CN115774928A - Improved Laplace model-based initial orbit optimization method for space debris short arc angle measurement only - Google Patents
Improved Laplace model-based initial orbit optimization method for space debris short arc angle measurement only Download PDFInfo
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Abstract
The invention provides a space debris short arc only angle measurement initial orbit optimization method based on an improved Laplace model, aiming at the problems that only angle measurement data short arc initial orbit is easy to fall into trivial solution and difficult to converge in space debris observation application, and the method comprises the following steps: receiving short arc observation data of the space debris acquired by an observation platform; constructing a measurement equation for representing a geometric relation and a differential equation for representing a dynamic relation based on short arc observation data; combining a measurement equation and a differential equation, and establishing an improved Laplace model by combining the observation condition of only measuring angles; constructing a solution model to be optimized by the initial value, the orbit dynamics model, the short arc observation data and the improved Laplace model; and optimizing an improved Laplace model in a model frame to be optimized and solved by utilizing an optimization algorithm, and determining the initial orbit of the space debris short arc only by measuring the angle. The method converts the initial orbit determination problem into a multivariable nonlinear over-determined equation optimization problem for improving the Laplace model, thereby being beneficial to the convergence of orbit determination results and obtaining the global optimal solution.
Description
Technical Field
The invention belongs to the technical field of space situation perception subdivision in the technical field of spaceflight, and relates to a space debris short arc only angle measurement initial orbit optimization method based on an improved Laplace model.
Background
The space-based space debris monitoring mode has better observation geometrical conditions and more observation windows, and becomes an astronomical sensing means which is competitively developed by various aerospace major countries and represented by the United states. However, the problems of orbit determination and cataloguing of focusing space debris and the problem of only angle measurement and orbit determination of short arcs are the common difficult problems faced by space-based space debris monitoring means. Generally speaking, observation within 1 ° of the central angle of orbit belongs to the short arc category, but in special cases, due to the faster relative motion speed and the non-follow-up tracking mode, the arc length of the low-orbit observation platform for observing the high-orbit target may be only 50 seconds to 100 seconds in extreme cases, and the arc section of the low-orbit observation platform for observing the low-orbit target may be only 10 seconds.
Initial Orbit Determination (IOD), hereinafter referred to as Initial Orbit Determination, is essentially to solve an optimal estimate of a target Orbit from a segment of noisy short arc observation data, with Laplace and Gauss methods being the most classical. Such as Curtis H D].Oxford:A number of studies were conducted in Butterworth-Heinemann,2013 283-325 and a series of classical monographs were written. Wherein, the Laplace method is to use a certain epoch t 0 Position vector r of time 0 And velocity vector v 0 Solving the number of target orbits, the Gauss rule is that two moments t are utilized 1 、t 2 Position vector r of 1 、r 2 The target track number is calculated. However, when the observation arc section is short, the problem of large error of the long half shaft cannot be avoided by both methods, and the method has intrinsic ill-conditioned nature, such as Lijun, a space target space-based optical monitoring and tracking key technology research [ D ]]Changsha, national defense science and technology university, 2009.
Aiming at the problem that iteration initial values are difficult to select, liulin spacecraft orbit determination theory and application [ M ]. Beijing: electronics industry Press, 2015. The method comprises the following steps of determining an initial orbit determination problem and a method [ J/OL ] of space-based optical space target monitoring in Zhao KeXin, gangqing Bo, liu Jing, tian base, aeronautics, 2022, 1-11.Http:// kns. Cnki.net/kcms/tail/11.1929. V.20228.1703.032. Html. In order to solve the initial Orbit Determination problem of some missing position vector information, christian J A, hollenberg C L.initial Orbit Determination from Three Velocity Vectors [ J ]. Journal of Velocity controls & Dynamics,2019,42 (4): 894-899, a primary Orbit Determination method for calculating the number of orbits by using Three Velocity Vectors is provided, but only a two-body Dynamics model is considered; a numerical calculation method for the Observability of three spacecrafts is provided in Y Hu, I Sharf, L Chen, three-space automatic estimation and observation Analysis with interferometric Angles-only Measurements [ J ]. Acta advanced, 2020, 170. However, as the number of observation arcs of space-based sensing system fragmentation increases, the above methods exhibit a common problem of being prone to trivial solutions and being less robust to goniometric data errors, e.g., wie B, ahn J.On Selecting the Correct Root of Angles-Only Initial estimate evaluation of Lagrange, laplace, and Gauss [ J ]. Journal of the advanced Sciences,2017,64 (1): 50-71.
In determining the trajectories for the small planets, the Italian student in Milani A, andrea, giovanni Gronchi. Theorality of Orbit Determination [ M ]. London, cambridge University Press,2010, 259-319, and Gronchi, giovanni F, giulio B, et al Keplerian integers, evaluation Theory and Identification of Very Short arms in a Large Database of Optical objectives [ J ]. Celestial Mechanics and dynamic analysis, 2017,127 (2): 211-232, studied the joint trajectories with 2-3 Short arc data and proposed the concept of the angular range of the sun as the center instead of the traditional angular range of the primer, and proposed the range of the target variation. Demars K J, jah M K, schumacher P W. Initial Orbit Determination Using Short-Arc Angle and Angle Rate Data [ J ]. IEEE Transactions on Aerospace and Electronic Systems,2012,48 (3): 2628-2637 studied an initial tracking method based on the concept of allowable domains Using information of assumed angles and angular rates without prior information. Aiming at the defect of large calculated amount, lidong, space-based observation target tracking, orbit determination and network routing algorithm research [ D ]. Changsha, the value range of allowable domain constraint distance and radial speed through a two-body orbit in the national defense science and technology university, 2012. But the premise of joint orbit determination is that clear multi-arc segment association and a determined target orbit range are used as prior, so that engineering application is restricted. The multi-arc combination substantially increases the length of the observation arc, without substantially addressing the bottleneck of the single extremely short arc orbit determination model.
