Workpiece geometry feature recognition and position relationship judgment method for automated assembly process

Abstract

The application provides a workpiece geometric feature recognition and position relationship judgment method of an automatic assembly process, and a systematic analysis and processing scheme is provided for geometric features and position information in a product design model.The application realizes accurate analysis of geometric features and position relationships by accurately recognizing geometric features of a three-dimensional design model, extracting relative position relationships of workpieces, and efficiently organizing and storing related data.The system obtains key process information in the assembly process through geometric topological information analysis and position relationship deduction, thereby supporting assembly process planning and optimization.The application aims to solve the technical problems of geometric feature recognition and position relationship analysis in the automatic assembly process, and provides an innovative solution for realizing intelligent and efficient assembly process reasoning and resource allocation.

Description

Technical Field

[0001] This invention relates to the field of intelligent manufacturing technology, and in particular to a method for identifying workpiece geometric features and determining positional relationships in an automated assembly process. Background Technology

[0002] In modern manufacturing, automated assembly is a crucial means to improve production efficiency and product consistency, especially in high-precision fields such as aerospace, automotive manufacturing, and precision electronics, where the requirements for assembly accuracy and reliability are particularly stringent. The key to achieving precision automated assembly lies in the accurate identification, rational layout, and precise judgment of workpiece geometric features and positional relationships. These tasks directly affect the operation path, workpiece positioning, and assembly method selection during the assembly process, thus impacting the quality and efficiency of the entire automated assembly system. However, traditional assembly processes typically rely on human experience and simple geometric relationship judgments, which cannot meet the needs of automated assembly systems for comprehensive extraction and refined analysis of workpiece geometric information. Especially when dealing with complex three-dimensional workpiece models, due to the diverse geometric features of the workpiece, such as cylinders, cones, spheres, toroidal coils, and polygonal planes, as well as various spatial positional relationships, manual analysis is difficult to achieve accurately and comprehensively within a limited time, and is inefficient.

[0003] To address this issue, the common approach is to manually identify geometric information and plan assembly based on the CAD design model of the workpiece using 3D modeling software. However, these methods have significant drawbacks in complex assembly processes: firstly, the lack of a systematic approach to geometric information extraction and structured storage makes the processing and analysis of workpiece geometric feature data complex and difficult; secondly, existing geometric analysis methods struggle to quickly determine the relative positional relationships between multiple workpieces, thus affecting the accuracy and controllability of automated assembly.

[0004] Therefore, existing technologies still need to be improved and enhanced. Summary of the Invention

[0005] In view of the shortcomings of the prior art, the purpose of this invention is to provide a method for identifying workpiece geometric features and determining positional relationships in an automated assembly process, aiming to solve the problem that existing geometric analysis methods are unable to quickly determine the relative positional relationships among multiple workpieces, thereby affecting the accuracy and controllability of automated assembly.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A method for identifying workpiece geometric features and determining positional relationships in an automated assembly process, the method comprising the following steps:

[0008] Step 1: Obtain the CAD design model of the workpiece, extract geometric information based on the STEP file of the workpiece design model, and extract the geometric topology information of entities, surfaces, lines and points in sequence according to the preset hierarchical structure to provide basic data for subsequent assembly process reasoning and analysis of positional relationships between workpieces;

[0009] Step 2: During the extraction of mating surface attribute information from the workpiece design model, the surfaces extracted from the workpiece geometry information are classified, and detailed identification and extraction are performed based on the specific geometric features of each type of mating surface. The system automatically identifies various mating surfaces, including cylindrical surfaces, conical surfaces, spherical surfaces, annular coil surfaces, circular planes, and polygonal planes, through keyword recognition and attribute discrimination technology. Finally, this structured data will be stored in a JSON file for further data processing and retrieval.

[0010] Step 3: When the geometric attribute information of the mating surfaces of the workpiece design model is incomplete, a complete geometric feature description is generated through analysis and completion techniques to ensure the accuracy and completeness of the geometric information of the mating surfaces. The high-level characterization information of the mating surfaces is extracted, and the attributes are calculated and derived based on preset rules. The completed workpiece mating surface information is classified and stored in a JSON file to form a complete workpiece geometric description database.

[0011] Step 4: When processing the workpiece data in the JSON file, since each workpiece has an independent local coordinate system, these local coordinate systems are standardized to uniformly perform spatial analysis and positional relationship judgment.

[0012] Step 5: After completing the transformation between the local and global coordinate systems, the relative positional relationships between different workpieces are calculated based on the unified global coordinate information, and the positional relationships are judged. These relative positional relationships include perpendicular, parallel, coaxial adjacency, coaxial separation, coaxial partial overlap, inclusion, and being included.

[0013] Furthermore, the method for identifying the geometric properties of the mating surfaces in step two is as follows:

[0014] Step 2-1: Extract and record the corresponding geometric attributes for different types of mating surfaces. Different surfaces require different attribute information to form a systematic mating surface data structure.

[0015] Step 2-2: Extract the attribute information of the cylindrical surface. The attribute information of the cylindrical surface includes the mating surface ID, the workpiece ID, the bottom radius, the height, the concavity and convexity, the coordinates of the axis endpoints, and the axis vector.

[0016] Steps 2-3: Extract the attribute information of the conical surface. The attribute information of the conical surface includes the mating surface ID, the workpiece ID, the bottom radius, the top radius, and the coordinates of the shaft endpoints.

[0017] Steps 2-4: Extract the attribute information of the sphere. The attribute information of the sphere includes the mating surface ID, the workpiece ID, the coordinates of the center point of the sphere, and the radius information.

[0018] Steps 2-5: Extract the attribute information of the ring coil surface. The attribute information of the ring coil surface includes the mating surface ID, the workpiece ID, the inner and outer radii, the center point, the axis vector, and the normal vector information.

[0019] Steps 2-6: Extract the attribute information of the annular plane. The attribute information of the annular plane includes the mating surface ID, the workpiece ID to which it belongs, the inner and outer radii, the center point, and the normal vector attribute.

[0020] Steps 2-7: Extract the attribute information of the polygonal plane. The attribute information of the polygonal plane includes the mating surface ID, the workpiece ID, the normal vector, and the constituent edge IDs, including the inner edge ID, outer edge ID, and total ID information. The edge ID, axis endpoint coordinates, and direction vector information are required for each edge.

