CN-122020769-A - Space circle determining method and system for shield steel ring installation position

CN122020769ACN 122020769 ACN122020769 ACN 122020769ACN-122020769-A

Abstract

The embodiment of the invention provides a method and a system for determining a space circle of a shield steel ring installation position, wherein the method comprises the following steps of firstly, acquiring coordinates of at least n space measuring points; the method comprises the steps of fitting based on coordinates of space measuring points to obtain a fitting plane, fitting based on coordinates of the space measuring points to obtain a fitting sphere, taking a plane circle intersected by the fitting plane and the fitting sphere as a target space circle, judging whether the target space circle is available or not based on flatness deviation and roundness error threshold values of each space measuring point, eliminating a worst space measuring point if the target space circle is unavailable, executing the steps from the second to the fifth by adopting residual space measuring points until the target space circle is available, and calculating by adopting the residual 3 space measuring points by utilizing three space points if the residual space measuring points are 3. The method does not need an initial value or iteration, and has simple principle and strong regularity.

Inventors

LIU ZHIMING
Tang Fashu
HUANG XIN
WANG PENG
HU CHENGXIANG
XIANG ZHIYUAN
TAO SEN
CHEN WEI
SONG YUEJUN
YU MENG
MA LIANGLIANG
DONG WEIDONG
LI XIANG
REN GAN
ZHAO YONG

Assignees

绍兴市轨道交通集团有限公司
北京城建勘测设计研究院有限责任公司

Dates

Publication Date: 20260512
Application Date: 20251204

Claims (10)

