Vibration prediction method for multi-degree-of-freedom mechanical arm system under support of thin plate based on Kirchhoff plate theory

Abstract

The invention discloses a Kirchhoff plate theory-based vibration prediction method for a multi-degree-of-freedom mechanical arm system under thin plate support, which comprises the following steps of: 1) establishing a coupling relationship between a thin plate and a mechanical arm based on a thin plate support-multi-degree-of-freedom mechanical arm coordinate system; 2) establishing a mapping relation of the end effector of the mechanical arm in a global coordinate system; (3) a multi-degree-of-freedom vibration partial differential equation of the thin plate supporting-multi-degree-of-freedom mechanical arm system is constructed; (4) the obtained partial differential vibration equation is subjected to numerical solution, and the vibration amplitude of an end effector of the mechanical arm is predicted; and (5) whether the vibration amplitude of the mechanical arm end effector meets the precision requirement of the mechanical arm or not is judged, and if not, the thin plate supporting-multi-degree-of-freedom mechanical arm coupling system is improved. The method can be used for multi-degree-of-freedom mechanical arm tail end pose error prediction, performance evaluation and structural parameter design under the support of different types of thin plates.

Description

Technical Field

[0001] The invention relates to the field of vibration prediction of mobile robots, in particular to a vibration prediction method for a multi-freedom mechanical arm system supported by a thin plate based on Kirchhoff plate theory. Background Art

[0002] Mobile robots have broad application prospects in shipbuilding industry, large spherical tank welding, etc. due to their simple structure, strong adaptability, and ability to operate in non-structural environments. The robot is installed on the box base and its control system is integrated in the box, which can realize the integrated design of the system, with the advantages of space saving, compact structure, convenient transportation, and strong resistance to external interference.

[0003] The robot arm is installed on the box, and the upper surface of the box can be regarded as a thin plate support. Therefore, the system can be equivalent to a typical thin plate support-robot arm coupling system. Unlike the traditional rigid base robot arm, when the robot arm is attached to the surface of a flexible thin plate structure to perform tasks, there will be a serious dynamic coupling effect between the two, resulting in low-frequency vibrations that affect the smooth progress of the robot arm tasks under such support. How to establish an effective prediction mechanism for this type of vibration behavior is urgently needed.

[0004] So far, there is still a lack of corresponding theory to realize the vibration prediction method of multi-degree-of-freedom robotic arm system supported by thin plates. Summary of the invention

[0005] The object of the present invention is to provide a vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory, comprising the following steps:

[0006] 1) The box support-multi-DOF robotic arm system is equivalent to a thin plate support-multi-DOF robotic arm coupling system;

[0007] 2) Establish a thin plate support-multi-degree-of-freedom robotic arm coordinate system;

[0008] 3) Based on the thin plate support-multi-DOF robotic arm coordinate system, establish the coupling relationship between the thin plate and the robotic arm;

[0009] 4) Establishing the mapping relationship between the end effector of the robot arm in the global coordinate system; the end of the robot arm is the end that holds the work object;

[0010] 5) Construct the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system;

[0011] 6) numerically solve the obtained partial differential vibration equation to predict the vibration amplitude of the end effector of the robot arm;

[0012] 7) Determine whether the vibration amplitude of the end effector of the robot arm meets the accuracy requirements of the robot arm. If not, improve the thin plate support-multi-degree-of-freedom robot arm coupling system and return to step 6).

[0013] Further, the box support-multi-DOF robotic arm system includes a box, a robotic arm support, and a robotic arm;

[0014] The mechanical arm support is fixed on the box body;

[0015] One end of the mechanical arm is fixed on the mechanical arm support, and the other end is used to clamp the work object;

[0016] The robotic arm has multiple degrees of freedom.

[0017] Further, the thin plate support-multi-DOF robotic arm coupling system comprises a rectangular thin plate, a robotic arm mounting base, and a robotic arm;

[0018] The length, width and height of the rectangular thin plate are determined by the length, width and height of the box body, which are denoted as a, b and h respectively;

[0019] The mechanical arm mounting base is fixed on a rectangular thin plate;

[0020] The mass of the robotic arm mounting base is m 1 , the distance from the center of mass of the robot arm mounting base to the upper surface of the rectangular thin plate is H 1 ′;

[0021] One end of the mechanical arm is fixed on the mechanical arm mounting base, and the other end is used to clamp the working object;

[0022] The mass of the robotic arm is m 2 , the installation position of the robot arm is (x 0 ,y 0 ), the azimuth angle of the robot is α, the rotation angle of the robot is θ, and the distance from the center of mass of the robot to the secondary axis of rotation of the robot is p.

[0023] Furthermore, the thin plate support-multi-DOF manipulator coordinate system includes the system global coordinate system o-xyz, the thin plate static coordinate system o l -x l y l z l , thin plate moving coordinate system o t -x t y t z t , the connected coordinate system o of the i-th rod of the robot pi -x pi y pi z pi .

[0024] Furthermore, the coupling relationship between the thin plate and the robot arm is expressed by the thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is represented;

[0025] Thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is as follows:

[0026]

[0027] In the formula, r 2 is the moving coordinate system o of the robot arm in the thin plate t -x t y t z t The position vector in ; is the coordinate system o t -x t y t z t In the coordinate system o l -x l y l z l The position vector in w o,xl and w o,yl for o t x t Axis and o l x l Axis, o t y t Axis and o l y l Angle between the axes; R 2 is a matrix; r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l Position vector in the global coordinate system;

[0028] Among them, the tangent line ξ of the material line u′, the tangent line ζ of the material line v′ and the normal line η on the neutral surface of the thin plate are as follows:

[0029] ξ=[10w 0,xl ] T (3)

[0030] ζ=[01w 0,yl ] T (4)

[0031] η=ξ×ζ=[-w 0,xl -w 0,yl 1] T (5).

