Heavy-load multi-degree-of-freedom envelope forming equipment error prediction method

Abstract

The present application relates to a kind of heavy load multi-degree-of-freedom envelope forming equipment error prediction method, heavy load multi-degree-of-freedom envelope forming equipment adopts six branch chain parallel configuration, servo motor is connected with ball screw through coupling, ball screw is connected with slider, slider is connected with upper ball seat, upper ball seat and the upper ball head of connecting rod form spherical pair, the lower ball head of connecting rod and lower ball seat form spherical pair;Error prediction method includes the following steps: S1, establish multi-degree-of-freedom envelope forming equipment kinematics model;S2, determine multi-degree-of-freedom envelope forming equipment error expression method;S3, determine multi-degree-of-freedom envelope forming equipment error transmission calculation method;S4, carry out multi-degree-of-freedom envelope forming equipment error sensitivity analysis;S5, determine multi-degree-of-freedom envelope forming equipment error calibration scheme;S6, determine multi-degree-of-freedom envelope forming equipment error prediction method.By the present application, the operation and control precision of equipment can be significantly improved, so as to realize the high-precision forming of thin-walled high-gusset extreme structure.

Description

Technical Field

[0001] This invention relates to the field of forming equipment technology, and more specifically, to an error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment. Background Technology

[0002] Multi-degree-of-freedom (DOF) envelope forming technology is an advanced continuous local loading metal forming technology characterized by low forming force, low energy consumption, high precision, and good flexibility. It can form large components with relatively small forces, making it particularly suitable for manufacturing complex, high-performance aerospace thin-walled, high-ribbed components. Parallel mechanisms can serve as a good carrier for multi-DOF envelope forming processes, achieving multi-DOF motion. They offer advantages such as high stiffness, low error accumulation, and good dynamic performance, meeting the requirements of precise mechanism control and heavy-load forming while achieving equipment lightweighting. The mold of multi-DOF parallel servo-driven envelope forming equipment requires real-time matching of the mold's envelope motion and force according to theoretical design for component envelope forming. However, the presence of machining and assembly errors in equipment components will significantly affect the motion trajectory of the envelope forming mold, thus impacting the high-precision forming of the component. Machining and assembly errors are unavoidable in equipment manufacturing; due to the multi-branch structure of parallel equipment, the transmission of machining and assembly errors becomes extremely complex. Currently, there are no reports on error prediction methods for heavy-duty multi-DOF envelope forming equipment. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide an error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment, which can significantly improve the operation and control accuracy of the equipment.

[0004] The technical solution adopted by this invention to solve its technical problem is as follows: A method for predicting errors in heavy-duty multi-degree-of-freedom envelope forming equipment is constructed. The heavy-duty multi-degree-of-freedom envelope forming equipment adopts a six-branch parallel configuration. A servo motor is connected to a ball screw via a coupling. The ball screw is connected to a slider. The slider is connected to an upper ball seat. The upper ball seat and the upper ball head of the connecting rod form a spherical pair. The lower ball head of the connecting rod and the lower ball seat form a spherical pair. Six identical branches are connected to an upper mold base according to a certain assembly angle relationship. The upper mold is connected to the upper mold base. The servo motor, ball screw, slider, and upper ball seat are mounted on the forming equipment frame. The error prediction method includes the following steps:

[0005] S1. Establish a kinematic model of a multi-degree-of-freedom envelope forming equipment;

[0006] S2. Determine the error expression method for multi-degree-of-freedom envelope forming equipment;

[0007] S3. Determine the error propagation calculation method for multi-degree-of-freedom envelope forming equipment;

[0008] S4. Conduct error sensitivity analysis on multi-degree-of-freedom envelope forming equipment;

[0009] S5. Determine the error calibration scheme for multi-degree-of-freedom envelope forming equipment;

[0010] S6. Determine the error prediction method for multi-degree-of-freedom envelope forming equipment.

[0011] According to the above scheme, in step S1, a coordinate system S is established on the static platform of the equipment frame. A (O A -x A y A z A The origin of the coordinate system is the center O of the static platform of the machine frame. A , z A The axis is perpendicular to the stationary platform of the frame; a coordinate system S is established on the mold base of the equipment. C (O C -x C y C z C The origin of the coordinate system is the center O of the upper mold base. C , z C The axis is perpendicular to the upper mold base; the initial positions of the slider on the stationary platform of the machine frame are A1, A2, A3, A4, A5, A6, and the coordinate system S is as follows when the slider is in its initial position. A S C The corresponding coordinate axes are parallel to each other; the center of the upper ball head-upper ball seat of the connecting rod is B1, B2, B3, B4, B5, B6, and the center of the lower ball head-lower ball seat of the connecting rod is C1, C2, C3, C4, C5, C6.

