CN103714563B - A Boundary Modeling Method of Curved Farmland Operation Area - Google Patents

A Boundary Modeling Method of Curved Farmland Operation Area Download PDF

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CN103714563B
CN103714563B CN201410002907.1A CN201410002907A CN103714563B CN 103714563 B CN103714563 B CN 103714563B CN 201410002907 A CN201410002907 A CN 201410002907A CN 103714563 B CN103714563 B CN 103714563B
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key point
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谭冠政
刘振焘
胡建中
阮启果
黄宇
张丹
罗倩慧
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Central South University
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Abstract

The invention discloses a kind of boundary of curve type farmland operation area modeling method, gather the key point of boundary of curve type farmland operation area, find out point and the point of abscissa value minimum that in all key points, abscissa value is maximum, utilize cubic spline function to set up coboundary model and lower boundary model.Barrier summit in the border in shaped form farmland operation region and farmland operation region can be recorded and draw out by the present invention accurately; thus the operating personnel for agricultural plant protection machine provides farmland operation region accurately, for the path of agricultural plant protection machine and trajectory planning and calculate farmland working area and provide reliable foundation.

Description

一种曲线形农田作业区域边界建模方法A Boundary Modeling Method of Curved Farmland Operation Area

技术领域technical field

本发明涉及一种曲线形农田作业区域边界建模方法。The invention relates to a method for modeling the boundary of a curved farmland operation area.

背景技术Background technique

随着科技的不断发展,采用现代化机械来替换人工劳动已成为各行各业的流行趋势。我国作为一个传统的农业大国,耕地面积十分广阔,然而,目前国内农田作业方面还是采用传统的人工作业方式。其中农药喷洒采取的是人工喷洒的方式,这种方式不仅效率低,而且对作业人员的身心有极大的伤害,因此,急需先进的技术来改变这一现象。农业植保机应运而生,然而,现今所有的植保机都是靠人工进行操作,对农田的喷洒区域及边界完全靠操作员的眼睛进行判断,难免产生误判、漏喷及多喷的现象。因此,如果操作员能够获得准确的农田作业区域的边界,在无人直升机进行作业时,实时的将路径和航迹在显示器上显示出来,操作员便可根据航迹判断出飞机的位置,从而很大程度的解决误判、漏喷及多喷的问题,且大大提高了农业喷洒的安全性。而这一问题的关键便在于要准确的获取农田的边界信息。因此,急需一种好的农田作业区域边界的建模方法来解决这一实际问题。With the continuous development of science and technology, the use of modern machinery to replace manual labor has become a popular trend in all walks of life. As a traditional large agricultural country, our country has a very large area of arable land. However, at present, traditional manual methods are still used in domestic farmland operations. Among them, pesticide spraying adopts the method of artificial spraying, which is not only inefficient, but also has great physical and mental harm to the workers. Therefore, advanced technology is urgently needed to change this phenomenon. Agricultural plant protection machines have emerged as the times require. However, all plant protection machines today are operated manually, and the spraying area and boundary of the farmland are judged entirely by the operator's eyes, which inevitably leads to misjudgment, missed spraying, and excessive spraying. Therefore, if the operator can obtain the accurate boundary of the farmland operation area, when the unmanned helicopter is operating, the path and track are displayed on the display in real time, and the operator can judge the position of the aircraft according to the track, thereby It solves the problems of misjudgment, missed spraying and excessive spraying to a great extent, and greatly improves the safety of agricultural spraying. The key to this problem is to accurately obtain the boundary information of the farmland. Therefore, there is an urgent need for a good modeling method for the boundaries of farmland operations to solve this practical problem.

北京农业信息技术研究中心的发明专利“采集农田关键顶点测绘成图的方法”提供了一种采集农田关键顶点测绘成图的方法,通过采集四种交界顶点、两种轮廓顶点一级一种辅助顶点后,根据关键顶点间的拓扑关系实时自动校验并查看测绘结果。包括以下步骤:S1:获取GPS位置信息;S2:勾勒待测区域轮廓;S3:测绘农田关键分界顶点,并标注地块名称和注释;S4:实时校验、提示与分割农田;S5:上传测绘数据;S6:获取矢量地图。The invention patent of Beijing Agricultural Information Technology Research Center "Method of collecting key vertices of farmland for surveying and mapping" provides a method of collecting key vertices of farmland for surveying and mapping into maps. After the vertices, according to the topological relationship between the key vertices, it will automatically verify and view the surveying and mapping results in real time. Including the following steps: S 1 : Obtain GPS location information; S 2 : Outline the area to be measured; S 3 : Survey and map the key boundary vertices of the farmland, and mark the plot name and notes; S 4 : Real-time verification, prompting and segmentation of the farmland; S 5 : upload surveying and mapping data; S 6 : obtain vector map.