[0032] Furthermore, the mapping relationship of the robot end effector in the global coordinate system is as follows:

[0033]

[0034] Among them, r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l The position vector in the global coordinate system, x p ,y p ,z p is the displacement parameter of the end effector of the robot arm in the x, y, and z directions in the global coordinate system; r p is the position vector of the robot end effector in the global coordinate system.

[0035] Furthermore, the method of constructing the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system is as follows: based on the Hamiltonian principle, the multi-degree-of-freedom vibration partial differential equation is constructed using Kirchhoff plate theory.

[0036] Furthermore, the steps of constructing the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system include:

[0037] a1) Based on the coordinate representation of the robot arm actuator in the global coordinate system, calculate the system kinetic energy T, that is:

[0038] T=T P +T m +T l (7)

[0039] Among them, the kinetic energy of the thin plate T P , motor and robot arm installation base kinetic energy T m 、Kinetic energy of robot arm T lThey are as follows:

[0040]

[0041] Where, ρ, m 1 , m 2 , are the volume density of the thin plate, the mass of the motor base, and the mass of the robotic arm respectively; J 1,x , J 1,y For the motor base about o t x t and t y t Moment of inertia; J 2,x , J 2,y For the robot arm about o t x t and t y t Moment of inertia; J m is the moment of inertia of the motor; a, b, h are the length, width, and height of the rectangular plate; is the differential of the rotation angle of the robot arm; θ r is the robot arm rotation angle. is the derivative of () in the time domain; The end effector of the robot arm is t x t and t y t The position component on ;

[0042] a2) Calculate the system potential energy V, that is:

[0043]

[0044] Among them, υ and D are the Poisson's ratio and bending stiffness of the thin plate respectively; θ r are the rotation angles of the robot arm; g is the acceleration of gravity; p is the distance from the center of mass of the robot arm to the secondary axis of rotation of the robot arm; w is the deflection of the thin plate, w 0 is the deflection of the thin plate at the installation position of the robot arm, H 1 H is the distance from the center of mass of the mounting base to the upper surface of the rectangular thin plate, 2 The distance from the motor center of mass to the upper surface of the rectangular plate.

[0045] a3) Calculate the virtual work W acting on the system, that is:

[0046]

[0047] Where τ is the torque applied to the motor; c r is the Rayleigh dissipation coefficient of the manipulator rotation joint, c p is the viscous damping coefficient of the thin plate; δ is the symbol of variation. r 、cp is the damping coefficient;

[0048] a4) Based on Hamilton’s principle, construct the equation:

[0049]

[0050] In the formula, δ(·) is the variation of (·), t 1 and t 2 are two time constants, t 1 <t<t 2 ; For the convenience of expression, the following variational expression of deformation related quantities is introduced:

[0051]

[0052] a5) Substitute the system kinetic energy T, system potential energy V, and virtual work W acting on the system into equation (13) to obtain the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system, namely:

[0053]

[0054] Where M(q,θ), Q(t), C, q, K, and F are the generalized coordinates, mass matrix, stiffness matrix, damping matrix, and external force terms of the system, which are determined by the system configuration and parameters.

[0055] Further, the step of predicting the vibration amplitude of the end effector of the robot arm includes:

[0056] b1) Set the torque applied to the motor Among them, M 0 and ω are the torque constant parameters applied to the manipulator;

[0057] b2) Solve the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system to obtain the vibration amplitude of the end effector of the manipulator.

[0058] Furthermore, methods for improving the thin plate support-multi-DOF robotic arm coupling system include: increasing the thickness of the thin plate and strengthening the rib plate.

[0059] The technical effect of the present invention is undoubted, and the beneficial effects of the present invention are as follows:

[0060] The present invention establishes a three-dimensional displacement field relationship between a thin plate support and a multi-degree-of-freedom robotic arm, and based on Kirchhoff plate theory calculations, can obtain a three-dimensional dynamic response solution model of the system, thereby realizing the end-point posture error prediction, performance evaluation, and structural parameter design of the multi-degree-of-freedom robotic arm under thin plate support, which has important theoretical significance and engineering practice value.

[0061] The three-dimensional coupling dynamic model between the thin plate and the robot arm established in the present invention can obtain the vibration displacement of the end effector of the robot arm under any installation posture;

[0062] The method described in the present invention can be extended to the dynamic modeling and analysis of more general multi-degree-of-freedom robotic arms or multiple eccentric motors and complex-shaped plates, composite plates, etc., and can be used for the prediction of end pose errors, performance evaluation and structural parameter design of multi-degree-of-freedom robotic arms supported by different types of thin plates. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 It is a schematic diagram of a physical model of the box support-multi-degree-of-freedom mechanical arm system of the present invention;

[0064] Figure 2 It is a schematic diagram of the physical model of the thin plate support-multi-degree-of-freedom mechanical arm system of the present invention;

[0065] Figure 3 It is a schematic diagram of establishing the coordinate system of the thin plate support and the mechanical arm coupling system of the present invention;

[0066] Figure 4 It is a flow chart of the thin plate support-multi-degree-of-freedom mechanical arm vibration prediction of the present invention.

