Space target collision early warning method based on semi-analytical orbit theory

Abstract

The invention provides a space target collision early warning method based on a semi-analytical orbit theory, and the method comprises the steps: carrying out the orbit prediction through large-step mean root integration based on the semi-analytical orbit theory, and calculating a relative position vector at an approaching moment in combination with a J2 perturbation first-order short period term; then, on the basis of a J2 perturbation semi-analysis first-order mean root equation, a mean root deviation state transition matrix equation without singular points is constructed, and the position error covariance at the approaching moment is calculated; and finally, based on the joint position error covariance of the space target, utilizing a Mahalanobis distance criterion to judge a collision risk. The collision early warning method provided by the invention can obtain good balance between the calculation speed and the early warning accuracy, can avoid the problem of artificially introducing the collision probability threshold, has the advantages of no singular point, simple calculation and convenient engineering application, and is suitable for rapid and high-precision collision early warning of a large number of space targets.

Description

Technical Field

[0001] The present invention belongs to the technical field of space situational awareness, and relates to a method for predicting the orbit of a space target and judging the collision risk based on semi-analytical orbit theory, which can provide a fast and accurate collision warning means for the situational awareness of a large number of space targets and space traffic management in the future. Background Art

[0002] Space targets refer to all man-made objects operating in outer space, including artificial satellites, space stations, spacecraft, space shuttles, and space debris generated by them or their waste. With the continuous development of human spaceflight and the increasing frequency of space activities, the number of space targets (spacecraft, rocket bodies, space debris, etc.) on orbit has continued to increase. As of January 2024, the total number of trackable targets on orbit catalogued by the US Space Surveillance Network (SSN) has reached 28,352, including 11,988 payloads and 16,364 other targets such as rocket bodies and space debris. The increase in the number of space targets has significantly increased the risk of collisions between spacecraft and other space targets such as space debris. In the future, there may even be chain collisions between space targets, causing long-term damage to the space environment. Reliable and efficient collision warnings for space targets have become the key to maintaining national space security.

[0003] At present, collision warning is mainly based on analytical or numerical orbit prediction methods combined with collision probability to evaluate the collision risk of space targets. In terms of orbit prediction, establishing an accurate and efficient orbit prediction model and quickly providing long-term accurate orbit prediction data for space targets is the basis for ensuring the effective implementation of collision warning. There are two main types of orbit prediction methods: analytical method and numerical method for different application requirements. The analytical method is based on perturbation theory and usually relies on the series expansion of perturbation acceleration to analytically integrate the long-term term, long-period term and short-period term of orbit element changes. Commonly used analytical methods at home and abroad include the analytical orbit theory of Kozai and Brouwer, the quasi-average root method proposed by Liu Lin, and the SGP4 / SDP4 model used by the US SSN. In contrast to the analytical method, the numerical method directly integrates the instantaneous orbit differential equation, such as the Cowell method and the Encke method. However, with the rapid growth of the number of space targets, the number of approach events of space targets has increased dramatically. Both analytical and numerical methods have serious deficiencies in the face of a large number of collision warning calculations. Reliable collision warning requires an orbit prediction accuracy of 10 to 100 meters. The advantage of analytical methods is high computational efficiency, but the disadvantage is insufficient prediction accuracy. The prediction error is on the order of 1 to 10 km per day, which is difficult to meet the needs of accurate collision warning and is prone to a large number of false alarms and missed alarms. In addition, the above analytical methods are based on classical orbital elements or Delaunay variables for orbit prediction. There are singularities in the case of near-circular orbits and zero-inclination orbits, which also leads to large orbit prediction errors. Although the numerical method has high accuracy, the amount of calculation is large, because the short-period terms of the instantaneous orbital elements severely limit the step size of numerical integration. Usually, one orbit needs to be integrated for more than 100 steps to obtain a relatively accurate result, so it is difficult to achieve timely collision warning. In addition to analytical and numerical methods, there is another type of orbit prediction method, namely semi-analytical method. Semi-analytical method combines the advantages of analytical and numerical methods to balance the calculation speed and accuracy of orbit prediction. Its essence is to remove the short-period terms from the instantaneous orbital elements, and the remaining average orbital elements (including long-term terms and long-period terms) can be numerically integrated with large steps (usually about 1 day). Therefore, facing the collision warning between the rapidly growing number of space targets, the semi-analytical method has great application prospects.

