A robust 13-parameter 3D datum transformation method

Abstract

This invention discloses a robust 13-parameter 3D datum transformation method. This method first uses translation to align the origins of two coordinate systems, eliminating the translation parameter and avoiding strong correlations between parameters. Then, the rotation matrix and scaling factor are treated as a whole and separated through matrix transformation, yielding nine rotation matrix elements and one scaling parameter. Finally, these ten parameters are used as known values ​​for back-substitution to recover three translation parameters. Based on these 13 parameters, the transformation between non-common point coordinates is completed. This method features a simple model, high transformation accuracy, and strong stability. It is applicable to coordinate transformations outside the control point range and transformations between left-handed and right-handed coordinate systems, significantly improving the accuracy of solving transformation parameters for small areas or few control points, and expanding the applicability of the parameters.

Description

Technical Field

[0001] This invention belongs to the field of communication technology, specifically relating to a robust 13-parameter three-dimensional reference transformation method. Background Technology

[0002] Three-dimensional datum transformation plays a crucial role in satellite geodesy, remote sensing and photogrammetry, and precision engineering surveying. It is widely used for coordinate transformation between different BeiDou / GNSS coordinate systems, establishing spatial resection and collinearity equations in photogrammetry, and transforming independent coordinate systems in precision engineering surveying. Especially with the rapid development of modern surveying technology, multi-mode, multi-system combined positioning, multi-source observation data fusion, and multi-platform, multi-framework processing coordinate systems are widespread. This often leads to the problem of transforming different coordinate systems in practical work, which will inevitably affect the unification of various surveying results to a significant extent. Therefore, research on three-dimensional datum transformation models under different surveying coordinate systems is of great significance.

[0003] Three-dimensional datum transformation essentially involves estimating the second set of coordinates for non-common points using two sets of coordinates for common points and the first set of coordinates for non-common points. Three-dimensional datum transformation typically employs three types of parameters—translation, rotation, and scaling—for similarity transformation, proceeding in two steps: first, calculating the transformation parameters using the common point coordinates, and then using these parameters to transform the coordinates of non-common points. Depending on the different definitions of the reference point for rotation and scaling parameters, commonly used three-dimensional datum transformation models include the Bursa-Wolf model, the Molodensky model, the Veis model, the Paradigm model, and the Wushang model. The prerequisite for these models is based on small-angle rotation. The transformation, at this point is approximately considered Therefore, the above model is a simplified linear model. However, when the rotation angle is large, using a linear approximation model will cause a large model error, and nonlinear problems need to be addressed. In particular, of the nine parameters of the rotation matrix, only three are independent, while the remaining six are not independent, and their relationship with the independent rotation parameters is complex, making it difficult to solve the nonlinear model. To address this, many scholars have studied methods for solving the reference transformation for large rotation angles, mainly divided into two categories: iterative methods and analytical methods. The iterative method constructs the rotation matrix in the form of an orthogonal matrix, a unit quaternion, and Euler angles, and then establishes a Newton-Raphson iterative scheme to solve for the three-dimensional coordinate transformation parameters. However, the iterative algorithm requires initial values, has low computational efficiency, and the results are heavily dependent on the selection of initial values. The analytical method adopts a distribution-optimal strategy, first solving for the scale parameter, and then using the SVD decomposition method, the orthogonal matrix method, and the unit quaternion method to solve for the rotation matrix, finally obtaining the translation parameters. However, this method is not entirely suitable for transformations between left-handed and right-handed coordinate systems. Furthermore, in 3D datum transformation, there is a strong correlation between translation parameters and rotation and scale change parameters, leading to ill-conditioned solution models and significant differences in the calculated parameters. This is especially true when the engineering control network covers only a few hundred to tens of square kilometers or even smaller; in such cases, the correlation between rotation and translation parameters is even stronger, and even small measurement errors can cause large errors in the estimated transformation parameters, particularly for translation parameters which can have errors on the order of meters or even hundreds of meters. To address the potential ill-conditioned nature of the transformation model, some scholars have proposed transformation methods such as regularization and singular value decomposition for local 3D datum transformation, but these are cumbersome to implement. Moreover, within the range of control points used to calculate the transformation parameters, the coordinates transformed by these parameters still have centimeter-level accuracy; beyond this range, the coordinate transformation results are unreliable. There is an urgent need for a 3D coordinate datum transformation method that can improve the accuracy of transformation parameter calculation for small-scale control points, ensuring the accuracy and reliability of coordinate transformation results even outside the control point range. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention aims to provide a robust 13-parameter three-dimensional datum transformation method. This method features a simple model, high transformation accuracy, and strong stability. It is also applicable to coordinate transformation outside the control point range and transformation between left-handed and right-handed coordinate systems. It can significantly improve the accuracy of solving transformation parameters for small ranges or few control points and expand the applicability range of the parameters.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A robust 13-parameter three-dimensional datum transformation method, characterized by the following steps:

[0007] 1) Eliminate translation parameters: Move the origin of the two coordinate systems to the centroid of the common point where the transformation parameters are obtained, and reconstruct the new transformation model with the centroid as the origin;

[0008] 2) Treat the rotation matrix and scaling factor as a whole, and solve them separately through matrix transformation;

[0009] 3) Using the 9 rotation matrix elements and 1 scaling parameter obtained from the solution as known values, the 3 translation parameters are recovered;

[0010] 4) Use the transformation parameters from step 3) above to transform the coordinates of non-common points.

[0011] Step 1) specifically includes the following steps:

[0012] For two different Cartesian coordinate systems, the coordinate transformation formula can be expressed as:

[0013] (1)

[0014] In the formula, X i Y i Z i Let represent the three-dimensional coordinates of the i-th point, with subscripts I and II indicating two coordinate systems; These are three translation factors; parameters m is the scaling parameter between the two coordinate systems; For coordinate rotation matrix, These are the three rotation angles of the coordinate axes;

[0015] Substituting the matrix elements and rearranging, we get

[0016] (2);

[0017] After translation, rotation, and scaling transformations, the set of common points in the first coordinate system completely coincides with the set of points in the second coordinate system. In other words, the centroids of both sets, like all common points, exhibit the same translation, rotation, and scaling relationships.

[0018] (3),

[0019] In the formula, , , Indicates the location of the center of gravity. , , n is the number of common points, and the subscripts I and II represent two coordinate systems;

[0020] Subtracting equation (3) from equation (2) yields:

[0021] (4),

[0022] In the formula , , This represents the difference between the coordinates of each point and the centroid;

[0023] If we assume

[0024] ,

[0025] , ,

[0026] The above model can then be further simplified to: (5).

[0027] Step 2) specifically includes the following steps:

[0028] Given at least three common points with different coordinates in two coordinate systems, according to the 3D coordinate transformation model above, we have:

[0029] (6)

[0030] In the above model, matrix T is 3×3, and the other two matrices are both 3×n;

[0031] Therefore, both sides of equation (6) are multiplied by... The transpose of T is used to construct a 3×3 matrix. After matrix transformation, T is separated to obtain...

[0032] (7),

[0033] The T matrix elements contain information on both rotation and scaling. Here, we assume the scaling factor is 1, so T is a rotation matrix with 9 matrix elements representing 9 rotation parameters.

[0034] Step 3) specifically includes the following steps:

[0035] Substituting the nine rotation matrix elements and one scaling parameter obtained above as known values ​​into equation (3), the three translation parameters can be recovered using the two sets of coordinates of one common point. .

[0036] Step 4) specifically includes: based on the 13 transformation parameters obtained in step 3), including 9 rotation parameters and 3 translation parameters, the second set of coordinates can be estimated from the first set of coordinates of the non-common points based on the benchmark transformation model of equation (2).