[0012] When there are no machining or assembly errors in the equipment, A1, A2, A3, A4, A5, and A6 are in coordinate system S. A The position vector in is Determined by equation (1):

[0013]

[0014] In the formula, r A Let A1, A2, A3, A4, A5, and A6 be the radius of the circle in the plane containing A1, A2, A3, A4, A5, and A6. Let A1, A2, A3, A4, A5, and A6 be the central angles corresponding to two adjacent center points. The displacement of the six sliders relative to the stationary platform of the frame;

[0015] When there are no machining or assembly errors in the equipment, C1, C2, C3, C4, C5, and C6 are in coordinate system S. C The position vector in is Determined by equation (2):

[0016]

[0017] In the formula, r C Let C1, C2, C3, C4, C5, and C6 be the radii of the circles in the plane containing them. The central angles are the angles between two adjacent center points of C1, C2, C3, C4, C5, and C6.

[0018] When there are no machining or assembly errors in the equipment, the center point of the upper die during the multi-degree-of-freedom envelope forming process is in coordinate system S. C (O C -x C y C z C The spatial position in ) is represented by equation (3):

[0019] g p =[α p ,β p γ p x p y p , z p ] T (3)

[0020] In the formula, α p ,β p γ p Indicates revolving around x C y C z C The angle values ​​of the coordinate axes, x p y p , z p Indicates along x C y C z C The position values ​​of the coordinate axes;

[0021] According to equation (3), when the equipment has no machining or assembly errors, S C To S A coordinate transformation matrix Equation (4) represents:

[0022]

[0023] When there are no machining or assembly errors in the equipment, the vector constraint conditions of the equipment branches are determined by equation (5):

[0024]

[0025] In the formula, l i It is the length of the connecting rod;

[0026] Based on equations (1)-(5), the displacements of each slider corresponding to the multi-degree-of-freedom motion of the upper mold can be solved.

[0027] According to the above scheme, in step S2, the machining and assembly errors of the equipment consider the machining and assembly errors of the lead screw-lead screw cylinder, upper ball seat-upper ball head, connecting rod, and lower ball head-lower ball seat, and are expressed by small displacement rotation. Each lead screw, upper ball seat, and lower ball seat has a fixed coordinate system, with the origin of the lead screw coordinate system being... Located on its axis, the origin of the upper spherical coordinate system Located at the center of its spherical seat, the origin of the lower spherical seat coordinate system Located at the center of its sphere; Each chain screw cylinder, upper ball head, and lower ball head has a fixed coordinate system, and the origin of the screw cylinder coordinate system is also fixed. Located on its axis, the origin of the upper spherical head coordinate system Located at its center, the origin of the lower spherical head coordinate system Located at its center; when the equipment has no machining or assembly errors, the origin of the coordinate system is... respectively with Overlap, when there are machining and assembly errors in the equipment, the coordinate system From origin to coordinate system The errors between the origins are respectively That is, the total machining and assembly error of the 6-branch lead screw-cylinder is The total machining and assembly error of the 6-branch ball seat-upper ball head is The total machining and assembly error of the 6-branch lower ball head and lower ball seat is The total machining error of the 6 branch links is The above four total errors are expressed by the small displacement spinor shown in equation (6):

[0028]

[0029] Furthermore, the unknowns in equation (6) are obtained from equation (7):

[0030]

[0031] In equations (6)-(7), These are the machining and assembly errors of each branch screw-screw cylinder. These are the machining and assembly errors of the ball seat and ball head on each branch chain, specifically the rotation amount. These are the machining and assembly errors of the lower ball head and lower ball seat for each branch. Indicates the magnitude of the error. The unit of error is rotation.

[0032] According to the above scheme, in step S3, the error spinor is transformed into a coordinate transformation matrix for error propagation calculation using the exponential product formula. Through continuous coordinate transformation, the equipment processing and assembly error is accumulated to the reference coordinate system of the frame static platform.

[0033] In equation (6) Convert them into exponential product form, as shown in equation (8):

[0034]

[0035] In the formula, These are the error transfer coordinate transformation matrices between the fixed coordinate system of lead screw-lead screw cylinder, the fixed coordinate system of upper ball seat-upper ball head, and the fixed coordinate system of lower ball head-lower ball seat;

[0036] When the equipment has machining and assembly errors, A1, A2, A3, A4, A5, and A6 are in coordinate system S. A The position vector in is Determined by equation (9):

[0037]

[0038] In the formula, It can be obtained from equation (8);

[0039] When the equipment has machining and assembly errors, C1, C2, C3, C4, C5, and C6 are in coordinate system S. C The position vector in is Determined by equation (10):

[0040]

[0041] In the formula, It can be obtained from equation (8);

[0042] When the equipment has machining and assembly errors, the vector constraint condition of the equipment branch is determined by equation (11):

[0043]

[0044] In the formula, T e When there are machining and assembly errors in the equipment, coordinate system S C To coordinate system S A The coordinate transformation matrix;

[0045] When the equipment has machining and assembly errors, the center point of the upper die in coordinate system S during the multi-degree-of-freedom envelope forming process C (O C -x C y C z C The spatial position in ) is represented by equation (12):

[0046] g p1 =[α p1 ,β p1 γ p1x p1 y p1 , z p1 ] T (12)

[0047] In the formula, α p1 ,β p1 γ p1 Indicates revolving around x C y C z C The angle values ​​of the coordinate axes, x p1 y p1 , z p1 Indicates along x C y C z C The position values ​​of the coordinate axes;

