A flexible hydraulic robot arm system multi-body dynamics modeling method

Abstract

The application discloses a flexible hydraulic mechanical arm system multi-body dynamics modeling method. The core comprises the following steps: different mechanical structures of each arm rod of the mechanical arm are respectively simplified into common beam types or combinations thereof according to the different mechanical structures; specific base functions of the arm frames with different structures are respectively derived according to the Rayleigh-Ritz method; the Hamilton action of the arm rod is solved according to the Hamilton principle, and a characteristic vector is calculated and substituted into the base function group; the flexible arm is decomposed into an imaginary rigid part and a flexible part by adopting the rigid and flexible virtual separation idea and is processed respectively; the hydraulic driving force and the driving torque of the mechanical arm under the flexible deformation condition are derived and calculated; and each description quantity is integrated into the Lagrange equation according to the Lagrange dynamics modeling method, and a complete system mathematical model is derived. The application has small calculation amount, achieves the purpose of simplifying calculation and improving efficiency, and has clear derivation idea and good portability.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of control of hydraulic mechanical arm, and particularly relates to a multi-body dynamics modeling method of flexible hydraulic mechanical arm system. BACKGROUND

[0002] The flexible hydraulic mechanical arm system is a liquid-solid coupled series mechanical arm system with flexible deformation driven by a hydraulic cylinder. The hydraulic mechanical arm is suitable for the harsh working environment in the fields of construction, power, nuclear industry, ocean engineering, emergency rescue and the like, and has the advantages of light weight, large load ratio, fast movement speed and low energy consumption.

[0003] The actual mechanical arm frame vibration and control involve multiple cross disciplines such as machinery, hydraulic pressure and control, the solid-liquid coupled dynamics model and vibration control are very complex, and the research work such as elastic vibration active damping control strategy and trajectory planning needs to be based on the accurate dynamics model. At present, many researches only consider the dynamics modeling of a certain flexible arm with some specific degrees of freedom, and the process of actually calculating the dynamics model is still very complex and has low calculation efficiency.

[0004] The mechanical system modeling method improves the traditional mechanical system Lagrange dynamics modeling method based on the assumption of modal method, simplifies the arm rods of different mechanical structures, selects appropriate base functions for the simplified arm rods, and then models the mechanical system by using the Lagrange dynamics modeling method. SUMMARY

[0005] The present application relates to the technical field of control of hydraulic mechanical arm, and particularly relates to a multi-body dynamics modeling method of flexible hydraulic mechanical arm system.

[0006] To achieve the above object, the present application provides a multi-body dynamics modeling method of flexible hydraulic mechanical arm system, which comprises the following steps:

[0007] S1, according to the different mechanical structures of each arm rod of the mechanical arm, the arm rod is simplified into a common beam type such as a cantilever beam and a simply supported beam or a combination of different types of beams;

[0008] S2, judging the deflection shape and curvature change of each arm rod of the mechanical arm according to the actual structure of the arm rod in the working state, selecting a function type capable of approximately describing the deformation of the arm rod, and respectively deriving specific basis functions of the arm rod of different structures according to the Rayleigh-Ritz method;

[0009] S3, according to the principle of minimum potential energy, when the potential energy is minimum, the system is in a stable state, and the calculated deflection of the arm rod is the stable value in the instantaneous stable state; according to the Hamilton principle, the Hamilton action of the arm rod is solved, and the characteristic vector is calculated and substituted into the basis function group;

[0010] S4, the modal coordinates of each part of the arm rod are obtained through the joint calculation of the basis function and the modal angle equation, and the actual deflection of the arm rod is further obtained, and the deflection of each arm rod is superimposed and calculated;

[0011] S5, the flexible arm is decomposed into an imaginary rigid part and a flexible part according to the idea of rigid-flexible virtual separation, and is processed respectively;

[0012] S6, the hydraulic driving force and driving torque of the mechanical arm in the flexible deformation condition are derived and calculated;

[0013] S7, the mathematical model of the flexible hydraulic mechanical arm system is derived: according to the Lagrange dynamics modeling method, the position vector, velocity vector and other description quantities of each arm rod of the hydraulic mechanical arm are integrated into the Lagrange equation, the complete system mathematical model is derived, and system dynamics modeling and analysis are carried out.