其中,在S2中,勾勒待测区域轮廓包括:顺序标定和无序标定,所述顺序标定,以轮廓顶点的编号为顺序,依次标定待测区域轮廓;所述无序标定,是指自动标定包含全部轮廓顶点的面积最大的多边形为区域轮廓。Wherein, in S 2 , drawing the outline of the region to be measured includes: sequential calibration and disorderly calibration, the sequential calibration, in order of the number of the contour vertices, sequentially demarcate the contour of the area to be measured; the disorderly calibration refers to automatic The polygon with the largest area containing all the vertices of the contour is designated as the contour of the region.

现有方法是以绘出农田的矢量地图为目的,而并非以画出农田作业区域边界为目的,因此,所绘出的农田轮廓包含两个部分:作业区域和非作业区域。在将此农田轮廓图用于农业植保机喷洒农药的路径和航迹规划等场合时,会出现非作业区域的多喷现象,浪费农药;在用于计算农田作业面积时,会出现作业面积的计算不准确现象;现有方法未对农田区域边界建立数学模型,从而无法为农田作业区域的路径和航迹规划提供可靠依据;没有将农田作业区域以内可能存在的障碍物顶点(如电线杆、树、信号发射塔等)标记出来,在将此农田轮廓图用于农业植保机喷洒农药的路径和航迹规划等场合时,可能导致农业植保机出现撞机或坠毁危险现象发生。The purpose of the existing method is to draw a vector map of the farmland, but not to draw the boundary of the farmland operation area. Therefore, the drawn farmland outline includes two parts: the operation area and the non-operation area. When this farmland outline map is used for the path and track planning of agricultural plant protection machine spraying pesticides, there will be excessive spraying in non-operation areas, which will waste pesticides; The calculation is inaccurate; the existing method does not establish a mathematical model for the boundary of the farmland area, so it cannot provide a reliable basis for the path and track planning of the farmland operation area; the vertices of obstacles that may exist in the farmland operation area (such as utility poles, Trees, signal towers, etc.), when the farmland outline map is used for agricultural plant protection aircraft spraying pesticide paths and track planning, it may cause the agricultural plant protection aircraft to collide or crash dangerously.

发明内容Contents of the invention

本发明所要解决的技术问题是,针对现有技术不足,提供一种曲线形农田作业区域边界建模方法,将曲线形农田作业区域的边界以及农田作业区域内的障碍物顶点精准地记录并绘制出来,从而为农业植保机的作业人员提供准确的农田作业区域,为农业植保机的路径和航迹规划以及计算农田作业面积提供可靠的依据。The technical problem to be solved by the present invention is to provide a method for modeling the boundary of a curved farmland operation area to accurately record and draw the boundary of the curved farmland operation area and the obstacle vertices in the farmland operation area. Come out, so as to provide accurate farmland operation area for the operator of agricultural plant protection machine, and provide a reliable basis for the path and track planning of agricultural plant protection machine and the calculation of farmland operation area.

为解决上述技术问题,本发明所采用的技术方案是:一种曲线形农田作业区域边界建模方法,该方法为:In order to solve the above-mentioned technical problems, the technical solution adopted in the present invention is: a method for modeling the boundary of a curved farmland operation area, the method is:

1)从曲线形农田作业区域边界的某一个关键点D1开始,以顺时针方向采集曲线形农田作业区域边界的所有关键点,设共有m个关键点,分别为D1(x1,y1),D2(x2,y2),D3(x3,y3),…,Dm(xm,ym);建立直角坐标系,将所述m个关键点(每隔一个采样周期(GPS本身的采样频率的倒数)就自动采集一个点,当回到初始位置,即关键点D1所在位置时结束采点)设置在直角坐标系的第一象限内;1) Starting from a certain key point D 1 on the border of the curved farmland operation area, collect all key points on the border of the curved farmland operation area in a clockwise direction. Suppose there are m key points in total, which are respectively D 1 (x 1 ,y 1 ),D 2 (x 2 ,y 2 ),D 3 (x 3 ,y 3 ),…,D m (x m ,y m ); establish a Cartesian coordinate system, and place the m key points (every One sampling period (the reciprocal of the sampling frequency of GPS itself) automatically collects a point, and when returning to the initial position, that is, the position where the key point D 1 is located, the collection point ends) and is set in the first quadrant of the Cartesian coordinate system;

