CN115326077B - Short arc optical measurement initial track determination method suitable for small eccentricity track - Google Patents
Short arc optical measurement initial track determination method suitable for small eccentricity track Download PDFInfo
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Abstract
The disclosed embodiment relates to a method for determining an initial track of a short arc optical measurement suitable for a track with small eccentricity. The method comprises the following steps: obtaining a semi-long axis and a true paraxial point angle by a single-point optical measurement method or a short arc multi-point measurement method; setting the eccentricity, the inclination angle, the ascension at the ascending intersection point and the argument of the perigee to be 0; and determining an initial orbit of an initial epoch moment according to the obtained semi-major axis and the true paraxial point angle. The embodiment of the disclosure realizes that the analytic equation determined by the geostationary satellite initial orbit at a single epoch moment is deduced by using the ground optical measurement data.
Description
Technical Field
The embodiment of the disclosure relates to the technical field of orbit calculation of artificial earth satellites, in particular to a method for determining initial orbit of short arc optical measurement suitable for an orbit with small eccentricity.
Background
At present, a plurality of mature and stable satellite initial orbit determination methods exist. Under the ground short arc optical measurement condition, due to the influence of measurement errors, the calculation accuracy of the common laplacian (Laplace) type and gaussian (Gauss) type initial orbit determination methods is poor, even if a precise orbit improvement method is utilized, the orbit calculation accuracy is still not high, and even an error orbit may occur. Initial orbit determination under the condition of short arc optical measurement is still a technical problem in the field of orbit calculation of artificial earth satellites.
In the related art, the traditional initial orbit determination method is difficult to calculate the initial orbit parameters of geostationary satellites by using short arc optical measurement.
Accordingly, there is a need to ameliorate one or more of the problems with the related art solutions described above.
It is to be noted that the information disclosed in the above background section is only for enhancement of understanding of the background of the present disclosure, and thus may include information that does not constitute prior art known to those of ordinary skill in the art.
Disclosure of Invention
The purpose of the embodiments of the present disclosure is to provide a method for determining an initial orbit of a short arc optical measurement suitable for a small eccentricity orbit, so as to at least solve the problem of calculating an initial orbit parameter of a geostationary satellite by using the short arc optical measurement.
The purpose of the invention is realized by adopting the following technical scheme:
the invention provides a method for determining an initial track of short arc optical measurement suitable for a track with small eccentricity, which comprises the following steps:
obtaining a semi-long axis and a true paraxial point angle by a single-point optical measurement method or a short arc multipoint measurement method;
setting the eccentricity, the inclination angle, the ascent intersection right ascension and the perigee argument to be 0;
and determining an initial orbit of an initial epoch moment according to the obtained semi-major axis and the obtained true near point angle.
Optionally, the step of obtaining the semi-major axis and the true paraxial point angle by a single-point optical measurement method or a short-arc multipoint measurement method further includes:
and constructing an observation equation of the semi-long axis and the true near point angle at the sequence epoch moment through auxiliary parameters in the earth-centered inertial coordinate system at the sequence epoch moment, and solving the semi-long axis and the true near point angle.
Optionally, the step of constructing an observation equation of the semi-major axis and the true proximal angle at the sequence epoch time through auxiliary parameters in the geocentric inertial coordinate system, and obtaining the semi-major axis and the true proximal angle further includes:
the expression of the auxiliary parameter in the earth-centered inertial coordinate system at the moment of the sequence epoch is as follows:
wherein, P j And Q j The auxiliary parameters are auxiliary parameters in the geocentric inertial coordinate system of the sequence epoch moment at the j moment; by
Formed matrix
Indicating the time of the sequence epoch at time jBy X, the line-of-sight vector j Y j Z j The composed matrix [ X ] j Y j Z j ] T Representing optic vignetting point coordinates at time j of the sequence epoch time;
the observation equation of the semi-major axis and the true paraxial point angle at the time of the sequence epoch is as follows:
wherein a is a semi-major axis;f j is a true proximal angle.
