Nonlinear least squares based surface body positioning reference plane adjustment method

Abstract

The application belongs to the field of curved surface body positioning reference plane adjustment and discloses a curved surface body positioning reference plane adjustment method based on nonlinear least squares, which aims to provide an improved curved surface body positioning reference plane adjustment method based on nonlinear least squares. After the curved surface body is fixed on a workbench, the coordinates of the registration control points on the curved surface body are measured and obtained by a laser tracker. In the Boolean sand model, the two coordinate system origins are translated to the center positions of the reference control points to establish a center auxiliary conversion coordinate system. Seven parameters are solved based on the nonlinear least squares method and the Gauss-Newton iteration method, and the adjustment is performed based on the offset and the scaling ratio. The corresponding displacement and deflection angle values are compensated to the reference plane, and the curved surface body is moved to the optimal working position. The application is applied to the field of curved surface body reference plane positioning.

Description

Technical Field

[0001] This invention relates to a method for adjusting the positioning reference plane of a curved body based on nonlinear least squares, and belongs to the field of adjusting the positioning reference plane of a curved body. Background Technology

[0002] Currently, when using machine tools to finish curved surfaces, it is difficult to position the reference plane of the curved surface. The curved surface will be placed on the machine tool table or attitude adjustment driver in any posture, and there is a certain positional deviation between its reference plane and the coordinate system of the table. Displacement compensation operations such as origin offset and coordinate axis rotation are required.

[0003] Currently, for adjusting the positioning datum of curved bodies, the Bursa model is usually used for coordinate transformation. In the process of calculating and solving the parameters, the transformation model omits second-order and higher-order components, or the trigonometric functions in the rotation matrix are approximated as angles. This will produce a large truncation error, which will impair the accuracy of coordinate transformation. Furthermore, the calculation process is cumbersome, and the selection of the datum point has a significant impact on the parameters. Since the higher-order terms of the expansion are omitted in the calculation of parameters, the accuracy is low. This adjustment method is only suitable for cases with small rotation angles, or for coordinate system transformation algorithms based on the seven-parameter nonlinear least squares method, which is cumbersome in calculation and increases the parameter deviation. Summary of the Invention

[0004] In order to overcome the shortcomings of the prior art, the technical problem to be solved by the present invention is to provide an improvement on the method for adjusting the positioning reference plane of a curved body based on nonlinear least squares.

[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for adjusting the positioning reference plane of a curved surface based on nonlinear least squares, comprising the following adjustment steps:

[0006] Step 1: Let the theoretical coordinate system of the curved surface be: The theoretical coordinates of its three registration points A, B, and C are as follows:

[0007] ;

[0008] After fixing the curved surface to the worktable, the coordinates of the registration control points on the curved surface are obtained by measuring with a laser tracker;

[0009] Let the actual coordinate system of the curved surface be: The actual coordinates of its three registration points D, E, and F are as follows:

[0010] ;

[0011] Assume the actual coordinate system of the surface is offset relative to the theoretical coordinate system by three translation parameters. , , Each parameter represents the translation along the X, Y, and Z axes, respectively. The three rotation parameters α, β, and γ represent the rotation angles around the X, Y, and Z axes, respectively. (Actual coordinate system) Relative theoretical coordinate system The scaling factor is k;

[0012] Step 2: In the Bursa model, establish a central auxiliary transformation coordinate system by translating the origins of the two coordinate systems to the center of their respective reference control points:

[0013] The coordinate transformation model based on the Bursa-Wolf model is constructed as follows:

[0014] ;

[0015] in:

[0016] ;

[0017] ;

[0018] Simplified to:

[0019] ;

[0020] in: Theoretical coordinate system The coordinates of the upper reference point;

[0021] They are the actual coordinate system The coordinates of the upper reference point;

[0022] These are the rotation matrices for rotating the actual coordinate system of the surface about its own Z-axis by an angle α, about its Y-axis by an angle β, and about its Z-axis by an angle γ, respectively.

