Adaptive path generation for CNC machining

By defining a feature point grid on the CNC machine and calculating the spatial mapping function, the machining inconsistency problem caused by the difference in nominal and actual workpiece shapes is solved, and precise workpiece adaptation and machining effects are achieved.

CN112748701BActive Publication Date: 2025-09-16FANUC LTD
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Patent Information

Application Number
CN202011181596.1
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Priority Date
2019-10-30
Filing Date
2020-10-29
Publication Date
2025-09-16
Estimated Expiration
2040-10-29

AI Technical Summary

Technical Problem

When CNC machines process actual workpieces, the difference between the nominal workpiece shape and the actual workpiece shape leads to inconsistent processing depth and failure to meet accuracy requirements. This is especially true when milling or etching surfaces, as existing technologies cannot effectively adapt to the unique shape of each actual workpiece.

Method used

By defining a grid of feature points on the nominal workpiece, using a probe to measure the position of the feature points on the actual workpiece, calculating the spatial mapping function, adapting the nominal tool path to the actual workpiece shape, and generating a new tool path to ensure machining accuracy.

Benefits of technology

It realizes the precise processing of CNC machines on actual workpieces, ensures the consistency of processing depth and surface finish, reduces tool vibration, and adapts to individual differences in workpiece shapes.

✦ Generated by Eureka AI based on patent content.

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Abstract

A method and system for adapting a CNC machine tool path from a nominal workpiece shape to an actual workpiece shape. The method includes defining a grid of feature points on the nominal workpiece shape, wherein the feature points encompass an area surrounding a machine tool path, but not necessarily points on the machine tool path. A probe is used to detect the positions of the feature points on the actual workpiece. A spatial mapping function is calculated as a transformation from the nominal feature points to the actual feature points, and the function is applied to the nominal tool path to calculate a new tool path. The new tool path is used by the CNC machine to operate on the actual workpiece. The feature points are used to characterize the three-dimensional shape of the working surface of the actual workpiece, rather than just curves or contours.
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Description

Technical Field

[0001] The present invention relates to the field of computational tool paths for computer numerical control (CNC) machines, and more particularly to a method for adapting a CNC machine tool path from a nominal workpiece shape to an actual workpiece shape, which defines a grid of a plurality of feature points on the nominal workpiece shape, measures positions of the plurality of feature points on the actual workpiece shape, and uses a spatial mapping function with the measured positions to transform the CNC machine path to the actual workpiece shape. Background Art

[0002] Computer numerical control (CNC) machines—such as drills, lathes, and milling machines—have been used for many years to automatically produce parts based on computer-aided design (CAD) files that define the part's shape. CNC machines provide the precision to reliably and repeatedly produce parts to the desired shape and specifications, and can produce parts quickly.

[0003] However, by their very nature, CNC machines perform their cutting operations relative to a designed nominal workpiece shape. In reality, each real workpiece has an actual shape that differs slightly from the nominal design shape. In some types of workpieces, the variation between the nominal and actual workpiece shapes is insignificant. However, in other types of workpieces, the variation between the nominal and actual workpiece shapes, as well as the variation between the shapes of each individual actual workpiece, can be significant.

[0004] For example, some applications require a CNC machine to mill or etch shapes onto the surface of a part. If the part is cast or forged, the actual shape of the surface can vary from part to part. In these applications, in order to etch the shape into the part's surface at a constant depth, the actual shape of the surface must be known. Without knowing the actual surface shape and contour, the CNC machine may etch too deep in some areas and / or not deep enough in others.

[0005] In view of the above, there is a need for a robust and accurate technique for adapting the CNC machine tool path to each potentially different actual workpiece shape. Summary of the Invention

[0006] According to the teachings of the present disclosure, a method for adapting a CNC machine tool path from a nominal workpiece shape to an actual workpiece shape is disclosed. The method includes defining a grid of multiple feature points on the nominal workpiece shape, wherein the multiple feature points encompass an area surrounding a machine tool path but do not necessarily represent points on the machine tool path. A probe is used to detect the positions of the multiple feature points on the actual workpiece. A spatial mapping function is calculated as a transformation from the multiple nominal feature points to the multiple actual feature points, and the function is applied to the nominal tool path to calculate a new tool path. The new tool path is used by the CNC machine to operate on the actual workpiece. The multiple feature points are used to characterize the three-dimensional shape of the working surface of the actual workpiece, rather than just a curve or contour.