[0004] In terms of collision risk judgment criteria, the early international space target collision warning was mainly based on the regional method (Box method), that is, a fixed warning area is preset near the satellite. When the target may enter the area, a collision warning is issued and the satellite is controlled to perform evasive maneuvers. This evaluation index is relatively conservative and easily leads to unnecessary orbital maneuvers. In order to overcome the limitations of the regional method, the collision risk judgment method based on collision probability (Pc method) is widely used internationally. It comprehensively considers the approaching geometric relationship of the two targets and the uncertainty of the position error. It evaluates the collision risk by calculating the collision probability when approaching and comparing it with the preset threshold value. The disadvantage is that the collision probability needs to be preset and it is more dependent on experience. If the threshold value is too high, it is easy to cause a large number of false alarms, while if the threshold value is too low, it may cause missed alarms and have serious consequences. In addition, there is another evaluation index, namely the Mahalanobis distance method, which issues a collision warning through the relationship between the relative position covariance ellipsoid and the relative position vector. The advantage is that it avoids the introduction of artificially set error thresholds and has important practical value.

[0005] The present invention proposes a space target collision warning method based on semi-analytical orbit theory, which performs fast and accurate J2 flat root prediction based on the semi-analytical orbit theory and the orbit elements of the vernal equinox to obtain the average root with high precision and no singularity; and calculates the instantaneous root and position vector of the space target at any time in combination with the J2 first-order short-period term, and then calculates the approach time and relative position vector; then, based on the J2 perturbation semi-analytical first-order flat root equation, constructs the flat root deviation state transfer matrix equation without singularity, propagates the uncertainty of the initial orbit element, and obtains the position error covariance at the approach time; finally, based on the joint position error covariance of the space target, the collision risk is judged using the Mahalanobis distance criterion. Summary of the invention

[0006] In response to the demand for rapid and high-precision collision warnings for a large number of space targets in the future, the present invention proposes a space target collision warning method based on semi-analytical orbit theory. Taking J2 perturbation into consideration, the orbit state and error covariance matrix of the space target are propagated rapidly and accurately based on the semi-analytical orbit theory. Based on the orbit state prediction results, the closest time and the corresponding relative position vector between the two space targets are calculated. Finally, based on the joint position error covariance of the two space targets, the collision risk is judged using the Mahalanobis distance criterion. The technical solutions adopted by the present invention are as follows:

[0007] The space target collision warning method based on semi-analytical orbit theory proposed in the present invention comprises the following steps:

[0008] Step 1: Flat root prediction based on semi-analytical orbit theory

[0009] Known initial time t 0 The corresponding mean vernal equinox orbital element is a(t0 )=(a, h, k, p, q, λ), based on the semi-analytical orbit theory, a high-precision flat-root differential equation considering the first-order and second-order perturbations of the Earth’s non-spherical gravitational field J2 is established:

[0010]

[0011] where a i represents the i-th component of a (i=1, 2, ..., 6), ∈=J 2 represents the J2 band harmonic coefficient, A i1 and A i2 are the first-order and second-order root-mean-square rates of J2, respectively. In addition, for i=1, 2, ..., 5, we have δ i6 =0, for i=6, we have δ i6 =1.

[0012] For space targets A and B, the above flat root equation is respectively calculated in the time interval [t 0 , t] By using large step-length numerical integration, the high-precision average root number a of the space targets A and B at time t can be obtained A and a B .

[0013] Step 2: Calculate the approach time and relative position vector based on the flat root and J2 first-order short-period term, and preliminarily screen the time when the collision may occur. Calculate the instantaneous orbital elements of targets A and B at time using J2 first-order short-period term:

[0014]

[0015] in and denote the instantaneous orbital elements of A and B respectively, J represents the orbital root of the vernal equinox 2 First-order short-period term;

[0016] The relative position vector of A and B is expressed as:

[0017] Δr(t)=r A (t)-r B (t)

[0018] where r A (t) and r B (t) represents the absolute position vectors of targets A and B converted from the instantaneous equinox orbit elements;

[0019] Finally, given a time interval [t 0 , t f], the time interval is divided into several sub-intervals according to the orbital period, and the time t corresponding to the minimum value of the relative distance Δr(t)=||Δr(t)|| is calculated in each sub-interval min and the relative position vector Δr min , and according to the minimum distance threshold Δr tol Preliminary screening of possible collision time sequences and corresponding relative position vectors, that is, when ||Δr min ||<Δr tol , then it is assumed that spacecraft A and B are at t min A collision may occur at time t, otherwise it is assumed that spacecraft A and B are at t min No collision will occur.