[0048] α in equation (12) is obtained by measurement. p1 ,β p1 γ p1 x p1 y p1 , z p1 Then the coordinate system S C To coordinate system S A Coordinate transformation matrix T e Equation (13) represents:

[0049]

[0050] Performing vector operations on equations (12) and (3) yields the spinor of the upper mold motion error, which is expressed by equation (14):

[0051] δ p =g p1 -g p =[α p1 -α p ,β p1 -β p γ p1 -γ p x p1 -x p y p1 -y p , z p1 -z p ] T (14)

[0052] Convert the upper module motion error spinor in equation (14) into the form of the error propagation Jacobian matrix:

[0053]

[0054] In the formula, J gThe Jacobian matrix is ​​used to transfer motion errors of the upper model. For connecting rod machining errors, For lead screw-cylinder machining and assembly errors, For the machining and assembly errors of the upper ball seat and upper ball head, This refers to the machining and assembly error of the lower ball head and lower ball seat.

[0055] Based on equations (14) and (15), calculate

[0056] According to the above scheme, in step S4, the Jacobian matrix J of the upper model motion error in equation (15) g The singular value decomposition (SVD) of the form is expressed as:

[0057]

[0058] In the formula, [u1, u2, u3, u4, u5, u6] are J g J g T and J g T J g The eigenvector matrices σ1, σ2, σ3, σ4, σ5, σ6 are J g The singular values ​​of σ1, σ2, and σ3 are three angular terms related to the upper mold motion error, and σ4, σ5, and σ6 are three positional terms related to the upper mold motion error.

[0059] According to equation (16), J is obtained. g The singular values ​​σ1, σ2, σ3, σ4, σ5, σ6 are obtained, and the following 12 singular value cases are analyzed: (1) Maximum angular error sensitivity σ a =max(σ1, σ2, σ3), (2) average angular error sensitivity (3) Sensitivity of isotropic angular error ξ a =σ a / min(σ1, σ2, σ3), (4) Maximum position error sensitivity σ p =max(σ4, σ5, σ6), (5) Average position error sensitivity (6) Sensitivity to isotropic position error ξ p =σ p / min(σ4, σ5, σ6), (7) Maximum first singular value (8) Maximum second singular value (9) Maximum third singular value (10) Maximum fourth singular value (11) Maximum fifth singular value (12) The largest sixth singular value

[0060] According to the above scheme, in step S5, the 12 singular value cases will change with the movement of the upper mold. When the singular value is the largest, the sensitivity of the upper mold movement error is the highest, and the corresponding upper mold movement position is taken as the optimal calibration position, that is, the calibration position is determined as θ. i (i = 1, 2, 3...m), where m represents the total number of optimal calibration positions; from equation (15), we know that each branch has 6 × 3 + 1 = 19 error terms, so there are a total of 114 error terms to be solved for equipping 6 branches; g under each optimal calibration position p1 The measurement data can be used to construct 6 error equations, and solving for 114 error terms requires measuring g. p1 The minimum number of times is: n j =114 / 6=19, j represents the measurement of g p1 The minimum number of times; if j≤m, then the required optimal calibration position is sufficient, and 19 calibration positions can be randomly selected to measure the motion error data of the upper mold, and then substituted into equation (15) to solve; if j>m, then the required optimal calibration position is insufficient, and the optimal calibration position is determined according to equation (17):

[0061]

[0062] In the formula, n represents the interval θ between adjacent sensitive locations. 1 -θ 2 θ 2 -θ 3 ,…θ m-1 -θ m The number of sub-segments, k = 1, 2, 3, ..., (n-1) represents the k-th sub-segment.

[0063] According to the above scheme, in step S6, g at the optimal calibration position is measured according to equation (17). p1 , will g p1 Substituting into equation (14) to calculate δ p δ p Converted to spatial vector form:

[0064] δ p =[δ p0 δ p1 ] T (18)

[0065] In the formula, δ p0 δ p1 These are the screw quantities of the upper mold motion error, δ. p The original part and the dual part;

[0066] The motion error values ​​of each point on the upper die are predicted and calculated using the screw instantaneous axis theory; the position vector of any point P on the upper die is defined, and its direction points towards the motion error δ of the upper die. p The spinor axis decomposes the velocity of point P into tangential and normal velocities. According to the spinor instantaneous axis theory, the velocity of point P is a vector synthesis of the tangential and normal velocities, expressed by equation (19):

[0067]

[0068] In the formula, P c Represents coordinate system S C Origin O C The position vector to point P;

[0069] By calculating the instantaneous error of any point P in the upper model using equation (19), the instantaneous error prediction cloud map of each point in the upper model can be obtained.

[0070] The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment of the present invention has the following beneficial effects:

[0071] 1. Based on heavy-duty multi-degree-of-freedom envelope forming equipment, this invention proposes a method for expressing the processing and assembly errors of complex multi-branch envelope forming equipment. The error calculation is simple and the form is uniform, which simplifies the transmission and cumulative calculation of equipment processing and assembly errors.

[0072] 2. This invention proposes a method for predicting upper die motion error based on the screw instantaneous axis theory. By expressing the screw of upper die motion error, the instantaneous error value of each point of the upper die is calculated, which solves the difficult problem of predicting upper die motion error in complex multi-branch envelope forming equipment.

