Three-coordinate measuring machine volume error modeling method based on Abbe principle

Abstract

The invention discloses a three-coordinate measuring machine volume error modeling method based on the Abbe principle, and aims to improve the CMM measurement precision. Firstly, Abbe arm parameters of the CMM are defined and initialized, and the parameters describe spatial deviation between a measuring point of a probe and a reading head of a grating ruler of each motion axis; secondly, on the basis of a Denavit-Hartenberg (DH) modeling idea, a sequential cascade coordinate system from a workpiece coordinate system to a probe tail end tool coordinate system is established, and a unified kinematic chain topological structure is formed; then, calculating an Abbe error introduced by single-axis motion according to the Abbe principle, and establishing a single-axis error homogeneous transformation matrix containing angle and position deviations in combination with a DH modeling method; and finally, solving the actual position of the probe in a workpiece coordinate system by cascading the error homogeneous transformation matrix of each axis, and finally establishing a CMM volume error model based on the Abbe principle by subtracting the actual position vector from the theoretical position vector.

Description

Technical Field

[0001] This invention belongs to the field of precision measurement and relates to a volume error modeling method for improving the measurement accuracy of a coordinate measuring machine (CMM). Background Technology

[0002] With advancements in manufacturing technology, CMMs (Computer Measurement Machines) have been widely applied in high-precision inspection fields such as aerospace. Their measurement accuracy directly impacts product quality, making higher precision measurement a core issue in the precision measurement field. To effectively improve the measurement accuracy of CMMs, error compensation technology has become the mainstream solution for achieving higher performance and is widely used in industrial measurement.

[0003] For error compensation technology, the accuracy of the error model is crucial to determining the compensation effect. To this end, scholars both domestically and internationally have proposed various modeling methods for volumetric errors, specifically addressing the error characteristics of CMMs. However, existing modeling methods generally fail to effectively consider the impact of Abbe error on volumetric errors. This deficiency results in error models that cannot fully reflect the true error characteristics of CMMs, thus limiting further improvements in the accuracy of CMM error compensation.

[0004] Therefore, it is necessary to establish a new CMM volume error modeling method. Based on the traditional quasi-rigid body model, this method effectively introduces the Abbe error propagation mechanism to more accurately describe the relationship between the geometric error and volume error of CMM, providing reliable theoretical support for improving the measurement accuracy of CMM. Summary of the Invention

[0005] Step 1: Define the Abbe arm parameters of the CMM. The Abbe arm is the spatial offset connecting the CMM stylus measurement point to the reading head of the grating ruler on each motion axis. Specifically, for each motion axis of the CMM, there are spatial offset components along the X, Y, and Z directions between the stylus measurement point and the reading head of the grating ruler on that axis.

[0006] , , These represent the Abbe arm components along the X, Y, and Z directions from the probe measuring point to the X-axis grating ruler reading head.

[0007] , , These represent the Abbe arm components along the X, Y, and Z directions from the probe measuring point to the Y-axis grating ruler reading head.

[0008] , , The Abbe arm components along the X, Y, and Z directions from the probe measuring point to the Z-axis grating ruler reading head are defined. Therefore, a total of 9 Abbe arm components are defined. The initial components of these Abbe arms are determined by the static mechanical structure of the CMM and the probe's mounting position. These initial values ​​are defined as follows: , , , , , , , , These nine parameters are the basic structural constants that describe the CMM's non-compliance with the Abbe principle, and can be obtained through theoretical design drawings or actual calibration.

[0009] During CMM operation, as the coordinate axes move, some Abbe arm components change due to variations in the relative position between the probe measurement point and the corresponding grating ruler reading head. It's important to note that different types of CMMs differ in the arrangement and installation position of the grating ruler reading heads, which directly determines which components of the Abbe arm change with axis movement and which remain constant. Therefore, the variations in the Abbe arm under different CMM structures require specific analysis considering the arrangement of the grating rulers and the kinematic chain topology.

