CN104197894A - Tower inclination measure method based on circle fitting - Google Patents

Abstract

A related tower inclination measure method based on circle fitting comprises the following steps: 1) fitting plane equations of all planes corresponding to a polygonal tower; 2) obtaining vertex coordinates at high positions of tower marks at all layers according to the plane equations of all planes; 3) determining the center coordinates of all layers of the tower by employing a circle fitting method, namely determining the plane circle center approximate value of planes at all layers of the tower by utilizing the vertex coordinates of all layers, and then finding the centers of all layers of the tower by utilizing the circle fitting method; and 4) finally obtaining a space straight line with smallest error by utilizing the center coordinates of all layers, and taking the space straight line as an axis of the tower, so as to obtain the inclination degree and direction of the tower. Compared with the prior art, the method has the advantages of precise measuring, usage convenience, wide application scope and the like.

Description

A kind of tower inclination measurement method based on circle matching

Technical field

The present invention relates to a kind of tower inclination assessment method, especially relate to a kind of tower inclination measurement method based on circle matching.

Background technology

Chinese Tower comes from India's pagoda, in appearance moulding and version, be mainly divided into loft-style tower, close eaves formula tower, pavilion formula tower, cover the types such as alms bowl formula tower, its essential structure is generally made up of underground palace, pedestal, tower body, tower four parts of stopping, mainly bringing into play commemorate that Buddhist patriarch's Buddhist ceremony, town goblin are eliminating evil, offer a sacrifice to gods or ancestors pray, the effect such as souvenir.In the majority with loft-style tower and close eaves formula tower in the Chinese Tower of having deposited, the feature of this two classes tower is contracture, sets of brackets on top of the columns cornice and the fuzzy state of corner angle and the deformation measurement unique point that causes is difficult to determine layer by layer, determines that by classic method (determining each layer of angular coordinate of observation station total station survey in the edges and corners of each floor) heeling condition of tower is difficult to realize.

Certainly, the heeling condition of tower can be determined by three-dimensional laser scanning technique,, but laser scanner is expensive, one more than 100 ten thousand; And the data volume of obtaining very large (mass data), is unfavorable for post-processed like this, so total station survey tower still has superiority.

Summary of the invention

Object of the present invention is exactly to provide a kind of tower inclination measurement method based on circle matching in order to overcome the defect that above-mentioned prior art exists.

Object of the present invention can be achieved through the following technical solutions:

Based on a tower inclination measurement method for circle matching, it is characterized in that, the method comprises the following steps:

1) plane equation on corresponding each face of matching polygon tower;

2) draw the apex coordinate of each layer of tower beacon eminence according to the plane equation on each face;

3) justify fitting process and ask the centre coordinate of each layer of tower: utilize the apex coordinate of each layer, obtain the plane center of circle approximate value on the each layer plane of tower, then find out Ta Geceng center with circle approximating method;

4) finally utilize each layer of centre coordinate to draw the space line of error minimum, as the axis of tower, can obtain degree and direction that tower tilts.

Described step 1) comprise following sub-step:

11) ask the almost plane equation of each face: choose respectively 3 A (x in the one side of polygon tower ₁, y ₁), B (x ₂, y ₂), C (x ₃, y ₃), according to plane equation formula:

ax+by+cz＝1 (1)

Try to achieve almost plane equation formula a ₀x+b ₀y+c ₀z=1

A ₀, b ₀, c ₀approximate value for a, b, c:

$a^0=-\frac{y_1{z}_2-y_1{z}_3-y_2{z}_1+y_2{z}_3+y_3{z}_1-y_3{z}_2}{x_1{y}_2{z}_3-x_1{y}_3{z}_2-x_2{y}_1{z}_3+x_3{y}_1{z}_2+x_2{y}_3{z}_1-x_{3}y}---\left(2\right)$

$b_0=-\frac{-x_1{z}_2+x_1{z}_3+x_2{z}_1-x_2{z}_3-x_3{z}_1+x_3{z}_2}{x_1{y}_2{z}_3-x_1{y}_3{z}_2-x_2{y}_1{z}_3+x_3{y}_1{z}_2+x_2{y}_3{z}_1-x_{3}y}---\left(3\right)$