[0021] Furthermore, the geometric information extraction based on the STEP file of the workpiece design model in step one includes:

[0022] Step 1-1: Parse each entity of the workpiece from the STEP file, and then extract its corresponding surface information. Each surface includes its geometric shape attributes, and further parse the edges and endpoints connected to the surface.

[0023] Steps 1-2: All extracted geometric data will be stored in a JSON file according to a hierarchical structure to ensure data readability and structured management.

[0024] Furthermore, step three, which involves generating a complete geometric feature description using completion techniques, includes:

[0025] Step 3-1: For a cylindrical surface, the completion process includes extracting the axis endpoints, calculating the cylinder height, and determining the surface concavity / convexity. First, one of the axis endpoints is obtained, i.e., the center point of the given cylindrical surface. Then, the maximum control point of the B-spline curve in the direction of the mating surface axis vector is extracted as the other axis endpoint. The height of the cylindrical surface is determined by performing rule calculations using the two known axis endpoints. In addition, due to the special representation of the cylindrical surface, the cylinder generatrix needs to be segmented based on the rules and data patterns to extract and complete the information of the upper and lower bottom surfaces of the cylinder.

[0026] Step 3-2: For other mating surfaces, the missing axis endpoints, height, and normal vector attributes are also filled in as in Step 3-1.

[0027] Furthermore, the step 3-1 for determining the unevenness of the workpiece is achieved by judging the range of the angle between the normal vector N of the mating surface and the direction vector N1, as follows:

[0028] First, convert the points P, center point P0, and axis direction Axis_direction on the mating surface into NumPy arrays. Then, use the right-hand rule to calculate N using the cross product of the normal vector N1 and the axis vector Vec. e :

[0029] N e =N1×Vec;

[0030] Calculate the vector P from point P to center P0. Vec And obtain the tangent vector R through the cross product. Vec It can accurately describe the geometric properties of point P in the local coordinate system and is used in subsequent steps to determine the concavity and convexity of the cylindrical surface, as well as other geometric calculations:

[0031] P Vec =P-P0;

[0032] R Vec =P Vec ×Axis_direction;

[0033] Next, based on the properties of the cylindrical surface, the direction vector N2 is determined, and the final direction vector N is calculated using the cross product. The cylindrical surface can be either the inner or outer surface. The inner surface refers to the surface facing inwards, with its normal vector pointing inwards; the outer surface refers to the surface facing outwards, with its normal vector pointing outwards. When calculating N2, the tangent vector R is used. Vec Given the cylinder's axial vector Axis_direction, the direction vector perpendicular to these two vectors is calculated using the cross product formula. Then, the direction vector perpendicular to the cylinder's axial vector N is calculated. e The cross product of N and N2 yields the final normal vector N, which is geometrically perpendicular to N. e N2 and N2 effectively describe the normal relationship at point P, providing a basis for subsequent concavity and convexity determination:

[0034] N2=R Vec ×Axis_direction;

[0035] N = N e ×N2;

[0036] Then, the angle between N and N1 is calculated using the dot product and converted into degrees;

[0037] Finally, the concavity or convexity is determined based on the range of the included angle θ: if the included angle θ is between 0 and 90 degrees, it is considered a convex surface; if the included angle θ is between 90 and 180 degrees, it is considered a concave surface.

[0038]

[0039] Furthermore, in step four, since the workpiece is modeled or designed in its own local coordinate system, which is standardized and reflects local geometric features, the actual position and orientation of the workpiece in the entire product will change due to differences in the assembly coordinate system. Therefore, it is necessary to unify the workpiece coordinate system, that is, to calculate the rotation matrix from the workpiece to the product coordinate system to achieve a complete coordinate system transformation, providing a basis for subsequent accurate position determination. Specifically, this includes:

[0040] First, obtain the origin of the coordinate system and its direction vectors, including the X, Y, and Z axes. These direction vectors describe the spatial positioning of the workpiece within its own reference frame. To ensure that these direction vectors are unit vectors, thereby improving the accuracy of the transformation and maintaining mathematical orthogonality, these direction vectors are normalized. The normalization formula is as follows:

[0041]

[0042] Where v is the direction vector to be normalized, v' is the normalized unit vector, and ||v|| represents the magnitude of the vector, i.e., the Euclidean norm, which is calculated using the following formula:

[0043]

[0044] Using these normalized direction vectors, construct the rotation matrix of the local coordinate system:

[0045] R local =[X local Y local Z local ];

[0046] X local Y local Z local These are the normalized direction vectors of the local coordinate system along the X, Y, and Z axes, respectively. This rotation matrix describes the directional relationship between the local and global coordinate systems, thereby enabling the transformation from the local to the global coordinate system.

[0047] Next, we also need to construct the rotation matrix for the global coordinate system:

[0048] R global =[X global Y global Zglobal ];

[0049] X global Y global Z global These are the normalized direction vectors of the global coordinate system along the X, Y, and Z axes, respectively, obtained by rotating the global coordinate system using the rotation matrix R. global Multiply by the inverse of the rotation matrix of the local coordinate system The complete transformation matrix T is obtained:

[0050]

[0051] This transformation matrix is ​​used to transform points in the local coordinate system to the global coordinate system, and can accurately describe how the workpiece in the local coordinate system is positioned and oriented in the global coordinate system.

[0052] Furthermore, the method for calculating the relative positional relationship of the workpieces in step five is as follows:

[0053] Step 5-1: First, determine which parts are in contact. For parts that are determined to be in contact, further analyze the positional relationship of each surface to ensure the accuracy of the position determination.

[0054] Step 5-2: For contacting parts and each surface within the same part, determine their positional relationships in detail. Positional relationships include: perpendicular, parallel, coaxial separation, coaxial adjacency, coaxial partial overlap, inclusion, and being included, in order to establish an accurate geometric positional description and provide reliable data support for subsequent assembly operations.

[0055] Furthermore, the method for determining the contact between workpiece parts in step 5-1 is as follows:

[0056] After obtaining complete set attribute information, it is necessary to first determine whether they are the same part. The positional relationship of the same part can be determined directly using rules, while the positional relationship of different parts requires first determining whether they are in contact. First, calculate the vector between the center points of the two workpieces:

[0057]

[0058] in and These are the coordinates of the center points of workpiece 1 and workpiece 2, respectively.

