A method for optical arc segment correlation of space targets based on control metrics

Abstract

A method for associating optical arc segments of space targets based on control metrics is proposed. The steps are as follows: 1. Obtain optical arc segment data and construct a constrained admissible domain; 2. Obtain an initial orbit guess; 3. Iteratively calculate the control metric value; 4. Calculate the control metric threshold; 5. Determine the type of association between the two arc segments. Through the above process, a method for associating optical arc segments of space targets based on control metrics is proposed. The initial values ​​of the two observation arc segments are obtained through the admissible domain. The actual control metric value and the relevant threshold are then iteratively calculated to determine the association between the two arc segments. Starting from the perspective of optimal control, this method adopts a high-precision dynamic model and can achieve accurate association and fine classification of optical observation arc segments of low-orbit space targets and maneuverable spacecraft.

Description

Technical Field

[0001] The present invention provides a method for associating optical arc segments of space targets based on control metrics, which involves using an allowed domain method and optimal control theory to judge the correlation between two optical observation arc segments and belongs to the field of space situational awareness. Background Art

[0002] In recent years, the outer space environment has become increasingly crowded. In particular, the development of mega-constellations like Starlink and OneWeb has led to an explosive growth in the number of spacecraft in Low Earth Orbit (LEO). Consequently, the problem of space target surveillance has become increasingly important and difficult. Space surveillance is typically achieved using observations from radar or optical sensors. Optical sensors can provide angle measurements but not distance information. Therefore, orbit determination using a single short optical segment is illogical, a problem also known as the angle-only initial orbit determination (IOD) problem. However, if the segment and corresponding time information are combined, the angle and angular rate can be extracted to form a four-dimensional normalized vector. Combined with the physically constrained distance and range rate of change, candidate orbits can be calculated in a two-dimensional allowed domain, known as the allowed domain method. However, multiple segments are still required to obtain accurate orbit estimates.

[0003] To use optical observation segments to determine orbits or update the orbital information of space targets, it is first necessary to determine which target the segment belongs to. This step is called segment association and can be divided into segment-to-orbit association and segment-to-segment association. Segment-to-orbit association involves associating the observed segment with the orbit in the space target catalog to update the orbital data in the catalog. Segment-to-segment association is used to determine whether two segments belong to the same space target. This is more common in space situational awareness scenarios because most observed optical segments belong to non-cooperative targets, are not in the catalog, and have no historical orbital information. Therefore, compared to segment-to-orbit association, segment-to-segment association is significantly more difficult.

[0004] For low-orbit space targets, arc segment correlation is even more difficult because various perturbations significantly impact orbit predictions. Using low-precision models can lead to rapid divergence of orbit prediction errors, resulting in the separation of two arc segments that originally belonged to the same target. Furthermore, in traditional observation scenarios, the correlation between arc segments only has two possible outcomes: whether or not they belong to the same target, which is relatively crude. However, with the increasing competition between maneuvering spacecraft and other countries in space situational awareness, it has become increasingly important to be able to determine whether non-cooperative targets are maneuvering through optical observation of arc segments, thereby identifying their intentions. Therefore, the arc segment correlation results must be able to identify and distinguish three possible scenarios as much as possible: one, that the two arc segments belong to the same target and the target has not maneuvered; two, that the two arc segments belong to the same target before and after the maneuver; and three, that the two arc segments are unrelated and do not belong to the same target.

[0005] Representative methods for arc association include covariance-based association and control metric-based association methods. The overall idea of ​​using control metrics to associate space maneuvering targets is to determine the maneuver strategy required for a space target to change from its historical orbit to its current orbit. If the control cost can be represented by the systematic error, then the observed arc belongs to the target. If the control cost exceeds the influence of the systematic error and is lower than the preset threshold of the spacecraft's maneuverability, then it is highly suspected that the target has maneuvered. Ideally, two arcs for the same target can be connected to the true orbit, which is the target orbit without control. However, various systematic errors such as uncertainty in the dynamic model and observation errors will lead to the estimation of additional control costs. Therefore, quantifying these effects is the key and difficulty in establishing arc association criteria and determining which association type the arc belongs to.

