Mirror machining wall thickness error unified modeling method based on spinor theory

Abstract

The invention belongs to the technical field of numerical control machine tool machining, and discloses a mirror image machining wall thickness error unified modeling method based on a spinor theory, which comprises the following steps: analyzing a mirror image machining constraint relationship, establishing a geometric constraint among cutter-wall thickness-support head, describing nominal motion and error motion of a motion axis of a mirror image machining system by using the spinor theory, and establishing a mirror image machining wall thickness error model. Based on a multi-body system theory and a spinor theory, a machining side kinematic chain unified model and a supporting side kinematic chain unified model of a mirror image machining system are established respectively, so that a nominal wall thickness kinematic unified model is established based on mirror image machining geometric constraints; the motion axis error matrix is introduced into a nominal wall thickness kinematics unified model to establish a mirror image machining actual wall thickness kinematics unified model, and the difference value between the actual wall thickness and the nominal wall thickness unified model is a mirror image machining wall thickness error unified model.

Description

Technical Field

[0001] The invention belongs to the technical field of CNC machine tool processing, and specifically is a unified modeling method for mirror processing wall thickness error based on screw theory. Background Art

[0002] Mirror processing technology is one of the effective means to solve the manufacturing problems of high-performance large thin-walled parts represented by rocket fuel tank bottoms and aircraft skins. The mirror processing system consists of two moving heads, one for support and one for processing. During the processing, the two heads follow each other synchronously and always maintain a mirror relationship. The mirror processing "tool-workpiece-support" forms a kinematic closed chain, and the wall thickness accuracy is guaranteed by the associated constraint relationship between the three. This is feasible in principle for the processing of thin-walled parts with the remaining wall thickness as the key dimension.

[0003] The whole motion chain of the mirror processing system consists of the processing side motion chain and the support side motion chain. The single-side motion chain is the machine tool motion chain. During the processing, it is affected by various internal and external factors. The actual support trajectory and the processing trajectory deviate from the ideal trajectory, and the spatial posture of the tool and the support head produce deviation errors, resulting in residual wall thickness errors. Among the various error sources of the mirror processing system, geometric errors, thermal errors, and cutting force errors account for 65% of the total error and are the main factors affecting the wall thickness accuracy. Establishing a mirror processing wall thickness kinematic model, accurately calculating the actual posture relationship between the tool and the support head, and obtaining the mirror processing wall thickness error are of great significance to the prediction and control of the mirror processing accuracy of thin-walled parts.

[0004] In 2017, Xiao Juliang proposed a coordinated motion synchronization control method for wall thickness processing such as dual robot mirror milling in patent 201710908288.6. This method establishes the change relationship between the corresponding coordinate systems according to the position requirements of the tool and the support head, calculates the actual wall thickness of the workpiece by the relative distance between the tool and the support end in the Z-axis direction in the support head coordinate system, and subtracts the actual wall thickness from the designed wall thickness to obtain the wall thickness error. However, this method only obtains the actual wall thickness by bringing the parameters of each active joint of the robot into the robot kinematics forward solution model, and does not consider the errors of each motion component of the robot. It cannot reflect the transmission process of multi-source errors in the mirror processing system and its influence on the remaining wall thickness. In addition, the modeling method proposed based on the specific dual robot mirror processing is not suitable for promotion and use on other machine tools and has great limitations. In 2018, Zhang Shenggui proposed a five-axis structural error measurement method, system and device for a mirror milling system in patent 201810376777.6. This method establishes a geometric model of the spatial positioning error of the tool and support head in the processing area during the motion process, uses the processing head displacement measurement device to perform XYZ three-dimensional displacement error detection, and calculates the five-axis structural error characteristic value of the mirror processing system. However, this method only obtains the wall thickness error through the measurement device, does not consider the structural type of the mirror processing system and the multi-source error elements, and does not indicate the specific mirror processing wall thickness error kinematic modeling process. The wall thickness error value directly obtained by measurement cannot fully reflect the wall thickness change of the workpiece during the processing process.