[0073] 3. By applying the upper mold motion error prediction method proposed in this invention, the motion error of the upper mold of the equipment can be accurately calculated, and then the control module can be used to accurately compensate for the motion error of the upper mold of the equipment, thereby realizing high-precision forming of thin-walled high-rib extreme structures. Attached Figure Description

[0074] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0075] Figure 1 A schematic diagram of the branch configuration of a multi-degree-of-freedom envelope forming equipment;

[0076] Figure 2 This is a schematic diagram of the overall configuration of a multi-degree-of-freedom envelope forming equipment;

[0077] Figure 3 A schematic diagram of the motion model of a multi-degree-of-freedom envelope forming equipment;

[0078] Figure 4 A schematic diagram illustrating the processing and assembly errors of multi-freedom envelope forming equipment;

[0079] Figure 5 A schematic diagram of error propagation modeling for a multi-degree-of-freedom envelope forming equipment;

[0080] Figure 6 The figure shows the calculation results of error prediction at various points of the mold for a multi-degree-of-freedom envelope forming equipment. Detailed Implementation

[0081] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0082] This invention provides a method for error prediction of heavy-duty multi-degree-of-freedom envelope forming equipment, comprising the following steps:

[0083] S1. Configuration Design of Multi-DOF Envelope Forming Equipment; To achieve heavy-duty multi-DOF envelope forming, this equipment adopts a six-branch parallel configuration. The six branches have identical structures, and the branch kinematic pair configuration is as follows: Figure 1 As shown: The servo motor is connected to the ball screw via a coupling, the ball screw is connected to the slider, the slider is connected to the upper ball seat, the upper ball seat and the upper ball head of the connecting rod form a spherical pair, and the lower ball head of the connecting rod and the lower ball seat form a spherical pair. The overall structure of the forming equipment is as follows. Figure 2 As shown: Six identical branches are connected to the upper mold base at specific assembly angles. The upper mold is connected to the upper mold base. Servo motors, ball screws, sliders, and upper ball seats are mounted on the forming equipment frame. Each branch motor and ball screw drives the corresponding slider to move up and down. The slider drives the upper ball seat-connecting rod-lower ball seat to move. The lower ball seat drives the upper mold base and the upper mold to move. Through the synthesis of the six branch movements, the upper mold achieves multi-degree-of-freedom motion. The lower mold drives the blank to move upward in a linear feed motion. Under the multi-degree-of-freedom loading action of the upper and lower molds, the blank undergoes continuous local plastic deformation, and the metal undergoes multi-directional orderly flow, ultimately forming a complex, thin-walled, highly ribbed extreme structure in one integral step.

[0084] S2. Establish a kinematic model of a multi-degree-of-freedom envelope forming equipment;

[0085] like Figure 3 As shown, a coordinate system S is established on the static platform of the equipment frame. A (O A -x A y A z A The origin of the coordinate system is the center O of the static platform of the machine frame. A , z A The axis is perpendicular to the stationary platform of the frame. A coordinate system S is established on the mold base of the equipment. C (O C -x C y C z CThe origin of the coordinate system is the center O of the upper mold base. C , z C The axis is perpendicular to the upper mold base. The initial positions of the slider on the stationary platform of the machine frame are A1, A2, A3, A4, A5, A6. When the slider is in its initial position, the coordinate system S... A S C The corresponding coordinate axes are parallel to each other. The center of the upper ball head to the upper ball seat of the connecting rod is B1, B2, B3, B4, B5, B6, and the center of the lower ball head to the lower ball seat of the connecting rod is C1, C2, C3, C4, C5, C6.

[0086] When there are no machining or assembly errors in the equipment, A1, A2, A3, A4, A5, and A6 are in coordinate system S. A The position vector in is Determined by equation (1):

[0087]

[0088] In the formula, r A Let A1, A2, A3, A4, A5, and A6 be the radius of the circle in the plane containing A1, A2, A3, A4, A5, and A6. Let A1, A2, A3, A4, A5, and A6 be the central angles corresponding to two adjacent center points. The displacement of the six sliders relative to the stationary platform of the frame.

[0089] When there are no machining or assembly errors in the equipment, C1, C2, C3, C4, C5, and C6 are in coordinate system S. C The position vector in is Determined by equation (2):

[0090]

[0091] In the formula, r C Let C1, C2, C3, C4, C5, and C6 be the radii of the circles in the plane containing them. Let be the central angles corresponding to two adjacent center points of C1, C2, C3, C4, C5, and C6.

[0092] When there are no machining or assembly errors in the equipment, the center point of the upper die during the multi-degree-of-freedom envelope forming process is in coordinate system S. C (O C -x C y C z C The spatial position in ) is represented by equation (3):

[0093] g p =[α p ,β p γ p x p y p , zp ] T (3)

[0094] In the formula, α p ,β p γ p Indicates revolving around x C y C z C The angle values ​​of the coordinate axes, x p y p , z p Indicates along x C y C z C The position values ​​of the coordinate axes.

[0095] According to equation (3), when the equipment has no machining or assembly errors, S C To S A coordinate transformation matrix Equation (4) represents:

[0096]

[0097] When there are no machining or assembly errors in the equipment, the vector constraint conditions of the equipment branches are determined by equation (5):

[0098]

[0099] In the formula, l i It is the length of the connecting rod.