[0059] Next, the vector is projected onto the axis where the workpiece is located. Calculate the distance along the workpiece axis:

[0060]

[0061] Let the heights of workpiece 1 and workpiece 2 be h1 and h2 respectively, and calculate the perpendicular distance from the center vector to the axis containing the workpiece:

[0062]

[0063] Combining the radii R1 and R2 of the mating surfaces of workpiece 1 and workpiece 2, and the heights h1 and h2 of workpiece 1 and workpiece 2, the maximum allowable distance is obtained:

[0064] Based on this, if d Per1 ≤d max1 d Per2 ≤d max2 d Axis1 <d max3 If R1 and R2 are equal, then it is determined that the workpieces are in contact.

[0065] Furthermore, in step 5-2, when determining the spatial relationship between workpieces, corresponding calculation rules are set to ultimately determine the positional relationship. The positional relationship includes perpendicularity, parallelism, coaxial separation, coaxial adjacency, coaxial partial overlap, inclusion, and being included, specifically:

[0066] 5-2-1. Determining Vertical Relationships:

[0067] When detecting the perpendicular relationships between surfaces, different vector calculation methods should be used depending on the geometric shape. Specifically, based on the dot product relationship of the axis vectors or normal vectors of the geometric object, the perpendicular relationship can be accurately determined. The following describes three cases for determining the perpendicular relationship:

[0068] Between two 3D solids: When both geometric solids are 3D solids, they can be distinguished by their respective axis vectors. Let the axis vectors of 3D solid A and 3D solid B be respectively... and The dot product of the two is defined as If |d A If |≈0, then the two geometric bodies are determined to be perpendicular to each other;

[0069] Between a 3D solid and a 2D plane: When one geometric object is a 3D solid and the other is a 2D plane, the perpendicular relationship is determined using the axis vector of the 3D solid and the normal vector of the 2D plane. Let the axis vector of the 3D solid be... The normal vector of the two-dimensional plane is The dot product of the two is defined as If |d|≈1, then the three-dimensional solid and the two-dimensional plane are determined to be perpendicular to each other;

[0070] Between two two-dimensional planes: When both geometric objects are two-dimensional planes, their perpendicular relationship is determined using their respective normal vectors. Let the normal vectors of plane A and plane B be... and Then its dot product is defined as If |d n If |≈0, then the two planes are determined to be perpendicular.

[0071] 5-2-2 Determining Parallel and Coaxial Relationships:

[0072] When detecting the parallelism between surfaces, different vector calculation methods should be used depending on the geometric shape. Similarly, the following describes three cases for determining parallelism:

[0073] Between two 3D solids: For the axis vector relationship between two 3D solids, first perform a cross product operation on the axis vectors of the two objects. If the cross product result is a zero vector, then... If the axis vectors of the two objects are parallel, then they are determined to be non-parallel; otherwise, they are determined to be non-parallel.

[0074] Between a 3D solid and a 2D plane: If one object is a 3D solid and the other is a 2D plane, calculate the dot product between the axis vector of the 3D solid and the normal vector of the plane. If the dot product... If the axis vector of the 3D solid is perpendicular to the plane's normal vector, it indicates that the 3D solid is parallel to the 2D plane; otherwise, it is determined that they are not parallel.

[0075] Between two two-dimensional planes: If both objects are two-dimensional planes, the determination is made by calculating the cross product of the normal vectors of the two planes. If the cross product... If the two planes are parallel, then they are considered parallel; otherwise, they are considered non-parallel.

[0076] After determining that they are parallel, it is also necessary to determine whether the two surfaces are coaxial, and to determine the perpendicular distance L of the projection of the vector connecting the center points onto the vector of axis R1 or R2. h Whether it is 0 or not, if it is not 0, it is simply parallel; if it is 0, it is coaxial.

[0077] For coaxial cases, the following methods are used to determine intersection, containment, partial overlap, or adjacency relationships:

[0078] Calculate half the sum of the projected distance L from the line connecting the centers of the two objects to an axis and the object height. Relationship:

[0079] like Then the two objects are determined to be coaxially separated; if Then the two objects are determined to be adjacent; if Further determine whether L is less than half of the object height difference.

[0080] like Then the higher-level object contains the lower-level object; if The lower-level object is contained within the higher-level object;

[0081] A predetermined error tolerance is introduced in all judgment steps to account for small errors in numerical calculations and ensure the accuracy of positional relationship judgment.

[0082] Furthermore, when determining the spatial relationship between workpieces, for spheres in an assembly, which are mostly hemispherical structures, the axial vectors and normal vectors of a sphere lack clear physical meaning. Therefore, the geometric relationship is determined by comparing the center point of the hemisphere with the corresponding axis endpoints: if the center point is exactly equal to an axis endpoint, it is determined to be coaxially adjacent; if the center point is inconsistent with only one of the two axis endpoints and is far apart, it is determined to be coaxially separated; if the center point is inconsistent with only one of the two axis endpoints and the value of the center point is between the two axis endpoints, it is determined to be coaxially partially overlapping.

[0083] The technical solution adopted in this invention has the following beneficial effects:

[0084] This invention organically organizes and stores the geometric features and positional information of a 3D design model in a knowledge graph, enabling effective data association and convenient retrieval. The system obtains the relative positional relationships of workpieces through precise geometric feature recognition, thereby supporting the reasoning of assembly processes. In this process, the product design model is analyzed to identify its geometric features and the relative positional relationships of workpieces, extracting key assembly information. This invention aims to provide a method for workpiece geometric feature recognition and positional relationship judgment to improve the intelligence level of automated assembly processes. Attached Figure Description

[0085] Figure 1 This is the overall flowchart of the present invention;

[0086] Figure 2 Flowchart of geometric information extraction in this invention;

[0087] Figure 3 Diagram of the part body and its properties in this invention;

[0088] Figure 4 The geometric feature recognition result diagram of the present invention;

[0089] Figure 5 The specific parameter diagram of the mating surface extracted in this invention;

[0090] Figure 6 A detailed parameter diagram of the positional relationship in this invention;

[0091] Figure 7Flowchart for determining complete geometric attribute information of a workpiece according to the present invention;

[0092] Figure 8 A schematic diagram illustrating the coaxiality determination of geometric bodies according to the present invention. Detailed Implementation

[0093] To make the objectives, technical solutions, and effects of this invention clearer and more explicit, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0094] The following is in conjunction with the appendix Figure 1 - Appendix Figure 8 The specific embodiments of the present invention will be further described below. The following examples are only used to more clearly illustrate the technical solutions of the present invention, and should not be used to limit the scope of protection of the present invention.

