Epicyclic path planning method for complex cavity machining

Abstract

The application discloses a cycloid path planning method for complex cavity processing, and comprises the following steps: obtaining a central axis guide line by using a central axis conversion method according to a graph; selecting a point on the central axis guide line as a circle center to draw a maximum inscribed circle; drawing a cycloid unit in each circle according to the properties of the circle; and connecting the cycloid units by using an over curve. Therefore, the cycloid unit in the maximum inscribed circle of the cavity is designed based on a cycloid equation, and an efficient cycloid unit design method and a simple transition line connection method are provided to realize the rapid transition between the cycloid units.

Description

Technical Field

[0001] This invention belongs to the field of machining, and specifically relates to a cycloidal path planning method for machining complex cavities. Background Technology

[0002] Machining is the process of cutting and shaping a workpiece using mechanical equipment to achieve the desired shape, size, and surface quality. Common machining methods include turning, milling, grinding, and drilling. With the development of the manufacturing industry, the demands for machining accuracy, efficiency, and the ability to process complex parts are constantly increasing, driving advancements in machining technology. Simultaneously, the manufacturing industry has begun to have machining needs for more complex parts, such as parts with complex cavities.

[0003] Complex cavities typically refer to cavities with complex internal shapes and structures. These cavities are irregular in shape and may contain various curved surfaces, concave and convex structures, and variations in angles and dimensions. Unlike cavities with simple geometric shapes (such as circles, squares, and other regular shapes), complex cavities have more complex geometric features that are difficult to describe using simple mathematical formulas or geometric models. There is a significant demand for complex cavity parts in fields such as mold manufacturing, aerospace, biomedicine, and electronics.

[0004] However, machining complex cavities presents numerous challenges: First, the high complexity of the shape makes toolpath planning difficult. The boundaries of complex cavities may be free curves, and there may be islands or holes inside, which makes toolpath planning extremely difficult. For example, when machining the cavity of an aero-engine blade, its complex curved surface and internal structure require the toolpath to precisely avoid these obstacles; otherwise, the tool will collide with the cavity, damaging both the tool and the workpiece.

[0005] Secondly, uneven tool wear reduces tool life. Machining complex cavities typically requires longer processing times and higher cutting loads, which easily leads to uneven tool wear. Worn tools affect machining accuracy and surface quality, necessitating timely tool replacement and increasing machining costs and time.

[0006] Third, ensuring precision is extremely difficult. Many complex cavity parts require very high dimensional accuracy. For example, some components in the aerospace field may have dimensional tolerances of only a few micrometers or even smaller. Achieving such high-precision machining requires precise toolpath planning, high-quality tools, and a stable machining environment. In addition to dimensional accuracy, the shape accuracy of complex cavities is also difficult to guarantee. For example, when machining cavities with complex curved surfaces, it is difficult to achieve the ideal shape accuracy in a single machining operation; multiple grinding and finishing processes are usually required.

[0007] With the increasing demands for processing efficiency and precision in modern manufacturing, traditional straight-line cutting methods are no longer sufficient to meet the processing needs of complex parts. The machining of complex cavities requires better machining strategies, and shorter machining times and longer tool life are also the industry's pursuit. In addition to improvements in materials and equipment, process improvements and better machining path planning are also important approaches, and trochoidal machining has emerged as a result.

[0008] Trochoidal machining is a highly efficient and precise CNC machining method. Its core principle is to achieve efficient cutting or etching of complex cavities by moving a tool or electrode along a trochoidal trajectory. Compared with traditional linear or circular interpolation machining, trochoidal machining offers advantages such as lower cutting forces, lower tool wear, and higher machining efficiency, making it particularly suitable for difficult-to-machine structures such as deep cavities, narrow grooves, and thin walls. In recent years, with the increasing demands for precision, surface quality, and machining efficiency in high-end equipment manufacturing, trochoidal machining technology has made significant progress in process optimization, material compatibility, and equipment intelligence. Current trochoidal machining technologies mainly include trochoidal milling, electrical discharge machining (EDM) trochoidal machining, and composite trochoidal machining. The following is a brief description of these three methods.

[0009] Trochoidal milling is currently the most commonly used method for machining complex cavities. Its characteristic is that the tool moves along a trochoidal trajectory and achieves efficient cutting through continuous helical feed. Compared with traditional milling, trochoidal milling has the following advantages: (1) More uniform cutting force: The trochoidal trajectory makes the cutting load of the tool change periodically, reducing local overheating and tool wear; (2) Smoother chip removal: The helical feed mode is conducive to chip removal and avoids the problem of chip accumulation in deep cavity machining; (3) Higher machining efficiency: Higher feed rates can be used while maintaining a lower cutting depth, improving the material removal rate.

