CN103309229A - Satellite-to-ground dynamic bidirectional time synchronization and ranging combined algorithm based on fitting method - Google Patents

Abstract

The invention discloses a satellite-to-ground dynamic bidirectional time synchronization and ranging combined algorithm based on a fitting method. Radio transmitters and receivers are installed on a satellite (A) and a ground station (B), when the satellite A and the ground station B transmit and receive time synchronization and ranging signals of each other simultaneously, the satellite-to-ground time synchronization and ranging can be achieved based on the fitting method. By means of the satellite-to-ground dynamic bidirectional time synchronization and ranging combined algorithm based on the fitting method, effects of satellite movements on the accuracy of the algorithm can be reduced, and high-accuracy satellite-to-ground ranging can be achieved while moving satellite satellite-to-ground high-accuracy time synchronization is completed.

Description

A kind of star ground dynamic bidirectional time synchronized and range finding unified algorithm based on fitting process

Technical field

The invention belongs to star ground dynamic bidirectional time synchronized and range finding unified algorithm technical field, particularly a kind of star ground dynamic bidirectional time synchronized and range finding unified algorithm based on fitting process.

Background technology

The two-way time synchronized algorithm in star ground is one of the highest method for synchronizing time of present precision, but this method is when carrying out time synchronized, the aircraft that time synchronized is carried out in requirement is in relative static state with land station, therefore have higher precision when this method is used for the time synchronized of static relatively aircraft over the ground at present, then precision is lower to the time synchronized of the aircraft of motion over the ground.This algorithm utilizes the two-way time synchronized data in star ground dynamically to generate star ground distance and clock correction polynomial fitting respectively, united by two polynomial expressions then and find the solution moving satellite star ground clock correction and distance, can reduce satellite motion to the influence of arithmetic accuracy, can realize that high precision star ground finds range in synchronous finishing moving satellite star ground split-second precision.

Summary of the invention

The objective of the invention is to, provide a kind of satellite motion that can reduce to the influence of two-way time synchronized and location algorithm precision, can finish moving satellite star ground split-second precision realize in synchronous the range finding of high precision star ground based on the star ground dynamic bidirectional time synchronized of fitting process and the unified algorithm of finding range.

For solving the problems of the technologies described above, the present invention adopts following technical scheme: a kind of star ground dynamic bidirectional time synchronized and range finding unified algorithm based on fitting process, it is characterized in that: in satellite (A), land station (B) radio transmitter and receiver are installed, when satellite A and the B of land station simultaneously to the other side launch time synchronously and distance measuring signal, and after receiving the other side's time synchronized and distance measuring signal, obtain following equation:

T ₁＝Δt+t ₂+τ _BA+r ₁+δ ₁ (1)

T ₂＝-Δt+t ₁+τ _AB+r ₂+δ ₂ (2)

Δ t is the clock correction of satellite A and the B of land station in the formula, T ₁Be the mistiming of satellite A emission timing signal with the timing signal that receives the B of land station emission, t ₂Be the B of land station transmitter time delay, τ _BABe the propagation delay of the B of land station to satellite A, r ₁Be satellite A receiving equipment time delay, δ ₁Be other time delay; T ₂Be the mistiming of the B of land station emission timing signal with the timing signal that receives satellite A emission, t ₁Be satellite A transmitter time delay, τ _ABBe the propagation delay of satellite A to the B of land station, r ₂Be the B of land station receiving equipment time delay, δ ₂Be other time delay.To can get after (1), (2) arrangement:

(τ _BA+τ _AB)＝(T ₁+T ₂)-(t ₁+t ₂)-(r ₁+r ₂)-(δ ₁+δ ₂) (3)

$\mathrm{\Δt}=\frac{T_1-T_2}{2}+\frac{t_1-t_2}{2}+\frac{r_2-r_1}{2}+\frac{{\mathrm{\τ}}_{\mathrm{AB}}-{\mathrm{\τ}}_{\mathrm{BA}}}{2}+\frac{{\mathrm{\δ}}_2-{\mathrm{\δ}}_1}{2}---\left(4\right)$

