A complex curved surface slow tool servo turning trajectory optimization method based on curvature gradient clustering and trajectory error constraint

Abstract

The application discloses a complex curved surface slow tool servo turning track optimization method based on curvature gradient clustering and track error constraint, and belongs to the technical field of ultra-precision cutting machining and numerical control track optimization. The method comprises the following steps: step 1, track parameterization and initial equi-angle sampling; step 2, track curvature and curvature gradient are obtained; step 3, automatic region classification is carried out based on the curvature gradient density; step 4, point deletion based on target error constraint is carried out in the flat area; step 5, global error verification and local encryption of the super error feedback are carried out; and step 6, the final optimized track point set is output. The application can automatically and robustly divide the target curved surface into a 'complex area' and a 'flat area' under the premise that the maximum sag error of the track is not exceeded, and adopts an error-driven local point deletion strategy in the flat area and carries out minimum local encryption in the super error area, so that the final numerical control track point number, the program volume and the track preparation time are significantly reduced.

Description

Technical Field

[0001] This invention belongs to the field of ultra-precision cutting and CNC trajectory optimization technology, and relates to a trajectory generation and point set optimization method in Slow Tool Servo (STS) ultra-precision turning. Specifically, it relates to a trajectory optimization method for complex curved surface slow tool servo turning based on curvature gradient clustering and trajectory error constraints. Background Technology

[0002] Slow-tool servo turning is a core manufacturing technology for machining non-rotationally symmetric freeform surface components, and its machining accuracy depends on high-precision trajectory planning. The typical implementation process of slow-tool servo turning is to map the target surface into a parametric curve of rotation angle and radial coordinate, and to obtain trajectory points by discrete sampling. Then, the trajectory points are converted into Z-X-C three-axis motion commands that can be executed by the CNC system.

[0003] To ensure nanometer-level surface elevation accuracy, industrial applications often employ high-density uniform sampling or iterative sampling based on trajectory target accuracy as the primary strategies for trajectory discretization. However, these strategies have limitations when dealing with surfaces with complex shapes (such as sinusoidal meshes, microlens arrays, and polynomial freeform surfaces). Firstly, high-density sampling with equal spacing (e.g., equal angles or equal arc lengths) generates numerous redundant points in large, flat areas, leading to increased machining program volume and increased complexity in interpolation tasks and program management. Secondly, while iterative sampling based on trajectory target accuracy can guarantee accurate trajectory sampling for nanometer-level precision requirements, it often significantly increases the computation time and resources required for trajectory generation and simulation, reducing engineering efficiency. Furthermore, existing methods for improving trajectory point generation and compression based on geometric features often rely on the absolute value of curvature as a criterion and manually set thresholds, making it difficult to robustly distinguish between truly complex regions that should maintain high-density sampling and flat regions where redundant points can be safely removed. This makes them unsuitable for surfaces with complex shapes. Furthermore, if large-scale global interpolation or verification is frequently performed during the process of deleting redundant points and encrypting the out-of-tolerance feedback point set, it will cause a sharp increase in computational load and memory usage, thereby reducing the engineering feasibility of the algorithm. Summary of the Invention

[0004] This invention provides a method for optimizing the trajectory of slow-speed servo turning of complex curved surfaces based on curvature gradient clustering and trajectory error constraints. It is used to generate and optimize the tool motion trajectory for three-axis (Z–X–C) slow-speed servo turning, and is applicable to ultra-precision machining objects such as optical freeform surfaces (e.g., aspherical surfaces, freeform surfaces, polynomial surfaces, and array surfaces with periodic microstructures).

