A geodesic-based free-form surface numerical control machining path planning method

Abstract

The present application relates to a kind of free-form surface numerical control machining path planning method based on geodesic line.The method constructs geodesic line on free-form surface, and continuously obtains point on geodesic line as tool contact point, and the tool contact point is sorted according to machining sequence, which is the required numerical control machining path;The tool contact point includes the following steps: step S1, based on NURBS modeling to build the free-form surface;Step S2, four boundary lines of the free-form surface are represented using NURBS curve.The present application innovatively plans the numerical control machining path of free-form surface by geodesic line, compared with traditional tool path planning method, the planning method is suitable for cutting NURBS surface, not dependent on the u, v direction of surface parameter domain, more adaptable, and there is no surface cusp in machining process, smoothness is good, meet the demand of high-precision numerical control machining of free-form surface.

Description

Technical Field

[0001] This invention relates to CNC machining, and in particular to a method for CNC machining path planning of freeform surfaces based on geodesics. Background Technology

[0002] With the rapid development of technology, freeform surfaces, with their excellent aerodynamic, fluid dynamic, and thermodynamic characteristics, are widely used in the design of many parts in aerospace, defense, transportation, power, and equipment manufacturing fields. Freeform surfaces are among the most complex and frequently encountered surfaces in engineering, and the requirements for their machining accuracy are becoming increasingly stringent. Toolpath planning for freeform surfaces is a crucial step in CNC machining, affecting both the machining quality and efficiency.

[0003] Traditional toolpath planning methods include the isoparametric line method and the cutting plane method. While the isoparametric line method is computationally simple, it suffers from inconsistent path density, and the generated toolpath may not match the cutting boundary of the surface. The cutting plane method generates toolpaths using intersecting lines; therefore, when the surface curvature changes, inconsistent path density still occurs, leading to inconsistent tool cutting bandwidth and varying cutting forces during machining. This negatively impacts tool life and the machining efficiency and quality of freeform surfaces.

[0004] With the development of CNC technology, traditional tool path planning methods can no longer meet the needs of high-precision CNC machining of freeform surfaces. Therefore, how to realize a tool path generation technology that can improve the machining quality and efficiency of freeform surfaces has become an urgent technical problem to be solved. Summary of the Invention

[0005] Therefore, it is necessary to provide a geodesic-based CNC machining path planning method for freeform surfaces, which addresses the fact that traditional tool path planning methods can no longer meet the requirements of high-precision CNC machining of freeform surfaces, leading to problems such as sharp points and unevenness during the machining process.

[0006] A method for planning CNC machining paths on freeform surfaces based on geodesics involves constructing geodesics on the freeform surface, continuously selecting points on the geodesics as tool contact points, and then arranging these tool contact points according to the machining sequence to obtain the desired CNC machining path. Obtaining the tool contact points includes the following steps:

[0007] Step S1: Construct the freeform surface based on NURBS modeling;

[0008] Step S2: Represent the four boundary lines of the freeform surface using NURBS curves, select one of the boundary lines, and discretize the selected boundary line according to the chord height constraint to obtain multiple first reference points;

[0009] Step S3: For each of the first reference points, numerical methods are used to construct and fit, resulting in multiple geodesics;

[0010] Step S4: Discretize points for each geodesic line according to the specified arc length or residual height constraint to obtain multiple second reference points;

[0011] Step S5: Arrange multiple second reference points with the same order on the geodesic lines into a group, and fit the second reference points in each group with NURBS curves to obtain multiple fitted curves;

[0012] Step S6: Take points continuously for each fitted curve according to the chord height constraint, and the obtained points are the knife contact points.

[0013] As a preferred example, in step 1, NURBS surfaces are used to represent the surface to be processed, and geometric parameter analysis is performed on the NURBS surfaces to obtain the first basic form and the second basic form;

[0014] The NURBS surface formula is as follows:

[0015]

[0016] Where, d ij ω represents the control points of the surface. ij N represents the weights of the surface. i,k (u) represents the k-th degree B-spline basis function of the surface, N j,l (v) represents the l-th order B-spline basis function of the surface.

[0017] As a preferred example, the geometric parameter analysis includes the normal vector N(u0,v0) of any point P(u0,v0) on the freeform surface and the normal curvature κ of the freeform surface in any direction at that point. n ;

[0018] The expression for the normal vector N(u0,v0) is as follows:

[0019]

[0020] In the formula, E, F, and G are the first type of basic quantities of the freeform surface.

