CN109708643B - Evaluation and selection method for asteroid surface optical navigation road sign - Google Patents
Evaluation and selection method for asteroid surface optical navigation road sign Download PDFInfo
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Abstract
The invention discloses a method for evaluating and selecting an optical navigation road sign on the surface of a minor planet, which is particularly suitable for a navigation system for autonomously determining the position and the posture of a deep space probe by using a meteor crater as a road sign, and belongs to the field of autonomous navigation. The implementation method of the invention comprises the following steps: detecting and extracting meteor crater information on the surface of the asteroid in the navigation camera image by using an image processing algorithm; and carrying out ellipse fitting and meteor crater center positioning on the detected and extracted meteor crater information. Solving the error covariance matrix of the meteorite crater according to the uncertainty propagation characteristic by utilizing the fitted elliptic equation
And then error ellipse major and minor semi-axes a and b for describing the uncertainty of the meteorite crater error are obtained and used as indexes for evaluating the observation quality of the meteorite crater, so that an accurate and feasible method for autonomously selecting a positioning road sign is provided for the deep space probe, the optimal navigation road sign is selected, the accuracy of selecting the navigation road sign of the deep space probe is improved, and the navigation accuracy of the position posture of the deep space probe is improved.
Description
Technical Field
The invention relates to an evaluation and selection method for a deep space probe optical navigation road sign on the surface of an irregular asteroid, which is particularly suitable for a navigation system for autonomously determining the position and the posture of a deep space probe by using a meteor crater as a road sign and belongs to the field of autonomous navigation.
Background
The near-target celestial body flight is one of the most core tasks of deep space exploration in the future, the deep space exploration has long navigation distance and long time, and the traditional measurement and control mode has larger communication delay. In addition, the deep space dynamic environment is complex, the traditional navigation and control mode based on ground remote control cannot meet the requirement of realizing high-precision detection, and the detector is required to have an autonomous navigation function. Although the traditional inertial navigation does not need the support of a ground deep space measurement and control network, the satellite-borne inertial device has different degrees of zero drift error, random error and calculation error accumulation in the use process, and the requirement of a precise navigation task of deep space exploration cannot be met. With the breakthrough of computer hardware technology and the development of optical sensitive devices, the autonomous optical navigation method based on the spaceborne computer and the optical navigation camera becomes a research hotspot. The meteorite crater morphological characteristics on the surface of the asteroid are used as natural geographic terrain road signs, the visibility and the distinguishability are high, a detector does not need to additionally carry road sign loads, the task complexity is effectively reduced, and the application prospect is wide.
The autonomous navigation method of the deep space probe based on meteorite pit optical information as a navigation road sign has become a research hotspot at present, wherein how to select a proper road sign from numerous navigation road signs so as to meet the expected performance requirement is a key technology based on road sign navigation, the calculation efficiency of a software algorithm and the autonomous positioning capability of the probe are directly influenced, and whether a detection task can be successfully completed is determined, so the autonomous selection method of road sign navigation is one of key problems concerned by current science and technology personnel.
In the developed navigation road sign autonomous selection method, in the prior art [1] (Y.Tian, M.Yu.A novel classifier based visual navigation approach for accurate prediction pin-point mapping, Aero.Sci.Technol.70(2017)1-9), interaction Fisher information is maximized based on a one-step greedy forward-looking strategy, a planet surface in a laser radar range is selected as a feasible region, then those meteorite pits farthest from the image center in the feasible region are selected as road signs, and the estimation uncertainty of a detector is reduced through the meteorite pit selection mechanism. However, the method considers all meteorites as equal-precision observation, and does not consider the imaging characteristics and error uncertainty of different meteorites.
In the prior art [2] (see Chi Pingyuan and the like, an observation matrix-based autonomous navigation road sign selection method for a deep space probe, namely ZL 201010103514.1 [ P ],2012-01-04), aiming at the problem that the current deep space probe based on road sign navigation does not have an accurate and feasible autonomous navigation road sign selection method, the influence of a navigation road sign and a position relationship between the navigation road sign and a detector on navigation precision is considered, pixel information of three road signs is selected to autonomously determine the position and the posture of the deep space probe, and the autonomous navigation road sign selection method for the deep space probe based on the observation matrix is provided. However, the method does not evaluate the quality of the navigation road sign, and is only suitable for the condition that the observation states of the navigation road signs are the same.
