Robot magneto-rheological polishing TCP calibration method based on three-coordinate measurement

Abstract

The invention relates to the technical field of magneto-rheological polishing calibration, in particular to a TCP calibration method for robot magneto-rheological polishing based on three-coordinate measurement, which comprises the following steps of: measuring different point positions on a polishing wheel in a magneto-rheological polishing module by using a three-coordinate measuring instrument to obtain a sphere center position of a spherical surface on which the polishing wheel is positioned; measuring a normal vector of any curved surface on the polishing wheel by using a three-coordinate measuring instrument, and determining a rotation matrix of the TCP based on the normal vector; the rotation matrix is converted into a quaternion, the quaternion is input into the robot, and TCP attitude correction is completed; according to the sphere center position and the spherical radius, the working coordinates of the TCP and the offset of the TCP relative to the center of the robot flange plate are determined; and the working coordinates and the offset are input into the robot, and TCP position correction is completed. According to the invention, dual calibration of the TCP position and attitude is realized, the whole process does not depend on artificial experience, no complex algorithm is needed, the operation is simple and convenient, the calibration is efficient, and the precision is reliable.

Description

Technical Field

[0001] The present invention belongs to the technical field of magnetorheological polishing calibration, and particularly relates to a TCP calibration method for robot magnetorheological polishing based on three-coordinate measurement. Background Art

[0002] Magnetorheological polishing technology is a commonly used processing technology in the field of high-precision optical processing. This processing technology has significant advantages such as a stable removal function, high certainty in the processing process, and rapid convergence. This technology is usually combined with industrial robots to give full play to the advantages of industrial robots, such as low cost, strong operation flexibility, and high scalability. The magnetorheological polishing equipment is connected to the end of the robot, and different processing tasks are executed through robot control.

[0003] Therefore, how to perform TCP calibration of the magnetorheological polishing equipment efficiently and accurately directly determines the application effect of this technology. At present, the existing TCP calibration methods mainly fall into two categories: four-point method calibration and external reference sensor calibration.

[0004] Four-point method calibration controls the robot to let its tool end carry a needle-like object, move to a predetermined spatial position in different postures, and make four contacts, thereby completing the calculation of TCP. However, this method is overly dependent on manual operation and has a low calibration accuracy.

[0005] External reference sensor calibration uses external devices such as laser trackers, vision cameras, or ballbar testers for positioning and completes TCP calibration through an external reference. However, this method has complex calculations and high equipment costs. Summary of the Invention

[0006] In view of this, the present invention aims to provide a TCP calibration method for robot magnetorheological polishing based on three-coordinate measurement. By using the measurement and fitting capabilities of a three-coordinate measuring instrument, accurately measure the spatial coordinates of key points of the magnetorheological polishing module, fit the key parts, and then through a calculation step, achieve dual calibration of the TCP position and posture. The whole process does not rely on manual experience, does not require complex algorithms, is easy to operate, and has high calibration efficiency and reliable accuracy.

[0007] To achieve the above object, the technical solution of the present invention is realized as follows: A TCP calibration method for robot magnetorheological polishing based on three-coordinate measurement, characterized by comprising: S1: Install and connect a connecting workpiece concentric with the robot flange to the magnetorheological polishing module; use a three-coordinate measuring instrument to measure different points on the polishing wheel in the magnetorheological polishing module to obtain the center position of the spherical surface where the polishing wheel is located; S2: Use a three-dimensional coordinate measuring machine to measure the normal vector of any curved surface on the polishing wheel, and determine the posture correction angle of the TCP of the magnetorheological polishing module based on the normal vector; S3: Determine the rotation matrix of the TCP based on the posture correction angle obtained in step S2; convert the rotation matrix into a quaternion, and input the quaternion into the robot to complete the posture correction of the TCP; S4: Connect the magnetorheological polishing module to the robot flange by connecting the workpiece, and determine the working coordinates of the TCP and the offset of the TCP relative to the center of the robot flange based on the sphere center position and spherical radius obtained in step S1; input the working coordinates and offset into the robot to complete the position correction of the TCP.

