CN111984917B - Calculation method of turning center in ball great circle track turning process - Google Patents
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Abstract
The invention provides a calculation method of turning center in the course of turning of a large sphere track, which respectively constructs small sphere with the spherical distance of turning radius to two sections of turning tracks, and takes the intersection point close to the turning track point in the intersection point of the two small sphere as the turning center. The method for calculating the turning center is simple and high in position accuracy, the problem that the calculation error of the turning center position is large when the aircraft turns with a large radius along the route is avoided, and the accurate realization of route alignment after the aircraft turns out is ensured.
Description
Technical Field
The invention belongs to the technical field of track control, and particularly relates to a calculation method of a turning center in a ball great circle track turning process.
Background
When the cruiser needs to set the route points and the turning radius of each route in advance in the process of moving according to the designated route, the cruiser makes a turn according to the set turning radius after cruising along one section of route is completed, and the cruiser aligns to the next route after completing the turn. When the set turning radius is smaller, the turning center can be obtained by utilizing the local plane coordinate system, but under the condition of larger turning radius, the turning center position obtained by using the plane coordinate system has larger access to the actual turning center position, so that a larger error exists between the cruiser and the original set route after turning is completed.
Disclosure of Invention
Aiming at the technical problem that a larger error exists between a cruise device and a set route after completing a turn with a larger radius in the prior art, the invention provides a calculation method of a turning center in the course of turning a great circle track of a ball, which can accurately calculate the turning center position of the great circle track of the ball and ensure that an aircraft sails according to the set route after turning.
The technical scheme adopted for solving the technical problems is as follows:
a calculation method of a turning center in a ball great circle track turning process comprises the following steps:
s1, acquiring position information and turning radius information of three route points of two sections of turning tracks AB and BC, and preprocessing data;
s2, respectively alongDirection(s) (i.e. the directions of the eyes)>The direction builds the ball small circle corresponding to the track AB, BC, the two ball small circles are parallel to the ball big circle corresponding to the track, the spherical distance from the ball big circle corresponding to the track is the turning radius, and the coordinates of the centers of the two ball small circles are calculated, wherein>The unit normal vectors are the great circles of the balls where the tracks AB and BC are located respectively;
s3, establishing a turning center calculation equation set according to the fact that the turning center is located on two small circles and the spherical surface of the earth;
s4, if the equation set has a singular phenomenon, singular value processing is carried out, otherwise, the next step is carried out;
s5, simplifying and solving a turning center equation set;
s6, checking the obtained two groups of feasible solutions, and taking the solution which is closer to the waypoint B as a turning center.
Further, the data preprocessing in step S1 includes:
angle adjustment, namely adjusting longitude and latitude information of three waypoints A, B, C to be within a longitude range (-180, 180) and a latitude range (-90, 90);
coordinate conversion, namely converting a longitude and latitude coordinate system into a rectangular coordinate system with a geocentric as an origin;
course conversion, if A, B, C three course points are arranged in a clockwise direction, the course conversion is not processed; otherwise, exchanging the position information of the A point and the C point to enable A, B, C to be arranged in a clockwise direction;
limiting the turning radius to make the turning radius r 0 Satisfy r min ≤r 0 ≤r max Wherein r is min R is the minimum turning radius of the aircraft max Is the upper limit of the turning radius; if r 0 ＞r max R is then 0 ＝r max The method comprises the steps of carrying out a first treatment on the surface of the If r 0 ＜r min R is then 0 ＝r min 。
Further, the upper limit of the turning radius is
Wherein O is the center of the earth.
Further, the method for calculating the coordinates of the centers of the two small spheres in the step S2 is as follows:
track AB corresponds to the center O of a small sphere 1 Coordinates are
Track BC corresponds to the center O of the small sphere 2 Coordinates are
Wherein lambda is AB Is the central angle corresponding to the track AB
λ BC For the central angle corresponding to the track BC
(x A 、y A 、z A )、(x B 、y B 、z B )、(x C 、y C 、z C ) Rectangular coordinates of the waypoints A, B, C respectively;
h r the center distance h from the small sphere to the large sphere with the corresponding track r ＝R 0 sinθ r ，R r Is a half R of a sphere r ＝R 0 cosθ r Wherein, the method comprises the steps of, wherein,r 0 for turning radius, R 0 Is the distance from the three waypoints to the center of the earth.