Utilizing (3) formula to multiply by light velocity c, can to obtain star ground as follows apart from the ρ computing formula:

$\mathrm{\ρ}=c\·\frac{({\mathrm{\τ}}_{\mathrm{BA}}+{\mathrm{\τ}}_{\mathrm{AB}})}{2}=\frac{(T_1+T_2)-(t_1+t_2)-(r_1+r_2)-({\mathrm{\δ}}_1+{\mathrm{\δ}}_2)}{2}---\left(5\right)$

(4), in (5) formula, T ₁, T ₂A, the B of land station measure acquisition, t separately via satellite ₁, r ₁And t ₂, r ₂The frequency that transmits according to satellite and land station is demarcated in advance respectively.The frequency of sending out time synchronized and distance measuring signal as satellite A and the B of land station mutually approaches, and the link symmetry, and the propagation delay approximately equal namely has τ _AB=τ _BA, ignore other time delay δ simultaneously ₁And δ ₂Influence, can try to achieve the star ground of satellite and land station apart from ρ and clock correction Δ t.

Suppose that clock correction Δ t remains unchanged in star ground time synchronized and the ranging process, the influence of transceiver time delay and other time delay is eliminated, and then star ground propagation delay and the clock correction of Jian Huaing is as follows:

${\mathrm{\τ}}_{\mathrm{AB}}\left({\mathrm{\τ}}_{\mathrm{BA}}\right)=\frac{(T_1+T_2)}{2}---\left(6\right)$

$\mathrm{\Δt}=\frac{T_1-T_2}{2}---\left(7\right)$

The star ground of simplifying apart from computing formula is:

$\mathrm{\ρ}=c\·{\mathrm{\τ}}_{\mathrm{AB}}=c\·\frac{T_1+T_2}{2}---\left(8\right)$

(7), (8) formula is star ground clock correction and the distance associating solution formula of deriving and obtaining.

By to moving satellite star ground apart from Changing Pattern emulation as can be known, because satellite is in motion state at a high speed, no longer satisfy τ when causing the two-way time synchronized in star ground _AB=τ _BAThis precondition, if still adopt formula (7), (8) carry out star ground time synchronized and range finding is resolved, the star ground distance of obtaining and star ground clock correction and actual result are inconsistent, must revise with the range finding result the two-way time synchronized that satellite obtains in motion process, just can obtain the star ground distance and the star ground clock correction that approach the most with actual result.By to the analysis of the two-way time synchronized data of Changing Pattern and star ground of star ground distance as can be known, in satellite and land station's distance process that changes, in the time of directly over moving satellite is in land station, minimum with the distance of land station, the propagation delay of this two-way time synchronizing signal in clock star ground is the most approaching, so by this star ground distance that calculates constantly and clock correction and actual range and clock correction error minimum.For this reason will be sometime the star ground that calculates of the two-way time synchronized in section star ground regard the function of time as apart from ρ and clock correction Δ t, consider that the two-way time synchronized in star ground and range finding are subjected to the star ground that the influence of various errors causes obtaining easily and have observational error usually apart from data and clock correction data, and the data volume of time synchronized is generally bigger, approaches the Changing Pattern that can obtain star ground distance and clock correction so only need carry out match to the variation tendency of the range data that has observational error and clock correction data.Can be in the hope of the moment of minimum star ground distance apart from polynomial fitting by star ground, this moment also is the moment of match clock correction and actual clock correction error minimum.