[0010] Step Two: To achieve unified modeling of the kinematic characteristics and error propagation process of a CMM, this invention establishes a sequential cascaded coordinate system from the workpiece coordinate system to the probe end-tool coordinate system based on the Denavit-Hartenberg (DH) modeling concept. This method follows the principle of "workpiece coordinate system priority, motion axes established sequentially according to motion order." For different types of CMMs, the establishment method of each level of coordinate system can be determined by their motion sequence. Specifically, the following coordinate systems are set:

[0011] Global workpiece coordinate system: If the CMM is a workpiece fixed type (such as FXYZ, FYXZ), the workpiece coordinate system is fixed on the base or worktable and does not change with the movement; if the CMM includes workpiece movement (such as XFYZ, XYZF), the workpiece coordinate system is established on the workpiece moving parts and changes with the first movement of the workpiece.

[0012] First Axis Coordinate System: The first motion direction is determined by the first letter after F in the CMM model number, and a coordinate system is established on the corresponding moving component. For example, if the first motion of the FYXZ type is the Y-axis, then a Y-axis coordinate system is established on the cable tray.

[0013] Second axis coordinate system: The second motion direction is determined by the second letter after F in the CMM model number, and a coordinate system is established on the corresponding moving part. For example, if the second motion of the FYXZ type is the X-axis, then an X-axis coordinate system is established on the slide.

[0014] The third axis (tool) coordinate system is determined by the third letter after F in the CMM model number, usually the Z-axis, corresponding to the linear motion of the spindle or probe. A tool coordinate system is established at the end of this component as a reference system for the probe tip.

[0015] The above rules form a complete kinetic chain topology:

[0016] Workpiece coordinate system → First axis coordinate system → Second axis coordinate system → Third axis (tool) coordinate system.

[0017] For any type of CMM, simply read the motion sequence according to its model number to determine the establishment method of each level of coordinate system, thereby forming a unified and standardized motion chain coordinate system sequence, which provides a foundation for subsequent pose transformation description and error modeling.

[0018] Step 3: Calculate the Abbe error introduced by single-axis motion. According to the error propagation mechanism of Abbe's principle, the Abbe error generated by the angular motion error of each axis can be obtained by multiplying the angular motion error of that axis by the Abbe arm component acting on that angular motion error. For example... Figure 1 As shown, the Abbe arm of the probe and the Y-axis grating ruler in the Z direction is... The pitch angle motion error of the Y-axis during motion is Therefore, the Abbe error along the X direction caused by the Y-axis pitch angle motion error at the probe is:

[0019] Define the three-dimensional Abbe errors introduced by the CMM's motion along the X, Y, and Z axes as follows: , , Their relationship can be expressed as the following formula (1):

[0020]

[0021] in , , Corresponding to CMM along the first

[0022] Abbe error components generated in three directions during 𝑖 axis motion.

[0023] Step 4: Establish the homogeneous transformation matrix of error for each single-axis motion. Based on the DH modeling method, encapsulate the movement of each motion axis of the CMM and the associated error terms into a 4×4 homogeneous transformation matrix. As shown in equation (2), this matrix completely describes the relative pose transformation between two adjacent local coordinate systems caused by the CMM axis motion. The homogeneous transformation matrix... The internal structure includes:

[0024] The 3x3 rotation matrix in the top left corner This matrix is ​​used to characterize the attitude deviation caused by angular motion errors. According to kinematic theory, this matrix represents the rotation operation required to transform vector coordinates from the current moving coordinate system to the previous fixed coordinate system.

[0025] The 3×1 translation vector A in the upper right corner is used to characterize the positional deviation. This deviation is composed of the ideal displacement, positioning error, straightness error, and the Abbe error calculated in step three.

[0026]

[0027] In addition, for axes that are not at the beginning of the motion chain, the influence of perpendicularity error between adjacent axes must be considered in their homogeneous transformation matrix. In the above way, the error homogeneous transformation moments of CMM in the X, Y and Z motion axis directions can be constructed respectively, and their forms are as follows (3).