[0005] In summary, the current mirror processing wall thickness error model is a specific error model established for a specific mirror processing machine tool structure. These models are only applicable to specific mirror processing systems. When facing mirror processing systems with different structures, they must be remodeled, which lacks universality. Therefore, it is necessary to establish a unified wall thickness error model applicable to any mirror processing system structure. Summary of the invention

[0006] The purpose of the present invention is to provide a unified modeling method for mirror processing wall thickness error based on screw theory, analyze the constraint relationship of mirror processing, establish the geometric constraint between "tool-wall thickness-support head", use screw theory to describe the nominal motion and error motion of the motion axis of the mirror processing system, and establish the unified kinematic chain model of the processing side and the support side of the mirror processing system based on the multi-body system theory and screw theory, so as to establish a unified kinematic model of nominal wall thickness based on the geometric constraint of mirror processing, introduce the motion axis error matrix into the unified kinematic model of nominal wall thickness to establish the unified kinematic model of actual wall thickness of mirror processing, and subtract the unified kinematic model of actual wall thickness from the unified model of nominal wall thickness to obtain the unified kinematic model of wall thickness error of mirror processing. When modeling the mirror processing system of a specified structure, it is only necessary to sort out the topological structure of the target mirror processing system and bring in the unified kinematic model of wall thickness error of mirror processing to obtain the wall thickness error model of the target mirror processing system. It can effectively solve the problem that modeling mirror processing systems of different structures is time-consuming and labor-intensive, and is conducive to analyzing the influence of the errors of each motion axis on the wall thickness accuracy.

[0007] The technical solution of the present invention:

[0008] The first step is to analyze the mirror processing constraints

[0009] The mirror processing constraint relationship requires that the tool vector and the support head vector always follow the normal vector of the workpiece surface, and the two remain collinear in the opposite direction; the tool center point and the support center point are on the common normal line of the workpiece surface, and the distance is equal to the wall thickness; then the mirror processing wall thickness d is expressed as:

[0010]

[0011] Where t1 represents the tool, t2 represents the support head; O(t1)=(O x (t1),O y (t1),O z (t1)) is the vector of the tool in the workpiece coordinate system along the xyz coordinate axis; P(t1) = (P x (t1),P y (t1),P z (t1)) and P(t2) = (P x (t2),P y (t2),P z (t2)) are the xyz coordinate values ​​of the tool center point in the workpiece coordinate system and the xyz coordinate value of the support head center point in the workpiece coordinate system; θ1(t1), θ2(t1), …θ m (t1) is the position of the motion axis on the machining side, θ1(t2),θ2(t2),…θ n(t2) is the position of the motion axis on the support side, and both are determined according to the specific topological structure of the mirror processing system; P(t1) = f1[θ1(t1),θ2(t1),…,θ m (t1)] represents the functional relationship between the coordinates of the tool center point and the position of the motion axis; P(t2) = f2[θ1(t2),θ2(t2),…,θ n (t2)] represents the functional relationship between the coordinates of the center point of the support head and the position of the motion axis; O(t1) = h[θ1(t1),θ2(t1),…,θ m (t1)] represents the functional relationship between the support head vector and the motion axis position;

[0012] The second step is to describe the nominal motion of the motion axis based on the screw theory

[0013] The basic unit of kinematic modeling based on screw theory is spiral motion The motion of a rigid body from one position to another is decomposed into rotation around a straight line and translation along this line, so spiral motion It consists of a spinor ξ and a momentum θ; the expression of the spinor ξ is:

[0014]

[0015] Among them, ω is the unit vector of the axis; ν is the translational momentum along the axis; q is an arbitrary point on the axis; h is the pitch of the spinor motion;

[0016] The homogeneous transformation matrix corresponding to the rotation motion of a certain axis is:

[0017]

[0018] Among them, θ i Represents the amount of rotation around the i-axis, i = x, y, z, a, b, c; I 3×3 represents the third-order identity matrix; 0 1×3 represents a 1×3 zero matrix;

[0019] ^

[0020] represents the skew-symmetric matrix obtained by operating the vector ω;

[0021] The third step is to analyze the motion axis error elements and establish the spinor transformation matrix

[0022] An object has 6 degrees of freedom in space. When each axis of the machine tool moves, 6 error elements will be generated. For geometric errors, there are 3 linear errors δ xi , δ yi , δ zi and 3 angular errors ε xi , εyi , ε zi In addition, there is a vertical error S between the axes. xy , S yz , S zx Similarly, there are three linear thermal errors for thermal errors: and 3 angle thermal errors There are 3 linear cutting force errors for cutting force errors And 3 angle cutting force errors Among them, i = x, y, z, a, b, c represents the motion axes of the machine tool, and the error term of each motion axis is measured by the instrument;

[0023] The error elements are described by using the screw theory. For a certain motion axis, the error screw transformation matrices corresponding to the 18 geometric errors, thermal errors and cutting force errors of the axis are multiplied together. The error matrix is ​​simplified based on the small error assumption. The error screw transformation matrix of the motion axis is:

[0024]

[0025] In addition, the verticality error S xy , S yz , S zx The error homogeneous transformation matrices are:

[0026]

[0027] The fourth step is to establish a unified kinematic model of wall thickness error in mirror processing

[0028] Firstly, the kinematic unified models of the machining side kinematic chain and the supporting side kinematic chain are established respectively;

[0029] For the machining side, assuming that there are m1 and n1 motion axes in the tool motion chain and the workpiece motion chain respectively, the homogeneous transformation relationship between the tool coordinate system and the machine tool coordinate system is:

[0030]

[0031] in, is the homogeneous transformation relationship between the tool coordinate system and the machine tool coordinate system;

[0032] and It is the motion rotation and motion distance or angle of the i1th motion axis in the tool motion chain in the machine tool coordinate system; the subscript t1 indicates that it is in the tool motion chain on the machining side; It is the homogeneous transformation matrix of the tool coordinate system to the machine tool coordinate system in the initial state.

[0033] The homogeneous transformation relationship between the workpiece coordinate system and the machine tool coordinate system is:

[0034]

[0035] in, and It is the motion rotation and motion distance or angle of the j1th motion axis in the workpiece motion chain on the machining side in the machine tool coordinate system; the subscript w1 indicates that it is in the workpiece coordinate system on the machining side; It is the homogeneous transformation matrix of the workpiece coordinate system on the machining side relative to the coordinate systems of several points in the initial state.

[0036] The homogeneous transformation relationship between the tool coordinate system and the workpiece coordinate system is:

[0037]

[0038] in, It is the inverse of the transformation matrix, and its value is the same as the inverse transformation matrix equal;

[0039] Based on the position vector in the tool coordinate system and the posture vector Calculate the nominal position vector of the tool in the workpiece coordinate system and the posture vector

[0040]

[0041] Similarly, the homogeneous transformation relationship between the support head coordinate system and the support workpiece coordinate system is obtained as follows:

[0042]

[0043] In order to establish the kinematic expression model of mirror machining wall thickness, it is necessary to transform the homogeneous transformation of the support head coordinate system relative to the support workpiece coordinate system into the homogeneous transformation relationship relative to the machining workpiece coordinate system:

[0044]

[0045] in, is the transformation matrix of the support side tool coordinate system relative to the machining workpiece coordinate system;

[0046] Based on the position vector in the support head coordinate system and the posture vector Calculate the nominal position vector of the support head in the workpiece coordinate system and the posture vector

[0047]

[0048] The unified kinematic model of the nominal remaining wall thickness of mirror processing is obtained based on the constraint relationship of mirror processing;

[0049]

[0050] Then, the error matrix of each motion axis is introduced into formulas (18) and (22) to calculate the unified kinematic model of the actual wall thickness of mirror machining;

[0051] When considering the influence of errors, the homogeneous transformation relationship between the tool coordinate system and the workpiece coordinate system is:

[0052]

[0053] Among them, k1(i1)=0or1, k2(i1)=0or1, k3(i1)=0or1, k1(j1)=0or1, k2(j1)=0or1, k3(j1)=0or1 are used to determine the placement position of the verticality error matrix on the processing side, which is determined according to the specific structure of the mirror processing system;

[0054] Based on the position vector in the tool coordinate system and the posture vector Calculate the actual position vector of the tool in the workpiece coordinate system and the posture vector

[0055]

[0056] When considering the influence of errors, the homogeneous transformation relationship between the support head coordinate system and the workpiece coordinate system is:

[0057]

[0058] Among them, k1(i2)=0or1,k2(i2)=0or1,k3(i2)=0or1,k1(j2)=0or1,k2(j2)=0or1,k3(j2=0or1 are used to determine the placement position of the support side verticality error matrix, which is determined according to the specific structure of the mirror processing system;

[0059] Based on the position vector in the support head coordinate system and the posture vector Calculate the actual position vector of the support head in the workpiece coordinate system and the posture vector

[0060]

[0061] The unified kinematic model of the actual remaining wall thickness in mirror machining is:

[0062]

[0063] Finally, the error model of the mirror processing residual wall thickness is obtained by subtracting the kinematic unified model of the actual residual wall thickness of the mirror processing from the kinematic unified model of the nominal residual wall thickness.