If star ground apart from function ρ and clock correction function Δ t at a time between between the synchronization zone [a, b] go up and exist and continuously, and at n different some a≤t ₁, t ₂..., t _nThe last value ρ respectively of≤b ₁, ρ ₂..., ρ _nWith Δ t ₁, Δ t ₂..., Δ t _n.Requirement is asked function in function class Φ

And ψ (t) makes and to satisfy following equation:

And at other t ≠ t _iThe point on,

Can realize and ρ approximate that ψ (t) can realize and being similar to of Δ t that (9) formula is fitting condition, function class Φ is the fitting function class,

And ψ (t) is respectively function ρ and Δ t at node t ₁, t ₂..., t _nThe fitting function at place, the fitting function of selection can make the quadratic sum minimum of deviation:

Formula (10) is the least square fitting polynomial solving formula of star ground distance and clock correction.

The discrete data that is obtained by the dynamic bidirectional time synchronized obtains star ground respectively apart from sequence (t _i, ρ _i) and clock correction sequence (t _i, Δ t _i) (i=1,2 ..., n) after, require at function class

With

The middle searching

Make

Namely

Set up.

Formula (13) is right respectively

With

Differentiate can get

Solution formula (14) is obtained

With

(k=1,2 ..., m) after, in substitution (11) formula, can obtain star ground apart from sequence (t _i, ρ _i) and clock correction sequence (t _i, Δ t _i) (i=1,2 ..., least square fitting polynomial expression n).By the Changing Pattern emulation of star ground distance and the analysis on change of star ground clock correction, star ground is apart from fitting function Adopt quadratic function, and star ground clock correction fitting function ψ ^*(t) can adopt linear function, the interval starting point of time synchronized is generally got a=0.If the star ground after the match is as follows apart from ρ and clock correction Δ t polynomial expression:

$\left\{\begin{array}{c}\mathrm{\ρ}=f_1\left(t\right)\\ \mathrm{\Δt}=f_2\left(t\right)\end{array}\right.---\left(15\right)$

In the following formula star ground can be tried to achieve on star ground apart from minimal value ρ apart from the time differentiate of ρ d ρ/dt=0 _MinThe time corresponding moment t ₃, this is constantly and calculates the most approaching moment of star ground distance and clock correction and actual range and clock correction, with t ₃Star ground is obtained apart from ρ apart from polynomial expression and clock correction polynomial expression in substitution star ground respectively _MinWith star ground clock correction Δ t _MinAs follows:

$\left\{\begin{array}{c}{\mathrm{\ρ}}_{\min }=f_1\left(t_3\right)\\ {\mathrm{\Δt}}_{\min }=f_2\left(t_3\right)\end{array}\right.---\left(16\right)$

By the simulation calculation to MEO satellite and station, Xi'an, the result shows and comprising under the condition of phantom error, algorithm timing tracking accuracy of the present invention in 5ns, distance accuracy in 3m, can be used for various moving satellites star ground split-second precision synchronously with range finding.For reducing fitting of a polynomial to the influence of arithmetic accuracy, time period is should get symmetry substantially with respect to starlet distance from the moment that occurs, choosing with comparatively suitable in 20～30 minutes of duration to guarantee the fitting precision of algorithm, improved star ground time synchronized and distance accuracy.

Simultaneously, in simulation example, owing in star ground time synchronized and in the range finding time period, fail to consider satellite fortune direction to the influence of algorithm, thereby cause phantom error bigger than normal.But in the star ground of reality dynamic bidirectional time synchronized and range finding, the satellite motion direction has lain in T to the influence of algorithm ₁And T ₂Measurement in, so this factor will reduce the shadow of algorithm, precision will further improve.

The present invention in actual applications in sum, handle star ground time synchronized and ranging data when adopting the least square fitting model, star ground adopts quadratic function apart from fitting function, and star ground clock correction fitting function adopts linear function, the interval starting point of time synchronized is taken as 0, and time synchronized and range finding time period with respect to starlet distance under the condition of basic symmetry of the moment that occurs, the precision of algorithm can be guaranteed, therefore actual measuring accuracy will can be used for Dynamic High-accuracy time synchronized and the range finding of various motion aircraft than simulation result height.

