CN115143936B - Method for measuring gradient of power transmission engineering pole tower based on laser point cloud - Google Patents
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Abstract
The invention discloses a method for measuring the inclination of a power transmission engineering pole tower based on laser point cloud, which comprises the following steps: separating out tower point cloud data; determining the centroid of the surrounding layer corresponding to each horizontal partition surrounding box, and taking four points, which are farthest from the centroid along the diagonal direction, in each horizontal partition surrounding box as outer contour points of the tower; fitting the outer contour points to obtain a three-dimensional space linear model of each side edge of the tower quadrangular; obtaining top surface outer contour point coordinates and bottom surface outer contour point coordinates; and determining the gradient of the tower according to the top surface center point coordinates and the bottom surface center point coordinates. The invention can efficiently and accurately realize the automatic measurement of the gradient of the transmission engineering tower, and can be widely applied to the inspection and acceptance work of the transmission engineering.
Description
Technical Field
The invention relates to the technical field of completion acceptance of power transmission and transformation engineering construction, in particular to an automatic measuring method for inclination of a power transmission engineering pole tower based on laser point cloud.
Background
Today, the three-dimensional laser scanning technology is widely applied to transmission line inspection and completion acceptance, and in the inspection and completion acceptance, the measurement of the inclination of a pole tower is a crucial work, and the measurement is related to whether the whole power transmission and transformation project can be normally put into operation.
At present, the transmission line tower gradient measuring method based on laser point cloud mainly adopts a manual point selection measuring mode, so that the method is low in working efficiency, and is extremely easy to cause misoperation due to the randomness of point selection, and therefore the improved tower gradient measuring method based on the laser point cloud data is significant.
Disclosure of Invention
In view of the foregoing, it is desirable to provide a method for measuring the inclination of a power transmission engineering pole based on a laser point cloud.
The embodiment of the invention provides a method for measuring the inclination of a power transmission engineering pole tower based on laser point cloud, which comprises the following steps:
separating out the tower point cloud data from the laser point cloud data in the unit span of the power transmission project;
Dividing the tower point cloud data into horizontal partitions layer by layer from bottom to top according to the height of the horizontal partition bounding box; determining the centroid of a surrounding layer corresponding to each horizontal partition bounding box for each horizontal partition bounding box before the partitioning is finished; taking four points farthest from the centroid along the diagonal direction in each horizontal partition bounding box as tower outer contour points;
Fitting the external contour points based on a random sampling consensus algorithm RANSAC to obtain a three-dimensional space linear model of each side edge of the tower quadrangular prism;
For a horizontal partition bounding box at the end of partition, obtaining the maximum value of the ordinate in the outer contour points corresponding to the horizontal partition, and obtaining the top surface outer contour point coordinates based on a three-dimensional space linear model; extending the straight lines of the four side edges to intersect with the XOY plane of the outer contour point of the lowest tower, and obtaining the intersection point coordinates of each straight line and the XOY plane based on a three-dimensional space straight line model, namely the bottom surface outer contour point coordinates; and determining the gradient of the tower according to the top surface center point coordinates and the bottom surface center point coordinates.
Further, the step of separating the tower point cloud data from the laser point cloud data in the unit span of the power transmission project specifically includes:
Acquiring laser point cloud data in a unit span of a power transmission project; the laser point cloud data includes: ground point cloud data, vegetation point cloud data, power line point cloud data and tower point cloud data;
Adopting an adaptive filtering method to separate ground point cloud data from laser point cloud data;
Separating vegetation point cloud data from the laser point cloud data according to geometrical dimension characteristics of vegetation, towers and power lines; the vegetation point cloud data are in spherical dimension characteristics, and the tower point cloud data and the power line point cloud data are in rod-shaped dimension characteristics;
Separating out the power line point cloud data from the laser point cloud data according to the direction characteristics of the tower and the power line; the rod-shaped point cloud data parallel to the horizontal plane are lead point cloud data, and the rod-shaped point cloud data perpendicular to the horizontal plane are tower point cloud data;
And judging the trend of the tower according to the coordinate maximum value of the tower point cloud data, and separating according to the trend of the tower to obtain tower point cloud data T 1 and tower point cloud data T 2.