[0100] Based on equations (1)-(5), the displacements of each slider corresponding to the multi-degree-of-freedom motion of the upper mold can be solved.

[0101] S3. Error Expression Method for Multi-Degree-of-Freedom Envelope Forming Equipment; The equipment machining and assembly errors consider the machining and assembly errors of the lead screw-lead screw cylinder, upper ball seat-upper ball head, connecting rod, and lower ball head-lower ball seat, and are expressed using small displacement rotation, such as... Figure 4 As shown. Each lead screw, upper ball seat, and lower ball seat has a fixed coordinate system, with the origin of the lead screw coordinate system being... It should be on its axis, at the origin of the upper spherical coordinate system. Located at the center of its spherical seat, the origin of the lower spherical seat coordinate system Located at the center of its sphere; Each chain screw cylinder, upper ball head, and lower ball head has a fixed coordinate system, and the origin of the screw cylinder coordinate system is also fixed. Located on its axis, the origin of the upper spherical head coordinate system Located at its center, the origin of the lower spherical head coordinate system It should be at its center. When the equipment has no machining or assembly errors, the origin of the coordinate system is... respectively with Overlap, when the equipment has machining and assembly errors, such as Figure 5 As shown, coordinate system From origin to coordinate system The errors between the origins are respectively That is, the total machining and assembly error of the 6-branch lead screw-cylinder is The total machining and assembly error of the 6-branch ball seat-upper ball head is The total machining and assembly error of the 6-branch lower ball head and lower ball seat is The total machining error of the 6 branch links is The above four total errors are expressed by the small displacement spinor shown in equation (6):

[0102]

[0103] Furthermore, the unknown quantity in equation (6) can be obtained from equation (7):

[0104]

[0105] In equations (6)-(7), These are the machining and assembly errors of each branch screw-screw cylinder. These are the machining and assembly errors of the ball seat and ball head on each branch chain, specifically the rotation amount. These are the machining and assembly errors of the lower ball head and lower ball seat for each branch. Indicates the magnitude of the error. The unit of error is rotation.

[0106] S4. Error propagation calculation method for multi-degree-of-freedom envelope forming equipment: The error spinor is transformed into a coordinate transformation matrix for error propagation calculation using the exponential product formula. Through continuous coordinate transformation, the equipment processing and assembly errors are accumulated to the reference coordinate system of the static platform of the machine frame.

[0107] In equation (6) Convert them into exponential product form, as shown in equation (8):

[0108]

[0109] In the formula, These are the error propagation coordinate transformation matrices between the lead screw-lead screw cylinder fixed coordinate system, the upper ball seat-upper ball head fixed coordinate system, and the lower ball head-lower ball seat fixed coordinate system, respectively.

[0110] When the equipment has machining and assembly errors, A1, A2, A3, A4, A5, and A6 are in coordinate system S. A The position vector in is Determined by equation (9):

[0111]

[0112] In the formula, It can be obtained from equation (8).

[0113] When the equipment has machining and assembly errors, C1, C2, C3, C4, C5, and C6 are in coordinate system S. C The position vector in is Determined by equation (10):

[0114]

[0115] In the formula, It can be obtained from equation (8).

[0116] When the equipment has machining and assembly errors, the vector constraint condition of the equipment branch is determined by equation (11):

[0117]

[0118] In the formula, T e When there are machining and assembly errors in the equipment, coordinate system S C To coordinate system S A The coordinate transformation matrix.

[0119] When the equipment has machining and assembly errors, the center point of the upper die in coordinate system S during the multi-degree-of-freedom envelope forming process C (O C -x C y C z C The spatial position in ) is represented by equation (12):

[0120] g p1 =[α p1 ,β p1 γ p1 x p1 y p1 , z p1 ] T (12)

[0121] In the formula, α p1 ,β p1 γ p1 Indicates revolving around x C y C z C The angle values ​​of the coordinate axes, x p1 y p1 , z p1 Indicates along x C y C z C The position values ​​of the coordinate axes.

[0122] α in equation (12) is obtained by measurement. p1 ,β p1 γ p1 x p1 y p1 , z p1 Then the coordinate system S C To coordinate system S A Coordinate transformation matrix T e Equation (13) represents:

[0123]

[0124] Performing vector operations on equations (12) and (3) yields the spinor of the upper mold motion error, which is expressed by equation (14):

[0125] δ p =g p1 -g p =[α p1 -α p ,β p1 -β p γ p1 -γ p x p1 -x p y p1 -y p , z p1 -z p ] T (14)

[0126] Convert the upper module motion error spinor in equation (14) into the form of the error propagation Jacobian matrix:

[0127]

[0128] In the formula, J g The Jacobian matrix is ​​used to transfer motion errors of the upper model. For connecting rod machining errors, For lead screw-cylinder machining and assembly errors, For the machining and assembly errors of the upper ball seat and upper ball head, This refers to the machining and assembly error of the lower ball head and lower ball seat.