[0095] This invention presents a framework method for accurately identifying workpiece geometric features and determining positional relationships. It proposes an improved geometric topology information extraction and global position analysis technique, sequentially performing geometric feature extraction, mating surface attribute analysis, and calculation of positional relationships between workpieces for complex workpiece structures. The entire process is as follows: Figure 1 As shown, the STEP file parsing of a typical workpiece design model is used as an example for illustration. The same applies to other models such as assembly models or structural component models.

[0096] This invention belongs to the field of computer-aided design and intelligent manufacturing, and relates to geometric information extraction and topology analysis methods. Specifically, it relates to a method for identifying workpiece geometric features and determining positional relationships in automated assembly processes, including geometric topology analysis, mating surface classification and attribute extraction, and calculation and determination of relative positional relationships between workpieces.

[0097] This invention first inputs the STEP file of the workpiece design model and parses its geometric topology information. Following the hierarchical structure, it progressively extracts the topology data of SHELL (solid), FACE (face), EDGE (line), and POINT (point) to describe the basic geometric features of the workpiece. Next, the extracted FACE geometric information is classified, identifying various mating surface types such as cylindrical, conical, multi-plane, spherical, and loop coils. Geometric attribute data is then completed based on the characteristics of each surface type, generating a complete geometric description of the workpiece. Regarding the assembly relationships between workpieces, this invention further unifies the local coordinate system of each workpiece to the global coordinate system, ensuring that different workpieces are analyzed within the same coordinate framework. By calculating relative positional relationships, it determines whether there are coaxial, adjacent, contained, or other spatial relationships between workpieces, thereby providing accurate data support for assembly process reasoning and path planning. This facilitates direct use in subsequent assembly simulation and digital twin optimization, effectively improving the efficiency of model application.

[0098] In one optional instance of the application, a method for workpiece geometric feature recognition and positional relationship determination is provided, including the following steps:

[0099] Step 1: Obtain the CAD design model of the workpiece. Based on the STEP file of the workpiece design model, extract geometric information. Extract the geometric topology information of entities, faces, lines, and points in sequence according to the preset hierarchical structure. This provides basic data for subsequent assembly process reasoning and analysis of the positional relationship between workpieces. Specifically, parse each shell of the workpiece from the STEP file, and then extract its corresponding face information. Each face includes its geometric shape attributes, and further parse the edges (LOOP / EDGE) and endpoints (POINT) connected to the face. All extracted geometric data will be stored in a JSON file according to the hierarchical structure to ensure data readability and structured management.

[0100] Step Two: During the extraction of mating surface attribute information from the workpiece design model, the surfaces extracted from the workpiece geometry are classified, and detailed identification and extraction are performed based on the specific geometric characteristics of each type of mating surface. The system automatically identifies various mating surfaces, including cylindrical surfaces, conical surfaces, spherical surfaces, annular coil surfaces, toroidal planes, and polygonal planes, using keyword recognition and attribute discrimination technology. Finally, this structured data is stored in a JSON file for further data processing and retrieval. The geometric information extraction process is as follows: Figure 2 Show.

[0101] Specifically, for different types of mating surfaces, corresponding geometric attributes are extracted and recorded. Different surfaces require different attribute information to form a systematic mating surface data structure. Specifically, the attribute information for a cylindrical surface includes the mating surface ID, the workpiece ID, the base radius, height, concavity / convexity, axis endpoint coordinates, and axis vector; the attribute information for a conical surface includes the mating surface ID, the workpiece ID, the base radius, the top radius, and axis endpoint coordinates; the attribute information for a spherical surface includes the mating surface ID, the workpiece ID, the center point coordinates, and radius information; the attribute information for a toroidal coil surface includes the mating surface ID, the workpiece ID, inner and outer radii, the center point, axis vector, and normal vector information; the attribute information for a toroidal plane includes the mating surface ID, the workpiece ID, inner and outer radii, the center point, and normal vector attributes; the attribute information for a polygonal plane includes the mating surface ID, the workpiece ID, the normal vector, and the component edge IDs, including inner edge IDs, outer edge IDs, and the overall ID. For each edge, edge ID, axis endpoint coordinates, and direction vector information are required. Specifically, as follows... Figure 5 Show.

[0102] Step 3: When the geometric attribute information of the mating surfaces of the workpiece design model is incomplete, a complete geometric feature description is generated through analysis and completion techniques to ensure the accuracy and completeness of the geometric information of the mating surfaces. The high-level characterization information of the mating surfaces is extracted, and the attributes are calculated and derived based on preset rules. The completed workpiece mating surface information is classified and stored in a JSON file to form a complete workpiece geometric description database.

[0103] The method for completing the geometric attribute information of different mating surfaces is as follows: For cylindrical surfaces, the completion process includes extracting the axis endpoints, calculating the cylinder height, and determining the surface concavity / convexity. First, one axis endpoint is obtained, i.e., the center point of the given cylindrical surface, and the maximum control point of the B-spline curve along its direction is extracted as the other axis endpoint based on the mating surface axis vector. The height of the cylindrical surface is then determined by rule-based calculation using the two known axis endpoints. In addition, for special representations of the cylinder, such as 10 circles or 10 lines, the cylinder generatrix needs to be segmented based on the rules and data patterns to extract and complete the information of the upper and lower base surfaces. Similar completion processing is performed for missing axis endpoints, height, and normal vector attributes in other mating surfaces.

[0104] Step three specifically includes:

[0105] Specifically, the concavity / convexity property can be determined by judging the range of the angle between the normal vector N and the direction vector N1 of the mating surface. The process is as follows: First, convert the point P, the center point P0, and the axis direction Axis_direction on the mating surface into NumPy arrays. Then, use the right-hand rule to calculate N using the cross product of the normal vector N1 and the axis vector Vec. e .