[0020] Step 3: Calculate the position error covariance at the time of possible collision based on the semi-analytical flat root deviation state transfer matrix

[0021] Calculate the position error covariance of A and B at the time when the collision may occur, that is, the first-order square root rate A of the orbital elements of the vernal equinox under the J2 perturbation 1 =(A 11 , A 21 , ..., A 61 ), construct the flat root deviation state transfer matrix equation without singular points based on the orbital elements of the vernal equinox:

[0022]

[0023] in is the square root deviation state transfer matrix of the orbital elements of the vernal equinox. The square root deviation state transfer matrix at any time can be obtained by numerically integrating the equation with a large step length.

[0024] Calculate the time t at which a collision may occur min The corresponding root mean square error covariance of targets A and B is:

[0025]

[0026] in and Respectively represent the square root error covariance matrix of targets A and B at the initial moment;

[0027] Then, the possible collision time t is calculated min The corresponding position error covariance of targets A and B is:

[0028]

[0029] At this point, the position error covariance of targets A and B at the moment when a collision may occur is obtained.

[0030] Step 4: Determine collision risk based on Mahalanobis distance

[0031] Calculate the time t at which a collision may occur min The corresponding joint position error covariance of targets A and B is:

[0032]

[0033] The collision warning criterion based on Mahalanobis distance is:

[0034]

[0035] Where k is a non-negative real number, and different k values ​​represent different collision risk tolerances. The final collision warning criterion is: when the above equation is true, a collision warning is issued; otherwise, no collision warning is issued.

[0036] Through the above steps, space target collision warning based on semi-analytical orbit theory can be realized.

[0037] The space target collision warning method based on semi-analytical orbit theory proposed in the present invention has the following advantages:

[0038] (1) The method proposed in the present invention predicts orbital state quantities and their error covariance based on semi-analytical orbit theory. By performing large-step numerical integration on the slowly changing average orbital elements, a good balance can be achieved between prediction speed and accuracy. It is suitable for rapid and high-precision collision warning of a large number of space targets.

[0039] (2) The method proposed in the present invention performs orbit prediction based on the orbit elements of the vernal equinox, thus avoiding the singularity problem in the case of zero eccentricity or zero inclination, and is applicable to the prediction of elliptical orbits with arbitrary eccentricity, and has the advantages of wide application range and high prediction accuracy.

[0040] (3) The method proposed in the present invention is based on the joint covariance matrix and Mahalanobis distance criterion to judge the collision risk, which can avoid the problem of artificially introducing a collision probability threshold and is convenient for engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 This is a flow chart of space target collision warning based on semi-analytical orbit theory. DETAILED DESCRIPTION

[0042] The specific implementation process of the present invention will be further described below with reference to the accompanying drawings.

[0043] The steps and specific implementation methods of the space target collision warning method based on semi-analytical orbit theory proposed in the present invention are as follows:

[0044] Step 1: Flat root prediction based on semi-analytical orbit theory

[0045] Let the mean vernal equinox orbital element be a = (a, h, k, p, q, λ), and the conversion relationship with the classical orbital element is as follows:

[0046] a=a,λ=M+ω+IΩ

[0047] h=e sin(ω+IΩ), k=e cos(ω+IΩ)

[0048] p=tan I (i / 2)sinΩ,q=tan I (i / 2)cosΩ

[0049] Among them, a, e, i, ω, Ω, and M represent the semi-major axis, eccentricity, orbit inclination, argument of perigee, right ascension of ascending node, and mean anomaly, respectively. I=1 represents a prograde orbit, and I=-1 represents a retrograde orbit.

[0050] Based on the semi-analytical orbit theory, a high-precision flat-root differential equation considering the first-order and second-order perturbations of the Earth's non-spherical gravitational field J2 is established:

[0051]

[0052] where a i represents the i-th component of the mean vernal equinox orbital element a (i=1, 2, ..., 6), represents the average orbital angular velocity, ∈ = J 2 represents the J2 band harmonic coefficient; for i = 1, 2, ..., 5, δ i6 =0, for i=6,δ i6 =1; in addition, A i1 and A i2 are the first and second order root mean square rate of J2, A i1 The display expression is as follows:

[0053] A 11 =0,

[0054]

[0055]

[0056] in,

[0057]

[0058] S=KH 4 a 3

[0059] C h = k(3γ 2-1+2γ(pα-Iqβ))

[0060] C k =-(3γ 2 -1+2γ(pα-Iqβ))

[0061] C p =-Cβγ

[0062] C q =-ICαγ

[0063] C λ =(1+H)(3γ 2 -1)+2γ(pa-Iqβ)

[0064] Where R is the radius of the earth, K = na 2 , C=1+p 2 +q 2 ,

[0065] In addition, A i2 The explicit expression for is:

[0066] A 12 =0

[0067]

[0068] in,

[0069] It is known that the initial average vernal equinox orbital elements of space targets A and B are a A (t 0 ) and a B (t 0 ), respectively, for formula (1) in the time interval [t 0 , t] By using large step-length numerical integration, the high-precision average root number a of the space targets A and B at time t can be obtained A (t) and a B (t).