[0005] The objective of this invention is achieved through the following technical solution:

[0006] A method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints includes the following steps:

[0007] Step 1: Trajectory parameterization and initial iso-angle sampling:

[0008] Step 1-1: Represent the complex surface to be processed as a vector height function. And using polar coordinate system parameters Describe the trajectory, where:

[0009]

[0010] in, For polar angle parameters, Polar angle radial coordinates at time and The polar angle in a rectangular coordinate system Coordinates at time;

[0011] Take a linear radial mapping:

[0012]

[0013] in, The maximum radius of the workpiece. The total angle of the spiral trajectory;

[0014] Steps 1-2: Under the condition of satisfying the trajectory target error Under the premise of equal angles Initial sampling is performed to obtain an initial trajectory point set. :

[0015]

[0016] in, The radial coordinates of the initial trajectory point. The polar angle of the initial trajectory point. Let the initial trajectory point be the sag. This represents the initial number of equal-angle sampling points.

[0017] Step 2: Calculate the trajectory curvature With curvature gradient :

[0018] Step 2-1, in trajectory parameters Define trajectory space vectors and define the derivative vector. and :

[0019]

[0020] Then the trajectory curvature for:

[0021]

[0022] Step 2-2: Denote the differential of the arc length as... , for arc length at Points earned Then the curvature gradient for:

[0023]

[0024] Steps 2-3: Based on the initial trajectory point set The curvature gradient set is calculated. ;

[0025] Step 3: Automatic region classification based on curvature gradient density:

[0026] Step 3-1: For the curvature gradient set Construct a one-dimensional kernel density estimate and calculate the probability density function estimate. :

[0027]

[0028] in, The value of the curvature gradient for the probability density currently being evaluated; This represents the total number of samples in the curvature gradient sequence. For the first Curvature gradient values ​​at each sample point; For kernel functions; For bandwidth;

[0029] Step 3-2: Use the derivative method in Detecting local valleys to determine one or more segmentation thresholds ;

[0030] Step 3-3: To perform complex / gradual partitioning, select the first significant valley. As a dividing line:

[0031]

[0032] Steps 3-4: Merge consecutive segments in the obtained category sequences: For segments shorter than the minimum length threshold... The segments are merged into the more similar side based on the median or mean of the adjacent segments, eliminating isolated short segments caused by noise;

[0033] Step 4: Delete points in the flat region based on the target error constraint.

[0034] Step 4-1: For each continuous interval judged as smooth... Construct a linear interpolation reconstruction function using the interval endpoints. Define the local interpolation error function for this interval:

[0035]

[0036] in, Polar angle Theoretical vector height of time, Polar angle The vector height is obtained through linear interpolation;

[0037] The criteria for determining whether to delete points are:

[0038]

[0039] Step 4-2: To efficiently find the largest deletable internal interval, the following numerical judgment logic is adopted:

[0040] (1) From the left end of the interval Initially, we attempted to quickly find a right-hand side using an exponential expansion method. , making the interval The local maximum error is less than the trajectory target error. If an expansion fails, a binary search is used to determine the maximum deletable right endpoint between the previous successful position and the failed position.

[0041] (2) Mark all the largest deletable internal points as "deletable"; then continue to repeat the above steps from the right end until the interval ends;

[0042] Step 5: Global error verification and out-of-tolerance feedback with local encryption:

[0043] Step 5-1: After deleting points in the flat areas, perform a global error verification on the entire optimized trajectory point set. If there are out-of-tolerance locations... These are extremely poor Location mapping to the corresponding optimization point range ;

[0044] Step 5-2: For each identified interval, calculate its local maximum deviation. Dynamically determine the number of insertion points These points are evenly inserted within the interval to densify the grid;

[0045] Step 5-3: Repeat the global verification-local encryption process until the global error is satisfied. ;

[0046] Step 6: Output the final optimized trajectory point set:

[0047] Output the final optimized trajectory point set. ,in, To optimize the radial coordinates of the trajectory points, To optimize the polar angle of the trajectory points, To optimize the elevation of the trajectory points in three-axis slow-speed servo turning, That is, the position of movement along the X-axis. The angular position of the C-axis. The Z-axis movement position is determined based on the actual machine tool dynamics constraints, the axis linkage speed is determined, and complete machining G-code is generated.

