A method for measuring the tilt of a prism bar

Abstract

The application relates to the technical field of engineering surveying, in particular to a prism rod inclination measuring method, which comprises the following steps: S1, fixing the bottom of a prism rod at a point to be measured, tilting the prism rod multiple times, and obtaining the coordinates of the observation points after each tilting; S2, constructing an original observation equation set and calculating an average equation according to the coordinates of the observation points obtained each time, taking the average equation as a reference, linearly processing each original observation equation, and obtaining a linear observation equation set; S3, solving the initial coordinate estimation value of the point to be measured based on the linear observation equation set through an equal-weight least square method; S4, calculating the deviation absolute values of the elevation values of the observation points and the initial elevation estimation value of the point to be measured, and completing the weight distribution of the coordinates of the observation points according to the deviation absolute values; and S5, solving the coordinates of the point to be measured through a weighted least square method, so that the measurement of the coordinates of the point to be measured is more reasonable and accurate.

Description

Technical Field

[0001] This application relates to the field of engineering surveying technology, and in particular to a method for measuring the tilt of a prism rod. Background Technology

[0002] In engineering surveying, using a total station in conjunction with a prism to obtain point coordinates is a conventional method, but it requires the prism to be perfectly level. In actual engineering projects, due to site constraints (such as building corners, narrow spaces, etc.), it is often difficult to level the prism. Existing tilt prism measurement methods establish and solve equations through multiple tilt measurements, but they have the following shortcomings: 1. Linearization methods suffer from benchmark dependency: Existing methods linearize by subtracting the first equation from all equations. If the first observation has gross errors or large errors, these errors will propagate to all equations, affecting the accuracy of the solution.

[0003] 2. Lack of geometric basis for weight allocation: Existing weighting methods fail to make full use of the geometric information of the measurement data itself, and the weight allocation lacks a direct correlation with the distribution of observation points.

[0004] Therefore, there is a need to provide an improved technical solution that addresses the shortcomings of the existing technology. Summary of the Invention

[0005] The purpose of this application is to provide a method for measuring the tilt of a prism rod, so as to solve or alleviate the problems existing in the prior art.

[0006] To achieve the above objectives, this application provides the following technical solution: A method for measuring the tilt of a prism rod, the method comprising: Step S1: Fix the bottom of the prism rod to the point to be measured, tilt the prism rod multiple times, and obtain the coordinates of the observation point after each tilt; Step S2: Based on the coordinates of the observation points obtained each time, construct the original observation equation system and calculate the average equation. Using the average equation as a benchmark, linearize each original observation equation to obtain the linear observation equation system. Step S3: Based on the linear observation equation system, solve for the estimated initial coordinates of the points to be measured using the equal-weighted least squares method; Step S4: Extract the elevation values ​​from the coordinates of the observation points and the elevation estimates from the initial coordinate estimates of the points to be measured. Calculate the absolute value of the deviation between the elevation values ​​of each observation point and the initial elevation estimates of the points to be measured. Based on the absolute values ​​of the deviations, complete the weight allocation of the coordinates of each observation point. Step S5: Calculate the coordinates of the point to be measured using the weighted least squares method.

[0007] Preferably, the measurement method further includes: Step S6: Use the coordinates of the test point calculated in step S5 as the new initial coordinate estimate of the test point in step S4. Repeat steps S4-S6 until the calculated coordinates of the test point obtained in step S5 pass the accuracy evaluation and reliability analysis.

[0008] Preferably, in step S6, the indicators used for accuracy evaluation include residual, unit weight error, cofactor matrix, parametric covariance matrix, coordinate component error, and point position error.

[0009] Preferably, in step S6, the reliability analysis includes internal reliability and external reliability.

[0010] Preferably, step S2 specifically includes: Step S2.1, construct the original observation equation set: , In the formula, The coordinates of the point to be measured; ; This represents the total number of observations. The length of the prism rod; Step S2.2, calculate the average coordinates of the observation points: , Step S2.3, construct the average equation: Adding the original observation equations and dividing by After unfolding and organizing, we get: , in, ; Step S2.4, Linearization: Subtract the average equation from each set of original observation equations to eliminate... The items, when expanded and rearranged, yield: , in, .

