parameters are introduced into the dynamic model. λ=min{K i }, i = 1, 2, ..., n, K i For each element in the stiffness matrix K, a new state variable K is introduced. s =μ 2 K, z = q / μ 2 And substitute it into the formula about N. and formula get: Transformation Formula 1: Transformation Formula 2: Setting the perturbation parameter μ = 0, we get from the transformation formula 2: Where the variable with subscript 's' represents the component of the slowly varying subsystem model, and then the formula z... s Substituting into the transformation formula 1, we obtain the slow-varying subsystem model, which is expressed as: in, N 11s N 12s N 21s N 22s For N s matrix elements, u is an estimate of θ. s Input for the slow-varying subsystem model.

4. The combined control method for the flexible robotic arm according to claim 3, characterized in that: In step S2, singular perturbation theory is used to decouple the dynamic model to obtain a fast-changing subsystem model. The method includes: Since θ and q have different time characteristics, a fast timescale t is introduced. p = t / μ, in the fast timescale t p =t / μ, the slow variable and z s Treat it as a constant: Let z f =zz s Substituting into the transformation formula 2, we obtain the fast-changing subsystem model, which is expressed as:

5. The combined control method for the flexible robotic arm according to claim 4, characterized in that: In step S3, the trajectory tracking controller based on a novel reaching law is designed for the slowly varying subsystem model. The method includes: Define the trajectory tracking error of the flexible robotic arm as: e(t)=θ d (t)-θ(t) Where, θ d θ(t) represents the desired angular displacement, and θ(t) represents the actual angular displacement. Let x1 = e(t), We can obtain: The sliding surface in sliding mode control is defined as follows: Where c > 0; The trajectory tracking controller, constructed by combining the aforementioned slow-varying subsystem model, novel reaching law, and sliding surface, is expressed as follows:

6. The combined control method for the flexible robotic arm according to claim 5, characterized in that: The method for designing a vibration suppression controller for the rapidly changing subsystem model in step S3 includes: According to the aforementioned slowly varying subsystem model, let The state-space expression of the fast-changing subsystem model is then: Solve for the control quantity u f Minimizing the objective function J is expressed as: Where Q and R are x and u, respectively. f The weighted matrix, and Q = Q T ≥0, R=R T ≥0; According to the principle of extrema, the vibration suppression controller that minimizes the objective function J is: breast f =-Kx=-R -1 BPx Where K is the optimal feedback gain, and P = P T >0 and satisfy PA+A T P-PBR -1 B T P+Q=0.

7. The combined control method for the flexible robotic arm according to claim 6, characterized in that: In step S4, the outputs of the trajectory tracking controller and the vibration suppression controller are superimposed, as shown in the following formula: u(t)=u s (t)+u f (t) Among them, u s (t) is the trajectory tracking controller, used to control the rigid motion of the flexible robotic arm, u f (t) is a vibration suppression controller used to control the elastic vibration of the flexible robotic arm.

8. A combined control system for a flexible robotic arm, used to implement the combined control method for the flexible robotic arm as described in any one of claims 1 to 7, characterized in that: include: Module: Used to construct the dynamic model of the flexible robotic arm based on the hypothetical modal method and the Lagrange method; Decoupling module: used to decouple the dynamic model using singular perturbation theory to obtain a slow-varying subsystem model and a fast-varying subsystem model; Design module: used to design a trajectory tracking controller based on a novel reaching law for the slow-varying subsystem model; and also used to design a vibration suppression controller for the fast-varying subsystem model; Superposition module: used to superimpose the outputs of the trajectory tracking controller and the vibration suppression controller to obtain the output of the combined control, and apply the output of the combined control to the flexible robotic arm.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, it implements the steps of the combined control method for the flexible robotic arm as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by the processor, it implements the steps of the combined control method for the flexible robotic arm as described in any one of claims 1 to 7.

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