Method and system for determining initial relative orbit of space non-cooperative target

Abstract

The embodiment of the invention provides a method and system for determining an initial relative orbit of a space non-cooperative target, and the method comprises the steps: shooting a sequence image of the space non-cooperative target through an optical camera carried by a service spacecraft for the space non-cooperative target; the sequence image is shot at the initial moment when the self-service spacecraft finds the space non-cooperative target; determining a kinetic equation and a measurement equation of a relative motion state of the spatial non-cooperative target under a spherical coordinate system based on the sequence image; and constructing a singular value of the matrix to solve the kinetic equation and the measurement equation of the relative motion state to obtain an initial relative orbit of the spatial non-cooperative target. Through singular value decomposition, an efficient approximate solution of an initial relative orbit determination problem constructed by shooting sequence images of the spatial non-cooperative target through an optical camera carried by a service spacecraft can be obtained, efficient calculation is achieved through solution, and high-precision initial relative orbit determination is achieved.

Description

Technical Field

[0001] The present invention relates to the field of space non-cooperative target positioning, and in particular to a method and system for determining the initial relative orbit of a space non-cooperative target. Background Art

[0002] Non-cooperative objects are space objects that are unable to proactively provide relative status information to spacecraft or tracking equipment. These objects lack recognition and response capabilities, and their sizes are often unknown. They include space debris, defunct satellites, and asteroids. With the increase in space activities, space debris removal, in-orbit servicing of defunct satellites, and asteroid exploration have become key issues in current space exploration. To ensure the smooth implementation of relevant space missions, spacecraft must accurately obtain the motion parameters of non-cooperative objects, including relative position and velocity.

[0003] Traditional ground-based orbit determination methods perform well in determining the relative states of non-cooperative targets. However, they rely heavily on ground-based observation stations, which are limited by observation resources, target visibility, and observation accuracy. Furthermore, faced with the need to monitor a large number of space targets, these methods have significant limitations in terms of timely tracking and efficient control.

[0004] Therefore, developing autonomous relative orbit determination methods for spacecraft (also called space servers) has become an important approach to solving this problem, especially autonomous methods based solely on angle measurements using optical cameras. Optical cameras, with their advantages of light weight, low power consumption, low cost, and long range, have become important measurement equipment in space target detection and tracking.

[0005] The relative navigation problem for non-cooperative spatial targets using monocular image sequences involves estimating the relative position and velocity of a non-cooperative target at a specific moment in time through navigation filtering using a series of images captured by an optical camera. However, the filtering process requires the predetermination of the target's initial relative orbital state. For non-cooperative spatial targets, the accuracy of the initial relative orbital parameters is extremely limited or even unknown, and the accuracy of the navigation filtering estimation is heavily dependent on the accuracy of the initial relative orbit determination. Therefore, obtaining high-precision initial relative orbit information of the target is extremely important for improving the accuracy and convergence rate of state estimation.

[0006] Scholars have proposed a variety of solutions to the problem of relative orbit determination IROD. Based on the linear Hill-Clohessy-Wiltshire (HCW) dynamic model, Woffinden pointed out that in the absence of orbital maneuvers or disturbance forces and the camera is located at the center of mass of the observer, it is impossible to determine a unique relative state by relying solely on angle measurements. In addition, camera offsets are introduced to increase geometric information, thereby improving the observability of the target state, and a simple IROD algorithm is proposed. However, this method is only applicable to close-range targets, and its scope of application is proportional to the camera offset. In order to expand the applicability, there is also a method based on a high-order dynamic model, using a second-order quadratic Volterra (QV) solution, and combining the Macaulay result expression to solve the polynomial equations, which improves the accuracy of relative state estimation. However, due to the high complexity of the solution, its practical application is still limited.

[0007] In order to expand the applicable scenarios, the researchers further extended the IROD method from near-circular orbits to elliptical orbits. Tschauner-Hempel et al. proposed relative motion equations applicable to elliptical reference orbits, which significantly expanded the scope of application of the HCW model. Based on elliptical orbits, the state transfer matrix was further derived, and a method to simplify the model through orbital integration was proposed. Based on the relative orbital element (ROE), the position and velocity differences between the two spacecraft were approximated as linear transformations of the orbital elements, and a simpler and easier-to-implement model was constructed. In addition, the introduction of time-explicit models and dimensionless variables makes the description of relative motion on eccentric orbits more accurate.

[0008] In complex dynamic environments, the study of perturbations, especially Earth oblateness (J2) perturbations, is of great significance. Schweighart et al. derived linearized equations under J2 perturbation conditions. The J2-invariant relative orbit model proposed by Schaub et al. further supports the long-term stability of formation flying. Furthermore, curvilinear modeling methods in cylindrical and spherical coordinates demonstrate significant advantages under conditions of long-distance separation and complex perturbations, providing new research directions for the IROD problem.

[0009] However, there are still deficiencies in efficient algorithms and theoretical models for non-circular orbits and complex disturbance environments. Summary of the Invention

[0010] The embodiments of the present invention provide a method and system for determining the initial relative orbit of a non-cooperative space target, which can solve the above-mentioned technical problems in the prior art.

[0011] To achieve the above objectives, in a first aspect, an embodiment of the present invention provides a method for determining an initial relative orbit of a non-cooperative space target, comprising:

[0012] For a non-cooperative space target, a sequence of images of the non-cooperative space target is captured by an optical camera carried by the servicing spacecraft; the sequence of images is captured starting from the initial moment when the servicing spacecraft discovers the non-cooperative space target;

[0013] Determining a dynamic equation and a measurement equation of a relative motion state of the space non-cooperative target in a spherical coordinate system based on the sequence of images;

[0014] The singular values ​​of the constructed matrix are used to solve the dynamic equations and measurement equations of the relative motion state to obtain the initial relative orbit of the space non-cooperative target.

[0015] In a second aspect, an embodiment of the present invention provides a system for determining an initial relative orbit of a non-cooperative space target, comprising:

[0016] a sequence image acquisition unit, configured to capture a sequence of images of a non-cooperative space target using an optical camera carried by a servicing spacecraft; the sequence of images is captured starting from the initial moment the servicing spacecraft discovers the non-cooperative space target;

[0017] a trajectory modeling unit, configured to determine a dynamic equation and a measurement equation of a relative motion state of the space non-cooperative target in a spherical coordinate system based on the sequence of images;

[0018] The solving unit is used to construct the singular values ​​of the matrix to solve the dynamic equations and measurement equations of the relative motion state, and obtain the initial relative orbit of the space non-cooperative target.

[0019] In a third aspect, in combination with an embodiment of the present invention, a computer-readable storage medium is provided, wherein the computer-readable storage medium stores one or more programs, and when the one or more programs are executed by a computer device, the computer device executes any one of the methods for determining the initial relative orbit of a non-cooperative space target.

[0020] The above technical solution has the following beneficial effects: through singular value decomposition, an efficient approximate solution to the initial relative orbit determination problem constructed by taking a sequence of images of the non-cooperative target in space by the optical camera carried by the service spacecraft can be obtained, and the solution realizes efficient calculation and high-precision initial relative orbit determination. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0022] Figure 1 This is a flow chart of a method for determining an initial relative orbit of a non-cooperative space target according to an embodiment of the present invention;

[0023] Figure 2 is a planar schematic diagram of a spatial non-cooperative target in a spherical coordinate system according to an embodiment of the present invention;

[0024] Figure 3 is a three-dimensional schematic diagram of a spatial non-cooperative target in a spherical coordinate system according to an embodiment of the present invention;

[0025] Figure 4 is the impact of different noises on the LEO error mean in the embodiment of the present invention;

[0026] Figure 5 is the impact of different noises on the LEO error mean in the embodiment of the present invention;

[0027] Figure 6 is the impact of different noises on the GEO error mean in the embodiment of the present invention;

[0028] Figure 7 This is the impact of different noises on the GEO error mean in the embodiment of the present invention. DETAILED DESCRIPTION

[0029] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0030] like Figure 1 As shown, in combination with an embodiment of the present invention, a method for determining an initial relative orbit of a non-cooperative space target is provided, comprising:

[0031] S101: capturing a sequence of images of a non-cooperative space target using an optical camera carried by a servicing spacecraft; the sequence of images is captured starting from the initial moment the servicing spacecraft discovers the non-cooperative space target;

[0032] S102: Determining a dynamic equation and a measurement equation of a relative motion state of the spatial non-cooperative target in a spherical coordinate system based on the sequence of images;

[0033] S103: Constructing the singular values ​​of the matrix to solve the dynamic equation and measurement equation of the relative motion state, and obtain the initial relative orbit of the space non-cooperative target.