[0028]

[0029] Step 5: Derivation of the actual position of the probe. To obtain the final volume error caused by the cumulative motion errors of the CMM along each axis, it is necessary to accurately solve for the actual position vector OP of the CMM's probe measuring point P in the workpiece coordinate system OXYZ when the CMM moves distances x, y, and z along the X, Y, and Z axes, respectively. First, the initial position vector of the probe in the tool coordinate system at the far end is used as the starting point. Based on the kinematic chain topology established in step two, the cascading order of the homogeneous transformation matrices of each single-axis error is determined. Finally, the homogeneous transformation matrices of each axis error constructed in step four are multiplied in sequence to obtain the actual position vector OP of the probe measuring point P in the workpiece coordinate system OXYZ.

[0030] Step Six: Establishing the CMM volume error model based on Abbe's principle. In Step Five, the actual position vector OP of the probe measuring point P in the workpiece coordinate system OXYZ has been obtained. Since the geometric errors of each axis of the CMM are all minute (usually at the micrometer or microradian level), when expanding the homogeneous transformation matrix of errors in Step Five, there are higher-order error terms involving the multiplication of two or more errors. Their magnitude is much smaller than that of the first-order error terms and can be ignored.

[0031] When the CMM moves along the X, Y, and Z axes (x, y, z), the theoretical position vector of the probe measuring point P in the workpiece coordinate system OXYZ under ideal conditions can be obtained. By subtracting the actual position vector OP from the theoretical position vector, the volume error model of the CMM based on Abbe's principle can be obtained. Attached Figure Description

[0032] Figure 1 Abbe error in CMM;

[0033] Figure 2 Abbe arm between the probe measuring point and the grating ruler reading head;

[0034] Figure 3 Schematic diagram of FYXZ type CMM; Detailed Implementation

[0035] To enable those skilled in the art to better understand and implement the present invention, the present invention will be further described in detail below with reference to the accompanying drawings. This specific embodiment uses a typical FYXZ type fixed bridge coordinate measuring machine (e.g., Figure 3 Using the example shown, this paper elaborates on the implementation process of a new CMM volume error modeling method based on Abbe's principle.

[0036] Step 1: Define geometric error and Abbe arm parameters (refer to...) Figure 2 )

[0037] according to Figure 2 The typical FYXZ type CMM structure shown defines the following 9 Abbe arm components:

[0038] , , ; , , ; , ,

[0039] When the CMM probe is at the origin, the values ​​of the nine Abbe arm components are uniquely determined by the CMM's static mechanical structure, the assembly positions of each component, and the probe configuration. These initial values ​​are defined as: , , , , , , , ,

[0040] When the CMM moves by x, y, and z distances along the X, Y, and Z directions respectively, the relative positions between the probe's measuring point and the reading heads of each axis change, causing 6 of the 9 Abbe arm components to change accordingly, while the other 3 remain unchanged. The Abbe arm components that change with motion can be expressed as follows (4):

[0041]

[0042] (Note: The plus and minus signs here depend on how the coordinate system is established. Please analyze the specific model accordingly.)

[0043] The other three Abbe arm components , , Due to the structure of the CMM and the mounting position of the grating ruler reading head, its length will not change as the CMM moves, and will always maintain its initial value, which is expressed as follows (5):

[0044]

[0045] Step 2: Establish the kinematic chain coordinate system (refer to...) Figure 3 )

[0046] In the FYXZ type moving bridge CMM structure, the Y-axis motion is the bridge frame displacement, the X-axis motion is the sliding seat displacement, and the Z-axis motion is the principal axis displacement. Based on this motion structure, the coordinate system is constructed as follows:

[0047] Global workpiece coordinate system OXYZ: This coordinate system is a global reference system and remains stationary. For moving bridge CMMs, it is typically established on the worktable, which serves as the workpiece bearing reference. The final position of all probes will be uniformly transformed to this coordinate system for evaluation.