[0014] Preferably, the flexible hydraulic mechanical arm is a solid-liquid coupled planar multi-link series arm driven by a hydraulic drive and having a flexible deformation. The present application takes a planar series three-link flexible hydraulic mechanical arm as an example to model the system.

[0015] Preferably, in the step S1, the actual mechanism of the arm rod is respectively simplified as different types of beam frames, and the example arm rod is simplified and equivalent, wherein the arm rod one is equivalent to a cantilever beam, the arm rod two is equivalent to a combination of a simply supported beam and a cantilever beam, and the arm rod three is equivalent to a cantilever beam. In particular, the length of the front end part of the arm rod one hinged with the hydraulic rod is very short relative to the actual length of the whole arm rod, and the deflection deformation produced has little effect on the overall deflection. Therefore, it is ignored and assumed as a rigid part.

[0016] Preferably, in the step S2, the geometric boundary conditions required to be met by the arm rod basis function of the mechanical arm system are as follows:

[0017] ,

[0018]

[0019]

[0020] ,

[0021] Basis functions of arm 1 based on the Rayleigh-Ritz method that satisfy the boundary conditions:

[0022]

[0023] Select complete front Using the superposition of the first two modes as basis functions is too complicated for the overall system calculation, and the improvement in accuracy is negligible. Therefore, the superposition of the first two modes, which have a greater impact on the overall system, is selected as the arm basis functions.

[0024]

[0025] in This is the length of the boom before a hinge point.

[0026] The last section of the boom at the end of the entire boom can be simplified into a typical cantilever beam structure. Similarly, only the superposition of the first two modes needs to be selected as the basis function, which can meet the accuracy requirements and greatly improve the calculation efficiency.

[0027] An example of this invention is a planar series three-bar flexible hydraulic manipulator system, whose basis function set is as follows:

[0028]

[0029]

[0030]

[0031] Preferably, in step S3, the Hamiltonian action of arm one can be expressed as:

[0032]

[0033] Define the variable as:

[0034]

[0035]

[0036] The Hamiltonian action can be expressed in matrix form as follows:

[0037]

[0038] Finding approximate solutions to the basis functions is transformed into solving the parameter problem, i.e., solving... The necessary and sufficient condition for each parameter to have a non-zero solution is that its coefficient determinant is zero.

[0039]

[0040] Thus, n eigenvalues and corresponding eigenvectors can be obtained as , and substituting the base function group into the equation facilitates subsequent calculation substitution.

[0041] Preferably, in the step S5, the arm rod is virtually decomposed into rigid and flexible parts. The deflection change of the planar serial mechanical arm system mainly reflects in the vertical direction, so the virtual rigid part is assumed in the driven coordinate system as the axis of the flexible arm , the axis of the flexible part. The virtual rigid motion generalized coordinates and the flexible deformation generalized coordinates are analyzed respectively to obtain the virtual rigid arm rotating joint coupling term and the coupling term reflecting the deformation and deformation speed of the flexible arm , .

[0042] Rigid joint coupling term:

[0043]

[0044] Flexible deformation coupling term:

[0045]

[0046]

[0047]

[0048] wherein is the inertia matrix of the arm rod , and and are the deformation and deformation speed coupling effects of the arm rod .

[0049] Preferably, in the step S6, the influence of the valve core and the hydraulic cylinder leakage on the rigid body is ignored, it is assumed that the oil pressure in the hydraulic cylinder is the same, and the pipe internal friction loss is ignored. The motion equation of the hydraulic cylinder piston is:

[0050]

[0051] Let , then the hydraulic cylinder driving force equation is:

[0052]

[0053] Let ,​ Let |OA| be the hinge point between the hydraulic cylinder and the two adjacent booms. |OB|= The position vector of hydraulic cylinder 1 between boom 1 and base can be expressed as:

[0054]

[0055] After differentiation, the velocity vector and acceleration vector of hydraulic cylinder 1 are obtained:

[0056]

