CN113702994A - Laser tracker measurement accuracy improving method based on rigid constraint - Google Patents

Laser tracker measurement accuracy improving method based on rigid constraint Download PDF

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CN113702994A
CN113702994A CN202110927134.8A CN202110927134A CN113702994A CN 113702994 A CN113702994 A CN 113702994A CN 202110927134 A CN202110927134 A CN 202110927134A CN 113702994 A CN113702994 A CN 113702994A
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laser tracker
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CN113702994B (en
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刘巍
伍嘉豪
逯永康
赵伟康
郑研
张洋
马建伟
周颖皓
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Dalian University of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S17/00—Systems using the reflection or reradiation of electromagnetic waves other than radio waves, e.g. lidar systems
    • G01S17/66—Tracking systems using electromagnetic waves other than radio waves
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
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Abstract

The invention belongs to the field of large-size/digital measurement, and relates to a method for improving measurement accuracy of a laser tracker based on rigid constraint. The method comprises the steps of firstly setting a plurality of common points and test points based on a stability principle, and collecting three-dimensional coordinates of the common points and the test points by utilizing a plurality of stations of a laser tracker. And then, obtaining the transformed common point coordinates through rough transformation of a coordinate system. Setting weight according to the distance deviation of the common points before and after coarse conversion relative to the centroid; and finally, the coordinate system conversion is carried out by taking the minimum weighted sum of squares of the matched residual errors as an optimality condition, so that the high-precision solution of the conversion parameters of the measurement coordinate system among different stations of the laser tracker is realized. The method fully considers the influence of the common point measurement error on the coordinate system conversion precision, and the weight is set relative to the centroid distance deviation before and after the common point conversion, so that the influence of the measurement error is effectively reduced, and the coordinate system conversion precision is improved.

Description

Laser tracker measurement accuracy improving method based on rigid constraint
Technical Field
The invention belongs to the field of large-size/digital measurement, and relates to a method for improving measurement accuracy of a laser tracker based on rigid constraint.
Background
The core component structure of the large aerospace equipment is more and more complex, the size is larger and larger, the precision requirement is higher and higher, and higher requirements are provided for the measurement range, precision and robustness in the manufacturing process. The laser tracker is one of the most widely used large-size measuring instruments in the field of aerospace assembly measurement at present, but due to the conditions of large size, complex shape, on-site shielding and the like of an assembly object, the measurement task is difficult to complete by single-station measurement, and the complete measurement of a large-size component must be realized by adopting a multi-station cooperative measurement mode of the laser tracker. In the measurement process of large-scale components, a global coordinate system is generally established as a uniform assembly reference, partial reference points are selected on the ground, the reference unification of measurement data of the laser tracker at different stations is completed, and then the pose of the current workpiece in the global coordinate system is calculated through coordinate conversion. In the actual assembly process, the common datum point is generally arranged on the ground, a tool or a component, the measurement error of the common datum point acquired by the laser tracker under the conditions of different distances, angles, environments and the like has typical non-uniformity and anisotropy, and the overall measurement accuracy is difficult to guarantee. In the traditional method, uncertainty is measured by a theoretical analysis system, and the overall measurement precision is improved by establishing a theoretical measurement uncertainty weight, but pure theoretical analysis is difficult to be suitable for field measurement. Therefore, in order to improve the accuracy of multi-station measurement of the laser tracker, a large-scale measurement coordinate system conversion method must be studied.
At present, methods for resolving a coordinate system transformation matrix mainly include a quaternion method, a least square method, a singular value decomposition method and a Rodrigues matrix method. In 9.2011, beijing aerospace university chenwuyi, on the 'rigid body pose error detection method based on singular value decomposition' in 9 th volume of 'computer integrated manufacturing system', a singular value decomposition method is adopted to determine the actual pose of the moving platform, the sum of squares of position errors of measurement points after pose transformation is taken as a target function, the elimination of dead spots in actual measurement data is realized through a target function value, and the correctness of a result is ensured. The method provides good idea reference for improving the conversion precision of the coordinate system. However, since the number of common points for registration is limited, the elimination of too many unstable common points will cause the loss of registration accuracy, which is not favorable for the improvement of assembly accuracy.