[0008] Furthermore, the magnetorheological polishing module also includes: A magnetorheological polishing module body, on which the connecting workpiece is arranged; The polishing motor drives the polishing wheel to rotate through the track.

[0009] Furthermore, the three-coordinate measuring instrument includes a measuring instrument body, a control center and a three-coordinate probe; wherein, the measuring instrument body includes a platform to be measured, a drive system and an XYZ structural frame, the three-coordinate probe is set on the XYZ structural frame, and the control center drives the XYZ structural frame through the drive system, driving the three-coordinate probe to contact the magnetorheological polishing module and the connected workpiece on the platform to be measured, and measures the spatial point positions of the magnetorheological polishing module and the connected workpiece according to the contact results.

[0010] Furthermore, step S1 also includes: measuring a connection surface of the workpiece connected to the robot flange using a three-coordinate measuring machine, and establishing a measurement coordinate system based on the connection surface.

[0011] Furthermore, the process of measuring the connection surface of the workpiece connected to the robot flange using the three-dimensional coordinate measuring machine includes: Use a three-coordinate measuring machine to randomly measure no less than 3 different position points on the connected workpiece, fit the connection surface through the three-coordinate measuring machine, and obtain the spatial position of the center point of the connection surface.

[0012] Furthermore, the process of establishing a measurement coordinate system based on the connection surface includes: The plane where the connection surface is located is the reference plane, which is defined as the X coordinate system of the measurement M Y M The plane is defined as the center point of the connection surface as the origin of the measurement coordinate system, and the direction from the polishing wheel to the connection workpiece is defined as the +Z of the measurement coordinate system. M direction.

[0013] Furthermore, step S2 includes: randomly measuring at least three positions on the polishing wheel using a three-coordinate measuring machine, and calculating the normal vector of the surface where the position point is located in the measurement coordinate system; and determining the posture correction angle by the following formula: ; Among them, (α', β', γ') represents the attitude correction angles in the three directions of the TCP measurement coordinate system, (α, β, γ) represents the actual angles between the normal vector and the three directions of the measurement coordinate system, and (α0, β0, γ0) represents the theoretical angles between the normal vector and the three directions of the measurement coordinate system.

[0014] Furthermore, the rotation matrix in step S3 is expressed as: ; Among them, R represents the rotation matrix, r 11 ~r 33 Represents the elements in the rotation matrix, R x (α') represents the rotation matrix of TCP around the X-axis in the measurement coordinate system, R y (β') represents the rotation matrix of TCP around the Y axis in the measurement coordinate system, R z (γ') represents the rotation matrix of TCP around the Z axis in the measurement coordinate system; Rotation matrix R x (α') is: ; Rotation matrix R y (β') is: ; Rotation matrix R z (γ') is: .

[0015] Furthermore, in step S3, the rotation matrix is converted into a quaternion using the following formula: ; Among them, q0~q3 represent quaternions.

[0016] Furthermore, in step S4: The working coordinates are obtained by the following formula: ; Among them, (x T ,y T ,z T ) represents the working coordinates, (x1, y1, z1) represents the center position of the sphere, and R' represents the radius of the sphere; The offset is obtained by the following formula: ; Among them, (Δx T , Δy T , Δz T ) represents the offset.