Further, in the step S3, the rotation center Q (x, y, z) calculates the equation set as
Further, the singular value processing method in step S4 is as follows: adding a small value to a variable which causes the singular value to appear to be 0, wherein if the turning radius is more than 10km, the small value is selected to be between 0.0001 and 0.001; if the turning radius is 10km or less, the small value is selected to be 10 -7 Magnitude.
Further, the singular value processing method in step S4 is as follows: and rotating the earth coordinate system along the y-axis and the z-axis by an angle to obtain a new earth coordinate system, solving the coordinate of the turning center in the new earth coordinate system, and then solving the coordinate of the turning center relative to the original earth coordinate system through a coordinate transformation method.
Further, the coordinate transformation is performed by using a direction cosine matrix or a quaternion method.
Further, the step S6 further comprises
Coordinate conversion, namely calculating longitude and latitude coordinates of a turning center:
if y < 0, then the output longitude lambda Q ＝-λ' Q Otherwise lambda Q ＝λ Q '。
Wherein x, y and z are right-angle coordinate values of a turning center, R 0 The distance from the radius of the earth, namely three waypoints, to the sphere center of the earth;
and angle adjustment, namely adjusting the output angle to be in a longitude range (-180, 180) and in a latitude range (-90, 90).
Compared with the prior art, the invention has the beneficial effects that:
according to the calculation method for the turning center in the turning process of the spherical great circle track, provided by the invention, the spherical small circles with the spherical distances of the two sections of turning tracks being the turning radius are respectively constructed, the intersection point, which is close to the turning route point, of the intersection points of the two spherical small circles is used as the turning center, the turning center is high in precision, the problem that the calculation error of the turning center position is large when the aircraft turns with a large radius along the (great circle) route is avoided, and the aim of aligning the route can be realized accurately after the aircraft goes out of the curve.
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The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention. It is evident that the drawings in the following description are only some embodiments of the present invention and that other drawings may be obtained from these drawings without inventive effort for a person of ordinary skill in the art.
FIG. 1 is a schematic diagram of a calculated turning center according to an embodiment of the present invention;
FIG. 2 is a schematic view of a building sphere according to an embodiment of the present invention;
FIG. 3 is a graph showing test results for calculating a turning center using the present invention; wherein, (a) the input waypoint information is A (144,46), B (150, 32) and C (138,18), and the turning radius is r 0 =500 km; (b) The input route point information is A (-160,0), B (0, 0), C (0, 20), and the turning radius is r 0 =1000 km; (c) The input route point information is A (-10, 0), B (0, 0), C (0, 20), and the turning radius is r 0 =2000 km; (d) The input route point information is A (-10, 0), B (10, 10), C (30, -10), and the turning radius is r 0 ＝700km。
Detailed Description
The invention calculates the sphere large circle track turning center from the perspective of solid geometry, and the information required by calculating the track turning center has two sections of track information and turning radius. As shown in fig. 1, the track information is given in the form of waypoints, and for convenience of explanation, the track before the turn is denoted by AB, and the track after the turn is denoted by BC. The current track AB and the subsequent track BC are large spherical tracks (the circle centers of the tracks coincide with the sphere center of the earth), and the spherical distances from the turning center Q to the two tracks AB and BC are the same, so that the turning center is symmetrical relative to the two tracks.
The invention provides a calculation method of a turning center in a ball great circle track turning process, which comprises the following steps:
s1, acquiring position information and turning radius information of three route points of two sections of turning tracks AB and BC, and preprocessing data.
The longitude and latitude coordinates of the route point A, B, C of the tracks AB and BC are respectively recorded as Turning radius r 0 。
The data preprocessing method comprises the following steps:
s1.1, angle adjustment: adjusting the range and format of the acquired data
The longitude range is (-180, 180), the east longitude is positive, the west longitude is negative, the latitude range is (-90, 90), the north latitude is positive, and the south latitude is negative.