[0023] Theoretical coordinate system G Transform to actual coordinate system L Rotation transformation matrix on;

[0024] The location of the center point of the registration point is calculated as follows:

[0025] ;

[0026] in: X These are the coordinates of the registration points in the coordinate system. N To determine the number of registration points, The coordinates of the center of the registration point set;

[0027] The center point of the three registration points on the actual coordinates The coordinates are:

[0028] ;

[0029] in: Center point The X-axis coordinate value, Center point Y-axis coordinate value, Center point Z-axis coordinate value;

[0030] The center point of the three registration points on the theoretical coordinate system The coordinates are:

[0031] ;

[0032] in: Center point The X-axis coordinate value, Center point Y-axis coordinate value, Center point Z-axis coordinate value;

[0033] The central auxiliary coordinate transformation model based on the Bursa-Wolf model is then constructed as follows:

[0034] ;

[0035] Abbreviated as: ;

[0036] in:

[0037] ;

[0038] ;

[0039] In the formula: These are the theoretical coordinates of the control point set. These are the actual coordinates of the control point set;

[0040] Step 3: Solve the center-auxiliary coordinate transformation model based on the nonlinear least squares method and the Gauss-Newton iteration method to obtain seven parameters, and adjust the position of the curved surface according to the seven parameters to reach the optimal working position:

[0041] First let ;

[0042] The three-dimensional coordinate transformation model According to the least squares principle, the minimum value is:

[0043] ;

[0044] Simplified versions include: for ;

[0045] Right now: ;

[0046] Solve again: ;

[0047] In the formula:

[0048] ;

[0049] The Gauss-Newton iteration method is used in the solution process, and the specific steps are as follows:

[0050] Step 3.1: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require Linearization, at the reference point right Performing a Taylor expansion, we get:

[0051] ;

[0052] Step 3.2: Substitute the above formula into the equation. In this process, the next iteration formula is obtained as follows:

[0053] ;

[0054] During the iteration process, the iteration stops when the norm of the difference between two adjacent seven-parameter vectors meets the accuracy requirement or is less than a certain threshold, and the optimal solution under the least squares rule can be obtained.

[0055] Step 3.3: The calculated x , y , z This represents the offset between the origin of the actual coordinate system and the origin of the theoretical coordinate system of the curved surface.

[0056] The calculated α, β, and γ are the deflection angles between the actual coordinate system and the theoretical coordinate system of the surface solid.

[0057] The calculated k is the scaling factor between the two coordinate systems;

[0058] Step 3.4: Adjust based on offset and scaling ratio, compensate for corresponding displacement and deflection angle values ​​on the reference plane, and move the surface to the optimal working position.

[0059] The advantages of this invention compared to existing technologies are as follows: This invention specifically provides a method for adjusting the reference plane of a curved body based on a center-assisted spatial coordinate transformation algorithm of nonlinear least squares. First, the center coordinates of the reference points on the theoretical model and the reference points measured by the laser tracker are solved. Then, the minimum value of the multivariate function is used to calculate the optimal values ​​of the seven parameters. This method achieves fast convergence speed while improving accuracy, and the calculation results are correct and stable, providing a new solution for adjusting the reference plane of a curved body. Experimental calculations show that this method can reduce the number of iterations and converge, obtaining equally accurate seven-parameter solutions, indicating that the algorithm in this paper is effective in solving large rotation angle problems and has the characteristic of fast convergence. This invention is not sensitive to translation parameters and can converge quickly, obtaining correct seven-parameter solutions. Detailed Implementation

[0060] In implementing this invention, registration reference points are pre-planned on the CAD model of the curved surface. High-precision measuring equipment, such as a laser tracker, is used to detect the position of these reference points on the worktable. After determining the position of the reference plane on the worktable, a more accurate mathematical model for the fitted pose of the reference plane is constructed by improving the seven-parameter Bursa coordinate transformation formula. This model solves for the positional relationship between the theoretical coordinates and the actual coordinates of the curved surface, thus determining the displacement and rotation angles that need to be adjusted for the positioning reference plane. This method effectively reduces the computational load and improves the efficiency of datum plane positioning for the curved surface. The method for adjusting the positioning reference plane of a curved surface provided by this invention has universal applicability and can be used in the adjustment process of reference planes for various complex curved surfaces.

[0061] The above refers to the registration control points for curved surfaces, which are the set of model registration control points pre-planned on the CAD or 3D model of the curved surface. Feature points of the curved surface are selected as model registration control points. The selection objectives are: to identify features as clearly as possible, to facilitate measurement, and to ensure stable registration. While meeting the requirements for registration reliability, the accuracy and efficiency of registration positioning should be maximized. Examples include feature points at the intersection of curved surfaces, support points under the curved surface, and feature points at special structures of the curved surface. Appropriate reference control points are selected, and three points that are not on the same straight line are taken as reference points. This serves as a reference plane, meaning that changes in the posture of the curved surface will be reflected in the displacement and deflection of the reference plane.

