Non-standard rotary body structural member modeling method based on laser scanning point cloud

By using hand-eye calibration and ICP algorithm alignment and registration processing, the problem of high-precision reverse modeling of non-standard rotating body structural parts was solved, and high-precision point cloud data stitching was achieved, meeting the automated operation requirements of non-standard rotating body structural parts.

CN121982195APending Publication Date: 2026-05-05DONGFANG ELECTRIC GROUP DIGITAL TECHNOLOGY CO LTD
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
DONGFANG ELECTRIC GROUP DIGITAL TECHNOLOGY CO LTD
Filing Date
2025-12-24
Publication Date
2026-05-05

AI Technical Summary

Technical Problem

Existing technologies make it difficult to achieve high-precision reverse modeling of non-standard rotating structural components, resulting in missing or inaccurate point cloud data, which affects the accuracy of robotic arm automated operations.

Method used

By integrating the 3D camera coordinate system with the working world coordinate system of the robotic arm through hand-eye calibration processing, and calibrating the rotation axis of the positioner with the 3D camera coordinate system, and combining the ICP algorithm to align, register and deduplicate adjacent two sets of point cloud data, high-precision stitching of point cloud data is achieved.

Benefits of technology

It has achieved high-precision surface splicing modeling of non-standard rotating structural parts, which has improved the accuracy of robotic arms in automated operations such as spraying, grinding, and welding.

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Abstract

The invention relates to the technical field of reverse modeling, and particularly discloses a non-standard rotary body structural member modeling method based on laser scanning point clouds, which comprises the following steps: S1, clamping a workpiece on a positioner; the 3D camera is clamped on a mechanical arm and calibrated; s2, calibrating a camera and a positioner at a photographing teaching position; s3, scanning the workpiece, and collecting point cloud data of a corresponding surface area; s4, sequentially rotating the position changing machine according to the set angle and the surface continuity; sequentially collecting point cloud data of the corresponding surface areas; s5, performing alignment and registration processing on two adjacent groups of point cloud data collected in sequence; s6, performing de-weighting optimization processing and fitting on the aligned and registered point cloud data; and S7, obtaining a complete point cloud of the workpiece, and completing modeling. The method is designed aiming at the particularity of the reverse modeling technology and the non-standard rotary body structural part, and the high-precision splicing modeling technology requirement of the non-standard rotary body structural part can be effectively met.
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Description

Technical Field

[0001] This invention relates to the field of reverse modeling technology, specifically a method for modeling non-standard rotating structural components based on laser scanning point clouds. Background Technology

[0002] With the development of automation and intelligent technologies, by modeling and designing workpieces, and then using trajectory point automatic path planning algorithms, robotic arms can perform automatic operations, which have wide applications in fields such as spraying, grinding, and welding.

[0003] Modeling of workpieces can be divided into reverse modeling and forward modeling. Forward modeling is the process from blueprint design to finished product production, while reverse modeling is the process of reconstructing a mathematical model of an existing physical object, and is a process of redesigning and recreating an existing product.

[0004] Currently, reverse modeling technology is relatively mature for fixed-posture or standardized structural components, enabling high-precision surface stitching modeling. However, for non-standard rotating structural components, due to their irregular shapes and variable dimensions, high-precision surface stitching in reverse modeling faces numerous technical challenges. These challenges include the fact that traditional modeling methods often suffer from missing or inaccurate point cloud data due to mechanical system errors, which in turn affects the accuracy of subsequent reverse modeling and automated robotic arm operations.

[0005] Therefore, developing a high-precision splicing modeling method that can effectively handle non-standard rotating structural components is of great practical significance. Summary of the Invention

[0006] The technical objective of this invention is to provide a high-precision splicing modeling method based on laser scanning point clouds that can effectively address the special characteristics of non-standard rotating body structural components, as well as the shortcomings of existing technologies, in light of the aforementioned reverse modeling techniques and the limitations of existing technologies.