1. The method for determining the space circle of the installation position of the shield steel ring is characterized by comprising the following steps of: the method comprises the steps of firstly, acquiring coordinates of n space measuring points in a preset installation range of the end part of an underground shield tunnel of a shield steel ring to be installed, wherein the space measuring points are not strictly positioned in the same plane, and n is at least 4; Fitting based on the coordinates of the space measuring points to obtain a fitting plane, wherein the fitting plane meets the minimum of objective functions constructed for the fitting plane, and the objective functions constructed for the fitting plane mean that the square sum of the distances from all the space measuring points to the fitting plane is minimum; fitting is carried out based on the coordinates of the space measuring points to obtain a fitted sphere, the fitted sphere meets the minimum objective function constructed for the fitted sphere, and the objective function constructed for the fitted sphere is obtained based on spherical fitting deviation of all the adopted space measuring points; Step four, taking a plane circle intersecting the fitting plane and the fitting sphere as a target space circle; Judging whether a target space circle is available or not based on the flatness deviation and roundness error threshold values of all the space measuring points, and if the target space circle is available, taking the target space circle as an optimal space circle of the shield steel ring installation position for installing the shield steel ring according to the optimal space circle; And step six, adopting the residual space measuring points, and executing the steps two to five until the target space circle is available, wherein if the number of the residual space measuring points is 3, the target space circle is obtained by adopting the residual 3 space measuring points through space three-point calculation, and the target space circle is used as the optimal space circle of the shield steel ring installation position.
2. The method for determining the space circle of the shield steel ring installation position according to claim 1, wherein the step two is characterized in that fitting is performed based on coordinates of the space measuring points to obtain a fitting plane, the fitting plane meets the minimum objective function constructed for the fitting plane, the objective function constructed for the fitting plane means that the square sum of distances from all the space measuring points to the fitting plane is minimum, and the method comprises the following steps: determining a fitting plane based on the coordinates of the spatial measurement points, and establishing an objective function Q1 of the fitting plane, wherein the objective function Q1 of the fitting plane refers to the square sum of distances d i from all the spatial measurement points to the fitting plane, and the objective function Q1 of the fitting plane is minimum: The equation for the fitted plane is expressed as: Ax+By+Cz-1=0 (1) a, B, C respectively represents the reciprocal of the intercept of the equation of the fitting plane on the coordinate axes corresponding to x, y and z; Coordinates of the ith spatial measurement point are P i (x i ,y i ,z i ), and a distance d i from the ith spatial measurement point to the fitting plane is expressed as: Then, there are: Linearizing equation (3) by substituting the following variables: A ', B', C 'respectively represent the normal vector of the fitting plane and the directional cosine of three coordinate axes, and D' represents the distance from the origin of the coordinate system to the fitting plane; then the equation for the fitted plane is calculated as: A′x+B′y+C′z-D′=0 (4) Wherein a ', B', C 'satisfy the condition a' 2 +B′ 2 +C′ 2 =1; Converting the objective function Q1 of the fitting plane into: According to the extremum solving principle of the multiple functions, the objective function Q1 of the formula (5) is respectively subjected to partial derivative solving of the coefficients A ', B ' and C ', and the value after partial derivative solving is equal to zero, so that the method is obtained: dividing each of the formulas (6), (7) and (8) by D' while canceling the common factor 2, and combining the formula (1): expanding each item, and arranging the items into a matrix form to obtain: MX=N (10) Wherein: X=(ABC) T When the spatial measuring points are not on the same straight line, M is a full rank symmetric square matrix, an inverse matrix exists, and the method can be obtained by solving: X=M -1 N (13) and solving the equation (13) to obtain a A, B, C value in the equation of the fitting plane, and solving the equation of the fitting plane.
3. The method for determining a space circle of a shield steel ring installation position according to claim 2, wherein the step three of fitting based on coordinates of the space measurement points to obtain a fitted sphere, the fitted sphere meeting that an objective function constructed for the fitted sphere is minimum, the objective function constructed for the fitted sphere being obtained based on spherical fitting deviations of all the space measurement points adopted, comprises: Determining a fitting sphere based on the coordinates of the spatial measurement points, and expressing the equation of the best fitting sphere of all the spatial measurement points as: (x-x 0 ) 2 +(y-y 0 ) 2 +(z-z 0 ) 2 ＝R 2 (14) wherein x, y and z represent the coordinates of any point on the fitting sphere, the sphere center of the fitting sphere is C (x 0 ,y 0 ,z 0 ), and R is the radius of the fitting sphere; For each spatial measurement point P i (x i ,y i ,z i , let the spherical fitting deviation ε i of the ith spatial measurement point be: ε i ＝(x i -x 0 ) 2 +(y i -y 0 ) 2 +(z i -z 0 ) 2 -R 2 (15) the formula (15) is developed to obtain: ε i ＝x i 2 +y i 2 +z i 2 -2x i x 0 -2y i y 0 -2z i z 0 -[R 2 -(x 0 2 +y 0 2 +z 0 2 )] (16) Wherein x 0 、y 0 、z 0 is