[0048] Y-axis coordinate system This local coordinate system is fixed to the bridge frame. When the Y-axis is driven, the entire bridge frame, along with all the components mounted on it (slides, spindles, etc.), will move in a direction parallel to the Y-axis.

[0049] X-axis coordinate system The local coordinate system is fixed to the slide. The slide is mounted on the bridge frame and can move along a direction parallel to the X-axis.

[0050] Z-axis coordinate system This local coordinate system, also known as the tool coordinate system, is fixed to the end of the spindle. The spindle is mounted on a slide and can extend and retract along a direction parallel to the Z-axis.

[0051] Based on the above definition, the motion chain topology from the workpiece to the probe in this CMM is clearly established. The motion and error transmission path from the workpiece to the probe can be described as a cascade of coordinate transformations, expressed as the following equation (6):

[0052]

[0053] Step 3: Calculate the Abbe error introduced by single-axis motion.

[0054] In this embodiment, the three-dimensional Abbe error introduced by the CMM's motion along the Y-axis, X-axis, and Z-axis is calculated respectively. , , .

[0055] 1. Abbe error introduced by CMM movement along the Y-axis

[0056] When the bridge frame of the CMM moves a distance y along the Y-axis, it first causes a transformation in the Abbe arm associated with the Y-axis grating ruler reading head. According to the derivation in step one, its Y-direction component... It changes with the movement of y, while its x and z directions change. Quantity and Then it remains unchanged. Therefore, the Abbe arm component related to the Y-axis grating ruler reading head can be expressed as the following equation (7):

[0057]

[0058] At the same time, including yaw angle error Roll angle error and pitch angle error The cross product of the angular motion error and the error introduced by the Y-axis motion yields the Abbe error. Its expression is as follows (8):

[0059]

[0060] In this expression (8), , and These represent the linear displacement error components along the X, Y, and Z directions at the probe when the CMM moves along the Y-axis. Together, they constitute the Abbe error introduced by the Y-axis movement. .

[0061] 2. Abbe error introduced by CMM movement along the X-axis

[0062] When the CMM slide moves a distance x along the X-axis, it will also cause a change in the Abbe arm component related to the X-axis grating ruler reading head. According to the derivation in step one, its X-direction component... It changes with the movement of x, while its Y and Z directions... Quantity and Then it remains unchanged. Therefore, the Abbe arm component related to the X-axis grating ruler reading head can be expressed as the following equation (9):

[0063]

[0064] At the same time, including yaw angle error Roll angle error and pitch angle error The cross product of the angular motion error and the error introduced by the X-axis motion yields the Abbe error. Its expression is as follows (10):

[0065]

[0066] In this expression, , and These represent the linear displacement error components along the X, Y, and Z directions at the probe when the CMM moves along the Y-axis. Together, they constitute the Abbe error introduced by the X-axis movement. .

[0067] 3. Abbe error introduced by CMM's movement along the Z-axis

[0068] When the CMM spindle moves a distance z along the Z-axis, it will also cause a change in the Abbe arm associated with the Z-axis grating ruler reading head. According to the derivation in step one, its Z-direction component... It will change with the movement of z, and its X and Y directions will change. Quantity and Then it remains unchanged. Therefore, the Abbe arm associated with the Z-axis grating ruler reading head can be expressed as the following equation (11):

[0069]

[0070] At the same time, including yaw angle error Roll angle error and pitch angle error The cross product of the angular motion error and the error introduced by the Z-axis motion yields the Abbe error. Its expression is as follows (12):

[0071]

[0072] In this expression, , and These represent the linear displacement error components along the X, Y, and Z directions at the probe when the CMM moves along the Z-axis. Together, they constitute the Abbe error introduced by the Z-axis movement. .