Optionally, the step of obtaining the semi-major axis and the true paraxial point angle by a single-point optical measurement method or a short-arc multipoint measurement method further includes:
and constructing an observation equation of a parameter vector through a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true paraxial point angle according to the parameter vector, and obtaining the semimajor axis and the true paraxial point angle which meet convergence conditions through batch processing.
Optionally, the method includes the steps of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle which satisfy a convergence condition by batch processing, and further includes:
setting an initial semi-long shaft, and calculating an initial orbital motion angular rate through the set initial semi-long shaft; and (3) constructing an observation equation of the parameter vector through the auxiliary parameters of the sequence epoch time:
wherein x is
Representing a vector of parameters; a is halfA long axis; f j 、G j 、P j And Q j The auxiliary parameter is the auxiliary parameter of the sequence epoch moment at the j moment; v. of j The velocity at time j of the sequence epoch; μ is the earth's gravitational constant.
Optionally, the method further includes the steps of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true apoint angle according to the parameter vector, and obtaining the semimajor axis and the true apoint angle satisfying a convergence condition by batch processing, and further including:
constructing a uniform beam least square estimation equation of the parameter vector:
constructing an unconstrained lagrange function:
wherein x is
Representing a vector of parameters;κandsthe lagrange multiplier is represented by a number of lagrange multipliers,fto representP j AndQ j a combinatorial matrix of (a);yto representF j AndG j the combination matrix of (1).
Optionally, the method includes the steps of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle which satisfy a convergence condition by batch processing, and further includes:
three conditional equations are established and respectively expressed as follows:
according to the 1 st KKT condition equation
And solving the state vector and the Lagrange multiplier by combining a Newton iteration method.
Optionally, the method includes the steps of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle which satisfy a convergence condition by batch processing, and further includes:
setting an initial Lagrange multiplier
；
Calculating the correction quantity of the kth parameter vector and the correction quantity of the Lagrange multiplier according to the kth parameter vector and the Lagrange multiplier;
wherein the content of the first and second substances,
is the parameter vector in the k-th cycle;
and
is the lagrange multiplier in the kth cycle;
correction amount of the parameter vector in the k-th cycle;
and
correction for lagrange multiplier in the kth cycle;
calculating a parameter vector and a Lagrange multiplier in the cycle of the (k + 1) th time:
wherein
Is the parameter vector in the cycle of the (k + 1) th time;
and
lagrange multiplier in the cycle of the (k + 1) th time;
calculating correction amount of parameter vector in k-th cycle
And correction of lagrange multiplier
And
and the vector of parameters in the (k + 1) th cycle
And lagrange multiplier
And
up to lagrange multiplier correction
And
the convergence condition is satisfied.
Optionally, the method includes the steps of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle which satisfy a convergence condition by batch processing, and further includes:
according to the calculated parameter vector satisfying the convergence condition
Calculating the true approximate point angle of the semi-major axis and the initial epoch time:
wherein a is a semi-major axis;f 0 the true anomaly is the true anomaly at 0 at the initial epoch time;
circularly calculating parameter vector satisfying convergence condition
And semi-major axis a and true proximal angle f 0 And obtaining the semimajor axis and the true proximal angle which meet the convergence condition until the semimajor axis a meets the convergence condition.
Optionally, the step of determining an initial trajectory of an initial epoch time according to the obtained semi-major axis and the obtained true paraxial point angle further includes:
when the semi-long axis and the true near point angle are obtained by a single-point optical measurement method, the initial orbit determination of the initial epoch time by using single-point optical measurement is completed;
and when the semi-long axis and the true paraxial point angle are obtained by the short arc multipoint optical measurement method, the initial orbit determination of the initial epoch time by using the short arc multipoint optical measurement is completed.
The technical scheme provided by the embodiment of the disclosure can have the following beneficial effects:
in the embodiment of the disclosure, by the method, the analytic equation determined by the initial orbit of the geostationary satellite at a single epoch moment is deduced by using the ground optical measurement data.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosure.