Some scholars attempt to solve the short arc convergence problem by applying an intelligent optimization algorithm. Li Xinran is an extremely short arc orbit determination method [ D ] based on evolutionary computation, 2018 is a China scientific and technical university establishes a layered optimization orbit determination model and introduces an intelligent optimization algorithm to solve, but the method has a short plate for large eccentricity orbit estimation and needs to improve universality of different orbit types. A method for approximating an angle-Only Orbit Determination Likelihood Function by a Gaussian Mixture Function is provided in Psiaki M L, ryan M W, moriba K J.Gaussian Mixture Approximation of Angles-Only Orbit Determination [ J ]. Journal of guide, control, and Dynamics 2017,40 (11): 2807-2819, and the obtained probability density Function can provide prior information for a Gaussian Mixture Orbit Determination filter to adapt to angle Determination data and change of constraint conditions. Baichun G, gong B, li W, li S, et al, angles-Only Initial Relative estimation Algorithm for Non-Cooperative space acceleration algorithms [ J ]. Astrodynamics,2018,2 (3): 217-231, a close-range Relative Orbit Determination method for Non-Cooperative targets was developed based on the state transition matrix of the kinetic models of Clohessy-Wiltshire and Tschauner-Hempel. Fei F researches a distribution regression orbit determination method of a mapping relation from observation data to a target orbit based on a data driving thought in Feng F, zhang Y, li H, et al.A novel space-based orbit determination method on distribution regression and its space solution [ J ]. IEEE Access,2019, 7. The partial intelligent optimization algorithm is helpful for solving the extremely short arc orbit determination problem, but the selection of optimization variables and the construction of the algorithm are difficult to realize universality on all space targets due to the excessive constraint on the preconditions.
In summary, based on the traditional Laplace method, the theory of the space target orbit determination method is relatively perfect after many years of research, but in practice, it is found that there are some difficulties to be solved urgently, such as: when the traditional or popularized Laplace iterative solution method is used for processing short arcs of GEO targets and only measuring angle measured data, convergence is difficult or local optimal solution is easy to fall into.
Disclosure of Invention
The problem of short arc only angle measurement and orbit determination is a common difficulty faced by space-based space debris monitoring means, and due to the fact that observation arc sections are small in proportion, the iterative solution method of the traditional Laplace model has the common problems of poor robustness to measurement noise and easiness in convergence to a trivial solution when processing extremely short arc only angle measurement observation data. Aiming at the problem, the invention provides a space debris short arc only angle measurement initial orbit optimization method based on an improved Laplace model, a solution model to be optimized is constructed, a track dynamics model, short arc observation data and the improved Laplace model in the solution model to be optimized are taken as a frame, namely, the whole process from the track dynamics model in the frame to a second improved Laplace initial orbit model is taken as an objective function to be optimized, the initial orbit problem is converted into an optimization problem of a multivariable nonlinear over-determined equation, optimization is directly applied to an optimization algorithm, so that a global optimal solution is obtained, and the convergence condition of the traditional or popularized Laplace iterative solution method when the short arc only angle measurement actual measurement data of a GEO target is processed is improved. Experiments based on simulation and actual measurement data show that the orbit determination result has better performance in the aspects of convergence characteristics and precision, and for short arc observation data comprising 2' angle measurement error and hectometer platform position error, the orbit determination errors in the X direction, the Y direction and the Z direction of the optimization solution method provided by the invention are respectively 2.9712km, 3.8663km and 0.4581km.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a space debris short arc only angle measurement initial orbit optimization method based on an improved Laplace model, which comprises the following steps of:
receiving short arc observation data of space debris acquired by an observation platform;
secondly, constructing a measurement equation for representing a geometric relationship and a differential equation for representing a dynamic relationship based on short arc observation data;
step three, combining a measurement equation and a differential equation, and establishing an improved Laplace model containing observed quantity and space debris state by combining with an observation condition of only measuring angles;
step four, constructing a solution model to be optimized by the initial value, the orbit dynamics model, the short arc observation data and the improved Laplace model; and (3) taking the track dynamics model, the short arc observation data and the improved Laplace model in the solution model to be optimized as a frame, optimizing the improved Laplace model in the solution model frame to be optimized by utilizing an optimization algorithm, outputting a orbit determination result, and completing the determination of the initial track of the space debris short arc only for angle measurement.
In the first step, the short arc observation data includes: angle measurement data sequence of the space debris and position coordinate sequence of the observation platform.
In the second step, a measurement equation representing the geometric relationship is constructed based on the short arc observation data and is expressed as follows:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0; p is a distance scalar of the observation platform to the space debris,
for observing the direction vector of the platform to the space debris, R s Is a position vector of the observation platform under the geocentric J2000.0 inertial coordinate system.
Further, the direction vector from the observation platform to the space debris
Expressed as:
wherein alpha is the declination of the angle measurement data of the space debris under the inertial coordinate system of the centroid J2000.0 of the observation platform; delta is declination of angle measurement data of the space debris under an inertial coordinate system of the centroid J2000.0 of the observation platform; [ l, m, n ]] T And l, m and n are respectively intermediate variables related to the angle measurement data of the space debris for observing the direction vector from the platform to the space debris.
Furthermore, only the observation condition of the angle is measured, and a measurement equation representing the geometric relationship is constructed based on the short arc observation data and expressed as follows:
wherein r is a position vector of the space debris in an inertia coordinate system of the geocenter J2000.0;
is a direction vector from the observation platform to the space debris; r s The position vector of the observation platform under the geocentric J2000.0 inertial coordinate system is shown.
In the second step, a differential equation representing a dynamic relationship is constructed based on short arc observation data and expressed as follows:
wherein ,
is a space debris gravitational acceleration vector; mu is a gravitational constant, r is a position vector of the space debris under an inertial coordinate system of the geocentric J2000.0; f. of ε (r, v, t) is perturbation acceleration.
In the third step, a measurement equation and a differential equation are combined, and an obtained combined model is as follows:
wherein r is a position vector of the space debris in an inertia coordinate system of the geocenter J2000.0; f. of r Is a position intermediate variable; r is a radical of hydrogen 0 For space debris at an initial time t 0 The position vector of (a); f. of v Is a speed intermediate variable; v. of 0 For space debris at an initial time t 0 The velocity vector of (2); f. of n Is a normal intermediate variable of the orbital plane; v is the velocity vector of the space debris in the inertial coordinate system of the geocenter J2000.0.