[0062] To determine the position of the reference plane of an actual curved surface, it is necessary to know the relative position between the actual coordinate system and the theoretical coordinate system of the curved surface. This requires determining the coordinate transformation relationship between the three actual registration points of the curved surface and the three registration points of the theoretical model. Therefore, it is necessary to first determine the transformation relationship between the actual coordinate system of the curved surface on the worktable and the coordinate system of the theoretical model, that is, to know the relative position of the reference plane of the curved surface on the worktable. The transformation parameters are the displacement and rotation angles that need to be adjusted. By adjusting the position of the reference plane, the optimal working position of the curved surface can be achieved. This invention assumes that the theoretical coordinate system and the working coordinate system of the curved surface coincide (the origin and coordinate axes coincide), that is, the reference plane on the theoretical coordinate system is the actual position that the positioning reference plane of the curved surface on the worktable needs to be adjusted to. In practical applications, the positional relationship between the theoretical coordinate system and the working coordinate system of the curved surface needs to be analyzed on a case-by-case basis.

[0063] This invention provides a center-aided spatial coordinate transformation algorithm based on nonlinear least squares for adjusting the datum of a curved surface. It solves for the center coordinates of the datum points on the theoretical model and the datum points measured by a laser tracker, and uses multivariate function minimization to calculate the optimal values ​​of seven parameters. This provides a new solution for adjusting the datum of a curved surface. The specific method is as follows:

[0064] First, let the theoretical coordinate system of the curved surface be: The theoretical coordinates of its three registration points A, B, and C are as follows:

[0065] ;

[0066] After the curved surface is fixed on the worktable, the coordinates of the registration control points on the curved surface are obtained by measuring with a laser tracker. Let the actual coordinate system of the curved surface be: The actual coordinates of its three registration points D, E, and F are as follows:

[0067] ;

[0068] The actual coordinate system of the surface relative to the theoretical coordinate system is offset by three translation parameters. , , The values ​​represent the translation along the X, Y, and Z axes, respectively, and the three rotation parameters α, β, and γ represent the rotation angles around the X, Y, and Z axes, respectively.

[0069] Actual coordinate system Relative theoretical coordinate system The scaling factor is k. The origins of the theoretical coordinate system G and the actual coordinate system L of the surface do not coincide, and the directions of each coordinate axis are not the same.

[0070] Then, a coordinate transformation model based on the Bursa-Wolf model is constructed:

[0071] ;

[0072] in:

[0073] ;

[0074] ;

[0075] Simplified to:

[0076] ;

[0077] In the Bursa model, a central auxiliary transformation coordinate system is established by translating the origins of the two coordinate systems to the center of the reference control points, so that the reference control points are evenly distributed within it.

[0078] First, calculate the position of the center point:

[0079] ;

[0080] The center point of the actual three registration points The coordinates are:

[0081] ;

[0082] Similarly, the center points of the three registration points on the theoretical coordinate system... coordinate:

[0083] ;

[0084] The central auxiliary coordinate transformation model based on the Bursa-Wolf model is then constructed as follows:

[0085] ;

[0086] Abbreviated as: ;

[0087] in:

[0088] ;

[0089] ;

[0090] These are the theoretical coordinates of the control point set. These are the actual coordinates of the control point set;

[0091] At this point, the X, Y, and Z axes of the central auxiliary transformation coordinate system are parallel to the X, Y, and Z axes of the theoretical coordinate system.

[0092] Then, using the nonlinear least squares method and the Gauss-Newton iteration method, the central auxiliary coordinate transformation model is solved, and seven parameters are obtained:

[0093] First let ;

[0094] The three-dimensional coordinate transformation model Using the least squares principle, we can find the minimum value:

[0095] ;

[0096] Simplified to: for ;

[0097] Solve again: ;

[0098] In the formula:

[0099] ;

[0100] The Gauss-Newton iteration method is used in the solution process:

[0101] First Linearization, in Point to point Performing a Taylor expansion, we get:

[0102] ;

[0103] Substituting the above equation into the equation If we can get the next iteration formula:

[0104] ;

[0105] During the iteration process, the iteration stops when the norm of the difference between two adjacent seven-parameter vectors is less than a certain threshold, thus obtaining the optimal solution under the least squares rule.