In the prior art [3] (see Chi Pingyuan and the like, an observation condition number-based autonomous positioning signpost selection method for a deep space probe, China, ZL 201010103515.6 [ P ],2011-11-09) considers the influence of a signpost position on navigation precision, selects two signposts to construct the position of the probe under a target celestial body fixed connection coordinate system based on condition numbers of an observation equation by calculating and comparing condition numbers of an observation matrix, and provides an accurate and feasible positioning signpost autonomous selection method for the deep space probe flying at a low orbit. However, the method neglects the own observation errors of different navigation road signs, and is not suitable for selecting the planet surface navigation road signs in the actual task of the deep space probe.
In the process of carrying out autonomous navigation on the deep space probe by using meteorite crater optical information as a navigation road sign, the edge detection of the meteorite crater inevitably has observation errors, so that the uncertainty of different meteorite craters is different. The existing autonomous optical navigation method for the deep space probe does not consider the influence of uncertainty of the meteorite pit center, does not evaluate the observation quality of different navigation road signs, and causes the autonomous selection method for the navigation road signs of the deep space probe to be not accurate enough, thereby causing the estimation precision of the position and pose of the probe to be low.
Disclosure of Invention
The method aims to solve the problem that the existing deep space probe navigation road sign selection method does not consider the influence of uncertainty of the meteorite pit center, and further the navigation road sign independent selection method is not accurate enough. The invention discloses a method for evaluating and selecting an optical navigation road sign on the surface of a minor planet, which aims to solve the technical problems that: the observation quality of different meteorite craters on the surface of the irregular asteroid is evaluated by considering the observation uncertainty of the different meteorite craters, the optimal navigation road sign is selected, the selection accuracy of the navigation road sign of the deep space detector is further improved, and therefore the navigation accuracy of the position posture of the deep space detector is improved.
The invention is realized by the following technical scheme.
The invention discloses a method for evaluating and selecting an optical navigation road sign on the surface of a minor planet. And carrying out ellipse fitting and meteor crater center positioning on the detected and extracted meteor crater information. Solving the error covariance matrix of the meteorite crater according to the uncertainty propagation characteristic by utilizing the fitted elliptic equation
And then error ellipse major and minor semi-axes a and b for describing the meteorite crater error uncertainty are obtained and used as indexes for evaluating the meteorite crater observation quality, so that an accurate and feasible method for autonomously selecting a positioning road sign is provided for the deep space detector, namely the evaluation and selection of the irregular asteroid surface navigation road sign are realized based on the meteorite crater uncertainty.
The invention discloses a method for evaluating and selecting an optical navigation road sign on the surface of a minor planet, which comprises the following steps:
step 1: and detecting, extracting, fitting and positioning the meteor crater on the surface of the asteroid by using an image processing algorithm in the navigation camera image.
After reading the topographic image of the surface of the target planet shot by the optical camera, detecting and extracting the meteor crater edge of the image based on an image processing algorithm, and obtaining the meteor crater information of the surface of the asteroid, namely the pixel value of the meteor crater edge point. When observing that the meteorite crater boundary points are not less than five, determining coefficients B, C, D, E and F of the meteorite crater fitting elliptic equation through an elliptic fitting algorithm, and further obtaining the fitting elliptic equation of the meteorite crater edge as x2+2Bxy+Cy2+2Dx +2Ey + F ═ 0. Determining the center O (x) of the fitted ellipse by centroid formula0,y0) And realizing the positioning of the meteorite crater.
Preferably, to facilitate the analytical solution, the coefficients B, C, D, E, F of the merle crate fitting elliptic equation are determined by the least squares method.
Step 2: solving an error covariance matrix according to the uncertainty propagation characteristic of the error by using the fitted elliptic equation in the step 1
Writing n elliptical equations for n edge points of each meteorite crater navigation road sign, solving a covariance matrix P of elliptical equation coefficients through an observation error V, and generating the observation error covariance matrix P of each meteorite crater from the covariance matrix P of the elliptical equation coefficients according to the covariance propagation law
H is a matrix formed by the centroids of the coefficients B, C, D, E and F of the elliptic equation in the step 1.
