An Adaptive Path Planning Method for Free-Form Surface Turning

Abstract

The present invention provides an adaptive path planning algorithm applicable to free-form surface turning, which is used to solve the technical problems that there are a large amount of redundant data when processing free-form surfaces using traditional path planning methods, resulting in long processing time, large amount of data, and limited machining accuracy. An adaptive path planning method applicable to free-form surface turning of the present invention is as follows: by calculating the tangential curvature radius of an Archimedean spiral at different polar angles in one period, the line density function of the Archimedean spiral in this period is obtained; then, according to the line density function and the number of selected data points, the positions of discrete data points on the Archimedean spiral in this period are determined, and further the position coordinates of each data point are obtained; according to the position coordinates of the data points obtained on the Archimedean spiral and the turning tool nose radius, the tool control points corresponding to each data point are calculated, and then the turning path of the free-form surface is generated, and the path planning is completed.

Description

Technical Field

[0001] The present invention relates to free-form surfaces, and in particular, to an adaptive path planning algorithm applicable to free-form surface turning. Background Art

[0002] Traditional spherical mirrors and aspherical mirror surface structures both have rotational symmetry, and relatively large off-axis aberrations will be generated in an off-axis system. Since free-form surfaces have a high degree of design freedom, they can better solve the aberration problem, and by introducing free-form surfaces, the effects of multiple traditional mirrors can be achieved, which enables the opto-mechanical structure with free-form surfaces to be more lightweight and miniaturized. Therefore, free-form surfaces have been widely used in various fields, including space exploration, microelectronic lithography, and vehicle-mounted lighting and HUD systems.

[0003] However, the high degree of design freedom of free-form surfaces increases the complexity of their surface shapes. Compared with traditional mirrors, free-form surfaces are more difficult to machine, and at the same time, their machining accuracy and machining aperture are also limited. Traditional machining methods can no longer achieve the machining of free-form surfaces, and more advanced three-axis servo systems need to be used to control single-point diamond turning for the machining of free-form surfaces. The three-axis servo system mainly includes a rotating axis C-axis and two translational axes X-axis and Z-axis. The workpiece to be machined is placed on the C-axis for rotation, and at the same time, the relative positions of the diamond tool head and the C-axis are controlled by the X-axis and Z-axis. A very crucial step in machining is path planning. Path planning is to control the three-axis servo system by selecting appropriate discrete data points according to the topography of the ideal surface. Therefore, the path planning algorithm directly determines the surface shape accuracy and machining time of the free-form surface. When using a reasonable path planning algorithm to obtain discrete points to control the servo system, it can not only improve the machining accuracy of the free-form surface, but also reduce the number of control points and machining time.

[0004] Traditional path planning methods include the equal-angle method, the equal-arc-length method, and the method combining equal-angle and equal-arc-length. These methods select data points in an Archimedean spiral on a horizontal plane in an equal-angle or equal-arc-length manner. The above are all general algorithms that can perform path planning for all types of surfaces. As long as there are enough data points, the required accuracy can always be achieved. However, these algorithms ignore the uniqueness of the surface shape of free-form surfaces, which will generate a large amount of redundant data, resulting in problems such as long machining time and large data volume. In addition, due to the limited accuracy of servo system control, the machining accuracy of the traditional method will be limited when machining parts with a relatively large machining aperture. Summary of the Invention

[0005] The object of the present invention is to solve the technical problems existing in the machining of free-form surfaces using traditional path planning methods, such as generating a large amount of redundant data, resulting in long machining time, large data volume, and limited machining accuracy, and to provide an adaptive path planning algorithm suitable for free-form surface turning.