[0006] In summary, focusing on quantifying the impact of system errors on spacecraft control costs and analytically deriving the criteria for the correlation between arc segments, the present invention proposes an arc-to-arc correlation method based on control metrics, which is suitable for low-orbit spacecraft and can incorporate the maneuverability of the spacecraft into relevant criteria, and can realize the correlation of observation arc segments belonging to the same maneuverable spacecraft. Summary of the Invention

[0007] (1) Purpose of the invention

[0008] This paper exploits the characteristic of spacecraft trajectory changes often requiring optimal energy control. Based on optimal control theory, it proposes a control-metric-based method for correlating optical arc segments of space targets. This method categorizes two arc segments into three categories: those belonging to the same space target, those belonging to the same target before and after maneuvering, and those belonging to different targets. This method is applicable to space targets at various orbital altitudes and with maneuvering capabilities. This method fully quantifies system errors and model errors, rigorously establishing correlation criteria between arc segments, and achieves high-accuracy correlation of low-orbit space targets. It also enables inter-segment correlation and maneuver detection for maneuverable spacecraft.

[0009] (2) Technical solution

[0010] The present invention relates to a method for associating optical arc segments of space targets based on control metrics. The method requires two optical observation arc segments observed by a low-orbit observation satellite or a ground-based optical observation station as input data to determine the correlation between the two segments. First, the corresponding right ascension and declination angles and angle change rates are extracted from the two observation arc segments. Two corresponding range-range change rate allowable domains are then generated from these angles. Two sets of distances and range change rates that bring the two orbits closest together are obtained as iterative initial values. The initial value of the co-state variable of the first segment is calculated based on optimal control theory. Next, the actual control metric value, the control metric threshold containing systematic error, and the control metric threshold containing systematic error and maneuverability are iteratively calculated. The actual control metric value is compared with the two thresholds to determine the correlation between the two segments.

[0011] The space target optical arc segment association method based on control measurement of the present invention is described in the attached Figure 1 , and its implementation steps are as follows:

[0012] Step 1: Obtain optical arc segment data and construct the constraint allowable domain

[0013] For a segment of optical observation arc data, extract the normalized vector of angle and angle change rate:

[0014]

[0015] Where α and β represent right ascension and declination; Represents the rate of change of the angle of right ascension and declination.

[0016] Refer to the attached Figure 2 The observation model of , the sight vector is:

[0017]

[0018] Let the coordinate vector of the observation satellite be r o , the velocity vector is v o , so the position vector and velocity vector of the target satellite are:

[0019]

[0020] where ρ is the relative distance, is the relative distance change rate, and:

[0021]

[0022] The orbital state can then be described as For two arc segments observed at time t1 and time t2, a set of At this point, the states of the first and last orbits are known. Space targets that are in low orbit and can be observed must meet the constraints of observation distance, semi-major axis, eccentricity, perigee height, etc., so it is possible to establish and The two-dimensional constraint allowed domain of , each value point in the allowed domain represents an orbit.

[0023] Step 2: Get an initial orbital guess

[0024] Two arcs can obtain two admissible domains, and the pair of orbits between the two admissible domains is used as the initial guess when the difference in their orbital elements is minimized. That is, the nth point in the first admissible domain and the mth point in the second admissible domain are selected to minimize the following objective function:

[0025]

[0026] in:

[0027]

[0028] where μ is the Earth's gravitational constant; a k (n),e k (n),i k (n),Ω k (n) and ω k (n) represents the semi-major axis, eccentricity, orbital inclination, right ascension of ascending node and argument of perigee calculated from the n-th value point in the k-th allowable domain; J2 = 0.0010826267 is the J2 perturbation constant, R E is the radius of the Earth.

[0029] After traversing the two distance-distance change rate allowable domains, the closest orbital state can be obtained as the initial guess of the distance and distance change rate, that is, the initial guess of the orbital position and velocity. At this time, the minimum energy control strategy for the two initial orbits can be calculated accordingly, which is described as:

[0030]

[0031] The indirect method is used to solve the optimal control strategy through the initial adjoint state. r (t), p v (t) Satisfy:

[0032]

[0033] in

[0034]

[0035] Integrating formula (10) yields:

[0036]

[0037] Where Φp(t,t1) is the adjoint state transfer matrix, which is given by the differential equation Φ p The unit matrix is ​​used as the initial value integration. Then the optimal control quantity given by Pontryagin's maximum principle is:

[0038] u * (t)=-p v (t) (13)

[0039] Combining equations (9), (12), and (13) and linearizing them, we get:

[0040]

[0041] in:

[0042]

[0043] Then integrate formula (14):

[0044]

[0045] where Φ x (t,t1) is the orbital state transfer matrix, which is given by the differential equation The integration is obtained by taking the unit matrix as the initial value; S p (t,t1) is the sensitivity matrix of the orbital state to the initial adjoint state, given by It is obtained by integrating with the zero matrix as the initial value.

