Flexible robotic arm control method and system based on finite element discretization and modal reduction

Abstract

The present invention discloses a flexible manipulator control method and system based on finite element discretization and modal order reduction, including: dividing the flexible manipulator connecting rod into a plurality of units; constructing the motion control equation of any unit of the flexible manipulator after discretization; establishing the finite element motion control equation of each unit, assembling the finite element motion control equation of each unit to obtain the global dynamic finite element equation of the flexible manipulator; solving the generalized eigenvalue problem of the global dynamic finite element equation to obtain the modal vibration shapes of each order of the flexible manipulator; performing order reduction transformation on the global dynamic finite element equation based on the modal vibration shapes of each order to obtain the reduced global dynamic finite element equation; and performing motion control on the flexible single-rod manipulator of the robot based on the global dynamic finite element equation. The present invention can realize the effective suppression and precise control of the vibration of the flexible manipulator.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible robotic arm control, and in particular to a flexible robotic arm control method and system based on finite element discretization and modal reduction. Background Art

[0002] The statements in this section merely provide background information related to the present invention and do not necessarily constitute prior art.

[0003] Robotic arms are widely used in the fields of intelligent manufacturing and intelligent construction equipment, and can effectively replace manual repetitive operations, thereby improving production efficiency and stability. As robotic arms develop towards high speed and light weight, a large number of robotic arms with long arm spans and light weight have been widely used. Such robotic arms often show a certain degree of flexibility due to insufficient connecting rod and joint stiffness, so they are also called flexible robotic arms. The elastic deformation and vibration generated by the flexible robotic arm during operation will reduce the positioning accuracy and trajectory accuracy of the robotic arm, especially when the load inertia is large and the operating speed is high, making it difficult to perform tasks with high precision requirements.

[0004] In order to improve the accuracy of flexible robotic arms and reduce the adverse effects of their elastic deformation and vibration, academia and industry have conducted in-depth research on the vibration control of flexible robotic arms and proposed a variety of elastic deformation and vibration suppression methods. Among these existing methods, the construction of the dynamic model of the flexible robotic arm plays an important role, which can be used for robotic arm motion trajectory planning, structural design and other tasks to achieve the suppression of robotic arm vibration.

[0005] At present, the assumed modal method and the finite element method are two commonly used methods in the field of flexible robotic arm dynamics modeling. However, the assumed modal method has limitations on the geometric shape and boundary conditions of the flexible robotic arm, and the dynamic model established by the finite element method has more degrees of freedom, which is not conducive to the real-time control of the flexible robotic arm based on the model. Summary of the invention

[0006] In order to solve the above problems, the present invention proposes a flexible robotic arm control method and system based on finite element discretization and modal reduction. Through the combination of finite element method and assumed modal method, a flexible robotic arm dynamic model with a wide range of applications and high solution efficiency is constructed, which can be used for model-based velocity / acceleration feedback control of the robotic arm, thereby achieving effective suppression and precise control of the vibration of the flexible single-rod robotic arm.

[0007] In some embodiments, the following technical solutions are adopted:

[0008] A flexible manipulator control method based on finite element discretization and modal reduction, comprising:

[0009] The flexible manipulator is spatially discretized using finite element grids, and the connecting rods of the flexible manipulator are divided into several units.

[0010] The Hamiltonian equation is used to construct the motion control equation of any unit of the discretized flexible manipulator.

[0011] The shape functions of beams and rods are used to interpolate the lateral and axial deformations of the flexible manipulator, respectively, to obtain the finite element matrix of the unit, and to establish the finite element motion control equations of each unit;

[0012] The finite element motion control equations of each unit are assembled to obtain the global dynamic finite element equation of the flexible manipulator;

[0013] Ignore the nonlinear terms in the global dynamics finite element equation, solve the generalized eigenvalue problem of the global dynamics finite element equation, and obtain the modal vibration shapes of each order of the flexible manipulator;

[0014] Based on the modal vibration shapes of each order, the global dynamic finite element equation is reduced to obtain the reduced-order global dynamic finite element equation;

[0015] The motion of the flexible single-rod manipulator of the robot is controlled based on the global dynamic finite element equation.