Wudan et al, university of Qinghua, 12.2020 published "Construction and incertation Evaluation of Large-scale Measurement System of Laser transceivers in Aircraft Assembly" on Measurement, which proposed a multi-station beam adjustment method. And (3) constructing an integral measurement coordinate system by adjusting all the measurement stations and the enhanced reference points through light beams, and then registering the integral measurement coordinate system to the assembly coordinate system by using the stable reference enhanced points. The method unifies all laser trackers and reference enhanced point points into an overall measurement coordinate system through beam adjustment, greatly increasing the number of common points available for registration. Thus, elimination of some reinforced reference points with poor stability does not cause loss of registration accuracy. However, in the method, reference values such as a reference scale are not set, and system errors possibly caused by beam adjustment are ignored.
Disclosure of Invention
The invention overcomes the defects of the prior art, and provides a method for improving the measurement accuracy of a laser tracker based on rigid constraint aiming at the problems of large conversion error, poor robustness and the like during multi-station conversion of the laser tracker. The method comprises the steps that firstly, a certain number of common points and test points are arranged on the basis of a stability and envelopment principle, and a laser tracker respectively collects space coordinates of the key points at two stations; then, the singular value decomposition method is adopted to solve the conversion parameter R among the multiple coordinate systems in an equal weight mode0、T0(ii) a Obtaining the coordinates of each common point in the global coordinate system by using the conversion parameters; calculating the rigid distance deviation between the common point and the centroid distance before and after conversion according to the rigid distanceSetting weights from invariance constraints; and finally, converting a coordinate system by using the optimality condition with the minimum weighted sum of squares of the matched residual errors as a solution, thereby realizing the optimal solution of the conversion parameters between different stations of the laser tracker. Experiments prove that the method can effectively improve the conversion precision of the multi-station cooperative measurement of the laser tracker, avoids the complex solving process of multi-source error transfer modeling, and has the characteristics of high robustness, high precision and the like.
The technical scheme adopted by the invention is a method for improving the measurement precision of a laser tracker based on rigid constraint, which is characterized in that the method is characterized in that a plurality of common points and test points are arranged based on a stability source and envelopment, and the laser tracker collects the coordinates of the test points under two station positions; then, the singular value decomposition method is adopted to solve the conversion parameter R among the multi-coordinate systems in an equal weight mode0、T0And solving the coordinates of the common points under the global coordinate system according to the conversion parameters; determining a weight model according to the constraint that the rigid distance between the common point and the centroid is unchanged before and after conversion; finally, based on the weight coefficients, the conversion parameters of the coordinate systems of different stations of the laser tracker are solved by taking the minimum weighted sum of squares of the matched residual errors as a condition for solving optimality, so that the conversion precision among multiple coordinate systems in a large-size range is improved; the method comprises the following specific steps:
firstly, obtaining the common point coordinate of the multi-station cooperative measurement of the laser tracker
Firstly, mounting a plurality of common points and test points on a cement foundation or an assembly member of a measuring field; then, sequentially arranging the laser tracker at two different stations S1 and S2, carrying out j times of repeated measurement on n common points in a measurement field, and simultaneously collecting coordinates of the test points; the i-th common point coordinate obtained by the laser tracker in the station positions S1 and S2 for j times of repeated measurement is recorded as { P }r,iAnd { P }M,iTaking an average value of the obtained point set coordinates, namely the nominal values of the ith point under S1 and S2, and recording the nominal values as Pr,iAnd PM,iNamely:
Figure BDA0003209655450000031
Figure BDA0003209655450000032
wherein, the subscript j represents the number of repetitive measurements of the laser tracker; the k test point coordinates collected by the laser tracker at the station positions S1 and S2 are respectively recorded as Qr,k、QM,k。
Secondly, solving the conversion parameters of the multi-station coordinate system by equal weight based on the singular value decomposition method
Setting the objective function as
Figure BDA0003209655450000033
When the objective function reaches the minimum value, solving the objective function by utilizing an SVD decomposition method to obtain a conversion parameter R0、T0:
Figure BDA0003209655450000041
T0＝[tx ty tz]T (4)
Where α, β, γ denote the rotation angle of the measurement coordinate system of the station S1 with respect to the measurement coordinate system of the station S2, tx,ty,tzIndicating that the station S1 measures the position of the coordinate system origin in the measurement coordinate system of the station S2.