[0046] Solving the above equations using the shooting method or the homology method can yield the initial adjoint state, and then obtaining the optimal control quantity and the minimum energy control metric from Equation (12). Note that the control quantity is calculated given the initial orbit guess, which is equivalent to obtaining the initial guess for the optimal control quantity.

[0047] Step 3: Iteratively calculate the control metric value

[0048] Equation (9) gives the calculation method of the optimal control strategy when the initial orbit state is determined. The initial orbit and the corresponding control quantity should be further optimized. The actual problem is the minimum energy control problem that satisfies the boundary value constraints:

[0049]

[0050] And the optimal control strategy described by the initial adjoint state and equations (12) and (13):

[0051]

[0052] Distance-distance change rate Applying a small disturbance, from equations (2) to (5), we can obtain:

[0053]

[0054] Combined with formula (16), we have:

[0055]

[0056] Combining the above formula with formula (17) and (18), we can get the perturbation of M:

[0057]

[0058] in:

[0059]

[0060] T2(t)=[0 3×3 -I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 [-Φ x (t2,t1)Λ(t1)Λ(t2)] (24)

[0061] It can be seen that formula (21) is quadratic. When δM is minimized, we have:

[0062]

[0063] Therefore, the initial distance, the initial distance change rate and the corresponding initial adjoint state should be iteratively updated as follows:

[0064]

[0065] It should be noted that due to the nonlinearity of the system, the updated initial adjoint state vector is not the exact minimum energy solution after the update. This can be further refined using a shooting method. The above iterative process is repeated until the obtained perturbation values ​​of the distance and the rate of change of the distance are less than a certain threshold. The iteration has converged and the solution has been completed.

[0066] Step 4: Calculate the control metric threshold

[0067] First, calculate the upper bound of the control metric value of the non-maneuvering target. The dynamic equation of the true trajectory without control is:

[0068]

[0069] in and Represent the true inertial position, velocity and acceleration respectively.

[0070] Compared with the orbital state equation with control in Equation (17), a first-order Taylor expansion is adopted, and u(t) is expressed by the initial adjoint state according to the optimal control quantity Equations (10) to (13). Then, an integration process similar to Equation (14) is performed from time t1 to t2 to obtain:

[0071]

[0072] in as well as The position and velocity deviations can be expressed based on the observation error of the normalized vector, the estimated error of the distance and the distance change rate, and the equations (3) and (4) can be linearized to small quantities:

[0073]

[0074] in:

[0075]

[0076] Substituting formula (30) into formula (29) and combining formula (17) and (18), we have:

[0077]

[0078] in:

[0079]

[0080] T1(t)=[0 3×3 -I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 [-Φ x (t2,t1)Ψ(t1)Ψ(t2)] (37)

[0081] T3(t)=[0 3×3 I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 (38)

[0082] The definition of T2(t) is given by Equation (24). It can be found that Z represents the orbit propagation error caused by the dynamic model error, and Equation (34) is also quadratic. Therefore, when the range and range change rate estimation errors are minimized, the control metric value is minimized, which should be:

[0083]

[0084] The above formula is the theoretical upper limit of the control metric value when there is no control, taking into account the dynamic model error and observation error. When the distribution of the model error is given, a set of distributions of the theoretical upper limit can be calculated by the Monte Carlo method, and the upper limit value with a higher associated probability is selected as the control metric threshold d for actual use. c .

[0085] Then calculate the control metric threshold considering the maneuverability of the spacecraft. In the maneuvering case, the minimum energy control metric is

[0086]

[0087] in The actual control acceleration of the maneuvering spacecraft, Δu is the additional control acceleration caused by the system error. Scaling and by the Cauchy-Schwarz inequality:

[0088]

[0089] Therefore, the control metric threshold of the correlation between the arc segments of the maneuverable spacecraft is taken as:

[0090]

[0091] where d s Represents a measure of the spacecraft's maneuverability, which varies depending on the spacecraft's situation.