$c^0=-\frac{x_1{y}_2-x_1{y}_3-x_2{y}_1+x_2{y}_3+x_3{y}_1-x_3{y}_2}{x_1{y}_2{z}_3-x_1{y}_3{z}_2-x_2{y}_1{z}_3+x_3{y}_1{z}_2+x_2{y}_3{z}_1-x_{3}y}---\left(4\right)$

12) according to least square method each point to the distance minimum of fit Plane, try to achieve the fit Plane equation of each of tower, have:

$\min \underset{1}{\overset{3}{\mathrm{\Σ}}}{V_i}^2=\min \underset{1}{\overset{3}{\mathrm{\Σ}}}{(B_i\hat{X}-I)}^2---\left(5\right)$

$V=\left(\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right),$ $B=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right),$ $I=\left(\begin{array}{c}1\\ 1\\ .\\ .\\ .\\ 1\end{array}\right)$

Wherein V is error, and for correction factor, the plane equation after adjustment is:

Described step 2) comprise following sub-step:

31) draw the corner angle straight-line equation of every layer of tower according to the equation on each face;

32) utilize each corner angle straight-line equation, obtain the polygon vertex at high building layer plane place, obtain n apex coordinate (X of tower beacon eminence section _i, Y _i)

Described step 3) comprise following sub-step:

31) calculate the approximate point coordinate of the centre point of tower k layer with approximate radius

$X_k^0=\frac{1}{n}\underset{1}{\overset{n}{\mathrm{\Σ}}}{X}_i,Y_k^0=\frac{1}{n}\underset{1}{\overset{n}{\mathrm{\Σ}}}{Y}_i---\left(7\right)$

$R_k^0=\frac{1}{n}\underset{1}{\overset{n}{\mathrm{\Σ}}}\sqrt{{(X_i-X_k^0)}^2+{(Y_i-Y_k^0)}^2}---\left(8\right)$

32) adopt circle fitting process to ask the centre coordinate O of tower k layer _k(X _k, Y _k) and radius R _k;

And setting modified value have:

$X_k=X_k^0+{\hat{x}}_k,Y_k=Y_k^0+{\hat{y}}_k,R_k=R_k^0+{\hat{r}}_k$

Utilize least square method to arrive the correction V in the best center of circle at each point _iin minimum situation, ask for three parameters the optimum evaluation of correction, that is:

$\min \underset{1}{\overset{n}{\mathrm{\Σ}}}{v}_i^2=\min \underset{1}{\overset{n}{\mathrm{\Σ}}}{[\sqrt{{(X_i-X_k^0)}^2+{\left(Y_i{-Y}_k^0\right)}^2}-R_k]}^2---\left(9\right)$

Obtain by formula (9) linearization and by the substitution of parameter approximate value:

$v_i=\left(\begin{array}{ccc}-\frac{X_i-X_k^0}{R_i}& -\frac{Y_i-Y_k^0}{R_i}& -1\end{array}\right)\left(\begin{array}{c}{\hat{x}}_k\\ {\hat{y}}_k\\ {\hat{r}}_k\end{array}\right)-(R_k^0-R_i)$

$V=\left(\begin{array}{c}{v}_1\\ v_2\\ .\\ .\\ .\\ v_n\end{array}\right)=\left(\begin{array}{ccc}-\frac{X_1-X_k^0}{R_1}& -\frac{Y_1-Y_k^0}{R_1}& -1\\ -\frac{X_2-X_k^0}{R_2}& -\frac{Y_2-Y_k^0}{R_2}& -1\\ .& .& .\\ .& .& .\\ .& .& .\\ -\frac{X_n-X_k^0}{R_n}& -\frac{Y_n-Y_k^0}{R_n}& -1\end{array}\right)\left(\begin{array}{c}{\hat{x}}_k\\ {\hat{y}}_k\\ {\hat{r}}_k\end{array}\right)-\left(\begin{array}{c}{R}_k^0-R_1\\ R_k^0-R_2\\ .\\ .\\ .\\ R_k^0-R_n\end{array}\right)-B\hat{X}-l---\left(10\right)$