[0057] Similarly, the position vectors of hydraulic cylinders 2 and 3 are respectively:

[0058]

[0059]

[0060]

[0061]

[0062] Similarly, the velocity and acceleration vectors of hydraulic cylinders two and three can be obtained. Substituting the above equations for the position, velocity, and acceleration vectors of the hydraulic cylinders into the equation for the driving force of the hydraulic cylinders, we get:

[0063]

[0064] The driving torque of the hydraulic cylinder in the flexible hydraulic robotic arm can be expressed as:

[0065]

[0066] Preferably, in step S7, the kinetic energy, gravitational potential energy, and elastic potential energy of the flexible boom are calculated as follows:

[0067]

[0068]

[0069]

[0070] Where the kinetic energy is Gravitational potential energy is The elastic potential energy of the deformable storage is .

[0071] Define the Lagrangian function for a flexible robotic arm:

[0072]

[0073] Select generalized coordinates wherein , , according to Lagrange equation,

[0074]

[0075] wherein, is generalized velocity, is generalized moment.

[0076] After operation and arrangement, the matrix form flexible hydraulic mechanical arm system multi-body dynamics equation is:

[0077]

[0078] Preferably, in the step S7, the dynamics mathematical model of the flexible hydraulic mechanical arm system is derived through the Mathematica scientific calculation software, and data simulation is carried out through the MATLAB simulation software.

[0079] Compared with the prior art, the present application has the following beneficial effects: (1) the present application proposes a multi-base function combination flexible hydraulic mechanical arm system multi-body dynamics modeling method based on the hypothesis modal method, respectively equivalent simplifies each arm rod mechanical structure, selects the appropriate base function according to the structure after simplification to describe the flexible deformation of the flexible arm rod, and improves the reliability and accuracy of the flexible hydraulic mechanical arm system dynamics modeling; (2) the present application adopts the idea of rigid and flexible virtual separation to decompose the flexible arm into an imaginary rigid part and a flexible part and process them respectively, reduces unnecessary coupling calculation to achieve the purpose of simplifying calculation and improving efficiency; (3) in the modeling method of the present application, the flexible characteristics of the rigid-flexible coupled mechanical arm are only reflected in the 、 and coupling terms, so when modeling the dynamics of different types of planar flexible mechanical arms, only the rigid-flexible coupling related terms need to be modified, and the dynamics model can be obtained. The present application has small calculation amount, clear derivation thought and good portability. BRIEF DESCRIPTION OF DRAWINGS

[0080] Figure 1 is a flexible mechanical arm system diagram of the present application;

[0081] Figure 2 is an equivalent simplified schematic diagram of arm rod one of the present application;

[0082] Figure 3 is an equivalent simplified schematic diagram of arm rod two of the present application;

[0083] Figure 4 is an equivalent simplified schematic diagram of arm rod three of the present application;

[0084] Figure 5 Coordinate assignment diagram for the flexible hydraulic robot arm system of the present application;

[0085] Figure 6 Structure diagram of the hydraulic cylinder of the present application;

[0086] Figure 7 Simplified position diagram of the hydraulic cylinder of the present application;

[0087] Figure 8 Modeling process of the flexible hydraulic robot arm system of the present application. DETAILED DESCRIPTION

[0088] The present application provides a multi-body dynamics modeling method for a flexible hydraulic robot arm system. In order to make the application more understandable, the present application is further described below in combination with the drawings and specific embodiments.

[0089] The flexible hydraulic robot arm system of the present application comprises a base, a planar serial robot arm, a hydraulic cylinder and a triangular connecting frame. The modeling method of the system is to first simplify each arm link respectively, then select base functions in combination with the assumed modal method and the actual deflection of the arm link, and then use the rigid and flexible virtual separation idea to process the imaginary rigid part and the flexible part of the arm frame respectively, and use the Lagrange dynamics modeling method to build a complete multi-body dynamics model. The main steps are as follows:

[0090] S1, according to the different mechanical structures of each arm link of the robot arm, the arm link is simplified into a common beam type such as a cantilever beam and a simply supported beam or a combination of different types of beams;

[0091] For example, Figure 2 Arm link one is simplified as a cantilever beam. In particular, the vertical flexibility deformation of the part before the hinge point of the hydraulic cylinder can be ignored, and is approximately equivalent to a rigid part, compared with the length of the whole arm link one.