[0011] Preferably, step S3 specifically comprises: , in, Let the estimated initial coordinates of the point to be measured be denoted as: ; ; .

[0012] Preferably, step S4 includes: Step S4.1: Calculate the absolute value of the deviation between the elevation values ​​of each observation point and the initial elevation estimate of the point to be measured. : ; Step S4.2: Calculate the coordinate weights of each observation point. : , in, ; For exponential parameters; This is the minimum weight threshold.

[0013] Preferably, step S5 includes: Step S5.1: Construct the objective function based on the weighted least squares method: According to the least squares principle, the weighted sum of squared residuals of the observation point coordinates satisfy: ; in, Let be the coordinate vector of the point to be measured, represented as ; ; Step S5.2, Solve the normal equation: Regarding the objective function Taking the derivative and setting it to zero, we get: ; Step S5.3: Obtain the coordinates of the point to be measured based on the normal equation. The solution value .

[0014] Compared with the closest prior art, the technical solution of this application has the following beneficial effects: This application uses the average equation as a benchmark to linearize the original observation equation set, which can eliminate the benchmark dependency problem caused by traditional linearization and avoid the propagation of gross errors at single points. At the same time, it also introduces the elevation deviation between the observation point and the point to be measured to participate in the weight allocation, so as to achieve a balanced treatment of the constraints of each observation point, making the coordinate measurement of the point to be measured more reasonable and accurate. Attached Figure Description

[0015] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. Wherein: Figure 1 This is a schematic flowchart of a prism rod tilt measurement method according to some embodiments of this application; Figure 2This is a schematic diagram of the prism rod arrangement according to some embodiments of this application.

[0016] Explanation of reference numerals in the attached figures: 1. The point to be measured; 2. The observation point; 3. The prism rod. Detailed Implementation

[0017] The present application will now be described in detail with reference to the accompanying drawings and embodiments. Various examples are provided by way of explanation and not by way of limitation. In fact, those skilled in the art will recognize that modifications and variations can be made to the present application without departing from the scope or spirit thereof. For example, a feature shown or described as part of one embodiment may be used in another embodiment to produce yet another embodiment. Therefore, it is desirable that the present application encompass such modifications and variations that fall within the scope of the appended claims and their equivalents.

[0018] In the following description, the terms "first / second / third" are used merely to distinguish similar objects and do not represent a specific order of objects. It is understood that "first / second / third" may be interchanged in a specific order or sequence where permitted, so that the embodiments of this application described herein can be implemented in an order other than that illustrated or described herein.

[0019] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. The terminology used herein is for the purpose of describing embodiments of this disclosure only and is not intended to limit this disclosure.

[0020] In the description of this application, the terms "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," and "bottom," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this application and do not require that this application be constructed and operated in a specific orientation, and therefore should not be construed as limiting this application. The terms "connected," "linked," and "set up" used in this application should be interpreted broadly. For example, they can refer to fixed connections or detachable connections; direct connections or indirect connections through intermediate components; wired connections, radio connections, or wireless communication signal connections. Those skilled in the art can understand the specific meaning of the above terms according to the specific circumstances.

[0021] The following will be combined with the appendix Figure 1-2 The present application provides a more detailed description of a method for measuring the tilt of a prism rod 3.