2)比较上述m个关键点的横坐标,找出横坐标值最大的关键点和最小的关键点;若有多个关键点的横坐标值均为最小值,且该多个关键点是第一个关键点D1与最后一个关键点Dm之间的关键点,则将次序最先的关键点作为横坐标值最小的关键点,若第一个关键点D1和最后一个关键点Dm的横坐标值都是最小值,则将最后一个关键点作为横坐标值最小的关键点;若有多个关键点的横坐标值均为最大值,且该多个关键点是第一个关键点D1与最后一个关键点Dm之间的关键点,则将次序最后的关键点作为横坐标值最大的关键点,若第一个关键点D1和最后一个关键点Dm的横坐标值都是最大值,则将第一个关键点D1作为横坐标值最大的关键点;将横坐标值最小的关键点Di和最大的关键点Dj分别作为所有的关键点在直角坐标系中的左端和右端;2) Compare the abscissas of the above m key points, and find out the key point with the largest abscissa value and the smallest key point; if there are multiple key points with the smallest abscissa values, and the multiple key points are the first For a key point between a key point D 1 and the last key point D m , the key point with the first order is taken as the key point with the smallest abscissa value, if the first key point D 1 and the last key point D The abscissa values of m are all minimum values, then the last key point is taken as the key point with the minimum abscissa value; if there are multiple key points whose abscissa values are all maximum values, and the multiple key points are the first For the key points between the key point D 1 and the last key point D m , the key point at the end of the order is taken as the key point with the largest abscissa value, if the abscissa of the first key point D 1 and the last key point D m If the coordinate values are all the maximum value, then the first key point D 1 is taken as the key point with the largest abscissa value; the key point D i with the smallest abscissa value and the largest key point D j are respectively regarded as all key points at right angles the left and right extremities in the coordinate system;

3)以Di为起始点,Dj为终点;以顺时针方向,令Di=B0,Di+1=B1,…,Dj=Bn,将从Di到Dj的各点依次即为B0(x0,y0),K,Bn(xn,yn);建立以下上边界函数模型St(x)3) Take D i as the starting point and D j as the end point; in a clockwise direction, set D i =B 0 ,D i+1 =B 1 ,…,D j =B n , and transfer from D i to D j Each point is B 0 (x 0 ,y 0 ), K, B n (x n ,y n ) in turn; establish the following upper boundary function model S t(x) :

SS tt (( xx )) == Mm tt (( xx tt ++ 11 -- xx )) 33 66 hh tt ++ Mm tt ++ 11 (( xx -- xx tt )) 33 66 hh tt ++ (( ythe y tt -- Mm tt hh tt 22 66 )) xx tt ++ 11 -- xx hh tt ++ (( ythe y tt ++ 11 -- Mm tt ++ 11 hh tt 22 66 )) xx -- xx tt hh tt ;;

其中,x∈[xt,xt+1,],t=0,1,...,n-1;xt、xt+1分别为B0、Bn之间的点Bt、Bt+1的横坐标,且Bt+1为Bt顺时针方向上的下一点;系数Mt、Mt+1通过追赶法求解;ht=xt+1-xt;yt、yt+1分别为点Bt、Bt+1的纵坐标;Among them, x∈[x t ,x t+1 ,],t=0,1,...,n-1; x t and x t+1 are the points B t and B between B 0 and B n respectively The abscissa of B t+ 1 , and B t+1 is the next point in the clockwise direction of B t ; the coefficients M t and M t+1 are solved by the pursuit method; h t =x t+1 -x t ; y t , y t+1 are the vertical coordinates of points B t and B t+1 respectively;

4)以Di为起始点,Dj为终点;以逆时针方向,令Di=C0,Di-1=C2,…,Dj=Cr,建立以下上边界函数模型Sr(x)4) Take D i as the starting point and D j as the end point; in the counterclockwise direction, let D i =C 0 ,D i-1 =C 2 ,…,D j =C r , establish the following upper boundary function model S r (x) :

SS rr (( xx )) == Mm rr (( xx rr ++ 11 -- xx )) 33 66 hh rr ++ Mm rr ++ 11 (( xx -- xx rr )) 33 66 hh rr ++ (( ythe y rr -- Mm rr hh rr 22 66 )) xx rr ++ 11 -- xx hh rr ++ (( ythe y rr ++ 11 -- Mm rr ++ 11 hh rr 22 66 )) xx -- xx rr hh rr ;;

其中,x∈[xr,xr+1,],r=0,1,...,m-n;xr、xr+1分别为C0、Cm-n之间的点Cr、Cr+1的横坐标,且Cr+1为Cr逆时时针方向上的下一点;系数Mr、Mr+1通过追赶法求解;hr=xr+1-xr;yr、yr+1分别为点Cr、Cr+1的纵坐标。Among them, x∈[x r ,x r+1 ,], r=0,1,...,mn; x r , x r+1 are the points C r and C r between C 0 and C mn respectively +1 , and C r+1 is the next point in the counterclockwise direction of C r ; the coefficients M r and M r+1 are solved by the pursuit method; h r =x r+1 -x r ; y r , y r+1 are the vertical coordinates of points C r and C r+1 respectively.

当所述曲线形农田作业区域内存在障碍物时,采集曲线形农田作业区域内障碍物顶点的坐标Z(x0,y0),建立以下障碍物警戒线函数方程:When there is an obstacle in the curved farmland operation area, the coordinates Z(x 0 , y 0 ) of the obstacle vertex in the curved farmland operation area are collected, and the following obstacle warning line function equation is established:

一级障碍物警戒线的函数方程为(x-x0)2+(y-y0)2=a2The functional equation of the first-level obstacle warning line is (xx 0 ) 2 +(yy 0 ) 2 =a 2 ;

二级障碍物警戒线的函数方程为(x-x0)2+(y-y0)2=b2The functional equation of the second-level obstacle warning line is (xx 0 ) 2 +(yy 0 ) 2 =b 2 ;

三级障碍物警戒线的函数方程为(x-x0)2+(y-y0)2=c2The functional equation of the three-level obstacle warning line is (xx 0 ) 2 +(yy 0 ) 2 =c 2 ;

其中,0<a<b<c<50。Among them, 0<a<b<c<50.