[0021] As a preferred example, the first fundamental form of the freeform surface P(u,v) is:

[0022] I = Edu 2 +2Fdudv+Gdv 2

[0023] The second fundamental form of the freeform surface P(u,v) is:

[0024] Π=Ldu 2 +2Mdudv+Ndv 2

[0025] In the formula, E, F, and G are the first type of basic quantities of the freeform surface; L, M, and N are the second type of basic quantities of the freeform surface.

[0026] As a preferred example, in step 2, the expression for representing the boundary line using NURBS curves is:

[0027] Where, d i ω represents the control points of the surface. i N represents the weight of the control points. i,k (u) represents the k-th degree B-spline basis function of the surface.

[0028] As a preferred example, step 2, where the boundary line is discretely selected based on the chord height constraint, includes the following steps:

[0029] Step 21: Take one of the boundary lines as a reference line, select the first reference point corresponding to u1=0 as the initial point, and set the target arc length s1=0;

[0030] Step 22: Calculate the arc length ds between two adjacent points on the reference line using the chord height error formula, and calculate the target arc length s based on the arc length ds. i =s i-1 +ds, and then calculate the next point of the initial point according to the arc length integral formula and Simpson's quadrature formula;

[0031] Step 23: Take the next point after the initial point as the new initial point, and determine the radius of curvature R of the curve and the normal curvature k of the surface at that initial point. n Substituting the formulas for chord height error, arc length integral, and Simpson's quadrature, we iteratively solve the problem until s... i ≥s total At this time, let s i =s total Solve for the last point, then the loop terminates;

[0032] Step 24: After the loop terminates, the point obtained is the first reference point.

[0033] As a preferred example, in step S3, constructing and fitting the geodesic includes the following steps:

[0034] Step S31: Taking the first reference point as a reference, each first reference point corresponds to a geodesic line. Solve the unit tangent vector at the first reference point and the next point of the first reference point on the same geodesic line according to the geodesic iterative formula.

[0035] Step S32: Correct the obtained unit tangent vector using the mean filtering algorithm, and project the corrected unit tangent vector onto the tangent plane of the surface at the first reference point to obtain a projection vector, and replace the unit tangent vector with the projection vector.

[0036] The geodesic iterative formula is as follows:

[0037]

[0038]

[0039] In the formula, T i j Let T be the j-th point on the i-th geodesic line, and let Δs be the value of point T. i j and T i j-1 The arc length between For a free surface at point T i j-1 along Normal curvature of direction, It is the i-th geodesic line at point T i j The unit tangent vector at that point is also the step direction. For a free surface at point T i j-1 The unit normal vector at that location;

[0040] Step S33: Repeat steps S31 and S32 in a loop, using the geodesic iterative formula to continuously solve the problem and obtain multiple third reference points on the i-th geodesic line until the third reference points are outside the freeform surface;

[0041] Step S34: Use NURBS curves to fit multiple third reference points to obtain the geodesic.

[0042] As a preferred example, the boundary line is replaced with the geodesic line.

[0043] As a preferred example, in step S4, for each geodesic line, starting from the starting point, points are discretely selected according to the specified arc length or residual height constraint; the projection point on the boundary line is taken as the last point.

[0044] As a preferred example, in step S6, the tool contacts are sorted according to the processing order and connected by straight lines to form the required CNC machining path.

[0045] The beneficial effects of this invention are as follows:

[0046] 1. This invention innovatively plans the CNC machining path of freeform surfaces using geodesics. Compared to traditional toolpath planning methods, this method is applicable to trimmed NURBS surfaces, is independent of the u and v directions of the surface parameter domain, and has greater adaptability. Simultaneously, it avoids surface spurs during machining, resulting in smoothness and improving machining efficiency and quality, thus meeting the requirements for high-precision CNC machining of freeform surfaces.

[0047] 2. The CNC machining path planning method proposed in this invention can generate tool paths with equal residual height by setting different parameters, thus solving the problem of inconsistent path density. It can also generate paths with consistent boundaries, which is suitable for machining surfaces that require extremely high surface finish or excellent aerodynamic effects. Attached Figure Description

[0048] Figure 1 This is a flowchart illustrating the path planning method for CNC machining of freeform surfaces based on geodesics.