[0010] The trochoidal electrical discharge machining (EDM) method is mainly used for machining complex cavities in high-hardness, difficult-to-cut materials (such as cemented carbide, titanium alloys, etc.). Its principle is to achieve efficient discharge erosion by moving the electrode along the trochoidal trajectory. The advantages of this method include: (1) it is suitable for superhard materials and can machine materials that are difficult to process by traditional milling, such as tungsten carbide, ceramics, etc.; (2) there is no mechanical cutting force, avoiding tool deformation and workpiece stress concentration, and it is suitable for machining thin-walled and micro-structured materials; (3) it produces high surface quality, and by optimizing the discharge parameters, a surface roughness of Ra below 0.2μm can be obtained.

[0011] The composite cycloidal machining methods mainly include: (1) milling-electrical discharge machining, which first uses cycloidal milling for roughing and then electrical discharge machining for finishing, taking into account both efficiency and accuracy; (2) laser-assisted cycloidal machining, which mainly uses laser to soften materials, reduce cutting force, and improve tool life; (3) ultrasonic vibration-assisted cycloidal machining, which improves chip removal conditions through high-frequency vibration and is suitable for deep cavity machining. The composite cycloidal machining methods can significantly improve machining quality. For example, laser-assisted cycloidal milling can increase the machining efficiency of titanium alloys by 30% and extend tool life by 50%.

[0012] In the aforementioned cycloidal machining method, the most critical issue is the planning of the cycloidal path. Relevant prior art can be found in Chinese Patent 202010684245.6, filed on July 16, 2020, which discloses a tool path planning method for elliptical cycloidal milling of cavities based on the contour centerline. This method describes the complex cavity boundary features using the contour centerline and calculates the elliptical tool path using the centerline point information. Under the cutting width threshold constraint, starting from the centerline point closest to the largest root node, all centerline points within the reachable area are traversed along the direction from root node to root node or from root node to leaf node, retaining the centerline point information and corresponding tool path that satisfy the cutting width threshold constraint. The problems with this method are: First, while the elliptical cycloidal trajectory is considered an improvement over the circular trajectory due to its flat shape and high cutting efficiency, it requires a custom parameter for each largest inscribed circle, resulting in high computational complexity. Additionally, the elliptical cycloidal trajectory sometimes exhibits excessive curvature changes, which is undesirable during machining. Secondly, the handling of the central axis guide line is a thorny issue in trochoidal machining. The central axis of complex shapes has many branches, and it's difficult to continuously calculate near these branches; otherwise, problems such as chaotic transition lines and inability to traverse all branches will arise. Current technology has not provided a perfect solution. Thirdly, the above problems, combined with material and process constraints, will create new issues. For example, it's necessary to meet the cutting characteristics of some difficult-to-machine materials (such as high-temperature alloys and composite materials). High-hardness materials (such as hardened steel and cemented carbide) will increase tool wear and reduce machining efficiency. High-toughness materials (such as titanium alloys and stainless steel) are prone to work hardening, affecting surface quality and tool life. Materials with poor thermal conductivity (such as titanium alloys) are prone to heat accumulation during cutting, leading to thermal deformation and tool wear. Highly chemically reactive materials (such as titanium alloys) are prone to chemical reactions with the tool at high temperatures, accelerating tool wear. Furthermore, machining complex cavities requires the use of small-diameter tools, but these tools have poor rigidity and wear resistance, making them prone to wear or breakage. Material requirements can be alleviated through better processes. Smooth trajectories with stable curvature reduce tool wear and enhance material adaptability. Designing cycloidal equations with stable curvature is a direction that the industry is striving for. Summary of the Invention

[0013] The purpose of this invention is to provide a cycloidal path planning method for machining complex cavities, in order to solve the problems of high computational complexity and excessive curvature variation in existing elliptic cycloidal methods.

[0014] To achieve the above objectives, the cycloidal path planning method for complex cavity machining of the present invention includes the following steps: Step 1: Obtain the central axis guide line based on the diagram using the central axis transformation method; Step 2: Select a point on the central axis as the center and construct the largest inscribed circle; Step 3: Construct cycloidal units within each circle based on its properties; Step 4: Connect the cycloidal units using transition curves.

[0015] Based on the above main characteristics, the median transformation method used in step one includes one of the following: median transformation method based on Voronoi diagram, distance transformation method, median transformation method based on thinning, and shrinking sphere algorithm.

[0016] Based on the above main features, step one also includes a step to determine the bifurcation point. The specific process is to traverse the central axis points, use the k-nn algorithm for each central axis point to obtain nearby central axis points, and obtain a point set; perform the PCA algorithm on the point set, let the first largest feature value be λ1 and the second largest feature value be λ2. If λ2 / λ1 > the preset threshold h, then the point is considered to be a bifurcation point.

[0017] Based on the above main features, after finding all the branching points, the central axis points are selected at equal intervals on each branch, and then the inscribed circle is extracted.