[0073] Step 4: Establish the homogeneous transformation matrix of errors for each single-axis motion.

[0074] For the three motion axes Y, X, and Z of the FYXZ type CMM, construct their error homogeneous transformation matrices respectively.

[0075] When the bridge frame moves a distance y along the Y-axis, its motion transformation is changed from the workpiece coordinate system OXYZ to the bridge frame coordinate system. The structure of the homogeneous error matrix T(y) of this transformation is as follows:

[0076]

[0077] The Abbe error calculated in step three Substituting expression (8) into equation (13) yields the following equation (14):

[0078]

[0079] When the slide moves a distance x along the X-axis, its motion transformation is determined by the bridge frame coordinate system. Transform to slide coordinate system In addition to the six geometric errors inherent to the X-axis and the Abbe error... In addition, the perpendicularity error between the X and Y axes must be considered in the translation vector part. Due to the influence of this, the structure of the homogeneous error matrix T(x) of this transformation is as follows:

[0080]

[0081] The Abbe error calculated in step three Substituting expression (10) into equation (15) yields the following equation (16):

[0082]

[0083] When the main shaft moves a distance z along the Z-axis, its motion transformation is determined by the slider coordinate system. Transform to principal coordinate system In addition to the six geometric errors inherent to the Z-axis and the Abbe error... In addition, the perpendicularity error between the XZ and YZ axes must be considered in the translation vector part. and Due to the influence of this, the structure of the homogeneous error matrix T(z) of this transformation is as follows:

[0084]

[0085] The Abbe error calculated in step three Substituting expression (12) into equation (17) yields the following equation (18):

[0086]

[0087] Step 5: Determine the actual position of the probe

[0088] For the FYXZ type CMM in this embodiment, the probe measuring point P is first defined in the principal axis coordinate system. The initial homogeneous position vector in the tool coordinate system

[0089]

[0090] Subsequently, along the direction opposite to the kinematic chain, the initial position vector is transformed from... by continuously left-multiplying the homogeneous transformation matrix of the error... The coordinate system is gradually transformed back to the global workpiece coordinate system OXYZ. This is to determine the coordinate system of point P in the slide block coordinate system. Position vector in It is necessary to Left-multiply by the homogeneous transformation matrix T(z) along the Z-axis:

[0091]

[0092] Similarly, to find the coordinate system of point P in the bridge frame Position vector in The result of the previous step needs to be... Left-multiply by the homogeneous transformation matrix T(x) of the X-axis:

[0093]

[0094] Finally, to obtain the final actual position vector of point P in the global workpiece coordinate system OXYZ. , need to Left-multiply by the homogeneous transformation matrix T(y) along the Y-axis:

[0095]

[0096] By combining the above transformation steps, we can obtain the final expression for the homogeneous vector OP of the actual position of the probe measuring point P in the workpiece coordinate system.

[0097]

[0098] Step Six: Establishment of CMM Volume Error Model Based on Abbe's Principle

[0099] First, based on step five, obtain the homogeneous vector OP of the actual position of the probe measuring point P in the workpiece coordinate system OXYZ. Simultaneously, after the CMM moves along the X, Y, and Z axes (x, y, z), the theoretical position of the probe measuring point P in the workpiece coordinate system should be... The difference between the two is the volume error of CMM.

[0100]

[0101] Finally, the volume error model based on Abbe's principle for the FYXZ type CMM can be obtained. Its specific expression is as follows:

[0102]

[0103]

[0104] .