[0006] To achieve the above object, the technical solution of the present invention is as follows:

[0007] An adaptive path planning method suitable for free-form surface turning, which selects an Archimedean spiral as the horizontal movement trajectory of the turning tool head. The special feature is that it includes the following steps:

[0008] 1. According to the ideal surface shape and machining accuracy requirements, determine the pitch d of the Archimedean spiral;

[0009] 2. Select data points on the Archimedean spiral and obtain the position coordinates of the data points

[0010] 2.1 Select one period of the Archimedean spiral, combine the pitch and the expression of the ideal surface shape, obtain the height function of the Archimedean spiral at different polar angles in this period, and then obtain the tangential curvature radius of the Archimedean spiral at different polar angles in this period;

[0011] 2.2 According to the tangential curvature radius of the Archimedean spiral at different polar angles in this period, obtain the line density function of the Archimedean spiral in this period;

[0012] 2.3 According to the machining accuracy requirements, select the number of data points on the Archimedean spiral in this period, and then combine the line density function of the Archimedean spiral in this period obtained in step 2.2 to determine the positions of the discrete data points on the Archimedean spiral in this period, and then obtain the corresponding polar angles of each data point;

[0013] 2.4 Combine the height function of the Archimedean spiral at different polar angles in this period obtained in step 2.1 and the corresponding polar angles of each data point obtained in step 2.3 to obtain the position coordinates of each data point;

[0014] 2.5 Repeat steps 2.1 - 2.4 until the position coordinates of the data points on all periods of the Archimedean spiral are obtained;

[0015] 3. According to the position coordinates of the data points obtained on all periods of the Archimedean spiral and the radius of the turning tool head, calculate the corresponding tool control points through the tool compensation algorithm; then generate the turning path of the free-form surface through the tool control points to complete the path planning.

[0016] Further, in step 2.2, the line density function λ of the Archimedean spiral in this period t(θ) is expressed as:

[0017]

[0018] where θ represents the polar angle, with a range of 0 to 2π; K t (θ) represents the tangent curvature of the Archimedean spiral at different polar angles in this period, R t (θ) represents the radius of the tangential curvature of the Archimedean spiral at different polar angles in this period; α represents the regulation amount of the linear density function; ∈ represents the tangential inclination angle of the Archimedean spiral at different polar angles.

[0019] Furthermore, in step 2.3, the polar angle corresponding to each data point is obtained through the following formula:

[0020]

[0021] where θ i represents the polar angle corresponding to the i-th data point; n represents the number of data points selected on the Archimedean spiral in this period; d represents the spiral pitch.

[0022] Furthermore, step 2.4 also includes:

[0023] According to the position coordinates of the obtained data points and the ideal surface shape expression, determine the distribution of the linear interpolation error δ on the Archimedean spiral in this period, and then determine the corresponding error peak-to-valley value H 0 ; Combining the concavity and convexity of the height function of the Archimedean spiral at different polar angles obtained in step 2.1, redistribute the data points to obtain the optimized position coordinates of the data points on the Archimedean spiral in this period.

[0024] Furthermore, in step 2.4, combining the concavity and convexity of the height function of the Archimedean spiral at different polar angles obtained in step 2.1, the redistribution of the data points is specifically as follows:

[0025] In the convex interval of the height function, reduce the height of the data points in this interval by H 0 / 2;

[0026] In the concave interval of the height function, increase the height of the data points in this interval by H 0 / 2.

[0027] Furthermore, step 1 specifically includes:

[0028] 1.1 Obtain the machining residual error of the Archimedean spiral according to the machining accuracy requirements;

[0029] 1.2】Calculate the limit value of the helix pitch of the ideal surface based on the machining residual error and the tool nose radius:

[0030] 1.3】Select a value smaller than the limit value of the helix pitch as the helix pitch d.

[0031] Furthermore, in step 2.5】, when repeating step 2.3】, when selecting data points on Archimedean spirals of different cycles, as the machining diameter of the Archimedean spiral increases or decreases, the number of selected data points increases or decreases accordingly.

[0032] The beneficial effects of the present invention compared with the prior art are as follows:

[0033] 1. An adaptive path planning algorithm suitable for free-form surface turning provided by the present invention can, compared with the traditional equal-angle path planning algorithm, select data points according to the shape transformation of the free-form surface, plan different paths for different shapes, effectively reduce redundant data, reduce machining time, and improve machining accuracy, providing a basis for the machining of large-diameter free-form surfaces.