Further, the determining of the geometric dimensional characteristics of the vegetation, towers and power lines includes:
Determining the optimal neighborhood radius of the laser point cloud data according to the self-adaptive neighborhood calculation method;
a matrix singular value decomposition method is adopted to obtain singular values in corresponding neighborhood laser point cloud data;
Geometric dimension features are calculated based on the singular values.
Further, the determining of the height of the horizontal partition bounding box includes:
Traversing the tower point cloud data to find the highest point of the tower And lowest point/>And determining the height bh of the horizontal partition bounding box:
Where num is the number of horizontally partitioned bounding boxes and z max and z min are the highest points of the towers, respectively And the lowest pointIs defined by the z-axis coordinate of (c).
Further, for each horizontal partition bounding box before the end of the partition, determining the centroid of the corresponding bounding layer of each horizontal partition bounding box specifically includes:
recording the number n i of point clouds in the ith horizontal partition, and calculating the characteristic elevation ce i of the ith horizontal partition; the feature elevations are represented as follows:
Determining each horizontal partition bounding box before the partition is finished according to the horizontal partition finishing condition; the horizontal partition end condition is expressed as follows:
when the horizontal partition ending condition is not satisfied, recording the centroid of the corresponding bounding layer of the horizontal partition bounding box
Wherein, B ixmax,Bixmin,Biymax and B iymin are the maximum x-axis coordinate, the minimum x-axis coordinate, the maximum y-axis coordinate and the minimum y-axis coordinate of all points in the ith horizontal partition; x ij,yij and z ij are the x-axis, y-axis and z-axis coordinates, respectively, of each point within the bounding box of the ith horizontal partition; n i+1 is the number of point clouds in the (i+1) th horizontal partition; ce i+1 is the characteristic elevation of the (i+1) th horizontal partition.
Further, the method takes four points farthest from the centroid along the diagonal direction in each horizontal partition bounding box as outer contour points of the tower, and specifically comprises the following steps:
Determining the distance between the j point and the centroid in the i-th horizontal partition bounding box:
traversing all points in the ith horizontal partition bounding box, if the point B ij meets the condition B ijx＞Bix,Bijy＞Biy, calculating dis, comparing dis meeting the condition, and taking the corresponding point of the maximum value of dis as the outer contour point of the tower.
Further, the method for obtaining the three-dimensional space linear model of each side edge of the tower quadrangular specifically comprises the following steps:
Forming four point sets {p11,p12,p13,...,p1m},{p21,p22,p23,...,p2m},{p31,p32,p33,...,p3m} and { p 41,p242,p43,...,p4m } by contour points of the same side edges, wherein m is the number of horizontal partition bounding boxes when the horizontal partition ending condition is met;
randomly selecting a group of subsets from the four point sets, and assuming the selected subsets as intra-office points;
Calculating various coefficients of the three-dimensional space linear model through the assumed local points; testing other data in the four point sets by using the obtained three-dimensional space linear model, and if a certain point is suitable for the estimated model, considering the test point as a local point;
re-estimating the model by using all assumed local points, and estimating the model by estimating the mean square deviation between the local points and the model to obtain a three-dimensional space linear model of four point sets; the straight line L 1 obtained by the { p 11,p12,p13,...,p1m } point set is:
Wherein A 11、A12、B11、B12、C11、C12、D11、D12 is each coefficient of the three-dimensional space linear model; p 1m is the point coordinates in the m-th horizontal partition in the first set of points; p 2m is the point coordinates in the m-th horizontal partition in the second set of points; p 3m is the point coordinates in the m-th horizontal partition in the third point set; p 4m is the point coordinates in the m-th horizontal partition in the fourth set of points.
Further, the top surface outline point coordinatesThe method comprises the following steps:
Where z m is the maximum value on the ordinate of the outer contour point.