Further, the step 2 is realized by the following specific method:
for each meteorite crater navigation road sign, n edge points exist and are xi＝(xi,yi) And i is 1,2 and 3 … n, writing n fitting elliptical equations for the n edge points by using the fitting elliptical equation in the step 1, and expressing an error equation as a matrix
Order to
The error equation is written as V ═ AX + Y. Observation error v of ith edge pointiHas a variance of
Wherein the content of the first and second substances,
i is the identity matrix of 2 × 2,
is the variance of the ith edge point of the meteorite crater,
variance matrix of observation error V
Is composed of
The covariance matrix P of the elliptic equation coefficients is defined as
Wherein A isTFinally, according to the covariance propagation law, an error covariance matrix of the centers of all meteorite craters is generated by P
Wherein, H is a matrix formed by the centroids of the coefficients B, C, D, E and F of the elliptic equation calculated by the centroid formula in the step 1;
and step 3: using error covariance matrices
And solving an error ellipse of each meteorite crater, and evaluating the observation uncertainty of the navigation signpost of the meteorite crater by using the major and minor semiaxes a and b of the error ellipse.
According to the error covariance matrix
The formula for obtaining the error ellipse parameter is as follows
Wherein a and b are respectively the major semi-axis and the minor semi-axis of the error ellipse,
is the major semi-axis azimuth of the error ellipse,
is the variance of the unit weight,
the magnitude of a and b represents the magnitude of the observation uncertaintyNamely, the short semi-axis a and the short semi-axis b of the error ellipse are used for evaluating the observation uncertainty of the navigation road sign of the meteorite crater.
And 4, step 4: and comparing the evaluation parameters a and b of different meteorite pits to select the optimal meteorite pit navigation road sign, thereby improving the accuracy of the selection of the navigation road sign of the deep space probe.
The selection criteria are: when m meteorite crater navigation beacons are detected and extracted in the step 1, the error ellipse parameter of the jth meteorite crater is (a)j,bj) The number of the expected selected meteorite crater navigation signposts is N, the objective function J randomly selects N meteorite craters from m meteorite craters extracted through detection, the sum of squares of the major and minor semi-axes a and b of the error ellipses of the N meteorite craters is the minimum, and the selected N meteorite craters are the optimal signposts with the minimum observation errors in the m meteorite crater navigation signposts. The mathematical model of the optimal road sign selection problem is shown below
aj<c2
bj<c3
Wherein, wjCalled decision variable, for randomly picking N merles from m merles. c. C1,c2And c3Is an observation accuracy constraint. The solution meeting the optimal road sign selection problem is the navigation road sign with the optimal planetoid surface, and the accuracy of the navigation road sign selection of the deep space probe is improved.
Further comprising the step 5: and (4) determining the position posture of the deep space detector based on the optimal meteorite pit navigation road sign selected in the step (4), so that the navigation precision of the position posture of the deep space detector is improved.
Has the advantages that:
the invention discloses a method for evaluating and selecting an optical navigation road sign on the surface of a minor planet. And carrying out ellipse fitting and meteor crater center positioning on the detected and extracted meteor crater information. Solving the error covariance matrix of the meteorite crater according to the uncertainty propagation characteristic by utilizing the fitted elliptic equation
And then obtaining error ellipse major and minor semi-axes a and b for describing meteorite pit error uncertainty, comparing evaluation parameters a and b of different meteorite pits to select the optimal meteorite pit navigation road sign, and improving the accuracy of selection of the navigation road sign of the deep space probe, thereby improving the navigation accuracy of the position posture of the deep space probe.
Drawings
FIG. 1 is a schematic flow chart of the evaluation and selection method of the asteroid surface optical navigation road sign of the present invention;
FIG. 2 is a schematic view of the navigation relationship of the deep space probe in the present invention for observing a target celestial body;
FIG. 3 is a schematic diagram of the image processing effect of step 1 in the example of the present invention, in which FIG. 3(a) is reading an original image, FIG. 3(b) is Gaussian noise filtering, FIG. 3(c) is edge detection, FIG. 3(d) is non-edge point rejection, FIG. 3(e) is false edge removal, and FIG. 3(f) is ellipse fitting and positioning;
FIG. 4 is a schematic diagram of image processing and error ellipses for step 3 meteor crater in the example of the present invention, wherein FIG. 4(a) is the original image and FIG. 4(b) is the image processing and error ellipses;
FIG. 5 is a diagram showing the result of selecting the meteor crater navigation signpost evaluation in step 4 in the example of the present invention.