[0018] Based on the above main characteristics, the cycloidal unit in step three is determined in the following way: The distance between the set point and the center of the circle is The radius of the rounding is t is the parameter of the cycloid, and the equation of the cycloid is:

[0019] For the largest incircle The corresponding cycloidal unit is defined by the largest inscribed circle. The radius is Let the radius of the cutting tool be... The starting and ending points of the cycloid and the range of values ​​for the parameter t of the cycloid need to be determined. The specific derivation process is as follows: (1) Shrink the largest inscribed circle inward by r to form a smaller circle, the point of tangency of the largest inscribed circle and The cycloid is tangent to the small circle at... , ,in correspond , correspond , and These are the starting and ending points of the cycloid; (2) Set as

[0020] (3) Determine the range of parameter t through the tangent to the curve. Let the cycloid segment be... arrive For the part where the tangent to the cycloid is located, the formula is as follows:

[0021] Then we can obtain the following equation:

[0022]

[0023] so:

[0024] Next, we find the parameter k, which is easy to obtain:

[0025] again:

[0026]

[0027] have to:

[0028] From the above, the cycloidal path from M1 to M2 can be completely determined by K and t0.

[0029] Based on the above main features, the cycloidal path planning method for complex cavity machining also provides a method for determining a secondary cycloid. Let the secondary cycloid's rounding radius be *a* and the fixed-point radius be *b*. The reasoning process for the secondary cycloid is as follows: The distance between the set point and the center of the circle is The radius of the rounding is t is the parameter of the cycloid, and the equation of the cycloid is:

[0030] For the largest incircle The corresponding cycloidal unit is defined by the largest inscribed circle. The radius is Let the radius of the cutting tool be... The starting and ending points of the cycloid and the range of values ​​for the parameter t of the cycloid need to be determined. The specific derivation process is as follows: (1) Shrink the largest inscribed circle inward by r to form a smaller circle, the point of tangency of the largest inscribed circle and The cycloid is tangent to the small circle at... , ,in correspond , correspond , and These are the starting and ending points of the cycloid; (2) Set as

[0031] (3) Determine the range of parameter t through the tangent to the curve. Let the cycloid segment be... arrive For the part where the tangent to the cycloid is located, the formula is as follows:

[0032] Then we can obtain the following equation:

[0033]

[0034] First, determine the shape of the cycloid you want to use, which is determined by m in the following formula:

[0035] have:

[0036]

[0037]

[0038]

[0039]

[0040] There are two possible scenarios at this point, but based on practical considerations, the following one should be chosen: .

[0041] Based on the above main features, the cycloidal path planning method for complex cavity machining also includes drawing a perpendicular straight line through the center of the circle in the direction of the guide line's advance, intersecting the small circle at two points, calling the generation formula of the cycloidal unit to use these two points as the cycloidal unit, and taking the arc line from these two points on the small circle to the corresponding point on the small circle of the actual feature point as another part of the cycloidal unit.

[0042] Based on the above main characteristics, for the design of the transition curve for closing the cycloidal unit in step four, for the transition curve inside the maximum inscribed circle, the following method is used to connect the beginning and end: First, make an arc tangent to the auxiliary circle on both sides of the cycloidal line with the same small radius r, ensuring that the two arcs are the same size and that the final normal vectors are on the same straight line. Then, connect the two ends of the two arcs with a straight line. For the transition curve outside the maximum inscribed circle, connect the two ends of the cycloidal line directly with a straight line.

[0043] Compared with existing technologies, this invention designs cycloidal elements within the largest inscribed circle of the cavity based on the cycloidal equation, presenting an efficient cycloidal element design method and a simple transition line connection method to achieve rapid transition between cycloidal elements. Furthermore, based on the relative positional relationship between the central axis point and its neighboring points, this paper utilizes Principal Component Analysis (PCA) to find bifurcation points on the central axis, decomposing the central axis into connected branches, thereby constructing the cavity graph structure. Additionally, the processing sequence problem is transformed into a minimum path coverage problem on the graph and solved accordingly. Attached Figure Description

[0044] Figure 1 A flowchart illustrating the cycloidal path planning method for machining complex cavities according to the present invention.

[0045] Figure 2A and Figure 2B A schematic diagram of the central axis and branch lines of the graphic identified in the cycloidal path planning method for complex cavity machining of the present invention.

[0046] Figure 3 A schematic diagram of the inscribed circle extracted by the cycloidal path planning method for complex cavity machining of the present invention.

[0047] Figure 4A This is a mathematical model of a cycloid unit.

[0048] Figure 4B and Figure 4C This is a schematic diagram of placing the cycloid in a coordinate system.

[0049] Figure 5 This is a schematic diagram of the cycloidal algorithm.

[0050] Figure 6 Images of three types of cycloids.

[0051] Figure 7 This refers to the cycloidal unit when the forward direction is the dominant arc direction.