[0037] The beneficial effects of this invention are as follows:

[0038] This invention features a simple calculation process, facilitating program implementation. It effectively addresses the issues of strong correlation between parameters, nonlinearity in large-angle coordinate transformations, and low accuracy in coordinate transformations outside the control point range. It demonstrates strong robustness and universality, significantly improving the accuracy of transformation parameter calculations for small areas or few control points, and expanding the applicability of the parameters. Even when transforming coordinates outside the control point range, its reliability remains guaranteed, which is the greatest advantage of this model. Attached Figure Description

[0039] Figure 1 This is a diagram showing the distribution of engineering example points in the simulation experiment of this invention;

[0040] Figure 2 This is a residual distribution diagram of the model of the present invention;

[0041] Figure 3 A residual distribution diagram of the model generated using existing techniques;

[0042] Figure 4 This is an error distribution diagram of the 14 checkpoints in this invention;

[0043] Figure 5 This is an error distribution map of 14 checkpoints generated using existing technology. Detailed Implementation

[0044] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are for illustrative purposes only and are not intended to limit the scope of the invention.

[0045] This invention proposes a robust 13-parameter 3D reference transformation method applicable to coordinate transformation between different BeiDou / GNSS coordinate systems. The method first uses translation to align the origins of the two coordinate systems, eliminating the translation parameters and avoiding strong correlations between parameters. Then, the rotation matrix and scaling factor are treated as a whole and separated through matrix transformation. Finally, the nine rotation matrix elements and one scaling parameter are used as known values ​​to recover the three translation parameters. Specifically, the method includes the following steps:

[0046] Step 1: Eliminate translation parameters: Move the origin of the two coordinate systems to the centroid of the common point where the transformation parameters are obtained, and rebuild the transformation model with this centroid as the origin.

[0047] For two different spatial rectangular coordinate systems, the coordinate transformation formula can be expressed as:

[0048] (1)

[0049] In the formula, X i Y i Z iLet I represent the three-dimensional coordinates of the i-th point, with subscripts I and II indicating two coordinate systems; These are three translation factors; parameters m is the scaling parameter between the two coordinate systems; For coordinate rotation matrix, These are the three rotation angles of the coordinate axes.

[0050] Substituting the matrix elements and rearranging, we get

[0051] (2)

[0052] The above model is a nonlinear model with dependent parameters, making the solution extremely complex and difficult to guarantee the accuracy and reliability of the coordinate transformation results outside the control point range. The translation parameters are strongly correlated with other parameters; the ill-conditioned problem of the normal equation matrix can be solved by first eliminating the coordinate translation parameters. Because the common point set in the first coordinate system completely coincides with the point set in the other coordinate system after translation, rotation, and scaling transformations, or in other words, the centroids of the two point sets and all common points have the same translation, rotation, and scaling relationships, i.e.

[0053] (3)

[0054] In the formula, , , Indicates the location of the center of gravity. , , , where n is the number of common points.

[0055] Subtracting equation (3) from equation (2) yields

[0056] (4)

[0057] In the formula , , This represents the difference between the coordinates of each point and the centroid.

[0058] The above model is essentially moving the origins of the two coordinate systems to the centroid of the common point used to calculate the transformation parameters, and reconstructing the new transformation model with this centroid as the origin. The new model is equivalent to simply translating the coordinate systems, eliminating the translation parameters, while the orientation of the three coordinate axes remains unchanged. Therefore, the rotation and scaling parameters of the three axes remain constant. This not only avoids the strong correlation between translation parameters and rotation and scaling parameters, but also simplifies the model by ensuring that the point sets only contain rotation and scaling information, exhibiting a purely linear relationship.

[0059] If we assume

[0060] ,

[0061] , ,

[0062] The above model can then be further simplified to: (5).

[0063] Step 2: Treat the rotation matrix and scaling factor as a whole and solve them separately through matrix transformation.

[0064] 3D datum transformation is performed in two steps: first, the transformation parameters are solved using the coordinates of common points; then, the transformation parameters are used to transform the coordinates of non-common points. Therefore, the transformation parameters are merely an intermediate product; the ultimate goal of obtaining them is to perform coordinate transformation. While the rotation angle parameter has the most obvious geometric meaning, its calculation involves complex nonlinear problems and tedious trigonometric function operations. If the scale and rotation matrix are treated as a whole, the model exhibits a simple linear relationship. Solving only on the matrix elements reduces computational complexity, eliminates singular solutions, is easy to understand and accept, and requires only three pairs of common points while still achieving the purpose of coordinate transformation.