[0049] Figure 2 This is a schematic diagram of the NURBS surface constructed in an embodiment of the present invention;

[0050] Figure 3 This is a schematic diagram of the process for obtaining the first reference point in an embodiment of the present invention;

[0051] Figure 4 This is a schematic diagram illustrating the use of NURBS curves to approximate the four boundaries of a surface in an embodiment of the present invention;

[0052] Figure 5 This is a schematic diagram showing the distribution of the first reference point on the curved surface in an embodiment of the present invention;

[0053] Figure 6 This is a schematic diagram illustrating the solution of the third reference point using the geodesic iterative formula in an embodiment of the present invention;

[0054] Figure 7 This is a schematic diagram of n geodesic lines on a freeform surface in an embodiment of the present invention;

[0055] Figure 8 This is a schematic diagram illustrating the relationship between the cutting tool and the tool contact point in an embodiment of the present invention;

[0056] Figure 9 This is a schematic diagram illustrating the distribution of points for each of the n geodesic lines discretely selected according to the residual height constraint in an embodiment of the present invention;

[0057] Figure 10 This is a schematic diagram of the fitting result of the second reference point in an embodiment of the present invention;

[0058] Figure 11 This is a schematic diagram of the distribution of the knife contact points in an embodiment of the present invention;

[0059] Figure 12 This is a schematic diagram of the processing path in an embodiment of the present invention. Detailed Implementation

[0060] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0061] It should be noted that when a component is said to be "installed on" another component, it can be directly on the other component or it may be in a component that is centered on it. When a component is said to be "set on" another component, it can be directly set on the other component or it may also be in a component that is centered on it. When a component is said to be "fixed to" another component, it can be directly fixed to the other component or it may also be in a component that is centered on it.

[0062] It should be understood that the terms "length", "width", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this application.

[0063] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the specification of this invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "or / and" as used herein includes any and all combinations of one or more of the associated listed items.

[0064] Please refer to Figure 1 A method for planning CNC machining paths on freeform surfaces based on geodesics is proposed. This method constructs geodesics on the freeform surface and continuously selects points on the geodesics as tool contact points. The tool contact points are then arranged according to the machining sequence to obtain the desired CNC machining path. The specific steps for obtaining these tool contact points are as follows:

[0065] Step S1: Construct a freeform surface based on NURBS modeling.

[0066] In this step, the geometric parameters of the freeform surface are analyzed as follows: During the freeform surface machining path planning process, NURBS surfaces are used to represent the surface to be machined. The NURBS surface formula is as follows:

[0067]

[0068] In the formula, d ij ω represents the control points of the surface. ij N represents the weights of the surface. i,k (u) represents the k-th degree B-spline basis function of the surface, N j,l (v) represents the l-order B-spline basis function of the surface. The NURBS surface constructed using this surface formula is shown below. Figure 2 As shown.

[0069] The first and second fundamental forms of the surface P(u,v) can be obtained from the curves on the surface, the metric properties of the surface, and the curvature properties of the surface, as follows:

[0070] I = Edu 2 +2Fdudv+Gdv 2

[0071] Π=Ldu 2 +2Mdudv+Ndv 2

[0072] In the formula, E, F, and G are the first type of basic quantities of the freeform surface; L, M, and N are the second type of basic quantities of the freeform surface.

[0073] The normal vector of any point P(u0,v0) on the surface can be obtained as follows:

[0074]

[0075] Then, the two principal curvatures κ1 and κ2 of a point on the surface are obtained by the following formula, where κ1 > κ2:

[0076]

[0077] The two principal directions of the two principal curvatures are determined by the following formula:

[0078]

[0079] Therefore, we can use Euler's formula:

[0080]

[0081] Find the normal curvature κ of the surface in any direction at that point. n ,in The angle between the selected direction and the principal direction where κ1 is located.

[0082] Step S2: Represent the four boundary lines of the freeform surface using NURBS curves. Select one boundary line and discretize it according to the chord height constraint to obtain multiple first reference points. Specific steps are as follows: Figure 3 As shown.

[0083] In this step, since the boundaries of NURBS surfaces are generally obtained after surface trimming, they cannot be precisely represented by mathematical expressions. For example... Figure 4 As shown, NURBS curves are therefore used to approximate the four boundary lines of the surface, denoted as C0, C1, C2, and C3, respectively, with a container V for storing points. The expression for the NURBS curve at this point is as follows:

[0084]

[0085] In the formula, d i ω represents the control points of the surface. i N represents the weight of the control points. i,k (u) represents the k-th degree B-spline basis function of the surface.