[0129] Based on equations (14) and (15), calculate

[0130] S5. Error sensitivity analysis of multi-degree-of-freedom envelope forming equipment; J is the Jacobian matrix of upper mold motion error in equation (15). g The singular value decomposition (SVD) of the form is expressed as:

[0131]

[0132] In the formula, [u1, u2, u3, u4, u5, u6] are J g J g T and J g T J g The eigenvector matrices σ1, σ2, σ3, σ4, σ5, σ6 are J g The singular values ​​of σ1, σ2, and σ3 are three angular terms related to the upper mold motion error, and σ4, σ5, and σ6 are three position terms related to the upper mold motion error.

[0133] According to equation (16), J is obtained. g The singular values ​​σ1, σ2, σ3, σ4, σ5, σ6 are obtained, and the following 12 singular value cases are analyzed: (1) Maximum angular error sensitivity σ a =max(σ1, σ2, σ3), (2) average angular error sensitivity (3) Sensitivity of isotropic angular error ξ a =σ a / min(σ1, σ2, σ3), (4) Maximum position error sensitivity σ p =max(σ4, σ5, σ6), (5) Average position error sensitivity (6) Sensitivity to isotropic position error ξ p =σ p / min(σ4, σ5, σ6), (7) Maximum first singular value (8) Maximum second singular value (9) Maximum third singular value (10) Maximum fourth singular value (11) Maximum fifth singular value (12) The largest sixth singular value

[0134] S6. Error calibration scheme for multi-degree-of-freedom envelope forming equipment; the 12 singular value cases in S5 will change with the movement of the upper die. When the singular value is the largest, the sensitivity of the upper die movement error is the highest, and the corresponding upper die movement position is taken as the optimal calibration position, that is, the calibration position is determined as θ. i (i = 1, 2, 3...m), where m represents the total number of optimal calibration positions. From equation (15), we know that each branch has 6 × 3 + 1 = 19 error terms, so equipping 6 branches results in a total of 114 error terms to be solved. At each optimal calibration position, g... p1 The measurement data can be used to construct 6 error equations, and solving for 114 error terms requires measuring g. p1 The minimum number of times is: n j=114 / 6=19, j represents the measurement of g p1 The minimum number of times. If j≤m, then the required optimal calibration position is sufficient, and 19 calibration positions can be randomly selected to measure the motion error data of the upper mold, and then substituted into equation (15) to solve. If j>m, then the required optimal calibration position is insufficient, and the optimal calibration position is determined according to equation (17):

[0135]

[0136] In the formula, n represents the interval θ between adjacent sensitive locations. 1 -θ 2 θ 2 -θ 3 ,…θ m-1 -θ m The number of sub-segments, k = 1, 2, 3, ..., (n-1) represents the k-th sub-segment.

[0137] S7. Error prediction method for multi-degree-of-freedom envelope forming equipment; according to equation (17), measure g at the optimal calibration position. p1 , will g p1 Substituting into equation (14) to calculate δ p δ p Converted to spatial vector form:

[0138] δ p =[δ p0 δ p1 ] T (18)

[0139] In the formula, δ p0 δ p1 These are the screw quantities of the upper mold motion error, δ. p The original part and the dual part.

[0140] Since the screw force of the upper die motion error cannot directly predict the motion error value of each point on the upper die, the screw instantaneous axis theory is used to predict and calculate the motion error value of each point on the upper die. The position vector of any point P on the upper die is defined, with its direction pointing towards the upper die motion error δ. p The spinor axis decomposes the velocity of point P into tangential and normal velocities. According to the spinor instantaneous axis theory, the velocity of point P is a vector synthesis of the tangential and normal velocities, expressed by equation (19):

[0141]

[0142] In the formula, P c Represents coordinate system S C Origin O C The position vector to point P.

[0143] By calculating the instantaneous error of any point P in the upper model using equation (19), the instantaneous error prediction cloud map of each point in the upper model can be obtained.

[0144] Based on the design constraints of the multi-degree-of-freedom envelope forming equipment given in Table 1, the kinematic modeling of the equipment and the error sensitivity of the Jacobian matrix for error propagation are analyzed. The results of the error sensitivity at different motion positions of the upper die within the motion cycle are shown in Table 2. From the results in Table 2, it can be seen that the error-sensitive positions of the upper die are [1.3195, 2.0735, 2.5761, 2.8274, 4.7124, 5.215, 5.592, 5.969] rad. The given eight error-sensitive positions of the upper die do not satisfy g. p1 The minimum requirement of 19 measurements is met, therefore, a calibration scheme is reasonably formulated according to equation (17), and the measurement results of the upper mold motion error are given, as shown in Table 3. Using the measurement data in the calibration scheme in Table 3, the 114 machining and assembly error terms of the equipment in equation (15) are solved, as shown in Table 4. Using the data in Table 4, the instantaneous error screw of the upper mold motion can be calculated, and then, using equation (19), the predicted cloud map of the motion error of each point of the upper mold at different motion moments can be obtained, as shown in Table 4. Figure 6 As shown.