Embodiment

Embodiments of the invention:

A kind of star ground dynamic bidirectional time synchronized and range finding unified algorithm based on fitting process, it is characterized in that: in satellite (A), land station (B) radio transmitter and receiver are installed, when satellite A and the B of land station simultaneously to the other side launch time synchronously and distance measuring signal, and after receiving the other side's time synchronized and distance measuring signal, obtain following equation:

T ₁＝Δt+t ₂+τ _BA+r ₁+δ ₁ (1)

T ₂＝-Δt+t ₁+τ _AB+r ₂+δ ₂ (2)

Δ t is the clock correction of satellite A and the B of land station in the formula, T ₁Be the mistiming of satellite A emission timing signal with the timing signal that receives the B of land station emission, t ₂Be the B of land station transmitter time delay, τ _BABe the propagation delay of the B of land station to satellite A, r ₁Be satellite A receiving equipment time delay, δ ₁Be other time delay; T ₂Be the mistiming of the B of land station emission timing signal with the timing signal that receives satellite A emission, t ₁Be satellite A transmitter time delay, τ _ABBe the propagation delay of satellite A to the B of land station, r ₂Be the B of land station receiving equipment time delay, δ ₂Be other time delay.To can get after (1), (2) arrangement:

(τ _BA+τ _AB)＝(T ₁+T ₂)-(t ₁+t ₂)-(r ₁+r ₂)-(δ ₁+δ ₂) (3)

$\mathrm{\Δt}=\frac{T_1-T_2}{2}+\frac{t_1-t_2}{2}+\frac{r_2-r_1}{2}+\frac{{\mathrm{\τ}}_{\mathrm{AB}}-{\mathrm{\τ}}_{\mathrm{BA}}}{2}+\frac{{\mathrm{\δ}}_2-{\mathrm{\δ}}_1}{2}---\left(4\right)$

Utilizing (3) formula to multiply by light velocity c, can to obtain star ground as follows apart from the ρ computing formula:

$\mathrm{\ρ}=c\·\frac{({\mathrm{\τ}}_{\mathrm{BA}}+{\mathrm{\τ}}_{\mathrm{AB}})}{2}=\frac{(T_1+T_2)-(t_1+t_2)-(r_1+r_2)-({\mathrm{\δ}}_1+{\mathrm{\δ}}_2)}{2}---\left(5\right)$

(4), in (5) formula, T ₁, T ₂A, the B of land station measure acquisition, t separately via satellite ₁, r ₁And t ₂, r ₂The frequency that transmits according to satellite and land station is demarcated in advance respectively.The frequency of sending out time synchronized and distance measuring signal as satellite A and the B of land station mutually approaches, and the link symmetry, and the propagation delay approximately equal namely has τ _AB=τ _BA, ignore other time delay δ simultaneously ₁And δ ₂Influence, can try to achieve the star ground of satellite and land station apart from ρ and clock correction Δ t.

Suppose that clock correction Δ t remains unchanged in star ground time synchronized and the ranging process, the influence of transceiver time delay and other time delay is eliminated, and then star ground propagation delay and the clock correction of Jian Huaing is as follows:

${\mathrm{\τ}}_{\mathrm{AB}}\left({\mathrm{\τ}}_{\mathrm{BA}}\right)=\frac{(T_1+T_2)}{2}---\left(6\right)$

$\mathrm{\Δt}=\frac{T_1-T_2}{2}---\left(7\right)$

The star ground of simplifying apart from computing formula is:

$\mathrm{\ρ}=c\·{\mathrm{\τ}}_{\mathrm{AB}}=c\·\frac{T_1+T_2}{2}---\left(8\right)$

(7), (8) formula is star ground clock correction and the distance associating solution formula of deriving and obtaining.