In the third step, the improved Laplace model is composed of a first component model, a second component model, a third component model, a first improved Laplace initial orbit determination model and a second improved Laplace initial orbit determination model, wherein:
first component model:
second component model:
the third component model:
wherein ,fr Is a position intermediate variable; f. of v Is a speed intermediate variable; f. of n Is a normal intermediate variable of the track surface;
is the derivative of the intermediate variable of position;
is the derivative of the intermediate variable of speed;
is the derivative of the normal intermediate variable of the orbital plane; r is 0 For space debris at an initial time t 0 A position vector of (a); v. of 0 For space debris at an initial time t 0 The velocity vector of (2); r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0; v is a velocity vector of the space debris under an inertial coordinate system of the geocenter J2000.0;
the first improved Laplace initial orbit determination model comprises the following steps: substituting the combined model into a measurement equation under the condition of only measuring angles, and obtaining a first improved Laplace initial orbit determination model as follows:
wherein ,Rs_j Observing the position vector of the platform under the geocentric J2000.0 inertial coordinate system at the moment J;
observing platform to space debris for time jA vector; r is 0 For space debris at an initial time t 0 Position vector of r 0 ＝[x 0 ,y 0 ,z 0 ] T ；v 0 For space debris at an initial time t 0 Velocity vector of v 0 ＝[v x0 ,v y0 ,v z0 ] T ；
The second improved Laplace initial orbit determination model comprises the following steps: the direction vector from the first improved Laplace initial orbit determination model to the space debris on the observation platform
And expanding on the basis of a formula to obtain a second improved Laplace initial orbit determination model as follows:
wherein ,x0 、y 0 、z 0 Respectively space debris at an initial time t 0 Each component of the position vector of (a); v. of x0 、v y0 、v z0 Respectively space debris at an initial time t 0 Each component of the velocity vector of (a); x s 、Y s 、Z s Respectively representing each component of a position vector of the observation platform under an inertia coordinate system of the geocentric J2000.0; l, m and n are respectively intermediate variables related to space debris angle measurement data (right ascension and declination).
In the fourth step, the step of constructing the solution model to be optimized by the initial value, the orbit dynamics model, the short arc observation data and the improved Laplace model specifically comprises the following steps:
(1) Randomly giving an initial value of short arc observation data of a group of space debris, wherein the initial value is the space debris at an initial time t 0 Position vector r of 0 And velocity vector v 0 ；
(2) Position vector r by orbit dynamics model 0 And velocity vector v 0 Extrapolate to each measurement instant t i I =1,.., N; n is the number of observation points in the observation arc section, and N is more than or equal to 1; and obtain the measurement time t i Lower position vector state quantity r i Sum velocity vector state quantity v i ；
(3) Based on the measurement time t i Lower position vector state quantity r i Sum velocity vector state quantity v i F is calculated by the first component model, the second component model and the third component model respectively r 、f v 、f n (ii) a Based on f r 、f v 、f n And writing the short arc observation data of the space debris acquired by the observation platform in the step one by the first improved Laplace initial orbit determination model into a nonlinear over-determined equation set of a second improved Laplace initial orbit determination model, namely the model to be optimized.
Further, the step of optimizing the improved Laplace model in the solution model to be optimized by using an optimization algorithm and outputting a orbit determination result by taking the orbit dynamics model, the short arc observation data and the improved Laplace model in the solution model to be optimized as a frame to complete the determination of the initial orbit of the space debris short arc only angle measurement comprises the following steps:
and (3) integrally considering the step (2) and the step (3) of the solution model to be optimized as an objective function to be optimized, after a group of initial values are given, solving and optimizing the multivariable nonlinear over-definite equation of the second improved Laplace initial orbit determination model listed in the step (3) by using an optimization algorithm through the step (2), outputting an orbit determination result, and completing the determination of the space debris short arc only angle measurement initial orbit.
Wherein the optimization algorithm includes, but is not limited to: non-linear least squares, confidence domain algorithms, or particle swarm algorithms.
The orbit dynamics model is a gravitational model to which the satellite flies in space, including but not limited to the earth's gravitational force, the lunar gravitational force, or the solar gravitational force; the earth gravitational field model can be an 8 x 8 order JGM-3 earth gravitational field model.
The invention discloses an improved Laplace model-based initial orbit optimization method for only angle measurement of short arcs of space fragments, and develops in-depth research on the improved Laplace model and a solving method thereof under the background of only angle measurement of extremely short arcs. Firstly, aiming at the problem that short arc observation data are not easy to converge under the actual measurement environment, an applicable optimization is providedThe improved Laplace model orbit determination optimization solving method of the (search) algorithm further simulates and verifies the iterative convergence characteristic of the method, and unlike the popularization of the Laplace method, the method can converge to an ideal result only through 2-3 steps of iteration under the noise-free condition. In addition, monteCarlo simulation data and measured data are respectively utilized to verify the orbit determination precision of the method under the influence of measured random noise. The result shows that the orbit determination precision of the invention is obviously better than that of the popularization Laplace method, for example, when the random noise level of the angle measurement data and the platform position are respectively (2') 2 1 sigma and (100 m) 2 And 1 sigma of zero mean Gaussian white noise, the Laplace method is popularized, and the standard deviation of the errors in the X, Y and Z directions is respectively 4.2260km, 5.1112km and 0.6001km, while the corresponding results of the method are respectively 3.2675km, 4.2499km and 0.5306km. As for the method, the optimization solution method provided by the invention is slightly better than the iterative method in orbit determination precision, but the calculation time is slightly longer than that of the iterative method. Meanwhile, for actually measured data with more complex noise, the optimization solution method shows more stable convergence characteristics than the iteration method, and the problem that the iteration solution method is easy to fall into a local optimal solution is effectively solved.
Drawings
The invention is explained in more detail below with reference to the figures and examples.
Fig. 1 is a geometric schematic diagram of space debris observation.
FIG. 2 is a schematic diagram of an iterative solution process for improving a Laplace model.
FIG. 3 is a schematic diagram of an optimization solution process of the improved Laplace model.
FIG. 4 is a position vector r 0 And (4) a schematic diagram of the convergence situation of the direction iteration error.
FIG. 5 is a velocity vector v 0 And (5) a schematic diagram of the convergence situation of the directional iteration error.
FIG. 6a is a schematic diagram of the tracking error in the X direction in the position vector by the improved Laplace iterative solution method.
FIG. 6b is a schematic diagram of the orbit determination error in the Y direction in the position vector by the improved Laplace iterative solution method.
FIG. 6c is a schematic diagram of the tracking error in the Z direction in the position vector by the improved Laplace iterative solution method.