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
In order to verify the feasibility of the invention, the shot meteorite crater image information on the surface of the minor planet Eros 433 is utilized, the result obtained by image processing of the meteorite crater and the fitting ellipse are drawn on an original image, as shown in FIG. 4(b), the three-dimensional coordinates of each meteorite crater navigation road sign in a minor celestial body fixed coordinate system are defined, and the initial position of a detector in the minor celestial body fixed coordinate system is [ 500; -300; 2000 m, initial attitude [ 5; -10; 30 deg.. The field angle is 30 degrees, the focal length of the navigation camera is f-8 mm, and mathematical simulation verification is carried out.
The method for evaluating and selecting the asteroid surface optical navigation road sign disclosed by the embodiment comprises the following specific implementation steps of:
step 1: and detecting, extracting, fitting and positioning the meteor crater on the surface of the asteroid by using an image processing algorithm in the navigation camera image.
After reading a topographic image of the surface of a target celestial body shot by an optical camera, detecting and extracting the meteor crater edge of the image through a Canny operator, and obtaining meteor crater information on the surface of the asteroid, namely the pixel value of a meteor crater edge point. When observing that the meteorite crater boundary points are not less than five, determining coefficients B, C, D, E and F of the meteorite crater fitting elliptic equation by the least square method, and further obtaining the fitting elliptic equation of the meteorite crater edge as x2+2Bxy+Cy2+2Dx +2Ey + F ═ 0. Determining the center O (x) of the fitted ellipse by centroid formula0,y0) And realizing the positioning of the meteorite crater.
The processing procedure and the result are shown in fig. 3, wherein the yellow point in fig. 3(f) is the edge point extracted by detection, the red ellipse is the fitting ellipse, and the green point is the positioning center.
Step 2: solving an error covariance matrix according to the uncertainty propagation characteristic of the error by using the fitted elliptic equation in the step 1
Writing n elliptical equations for n edge points of each meteorite crater navigation road sign, obtaining a covariance matrix P of elliptical equation coefficients through observation errors V, and calculating the covariance matrix P according to the resultsCovariance propagation law, the covariance matrix P of the elliptic equation coefficients is used to generate the covariance matrix of the observed error for each meteor crater
H is a matrix formed by solving partial derivatives of elliptic equation coefficients B, C, D, E and F by the centroid formula in the step 1;
the step 2 is realized by the following specific method:
for each meteorite crater navigation road sign, n edge points exist and are xi＝(xi,yi) And i is 1,2 and 3 … n, writing n fitting elliptical equations for the n edge points by using the fitting elliptical equation in the step 1, and expressing an error equation as a matrix
Order to
The error equation is written as V ═ AX + Y. Observation error v of ith edge pointiHas a variance of
Wherein the content of the first and second substances,
i is the identity matrix of 2 × 2,
is the variance of the ith edge point of the meteorite crater,
variance matrix of observation error V
Is composed of
The covariance matrix P of the elliptic equation coefficients is defined as
Wherein A isTFinally, according to the covariance propagation law, an error covariance matrix of the centers of all meteorite craters is generated by P
Wherein, H is a matrix formed by the centroids in the step 1 to calculate the partial derivatives of the elliptic equation coefficients B, C, D, E and F.
And step 3: using error covariance matrices
And solving an error ellipse of each meteorite crater, and evaluating the observation uncertainty of the navigation signpost of the meteorite crater by using the major and minor semiaxes a and b of the error ellipse.
The formula for obtaining the error ellipse parameter according to the error covariance matrix is as follows
Wherein a and b are respectively the major semi-axis and the minor semi-axis of the error ellipse,
is the major semi-axis azimuth of the error ellipse,
is the variance of the unit weight,
the sizes of a and b represent the size of observation uncertainty, namely the observation uncertainty of the navigation road sign of the meteorite crater is evaluated by using the short semi-axis a and the short semi-axis b of the error ellipse. The error ellipse obtained by calculating each meteorite crater is drawn on the shot navigation image, as shown in figure 4, the observation uncertainty of different meteorite craters can be visually displayed, and further basis is provided for subsequent navigation road sign selection.