[0007] Additional features of the presently disclosed apparatus and methods will become apparent from the following description and appended claims, taken in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Figure 1 is an illustration of what happens when a CNC machine tool path designed for a nominal workpiece shape is applied to an actual workpiece having a shape different from the nominal;

[0009] Figure 2 is an illustration of a technique for adaptive path generation for CNC machining according to an embodiment of the present disclosure;

[0010] Figure 3 is an illustration of a CNC machine tool path applied to an actual part that differs from a nominal part with and without adaptation according to an embodiment of the present disclosure;

[0011] Figure 4 is a flow chart of a method for adaptive path generation for CNC machining according to an embodiment of the present disclosure; and

[0012] Figure 5 is a schematic diagram of a system for adaptive path generation for CNC machining according to an embodiment of the present disclosure. DETAILED DESCRIPTION

[0013] The following discussion of embodiments of the present disclosure directed to methods and systems for adaptive path generation for CNC machining is merely exemplary in nature and is in no way intended to limit the disclosed apparatus and techniques or applications or uses thereof.

[0014] Computer numerical control (CNC) machines have been used for many years to automatically produce parts based on computer-aided design (CAD) files or other data that defines the part's shape. There are several different types of machines that can be mathematically or numerically controlled, such as drill presses, lathes, and milling machines. Discussion of CNC machines in this disclosure should be understood to include the aforementioned types of machines and any other type of machine tool that is automatically controlled by a computer or processor to perform a machining function based on a mathematical model or digital data file representing a shape.

[0015] CNC machines perform their cutting operations according to design relative to a nominal workpiece shape. Specifically, the part being operated on has some nominal pre-machined shape, such as defined in a CAD system, and the machine tool path is calculated based on this nominal shape. However, in reality, each real workpiece has an actual shape that may differ slightly from the nominal design shape. For some types of workpieces, the variation between the nominal and actual workpiece shapes is insignificant. However, for other types of workpieces, the variation between the nominal and actual workpiece shapes, as well as the variation between the shapes of each individual actual workpiece, can be significant.

[0016] Figure 1 is an illustration of what happens when a CNC machine tool path designed for a nominal workpiece shape is applied to an actual workpiece having a shape that is different from the nominal. The nominal workpiece 110 has a shape that is considered a standard or nominal shape. The nominal workpiece 110 may be a designed shape from a CAD model, or it may be a measured shape of a prototype or production part that is considered nominal or a "benchmark." The actual workpiece 120 has a shape that is different to some extent from the nominal workpiece 110. Figure 1 , it can be seen that the actual workpiece 120 has a different cross-sectional curvature than the nominal workpiece 110 .

[0017] Consider an application where a CNC machine is tasked with milling or etching a pattern 112 onto the upper surface of a part, represented by a nominal workpiece 110. Consider also the requirement that pattern 112 have a constant depth of cut into the surface of the part, represented by nominal workpiece 110. The CNC machine is programmed with a tool path based on the shape of nominal workpiece 110. When this tool path is applied to an actual workpiece 120, the results will be completely unacceptable; some portions of pattern 112 will have a depth of cut that is too shallow or completely miss the actual workpiece 120, while other portions of pattern 112 will have a depth of cut that is too great. Many other scenarios can be envisioned where a CNC tool path based on the nominal workpiece shape would be unsatisfactory when applied to the actual workpiece shape. This is the problem addressed by the techniques of the present disclosure.

[0018] Other known path-fitting techniques measure points along the perimeter of the part or along the intended tool path, fit a spline through the measured points on the actual part, then attempt to estimate the actual part's profile compared to the nominal part and compensate the tool path based on the offsets. However, these techniques do not fully account for the entire working surface of the part, as they are based on interpolation of curves rather than surfaces. Furthermore, these techniques require a large number of feature points along the length of the tool path in order to define a good-quality spline.

[0019] Figure 2 FIG is an illustration of a technique for adaptive path generation for CNC machining according to an embodiment of the present disclosure. Figure 2 In Greek mythology, the egg is used as a symbol for a workpiece, such as a molded or cast part. Figure 2 The situation is that a pattern is to be milled out on the outer surface of an egg shell. Obviously, the pattern must be milled at a very shallow and consistent depth to avoid penetrating the shell. Because the size and shape of each egg varies slightly, it is clear that a CNC machine tool path cannot be defined based on one "nominal" egg and then satisfactorily applied to every other egg. Instead, according to the technology of the present disclosure, the machine tool path is defined relative to a grid of measured feature points on the nominal egg, and then the corresponding grid of measured feature points on each actual egg is used to calculate a spatial mapping function that enables an adjusted tool path to be calculated for each actual egg. Note that machining on an egg shell is only an example. The disclosed technology can be applied to the general case of part machining beyond this specific example.