[0092] Step 5: Determine the association type between the two arcs

[0093] Compare the actual control metric value M with the two thresholds d c d sc , the association results are divided into the following three cases:

[0094] (1) When M<d c When , the estimated control metric value is less than the control metric threshold containing systematic error, so it is considered that the two arcs are likely to belong to the same space target, and the space target has not maneuvered;

[0095] (2) When d c <M<d sc When , the estimated control metric value exceeds the control metric threshold containing systematic errors, but is less than the control metric threshold considering maneuverability. Therefore, it is believed that the two arcs are likely to belong to the same spatial target before and after the maneuver.

[0096] (3) When M>d sc When , the estimated control metric value has exceeded the preset maneuverability, so it is considered that the two arcs are unrelated and most likely belong to different space targets.

[0097] In summary, the implementation steps of the present invention are as follows: Figure 1 As shown in the figure. The arc segment association method proposed through the above process is derived from the perspective of optimal control theory and is fundamentally different from traditional methods. The idea is to find an orbit connecting two arc segments so that its energy control metric is minimized. By combining the angle, angle change rate, distance, and distance change rate to describe the orbital state during the two observation arc segments, the two-point boundary value problem is solved by optimizing the distance and distance change rate. The impact of the system error is theoretically quantified, and the criteria for the association between arc segments are given. This method is also applicable to situations where the spacecraft maneuvers between two arc segments, and can achieve high-accuracy arc segment association of low-orbit space targets.

[0098] (3) Advantages

[0099] The advantages of the space target optical arc segment association method based on control measurement provided by the present invention are:

[0100] ① The optical arc segment correlation method proposed in this invention is applicable to low-orbit space targets and maneuverable spacecraft, has a wide range of applications, adopts a high-precision dynamic model, and has a high accuracy rate of correlation results;

[0101] ② The initial value guessing method proposed in this invention uses the grid method of the allowed domain to obtain the two closest points of the orbit in the two constraint allowed domains. The speed is extremely fast, and the initial value guess is close to the true value, which facilitates rapid convergence.

[0102] ③ The iterative calculation method of the control metric value proposed in this invention obtains the iterative formula by applying disturbance.

[0103] Only three to four iterations are needed to converge, and the convergence effect is good

[0104] ④ The control metric threshold calculation method proposed in this invention theoretically quantifies the impact of system errors and provides the correlation criteria between arc segments. It also incorporates the maneuverability of the spacecraft into the calculation method, and can classify the correlation types between the two arc segments in a relatively fine manner. BRIEF DESCRIPTION OF THE DRAWINGS

[0105] Figure 1 It is a flow chart of the implementation steps of the present invention.

[0106] Figure 2 This is the observation model in this method.

[0107] Figure 3 This is a diagram of the constraint allowed domain. Each point in the diagram represents a track.

[0108] Figure 4 This is the first result of the association between two arcs. This situation indicates that the two arcs belong to the same space target and the target does not maneuver.

[0109] Figure 5 This is the second result of the association between the two arcs, which means that the two arcs belong to the same spatial target before and after the maneuver.

[0110] Figure 6 This is the third result of the association between two arcs, which means that the two arcs do not belong to the same target. DETAILED DESCRIPTION

[0111] The specific implementation process of the present invention will be further described in detail below in conjunction with the technical solution.

[0112] The present invention relates to a method for associating optical arc segments of space targets based on control metrics. The method requires two optical observation arc segments observed by a low-orbit observation satellite or a ground-based optical observation station as input data to determine the correlation between the two segments. First, the corresponding right ascension and declination angles and angle change rates are extracted from the two observation arc segments. Two corresponding range-range change rate allowable domains are then generated from these angles. Two sets of distances and range change rates that bring the two orbits closest together are obtained as iterative initial values. The initial value of the co-state variable of the first segment is calculated based on optimal control theory. Next, the actual control metric value, the control metric threshold containing systematic error, and the control metric threshold containing systematic error and maneuverability are iteratively calculated. The actual control metric value is compared with the two thresholds to determine the correlation between the two segments.