[0022] A method for measuring the tilt of a prism rod 3, the method comprising: Step S1: Fix the bottom of the prism rod 3 to the point to be measured 1, tilt the prism rod 3 multiple times, and obtain the coordinates of the observation point 2 after each tilt; Step S2: Based on the coordinates of observation point 2 obtained each time, construct the original observation equation system and calculate the average equation. Using the average equation as the benchmark, linearize each original observation equation to obtain the linear observation equation system. Step S3: Based on the linear observation equation system, solve for the initial coordinate estimate of the point to be measured 1 using the equal-weight least squares method; Step S4: Extract the elevation values ​​from the coordinates of observation point 2 and the elevation estimates from the initial coordinate estimates of the point to be measured 1. Calculate the absolute value of the deviation between the elevation values ​​of each observation point 2 and the initial elevation estimates of the point to be measured 1. Based on the absolute values ​​of the deviations, complete the weight allocation of the coordinates of each observation point 2. Step S5: Solve for the coordinates of the point to be measured 1 using the weighted least squares method.

[0023] In a specific embodiment of this application, Step S1 includes: Step S1.1: Precisely fix the bottom of the prism rod 3 to the point to be measured 1, ensuring the rod length is correct. Known and fixed; Step S1.2: Tilt prism rod 3 arbitrarily and use a total station to measure the coordinates of the prism center. (This step is repeated once.) Second measurement, record the coordinates of observation point 2; routine measurement: High precision requirements: Ensure that at least one measurement shows prism rod 3 is away from the horizontal plane (estimated tilt angle is not in the range of 70°~110°); otherwise, increase the number of observations. .

[0024] Step S1.3: Check data quality and remove obvious outliers.

[0025] This application uses the average equation as a benchmark to linearize the original observation equation set, which can eliminate the benchmark dependency problem caused by traditional linearization and avoid the propagation of gross errors at single points. At the same time, it also introduces the elevation deviation between observation point 2 and the measured point 1 to participate in the weight allocation, so as to achieve a balanced treatment of the constraints of each observation point 2, making the coordinate measurement of the measured point 1 more reasonable and accurate.

[0026] To further improve the measurement accuracy of the coordinates of point 1, the measurement method also includes: Step S6: Use the coordinates of the test point 1 calculated in step S5 as the new initial coordinate estimate of the test point 1 in step S4, and repeat steps S4-S6 until the calculated coordinates of the test point 1 pass the accuracy evaluation and reliability analysis.

[0027] In step S6, the indicators used for accuracy assessment include residuals, unit weight error, cofactor matrix, parametric covariance matrix, coordinate component error, and point position error.

[0028] In step S6, the reliability analysis includes internal reliability and external reliability.

[0029] According to the "Code for Engineering Surveying" (GB50026-2020) and surveying industry practice, in this embodiment, the qualified threshold for unit weight error is set to 2.0 mm, and the coordinate component error is no more than 3.0 mm for plane and no more than 5.0 mm for elevation.

[0030] Step S2 specifically includes: Step S2.1, construct the original observation equation set: Observation point 2 (prism center) can be considered as distributed around point 1 as the center of the sphere, with a rod length of [missing information]. A point on a sphere with radius .

[0031] According to the principles of spherical geometry, each observation point 2 satisfies the following equation: , In the formula, Let the coordinates of point 1 be the coordinates of the point to be measured. ; This represents the total number of observations. The prism rod is 3 rods long; Step S2.2, calculate the average coordinates of observation point 2: , Step S2.3, construct the average equation: Will Adding the original observation equations and dividing by have to: , After unfolding and organizing, we get: , in, ; Step S2.4, Linearization: Subtract the average equation from each set of original observation equations to eliminate... item; As in the first Subtracting the average equation from the original observation equation yields: ; After unfolding and organizing, we get: , in, .

[0032] Step S3 is as follows: , in, Let the estimated initial coordinates of point 1 be denoted as: ; ; .

[0033] In the spherical geometric model, the tilt angle of prism rod 3 Elevation value of observation point 2 The following relationship exists:

[0034] in Let be the elevation value of point 1 to be measured. Therefore, we can obtain: when When (prism rod 3 is vertically upward), It reaches its maximum value; when When (prism rod 3 is horizontal), ; when When (prism rod 3 is vertically downward), Reaching the minimum value The position of each observation point 2 on the sphere determines the constraint direction of the measurement point 1: Observations in near-vertical direction ( or Observation point 2 is located near the "two poles" of the sphere. Its position has a strong constraint on the horizontal coordinate of observation point 2, but a weak constraint on its elevation. Near-horizontal observation Observation point 2 is located near the "equator" of the sphere. Its location has a strong constraint on elevation but a weak constraint on horizontal coordinates.