[0067] Figure 5 It is the thin plate support-multi-degree-of-freedom mechanical arm end vibration prediction displacement of the present invention; Figure 5 (a) Vibration displacement of the thin plate installation position; Figure 5 (b) ～ Figure 5 (d) are the vibration displacements of the end of the robot arm in the global coordinate x, y, and z directions, respectively.

[0068] In the figure: 1 is a box body, 2 is a robot arm support, 3 is a robot arm, and 4 is a working object. DETAILED DESCRIPTION

[0069] The present invention is further described below in conjunction with the embodiments, but it should not be understood that the above subject matter of the present invention is limited to the following embodiments. Without departing from the above technical ideas of the present invention, various substitutions and changes are made according to the common technical knowledge and customary means in the art, which should all be included in the protection scope of the present invention.

[0070] Embodiment 1:

[0071] See also Figures 1 to 5 A vibration prediction method for a single-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory comprises the following steps:

[0072] 1) The box support-multi-DOF robotic arm system is equivalent to a thin plate support-multi-DOF robotic arm coupling system;

[0073] 2) Establish a thin plate support-single degree of freedom robotic arm coordinate system;

[0074] 3) Based on the thin plate support-single degree of freedom robotic arm coordinate system, establish the coupling relationship between the thin plate and the robotic arm;

[0075] 4) Establishing the mapping relationship between the end effector of the robot arm in the global coordinate system; the end of the robot arm is the end that holds the work object;

[0076] 5) Construct the multi-degree-of-freedom vibration partial differential equation of the thin plate support-single-freedom manipulator system;

[0077] 6) numerically solve the obtained partial differential vibration equation to predict the vibration amplitude of the end effector of the robot arm;

[0078] 7) Determine whether the vibration amplitude of the end effector of the robot arm meets the accuracy requirements of the robot arm. If not, improve the thin plate support-single degree of freedom robot arm coupling system and return to step 6).

[0079] The box support-single degree of freedom robotic arm system comprises a box 1, a robotic arm support 2, and a robotic arm 3;

[0080] The mechanical arm support 2 is fixed on the box body 1;

[0081] One end of the robot arm 3 is fixed on the robot arm support 2, and the other end is used to clamp the work object 4;

[0082] The robot arm 3 has multiple degrees of freedom.

[0083] The thin plate support-single degree of freedom mechanical arm coupling system comprises a rectangular thin plate, a mechanical arm mounting base, and a mechanical arm;

[0084] The length, width and height of the rectangular thin plate are determined by the length, width and height of the box 1, which are denoted as a, b and h respectively;

[0085] The mechanical arm mounting base is fixed on a rectangular thin plate;

[0086] The mass of the robotic arm mounting base is m 1 , the distance from the center of mass of the robot arm mounting base to the upper surface of the rectangular thin plate is H 1 ′;

[0087] One end of the mechanical arm is fixed on the mechanical arm mounting base, and the other end is used to clamp the working object;

[0088] The mass of the robotic arm is m 2 , the installation position of the robot arm is (x 0 ,y 0), the azimuth angle of the robot is α, the rotation angle of the robot is θ, and the distance from the center of mass of the robot to the secondary axis of rotation of the robot is p.

[0089] The thin plate support-single degree of freedom manipulator coordinate system includes the system global coordinate system o-xyz, the thin plate static coordinate system o l -x l y l z l , thin plate moving coordinate system o t -x t y t z t , the connected coordinate system o of the i-th rod of the robot pi -x pi y pi z pi .

[0090] The method for establishing the thin plate support-multi-DOF manipulator coordinate system is as follows: Based on Kirchhoff plate theory and space description method, considering the manipulator at any installation position (x 0 ,y 0 ), installation angle α and plate inclination In this case, the Figure 3 The mixed coordinate system shown in Figure 2 is used for the convenience of description. The following expression is defined: Material lines u and v are the projections of the motor shaft axis and the robot arm axis on the neutral layer of the thin plate when the thin plate is not deformed. l is the intersection of material lines u and v, and there exists: u⊥v. When the thin plate is deformed, the material lines u, v become u′ and v′, o l becomes o t , o l o T =w(x 0 ,y 0 ,t). The system coordinate system is established as follows: o-xyz is the system global coordinate system. l -x l y l z l The static coordinate system of the thin plate is Figure 3 (b) shows that the origin of the coordinate system is at the o-xyz position (x 0 ,y 0 ), x l and l A rectangular coordinate system whose axes coincide with the material lines u and v respectively. Figure 3 (c) as shown, o t -x t y t z t is the moving rectangular coordinate system of the thin plate, y t The axis is the tangent line of the material line v′, z tThe neutral surface of the thin plate after deformation t The normal of a point. Figure 3 (d) as shown, o 2 -x m y m z m is the local rectangular coordinate system of the robot base, o 2 y m Coincident with the motor axis, when the thin plate does not move, there exists: o 2 -x m y m z m ∥o l -x l y l z l . o p -x p y p z p is the connected coordinate system of the robot arm, which is used to describe the rotation angle of the robot arm. 2 -x m y m z m Along o 2 z m Rotate θ to get .