[0092] For example, Figure 3 Arm link two is simplified as a combination of a simply supported beam and a cantilever beam, wherein the part before the hinge point is simplified as a simply supported beam, and the part after the hinge point is simplified as a cantilever beam.

[0093] For example, Figure 4 Arm link three is simplified as a cantilever beam.

[0094] S2, according to the actual structure of each arm link of the robot arm, the deflection shape and curvature change in the working state are judged, the function type capable of approximately describing the deformation is selected, and the specific base functions of different structure arm frames are derived according to the Rayleigh-Ritz method;

[0095] According to the change of bending degree and curvature in actual application, the deflection of the arm increases with the length of the arm. The geometric boundary conditions that the arm link base functions need to satisfy are as follows:

[0096]

[0097]

[0098]

[0099]

[0100] For the mechanical arm system, the arm rod base function satisfying the boundary condition based on the Rayleigh-Ritz method is:

[0101]

[0102] The function of superposition of complete first-order modal is selected as the base function, which is too complex for the overall calculation of the system, and the improvement in accuracy is very small, therefore the superposition of the first two orders of modal is selected as the arm rod base function.

[0103] Wherein, L1 is the length before the arm rod one-hinged point;

[0104] The last arm rod of the whole arm frame end can be simplified as a typical cantilever beam structure, and only the superposition of the first two orders of modal is selected as the base function, which can meet the accuracy requirement and greatly improve the calculation efficiency.

[0105] The example of the application is a planar series three-link flexible hydraulic mechanical arm system, and the base function group is as follows:

[0106]

[0107]

[0108]

[0109] S3, according to the principle of minimum potential energy, when the potential energy is minimum, the system is in a stable state, and the arm rod deflection calculated is the stable value under the instantaneous stable state; according to the Hamilton principle, the Hamilton action of the arm rod is solved, and the characteristic vector is calculated and substituted into the base function group. The application takes a planar series three-link flexible hydraulic mechanical arm system as an example, therefore in the example, the Hamilton action of the arm rod one can be expressed as:

[0110]

[0111] Define variables

[0112]

[0113] ​​

[0114]

[0115] The Hamilton action is expressed in matrix form as:

[0116]

[0117] Solving the approximate solution of the base function is converted into solving the parameter problem, that is, solving The necessary and sufficient condition for each parameter to have a non-zero solution is that its coefficient determinant is zero,

[0118]

[0119] Therefore, n eigenvalues and corresponding eigenvectors can be obtained as , and substituting into the base function set facilitates subsequent calculation substitution.

[0120] S4, the modal coordinates of each arm are obtained by common calculation of the base function and the modal angle equation, and the actual deflection of the arm can be further obtained. The deflection of each arm is calculated by superposition;

[0121] S5, the flexible arm is decomposed into an imaginary rigid part and a flexible part by adopting the idea of rigid and flexible virtual separation and is processed respectively. The deflection change of the planar serial mechanical arm system is mainly reflected in the vertical direction, so the imaginary in the follow-up coordinate system is the virtual rigid part of the flexible arm , the shaft is the flexible part. The generalized coordinates of the virtual rigid motion and the generalized coordinates of the flexible deformation are analyzed respectively, and the virtual rigid arm rotation joint coupling term and the coupling term , reflecting the deformation and the speed of the flexible arm are obtained.

[0122] Rigid joint coupling term:

[0123]

[0124] Flexible deformation coupling term:

[0125]

[0126]

[0127]

[0128] wherein is the arm the inertia matrix of the rigid body, and is the arm rod of the deformation and the coupling effect of the deformation speed.