Further, the bottom surface outline point coordinatesThe method comprises the following steps:
further, determining the tower inclination according to the top surface center point coordinates and the bottom surface center point coordinates specifically includes:
According to four top surface outline points And/>Determination of Top surface center Point/>
Cmz＝zm
According to four bottom surface outline pointsAnd/>Determining a bottom surface center point
Cminz＝zmin
Determining tower inclination by:
Compared with the prior art, the automatic measuring method for the inclination of the power transmission engineering pole tower based on the laser point cloud has the following beneficial effects:
According to the method, laser point cloud data in a unit span of a power transmission project is input first, and the laser point cloud data are divided into four types of ground points, vegetation points, tower points and ground points according to dimension and direction characteristics; then, utilizing the horizontal partition of the tower point cloud to obtain outer contour points; fitting the outer contour points by using a random sampling coincidence algorithm to obtain a linear model of four side edges of the pole tower; and finally, correcting the top surface and bottom surface coordinates of the quadrangular prism based on the linear model of each side edge, and obtaining the gradient of the tower. The method has the advantages of no need of manual intervention, high working efficiency and high practical value.
Drawings
FIG. 1 is a flowchart of an implementation of a method for automatically measuring inclination of a power transmission project tower based on a laser point cloud according to one embodiment;
fig. 2 is a schematic diagram of classification of data of power transmission engineering point cloud based on a method for automatically measuring inclination of a power transmission engineering pole tower according to the laser point cloud in one embodiment;
Fig. 3 is a schematic tower diagram of a method for automatically measuring inclination of a power transmission engineering tower based on a laser point cloud according to an embodiment;
Fig. 4 is a schematic diagram of a tower horizontal zonal bounding box of a method for automatically measuring the inclination of a power transmission engineering tower based on a laser point cloud according to an embodiment;
Fig. 5 is a schematic diagram of extracting an outer contour point of a tower quadrangular prism of a power transmission engineering tower inclination automatic measurement method based on laser point cloud according to an embodiment.
Fig. 6 is a schematic diagram of automatic tower inclination measurement of a method for automatic measurement of tower inclination of a power transmission project based on laser point cloud according to an embodiment.
Detailed Description
The present application will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present application more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the application.
Referring to fig. 1, in one embodiment, a method for automatically measuring inclination of a power transmission engineering pole tower based on a laser point cloud is provided, and the method includes:
(1) Inputting a power transmission engineering laser point cloud file by taking a span as a unit, and dividing the power transmission engineering laser point cloud file into four types, namely a ground point, a vegetation point, a tower point and a power line point, wherein the step (1) comprises the following steps:
1-1) inputting point cloud data of an original airborne three-dimensional laser scanner, dividing the point cloud data by taking a span as a unit, and removing rough differences by adopting a statistical analysis method.
1-2) Adopting an adaptive filtering method to separate the ground data from the laser point cloud data. Dividing according to different geometrical characteristics of vegetation, towers and power lines: and separating vegetation points by using dimension characteristics, and separating tower points and power line points by using direction characteristics.
1-3) Firstly determining the optimal neighborhood radius of the laser point according to a self-adaptive neighborhood calculation method, then adopting a matrix singular value decomposition method to obtain singular values in the corresponding neighborhood laser radar point cloud data, calculating dimension characteristics based on the singular values, wherein the vegetation point is a spherical target as a whole due to the irregularity of the vegetation point, and the tower and the power line are rod-shaped targets.
1-4) Analyzing and judging the direction of the feature vector of the rod-shaped target point, reserving the rod-shaped target point parallel to the horizontal plane as a conducting wire, and separating the rod-shaped target point perpendicular to the horizontal plane as a tower, wherein the classification result of the span point cloud is shown in fig. 2.
1-5) Separating the tower point clouds TowerCloud from the tower point clouds T 1 and T 2. The trend is judged by utilizing the coordinate maximum value of the tower point cloud TowerCloud, and the two tower point clouds are separated according to the main trend T 1 and T 2, and the separated tower T 1 point cloud is shown in fig. 3.
(2) Obtaining an outer contour point of the tower based on the characteristic elevation of the horizontal partition; the step (2) comprises the following steps:
2-1) firstly inputting a tower T 1 point cloud, traversing the tower T 1 point cloud, and searching the highest point of the tower And lowest point/>And determining the height bh of the horizontal partition bounding box:
Where num is the number of horizontally partitioned bounding boxes and z max and z min are the highest points of the towers, respectively And the lowest pointIs defined by the z-axis coordinate of (c).
Calculating to obtain the highest point of the tower T 1 And the lowest pointThe height bh= 0.731081m of the horizontal partition bounding box.