FIG. 6d is a diagram of V in velocity vectors for an improved Laplace iterative solution method x Schematic diagram of the tracking error of direction.
FIG. 6e is a diagram of V in velocity vector for the improved Laplace iterative solution method y Schematic diagram of the tracking error of direction.
FIG. 6f is a diagram of V in velocity vectors for an improved Laplace iterative solution method z Schematic diagram of the tracking error of direction.
FIG. 7a is a schematic diagram of the orbit determination error in the X direction in the position vector by the improved Laplace optimization solution method.
FIG. 7b is a schematic diagram of the orbit determination error in the Y direction in the position vector by the improved Laplace optimization solution method.
FIG. 7c is a schematic diagram of the tracking error in the Z direction in the position vector by the improved Laplace optimization solution method.
FIG. 7d is a diagram of V in a velocity vector for the improved Laplace optimization solution method x Schematic diagram of the tracking error of the direction.
FIG. 7e is a diagram of V in velocity vectors for an improved Laplace optimization solution method y Schematic diagram of the tracking error of the direction.
FIG. 7f is a diagram of V in a velocity vector by the improved Laplace optimization solution method z Schematic diagram of the tracking error of the direction.
Fig. 8 is a schematic diagram illustrating the convergence effect of model residuals based on measured data.
Detailed Description
Example one
The embodiment of the invention provides an improved Laplace model-based initial orbit determination optimization method for only measuring angles of space debris short arcs, which comprises the following steps:
receiving short arc observation data of space debris acquired by an observation platform;
secondly, constructing a measurement equation for representing a geometric relationship and a differential equation for representing a dynamic relationship based on short arc observation data;
step three, combining a measurement equation and a differential equation, and establishing an improved Laplace model containing observed quantity and space debris state by combining an observation condition of only measuring angles;
step four, constructing a solution model to be optimized by the initial value, the orbit dynamics model, the short arc observation data and the improved Laplace model; and (3) taking the track dynamics model, the short arc observation data and the improved Laplace model in the solution model to be optimized as a frame, optimizing the improved Laplace model in the solution model frame to be optimized by utilizing an optimization algorithm, outputting a orbit determination result, and completing the determination of the initial track of the space debris short arc only for angle measurement.
In the first step, the short arc observation data includes: angle measurement data sequence of the space debris and position coordinate sequence of the observation platform.
Space debris T, observation platform S and geocentric O E The observation geometry in between is shown in figure 1. In the second step, a measurement equation representing the geometric relationship is constructed based on the short arc observation data and expressed as follows:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0; p is a distance scalar of the observation platform to the space debris,
for the direction vector of the observation platform to the space debris, R s The position vector of the observation platform under the geocentric J2000.0 inertial coordinate system is shown. R is s ＝[X s ,Y s ,Z s ] T 。
Further, the direction vector from the observation platform to the space debris
Expressed as:
wherein alpha is space debris on observation platformThe angle measurement data right ascension under the mass center J2000.0 inertial coordinate system; delta is declination of angle measurement data of the space debris under an inertial coordinate system of the centroid J2000.0 of the observation platform; [ l, m, n ]] T And l, m and n are respectively intermediate variables related to the angle measurement data of the space debris for observing the direction vector from the platform to the space debris.
Because the distance rho between the observation platform and the target is unknown under the orbit determination condition of only measuring the angle, in order to eliminate the term, cross multiplication vectors are arranged at two sides of a measurement equation
Furthermore, only the observation condition of the angle is measured, and a measurement equation representing the geometric relationship is constructed based on the short arc observation data and expressed as follows:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0;
is a direction vector from the observation platform to the space debris; r s The position vector of the observation platform under the geocentric J2000.0 inertial coordinate system is shown.
In the second step, a differential equation representing a dynamic relationship is constructed based on short arc observation data and expressed as follows:
wherein ,
is a space debris gravitational acceleration vector; mu is a gravitational constant, r is a position vector of the space debris under an inertial coordinate system of the geocentric J2000.0; f. of ε (r, v, t) is perturbation acceleration.
In step three, assume that the position vector r of the space fragment is defined by r 0 and v0 In a plane spanned by r 0 、v 0 Respectively representing the space debris at an initial time t 0 A position vector and a velocity vector. In fact, the orbital plane of the spacecraft is also subject to variations due to earth's non-spherical perturbations and tripartite gravitational forces. Introducing orbit normal vector constraint, combining a measurement equation and a differential equation to obtain a combined model:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0; f. of r Is a position intermediate variable; r is 0 For space debris at an initial time t 0 A position vector of (a); f. of v Is a speed intermediate variable; v. of 0 For space debris at an initial time t 0 The velocity vector of (2); f. of n Is a normal intermediate variable of the orbital plane; v is the velocity vector of the space debris in the inertial coordinate system of the geocenter J2000.0.
Preferably, in the third step, the measurement equation and the differential equation are combined, and the principle of the obtained combined model is as follows:
an implicit function containing observed quantity and space debris state can be obtained by combining a measurement equation and a differential equation, and the specific form of the implicit function is as follows:
Φ(t,X,Y)＝0 (5)
wherein, X = [ t = i ,α i ,δ i ,X si ,Y si ,Z si ]For short arc observation data within a finite arc length, i = 1.., N, Y = [ r, v =] T Is in a space debris state. In practice, the implicit function cannot be absolutely equal to 0, since both the short arc observation data and the orbit dynamics model have errors. Whether the initial orbit determination or the precise orbit determination is carried out, the essence is to find a target orbit at an initial time t 0 Time-optimal space debris state Y 0 ＝[r 0 ,v 0 ] T So that it becomes a solution of the above implicit functions, i.e.
In the conventional Laplace method, the differential equation does not contain perturbation term f ε (r, v, t) but instead by a two-body model, this obviously does not meet the accuracy requirements. In the background technology, a generalized Laplace method (collectively called the generalized Laplace method) is proposed in Liulin, the shooting condition of space debris is considered, and the right term of a differential equation is divided into time intervals delta t = t-t 0 Has a power series form expansion of:
wherein ,
the k-th derivative of r at t 0 The value of (c). Since the derivatives of r of more than the second order can be represented by r 0 、v 0 Then the above formula can be further expressed as:
r＝F(r 0 ,v 0 ,Δt)·r 0 +G(r 0 ,v 0 ,Δt)·v 0 (8)
wherein ,F(r0 ,v 0 ,Δt)、G(r 0 ,v 0 Δ t) are each independently of r 0 、v 0 And Δ t. Thus, the state of the space debris at any position can be converted into the state of r 0 、v 0 By using an analytical expression of (2), can be expressed by (0) ＝1、G (0) And = Δ t is solved iteratively as an initial value.