[0064] E=d e -d (21)

[0065] Beneficial effects of the present invention: The method of the present invention analyzes the constraint relationship of mirror processing, establishes a geometric constraint model between "tool-wall thickness-support head", uses the screw theory to establish the nominal motion and error matrix of the motion axis of the mirror processing system, and establishes the unified kinematic model of the nominal wall thickness and actual wall thickness of the mirror processing based on the multi-body system theory and the screw theory, thereby obtaining the unified kinematic model of the wall thickness error of the mirror processing. Based on the unified kinematic model of the wall thickness error of the mirror processing, when modeling the mirror processing system of a specified structure, it is only necessary to sort out the topological structure of the target mirror processing system to obtain the wall thickness error model of the target mirror processing system. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 The figure is a flow chart of the modeling method of the present invention.

[0067] Figure 2 A schematic diagram of the structure of a mirror processing system provided in one embodiment of the present invention.

[0068] Figure 3 A schematic diagram of a kinematic chain of a mirror processing system provided in one embodiment of the present invention. DETAILED DESCRIPTION

[0069] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0070] Figure 1 The flowchart of the unified modeling method of mirror processing wall thickness error based on screw theory of the present invention is shown. Figure 2 The figure shows the structure diagram of a vertical dual five-axis mirror machining system. The topological structure of the mirror machining system is T2RY2Z2X2C2FX1Y1Z1C1A1T1, where the X2 axis and the Z2 axis form an angle of 45°; the 3-RCU parallel mechanism has 1 translation and 2 rotational degrees of freedom. To simplify the model, the 3-RCU parallel mechanism is regarded as 3 motion axes: A r , B r and Z r Taking the mirror processing system as an example, the implementation mode of the present invention is described in detail.

[0071] The first step is to analyze the mirror processing constraints

[0072] The mirror processing constraint relationship requires that the tool vector and the support head vector always follow the normal vector of the workpiece surface, and the two remain collinear in opposite directions; the tool center point and the support center point are on the common normal line of the workpiece surface, and the distance is equal to the wall thickness. The mirror processing wall thickness d can be expressed as:

[0073]

[0074] Among them, X1, Y1, Z1, A1, C1, X2, Y2, Z2, C2, and R are the positions of the motion axes on the processing side and the supporting side.

[0075] The second step is to describe the nominal motion of the motion axis based on the screw theory

[0076] The rotational motion transformation matrix of each linear axis and rotary axis of the vertical dual five-axis mirror machining system is:

[0077]

[0078] The third step is to analyze the error elements of the motion axis and establish the rotation transformation matrix. The vertical dual five-axis mirror machining system has 6 linear axes, 3 rotation axes, and 1 parallel mechanism. Each axis has 6 geometric errors δ xi , δ yi , δ zi ε xi , ε yi , ε zi , 6 thermal errors 3 force errors Among them, i = X1, Y1, Z1A1, C1, X2, Y2, Z2, C2, R. There is also a vertical error between the X1, Y1, and Z1 axes. There is also a verticality error between the X2, Y2, and Z2 axes A total of 186 errors.

[0079] Using the screw theory to describe the error elements, the error screw homogeneous transformation matrix of each motion axis is expressed as:

[0080]

[0081] Where i = X1, Y1, Z1A1, C1, X2, Y2, Z2, C2, R

[0082] In addition, the verticality error The error spinor homogeneous transformation matrices are:

[0083]

[0084]

[0085] Step 4: Establish the kinematic model of wall thickness error in mirror processing

[0086] For the machining side, assuming that there are 5 and 0 motion axes in the tool motion chain and the workpiece motion chain respectively, the homogeneous transformation matrix of the tool coordinate system relative to the machine tool coordinate system is:

[0087]

[0088] The homogeneous transformation matrix of the workpiece coordinate system relative to the machine tool coordinate system is:

[0089]

[0090] The homogeneous transformation of the tool coordinate system relative to the workpiece coordinate system is:

[0091]

[0092] in,

[0093] Based on the position vector in the tool coordinate system and the posture vector Calculate the tool position vector in the workpiece coordinate system and the posture vector

[0094]

[0095] in,

[0096] Similarly, the homogeneous transformation of the support head coordinate system relative to the support workpiece coordinate system can be obtained as:

[0097]

[0098] in,

[0099] In order to establish the kinematic expression model of mirror machining wall thickness, it is necessary to transform the homogeneous transformation of the support head coordinate system relative to the support workpiece coordinate system into the homogeneous transformation relationship relative to the machining workpiece coordinate system:

[0100]

[0101] in,

[0102] Based on the position vector in the support head coordinate system and the posture vector Calculate the support head pose vector in the workpiece coordinate system and the posture vector

[0103]

[0104] in,

[0105] The unified kinematic model of nominal remaining wall thickness in mirror machining is obtained based on the constraint relationship of mirror machining.