[0052] Figure 8 This is a schematic diagram of the transition curve generation.

[0053] Figures 9A to 9D This is a schematic diagram illustrating the identification of bifurcation points and branches of the central axis in different samples using the present invention.

[0054] Figures 10A to 10C This is a schematic diagram of the cycloidal processing path, where red represents the cycloidal unit and green and blue represent transition curves. Detailed Implementation

[0055] To facilitate understanding, we will first introduce some concepts and their definitions used in cycloidal machining trajectory planning.

[0056] (1) Central axis guide line: The central axis guide line is an auxiliary curve that guides the direction of machining path generation in cycloidal machining trajectory planning. It is generally the central axis of the graphic. The central axis is the product of a special image processing method (central axis transformation method). The distance between the point on the central axis and at least two boundary points is the minimum, so it can be regarded as the central axis of the graphic.

[0057] (2) Maximum inscribed circle: Based on the properties of the central axis mentioned above, a circle can be drawn with a point on the central axis as the center and the minimum distance from this point to the boundary of the figure as the radius. Obviously, this circle is tangent to the boundary of the figure and has at least two points of tangency. This circle is called the maximum inscribed circle. This circle can reflect the local geometric features of the figure. The path planning of cycloidal processing is mainly carried out in these circles. The points of tangency of the maximum inscribed circle are also called feature points.

[0058] (3) Cycloidal unit: The cycloidal unit refers to the machining path curve generated in the largest inscribed circle based on the geometric characteristics of the largest inscribed circle. This is also the most important part of cycloidal path machining. A good design method can make the calculation simpler, the machining smoother, and the machining more accurate.

[0059] (4) Transition curves: Cycloidal units are often isolated in the largest inscribed circles, and often they are not even closed. Therefore, it is necessary to use transition curves to connect different cycloidal units, and sometimes it is also necessary to use transition curves to close the cycloidal units.

[0060] Please see Figure 1 The diagram shows a flow chart of the cycloidal path planning method for machining complex cavities according to the present invention. The cycloidal path planning method for machining complex cavities according to the present invention includes the following steps: Step 1: Obtain the central axis guide line based on the diagram using the central axis transformation method; Step 2: Select a point on the central axis as the center and construct the largest inscribed circle; Step 3: Construct cycloidal units within each circle based on its properties; Step 4: Connect the cycloidal units using transition curves.

[0061] The following provides a detailed explanation of each step.

[0062] In step one, before designing the cycloidal machining path, it is often necessary to generate a central axis guide line using the central axis transformation method, which serves as a guide for path generation. The central axis transformation (also known as skeletonization or symmetry axis transformation) is a mathematical method that simplifies a two-dimensional or three-dimensional shape to its central axis (skeleton). The central axis is the locus of the centers of all the largest inscribed circles within the shape, reflecting the topological structure and key features of the geometry. This is because the central axis guide line remains homotopic with the shape, allowing the extraction of topological features. The largest inscribed circles also reflect the local characteristics of the shape, thus enabling the relatively easy design of cycloidal units with the aid of the central axis guide line.

[0063] Hotopy is defined as follows: Let... If there are two continuous mappings, then... satisfy: (2.1) Then it is called a function and Hotopy, function This is called deformation.

[0064] Hotopism of two functions It can change continuously as This process preserves all original geometric features, such as the number of bends, stretches, and holes, including topological features like connectivity.

[0065] As mentioned earlier, in a central axis transformation, every point on the central axis has at least two feature points on the boundary, and every point on the boundary can be matched with a point on the central axis. Therefore, if we consider the boundary of the figure as... Viewing the central axis as , Each point above can have The upper point is derived from the largest inscribed circle, which is also called the intuitive reflection of deformation information. Therefore, the central axis is homotopic with the boundary of the figure.

[0066] The central axis extracts the topological features of the graphic, and the largest inscribed circle extracts the changes from the graphic boundary to the central axis. Therefore, by using the central axis point, the central axis line, and the largest inscribed circle, a cycloidal unit that meets the processing requirements here can be created.

[0067] The median transformation method includes the median transformation method based on Voronoi diagram, the method based on distance transformation, the median transformation calculation method based on thinning, and the shrinking sphere algorithm. All of the above methods are existing technologies and will not be described in detail here.

[0068] Center-axis transformation has significant applications in machining path planning. It can be used to optimize toolpaths, especially in scenarios such as narrow slot machining and equidistant offset machining. The center-axis transformation method is particularly suitable for optimizing complex contours and narrow areas in machining path planning, but the calculation method should be selected based on specific requirements (e.g., the Voronoi method is suitable for CAD models, while the distance transformation method is suitable for images).

