CN110044362B - Method for rapidly calculating minimum value of relative distance between space targets - Google Patents

Method for rapidly calculating minimum value of relative distance between space targets Download PDF

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CN110044362B
CN110044362B CN201910357088.5A CN201910357088A CN110044362B CN 110044362 B CN110044362 B CN 110044362B CN 201910357088 A CN201910357088 A CN 201910357088A CN 110044362 B CN110044362 B CN 110044362B
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relative distance
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minimum value
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崔文
祝开建
张炜
张育卫
杨洋
游经纬
田鑫
郭建宏
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Chinese People's Liberation Army 32035
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
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Abstract

The invention discloses a method for quickly calculating a minimum value of a relative distance between two space targets, aiming at solving the problems of high calculation precision requirement of a local minimum value between the two space targets and long time consumption of a traditional direct interpolation calculation method.

Description

Method for rapidly calculating minimum value of relative distance between space targets
Technical Field
The invention relates to the field of space measurement and control, in particular to a method for quickly calculating a minimum value of a relative distance between space targets.
Background
With the continuous progress and development of aerospace technology, the space activities of human beings are continuously increased, and the space debris tends to be continuously and violently increased. According to statistics, tens of millions of space debris with the diameter smaller than 1 cm exist at present, hundreds of thousands of space debris with the diameter between 1 cm and 10 cm exist, more than 20000 space debris with the diameter larger than 10 cm exist at present, and the total number of space targets which can be observed on the ground and have the area reaching or exceeding the sub-square decimeter is about 16000 at present. On 11 days 2 and 2009, commercial communication spacecraft No. iridium 33 emitted by the American Iridium spacecraft LLC company in 1997 collides with military communication spacecraft No. universe 2251 emitted by Russia in 1993 in space, which is the first collision event of complete in-orbit spacecrafts in human history, causes the attention of each space nation to the space collision event, and is confirmed that more than 1200 detectable fragments are generated by the collision. The average collision speed of the space debris and the spacecraft is 10 kilometers per second, the conversion is carried out according to a formula that the kinetic energy is equal to mass multiplied by the square of the speed, the kinetic energy generated when the space debris with the weight of 10 grams collides with the spacecraft is equivalent to the kinetic energy generated when a car on a highway collides with a speed of 100 kilometers per hour, the result is considered to be catastrophic, and the space debris with the size of centimeter or more can cause the complete damage of the spacecraft.
In order to ensure the on-orbit safe operation of the spacecraft, for the space target which can be tracked at present, the orbit prediction can be carried out by utilizing observation data, whether the space target and the spacecraft have the possibility of collision or not is predicted, and the orbit of the spacecraft is adjusted timely to ensure that the space target is not collided, so that the orbit prediction method becomes a 'conventional action' of the safety guarantee of the spacecraft in the state of advanced foreign aerospace technology.
The most important work in the collision prediction analysis between the spacecraft and the space target is to calculate all local minimum values of the relative distance between the two space targets in a period of time. Since the relative speed between two space targets is generally in the order of 10 kilometers per second, in order to make the calculation accuracy of the relative distance minimum reach the meter order, the time corresponding to the minimum should be calculated to be below the millisecond order. According to the traditional calculation method, if the calculation is carried out below the millisecond level, the orbit data of the volume space target needs to be interpolated below the millisecond level. At present, the total number of the targets in the on-orbit space is about 1.6 ten thousand, if the targets are all calculated to the millisecond level according to the traditional direct interpolation one by one, the calculation amount is large and hard to imagine, and even in the current era of high-speed computer development, the results are unlikely to be obtained in a short time.
In view of the above reasons, the invention provides a method for rapidly calculating all local minimum values of the relative distance between two space targets based on the combination of a numerical method and an analytical method aiming at the characteristics of the operation orbits of spacecraft and space targets.
Disclosure of Invention
The present invention is directed to solving the above problems by providing a method for rapidly calculating a minimum value of a relative distance between spatial objects.
The invention realizes the purpose through the following technical scheme:
the invention comprises the following steps:
the method comprises the following steps: at a given time period [ t0, t1 ]]In the J2000 inertial coordinate system, the positions of two space targets at one minute are known as [ x1i y1i z1i ] respectively]，[x2i y2i z2i]Where i is 1,2, … n, n is an integer part of (t1-t0) x 1440; according to the position components of the two targets, the relative distance between the two targets can be calculated
Figure GDA0003492973900000021
Wherein i is 1,2, … n; will all satisfy the condition di＜di-1And d isi＜di+1The time corresponding to the relative distance of i 1,2, … n-1 is denoted as TkA total of m, i.e., k ═ 1,2, … m;
step two: at each time interval Tk1 minute, Tk+1 minute]In accordance with step one, known as [ x1i y1i z1i]And [ x2i y2i z2i]Each is taken [ Tk-3 minutes, Tk+4 minutes]Corresponding 8 position data are interpolated into data with interval of 0.5 s by adopting Lagrange interpolation method, and the data are marked as
Figure GDA0003492973900000028
And
Figure GDA00034929739000000211
wherein k is 1,2, … m, j is 1,2, … 241;
step three: obtained according to step two
Figure GDA0003492973900000029
And
Figure GDA00034929739000000210
calculating a relative distance value
Figure GDA0003492973900000022
Note DkjThe time corresponding to the minimum value is TkaWhich isWherein k is 1,2, … m, j is 1,2, … 241;