[0034] In conjunction with an embodiment of the present invention, a system for determining an initial relative orbit of a non-cooperative space target is provided, comprising:

[0035] a sequence image acquisition unit, configured to capture a sequence of images of a non-cooperative space target using an optical camera carried by a servicing spacecraft; the sequence of images is captured starting from the initial moment the servicing spacecraft discovers the non-cooperative space target;

[0036] a trajectory modeling unit, configured to determine a dynamic equation and a measurement equation of a relative motion state of the space non-cooperative target in a spherical coordinate system based on the sequence of images;

[0037] The solving unit is used to construct the singular values ​​of the matrix to solve the dynamic equations and measurement equations of the relative motion state, and obtain the initial relative orbit of the space non-cooperative target.

[0038] Preferably, S102, determining the dynamic equation and measurement equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system based on the sequence of images, includes:

[0039] For non-cooperative space targets, a fixed reference orbital plane is selected. The reference orbital plane coincides with the initial orbital plane of the service spacecraft. The reference orbital plane takes the center of the Earth as its coordinate origin and the coordinates i ρ -i θ As the initial orbital plane for service spacecraft and non-cooperative space targets, i φ Direction perpendicular to i ρ -i θ The direction of the plane establishes the spherical coordinate system of the space target, i ρ Represents a unit vector along the radial direction, pointing from the origin to the position of the point, i θ represents the unit vector perpendicular to the radial direction and along the motion direction of the service spacecraft in the initial orbital plane, i φ Represents the unit vector traced to the plane formed by the radial and tangential directions, which conforms to the right-hand rule;

[0040] Assuming that the distance between the service spacecraft and the non-cooperative space target is close and the service spacecraft is in a near-circular orbit, δ ρ ,δ φ <<R, ρ s ≡R, R represents the orbital radius of the service spacecraft, δ ρ represents the radial relative distance between the non-cooperative target and the service spacecraft, δ ρ ,δ θ ,δ φ Indicates that the spatial non-cooperative target is in i ρ、i θ 、i φ Relative motion state in direction; They are δ ρ ,δ θ ,δ φ The corresponding first-order derivative with respect to time, ρ s ,θ s ,φ s Indicates that the service spacecraft is in i ρ 、i θ 、i φ Inertial state in direction, They are ρ s ,θ s The corresponding first-order derivative with respect to time, the dynamic equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system is:

[0041]

[0042] Among them, δ ρ represents the radial relative position of the non-cooperative space target and the service spacecraft, represents the radial relative velocity of the non-cooperative target in space, represents the radial relative acceleration of the non-cooperative target in space, δ θ represents the relative azimuth of the non-cooperative target in space, Indicates the azimuth change rate of the non-cooperative target in space, represents the azimuth angular acceleration of the non-cooperative target in space, δ φ represents the relative pitch angle of the non-cooperative target in space, represents the pitch angle change rate of the non-cooperative target in space, represents the pitch angle angular acceleration of the non-cooperative target in space, represents the average angular velocity of the service spacecraft, μ represents the gravitational constant, and a represents the semi-major axis of the orbit of the non-target space;

[0043] Let the relative orbital state vector of the non-cooperative target in the spherical coordinate system be Then the dynamic equation (1) of the relative orbital state of the non-cooperative space target is written as a differential equation:

[0044]

[0045] in, Indicates the pitch angle change rate of the non-cooperative target in space;

[0046]

[0047] If the initial relative orbital state vector of the known non-cooperative space target is Then the solution of differential equation (2) is:

[0048] X(t)=e At X0=Φ(t k ,t0)X0 (5)

[0049] The state transfer matrix is ​​expressed as:

[0050]

[0051] t=t k -t0,τ=n·t,s=sinτ,c=cosτ (6)

[0052] Among them, t represents the time interval between the current time and the initial time, t k Indicates t k time, t0 represents the initial time, Φ ρ (t k ,t0) represents the state transfer coefficient at the radial position, Φ θ (t k ,t0) represents the state transfer coefficient at the radial position, Φ φ (t k ,t0) represents the state transfer coefficient of the spatial non-cooperative target in the radial position.

[0053] The above technical means are also the specific technical means of the track modeling unit.

[0054] Preferably, S102: determining the dynamic equation and measurement equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system based on the sequence of images, further includes:

[0055] Assume that the relative azimuth and relative pitch angles of the space non-cooperative target relative to the service spacecraft measured from the images taken by the optical camera are α and β respectively, and the position of the space non-cooperative target relative to the service spacecraft is expressed as:

[0056]

[0057] Among them, r m Represents the position vector of the non-cooperative target in space, r s represents the position vector of the service spacecraft, ρ m and ρ s denote the radial distances of the non-cooperative space target and the service spacecraft, respectively; Indicates three directions: i ρ 、i θ 、i φ The corresponding unit vector;

[0058] The line-of-sight measurement based on an optical camera is called LOS measurement, and the LOS measurement equation of a non-cooperative target in space is expressed as:

[0059]

[0060] By transforming equation (8), we can obtain the nonlinear measurement equation of the spatial non-cooperative target:

[0061]

[0062] Where z represents the true value combination of the measurement angle of the spatial non-cooperative target, v represents the measurement noise, and the measurement noise v is usually assumed to be zero-mean Gaussian white noise with a variance of R.

[0063] The above technical means are also the specific technical means of the track modeling unit.

[0064] Preferably, S102: determining the dynamic equation and measurement equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system based on the sequence of images, further includes:

[0065] Due to the relative motion state (δ ρ ,δ θ ,δ φ ) is a small quantity. The nonlinear measurement equation (9) of the spatial non-cooperative target is expanded by a first-order Taylor series to obtain the first-order measurement equation:

[0066] ρ s δ θ -δ ρ tanα=0 (10)

[0067] δ φ -δ ρ tanβ=0 (11)

[0068] Assuming there are n measured values, for the first-order measurement equation at each measurement time t k ,k=0,1,...,n-1, and evaluate the equation (6) corresponding to the state transfer matrix and the initial relative orbit state vector Correlated, we get:

[0069] δ ρ =Φ ρ (t k )X0,δ θ =Φ θ (t k )X0,δ φ =Φ φ (t k )X0 (12)

[0070] Among them, Φρ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ ρ The relevant part, which describes the initial state X0 at the initial time t0 to the time t k The propagation relationship in the radial direction when Φ θ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ θ The relevant part, which describes the initial state X0 at the initial time t0 to the time t k The propagation relationship in the radial direction when Φ φ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ φ The relevant part, which describes the initial state X0 at the initial time t0 to the time t k The propagation relationship in the radial direction when

[0071] Combining equations (10), (11) and (12) corresponding to the first-order measurement equation, 2n equations can be generated in the following form:

[0072] BX0=0(13)

[0073] in

[0074]

[0075] B k,α =ρ s Φ θ (t k )-Φ ρ (t k )tanα(t k ),k=0,1,...,n-1 (15)

[0076] B k,β =Φ φ (t k )-Φ ρ (t k )tanβ(t k ),k=0,1,...,n-1 (16)

[0077] Among them, tanα(tk ) Indicates that at time t k , the tangent value of the relative azimuth angle between the service spacecraft and the non-cooperative target in space, tanβ(t k ) indicates that at time t k, the tangent value of the relative pitch angle from the service spacecraft to the non-cooperative target in space.

[0078] The above technical means are also the specific technical means of the track modeling unit.

[0079] Preferably, S102: determining the dynamic equation and measurement equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system based on the sequence of images, further includes:

[0080] Assuming the relative motion state (δ ρ ,δ θ ,δ φ ) is a small quantity, and the nonlinear measurement equation (9) of the spatial non-cooperative target is expanded by the second-order Taylor to obtain the second-order measurement equation:

[0081]

[0082] Assuming there are n measured values, for the second-order measurement equation at each measurement time t k ,k=0,1,...,n-1, and evaluate the equation (6) corresponding to the state transfer matrix and the initial relative orbit state vector Associated with each other, we get the second-order Taylor series expansion:

[0083]

[0084] in:

[0085]

[0086] B k,α =ρ s Φ θ (t k )-Φ ρ (t k )tanα(t k ),k=0,1,...,n-1 (15)

[0087] B k,β =Φ φ (t k )-Φ ρ (t k )tanβ(t k ), k=0,1,...,n-1 (16).