[0091] The coupling relationship between the thin plate and the robot arm is determined by the thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is represented;

[0092] Thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is as follows:

[0093]

[0094] In the formula, r 2 is the moving coordinate system o of the robot arm in the thin plate t -x t y t z t The position vector in ; is the coordinate system o t -xt y t z t In the coordinate system o l -x l y l z l The position vector in w o,xl and w o,yl for o t x t Axis and o l x l Axis, o t y t Axis and o l y l The angle between the axes; R 2 is a matrix; r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l Position vector in the global coordinate system;

[0095] Among them, the tangent line ξ of the material line u′, the tangent line ζ of the material line v′ and the normal line η on the neutral surface of the thin plate are as follows:

[0096] ξ=[10w 0,xl ] T (3)

[0097] ζ=[01w 0,yl ] T (4)

[0098] η=ξ×ζ=[-w 0,xl -w 0,yl 1] T (5).

[0099] The mapping relationship of the robot arm end effector in the global coordinate system is as follows:

[0100]

[0101] Among them, r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l The position vector in the global coordinate system, x p ,y p ,z pis the displacement parameter of the end effector of the robot arm in the x, y, and z directions in the global coordinate system; r p is the position vector of the robot end effector in the global coordinate system.

[0102] The method of constructing the multi-degree-of-freedom vibration partial differential equation of the thin plate support-single-freedom manipulator system is: constructing the multi-degree-of-freedom vibration partial differential equation based on the Hamiltonian principle.

[0103] The steps of constructing the multi-degree-of-freedom vibration partial differential equation of the thin plate support-single-freedom manipulator system include:

[0104] a1) Based on the coordinate representation of the robot arm actuator in the global coordinate system, calculate the system kinetic energy T, that is:

[0105] T=T P +T m +T l (7)

[0106] Among them, the kinetic energy of the thin plate T P , motor and robot arm installation base kinetic energy T m 、Kinetic energy of robot arm T l They are as follows:

[0107]

[0108] Where, ρ, m 1 , m 2 , are the volume density of the thin plate, the mass of the motor base, and the mass of the robot arm respectively; J 1,x , J 1,y For the motor base about o t x t and t y t Moment of inertia; J 2,x , J 2,y For the robot arm about o t x t and t y t Moment of inertia; J m is the moment of inertia of the motor; a, b, h are the length, width, and height of the rectangular plate; is the differential of the rotation angle of the robot arm; θ r is the robot arm rotation angle. is the derivative of () in the time domain; The end effector of the robot arm is t x t and t y t The position component on ;

[0109] a2) Calculate the system potential energy V, that is:

[0110]

[0111] Among them, υ and D are the Poisson's ratio and bending stiffness of the thin plate respectively; θ r are the rotation angles of the robot arm; g is the acceleration of gravity; p is the distance from the center of mass of the robot arm to the secondary axis of rotation of the robot arm; w is the deflection of the thin plate, w 0 is the deflection of the thin plate at the installation position of the robot arm, H 1 H is the distance from the center of mass of the mounting base to the upper surface of the rectangular thin plate, 2 The distance from the motor center of mass to the upper surface of the rectangular plate.

[0112] a3) Calculate the virtual work W acting on the system, that is:

[0113]

[0114] Where τ is the torque applied to the motor; c r is the Rayleigh dissipation coefficient of the manipulator rotation joint, c p is the viscous damping coefficient of the thin plate; δ is the symbol of variation. r 、c p is the damping coefficient;

[0115] a4) Based on Hamilton’s principle, construct the equation:

[0116]

[0117] In the formula, δ(·) is the variation of (·), t 1 and t 2 are two time constants, t 1 <t<t 2 ; For the convenience of expression, the following variational expression of deformation related quantities is introduced:

[0118]

[0119] a5) Substitute the system kinetic energy T, system potential energy V, and virtual work W acting on the system into equation (13) to obtain the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system, namely:

[0120]

[0121] Where Q(t), M(q,θ), K, C, and F are the generalized coordinates, mass matrix, stiffness matrix, damping matrix, and external force terms of the system, which are determined by the system configuration and parameters.

[0122] Q(t)=[q(t)θθ r ] T (m+2)×1

[0123]

[0124] F=[f 1 f 2 τ] T (m+2)×1

[0125]

[0126] f 2 =-m 2 gp(-sθψ 0,x q+cθ)

[0127] The steps to predict the vibration amplitude of the robot end effector include:

[0128] b1) Set the torque applied to the motor Among them, M 0 and ω are the torque constant parameters applied to the manipulator;

[0129] b2) Solve the multi-degree-of-freedom partial differential vibration equation of the thin plate support-single-degree-of-freedom robotic arm system to obtain the vibration amplitude of the robotic arm end effector.

[0130] Methods for improving the thin plate support-multi-degree-of-freedom robotic arm coupling system include: increasing the thickness of the thin plate and strengthening the rib plate.

[0131] Embodiment 2:

[0132] A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on a Kirchhoff plate comprises the following steps:

[0133] 1) The box support-multi-DOF robotic arm system is equivalent to a thin plate support-multi-DOF robotic arm coupling system;

[0134] 2) Establish a thin plate support-multi-degree-of-freedom robotic arm coordinate system;

[0135] 3) Based on the thin plate support-multi-DOF robotic arm coordinate system, establish the coupling relationship between the thin plate and the robotic arm;

[0136] 4) Establishing the mapping relationship between the end effector of the robot arm in the global coordinate system; the end of the robot arm is the end that holds the work object;

[0137] 5) Construct the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system;

[0138] 6) numerically solve the obtained partial differential vibration equation to predict the vibration amplitude of the end effector of the robot arm;

[0139] 7) Determine whether the vibration amplitude of the end effector of the robot arm meets the accuracy requirements of the robot arm. If not, improve the thin plate support-multi-degree-of-freedom robot arm coupling system and return to step 6).

