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

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

A Boundary Modeling Method of Curved Farmland Operation Area

technical field

The invention relates to a method for modeling the boundary of a curved farmland operation area.

Background technique

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.

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.

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.

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.

Contents of the invention

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.

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) 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) 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) 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) :

${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}\frac{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; )</span>\frac{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; )</span>\frac{{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; ;</span>$

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) 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)} :

${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}\frac{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; )</span>\frac{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; )</span>\frac{{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; ;</span>$

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.

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:

The functional equation of the first-level obstacle warning line is (xx ₀ ) ² +(yy ₀ ) ² =a ² ;

The functional equation of the second-level obstacle warning line is (xx ₀ ) ² +(yy ₀ ) ² =b ² ;

The functional equation of the three-level obstacle warning line is (xx ₀ ) ² +(yy ₀ ) ² =c ² ;

Among them, 0<a<b<c<50.

The steps to solve the coefficients M _t and M _t+1 by the chasing method are:

1) Let β ₁ =b ₁ ,y ₁ =d ₁ ; where, $b_1=2+{\mathrm{\&mu;}}_1+\frac{{\mathrm{\&mu;}}_1{h}_0}{h_1},b_2=2,...,b_{\mathrm{no}-2}=2,$ $b_{\mathrm{no}-1}=2+{\mathrm{\&lambda;}}_{\mathrm{no}-1}+\frac{{\mathrm{\&lambda;}}_{\mathrm{no}-1}{h}_{\mathrm{no}-1}}{h_{\mathrm{no}-2}};$ ${\mathrm{\&mu;}}_p=\frac{h_{p-1}}{h_{p-1}+h_p};$ $d_p=6\frac{f[x_p,x_{p+1}]-f[x_{p-1},x_p]}{h_{p-1}+h_p}=6\frac{\frac{{\mathrm{the\; y}}_{p+1}-{\mathrm{the\; y}}_p}{x_{p+1}-x_p}-\frac{{\mathrm{the\; y}}_p-{\mathrm{the\; y}}_{p-1}}{x_p-x_{p-1}}}{h_{p-1}+h_p};$ λ _p =1-μ _p ; p=1,2,...,n-1;

2) 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; $a_2={\mathrm{\&mu;}}_2,$ $a_3={\mathrm{\&mu;}}_3,...,a_{\mathrm{no}-1}={\mathrm{\&mu;}}_{\mathrm{no}-1}-\frac{{\mathrm{\&lambda;}}_{\mathrm{no}-1}{h}_{\mathrm{no}-1}}{h_{\mathrm{no}-2}};$ $c_1={\mathrm{\&lambda;}}_1-\frac{{\mathrm{\&mu;}}_1{h}_0}{h_1},c_2={\mathrm{\&lambda;}}_2,c_3={\mathrm{\&lambda;}}_3,...,c_{\mathrm{no}-2}={\mathrm{\&lambda;}}_{\mathrm{no}-2};$

3) Solve for M _n-1 using the following formula:

4) Use the following formula to calculate M _n-2 , M _n-3 , L, M ₁ : s=n-2,n-3,...,1;

5) Calculate M ₀ and M _n by the following formula: $m_0=m_1-\frac{h_0}{h_1}(m_2-m_1),$ $m_{\mathrm{no}}=m_{\mathrm{no}-1}+\frac{h_{\mathrm{no}-1}}{h_{\mathrm{no}-2}}(m_{\mathrm{no}-1}-m_{\mathrm{no}-2}).$

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.

Description of drawings

Fig. 1 is a schematic diagram of a curved boundary model according to an embodiment of the present invention;

Fig. 2 is an obstacle warning line model of a farmland operation area with obstacles according to an embodiment of the present invention.

detailed description

Specific embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings.

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) 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) 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 .

Then the upper boundary function S(x) satisfies:

(1) S(x _t )=y _t (j=0,1,...,n);

(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) has a continuous second-order derivative on [x ₀ , x _n ];

(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.

According to the three-bending-moment equation method for solving cubic spline functions, let S″(x _t )=M _t , t=0,1,...,n,

Remember h _t =x _t+1 -x _t ,t=0,1,...,n-1.