[0088] The above technical means are also the specific technical means of the track modeling unit.

[0089] Preferably, S103: constructing the singular values ​​of the matrix to solve the dynamic equation and the measurement equation of the relative motion state to obtain the initial relative orbit of the space non-cooperative target includes:

[0090] S103-1: Calculate the approximate unit vector of the initial relative orbital state vector;

[0091] S103-2: Using the second-order Taylor series expansion of the nonlinear measurement equation, solve the scalar unknown factor used to scale the approximate unit vector to obtain the initial relative orbit vector of the space non-cooperative target.

[0092] The above technical means are also specific technical means for solving the unit.

[0093] Preferably, S103-1: calculating the approximate unit vector of the initial relative orbital state vector specifically includes:

[0094] Given at least three measurements, the approximate unit vector of the initial relative orbital state vector in equations (10) and (11) corresponding to the first-order measurement equation is determined by singular value decomposition of the matrix; specifically, the singular value decomposition of equation (13) is performed to obtain:

[0095]

[0096] Among them, the matrix U = [u0,u1,u2,u3,u4,u5], U contains 6 left singular vectors; the matrix V = [v0,v1,v2,v3,v4,v5], V contains 6 right singular vectors; Σ represents the singular value (σ i ,i=0,1,...,5) diagonal matrix; Since the homogeneous equation BX0=0 has a minimum singular value, denoted as σ i , and the corresponding right singular vector is denoted as v i , then Bv i =σ i u i ; Therefore, the approximate unit vector of formula (13) is the smallest singular value σ i The associated right singular vector ν i ;

[0097] Calculate the dot product of the initial measurement angle and the right singular vector corresponding to each minimum singular value, and select the right singular vector that produces the maximum dot product result as the best approximate unit vector of equation (13); let the best right singular vector be ν0, then the initial relative orbit state vector is expressed as:

[0098] X0≈d0ν0 (24)

[0099] Where d0 is the scalar unknown factor that best approximates the unit vector;

[0100] The complete solution space of the initial relative orbital state vector is spanned by all column vectors of the matrix V = [v0,v1,v2,v3,v4,v5], and we get:

[0101] X0=d0ν0+ν 1:5 d 1:5 (25)

[0102] Among them, d 1:5 represents the remaining scalar unknown factor; ν 1:5 Represents the remaining right singular vectors v1, v2, v3, v4, v5 of the matrix V;

[0103] Through singular value decomposition, X0 can be decomposed into two parts: d0ν0 and ν 1:5 d 1:5 .

[0104] The above technical means are also specific technical means for solving the unit.

[0105] Preferably, S103-2: using a second-order Taylor series expansion of the nonlinear measurement equation to solve a scalar unknown factor for scaling the approximate unit vector to obtain an initial relative orbit vector of the space non-cooperative target includes:

[0106] Substitute equation (25) into equations (17), (18) and (20) corresponding to the second-order measurement equation, and ignore d 1:5 The quadratic term of , we get:

[0107]

[0108] At this point, there are 2n equations and 6 unknowns (d0 and d 1:5 ), then select the remaining 2n-1 equations, which are as follows:

[0109]

[0110] Among them, R and S play the role of variable replacement;

[0111] Solving equation (28) using the least squares method yields:

[0112]

[0113] Substituting equation (30) into equation (26), when k = 0, we get

[0114]

[0115] Divide both sides of equation (31) by get:

[0116]

[0117] make

[0118]

[0119]

[0120] According to the matrix determinant lemma, equation (32) can be written as:

[0121]

[0122] Multiplying both sides of equation (36) by Det[Υ], we get:

[0123]

[0124] Simplifying equation (37) yields:

[0125]

[0126] Extracting d0 from equation (38) yields the polynomial equation for d0:

[0127]

[0128] By solving equation (39), we can get the solution of unknown scalar factor d0. Substituting the solution of unknown scalar factor d0 into equation (30), we can get d 1:5 For each solution d0, according to formula (25), the solution of the initial relative orbital state vector is expressed as:

[0129] X0=d0ν0+ν 1:5 d 1:5 (40)

[0130] There are at most 15 solutions to the 15th-degree polynomial of d0. According to the physical meaning of d0, which is the distance between the service spacecraft and the non-cooperative space target, d0 is a positive number, and d0 is greater than 100m and less than 10,000km, which can ensure the boundary of the distance between the service spacecraft and the non-cooperative space target formed by the dynamic equation of the relative motion state; by retaining the real part of the solution, a valid solution is obtained as The specific value in .

[0131] The above technical means are also specific technical means for solving the unit.

[0132] In combination with an embodiment of the present invention, a computer-readable storage medium is provided, which stores one or more programs. When the one or more programs are executed by a computer device, the computer device executes any one of the methods for determining the initial relative orbit of a non-cooperative space target.

[0133] The above technical solutions of the embodiments of the present invention are described in detail below with reference to specific application examples. For technical details not introduced during the implementation process, please refer to the relevant description above.

[0134] A method for determining the initial relative orbit of a non-cooperative space target is developed. Based on monocular image sequences, the dynamic equations and measurement equations for the relative motion state in a spherical coordinate system are derived, providing a theoretical basis for the IROD algorithm under near-circular orbit conditions. A non-iterative IROD algorithm is developed. Through singular value decomposition (singular value decomposition of a 6×6 matrix) and 15th-order polynomials, an efficient approximate solution to the IROD problem for monocular image sequences is given, achieving efficient computation. Numerical simulations are used to evaluate the performance of the IROD algorithm in non-circular low-Earth orbit (LEO) and near-circular geostationary orbit (GEO) scenarios, both with and without measurement errors. The method demonstrates good performance in LEO and GEO scenarios, providing effective support for high-precision initial orbit determination under complex conditions.

[0135] 1. Relative motion state equation of a non-cooperative target in space in spherical coordinates

[0136] For non-cooperative targets in space, when determining the initial relative orbit, the nonlinear equations of motion in spherical coordinates are not functions of orbital angular displacement, so their linearized equations remain valid for arbitrarily large relative orbital spacings. Therefore, using spherical coordinates to describe the initial relative orbit of spacecraft is more advantageous than using traditional rectangular coordinates.

[0137] The motion of service spacecraft and non-cooperative space targets orbiting the Earth in the spherical coordinate system is as follows: Figure 2 and Figure 3 To better describe the relative motion of space targets, a fixed reference orbital plane is selected, which coincides with the initial orbital plane of the service spacecraft, and the relative motion equation of the space non-cooperative target in the spherical coordinate system is derived. The center of the earth is selected as the coordinate origin, and i ρ -i θ As the initial orbital plane for service spacecraft and non-cooperative space targets, i φ Direction perpendicular to i ρ -i θ The direction of the plane establishes the spherical coordinate system of the space target, i ρ Represents the unit vector along the radial direction (radius direction), pointing from the origin to the position direction of the point, i θ represents the unit vector perpendicular to the radial direction and along the motion direction of the service spacecraft in the initial orbital plane, i φ Represents the unit vector perpendicular to the plane formed by the radial and tangential directions, forming a right-handed coordinate system, such as Figure 2 and Figure 3 As shown, Figure 2 and Figure 3 The subscripts s and m in the symbols represent the service spacecraft and space non-cooperative targets, respectively.

[0138] Furthermore, assuming that the distance between the service spacecraft and the non-cooperative space target is close and the service spacecraft is in a near-circular orbit: δ ρ ,δ φ <<R, ρ s ≡R, R represents the orbital radius of the service spacecraft, δ ρ represents the radial relative distance between the non-cooperative target and the service spacecraft, δ ρ ,δ θ ,δ φ represents the relative motion state of non-cooperative targets in space, They are δ ρ ,δ θ ,δ φ The corresponding first-order derivative with respect to time, ρ s ,θ s ,φ s Indicates that the service spacecraft is in i ρ 、i θ 、i φ Inertial state in direction, They are ρ s ,θ s The corresponding first-order derivative with respect to time, the dynamic equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system is:

[0139]

[0140] Among them, δ ρ represents the radial relative position of the non-cooperative space target and the service spacecraft, represents the radial relative velocity of the non-cooperative target in space, represents the radial relative acceleration of the non-cooperative target in space, δ θ represents the relative azimuth of the non-cooperative target in space, δ θ Indicates the azimuth change rate of the non-cooperative target in space, represents the azimuth angular acceleration of the non-cooperative target in space, δ φ represents the relative pitch angle of the non-cooperative target in space, represents the pitch angle change rate of the non-cooperative target in space, represents the pitch angle angular acceleration of the non-cooperative target in space; represents the average angular velocity of the service spacecraft, μ represents the gravitational constant, and a represents the semi-major axis of the orbit of the non-target space.