[0140] Embodiment 3:

[0141] A vibration prediction method for a multi-degree-of-freedom robotic arm system supported by a thin plate based on Kirchhoff plate theory, wherein the calculation content is the same as that of Example 2, and further, the box support-multi-degree-of-freedom robotic arm system comprises a box 1, a robotic arm support 2, and a robotic arm 3;

[0142] The mechanical arm support 2 is fixed on the box body 1;

[0143] One end of the robot arm 3 is fixed on the robot arm support 2, and the other end is used to clamp the work object 4;

[0144] The robot arm 3 has multiple degrees of freedom.

[0145] Embodiment 4:

[0146] A vibration prediction method for a multi-degree-of-freedom robotic arm system under a thin plate support based on a Kirchhoff plate, wherein the calculation content is the same as any one of Embodiments 2-3, and further, the thin plate support-multi-degree-of-freedom robotic arm coupling system comprises a rectangular thin plate, a robotic arm mounting base, and a robotic arm;

[0147] The length, width and height of the rectangular thin plate are determined by the length, width and height of the box 1, which are denoted as a, b and h respectively;

[0148] The mechanical arm mounting base is fixed on a rectangular thin plate;

[0149] The mass of the robotic arm mounting base is m 1 , the distance from the center of mass of the robot arm mounting base to the upper surface of the rectangular thin plate is H 1 ′;

[0150] One end of the mechanical arm is fixed on the mechanical arm mounting base, and the other end is used to clamp the working object;

[0151] The mass of the robotic arm is m 2 , the installation position of the robot arm is (x 0 ,y 0 ), the azimuth angle of the robot is α, the rotation angle of the robot is θ, and the distance from the center of mass of the robot to the secondary axis of rotation of the robot is p.

[0152] Embodiment 5:

[0153] A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory, wherein the calculation content is the same as any one of embodiments 2-4, and further, the thin plate support-multi-degree-of-freedom manipulator coordinate system includes a system global coordinate system o-xyz, a thin plate static coordinate system o l -x l y l z l , thin plate moving coordinate system o t -x t y t z t , the connected coordinate system o of the i-th rod of the robot pi -x pi y pi z pi .

[0154] Embodiment 6:

[0155] A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory, wherein the calculation content is the same as any one of Embodiments 2-5, and further, the coupling relationship between the thin plate and the manipulator is expressed by the thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is represented;

[0156] Thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is as follows:

[0157]

[0158] In the formula, r 2 is the moving coordinate system o of the robot arm in the thin plate t -x t y t z t The position vector in ; is the coordinate system o t -x t y t z t In the coordinate system o l -x l y l zl The position vector in w o,xl and w o,yl for o t x t Axis and o l x l Axis, o t y t Axis and o l y l The angle between the axes; R 2 is a matrix; r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l Position vector in the global coordinate system;

[0159] Among them, the tangent line ξ of the material line u′, the tangent line ζ of the material line v′ and the normal line η on the neutral surface of the thin plate are as follows:

[0160] ξ=[10w 0,xl ] T (3)

[0161] ζ=[01w 0,yl ] T (4)

[0162] η=ξ×ζ=[-w 0,xl -w 0,yl 1] T (5).

[0163] Embodiment 7:

[0164] A vibration prediction method for a multi-degree-of-freedom robotic arm system supported by a thin plate based on Kirchhoff plate theory, wherein the calculation content is the same as any one of Embodiments 2-6, and further, the mapping relationship of the robotic arm end effector in the global coordinate system is as follows:

[0165]

[0166] Among them, r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l The position vector in the global coordinate system, x p ,y p ,z pis the displacement parameter of the end effector of the robot arm in the x, y, and z directions in the global coordinate system; r p is the position vector of the robot end effector in the global coordinate system.

[0167] Embodiment 8:

[0168] A vibration prediction method for a multi-degree-of-freedom robotic arm system under a thin plate support based on Kirchhoff's plate theory, wherein the calculation content is the same as any one of Examples 2-7. Furthermore, a method for constructing a multi-degree-of-freedom vibration partial differential equation for a thin plate support-multi-freedom robotic arm system is as follows: based on the Hamiltonian principle, Kirchhoff's plate theory is used to construct a multi-degree-of-freedom vibration partial differential equation.

[0169] Embodiment 9:

[0170] A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory, wherein the calculation content is the same as any one of Embodiments 2-8, and further, the step of constructing a multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-degree-of-freedom manipulator system comprises:

[0171] a1) Based on the coordinate representation of the robot arm actuator in the global coordinate system, calculate the system kinetic energy T, that is:

[0172] T=T P +T m +T l (7)

[0173] Among them, the kinetic energy of the thin plate T P , motor and robot arm installation base kinetic energy T m 、Kinetic energy of robot arm T l They are as follows:

[0174]

[0175] Where, ρ, m 1 , m 2 , are the volume density of the thin plate, the mass of the motor base, and the mass of the robotic arm respectively; J 1,x , J 1,y For the motor base about o t x t and t y t Moment of inertia; J 2,x , J 2,y For the robot arm about o t x t and t y t Moment of inertia; J m is the moment of inertia of the motor; a, b, h are the length, width, and height of the rectangular plate; is the differential of the rotation angle of the robot arm; θ r is the robot arm rotation angle. The end effector of the robot arm is t x t and t y t The position component on ;

[0176] a2) Calculate the system potential energy V, that is:

[0177]

[0178] Among them, υ and D are the Poisson's ratio and bending stiffness of the thin plate respectively; θ r are the rotation angles of the robot arm; g is the acceleration of gravity; p is the distance from the center of mass of the robot arm to the secondary axis of rotation of the robot arm; w is the deflection of the thin plate, w 0 is the deflection of the thin plate at the installation position of the robot arm, H 1 H is the distance from the center of mass of the mounting base to the upper surface of the rectangular thin plate, 2 The distance from the motor's center of mass to the upper surface of the rectangular plate.