[0020] The nominal workpiece 210 represents a part having a specified or theoretical size and shape and is shown in step 1. A nominal tool path 212 is defined relative to the shape of the nominal workpiece 210. The nominal tool path 212 has a prescribed shape or pattern (in this example, the letter "F" is a certain size) and can have any suitable or desired characteristics, such as a constant depth of cut into the surface of the nominal workpiece 210, a variable depth of cut, and / or a surface normal condition. A grid of nominal feature points 214 is defined in an area surrounding the nominal tool path 212 on the surface of the nominal workpiece 210. The individual feature points in the grid of nominal feature points 214 need not be superimposed on the nominal tool path 212; rather, the grid of nominal feature points 214 is merely required to define an area containing or surrounding the nominal tool path 212. The grid of nominal feature points 214 can have any size suitable for achieving the desired machining accuracy. Figure 2In the example shown, a 6x8 point grid is used. However, the grid of nominal feature points 214 can be square, rectangular, or any other shape suitable for appropriately fitting the nominal tool path 212. The grid of nominal feature points 214 is preferably fairly evenly spaced, with approximately equal distances between points in rows and columns. In steps 2 and 3, the grid of nominal feature points 214 and the nominal tool path 212 are provided to the next row of process steps for application to the actual workpiece.

[0021] The new workpiece 220 (step 4) is the part on which the machine tool will actually operate. The new workpiece 220 has a slightly different shape than the nominal workpiece 210; therefore, a new tool path 222 must be calculated for the new workpiece 220. In step 5, a probe 230 is used to measure a grid of actual feature points 224 on the new workpiece 220. The probe 230 can be any suitable point position measurement device, such as a mechanical probe, a laser probe, or other type. The probe 230 can be part of a CNC machine. The probe 230 measures the three-dimensional position of each individual feature point in the grid of actual feature points 224. The overall position of the grid of actual feature points 224 can be established by any suitable means, such as corner grid points or a central grid point having positions established relative to both ends and both side edges of the new workpiece 220, or a central grid point located at the highest point on the new workpiece 220, where the new workpiece 220 is held in a fixture during the probe measurement. Rules for measuring the grid of actual feature points 224 may be established appropriately—for example, requiring the probe 230 to approach each individual grid point in a direction perpendicular to a local tangent plane in the grid of known nominal feature points 214 .

[0022] After measuring the grid of actual feature points 224, a mapping function f is constructed in step 6 that relates the grid of actual feature points 224 to the grid of nominal feature points 214. In other words, the function f is constructed to align the two sets of points in space so that f applied to each nominal feature point 214 is equal to or close to the corresponding one in the grid of actual feature points 224. The function f can be any suitable spatial mapping function that describes the mathematical relationship between the first surface patch (the grid of nominal feature points 214) and the second surface patch (the grid of actual feature points 224). For example, the function f can be a combination of multiple independent functions of order 1, 2, or 3, each of which defines a spatial mapping in a local area of ​​the grid of nominal / actual feature points, where each of the independent functions of order 1, 2, or 3 uses a subset of the points in the grid as input.

[0023] After calculation, the function f is applied (step 7) to the nominal tool path 212 to obtain the new tool path 222. It should be understood that the nominal tool path 212 and the new tool path 222 are generally three-dimensional shapes. That is, although the letter F of the nominal tool path 212 appears somewhat two-dimensional in the figure, it is actually twisted or warped to conform to the surface shape of the nominal workpiece 210. Similarly, the new tool path 222 is warped to conform to the surface shape of the new workpiece 220. In addition, the nominal tool path 212 and the new tool path 222 are modeled as discrete point sets so that each point of the nominal tool path 212 can be inserted into the function f to obtain the corresponding point in the new tool path 222. After calculating the new tool path 222 using the function f, the CNC machine performs a machining operation on the new workpiece 220 using the new tool path 222. Steps 4-7 of the process are repeated for each new actual workpiece.