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The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the present disclosure and together with the description, serve to explain the principles of the disclosure. It is to be understood that the drawings in the following description are merely exemplary of the disclosure, and that other drawings may be derived from those drawings by one of ordinary skill in the art without the exercise of inventive faculty.
Fig. 1 shows a flow diagram of an initial track determination method in an exemplary embodiment of the disclosure;
FIG. 2 shows a flow diagram of an initial track determination method by single point optical measurement in an exemplary embodiment of the present disclosure;
fig. 3 shows a flowchart of an initial orbit determination method by short arc multipoint measurement in an exemplary embodiment of the present disclosure.
Detailed Description
Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments may, however, be embodied in many different forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of example embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.
Furthermore, the drawings are merely schematic illustrations of the present disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and thus their repetitive description will be omitted. Some of the block diagrams shown in the figures are functional entities and do not necessarily correspond to physically or logically separate entities. These functional entities may be implemented in the form of software, or in one or more hardware modules or integrated circuits, or in different networks and/or processor devices and/or microcontroller devices.
First, in the present exemplary embodiment, a method for determining an initial track of a short arc optical measurement suitable for a track with small eccentricity is provided, and referring to fig. 1, the method may include the following steps:
step S101: obtaining a semi-long axis and a true paraxial point angle by a single-point optical measurement method or a short arc multi-point measurement method;
step S102: setting the eccentricity, the inclination angle, the ascension at the ascending intersection point and the argument of the perigee to be 0;
step S103: and determining an initial orbit of the initial epoch moment according to the obtained semi-major axis and the true paraxial point angle.
It is to be understood that, in the geocentric inertial coordinate system, the small eccentricity and the small inclination of the geostationary satellite are approximated to 0, and under the condition that 2 parameters are approximated to 0, the ascension point and the argument of the perigee are also degraded to 0 in the geocentric inertial coordinate system. Therefore, the conventional initial orbit determination method needs to estimate 6 independent parameters, and degrades into only 2 independent parameters of the semi-major axis and the true paraxial point angle. In addition, the small eccentricity track refers to a track having an eccentricity of approximately 0.
It should also be understood that the step S101 includes two methods for estimating the semi-major axis and the true proximal angle, respectively. The first method is a sequence epoch time initial orbit determination method by using single-point optical measurement. The single-point optical measurement can provide independent parameters of 2 degrees of freedom, and meets observability conditions for solving the independent parameters of 2 items of semi-major axis and true anomaly angle, thereby realizing initial orbit determination of sequence epoch time. The second method is an initial track determination method by using the initial epoch time of the short arc multipoint optical measurement. And establishing a constraint observation equation of a semi-major axis and a true near point angle at the initial epoch time, solving a least square batch estimation problem of nonlinear constraint by using a sequence quadratic programming algorithm, and overcoming the influence of random noise of single-point optical measurement in initial orbit determination of a single-point optical measurement method on the orbit calculation precision.
It will also be appreciated that the precise calculation of the satellite orbit requires the determination of the initial input data for the calculation, i.e. the initial orbit of the satellite. The tracking measurement of the on-orbit satellite on the ground mainly comprises two ways of radio external measurement and optical measurement. The radio external measurement refers to tracking measurement of a satellite by using radio signals sent by equipment such as a medium-long range phased array radar or a precision tracking radar, and the like, so as to determine parameters such as the orbit and target characteristics of the satellite. The basic principle is that a ground transmitter generates a radio signal, the radio signal is transmitted to a target through an antenna, ground equipment receives a target reflection signal or a transponder retransmission signal, the target reflection signal or the transponder retransmission signal is processed by a receiver, and finally a terminal gives measurement parameters such as target distance, angle, distance change rate and the like. The optical measurement refers to the measurement of flight path parameters of a satellite by using an optical signal so as to obtain laser ranging and infrared angle measurement, including azimuth and elevation. The radio external measurement and optical measurement parameters are processed, and satellite state parameters including time, radial vectors, angles, velocity vectors and the like can be obtained in an inertial coordinate system.