[0016] In other embodiments, the following technical solutions are adopted:

[0017] A flexible manipulator control system based on finite element discretization and modal reduction, comprising:

[0018] A unit division module is used to spatially discretize the flexible manipulator using a finite element grid and divide the connecting rod of the flexible manipulator into a number of units;

[0019] The motion control equation construction module is used to construct the motion control equation of any unit of the discretized flexible manipulator using the Hamiltonian equation;

[0020] The motion control equation discretization module is used to interpolate the lateral and axial deformations of the flexible manipulator using the shape functions of the beam and the rod, respectively, to obtain the finite element matrix of the unit, and to establish the finite element motion control equations of each unit;

[0021] The global dynamics finite element equation building module is used to assemble the finite element motion control equations of each unit to obtain the global dynamics finite element equations of the flexible manipulator;

[0022] A global dynamics finite element equation solving module is used to ignore the nonlinear terms in the global dynamics finite element equation, solve the generalized eigenvalue problem of the global dynamics finite element equation, and obtain the modal vibration shapes of each order of the flexible manipulator; based on the modal vibration shapes of each order, the global dynamics finite element equation is reduced in order to obtain the reduced-order global dynamics finite element equation;

[0023] The robot arm control module is used to control the motion of the robot flexible single-rod robot arm based on the global dynamic finite element equation.

[0024] In other embodiments, the following technical solutions are adopted:

[0025] A robot controller comprises a processor and a memory, wherein the processor is used to implement instructions; the memory is used to store a plurality of instructions, wherein the instructions are suitable for being loaded by the processor and executing the above-mentioned flexible robot arm control method based on finite element discretization and modal reduction.

[0026] In other embodiments, the following technical solutions are adopted:

[0027] A robot adopts the flexible robotic arm control method based on finite element discretization and modal order reduction to realize motion control of a single-rod flexible robotic arm.

[0028] Compared with the prior art, the present invention has the following beneficial effects:

[0029] (1) The present invention effectively integrates the advantages of the finite element method and the assumed modal method in the field of flexible robotic arm modeling. That is, similar to the finite element method, it can be applied to flexible robotic arms of various complex shapes. At the same time, similar to the assumed modal method, it can establish dynamic control equations with smaller dimensions, which is conducive to the realization of real-time control of the flexible robotic arm.

[0030] (2) The present invention can obtain the global modal vibration shape of the flexible robotic arm and obtain a reduced-order robotic arm dynamics model. By embedding the robotic arm dynamics model into the controller as a feedforward signal, it can provide guidance for the suppression of vibration of the flexible robotic arm, path planning, speed / acceleration planning, etc., thereby achieving effective suppression and precise control of the vibration of the flexible robotic arm.

[0031] Other features and advantages of additional aspects of the present invention will be given in part in the following description, and in part will become obvious from the following description, or will be learned through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 Schematic diagram of a single-rod flexible mechanical arm model and its deformation in an embodiment of the present invention;

[0033] Figure 2is the change in the time domain of the driving torque of the single-rod flexible mechanical arm in the embodiment of the present invention;

[0034] Figure 3 is a flow chart of a flexible robotic arm control method based on finite element discretization and modal order reduction in an embodiment of the present invention;

[0035] Figure 4 (a)-(f) are the first 6 non-zero global modes of the single-rod flexible manipulator in an embodiment of the present invention;

[0036] Figure 5 This is an example of the robot control structure in the embodiment of the present invention;

[0037] Figure 6 and Figure 7 They are curve diagrams of the elastic deformation and rotation angle of the end of the single-rod flexible robotic arm driven by a motor in the embodiment of the present invention in the time domain. DETAILED DESCRIPTION

[0038] It should be noted that the following detailed descriptions are illustrative and are intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in the present invention have the same meanings as those commonly understood by those skilled in the art to which the present application belongs.

[0039] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0040] Embodiment 1

[0041] Figure 1 A single-rod flexible manipulator model with a curved profile is presented. The material density of the manipulator is 7800 kg / m 3 , elastic modulus is 210 GPa, cross section is a hollow circle with outer diameter R1 = 10 mm, R2 = 5 mm, span is 1 m, profile curve is y = 0.1 sinπx, proximal hub diameter is 5 cm, moment of inertia is 1×10 -3 kg·m 2 The motor rotor moment of inertia is 1×10 -5 kg·m 2 , the reduction ratio is 100, the torsional joint stiffness is 800N·m / rad, the end load mass is 1kg, and the moment of inertia relative to the center of mass is 5×10 -3 kg·m 2, the driving torque of the motor is Figure 2 shown.