[0069] In step two, the generated central axis and the largest inscribed circle corresponding to its central axis point cannot be used directly. Firstly, for example... Figure 2A As shown, the central axis has a bifurcation, and the above algorithm cannot be directly used at the bifurcation point. Figure 2A The small blue circles in the diagram represent the nearest neighbors of the bifurcation point. On the other hand, the central axis points are generated very densely in the algorithm for generating the central axis, but such density is not required in actual processing. Therefore, it is necessary to select some points from them, and points near the bifurcation (hereinafter referred to as bifurcation points) should also be avoided when selecting them.

[0070] Bifurcations are points where the guide line branches out in different directions. These points can cause inconvenience from both a manufacturing design and mathematical calculation perspective, so it is necessary to identify and mark them in advance.

[0071] First, k-neighbor identification (KNN algorithm) is performed on each point on the guide line. The KNN (K-Nearest Neighbor) algorithm is based on the assumption of "nearest neighbor similarity." It calculates the distance between the sample to be predicted and all samples in the training set, selecting the K nearest samples (i.e., "nearest neighbors"). In this way, it can find which points on the guide line are near each point, determine whether these points are on the same line, and thus determine whether the point is near a bifurcation point, adding the neighboring points to the point set.

[0072] The KNN algorithm finds the k nearest neighbors with the same features. This method can be used to find the central point on the central axis of the algorithm (where k is 3 or 5, which may be better).

[0073] PCA, or Principal Component Analysis, can be used to determine whether neighboring points found using the KNN algorithm lie on a straight line (if they are approximately on a straight line, only one eigenvector will have an eigenvalue much greater than 0 and close to 1). PCA calculates the principal component eigenvalues ​​and eigenvectors, yielding two eigenvalues. If all points are approximately on a straight line, λ1 is close to 1 and λ2 is close to 0; if they are not on a straight line, neither will be close to 0. Therefore, it can be used... Make a judgment, set a threshold of 0.1, if If the value is greater than 0.1, then the point is considered a bifurcation point or very close to a bifurcation point and needs to be marked.

[0074] The specific process is as follows: Traverse the central axis points, use the KNN algorithm on the central axis point 1 to obtain the nearby central axis points, and obtain a point set A1; perform the PCA algorithm on the point set A1, let the first largest feature value be λ1 and the second largest feature value be λ2. If λ2 / λ1 > the preset threshold h (e.g., 0.1), then the point is considered to be a bifurcation point.

[0075] Please see Figure 2A and Figure 2B As shown, Figure 2A The red line represents the central axis of the graph, and the small blue circles represent the k-domain of the bifurcation points; in Figure 2B In the diagram, the lines of different colors represent different branches of the identified central axis.

[0076] During the processing, it is obviously impossible to generate paths for all the largest inscribed circles. Therefore, it is necessary to select certain central axis points, generally at equal intervals. This invention, after finding all branch points, selects central axis points at equal intervals on each branch, and then extracts the inscribed circles, specifically as follows... Figure 3 As shown. Using the above method, this invention finds the bifurcation points on the central axis based on the relative positional relationship between the central axis point and its neighboring points, decomposes the central axis into connected branches, and then constructs the cavity graph structure. Furthermore, it transforms the processing sequence problem into a minimum path coverage problem on the graph and solves it.

[0077] In step three, the equation of the cycloid is used as the basic expression of the cycloid unit. The planning method of the cycloid unit will be introduced below, and a mathematical model will be established based on it. Then, based on the geometric information of the largest inscribed circle, the shape and size of the cycloid can be determined through mathematical derivation.

[0078] like Figure 4A As shown, this is a mathematical model of a cycloid unit. The black circle is the largest inscribed circle along the central axis, the small circle is the model of the tool, and the green line is a cycloid trajectory, denoted as a cycloid unit. The tool at both ends of the cycloid unit is tangent to the workpiece contour. At the same time, the tangents of the cycloid at these two points should be parallel to the tangents at the points where they are tangent to the outer surface. This cycloid unit is the cycloid model corresponding to the inscribed circle.

[0079] In a maximally inscribed circle, the cycloidal element needs to be derived using its geometric information (center coordinates, tangent coordinates, radius).

[0080] First, the equation of the cycloid is given: The distance between the set point and the center of the circle is The radius of the rounding is t is the parameter of the cycloid, and the equation of the cycloid is: (3.1) The following discussion only covers the generation method of cycloidal elements for ordinary cycloids (i.e., cycloids with fixed points located on the rolling circle). To calculate the maximum inscribed circle... The corresponding cycloidal unit, the largest inscribed circle The radius is Let the radius of the cutting tool be... The solution needs to determine the starting and ending points of the cycloid, the range of values ​​for the cycloid's parameters (i.e., t in equation 3.1), and the constants of the equation. The specific derivation process is as follows: (1) Shrink the largest inscribed circle inward by r to obtain Figure 5 The smaller circle in the circle, where the point of tangency of the largest incircle. and ,in correspond , correspond ,but and The starting and ending points of the cycloid are located at points where the cycloid is tangent to the small circle. , (If the blade does not cut tangent, and the blade continues to move smoothly downwards, the blade will inevitably cut outside the boundary.) (2) Set as ; (3) such as Figure 4B and Figure 4C As shown, this is a schematic diagram of placing the required cycloid in a coordinate system, with the angle between the two tangents being... The range of parameter t can be determined by the tangent to the curve. Let the cycloid segment be... arrive For the part where the tangent to the cycloid is located, the formula is as follows: (3.2) Then we can obtain the following equation: (3.3) (3.4) so: (3.5) Next, we find the parameter k, which is easy to obtain: (3.6) again: (3.7) 3.8) have to: (3.9) In summary, K and t0 can completely determine the cycloidal path from M1 to M2.