[0141] Compared with the classical CW equation, the dynamic equation of the relative motion state in the spherical coordinate system is not δ θ and function, so the linearized dynamic equation of the relative motion state is θ and This is significantly different from the scope of application of the CW equation, which is usually limited to a small range of relative displacement conditions.

[0142] Let the relative orbital state vector of the non-cooperative space target in the spherical coordinate system be Then the dynamic equation (1) of the relative motion state of the non-cooperative target in space can be written in the form of a differential equation:

[0143]

[0144] in, Indicates the pitch angle change rate of the non-cooperative target in space;

[0145]

[0146] If the initial relative orbital state vector of the known non-cooperative space target is Then the solution of differential equation (2) is:

[0147] X(t)=e At X0=Φ(t k ,t0)X0 (5)

[0148] The state transfer matrix is ​​expressed as:

[0149]

[0150] t=t k -t0,τ=n·t,s=sinτ,c=cosτ

[0151] Among them, t represents the time interval between the current time and the initial time, t k Indicates t k time, t0 represents the initial time, Φ ρ (t k ,t0) represents the state transfer coefficient at the radial position, Φ θ (t k ,t0) represents the state transfer coefficient at the radial position, Φ φ (t k ,t0) represents the state transfer coefficient of the spatial non-cooperative target in the radial position.

[0152] 2. Measurement equations for spatial non-cooperative targets in spherical coordinates

[0153] In the relative motion of non-cooperative space targets, line-of-sight measurement (LOS measurement) based on optical cameras is a key means of obtaining the relative position and orbital state of non-cooperative space targets. Sequential images are a series of images of a non-cooperative space target acquired continuously over a certain time interval. Sequential images can capture the time-varying trajectory of the non-cooperative space target, thereby extracting the relative motion state of the non-cooperative space target. A spherical coordinate LOS measurement equation based on monocular sequential images is used to more accurately describe the relative position of the non-cooperative space target with respect to the service spacecraft.

[0154] according to Figure 2 and Figure 3 The geometric relationship given is used to introduce the LOS measurement equation based on monocular sequence images in the spherical coordinate system. The optical camera is installed on the service spacecraft. Let the relative azimuth and relative pitch angles of the space non-cooperative target relative to the service spacecraft obtained from the images taken by the optical camera be α and β respectively. Then the position of the space non-cooperative target relative to the service spacecraft is:

[0155]

[0156] The subscripts s and m in the formula represent the service spacecraft and the space non-cooperative target respectively; r represents the position vector, r m Represents the position vector of the non-cooperative target in space, r s represents the position vector of the service spacecraft, ρ m and ρ s denote the radial distances of the non-cooperative space target and the service spacecraft, respectively; Indicates three directions: i ρ 、i θ 、i φ The three unit vectors corresponding to the service spacecraft.

[0157] Further, if Figure 2 and Figure 3 As shown, the LOS measurement equation of the non-cooperative target in space can be obtained as:

[0158]

[0159] Based on equation (8), the nonlinear measurement equation of the spatial non-cooperative target is expressed as:

[0160]

[0161] Where α and β represent the azimuth and pitch angles (true values) of the space non-cooperative target relative to the service spacecraft measured by the optical camera, respectively; z represents the combination of the true values ​​of the measured angles of the space non-cooperative target; v represents the measurement noise, which is usually assumed to be zero-mean Gaussian white noise with a variance of R.

[0162] The nonlinear measurement equation (9) of non-cooperative targets in space is an accurate nonlinear measurement equation in spherical coordinates derived without any assumptions, which ensures high accuracy and reliability of measurement and is suitable for complex relative motion analysis of non-cooperative targets in space.

[0163] 2.1 First-order measurement equation

[0164] Due to the relative motion state (δ ρ ,δ θ ,δ φ ) is a small quantity (a mathematical term referring to a very small number). The nonlinear measurement equation (9) of the spatial non-cooperative target is expanded by a first-order Taylor series to obtain the first-order measurement equation:

[0165] ρ s δ θ -δ ρ tanα=0 (10)

[0166] δ φ -δ ρ tanβ=0 (11)

[0167] Assuming there are n measurement values ​​(obtained from the images taken by the optical camera: α and β), the first-order measurement equation is used at each moment, that is, the measurement time t k ,k=0,1,...,n-1, and evaluate the equation (6) corresponding to the state transfer matrix and the initial relative orbit state vector Correlated, we get:

[0168] δ ρ =Φ ρ (t k )X0,δ θ =Φ θ (t k )X0,δ φ =Φ φ (t k )X0 (12)

[0169] Among them, Φ ρ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ ρ The relevant part, which describes the initial state X0 at the initial time t0 to the time tk The propagation relationship in the radial direction when Φ θ (tk) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ θ The relevant part, which describes the initial state X0 at the initial time t0 to the time t k The propagation relationship in the radial direction when Φ φ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ φ The relevant part, which describes the initial state X0 at the initial time t0 to the time t k The propagation relationship in the radial direction when

[0170] Combining equations (10), (11) and (12) corresponding to the first-order measurement equation, 2n equations can be generated in the following form:

[0171] BX0=0 (13)

[0172] in

[0173]

[0174] B k,α =ρ s Φ θ (t k )-Φ ρ (t k )tanα(t k ),k=0,1,...,n-1 (15)

[0175] B k,β =Φ φ (t k )-Φ ρ (t k )tanβ(t k ),k=0,1,...,n-1 (16)

[0176] Among them, tanα(t k ) indicates that at time t k , the tangent value of the horizontal line of sight angle (relative azimuth) between the service spacecraft and the non-cooperative target in space, that is, the relative horizontal azimuth measurement value, tanβ(t k ) indicates that at time t k , the tangent value of the line of sight angle (relative pitch angle) in the vertical direction of the line connecting the service spacecraft and the non-cooperative target in space, that is, the measurement value of the relative pitch angle;

[0177] Since Equation (13) is a homogeneous system of equations, although it contains 2n equations, the relative motion state of the non-cooperative target in space is only 6-dimensional, namely The six parameters in it cannot be uniquely solved. This is because the dynamic equation (1) of the relative motion state and the equations (10) and (11) corresponding to the first-order measurement equation lose some nonlinear information during the linearization process, which makes the solution of the relative motion state of the space non-cooperative target uncertain. To solve this problem, the introduction of the second-order measurement equation can partially compensate for the information loss caused by the linearization of the equations (10) and (11) corresponding to the first-order measurement equation, thereby improving the uniqueness of the solution and the observability of the relative motion state.

[0178] 2.2 Second-order measurement equation

[0179] The same assumptions as the first-order measurement equation, assuming the relative motion state (δ ρ ,δ θ ,δ φ ) is a small quantity, and the nonlinear measurement equation (9) of the spatial non-cooperative target is expanded by the second-order Taylor to obtain the second-order measurement equation:

[0180]

[0181] Assuming there are n measured values, for the second-order measurement equation at each measurement time t k ,k=0,1,...,n-1, and evaluate the equation (6) corresponding to the state transfer matrix and the initial relative orbit state vector Associated with, we can get the second-order Taylor series expansion as:

[0182]

[0183] in

[0184]

[0185] B k,α =ρ s Φ θ (t k )-Φ ρ (t k )tanα(t k ),k=0,1,...,n-1 (15)

[0186] B k,β =Φ φ (t k )-Φ ρ (t k )tanβ(t k ),k=0,1,...,n-1(16)

[0187] Combining the previously derived relative motion equations (dynamic equations of the relative motion state (1)), the first-order measurement equations, and the second-order measurement equations, an effective algorithm is needed to solve the initial relative orbital state of a non-cooperative space target. The IROD algorithm is derived below, with the ultimate goal of determining the solution X0 to these 2n quadratic equations.