[0042] Based on this, in one or more embodiments, a flexible manipulator control method based on finite element discretization and modal reduction is disclosed, combined with Figure 3 , specifically including the following process:

[0043] S101: Use finite element grids to spatially discretize the flexible robotic arm and divide the flexible robotic arm connecting rod into several units; use Hamiltonian equations to construct the motion control equations of any unit of the discretized flexible robotic arm.

[0044] Specifically, Figure 1 The single-rod robot link in the figure is divided into several units. Based on the basic assumption of small deformation of the flexible robot, the floating frame method and Hamilton principle are used to establish the motion control equation of the flexible robot of any unit:

[0045]

[0046]

[0047]

[0048] Where ρ is the material density of the connecting rod, A is the cross-sectional area of ​​the connecting rod unit, θ is the rigid angular displacement of the connecting rod, u is the axial elastic displacement of the connecting rod unit, and w is the lateral elastic displacement of the connecting rod unit. The integrals in the above formula are all performed within the unit range. m and m are the horizontal and vertical coordinates of the unit mass center in the unit local coordinate system, respectively. The origin of the unit local coordinate system is located at the center of the unit. The x-direction is along the tangent direction of the unit, the y-direction is along the normal direction of the unit, ξ is the x-coordinate of the point on the connecting rod in the unit local coordinate system, and t represents time. u , F w and F θ are the nonlinear forces of the unit in the axial, lateral and rigid body motion directions respectively. The corresponding expressions can be derived from the Hamilton principle:

[0049]

[0050]

[0051]

[0052] S102: interpolating the lateral and axial deformations of the flexible manipulator using the shape function of the beam and the shape function of the rod respectively to obtain the finite element matrix of the unit, and establishing the finite element motion control equation of each unit; assembling the finite element motion control equation of each unit to obtain the global dynamic finite element equation of the flexible manipulator.

[0053] Specifically, the bending deformation of the flexible manipulator is simulated using beam units, the telescopic deformation of the flexible manipulator is simulated using rod units, and the finite element motion control equation of the flexible manipulator unit is derived using the Lagrange equation:

[0054]

[0055] Among them, M i , C i and K i are the mass matrix, damping matrix and stiffness matrix of the ith unit, G i (t) is the nonlinear force vector, X i They represent the acceleration, velocity and displacement vector of the unit node respectively.

[0056] Assemble the above equations according to the standard finite element process to obtain the global dynamics nonlinear finite element equation of the flexible manipulator:

[0057]

[0058] Among them, M is the overall mass matrix of the flexible manipulator, which is a linear constant matrix; C(t) is the damping matrix of the flexible manipulator, which is a nonlinear time-varying matrix; τ is the external force vector of the manipulator, X, are the global position, velocity, and acceleration vectors after assembly. In the above formula, K is the overall stiffness matrix of the flexible manipulator, which is a linear constant matrix; G(t) is the column vector corresponding to the nonlinear force, which is a nonlinear time-varying vector. By considering the nonlinear force as the generalized force of the unit, the linear stiffness matrix K and mass matrix M are obtained.

[0059] S103: Ignore the nonlinear terms in the global dynamics finite element equation, solve the generalized eigenvalue problem of the global dynamics finite element equation, and obtain the modal vibration shapes of each order of the flexible manipulator;

[0060] Specifically, considering that the rotation speed of the robot arm is generally much smaller than the critical speed, and the deformation does not reach the geometric large deformation level, the nonlinear force term in the global dynamics equation is ignored, and the following equation is obtained:

[0061]

[0062] Then we get the generalized eigenvalue equation of the above formula:

[0063] (K-ω 2 M)Y=0

[0064] Among them, ω is the natural frequency of the flexible manipulator after linearization, and Y is the natural vibration mode of the flexible manipulator after linearization. Since both rigid motion and flexible motion are considered at the same time, the first-order natural frequency of the above formula will correspond to the rigid body motion and will be zero. Since the finite element method is used to discretize the elastic deformation of the flexible rod, it is necessary to create enough unit grids to make the displacement field of the shape function piecewise interpolation smooth enough. Figure 4 (a)-(f) in the figure are respectively the grid pairs with different numbers. Figure 1 The modes corresponding to the first six non-zero eigenvalues ​​obtained when meshing the flexible robotic arm in .