By to moving satellite star ground apart from Changing Pattern emulation as can be known, because satellite is in motion state at a high speed, no longer satisfy τ when causing the two-way time synchronized in star ground _AB=τ _BAThis precondition, if still adopt formula (7), (8) carry out star ground time synchronized and range finding is resolved, the star ground distance of obtaining and star ground clock correction and actual result are inconsistent, must revise with the range finding result the two-way time synchronized that satellite obtains in motion process, just can obtain the star ground distance and the star ground clock correction that approach the most with actual result.By to the analysis of the two-way time synchronized data of Changing Pattern and star ground of star ground distance as can be known, in satellite and land station's distance process that changes, in the time of directly over moving satellite is in land station, minimum with the distance of land station, the propagation delay of this two-way time synchronizing signal in clock star ground is the most approaching, so by this star ground distance that calculates constantly and clock correction and actual range and clock correction error minimum.For this reason will be sometime the star ground that calculates of the two-way time synchronized in section star ground regard the function of time as apart from ρ and clock correction Δ t, consider that the two-way time synchronized in star ground and range finding are subjected to the star ground that the influence of various errors causes obtaining easily and have observational error usually apart from data and clock correction data, and the data volume of time synchronized is generally bigger, approaches the Changing Pattern that can obtain star ground distance and clock correction so only need carry out match to the variation tendency of the range data that has observational error and clock correction data.Can be in the hope of the moment of minimum star ground distance apart from polynomial fitting by star ground, this moment also is the moment of match clock correction and actual clock correction error minimum.

If star ground apart from function ρ and clock correction function Δ t at a time between between the synchronization zone [a, b] go up and exist and continuously, and at n different some a≤t ₁, t ₂..., t _nThe last value ρ respectively of≤b ₁, ρ ₂..., ρ _nWith Δ t ₁, Δ t ₂..., Δ t _n.Requirement is asked function in function class Φ

And ψ (t) makes and to satisfy following equation:

And at other t ≠ t _iThe point on,

Can realize and ρ approximate that ψ (t) can realize and being similar to of Δ t that (9) formula is fitting condition, function class Φ is the fitting function class,

And ψ (t) is respectively function ρ and Δ t at node t ₁, t ₂..., t _nThe fitting function at place, the fitting function of selection can make the quadratic sum minimum of deviation:

Formula (10) is the least square fitting polynomial solving formula of star ground distance and clock correction.

The discrete data that is obtained by the dynamic bidirectional time synchronized obtains star ground respectively apart from sequence (t _i, ρ _i) and clock correction sequence (t _i, Δ t _i) (i=1,2 ..., n) after, require at function class

And Φ ₂=(ψ ₁, ψ ₂..., ψ _n) the middle searching

Make

Namely

Set up.

Formula (13) is right respectively

With Differentiate can get

Solution formula (14) is obtained

With (k=1,2 ..., m) after, in substitution (11) formula, can obtain star ground apart from sequence (t _i, ρ i) and clock correction sequence (t _i, Δ t _i) (i=1,2 ..., least square fitting polynomial expression n).By the Changing Pattern emulation of star ground distance and the analysis on change of star ground clock correction, star ground is apart from fitting function

Adopt quadratic function, and star ground clock correction fitting function ψ ^*(t) can adopt linear function, the interval starting point of time synchronized is generally got a=0.If the star ground after the match is as follows apart from ρ and clock correction Δ t polynomial expression:

$\left\{\begin{array}{c}\mathrm{\ρ}=f_1\left(t\right)\\ \mathrm{\Δt}=f_2\left(t\right)\end{array}\right.---\left(15\right)$

In the following formula star ground can be tried to achieve on star ground apart from minimal value ρ apart from the time differentiate of ρ d ρ/dt=0 _MinThe time corresponding moment t ₃, this is constantly and calculates the most approaching moment of star ground distance and clock correction and actual range and clock correction, with t ₃Star ground is obtained apart from ρ apart from polynomial expression and clock correction polynomial expression in substitution star ground respectively _MinWith star ground clock correction Δ t _MinAs follows:

$\left\{\begin{array}{c}{\mathrm{\ρ}}_{\min }=f_1\left(t_3\right)\\ {\mathrm{\Δt}}_{\min }=f_2\left(t_3\right)\end{array}\right.---\left(16\right)$

By the simulation calculation to MEO satellite and station, Xi'an, the result shows and comprising under the condition of phantom error, algorithm timing tracking accuracy of the present invention in 5ns, distance accuracy in 3m, can be used for various moving satellites star ground split-second precision synchronously with range finding.For reducing fitting of a polynomial to the influence of arithmetic accuracy, time period is should get symmetry substantially with respect to starlet distance from the moment that occurs, choosing with comparatively suitable in 20～30 minutes of duration to guarantee the fitting precision of algorithm, improved star ground time synchronized and distance accuracy.