2-2) Dividing the tower T 1 layer by layer according to the height bh of the horizontal partition bounding box from bottom to top to obtain an elevation histogram of the point cloud of the tower, recording the number n i of the point cloud in each horizontal partition, and calculating the characteristic elevation ce i of the point cloud.
Where B ixmax,Bixmin,Biymax and B iymin are the maximum, minimum x-axis and y-axis coordinates of all points in the ith horizontal partition.
The number of point clouds in the 3 rd horizontal partition bounding box n 3 =30 is calculated, and the characteristic elevation ce 3 = 890.788m.
Setting a threshold value, and discarding bounding boxes with the number of point clouds in the horizontal partition being smaller than the threshold value. Judging a horizontal partition ending mark according to the characteristics of the tower to obtain a complete tower quadrangular prism:
And (5) finishing the calculation when the tower T 1 in the span is divided into 46 th horizontal partition bounding boxes.
2-3) If the ending condition is not satisfied, recording the centroid of the surrounding layer of the current horizontal partition
Where x ij,yij and z ij are the x-, y-and z-axis coordinates, respectively, of points within the bounding box of the ith horizontal partition.
The centroid B 3 (740.489 m, -1291.79m, -4.33159 m) of the 3 rd horizontal partition bounding box is calculated.
2-4) For useAnd/>Respectively expressed in the ith horizontal partition bounding box, from centroid/>, along its diagonal directionThe four furthest points are judged as follows:
Where B ijx,Bijy and B ijz are the x-, y-and z-axis coordinates of the j-th point in the i-th horizontal partition bounding box.
To be used forFor example, traversing all points in the ith horizontal partition bounding box, if the point B ij meets the condition B ijx＞Bix,Bijy＞Biy, calculating dis, comparing all dis meeting the condition, and taking the corresponding point of the maximum value of dis as/>And (5) a dot.
To be used forFor example, traversing all points in the ith horizontal partition bounding box, if the point B ij meets the condition B ijx＞Bix,Bijy＜Biy, calculating dis, comparing all dis meeting the condition, and taking the corresponding point of the maximum value of dis as/>And (5) a dot.
The four outline points of the 3 rd horizontal partition bounding box are calculated as follows:
(3) Based on random sampling consensus algorithm (RANSAC) to fit the external contour point cloud, a three-dimensional space straight line model of each side edge of the tower quadrangular is obtained, and the step (3) comprises the following steps:
In the process of extracting the outer contour points from the horizontal subareas, certain out-of-position noise points are contained due to the extraction and separation of the tower point clouds TowerCloud. The implementation process of the least square method is suitable for all points including the outlier as much as possible, so that the fitting straight line tends to deviate greatly. In contrast, the RANSAC can obtain a model which is calculated only by using local points, and the accuracy of the model is high, so that the RANSAC algorithm is selected for straight line fitting.
3-1) The contour points belonging to the same side edge form four point sets {p11,p12,p13,...,p1m},{p21,p22,p23,...,p2m},{p31,p32,p33,...,p3m} and { p 41,p242,p43,...,p4m }, and m is the number of the horizontal partition bounding boxes when the end condition is met. The goal is achieved by selecting a set of random subsets of data multiple times, the selected subsets being assumed to be intra-office points.
3-2) Calculating various unknown parameters of the three-dimensional space straight line model through the assumed local points. All other data is tested with the resulting three-dimensional spatial linear model, and if a point is suitable for the estimated model, it is considered to be an intra-local point as well. If there are enough points to be classified as hypothetical local points, then the estimated model is reasonable enough.
3-3) Re-estimating the model with all assumed local points, and evaluating the model by estimating the mean square deviation of the local points from the model. This process is repeated a fixed number of times, and each generated model is either discarded because of too few local points or selected for use because it is better than the existing models.