[0048] Compared with the prior art, the present invention has the following advantages:

[0049] 1. This invention addresses the problem of numerous redundant points generated by high-density sampling of equal angles or equal arc lengths in flat regions. It provides a method to significantly compress the number of trajectory points without reducing machining accuracy (keeping the maximum sag error below a preset threshold), thereby reducing the size of the CNC program. The desired goal is to achieve a significant point reduction rate while maintaining or exceeding the maximum interpolation error limit of the original sampling.

[0050] 2. Existing methods using absolute curvature as the criterion lack sufficient discriminative power on multi-directional or periodically changing surfaces, and cannot reliably distinguish between locally abruptly changing regions that "need to be retained with high density" and gently changing regions that "can be deleted". This invention introduces a more discriminative geometric quantity (the rate of change of curvature along the arc length—curvature gradient) and combines it with data-driven clustering or density estimation methods to achieve automatic segmentation of segments with different complexities on the trajectory, thereby providing a reliable basis for targeted point deletion or retention decisions.

[0051] 3. The decision to delete points must be based on error constraints, avoiding the risk of individual locations exceeding the preset elevation limit while keeping verification and recovery costs within acceptable engineering limits. Therefore, this invention designs an error-driven deletion and recovery process: In smooth regions, efficient local error detection (using strategies such as vectorized local sampling, exponential expansion, and binary search) is employed for rapid determination; after deletion, a one-time global fine-check is performed; and for any discovered out-of-tolerance segments, local densification interpolation is implemented until convergence, thus forming a reliable error closed loop.

[0052] 4. This invention can automatically and robustly divide the target surface into "complex area" and "smooth area" while ensuring that the maximum allowable sag error of the trajectory is not exceeded. In the smooth area, an error-driven local point deletion strategy is adopted, and in the out-of-tolerance area, local encryption is minimized, which significantly reduces the number of points in the final CNC trajectory, the program size and the trajectory preparation time. Attached Figure Description

[0053] Figure 1 This is an overall flowchart of a method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints.

[0054] Figure 2 It is a sinusoidal mesh surface model;

[0055] Figure 3 The original equal-angle trajectory theory error distribution;

[0056] Figure 4 To optimize the error distribution of the trajectory theory. Detailed Implementation

[0057] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0058] This invention provides a method for optimizing the trajectory of slow-tool servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints. The method maps the target surface to polar coordinates and constructs an initial equiangular spiral trajectory. It calculates the spatial curvature gradient of the trajectory and automatically divides the trajectory into complex and smooth regions using clustering. In the smooth regions, an exponential expansion combined with binary search for point deletion based on error constraints is employed. Subsequently, a one-time global verification is performed, and local densification is implemented for a few out-of-tolerance regions. Finally, significant trajectory point compression is achieved while ensuring that the maximum elevation error does not exceed a preset threshold. Figure 1 As shown, the specific steps include the following:

[0059] Step 1: Trajectory parameterization and initial iso-angle sampling:

[0060] Representing the complex surface to be processed as a vector height function And using polar coordinate system parameters Describe the trajectory, where:

[0061] (1)

[0062] in, For polar angle parameters, Polar angle radial coordinates at time and The polar angle in a rectangular coordinate system The coordinates at that time.

[0063] Take a linear radial mapping:

[0064] (2)

[0065] in, The maximum radius of the workpiece. The total angle of the spiral trajectory.

[0066] In order to meet the trajectory target error Under the premise of equal angles Initial sampling is performed to obtain an initial trajectory point set. :

[0067] (3)

[0068] in, The radial coordinates of the initial trajectory point. The polar angle of the initial trajectory point. is the initial trajectory point's elevation, and is the initial number of equal-angle sampling points.

[0069] Step 2: Calculate the trajectory curvature With curvature gradient :

[0070] In trajectory parameters Define trajectory space vectors and define the derivative vector. and :

[0071] (4)

[0072] The trajectory curvature is:

[0073] (5)

[0074] Let the differential of arc length be denoted as , for arc length at Points obtained Define curvature gradient :

[0075] (6)

[0076] Based on the initial trajectory point set The set of curvature gradients can be calculated. This information is used for subsequent cluster analysis.