[0113] The space target optical arc segment association method based on control measurement of the present invention is described in the attached Figure 1 , and its implementation steps are as follows:

[0114] Step 1: Obtain optical arc segment data and construct the constraint allowable domain

[0115] For a segment of optical observation arc data, extract the normalized vector of angle and angle change rate:

[0116]

[0117] Where α and β represent right ascension and declination; Represents the rate of change of the angle of right ascension and declination.

[0118] Refer to the attached Figure 2 The observation model of , the sight vector is:

[0119]

[0120] Let the coordinate vector of the observation satellite be r o , the velocity vector is v o , so the position vector and velocity vector of the target satellite are:

[0121]

[0122] where ρ is the relative distance, is the relative distance change rate, and:

[0123]

[0124] The orbital state can then be described as For two arc segments observed at time t1 and time t2, a set of At this point, the states of the first and last orbits are known. Space targets that are in low orbit and can be observed must meet the constraints of observation distance, semi-major axis, eccentricity, perigee height, etc., so it is possible to establish and The two-dimensional constraint allowed domain of , each value point in the allowed domain represents a track. Figure 3 To constrain the allowable domain, the sampling intervals for distance and distance change rate are 20 km and 20 m / s respectively. The constraints are: minimum observation distance 50 km, maximum observation distance 15,000 km; minimum semi-major axis 6,700 km, maximum semi-major axis 8,400 km; maximum eccentricity 0.2; minimum perigee height 300 km.

[0125] Step 2: Get an initial orbital guess

[0126] Two arcs can obtain two admissible domains, and the pair of orbits between the two admissible domains is used as the initial guess when the difference in their orbital elements is minimized. That is, the nth point in the first admissible domain and the mth point in the second admissible domain are selected to minimize the following objective function:

[0127]

[0128] in:

[0129]

[0130] where μ is the Earth's gravitational constant; a k (n),e k (n),i k (n),Ω k (n) and ω k (n) represents the semi-major axis, eccentricity, orbital inclination, right ascension of ascending node and argument of perigee calculated from the n-th value point in the k-th allowable domain; J2 = 0.0010826267 is the J2 perturbation constant, R E is the radius of the Earth.

[0131] After traversing the two distance-distance change rate allowable domains, the closest orbital state can be obtained as the initial guess of the distance and distance change rate, that is, the initial guess of the orbital position and velocity. At this time, the minimum energy control strategy for the two initial orbits can be calculated accordingly, which is described as:

[0132]

[0133] The indirect method is used to solve the optimal control strategy through the initial adjoint state. r (t), p v (t) Satisfy:

[0134]

[0135] in

[0136]

[0137] Integrating equation (53) yields:

[0138]

[0139] Where Φp(t,t1) is the adjoint state transfer matrix, which is given by the differential equation Φ p The unit matrix is ​​used as the initial value integration. Then the optimal control quantity given by Pontryagin's maximum principle is:

[0140] u * (t)=-p v (t) (56)

[0141] Combining equations (52), (55), and (56) and linearizing them, we obtain:

[0142]

[0143] in:

[0144]

[0145] Then integrate formula (57):

[0146]

[0147] where Φ x (t,t1) is the orbital state transfer matrix, which is given by the differential equation The integration is obtained by taking the unit matrix as the initial value; S p (t,t1) is the sensitivity matrix of the orbital state to the initial adjoint state, given by It is obtained by integrating with the zero matrix as the initial value.

[0148] Solving the above equations using the shooting method or the homology method can yield the initial adjoint state, and then obtaining the optimal control quantity and the minimum energy control metric from Equation (12). Note that the control quantity is calculated given the initial orbit guess, which is equivalent to obtaining the initial guess for the optimal control quantity.

[0149] Step 3: Iteratively calculate the control metric value

[0150] Equation (52) gives the calculation method of the optimal control strategy when the initial orbit state is determined. The initial orbit and the corresponding control quantity should be further optimized. The actual problem is the minimum energy control problem that satisfies the boundary value constraints:

[0151]

[0152] And the optimal control strategy described by the initial adjoint state and equations (55) and (56):

[0153]

[0154] Distance-distance change rate Applying a small disturbance, from equations (45) to (48), we can obtain:

[0155]

[0156] Combined with formula (59), we have:

[0157]

[0158] Combining the above formula with formula (60) and (61), we can get the perturbation of M:

[0159]