[0081] like Figure 6 The image shows three types of cycloids. Although the problem can be solved using only cycloid units, for the sake of flexibility in toolpath planning, this invention also provides a method for generating subcycloid units. Let the radius of the subcycloid's rounding be a, and the radius of the fixed point be b.

[0082] The reasoning for the secondary cycloid is the same as that for the cycloid in the preceding part, but equation 3.4 is transformed as follows: (3.10) First, determine the shape of the cycloid you want to use, which is determined by m in the following formula: (3.11) have: (3.12) (3.13) (3.14) (3.15) (3.16) There are two possible scenarios at this point, but based on practical considerations, the following one should be chosen: (3.17) Regarding the direction of travel, if the largest inscribed circle is divided into two arcs by two feature points, the cycloidal unit will point in the direction of the minor arc (shorter arc). In most scenarios, the path generation direction is indeed on the side of the minor arc, but there are still many times when the guide line will point on the side of the major arc.

[0083] At this point, the algorithm needs to be improved, such as... Figure 7 As shown, the blue arrow indicates the direction of the guide line. A perpendicular line is drawn through the center of the circle, intersecting the small circle at two points. The cycloidal element generation formula is used again, with these two points as "pseudo-feature points" to create the cycloidal element, i.e., the blue curve. At the same time, the arc from these two points on the small circle to the corresponding points on the small circle of the true feature point is taken, i.e., the red part, as the other part of the cycloidal element.

[0084] For the design of the transition curve for closing the cycloidal unit in step four, please refer to [link to relevant documentation] for the transition curve within the largest inscribed circle. Figure 8 As shown in the figure, the red part represents the cycloidal unit, which is connected end to end using the following method: First, draw an arc with the same small radius r on both sides of the cycloid, tangent to the auxiliary circle (or the small circle tangent to the cycloid unit). Ensure that the two arcs are the same size and that their normal vectors lie on the same straight line. Figure 8 The blue section will then connect the two ends of the two arcs with a straight line, as shown below. Figure 8 The green part.

[0085] For the transition curve outside the largest inscribed circle, such as Figure 8 The light blue section connects the two ends of the front and rear cycloids directly with a straight line.

[0086] As shown in Figure 2, for complex cavities, the guide line is not a single line but consists of many branches. Therefore, even with a complete method for generating cycloidal units and transition lines, it is impossible to generate them directly from beginning to end. In path generation, it is necessary to ensure that generation is completed in one branch before proceeding to the next; otherwise, problems arise in transition line generation and the design of the processing sequence. The preceding description revealed that each branch was identified by finding its bifurcation points, and a certain number of central axis points were selected at equal intervals from each branch. If these central axis points and their connections are considered as a graph, each branch becomes a connected component. Generating cycloidal units within each inscribed circle is considered as traversing these points. Therefore, the processing sequence problem can be viewed as a minimum path coverage problem for this graph. This invention solves this problem using the following method: (1) Construct a graph structure. For points on the same branch, connect adjacent points one by one. There are no connections between points on different branches. (2) Find the longest path in the graph, that is, the path that passes through the most points. Since the points in the graph are linearly connected one by one, the time complexity of this step is not high. Generate the processing path from the points of one segment of the path to another. (3) Remove all points in the path in the graph and repeat the above process to find the maximum path in the remaining graph.

[0087] Please see Figures 9A to 9D The diagram shown illustrates the identification of bifurcation points and branches of the central axis in different samples using this invention. Figure 9A and Figure 9C The red line represents the central axis of the graph, and the small blue circles represent the k-domain of the bifurcation point. Figure 9B and Figure 9D The different colored lines represent different branches of the identified central axis. (Combined) Figure 2A and Figure 2B As can be seen from the results, the method disclosed in this invention has a good recognition effect on trifling and bifling, as well as short branches and long branches.

[0088] Please see Figures 10A to 10CAs shown, this is a cycloidal processing path planning using the method of the present invention, where red represents cycloidal units, and green and blue represent transition curves, with blue indicating the small arc portion of the transition curve.