[0065] By solving the above standard eigenvalue equation, we can obtain the values ​​of each order natural frequency ω and natural vibration mode Y of the manipulator after ignoring the nonlinear term, where the natural frequency ω is a positive real number and Y is a column vector. The latter can be used to reduce the order of the nonlinear dynamic finite element equation of the flexible manipulator.

[0066] S104: performing order reduction transformation on the global dynamics finite element equation based on the modal vibration shapes of each order to obtain a reduced-order global dynamics finite element equation;

[0067] Specifically, the modal vibration shape of the flexible manipulator is used for order reduction. According to the calculation accuracy requirements, Figure 1 The first three modal vectors of the flexible manipulator in the figure form a matrix Φ. Each term of the global dynamic nonlinear finite element equation of the flexible manipulator is multiplied by the modal matrix Φ on the right and by the matrix Y on the left. n The transpose Φ T , we can get:

[0068]

[0069] Among them, M r , C r , K r and G r They correspond to the reduced mass matrix, damping matrix, stiffness matrix and generalized force vector, respectively, and can be expressed as

[0070] M r =Φ T MΦ C r =Φ T CΦ K r =Φ T KΦ

[0071] G r =Φ T G τ r =Φ T τ

[0072] Among them, the dimension of each reduced-order matrix is ​​only determined by the number of selected modes. Since the low-order modes play a dominant role in the elastic deformation of the flexible manipulator, only a few modes are selected to ensure the accuracy and convergence of the motion control equation after the reduction. For example: when only the first three modes of the flexible manipulator are selected for reduction, the reduced-order mass matrix, damping matrix and stiffness matrix will be only 3×3 dimensions respectively, and the generalized force vector and the driving force vector will be 3×1 column vectors, thereby achieving effective reduction of the finite element motion control equation, effectively improving the calculation speed, reducing the memory usage and other computing consumption, and facilitating the realization of real-time control of the flexible manipulator based on the dynamic model. The dynamic equation of the flexible manipulator after the reduction can be numerically simulated by time integration and other methods, providing theoretical reference for the path planning of the flexible manipulator.

[0073] S105: Motion control of the robot's flexible single-rod manipulator based on the global dynamics finite element equation.

[0074] Specifically, the number of degrees of freedom of the reduced-order robot motion control equation is much smaller than that before the reduction due to the use of modal reduction, so it can be used for model-based real-time control of the robot; Figure 5 An example of robot motion control is given. The reduced-order dynamic equations obtained can be embedded in the robot controller as velocity / acceleration feedforward signals to adjust the robot servo motor output signals, thereby reducing the vibration of the robot during operation and the residual vibration after the motion is terminated.

[0075] In this embodiment, the time integration method is used to Figure 2 The torque in Figure 1 The dynamic response of the flexible manipulator is solved, and the end lateral deformation and end axial deformation of the flexible manipulator are obtained as follows: Figure 6 As shown, the joint angle is Figure 7 shown.

[0076] Embodiment 2

[0077] In one or more embodiments, a flexible robotic arm control system based on finite element discretization and modal order reduction is disclosed, specifically comprising:

[0078] A unit division module is used to spatially discretize the flexible manipulator using a finite element grid and divide the connecting rod of the flexible manipulator into a number of units;

[0079] The motion control equation construction module is used to construct the motion control equation of any unit of the discretized flexible manipulator using the Hamiltonian equation;

[0080] The motion control equation discretization module is used to interpolate the lateral and axial deformations of the flexible manipulator using the shape functions of the beam and the rod, respectively, to obtain the finite element matrix of the unit, and to establish the finite element motion control equations of each unit;

[0081] The global dynamics finite element equation building module is used to assemble the finite element motion control equations of each unit to obtain the global dynamics finite element equations of the flexible manipulator;

[0082] A global dynamics finite element equation solving module is used to ignore the nonlinear terms in the global dynamics finite element equation, solve the generalized eigenvalue problem of the global dynamics finite element equation, and obtain the modal vibration shapes of each order of the flexible manipulator; based on the modal vibration shapes of each order, the global dynamics finite element equation is reduced in order to obtain the reduced-order global dynamics finite element equation;

[0083] The robot arm control module is used to control the motion of the robot flexible single-rod robot arm based on the global dynamic finite element equation.