different variable, R 2 -(x 0 2 +y 0 2 +z 0 2 ) is regarded as variable H, and the following substitution is carried out 2x 0 ＝E,2y 0 ＝F,2z 0 ＝G,R 2 -(x 0 2 +y 0 2 +z 0 2 )＝H (17) Wherein E, F, G, H are three different coefficients in the fitted sphere equation; using equation (17), the formula (16) can be developed: ε i ＝x i 2 +y i 2 +z i 2 -Ex i -Fy i -Gz i -H (18) establishing an objective function Q2 of the fitting sphere, and minimizing the objective function Q2 of the fitting sphere: according to the extremum solving principle of the multiple functions, the objective function Q2 of the fitting sphere is respectively subjected to the polarization solving of E, F, G, H, and the value of each polarization solving is set to be zero, so that the method is obtained: And (3) eliminating common factors in the formulas (20) to (23), respectively expanding and combining, and listing the common factors into a matrix equation form: M′X′=N’ (24) Wherein: X′=(E F G H) T The method can be solved as follows: X′=M′ -1 N′ (27) Solving according to formula (27) to obtain a coefficient E, F, G, H, substituting the solved coefficient E, F, G, H into formula (17) to calculate to obtain a spherical center coordinate C (x 0 ,y 0 ,z 0 ) and a radius R of the fitting sphere, and obtaining:
4. The method for determining a space circle for a mounting position of a shield steel ring according to claim 3, wherein the step four of taking a plane circle where the fitting plane intersects the fitting sphere as a target space circle comprises: Obtaining the sag distance from the sphere center C (x 0 ,y 0 ,z 0 ) of the fitting sphere to the fitting plane by using a point-to-plane distance formula: Taking a plane circle intersecting the fitting plane and the fitting sphere as a target space circle, wherein the radius of the target space circle is as follows: and as the direction vector from the spherical center C of the fitting sphere to the perpendicular of the fitting plane is the normal vector of the fitting plane, the point-slope equation from the spherical center C of the fitting sphere to the perpendicular of the fitting plane is expressed as: Wherein x, y and z represent the coordinates of any point on the vertical line; Combining the formula (31) and the formula (1) to obtain: the center C (x C ,y C ,z C ) of the target space circle is obtained as And solving according to the steps to obtain the target space circle.
5. The method for determining the space circle of the installation position of the shield steel ring according to claim 4, further comprising the steps of: calculating by adopting an arithmetic formula (2) to obtain the flatness deviation of each spatial measuring point relative to the fitting plane; Calculating a projection distance s i of each spatial measuring point to the fitting plane, and taking the difference between the projection distance s i and the radius r of the target spatial circle as a roundness error e i of the spatial measuring point:
6. The method for determining a space circle of a shield steel ring installation position according to claim 5, wherein the step five of judging whether a target space circle is available based on the flatness deviation and roundness error threshold values of all the space measurement points, if the target space circle is available, using the target space circle as an optimal space circle of the shield steel ring installation position for installing the shield steel ring according to the optimal space circle, and if the target space circle is not available, removing a worst space measurement point comprises: Setting a flatness deviation threshold value, and judging whether the absolute value of the flatness deviation of each space measuring point is smaller than the flatness deviation threshold value; Setting a roundness error threshold value, and judging whether the roundness error e i of each spatial measuring point is smaller than the roundness error threshold value; if the absolute value of the flatness deviation of each space measuring point is smaller than a flatness deviation threshold value and the roundness error e i of each space measuring point is smaller than a roundness error threshold value, indicating that the target space circle is available, taking the target space circle as an optimal space circle of the shield steel ring installation position, and installing the shield steel ring according to the optimal space circle; When a space measuring point with the absolute value of the flatness deviation being larger than or equal to a flatness deviation threshold value or a space measuring point with the roundness error e i being larger than or equal to a roundness error threshold value exists, the target space circle is not available, and the space measuring point with the largest roundness error e i is removed.
7. The method for determining the space circle of the shield steel ring installation position according to claim 6, wherein the step six is characterized in that the remaining space measuring points are adopted, and the steps two to five are executed until the target space circle is available, wherein if the remaining space measuring points are 3, the target space circle is obtained by using the three space measuring points to calculate, by using the three space measuring points, the remaining 3 space measuring points, and the method is used as the optimal space circle of the shield steel ring installation position, and comprises the following steps: adopting the rest space measuring points, and executing the second to fifth steps until the adopted space measuring points all meet that the absolute value of the flatness deviation is smaller than the flatness deviation threshold value and all meet that the roundness error e i is smaller than the roundness error threshold value, so that the corresponding target space circle is available at the moment; When the number of the remaining space measuring points is 3, a unique plane is determined by adopting a triangle formed by connecting the remaining 3 space measuring points, and the circumscribed circle of the triangle in the unique plane is taken as a target space circle.