With the data substitution having calculated, obtain the value of matrix B, l above

Use again the principle of least square act on (10) formula, obtain equation:

$B^{T}B\hat{X}-B^{T}l=0,$ Then can obtain parametric solution is $\hat{X}={\left(B^{T}B\right)}^{-1}{B}^{T}l,$ Have

Obtaining parameter corrected value is

Further obtain the matching central coordinate of circle of k layer radius

Described step 4) comprise following sub-step:

41) the matching central coordinate of circle that the tower that obtains according to matching is each layer, calculates the straight-line equation of the central axis of tower;

$\frac{x-x_0}{p}=\frac{y-y_0}{q}=\frac{z-z_0}{r}---\left(11\right)$

Wherein p, q, r are direction numbers, (x ₀, y ₀, z ₀) be origin value;

Obtain error equation by this straight-line equation

$\left\{\begin{array}{c}x=\frac{p}{r}(z-z_0)+x_0=\mathrm{az}+b\\ y=\frac{q}{r}(z-z_0)+y_0=\mathrm{cz}+d\end{array}\right.---\left(12\right)$

42) ask for angle and the deflection degree between this straight line and each axle according to the straight-line equation of the central axis of tower;

Can be obtained by error equation: $\left\{\begin{array}{c}a=\frac{p}{r}\→p=\mathrm{ar}\\ c=\frac{q}{r}\→q=\mathrm{cr}\end{array}\right.---\left(13\right)$

cos ²α+cos ²β+cos ²γ＝1 (14)

By formula (13) (14) can obtain respectively this axis respectively with angle α, the β of X, Y, Z axis, the value of γ

$\cos \mathrm{\α}=\frac{p}{\sqrt{p^2+q^2+r^2}}=\frac{\mathrm{ar}}{\sqrt{{\left(\mathrm{ar}\right)}^2+{\left(\mathrm{cr}\right)}^2+r^2}}=\frac{a}{\sqrt{a^2+c^2+1}}$

$\cos \mathrm{\β}=\frac{q}{\sqrt{p^2+q^2+r^2}}=\frac{\mathrm{cr}}{\sqrt{{\left(\mathrm{ar}\right)}^2+{\left(\mathrm{cr}\right)}^2+r^2}}=\frac{c}{\sqrt{a^2+c^2+1}}$

$\cos \mathrm{\γ}=\frac{r}{\sqrt{p^2+q^2+r^2}}=\frac{r}{\sqrt{{\left(\mathrm{ar}\right)}^2+{\left(\mathrm{cr}\right)}^2+r^2}}=\frac{1}{\sqrt{a^2+c^2+1}}.$

Compared with prior art, the present invention has the following advantages.

1, measurement is accurate, error is little;

2, easy to use, good economy performance;

3, applied widely, also can be used for the inclination test and appraisal of the ancient tower of modern imitations.

Brief description of the drawings

Fig. 1 is method flow diagram of the present invention.

Fig. 2 is the each matching schematic cross-section of ancient tower.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

Embodiment:

Longevity Pagoda is located in Chinese Anhui Province and county, is built in the period of Three Kingdoms.Tower body plane is sexangle, and the sexangle length of side increases with tower body height at the bottom of tower, and the plane sexangle length of side is progressively dwindled.

Based on the moulding of Longevity Pagoda, tower body does not have ready-made vertical plane or perpendicular line to can be used as the characteristic indication of inclination measurement, therefore inclination measurement: the three-dimensional coordinate that first gathers unique point on six faces of tower body appearance, then make the position of each metope according to unique point, ask for again the cross section of surveyed position wall hexahedron height, obtain hexagon plane geometry center, and try to achieve tower body central axis by tower body upper and lower part hexagon geometric center, obtain again the slope of tower body central axis, be the slope of tower body.