[0086] In this embodiment, one of the four fitted boundary lines is arbitrarily selected as the reference line. Taking curve C0 as an example, the point corresponding to u1 = 0 is selected as the initial point, and the target arc length s1 = 0. According to the chord height error formula:

[0087]

[0088]

[0089] Calculate the arc length ds between two adjacent points. In the formula, e represents the maximum allowable chord height error, R is the radius of curvature of the curve at that point, f is the straight-line distance between two adjacent points, and k... n Let be the normal curvature of the surface at that point along the tangent direction of the reference line.

[0090] Calculate the target arc length s based on the arc length ds. i =s i-1 +ds, then apply the integral formula for arc length:

[0091]

[0092] Then, using Simpson's quadrature formula, the bisection method is used to iterate u to solve L(u) until the value of L(u) is within the error range of s2, thus obtaining u2. Calculate C0(u2) to get the next point.

[0093] Solve in a loop until s i ≥s total At this time, let s i =stotal The process continues until the last point is outside the surface, at which point the loop terminates. The last point in container V (i.e., the point outside the surface) is then deleted. The solved points become the first reference points, and all are stored in container V. Figure 5 As shown, the first reference points in container V are output sequentially, and these first reference points are used as the starting points for the geodesic lines generated in the next step. Calculate the parameter u. i When, the iteration interval of parameter u is changed to (u i-1 [1] can reduce the number of times the bisection method is used.

[0094] Step S3: For each first reference point, numerical methods are used to construct and fit multiple geodesics. Specifically, please refer to... Figure 6 T i j Let j be the point on the i-th geodesic line. It is the feed direction, n i j For the surface at point T i j The unit normal vector at that location, It is the i-th geodesic line at point T i j The unit tangent vector at point T is also the step direction. i 1 Let i be the starting point of the i-th geodesic line. That is, curve C0 at point T i 1 The unit tangent vector at that point is also the feed direction. For the surface at point T i 1 The unit normal vector at that location, It is the i-th geodesic line at point T i 1 The unit tangent vector at that point is also the step direction, and its value is:

[0095]

[0096] Please refer to Figure 4 and Figure 6 ,in, For curve C3 at point T1 1 The unit tangent vector at that point, For curve C1 at point The unit tangent vector at that point. (This is followed by a list of all vectors.) After obtaining the result, a mean filtering algorithm is used for correction. The revised version Projected onto the surface at a point On the tangent plane at the point, obtain the projection vector, and then... Replace it with the corresponding normalized projection vector.

[0097] Iterative formula for geodesics:

[0098]

[0099]

[0100] In the formula, Δs represents point T. i j and T i j-1 The arc length between For the surface at point T i j-1 along Normal curvature of direction.

[0101] The next point T can be determined using the geodesic iteration formula. i 2 and the geodesic at point T i 2 The unit tangent vector b at point i 2 T i 2 (2≤i≤n-1) is replaced with the projection points on the surface, T1 2 Replace it with its projection point on curve C3. Replace it with its projection point on curve C1. Then change b1. 2 Replace with curve C3 at point T1 2 The unit tangent vector at that point will Replace with curve C1 at point The unit tangent vector at that point is corrected using the mean filtering algorithm. The revised version Projected onto the surface at a point On the tangent plane at the point, obtain the projection vector, and then... Replace it with the corresponding normalized projection direction. Then calculate the surface at the point. unit normal phasor and the surface at that point along normal curvature of direction

[0102] By analogy, the geodesic iterative formula and mean filtering algorithm can be used again to calculate... Iterate until Outside the curved surface. Then Replace with The projection point on curve C2. These points are located on the i-th geodesic line. A NURBS curve is used to fit these points (the third reference point), and the resulting spline curve is the desired geodesic line. Then, the 1st and nth geodesic lines are replaced with curves C3 and C1, respectively. For example... Figure 7 The diagram shown is a schematic of n geodesic lines, denoted as . The starting points of these n geodesic lines all fall on curve C0.

[0103] Step S4: Discretize points for each geodesic line according to the specified arc length or residual height (residual height) constraint to obtain multiple second reference points.