[0160] in:

[0161]

[0162] T2(t)=[0 3×3 -I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 [-Φ x (t2,t1)Λ(t1)Λ(t2)] (67)

[0163] It can be seen that formula (64) is quadratic. When δM is minimized, we have:

[0164]

[0165] Therefore, the initial distance, the initial distance change rate and the corresponding initial adjoint state should be iteratively updated as follows:

[0166]

[0167] It should be noted that due to the nonlinearity of the system, the updated initial adjoint state vector is not the exact minimum energy solution after the update. This can be further refined using a shooting method. The above iterative process is repeated until the obtained perturbation values ​​of the distance and the rate of change of the distance are less than a certain threshold. The iteration has converged and the solution has been completed.

[0168] Step 4: Calculate the control metric threshold

[0169] First, calculate the upper bound of the control metric value of the non-maneuvering target. The dynamic equation of the true trajectory without control is:

[0170]

[0171] in and Represent the true inertial position, velocity and acceleration respectively.

[0172] Compared with the orbital state equation with control in Equation (60), a first-order Taylor expansion is adopted, and u(t) is expressed by the initial adjoint state according to the optimal control quantity Equations (53) to (56). Then, an integration process similar to Equation (57) is performed from time t1 to t2 to obtain:

[0173]

[0174] in as well as The position and velocity deviations can be expressed based on the observation error of the normalized vector, the estimated error of the distance and the distance change rate, and the equations (46) and (47) can be linearized to small quantities:

[0175]

[0176] in:

[0177]

[0178] Substituting equation (73) into equation (72) and combining equations (60) and (61), we have:

[0179]

[0180] in:

[0181]

[0182] T1(t)=[0 3×3 -I3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 [-Φ x (t2,t1)Ψ(t1)Ψ(t2)] (80)

[0183] T3(t)=[0 3×3 I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 (81)

[0184] The definition of T2(t) is given by Equation (67). It can be found that Z represents the orbit propagation error caused by the dynamic model error, and Equation (77) is also quadratic. Therefore, when the range and range change rate estimation errors are minimized, the control metric value is minimized, which should be:

[0185]

[0186] The above formula is the theoretical upper limit of the control metric value when there is no control, taking into account the dynamic model error and observation error. When the distribution of the model error is given, a set of distributions of the theoretical upper limit can be calculated by the Monte Carlo method, and the upper limit value with a higher associated probability is selected as the control metric threshold d for actual use. c .

[0187] Then calculate the control metric threshold considering the maneuverability of the spacecraft. In the maneuvering case, the minimum energy control metric is

[0188]

[0189] in The actual control acceleration of the maneuvering spacecraft, Δu is the additional control acceleration caused by the system error. Scaling and by the Cauchy-Schwarz inequality:

[0190]

[0191] Therefore, the control metric threshold of the correlation between the arc segments of the maneuverable spacecraft is taken as:

[0192]

[0193] where d s Represents a measure of the spacecraft's maneuverability, which varies depending on the spacecraft's situation.

[0194] Step 5: Determine the association type between the two arcs

[0195] Compare the actual control metric value M with the two thresholds d cd sc , the association results are divided into the following three cases:

[0196] (1) When M<d c When the estimated control metric value is less than the control metric threshold value containing system error, it is considered that the two arcs are likely to belong to the same space target, and the space target has not maneuvered. Figure 4 ;

[0197] (2) When d c <M<d sc When the estimated control metric value exceeds the control metric threshold including the system error, but is less than the control metric threshold considering the maneuverability, it is believed that the two arcs are likely to belong to the same space target before and after the maneuver. Figure 5 ;

[0198] (3) When M>d sc When the estimated control metric value exceeds the preset maneuverability, it is considered that the two arcs are unrelated and most likely belong to different space targets. Figure 6 .

[0199] In summary, the implementation steps of the present invention are as follows: Figure 1 As shown in the figure. The arc segment association method proposed through the above process is derived from the perspective of optimal control theory and is fundamentally different from traditional methods. The idea is to find an orbit connecting two arc segments so that its energy control metric is minimized. By combining the angle, angle change rate, distance, and distance change rate to describe the orbital state during the two observation arc segments, the two-point boundary value problem is solved by optimizing the distance and distance change rate. The impact of the system error is theoretically quantified, and the criteria for the association between arc segments are given. This method is also applicable to situations where the spacecraft maneuvers between two arc segments, and can achieve high-accuracy arc segment association of low-orbit space targets.