[0145] Table 1 Configuration parameters of multi-degree-of-freedom envelope forming equipment

[0146]

[0147] Table 2 Results of Error Sensitivity

[0148]

[0149]

[0150] Table 3 Calibration scheme and error measurement results

[0151]

[0152] Table 4 Calibration calculation error results

[0153]

[0154]

[0155] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A method for error prediction in heavy-duty multi-degree-of-freedom envelope forming equipment, wherein the heavy-duty multi-degree-of-freedom envelope forming equipment adopts a six-branch parallel configuration, a servo motor is connected to a ball screw via a coupling, the ball screw is connected to a slider, the slider is connected to an upper ball seat, the upper ball seat and the upper ball head of a connecting rod form a spherical pair, and the lower ball head of the connecting rod and the lower ball seat form a spherical pair; six identical branches are connected to an upper mold base according to a certain assembly angle relationship, the upper mold is connected to the upper mold base, and the servo motor, ball screw, slider, and upper ball seat are mounted on the frame of the forming equipment; characterized in that, Error prediction methods include the following steps: S1. Establish a kinematic model of a multi-degree-of-freedom envelope forming equipment; S2. Determine the error expression method for multi-degree-of-freedom envelope forming equipment; S3. Determine the error propagation calculation method for multi-degree-of-freedom envelope forming equipment; S4. Conduct error sensitivity analysis on multi-degree-of-freedom envelope forming equipment; S5. Determine the error calibration scheme for multi-degree-of-freedom envelope forming equipment; S6. Determine the error prediction method for multi-degree-of-freedom envelope forming equipment; In step S2, the machining and assembly errors of the equipment are considered in terms of the machining and assembly errors of the lead screw-lead screw cylinder, upper ball seat-upper ball head, connecting rod, and lower ball head-lower ball seat, and are expressed by small displacement rotation. In step S3, the error spinor is transformed into a coordinate transformation matrix for error propagation calculation using the exponential product formula. Through continuous coordinate transformation, the equipment processing and assembly errors are accumulated to the reference coordinate system of the static platform of the frame. An error propagation Jacobian matrix is ​​established and its singular value decomposition is performed for sensitivity analysis. The motion error values ​​of each point of the upper mold are predicted and calculated using the spinor instantaneous axis theory.

2. The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment according to claim 1, characterized in that, In step S1, a coordinate system is established on the static platform of the equipment frame. The origin of the coordinate system is the center of the static platform of the machine frame. , The axis is perpendicular to the stationary platform of the frame; a coordinate system is established on the mold base of the equipment. The origin of the coordinate system is the center of the upper mold base. , The axis is perpendicular to the upper mold base; the initial position of the slider on the stationary platform of the machine frame is... The coordinate system when the slider is in its initial position The corresponding coordinate axes are parallel to each other; the center of the upper ball head and the upper ball seat on the connecting rod is... The center of the lower ball head and the lower ball seat of the connecting rod is ; When the equipment has no machining or assembly errors, In coordinate system The position vector in is It is determined by equation (1): （1） In the formula, for The radius of the circle in the plane, for The central angles corresponding to two adjacent center points, The displacement of the six sliders relative to the stationary platform of the frame; When the equipment has no machining or assembly errors, In coordinate system The position vector in is It is determined by equation (2): （2） In the formula, for The radius of the circle in the plane, for The central angle between two adjacent center points; When there are no machining or assembly errors in the equipment, the center point of the upper die in the coordinate system during the multi-degree-of-freedom envelope forming process The spatial position in is represented by equation (3): （3） In the formula, Indicates circling The angle values ​​of the coordinate axes, Indicates along The position values ​​of the coordinate axes; According to equation (3), when the equipment has no machining or assembly errors... arrive coordinate transformation matrix Equation (4) represents: （4） When there are no machining or assembly errors in the equipment, the vector constraint conditions of the equipment branches are determined by equation (5): （5） In the formula, It is the length of the connecting rod; Based on equations (1)-(5), the displacements of each slider corresponding to the multi-degree-of-freedom motion of the upper mold can be solved. .

3. The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment according to claim 2, characterized in that, In step S2, the machining and assembly errors of the equipment consider the machining and assembly errors of the lead screw-lead screw cylinder, upper ball seat-upper ball head, connecting rod, and lower ball head-lower ball seat, and are expressed by small displacement rotation. Each lead screw, upper ball seat, and lower ball seat has a fixed coordinate system, with the origin of the lead screw coordinate system being... Located on its axis, the origin of the upper spherical coordinate system Located at the center of its spherical seat, the origin of the lower spherical seat coordinate system Located at the center of its sphere; Each chain screw cylinder, upper ball head, and lower ball head has a fixed coordinate system, and the origin of the screw cylinder coordinate system is also fixed. Located on its axis, the origin of the upper spherical head coordinate system Located at its center, the origin of the lower spherical head coordinate system Located at its center; when the equipment has no machining or assembly errors, the origin of the coordinate system is... respectively with Overlap, when there are machining and assembly errors in the equipment, the coordinate system From origin to coordinate system The errors between the origins are respectively That is, the total machining and assembly error of the 6-branch lead screw-cylinder is The total machining and assembly error of the 6-branch ball seat-ball head is The total machining and assembly error of the 6-branch lower ball head-lower ball seat is The total machining error of the 6 branch links is The total errors of the above four terms are expressed by the small displacement spinor shown in equation (6): （6） Furthermore, the unknowns in equation (6) are obtained from equation (7): （7） In equations (6)-(7), These are the machining and assembly errors of each branch screw-screw cylinder. These are the machining and assembly errors of the ball seat and ball head on each branch chain, specifically the rotation amount. These are the machining and assembly errors of the lower ball head and lower ball seat for each branch. Indicates the magnitude of the error. The unit of error is rotation.