[0106]

[0107] in,

[0108]

[0109] L1=289.940mm is the distance from the tool center to the A1 axis rotation center. is the X1 axis Z-direction offset distance, L2 is the distance from the center point of the support head to the rotation center of the parallel mechanism, B zx =45° is the angle between the Z2 axis and the X2 axis, is the X-axis offset distance of C2, is the Z-direction offset distance of the C2 axis,

[0110] Then, the error matrix of each motion axis is introduced into formulas (34) and (36) to calculate the unified kinematic model of the actual wall thickness of mirror machining.

[0111] When considering the influence of errors, the homogeneous transformation of the tool coordinate system relative to the workpiece coordinate system is:

[0112]

[0113] Based on the position vector in the tool coordinate system and the posture vector Tool pose vector in the workpiece coordinate system under the influence of calculation error and the posture vector

[0114]

[0115] When considering the influence of errors, the homogeneous transformation relationship between the support head coordinate system and the workpiece coordinate system is:

[0116]

[0117] Based on the position vector in the support head coordinate system and the posture vector The support head pose vector in the workpiece coordinate system under the influence of calculation error and the posture vector

[0118]

[0119] The unified kinematic model of the actual remaining wall thickness in mirror machining is:

[0120]

[0121] in,

[0122]

[0123]

[0124]

[0125]

[0126]

[0127]

[0128]

[0129]

[0130] Finally, the kinematic model of the actual remaining wall thickness of the mirror processing and the kinematic model of the nominal remaining wall thickness are subtracted to obtain the error model of the remaining wall thickness of the mirror processing.

[0131] E=d e -d (56)