通过追赶法求解系数Mt、Mt+1的步骤为:The steps to solve the coefficients M t and M t+1 by the chasing method are:

1)令β1=b1,y1=d1;其中, b 1 = 2 + &mu; 1 + &mu; 1 h 0 h 1 , b 2 = 2 , . . . , b n - 2 = 2 , b n - 1 = 2 + &lambda; n - 1 + &lambda; n - 1 h n - 1 h n - 2 ; &mu; p = h p - 1 h p - 1 + h p ; d p = 6 f [ x p , x p + 1 ] - f [ x p - 1 , x p ] h p - 1 + h p = 6 y p + 1 - y p x p + 1 - x p - y p - y p - 1 x p - x p - 1 h p - 1 + h p ; λp=1-μp;p=1,2,...,n-1;1) Let β 1 =b 1 ,y 1 =d 1 ; where, b 1 = 2 + &mu; 1 + &mu; 1 h 0 h 1 , b 2 = 2 , . . . , b no - 2 = 2 , b no - 1 = 2 + &lambda; no - 1 + &lambda; no - 1 h no - 1 h no - 2 ; &mu; p = h p - 1 h p - 1 + h p ; d p = 6 f [ x p , x p + 1 ] - f [ x p - 1 , x p ] h p - 1 + h p = 6 the y p + 1 - the y p x p + 1 - x p - the y p - the y p - 1 x p - x p - 1 h p - 1 + h p ; λ p =1-μ p ; p=1,2,...,n-1;

2)计算βq=bq-lqcq-1,yq=dq-lqyq-1;其中,q=2,3,…,n-1; a 2 = &mu; 2 , a 3 = &mu; 3 , . . . , a n - 1 = &mu; n - 1 - &lambda; n - 1 h n - 1 h n - 2 ; c 1 = &lambda; 1 - &mu; 1 h 0 h 1 , c 2 = &lambda; 2 , c 3 = &lambda; 3 , . . . , c n - 2 = &lambda; n - 2 ; 2) calculate β q =b q -l q c q-1 ,y q =d q -l q y q-1 ; where, q=2,3,…,n-1; a 2 = &mu; 2 , a 3 = &mu; 3 , . . . , a no - 1 = &mu; no - 1 - &lambda; no - 1 h no - 1 h no - 2 ; c 1 = &lambda; 1 - &mu; 1 h 0 h 1 , c 2 = &lambda; 2 , c 3 = &lambda; 3 , . . . , c no - 2 = &lambda; no - 2 ;

3)利用下式求解Mn-1 3) Solve for M n-1 using the following formula:

4)利用下式计算Mn-2,Mn-3,L,M1s=n-2,n-3,…,1;4) Use the following formula to calculate M n-2 , M n-3 , L, M 1 : s=n-2,n-3,...,1;

5)通过下式计算M0和Mn M 0 = M 1 - h 0 h 1 ( M 2 - M 1 ) , M n = M n - 1 + h n - 1 h n - 2 ( M n - 1 - M n - 2 ) . 5) Calculate M 0 and M n by the following formula: m 0 = m 1 - h 0 h 1 ( m 2 - m 1 ) , m no = m no - 1 + h no - 1 h no - 2 ( m no - 1 - m no - 2 ) .

与现有技术相比,本发明所具有的有益效果为:本发明可以将曲线形农田作业区域的边界以及农田作业区域内的障碍物顶点精准地记录并绘制出来,从而为农业植保机的作业人员提供准确的农田作业区域,为农业植保机的路径和航迹规划以及计算农田作业面积提供可靠的依据。Compared with the prior art, the present invention has the beneficial effects that: the present invention can accurately record and draw the boundary of the curved farmland operation area and the obstacle vertices in the farmland operation area, so as to facilitate the operation of the agricultural plant protection machine. The personnel provide accurate farmland operation area, and provide a reliable basis for the path and track planning of the agricultural plant protection machine and the calculation of the farmland operation area.

附图说明Description of drawings

图1为本发明一实施例曲线型边界模型示意图;Fig. 1 is a schematic diagram of a curved boundary model according to an embodiment of the present invention;

图2为本发明一实施例有障碍物的农田作业区域障碍物警戒线模型。Fig. 2 is an obstacle warning line model of a farmland operation area with obstacles according to an embodiment of the present invention.

具体实施方式detailed description

以下结合附图详细说明本发明的具体实施方式。Specific embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings.