[0017] Compared with the prior art, the present invention can achieve the following beneficial effects: The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to the present invention simplifies the TCP calibration operation and calculation method, improves its accuracy and efficiency. By using the measurement and fitting functions of a three - coordinate measuring instrument, the TCP position correction of magnetorheological polishing is completed, and through a simple calculation method, the TCP attitude correction of magnetorheological polishing is completed. Compared with the traditional process, this process is simple to operate, does not rely on the operating experience of operators, and does not require complex calculation methods, having obvious advantages. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The accompanying drawings forming a part of the present invention are used to provide a further understanding of the present invention. The schematic embodiments and descriptions thereof of the present invention are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings: Figure 1 is a schematic flow chart of the TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to the embodiment of the present invention; Figure 2 is a flow block diagram of the TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to the embodiment of the present invention; Figure 3 is a combined structural schematic diagram of the magnetorheological polishing module and the three - coordinate measuring instrument according to the embodiment of the present invention; Figure 4 is a structural schematic diagram of the magnetorheological polishing module according to the embodiment of the present invention.

[0019] Description of the reference numerals in the drawings: 1. Three - coordinate probe; 2. Platform to be measured; 3. XYZ structural framework; 4. Main body of the magnetorheological polishing module; 5. Polishing wheel; 6. Connected workpiece; 7. Robot flange; 8. Crawler. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0020] In order to make the purpose, technical solutions and advantages of the present invention clearer, the following further elaborates on the present invention in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation to the present invention.

[0021] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0022] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by the terms "center", "longitudinal", "transverse", "up", "down", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first", "second", etc. may explicitly or implicitly include one or more of such features. In the description of the present invention, unless otherwise specified, the meaning of "a plurality" is two or more.

[0023] In the description of the present invention, it should be noted that unless otherwise clearly specified and limited, the terms "installation", "connection", and "coupling" should be understood in a broad sense. For example, it may be a fixed connection, a detachable connection, or an integral connection; it may be a mechanical connection or an electrical connection; it may be directly connected or indirectly connected through an intermediate medium, and it may be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood through specific situations.

[0024] The present invention will be described in detail below with reference to the drawings and in conjunction with embodiments.

[0025] As Figure 1 and Figure 2 shown, the TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to the embodiment of the present invention includes: S1: Install and connect a connecting workpiece concentric with the robot flange to the magnetorheological polishing module; use a three - coordinate measuring instrument to measure different points on the polishing wheel in the magnetorheological polishing module to obtain the center position of the spherical surface where the polishing wheel is located.

[0026] In some embodiments, as Figure 3As shown in the figure, the coordinate measuring machine includes a measuring machine body, a coordinate measuring probe 1, and a control center. Among them, the measuring machine body includes a platform to be measured 2, a drive system, and an XYZ structural frame 3. The coordinate measuring probe 1 is arranged on the XYZ structural frame 3. The control center drives the XYZ structural frame 3 through the drive system, drives the coordinate measuring probe 1 to contact the magnetorheological polishing module and the connecting workpiece 6 on the platform to be measured, and measures the spatial positions of the magnetorheological polishing module and the connecting workpiece 6 according to the contact results. In the embodiment of the present invention, a coordinate measuring machine of ZEISS PRISMO navigator is adopted.

[0027] In some embodiments, as Figure 4 shown, Figure 4 Figures (a) and (b) in show the structures of the magnetorheological polishing module from two different perspectives respectively. The magnetorheological polishing module includes a magnetorheological polishing module body 4, a polishing motor, and a polishing wheel 5. The connecting workpiece 6 is arranged on the magnetorheological polishing module body 4, so that the robot flange 7 is installed and connected to the magnetorheological polishing module body 4. The polishing motor is located in the magnetorheological polishing module body 4, and the output end of the polishing motor drives the polishing wheel 5 to rotate through a crawler 8. During processing, the lowest point of the polishing wheel 5 cooperates with the magnetorheological polishing medium to process the workpiece to be processed, that is, the lowest point of the polishing wheel 5 is the TCP point. By randomly measuring different points on the polishing wheel 5 with a coordinate measuring machine, the center position of the spherical surface where the polishing wheel 5 is located can be obtained. It can be understood that the coordinate measuring probe 1 in the coordinate measuring machine randomly contacts different points on the polishing wheel 5, and the center position of the spherical surface where the polishing wheel 5 is located is obtained. In the embodiment of the present invention, the magnetorheological processing module and the magnetorheological fluid circulation system in the invention patent application with the Chinese patent publication number CN118123694A, publication date August 16, 2024, and patent name "Processing capacity evaluation method for magnetorheological optical processing equipment based on hybrid robot" are integrated to obtain the magnetorheological polishing module. Specifically, the components of the magnetorheological processing module and the magnetorheological fluid circulation system other than the polishing wheel are integrated into the magnetorheological polishing module body 4.