[0007] The technical objective of this invention is achieved through the following technical solution: a method for modeling non-standard rotating structural components based on laser scanning point clouds, the modeling method comprising: S1. Clamp the workpiece on the positioner; The 3D camera is mounted on the robotic arm and hand-eye calibration is performed. S2. Move the robotic arm to the set photo teaching position; The axis of rotation of the positioner is calibrated relative to the coordinate system of the 3D camera; S3. Scan the workpiece clamped by the positioner at its current position and collect point cloud data of the corresponding surface area; S4. Rotate the positioner sequentially according to the set rotation angle and the continuity of the workpiece surface; After the current rotation is completed, the workpiece clamped by the positioner is scanned at the current position to collect point cloud data of the corresponding surface area. Continue collecting point cloud data for all areas of the workpiece surface that need to be collected, according to the areas that need to be collected on the workpiece surface. S5. Align and register adjacent sets of point cloud data according to the order of point cloud data collected in the workpiece rotation sequence. S6. Perform deduplication and optimization on the aligned and registered point cloud data, and fit the overlapping parts; S7. Continue until each group of point cloud data is aligned, registered, deduplicated, and fitted in sequence to obtain the complete point cloud of the workpiece and complete the modeling.

[0008] Furthermore, in S1, the hand-eye calibration process adopts the method of "eye on hand" to achieve the fusion of the 3D camera coordinate system and the world coordinate system of the robotic arm's operation.

[0009] Furthermore, the hand-eye calibration process satisfies the following relationship: AX = XB; In the formula, A is the homogeneous spatial transformation matrix of the 3D camera during two adjacent motions; B is the homogeneous transformation matrix of the end-effector coordinate system during two consecutive movements; X is the hand-eye matrix to be solved; The process of solving for A satisfies the following relationship: A1=H c2 *H c1 -1 ; A2=H c3 *H c2 -1 ; In the formula, A1 is the spatial transformation homogeneous matrix of the 3D camera during its first motion; A2 is the homogeneous spatial transformation matrix of the 3D camera during its second motion; H c1 H c2 H c3 These are the extrinsic parameter matrices of the calibration images used in the calibration of the 3D camera; The process of solving for B satisfies the following relationship: B1 = Hc2-1 * Hc1; B2 = Hc3 - 1 * Hc2; B1 is the homogeneous transformation matrix of the robot arm's end-effector coordinate system during the first motion; B1 is the homogeneous transformation matrix of the robot arm's end-effector coordinate system during the second motion; Hg1 is the calibration image H c1 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system; Hg2 is the calibration image H c2 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system; Hg3 is the calibration image H c3 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system.

[0010] Furthermore, in S2, the photographic teaching position is directly above the workpiece, and the 3D camera scans the workpiece in a downward direction perpendicular to the ground to capture the image.

[0011] Furthermore, in S2, the calibration of the positioner's rotation axis centerline and the 3D camera coordinate system is performed by teaching and calibrating the positioner's rotation axis centerline based on the 3D camera coordinate system.

[0012] Furthermore, the teaching calibration of the rotation axis of the positioner satisfies the following relationship: = ; In the formula, This is the transformation matrix for the rotation axis of the positioner; The axis of the coordinate system The normal vector of the direction; The axis of the coordinate system The normal vector of the direction; The axis of the coordinate system The normal vector of the direction; K=1- ; M= + + .

[0013] Furthermore, in S4, the rotation angle of the positioner is 90°.

[0014] Furthermore, in S3 and S4, the collected point cloud data undergoes preprocessing to remove invalid points, set ROI regions, remove outliers, and estimate normal vectors.

[0015] Furthermore, in S5, the alignment and registration process for two adjacent sets of point cloud data is performed by registering the overlapping part of the two sets of point cloud data. The overlapping part is obtained by the distance between the two sets of point cloud data. If a point in the first set of point cloud data exists within the neighborhood of a point in the second set of point cloud data, it is considered as the overlapping part. The two sets of point cloud data in the overlapping part are registered using the ICP algorithm to obtain the transformation matrix; The transformation matrix is ​​applied to two adjacent sets of complete point cloud data, resulting in a second matching of these two sets of point cloud data after rotation.