1)假设采点工作者以第一个关键点开始,以顺时针方向依次采集m个关键点(以GPS的周期Ts作为采集农田作业区域边界点的采样周期),分别为D1(x1,y1),D2(x2,y2),D3(x3,y3),…,Dm(xm,ym),如图1所示;1) Assume that the point collector starts with the first key point, and collects m key points in a clockwise direction (the period T s of GPS is used as the sampling period for collecting the boundary points of the farmland operation area), respectively D 1 (x 1 ,y 1 ), D 2 (x 2 ,y 2 ),D 3 (x 3 , y3 ),…,D m (x m ,y m ), as shown in Figure 1;

2)比较这m个点的横坐标,找出其中横坐标最小和最大的点。假设横坐标最小的点为Di(xi,yi),横坐标最大的点为Dj(xj,yj)。(若遇到多个点的横坐标都为最小值,则把取点次序最先的点作为最小点,在此有个特例:如果D1和Dm的横坐标都相等,但是在此取Dm作为最小点;若遇到多个点的横坐标都为最大值,则把取点次序最后的点作为最大点,在此有个特例:如果Dm和D1的横坐标都相等,但是在此取D1作为最大点),那么Di(xi,yi)即为左端,Dj(xj,yj)即为右端;2) Compare the abscissas of these m points and find out the points with the smallest and largest abscissas. Assume that the point with the smallest abscissa is D i (x i , y i ), and the point with the largest abscissa is D j (x j ,y j ). (If the abscissas of multiple points are all the minimum value, then take the first point of the point order as the minimum point, here is a special case: if the abscissas of D 1 and D m are all equal, but take the point here D m is taken as the minimum point; if the abscissa of multiple points is the maximum value, the point at the end of the point order is taken as the maximum point. Here is a special case: if the abscissa of D m and D 1 are all equal, But take D 1 as the maximum point here), then D i ( xi , y i ) is the left end, and D j (x j , y j ) is the right end;

3)假设由左端Di与右端Dj所确定的上边界有|j-i|+1个点,令n=|j-i|(包括左端Di与右端Dj),为便于接下来的计算,将上边界上各点(包括左端Di与右端Dj)从左至右按顺时方向依次记为B0(x0,y0),K,Bn(xn,yn)(即令Di=B0,Di+1=B1,…,Dj=Bn若当j<i,则在顺时针方向移点的过程中,如果Dn出现在Dj的左边(即xn<xj),令Dn的下一个点为D1,如果Dn=Bk(k<n)那么D1=Bk+1,D2=Bk+2,K,Dj=Bn3) Suppose there are |ji|+1 points on the upper boundary determined by the left end D i and the right end D j , let n=|ji| (including the left end D i and the right end D j ), for the convenience of the following calculation, Each point on the upper boundary (including the left end D i and the right end D j ) is recorded as B 0 (x 0 , y 0 ), K, B n (x n , y n ) in the clockwise direction from left to right (that is, let D i =B 0 ,D i+1 =B 1 ,…,D j =B n If j<i, then in the process of clockwise moving, if D n appears on the left of D j (that is, x n <x j ), let the next point of D n be D 1 , if D n =B k (k<n) then D 1 =B k+1 , D 2 =B k+2 ,K,D j =B n .

则上边界函数S(x)满足:Then the upper boundary function S(x) satisfies:

(1)S(xt)=yt(j=0,1,…,n);(1) S(x t )=y t (j=0,1,...,n);

(2)S(x)在每个小区间[xt,xt+1](t=0,1,…,n-1)上是3次多项式;(2) S(x) is a 3rd degree polynomial in each small interval [x t ,x t+1 ](t=0,1,…,n-1);

(3)S(x)在[x0,xn]上有连续2阶导数;(3) S(x) has a continuous second-order derivative on [x 0 , x n ];

(4)端点条件:强迫第一个和第二的三次多项式的三阶导数相等,对最后一个和倒数第二个三次多项式也做同样处理。(4) Endpoint condition: Force the third-order derivatives of the first and second cubic polynomials to be equal, and do the same for the last and penultimate cubic polynomials.

按求解3次样条函数的三弯矩方程方法,设S″(xt)=Mt,t=0,1,...,n,According to the three-bending-moment equation method for solving cubic spline functions, let S″(x t )=M t , t=0,1,...,n,

记ht=xt+1-xt,t=0,1,...,n-1。Remember h t =x t+1 -x t ,t=0,1,...,n-1.

(1)整理三次样条函数的推导过程得μtMt-1+2MttMt+1=dt,t=1,2,...,n-1,其中, &mu; t = h t - 1 h t - 1 + h t , λt=1-μt, d t = 6 f [ x t - 1 , x t , x t + 1 ] = 6 f [ x t , x t + 1 ] - f [ x t - 1 , x t ] h t - 1 + h t = 6 y t + 1 - y t x t + 1 - x t - y t - y t - 1 x t - x t - 1 h t - 1 + h t . (1) Arrange the derivation process of the cubic spline function to get μ t M t-1 +2M tt M t+1 =d t , t=1,2,...,n-1, where, &mu; t = h t - 1 h t - 1 + h t , λ t =1-μ t , d t = 6 f [ x t - 1 , x t , x t + 1 ] = 6 f [ x t , x t + 1 ] - f [ x t - 1 , x t ] h t - 1 + h t = 6 the y t + 1 - the y t x t + 1 - x t - the y t - the y t - 1 x t - x t - 1 h t - 1 + h t .