Calculating the mass center and covariance matrix of the two point sets by using a singular value decomposition method:
Figure BDA0003209655450000042
Figure BDA0003209655450000043
in the formula, mur、μMThe centroid of the common point coordinate point set measured by the first station and the second station is respectively, and the matrix H is a covariance matrix of the two point sets.
SVD is carried out on the covariance matrix H, so that H is UDVTWhere D is a diagonal matrix and U and V are orthogonal matrices, we can obtain:
Figure BDA0003209655450000044
if det R0When is +1, then R0For the solved rotation matrix, if det R0When it is 1, then let V0＝[v1 v2 -v3]Substituting to obtain: r0＝V0UT。
Thirdly, constructing a weight coefficient based on rigid distance constraint of common points
Rotation matrix R solved according to the second step0Translation vector T0And the formula (8),
Figure BDA0003209655450000047
converting the common point coordinate measured under the station position S2 to the position S1 to obtain a conversion point group
Figure BDA0003209655450000048
Measuring the point group { P) according to the geometric invariance of the rigid body distancer,iAnd set of switching points
Figure BDA0003209655450000051
Should have the same centroid, find the respective centroid coordinates:
Figure BDA0003209655450000052
the distance from the ith common point in the measurement point group and the conversion point group to the respective centroid is as follows:
Figure BDA0003209655450000053
due to the effect of common point measurement errors,
Figure BDA0003209655450000054
i.e. there is a non-ideal stiffness distance deviation:
Figure BDA0003209655450000055
this deviation is used as an evaluation reference index for the coordinate accuracy of the point. Calculating a weight coefficient according to the non-ideal rigid distance deviation of the distance between the common point and the center of mass before and after conversion according to the principle that the deviation and the weight are negatively related:
order to
Figure BDA0003209655450000056
The weight is set to:
Figure BDA0003209655450000057
after homogenization, the following steps are performed:
Figure BDA0003209655450000058
the fourth step, coordinate conversion with the minimum of the weighted sum of squares of the matching residuals as the optimal solution
According to the proposed weight coefficient calculation method and the collected common point coordinates { P ] under the measuring coordinate system of the two laser tracker stations S1 and S2r,iAnd { P }M,iAnd the corresponding optimized objective function with the minimum weighted square sum of the matching residuals is as follows:
Figure BDA0003209655450000059
when the objective function value is minimum, the objective function is solved by using an SVD decomposition method to obtain the conversion parameters R, T.
According to the set weight, solving the weighted centroid of the common point coordinates acquired under the laser tracker stations S1 and S2:
Figure BDA00032096554500000510
to translate a vector from Pr,i＝(RPM,i+ T), simplifying the calculation and solving process, and respectively centroiding the two groups of coordinates:
Figure BDA0003209655450000061
the objective function for which the weighted sum of squares of the corresponding matching residuals is minimal is:
Figure BDA0003209655450000062
solving the objective function minimization problem to solve
Figure BDA0003209655450000063
In the formula:
Figure BDA0003209655450000064
SVD decomposition is carried out on H to obtain H ═ U Λ VTWhere Λ is a diagonal matrix, U and V are orthogonal matrices, then the rotation matrix is: r ═ VUTThe translation vector is:
Figure BDA0003209655450000065
through the steps, high-precision conversion of different coordinate systems in a large-size range can be realized.
The method has the advantages that the target function is constructed by using the optimality condition of the minimum solution of the weighted square sum of the matching residuals, the weight is determined by using the deviation of the distance between the common points before and after the rough conversion relative to the centroid, the high-precision solution of R, T for measuring the conversion parameters between coordinate systems under different station positions of the laser tracker is realized, the coordinate conversion of the common points is completed, the conversion precision between the coordinate systems is improved, and the influence of the matching residuals on the conversion result is reduced. Practice proves that: the method has the advantages of good practicability and universality, good robustness, high precision and high stability. The method can be used for the assembly process of large mechanical products such as airplanes in the aerospace field, can effectively reduce the influence of common point measurement errors in the installation process, and has wide application prospect.
Drawings
FIG. 1 is a flow chart of a method for improving measurement accuracy of a laser tracker based on rigid constraint;
FIG. 2 is a schematic view of a laser tracker multi-site measurement; wherein S1 and S2 are respectively different station positions of the laser tracker; p1～P5The black dots are common dots; q1～Q3The black triangle is a test point; O-XrYrZrFor the measuring coordinate system of the laser tracker station S1, O-XMYMZMIs the measurement coordinate system of the laser tracker station S2.