Simultaneously, in simulation example, owing in star ground time synchronized and in the range finding time period, fail to consider satellite fortune direction to the influence of algorithm, thereby cause phantom error bigger than normal.But in the star ground of reality dynamic bidirectional time synchronized and range finding, the satellite motion direction has lain in T to the influence of algorithm ₁And T ₂Measurement in, so this factor will reduce the shadow of algorithm, precision will further improve.

The present invention in actual applications in sum, handle star ground time synchronized and ranging data when adopting the least square fitting model, star ground adopts quadratic function apart from fitting function, and star ground clock correction fitting function adopts linear function, the interval starting point of time synchronized is taken as O, and time synchronized and range finding time period with respect to starlet distance under the condition of basic symmetry of the moment that occurs, the precision of algorithm can be guaranteed, therefore actual measuring accuracy will can be used for Dynamic High-accuracy time synchronized and the range finding of various motion aircraft than simulation result height.

Claims (1)

1. one kind based on the star ground dynamic bidirectional time synchronized of fitting process and range finding unified algorithm, it is characterized in that: in satellite (A), land station (B) radio transmitter and receiver are installed, when satellite A and the B of land station simultaneously to the other side launch time synchronously and distance measuring signal, and after receiving the other side's time synchronized and distance measuring signal, obtain following equation:

T ₁＝Δt+t ₂+τ _BA+r ₁+δ ₁ (1)

T ₂＝-Δt+t ₁+τ _AB+r ₂+δ ₂ (2)

Δ t is the clock correction of satellite A and the B of land station in the formula, T ₁Be the mistiming of satellite A emission timing signal with the timing signal that receives the B of land station emission, t ₂Be the B of land station transmitter time delay, τ _BABe the propagation delay of the B of land station to satellite A, r ₁Be satellite A receiving equipment time delay, δ ₁Be other time delay; T ₂Be the mistiming of the B of land station emission timing signal with the timing signal that receives satellite A emission, t ₁Be satellite A transmitter time delay, τ _ABBe the propagation delay of satellite A to the B of land station, r ₂Be the B of land station receiving equipment time delay, δ ₂Be other time delay.To can get after (1), (2) arrangement:

(τ _BA+τ _AB)＝(T ₁+T ₂)-(t ₁+t ₂)-(r ₁+r ₂)-(δ ₁+δ ₂) (3)

$\mathrm{\Δt}=\frac{T_1-T_2}{2}+\frac{t_1-t_2}{2}+\frac{r_2-r_1}{2}+\frac{{\mathrm{\τ}}_{\mathrm{AB}}-{\mathrm{\τ}}_{\mathrm{BA}}}{2}+\frac{{\mathrm{\δ}}_2-{\mathrm{\δ}}_1}{2}---\left(4\right)$

Utilizing (3) formula to multiply by light velocity c, can to obtain star ground as follows apart from the ρ computing formula:

$\mathrm{\ρ}=c\·\frac{({\mathrm{\τ}}_{\mathrm{BA}}+{\mathrm{\τ}}_{\mathrm{AB}})}{2}=\frac{(T_1+T_2)-(t_1+t_2)-(r_1+r_2)-({\mathrm{\δ}}_1+{\mathrm{\δ}}_2)}{2}---\left(5\right)$

(4), in (5) formula, T ₁, T ₂A, the B of land station measure acquisition, t separately via satellite ₁, r ₁And t ₂, r ₂The frequency that transmits according to satellite and land station is demarcated in advance respectively.The frequency of sending out time synchronized and distance measuring signal as satellite A and the B of land station mutually approaches, and the link symmetry, and the propagation delay approximately equal namely has τ _AB=τ _BA, ignore other time delay δ simultaneously ₁And δ ₂Influence, can try to achieve the star ground of satellite and land station apart from ρ and clock correction Δ t.