Claims

1. A method for modeling volume error of a three-coordinate measuring machine based on Abbe's principle, characterized in that, The method comprises the following steps: step Step one: define the Abbe arm parameters of each motion axis of the CMM, and take the spatial offset between the measurement point of the probe and the reading head of the X, Y and Z axis grating scales as the Abbe arm component; Step two: based on the DH method, establish a sequential cascade coordinate system from the workpiece coordinate system to the end tool coordinate system of the probe, and the cascade sequence is determined according to the motion sequence of each motion axis; Step three: calculate the Abbe error introduced by each single-axis motion; for each motion axis, multiply the angular motion error of the axis by the corresponding Abbe arm vector component to obtain the Abbe error in the X, Y and Z directions generated by the motion of the axis at the probe; Step four: construct the homogeneous transformation matrix of each axis error containing the geometric error and the Abbe error term; the homogeneous matrix includes a rotation matrix composed of angular motion errors and a translation vector composed of positioning errors, straightness errors and Abbe errors calculated in step three; for the translation vector of the non-first-end axis, the perpendicularity error term between adjacent axes is also included; Step five: according to the coordinate chain topology structure determined in step two, multiply the homogeneous transformation matrix of each axis error constructed in step four by the initial homogeneous coordinate vector of the measurement point of the probe in turn to obtain the actual position vector of the measurement point of the probe in the global workpiece coordinate system; Step six: according to the difference between the actual position vector and the ideal position vector, establish a CMM volume error model based on the Abbe principle.

2. The method of claim 1, wherein, In step one, the Abbe arm is defined as the spatial offset between the measurement point of the probe and the reading head of each motion axis grating scale, and the initial component is determined by the static structure of the CMM. The defined Abbe arm component is composed of the Abbe arm component whose relative position changes with the movement of the CMM motion axis and the Abbe arm component whose value remains fixed.

3. The method of claim 1, wherein, In step two, when establishing the sequential cascade coordinate system from the workpiece coordinate system to the end tool coordinate system of the probe, the principle of giving priority to the workpiece coordinate system and sequentially establishing the motion axes according to the motion sequence is followed, and the establishment method of each coordinate system is determined according to the model and motion characteristics of the CMM.

4. The volume error modeling method of a three-coordinate measuring machine based on Abbe principle according to claim 1 or 2, characterized in that, In the third step, the Abbe error introduced by each single-axis motion is calculated according to the error propagation mechanism of Abbe principle, and the Abbe error is obtained by multiplying the angular motion error of the axis with the corresponding Abbe arm component; the Abbe errors introduced by the motions of the CMM along the X-axis, Y-axis and Z-axis are defined as , , .

5. The method of claim 1 or 4, wherein, The homogeneous transformation matrix of each axis error in step four is a 4x4 matrix, and its form is: ， where the 3x3 rotation matrix in the top-left corner , which represents the attitude deviation caused by the angular motion error, and the 3x1 translation vector A in the top-right corner, which represents the position deviation; the position deviation is composed of the ideal displacement, the positioning error, the straightness error, and the Abbe error calculated in step three; for the shaft at the first end of the motion chain, the influence of the perpendicularity error between adjacent shafts on the translation vector A also needs to be considered in the homogeneous transformation matrix.

6. The method of claim 1, 3, 4 or 5, wherein, In step five, to obtain the final volumetric error caused by the cumulative errors of each axis of the CMM, the actual position vector OP of the probe measurement point P in the workpiece coordinate system OXYZ is solved, with the initial position vector of the probe in the last tool coordinate system As a reference, the order of the homogeneous transformation matrix of each single-axis error is determined according to the motion chain topology established in step two, and the initial position vector is gradually converted from the last tool coordinate system back to the global workpiece coordinate system OXYZ in the opposite direction of the motion chain by successively left multiplying the homogeneous transformation matrix of each axis error constructed in step four, to obtain the actual position vector OP of the probe measurement point P.

7. The method of claim 1 or 6, wherein In step six, the CMM volume error model is obtained by subtracting the theoretical position vector of the measurement point P of the probe in the ideal case from the actual position vector OP of the measurement point P of the probe in the workpiece coordinate system OXYZ when the CMM moves along the X, Y and Z axes by x, y and z; when solving the actual position vector OP, the high-order error term involving the multiplication of two or more errors in the expansion process of the error homogeneous transformation matrix can be ignored.

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