[0070] Step 2: Calculate the approach time and relative position vector based on the flat root and J2 first-order short-period term, and preliminarily screen the time when collision may occur

[0071] Using the first-order short-period term of J2, calculate the instantaneous orbital elements of targets A and B at time t:

[0072]

[0073] in and denote the instantaneous orbital elements of A and B respectively, J represents the orbital elements (a, h, k, p, q, λ) of the vernal equinox 2 First-order short-period term. It can be expressed as follows:

[0074]

[0075] Among them, L = ω + Ω + θ is the true longitude, θ is the true anomaly, Both Di can be expressed as functions of a, h, k, p, q.

[0076] Then, the relative position vector of A and B is expressed as:

[0077] Δr(t)=r A (t)-r B (t)

[0078] where r A (t) and r B (t) respectively represent the orbital elements of the instantaneous vernal equinox and The absolute position vectors of the transformed targets A and B;

[0079] Finally, given a time interval [t 0 , t f ], the time interval is evenly divided into several sub-intervals according to the orbital period, and the time and relative position vector corresponding to the minimum value of the relative distance Δr(t)=||Δr(t)|| are calculated in each sub-interval, and the time and relative position vector of possible collision are preliminarily selected according to the minimum distance threshold. The specific method is as follows:

[0080] (1) Given a minimum distance threshold Δr tol , let s = 0;

[0081] (2) Based on Calculate the orbital periods of targets A and B to be T A and T B , take the minimum orbital period T min =min(T A , T B );

[0082] (3) The time interval [t 0 , t f ] is evenly divided into N subintervals [t 0 , t 1 ],[t 1 , t 2 ],...,[t N-1 , t N ], where N = ceil((t f-t 0 ) / T min ), ceil means round up, t N =t f ;

[0083] (4) For each subinterval [t k , t k+1 ](k=0,1,…,N-1), and iteratively calculate the minimum value of Δr in this subinterval according to the gradient descent method The corresponding time Right now:

[0084]

[0085] where ρ∈[0,1], represents the jth iteration The value of the iteration is The stopping condition for the iteration is where ε is the iteration tolerance.

[0086] After the iteration, calculate The corresponding minimum distance between targets The corresponding relative position vector is

[0087] (5) Obtain the time interval [t 0 , t f ] possible collision moments and corresponding relative position vector sequences: in

[0088] Step 3: Calculate the position error covariance at the time of possible collision based on the semi-analytical flat root deviation state transfer matrix

[0089] The first-order square root rate A of the orbital elements of the vernal equinox under the known J2 perturbation 1 =(A 11 , A 21 , …, A 61 ), construct the flat root deviation state transfer matrix equation without singular points based on the orbital elements of the vernal equinox:

[0090]

[0091] in is the state transfer matrix of the square root deviation of the orbit elements of the vernal equinox. The display expression is:

[0092]

[0093]

[0094]

[0095] By numerically integrating formula (2) with a large step length, the flat root deviation state transfer matrix at any time can be obtained.

[0096] Then for the sequence Each approach moment and the corresponding relative position vector Calculate the approach time t min The corresponding root mean square error covariance of targets A and B is:

[0097]

[0098] in and Respectively represent the square root error covariance matrices of targets A and B at the initial moment.

[0099] Then, calculate the approach time t min The corresponding position error covariance of targets A and B is:

[0100]

[0101] in, The explicit expression for is:

[0102]

[0103] Where f, g and w represent the unit vectors of the x, y and z axes of the vernal equinox orbital coordinate system, respectively, and X and Y represent the projections of the target position vector on the x and y axes of the vernal equinox orbital coordinate system, respectively.

[0104] Step 4: Determine collision risk based on Mahalanobis distance

[0105] Calculate t min The corresponding joint position error covariance of targets A and B is:

[0106]

[0107] The collision warning criterion based on Mahalanobis distance is:

[0108]

[0109] Wherein k is a non-negative real number, and different k values ​​represent different collision risk tolerances. Here, k=5.