[0034] 2. An adaptive path planning algorithm suitable for free-form surface turning provided by the present invention can achieve the same shape accuracy with about 30%-40% less data volume when using the adaptive algorithm for machining the same diameter.

[0035] 3. An adaptive path planning algorithm suitable for free-form surface turning provided by the present invention has better error uniformity in machining the shape.

[0036] 4. An adaptive path planning algorithm suitable for free-form surface turning provided by the present invention selects data points that are non-uniformly distributed according to the shape of the ideal surface, making the displacement of the tool nose between data points smoother during machining, and is very suitable for free-form surfaces with large shape changes at different diameters. Description of the Drawings

[0037] Figure 1 It is a three-dimensional structure diagram of the ideal surface to be machined in the embodiment of the present invention;

[0038] Figure 2 It is a schematic diagram of selecting data points on a spiral using the existing equal-angle algorithm on a plane;

[0039] Figure 3 It is a schematic diagram of the distribution of data points selected using the existing equal-angle algorithm at a machining diameter of 20 mm;

[0040] Figure 4Schematic diagram of the linear interpolation error distribution obtained after 200 data points are taken on the spiral line at a machining diameter of 20 mm using the existing equal-angle algorithm;

[0041] Figure 5 In the embodiment of the present invention, the variation of the spiral line trajectory height z with the polar angle θ at a machining diameter of 20 mm;

[0042] Figure 6 Schematic diagram of the line density function distribution and curvature distribution when α = 0.5 in the embodiment of the present invention;

[0043] Figure 7 Schematic diagram of the data point distribution selected on the spiral line at a machining diameter of 20 mm using the adaptive algorithm in the embodiment of the present invention;

[0044] Figure 8 Schematic diagram of the linear interpolation error distribution obtained after 200 data points are taken on the spiral line at a machining diameter of 20 mm using the adaptive algorithm in the embodiment of the present invention;

[0045] Figure 9 Schematic diagram of the machining error distribution obtained by performing path planning on the ideal surface shape using the existing equal-angle algorithm;

[0046] Figure 10 Schematic diagram of the machining error distribution obtained by performing path planning on the ideal surface shape using the adaptive algorithm in the embodiment of the present invention. Detailed implementation manner

[0047] To make the advantages and features of the present invention clearer, the following further describes the present invention in detail with reference to the accompanying drawings and specific embodiments.

[0048] As Figure 1 shown, in this embodiment, the ideal surface to be machined is a free surface with a quadratic surface as the base surface and an additional term as an XY polynomial, and its surface shape expression is:

[0049]

[0050] wherein, R 0 represents the base surface curvature radius of the free surface, r 0 represents the base surface diameter of the free surface, k represents the conic coefficient, i represents the degree of X in the XY polynomial, j represents the degree of Y in the XY polynomial, m represents the maximum degree of X in the XY polynomial, and n represents the maximum degree of Y in the XY polynomial.

[0051] As shown in Table 1, in this embodiment, the base surface curvature radius R 0 , conic coefficient k, and XY polynomial coefficients of the free surface are as follows:

[0052] Table 1 Base Curvature Radius R of Freeform Surface 0 , cone coefficient k, and XY polynomial coefficients

[0053]

[0054]

[0055] An adaptive path planning method applicable to freeform surface turning provided by the present invention, in the cylindrical coordinate system where the workpiece to be machined is located, according to the machining aperture of the ideal surface, an Archimedean spiral projected on the horizontal plane is selected as the horizontal direction movement trajectory of the turning tool head during the turning process, which specifically includes the following steps:

[0056] 1. Determine the spiral pitch d of the Archimedean spiral

[0057] 2.1 Obtain the machining residual error h of the Archimedean spiral according to the machining accuracy requirements. In this embodiment, the machining accuracy RMS is less than 1 / 30 of a wavelength, where the wavelength is 632.8 nm, and the machining residual error h is obtained by experience as 30 nm;