[0129] S6, the hydraulic drive force and drive torque of the flexible deformation of the mechanical arm are derived and calculated;

[0130] Ignoring the influence of the valve core and the hydraulic cylinder leakage on the rigid body, assuming that the oil pressure in the hydraulic cylinder is the same, ignoring the pipe internal friction loss. The motion equation of the hydraulic cylinder piston is:

[0131]

[0132] Let , then the hydraulic cylinder drive force equation is:

[0133]

[0134] Let , Let OA and OB be the hinge points of the hydraulic cylinder and the adjacent two arm rods, respectively, |OA|= , |OB|= . Then the position vector of the hydraulic cylinder 1 between the arm rod 1 and the base can be expressed as:

[0135]

[0136] After derivation, the velocity vector and acceleration vector of the hydraulic cylinder 1 are obtained:

[0137]

[0138] Similarly, the position vectors of hydraulic cylinders 2 and 3 are:

[0139]

[0140] -

[0141]

[0142]

[0143] Similarly, the velocity vector and acceleration vector of the hydraulic cylinder two and three can be obtained. Substitute the above hydraulic cylinder position, velocity, and acceleration vector equations into the hydraulic cylinder drive force equation to obtain:

[0144]

[0145] Then the hydraulic cylinder drive torque of the flexible hydraulic mechanical arm is expressed as:

[0146]

[0147] S7, derive flexible hydraulic mechanical arm system mathematical model: according to the Lagrange dynamics modeling method, the position vector, velocity vector and other description quantities of each arm rod of the hydraulic mechanical arm are integrated into the Lagrange equation, the complete system mathematical model is derived, and the system dynamics modeling and analysis are carried out.

[0148] The kinetic energy of the flexible arm frame is The gravitational potential energy is The elastic potential energy stored due to deformation is , which is expressed as follows.

[0149]

[0150]

[0151]

[0152] Define the Lagrange function of the flexible mechanical arm:

[0153]

[0154] Select the generalized coordinates , wherein , According to the Lagrange equation

[0155]

[0156] wherein, is the generalized velocity, is the generalized torque

[0157] After operation and arrangement, the matrix form of the flexible hydraulic mechanical arm system multi-body dynamics equation is:

[0158] .

Claims

1. A method for multi-body dynamics modeling of a flexible hydraulic manipulator system, characterized in that, The method comprises the following steps: S1, different mechanical structures of each arm rod of the mechanical arm are respectively equivalent simplified into common beam types, and the common beam types include cantilever beam, simply supported beam or combination of cantilever beam and simply supported beam; S2, the actual structure of each arm rod of the mechanical arm is judged to determine the deflection shape and curvature change in the working state, a function type capable of approximately describing the deformation is selected, and specific basis functions of the arm frame of different structures are respectively derived according to Rayleigh-Ritz method; S3, according to the principle of minimum potential energy, when the potential energy is minimum, the system is in a stable state, and the calculated arm rod deflection is the stable value in the instantaneous stable state; according to Hamilton principle, the Hamilton action of the arm rod is solved, and the characteristic vector is calculated and substituted into the basis function group; S4, the modal coordinates of each part of the arm rod are obtained through common calculation of the basis function and the modal angle equation, and the actual deflection of the arm rod is further obtained, and the deflection of each arm rod is superimposed and calculated; S5, the idea of rigid and flexible virtual separation is adopted to decompose the flexible arm into an imaginary rigid part and a flexible part and process them respectively; S6, the hydraulic driving force and driving torque of the mechanical arm under the condition of flexible deformation are derived and calculated; S7, the mathematical model of the flexible hydraulic mechanical arm system is derived: according to Lagrange dynamics modeling method, the position vector and velocity vector of each arm rod of the hydraulic mechanical arm are integrated into Lagrange equation, the complete system mathematical model is derived, and system dynamics modeling and analysis are carried out.

2. The method of claim 1, wherein, In step S1, specifically comprising: According to the actual structure of the arm rod, the arm rod one is equivalent simplified in the planar series three-link flexible hydraulic mechanical arm, the part before the hinge joint with the hydraulic cylinder is regarded as a rigid part, and the part after the hinge joint of the arm rod one is equivalent simplified into a cantilever beam; the arm rod two is equivalent simplified into a combination of simply supported beam and cantilever beam, the part before the hinge joint of the arm rod two is a simply supported beam, and the part after the hinge joint is a cantilever beam; the arm rod three is equivalent simplified into a cantilever beam.

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