(2)由端点条件知, M n - M n - 1 h n - 1 = M n - 1 - M n - 2 h n - 2 &DoubleRightArrow; M n = M n - 1 + h n - 1 h n - 2 ( M n - 1 - M n - 2 ) M 1 - M 0 h 0 = M 2 - M 1 h 1 &DoubleRightArrow; M 0 = M 1 - h 0 h 1 ( M 2 - M 1 ) , (2) Known from the endpoint conditions, m no - m no - 1 h no - 1 = m no - 1 - m no - 2 h no - 2 &DoubleRightArrow; m no = m no - 1 + h no - 1 h no - 2 ( m no - 1 - m no - 2 ) m 1 - m 0 h 0 = m 2 - m 1 h 1 &DoubleRightArrow; m 0 = m 1 - h 0 h 1 ( m 2 - m 1 ) ,

由μtMt-1+2MttMt+1=dt,t=1,2,...,n-1知μ1M0+2M11M2=d1 From μ t M t-1 +2M tt M t+1 =d t ,t=1,2,...,n-1, μ 1 M 0 +2M 11 M 2 =d 1

( 2 + &mu; 1 + &mu; 1 h 0 h 1 ) M 1 + ( &lambda; 1 - &mu; 1 h 0 h 1 ) M 2 = d 1 , ( 2 + &mu; 1 + &mu; 1 h 0 h 1 ) m 1 + ( &lambda; 1 - &mu; 1 h 0 h 1 ) m 2 = d 1 ,

由μtMt-1+2MttMt+1=dt,t=1,2,...,n-1知μn-1Mn-2+2Mn-1n-1Mn=dn-1,From μ t M t-1 +2M tt M t+1 =d t ,t=1,2,...,n-1, μ n-1 M n-2 +2M n-1n-1 M n =d n-1 ,

( &mu; n - 1 - &lambda; n - 1 h n - 1 h n - 2 ) M n - 2 + ( 2 + &lambda; n - 1 + &lambda; n - 1 h n - 1 h n - 2 ) M n - 1 = d n - 1 . ( &mu; no - 1 - &lambda; no - 1 h no - 1 h no - 2 ) m no - 2 + ( 2 + &lambda; no - 1 + &lambda; no - 1 h no - 1 h no - 2 ) m no - 1 = d no - 1 .

(3)结合(1)和(2)得矩阵(3) Combine (1) and (2) to get the matrix

求出三弯矩阵的系数阵和右端项。Find the coefficient matrix and right-hand term of a three-curved matrix.

(4)另 b 1 = 2 + &mu; 1 + &mu; 1 h 0 h 1 , b 2 = 2 , . . . , b n - 2 = 2 , b n - 1 = 2 + &lambda; n - 1 + &lambda; n - 1 h n - 1 h n - 2 , (4) another b 1 = 2 + &mu; 1 + &mu; 1 h 0 h 1 , b 2 = 2 , . . . , b no - 2 = 2 , b no - 1 = 2 + &lambda; no - 1 + &lambda; no - 1 h no - 1 h no - 2 ,

cc 11 == &lambda;&lambda; 11 -- &mu;&mu; 11 hh 00 hh 11 ,, cc 22 == &lambda;&lambda; 22 ,, cc 33 == &lambda;&lambda; 33 ,, .. .. .. ,, cc nno -- 22 == &lambda;&lambda; nno -- 22 ,,

aa 22 == &mu;&mu; 22 ,, aa 33 == &mu;&mu; 33 ,, .. .. .. ,, aa nno -- 11 == &mu;&mu; nno -- 11 -- &lambda;&lambda; nno -- 11 hh nno -- 11 hh nno -- 22 ,,

则(3)中的矩阵变为 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 &CenterDot; &CenterDot; &CenterDot; &CenterDot; &CenterDot; &CenterDot; a n - 2 b n - 2 c n - 2 a n - 1 b n - 1 M 1 M 2 M 3 &CenterDot; &CenterDot; M n - 2 M n - 1 = d 1 d 2 d 3 &CenterDot; &CenterDot; d n - 2 d n - 1 Then the matrix in (3) becomes b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 &CenterDot; &CenterDot; &Center Dot; &Center Dot; &Center Dot; &Center Dot; a no - 2 b no - 2 c no - 2 a no - 1 b no - 1 m 1 m 2 m 3 &Center Dot; &CenterDot; m no - 2 m no - 1 = d 1 d 2 d 3 &Center Dot; &Center Dot; d no - 2 d no - 1

(5)用追赶法求出Mt,t=1,2,…,n-1。(5) Calculate M t , t=1,2,...,n-1 by the chasing method.