8. The method for determining a space circle of a shield steel ring installation position according to claim 7, wherein in the sixth step, when the number of remaining space measuring points is 3, a unique plane determined by a triangle formed by connecting the remaining 3 space measuring points is adopted, and a circumscribed circle of the triangle in the unique plane is taken as a target space circle, which comprises: Taking two sides of the triangle, wherein the intersection line of the respective middle vertical surfaces of the two sides is a vertical line which is perpendicular to the triangle and passes through the circle center of the circumscribing circle, wherein the middle vertical surface is a plane which is perpendicular to the sides and passes through the midpoint of the sides; And solving the distance from any vertex of the triangle to the circle center of the circumscribed circle according to a space two-point distance formula to obtain the radius of the target space circle.
9. The method for determining the space circle of the shield steel ring installation position according to claim 8, wherein two sides of the triangle are taken, wherein an intersection line of each of the perpendicular planes of the two sides is a perpendicular line perpendicular to the triangle and passing through the circle center of the circumscribed circle, an intersection point of the intersection line and the unique plane is used as the circle center of the circumscribed circle, and the method for determining the distance from any vertex of the triangle to the circle center of the circumscribed circle according to a space two-point distance formula to obtain the radius of the target space circle comprises the following steps: the unique plane equation for 3 spatial points P i (x i ,y i ,z i ) (i=1, 2, 3) is: the equation for the mid-perpendicular plane of the corresponding edge P 1 P 3 for the spatial survey point P 1 、P 3 is: (x 3 -x 1 )(x-x 13 )+(y 3 -y 1 )(y-y 13 )+(z 3 -z 1 )(z-z 13 )＝0 (41) Where x 13 、y 13 、z 13 represents the coordinates of the midpoint of the edge P 1 P 3 ; The equation for the mid-perpendicular plane of the corresponding edge P 1 P 2 for the spatial survey point P 1 、P 2 is: (x 2 -x 1 )(x-x 12 )+(y 2 -y 1 )(y-y 12 )+(z 2 -z 1 )(z-z 12 )＝0 (42) wherein x 12 、y 12 、z 12 represents the coordinates of the midpoint of P 1 P 2 ; In the formulas (41) and (42), The combined type (40), (41) and (42) are unfolded into a matrix form: Order the a 1 ＝x 3 -x 1 ,b 1 ＝y 3 -y 1 ,c 1 ＝z 3 -z 1 ; a 2 ＝x 2 -x 1 ,b 2 ＝y 2 -y 1 ,c 2 ＝z 2 -z 1 Obtaining the product n x ＝b 1 c 2 -b 2 c 1 ,n y ＝a 2 c 1 -a 1 c 2 ,n z ＝a 1 b 2 -a 2 b 1 N x 、n y 、n z represents the component of the space vector on each coordinate axis of the space rectangular coordinate system formed by any point connecting line from the space measuring point P 1 to the unique plane corresponding to the formula (40); If it is provided with Then there is M′′X′′=N′′ (46) The center coordinates C (x c ,y c ,z c ) of the circumscribed circles of the triangle are obtained X′′=M′′ -1 N′′ (47) Wherein the method comprises the steps of X′′=(x c y c z c ) T Thereby obtaining the radius of the triangle circumscribing circle: And solving the steps to obtain a target space circle which is used as an optimal space circle of the shield steel ring installation position.
10. The space circle determining system for the shield steel ring installation position is characterized by comprising a measuring point obtaining unit, a fitting plane constructing unit, a fitting sphere constructing unit, a space circle constructing unit, a judging unit and a space three-point solution space circle unit, wherein: the measuring point obtaining unit is used for obtaining coordinates of n space measuring points in a preset installation range of the end part of the underground shield tunnel of the shield steel ring to be installed, wherein the space measuring points are not strictly positioned in the same plane, and n is at least 4; The fitting plane constructing unit is used for carrying out fitting based on the coordinates of the space measuring points to obtain a fitting plane, the fitting plane meets the minimum objective function constructed for the fitting plane, and the objective function constructed for the fitting plane means that the square sum of the distances from all the space measuring points to the fitting plane is minimum; The fitting sphere construction unit is used for carrying out fitting based on the coordinates of the space measuring points to obtain a fitting sphere, the fitting sphere meets the minimum objective function constructed for the fitting sphere, and the objective function constructed for the fitting sphere is obtained based on the spherical fitting deviation of all the adopted space measuring points; the space circle construction unit is used for taking a plane circle intersecting the fitting plane and the fitting sphere as a target space circle; The judging unit is used for judging whether the target space circle is available or not based on the flatness deviation and roundness error threshold values of all the space measuring points, if the target space circle is available, the target space circle is used as an optimal space circle of the shield steel ring installation position and is used for installing the shield steel ring according to the optimal space circle; The fitting plane construction unit, the fitting sphere construction unit, the space circle construction unit and the judgment unit work sequentially for the rest space measuring points until a target space circle is available; And the space three-point solution space circle unit is used for obtaining a target space circle by adopting the space three-point solution of the remaining 3 space measuring points if the number of the remaining space measuring points is 3, and the target space circle is used as an optimal space circle of the shield steel ring installation position.