And 4, step 4: and comparing the evaluation parameters a and b of different meteorite pits to select the optimal meteorite pit navigation road sign, thereby improving the accuracy of the selection of the navigation road sign of the deep space probe.
The selection criteria are: when m meteorite crater navigation beacons are detected and extracted in the step 1, the error ellipse parameter of the jth meteorite crater is (a)j,bj) The number of the expected selected meteorite crater navigation signposts is N, the objective function J randomly selects N meteorite craters from m meteorite craters extracted through detection, the sum of squares of the major and minor semi-axes a and b of the error ellipses of the N meteorite craters is the minimum, and the selected N meteorite craters are the optimal signposts with the minimum observation errors in the m meteorite crater navigation signposts. The mathematical model of the optimal road sign selection problem is shown below
aj<c2
bj<c3
Wherein, wjCalled decision variable, for randomly picking N merles from m merles. c. C1,c2And c3Is an observation accuracy constraint. The solution meeting the optimal road sign selection problem is the navigation road sign with the optimal planetoid surface, and the accuracy of the navigation road sign selection of the deep space probe is improved.
Taking the surface image of the planetoid minor Eros 433 shot in FIG. 4(a) as an example, the image processing and detection in step 1 extracts 24 meteorite craters, 6 meteorite craters are required to be selected as navigation landmarks for subsequent pose determination, and different error ellipses of 24 meteorite craters are calculated based on step 3 and are shown in the following table
TABLE 1 results of error ellipse parameter calculation
C, specifying the accuracy of the navigation signpost in the deep space exploration task1＝4,c2＝2,c3And (4) screening out 6 meteor crater navigation signposts with the minimum observation error by solving, wherein the preferred signposts are No. 1,9,12,13,15 and 23 meteor craters, and the calculation time is 2.3406 seconds.
Further comprising the step 5: and (4) determining the position posture of the deep space detector based on the optimal meteorite pit navigation road sign selected in the step (4), so that the navigation precision of the position posture of the deep space detector is improved.
The 6 meteorite craters selected by the asteroid surface optical navigation landmark evaluation and selection method can be used for determining the posture of the position of the detector. In order to verify the superiority of the meteorite crater selected by the method, the influence of the preferred meteorite crater signposts in the method and the influence of randomly selecting 6 meteorite crater signposts from 24 meteorite craters on the pose estimation precision are compared through 1000 Monte Carlo simulations. The comparative results are shown in the following table.
Table 2 pose determination results for preferred landmarks and random landmarks
It can be seen that the 6 meteorite craters selected by the method have higher pose estimation accuracy, so that the effectiveness of the selection result is verified. Therefore, the larger the observation uncertainty of the navigation road sign is, the poorer the observation quality of the navigation road sign is, and the road sign can be removed in the selection process through the restriction of the observation precision of the road sign, so that the selected meteorite pit navigation road sign has higher observation quality, and the pose estimation precision of the deep space detector is improved.
And then, finishing the evaluation and selection of the optical navigation road sign on the surface of the minor planet in the process of autonomously determining the position and the posture of the deep space probe.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (7)
1. The asteroid surface optical navigation road sign evaluation selection method is characterized by comprising the following steps of: comprises the following steps of (a) carrying out,
step 1: detecting, extracting, fitting and positioning meteor craters on the surfaces of the minor planets in the navigation camera image by using an image processing algorithm to obtain a fitted elliptic equation;
step 2: using fitting in step 1An ellipse equation for solving the error covariance matrix according to the propagation characteristics of the uncertainty of the error
And step 3: using error covariance matrices
Solving an error ellipse of each meteorite crater, and evaluating the observation uncertainty of the navigation signpost of the meteorite crater by using the major and minor semiaxes a and b of the error ellipse;
and 4, step 4: comparing the evaluation parameters a and b of different meteorite pits to select the optimal meteorite pit navigation road sign, and improving the accuracy of the deep space probe navigation road sign selection;
the selection criterion in the step 4 is as follows: when m meteorite crater navigation beacons are detected and extracted in the step 1, the error ellipse parameter of the jth meteorite crater is (a)j,bj) The number of the expected selected meteorite crater navigation signposts is N, the objective function J randomly selects N meteorite craters from m meteorite craters extracted by detection, the sum of squares of the major and minor semi-axes a and b of the error ellipses of the N meteorite craters is the minimum, and the selected N meteorite craters are the optimal signposts with the minimum observation errors in the m meteorite crater navigation signposts; the mathematical model of the optimal road sign selection problem is shown below
aj＜c2
bj＜c3
Wherein, wjCalled decision variable, for randomization from m merle cratesSelecting N meteorite craters; c. C1,c2And c3Is an observation accuracy constraint; the solution meeting the optimal road sign selection problem is the navigation road sign with the optimal planetoid surface, and the accuracy of the navigation road sign selection of the deep space probe is improved.