[0089] The experimental results show that the proposed method can generate reasonable machining paths for various complex cavities, and it also handles relatively complex parts such as bifurcations, trifurcations, curved boundaries, and concave curved boundaries well, demonstrating sufficient robustness. The computation time for path generation is very short and negligible in actual machining, highlighting its high efficiency. Compared to circular trajectories, in areas with large edge curvature, the angle between the two feature points of the largest inscribed circle becomes smaller, resulting in a smoother cycloidal portion captured by the cycloidal unit, offering better adaptability.

[0090] The data for the two trajectories in the three experimental samples are collected and compared below. Tables 4.1 and 4.2 show the data for the cycloidal trajectory and the circular trajectory, respectively, including the total length of the cycloidal unit, the length of the transition line, and the total length of the processing path.

[0091] Table 4.1 Cycloidal Trajectory Data

[0092] Table 4.2 Circular trajectory data

[0093] As can be seen from the table, for the same experimental sample, the shorter lengths of each part of the cycloidal trajectory allow it to complete the same task as the circular trajectory, indicating that the processing efficiency of this invention is higher. In actual processing, a shorter processing path length means less wear on the tool, resulting in a longer tool life. Table 4.3 records the percentage increase in trajectory distance for each part of the cycloidal trajectory relative to the circular trajectory (the percentage decrease in trajectory distance is decreasing).

[0094] Table 4.3 Comparison of the two trajectory planning methods

[0095] Table 4.4 records the total number of cycloidal units for the two trajectories. Both experiments used the same central axis processing method, but the circular trajectory may have fewer cycloidal units than the cycloidal trajectory. As can be seen in the two sets of figures in 4.2, at some boundaries with small radii of curvature, the circular trajectory struggles to produce cycloidal units. This indicates that the present invention handles these boundaries more precisely.

[0096] Table 4.4 Comparison of the number of cycloidal units

[0097] Like other cycloidal unit design methods, cycloidal trajectories have their own unique advantages and limitations. Combining the cycloidal formula, cycloidal trajectories have the following advantages: (1) Reduced tool wear: The mathematical model of a cycloid is generated by a circle rolling along a straight line or another circle. Its trajectory is a continuous and smooth curve without sharp inflection points. This smoothness makes the tool uniformly stressed during movement, avoiding the impact load caused by sudden changes in direction in traditional straight paths, thereby reducing tool wear. (2) Adaptable to high-speed machining: The curvature of the cycloidal path changes gently, making it suitable for high-speed machining. The tool will not vibrate or become unstable due to sudden changes in path during high-speed movement. (3) Improved machining quality: The smooth cycloidal machining path reduces vibration and cutting marks, improving surface finish. At the same time, it reduces tool deformation and improves machining accuracy. (4) Adaptable to complex geometries: Cycloidal machining is flexible. Compared with circular paths, cycloidal paths can generate different curve shapes by adjusting the radius ratio of the rolling circle and the base circle, adapting to various complex contours.

[0098] Cycloidal machining also has the following limitations: (1) High computational complexity. The cycloidal path generation algorithm is complex and computationally intensive, especially under high precision requirements. Path optimization needs to consider multiple factors, making optimization difficult. (2) High requirements for machine tools. The machine tool needs to have high dynamic performance to cope with rapidly changing cutting forces. A high-precision control system is required to ensure path accuracy.

[0099] In summary, this invention proposes a new cycloidal unit design method based on circular and elliptical trajectories, providing engineers in related industries with more options.

[0100] This invention also provides a new approach to the design of cycloidal units, namely, that some seemingly complex non-natural curves can well meet the requirements of the cycloidal equation, and the calculation and processing are not complicated in practice. The derivation method disclosed in this invention can sometimes also be applied to such equations. The slope difference between the two endpoints is obtained by using the relationship of the tangent points, the abscissa difference is obtained by the distance between the two tangent points, and the parameters and range of the equation can be solved by using the requirement of symmetry. In the above-mentioned cycloidal processing method, the most critical issue is the planning of the cycloidal path. Therefore, the above only describes the planning of the cycloidal path. Other contents can be found in the description of Chinese Patent 202010684245.6 filed on July 16, 2020, and will not be described in detail here.

[0101] Compared with existing technologies, this invention designs cycloidal elements within the largest inscribed circle of the cavity based on the cycloidal equation, presenting an efficient cycloidal element design method and a simple transition line connection method to achieve rapid transition between cycloidal elements. Furthermore, based on the relative positional relationship between the central axis point and its neighboring points, this paper utilizes Principal Component Analysis (PCA) to find bifurcation points on the central axis, decomposing the central axis into connected branches, thereby constructing the cavity graph structure. Additionally, the processing sequence problem is transformed into a minimum path coverage problem on the graph and solved accordingly.

[0102] It is understood that those skilled in the art can make equivalent substitutions or modifications to the technical solution and inventive concept of the present invention, and all such substitutions or modifications should fall within the protection scope of the appended claims.