具体过程如下:The specific process is as follows:

①β1=b1,y1=d1①β 1 =b 1 ,y 1 =d 1 ;

②对于i=2,3,…,n-1,计算βi=bi-lici-1,yi=di-liyi-1②For i=2,3,…,n-1, calculate β i =b i -l i c i-1 ,y i =d i -l i y i-1 ;

M n - 1 = y n - 1 &beta; n - 1 ; m no - 1 = the y no - 1 &beta; no - 1 ;

④对于i=n-2,n-3,…,1,计算 ④For i=n-2,n-3,...,1, calculate

⑤计算 M 0 = M 1 - h 0 h 1 ( M 2 - M 1 ) , M n = M n - 1 + h n - 1 h n - 2 ( M n - 1 - M n - 2 ) ; ⑤ calculation m 0 = m 1 - h 0 h 1 ( m 2 - m 1 ) , m no = m no - 1 + h no - 1 h no - 2 ( m no - 1 - m no - 2 ) ;

(6)按下列公式求出S(x)在各个区间[xt,xt+1]上表达式St(x),其为 S t ( x ) = M t ( x t + 1 - x ) 3 6 h t + M t + 1 ( x - x t ) 3 6 h t + ( y t - M t h t 2 6 ) x t + 1 - x h t + ( y t + 1 - M t + 1 h t 2 6 ) x - x t h t ( x &Element; [ x t , x t + 1 , ] , t = 0,1 , . . . , n - 1 ) (6) Find the expression S t(x) of S (x) on each interval [x t ,x t+1 ] according to the following formula, which is S t ( x ) = m t ( x t + 1 - x ) 3 6 h t + m t + 1 ( x - x t ) 3 6 h t + ( the y t - m t h t 2 6 ) x t + 1 - x h t + ( the y t + 1 - m t + 1 h t 2 6 ) x - x t h t ( x &Element; [ x t , x t + 1 , ] , t = 0,1 , . . . , no - 1 )

4)下边界函数模型4) Lower boundary function model

左端Di与右端Dj所确定的下边界上有n+1=m-|j-i|+1个点(包括左端Di与右端Dj),为便于后来计算,将下边界上各点(包括左端Di与右端Dj)从左至右按逆时针方向依次记为C0(x1,y1),K,Cn(xn,yn)(即令Di=C0,Di-1=C2,…,Dj=Cn,若当j>i,则在逆时针方向移点的过程中,如果D1出现在Dj的左边(即X1<Xj),那么令D1的下一个点为Dn;如果D1=Ck(k<n),那么Dn=Ck+1,Dn-1=Ck+2,...,Dj=Cn)。There are n+1=m-|ji|+1 points on the lower boundary determined by the left end D i and the right end D j (including the left end D i and the right end D j ), for later calculation, the points on the lower boundary ( Including the left end D i and the right end D j ) From left to right, it is recorded as C 0 (x 1 ,y 1 ),K,C n (x n ,y n ) in the counterclockwise direction (that is, let D i =C 0 ,D i-1 =C 2 ,…,D j =C n , if j>i, then in the process of moving points counterclockwise, if D 1 appears on the left of D j (that is, X 1 <X j ), Then let the next point of D 1 be D n ; if D 1 =C k (k<n), then D n =C k+1 , D n-1 =C k+2 ,...,D j = C n ).

下边界函数计算与上边界函数计算过程一样。The lower boundary function calculation is the same as the upper boundary function calculation process.

如果农田中有障碍物,则由采点工作者采集到障碍物点的坐标为Z(x0,y0),假设一级警戒线为离障碍物点a米的圆,二级警戒线为离障碍物点b米的圆,三级警戒线为离障碍物点c米的圆(50>c>b>a>0),如图2所示,图2中三个圆从内至外分别为一级警戒线、二级警戒线、三级警戒线。If there is an obstacle in the farmland, the coordinates of the obstacle point collected by the worker is Z(x 0 , y 0 ), assuming that the first-level warning line is a circle a meter away from the obstacle point, and the second-level warning line is A circle of b meters away from the obstacle point, the third-level warning line is a circle of c meters away from the obstacle point (50>c>b>a>0), as shown in Figure 2, the three circles in Figure 2 are from inside to outside They are the first-level warning line, the second-level warning line, and the third-level warning line.

则一级警戒线的函数方程为(x-x0)2+(y-y0)2=a2Then the functional equation of the first-level warning line is (xx 0 ) 2 +(yy 0 ) 2 =a 2 ;

二级警戒线的函数方程为(x-x0)2+(y-y0)2=b2The functional equation of the secondary warning line is (xx 0 ) 2 +(yy 0 ) 2 =b 2 ;

三级警戒线的函数方程为(x-x0)2+(y-y0)2=c2The functional equation of the three-level warning line is (xx 0 ) 2 +(yy 0 ) 2 =c 2 .