2. The asteroid surface optical navigation landmark evaluation and selection method of claim 1, wherein: and 5, determining the position posture of the deep space probe based on the optimal meteor crater navigation signpost selected in the step 4, thereby improving the navigation precision of the position posture of the deep space probe.
3. The asteroid surface optical navigation landmark evaluation and selection method of claim 1 or 2, characterized in that: the step 1 is realized by the method that,
after reading a topographic image of the surface of a target planet shot by an optical camera, detecting and extracting the meteor crater edge of the image based on an image processing algorithm to obtain meteor crater information of the surface of the asteroid, namely the pixel value of the meteor crater edge point; when observing that the meteorite crater boundary points are not less than five, determining coefficients B, C, D, E and F of the meteorite crater fitting elliptic equation through an elliptic fitting algorithm, and further obtaining the fitting elliptic equation of the meteorite crater edge as x2+2Bxy+Cy2+2Dx +2Ey + F ═ 0; determining the center O (x) of the fitted ellipse by centroid formula0,y0) Realizing the positioning of the meteorite crater;
4. the asteroid surface optical navigation landmark evaluation and selection method of claim 3, wherein: the step 2 is realized by the method that,
writing n elliptical equations for n edge points of each meteorite crater navigation road sign, solving a covariance matrix P of elliptical equation coefficients through an observation error V, and generating the observation error of each meteorite crater by the covariance matrix P of the elliptical equation coefficients according to a covariance propagation lawCovariance matrix
H is a matrix formed by solving partial derivatives of elliptic equation coefficients B, C, D, E and F by the centroid formula in the step 1;
5. the asteroid surface optical navigation landmark evaluation and selection method of claim 4, wherein: the specific implementation method of the step 2 is that,
for each meteorite crater navigation road sign, n edge points exist and are xi＝(xi,yi) And i is 1,2,3 … n, writing n fitting elliptical equations for the n edge points by using the fitting elliptical equation in the step 1, and expressing error equations by using a matrix as follows:
order to
The error equation is written as V ═ AX + Y; observation error v of ith edge pointiHas a variance of
Wherein the content of the first and second substances,
I2×2is an identity matrix of 2 × 2,
for i-th marginal point of meteorite craterThe variance of the measured values is calculated,
variance matrix of observation error V
Is composed of
The covariance matrix P of the elliptic equation coefficients is defined as
Wherein A isTA is called autocorrelation matrix, In×nIs the unit matrix of n × n, and finally, the error covariance matrix of each meteorite crater center is generated by P according to the covariance propagation law
Wherein, H is a matrix formed by the centroids of the coefficients B, C, D, E and F of the elliptic equation calculated by the centroid formula in the step 1;
6. the asteroid surface optical navigation landmark evaluation and selection method of claim 5, wherein: the specific implementation method of the step 3 is that,
according to the error covariance matrix
The formula for obtaining the error ellipse parameter is as follows
Wherein a and b are respectively the major semi-axis and the minor semi-axis of the error ellipse,
is the major semi-axis azimuth of the error ellipse,
is the variance of the unit weight,
the sizes of a and b represent the size of observation uncertainty, namely the observation uncertainty of the navigation road sign of the meteorite crater is evaluated by using the short semi-axis a and the short semi-axis b of the error ellipse.
7. The asteroid surface optical navigation landmark evaluation and selection method of claim 3, wherein: to facilitate the analytical solution, coefficients B, C, D, E, F of the merle crate fitting elliptic equation are determined by the least squares method.
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