[0024] In some embodiments, when the function f is applied to the nominal tool path 212 to obtain the new tool path 222, the new tool path 222 includes not only the position (x, y, z), but also the tool orientation angles (e.g., yaw, pitch, and roll or W, P, R rotations). Differences in tool orientation angles can be significant for defining the surface normal of the machining area, reducing tool vibration / chatter, and for other reasons. One technique for determining the orientation angle is to use the positions of neighboring points in the grid of actual feature points 224 to estimate the surface tangent plane and surface normal direction for a particular grid point. The surface normal direction of each point in the grid of actual feature points 224 can then be applied to the new tool path 222, where the CNC machine tool can then be oriented to the surface normal direction of each point in the new tool path 222.

[0025] Figure 3 is an illustration of a CNC machine tool path without and with adaptation applied to an actual part that differs from the nominal part according to an embodiment of the present disclosure. Figure 3 As shown on the left side of FIG, the nominal tool path 212 and the grid of nominal feature points 214 are as shown in FIG. Figure 2 As shown in step 1, Figure 3 To the right of the figure, a portion of a new workpiece 220 is shown in a magnified view. At the top right, the new workpiece 220 is shown, where the nominal tool path 212 and the grid of nominal feature points 214 are overlapped; that is, the grid of feature points and the tool path have not yet been adapted to the shape of the new workpiece 220. It can be seen that, without adaptation, both the nominal tool path 212 and the grid of nominal feature points 214 deviate from the surface of the new workpiece 220. If a CNC machine were to attempt to operate on the new workpiece 220 using the nominal tool path 212, it would be clear that for most operations, there would be no tool-workpiece contact. This is precisely the problem that the techniques of the present disclosure are designed to address.

[0026] exist Figure 3 In the lower right corner of the figure, a new workpiece 220 is shown with the new tool path 222 and the grid of actual feature points 224 overlaid; that is, the grid of feature points and the tool path have been adapted to the shape of the new workpiece 220. As can be seen, through the adaptation according to the present disclosure, both the new tool path 222 and the grid of actual feature points 224 faithfully follow the surface of the new workpiece 220. When the CNC machine operates on the new workpiece 220 using the new tool path 222, the tool will maintain contact with the surface of the new workpiece 220 throughout the operation, and the machining operation will provide the desired results (depth of cut, etc.) throughout the tool path. Correctly adapting the new tool path 222 to the new workpiece 220 also results in improved surface finish quality and reduced tool vibration.

[0027] The adaptive toolpath generation techniques disclosed above have been shown to work well for all types of nominal-to-actual part shape differences. These differences include: two parts having the same characteristic shape but different curvatures (i.e., the egg example); one part having a concave shape while the other has a convex shape; different amounts of concave and convex areas (depressions and ridges) between one part and the other; and any other type of arbitrary difference in shape between the nominal part and actual part surfaces in the area of ​​the toolpath. The disclosed adaptive toolpath generation techniques are able to handle all of these types of part-to-part shape differences because a grid of points is used to characterize the entire surface, not just the outline or a single path.

[0028] Figure 4 4 is a flow chart of a method for adaptive path generation for CNC machining according to an embodiment of the present disclosure. In block 402, a nominal workpiece and a nominal tool path are provided. The nominal tool path is designed to perform a desired operation on the nominal workpiece. In block 404, a grid of nominal feature points is defined on the surface of the nominal workpiece for an area surrounding the nominal tool path. The activities of blocks 402 and 404 are Figure 2 Steps 1-3 are shown and described in the corresponding discussion.

[0029] At block 406, Figure 2 As shown in step 4 of , a new workpiece is provided, wherein the new workpiece is the actual part to be operated by the CNC machine. Figure 2 As shown in step 5 of and described in the corresponding discussion, a grid of actual feature points is measured on the new workpiece by the probe. At block 410, a spatial mapping function f is constructed. Figure 2As discussed in the discussion of step 6 of FIG4 , the spatial mapping function f is a function that, when applied to the mesh of nominal feature points (from block 404), produces a mesh of actual feature points (from block 408). At block 412, a new tool path is calculated by applying the spatial mapping function f to the nominal tool path. The CNC machine then performs a machining operation on the new workpiece using the new tool path. The process of flowchart 400 then loops back to block 406, where another new workpiece is provided. The nominal workpiece shape, nominal tool path, and mesh of nominal feature points are used as a baseline for all new / actual workpieces.