[0106] The solution x , y , z This represents the offset between the origin of the actual coordinate system and the origin of the theoretical coordinate system of the curved surface.

[0107] α, β, γ are the deflection angles between the actual coordinate system axes and the theoretical coordinate system axes of the surface solid;

[0108] k is the scaling factor between the two coordinate systems;

[0109] Finally, the optimal working position of the curved surface is achieved by adjusting the corresponding displacement and deflection angle of the reference plane.

[0110] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for adjusting a reference plane of a curved body based on nonlinear least squares, characterized by: The following adjustment steps are included: Step one: Set the surface body theory coordinate system as: The theoretical coordinates of its three registration points A, B, and C are respectively: ； After fixing the curved surface to the worktable, the coordinates of the registration control points on the curved surface are obtained by measuring with a laser tracker; Let the actual coordinate system of the curved surface body be: The actual coordinates of its three registration points D, E, and F are respectively: ； The actual coordinate system of the curved surface body is offset from the theoretical coordinate system by three translation parameters , , Each parameter represents a translation along the X, Y, and Z axes, respectively. The three rotation parameters α, β, and γ represent the rotation angles around the X, Y, and Z axes, respectively. The scale of the actual coordinate system relative to the theoretical coordinate system is k. Step 2: In the Bursa model, establish a central auxiliary transformation coordinate system by translating the origins of the two coordinate systems to the center of their respective reference control points: The coordinate transformation model based on the Bursa-Wolf model is constructed as follows: ； in: ； ； Simplified to: ； wherein: respectively the theoretical coordinate system the coordinate value of the upper reference point; actual coordinate system coordinate value of the upper reference point These are the rotation matrices for rotating the actual coordinate system of the surface about its own Z-axis by an angle α, about its Y-axis by an angle β, and about its Z-axis by an angle γ, respectively. Theoretical coordinate system G Transform to actual coordinate system L Rotation transformation matrix on; The location of the center point of the registration point is calculated as follows: ； in: X These are the coordinates of the registration points in the coordinate system. N To determine the number of registration points, The coordinates of the center of the registration point set; The center point of the three registration points on the actual coordinates The coordinates are: ； in: Center point The X-axis coordinate value, Center point Y-axis coordinate value, Center point Z-axis coordinate value; The center point of the three registration points on the theoretical coordinate system The coordinates are: ； in: Center point The X-axis coordinate value, Center point Y-axis coordinate value, Center point Z-axis coordinate value; The central auxiliary coordinate transformation model based on the Bursa-Wolf model is then constructed as follows: ； Abbreviated as: ; in: ； ； In the formula: These are the theoretical coordinates of the control point set. These are the actual coordinates of the control point set; Step 3: Solve the center-auxiliary coordinate transformation model based on the nonlinear least squares method and the Gauss-Newton iteration method to obtain seven parameters, and adjust the position of the curved surface according to the seven parameters to reach the optimal working position: First let ; The three-dimensional coordinate transformation model According to the least squares principle, the minimum value is: ； Simplified versions include: for ; Right now: ; Solve again: ; In the formula: ； The Gauss-Newton iteration method is used in the solution process, and the specific steps are as follows: Step 3.1: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require Linearization, at the reference point right Performing a Taylor expansion, we get: ； Step 3.2: Substitute the above formula into the equation. In this process, the next iteration formula is obtained as follows: ； During the iteration process, the iteration stops when the norm of the difference between two adjacent seven-parameter vectors meets the accuracy requirement or is less than a certain threshold, and the optimal solution under the least squares rule can be obtained. Step 3.3: The calculated x , y , z This represents the offset between the origin of the actual coordinate system and the origin of the theoretical coordinate system of the curved surface. The calculated α, β, and γ are the deflection angles between the actual coordinate system and the theoretical coordinate system of the surface solid. The calculated k is the scaling factor between the two coordinate systems; Step 3.4: Adjust based on offset and scaling ratio, compensate for corresponding displacement and deflection angle values ​​on the reference plane, and move the surface to the optimal working position.

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