[0077] Step 3: Automatic region classification based on curvature gradient density:

[0078] For the set of curvature gradients Construct a one-dimensional kernel density estimate, and obtain the probability density function estimate according to formula (7). :

[0079] (7)

[0080] in, The value of the curvature gradient for the probability density currently being evaluated; This represents the total number of samples in the curvature gradient sequence. For the first Curvature gradient values ​​at each sample point; Here, we take the kernel function (a Gaussian kernel). ; The bandwidth is used to control the smoothness of the kernel density estimation. denoted as the standard deviation of the curvature gradient samples.

[0081] Using derivative methods in Detecting local valleys to determine one or more segmentation thresholds .

[0082] To perform binary classification (complex / flat), the first significant valley is taken. As a dividing line:

[0083] (8)

[0084] The resulting category sequences are then merged into consecutive segments: segments shorter than the minimum length threshold are merged. The segments are merged into a more similar side based on the median or mean of adjacent segments, eliminating isolated short segments caused by noise.

[0085] Step 4: Delete points in the flat region based on the target error constraint.

[0086] For each continuous interval judged as smooth Construct a linear interpolation reconstruction function using the interval endpoints. (The endpoints are determined by the maintained trajectory points). Define the local interpolation error function for this interval:

[0087] (9)

[0088] in, Polar angle Theoretical vector height of time, Polar angle The height of the vector is obtained through linear interpolation.

[0089] The criteria for determining whether to delete points are:

[0090] (10)

[0091] To efficiently find the largest deletable internal interval, the following numerical judgment logic is adopted:

[0092] (1) From the left end of the interval Initially, we attempted to quickly find a right-hand side using an exponential expansion method. , making the interval The local maximum error is less than the trajectory target error. If an expansion fails, a binary search is used to determine the maximum deletable right endpoint between the previous successful position and the failed position.

[0093] (2) Mark all the largest deletable internal points as "deletable"; then continue to repeat the above steps from the right end until the interval ends.

[0094] Step 5: Global error verification and out-of-tolerance feedback with local encryption:

[0095] After removing points from the flat areas, a global error verification is performed on the entire optimized trajectory point set. If a set of locations with out-of-tolerance errors exists... These are extremely poor Location mapping to the corresponding optimization point range For each identified interval, calculate its local maximum deviation. Dynamically determine the number of insertion points (For example, proportionally or using a hierarchical function), uniformly insert these points within the interval to refine the mesh; repeat the global verification-local refinement process until the global error is satisfied. .

[0096] This closed-loop mechanism ensures that the final output trajectory meets the target error requirements, while only locally refining the data where necessary to avoid excessive global addition of points.

[0097] Step 6: Output the final optimized trajectory point set:

[0098] Output the final optimized trajectory point set. ,in, To optimize the radial coordinates of the trajectory points, To optimize the polar angle of the trajectory points, To optimize the elevation of the trajectory points. In three-axis slow-tool servo turning, That is, the position of movement along the X-axis. The angular position of the C-axis. By determining the Z-axis movement position and the axis linkage speed based on the actual machine tool dynamics constraints, complete machining G-code can be generated.

[0099] Example:

[0100] In this embodiment, the sinusoidal mesh surface after tool compensation Optimize the trajectory points, with amplitude A=100. Spatial period ; radius of circular workpiece Maximum allowable elevation error The number of spiral turns is 200; the model of the sinusoidal mesh surface is shown below. Figure 2 .

[0101] First, based on the workpiece radius and the required number of spiral turns, the maximum angle range required on the polar angle is determined. The maximum step size for equal angles is calculated to be 0.0183 degrees based on the maximum sag error constraint. The initial trajectory point set is then calculated using formulas (1), (2), and (3). As input for subsequent optimization, the number of trajectory points at this point is 3,971,429, and the maximum elevation error is 0.00498. The original equal-angle trajectory theory error distribution is as follows: Figure 3 As shown in the diagram. This step ensures that the initial trajectory theoretically satisfies the given upper bound of the global target elevation error, thus providing a reasonable starting point for subsequent point deletion.