[0030] Figure 5 is a schematic diagram of a system 500 for adaptive path generation for CNC machining according to an embodiment of the present disclosure. The system 500 includes at least one computer, controller, or server, which is represented by computer 502. Computer 502 includes a processor and memory 504, and is capable of receiving and storing data, performing calculations, and optionally controlling a machine tool. Memory 504 contains data about a nominal workpiece as previously described, including Figure 2 The nominal path and the mesh of nominal feature points are shown in steps 1 to 3. Data about the nominal workpiece can be provided to the memory 504 of the computer 502 from a different computer, such as a CAD system or a 3D modeling system.

[0031] System 500 also includes Figure 2 The probe 230 shown and previously discussed is shown. The probe 230 can be a mechanical probe, a laser probe, or other device capable of measuring a plurality of feature points on the actual / new workpiece. The probe 230 communicates with the computer 502 and, after measuring the points, provides data defining a grid of actual feature points. The computer 502 calculates or constructs a spatial mapping function f based on the grid of nominal feature points and the grid of actual feature points. The computer 502a also calculates a new tool path by applying the spatial mapping function f to the nominal tool path.

[0032] The system 500 may also include a CNC machine 506 that receives the new tool path from the computer 502 and performs the machining operation on the actual / new workpiece. The CNC machine 506 itself is not a necessary part of the adaptive path generation process of the present disclosure, but rather receives the output of the adaptive path generation. As previously mentioned, the CNC machine 506 can be any type of CNC machine, such as a drill, lathe, or milling machine, or any other type of machine tool that is automatically controlled by a computer or processor to perform machining functions based on a model or data file that defines the tool path. In practice, the probe 230 can be part of the CNC machine 506, and the computer 502 can be the controller of the CNC machine 506.

[0033] In some embodiments, the computer 502 can be a plurality of computers communicating on a network (wired or wireless). As will be appreciated by those skilled in the art, several computing functions, including providing CAD data describing the nominal workpiece, controlling the probe 230, calculating the spatial mapping function f and calculating the new tool path using f, and controlling the CNC machine 506, can be performed by any combination of one or more computing devices without changing the scope or nature of the presently disclosed technology. These devices can be general-purpose computers, custom controllers or processors, or any other computing device suitable for the purpose. In the preferred embodiment described above, the computer 502 is a CNC machine controller; that is, the data describing the nominal workpiece is provided to the CNC machine controller, the probe 230 is also part of the CNC machine 506 and provides measurement data to the controller, the CNC machine controller calculates the spatial mapping function f, and calculates the new tool path using f.

[0034] As described above, the disclosed techniques for adaptive path generation for CNC machining improve the performance of machining operations for applications where part-to-part variations are significant or where machining operations are sensitive to even small variations in part shape.

[0035] Although various exemplary aspects and embodiments of the method and system for adaptive path generation for CNC machining have been discussed above, those skilled in the art will recognize modifications, permutations, additions, and sub-combinations thereof. It is therefore intended that the appended claims and claims hereafter introduced be interpreted as including all such modifications, permutations, additions, and sub-combinations as fall within their true spirit and scope.

Claims

1. A method for adapting a machine tool path to an actual part shape, the method comprising: providing a nominal tool path defining a machining operation on a surface of a nominal workpiece; defining a grid of a plurality of nominal feature points, wherein each of the plurality of nominal feature points is located on the surface of the nominal workpiece, and the grid of the plurality of nominal feature points covers an area containing the nominal tool path; Provide new artifacts; measuring a grid of a plurality of actual feature points on the new workpiece, wherein measuring the grid of the plurality of actual feature points comprises measuring the plurality of actual feature points using a probe, the probe comprising a mechanical probe or a laser probe, wherein each of the plurality of actual feature points is measured using the probe by approaching a position of a corresponding nominal feature point in a direction perpendicular to a local surface section plane, wherein the local surface section plane is determined based on adjacent points in the grid of the plurality of nominal feature points; constructing, using a computer having a processor and a memory, a spatial mapping function, wherein the spatial mapping function defines a spatial transformation from the grid of the plurality of nominal feature points to the grid of the plurality of actual feature points; and using the computer, calculating a new tool path by applying the spatial mapping function to the nominal tool path; The new tool path includes a position difference and a rotation difference from the nominal tool path. 2 . The method of claim 1 , wherein the nominal tool path and the grid of the plurality of nominal feature points are defined in a computer-aided design system.