Claims (3)

1. A method for modeling the boundary of a curved farmland operation area is characterized by comprising the following steps:
1) a certain key point D from the boundary of curved farmland operation area1Starting, collecting all key points of the boundary of the curved farmland operation area in the clockwise direction, and setting a total of m key points as D1(x1,y1),D2(x2,y2),D3(x3,y3),…,Dm(xm,ym) (ii) a Establishing a rectangular coordinate system, and setting the m key pointsThe device is arranged in a first quadrant of a rectangular coordinate system;
2) comparing the abscissa of the m key points to find out the key point with the maximum abscissa value and the key point with the minimum abscissa value; if the abscissa values of the plurality of key points are all the minimum values, and the plurality of key points are the first key point D1And the last key point DmThe key point with the first order is used as the key point with the minimum abscissa value if the key point D is the first key point1And the last keypoint DmIf the abscissa values are all the minimum values, the last key point is taken as the key point with the minimum abscissa value; if the abscissa values of the plurality of key points are all the maximum values, and the plurality of key points are the first key point D1And the last key point DmThe key point with the last order is taken as the key point with the maximum abscissa value, if the first key point D1And the last keypoint DmIf the abscissa values of the first and second points are all maximum values, the first key point D is determined1As the key point with the maximum horizontal coordinate value; key point D with smallest horizontal coordinate valueiAnd the largest key point DjRespectively serving as the left end and the right end of all key points in a rectangular coordinate system; if the abscissa values of a plurality of key points are all minimum values, one key point is a first key point or a last key point, and the other key points are key points between the first key point and the last key point, taking a point with the first order in a closed interval from the first key point to the last key point as a minimum point; if the abscissa values of a plurality of key points are the maximum values, one key point is a first key point or a last key point, and the other key points are key points between the first key point and the last key point, taking a point with the last order in a closed interval from the first key point to the last key point as a maximum point;
3) with DiAs a starting point, DjIs the end point; in the clockwise direction, order Di=B0,Di+1=B1,…,Dj=BnWill be driven from DiTo DjEach point in turn is B0(x0,y0),...,Bn(xn,yn) (ii) a Establishing the following upper boundary function model St(x)
S t ( x ) = M t ( x t + 1 - x ) 3 6 h t + M t + 1 ( x - x t ) 3 6 h t + ( y t - M t h t 2 6 ) x t + 1 - x h t + ( y t + 1 - M t + 1 h t 2 6 ) x - x t h t ;
Wherein, x ∈ [ xt,xt+1,],t=0,1,...,n-1;xt、xt+1Are respectively B0、BnPoint B in betweent、Bt+1On the abscissa of (a) and Bt+1Is BtThe next point in the clockwise direction; coefficient Mt、Mt+1Solving by a catch-up method;
ht=xt+1-xt,t=0,1,...,n-1;yt、yt+1are respectively point Bt、Bt+1The ordinate of (a);
4) with DiAs a starting point, DjIs the end point; in the counterclockwise direction, let Di=C0,Di-1=C1,…,Dj=CrEstablishing the following upper boundary function model Sr(x)
S r ( x ) = M r ( x r + 1 - x ) 3 6 h r + M r + 1 ( x - x r ) 3 6 h r + ( y r - M r h r 2 6 ) x r + 1 - x h r + ( y r + 1 - M r + 1 h r 2 6 ) x - x r h r ;
Wherein, x ∈ [ xr,xr+1,],r=0,1,...,m-n;xr、xr+1Are respectively C0、Cm-nPoint C in betweenr、Cr+1On the abscissa of (a), and Cr+1Is CrThe next point in the counterclockwise direction; coefficient Mr、Mr+1Solving by a catch-up method; h isr=xr+1-xr;yr、yr+1Are respectively point Cr、Cr+1The ordinate of (c).
2. The method of modeling the boundary of a curved farmland operation area according to claim 1, wherein when an obstacle exists in the curved farmland operation area, the coordinates Z (x) of the apex of the obstacle in the curved farmland operation area are collected0,y0) Establishing the following function equation of the barrier warning line:
the function equation of the first-level obstacle warning line is (x-x)0)2+(y-y0)2=a2
The function equation of the secondary obstacle warning line is (x-x)0)2+(y-y0)2=b2
Three-level barrier warningThe functional equation of the line is (x-x)0)2+(y-y0)2=c2
Wherein a is more than 0 and less than b and less than c and less than 50.
3. The method for modeling the boundary of a curved farmland working area according to claim 1 or 2, wherein the coefficient M is solved by a catch-up methodt、Mt+1Comprises the following steps:
1) let β1=b1,y1=d1(ii) a Wherein,b2=2,...,bn-2=2, λp=1-μp;p=1,2,...,n-1;
2) computingβq=bq-lqcq-1,yq=dq-lqyq-1(ii) a Wherein q is 2,3, …, n-1; a is2=μ2c2=λ2,c3=λ3,...,cn-2=λn-2
3) Solving for M usingn-1
4) Calculate M using the following equationn-2,Mn-3,…,M1s=n-2,n-3,...,1;
5) M is calculated by0And Mn M n = M n - 1 + h n - 1 h n - 2 ( M n - 1 - M n - 2 ) .
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