It is also understood that the motion of the satellite in the inertial frame can be represented by the orbital semi-major axis, the orbital first eccentricity, the orbital inclination, the ascension, the perigee angular separation, and the ascension angular separation. Wherein the inclination angle of the orbit and the right ascension angle of the ascending intersection determine the position of the orbit plane in the inertial space. The angular distance of the near point determines the position of the near point of the track in the track plane; the semi-major axis and the first eccentricity define the size and shape of the track; the elevation angle distance determines the position of the satellite at a given time on the orbit. In some cases, the 6 th track element may be a tangent angle or a decentering angle instead of the elevation angle distance.
By the method, an analytic equation for determining the initial orbit of the geostationary satellite at a single epoch moment is deduced by using the optical measurement data of the single point azimuth/pitch or right ascension/declination angle type on the ground. In order to overcome random errors existing in single-point optical measurement, a method for estimating initial epoch time track parameters by batch processing of optical measurement of a plurality of short-arc epoch times is provided, and an initial track parameter iterative calculation method for solving nonlinear constraint least square estimation is deduced.
Next, the respective steps of the above-described method in the present exemplary embodiment will be described in more detail with reference to fig. 1 to 3.
In one embodiment, referring to fig. 2, step S101 may include the following steps:
s104: and (3) constructing an observation equation of the semi-long axis and the true near point angle at the sequence epoch moment through auxiliary parameters in the sequence epoch moment geocentric inertial coordinate system, and solving the semi-long axis and the true near point angle.
It is to be understood that the observation equation of the semi-major axis and the true paraxial point angle at the time of the sequence epoch is constructed by a single-point optical measurement method, the calculation amount is small, and the initial orbit of the satellite can be preliminarily determined. A solution formula for calculating the semi-major axis and true paraxial point angles using single point optical measurements is derived. The method is suitable for realizing initial orbit determination by using single-point optical measurement of sequence epoch time under the constraint that eccentricity, inclination, ascension at ascending intersection point and near field blessing angle are all zero, and the problem that 6 parameters need to be solved in the traditional initial orbit determination method is avoided.
In one embodiment, referring to fig. 2, step S104 may include the following steps:
time t of sequence epoch j Auxiliary parameter P in geocentric inertial coordinate system j And Q j The expression of (a) is:
wherein the content of the first and second substances,
representing the time t of the sequence epoch j Optically measuring the line-of-sight vector, [ X ] j Y j Z j ] T Representing the time t of the sequence epoch j Optic vignetting point coordinates of;
time t of sequence epoch j Semi-major axis a and true proximal angle f j The observation equation of (a) is:
it is to be understood that the sequence epoch time t is calculated from the constructed observation equation j Semi-major axis a and true proximal angle f j ：
Semi-major axis a and true paraxial angle f calculated according to equation (3) j Since the parameters of 4 items of eccentricity, inclination angle, ascension at ascending intersection point and argument of perigee are all 0, the time t of sequence epoch can be solved j 6 parameters of the initial track to complete the sequence epoch time t by single-point optical measurement j The initial orbit of (2) is determined.
In one embodiment, referring to fig. 3, step S101 may include the steps of:
step S105: and constructing an observation equation of the parameter vector through a sequential quadratic programming algorithm, calculating the parameter vector through a nonlinear constrained least square method, calculating a semimajor axis and a true paraxial point angle through the parameter vector, and obtaining the semimajor axis and the true paraxial point angle which meet a convergence condition through batch processing.
It is to be understood that, in order to overcome random errors existing in single-point optical measurement, a method for estimating initial epoch time track parameters by batch processing of optical measurements at a plurality of short arcs and epoch times is provided, and an initial track parameter iterative calculation method for deriving and solving nonlinear constraint least square estimation can calculate a more accurate initial track compared with single-point optical measurement.
In one embodiment, referring to fig. 3, step S105 may include the following steps:
firstly, setting an initial semi-long axis a, and calculating an initial orbit motion angular speed n through the set initial semi-long axis; by the time t of the sequence epoch j Auxiliary parameter F of j 、G j 、P j And Q j Constructing an observation equation of the parameter vector x:
it is to be understood that the initial semi-major axis a can be set at a =42164000.0. The semimajor axis may be obtained as the initial semimajor axis a by single-point optical measurement according to a single-point optical measurement method.