3. The method of claim 1, wherein the probe is part of a machine tool that performs the machining operation.

4. The method of claim 3, wherein the computer is a controller of the machine tool that performs the machining operation.

5. The method according to claim 1, wherein The grid of the plurality of actual feature points has the same number of rows and columns of points as the grid of the plurality of nominal feature points.

6. The method according to claim 1, wherein The spatial mapping function specifies a mathematical relationship between points in the grid of the plurality of nominal feature points and points in the grid of the plurality of actual feature points, and wherein the spatial mapping function comprises a plurality of independent functions of order 1, 2 or 3, each independent function defining a spatial mapping in a local area of ​​the grid of the plurality of nominal feature points and the plurality of actual feature points, wherein each of the plurality of independent functions uses a subset of the points in the grid as input.

7. The method according to claim 6, wherein: The new toolpath includes x, y, and z position differences from the nominal toolpath, as well as yaw, pitch, and roll rotational differences from the nominal toolpath. 8 . The method of claim 1 , further comprising performing the machining operation on the new workpiece using the new tool path by a machine tool.

9. The method of claim 8, wherein the machine tool is a computer controlled milling machine, drilling machine, or lathe.

10. A method for adapting a machine tool path to an actual part shape, comprising: defining a grid of a plurality of nominal feature points on a surface of a nominal workpiece, the grid of the plurality of nominal feature points covering an area containing a nominal tool path; measuring a grid of a plurality of actual feature points on a new workpiece, wherein measuring the grid of the plurality of actual feature points comprises measuring the plurality of actual feature points using a probe, the probe comprising a mechanical probe or a laser probe, wherein each of the plurality of actual feature points is measured using the probe by approaching a position of a corresponding nominal feature point in a direction perpendicular to a local surface tangent plane, wherein the local surface tangent plane is determined based on neighboring points in the grid of the plurality of nominal feature points; constructing a spatial mapping function using a computer having a processor and a memory, wherein the spatial mapping function defines a spatial transformation from the grid of the plurality of nominal feature points to the grid of the plurality of actual feature points; and calculating a new tool path by applying the spatial mapping function to the nominal tool path; The new tool path includes a position difference and a rotation difference from the nominal tool path.

11. A system for adapting a machine tool path to an actual part shape, the system comprising: a measurement probe configured to measure positions of a grid of a plurality of actual feature points on an actual workpiece, wherein measuring the grid of the plurality of actual feature points comprises measuring the plurality of actual feature points using a probe, the probe comprising a mechanical probe or a laser probe, wherein each of the plurality of actual feature points is measured using the probe by approaching a position of a corresponding nominal feature point in a direction perpendicular to a local surface tangent plane, wherein the local surface tangent plane is determined based on neighboring points in the grid of the plurality of nominal feature points; and a computer having a processor and a memory, the computer communicating with the probe and receiving the positions of the grid of the plurality of actual feature points, the computer being configured to construct a spatial mapping function defining a spatial transformation from a grid of a plurality of nominal feature points to the grid of the plurality of actual feature points, the grid of the plurality of nominal feature points being on a surface of a nominal workpiece and covering an area containing a nominal tool path, and to calculate a new tool path by applying the spatial mapping function to the nominal tool path, wherein the new tool path includes a positional difference and a rotational difference from the nominal tool path.

12. The system of claim 11, wherein the nominal tool path and the mesh of the plurality of nominal feature points are defined in a computer-aided design system and provided to the computer.

13. The system according to claim 11, wherein: The grid of the plurality of actual feature points has the same number of rows and columns of points as the grid of the plurality of nominal feature points.

14. The system according to claim 11, wherein: The spatial mapping function specifies a mathematical relationship between points in the grid of the plurality of nominal feature points and points in the grid of the plurality of actual feature points, and wherein the spatial mapping function comprises a plurality of independent functions of order 1, order 3, or order 3, each independent function defining a spatial mapping in a local area of ​​the grid of the plurality of nominal feature points and the plurality of actual feature points, wherein each of the plurality of independent functions uses a subset of the points in the grid as input.

15. The system according to claim 14, wherein: The new toolpath includes x, y, and z position differences from the nominal toolpath, as well as yaw, pitch, and roll rotational differences from the nominal toolpath.

16. The system of claim 11, wherein the probe and the computer are included in a machine tool that uses the new tool path to perform a machining operation on the actual workpiece, wherein the machine tool is a computer-controlled milling machine, drilling machine, or lathe.

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