[0028] In some embodiments, step S1 further includes: measuring the connection surface of the connecting workpiece connected to the robot flange with a coordinate measuring machine, and establishing a measurement coordinate system based on the connection surface.

[0029] In some embodiments, the process of measuring the connection surface of the connecting workpiece connected to the robot flange with a coordinate measuring machine includes: randomly measuring no less than 3 different position points on the connecting workpiece with a coordinate measuring machine, fitting out the connection surface through the coordinate measuring machine, and obtaining the spatial position of the center point of the connection surface. The process of establishing a measurement coordinate system based on the connection surface includes: taking the plane where the connection surface is located as the reference plane, and defining it as the X of the measurement coordinate system M Y MThe plane, with the center point of the connection surface defined as the origin of the measurement coordinate system, and the direction from the polishing wheel to the connected workpiece defined as the +Z direction of the measurement coordinate system. M In the embodiment of the present invention, the three-coordinate probe 1 of the three-coordinate measuring instrument randomly contacts 4 different position points on the connection surface of the connected workpiece 6 (such as a~d in Figure 3 ), and 4 different position points on the side surface of the connected workpiece 6 (such as e~g in Figure 3 ). Using the fitting function of the three-coordinate measuring instrument, the connection surface of the connected workpiece 6 is fitted to obtain the spatial position of the center point O of the connection surface of the connected workpiece 6. The process of establishing the measurement coordinate system based on the connection surface includes: taking the plane where the connection surface of the connected workpiece 6 is located as the reference plane, defined as the X M Y M plane of the measurement coordinate system M. With the center point O of the connection surface defined as the origin (0, 0, 0) of the measurement coordinate system M, and the direction from the polishing wheel 5 to the connected workpiece 6 defined as the +Z M direction.

[0030] In the embodiment of the present invention, the process of obtaining the position of the center of the sphere where the polishing wheel 5 is located includes: randomly measuring 12 different position points on the polishing wheel 5 by the three-coordinate measuring instrument, and the distribution of the 12 position points should cover the entire polishing wheel 5 as much as possible (such as i~t in Figure 3 ). Based on the fitting function of the three-coordinate measuring instrument, the sphere where the surface of the polishing wheel 5 is located is fitted, and the position of the center of the sphere S is obtained. The spatial coordinates of the center of the sphere S are (x1, y1, z1), and its sphere radius is R'.

[0031] S2: Use the three-coordinate measuring instrument to measure the normal vector of any curved surface on the polishing wheel, and determine the attitude correction angle of the TCP of the magnetorheological polishing module based on the normal vector.

[0032] In some embodiments, step S2 includes: randomly measuring no less than 3 position points on the polishing wheel by the three-coordinate measuring instrument, and calculating the normal vector of the curved surface where the position points are located in the measurement coordinate system; determining the attitude correction angle through the following formula: ; where, (α’, β’, γ’) represents the attitude correction angles in the three directions of the TCP measurement coordinate system, (α, β, γ) represents the actual angles between the normal vector and the three directions of the measurement coordinate system respectively, (α0, β0, γ0) represents the theoretical angles between the normal vector and the three directions of the measurement coordinate system respectively. Among them, the theoretical angle is based on the theoretical use attitude of the magnetorheological polishing module, that is, in the model of the magnetorheological module, combined with the actual use pose requirements, the direction of the theoretical normal vector is set and converted into the angle with the measurement coordinate system.