[0179] a3) Calculate the virtual work W acting on the system, that is:

[0180]

[0181] Where, τ is the torque applied to the motor; τ is the torque applied to the motor; c r is the Rayleigh dissipation coefficient of the manipulator rotation joint, c p is the viscous damping coefficient of the plate.

[0182] a4) Based on Hamilton’s principle, construct the equation:

[0183]

[0184] In the formula, δ(·) is the variation of (·), t 1 and t 2 are two time constants, t 1 <t<t 2 ; For the convenience of expression, the following variational expression of deformation related quantities is introduced:

[0185]

[0186] a5) Substitute the system kinetic energy T, system potential energy V, and virtual work W acting on the system into equation (13) to obtain the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system, namely:

[0187]

[0188] Where M(q,θ), Q(t), C, q, K, and F are the generalized coordinates, mass matrix, stiffness matrix, damping matrix, and external force terms of the system.

[0189] Embodiment 10:

[0190] A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on a Kirchhoff plate, wherein the calculation content is the same as any one of Embodiments 2-9, and further, the step of predicting the vibration amplitude of the end effector of the manipulator comprises:

[0191] b1) Set the torque applied to the motor Among them, M 0 and ω are the torque constant parameters applied to the manipulator;

[0192] b2) Solve the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system to obtain the vibration amplitude of the end effector of the manipulator.

[0193] Embodiment 11:

[0194] A vibration prediction method for a multi-degree-of-freedom robotic arm system under a thin plate support based on Kirchhoff plate theory, wherein the calculation content is the same as any one of Examples 2-10. Furthermore, a method for improving the thin plate support-multi-degree-of-freedom robotic arm coupling system includes: increasing the thickness of the thin plate and strengthening the rib plate.

[0195] Embodiment 12:

[0196] A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory, the steps are as follows:

[0197] Step 1: Based on the structural design parameters and considering the main influencing factors, the box support-multi-DOF robotic arm system is equivalent to a thin plate support-multi-DOF robotic arm coupling system.

[0198] Step 2: Establish a thin plate support-multi-degree-of-freedom robotic arm coordinate system.

[0199] Step 3: Establish the coupling relationship between the thin plate and the robotic arm.

[0200] Step 4: Establish a mapping relationship between the end position of the robot arm and the global coordinate system.

[0201] Step 5, based on the Hamiltonian principle, Kirchhoff plate theory is used to obtain the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system.

[0202] Step 6: numerically solve the obtained partial differential vibration equation to predict the vibration amplitude of the end effector of the robot arm.

[0203] Step 7: Make a decision based on the vibration amplitude of the end effector of the robot arm to determine whether the robot arm accuracy requirements are met.

[0204] Step 1, such as Figure 1 As shown in Figure 1, according to the structural design parameters of the supporting thin plate and the multi-DOF manipulator on the box, the system is simplified into a rectangular thin plate support-multi-DOF manipulator coupling system. The specific structural parameters are as follows: Figure 2 As shown, the length, width, and height of the rectangular plate are a, b, and h respectively; the mass of the robot arm is m 2 , the distance from the center of mass of the robot arm to the rotation axis is p, and the installation position and azimuth of the robot arm on the plate are (x 0 ,y 0 ) and α; the mass of the robot arm installation base is m 1 , H 1 ′ is the distance from the center of mass of the mounting base to the upper surface of the rectangular thin plate; the rotation angle of the robot arm is θ.

[0205] Step 2 is to establish the thin plate support-multi-DOF manipulator coordinate system. The system coordinate system is established as follows: o-xyz is the global coordinate system of the system. l -x l y l z l The static coordinate system of the thin plate, o t -x t y t z t is the moving coordinate system of the thin plate, o pi -x pi y pi z pi is the connected coordinate system of the i-th rod of the robot arm.

[0206] Step 3 is to establish the coupling relationship between the thin plate and the robot arm. Assume that the deflection of the thin plate is w, w o,xl and w o,yl Describe o t x t ' axis and o l x l Axis, o t y t Axis and o l y l In order to establish the coupling relationship between the thin plate and the robot arm, it is necessary to establish the thin plate moving coordinate system o t -x t y t z t With the static coordinate system o l -x l y l z lThe coupled kinematic relationship between them can be described by establishing a mapping relationship between the two coordinate systems through the attitude description method. This mapping relationship represents the moving coordinate system of the thin plate. t -x t y t z t The unit vectors of each principal axis and the static coordinate system o of the plate l -x l y l z l The mapping relationship between the principal axes (3×3 unit rotation matrix). Assume that ξ′, ζ′ and η′ are the tangent lines of the material lines u′ and v′ on the neutral surface of the thin plate and the T The normal of l -x l y l z l The following can be expressed as:

[0207] ξ=[1 0 w 0,xl ] T (1)

[0208] ζ=[0 1 w 0,yl ] T (2)

[0209] η=ξ×ζ=[-w 0,xl -w 0,yl 1] T (3)

[0210] Assume that the thin plate is a small deflection plate, that is, w < < h. By normalization and ignoring high-order terms, the moving coordinate system of the thin plate is o t -x t y t z t With Jing l -x l y l z l The following relationship exists:

[0211]

[0212] Among them, r 2 is the robot arm in the moving coordinate system o t -x t y t z t Description in; is the coordinate system o t -x t y t z t In the coordinate system o l -x l y l zl The position vector in

[0213] Step 4: Establish the mapping relationship between the end position of the robot arm and the global coordinate system. The end effector of the robot arm can be expressed in the global coordinate system as:

[0214]

[0215] Among them, r 1 =[x 0 ,y 0 ,0] T is the coordinate system o l -x l y l z l In the global coordinate system, x p ,y p ,z p It represents the displacement of the robot end effector in the global coordinate system in the x, y, and z directions.