[0058] 2.2 Calculate the limit value of the spiral pitch according to the surface shape of the ideal surface

[0059] Since the freeform surface may simultaneously have a plane and a concave surface, or a plane and a convex surface, or a plane, a concave surface, and a convex surface, it is necessary to calculate the corresponding spiral pitch reference values for the corresponding plane, concave surface, and convex surface respectively;

[0060] Specifically, the spiral pitch reference value d when the machining surface is a plane, the spiral pitch reference value d when the machining surface is a concave surface, and the spiral pitch reference value d when the machining surface is a convex surface are calculated respectively through the following three formulas: flt , the spiral pitch reference value d when the machining surface is a concave surface cav , and the spiral pitch reference value d when the machining surface is a convex surface vex :

[0061]

[0062]

[0063]

[0064] Among them, r represents the tool tip radius; R represents the curvature radius of the machining surface;

[0065] When the freeform surface simultaneously has a plane and a concave surface, the limit value of the spiral pitch is taken as the smaller value of d flt , d cav ; when the freeform surface simultaneously has a plane and a convex surface, the limit value of the spiral pitch is taken as the smaller value of d flt , d vexThe smaller value among them; when the freeform surface has planes, concave surfaces, and convex surfaces at the same time, the limited value of the helix pitch is taken as d flt , d cav , d vex The minimum value among them.

[0066] Since the ideal surface in this embodiment is a concave surface and a plane as a whole, and the cutter head radius r = 0.5 mm, the helix pitch d flt = 10.95 μm, d cav = 11.02 μm. Therefore, when designing, taking the smaller value of 10.95 μm for the limited value of the helix pitch of the ideal surface can meet the accuracy requirements.

[0067] 2.3】Select a value smaller than the limited value of the helix pitch as the helix pitch d. Since the limited value of the helix pitch in this embodiment is 10.95 μm, the helix pitch d in this embodiment is set to 10 μm for machining experiments. After determining the helix pitch d, data points are selected on the Archimedean spiral according to certain rules, and the servo system can be controlled to complete turning machining; the rule for selecting data points on the Archimedean spiral is the core of the present invention and will be introduced in detail below.

[0068] 2】Select data points on the Archimedean spiral and obtain the position coordinates of the data points

[0069] 2.1】Obtain the variation of the tangential curvature radius of the surface corresponding to the Archimedean spiral with the polar angle

[0070] 2.1.1】Select the j-th period of the Archimedean spiral for analysis. Combining the helix pitch d obtained in step 2】, the equation of the j-th period of the Archimedean spiral in the horizontal polar coordinates is obtained as follows:

[0071]

[0072] where θ represents the polar angle, and the range is 0 to 2π;

[0073] Combined with the ideal surface shape expression f(x, y), calculate the height function of the Archimedean spiral at different polar angles in this period, that is, the variation of the height of the Archimedean spiral with the polar angle, and its expression is as follows:

[0074] z(ρ, θ) = f(ρcos(θ), ρsin(θ))

[0075] 2.1.2】Combined with the height function of the Archimedean spiral at different polar angles in this period, calculate the tangential curvature radius R of the Archimedean spiral at different polar angles through the following formula t(θ), i.e., the variation of the tangential curvature radius of the Archimedean spiral in this period with the polar angle:

[0076]

[0077] 2.2】 Calculate the line density function of the Archimedean spiral in the j-th period