[0188] The “combination” here refers to the simultaneous use of the relative motion equations derived in the previous article (i.e., Equation (1) or the state transfer matrix Equation (6)) and the measurement equations (such as Equation (9)) to construct a set of polynomial constraint equations for solving the initial relative orbital state of the non-cooperative space target.

[0189] 3. Determination of the initial relative motion state vector

[0190] The complete navigation filtering process must start with the estimation of the relative motion state and the initialization of the covariance matrix. The accurate initial relative motion state is crucial for the relative navigation of the spacecraft (here is the relative navigation, not the service spacecraft). The method for determining the initial relative motion state is to estimate the initial relative orbit state vector by a batch of angle measurements. First, the approximate unit vector of the initial relative orbital state vector is calculated. Then, the second-order Taylor series expansions (19) and (20) obtained by expanding the nonlinear measurement model (9) are used to solve the scalar unknown factor d0 used to correctly scale the approximate unit vector, thereby obtaining an approximate solution to the IROD problem.

[0191] 3.1 Solving the approximate unit vector

[0192] First, given three or more measurement values ​​(i.e., three or more pairs of measurement angles (α and β, obtained from the optical camera image), the approximate unit vector of the initial relative orbit state vector in the first-order measurement equations (10) and (11) is determined by singular value decomposition of the matrix. Singular value decomposition of equation (13) yields:

[0193]

[0194] Among them, the matrix U = [u0,u1,u2,u3,u4,u5], U contains 6 left singular vectors; the matrix V = [v0,v1,v2,v3,v4,v5], V contains 6 right singular vectors; Σ represents the singular value (σ i ,i=0,1,...,5). Since the homogeneous equation BX0=0 has a minimum singular value, denoted as σ i , and the corresponding right singular vector is denoted as v i , then Bv i =σ i u iTherefore, the approximate unit vector of equation (13) is the smallest singular value σ i The associated right singular vector ν i . However, there may be multiple minimum singular values, so the right singular vector associated with the minimum singular value is not necessarily the optimal solution. The best right singular vector should be the vector closest to the initial LOS direction. In order to find the best right singular vector, it is necessary to calculate the dot product of the initial LOS unit vector (that is, the initial measurement angle: α and β at the initial time t0) and the right singular vector corresponding to each minimum singular value, and select the right singular vector that produces the largest dot product result as the best approximate unit vector of equation (13). Let the best right singular vector be ν0, then the initial relative orbit state vector can be expressed as:

[0195] X0≈d0ν0(24)

[0196] where d0 is the scalar unknown factor that best approximates the unit vector.

[0197] The complete solution space of the initial relative orbital state vector is spanned by all column vectors in the matrix V = [v0,v1,v2,v3,v4,v5], so:

[0198] X0=d0ν0+ν 1:5 d 1:5 (25)

[0199] where d 1:5 represents the remaining scalar unknown factor, which is very small compared to d0; ν 1:5 Represents the remaining right singular vectors v1,v2,v3,v4,v5 of the matrix V.

[0200] By singular value decomposition, X0 can be decomposed into two parts: one is the main part d0ν0, and the other is a smaller secondary part ν0 orthogonal to ν0 1:5 d 1:5 The six unknown parameters originally required to solve X0 are replaced by unknown scalar factors d0 and d 1:5 Replacement, thus simplifying the complexity of the problem.

[0201] 3.2 Determining scalar unknown factors

[0202] Since the equations (10) and (11) corresponding to the first-order measurement equation are homogeneous linear equations, the unknown scalar factors d0 and d 1:5 can be extracted, resulting in the non-uniqueness of the solution. In order to solve the ambiguity of the solution, the equations (17) and (18) corresponding to the second-order measurement equation are used to eliminate the ambiguity and obtain a non-trivial solution, thereby determining the unknown scalar factor.

[0203] Substituting equation (25) into equations (17), (18) and (20) corresponding to the second-order measurement equation, since d 1:5 is a small quantity, ignoring the quadratic term, we can get:

[0204]

[0205] At this point, there are 2n equations and 6 unknowns (d0 and d 1:5 ), then select the remaining 2n-1 equations, which are as follows:

[0206]

[0207] Among them, R and S play the role of variable replacement, R represents the vector in the first bracket on the left side of the equation, and S represents the vector in the second bracket on the left side of the equation;

[0208] Solving equation (28) using the least squares method yields:

[0209]

[0210] Substituting equation (30) into equation (26), when k = 0, we have

[0211]

[0212] Divide both sides of equation (31) by have

[0213]

[0214] make

[0215]

[0216]

[0217] According to the matrix determinant lemma, equation (32) can be written as:

[0218]

[0219] Multiplying both sides of equation (36) by Det[Υ], we get:

[0220]

[0221] Further simplifying equation (37) yields:

[0222]

[0223] From equation (38), we can extract (i.e. divide) d0 and obtain the polynomial equation about d0:

[0224]

[0225] Formula (39) is a polynomial about d0. By solving it, we can get the solution of the unknown scalar factor d0. Substituting the solution of the unknown scalar factor d0 into formula (30), we can get d 1:5 For each solution d0, according to formula (25), the solution of the initial relative orbital state vector is expressed as:

[0226] X0=d0ν0+ν 1:5 d 1:5 (40)

[0227] There are at most 15 solutions to the 15th-degree polynomial of d0. However, the physical meaning of d0 is the distance between the service spacecraft and the non-cooperative target in space. Therefore, d0 must be a positive number. In order not to exceed the boundary of equation (1) corresponding to the linearized relative motion model, the distance between the service spacecraft and the non-cooperative target in space should not be too close or too far. Therefore, d0 must be greater than 100m and less than 10,000km. Retaining only the real part of these solutions, there is usually one valid solution:

[0228] The specific value in .

[0229] The beneficial technical effects achieved by the embodiments of the present invention are as follows:

[0230] An efficient approximate solution to the IROD problem for monocular image sequences is presented using singular value decomposition (Singular Value Decomposition of a 6×6 matrix) and a 15th-order polynomial, achieving computational efficiency. Numerical simulations evaluate the performance of the IROD algorithm in both non-circular low Earth orbit (LEO) and near-circular geostationary orbit (GEO) scenarios, with and without measurement errors, achieving high-precision initial relative orbit determination.

[0231] Based on numerical simulation, the performance and applicability of the non-cooperative target initial relative orbit determination (IROD) method in spherical coordinates under different orbital conditions are analyzed. The study shows that in the GEO scenario, in the Leader-Follower configuration with low inclination and low eccentricity, the IROD algorithm has high estimation accuracy. However, under the conditions of high inclination, high eccentricity orbits or complex configurations (such as Flyby and Football configurations), the algorithm performance is significantly affected by nonlinear effects, which may lead to a decrease in estimation accuracy or even non-convergence. In addition, the initial relative distance is sensitive to the estimation accuracy. In the LEO scenario, when the initial relative distance is 5km or 500km, the algorithm can meet the mission requirements; but in the GEO scenario, it can only meet the mission requirements when the measurement error is strictly controlled (such as σ=10 -7rad), the algorithm can meet the requirements of long-distance orbit determination. Future work should further study the impact of factors such as the number of observations, target maneuvers, and non-spherical perturbations of the Earth on the algorithm performance, in order to improve the applicability and robustness of the algorithm under complex orbital conditions.

[0232] 4. Simulation Analysis of IROD

[0233] Through observability simulation analysis, the changing patterns of the rank and condition number of the observation matrix of the system under different time intervals and initial conditions are studied, and the observability of the system and the numerical stability of the observation matrix are evaluated. Through IROD simulation analysis, the accuracy performance of the IROD algorithm under different observation conditions is further verified, providing theoretical support and optimization direction for improving the relative orbit determination of non-cooperative targets.