The method breaks through the limit of a two-body model in the traditional Laplace method, and converts the nonlinear problem into a simple linear equation set iterative solution process in a power series expansion mode. However, this method has the following problems:
(1) The form of power series expansion makes the truncation error unavoidable, and causes certain loss to the orbit determination precision;
(2) The device isThe method requires that perturbation terms exist in an analytic form, and the complexity of F and G expressions is also dependent on F ε (r, v, t). The dynamic model cannot be too complex, and the requirement of representation in an analytic form is a bottleneck for further improving the orbit determination precision of the method. In engineering, a high-precision dynamic model generally adopts a numerical integration method, and a target state at any moment is directly obtained from the state of a target in an initial epoch.
Therefore, without changing the angle observation formula (8), it can be regarded that r is limited to r 0 and v0 In the plane of the flare, but in fact, the orbital plane of the spacecraft is also subject to variations due to earth non-spherical perturbations and three-body forces. By introducing the constraint of the normal direction of the orbital plane, the formula (8) can be further expanded into:
in the third step, the improved Laplace model is composed of a first component model, a second component model, a third component model, a first improved Laplace initial orbit determination model and a second improved Laplace initial orbit determination model, wherein:
first component model:
second component model:
the third component model:
wherein ,fr Is a position intermediate variable; f. of v Is a velocity intermediate variable; f. of n Is a normal intermediate variable of the orbital plane;
being a position intermediate variableA derivative;
is the derivative of the intermediate variable of speed;
is the derivative of the normal intermediate variable of the orbital plane; r is 0 For space debris at an initial time t 0 A position vector of (a); v. of 0 For space debris at an initial time t 0 The velocity vector of (2); r is a position vector of the space debris under an inertia coordinate system of the geocentric J2000.0; v is a velocity vector of the space debris under an inertia coordinate system of the geocenter J2000.0;
the first improved Laplace initial orbit determination model: substituting the combined model into a measurement equation under the condition of only measuring angles, and obtaining a first improved Laplace initial orbit determination model as follows:
wherein ,Rs_j Observing the position vector of the platform under the geocentric J2000.0 inertial coordinate system at the moment J;
observing the direction vector from the platform to the space debris for the moment j; r is 0 For space debris at an initial time t 0 Position vector of r 0 ＝[x 0 ,y 0 ,z 0 ] T ；v 0 For space debris at an initial time t 0 Velocity vector of v 0 ＝[v x0 ,v y0 ,v z0 ] T ；
The second improved Laplace initial orbit determination model comprises the following steps: the direction vector from the first improved Laplace initial orbit determination model to the space debris on the observation platform
And (3) expanding on the basis of the formula to obtain a second improved Laplace initial orbit determination model as follows:
wherein ,x0 、y 0 、z 0 Respectively space debris at an initial time t 0 Each component of the position vector of (a); v. of x0 、v y0 、v z0 Respectively space debris at an initial time t 0 Each component of the velocity vector of (a); x s 、Y s 、Z s Respectively representing each component of a position vector of the observation platform under an inertial coordinate system of the geocenter J2000.0; l, m and n are respectively intermediate variables related to space debris angle measurement data (right ascension and declination).
In the fourth step, the step of constructing the solution model to be optimized by the initial value, the orbit dynamics model, the short arc observation data and the improved Laplace model specifically comprises the following steps:
(1) Randomly giving an initial value of short arc observation data of a group of space debris, wherein the initial value is the space debris at an initial time t 0 Position vector r of 0 And velocity vector v 0 ；
(2) Position vector r by orbit dynamics model 0 And velocity vector v 0 Extrapolate to each measurement instant t i I =1,.., N; n is the number of observation points in the observation arc section, and N is more than or equal to 1; and obtain the measurement time t i Lower position vector state quantity r i Sum velocity vector state quantity v i ；
(3) Based on the measuring time t i Lower position vector state quantity r i Sum velocity vector state quantity v i F is calculated by the first component model, the second component model and the third component model respectively r 、f v 、f n (ii) a Based on f r 、f v 、f n And writing the short arc observation data of the space debris acquired by the observation platform in the step one by the first improved Laplace initial orbit determination model into a nonlinear over-determined equation set of a second improved Laplace initial orbit determination model, namely the model to be optimized.
Further, the step of optimizing the improved Laplace model in the solution model to be optimized by using an optimization algorithm and outputting a orbit determination result by taking the orbit dynamics model, the short arc observation data and the improved Laplace model in the solution model to be optimized as a frame to complete the determination of the initial orbit of the space debris short arc only angle measurement comprises the following steps:
and (3) integrally considering the step (2) and the step (3) of the solution model to be optimized as an objective function to be optimized, after a group of initial values are given, solving and optimizing the multivariable nonlinear over-definite equation of the second improved Laplace initial orbit determination model listed in the step (3) by using an optimization algorithm through the step (2), outputting an orbit determination result, and completing the determination of the space debris short arc only angle measurement initial orbit.
Wherein the optimization algorithm includes, but is not limited to: non-linear least squares, confidence domain algorithms, or particle swarm algorithms.
The orbit dynamics model is a gravitational model to which the satellite flies in space, including but not limited to the earth's gravitational force, the lunar gravitational force, or the solar gravitational force; the earth gravitational field model is preferably a JGM-3 earth gravitational field model of 8 × 8 orders, and in addition, any one of 1 × 1 order, 21 × 21 order or other earth gravitational field models can be selected according to the precision requirement on the result.
The principle of the fourth step is as follows: wherein, the second improved Laplace initial orbit determination model in the improved Laplace model is the concrete form of the implicit function phi (t, X, Y) in the improved Laplace model, and the essence of orbit determination is to solve the Y which minimizes phi (t, X, Y) 0 ＝[r 0 ,v 0 ] T And if N groups of observation data points are arranged in the observation arc section, the number of equations is 3N, but only 2N are mutually independent, and the number of variables to be solved is 6, so that at least 3 groups of observation data are needed to solve, and more observed quantities are beneficial to improving the orbit determination precision.