[0104] like Figure 8 As shown, point It is the m-th tool contact point on the i-th geodesic line. The chord length l in the toolpath step direction can be obtained by the following formula based on the maximum allowable residual height h. i m :

[0105]

[0106] In the formula, Is the curved surface at a point The normal curvature along the lateral step direction, when When the curved surface is a plane, when When the curved surface is concave, When the surface is convex, r is the radius of the ball end mill. It is a geodesic at point The radius of curvature at that point.

[0107] The arc length in the lateral step direction can be further obtained from geometric relationships.

[0108]

[0109] Starting from the geodesic starting point, s1 = 0, according to Calculate the target arc length Based on the arc length integral formula:

[0110]

[0111] Using Simpson's quadrature formula, the bisection method is used to iteratively solve for L(u) until the value of L(u) is within the error range of s2, thus obtaining u2, and F is calculated. i (u2) is the point below the side step direction.

[0112] Solve in a loop until s m ≥s total At this time, let s m =s totalSolve for the last point, the loop terminates, and the point obtained is the second reference point. The second reference point outside the surface will be deleted, and F will be set to... i (u m-1 The projection point of curve C2 is taken as the last point. Points are discretized for each of the n geodesics according to the residual height constraint. The final output of the second reference point is as follows: Figure 9 As shown. In calculating the parameter u... m When, the iteration interval of parameter u is changed to (u m-1 [1] can reduce the number of times the bisection method is used.

[0113] In the above point selection process, when points are selected uniformly on each geodesic line, that is, when the arc lengths between the second reference points on each geodesic line are the same and the number of second reference points on each geodesic line is the same, a toolpath with consistent boundaries will be generated. When points are selected discretely on each geodesic line with the same residual height, a toolpath with equal residual height will be generated.

[0114] Step S5: Arrange the second reference points with the same order on the n geodesic lines into a group. Fit the second reference points in each group using NURBS curves. The NURBS curves pass through the first and last second reference points in the same group, resulting in multiple fitted curves. The fitting results are as follows: Figure 10 As shown;

[0115] Step S6: Continuously select points on the fitted curve obtained in the above steps according to the chord height constraint. The points obtained are the knife contact points, such as... Figure 11 As shown. Arranging these tool contacts in the machining sequence and connecting them with straight lines forms the required CNC machining path, such as... Figure 12 As shown.

[0116] In summary, the freeform surface CNC machining path planning method provided by this invention innovatively plans the CNC machining path of freeform surfaces using geodesics. Compared with traditional tool path planning methods, this planning method is applicable to trimmed NURBS surfaces, does not depend on the u and v directions of the surface parameter domain, has stronger adaptability, and avoids surface spurs during machining, resulting in good smoothness and meeting the requirements of high-precision CNC machining of freeform surfaces. This freeform surface CNC machining path planning method is also applicable to freeform surfaces expressed by NURBS surfaces and ruled surfaces. In the application of this invention, points are taken on the boundary line (reference line) according to the chord height before generating geodesics, which minimizes the number of geodesics and improves algorithm efficiency. At the same time, during geodesic iteration, a mean filtering algorithm is introduced to reduce the influence of small perturbations of the reference edge on the geodesic iteration direction, reduce the probability of geodesic intersection, and improve the smoothness of the path.

[0117] This invention also provides a computer-readable storage medium storing a computer program thereon. When executed by a processor, the program implements the steps of a geodesic-based freeform surface CNC machining path planning method. The geodesic-based freeform surface CNC machining path planning method can be applied in software form, such as by designing a program that can run independently on a computer-readable storage medium. The computer-readable storage medium includes at least one type of readable storage medium, such as flash memory, hard disk, multimedia card, card-type memory (e.g., SD or DX memory), magnetic memory, disk, optical disk, etc. It is used to store and install application software and executable computer programs for the geodesic-based freeform surface CNC machining path planning method. This geodesic-based freeform surface CNC machining path planning method can also be designed as an embedded program and installed on relevant devices.