Claims

1. A method for associating optical arc segments of space targets based on control metrics, characterized by: The steps are as follows: Step 1: Obtain optical arc segment data and construct the constraint allowable domain For a segment of optical observation arc data, extract the normalized vector of angle and angle change rate: Where α and β represent right ascension and declination; represents the rate of change of the angle of right ascension and declination; The sight vector is: Let the coordinate vector of the observation satellite be r o , the velocity vector is v o , so the position vector and velocity vector of the target satellite are: where ρ is the relative distance, is the relative distance change rate, and: The orbital state can then be described as function; for two arc segments observed at time t1 and t2 respectively, a set of At this point, the states of the first and last orbits are known; the space targets that are running in low orbit and can be observed must meet the constraints of observation distance, semi-major axis, eccentricity, perigee height, etc., so it is possible to establish and The two-dimensional constraint allowed domain, each value point in the allowed domain represents an orbit; Step 2: Get an initial orbital guess Two arc segments can obtain two admissible domains, and a pair of orbits between the two admissible domains is used as the initial guess when the difference between their orbital elements is the smallest; After the two distance-distance change rate admissible domains are traversed, the closest orbital state can be obtained as the initial guess value of the distance and distance change rate, that is, the initial guess value of the orbital position and velocity is obtained; at this time, the minimum energy control strategy of the two initial orbits can be calculated accordingly, which is described as: Using the indirect method, through the initial adjoint state p r (t), p v (t) is used to solve the optimal control strategy using the shooting method; the optimal control quantity given by the Pontryagin maximum principle is: u * (t)=-p v (t) (7) Note that the control amount at this time is calculated given the initial trajectory guess, which is equivalent to obtaining the initial guess of the optimal control amount; Step 3: Iteratively calculate the control metric value Equation (6) gives the calculation method of the optimal control strategy when the initial orbit state is determined. The initial orbit and the corresponding control quantity should be further optimized. The actual problem is the minimum energy control problem that satisfies the boundary value constraints: And the optimal control strategy described by the initial adjoint state: Distance-distance change rate Apply a small perturbation and obtain the iterative update equation. Iterate until the obtained distance and distance change rate perturbation values ​​are less than a certain threshold, then the iteration has converged and the optimal orbit state and control quantity, as well as the control metric value M, have been obtained. Step 4: Calculate the control metric threshold Quantify the system error, calculate the upper bound of the control metric value without maneuvering targets, and obtain the control metric threshold d containing the system error c The control metric threshold d considering maneuverability sc ; Step 5: Determine the association type between the two arcs Compare the actual control metric value M with the two thresholds d c d sc , the association results are divided into the following three cases: (1) When M<d c When , the estimated control metric value is less than the control metric threshold containing systematic error, so it is considered that the two arcs are likely to belong to the same space target, and the space target has not maneuvered; (2) When d c <M<d sc When , the estimated control metric value exceeds the control metric threshold containing systematic errors, but is less than the control metric threshold considering maneuverability. Therefore, it is believed that the two arcs are likely to belong to the same spatial target before and after the maneuver. (3) When M>d sc When , the estimated control metric value has exceeded the preset maneuverability, so it is considered that the two arcs are unrelated and most likely belong to different space targets.