[0078] During ultra-precision machining, since there are many data points on the Archimedean spiral in each period and the spacing between data points is relatively close, therefore, the ideal curve segment between data points can be approximately regarded as an arc. During actual machining, since these data points are used to control the tool's motion trajectory and the tool moves linearly between adjacent discrete data points, this will cause a certain deviation between the ideal surface and the actual surface, that is, the linear interpolation error. By observing the distribution law of the linear interpolation error, it can be seen that at the peak and valley points of the surface height change, the overall error will be relatively large. At the inflection points, the overall error will be relatively small; at the same time, by analyzing the properties of the ideal surface shape itself, at the peak and valley points of the surface height change, the tangential curvature of the surface shape will be relatively large, and at the inflection points, the tangential curvature of the surface shape will be relatively small. Therefore, the distribution trend of the linear interpolation error is basically the same as the distribution trend of the tangential curvature. Therefore, in the present invention, the tangential curvature is used as the evaluation criterion to calculate the line density function of the Archimedean spiral in the corresponding period.

[0079] According to the linear interpolation error δ between the ideal surface and the actual surface, the maximum spacing L(θ) between adjacent data points is determined as:

[0080]

[0081] where the magnitude of L(θ) indicates the flatness of the surface at different polar angles and represents the surface morphology characteristics during machining;

[0082] Furthermore, the projection x of the maximum spacing between adjacent data points on the tangential direction x-axis is obtained t (θ), i.e., the distribution of the maximum spacing between adjacent data points:

[0083]

[0084] where ∈ represents the tangential inclination angle of the Archimedean spiral in this period at different polar angles;

[0085] Since during path planning, the curve is not a strict arc, it is necessary to adjust the line density function by α. Combining the distribution of the maximum spacing of data points, the distribution of data points, that is, the line density function of data points, is obtained through the following formula:

[0086]

[0087] Among them, α represents the regulation amount of the linear density function; K t (θ) represents the tangent curvature of the Archimedean spiral in this period at different polar angles,

[0088] For Archimedean spirals of different periods, the respective linear density functions can be calculated according to different surface change situations, and then discrete data points can be selected. This is very applicable to free-form surfaces with large surface shape changes at different apertures.

[0089] 2.3】Select the number of data points n on the Archimedean spiral of this period. According to the linear density function of the Archimedean spiral of this period, that is, the change relationship between the linear density and the polar angle, determine the positions of the n discrete data points on the Archimedean spiral of this period, and then obtain the polar angles corresponding to each data point;

[0090] Among them, the polar angle θ corresponding to the i-th data point i is determined by the following formula:

[0091]

[0092] 2.4】Combine the height function of the Archimedean spiral of this period at different polar angles obtained in step 2.1】 and the polar angles corresponding to each data point obtained in step 2.3】 to obtain the preliminary position coordinates of each data point; Preferably, in this embodiment, according to the position coordinates of the data points obtained in step 3.3】 and the ideal surface shape expression, determine the distribution of the linear interpolation error δ on the Archimedean spiral of this period, and then determine the corresponding error peak-to-valley value H 0 ; Combine the concavity and convexity of the height function of the Archimedean spiral of this period obtained in step 2.1】, redistribute the data points, and obtain the optimized position coordinates of the data points on the Archimedean spiral of this period.

[0093] Specifically, in the convex interval of the height function, that is, the second derivative of the function is less than zero, reduce the height of the data points in this interval by H 0 / 2;

[0094] In the concave interval of the height function, that is, the second derivative of the function is greater than zero, increase the height of the data points in this interval by H 0 / 2.

[0095] 2.5】Repeat steps 2.1】-2.4】 until the position coordinates of the data points on all period Archimedean spirals are obtained;

[0096] Among them, when selecting data points on Archimedean spirals of different periods, as the machining diameter of the Archimedean spiral increases or decreases, the arc length corresponding to the same central angle becomes longer or shorter. Therefore, in order to meet the machining accuracy, it is necessary to correspondingly increase or decrease the number of data points on the corresponding Archimedean spiral.

[0097] 3】According to the position coordinates of the data points obtained on all-period Archimedean spirals and the turning tool nose radius, calculate the tool control points corresponding to each data point through the tool compensation algorithm; then generate the turning path of the free form surface through the tool control points to complete the path planning.