[0102] Secondly, regarding the above initial trajectory point sequence The trajectory is calculated using formulas (4), (5), and (6) based on the parameters. Curvature sequence on and the corresponding curvature gradient sequence For the effective sample set of curvature gradients after removing several initial points (used to remove computational instability factors at the sampling start point), the probability density function of the curvature gradient distribution is estimated using one-dimensional kernel density estimation according to formula (7). The location of local valleys is automatically detected based on this density function, and the first significant valley is used as the dividing threshold. If no significant valley is detected, the median of the samples is used as the backup threshold. The trajectory points are automatically divided into "complex areas" and "flat areas" based on this threshold. To eliminate short-term label fluctuations caused by noise, a short segment merging strategy is adopted for the label sequence: for short segments with a length less than the predetermined minimum length (0.5% of the sequence length and not less than 3 points), they are merged into more representative neighboring categories based on the majority of labels of adjacent segments or the approximate distance of the curvature gradient within the segment. After completing this step, a set of continuous "flat segment" index intervals is obtained for subsequent point deletion operations.

[0103] Next, for each continuous segment identified as a "smooth area," the following efficient strategy is used to attempt point deletion: within the segment, starting from the left endpoint, the interval to be deleted is gradually expanded: an "exponential expansion" method is used to gradually increase the right-end index (i.e., doubling in steps of 1, 2, 4, 8...) to quickly find the boundary exceeding the maximum elevation error (if it does not exceed the tolerance, it is expanded to the end of the segment). To control local detection overhead, the maximum number of local detection points is set to 600, and long intervals are evaluated in a segmented manner. After the exponential expansion terminates, a binary search is used between the last index that meets the tolerance and the first index that does not meet the tolerance to determine the maximum deletable right-end index, thereby determining the maximum safe deletion interval. For the found deletable interval, points inside the interval (excluding the endpoints) are marked for deletion; then the process continues from the next unprocessed point until the end of the segment. After the entire segment is processed, the left and right endpoints of the segment are preserved to ensure the connectivity and boundary safety of the trajectory segment. After completing the initial point deletion for all smooth segments, an initial optimized point set is obtained. For this set of points, if the global maximum error does not exceed the given tolerance, the optimization ends; otherwise, all out-of-tolerance locations are extracted and each out-of-tolerance point is mapped to its corresponding optimization interval for local encryption.

[0104] Then, for each interval deemed out of tolerance, local encryption is performed according to the following rules: Within the interval, several equidistant test points are taken to calculate the maximum true error within the interval, and based on this, the number of points to be inserted is determined according to empirical rules (in this embodiment, the ratio of interval error to tolerance is rounded down and limited to 1-5 points). These new points are inserted within the interval and incorporated into the current optimized point sequence. After encryption of all out-of-tolerance intervals, global verification is performed again. If the global error still exceeds the tolerance, the above positioning and encryption process is repeated until the error meets the requirements. This embodiment underwent two local encryption iterations, ultimately meeting the maximum sag error requirement. The optimized trajectory points numbered 1,819,587, achieving a compression rate of 54.18%.

[0105] Finally, output the final optimized trajectory point set. In three-axis slow-tool servo turning, That is, the position of movement along the X-axis. The angular position of the C-axis. By determining the Z-axis movement position and the axis linkage speed based on the actual machine tool dynamics constraints, complete machining G-code can be generated.

[0106] The decision criteria, number of local detection points, short segment merging threshold, and local insertion strategy used in this embodiment can all be adjusted according to resource and rate requirements in engineering applications to control the trade-off between compression ratio and computational overhead.