S1.2, coordinate conversion: converting longitude and latitude coordinate system into rectangular coordinate system with earth center as origin
In the invention, the rectangular coordinate system takes the sphere center of the earth as the origin, the intersection point of the 0-degree meridian and the equator is the positive point of the x-axis pointing to the 0-degree meridian, the north of the z-axis pointing from the origin of the coordinate system is the positive point, and the y-axis meets the right-hand rule.
According to the relation from the spherical coordinate system to the rectangular coordinate system, calculating A, B, C Cartesian coordinates of each waypoint:
s1.3, course conversion: if A, B, C three waypoints are arranged in a clockwise direction, not processing; otherwise, exchanging the position information of the A point and the C point, arranging A, B, C in a clockwise direction, and entering the next step.
The calculation process of the present invention is processed in a clockwise course order, i.e., A, B, C in a clockwise direction (i.e., left turn). When A, B, C is anticlockwise, the information of the points A and C is exchanged, the course is changed into the clockwise direction and is calculated according to the converted course, and the position of the turning center is unchanged in consideration of the symmetry of the course.
S1.4, limiting the turning radius:
calculating unit normal vector of sphere large circle where AB track isAnd the unit normal vector of the sphere where the BC track is located +.>
Wherein O is the center of the earth.
Calculating the included angle of two unit normal vectorsCalculating the upper limit of the turning radius +.>
The turning radius is made to satisfy: r is (r) min ≤r 0 ≤r max ，r min R is the minimum turning radius of the aircraft max For the upper limit of the turning radius, if the inputted turning radius is larger than the turningUpper limit of radius of curvature r 0 ＞r max R is then 0 ＝r max The method comprises the steps of carrying out a first treatment on the surface of the If the input turning radius is smaller than the minimum turning radius r of the aircraft 0 ＜r min R is then 0 ＝r min 。
The turning radius is positive in the invention.
S2, respectively alongDirection(s) (i.e. the directions of the eyes)>The directions construct small circles corresponding to the tracks AB and BC, the two small circles are parallel to the large circle of the ball where the corresponding track is located, and the spherical distances from the large circle of the ball where the corresponding track is located are both the turning radius r 0 And calculating the coordinates of the centers of the two small spheres.
Let the spherical distance to the sphere big circle where the track AB, BC is located be r 0 Corresponding sphere, small circle and center of (2) are O respectively 1 And O 2 The track turning center is Q, the sphere center of the earth is O, and the absolute value of QO is given by 1 |＝|QO 2 |＝R r Let the distance from the three waypoints to the sphere center of the earth (namely the radius of the earth) be R 0 。
As shown in fig. 2, < a- 1 OB 1 ＝∠AOB＝λ AB ,OO 1 ＝h r ，O 1 B 1 ＝O 1 A 1 ＝R r Cambered surface distance AA 1 ＝BB 1 ＝r 0 Calculating central angle lambda corresponding to track AB AB ：
Calculating the center distance h between a small sphere and a large sphere corresponding to the track r Radius R of sphere r
h r ＝R 0 sinθ r
R r ＝R 0 cosθ r
Calculating the center coordinates of the small sphere corresponding to the track AB
I.e. O 1 The coordinates of (2) are:
similarly, calculating the central angle lambda corresponding to the track BC BC The track BC corresponds to the center coordinates of the small sphere
I.e. O 2 The coordinates of (2) are:
and S3, establishing a turning center calculation equation set according to the fact that the turning center is located on two small circles and the spherical surface of the earth.
The turning centers Q (x, y, z) are located on two small circles and on the sphere of the earth, then there are
S4, if the equation set has the singular phenomenon, singular value processing is carried out, and otherwise, the next step is carried out.
In solving equation set (1), due to x o1 、y o1 、x o2 、y o2 When equal to 0, singular results in subsequent calculation, resulting in no solution to the equation, but this phenomenon occurs due to limitations in the equation representation and not in the real physical case. The singular values can be eliminated by:
the method comprises the following steps: when the calculation does not require higher position accuracy and the equation is singular, x is given o1 、y o1 、x o2 、y o2 The equation is solved after a small value is added, the processing mode of the method is simpler, and the accuracy within 2 meters can be ensured. For example, the large sphere is the earth, and when the turning radius is more than 10km, the small value is preferably between 0.0001 and 0.001; the small value is selected to be 10 when the turning radius is below 10km -7 Is more suitable.