[0106] N e =N1×Vec

[0107] Calculate the vector P from point P to center P0. Vec And obtain the tangent vector R through the cross product. Vec It can accurately describe the geometric properties of point P in the local coordinate system and is used in subsequent steps to determine the concavity and convexity of the cylindrical surface, as well as other geometric calculations.

[0108] P Vec =P-P0

[0109] R Vec =P Vec ×Axis_direction

[0110] Next, based on the properties of the cylinder surface (inner or outer surface), the direction vector N2 is determined. The final direction vector N is then calculated using the cross product. The inner surface refers to the surface facing inwards from the cylinder, with its normal vector pointing inwards; the outer surface refers to the surface facing outwards from the cylinder, with its normal vector pointing outwards. The tangent vector R is used when calculating N2. Vec Given the cylinder's axial vector Axis_direction, the direction vector perpendicular to these two vectors is calculated using the cross product formula. Then, the normal vector N is used... e The cross product of N and N2 yields the final normal vector N. This vector is geometrically perpendicular to N. e N2 and N2 effectively describe the normal relationship at point P, providing a basis for subsequent concavity and convexity determination.

[0111] N2=R Vec ×Axis_direction

[0112] N = N e ×N2

[0113] Then, the angle between N and N1 is calculated using the dot product and converted into degrees. Finally, the concavity or convexity is determined based on the range of the angle: if the angle is between 0 and 90 degrees, it is considered a convex surface; if the angle is between 90 and 180 degrees, it is considered a concave surface.

[0114]

[0115] Finally, the geometric feature recognition results achieved using the method in step three are as follows: Figure 4 Show.

[0116] Step 4: When processing workpiece data in JSON files, since each workpiece usually has an independent local coordinate system, these local coordinate systems need to be standardized in order to perform unified spatial analysis and positional relationship judgment.

[0117] Step four specifically includes:

[0118] Since workpieces are typically modeled or designed in their own local coordinate system, which is standardized and reflects local geometric features, the actual position and orientation of the workpiece within the entire product can vary depending on the coordinate system of the assembly. Therefore, it is necessary to unify the workpiece coordinate system. This involves calculating the rotation matrix from the workpiece to the product coordinate system to achieve a complete coordinate system transformation, providing a foundation for accurate subsequent position determination. Specifically, it is first necessary to obtain the origin and its direction vectors (X-axis, Y-axis, and Z-axis) of this coordinate system. These direction vectors describe the spatial positioning of the workpiece in its own reference frame. To ensure that these direction vectors are unit vectors, thereby improving the accuracy of the transformation and maintaining mathematical orthogonality, they need to be normalized. The normalization formula is:

[0119]

[0120] Where v is the direction vector to be normalized, v′ is the normalized unit vector, and ||v|| represents the magnitude of the vector (i.e., the Euclidean norm), calculated using the following formula:

[0121]

[0122] Using these normalized direction vectors (X-axis, Y-axis, and Z-axis), a rotation matrix R is constructed for the local coordinate system. local =[X local Y local Z local The rotation matrix describes the directional relationship between the local and global coordinate systems, thus enabling the transformation from the local to the global coordinate system. Next, we also need to construct the rotation matrix for the global coordinate system, R. global =[X global Y global Z globa Then, by multiplying the rotation matrix of the global coordinate system by the inverse of the rotation matrix of the local coordinate system, the complete transformation matrix T is obtained.

[0123]

[0124] This transformation matrix is ​​used to transform points in the local coordinate system to the global coordinate system, and can accurately describe how the workpiece in the local coordinate system is positioned and oriented in the global coordinate system.

[0125] Step 5: After completing the transformation between the local and global coordinate systems, the relative positional relationships between different workpieces can be calculated based on the unified global coordinate information. These relative positional relationships include various geometric relationships such as perpendicular, parallel, coaxial adjacency, coaxial separation, coaxial partial overlap, inclusion, and being included.

[0126] Step five specifically includes:

[0127] The part body and attribute information diagram of the present invention are shown below. Figure 3 The process begins by determining which parts are in contact. For parts identified as in contact, we further analyze the positional relationships of their various surfaces to ensure the accuracy of the positional determination. For both contacting parts and individual surfaces within the same part, we meticulously determine their positional relationships. These relationships include: perpendicularity, parallelism, coaxial separation, coaxial adjacency, coaxial partial overlap, inclusion, and being included, to establish a precise geometrical positional description and provide reliable data support for subsequent assembly operations. The flowchart for determining the complete geometrical attribute information of the workpiece is shown below. Figure 7 The specific methods include the following:

[0128] The method for determining contact between workpiece parts is as follows:

[0129] After obtaining complete set attribute information, it is necessary to first determine whether they are the same part. The positional relationship of the same part can be determined directly using rules, while the positional relationship of different parts requires first determining whether they are in contact. First, the vector between the center points of the two workpieces is calculated.

[0130]

[0131] in and These are the coordinates of the center points of workpiece 1 and workpiece 2, respectively. Next, this vector is projected onto the axis along which the workpiece is located. Calculate the distance along the workpiece axis.

[0132]

[0133] Let the heights of workpiece 1 and workpiece 2 be h1 and h2 respectively, and calculate the perpendicular distance from the center vector to the axis containing the workpiece.

[0134]

[0135] By combining the workpiece mating surface radii R1 and R2, and the workpiece heights h1 and h2, the maximum allowable distance can be obtained. Based on this, if d Per1 ≤d max1 d Per2 ≤d max2 d Axis1 <d max3 If R1 and R2 are equal, then the workpieces are considered to be in contact.

[0136] The method for determining the positional relationship between workpiece parts is as follows:

[0137] When determining the spatial relationship between workpieces, corresponding calculation rules are set to ultimately determine the positional relationship. The positional relationship includes perpendicularity, parallelism, coaxial separation, coaxial adjacency, coaxial partial overlap, inclusion, and being included, as well as the rules for determining the relative positional relationship of workpieces.

[0138] 1. Determining vertical relationships

[0139] When detecting the perpendicular relationships between surfaces, different vector calculation methods should be used depending on the geometric shape. Specifically, the dot product relationship of the axis vectors or normal vectors of the geometric object can achieve accurate determination of perpendicular relationships. The following describes three cases for determining perpendicular relationships:

[0140] 1) Between two 3D solids: When both geometric solids are 3D solids, they can be distinguished by their respective axis vectors. Let the axis vectors of 3D solid A and 3D solid B be respectively... and The dot product of the two is defined as If |d A If |≈0, then the two geometric bodies are determined to be perpendicular to each other.