[0234] 4.1 Low Earth Orbit IROD Performance

[0235] Using a dynamic model (Equation (12)) and a measurement model (Equation (19)), the IROD performance is evaluated under three different orbital configurations (Leader-Follower, Flyby, and 2×1 Football) in low Earth orbit (LEO) by varying parameters such as out-of-plane lateral separation, orbital tangential separation, and the service spacecraft orbital inclination and eccentricity. The only source of error in the simulation is the model error (measurement noise is described in 4.3). The initial position error of the IROD estimate is used as the evaluation accuracy indicator, and the calculation formula is as follows:

[0236]

[0237] Among them, (x est ,y est ,z est ) is the position vector obtained by converting the estimated spherical coordinates (δρ, δθ, δφ) into a three-dimensional Cartesian coordinate system, (x true ,y true ,z true ) is the corresponding vector obtained by converting the real spherical coordinates (δρ, δθ, δφ). The simulation parameters are set as follows: the time interval is Δt = 250s, the end time is t f = 1000s, with 4 measurements. The initial relative distance, r, is 5e2m, 5e3m, 5e4m, and 5e5m. The initial orbital elements of the service spacecraft and target are shown in Table 1. For the Flyby configuration, the corresponding heights h for each r are 70m, 7e2m, 7e3m, and 7e4m, respectively.

[0238] Table 1 Orbital elements of the service spacecraft and target

[0239]

[0240] The format of Table 2 is as follows: The first column shows the orbital inclination of the service spacecraft (i s =0°,i s =10°,i s =45°,i s =60°); the second column (5e2m, 5e3m, 5e4m, 5e5m) indicates the initial relative distance; the next three columns show the relative distances of the service spacecraft at different initial eccentricities (e s =0, 0.01, 0.1), the error sub-column of each eccentricity represents the position error, and N indicates that the IROD algorithm has no solution to the estimated state.

[0241] Table 2 Leader-Follower position error

[0242]

[0243] Table 20 shows the IROD accuracy of the Leader-Follower orbit in LEO. According to the experimental results in Table 2, it can be seen that when the eccentricity or orbital inclination is small, the position error of the Leader-Follower orbit is small and the IROD accuracy is good. However, when the eccentricity or orbital inclination is large, the IROD algorithm cannot accurately estimate the correct initial relative orbit state. In some cases, the IROD algorithm cannot even find a feasible solution. This is because the position error is based on the linearization assumption of the state equation. Obtained due to At high orbital inclinations, the error can become arbitrarily large, violating the linearization assumption. Therefore, the IROD algorithm performs poorly at high orbital inclinations. Furthermore, it can be observed that the error increases as the initial relative distance increases. This indicates that a larger initial relative distance reduces the estimation accuracy.

[0244] Table 3 Flyby position error

[0245]

[0246] Table 3 shows the initial relative orbit determination accuracy of the Flyby orbit in LEO. According to the experimental results in Table 3, it can be seen that: ① Under low eccentricity and low inclination conditions, the estimation accuracy of the IROD algorithm is high. However, as the initial relative distance increases, the estimation accuracy decreases significantly. This is because the angular velocity of the relative motion is ① The eccentricity is small, the observation angle changes are limited, and the geometric configuration of long-distance observations weakens the observability of the system; ② Under low eccentricity and high inclination conditions, the estimation accuracy of the IROD algorithm is poor overall. This is because the high inclination makes the relative motion trajectory more complex, amplifies the nonlinear effects of orbital dynamics, and increases the observation error, thereby reducing the estimation accuracy. ③ Under high eccentricity conditions, the IROD estimation accuracy at low inclination is very poor, and even some scenarios have no solution. This is because in high eccentricity orbits, the nonlinear effects are significantly enhanced and the linearization assumptions of the state equations are no longer applicable. However, under high inclination conditions, the estimation accuracy is significantly improved overall. This is because high inclination increases the range of observation angle changes, improves the observability of the geometric configuration, compensates for the modeling errors of some linearized equations to a certain extent, and improves the IROD estimation accuracy.

[0247] Table 4 Position error of Football

[0248]

[0249]

[0250] Table 4 shows the initial relative orbit determination accuracy for football orbits in LEO. The experimental results in Table 4 show that the IROD algorithm's estimation accuracy is generally poor in football orbit scenarios, making it difficult to accurately obtain the initial relative orbit state. This indicates that the IROD algorithm is not suitable for football orbits.

[0251] 4.2 Geostationary Orbit IROD Performance

[0252] The orbits will be collectively referred to as GEO. The accuracy evaluation indicators are the same as those in 4.1. The simulation parameters are set as follows: the time interval is Δt = 4800s, and the end time is t f = 18000s, with four measurements. The initial along-track distance, r, is 5e3m, 5e4m, and 5e5m. The initial orbital elements of the service and target spacecraft are shown in Table 5. For the Flyby configuration, the corresponding heights h for each r are 7e2m, 7e3m, and 7e4m, respectively.

[0253] Table 5 Orbital elements of service spacecraft and target

[0254]

[0255] Table 6 shows the IROD performance data for three different relative motion trajectory scenarios: Leader-Follower, Flyby, and 2×1 Football. The performance data are shown for three different relative distances (5e3, 5e4, and 5e5m) with an orbit inclination of i. s =0°,5, eccentricity is es =0, 0.001, Error represents the position error, and N represents that there is no solution for the initial relative orbit state in the IROD algorithm.

[0256] Table 6 Leader-Follower position error, e s =0

[0257]

[0258] Table 6 shows the different relative orbital configurations in GEO orbits with eccentricity e s = 0. The leader-follower configuration has the best applicability for the IROD algorithm in GEO orbits, especially when the initial relative distance is close. However, the Flyby and Football configurations have lower estimation accuracy due to significant nonlinear dynamic effects and limited observation geometry.

[0259] Table 7 Leader-Follower position error, e s =0.001

[0260]

[0261] Table 7 shows the different relative orbital configurations in GEO orbit with eccentricity e s = 0.001, the estimation performance of the IROD algorithm. The experimental results in Table 7 further verify that the IROD algorithm is suitable for the Leader-Follower configuration in low eccentricity, low inclination, and close-range scenarios, while the Flyby and Football configurations perform poorly in long-range and nonlinear scenarios.

[0262] 4.3 Measurement Noise Analysis

[0263] Previous IROD algorithm research assumed the absence of LOS measurement error. However, in practical applications, LOS measurement error has a significant impact on the IROD algorithm and navigation. The magnitude of this measurement error is closely related to camera accuracy: higher camera accuracy reduces the measurement error, but also increases the cost. This section will compare the performance of the IROD algorithm under varying measurement error (Gaussian white noise) conditions using several case studies.

[0264] A. Measurement Noise Analysis in LEO

[0265] The following example shows that at an orbital inclination of i s =5, eccentricity e s= 0.01, time interval Δt = 1000s, and number of measurements N = 4. Considering different degrees of measurement noise, the different initial relative distances (r = 5×10 3 m and r = 5 × 10 5 m) and obtained the average position error and standard deviation of IROD through 100 Monte Carlo experiments. The results are shown in Table 8 and Figure 4 (,r=5×10 3 / (m)).

[0266] Each column in Table 8 corresponds to the mean and standard deviation of 100 IROD position errors under different measurement noise levels. The noise level is expressed in the form of 1-σ (from σ = 10 -3 rad to σ=10 -8 The last column shows the IROD position error in the absence of measurement noise.

[0267] Table 8 Error mean and standard deviation at different noise levels, r = 5 × 10 3 m

[0268]

[0269] Table 8 shows the leader-follower scenario in LEO, with an initial relative distance of 5×10 3 m(i s =5,e s =0.01, Δt=1000s, N=4). Figure 4 This is a visual presentation of the results in Table 8. Figure 4 It can be seen that with the increase of noise level, the mean error of LEO relative orbit determination at the initial distance of 5 km shows an overall increasing trend. -7 rad), the mean error is maintained at about 1.27m with a small variation, showing extremely high accuracy and stability. -5 rad), the mean error increases slightly but remains within a high accuracy range. This demonstrates the robustness and accuracy of the IROD algorithm for handling short-range initial relative orbit determination, particularly in low-noise conditions. The algorithm's performance is well-suited to LEO satellite mission requirements, such as formation flying and rendezvous and docking, providing support for the implementation of these tasks.

[0270] Table 9 Error mean and standard deviation at different noise levels, r = 5 × 10 5 m

[0271]

[0272] Table 9 shows the leader-follower scenario in LEO, with an initial relative distance of 5×10 5 m(i s =5,e s =0.01, Δt=1000s, N=4). Figure 5 (,r=5×10 5 / (m)) is a visual presentation of the results in Table 9. From Table 9 and Figure 5 It can be seen that the error mean and standard deviation of IROD are less than σ=10 -4 Radar is usually not sensitive to measurement noise. In addition, when the two spacecraft are initially far apart, the accuracy of IROD is mainly affected by nonlinear factors.