The second method is as follows: the singular problem is solved from the root, a coordinate transformation method can be adopted, when the singular phenomenon occurs, the earth coordinate system is rotated along the y axis and the z axis by an angle to obtain a new earth coordinate system, the equation set (1) in the new earth coordinate system is not singular any more, the coordinate of the Q point in the new earth coordinate system is solved, and the coordinate of the Q point relative to the original earth coordinate system is obtained through the coordinate transformation method, so that the absolute Q point coordinate can be obtained. The coordinate transformation may use a direction cosine matrix or quaternion, etc.
S5, simplifying and solving a turning center equation set
The expansion of equation set (1) is simplified to
Wherein A is 11 ＝-2x o1 ；A 12 ＝-2y o1 ；A 13 ＝-2z o1 ；x o1 2 +y o1 2 +z o1 2 +R 0 2 -R r 2 ＝B 1 ，A 21 ＝-2x o2 ；A 22 ＝-2y 02 ；A 23 ＝-2z o2 ；x o2 2 +y o2 2 +z o2 2 +R 0 2 -R r 2 ＝B 2 。
Solving equation (2) to obtain the position of the point Q (x, y, z), firstly, for A 11 、A 21 Carry out zero removal protection and make
Then there are:
y＝K 11 z+K 12 (3)
similarly, for A 12 And A 22 Zero removal protection is carried out, and the instruction
The method comprises the following steps:
x＝K 21 y+K 22 (4)
bringing the above results (3) (4) into x 2 +y 2 +z 2 ＝R 0 2 Obtaining:
az 2 +bz+c＝0
wherein:
b＝2(K 21 K 22 +K 11 K 12 )
calculate Δ=b 2 -4ac, a feasible solution of two sets of Q (x, y, z) points is obtained as follows:
x 1 ＝K 21 z 1 +K 22
y 1 ＝K 11 z 1 +K 12
x 2 ＝K 21 z 2 +K 22
y 2 ＝K 11 z 2 +K 12
s6, result inspection and data processing
And checking the obtained two sets of feasible solutions, taking the solution which is closer to the waypoint B as a turning center, and converting the form of the turning center according to the requirement.
S6.1, feasible solution inspection:
the calculated distances from the two turning center positions to the point B,
S QB1 ＝(x B -x 1 ) 2 +(y B -y 1 ) 2 +(z B -z 1 ) 2
S QB2 ＝(x B -x 2 ) 2 +(y B -y 2 ) 2 +(z B -z 2 ) 2
the solution closer to the point B is taken as the center of the turn Q (x, y, z).
S6.2, coordinate conversion: calculating the longitude and latitude of the turning center according to the relation from the rectangular coordinate system to the spherical coordinate system
If y < 0, then the output longitude lambda Q ＝-λ' Q Otherwise lambda Q ＝λ Q '。
S6.3, protection of an output angle range: the output angle is adjusted to be within a longitude range (-180, 180), the east longitude is positive, the west longitude is negative, the latitude range (-90, 90), the north latitude is positive, and the south latitude is negative.
The calculation method of the turning center in the course of turning of the large circle track of the ball is adopted for testing, as shown in figure 3, the earth radius is 6371km, and the algorithm can be seen to successfully give the position of the turning center. Through verification, the accuracy is within 2m, and the turning center position deviation obtained by using the traditional method for calculating the turning radius by using the local coordinates is in km level.
In addition, the 'larger turning radius' in the invention is aimed at the radius of the sphere where the aircraft is located, if the radius of the sphere where the aircraft runs is smaller, such as moon, the phenomenon that the turning center position error calculated by the plane coordinates is larger easily occurs even if the turning radius is not large in magnitude. The invention is therefore also applicable to track calculation and control of aircraft on certain small stars. The spherical distance from the point to the great circle track can be calculated by using the method reversely.
The above description is only of the preferred embodiments of the present invention and is not intended to limit the present invention, but various modifications and variations can be made to the present invention by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
The invention is not described in detail in a manner known to those skilled in the art.