[0141] 2) Between a 3D solid and a 2D plane: When one geometric solid is a 3D solid and the other is a 2D plane (such as a polygonal plane), the perpendicular relationship is determined using the axis vector of the 3D solid and the normal vector of the 2D plane. Let the axis vector of the 3D solid be... The normal vector of the two-dimensional plane is The dot product of the two is defined as If |d|≈1, then the three-dimensional solid and the two-dimensional plane are determined to be perpendicular to each other.

[0142] 3) Between two two-dimensional planes: When both geometric objects are two-dimensional planes, their perpendicular relationship is determined using their respective normal vectors. Let the normal vectors of plane A and plane B be... and Then its dot product is defined as If |d n If |≈0, then the two planes are determined to be perpendicular.

[0143] 2. Determining Parallel and Coaxial Relationships

[0144] When detecting the parallelism between surfaces, different vector calculation methods should be used depending on the geometric shape. Similarly, the following describes three cases for determining parallelism:

[0145] 1) Between two 3D solids: For the axial vector relationship between two 3D solids, first perform a cross product operation on the axial vectors of the two objects. If the cross product result is a zero vector, that is... If the axis vectors of the two objects are parallel, then they are determined to be non-parallel; otherwise, they are determined to be non-parallel.

[0146] 2) Between a 3D solid and a 2D plane: If one object is a 3D solid and the other is a 2D plane, calculate the dot product between the axis vector of the 3D solid and the normal vector of the plane. If the dot product... If the axis vector of the 3D solid is perpendicular to the plane's normal vector, it indicates that the 3D solid is parallel to the 2D plane; otherwise, it is determined that they are not parallel.

[0147] 3) Between two two-dimensional planes: If both objects are two-dimensional planes, the determination is made by calculating the cross product of the normal vectors of the two planes. If the cross product... If the two planes are parallel, then they are considered parallel; otherwise, they are considered non-parallel.

[0148] After parallelization, it is necessary to determine whether the two surfaces are coaxial, and to determine the perpendicular distance L of the projection of the vector connecting the center points onto the vector of axis R1 or R2. h Whether it is 0 or not, it is simply parallel, such as... Figure 8 As shown in A; when it is zero, it is coaxial, as shown in... Figure 8 As shown in B, for the coaxial case, the intersection, containment, partial overlap, or adjacency relationship is determined in the following way:

[0149] Calculate half the sum of the projected distance L from the line connecting the centers of the two objects to an axis and the object height. Relationship:

[0150] like Then the two objects are determined to be coaxially separated; if Then the two objects are determined to be adjacent; if Further determine whether L is less than half of the object height difference. like Then the higher-level object contains the lower-level object; if The lower-order object is then contained within the higher-order object. A predetermined error tolerance is introduced in all decision-making steps to account for minor errors in numerical calculations, ensuring the accuracy of positional relationship determination.

[0151] Specifically, for spheres in an assembly, which are often hemispherical structures, the axis vectors and normal vectors of a sphere usually lack clear physical meaning. Therefore, the geometric relationship is determined by comparing the center point of the hemisphere with the corresponding axis endpoints: if the center point is exactly equal to an axis endpoint, it is considered coaxially adjacent; if the center point is not aligned with only one axis endpoint and the two axes endpoints are far apart, it is considered coaxially separated; if the center point is not aligned with only one axis endpoint and the value of the center point lies between the two axis endpoints, it is considered coaxially partially overlapping. The results of the positional relationship determination are as follows: Figure 6 As shown, the first row and first column are the ID numbers of all faces, and each positional relationship is the relationship between its corresponding row ID and column ID.

[0152] The beneficial technical effects of the present invention include at least the following:

[0153] This invention presents a framework method for accurately identifying workpiece geometric features and determining positional relationships. It proposes a method for workpiece geometric feature identification and positional relationship determination based on geometric topology information analysis and mating surface attribute analysis. By parsing the STEP file of the workpiece design model, geometric topology information of SHELL (solid), FACE (face), EDGE (line), and POINT (point) is extracted layer by layer, and then classified and attribute-completed to solve the problem of identifying complex workpiece geometric features and determining positional relationships. This invention also combines unified processing of local and global coordinate systems to ensure accurate analysis of spatial positional relationships between workpieces.

[0154] The main improvement of this invention lies in the accurate extraction of workpiece geometric information and the method for determining positional relationships. Based on existing geometric analysis and topology analysis techniques, it combines mating surface classification and attribute completion techniques to unify the processing methods for local and global coordinate systems. By improving and optimizing the existing process, this invention proposes a new framework capable of efficiently completing geometric feature recognition and positional relationship determination. Experimental verification shows that the method provided in this application exhibits excellent performance in terms of the accuracy of geometric information extraction and the reliability of positional relationship determination, providing efficient data support for assembly process planning and path optimization.

[0155] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the solutions disclosed herein. This invention is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the claims.