Description

Space circle determining method and system for shield steel ring installation position Technical Field The invention relates to the field of shield steel ring installation, in particular to a space circle determining method and system for a shield steel ring installation position. Background The section of the urban rail transit shield tunnel is circular, a shield steel ring is required to be installed at the tunnel portal, preset installation positions are required to be set in advance, measurement and calculation are performed in the approximate installation range of the tunnel portal, and the specific installation positions (space circles) of the tunnel portal are determined, so that the specific installation positions are matched with the preset installation positions which are set in advance. In carrying out the present invention, the applicant has found that at least the following problems exist in the prior art: The method comprises the steps of measuring and calculating in the approximate installation range of a tunnel portal, determining the specific installation position (space circle) of the tunnel portal, and calculating the calculation targets including the three-dimensional coordinates and the radius of the space circle center, the roundness deviation and the flatness deviation of each measuring point, the elimination of a small amount of outlier out-of-tolerance measuring points and the like in the space circle settlement process. The method has the characteristics of large resolving difficulty, more steps and large data volume, and is a focus of long-term attention in the industry. The problem of solving the space circle is divided into two problems of solving the three-point space circle and solving the multi-point space circle, wherein the difficulty of solving the multi-point space circle is the largest. The prior research results are concentrated on solving the multi-point space circle problem and can be divided into three categories, wherein the first category is that a space circle is firstly converted into a two-dimensional plane, then is converted into three dimensions after being fitted, is required to have richer space imagination, the resolving steps are complicated, the second category is that the space circle is regarded as an intersection line of the space plane and a space sphere, the space sphere center passes through the space plane as a limiting condition, indirect adjustment solving with the limiting condition is used, whether direct solving or rough removing and adjustment solving are firstly carried out, the approximation value of the space circle center coordinates and the radius and iterative calculation are required, the difficulty is higher, and the third category is that the space circle is regarded as an intersection line of the space sphere which is not limited by the space plane and the sphere center position, and the space circle is solved through the geometric relation between the space sphere and the space plane. For example, the space sphere center is projected to a space plane to obtain a space circle center, but a specific algorithm of space sphere fitting is not mentioned, for example, a coefficient matrix of a plane equation and a coefficient matrix of a spherical equation are constructed, singular value decomposition is carried out on the two coefficient matrices respectively, unit normal vectors of the space plane are extracted from singular value decomposition of the coefficient matrix of the spherical equation, and then the space circle is calculated through the projection relation between the sphere center and the plane. The theory is obscure and the steps are complicated. Disclosure of Invention The embodiment of the invention provides a method and a system for determining a space circle of a shield steel ring installation position, which can solve the technical problems of the prior art, namely, the theoretical confusion and the complicated steps. In order to achieve the above object, in one aspect, an embodiment of the present invention provides a method for determining a space circle of an installation position of a shield steel ring, including: the method comprises the steps of firstly, acquiring coordinates of n space measuring points in a preset installation range of the end part of an underground shield tunnel of a shield steel ring to be installed, wherein the space measuring points are not strictly positioned in the same plane, and n is at least 4; Fitting based on the coordinates of the space measuring points to obtain a fitting plane, wherein the fitting plane meets the minimum of objective functions constructed for the fitting plane, and the objective functions constructed for the fitting plane mean that the square sum of the distances from all the space measuring points to the fitting plane is minimum; fitting is carried out based on the coordinates of the space measuring points to obtain a fitted sphere, the fitted sphere meets the min