[0016] Furthermore, the ICP algorithm performs registration and obtains the transformation matrix, which satisfies the following relationship: ; In the formula, The transformation matrix represents the overlapping portion of two adjacent sets of point cloud data. The last index of the points in the overlapping point cloud data; The starting sequence number of the points in the overlapping point cloud data; It is a rotation matrix; It is a translation matrix; The i-th point in the first set of point cloud data; Let i be the i-th point in the second set of point cloud data.

[0017] The beneficial technical effects of this invention are as follows: The above-mentioned technical measures, addressing the specific characteristics of reverse modeling technology and non-standard rotating structural parts, achieve high-precision surface splicing modeling of non-standard rotating structural parts by performing hand-eye calibration between the 3D camera and the robot's robotic arm, calibrating the axis of rotation of the positioner with the coordinate system of the 3D camera, and aligning and registering adjacent sets of point cloud data in the workpiece rotation sequence. Therefore, based on laser scanning point cloud technology, this enables high-precision surface splicing modeling of non-standard rotating structural parts. It is evident that these technical measures, designed specifically for the unique characteristics of reverse modeling technology and non-standard rotating structural parts, effectively address the high-precision splicing modeling requirements of non-standard rotating structural parts, thereby facilitating automated robotic arm operations for non-standard rotating structural parts in processes such as spraying, grinding, and welding. Attached Figure Description

[0018] Figure 1 This is a process flow diagram of the present invention.

[0019] Figure 2 This is a schematic diagram of point cloud rotation during the process of the present invention.

[0020] Figure 3 This is a schematic diagram of state one of the alignment and registration of the rotated point cloud during the process of the present invention.

[0021] Figure 4 This is a schematic diagram of state two of the alignment and registration of the rotated point cloud during the process of the present invention.

[0022] Figure 5 This is a schematic diagram of the transformation matrix in the process of the present invention.

[0023] Figure 6 This is a schematic diagram of the splicing effect during the process of this invention. Detailed Implementation

[0024] This invention relates to the field of reverse modeling technology, specifically a reverse modeling method for non-standard rotating structural components based on laser scanning point clouds. The following description, in conjunction with the accompanying drawings, illustrates this method. Figure 1 , Figure 2 , Figure 3 , Figure 4 , Figure 5 and Figure 6 The technical solution of this invention will be clearly and thoroughly explained.

[0025] It should be noted that the accompanying drawings of this invention are schematic, and unnecessary details have been simplified to clarify the technical objectives of this invention, so as to avoid obscuring the technical solutions contributed by this invention to the prior art. Furthermore, the terms "approximately" or "basically" used below to refer to quantities or fit relationships mean that reasonable assembly and processing errors are allowed in the industry, and do not literally describe absolute quantities or fit relationships.

[0026] See Figure 1 As shown, the present invention is a reverse modeling method for non-standard rotating structural components, which is based on laser scanning point cloud technology and specifically includes the following process.

[0027] S1. Mount the workpiece (i.e., non-standard rotating structural parts, hereinafter the same) that needs to be modeled onto the positioner.

[0028] An industrial 3D camera is mounted on the end effector of a robot's robotic arm. To accommodate the 3D camera's axial translation along the workpiece clamped by the positioner, the robot has a linear guide rail. The robotic arm holding the 3D camera is mounted on the linear guide rail and driven by a servo motor for linear translation. Simultaneously, to collect the data scanned by the 3D camera, a signal connection is established between the 3D camera and a surface data acquisition unit. In other words, the process of this invention is based on a laser scanning system constructed from a positioner, an industrial robot, a 3D camera, a surface data acquisition unit, and necessary controllers. Of course, these hardware devices are all existing mature technologies, and the technological contribution of this invention does not lie in the individual hardware devices themselves.

[0029] With the workpiece clamping and 3D camera clamping already implemented, hand-eye calibration of the 3D camera and robotic arm is performed using an eye-to-hand approach. This achieves the fusion of the 3D camera coordinate system and the world coordinate system of the robotic arm's operation. The specific hand-eye calibration process satisfies the following relationship: AX = XB; In the formula, A is the homogeneous spatial transformation matrix of the 3D camera during two adjacent motions; B is the homogeneous transformation matrix of the end-effector coordinate system during two consecutive movements; X is the hand-eye matrix to be solved.