Suppose that clock correction Δ t remains unchanged in star ground time synchronized and the ranging process, the influence of transceiver time delay and other time delay is eliminated, and then star ground propagation delay and the clock correction of Jian Huaing is as follows:

${\mathrm{\τ}}_{\mathrm{AB}}\left({\mathrm{\τ}}_{\mathrm{BA}}\right)=\frac{(T_1+T_2)}{2}---\left(6\right)$

$\mathrm{\Δt}=\frac{T_1-T_2}{2}---\left(7\right)$

The star ground of simplifying apart from computing formula is:

$\mathrm{\ρ}=c\·{\mathrm{\τ}}_{\mathrm{AB}}=c\·\frac{T_1+T_2}{2}---\left(8\right)$

(7), (8) formula is star ground clock correction and the distance associating solution formula of deriving and obtaining.

If star ground apart from function ρ and clock correction function Δ t at a time between between the synchronization zone [a, b] go up and exist and continuously, and at n different some a≤t ₁, t ₂..., t _nThe last value ρ respectively of≤b ₁, ρ ₂..., ρ _nWith Δ t ₁, Δ t ₂..., Δ t _n.Requirement is asked function in function class Φ And ψ (t) makes and to satisfy following equation:

And at other t ≠ t _iThe point on,

Can realize and ρ approximate that ψ (t) can realize and being similar to of Δ t that (9) formula is fitting condition, function class Φ is the fitting function class,

And ψ (t) is respectively function ρ and Δ t at node t ₁, t ₂..., t _nThe fitting function at place, the fitting function of selection can make the quadratic sum minimum of deviation:

Formula (10) is the least square fitting polynomial solving formula of star ground distance and clock correction.

The discrete data that is obtained by the dynamic bidirectional time synchronized obtains star ground respectively apart from sequence (t _i, ρ _i) and clock correction sequence (t _i, Δ t _i(i=1,2 ..., n) after, require at function class

With

The middle searching

Make

Namely

Set up.

Formula (13) is right respectively

With

Differentiate can get

Solution formula (14) is obtained With

(k=1,2 ..., m) after, in substitution (11) formula, can obtain star ground apart from sequence (t _i, ρ _i) and clock correction sequence (t _i, Δ t _i) (i=1,2 ..., least square fitting polynomial expression n).By the Changing Pattern emulation of star ground distance and the analysis on change of star ground clock correction, star ground is apart from fitting function

Adopt quadratic function, and star ground clock correction fitting function ψ ^*(t) can adopt linear function, the interval starting point of time synchronized is generally got a=0.If the star ground after the match is as follows apart from ρ and clock correction Δ t polynomial expression:

$\left\{\begin{array}{c}\mathrm{\ρ}=f_1\left(t\right)\\ \mathrm{\Δt}=f_2\left(t\right)\end{array}\right.---\left(15\right)$

In the following formula star ground can be tried to achieve on star ground apart from minimal value ρ apart from the time differentiate of ρ d ρ/dt=0 _MinThe time corresponding moment t ₃, this is constantly and calculates the most approaching moment of star ground distance and clock correction and actual range and clock correction, with t ₃Star ground is obtained apart from ρ apart from polynomial expression and clock correction polynomial expression in substitution star ground respectively _MinWith star ground clock correction Δ t _MinAs follows:

$\left\{\begin{array}{c}{\mathrm{\ρ}}_{\min }=f_1\left(t_3\right)\\ {\mathrm{\Δt}}_{\min }=f_2\left(t_3\right)\end{array}\right.---\left(16\right)$