[0098] In order to verify the advantages of the path planning method adapted to the present invention, the following will be further described by comparing with the traditional equal-angle method.

[0099] As Figure 2 shown, it is a schematic diagram of selecting data points on the Archimedean spiral using the existing equal-angle algorithm on a plane. First, use the traditional equal-angle algorithm to select 200 data points on the Archimedean spiral at a machining diameter of 20 mm, and its distribution schematic diagram is as Figure 3 shown. It can be seen from the figure that the distribution of all data points is uniform in the x-axis direction. According to these data points, the distribution of the linear interpolation error on the Archimedean spiral in this period can be calculated as Figure 4 shown, with the error value PV = 78.2 nm and RMS = 18.9 nm.

[0100] The path planning method provided by the present invention takes the tangential curvature as the evaluation criterion and calculates the line density function on the Archimedean spiral at a machining diameter of 20 mm. As Figure 5 shown, it is the curve of the height of the Archimedean spiral at a machining diameter of 20 mm in this embodiment changing with the polar angle. When setting α = 0.47, the distribution schematic diagram of the curvature and the line density function changing with the polar angle is as Figure 6 shown. Then, according to the line density function and the formula for determining the polar angle corresponding to the data point, 200 data points are selected, which is the same as the number of data points used in the above equal-angle algorithm. The distribution of these data points on the Archimedean spiral is as Figure 7 shown. Comparing with the data point distribution of the equal-angle algorithm, the data point density at the peaks and valleys of the data points selected by the adaptive path planning method of the present invention is relatively larger than that at the inflection points; according to these data points, the distribution of the linear interpolation error on the Archimedean spiral in this period is calculated as Figure 8 shown. Compared with the error distribution of the traditional equal-angle algorithm, the error of the adaptive path planning method is smaller as a whole, with the error value PV = 21.27 nm and RMS = 6.42 nm, and the overall error is more uniform.

[0101] By selecting the same number of data using different methods for the Archimedean spiral in the same period, the feasibility of the adaptive algorithm is preliminarily verified. Then, independent data points are selected for each period of the Archimedean spiral on the entire ideal surface shape, and the linear interpolation error and the residual error between the Archimedean spirals are considered simultaneously. When using the equal-angle algorithm, 340 data points are selected on one turn of the spiral, and the simulated error values are PV = 49.3 nm and RMS = 8.1 nm. The machining error results are as Figure 9 shown. For the same surface shape, when using the adaptive path planning method of the present invention, 200 data points are selected on the Archimedean spiral in the same period, and the simulated error values are PV = 47.7 nm and RMS = 8.1 nm. The machining error results are as Figure 10 shown. By comparing the overall simulation results, it can be seen that the adaptive path planning method of the present invention makes the error uniformity of the machined surface shape better, thus reflecting the improvement effect on the overall surface shape in actual machining.

[0102] In summary, the advantages of the adaptive path planning method for free-form surface turning provided by the present invention are as follows: 1. When machining the same aperture, using the adaptive algorithm can achieve the same surface shape accuracy with a reduction of about 30%-40% in the amount of data. 2. The error uniformity of the machined surface shape using the adaptive algorithm is relatively good. 3. It makes the displacement of the tool tip between data points smoother during machining. In conclusion, using the adaptive algorithm for machining can reduce the machining time, improve the machining accuracy, make the error uniformity of the machined surface shape better, and provide a basis for the machining of large-aperture free-form surfaces.

[0103] The above is only used to illustrate the technical solutions of the present invention and is not intended to limit it. For ordinary professional technicians in the field, the specific technical solutions recorded in the above embodiments can be modified, or some of the technical features can be equivalently replaced. These modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions protected by the present invention.