[0033] In an embodiment of the present invention, four different position points on the surface of the polishing wheel 5 are randomly measured by a coordinate measuring machine (such as Figure 3 u-x in ; The normal vector is obtained. The normal vector is normalized to obtain the unit normal vector : The actual angles (α, β, γ) between the unit normal vector and the three directions of the measurement coordinate system M are obtained by the following formula: .

[0034] S3: Determine the rotation matrix of the TCP according to the attitude correction angle obtained in step S2; convert the rotation matrix into a quaternion and input the quaternion into the robot to complete the attitude correction of the TCP.

[0035] In some embodiments, the rotation matrix in step S3 is expressed as: ; where R represents the rotation matrix, and r 11 ~r 33 represent the elements in the rotation matrix, R x (α’) represents the rotation matrix of the TCP rotating around the X-axis in the measurement coordinate system, R y ⏐(β’) represents the rotation matrix of the TCP rotating around the Y-axis in the measurement coordinate system, and R z (γ’) represents the rotation matrix of the TCP rotating around the Z-axis in the measurement coordinate system; The rotation matrix R x (α’) is: ; The rotation matrix R y (β’) is: ; The rotation matrix R z (γ’) is: .

[0036] In step S3, the rotation matrix is converted into a quaternion by the following formula: ; where q0~q3 represent the quaternion.

[0037] Furthermore, the rotation matrix R can be obtained by the following formula: .

[0038] S4: Connect the magnetorheological polishing module to the workpiece and the robot flange, and determine the working coordinates of the TCP and the offset of the TCP relative to the center of the robot flange according to the center position of the spherical surface and the spherical radius obtained in step S1; input the working coordinates and the offset into the robot to complete the position correction of the TCP.

[0039] In some embodiments, the working coordinates are obtained by the following formula: ; where (x T , y T , z T ) represents the working coordinates; The offset is obtained by the following formula: ; where (Δx T , Δy T , Δz T ) represents the offset; In the embodiments of the present invention, the magnetorheological polishing module body 4 in the magnetorheological polishing module is combined with the workpiece 6 and the robot flange. At this time, the measurement coordinate system M is parallel to the robot coordinate system R, that is, the X M direction in the measurement coordinate system M is parallel to the X R direction in the robot coordinate system R, the Y M direction in the measurement coordinate system M is parallel to the Y R direction in the robot coordinate system R, and the Z M direction in the measurement coordinate system M is parallel to the Z R direction in the robot coordinate system R. At this time, in the robot coordinate system R, the position coordinate T of the working position of the magnetorheological polishing TCP is (x T , y T , z T ) as shown in the above formula, and the offset of the TCP relative to the center of the robot flange 7 is (Δx T , Δy T , Δz T ) as shown in the above formula. Input the calculated offset into the robot teach pendant to complete the position correction of the TCP. Thus, the dual calibration of the position and attitude of the TCP for robot magnetorheological polishing based on three-coordinate measurement is completed.

[0040] It should be understood that the various forms of processes shown above can be used, with steps reordered, added or deleted. For example, the steps described in the disclosure of the present invention can be executed in parallel, sequentially or in different orders, as long as the desired results of the technical solution disclosed in the present invention can be achieved, and no limitation is imposed herein.

[0041] The above specific embodiments do not constitute a limitation on the protection scope of the present invention. Those skilled in the art should understand that various modifications, combinations, sub - combinations and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement, characterized in that, Including: S1: Mount and connect a connecting workpiece concentric with the robot flange to the magnetorheological polishing module; Use a coordinate measuring machine to measure different points on the polishing wheel in the magnetorheological polishing module to obtain the center position of the spherical surface where the polishing wheel is located; S2: Use the coordinate measuring machine to measure the normal vector of any curved surface on the polishing wheel, and determine the attitude correction angle of the TCP of the magnetorheological polishing module based on the normal vector; S3: Determine the rotation matrix of the TCP according to the attitude correction angle obtained in step S2; convert the rotation matrix into a quaternion, and input the quaternion into the robot to complete the attitude correction of the TCP; S4: Connect the magnetorheological polishing module to the robot flange through the connecting workpiece, and determine the working coordinates of the TCP and the offset of the TCP relative to the center of the robot flange according to the center position and spherical radius of the spherical surface obtained in step S1; input the working coordinates and the offset into the robot to complete the position correction of the TCP.

2. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 1, characterized in that, The magnetorheological polishing module further includes: A magnetorheological polishing module main body, and the connecting workpiece is arranged on the magnetorheological polishing module main body; A polishing motor drives the polishing wheel to rotate through a track.

3. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 1, characterized in that, The coordinate measuring machine includes a measuring machine main body, a control center and a three-coordinate probe; wherein, the measuring machine main body includes a platform to be measured, a driving system and an XYZ structural frame, the three-coordinate probe is arranged on the XYZ structural frame, and the control center drives the XYZ structural frame through the driving system to drive the three-coordinate probe to contact the magnetorheological polishing module and the connecting workpiece on the platform to be measured, and measure the spatial points of the magnetorheological polishing module and the connecting workpiece according to the contact result.

4. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 1, wherein Step S1 further includes: Use the coordinate measuring machine to measure the connecting surface of the connecting workpiece connected to the robot flange, and establish a measurement coordinate system based on the connecting surface.

5. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 4, wherein The process of using the coordinate measuring machine to measure the connecting surface of the connecting workpiece connected to the robot flange includes: Use the coordinate measuring machine to randomly measure no less than 3 different position points on the connecting workpiece, and fit the connecting surface through the coordinate measuring machine to obtain the spatial position of the center point of the connecting surface.

6. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 5, wherein, The process of establishing a measurement coordinate system based on the connecting surface includes: Taking the plane where the connecting surface is located as the reference plane, it is defined as the X M Y M plane of the measurement coordinate system. Taking the center point of the connecting surface as the origin of the measurement coordinate system, the direction from the polishing wheel to the connecting workpiece is defined as the +Z M direction.

7. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 4, characterized in that, Step S2 includes: Use the coordinate measuring machine to randomly measure no less than 3 position points on the polishing wheel, and calculate the normal vector of the curved surface where the position points are located in the measurement coordinate system; Determine the attitude correction angle through the following formula: ； Wherein, (α’, β’, γ’) represents the attitude correction angles of the three directions of the TCP in the measurement coordinate system, (α, β, γ) represents the actual included angles of the normal vector with the three directions of the measurement coordinate system respectively, and (α0, β0, γ0) represents the theoretical included angles of the normal vector with the three directions of the measurement coordinate system respectively.

8. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 7, wherein, The rotation matrix in step S3 is expressed as: ； wherein, R represents the rotation matrix, and r 11 ~r 33 represents the elements in the rotation matrix, and R x (α’) represents the rotation matrix of the TCP rotating about the X-axis in the measurement coordinate system, and R y (β’) represents the rotation matrix of the TCP rotating about the Y-axis in the measurement coordinate system, and R z (γ’) represents the rotation matrix of the TCP rotating about the Z-axis in the measurement coordinate system; Rotation matrix R x (α’) is as follows: ； Rotation matrix R y (β’) is as follows: ； Rotation matrix R z (γ’) is as follows: 。 9. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 8, wherein, In step S3, the rotation matrix is converted into the quaternion through the following formula: ； Among them, q0~q3 represent the quaternion.

10. The TCP calibration method for robot magnetorheological polishing based on three - coordinate measurement according to claim 8, wherein In step S4: The working coordinate is obtained by the following formula: ； Among them, (x T , y T , z T ) represents the working coordinates, (x1, y1, z1) represents the position of the center of the sphere, and R' represents the radius of the spherical surface; The offset is obtained by the following formula: ； where (Δx T , Δy T , Δz T ) represents the said offset.

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