3-4) Obtaining a three-dimensional space linear model of four point sets, taking a straight line L 1 obtained by { p 11,p12,p13,...,p1m } point sets as an example:
The expression of the calculated straight line L 1 is:
(4) Correcting the coordinates of the top surface and the bottom surface of the quadrangular prism of the tower based on the linear model of each side edge, and solving the gradient of the tower, wherein the step (4) comprises the following steps:
4-1) correcting the outer contour points of the current horizontal partition surrounding layer according to the partition ending condition when the ending condition is met. Taking the maximum value of the z coordinate in the outer contour points of the current horizontal partition, and marking the maximum value as z m to obtain a correction point And/>To solve/>The point coordinates are for example:
calculated and corrected And/>The point coordinates are:
4-2) extending the straight line where the four side edges are located and the lowest point of the same tower The located XOY planes intersect to solve the intersection point/>, of each straight line and the planeAnd/>To solve/>The coordinates are as examples:
The intersection point coordinates of the four side edge fitting straight lines and the XOY plane where the lowest point is located are calculated as follows:
4-3) Top surface of known Tower T 1 Point cloud quadrangular prism And/>Four point coordinates, find the center point/>, of the top surface
Cmz＝zm
Calculating the center coordinates of the top surface
4-4) Bottom surface of known tower T 1 point cloud quadrangularAnd/>Four point coordinates, find the center point/>, of the bottom surface
Cminz＝zmin
Calculating to obtain the center coordinates of the bottom surface
4-5) The calculation formula of the inclination of the tower T 1 is as follows:
H＝|Cminz-Cmz|
calculated as s=0.00967407 m, h= 12.0717m, tower inclination of According to the tower gradient requirement of 110 kV-750 kV overhead transmission line construction and acceptance Specification GB50233-2014, the tower T 1 is known:
so the gradient of the tower meets the safety standard.
The above examples illustrate only a few embodiments of the application, which are described in detail and are not to be construed as limiting the scope of the application. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the application, which are all within the scope of the application. Accordingly, the scope of protection of the present application is to be determined by the appended claims.
Claims (4)
1. The utility model provides a transmission engineering shaft tower gradient measurement method based on laser point cloud which characterized in that includes:
separating out the tower point cloud data from the laser point cloud data in the unit span of the power transmission project;
dividing the tower point cloud data into horizontal partitions layer by layer from bottom to top according to the height of the horizontal partition bounding box; determining the centroid of a surrounding layer corresponding to each horizontal partition bounding box for each horizontal partition bounding box before the partitioning is finished; four points farthest from the centroid along the diagonal direction in each horizontal partition bounding box are used as outer contour points of the tower;
Fitting the external contour points based on a random sampling consensus algorithm RANSAC to obtain a three-dimensional space linear model of each side edge of the tower quadrangular prism;
For a horizontal partition bounding box at the end of partition, taking the maximum value of the ordinate in the outer contour points corresponding to the horizontal partition, and obtaining the top surface outer contour point coordinates based on a three-dimensional space linear model; extending the straight lines of the four side edges to intersect with the XOY plane of the outer contour point of the lowest tower, and obtaining the intersection point coordinates of each straight line and the XOY plane based on a three-dimensional space straight line model, namely the bottom surface outer contour point coordinates; determining the gradient of the tower according to the top surface center point coordinates and the bottom surface center point coordinates;
the method for obtaining the three-dimensional space linear model of each side edge of the tower quadrangular specifically comprises the following steps:
Forming four point sets {p11,p12,p13,...,p1m},{p21,p22,p23,...,p2m},{p31,p32,p33,...,p3m} and { p 41,p42,p43,...,p4m } by contour points of the same side edges, wherein m is the number of horizontal partition bounding boxes when the horizontal partition ending condition is met;
randomly selecting a group of subsets from the four point sets, and assuming the selected subsets as intra-office points;
Calculating various coefficients of the three-dimensional space linear model through the assumed local points; testing other data in the four point sets by using the obtained three-dimensional space linear model, and if a certain point is suitable for the estimated model, considering the test point as a local point;
re-estimating the model by using all assumed local points, and estimating the model by estimating the mean square deviation between the local points and the model to obtain a three-dimensional space linear model of four point sets; the straight line L 1 obtained by the { p 11,p12,p13,...,p1m } point set is:
Wherein A 11、A12、B11、B12、C11、C12、D11、D12 is each coefficient of the three-dimensional space linear model; p 1m is the point coordinates in the m-th horizontal partition in the first set of points; p 2m is the point coordinates in the m-th horizontal partition in the second set of points; p 3m is the point coordinates in the m-th horizontal partition in the third point set; p 4m is the point coordinates in the m-th horizontal partition in the fourth point set;