The method has the greatest advantage that an analytic expression of a perturbation equation does not need to be written, so that the existing more accurate orbit dynamics model with any order can be used as a single orbit dynamics module to be embedded into an iterative loop. In addition, although the method requires the initial time t to be given 0 A priori information r of 0 、v 0 But accuracy of initial valueThe method has low sexual requirements, can be converged only by a few iterations, generally takes the position and the speed of any spacecraft near the target orbit height as initial values, and refers to the simulation verification part in this chapter on specific convergence characteristics.
First a set of t is given 0 Initial value of time
Extrapolation of the initial values to each measurement instant t by means of a model of the dynamics of the orbit i I = 1.. N and obtains the position vector state quantity and the velocity vector state quantity at the moment
F is calculated by the first component model, the second component model and the third component model respectively r 、f v 、f n Substituting short arc observation data of the space debris acquired by the observation platform in the step one, writing a nonlinear over-determined equation set of a second improved Laplace initial orbit model through the first improved Laplace initial orbit model, and solving the nonlinear over-determined equation set, namely the solution model to be optimized.
The improved Laplace model can be solved through iterative computation, the iterative process is shown in figure 2, and the solved state quantities
Bring back into the iterative initial value again, repeatedly calculate in order, Q =0,1 max Is the number of iterations. Up to
And the calculation is finished when the error is smaller than the error epsilon or the set iteration number is exceeded. The orbit dynamics model exists as an independent module, and can be solved by utilizing the existing high-order numerical method without listing analytical expressions like the method for popularizing Laplace.
The iterative solution method has high calculation speed and only needs a few timesThe iterations of (a) may converge. However, in experiments, it is found that when the iteration method is used for processing only angle measurement actual measurement data of a short arc of a GEO target, convergence is difficult or local optimal solution is easy to fall into, and the reason is that when a core step of an iteration process, namely a nonlinear over-definite equation, is solved, r which is not solved every time under the condition of complex noise is solved 0 、v 0 The target state is converged towards the global optimum direction, and instead, the target state is trapped into a local optimum solution, so that the target state obtained by the solution has large deviation.
Aiming at the problem, the optimization algorithm is utilized to optimize the improved Laplace model in a frame of a solution model to be optimized, the whole process from the orbit dynamics model to the second improved Laplace initial orbit model is regarded as an objective function to be optimized, the optimization (search) algorithm can be directly applied to optimize, and the optimization objective is to minimize the second improved Laplace initial orbit model. Therefore, the initial orbit determination problem is converted into a multivariable nonlinear over-determined equation optimization problem of a second improved Laplace initial orbit determination model. The calculation flow is shown in fig. 3: at a given set of t 0 Initial value of time
Then, the target function is solved by using an optimization method by directly taking the initial value as an initial value, an iteration step is skipped, and the target state meeting the improved Laplace model constraint can be directly solved, so that the initial orbit determination is completed, and an orbit determination result r is output 0 、v 0 . The solution method not only retains the advantages of the improved Laplace model and can solve without good initial values, but also can solve the problem that the iterative solution process is easy to fall into a local solution under the complex noise condition because the orbit dynamics model still exists in a single module and an analytical expression does not need to be listed. The available mature optimization algorithms for solving such problems are many, such as nonlinear least squares, confidence domain algorithms, particle swarm optimization, and the like, which are not described in detail herein. In the method, the particle swarm algorithm has the advantages of better overall convergence and fewer parameters to be adjusted, and in the simulation verification, the particle swarm algorithm is taken as an example to perform calculation analysis on an improved Laplace model.
It should be noted that the optimization solution method has a disadvantage that, since the optimization method needs to frequently call the dynamical model each time the value of the objective function is calculated, when a high-order orbit dynamical model is adopted, the calculation time of the method is slightly longer than that of the iterative method.
2. Method verification
1. Simulation data verification
Setting a simulation scene as UTC time 2022, 3 months and 27 days, simulating an observation scene of a low-orbit observation platform on GEO space debris, wherein the observation platform runs on a sun synchronous orbit with the height of 700km, the long half shaft and the half shaft of a GEO target orbit are 42164km, and the orbit inclination angle is 1 degree. Firstly, calculating orbit determination results of two Laplace methods under the condition that short arc observation data of space debris acquired by an observation platform are noiseless, randomly extracting short arc observation data with the time of 80-120 seconds in an observable period, wherein the short arc observation data comprise the right ascension and the declination of the space debris and the position vector of the observation platform, and taking the initial moment of an observation arc segment as t 0 The time of day, the position vector r of the space debris at that time of day is calculated 0 And velocity vector v 0 。
Experiments show that under the condition of no noise, the time lengths of different arc sections can be converged to an ideal observation result, and under the condition, the arc length has little influence on the orbit determination result, so that the effectiveness of two method models (a Laplace method is popularized and a Laplace model is improved) is demonstrated. The convergence of the method is also faster, as shown in fig. 4-5, and can converge to near the true value after only 2-3 iterations.
And then, randomly generating 200 observation arc sections of the observation platforms for the space debris at different positions on the geosynchronous orbit zone in a simulation period by adopting a Monte Carlo simulation mode, so that the observation geometry between the space debris and the observation platforms has randomness, the observation arc length is different from 80 seconds to 120 seconds, and the sampling interval is different from 1 second to 5 seconds. According to the capability of the existing observation equipment, random noise is introduced into angle measurement data and an observation platform position vector, and variance of (2') 2 1 sigma and (100 m) 2 1 σ zero mean white gaussian noise. The orbital dynamics model adopted by the invention is a JGM-3 earth with 8 multiplied by 8 ordersAnd (3) setting a target Cr =1 and a surface-to-mass ratio of 0.015m2/kg by taking the gravity of the day and the month and the perturbation of the sunlight pressure into consideration in a gravitational field model. Table 1 shows the statistical results of the tracking errors of the three methods:
TABLE 1
Orbit determination result position vector r 0 Velocity vector v 0 The errors from the true target in each vector direction are shown in fig. 6a, 6b, 6c, 6d, 6e, 6f, 7a, 7b, 7c, 7d, 7e, and 7f, and the statistical results are shown in table 1. It can be noted that, in general, the tracking accuracy of the improved Laplace model is higher than that of the generalized Laplace method, and is more obvious in the position vector, for example, the standard deviation of the error of the generalized Laplace method in the X, Y and Z directions is 4.2260km, 5.1112km and 0.6001km respectively, while the corresponding results of the iterative method of the improved Laplace model are 3.2675km, 4.2499km and 0.5306km respectively. For the two solutions for improving the Laplace model, the accuracy of the optimization method is slightly higher than that of the iteration method, and taking standard deviations of errors in the X direction, the Y direction and the Z direction as examples, the results of the optimization method are respectively 2.9712km, 3.8663km and 0.4581km, but the accuracy of the three methods is basically in an order of magnitude level.