step four: based on D in step threekjJ is 1,2, … 241, and is [ Tka-1.5 seconds, Tka +2.0 seconds ]]The inner corresponding 8 relative distance values are marked as [ Dka1, Dka2, Dka3, Dka4, Dka5, Dka6, Dka7, Dka8]The corresponding times are [ Tka1, Tka2, Tka3, Tka4, Tka5, Tka6, Tka7, Tka8 ]]According to the lagrange interpolation formula, a 7-order polynomial dk (t) can be obtained as follows:
Figure GDA0003492973900000023
wherein
Figure GDA0003492973900000024
Step five: for 7 th order polynomial D in step fourk(T), where k is 1,2, … m, and the first derivative is obtained to obtain a6 th order polynomial D' k (T), as shown below:
Figure GDA0003492973900000025
wherein
Figure GDA0003492973900000026
Solving the 6-order polynomial D 'by adopting a dichotomy'k(T) in the interval [ Tka1.5 seconds, Tka+2.0 seconds]The only zero point in the interior, the convergence threshold is set as 1 x 10-4Second, the time T corresponding to the local minimum value of the relative distance can be obtainedkminWherein k is 1,2, … m, and m are total;
step six: according to the T obtained by calculation in the step fivekminCombining with [ x1i y1i z1i known in step one]And [ x2i y2i z2i]Each is taken [ Tkmin-3 minutes, Tkmin+4 minutes]Corresponding 8 position data can be obtained by adopting Lagrange interpolation rule to obtain TkminPosition data [ x ] corresponding to two targets at time1kmin y1kmin z1kmin]And [ x ]2kmin y2kmin z2kmin]So as to obtain the minimum value of the relative distance
Figure GDA0003492973900000027
Wherein k is 1,2, … m, and the total number of k is m.
The invention has the beneficial effects that:
the invention relates to a method for quickly calculating a minimum value of a relative distance between space targets, which is characterized in that a numerical method and an analytical method are combined to quickly calculate all local minimum values of the relative distance between two space targets aiming at the characteristics of a spacecraft and a space target operation orbit, and effectively solves the problems of high calculation precision requirement of the local minimum values between the two space targets and long time consumption of the traditional direct interpolation calculation method.
Detailed Description
The invention is further illustrated below:
the invention comprises the following steps:
the method comprises the following steps: at a given time period [ t0, t1 ]]In the J2000 inertial coordinate system, the positions of two space targets at one minute are known as [ x1i y1i z1i ] respectively]，[x2i y2i z2i]Where i is 1,2, … n, n is an integer part of (t1-t0) x 1440; according to the position components of the two targets, the relative distance between the two targets can be calculated
Figure GDA0003492973900000031
Wherein i is 1,2, … n; will all satisfy the condition di＜di-1And d isi＜di+1The time corresponding to the relative distance of i 1,2, … n-1 is denoted as TkA total of m, i.e., k ═ 1,2, … m;
step two: at each time interval Tk1 minute, Tk+1 minute]In accordance with step one, known as [ x1i y1i z1i]And [ x2i y2i z2i]Each is taken [ Tk-3 minutes, Tk+4 minutes]Corresponding 8 position data are interpolated into data with an interval of 0.5 second by adopting a Lagrange interpolation method, and the data are marked as
Figure GDA0003492973900000037
And
Figure GDA00034929739000000310
wherein k is 1,2, … m, j is 1,2, … 241;
step three: obtained according to step two
Figure GDA0003492973900000038
And
Figure GDA0003492973900000039
calculating a relative distance value
Figure GDA0003492973900000032
Note DkjThe time corresponding to the minimum value is TkaWherein k is 1,2, … m, j is 1,2, … 241;
step four: based on D in step threekjJ is 1,2, … 241, and is [ Tka-1.5 seconds, Tka +2.0 seconds ]]The inner corresponding 8 relative distance values are marked as [ Dka1, Dka2, Dka3, Dka4, Dka5, Dka6, Dka7, Dka8]The corresponding times are [ Tka1, Tka2, Tka3, Tka4, Tka5, Tka6, Tka7, Tka8 ]]According to the lagrange interpolation formula, a 7-order polynomial dk (t) can be obtained as follows:
Figure GDA0003492973900000033
wherein
Figure GDA0003492973900000034
Step five: for 7 th order polynomial D in step fourk(T), where k is 1,2, … m, and the first derivative is obtained to obtain a6 th order polynomial D' k (T), as shown below:
Figure GDA0003492973900000035
wherein
Figure GDA0003492973900000036
Solving the 6-order polynomial D 'by adopting a dichotomy'k(T) in the interval [ Tka1.5 seconds, Tka+2.0 seconds]The only zero point in the interior, the convergence threshold is set as 1 x 10-4Second, the time T corresponding to the local minimum value of the relative distance can be obtainedkminWherein k is 1,2, … m, and m are total;
step six: according to the T obtained by calculation in the step fivekminCombining with [ x1i y1i z1i known in step one]And [ x2i y2i z2i]Each is taken [ Tkmin-3 minutes, Tkmin+4 minutes]Corresponding 8 position data can be obtained by adopting Lagrange interpolation rule to obtain TkminPosition data [ x ] corresponding to two targets at time1kmin y1kmin z1kmin]And [ x ]2kmin y2kmin z2kmin]So as to obtain the minimum value of the relative distance
Figure GDA0003492973900000041
Wherein k is 1,2, … m, and the total number of k is m.
One target is arbitrarily selected as a main target from TLE data of 16932 space targets published in 11/5/11/2018, and local minimum values of relative distances between the selected target and the rest 1699 space targets are calculated. The number of the selected main target NORAD is 40059, the selected calculation time period is from 11 month, 5 days and 8 days in 2018 to 11 month, 8 days and 8 days in 2018, the threshold output by the local minimum value is temporarily set to be 10 kilometers due to space limitation, and the selected computer is configured as follows: under the conditions of Interi3-6300CPU @3.7GHz, a 4.0G memory and a Windows 732-bit operating system, the calculation result is shown in Table 1, and the time required for completing the calculation is 1 minute and 34 seconds.
TABLE 1 local minima of relative distances between object 40059 and other objects
Figure GDA0003492973900000042
As can be seen from table 1, the method for rapidly calculating the minimum value of the relative distance between the spatial targets is used, and for the calculation of the minimum value of the distance between one main target and over 16000 spatial targets, the calculation can be completed within 1 minute and 34 seconds for 3 days, and the calculation accuracy reaches more than millisecond. This shows that the calculation results using this method can satisfy both the time requirement and the accuracy requirement.
The foregoing shows and describes the general principles and features of the present invention, together with the advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (1)