Claims

1. An adaptive path planning method applicable to free-form surface turning, which selects an Archimedean spiral as the horizontal movement trajectory of the turning tool head. Characterized in that: It includes the following steps: 1】According to the ideal surface shape and machining accuracy requirements, determine the spiral pitch d of the Archimedean spiral. 2】Select data points on the Archimedean spiral and obtain the position coordinates of the data points. 2.1】Select one period of the Archimedean spiral, combine the spiral pitch and the expression of the ideal surface shape, obtain the height function of the Archimedean spiral at different polar angles in this period, and then obtain the tangential curvature radius of the Archimedean spiral at different polar angles in this period. 2.2】According to the tangential curvature radius of the Archimedean spiral at different polar angles in this period, obtain the line density function of the Archimedean spiral in this period. 2.3】According to the machining accuracy requirements, select the number of data points on the Archimedean spiral in this period, and then combine the line density function of the Archimedean spiral in this period obtained in step 2.2】 to determine the positions of the discrete data points on the Archimedean spiral in this period, and then obtain the polar angles corresponding to each data point. 2.4】Combine the height function of the Archimedean spiral at different polar angles in this period obtained in step 2.1】 and the polar angles corresponding to each data point obtained in step 2.3】 to obtain the position coordinates of each data point. 2.5】Repeat steps 2.1】-2.4】 until the position coordinates of the data points on all periods of the Archimedean spiral are obtained. 3】According to the position coordinates of the data points obtained on all periods of the Archimedean spiral and the turning tool head radius, calculate the tool control points corresponding to each data point through the tool compensation algorithm; then generate the turning path of the free-form surface through the tool control points to complete the path planning.

2. An adaptive path planning method applicable to free-form surface turning according to claim 1. Characterized in that: In [Step 2.2], the linear density function λ t (θ) of the Archimedean spiral in this period is expressed as: Among them, θ represents the polar angle, and the range is 0 to 2π; K t (θ) represents the tangent curvature of the Archimedean spiral of this period at different polar angles, R t (θ) represents the tangential curvature radius of the Archimedean spiral of this period at different polar angles; α represents the regulation amount of the linear density function; ∈ represents the tangential inclination angle of the Archimedean spiral of this period at different polar angles.

3. An adaptive path planning method applicable to free-form surface turning according to claim 2. Characterized in that: In step 2.3】, the polar angles corresponding to each data point are obtained through the following formula: where θ i represents the polar angle corresponding to the i-th data point; n represents the number of data points selected on the Archimedean spiral in this period; d represents the spiral pitch.

4. An adaptive path planning method applicable to free-form surface turning according to any one of claims 1-3. Characterized in that: Step 2.4】 further includes: Based on the position coordinates of the obtained data points and the expression of the ideal surface shape, determine the distribution of the linear interpolation error δ on the Archimedean spiral in this period, and then determine the corresponding peak-to-valley error value H. 0 ; Combine the concavity and convexity of the height function of the Archimedean spiral in this period obtained in step 2.1 to redistribute the data points and obtain the optimized position coordinates of the data points on the Archimedean spiral in this period.

5. An adaptive path planning method applicable to free-form surface turning according to claim 4. Characterized in that: In step 2.4】, in combination with the concavity and convexity of the height function of the Archimedean spiral at different polar angles obtained in step 2.1】, the redistribution of the data points is specifically as follows: In the convex interval of the height function, reduce the height of the data points in this interval by H 0 / 2; In the concave interval of the height function, increase the height of the data points in this interval by H 0 / 2.

6. An adaptive path planning method applicable to free-form surface turning according to claim 5. Characterized in that: Step 1】 specifically includes: 1.1】Obtain the machining residual error of the Archimedean spiral according to the machining accuracy requirements. 1.2】According to the machining residual error and the tool head radius, calculate the limit value of the spiral pitch of the ideal surface: 1.3】Select a value less than the limit value of the spiral pitch as the spiral pitch d.

7. An adaptive path planning method applicable to free-form surface turning according to claim 6. Characterized in that: In [Step 2.5], when repeating [Step 2.3], when selecting data points on Archimedean spirals of different periods, as the machining diameter of the Archimedean spiral increases or decreases, the number of selected data points increases or decreases accordingly.

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