2. Verification of measured data
Because the actual measurement data of the space-based platform is difficult to obtain, in order to verify the orbit determination effect of the algorithm on the actual measurement data, the convergence characteristics and the orbit determination precision of the iterative solution method and the optimization solution method of the improved Laplace model are tested by using only angle measurement observation data of a certain domestic astronomical platform for the GEO spacecraft. Due to the limited capability of the observation equipment, the wide difference of the angle measurement data of the target is large due to the long-exposure observation mode, and some abnormal jump data exist in the observation arc section, so that the data needs to be subjected to jump point elimination and adjustment pretreatment before orbit determination. Firstly, the jumping points are removed, and in the observation data of the short arc, the time sequence truth values of the right ascension and the declination can be regarded as a monotonous function, so that the jumping points can be removed by utilizing first-order difference. The adjustment process is then carried out to make the observed data as smooth as possible without distortion.
In the optimization solving method, a particle swarm method is adopted to solve the objective function to be optimized, which is constructed by the orbit dynamics model and the improved Laplace model. In the particle swarm optimization, each particle is a potential optimal solution, the motion of the particle in space is determined by the velocity of the particle, and the updating process of the velocity of the particle is the searching process of the optimal solution. The velocity of the particles is updated herein using the following equation: du K L, swamy M N S.search and Optimization by Metaerrors Techniques and Algorithms Rapid by Nature [ M].Basel:
2016:153-173.：
wherein ,
the (k + 1) th particle velocity, ω is a weight coefficient,
is the particle velocity at the kth time, r 1 、r 2 Is [0,1 ]]Random number in between, c 1 、c 2 As a factor of the acceleration, the acceleration is,
and
the k-th individual extremum and the group extremum,
the kth particle position. Inventive arrangement c 1 ＝c 2 =1.42, dynamic adjustment range of ω is [0.1,1.1]The particle size was 50.
The requirement of the invention on the initial value is very weak, the experiment takes a random GEO target as the initial value, and the calculation result is shown in FIG. 8. In the figure, the residual error of a vertical axis represents the mode of each equation value in the second improved Laplace initial orbit determination model, and the essence of the invention is to search a group of optimal target initial states [ r [ ] 0 ,v 0 ]Such that the residual error is minimized. It can be seen that, starting from the initial value of the same residual error, the optimal solution method quickly finds the optimal solution in a short time, and stabilizes the residual error at about 5.62, while the iterative solution method stabilizes the residual error at about a local solution of 2854.56 after several iterations. It should be noted that the iterative method does not converge to the optimal solution for each calculation of the measured data, and it is found in experiments that the stability of the method is related to noise distribution and the size of the observed data, and it is necessary to specifically analyze the method in combination with the actual observed data. Because the grasped actual measurement data is limited, the stability of the proposed optimization solving method can not be verified through a large amount of calculation, but from the orbit determination result of the grasped 66 groups of actual measurement data, the optimization solving method can stably output the orbit determination result of 55 groups of data, accounting for 83.3 percent, and the corresponding result of the iterative solving method is 28 groups, accounting for 42.4 percent.
In conclusion, the embodiment of the invention develops deep research on an improved Laplace model with extremely short arcs only under the background of angle measurement and a solving method thereof. Firstly, aiming at the problem that short arc observation data under an actual measurement environment are not easy to converge, an orbit determination optimization solving method of an improved Laplace model applicable to a search algorithm is provided, and further simulation verification is carried out on the iterative convergence characteristic of a space debris short arc only angle measurement initial orbit determination optimization method based on the improved Laplace model, different from the popularization of the Laplace method, under the noise-free condition, the method can converge to an ideal result only through 2-3 steps of iteration. In addition, monte Carlo simulation data and actual measurement data are respectively utilized to verify that the method is under the influence of random noise of measurementThe orbit determination precision of (2). The result shows that the orbit determination precision of the invention is obviously better than that of the popularization Laplace method, for example, when the random noise levels of angle measurement data and platform positions are respectively (2') 2 1 sigma and (100 m) 2 And when the zero mean Gaussian white noise of 1 sigma exists, the error standard deviations of the Laplace model in the X direction, the Y direction and the Z direction are respectively 4.2260km, 5.1112km and 0.6001km, and the corresponding results of the iteration method for improving the Laplace model are respectively 3.2675km, 4.2499km and 0.5306km. In terms of improving the Laplace model, the optimization solving method provided by the invention is slightly better than an iteration method in orbit determination precision, but the calculation time is slightly longer than that of the iteration method. Meanwhile, for actually measured data with more complex noise, the optimization solution method shows more stable convergence characteristics than the iteration method, and the problem that the iteration solution method is easy to fall into a local optimal solution is effectively solved.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily think of the changes or substitutions within the technical scope of the present invention, and shall cover the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the appended claims.
Claims (10)
1. A space debris short arc only angle measurement initial orbit optimization method based on an improved Laplace model is characterized by comprising the following steps:
receiving short arc observation data of space debris acquired by an observation platform;
secondly, constructing a measurement equation for representing a geometric relationship and a differential equation for representing a dynamic relationship based on short arc observation data;
step three, combining a measurement equation and a differential equation, and establishing an improved Laplace model containing observed quantity and space debris state by combining an observation condition of only measuring angles;
step four, constructing a solution model to be optimized by the initial value, the orbit dynamics model, the short arc observation data and the improved Laplace model; and (3) taking the track dynamics model, the short arc observation data and the improved Laplace model in the solution model to be optimized as a frame, optimizing the improved Laplace model in the solution model frame to be optimized by utilizing an optimization algorithm, outputting a orbit determination result, and completing the determination of the initial track of the space debris short arc only for angle measurement.