1, master data

10NM20111211 121111

06NM1.00000000

01NM：SET1130R V41-14 140784SET1130R V41-14 14078431

0.000

03NM0.0100

2, plane equation matching

Be illustrated in figure 2 each matching schematic cross-section;

3, justify matching

As shown in table 1 by hexagonal cross-section fitting circle Hou center.

Table 1

4, space line matching

Owing to there is no shared variable in two error equations, so can obtain separately the regression equation about x and y.

About V _xi=az _i+ b-x _iand V _yi=cz _i+ d-y _i, the factor arrays of their error equation and normal equation is all identical, but constant term difference,

$B=\left[\begin{array}{cc}1.183& 1\\ 3.763& 1\\ 6.840& 1\\ 9.142& 1\\ 10.644& 1\\ 14.535& 1\end{array}\right],$ V＝Bx+l， $l_x=\left[\begin{array}{c}464.367\\ 464.388\\ 464.402\\ 464.444\\ 464.427\\ 464.486\end{array}\right],$ $l_y=\left[\begin{array}{c}492.523\\ 492.647\\ 492.907\\ 493.099\\ 493.158\\ 493.392\end{array}\right]$

Compensating computation obtains:

$\left\{\begin{array}{c}a=0.0085\\ b=464.3533\end{array}\right.,\left\{\begin{array}{c}c=0.06747\\ d=492.4343\end{array}\right.$

Two straight-line equations are:

$\left\{\begin{array}{c}x=0.0085z+464.3533\\ y=0.06747z+492.4343\end{array}\right.$

Carry out subsequently the validity check of equation.

Now equation x=0.0085z+464.3533 is carried out to validity check:

As calculated, obtain total sum of squares of deviations SST=9.132 × 10 of equation ^-3, regression sum of square SSR=8.4894 × 10 ^-5, residual sum of squares (RSS) SSE=6.4257 × 10 ^-4

With F inspection, null hypothesis H ₀: regression equation is invalid.

Structure test statistics: $F=\frac{\mathrm{SSR}/1}{\mathrm{SE}/(n-2)}=\frac{8.4894\&times;{10}^{-3}/1}{6.4257\&times;{10}^{-4}/(6-2)}=52.85,$ If level of signifiance α=0.05, critical value F tables look-up to obtain _0.05(Isosorbide-5-Nitrae)=7.71 < F=52.85, so refusal null hypothesis thinks that x=0.0085z+464.3533 is effective.

In like manner, equation y=0.06747z+492.4343 is carried out to validation checking, result identification is effective.

Next verify whether x=0.0085z+464.3533 and the y=0.06747z+492.4 projection on XOY face in XOZ and YOZ face overlaps, two straight-line equation cancellation z are obtained to the EQUATION x=0.125554y+402.5260 on XOY face, and the equation that has the direct matching of observation method to obtain is x=0.1255y+402.526, above the angle of two lines be $\mathrm{tg\&beta;}=\left|\frac{0.125554-0.1255}{1+0.125554\&times;0.1255}\right|=0.000053<0.001$ Assert consistent.So straight-line equation $\left\{\begin{array}{c}x=0.0085z+464.3533\\ y=0.06747z+492.4343\end{array}\right.$ Believable.

5, tilt to test and assess

$\frac{x-x_0}{p}=\frac{y-y_0}{q}=\frac{z-z_0}{r}$

Middle p, q, r are respectively the direction number of straight line.

Obtained by error equation:

$\left\{\begin{array}{c}a=\frac{p}{r}\&RightArrow;p=\mathrm{ar}\\ c=\frac{q}{r}\&RightArrow;q=\mathrm{cr}\end{array}\right.$

Calculated angular separation cosine value α, β, the γ of this straight line and XYZ axle by error equation

cos ²α+cos ²β+cos ²γ＝1

$\cos \mathrm{\&alpha;}=\frac{p}{\sqrt{p^2+q^2+r^2}}=\frac{\mathrm{ar}}{\sqrt{{\left(\mathrm{ar}\right)}^2+{\left(\mathrm{cr}\right)}^2+r^2}}=\frac{a}{\sqrt{a^2+c^2+1}}=0.00848$