[0035] Therefore, the elevation deviation of observation point 2 This directly reflects the latitudinal position of observation point 2 on the sphere, that is, The larger the value, the closer observation point 2 is to the poles, the stronger the constraint on the horizontal coordinates, and the weaker the constraint on the elevation coordinates. The smaller the value, the closer observation point 2 is to the equator, the weaker the constraint on the horizontal coordinates, and the stronger the constraint on the elevation. This relationship can be rigorously proven by partial derivatives: the observation equations for... The absolute value of the partial derivative is ,right The absolute value of the partial derivative and Proportional.

[0036] Therefore, the weight allocation of the coordinates of observation point 2 can be made based on the absolute value of the elevation deviation between observation point 2 and the point to be measured 1.

[0037] Step S4 includes: Step S4.1: Calculate the absolute value of the deviation between the elevation values ​​of each observation point 2 and the initial elevation estimate of the point to be measured 1. : ; Step S4.2: Calculate the coordinate weights of each observation point. : , in, ; For exponential parameters; This is the minimum weight threshold.

[0038] In a specific embodiment of this application, the exponential parameter (Square weighting effectively highlights the contribution of observations near the poles and prioritizes locking in horizontal coordinates); minimum weight threshold (The best balance can be struck between preserving directional information and avoiding overweighting).

[0039] Step S5 includes: Step S5.1: Construct the objective function based on the weighted least squares method: According to the least squares principle, the weighted sum of squared residuals of the coordinates of observation point 2... satisfy: , in, This is the residual vector, with the same dimension as the number of original observation equations, representing the difference between the calculated values ​​and the observed values ​​of the original observation equation set.

[0040] ; in, Let be the coordinate vector of point 1 to be measured, denoted as: ; ; Step S5.2, Solve the normal equation: Regarding the objective function Taking the derivative and setting it to zero, we get: ; Step S5.3: Obtain the coordinates of the point to be measured 1 based on the normal equation. The solution value : matrix The weighted least squares solution for the coordinates of the point to be measured (point 1) is reversible and can be obtained as follows: , When all two observation points are coplanar, their elevations remain unchanged, or the number of observations is less than three, the matrix... Irreversible situations may occur, but this application has prevented such situations from happening by avoiding all observation points 2 being coplanar, setting changes in the elevation value of observation point 2, or increasing the number of observations.

[0041] In step S6, the coordinates of the point to be measured 1 are... The solution value Accuracy assessment and reliability analysis shall be performed according to the following formula: The formula for calculating residuals is: .

[0042] The formula for calculating the unit weight mean square error (representing the accuracy estimate of a linear observation equation) is as follows: ; This is used to verify the overall internal consistency accuracy of the observation sequence formed by all linear observation equations in this study. 3 represents the number of observation equations, and 3 represents the number of unknown parameters. These are redundant observations.

[0043] The formula for calculating the cofactor matrix is: ; This matrix represents the coordinates of point 1 to be measured. The cofactor matrix or weight inverse matrix is ​​the basis for calculating the final accuracy index.

[0044] The formula for calculating the parametric covariance matrix is: ; This matrix fully expresses the coordinates of the point to be measured, point 1. The solution value The accuracy of the coordinates and the correlation between their components are given. The diagonal elements represent the variance of each coordinate component, and the off-diagonal elements represent the covariance.

[0045] The formula for calculating the mean square error of coordinate components is: ; in, , For matrix The main diagonal elements are direct indicators for evaluating the accuracy of the components of the coordinate solution of the measured point 1 in each coordinate axis direction. They are used to compare with the tolerance requirements of the engineering design to determine whether the accuracy requirements are met.