Claims

1. A method for identifying workpiece geometric features and determining positional relationships in an automated assembly process, characterized in that, The method includes the following steps: Step 1: Obtain the CAD design model of the workpiece, extract geometric information based on the STEP file of the workpiece design model, and extract the geometric topology information of entities, surfaces, lines and points in sequence according to the preset hierarchical structure to provide basic data for subsequent assembly process reasoning and analysis of positional relationships between workpieces; Step 2: During the extraction of mating surface attribute information from the workpiece design model, the surfaces extracted from the workpiece geometry information are classified, and detailed identification and extraction are performed based on the specific geometric features of each type of mating surface. The system automatically identifies various mating surfaces, including cylindrical surfaces, conical surfaces, spherical surfaces, annular coil surfaces, circular planes, and polygonal planes, through keyword recognition and attribute discrimination technology. Finally, this structured data will be stored in a JSON file for further data processing and retrieval. Step 3: When the geometric attribute information of the mating surfaces of the workpiece design model is incomplete, a complete geometric feature description is generated through analysis and completion techniques to ensure the accuracy and completeness of the geometric information of the mating surfaces. The high-level characterization information of the mating surfaces is extracted, and the attributes are calculated and derived based on preset rules. The completed workpiece mating surface information is classified and stored in a JSON file to form a complete workpiece geometric description database. Step 4: When processing the workpiece data in the JSON file, since each workpiece has an independent local coordinate system, these local coordinate systems are standardized to uniformly perform spatial analysis and positional relationship judgment. Step 5: After completing the transformation between the local and global coordinate systems, the relative positional relationships between different workpieces are calculated based on the unified global coordinate information, and the positional relationships are judged. These relative positional relationships include perpendicular, parallel, coaxial adjacency, coaxial separation, coaxial partial overlap, inclusion, and being included. The method for identifying the geometric properties of the mating surfaces in step two is as follows: Step 2-1: Extract and record the corresponding geometric attributes for different types of mating surfaces. Different surfaces require different attribute information to form a systematic mating surface data structure. Step 2-2: Extract the attribute information of the cylindrical surface. The attribute information of the cylindrical surface includes the mating surface ID, the workpiece ID, the bottom radius, the height, the concavity and convexity, the coordinates of the axis endpoints, and the axis vector. Steps 2-3: Extract the attribute information of the conical surface. The attribute information of the conical surface includes the mating surface ID, the workpiece ID, the bottom radius, the top radius, and the coordinates of the shaft endpoints. Steps 2-4: Extract the attribute information of the sphere. The attribute information of the sphere includes the mating surface ID, the workpiece ID, the coordinates of the center point of the sphere, and the radius information. Steps 2-5: Extract the attribute information of the ring coil surface. The attribute information of the ring coil surface includes the mating surface ID, the workpiece ID, the inner and outer radii, the center point, the axis vector, and the normal vector information. Steps 2-6: Extract the attribute information of the annular plane. The attribute information of the annular plane includes the mating surface ID, the workpiece ID to which it belongs, the inner and outer radii, the center point, and the normal vector attribute. Steps 2-7: Extract the attribute information of the polygonal plane. The attribute information of the polygonal plane includes the mating surface ID, the workpiece ID, the normal vector, and the constituent edge IDs, including the inner edge ID, outer edge ID, and total ID information. The edge ID, axis endpoint coordinates, and direction vector information are required for each edge.

2. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 1, characterized in that, The geometric information extraction based on the STEP file of the workpiece design model in step one includes: Step 1-1: Parse each entity of the workpiece from the STEP file, and then extract its corresponding surface information. Each surface includes its geometric shape attributes, and further parse the edges and endpoints connected to the surface. Steps 1-2: All extracted geometric data will be stored in a JSON file according to a hierarchical structure to ensure data readability and structured management.

3. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 1, characterized in that, Step three, which involves generating a complete geometric feature description using completion techniques, includes: Step 3-1: For a cylindrical surface, the completion process includes extracting the axis endpoints, calculating the cylinder height, and determining the surface concavity / convexity. First, one of the axis endpoints is obtained, i.e., the center point of the given cylindrical surface. Then, the maximum control point of the B-spline curve in the direction of the mating surface axis vector is extracted as the other axis endpoint. The height of the cylindrical surface is determined by performing rule calculations using the two known axis endpoints. In addition, due to the special representation of the cylindrical surface, the cylinder generatrix needs to be segmented based on the rules and data patterns to extract and complete the information of the upper and lower bottom surfaces of the cylinder. Step 3-2: For other mating surfaces, the missing axis endpoints, height, and normal vector attributes are also filled in as in Step 3-1.

4. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 3, characterized in that, The step 3-1 for determining the unevenness of the workpiece involves judging the relationship between the normal vector N of the mating surface and the direction vector. This is achieved by controlling the range of the included angle, and the process is as follows: First, point P and center point on the mating surface. The axis direction (Axis_direction) is converted to a NumPy array, and then the right-hand rule is used to transform the normal vector. The cross product of the vector Vec and the axis vector is calculated. : ； Calculate from point P to the center vector And obtain the tangent vector through the cross product. It can accurately describe the geometric properties of point P in the local coordinate system and is used in subsequent steps to determine the concavity and convexity of the cylindrical surface, as well as other geometric calculations: ； ； Next, based on the properties of the cylindrical surface, determine the direction vector. The final direction vector N is calculated using the cross product. The cylindrical surface can be either the inner or outer surface. The inner surface refers to the surface facing inwards, with its normal vector pointing inwards; the outer surface refers to the surface facing outwards, with its normal vector pointing outwards. In the calculation... When using tangent vectors Given the cylinder's axial vector Axis_direction, the direction vector perpendicular to these two vectors is calculated using the cross product formula. Then, the normal vector is used... and The cross product yields the final normal vector N, which is geometrically perpendicular to the x-axis. and This effectively describes the normal relationship at point P, providing a foundation for subsequent concavity / convexity determination: ； ； Then, N and N are calculated using the dot product. The included angle between them, and converted to degrees; Finally, based on the included angle Determine the concavity / convexity within the range: if the included angle If the angle is between 0 and 90 degrees, it is represented as a convex surface; if the included angle is... A value between 90 and 180 degrees is considered a concave surface. 。 5. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 1, characterized in that, In step four, since the workpiece is modeled or designed in its own local coordinate system, which is standardized and reflects local geometric features, the actual position and orientation of the workpiece in the entire product will change due to differences in the coordinate system of the assembly. Therefore, it is necessary to unify the workpiece coordinate system, that is, to calculate the rotation matrix from the workpiece to the product coordinate system to achieve a complete coordinate system transformation, providing a basis for subsequent accurate position determination. Specifically, this includes: First, obtain the origin of the coordinate system and its direction vectors, including the X, Y, and Z axes. These direction vectors describe the spatial positioning of the workpiece within its own reference frame. To ensure that these direction vectors are unit vectors, thereby improving the accuracy of the transformation and maintaining mathematical orthogonality, these direction vectors are normalized. The normalization formula is as follows: ； Where v is the direction vector to be normalized. It is the normalized unit vector. The magnitude of a vector, also known as the Euclidean norm, is calculated using the following formula: ； Using these normalized direction vectors, construct the rotation matrix of the local coordinate system: ； These are the normalized direction vectors of the local coordinate system along the X, Y, and Z axes, respectively. This rotation matrix describes the directional relationship between the local and global coordinate systems, thereby enabling the transformation from the local to the global coordinate system. Next, we also need to construct the rotation matrix for the global coordinate system: ； These are the normalized direction vectors of the global coordinate system along the X, Y, and Z axes, respectively, obtained by rotating the global coordinate system. Multiply by the inverse of the rotation matrix of the local coordinate system The complete transformation matrix T is obtained: ； This transformation matrix is ​​used to transform points in the local coordinate system to the global coordinate system, and can accurately describe how the workpiece in the local coordinate system is positioned and oriented in the global coordinate system.

6. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 1, characterized in that, The method for calculating the relative positional relationship of the workpieces in step five is as follows: Step 5-1: First, determine which parts are in contact. For parts that are determined to be in contact, further analyze the positional relationship of each surface to ensure the accuracy of the position determination. Step 5-2: For contacting parts and each surface within the same part, determine their positional relationships in detail. Positional relationships include: perpendicular, parallel, coaxial separation, coaxial adjacency, coaxial partial overlap, inclusion, and being included, in order to establish an accurate geometric positional description and provide reliable data support for subsequent assembly operations.

7. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 6, characterized in that, The method for determining the contact between workpiece parts in step 5-1 is as follows: After obtaining complete set attribute information, it is necessary to first determine whether they are the same part. The positional relationship of the same part can be determined directly using rules, while the positional relationship of different parts requires first determining whether they are in contact. First, calculate the vector between the center points of the two workpieces: ； in and These are the coordinates of the center points of workpiece 1 and workpiece 2, respectively. Next, the vector is projected onto the axis where the workpiece is located. Calculate the distance along the workpiece axis: ， ； Let the heights of workpiece 1 and workpiece 2 be respectively and Calculate the perpendicular distance from the center vector to the axis containing the workpiece: ， ； Radius of the mating surfaces of workpiece 1 and workpiece 2 and and the height of workpiece 1 and workpiece 2 and To obtain the maximum allowed distance: , On this basis, if , , And exist and If they are equal, it is determined that the workpieces are in contact.

8. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 6, characterized in that, In step 5-2, when determining the spatial relationship between workpieces, corresponding calculation rules are set to ultimately determine the positional relationship. The positional relationship includes perpendicularity, parallelism, coaxial separation, coaxial adjacency, coaxial partial overlap, inclusion, and being included, specifically: 5-2-1. Determining Vertical Relationships: When detecting the perpendicular relationships between surfaces, different vector calculation methods should be used depending on the geometric shape. Specifically, based on the dot product relationship of the axis vectors or normal vectors of the geometric object, the perpendicular relationship can be accurately determined. The following describes three cases for determining the perpendicular relationship: Between two 3D solids: When both geometric solids are 3D solids, they can be distinguished by their respective axis vectors. Let the axis vectors of 3D solid A and 3D solid B be respectively... and The dot product of the two is defined as ,like If the two geometric bodies are perpendicular, then they are determined to be perpendicular to each other. Between a 3D solid and a 2D plane: When one geometric object is a 3D solid and the other is a 2D plane, the perpendicular relationship is determined using the axis vector of the 3D solid and the normal vector of the 2D plane. Let the axis vector of the 3D solid be... The normal vector of the two-dimensional plane is The dot product of the two is defined as ,like If so, then the three-dimensional solid and the two-dimensional plane are determined to be perpendicular to each other; Between two two-dimensional planes: When both geometric objects are two-dimensional planes, their perpendicular relationship is determined using their respective normal vectors. Let the normal vectors of plane A and plane B be... and Then its dot product is defined as ,like If the two planes are perpendicular, then they are determined to be perpendicular to each other. 5-2-2 Determining Parallel and Coaxial Relationships: When detecting the parallelism between surfaces, different vector calculation methods should be used depending on the geometric shape. Similarly, the following describes three cases for determining parallelism: Between two 3D solids: For the axis vector relationship between two 3D solids, first perform a cross product operation on the axis vectors of the two objects. If the cross product result is a zero vector, then... If the axial vectors of the two objects are parallel, then they are determined to be non-parallel; otherwise, they are determined to be non-parallel. Between a 3D solid and a 2D plane: If one object is a 3D solid and the other is a 2D plane, calculate the dot product between the axis vector of the 3D solid and the normal vector of the plane. If the dot product... If the axis vector of the 3D solid is perpendicular to the plane normal vector, it indicates that the 3D solid is parallel to the 2D plane; otherwise, it is determined that the two-dimensional solid is not parallel. Between two two-dimensional planes: If both objects are two-dimensional planes, the determination is made by calculating the cross product of the normal vectors of the two planes. If the cross product... If the two planes are parallel, then they are considered parallel; otherwise, they are considered non-parallel. After determining that they are parallel, it is also necessary to determine whether the two surfaces are coaxial, and to determine the vector of the line connecting the center points to the planes. or perpendicular distance of the projection onto the axial vector Whether it is 0 or not, if it is not 0, it is simply parallel; if it is 0, it is coaxial. For coaxial cases, the following methods are used to determine intersection, containment, partial overlap, or adjacency relationships: Calculate the projected distance of the line connecting the centers of two objects to an axis. Half the sum of the object's height Relationship: like If the two objects are coaxially separated, then the two objects are determined to be coaxially separated; if If the two objects are adjacent, then they are considered to be adjacent; if Further judgment Is it less than half the height difference of the objects? : like If the higher-level object contains the lower-level object; In this case, the lower-level object is contained within the higher-level object; A predetermined error tolerance is introduced in all judgment steps to account for small errors in numerical calculations and ensure the accuracy of positional relationship judgment.

9. The method for workpiece geometric feature identification and positional relationship determination in automated assembly process according to claim 8, characterized in that, When determining the spatial relationship between workpieces, for spheres in an assembly, which are mostly hemispherical structures, the axis vectors and normal vectors of a sphere lack clear physical meaning. Therefore, the geometric relationship is determined by comparing the center point of the hemisphere with the corresponding axis endpoints: if the center point is exactly equal to an axis endpoint, it is determined to be coaxially adjacent; if the center point is inconsistent with only one of the two axis endpoints and is far apart, it is determined to be coaxially separated; if the center point is inconsistent with only one of the two axis endpoints and the value of the center point is between the two axis endpoints, it is determined to be coaxially partially overlapping.

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