。
Calculating angular rate of orbital motion:
,where μ represents a gravitational constant.
Calculating a sequence epoch time t j Auxiliary parameter F of j 、G j 、P j And Q j ：
Constructing an observation equation (4) of a parameter vector x by using the auxiliary parameters in the formula (5); and solving the initial parameter vector x according to the observation equation of the formula (4) 0 ：
In one embodiment, referring to fig. 3, step S105 may include the steps of:
constructing a beam-averaging least square estimation equation of the parameter vector x:
constructing an unconstrained lagrange function:
wherein, κ andsrepresenting the lagrange multiplier.
It is to be understood that least squares (also known as the least squares method) is a mathematical optimization technique. It finds the best functional match of the data by minimizing the sum of the squares of the errors. Unknown data can be easily obtained by the least square method, and the sum of squares of errors between these obtained data and actual data is minimized. In addition, optimization without constraints can be continued to approach the optimal value by constructing a sequence of numbers according to the requirement of an extremum, such as a first derivative of 0. However, there are many practical problems that are constrained, such as the observation equation of the parameter vector, and the lagrange multiplier method is a common method for solving constrained optimization.
In one embodiment, referring to fig. 3, step S105 may include the following steps:
three conditional equations are established and respectively expressed as follows:
wherein the 1 st KKT conditional equation shows
And solving the state vector and the Lagrange multiplier by a Newton iteration method.
It should be understood that newton's method is an iterative algorithm that solves for the independent variable values with a function value equal to 0.
In one embodiment, referring to fig. 3, step S105 may include the following steps:
setting an initial Lagrange multiplier
；
According to the k-th parameter vector x, and the Lagrange multiplier
And
calculating the correction amount of the k-th parameter vector
And correction of lagrange multiplier
And
；
calculate the (k + 1) th parameter vector x k+1 And lagrange multiplier
And
：
circularly calculating correction quantity of k-th parameter vector
And lagrangeCorrection amount of multiplier
And
and the (k + 1) th parameter vector x k+1 And lagrange multiplier
And
up to lagrange multiplier correction
And
the convergence condition is satisfied.
It is to be understood that the optimal solution satisfying the convergence condition is derived through repeated iteration. And then determining the lagrange multiplier. Further determining a semi-major axis a and a true proximal angle f 0。
In one embodiment, referring to fig. 3, step S105 may include the following steps:
according to the calculated parameter vector x meeting the convergence condition k+1 Calculating the semimajor axis a and the initial epoch time t 0 True periapical angle of (f) 0 ：
Circularly calculating parameter vector x meeting convergence condition k+1 Semi-major axis a and true proximal angle f 0 And obtaining the semimajor axis and the true paraxial point angle which meet the convergence condition until the semimajor axis a meets the convergence condition.
It should be understood that the loop calculates the parameter vector x satisfying the convergence condition k+1 Semi-major axis a and true proximal angle f 0 Up toThe semimajor axis a meets the convergence condition and completes the initial epoch time t 0 Semi-major axis a and true proximal angle f 0 And (4) calculating. Meanwhile, 4 parameters of eccentricity, inclination angle, ascension at ascending intersection point and argument of perigee are all 0, so that the realization of initial epoch time t by using short arc multipoint optical measurement is completed 0 The initial orbit of (2) is determined.
In one embodiment, as shown with reference to fig. 2 and 3, step S103 may include the steps of:
when the semi-long axis and the true paraxial point angle are obtained by a single-point optical measurement method, the initial orbit determination of the initial epoch time by using single-point optical measurement is completed;
and when the semi-long axis and the true paraxial point angle are obtained by the short arc multipoint optical measurement method, the initial orbit determination of the initial epoch time by using the short arc multipoint optical measurement is completed.