FIG. 3 is a diagram of common point coordinate weight construction. Wherein P is1～P5Measuring a common point of the laser tracker under a coordinate system at a station position S1; the black rhombus is the centroid of the common point group; d1～d5Distance of the common point to its centroid; p'1～P′5Is a common point, d 'in the measurement coordinate system converted to the station site S2 after the rough conversion of the coordinate system'1～d′5Is the distance of the common point to the centroid after the coarse transformation.
Detailed Description
The following detailed description of the invention refers to the accompanying drawings.
The laser tracker selected by the embodiment is of a type of Leica AT960MR, and the maximum allowable measurement error MPE is ± (15 μm +6 μm/m); the target ball and the target ball holder used as the measurement point were 0.5 inches in size.
As shown in the attached figure 1, the method comprises the steps of firstly setting a plurality of common points and test points based on a stability principle, and repeatedly measuring the coordinates of the common points at two different station positions by using a laser tracker; then, coarse conversion of the measurement coordinate system under the two stations is carried out based on a singular value decomposition method to obtain a conversion parameter R between the two stations0、T0(ii) a Using the obtained coarse transformation parameter R0、T0Calculating the transformed common point coordinate set,determining the weight according to the deviation of the distance between the common points and the centroid before and after the rough transformation, as shown in figure 3; finally, according to the set weight coefficient, the coordinate system conversion is carried out by taking the minimum weighted sum of squares of the matched residual errors as a solution optimality condition, so that the solution of coordinate system conversion parameters of the laser tracker at different stations is realized, and the conversion precision of the coordinate system measured in a large-size range is improved; in order to meet the processing and measurement requirements of large-size aerospace core components, a laser tracker needs to be adopted for multi-station cooperative measurement. In order to realize the reference unification of the measurement data of the laser tracker under different station positions, the station position s1 is set as a reference station position, and the coordinates of a common point are collected by adopting multiple stations of the laser tracker; theoretically, the coordinates of the common point acquired by the laser tracker at different stations should be matched with each other after coordinate system conversion, but due to the influence of common point measurement error and random error, the coordinates of the common point acquired at station s2 are not matched with the coordinates of the common point acquired at station s1 after coordinate conversion. As shown in the third figure, based on the principle that the common point centroid is not changed before and after conversion, P4After the point is subjected to coordinate transformation, the distance of the point obviously changes relative to the centroid, and then P4The point is a point with a larger error, and the discovery is utilized to firstly set a coarse conversion, and the weight is set according to the deviation of the distance of the center of mass before and after the coarse conversion of the common point, so that the high-precision solution of the coordinate system conversion parameters of the laser tracker at different station positions is realized.
The method comprises the following specific steps:
first, common point coordinate acquisition based on multi-station laser tracker measurement
Firstly, mounting 8 common points and 3 test points at a position with better rigidity of a cement foundation or an assembly member of a measuring field; the laser trackers are then positioned at two different stations S1, S2 in sequence, with 8 common points P in the field of view1～P8Repeat measurement 100 times and simultaneously collecting test point Q1～Q3The coordinates of (a); defining a point set obtained by respectively carrying out 100 times of repeated measurement on the ith common point coordinate by the laser tracker at station positions S1 and S2 as Pr,iAnd { P }M,iIts average value is defined asNominal values of point i at S1, S2, denoted as Pr,iAnd PM,i：
Figure BDA0003209655450000081
Figure BDA0003209655450000091
The k test point coordinates collected by the laser tracker at the station positions S1 and S2 are respectively recorded as Qr,k、QM,k：
Figure BDA0003209655450000092
Secondly, solving the multi-station coordinate system rough conversion parameters based on the singular value decomposition method
And (3) calculating the mean value and covariance matrix of the common point coordinates under the two stations according to the formula (5) and the formula (6) by using an SVD decomposition method:
Figure BDA0003209655450000093
Figure BDA0003209655450000094
SVD is carried out on the covariance matrix H, so that H is UDVTFrom equation (7), we can obtain:
Figure BDA0003209655450000095
T0＝μM-R0μr＝(11863.4159 3933.2543 -139.3727)