α＝89°30′51″

$\cos \mathrm{\&beta;}=\frac{q}{\sqrt{p^2+q^2+r^2}}=\frac{\mathrm{cr}}{\sqrt{{\left(\mathrm{ar}\right)}^2+{\left(\mathrm{cr}\right)}^2+r^2}}=\frac{c}{\sqrt{a^2+c^2+1}}=0.06731$

β＝86°08′25″

$\cos \mathrm{\&gamma;}=\sqrt{1-{\cos }^2\mathrm{\&alpha;}-{\cos }^2\mathrm{\&beta;}}=\sqrt{\frac{1}{a^2+c^2+1}}=0.997696$

γ＝3°53′25″

Can obtain x=0.125527y+402.5512 by the plane equation of XOY again, slope tan θ=0.125527 of this straight line, so the position angle of this straight line in XOY plane is λ=90 °-θ=82 ° 50 ' 43 ".

Claims (5)

1. the tower inclination measurement method based on circle matching, is characterized in that, the method comprises the following steps:

1) plane equation on corresponding each face of matching polygon tower;

2) draw the apex coordinate of each layer of tower beacon eminence according to the plane equation on each face;

3) justify fitting process and ask the centre coordinate of each layer of tower: utilize the apex coordinate of each layer, obtain the plane center of circle approximate value on the each layer plane of tower, then find out Ta Geceng center with circle approximating method;

4) finally utilize each layer of centre coordinate to draw the space line of error minimum, as the axis of tower, can obtain degree and direction that tower tilts.

2. a kind of tower inclination measurement method based on circle matching according to claim 1, is characterized in that described step 1) comprise following sub-step:

11) ask the almost plane equation of each face: choose respectively 3 A (x in the one side of polygon tower ₁, y ₁), B (x ₂, y ₂), C (x ₃, y ₃), according to plane equation formula:

ax+by+cz＝1 (1)

Try to achieve almost plane equation formula a ₀x+b ₀y+c ₀z=1

A ₀, b ₀, c ₀approximate value for a, b, c:

12) according to least square method each point to the distance minimum of fit Plane, try to achieve the fit Plane equation of each of tower, have:

Wherein V is error, and for correction factor, the plane equation after adjustment is:

3. a kind of tower inclination measurement method based on circle matching according to claim 2, is characterized in that described step 2) comprise following sub-step:

31) draw the corner angle straight-line equation of every layer of tower according to the equation on each face;

32) utilize each corner angle straight-line equation, obtain the polygon vertex at high building layer plane place, obtain n apex coordinate (X of tower beacon eminence section _i, Y _i).

4. a kind of tower inclination measurement method based on circle matching according to claim 3, is characterized in that described step 3) comprise following sub-step:

31) calculate the approximate point coordinate of the centre point of tower k layer with approximate radius

32) adopt circle fitting process to ask the centre coordinate O of tower k layer _k(X _k, Y _k) and radius R _k;

And setting modified value have:

Utilize least square method to arrive the correction V in the best center of circle at each point _iin minimum situation, ask for three parameters the optimum evaluation of correction, that is:

Obtain by formula (9) linearization and by the substitution of parameter approximate value:

With the data substitution having calculated, obtain the value of matrix B, l above

Use again the principle of least square act on (10) formula, obtain equation:

then can obtain parametric solution is have

Obtaining parameter corrected value is

Further obtain the matching central coordinate of circle of k layer radius

5. a kind of tower inclination measurement method based on circle matching according to claim 4, is characterized in that described step 4) comprise following sub-step:

41) the matching central coordinate of circle that the tower that obtains according to matching is each layer, calculates the straight-line equation of the central axis of tower;

Wherein p, q, r are direction numbers, (x ₀, y ₀, z ₀) be origin value;

Obtain error equation by this straight-line equation

42) ask for angle and the deflection degree between this straight line and each axle according to the straight-line equation of the central axis of tower;

Can be obtained by error equation:

cos ²α+cos ²β+cos ²γ＝1(14)

By formula (13) (14) can obtain respectively this axis respectively with angle α, the β of X, Y, Z axis, the value of γ