[0030] The process of solving for A satisfies the following relationship: A1=H c2 *H c1 -1 ; A2=H c3 *H c2 -1 ; In the formula, A1 is the spatial transformation homogeneous matrix of the 3D camera during its first motion; A2 is the homogeneous spatial transformation matrix of the 3D camera during its second motion; H c1 H c2 H c3 These are the extrinsic parameter matrices of the calibration images for the 3D camera during calibration.

[0031] The process of solving for B satisfies the following relationship: B1 = Hc2-1 * Hc1; B2 = Hc3 - 1 * Hc2; B1 is the homogeneous transformation matrix of the robot arm's end-effector coordinate system during the first motion; B1 is the homogeneous transformation matrix of the robot arm's end-effector coordinate system during the second motion; Hg1 is the calibration image H c1 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system; Hg2 is the calibration image H c2 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system; Hg3 is the calibration image H c3 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system.

[0032] After obtaining the two sets of A and B mentioned above, substituting them into the above relation AX=XB will yield the unique solution X.

[0033] S2. Using the area directly above the workpiece clamped by the positioner as the teaching position for image capture, move the robotic arm to the set teaching position so that the 3D camera scans the workpiece perpendicular to the ground and downwards. In this state, the obtained point cloud of the workpiece surface is the point cloud of the surface directly above the workpiece within the field of view of the 3D camera, such as... Figure 2 As shown.

[0034] Based on the structural characteristics of the workpiece's rotating body, a single scan can cover at least 1 / 4 of the workpiece's surface area.

[0035] Before acquiring point cloud data of the workpiece surface, it is necessary to calibrate the rotation axis of the positioner relative to the coordinate system of the 3D camera. This calibration is based on the 3D camera coordinate system and involves teaching and calibrating the rotation axis of the positioner, specifically satisfying the following relationship: = ; In the formula, This is the transformation matrix for the rotation axis of the positioner; The axis of the coordinate system The normal vector of the direction; The axis of the coordinate system The normal vector of the direction; The axis of the coordinate system The normal vector of the direction; K=1- ; M= + + .

[0036] S3. After completing the calibration between the rotation axis of the positioner and the coordinate system of the 3D camera, the workpiece clamped by the positioner is scanned at the current position to collect point cloud data of the corresponding surface area.

[0037] S4. Setting the positioner's single rotation angle to 90° and its rotation around the clamped workpiece in the circumferential direction, a single 90° rotation, repeated four times sequentially, will allow the point cloud data collected by the 3D camera above to cover all circumferential surfaces of the workpiece. Thus, the four sets of point cloud data collected by the 3D camera in the circumferential direction of the workpiece will form a pairwise perpendicular relationship in space. That is, after rotating the workpiece four times around the positioner's axis, a complete workpiece point cloud can be assembled using the 3D camera.

[0038] According to the aforementioned rotation rules, the positioner is rotated sequentially after the previous point cloud data acquisition is completed; After the workpiece is rotated into position, the workpiece clamped by the positioner is scanned at the current position to collect point cloud data of the corresponding surface area.

[0039] Continue collecting point cloud data for all areas of the workpiece surface that require data collection, according to the areas that need to be collected.

[0040] S5. Based on the calibration relationship between the positioner's rotation axis and the 3D camera coordinate system, theoretically, when the positioner has no errors, the point cloud data collected during the positioner's rotation will sequentially form the complete surface point cloud of the workpiece. However, due to the existence of rotational errors in the positioner, the point cloud data collected during the positioner's rotation cannot be perfectly aligned. Therefore, it is necessary to perform alignment and registration processing on each set of rotated point clouds, such as... Figure 3 , Figure 4 As shown.

[0041] According to the order of the point cloud data collected in the workpiece rotation sequence, the adjacent two sets of point cloud data are aligned and registered to ensure high precision of point cloud stitching.