2. The method of claim 1, wherein in step one, the short arc observation data comprises: angle measurement data sequence of the space debris and position coordinate sequence of the observation platform.
3. The method of claim 1, wherein in the second step, the measurement equation characterizing the geometric relationship is constructed based on the short arc observation data and is expressed as:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0; p is a distance scalar of the observation platform to the space debris,
for observing the direction vector of the platform to the space debris, R s The position vector of the observation platform under the geocentric J2000.0 inertial coordinate system is shown.
4. The method of claim 3, wherein the direction vector of the platform to the space debris is observed
Expressed as:
wherein alpha is the declination of the angle measurement data of the space debris under the inertial coordinate system of the centroid J2000.0 of the observation platform; delta is the space debris in the observation planeDeclination of angle measurement data under an inertial coordinate system of a platform center of mass J2000.0; [ l, m, n ]] T And l, m and n are respectively intermediate variables related to the angle measurement data of the space debris for the direction vector from the observation platform to the space debris.
5. The method of claim 3, wherein the angle-only observation condition, the construction of the measurement equation characterizing the geometric relationship based on the short arc observation data, is represented as:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0;
is a direction vector from the observation platform to the space debris; r is s The position vector of the observation platform under the geocentric J2000.0 inertial coordinate system is shown.
6. The method according to claim 1, wherein in the second step, a differential equation representing the dynamic relationship is constructed based on the short arc observation data as:
wherein ,
is a space debris gravitational acceleration vector; mu is a gravitational constant, r is a position vector of the space debris under an inertial coordinate system of the geocentric J2000.0; f. of ε (r, v, t) is perturbation acceleration.
7. The method according to claim 1, 3 or 6, wherein in step three, the measurement equation and the differential equation are combined to obtain a combined model:
wherein r is a position vector of the space debris in an inertial coordinate system of the geocenter J2000.0; f. of r Is a position intermediate variable; r is a radical of hydrogen 0 For space debris at an initial time t 0 A position vector of (a); f. of v Is a speed intermediate variable; v. of 0 For space debris at an initial time t 0 The velocity vector of (2); f. of n Is a normal intermediate variable of the track surface; v is the velocity vector of the space debris in the inertial coordinate system of the geocenter J2000.0.
8. The method of claim 1, wherein in step three, the improved Laplace model consists of a first component model, a second component model, a third component model, a first improved Laplace initial orbit model, and a second improved Laplace initial orbit model, wherein:
first component model:
a second component model:
the third component model:
wherein ,fr Is a position intermediate variable; f. of v Is a velocity intermediate variable; f. of n Is a normal intermediate variable of the track surface;
is the derivative of the intermediate variable of position;
is the derivative of the intermediate variable of speed;
is the derivative of the normal intermediate variable of the orbital plane; r is a radical of hydrogen 0 For space debris at an initial time t 0 A position vector of (a); v. of 0 For space debris at an initial time t 0 The velocity vector of (a); r is a position vector of the space debris under an inertia coordinate system of the geocentric J2000.0; v is a velocity vector of the space debris under an inertia coordinate system of the geocenter J2000.0;
the first improved Laplace initial orbit determination model comprises the following steps: substituting the combined model into a measurement equation under the condition of only measuring angles, and obtaining a first improved Laplace initial orbit determination model as follows:
wherein ,Rs_j Observing the position vector of the platform under the geocentric J2000.0 inertial coordinate system at the moment J;
observing the direction vector from the platform to the space debris for the moment j; r is a radical of hydrogen 0 For space debris at an initial time t 0 Position vector of (a), r 0 ＝[x 0 ,y 0 ,z 0 ] T ；v 0 For space debris at an initial time t 0 Velocity vector of v 0 ＝[v x0 ,v y0 ,v z0 ] T ；
The second improved Laplace initial orbit determination model comprises the following steps: the direction vector from the first improved Laplace initial orbit determination model to the space debris on the observation platform
And expanding on the basis of a formula to obtain a second improved Laplace initial orbit determination model as follows:
wherein ,x0 、y 0 、z 0 Respectively, space debris at an initial time t 0 Each component of the position vector of (a); v. of x0 、v y0 、v z0 Respectively space debris at an initial time t 0 Each component of the velocity vector of (a); x s 、Y s 、Z s Respectively representing each component of a position vector of the observation platform under an inertia coordinate system of the geocentric J2000.0; l, m and n are respectively intermediate variables related to the right ascension and declination of the space debris angle measurement data.
9. The method of claim 1, wherein in the fourth step, the step of constructing the solution model to be optimized from the initial values, the orbit dynamics model, the short arc observation data and the improved Laplace model specifically includes:
(1) Randomly giving an initial value of short arc observation data of a group of space debris, wherein the initial value is the initial time t of the space debris 0 Position vector r of 0 And velocity vector v 0 ；
(2) Position vector r by orbit dynamics model 0 And velocity vector v 0 Extrapolating to each measurement time t i I =1,.., N; n is the number of observation points in the observation arc section, and N is more than or equal to 1; and obtain the measurement time t i Lower position vector state quantity r i Sum velocity vector state quantity v i ；
(3) Based on the measuring time t i Lower position vector state quantity r i Sum velocity vector state quantity v i F is calculated by the first component model, the second component model and the third component model respectively r 、f v 、f n (ii) a Based on f r 、f v 、f n And writing the short arc observation data of the space debris acquired by the observation platform in the step one by the first improved Laplace initial orbit determination model into a nonlinear over-determined equation set of a second improved Laplace initial orbit determination model, namely the model to be optimized.
10. The method of claim 9, wherein the step of optimizing the improved Laplace model in the solution model to be optimized by using an optimization algorithm and outputting a orbit determination result by using the orbit dynamics model, the short arc observation data and the improved Laplace model as frames in the solution model to be optimized and the optimization algorithm to complete the determination of the initial orbit of the space debris short arc only by angle measurement comprises the steps of:
and (3) integrally regarding the step (2) and the step (3) of the solution model to be optimized as an objective function to be optimized, after a group of initial values are given, solving and optimizing a multivariable nonlinear over-definite equation of the second improved Laplace initial orbit model listed in the step (3) by using an optimization algorithm through the step (2), outputting an orbit determination result, and completing the determination of the space debris short arc only angle measurement initial orbit.
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