4. The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment according to claim 3, characterized in that, In step S3, the error spinor is transformed into a coordinate transformation matrix for error propagation calculation using the exponential product formula. Through continuous coordinate transformation, the equipment processing and assembly error is accumulated to the reference coordinate system of the static platform of the frame. In equation (6) , , Convert them into exponential product form, as shown in equation (8): （8） In the formula, These are the error transfer coordinate transformation matrices between the fixed coordinate system of lead screw-lead screw cylinder, the fixed coordinate system of upper ball seat-upper ball head, and the fixed coordinate system of lower ball head-lower ball seat; When the equipment has machining and assembly errors, In coordinate system The position vector in is It is determined by equation (9): （9） In the formula, It can be obtained from equation (8); When the equipment has machining and assembly errors, In coordinate system The position vector in is It is determined by equation (10): （10） In the formula, It can be obtained from equation (8); When the equipment has machining and assembly errors, the vector constraint condition of the equipment branch is determined by equation (11): （11） In the formula, When there are machining and assembly errors in the equipment, the coordinate system To coordinate system The coordinate transformation matrix; When the equipment has machining and assembly errors, the center point of the upper die in the coordinate system during the multi-degree-of-freedom envelope forming process The spatial position in the middle is represented by equation (12): （12） In the formula, Indicates circling The angle values ​​of the coordinate axes, Indicates along The position values ​​of the coordinate axes; The value in equation (12) is obtained by measurement. Then the coordinate system To coordinate system coordinate transformation matrix Equation (13) represents: （13） Performing vector operations on equations (12) and (3) yields the spinor of the upper mold motion error, which is expressed by equation (14): （14） Convert the upper module motion error spinor in equation (14) into the form of the error propagation Jacobian matrix: （15） In the formula, The Jacobian matrix is ​​used to transfer motion errors of the upper model. For connecting rod machining errors, For lead screw-cylinder machining and assembly errors, For the machining and assembly errors of the upper ball seat and upper ball head, This refers to the machining and assembly error of the lower ball head and lower ball seat. Based on equations (14) and (15), calculate , , , .

5. The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment according to claim 4, characterized in that, In step S4, the Jacobian matrix of the upper model motion error in equation (15) The singular value decomposition (SVD) of the form is expressed as: （16） In the formula, yes and eigenvector matrix, yes The singular values ​​of, where These are three angular terms related to the motion error of the upper mold. These are three positional terms related to the motion error of the upper mold; According to equation (16), we obtain singular values And analyze the following 12 singular value cases: (1) Maximum angular error sensitivity (2) Sensitivity to average angle error (3) Sensitivity to isotropic angular error (4) Maximum position error sensitivity (5) Sensitivity to average position error (6) Sensitivity to isotropic position error (7) Maximum first singular value (8) Maximum second singular value (9) Maximum third singular value (10) The largest fourth singular value (11) The largest fifth singular value (12) The largest sixth singular value .

6. The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment according to claim 5, characterized in that, In step S5, the 12 singular value cases will change with the movement of the upper mold. When the singular value is the largest, the sensitivity of the upper mold movement error is the highest, and the corresponding upper mold movement position is taken as the optimal calibration position, that is, the calibration position is determined as... m represents the total number of optimal calibration positions; as shown in equation (15), each branch has If there are 114 error terms to be solved, then the equipment with 6 branches has a total of 114 error terms to be solved. At each optimal calibration position The measurement data can be used to construct 6 error equations, and solving for 114 error terms requires measuring... The minimum number of times is: j represents measurement The minimum number of times; if If the required optimal calibration position is sufficient, 19 calibration positions can be randomly selected to measure the motion error data of the upper mold, and then substituted into equation (15) to solve; if If the required optimal calibration position is insufficient, the optimal calibration position shall be determined according to formula (17): （17） In the formula, n represents the interval between adjacent sensitive locations. The number of sub-segments, Indicates the first Each segment.

7. The error prediction method for heavy-duty multi-degree-of-freedom envelope forming equipment according to claim 6, characterized in that, In step S6, the optimal calibration position is measured according to equation (17). ,Will Substituting into equation (14) yields the result. , Converted to spatial vector form: （18） In the formula, , These are the screw quantities of the upper mold motion error. The original part and the dual part; The motion error values ​​of each point on the upper die are predicted and calculated using the screw instantaneous axis theory; the position vector of any point P on the upper die is defined, and its direction points towards the motion error of the upper die. The spinor axis decomposes the velocity of point P into tangential and normal velocities. According to the spinor instantaneous axis theory, the velocity of point P is a vector synthesis of the tangential and normal velocities, expressed by equation (19): （19） In the formula, Representing the coordinate system origin The position vector to point P; By calculating the instantaneous error of any point P in the upper model using equation (19), the instantaneous error prediction cloud map of each point in the upper model can be obtained.

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