[0132] The above is only a specific embodiment of the present invention. The above embodiment is only for describing the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A unified modeling method for mirror machining wall thickness error based on screw theory, characterized in that: Here are the steps: The first step is to analyze the mirror processing constraints The mirror processing constraint relationship requires that the tool vector and the support head vector always follow the normal vector of the workpiece surface, and the two remain collinear in the opposite direction; the tool center point and the support center point are on the common normal line of the workpiece surface, and the distance is equal to the wall thickness; then the mirror processing wall thickness d is expressed as: Where t1 represents the tool, t2 represents the support head; O(t1)=(O x (t1),O y (t1),O z (t1)) is the vector of the tool in the workpiece coordinate system along the xyz coordinate axis; P(t1) = (P x (t1),P y (t1),P z (t1)) and P(t2) = (P x (t2),P y (t2),P z (t2)) are the xyz coordinate values ​​of the tool center point in the workpiece coordinate system and the xyz coordinate value of the support head center point in the workpiece coordinate system; θ1(t1), θ2(t1), …θ m (t1) is the position of the motion axis on the machining side, θ1(t2),θ2(t2),…θ n (t2) is the position of the motion axis on the support side, and both are determined according to the specific topological structure of the mirror processing system; P(t1) = f1[θ1(t1),θ2(t1),…,θ m (t1)] represents the functional relationship between the coordinates of the tool center point and the position of the motion axis; P(t2) = f2[θ1(t2),θ2(t2),…,θ n (t2)] represents the functional relationship between the coordinates of the center point of the support head and the position of the motion axis; O(t1) = h[θ1(t1),θ2(t1),…,θ m (t1)] represents the functional relationship between the support head vector and the motion axis position; The second step is to describe the nominal motion of the motion axis based on the screw theory The basic unit of kinematic modeling based on screw theory is spiral motion The motion of a rigid body from one position to another is decomposed into rotation around a straight line and translation along this line, so spiral motion It consists of a spinor ξ and a momentum θ; the expression of the spinor ξ is: Among them, ω is the unit vector of the axis; ν is the translational momentum along the axis; q is an arbitrary point on the axis; h is the pitch of the spinor motion; The homogeneous transformation matrix corresponding to the rotation motion of a certain axis is: Among them, θ i Represents the amount of rotation around the i-axis, i = x, y, z, a, b, c; I 3×3 represents the third-order identity matrix; 0 1×3 represents a 1×3 zero matrix; represents the skew-symmetric matrix obtained by the ^ operation of vector ω; The third step is to analyze the motion axis error elements and establish the spinor transformation matrix An object has 6 degrees of freedom in space. When each axis of the machine tool moves, 6 error elements will be generated. For geometric errors, there are 3 linear errors δ xi , δ yi , δ zi and 3 angular errors ε xi , ε yi , ε zi In addition, there is a vertical error S between the axes. xy , S yz , S zx Similarly, there are three linear thermal errors for thermal errors: and 3 angle thermal errors There are 3 linear cutting force errors for cutting force errors And 3 angle cutting force errors Among them, i = x, y, z, a, b, c represents the motion axes of the machine tool, and the error term of each motion axis is measured by the instrument; The error elements are described by using the screw theory. For a certain motion axis, the error screw transformation matrices corresponding to the 18 geometric errors, thermal errors and cutting force errors of the axis are multiplied together. The error matrix is ​​simplified based on the small error assumption. The error screw transformation matrix of the motion axis is: In addition, the verticality error S xy , S yz , S zx The error homogeneous transformation matrices are: The fourth step is to establish a unified kinematic model of wall thickness error in mirror processing Firstly, the kinematic unified models of the machining side kinematic chain and the supporting side kinematic chain are established respectively; For the machining side, assuming that there are m1 and n1 motion axes in the tool motion chain and the workpiece motion chain respectively, the homogeneous transformation relationship between the tool coordinate system and the machine tool coordinate system is: in, is the homogeneous transformation relationship between the tool coordinate system and the machine tool coordinate system; and It is the motion rotation and motion distance or angle of the i1th motion axis in the tool motion chain in the machine tool coordinate system; the subscript t1 indicates that it is in the tool motion chain on the machining side; is the homogeneous transformation matrix of the tool coordinate system to the machine tool coordinate system in the initial state; The homogeneous transformation relationship between the workpiece coordinate system and the machine tool coordinate system is: in, and It is the motion rotation and motion distance or angle of the j1th motion axis in the workpiece motion chain on the machining side in the machine tool coordinate system; the subscript w1 indicates that it is in the workpiece coordinate system on the machining side; g fw1 (0) is the homogeneous transformation matrix of the workpiece coordinate system on the processing side relative to the coordinate system of the several points in the initial state; The homogeneous transformation relationship between the tool coordinate system and the workpiece coordinate system is: in, It is the inverse of the transformation matrix, and its value is the same as the inverse transformation matrix equal; Based on the position vector in the tool coordinate system and the posture vector Calculate the nominal position vector of the tool in the workpiece coordinate system and the posture vector Similarly, the homogeneous transformation relationship between the support head coordinate system and the support workpiece coordinate system is obtained as follows: In order to establish the kinematic expression model of mirror machining wall thickness, it is necessary to transform the homogeneous transformation of the support head coordinate system relative to the support workpiece coordinate system into the homogeneous transformation relationship relative to the machining workpiece coordinate system: in, is the transformation matrix of the support side tool coordinate system relative to the machining workpiece coordinate system; Based on the position vector in the support head coordinate system and the posture vector Calculate the nominal position vector of the support head in the workpiece coordinate system and the posture vector The unified kinematic model of the nominal remaining wall thickness of mirror processing is obtained based on the constraint relationship of mirror processing; Then, the error matrix of each motion axis is introduced into formulas (18) and (22) to calculate the unified kinematic model of the actual wall thickness of mirror machining; When considering the influence of errors, the homogeneous transformation relationship between the tool coordinate system and the workpiece coordinate system is: Among them, k1(i1)=0or1, k2(i1)=0or1, k3(i1)=0or1, k1(j1)=0or1, k2(j1)=0or1, k3(j1)=0or1 are used to determine the placement position of the verticality error matrix on the processing side, which is determined according to the specific structure of the mirror processing system; Based on the position vector in the tool coordinate system and the posture vector Calculate the actual position vector of the tool in the workpiece coordinate system and the posture vector When considering the influence of errors, the homogeneous transformation relationship between the support head coordinate system and the workpiece coordinate system is: Among them, k1(i2)=0or1,k2(i2)=0or1,k3(i2)=0or1,k1(j2)=0or1,k2(j2)=0or1,k3(j2=0or1 are used to determine the placement position of the support side verticality error matrix, which is determined according to the specific structure of the mirror processing system; Based on the position vector in the support head coordinate system and the posture vector Calculate the actual position vector of the support head in the workpiece coordinate system and the posture vector The unified kinematic model of the actual remaining wall thickness in mirror machining is: Finally, the error model of the mirror processing residual wall thickness is obtained by subtracting the kinematic unified model of the actual residual wall thickness of the mirror processing from the kinematic unified model of the nominal residual wall thickness. E=d e -d (21)。

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