It should be noted that although the various steps of the methods of the present disclosure are depicted in the drawings in a particular order, this does not require or imply that these steps must be performed in this particular order, or that all of the depicted steps must be performed, to achieve desirable results. Additionally or alternatively, certain steps may be omitted, multiple steps combined into one step execution, and/or one step broken down into multiple step executions, etc. Additionally, it will also be readily appreciated that the steps may be performed synchronously or asynchronously, e.g., among multiple modules/processes/threads.
Hereinafter, more specific examples will be presented for a more detailed description in conjunction with the above-described examples.
Common preparation steps of the first embodiment and the second embodiment.
In the geocentric inertial coordinate system, define
Representing the time t of the sequence epoch j Optically measuring the line-of-sight vector, [ X ] j Y j Z j ] T Representing the time t of the sequence epoch j Calculating the time t of the sequence epoch j Auxiliary parameters in geocentric inertial coordinate systemP j AndQ j :
establishing an initial epoch time t 0 The basic equation for determining the Laplace type initial orbit is as follows:
in the formula, r 0 =[x 0 y 0 z 0 ] T And v 0 =[ẋ 0 ẏ 0 ẑ 0 ] T Representing the time t of the sequence epoch 0 The position and speed of (c).
In the geocentric inertial coordinate system, the eccentricity and the inclination can be approximately zero for the geostationary satellite, and the eccentricity, the inclination, the argument of the perigee and the ascension of the ascending intersection point are all zero for the orbit with small eccentricity and small inclination, then
The sorted initial orbit determination observation equation is expressed as:
in the formula, F j And G j The closed expression is expressed as follows:
the first embodiment is as follows: the method for determining the initial track of the sequence epoch time by using single-point optical measurement.
Step 1: for sequence epoch time t j Can measure each epoch time t j Is regarded as t 0 Then F is j =1.G j =0, the initial orbit determination basic observation equation is obtained:
optically measuring the line-of-sight vector, representing the time t of the sequence epoch j Optic vignetting point coordinates.
Step 2: constructing a sequence epoch time t j Semi-major axis a and true proximal angle f j Observation equation of (c):
and step 3: calculating a sequence epoch time t j Semi-major axis a and true proximal angle f j ：
Semi-major axis a and true paraxial angle f calculated according to the above formula j Since the parameters of 4 items of eccentricity, inclination angle, ascension at ascending intersection point and argument of perigee are all 0, the time t of sequence epoch can be solved j 6 parameters of the initial track to complete the sequence epoch time t by single-point optical measurement j The initial orbit of (2) is determined.
The second embodiment is as follows: an initial epoch time initial track determination method by using short arc multipoint optical measurement.
In order to overcome the random error of single-point optical measurement to sequence epoch time t j The influence of initial orbit determination can be selected from a batch processing estimation method of multipoint optical measurement, and the sequence epoch time t is required to be determined j Is converted to the initial epoch time t 0 The geocentric inertial coordinate system of (a).
The initial value of the semimajor axis a is 42164000.0, and the angular velocity of the orbit motion is calculated in half
,Auxiliary parameter F of sequence epoch time j 、G j 、P j And Q j The auxiliary parameters are respectively expressed as follows:
let x = [ x ] 1 x 2 x 3 x 4 ] T ，
Then, an observation equation for the parameter vector x is established as follows:
constructing a least squares estimation equation expressed as:
accordingly, an unconstrained least squares solution of the solvable parameter vector x is represented as
。
Meanwhile, the parameter vector x must satisfy the following constraint relationship:
the least square estimation equation constrained by the nonlinear equation is formed, and the estimation of the parameter vector x can be completed through a sequential quadratic programming algorithm.
A constrained optimization equation of a standard single objective function nonlinear equation is established, expressed as follows:
the following lagrange function is constructed:
in the formula (I), the compound is shown in the specification,
and
representing the lagrange multiplier.