[0273] contrast Figure 5 As shown in Table 9, when the two spacecraft are initially far apart, measurement noise has little impact on the accuracy of the LEO IROD algorithm. However, when the initial separation is close, even relatively small measurement noise can significantly affect the accuracy of the IROD algorithm. This is because for short-range relative motion in near-circular orbits, the relative state equation in spherical coordinates is equivalent to the CW equation, which is essentially unobservable. At long distances, the amplification effect of position errors is weakened, and the impact of measurement noise on the relative state estimate is reduced, resulting in higher accuracy for the IROD algorithm.

[0274] B. Analysis of Measurement Noise in GEO

[0275] The following is an example of GEO, which is 5×10 3 m and 5×10 5 m's Leader-Follower track (i s =5,e s =0, Δt=4800s, N=4) under the conditions of different degrees of measurement noise, the average value and standard deviation of the position error of 100 Monte Carlo experiments were obtained, and the results are shown in Tables 10 and 11.

[0276] Table 10 Error mean and standard deviation at different noise levels, r = 5 × 10 3 m

[0277]

[0278] Table 10 shows the leader-follower scenario in GEO, where the initial relative distance is 5×10 3 m(i s =5,e s=0, Δt=4800s, N=4). Figure 6 (r=5×10 3 / (m)) is a visual presentation of the results in Table 10. According to Table 10 and Figure 6 It can be seen that only when the measurement noise is less than σ=10 -7 rad, the accuracy of the IROD algorithm is relatively ideal. However, this low noise condition is often difficult to achieve in actual engineering applications. This shows that when the distance is 5×10 3 In GEO with m, relative navigation using only a monocular camera is not feasible.

[0279] Table 11 Error mean and standard deviation at different noise levels, r = 5 × 10 5 m

[0280]

[0281] Table 11 shows the leader-follower scenario in GEO, where the initial relative distance is 5×10 5 m(i s =5,e s =0, Δt=4800s, N=4). Figure 7 (r=5×10 5 / (m)) is a visual representation of the results in Table 11. According to Table 11 and Figure 7 It can be seen that the current IROD algorithm of monocular cameras is difficult to adapt to the relative orbit determination requirements of GEO long distances, and its accuracy is insufficient to support high-precision mission operations.

[0282] Comparing the results in Tables 10 and 11, the estimation accuracy of the IROD algorithm for GEO is more dependent on measurement noise than that for LEO. Therefore, in order to achieve high-precision initial relative orbit determination on GEO, it is necessary to use a more accurate optical camera or increase the relative distance.

[0283] It should be understood that the specific order or hierarchy of steps in the disclosed processes is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of the present disclosure. The accompanying method claims present elements of the various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy described.

[0284] In the foregoing detailed description, various features are grouped together in a single embodiment to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that embodiments of the claimed subject matter require more features than are expressly recited in each claim. On the contrary, as reflected in the appended claims, the invention comprises less than all the features of any individual disclosed embodiment. The appended claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.

[0285] The above description of the disclosed embodiments is intended to enable any person skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments without departing from the spirit and scope of the present disclosure. Therefore, the present disclosure is not limited to the embodiments presented herein but is intended to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0286] The foregoing description includes examples of one or more embodiments. Of course, it is not possible to describe all possible combinations of components or methods for the purposes of describing the above embodiments, but one of ordinary skill in the art will recognize that the various embodiments may be further combined and arranged. Therefore, the embodiments described herein are intended to encompass all such changes, modifications and variations that fall within the scope of the appended claims. Furthermore, to the extent the term "comprising" is used in the specification or claims, the term is intended to be encompassed in a manner similar to the term "including," as explained in terms of "including," used as a transitional word in the claims. Furthermore, any use of the term "or" in the specification of the claims is intended to mean a "non-exclusive or."

[0287] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for determining the initial relative orbit of a non-cooperative space target, characterized in that: include: For a non-cooperative space target, taking a sequence of images of the non-cooperative space target by an optical camera carried by the service spacecraft; The sequence of images is taken starting from the initial moment when the servicing spacecraft discovers the non-cooperative space target; Determining a dynamic equation and a measurement equation of a relative motion state of the space non-cooperative target in a spherical coordinate system based on the sequence of images; The singular values ​​of the constructed matrix are used to solve the dynamic equations and measurement equations of the relative motion state to obtain the initial relative orbit of the space non-cooperative target.

2. The method for determining the initial relative orbit of a non-cooperative space target according to claim 1, characterized in that: The method further comprises determining a dynamic equation and a measurement equation for a relative motion state of the non-cooperative space target in a spherical coordinate system based on the sequence of images, comprising: For non-cooperative space targets, a fixed reference orbital plane is selected. The reference orbital plane coincides with the initial orbital plane of the service spacecraft. The reference orbital plane takes the center of the Earth as its coordinate origin and the coordinates i ρ -i θ As the initial orbital plane for service spacecraft and non-cooperative space targets, i φ Direction perpendicular to i ρ -i θ The direction of the plane establishes the spherical coordinate system of the space target, i ρ Represents a unit vector along the radial direction, pointing from the origin to the position of the point, i θ represents the unit vector perpendicular to the radial direction and along the motion direction of the service spacecraft in the initial orbital plane, i φ Represents the unit vector traced to the plane formed by the radial and tangential directions, which conforms to the right-hand rule; Assuming that the distance between the service spacecraft and the non-cooperative space target is close and the service spacecraft is in a near circular orbit, R represents the orbital radius of the service spacecraft, δ ρ represents the radial relative distance between the non-cooperative target and the service spacecraft, δ ρ ,δ θ ,δ φ Indicates that the spatial non-cooperative target is in i ρ 、i θ 、i φ Relative motion state in direction; They are δ ρ ,δ θ ,δ φ The corresponding first-order derivative with respect to time, ρ s ,θ s ,φ s Indicates that the service spacecraft is in i ρ 、i θ 、i φ Inertial state in direction, They are ρ s ,θ s The corresponding first-order derivative with respect to time, the dynamic equation of the relative motion state of the spatial non-cooperative target in the spherical coordinate system is: Among them, δ ρ represents the radial relative position of the non-cooperative space target and the service spacecraft, represents the radial relative velocity of the non-cooperative target in space, represents the radial relative acceleration of the non-cooperative target in space, δ θ represents the relative azimuth of the non-cooperative target in space, Indicates the azimuth change rate of the non-cooperative target in space, represents the azimuth angular acceleration of the non-cooperative target in space, δ φ represents the relative pitch angle of the non-cooperative target in space, represents the pitch angle change rate of the non-cooperative target in space, represents the pitch angle angular acceleration of the non-cooperative target in space, represents the average angular velocity of the service spacecraft, μ represents the gravitational constant, and a represents the semi-major axis of the orbit of the non-target space; Let the relative orbital state vector of the non-cooperative target in the spherical coordinate system be Then the dynamic equation (1) of the relative orbital state of the non-cooperative space target is written as a differential equation: in, Indicates the pitch angle change rate of the non-cooperative target in space; If the initial relative orbital state vector of the known non-cooperative space target is Then the solution of differential equation (2) is: X(t)=e At X0=Φ(t k ,t0)X0 (5) The state transfer matrix is ​​expressed as: Among them, t represents the time interval between the current time and the initial time, t k Indicates t k time, t0 represents the initial time, Φ ρ (t k ,t0) represents the state transfer coefficient at the radial position, Φ θ (t k ,t0) represents the state transfer coefficient at the radial position, Φ φ (t k ,t0) represents the state transfer coefficient of the spatial non-cooperative target in the radial position.

3. The method for determining the initial relative orbit of a non-cooperative space target according to claim 2, wherein: Determining the dynamic equation and measurement equation of the relative motion state of the space non-cooperative target in the spherical coordinate system based on the sequence of images also includes: Assume that the relative azimuth and relative pitch angles of the space non-cooperative target relative to the service spacecraft measured from the images taken by the optical camera are α and β respectively, and the position of the space non-cooperative target relative to the service spacecraft is expressed as: Among them, r m Represents the position vector of the non-cooperative target in space, r s represents the position vector of the service spacecraft, ρ m and ρ s denote the radial distances of the non-cooperative space target and the service spacecraft, respectively; Indicates three directions: i ρ 、i θ 、i φ The corresponding unit vector; The line-of-sight measurement based on an optical camera is called LOS measurement, and the LOS measurement equation of a non-cooperative target in space is expressed as: By transforming equation (8), we can obtain the nonlinear measurement equation of the spatial non-cooperative target: Where z represents the true value combination of the measurement angle of the spatial non-cooperative target, v represents the measurement noise, and the measurement noise v is usually assumed to be zero-mean Gaussian white noise with a variance of R.