Claims

1. A cycloidal path planning method for machining complex cavities, characterized in that... The cycloidal path planning method for machining complex cavities includes the following steps: Step 1: Obtain the central axis guide line based on the diagram using the central axis transformation method; Step 2: Select a point on the central axis as the center and construct the largest inscribed circle; Step 3: Construct cycloidal units within each circle based on its properties; Step 4: Connect the cycloidal units using transition curves.

2. The cycloidal path planning method for machining complex cavities as described in claim 1, characterized in that: The median transformation method used in step one includes one of the following: median transformation based on Voronoi diagram, distance transformation based method, median transformation based on thinning, and shrinking sphere algorithm.

3. The cycloidal path planning method for machining complex cavities as described in claim 1, characterized in that: Step one also includes the step of determining the bifurcation point. The specific process is to traverse the central axis points, use the KNN algorithm for each central axis point to obtain the nearby central axis points, and obtain a point set; perform the PCA algorithm on the point set, let the first largest feature value be λ1 and the second largest feature value be λ2. If λ2 / λ1 > the preset threshold h, then the point is considered to be a bifurcation point.

4. The cycloidal path planning method for machining complex cavities as described in claim 3, characterized in that: After finding all the branching points, the central axis points are selected at equal intervals on each branch, and then the inscribed circle is extracted.

5. The cycloidal path planning method for machining complex cavities as described in claim 1, characterized in that: In step three, the cycloidal unit is determined as follows: The distance between the set point and the center of the circle is The radius of the rounding is t is the parameter of the cycloid, and the equation of the cycloid is: ； For the largest incircle The corresponding cycloidal unit is defined by the largest inscribed circle. The radius is Let the radius of the cutting tool be... The starting and ending points of the cycloid and the range of values ​​for the parameter t of the cycloid need to be determined. The specific derivation process is as follows: (1) Shrink the largest inscribed circle inward by r to form a smaller circle, the point of tangency of the largest inscribed circle and The cycloid is tangent to the small circle at... , ,in correspond , correspond , and These are the starting and ending points of the cycloid; (2) Set as ; (3) Determine the range of parameter t by using the tangent to the curve. Let the cycloid segment be... arrive For the part where the tangent to the cycloid is located, the formula is as follows: ； Then we can obtain the following equation: ， ； so: ； Next, we find the parameter k, which is easy to obtain: ； again: ， ； have to: ； From the above, the cycloidal path from M1 to M2 can be completely determined by K and t0.

6. The cycloidal path planning method for machining complex cavities as described in claim 1, characterized in that: The cycloidal path planning method for machining complex cavities also provides a method for determining a secondary cycloid. Let the secondary cycloid's rounding radius be *a* and the fixed-point radius be *b*. The reasoning process for the secondary cycloid is as follows: The distance between the set point and the center of the circle is The radius of the rounding is t is the parameter of the cycloid, and the equation of the cycloid is: ； For the largest incircle The corresponding cycloidal unit is defined by the largest inscribed circle. The radius is Let the radius of the cutting tool be... The starting and ending points of the cycloid and the range of values ​​for the parameter t of the cycloid need to be determined. The specific derivation process is as follows: (1) Shrink the largest inscribed circle inward by r to form a smaller circle, the point of tangency of the largest inscribed circle and The cycloid is tangent to the small circle at... , ,in correspond , correspond , and These are the starting and ending points of the cycloid; (2) Set as ; (3) Determine the range of parameter t through the tangent to the curve. Let the cycloid segment be... arrive For the part where the tangent to the cycloid is located, the formula is as follows: ； Then we can obtain the following equation: ， ； First, determine the shape of the cycloid you want to use, which is determined by m in the following formula: ； have: ， ， ， ， ； There are two possible scenarios at this point, but based on practical considerations, the following one should be chosen: 。 7. The cycloidal path planning method for machining complex cavities as described in claim 1, characterized in that: The cycloidal path planning method for complex cavity machining also includes drawing a perpendicular straight line through the center of the circle in the direction of the guide line's advance, intersecting the small circle at two points, calling the generation formula of the cycloidal unit to use these two points as the cycloidal unit, and taking the arc line from these two points on the small circle to the corresponding point on the small circle of the true feature point as another part of the cycloidal unit.

8. The cycloidal path planning method for machining complex cavities as described in claim 1, characterized in that: For the design of the transition curve that closes the cycloidal unit in step four, for the transition curve inside the maximum inscribed circle, the following method is used to connect the beginning and end: First, make an arc tangent to the auxiliary circle on both sides of the cycloidal line with the same small radius r, ensuring that the two arcs are the same size and that the final normal vectors are on the same straight line. Then, connect the two ends of the two arcs with a straight line. For the transition curve outside the maximum inscribed circle, connect the two ends of the cycloidal line directly with a straight line.

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