[0110] Finally, the collision warning criterion is: when formula (3) is established, a collision warning is issued; otherwise, no collision warning is issued.

[0111] Through the above steps, space target collision warning based on semi-analytical orbit theory can be realized.

Claims

1. A space target collision warning method based on semi-analytical orbit theory, characterized in that: The method steps include: Step 1: Perform flat root prediction based on semi-analytical orbit theory; Step 2: Calculate the approach time and relative position vector based on the flat root and the J2 first-order short-period term, and preliminarily select the time when the collision may occur; Step 3: Calculate the position error covariance at the time when the collision may occur based on the semi-analytical flat root deviation state transfer matrix; Step 4: Determine the collision risk based on the Mahalanobis distance.

2. A space target collision warning method based on semi-analytical orbit theory as claimed in claim 1, characterized in that: In the step 1, the process of performing flat root prediction based on semi-analytical orbit theory includes: It is known that the number of orbital elements of the mean vernal equinox corresponding to the initial time t0 is a(t0) = (a, h, k, p, q, λ). Based on the semi-analytical orbital theory, a high-precision flat-root differential equation considering the first-order and second-order perturbations of the Earth's non-spherical gravitational field J2 is established: where a i represents the i-th component of a (i=1, 2, ..., 6), ∈=J2 represents the J2 band harmonic coefficient, A i1 and A i2 are the first-order and second-order root mean square rates of J2, and for i=1, 2, ..., 5, δ i6 =0, for i=6,δ i6 =1; For space targets A and B, the above equations are numerically integrated with large steps in the time interval [t0, t] to obtain the high-precision average root number a of space targets A and B at time t. A and a B .

3. A space target collision warning method based on semi-analytical orbit theory as claimed in claim 1, characterized in that: In the step 2, the process of calculating the approach time and relative position vector based on the flat root and the J2 first-order short-period term, and preliminarily screening the time when the collision may occur includes: Using the first-order short-period term of J2, calculate the instantaneous orbital elements of targets A and B at time t: in and denote the instantaneous orbital elements of A and B respectively, represents the first-order short-period term of J2 under the orbital root of the vernal equinox; The relative position vector of A and B is expressed as: Δr(t)=r A (t)-r B (t) where r A (t) and r B (t) represents the absolute position vectors of targets A and B converted from the instantaneous equinox orbit elements; Finally, given a time interval [t0, t f ], the time interval is divided into several sub-intervals according to the orbital period, and the time t corresponding to the minimum value of the relative distance Δr(t)=||Δr(t)|| is calculated in each sub-interval min and the relative position vector Δr min , and according to the minimum distance threshold Δr tol Preliminary screening of possible collision time sequences and corresponding relative position vectors, that is, when |Δr min ||<Δr tol , then it is assumed that spacecraft A and B are at t min A collision may occur at time t, otherwise it is assumed that spacecraft A and B are at t min No collision will occur.

4. A space target collision warning method based on semi-analytical orbit theory as claimed in claim 1, characterized in that: In step 3, the process of calculating the position error covariance at the time when a collision may occur based on the semi-analytical flat root deviation state transfer matrix includes: Calculate the position error covariance of A and B at the time of possible collision, that is, according to the first-order square root rate A1 of the orbital elements of the vernal equinox under J2 perturbation = (A 11 , A 21 , ..., A 61 ), construct the flat root deviation state transfer matrix equation without singular points based on the orbital elements of the vernal equinox: in is the square root deviation state transfer matrix of the orbital elements of the vernal equinox. The square root deviation state transfer matrix at any time can be obtained by numerically integrating the equation with a large step length. Calculate the time t at which a collision may occur min The corresponding root mean square error covariance of targets A and B is: in and Respectively represent the square root error covariance matrix of targets A and B at the initial moment; Then, the possible collision time t is calculated min The corresponding position error covariance of targets A and B is: At this point, the position error covariance of targets A and B at the moment when a collision may occur is obtained.

5. A space target collision warning method based on semi-analytical orbit theory as claimed in claim 1, characterized in that: In step 4, the process of determining the collision risk based on the Mahalanobis distance includes: Calculate the time t at which a collision may occur min The corresponding joint position error covariance of targets A and B is: The collision warning criterion based on Mahalanobis distance is: Where k is a non-negative real number, and different k values ​​represent different collision risk tolerances; Finally, the collision warning criterion is: when the above equation is true, a collision warning is issued; otherwise, no collision warning is issued.

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