[0065] Given at least three common points with different coordinates in two coordinate systems, according to the 3D coordinate transformation model above, we have:

[0066] (6)

[0067] In the above model, matrix T is 3×3, and the other two matrices are both 3×n.

[0068] Therefore, both sides of equation (6) are multiplied by... The transpose of T is used to construct a 3×3 matrix. After matrix transformation, T is separated to obtain...

[0069] (7)

[0070] The T matrix elements contain information on both rotation and scaling. Here, we can assume that the scaling factor is 1, so T is a rotation matrix with 9 matrix elements representing 9 rotation parameters.

[0071] Step 3: Using the 9 rotation matrix elements and 1 scaling parameter obtained from the solution as known values, recover the 3 translation parameters.

[0072] Substituting the nine rotation matrix elements and one scaling parameter obtained above into equation (3), the three translation parameters can be recovered using the two sets of coordinates of one common point. .

[0073] Step 4: Use the above 13 transformation parameters to transform between coordinates of non-common points.

[0074] Based on the 13 transformation parameters, including 9 rotation parameters and 3 translation parameters, and based on the benchmark transformation model (2), the second set of coordinates of the non-common points can be estimated from the first set of coordinates.

[0075] This invention aims to provide a three-dimensional datum transformation method with a simple calculation process that is easy to implement in a program. It effectively solves the problems of strong correlation between parameters, nonlinearity in large-angle coordinate transformation, and low accuracy of coordinate transformation outside the control point range. It exhibits strong robustness and universality, and can significantly improve the accuracy of solving transformation parameters in small ranges or with few control points, expand the applicable range of parameters, and ensure its reliability even when transforming coordinates of points outside the control point range.

[0076] Simulation Experiment

[0077] To verify the reliability of the model, this invention compared actual data from an engineering case with existing large rotation angle 3D transformation methods. The data came from two sets of coordinate data for 17 nodes of a component in the Guangzhou New Airport terminal building space frame: the original coordinate system (measurement coordinate system) and the target coordinate system (design coordinate system). Figure 1 This shows the distribution of points in the engineering example.

[0078] In the simulation experiment, the origin of the target coordinate system is the center of the hinge at the end of the component, and the origin of the original coordinate system is arbitrarily assumed. The coordinate system axes are all positive with X (north), Z (east), and Y (vertical upward).

[0079] Based on the model proposed in this invention and the existing large rotation angle reference transformation model, all 17 points are used as common points to participate in the solution of model parameters. Regardless of the model residual distribution (e.g. Figure 2 and Figure 3 As shown in Table 3, the difference between the two is not significant in terms of internal consistency accuracy or the calculated model parameters. Regarding the statistical model accuracy, the in-between accuracy of this invention is 7.46 mm, slightly better than the in-between accuracy of the prior art (7.74 mm). The maximum residual value for a single component in this invention is 19.83 mm, slightly lower than the existing large rotation angle coordinate transformation model (23.6 mm).

[0080] Table 3. 13 transformation parameters and internal consistency accuracy of the present invention method and existing methods when all 17 points are solved as common points.

[0081]

[0082] However, using three points within a small area, and still using the other 14 non-common points as check points, the model accuracy was evaluated. The model of this invention differed significantly from existing large rotation angle datum transformation models. Table 2 and... Figure 4 , Figure 5 The results of solving the baseline transformation model parameters, external compliance accuracy, and three-component error distribution of 14 check points are presented, using the first three points s1, s5, and s6 as common points. Significant differences exist between the two methods in the solved transformation parameters, especially in the translation parameters. Compared to existing methods, the transformation parameters solved by the proposed model using only three points are closer to those solved using all points (see Table 2). Furthermore, compared to existing methods, the external compliance accuracy of the model at check points is greatly improved, with the maximum single-component error decreasing from 1470.107 mm to 43.641 mm, and the mean square error decreasing from 525.454 mm to 15.916 mm, representing a 97% improvement in accuracy.