Claims (5)
1. A calculation method of a turning center in a ball great circle track turning process is characterized by comprising the following steps:
s1, acquiring position information and turning radius information of three route points of two sections of turning tracks AB and BC, and preprocessing data;
the data preprocessing in step S1 includes:
angle adjustment, namely adjusting longitude and latitude information of three waypoints A, B, C to be within a longitude range (-180, 180) and a latitude range (-90, 90);
coordinate conversion, namely converting a longitude and latitude coordinate system into a rectangular coordinate system with a geocentric as an origin;
course conversion, if A, B, C three course points are arranged in a clockwise direction, the course conversion is not processed; otherwise, exchanging the position information of the A point and the C point to enable A, B, C to be arranged in a clockwise direction;
limiting the turning radius to make the turning radius r 0 Satisfy r min ≤r 0 ≤r max Wherein r is min For sailingMinimum turning radius of the device, r max Is the upper limit of the turning radius; if r 0 ＞r max R is then 0 ＝r max The method comprises the steps of carrying out a first treatment on the surface of the If r 0 ＜r min R is then 0 ＝r min ；
S2, respectively alongDirection(s) (i.e. the directions of the eyes)>The direction builds the ball small circle corresponding to the track AB, BC, the two ball small circles are parallel to the ball big circle corresponding to the track, the spherical distance from the ball big circle corresponding to the track is the turning radius, and the coordinates of the centers of the two ball small circles are calculated, wherein>The unit normal vectors are the great circles of the balls where the tracks AB and BC are located respectively;
the method for calculating the coordinates of the centers of the two small spheres in the step S2 is as follows:
track AB corresponds to the center O of a small sphere 1 Coordinates are
Track BC corresponds to the center O of the small sphere 2 Coordinates are
Wherein lambda is AB Is the central angle corresponding to the track AB
λ BC For the central angle corresponding to the track BC
(x A 、y A 、z A )、(x B 、y B 、z B )、(x C 、y C 、z c ) Rectangular coordinates of the waypoints A, B, C respectively;
h r the center distance h from the small sphere to the large sphere with the corresponding track r ＝R 0 sinθ r ，R r Is a half R of a sphere r ＝R 0 cosθ r Wherein, the method comprises the steps of, wherein,r 0 for turning radius, R 0 The distance from the three route points to the sphere center of the earth;
s3, establishing a turning center calculation equation set according to the fact that the turning center is located on two small circles and the spherical surface of the earth;
the rotation center Q (x, y, z) in the step S3 is calculated as the equation set
S4, if the equation set has a singular phenomenon, singular value processing is carried out, otherwise, the next step is carried out;
s5, simplifying and solving a turning center equation set;
s6, checking the obtained two groups of feasible solutions, and taking the solution which is closer to the waypoint B as a turning center;
the step S6 also comprises
Coordinate conversion, namely calculating longitude and latitude coordinates of a turning center:
if y < 0, then the output longitude lambda Q ＝-λ' Q Otherwise lambda Q ＝λ Q '；
Wherein x, y and z are right-angle coordinate values of a turning center, R 0 The distance from the radius of the earth, namely three waypoints, to the sphere center of the earth;
and angle adjustment, namely adjusting the output angle to be in a longitude range (-180, 180) and in a latitude range (-90, 90).
2. The method for calculating a turning center during a course of a great circle of balls according to claim 1, wherein the upper limit of the turning radius is
Wherein O is the center of the earth.
3. The method for calculating the turning center in the course of turning a great circle of balls according to claim 1, wherein the singular value processing method in step S4 is as follows: adding a small value to a variable which causes the singular value to appear to be 0, wherein if the turning radius is more than 10km, the small value is selected to be between 0.0001 and 0.001; if the turning radius is 10km or less, the small value is selected to be 10 -7 Magnitude.
4. The method for calculating the turning center in the course of turning a great circle of balls according to claim 1, wherein the singular value processing method in step S4 is as follows: and rotating the earth coordinate system along the y-axis and the z-axis by an angle to obtain a new earth coordinate system, solving the coordinate of the turning center in the new earth coordinate system, and then solving the coordinate of the turning center relative to the original earth coordinate system through a coordinate transformation method.
5. The method for calculating the turning center during the course of a spherical great circle according to claim 4, wherein the coordinate transformation is performed using a direction cosine matrix or a quaternion method.
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