The top surface outline point coordinates The method comprises the following steps:
wherein, the maximum value of the ordinate in the z m outline point;
The coordinates of the outline points of the bottom surface The method comprises the following steps:
The determining of the height of the horizontal partition bounding box comprises the following steps:
Traversing the tower point cloud data to find the highest point of the tower And lowest point/>And determining the height bh of the horizontal partition bounding box:
Where num is the number of horizontally partitioned bounding boxes and z max and z min are the highest points of the towers, respectively And lowest point/>Is the z-axis coordinate of (2);
For each horizontal partition bounding box before the end of the partition, determining the centroid of the corresponding bounding layer of each horizontal partition bounding box specifically comprises the following steps:
recording the number n i of point clouds in the ith horizontal partition, and calculating the characteristic elevation ce i of the ith horizontal partition; the feature elevations are represented as follows:
Determining each horizontal partition bounding box before the partition is finished according to the horizontal partition finishing condition; the horizontal partition end condition is expressed as follows:
when the horizontal partition ending condition is not satisfied, recording the centroid of the corresponding bounding layer of the horizontal partition bounding box
Wherein, B ixmax,Bixmin,Biymax and B iymin are the maximum x-axis coordinate, the minimum x-axis coordinate, the maximum y-axis coordinate and the minimum y-axis coordinate of all points in the ith horizontal partition; x ij,yij and z ij are the x-axis, y-axis and z-axis coordinates, respectively, of each point within the bounding box of the ith horizontal partition; n i+1 is the number of point clouds in the (i+1) th horizontal partition; ce i+1 is the feature elevation of the (i+1) th horizontal partition;
The method for using four points farthest from the centroid along the diagonal direction in each horizontal partition bounding box as the outer contour points of the tower specifically comprises the following steps:
Determining the distance between the j point and the centroid in the i-th horizontal partition bounding box:
Traversing all points in the ith horizontal partition bounding box, if the point B ij meets the condition B ijx＞Bix,Bijy＞Biy, calculating dis, comparing dis meeting the condition, and taking the corresponding point of the maximum value of dis as the outer contour point of the tower; where B ijx,Bijy and B ijz are the x-, y-and z-axis coordinates of the j-th point in the i-th horizontal partition bounding box.
2. The method for measuring the inclination of the power transmission engineering pole tower based on the laser point cloud according to claim 1, wherein the step of separating the pole tower point cloud data from the laser point cloud data in the unit span of the power transmission engineering specifically comprises the steps of:
Acquiring laser point cloud data in a unit span of a power transmission project; the laser point cloud data includes: ground point cloud data, vegetation point cloud data, power line point cloud data and tower point cloud data;
Adopting an adaptive filtering method to separate ground point cloud data from laser point cloud data;
Separating vegetation point cloud data from the laser point cloud data according to geometrical dimension characteristics of vegetation, towers and power lines; the vegetation point cloud data are in spherical dimension characteristics, and the tower point cloud data and the power line point cloud data are in rod-shaped dimension characteristics;
Separating out the power line point cloud data from the laser point cloud data according to the direction characteristics of the tower and the power line; the rod-shaped point cloud data parallel to the horizontal plane are lead point cloud data, and the rod-shaped point cloud data perpendicular to the horizontal plane are tower point cloud data;
And judging the trend of the tower according to the coordinate maximum value of the tower point cloud data, and separating according to the trend of the tower to obtain tower point cloud data T 1 and tower point cloud data T 2.
3. The method for measuring the inclination of the power transmission engineering pole tower based on the laser point cloud according to claim 2, wherein the determination of the geometrical dimension characteristics of vegetation, poles and power lines comprises:
Determining the optimal neighborhood radius of the laser point cloud data according to the self-adaptive neighborhood calculation method;
a matrix singular value decomposition method is adopted to obtain singular values in corresponding neighborhood laser point cloud data;
Geometric dimension features are calculated based on the singular values.
4. The method for measuring the inclination of the power transmission engineering pole tower based on the laser point cloud according to claim 1, wherein the determining the inclination of the pole tower according to the top surface center point coordinates and the bottom surface center point coordinates specifically comprises:
According to four top surface outline points And/>Determination of Top surface center Point/>
Cmz＝zm
According to four bottom surface outline pointsAnd/>Determining a bottom surface center point
Cminz＝zmin
Determining tower inclination by:
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