(1) Arrange the derivation process of the cubic spline function to get μ _t M _t-1 +2M _t +λ _t M _t+1 =d _t , t=1,2,...,n-1, where, ${\mathrm{\&mu;}}_t=\frac{h_{t-1}}{h_{t-1}+h_t},$ λ _t =1-μ _t , $d_t=6f[x_{t-1},x_t,x_{t+1}]=6\frac{f[x_t,x_{t+1}]-f[x_{t-1},x_t]}{h_{t-1}+h_t}=6\frac{\frac{{\mathrm{the\; y}}_{t+1}-{\mathrm{the\; y}}_t}{x_{t+1}-x_t}-\frac{{\mathrm{the\; y}}_t-{\mathrm{the\; y}}_{t-1}}{x_t-x_{t-1}}}{h_{t-1}+h_t}.$

(2) Known from the endpoint conditions, $\begin{array}{c}\frac{m_{\mathrm{no}}-m_{\mathrm{no}-1}}{h_{\mathrm{no}-1}}=\frac{m_{\mathrm{no}-1}-m_{\mathrm{no}-2}}{h_{\mathrm{no}-2}}\&DoubleRightArrow;{m_{\mathrm{no}}=m}_{\mathrm{no}-1}+\frac{h_{\mathrm{no}-1}}{h_{\mathrm{no}-2}}(m_{\mathrm{no}-1}-m_{\mathrm{no}-2})\\ \frac{m_{\text{1}}-m_{\text{0}}}{h_0}=\frac{m_2-m_1}{h_1}\&DoubleRightArrow;m_0=m_1-\frac{h_0}{h_1}(m_2-m_1)\end{array},$

From μ _t M _t-1 +2M _t +λ _t M _t+1 =d _t ,t=1,2,...,n-1, μ ₁ M ₀ +2M ₁ +λ ₁ M ₂ =d ₁

∴ $(2+{\mathrm{\&mu;}}_1+\frac{{\mathrm{\&mu;}}_1{h}_0}{h_1})m_1+({\mathrm{\&lambda;}}_1-\frac{{\mathrm{\&mu;}}_1{h}_0}{h_1})m_2=d_1,$

From μ _t M _t-1 +2M _t +λ _t M _t+1 =d _t ,t=1,2,...,n-1, μ _n-1 M _n-2 +2M _n-1 +λ _n-1 M _n =d _n-1 ,

∴ $({\mathrm{\&mu;}}_{\mathrm{no}-1}-\frac{{\mathrm{\&lambda;}}_{\mathrm{no}-1}{h}_{\mathrm{no}-1}}{h_{\mathrm{no}-2}})m_{\mathrm{no}-2}+(2+{\mathrm{\&lambda;}}_{\mathrm{no}-1}+\frac{{\mathrm{\&lambda;}}_{\mathrm{no}-1}{h}_{\mathrm{no}-1}}{h_{\mathrm{no}-2}})m_{\mathrm{no}-1}=d_{\mathrm{no}-1}.$

(3) Combine (1) and (2) to get the matrix

Find the coefficient matrix and right-hand term of a three-curved matrix.

(4) another $b_1=2+{\mathrm{\&mu;}}_1+\frac{{\mathrm{\&mu;}}_1{h}_0}{h_1},b_2=2,...,b_{\mathrm{no}-2}=2,b_{\mathrm{no}-1}=2+{\mathrm{\&lambda;}}_{\mathrm{no}-1}+\frac{{\mathrm{\&lambda;}}_{\mathrm{no}-1}{h}_{\mathrm{no}-1}}{h_{\mathrm{no}-2}},$

${\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>$

${\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; ,</span>$

Then the matrix in (3) becomes $\left(\begin{array}{ccccccc}{b}_1& c_1& & & & & \\ a_2& b_2& c_2& & & & \\ & a_3& b_3& c_3& & & \\ & & \&CenterDot;& \&CenterDot;& \&Center\; Dot;& & \\ & & & \&Center\; Dot;& \&Center\; Dot;& \&Center\; Dot;& \\ & & & & a_{\mathrm{no}-2}& b_{\mathrm{no}-2}& c_{\mathrm{no}-2}\\ & & & & & a_{\mathrm{no}-1}& b_{\mathrm{no}-1}\end{array}\right)\left(\begin{array}{c}{m}_1\\ m_2\\ m_3\\ \&Center\; Dot;\\ \&CenterDot;\\ m_{\mathrm{no}-2}\\ m_{\mathrm{no}-1}\end{array}\right)=\left(\begin{array}{c}{d}_1\\ d_2\\ d_3\\ \&Center\; Dot;\\ \&Center\; Dot;\\ d_{\mathrm{no}-2}\\ d_{\mathrm{no}-1}\end{array}\right)$

(5) Calculate M _t , t=1,2,...,n-1 by the chasing method.