[0046] The formula for calculating the positional error is: ; The standard deviation of the spatial distance between the actual position of the measured point 1 and its calculated value directly reflects the overall spatial positioning accuracy of the calculated coordinates of the measured point 1 and can be directly used as the basis for judging whether the measurement results are qualified.

[0047] Internal reliability analysis: Calculating the redundant observation components of the two coordinates of each observation point is an indicator for evaluating the ability of a measurement system to detect gross errors, reflecting the inherent ability of the adjustment system to resist observation errors. ; in, Coefficient matrix The OK, For the first The weights of the coordinates of each observation point.

[0048] External reliability analysis: Calculating the undetectable gross error limits for the coordinates of each observation point is an indicator for assessing the impact of undetected gross errors on the final results, reflecting the sensitivity of the observation system to external disturbances. ; in, This is a non-centralized parameter, typically set to 3.0~4.0; For the first The weights of the coordinates of each observation point.

[0049] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A method for measuring the tilt of a prism rod, characterized in that, The measurement method includes: Step S1: Fix the bottom of the prism rod to the point to be measured, tilt the prism rod multiple times, and obtain the coordinates of the observation point after each tilt; Step S2: Based on the coordinates of the observation points obtained each time, construct the original observation equation system and calculate the average equation. Using the average equation as a benchmark, linearize each original observation equation to obtain the linear observation equation system. Step S3: Based on the linear observation equation system, solve for the estimated initial coordinates of the points to be measured using the equal-weighted least squares method; Step S4: Extract the elevation values ​​from the coordinates of the observation points and the elevation estimates from the initial coordinate estimates of the points to be measured. Calculate the absolute value of the deviation between the elevation values ​​of each observation point and the initial elevation estimates of the points to be measured. Based on the absolute values ​​of the deviations, complete the weight allocation of the coordinates of each observation point. Step S5: Calculate the coordinates of the point to be measured using the weighted least squares method; Step S2 specifically includes: Step S2.1, construct the original observation equation set: ， In the formula, The coordinates of the point to be measured; ; This represents the total number of observations. The length of the prism rod; Step S2.2, calculate the average coordinates of the observation points: ， Step S2.3, construct the average equation: Adding the original observation equations and dividing by After unfolding and organizing, we get: ， in, ； Step S2.4, Linearization: Subtract the average equation from each set of original observation equations to eliminate... The items, when expanded and rearranged, yield: ， in, ； Step S3 specifically involves: ， in, Let the estimated initial coordinates of the point to be measured be denoted as: ; ； ； Step S4 includes: Step S4.1: Calculate the absolute value of the deviation between the elevation values ​​of each observation point and the initial elevation estimate of the point to be measured. : ； Step S4.2: Calculate the coordinate weights of each observation point. : ， in, ; For exponential parameters; This is the minimum weight threshold.

2. The method for measuring the tilt of a prism rod as described in claim 1, characterized in that, The measurement method further includes: Step S6: Use the coordinates of the test point calculated in step S5 as the new initial coordinate estimate of the test point in step S4. Repeat steps S4-S6 until the calculated coordinates of the test point obtained in step S5 pass the accuracy evaluation and reliability analysis.

3. The method for measuring the tilt of a prism rod as described in claim 2, characterized in that, In step S6, the indicators used for accuracy evaluation include residual, unit weight error, cofactor matrix, parametric covariance matrix, coordinate component error, and point position error.

4. The method for measuring the tilt of a prism rod as described in claim 2, characterized in that, In step S6, the reliability analysis includes internal reliability and external reliability.

5. The method for measuring the tilt of a prism rod as described in claim 1, characterized in that, Step S5 includes: Step S5.1: Construct the objective function based on the weighted least squares method: According to the least squares principle, the weighted sum of squared residuals of the observation point coordinates satisfy: ； in, Let be the coordinate vector of the point to be measured, represented as ; ； Step S5.2, Solve the normal equation: Regarding the objective function Taking the derivative and setting it to zero, we get: ； Step S5.3: Obtain the coordinates of the point to be measured based on the normal equation. The solution value .

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