[0083] Because there are only three common points for parameter calculation, and the check points are all distributed outside the common point survey area, the accuracy of coordinate transformation is difficult to guarantee. However, the model of this invention exhibits strong adaptability, significantly improving the accuracy of parameter calculation for small-scale control points, and ensuring that the accuracy and reliability of coordinate transformation results are still improved even outside the control point range. It is evident that the two models show little difference in coordinate transformation for points within the control point range, but for coordinate transformation of points outside the survey area, the model of this invention has a clear advantage and demonstrates strong robustness.

[0084] Table 4 shows the 13 transformation parameters obtained by the existing method and the method of this invention, using three points as common points, and their out-of-point compliance accuracy.

[0085]

[0086] Obviously, the embodiments described above are only some embodiments of this application, not all embodiments. The accompanying drawings show preferred embodiments of this application, but do not limit the patent scope of this application. This application can be implemented in many different forms; rather, the purpose of providing these embodiments is to provide a more thorough and comprehensive understanding of the disclosure of this application. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing specific embodiments, or make equivalent substitutions for some of the technical features. Any equivalent structures made using the content of this application's specification and drawings, directly or indirectly applied to other related technical fields, are similarly within the scope of patent protection of this application.

Claims

1. A robust 13-parameter three-dimensional datum transformation method, characterized in that: Specifically, the following steps are included: 1) Eliminate translation parameters: Move the origin of the two coordinate systems to the centroid of the common point where the transformation parameters are obtained, and reconstruct the new transformation model with the centroid as the origin; 2) Treat the rotation matrix and scaling factor as a whole, and solve them separately through matrix transformation; 3) Using the 9 rotation matrix elements and 1 scaling parameter obtained from the solution as known values, the 3 translation parameters are recovered; 4) Use the transformation parameters from step 3) to transform the coordinates of non-common points.

2. The robust 13-parameter three-dimensional reference transformation method according to claim 1, characterized in that: Step 1) specifically includes the following steps: For two different Cartesian coordinate systems, the coordinate transformation formula can be expressed as: (1) In the formula, X i Y i Z i Let represent the three-dimensional coordinates of the i-th point, with subscripts I and II indicating two coordinate systems; These are three translation factors; parameters m is the scaling parameter between the two coordinate systems; For coordinate rotation matrix, These are the three rotation angles of the coordinate axes; Substituting the matrix elements and rearranging, we get (2)； After translation, rotation, and scaling transformations, the set of common points in the first coordinate system completely coincides with the set of points in the second coordinate system. In other words, the centroids of both sets, like all common points, exhibit the same translation, rotation, and scaling relationships. (3)， In the formula, , , Indicates the location of the center of gravity. , , n is the number of common points, and the subscripts I and II represent two coordinate systems; Subtracting equation (3) from equation (2) yields: (4)， In the formula , , This represents the difference between the coordinates of each point and the centroid; If we assume ， ， ， The above model can then be further simplified to: (5).

3. The robust 13-parameter three-dimensional reference transformation method according to claim 2, characterized in that: Step 2) specifically includes the following steps: Given at least three common points with different coordinates in two coordinate systems, according to the 3D coordinate transformation model above, we have: (6) In the above model, matrix T is 3×3, and the other two matrices are both 3×n; Therefore, both sides of equation (6) are multiplied by... The transpose of T is used to construct a 3×3 matrix. After matrix transformation, T is separated to obtain... (7)， The T matrix elements contain information on both rotation and scaling. Here, we assume the scaling factor is 1, so T is a rotation matrix with 9 matrix elements representing 9 rotation parameters.

4. The robust 13-parameter three-dimensional datum transformation method according to claim 3, characterized in that: Step 3) specifically includes the following steps: Substituting the nine rotation matrix elements and one scaling parameter obtained above as known values ​​into equation (3), the three translation parameters can be recovered using the two sets of coordinates of one common point. .

5. The robust 13-parameter three-dimensional reference transformation method according to claim 3, characterized in that: Step 4) specifically includes the following steps: Based on the 13 transformation parameters, including 9 rotation parameters and 3 translation parameters, obtained in step 3), the second set of coordinates can be estimated from the first set of coordinates of the non-common points using the benchmark transformation model of equation (2).

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