[0042] Alignment and registration of two adjacent sets of point cloud data is performed by registering the overlapping part of the two sets of point cloud data. This overlapping part is obtained by the distance between the two sets of point cloud data. If a point in the first set of point cloud data exists within the neighborhood of a point in the second set of point cloud data, it is considered to be the overlapping part. The two sets of point cloud data in the overlapping area are registered using the ICP algorithm to obtain the transformation matrix; The transformation matrix is ​​applied to two adjacent sets of complete point cloud data, resulting in a second matching of these two sets of point cloud data after rotation.

[0043] The ICP algorithm performs registration and obtains the transformation matrix according to the following relationship: ; In the formula, The transformation matrix represents the overlapping portion of two adjacent sets of point cloud data. The last index of the points in the overlapping point cloud data; The starting sequence number of the points in the overlapping point cloud data; It is a rotation matrix; It is a translation matrix; The i-th point in the first set of point cloud data; Let i be the i-th point in the second set of point cloud data.

[0044] The aforementioned ICP algorithm iteratively optimizes the transformation matrices of two point cloud groups, A (the first group) and B (the second group), to minimize the distance error between the two groups after transformation. Its core is to obtain the transformation matrix for the overlapping portion of the point clouds. Then, the transformation matrix is ​​applied to the two complete point clouds, thereby performing a second matching of the two point clouds after rotation, eliminating some of the errors caused by the positioner's accuracy, such as... Figure 5 As shown.

[0045] S6. After the above transformation, the overlapping portions of two adjacent sets of point cloud data will be closer. Point cloud deduplication optimization is used to fit the overlapping portions, that is, deduplication optimization is performed on the aligned and registered point cloud data, and the overlapping portions are fitted, as follows: Figure 6 As shown.

[0046] S7. Continue until each group of point cloud data is aligned, registered, deduplicated, and fitted in sequence to obtain the complete point cloud of the workpiece and complete the modeling.

[0047] In the above method, before aligning and registering adjacent sets of point cloud data, preprocessing is required for each acquired point cloud data, including removing invalid points, setting ROI regions, removing outliers, and estimating normal vectors. This is because the raw point cloud data acquired by 3D cameras has technical characteristics such as excessively high resolution, numerous invalid points, and cluttered point clouds. Therefore, before stitching the acquired point cloud data, preprocessing is necessary to remove invalid points (NaN points), set ROI regions, remove outliers, and estimate normal vectors. Invalid points affect subsequent point cloud processing and increase computational consumption; therefore, removing invalid points is the first step in point cloud preprocessing. Setting ROI regions ensures that the acquired point clouds contain only the workpiece portion, preventing the appearance of other interfering point clouds. Using a pass-through filtering algorithm, the point cloud is cropped according to the coordinate axis range to retain the ROI regions. Outlier removal and noise removal are achieved using statistical filtering algorithms. Outliers are removed by calculating the distances between each point and other points in its local neighborhood. Points with excessively large mean deviations are deleted by calculating the mean distances from each point to its neighbors and setting a mean threshold.

[0048] The above specific technical solutions are only used to illustrate the present invention, and are not intended to limit it.

[0049] Although the present invention has been described in detail with reference to the specific technical solutions described above, those skilled in the art should understand that modifications can still be made to the specific technical solutions described above, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the present invention.

Claims

1. A method for modeling non-standard rotating structural components based on laser scanning point clouds, characterized in that, The modeling method includes: S1. Clamp the workpiece on the positioner; The 3D camera is mounted on the robotic arm and hand-eye calibration is performed. S2. Move the robotic arm to the set photo teaching position; The axis of rotation of the positioner is calibrated relative to the coordinate system of the 3D camera; S3. Scan the workpiece clamped by the positioner at its current position and collect point cloud data of the corresponding surface area; S4. Rotate the positioner sequentially according to the set rotation angle and the continuity of the workpiece surface; After the current rotation is completed, the workpiece clamped by the positioner is scanned at the current position to collect point cloud data of the corresponding surface area. Continue collecting point cloud data for all areas of the workpiece surface that need to be collected, according to the areas that need to be collected on the workpiece surface. S5. Align and register adjacent sets of point cloud data according to the order of point cloud data collected in the workpiece rotation sequence. S6. Perform deduplication and optimization on the aligned and registered point cloud data, and fit the overlapping parts; S7. Continue until each group of point cloud data is aligned, registered, deduplicated, and fitted in sequence to obtain the complete point cloud of the workpiece and complete the modeling.

2. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 1, characterized in that, In S1, the hand-eye calibration process uses the method of "eye on hand" to achieve the fusion of the 3D camera coordinate system and the world coordinate system of the robotic arm's operation.

3. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 2, characterized in that, The hand-eye calibration process satisfies the following relationship: AX = XB; In the formula, A is the homogeneous spatial transformation matrix of the 3D camera during two adjacent motions; B is the homogeneous transformation matrix of the end-effector coordinate system during two consecutive movements; X is the hand-eye matrix to be solved; The process of solving for A satisfies the following relationship: A1=H c2 *H c1 -1 ; A2=H c3 *H c2 -1 ; In the formula, A1 is the spatial transformation homogeneous matrix of the 3D camera during its first motion; A2 is the homogeneous spatial transformation matrix of the 3D camera during its second motion; H c1 H c2 H c3 These are the extrinsic parameter matrices of the calibration images used in the calibration of the 3D camera; The process of solving for B satisfies the following relationship: B1 = Hc2-1 * Hc1; B2 = Hc3 - 1 * Hc2; B1 is the homogeneous transformation matrix of the robot arm's end-effector coordinate system during the first motion; B1 is the homogeneous transformation matrix of the robot arm's end-effector coordinate system during the second motion; Hg1 is the calibration image H c1 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system; Hg2 is the calibration image H c2 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system; Hg3 is the calibration image H c3 The coordinate system description matrix of the corresponding robotic arm coordinate system in the base coordinate system.

4. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 1, characterized in that, In S2, the photographic teaching position is directly above the workpiece, and the 3D camera scans the workpiece in a downward direction perpendicular to the ground to capture the image.

5. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 1, characterized in that, In S2, the calibration of the positioner's rotation axis centerline and the 3D camera coordinate system is performed by teaching and calibrating the positioner's rotation axis centerline based on the 3D camera coordinate system.

6. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 5, characterized in that, The teaching calibration of the rotation axis centerline of the positioner satisfies the following relationship: = ; In the formula, This is the transformation matrix for the rotation axis of the positioner; The axis of the coordinate system The normal vector of the direction; The axis of the coordinate system The normal vector of the direction; The axis of the coordinate system The normal vector of the direction; K=1- ; M= - + 。 7. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 1, characterized in that, In S4, the rotation angle of the positioner is 90°.

8. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 1, characterized in that, In S3 and S4, the collected point cloud data is preprocessed by removing invalid points, setting ROI regions, removing outliers, and estimating normal vectors.

9. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 1, characterized in that, In S5, the alignment and registration process for two adjacent sets of point cloud data is performed by registering the overlapping part of the two sets of point cloud data. The overlapping part is obtained by the distance between the two sets of point cloud data. If there are points in the second set of point cloud data within the neighborhood of a point in the first set of point cloud data, then it is considered as the overlapping part. The two sets of point cloud data in the overlapping part are registered using the ICP algorithm to obtain the transformation matrix; The transformation matrix is ​​applied to two adjacent sets of complete point cloud data, resulting in a second matching of these two sets of point cloud data after rotation.

10. The method for modeling non-standard rotating structural components based on laser scanning point clouds according to claim 9, characterized in that, The ICP algorithm performs registration and obtains the transformation matrix, which satisfies the following relationship: ; In the formula, The transformation matrix represents the overlapping portion of two adjacent sets of point cloud data. The last index of the points in the overlapping point cloud data; The starting sequence number of the points in the overlapping point cloud data; It is a rotation matrix; It is a translation matrix; The i-th point in the first set of point cloud data; Let i be the i-th point in the second set of point cloud data.