thirdly, constructing a weight coefficient based on rigid constraint of common points
Using the rotation matrix obtained in the second stepR0Translation vector T0Converting the coordinates of the common point measured at the station position S2 to the position S1 according to the formula (8) to obtain a conversion point group
Figure BDA0003209655450000101
Measurement point set { Pr,iAnd set of switching points
Figure BDA0003209655450000102
Should have the same centroid, the respective centroid coordinates are found according to equation (9):
Figure BDA0003209655450000103
calculating the distances from the common points in the measurement point group and the theoretical point group to the respective centroids according to the formula (10) as follows:
Figure BDA0003209655450000104
the deviation exists:
Figure BDA0003209655450000105
this deviation is used as an evaluation index of the coordinate accuracy of the point. Order to
Figure BDA0003209655450000106
The weight coefficient is set by equation (11):
W＝(w1,w2,w3,w4,w5,w6,w7,w8)
＝(0.1444 0.0261 0.0178 0.1100 0.0985 0.2270 0.0799 0.2963)
the fourth step, coordinate conversion with the minimum of the weighted sum of squares of the matching residuals as the optimal solution
According to the determined weight and the common point coordinate set { P } measured by the laser tracker under the station S1, S2 measurement coordinate systemr,iAnd { P }M,iGet the corresponding matching residual weighted averageThe minimum and the square optimization objective function is:
Figure BDA0003209655450000107
when the objective function reaches a minimum, the conversion parameters R, T are obtained using SVD decomposition.
According to the set weight, solving the weighted centroid of the common point coordinates acquired under the laser tracker stations S1 and S2:
Figure BDA0003209655450000111
Figure BDA0003209655450000112
to translate a vector from Pr,i＝(RPM,i+ T), simplifying the calculation and solving process, and respectively centroiding the two groups of coordinates:
Figure BDA0003209655450000113
Figure BDA0003209655450000114
the objective function can be rewritten as:
Figure BDA0003209655450000121
solving the objective function minimization translates to solving:
Figure BDA0003209655450000122
in the formula:
Figure BDA0003209655450000123
SVD of HSolution, i.e. H ═ U Λ VTThe rotation matrix can be obtained:
Figure BDA0003209655450000124
the translation vector is:
Figure BDA0003209655450000125
through the steps, high-precision conversion of different measurement coordinate systems in a large-size range can be realized.
According to the invention, the target function is constructed by matching the optimality condition of the minimum solution of the weighted sum of squares of the residual errors, and the weight is determined by using the deviation of the distance relative to the centroid before and after the rough conversion of the common point, so that the R, T high-precision solution of the conversion parameters between the measurement coordinate systems of the laser tracker at different stations is realized, and the influence of the measurement error of the common point of the laser tracker is reduced. The method has guiding significance for improving the conversion precision of the multi-station coordinate system of the laser tracker, and has the characteristics of good robustness, high precision, high stability and higher application value.

Claims (1)

1. A method for improving the measurement precision of a laser tracker based on rigid constraint is characterized in that a plurality of common points and test points are set based on a stability principle, and the coordinates of the common points and the test points at two different station positions are collected by the laser tracker; then, coarse conversion of the measurement coordinate system under two stations is carried out by using a singular value decomposition method to obtain a coarse conversion parameter R between the two stations0、T0And according to R0、T0Calculating coordinates of the common points after conversion, and determining the weight according to the deviation of the distances of the common points before and after coarse conversion relative to the centroid; finally, based on the set weight, the coordinate system conversion is carried out by taking the minimum weighted sum of squares of the matched residual errors as a solution optimality condition, the solution of coordinate system conversion parameters of the laser tracker at different station positions is completed, and the conversion precision of the large-size range measurement coordinate system is improved; the method comprises the following specific steps:
first, common point coordinate acquisition based on multi-station laser tracker measurement
Firstly, mounting a plurality of common points and test points on a cement foundation or an assembly member of a measuring field; then, sequentially arranging the laser tracker at two different stations S1 and S2, carrying out j times of repeated measurement on n common points in a measurement field, and simultaneously collecting coordinates of the test points; the i-th common point coordinate obtained by the laser tracker in the station positions S1 and S2 for j times of repeated measurement is recorded as { P }r,iAnd { P }M,iTaking an average value of the obtained point set coordinates, namely the nominal values of the ith point under S1 and S2, and recording the nominal values as Pr,iAnd PM,iNamely:
Figure FDA0003209655440000011
Figure FDA0003209655440000012
wherein, the subscript j represents the number of repetitive measurements of the laser tracker; the k test point coordinates collected by the laser tracker at the station positions S1 and S2 are respectively recorded as Qr,k、QM,k；
Secondly, solving the multi-station coordinate system rough conversion parameters based on the singular value decomposition method
Setting the objective function as
Figure FDA0003209655440000021
When the objective function reaches the minimum value, solving the objective function by utilizing an SVD decomposition method to obtain a coarse conversion parameter R0、T0:
Figure FDA0003209655440000022
T0＝[tx ty tz]T (4)
Wherein, alpha, beta and gamma represent a measuring coordinate system of the station S1Rotation angle, t, of the measuring coordinate system relative to the station position S2x,ty,tzIndicating the position of the origin of the measurement coordinate system of the station S1 in the measurement coordinate system of the station S2;
calculating the mean value and covariance matrix of the two point sets by adopting an SVD decomposition method:
Figure FDA0003209655440000023
Figure FDA0003209655440000024
in the formula, mur、μMThe mean value of the common point coordinate point sets measured by the first station and the second station is respectively, and the matrix H is a covariance matrix of the two point sets;
SVD is carried out on the covariance matrix H, so that H is UDVTIn the formula, D is a diagonal matrix, and U and V are orthogonal matrices, and the following are obtained:
Figure FDA0003209655440000025
if det R0When is +1, then R0For the solved rotation matrix, if det R0When the result is-1, then let V0＝[v1 v2 -v3]Substituting to obtain: r0＝V0UT；
Thirdly, constructing a weight coefficient based on rigid constraint of common points
Rotation matrix R obtained from the coarse conversion0Translation vector T0And the formula (8),
Figure FDA0003209655440000031
converting the common point coordinate measured under the station position S2 to the position S1 to obtain a conversion point group
Figure FDA0003209655440000032
Measuring the set of points according to the geometric invariance of the rigid body { P }r,iAnd set of switching points
Figure FDA0003209655440000033
Should have the same centroid, find the respective centroid coordinates:
Figure FDA0003209655440000034
the distance from the ith common point in the measurement point group and the theoretical point group to the respective centroid is as follows:
Figure FDA0003209655440000035
due to the effect of common point measurement errors,
Figure FDA0003209655440000036
namely, there is a deviation:
Figure FDA0003209655440000037
taking the deviation as an evaluation index of the coordinate precision of the point; according to the principle that the larger the deviation is, the smaller the weight is, and the smaller the deviation is, the larger the weight is, the deviation before and after the rough conversion from the common point relative to the centroid distance is the set weight:
order to
Figure FDA0003209655440000038
The weight is set to:
Figure FDA0003209655440000039
after homogenization, the following steps are performed:
Figure FDA00032096554400000310
the fourth step, coordinate conversion with the minimum of the weighted sum of squares of the matching residuals as the optimal solution
According to the determined weight and the collected common point coordinate set { P ] under the measuring coordinate system of the two laser tracker stations S1, S2r,iAnd { P }M,iAnd the corresponding optimized objective function with the minimum weighted square sum of the matching residuals is as follows:
Figure FDA00032096554400000311
when the objective function value is minimum, solving the objective function by utilizing SVD decomposition to obtain fine conversion parameters R, T;
according to the set weight, solving the weighted centroid of the common point coordinates acquired under the laser tracker stations S1 and S2:
Figure FDA0003209655440000041
to derive a translation vector from the formula Pr,i＝(RPM,i+ T), simplifying the calculation and solving process, and respectively centroiding the two groups of coordinates:
Figure FDA0003209655440000042
the objective function for which the weighted sum of squares of the corresponding matching residuals is minimal is:
Figure FDA0003209655440000043
solving the objective function minimization problem to solve
Figure FDA0003209655440000044
In the formula:
Figure FDA0003209655440000045
SVD decomposition is carried out on H to obtain H ═ U Λ VTWhere Λ is a diagonal matrix, U and V are orthogonal matrices, then the rotation matrix is: r ═ VUTAccording to the formula Pr,i＝(RPM,i+ T) finding the translation vector T of each common pointiThen the translation vector is:
Figure FDA0003209655440000046
through the steps, high-precision conversion of different coordinate systems in a large-size range is achieved.
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