[0084] It should be noted that the specific implementation methods of the above modules have been described in Example 1 and will not be described in detail here.

[0085] Embodiment 3

[0086] In one or more embodiments, a robot controller is disclosed, including a server, the server including a memory, a processor, and a computer program stored in the memory and executable on the processor, and the processor implements the flexible manipulator control method based on finite element discretization and modal order reduction in embodiment 1 when executing the program. For the sake of brevity, it will not be described in detail here.

[0087] It should be understood that in this embodiment, the processor may be a central processing unit CPU, and the processor may also be other general-purpose processors, digital signal processors DSP, application-specific integrated circuits ASIC, off-the-shelf programmable gate arrays FPGA or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or the processor may also be any conventional processor, etc.

[0088] The memory may include a read-only memory and a random access memory, and provide instructions and data to the processor. A portion of the memory may also include a non-volatile random access memory. For example, the memory may also store information about the device type.

[0089] In the implementation process, each step of the above method can be completed by an integrated logic circuit of hardware in a processor or an instruction in the form of software.

[0090] Embodiment 4

[0091] In one or more embodiments, a robot is disclosed, which uses the flexible robotic arm control method based on finite element discretization and modal reduction described in Example 1 to achieve motion control of a single-rod flexible robotic arm.

[0092] Although the above describes the specific implementation mode of the present invention in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art on the basis of the technical solution of the present invention without creative work are still within the scope of protection of the present invention.

Claims

1. A flexible manipulator control method based on finite element discretization and modal reduction, characterized in that: include: The flexible manipulator is spatially discretized using finite element grids, and the connecting rods of the flexible manipulator are divided into several units. The Hamiltonian equation is used to construct the motion control equation of any unit of the discretized flexible manipulator. The shape functions of beams and rods are used to interpolate the lateral and axial deformations of the flexible manipulator, respectively, to obtain the finite element matrix of the unit, and to establish the finite element motion control equations of each unit; The finite element motion control equations of each unit are assembled to obtain the global dynamic finite element equation of the flexible manipulator; Ignore the nonlinear terms in the global dynamics finite element equation, solve the generalized eigenvalue problem of the global dynamics finite element equation, and obtain the modal vibration shapes of each order of the flexible manipulator; Based on the modal vibration shapes of each order, the global dynamic finite element equation is reduced to obtain the reduced-order global dynamic finite element equation; The motion of the flexible single-rod manipulator of the robot is controlled based on the global dynamic finite element equation.

2. A flexible manipulator control method based on finite element discretization and modal reduction as claimed in claim 1, characterized in that: The Hamiltonian equation is used to construct the motion control equation of any unit of the discrete flexible manipulator, which is: Where ρ is the material density of the connecting rod, A is the cross-sectional area of ​​the connecting rod unit, θ is the rigid angular displacement of the connecting rod, u is the axial elastic displacement of the connecting rod unit, w is the lateral elastic displacement of the connecting rod unit, and x m and m are the horizontal and vertical coordinates of the unit mass center in the unit local coordinate system, respectively. The origin of the unit local coordinate system is located at the center of the unit, its x-direction is along the tangent direction of the unit, and its y-direction is along the normal direction of the connecting rod. ξ is the x-coordinate of the point on the connecting rod in the unit local coordinate system, and t represents time. u , F w and F θ They are the nonlinear forces of the unit in the axial, lateral and rigid body motion directions respectively.

3. A flexible manipulator control method based on finite element discretization and modal reduction as claimed in claim 1, characterized in that: The shape functions of the beam and the rod are used to interpolate the lateral and axial deformations of the flexible manipulator, respectively, to obtain the finite element matrix of the discretized unit, and to establish the finite element motion control equations of each unit, which are as follows: Among them, M i , C i and K i are the mass matrix, damping matrix and stiffness matrix of the ith unit respectively, Gi(t) is the nonlinear force vector, X i They represent the acceleration, velocity and displacement vector of the unit node respectively.