2. The method for associating optical arc segments of space targets based on control metrics according to claim 1, characterized in that: The specific implementation method of step 2 to obtain the initial orbit guess is: Two arcs can obtain two admissible domains, and the pair of orbits between the two admissible domains is used as the initial guess when the difference in their orbital elements is minimized; that is, the nth point in the first admissible domain and the mth point in the second admissible domain are selected to minimize the following objective function: in: where μ is the Earth's gravitational constant; a k (n),e k (n),i k (n),Ω k (n) and ω k (n) represents the semi-major axis, eccentricity, orbital inclination, right ascension of ascending node and argument of perigee calculated from the n-th value point in the k-th allowable domain; J2 = 0.0010826267 is the J2 perturbation constant, R E is the radius of the Earth; After the two distance-distance change rate admissible domains are traversed, the closest orbital state can be obtained as the initial guess value of the distance and distance change rate, that is, the initial guess value of the orbital position and velocity is obtained; at this time, the minimum energy control strategy of the two initial orbits can be calculated accordingly, which is described as: Using the indirect method, through the initial adjoint state p r (t), p v (t) to solve the optimal control strategy; the accompanying state satisfies: in Integrating formula (14) yields: Where Φp(t,t1) is the adjoint state transfer matrix, which is given by the differential equation Φ p The unit matrix is ​​used as the initial value integration; therefore, the optimal control quantity given by the Pontryagin maximum principle is: u * (t)=-p v (t) (17) Combining equations (13), (16), and (17) and linearizing them, we get: in: Then integrate formula (18): where Φ x (t,t1) is the orbital state transfer matrix, which is given by the differential equation The integration is obtained by taking the unit matrix as the initial value; S p (t,t1) is the sensitivity matrix of the orbital state to the initial adjoint state, given by The integration is obtained by taking the zero matrix as the initial value; The initial adjoint state can be solved by using the shooting method or the homology method to solve the above equations, and then the optimal control quantity and the minimum energy control quantity value can be obtained by equation (16).

3. The method for associating optical arc segments of space targets based on control metrics according to claim 1, characterized in that: The specific implementation method of step 3 to obtain the actual control measurement value is as follows: The actual problem is the minimum energy control problem that satisfies the boundary value constraints: And the optimal control strategy described by the initial adjoint state and equations (16) and (17): Distance-distance change rate Applying a small disturbance, from equations (2) to (5), we can obtain: Combined with formula (20), we have: Combining the above formula with formula (21) and (22), we can get the perturbation of M: in: T2(t)=[0 3×3 -I 3×3 ]F p (t,t1)(S p (t2,t1)) -1 [-F x (t2,t1)Λ(t1) Λ(t2)] (28) It can be seen that formula (25) is quadratic. When δM is minimized, we have: Therefore, the initial distance, the initial distance change rate and the corresponding initial adjoint state should be iteratively updated as follows: This is the iterative formula; due to the nonlinearity of the system, the updated initial adjoint state vector is not the updated exact minimum energy solution, which can be further precisely solved using the shooting method; repeat the above iterative process until the obtained distance and distance change rate disturbance values ​​are less than a certain threshold, then the iteration has converged and the solution has been completed.

4. The method for associating optical arc segments of space targets based on control metrics according to claim 1, characterized in that: The specific implementation method of step 4 to obtain the control metric threshold is as follows: First, calculate the upper bound of the control metric value of the non-maneuvering target; the dynamic equation of the true trajectory without control is: in and Represent the true inertial position, velocity and acceleration respectively; Compared with the orbital state equation with control in Equation (21), a first-order Taylor expansion is adopted, and u(t) is represented by the initial adjoint state according to the optimal control quantity. Then, an integration process similar to Equation (18) is performed from time t1 to t2 to obtain: in as well as The position and velocity deviations can be expressed based on the observation error of the normalized vector, the estimated error of the distance and the distance change rate, and the equations (3) and (4) can be linearized to small quantities: in: Substituting equation (34) into equation (33) and combining equations (21) and (22), we have: in: T1(t)=[0 3×3 -I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 [-Φ x (t2,t1)Ψ(t1) Ψ(t2)] (41) T3(t)=[0 3×3 I 3×3 ]Φ p (t,t1)(S p (t2,t1)) -1 (42) The above formula is also quadratic, so when the distance and distance change rate estimation errors are minimized, the control metric value is minimized, and we should have: The above formula is the theoretical upper limit of the control metric value when there is no control, taking into account the dynamic model error and the observation error; when the distribution of the model error is given, a set of distributions of the theoretical upper limit can be calculated by the Monte Carlo method, and the upper limit value with a higher associated probability can be selected as the control metric threshold d for actual use c ; Then calculate the control metric threshold considering the maneuverability of the spacecraft; in the case of maneuvering, the minimum energy control metric is Where u is the actual control acceleration of the maneuverable spacecraft, and Δu is the additional control acceleration caused by the system error; scaled and expressed by the Cauchy-Schwarz inequality: Therefore, the control metric threshold of the correlation between the arc segments of the maneuverable spacecraft is taken as: where d s Represents a measure of the spacecraft's maneuverability, which varies depending on the spacecraft's situation.

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