4. The method for determining the initial relative orbit of a non-cooperative space target according to claim 3, wherein: Determining the dynamic equation and measurement equation of the relative motion state of the space non-cooperative target in the spherical coordinate system based on the sequence of images also includes: Due to the relative motion state (δ ρ ,δ θ ,δ φ ) is a small quantity. The nonlinear measurement equation (9) of the spatial non-cooperative target is expanded by a first-order Taylor series to obtain the first-order measurement equation: r s d θ -d ρ tanα=0(10) d φ -d ρ tanβ=0(11) Assuming there are n measured values, for the first-order measurement equation at each measurement time t k ,k=0,1,...,n-1, and evaluate the equation (6) corresponding to the state transfer matrix and the initial relative orbit state vector Correlated, we get: d ρ =Φ ρ (t k )X0,δ θ =Φ θ (t k )X0,δ φ =Φ φ (t k )X0(12) Among them, Φ ρ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ ρ The relevant part is used to describe the initial state X0 from the initial time t0 to the time t k The propagation relationship in the radial direction when Φ θ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ θ The relevant part is used to describe the initial state X0 from the initial time t0 to the time t k The propagation relationship in the radial direction when Φ φ (t k ) represents the state transfer matrix Φ(t k ,t0) and the radial relative position δ φ The relevant part is used to describe the initial state X0 from the initial time t0 to the time t k The propagation relationship in the radial direction when Combining equations (10), (11) and (12) corresponding to the first-order measurement equation, 2n equations can be generated in the following form: BX0=0(13) in B k,α =ρ s F θ (t k )-F ρ (t k )tanα(t k ),k=0,1,...,n-1(15) B k,β =Φ φ (t k )-Φ ρ (t k )tanβ(t k ),k=0,1,...,n-1(16) Among them, tanα(t k ) indicates that at time t k , the tangent value of the relative azimuth angle between the service spacecraft and the non-cooperative target in space, tanβ(t k ) indicates that at time t k , the tangent value of the relative pitch angle from the service spacecraft to the non-cooperative target in space.

5. The method for determining the initial relative orbit of a non-cooperative space target according to claim 4, characterized in that: Determining the dynamic equation and measurement equation of the relative motion state of the space non-cooperative target in the spherical coordinate system based on the sequence of images also includes: Assuming the relative motion state (δ ρ ,δ θ ,δ φ ) is a small quantity, and the nonlinear measurement equation (9) of the spatial non-cooperative target is expanded by the second-order Taylor to obtain the second-order measurement equation: Assuming there are n measured values, for the second-order measurement equation at each measurement time t k ,k=0,1,...,n-1, and evaluate the equation (6) corresponding to the state transfer matrix and the initial relative orbit state vector Associated with each other, we get the second-order Taylor series expansion: in: B k,α =ρ s F θ (t k )-F ρ (t k )tanα(t k ),k=0,1,...,n-1(15) B k,β =Φ φ (t k )-Φ ρ (t k )tanβ(t k ),k=0,1,...,n-1(16)。 6. The method for determining the initial relative orbit of a non-cooperative space target according to claim 5, characterized in that: The singular values ​​of the matrix are constructed to solve the dynamic equations and measurement equations of the relative motion state, and the initial relative orbit of the non-cooperative target in space is obtained, including: Calculate the approximate unit vector of the initial relative orbital state vector; The second-order Taylor series expansion of the nonlinear measurement equation is used to solve the unknown scalar factor used to scale the approximate unit vector and obtain the initial relative orbit vector of the space non-cooperative target.

7. The method for determining the initial relative orbit of a non-cooperative space target according to claim 6, wherein: Calculate the approximate unit vector of the initial relative orbital state vector, specifically including: Given at least three measurements, the approximate unit vector of the initial relative orbital state vector in equations (10) and (11) corresponding to the first-order measurement equation is determined by singular value decomposition of the matrix; specifically, the singular value decomposition of equation (13) is performed to obtain: Among them, the matrix U = [u0,u1,u2,u3,u4,u5], U contains 6 left singular vectors; the matrix V = [v0,v1,v2,v3,v4,v5], V contains 6 right singular vectors; Σ represents the singular value (σ i ,i=0,1,...,5) diagonal matrix; Since the homogeneous equation BX0=0 has a minimum singular value, denoted as σ i , and the corresponding right singular vector is denoted as v i , then Bv i =σ i u i ; Therefore, the approximate unit vector of formula (13) is the smallest singular value σ i The associated right singular vector ν i ; Calculate the dot product of the initial measurement angle and the right singular vector corresponding to each minimum singular value, and select the right singular vector that produces the maximum dot product result as the best approximate unit vector of equation (13); let the best right singular vector be ν0, then the initial relative orbit state vector is expressed as: X0≈d0ν0(24) Where d0 is the scalar unknown factor that best approximates the unit vector; The complete solution space of the initial relative orbital state vector is spanned by all column vectors of the matrix V = [v0,v1,v2,v3,v4,v5], and we get: X0=d0ν0+ν 1:5 d 1:5 (25) Among them, d 1:5 represents the remaining scalar unknown factor; ν 1:5 Represents the remaining right singular vectors v1, v2, v3, v4, v5 of the matrix V; Through singular value decomposition, X0 can be decomposed into two parts: d0ν0 and ν 1:5 d 1:5 .

8. The method for determining the initial relative orbit of a non-cooperative space target according to claim 6, wherein: The second-order Taylor series expansion of the nonlinear measurement equation is used to solve the unknown scalar factor used to scale the approximate unit vector, and the initial relative orbit vector of the space non-cooperative target is obtained, including: Substitute equation (25) into equations (17), (18) and (20) corresponding to the second-order measurement equation, and ignore d 1:5 The quadratic term of , we get: At this point, there are 2n equations and 6 unknowns (d0 and d 1:5 ), then select the remaining 2n-1 equations, which are as follows: Among them, R and S play the role of variable replacement; Solving equation (28) using the least squares method yields: Substituting equation (30) into equation (26), when k = 0, we get Divide both sides of equation (31) by get: make According to the matrix determinant lemma, equation (32) can be written as: Multiplying both sides of equation (36) by Det[Υ], we get: Simplifying equation (37) yields: Extracting d0 from equation (38) yields the polynomial equation for d0: By solving equation (39), we can get the solution of unknown scalar factor d0. Substituting the solution of unknown scalar factor d0 into equation (30), we can get d 1:5 For each solution d0, according to formula (25), the solution of the initial relative orbital state vector is expressed as: X0=d0ν0+ν 1:5 d 1:5 (40) There are at most 15 solutions to the 15th-degree polynomial of d0. According to the physical meaning of d0, which is the distance between the service spacecraft and the non-cooperative space target, d0 is a positive number, and d0 is greater than 100m and less than 10,000km, which can ensure the boundary of the distance between the service spacecraft and the non-cooperative space target formed by the dynamic equation of the relative motion state; by retaining the real part of the solution, a valid solution is obtained as The specific value in .

9. A system for determining the initial relative orbit of a non-cooperative space target, characterized in that: include: a sequence image acquisition unit, configured to capture sequence images of a non-cooperative space target by using an optical camera carried by the service spacecraft; The sequence of images is taken starting from the initial moment when the servicing spacecraft discovers the non-cooperative space target; a trajectory modeling unit, configured to determine a dynamic equation and a measurement equation of a relative motion state of the space non-cooperative target in a spherical coordinate system based on the sequence of images; The solving unit is used to construct the singular values ​​of the matrix to solve the dynamic equations and measurement equations of the relative motion state, and obtain the initial relative orbit of the space non-cooperative target.

10. A computer-readable storage medium, characterized in that The computer-readable storage medium stores one or more programs, which, when executed by a computer device, enable the computer device to execute the method for determining the initial relative orbit of a non-cooperative space target as described in any one of claims 1 to 8.

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