1. A method for quickly calculating the minimum value of the relative distance between space targets is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: at a given time period [ t0, t1 ]]In the following, the positions of two space targets at one minute and one point in the J2000 inertial coordinate system are known as [ xli yli zli]，[x2i y2i z2i]Where i is 1,2, … n, n is an integer part of (t1-t0) x 1440; according to the position components of the two targets, the relative distance between the two targets can be calculated
Figure FDA0003492973890000011
Wherein i is 1,2, … n; will all satisfy the condition di＜di-1And d isi＜di+1The time corresponding to the relative distance of i 1,2, … n-1 is denoted as TkA total of m, i.e., k ═ 1,2, … m;
step two: at each time interval Tk1 minute, Tk+1 minute]According to [ xli yli zli known in step one]And [ x2i y2i z2i]Each is taken [ Tk-3 minutes, Tk+4 minutes]Corresponding 8 position data are interpolated into data with an interval of 0.5 second by adopting a Lagrange interpolation method, and the data are marked as
Figure FDA0003492973890000017
And
Figure FDA0003492973890000018
wherein k is 1,2, … m, j is 1,2, … 241;
step three: obtained according to step two
Figure FDA0003492973890000019
Calculating a relative distance value
Figure FDA0003492973890000012
Note Dj kThe time corresponding to the minimum value is TkaWherein k is 1,2, … m, j is 1,2, … 241;
step four: based on D in step threej kJ is 1,2, … 241, and is [ Tka-1.5 seconds, Tka +2.0 seconds ]]The inner corresponding 8 relative distance values are marked as [ Dka1, Dka2, Dka3, Dka4, Dka5, Dka6, Dka7, Dka8]The corresponding times are [ Tka1, Tka2, Tka3, Tka4, Tka5, Tka6, Tka7, Tka8 ]]According to the lagrange interpolation formula, a 7-order polynomial dk (t) can be obtained as follows:
Figure FDA0003492973890000013
wherein
Figure FDA0003492973890000014
Step five: for 7 th order polynomial D in step fourk(T), where k is 1,2, … m, and the first derivative is obtained to obtain a6 th order polynomial D' k (T), as shown below:
Figure FDA0003492973890000015
solving the 6-order polynomial D 'by adopting a dichotomy'k(T) in the interval [ Tka1.5 seconds, Tka+2.0 seconds]The only zero point in the interior, the convergence threshold is set as 1 x 10-4Second, the time corresponding to the local minimum value of the relative distance can be obtainedTkminWherein k is 1,2, … m, and m are total;
step six: according to the T obtained by calculation in the step fivekminCombining [ xli zli ] known in step one]And [ x2i y2i z2i]Each is taken [ Tkmin-3 minutes, Tkmin+4 minutes]Corresponding 8 position data can be obtained by adopting Lagrange interpolation rule to obtain TkminPosition data [ x ] corresponding to two targets at timelkmin y1kmin zlkmin]And [ x ]2kmin y2kmin z2kmin]So as to obtain the minimum value of the relative distance
Figure FDA0003492973890000016
Wherein k is 1,2, … m, and the total number of k is m.
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