The specific process is as follows:

①β ₁ =b ₁ ,y ₁ =d ₁ ;

②For i=2,3,…,n-1, calculate β _i =b _i -l _i c _i-1 ,y _i =d _i -l _i y _i-1 ;

③ $m_{\mathrm{no}-1}=\frac{{\mathrm{the\; y}}_{\mathrm{no}-1}}{{\mathrm{\&beta;}}_{\mathrm{no}-1}};$

④For i=n-2,n-3,...,1, calculate

⑤ calculation $m_0=m_1-\frac{h_0}{h_1}(m_2-m_1),$ $m_{\mathrm{no}}=m_{\mathrm{no}-1}+\frac{h_{\mathrm{no}-1}}{h_{\mathrm{no}-2}}(m_{\mathrm{no}-1}-m_{\mathrm{no}-2});$

(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 $S_{t\left(x\right)}=m_t\frac{{(x_{t+1}-x)}^3}{{6h}_t}+m_{t+1}\frac{{(x-x_t)}^3}{{6h}_t}+({\mathrm{the\; y}}_t-\frac{m_t{h}_t^2}{6})\frac{x_{t+1}-x}{h_t}+({\mathrm{the\; y}}_{t+1}-\frac{m_{t+1}{h}_t^2}{6})\frac{{x-x}_t}{h_t}$ $(x\&Element;[x_t,x_{t+1,}],t=\mathrm{0,1},...,\mathrm{no}-1)$

4) Lower boundary function model

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 ).

The lower boundary function calculation is the same as the upper boundary function calculation process.

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.

Then the functional equation of the first-level warning line is (xx ₀ ) ² +(yy ₀ ) ² =a ² ;

The functional equation of the secondary warning line is (xx ₀ ) ² +(yy ₀ ) ² =b ² ;

The 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),…,D_m(x_m,y_m) (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 D_mThe 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 D_mIf 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 D_mThe 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 D_mIf 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 value_iAnd the largest key point D_jRespectively 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 D_iAs a starting point, D_jIs the end point; in the clockwise direction, order D_i＝B₀,D_i+1＝B₁,…,D_j＝B_nWill be driven from D_iTo D_jEach point in turn is B₀(x₀,y₀),...,B_n(x_n,y_n) (ii) a Establishing the following upper boundary function model S_t(x)：

$S_{t\left(x\right)}=M_t\frac{{(x_{t+1}-x)}^3}{6h_t}+M_{t+1}\frac{{(x-x_t)}^3}{6h_t}+(y_t-\frac{M_t{h_t}^2}{6})\frac{x_{t+1}-x}{h_t}+(y_{t+1}-\frac{M_{t+1}{h_t}^2}{6})\frac{x-x_t}{h_t};$

Wherein, x ∈ [ x_t,x_t+1,],t＝0,1,...,n-1；x_t、x_t+1Are respectively B₀、B_nPoint B in between_t、B_t+1On the abscissa of (a) and B_t+1Is B_tThe next point in the clockwise direction; coefficient M_t、M_t+1Solving by a catch-up method;

h_t＝x_t+1-x_t,t＝0,1,...,n-1；y_t、y_t+1are respectively point B_t、B_t+1The ordinate of (a);

4) with D_iAs a starting point, D_jIs the end point; in the counterclockwise direction, let D_i＝C₀,D_i-1＝C₁,…,D_j＝C_rEstablishing the following upper boundary function model S_r(x)：

$S_{r\left(x\right)}=M_r\frac{{(x_{r+1}-x)}^3}{6h_r}+M_{r+1}\frac{{(x-x_r)}^3}{6h_r}+(y_r-\frac{M_r{h_r}^2}{6})\frac{x_{r+1}-x}{h_r}+(y_{r+1}-\frac{M_{r+1}{h_r}^2}{6})\frac{x-x_r}{h_r};$

Wherein, x ∈ [ x_r,x_r+1,],r＝0,1,...,m-n；x_r、x_r+1Are respectively C₀、C_m-nPoint C in between_r、C_r+1On the abscissa of (a), and C_r+1Is C_rThe next point in the counterclockwise direction; coefficient M_r、M_r+1Solving by a catch-up method; h is_r＝x_r+1-x_r；y_r、y_r+1Are respectively point C_r、C_r+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 method_t、M_t+1Comprises the following steps:

1) let β₁＝b₁,y₁＝d₁(ii) a Wherein,b₂＝2，...，b_n-2＝2， λ_p＝1-μ_p；p＝1，2，...，n-1；

2) computingβ_q＝b_q-l_qc_q-1,y_q＝d_q-l_qy_q-1(ii) a Wherein q is 2,3, …, n-1; a is₂＝μ₂，c₂＝λ₂，c₃＝λ₃，...，c_n-2＝λ_n-2；

3) Solving for M using_n-1：

4) Calculate M using the following equation_n-2,M_n-3,…,M₁：s＝n-2，n-3，...，1；

5) M is calculated by₀And M_n： $M_n=M_{n-1}+\frac{h_{n-1}}{h_{n-2}}(M_{n-1}-M_{n-2}).$