Three conditional equations are established and respectively expressed as follows:
in the above formula, the 1 st KKT conditional equation shows
The state vector and the lagrangian multiplier can be solved by a newton iteration method, and the iterative computation process is expressed as:
solving for the parameter correction by:
in the above formula, the matrix parameters are respectively expressed as follows:
in addition to this, the present invention is,
for a second-order symmetric hessian matrix, the initial value of the parameter vector x can be an unconstrained least square estimation value, a Lagrange multiplier
And
can be a large positive integer, such as 1.0 × 10 6 。
Calculated parameter vector x k+1 Calculating the semimajor axis a and the initial epoch time t 0 True proximal angle f 0 Respectively, as follows:
circularly executing the auxiliary parameter expression at the time of the sequence epoch to the semimajor axis a and the initial epoch time t 0 True proximal angle f 0 Until the semi-major axis a meets the convergence condition, the initial epoch time t is completed 0 Semi-major axis a and true proximal angle f 0 And (4) calculating. Meanwhile, 4 parameters of eccentricity, inclination angle, ascension at ascending intersection point and argument of perigee are all 0, so that the realization of initial epoch time t by using short arc multipoint optical measurement is completed 0 The initial orbit of (2) is determined.
It should be noted that all the directional indicators (such as upper, lower, left, right, front, and rear … …) in the embodiment of the present invention are only used to explain the relative position relationship between the components, the motion situation, and the like in a specific posture (as shown in the drawing), and if the specific posture is changed, the directional indicator is changed accordingly.
In addition, the descriptions related to "first", "second", etc. in the present invention are only for descriptive purposes and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present invention, "a plurality" means at least two, e.g., two, three, etc., unless specifically limited otherwise.
In the present invention, unless otherwise expressly stated or limited, the terms "connected," "secured," and the like are to be construed broadly, and for example, "secured" may be a fixed connection, a removable connection, or an integral part; the connection can be mechanical connection, electrical connection, physical connection or wireless communication connection; they may be directly connected or indirectly connected through intervening media, or they may be connected internally or in any other suitable relationship, unless expressly stated otherwise. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
In addition, the technical solutions in the embodiments of the present invention may be combined with each other, but it must be based on the realization of the technical solutions by those skilled in the art, and when the technical solutions are contradictory to each other or cannot be realized, such a combination of the technical solutions should not be considered to exist, and is not within the protection scope of the present invention.
Other embodiments of the disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the disclosure disclosed herein. This application is intended to cover any variations, uses, or adaptations of the disclosure following, in general, the principles of the disclosure and including such departures from the present disclosure as come within known or customary practice within the art to which the disclosure pertains. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the disclosure being indicated by the following claims.
Claims (7)
1. A method for determining an initial track of a short arc optical measurement suitable for a track with small eccentricity is characterized by comprising the following steps:
obtaining a semi-long axis and a true paraxial point angle by a single-point optical measurement method or a short arc multipoint measurement method;
setting the eccentricity, the inclination angle, the ascension at the ascending intersection point and the argument of the perigee to be 0;
determining an initial orbit of an initial epoch moment according to the obtained semi-major axis and the obtained true paraxial point angle;
wherein the single-point optical measurement method comprises: constructing an observation equation of the semi-long axis and the true near point angle at the sequence epoch moment through auxiliary parameters in the geocentric inertial coordinate system at the sequence epoch moment, and solving the semi-long axis and the true near point angle; the short arc multipoint measuring method comprises the following steps: constructing an observation equation of a parameter vector through a sequential quadratic programming algorithm, calculating the parameter vector by combining a nonlinear constrained least square method, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle which meet convergence conditions through batch processing;
the process of constructing the observation equation of the semi-major axis and the true paraxial point angle at the sequence epoch moment through the auxiliary parameters in the sequence epoch moment geocentric inertial coordinate system is as follows:
the expression of the auxiliary parameter in the earth-centered inertial coordinate system at the moment of the sequence epoch is as follows:
wherein, P j And Q j The auxiliary parameters are auxiliary parameters in the geocentric inertial coordinate system of the sequence epoch time at the j time; by
The formed matrix
An optically measured line-of-sight vector representing the sequence epoch time at time j, denoted by X j Y j Z j The composed matrix [ X ] j Y j Z j ] T Representing the optical measurement point coordinate of the sequence epoch time at the j time;
the observation equation of the semi-major axis and the true paraxial point angle at the time of the sequence epoch is as follows:
wherein a is a semi-major axis;f j is a true proximal angle.