4. A flexible manipulator control method based on finite element discretization and modal reduction as claimed in claim 3, characterized in that: The finite element motion control equations of each unit are assembled to obtain the global dynamic finite element equation of the flexible manipulator, which is: Among them, M is the overall mass matrix of the flexible manipulator, C(t) is the damping matrix of the flexible manipulator, K is the overall stiffness matrix of the flexible manipulator, G(t) is the column vector corresponding to the nonlinear force, τ is the external force vector of the manipulator, X, and are the assembled global position, velocity, and acceleration vectors respectively.

5. A flexible manipulator control method based on finite element discretization and modal reduction as claimed in claim 4, characterized in that: The nonlinear terms in the global dynamic finite element equation are the damping matrix C(t) of the flexible manipulator and the column vector G(t) corresponding to the nonlinear force.

6. A flexible manipulator control method based on finite element discretization and modal reduction as claimed in claim 1, characterized in that: After ignoring the nonlinear terms in the global dynamics finite element equation, the generalized eigenvalue equation of the global dynamics finite element equation is obtained as follows: (K-ω 2 M)Y=0 Among them, ω is the natural frequency of the flexible manipulator after linearization, Y is the natural vibration mode of the flexible manipulator after linearization; M is the overall mass matrix of the flexible manipulator, and K is the overall stiffness matrix of the flexible manipulator; By solving the generalized eigenvalue equation, the values ​​of the natural frequencies ω and natural vibration modes Y of each order of the manipulator after ignoring the nonlinear term are obtained, wherein the natural frequency ω is a positive real number and Y is a column vector.

7. A flexible manipulator control method based on finite element discretization and modal reduction as claimed in claim 1, characterized in that: Based on the modal vibration shapes of each order, the global dynamic finite element equation is reduced to obtain the reduced-order global dynamic finite element equation, which is specifically: The first n-order modal vectors of the flexible manipulator are selected to form a matrix Φ, wherein the i-th column of the matrix Φ is the i-th order natural vibration mode vector of the flexible manipulator; Each term of the global dynamics nonlinear finite element equation of the flexible manipulator is multiplied by the modal matrix Φ on the right and the transpose Φ of the matrix Φ on the left. T , and the reduced-order global dynamic finite element equation is obtained: Among them, M, C, and K are the overall mass matrix, damping matrix, and overall stiffness matrix of the flexible manipulator, respectively; G is the column vector corresponding to the nonlinear force; τ is the external force vector acting on the manipulator.

8. A flexible manipulator control system based on finite element discretization and modal reduction, characterized in that: include: A unit division module is used to spatially discretize the flexible manipulator using a finite element grid and divide the connecting rod of the flexible manipulator into a number of units; The motion control equation construction module is used to construct the motion control equation of any unit of the discretized flexible manipulator using the Hamiltonian equation; The motion control equation discretization module is used to interpolate the lateral and axial deformations of the flexible manipulator using the shape functions of the beam and the rod, respectively, to obtain the finite element matrix of the unit, and to establish the finite element motion control equations of each unit; The global dynamics finite element equation building module is used to assemble the finite element motion control equations of each unit to obtain the global dynamics finite element equations of the flexible manipulator; A global dynamics finite element equation solving module is used to ignore the nonlinear terms in the global dynamics finite element equation, solve the generalized eigenvalue problem of the global dynamics finite element equation, and obtain the modal vibration shapes of each order of the flexible manipulator; based on the modal vibration shapes of each order, the global dynamics finite element equation is reduced in order to obtain the reduced-order global dynamics finite element equation; The robot arm control module is used to control the motion of the robot flexible single-rod robot arm based on the global dynamic finite element equation.

9. A robot controller, comprising a processor and a memory, wherein the processor is used to implement instructions; and the memory is used to store multiple instructions, characterized in that: The instructions are suitable for being loaded by a processor and executing the flexible robotic arm control method based on finite element discretization and modal reduction as described in any one of claims 1 to 7.

10. A robot, characterized in that: The flexible robotic arm control method based on finite element discretization and modal reduction as described in any one of claims 1 to 7 is adopted to realize the motion control of the single-rod flexible robotic arm.

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