[0216] Step 5 is to establish the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system. The kinematic analysis of step 4 can obtain the coordinate representation of the manipulator actuator in the global coordinate system, and the kinetic energy of the system can be expressed as:

[0217] T=T P +T m +T l (7)

[0218] Among them, T P , T m and T l They are the kinetic energy of the thin plate, the motor base (including the motor) and the robotic arm, respectively, as shown below:

[0219]

[0220] In the above formula, ρ, m 1 , m 2 , J r , J 1,x , J 1,y , J 2,x , J 2,y They are the volume density of the thin plate, the mass of the motor base, the mass of the robotic arm, the motor base, and the robotic arm about o t x t and t y t moment of inertia.

[0221] The potential energy of the system is:

[0222]

[0223] Among them, υ and D are the Poisson's ratio and bending stiffness of the thin plate respectively; θ r are the rotation angles of the robotic arms; g is the acceleration due to gravity.

[0224] Assume that the viscous damping force of the thin plate is The damping force at the revolving joint is given by the Rayleigh dissipation function Considering the torque τ acting on the robot arm, the virtual work acting on the system is:

[0225]

[0226] Using Hamilton's principle:

[0227]

[0228] Substituting (7), (11), and (12) into (13), for the multi-freedom manipulator system, the derivation of the equation is cumbersome and prone to errors. By performing symbolic operations on equation (13), the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system is obtained as follows:

[0229]

[0230] Step 6, numerically solve the obtained partial differential vibration equation to predict the vibration amplitude of the end effector of the robot arm. The present invention assumes that the torque τ applied to the motor is a linear function of the angular velocity of the robot arm: Among them, M 0 and ω are the torque constant parameters applied to the manipulator, which define the relationship between the torque input and the manipulator response θ. The simulation parameters in this example are: M 0 =0.4,ω=1.5rad / s.

[0231] Equation (14) is a nonlinear equation. The dynamic response of the thin plate and the robot arm can be solved by numerical methods such as the Runge-Kutta method. Figure 5 (a~d) are the vibration displacement of the thin plate and the displacement of the end effector of the robot arm in the x, y, and z directions, respectively. It can be found that the displacement errors of the end effector of the robot arm in the x, y, and z directions are 0.94, 0.18, and 4.62 mm, respectively, indicating that there is a serious dynamic coupling effect between the thin plate and the robot arm, that is, the execution accuracy of the robot arm is greatly affected by the vibration displacement of the thin plate.

[0232] Step 7, based on the vibration amplitude of the end effector of the robot arm obtained in step 6, make a decision whether it meets the accuracy requirements of the robot arm. If not, redesign the weak structure of the system, such as increasing the thickness of the thin plate or using a reinforcing rib plate. Repeat step 6 until the posture of the end effector of the robot arm meets the accuracy requirements of the system. At this time, the structural parameters are reasonable design parameters of the system.

[0233] In the process of system dynamics modeling, since the dynamics modeling method of multi-degree-of-freedom rigid robotic arms is already very mature, the multi-rod robotic arm in the modeling method proposed in the present invention only takes the first robotic arm and its base. The corresponding modeling method can be extended to the dynamics modeling, vibration prediction and structural parameter design of more general thin plate support-multi-degree-of-freedom robotic arm systems.

Claims

1. A vibration prediction method for a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory, characterized in that: The following steps are involved: 1) The box support-multi-DOF robotic arm system is equivalent to a thin plate support-multi-DOF robotic arm coupling system; 2) Establish a thin plate support-multi-degree-of-freedom robotic arm coordinate system; 3) Based on the thin plate support-multi-DOF robotic arm coordinate system, establish the coupling relationship between the thin plate and the robotic arm; 4) Establishing the mapping relationship between the end effector of the robot arm in the global coordinate system; the end of the robot arm is the end that holds the work object; 5) Construct the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system; 6) numerically solve the obtained partial differential vibration equation to predict the vibration amplitude of the end effector of the robot arm; 7) Determine whether the vibration amplitude of the end effector of the robot arm meets the accuracy requirements of the robot arm. If not, improve the thin plate support-multi-degree-of-freedom robot arm coupling system and return to step 6).

2. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1 is characterized in that: The box support-multi-degree-of-freedom robotic arm system comprises a box (1), a robotic arm support (2), and a robotic arm (3); The mechanical arm support (2) is fixed on the box body (1); One end of the mechanical arm (3) is fixed on the mechanical arm support (2), and the other end is used to clamp the work object (4); The mechanical arm (3) has multiple degrees of freedom.

3. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 2 is characterized in that: The thin plate support-multi-DOF robotic arm coupling system comprises a rectangular thin plate, a robotic arm mounting base, and a robotic arm; The length, width and height of the rectangular thin plate are determined by the length, width and height of the box body (1), and are denoted as a, b and h respectively; The mechanical arm mounting base is fixed on a rectangular thin plate; The mass of the robot arm mounting base is m1, and the distance from the center of mass of the robot arm mounting base to the upper surface of the rectangular thin plate is H1′; One end of the mechanical arm is fixed on the mechanical arm mounting base, and the other end is used to clamp the working object; The mass of the robotic arm is m2, the installation position of the robotic arm is (x0, y0), the azimuth angle of the robotic arm is α, the rotation angle of the robotic arm is θ, and the distance from the center of mass of the robotic arm to the secondary axis of rotation of the robotic arm is p.

4. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1, characterized in that: The thin plate support-multi-DOF manipulator coordinate system includes a system global coordinate system o-xyz; a thin plate static coordinate system o l -x l y l z l , thin plate moving coordinate system o t -x t y t z t , the connected coordinate system o of the i-th rod of the robot pi -x pi y pi z pi .

5. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 4, characterized in that: The coupling relationship between the thin plate and the robot arm is determined by the thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is represented; Thin plate moving coordinate system o t -x t y t z t With the thin plate static coordinate system o l -x l y l z l The mapping relationship between them is as follows: In the formula, r2 is the coordinate system of the robot arm in the thin plate o t -x t y t z t The position vector in ; is the coordinate system o t -x t y t z t In the coordinate system o l -x l y l z l The position vector in w o,xl and w o,yl for o t x t Axis and o l x l Axis, o t y t Axis and o l y l The angle between the axes; R2 is a matrix; r1 = [x0, y0, 0] T is the coordinate system o l -x l y l z l Position vector in the global coordinate system; Among them, the tangent line ξ of the material line u′, the tangent line ζ of the material line v′ and the normal line η on the neutral surface of the thin plate are as follows: ξ=[10w 0,xl ] T (3) ζ=[01w 0,yl ] T (4) η=ξ×ζ=[-w 0,xl -w 0,yl 1] T (5)。 6. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1, characterized in that: The mapping relationship of the robot end effector in the global coordinate system is as follows: Where r1=[x0,y0,0] T is the coordinate system o l -x l y l z l The position vector in the global coordinate system, x p ,y p ,z p is the displacement parameter of the end effector of the robot arm in the x, y, and z directions in the global coordinate system; r p is the position vector of the robot end effector in the global coordinate system.

7. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1, characterized in that: The method for constructing the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system is: constructing the multi-degree-of-freedom vibration partial differential equation based on the Hamiltonian principle.

8. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1, characterized in that: The steps of constructing the multi-degree-of-freedom vibration partial differential equation of the thin plate support-multi-freedom manipulator system include: 1) Based on the coordinate representation of the robot arm actuator in the global coordinate system, calculate the system kinetic energy T, that is: T=T P +T m +T l (7) Among them, the kinetic energy of the thin plate T P , motor and robot arm installation base kinetic energy T m 、Kinetic energy of robot arm T l They are as follows: Where ρ, m1, and m2 are the volume density of the thin plate, the mass of the motor base, and the mass of the robotic arm, respectively; J 1,x , J 1,y For the motor base about o t x t and t y t Moment of inertia; J 2,x , J 2,y For the robot arm about o t x t and t y t Moment of inertia; J m is the moment of inertia of the motor; a, b, h are the length, width, and height of the rectangular plate; is the differential of the rotation angle of the robot arm; θ r is the robot arm rotation angle; is the derivative of () in the time domain; The end effector of the robot arm is t x t and t y t The position component on ; 2) Calculate the system potential energy V, that is: Among them, υ and D are the Poisson's ratio and bending stiffness of the thin plate respectively; θ r are the rotation angles of the robot arm respectively; g is the acceleration of gravity; p is the distance from the center of mass of the robot arm to the secondary axis of rotation of the robot arm; w is the deflection of the thin plate, w0 is the deflection of the thin plate at the installation position of the robot arm, H1 is the distance from the center of mass of the mounting base to the upper surface of the rectangular thin plate, and H2 is the distance from the center of mass of the motor to the upper surface of the rectangular thin plate. 3) Calculate the virtual work W acting on the system, assuming that the viscous damping force of the thin plate is The damping force at the revolving joint is given by the Rayleigh dissipation function Get, that is: Where τ is the torque applied to the motor; δ is the symbol of the variation; c r 、c p is the damping coefficient; 4) Based on Hamilton's principle, construct the equation: Where δ(·) is the variation of (·), t1 and t2 are two time constants, t1<t<t2; 5) Substituting the system kinetic energy T, system potential energy V, and virtual work W acting on the system into equation (13), we can obtain the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system, namely: Where Q(t), M(q,θ), K, C, and F are the generalized coordinates, mass matrix, stiffness matrix, damping matrix, and external force terms of the system.

9. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1, characterized in that: The steps to predict the vibration amplitude of the robot end effector include: 1) Set the torque applied to the motor Where M0 and ω are the torque constant parameters applied to the manipulator; 2) Solve the multi-degree-of-freedom partial differential vibration equation of the thin plate support-manipulator system to obtain the vibration amplitude of the end effector of the manipulator.

10. The vibration prediction method of a multi-degree-of-freedom manipulator system supported by a thin plate based on Kirchhoff plate theory according to claim 1, characterized in that: Methods for improving the thin plate support-multi-degree-of-freedom robotic arm coupling system include: increasing the thickness of the thin plate and strengthening the rib plate.

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