Claims

1. A method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints, characterized in that... The method includes the following steps: Step 1: Perform trajectory parameterization and initial iso-angle sampling; Step 2: Calculate the trajectory curvature With curvature gradient The specific steps are as follows: Step 2-1, in trajectory parameters Define trajectory space vectors and define the derivative vector. and : in, and The polar angle in a rectangular coordinate system Coordinates at time; Then the trajectory curvature for: Step 2-2: Denote the differential of the arc length as... , for arc length at Points earned Then the curvature gradient for: Steps 2-3: Based on the initial trajectory point set The curvature gradient set is calculated. ; Step 3: Perform automatic region classification based on curvature gradient density. The specific steps are as follows: Step 3-1: For the curvature gradient set Construct a one-dimensional kernel density estimate and calculate the probability density function estimate. ; Step 3-2: Using the derivative method in Detecting local valleys to determine one or more segmentation thresholds ; Step 3-3: To perform complex / gradual partitioning, select the first significant valley. As a dividing line: Steps 3-4: Merge consecutive segments in the obtained category sequences: For segments shorter than the minimum length threshold... The segments are merged into the more similar side based on the median or mean of the adjacent segments, eliminating isolated short segments caused by noise; Step 4: Perform point deletion based on target error constraints in the flat region. The specific steps are as follows: Step 4-1: For each continuous interval judged as smooth... Construct a linear interpolation reconstruction function using the interval endpoints. Define the local interpolation error function for this interval: in, Polar angle Theoretical vector height of time, Polar angle The vector height is obtained through linear interpolation; The criteria for determining whether to delete points are: in, For trajectory target error; Step 4-2: To efficiently find the largest deletable internal interval, the following numerical judgment logic is adopted: From the left end of the interval Initially, we attempted to quickly find a right-hand side using an exponential expansion method. , making the interval The local maximum error is less than the trajectory target error. If an expansion fails, a binary search is used to determine the maximum deletable right endpoint between the previous successful position and the failed position. Mark all the largest deletable internal points as "deletable"; then continue the above steps to the right from the right end until the interval ends; Step 5: Perform global error verification and out-of-tolerance feedback with local encryption; Step 6: Output the final optimized trajectory point set.

2. The method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints as described in claim 1, characterized in that... The specific steps of step 1 are as follows: Step 1-1: Represent the complex surface to be processed as a vector height function. And using polar coordinate system parameters Describe the trajectory, where: in, For polar angle parameters, Polar angle radial coordinates at time and The polar angle in a rectangular coordinate system Coordinates at time; Take a linear radial mapping: in, The maximum radius of the workpiece. The total angle of the spiral trajectory; Steps 1-2: Under the condition of satisfying the trajectory target error Under the premise of equal angles Initial sampling is performed to obtain an initial trajectory point set. : in, The radial coordinates of the initial trajectory point. The polar angle of the initial trajectory point. Let the initial trajectory point be the sag. This represents the initial number of equal-angle sampling points.

3. The method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints according to claim 1, characterized in that... The probability density function estimation The calculation formula is: in, The value of the curvature gradient for the probability density currently being evaluated; This represents the total number of samples in the curvature gradient sequence. For the first Curvature gradient values ​​at each sample point; For kernel functions; For bandwidth.

4. The method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints according to claim 3, characterized in that... The , , denoted as the standard deviation of the curvature gradient samples.

5. The method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints according to claim 1, characterized in that... The specific steps of step 5 are as follows: Step 5-1: After deleting points in the flat areas, perform a global error verification on the entire optimized trajectory point set. If there are out-of-tolerance locations... These are extremely poor Location mapping to the corresponding optimization point range ; Step 5-2: For each identified interval, calculate its local maximum deviation. Dynamically determine the number of insertion points These points are evenly inserted within the interval to densify the grid; Step 5-3: Repeat the global verification-local encryption process until the global error is satisfied. .

6. The method for optimizing the trajectory of slow-speed servo turning of complex surfaces based on curvature gradient clustering and trajectory error constraints according to claim 1, characterized in that... In step 6, the final optimized trajectory point set is output. ,in, To optimize the radial coordinates of the trajectory points, To optimize the polar angle of the trajectory points, The sag of the optimized trajectory points.

Images