[0118] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0119] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. A geodesic-based free-form surface NC machining path planning method, characterized in that, It constructs geodesic lines on the free surface, and continuously takes points on the geodesic lines as tool contact points, and sorts the tool contact points in machining order to obtain the required numerical control machining path; the tool contact points are obtained by the following steps: Step S1, constructing the free surface based on NURBS modeling; Step S2, using NURBS curve to represent four boundary lines of the free-form surface, selecting one of the boundary lines, and discretely taking points according to chord height constraint to obtain a plurality of first reference points; the discretely taking points according to chord height constraint comprises the following steps: step 21, taking one of the boundary lines as a reference line, selecting a first reference point corresponding to u1=0 as an initial point, and setting a target arc length s1=0; step 22, calculating an arc length ds between two adjacent points on the reference line through a chord height error formula, calculating a target arc length s i = s i-1 +ds according to the arc length ds, and calculating a next point of the initial point according to an arc length integral formula and a Simpson integral formula; step 23, taking the next point of the initial point as a new initial point, and substituting a curve curvature radius R and a surface normal curvature k n at the initial point into the chord height error formula, the arc length integral formula and the Simpson integral formula to perform cyclic calculation until s i ≥ s total , at this time, setting s i = s total , solving a last point, and terminating the cycle; step 24, after the cycle is terminated, the solved point is a first reference point; Step S3, using a numerical method to construct and fit each first reference point to obtain a plurality of geodesic lines; the construction and fitting of the geodesic lines include the following steps: step S31, taking the first reference point as the reference, each first reference point corresponds to a geodesic line, and the unit tangent vector at the first reference point and the next point of the first reference point on the same geodesic line are solved according to the geodesic line iteration formula; step S32, using the mean filter algorithm to correct the obtained unit tangent vector, and projecting the corrected unit tangent vector onto the tangent plane of the first reference point on the surface to obtain a projection vector, and replacing the unit tangent vector with the projection vector; step S33, repeatedly performing step S31 and step S32 to continuously solve the geodesic line iteration formula to obtain a plurality of third reference points on the i-th geodesic line until the third reference point is outside the free surface; step S34, fitting a plurality of third reference points using a NURBS curve to obtain a geodesic line; wherein the geodesic line iteration formula is: wherein is the jth point on the ith geodesic, and Δs is the arc length between points and , is the normal curvature of the free-form surface at point in the direction , is both the unit tangent vector of the ith geodesic at point and the side step direction, is the unit normal vector of the free-form surface at point ; Step S4, discretely taking points according to the specified arc length or residual height constraint for each geodesic line to obtain a plurality of second reference points; Step S5, arranging the second reference points on the same order on the plurality of geodesic lines into a group, and fitting the second reference points in each group using a NURBS curve to obtain a plurality of fitting curves; Step S6, continuously taking points from each fitting curve according to the chord height constraint to obtain the tool contact points.

2. The geodesic-based free-form surface NC machining path planning method according to claim 1, characterized in that, In step 1, a NURBS surface is used to represent the machined surface, and the first fundamental form and the second fundamental form are obtained by geometric parameter analysis of the NURBS surface; The NURBS surface formula is as follows: where d ij denotes the control points of the surface, ω ij denotes the weights of the surface, N i,k (u) denotes the k-th B-spline basis function of the surface, N j,l (v) denotes the l-th B-spline basis function of the surface.

3. The geodesic-based free-form surface NC machining path planning method according to claim 2, characterized in that, The geometric parameter analysis includes a normal vector N(u0, v0) of any point P(u0, v0) on the free-form surface and a normal curvature κ of any direction on the free-form surface at the point n ; Wherein, the expression of the normal vector N(u0, v0) is: In the formula, E, F, and G are the first basic quantities of the free surface.

4. The geodesic-based free-form surface NC machining path planning method according to claim 2, characterized in that, The first fundamental form of the free surface P(u, v) is: I = Edu 2 + 2Fdudv + Gdv 2 The second fundamental form of the free surface P(u, v) is: Pi = Ldu 2 + 2Mdudv + Ndv 2 In the formula, E, F, and G are the first basic quantities of the free surface; L, M, and N are the second basic quantities of the free surface.

5. The geodesic-based free-form surface NC machining path planning method according to claim 1, wherein, In step 2, the expression for representing the boundary line using a NURBS curve is: where d i denotes the control points of the surface, ω i denotes the weight of the control points, N i,k (u) denotes the kth B-spline basis function of the surface.

6. The geodesic-based free-form surface NC machining path planning method according to claim 1, characterized by, The boundary line is replaced by the geodesic line.

7. The geodesic-based free-form surface NC machining path planning method according to claim 1, wherein, In step S4, discretely taking points according to the specified arc length or residual height constraint for each geodesic line from the starting point; the projection point on the boundary line is taken as the last point.

8. The geodesic-based free-form surface NC machining path planning method according to claim 1, characterized by, In step S6, sorting the tool contact points in machining order and connecting them with straight lines to obtain the required numerical control machining path.

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