2. The method according to claim 1, wherein the step of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a least square method of nonlinear constraint, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle satisfying a convergence condition by batch processing further comprises:
setting an initial semi-long shaft, and calculating an initial orbital motion angular rate through the set initial semi-long shaft; and (3) constructing an observation equation of the parameter vector through the auxiliary parameters of the sequence epoch time:
wherein x is
Representing a vector of parameters; a is a semimajor axis;F j 、G j 、P j andQ j the auxiliary parameter is the auxiliary parameter of the sequence epoch moment at the j moment; v. of j The velocity at time j of the sequence epoch; μ is the earth's gravitational constant.
3. The method according to claim 2, wherein the step of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a least square method of nonlinear constraint, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle satisfying a convergence condition by batch processing further comprises:
constructing a uniform beam least square estimation equation of the parameter vector:
constructing an unconstrained lagrange function:
wherein x is
Representing a vector of parameters;κand s represents the lagrange multiplier,fto representP j AndQ j a combination matrix of (a);yto representF j AndG j the combination matrix of (1).
4. The method according to claim 3, wherein the step of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a least square method of nonlinear constraint, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle satisfying a convergence condition by batch processing further comprises:
three conditional equations are established and respectively expressed as follows:
according to the 1 st KKT condition equation
And solving the state vector and the Lagrange multiplier by combining a Newton iteration method.
5. The method according to claim 4, wherein the step of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a least square method of nonlinear constraint, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle satisfying a convergence condition by batch processing further comprises:
setting an initial Lagrange multiplier
；
Calculating the correction quantity of the parameter vector in the k-th cycle and the correction quantity of the Lagrange multiplier according to the parameter vector in the k-th cycle and the Lagrange multiplier;
wherein the content of the first and second substances,
is the parameter vector in the k-th cycle;
and
is the lagrange multiplier in the kth cycle;
correction amount of the parameter vector in the k-th cycle;
and
correction for lagrange multiplier in the kth cycle;
calculating a parameter vector and a Lagrange multiplier in the cycle of the (k + 1) th time:
wherein the content of the first and second substances,
is the parameter vector in the cycle of the (k + 1) th time;
and
lagrange multiplier in the cycle of the (k + 1) th time;
calculating correction amount of parameter vector in k-th cycle
And correction of lagrange multiplier
And
and the parameter vector in the (k + 1) th cycle
And lagrange multiplier
And
up to lagrange multiplier correction
And
the convergence condition is satisfied.
6. The method according to claim 5, wherein the step of constructing an observation equation of a parameter vector by a sequential quadratic programming algorithm, calculating the parameter vector by combining a least square method of nonlinear constraint, calculating the semimajor axis and the true proximal angle according to the parameter vector, and obtaining the semimajor axis and the true proximal angle satisfying a convergence condition by batch processing further comprises:
according to the calculated parameter vector satisfying the convergence condition
Calculating the true paraxial point angle and the true paraxial point angle of the initial epoch time:
wherein a is a semi-major axis;f 0 the true anomaly is the true anomaly at 0 at the initial epoch time;
loop calculation satisfies the convergence barVector of parameters of a piece
Semi-major axis a and true proximal anglef 0 And obtaining the semimajor axis and the true proximal angle which meet the convergence condition until the semimajor axis a meets the convergence condition.
7. The method according to any one of claims 1-6, wherein the step of determining an initial trajectory of an initial epoch time based on the determined semi-major axis and the true paraxial angle further comprises:
when the semi-long axis and the true paraxial point angle are obtained by a single-point optical measurement method, the initial orbit determination of the initial epoch time by using single-point optical measurement is completed;
and when the semi-long axis and the true paraxial point angle are obtained by the short arc multipoint optical measurement method, the initial orbit determination of the initial epoch time by using the short arc multipoint optical measurement is completed.
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