CN101608921B - Pulsar/CNS integrated navigation method - Google Patents

Abstract

The invention relates to a pulsar/CNS integrated navigation method, belonging to the determination domain of the autonomous orbit of a spacecraft. By using the method, a UKF/H (infinity) filter is introduced, and navigation and position data provided by celestial navigation and pulsar navigation systems (CNS) is blended; and in the pulsar navigation system, an epoch difference method is provided so for eliminating clock errors and pulsar position errors. The method can obtain position information of the spacecraft with high precision, provides autonomous navigation with long time and high precision for the spacecraft so that the position precision of the spacecraft is not limited by a satellite-borne clock and a measuring device.

Description

A kind of pulsar/CNS Combinated navigation method

Technical field

The present invention relates to a kind of aerospacecraft autonomous navigation method.

Background technology

At present, be that the track of the aerospacecraft of representative is determined and forecast needs to rely on the support of ground observing and controlling net with the satellite, along with the rapid increase of the number of satellite that enters space, the burden of ground control station is more and more heavier.Support cost in order to reduce ground, or do not rely on the ground support, just require satellite to realize independent navigation.Autonomous navigation technology is to realize an autonomous importance of satellite, be the development trend of current satellite control technology, it to alleviate the ground observing and controlling burden, reduce the satellite transit expense, especially the aspects such as viability of deep space probe are all significant to improve the satellite viability.

In existing spacecraft navigate mode, having only celestial navigation system (CNS, CelestialNavigation System) and pulsar navigation is complete independent navigation mode.CNS relies on the azimuth information of heavenly body sensor observation celestial body to position navigation, and its positioning error and time are irrelevant, and the filtering cycle is short, but because the horizon instrument precision is low, this method positioning performance is relatively poor.Pulsar navigation is based on the pulse signal of pulsar emission and known pulsar positional information realizes the technology that spacecraft orbit is determined, the bearing accuracy height, but, measured the restriction of means, there is error in the pulsar positional information, and this error has a strong impact on the pulsar navigation system performance, this method bearing accuracy and pulse arrival time (Time of Arrival in addition, TOA) precision is relevant, and the TOA precision is influenced by clocking error, and clocking error increased along with time lapse.

CNS and pulsar navigation respectively have relative merits, if these two kinds and other multiple navigational system are combined, use the optimal estimation theory, and integrated navigation system is increased aspect precision.At application number is that 200710191527.7 Chinese patent file has been announced a kind of Combinated navigation method based on many information fusion, and this method has merged four kinds of different navigation information, wherein just includes CNS and pulsar navigation information.This method has improved bearing accuracy to a certain extent, but does not consider that the pulsar site error of pulsar and clocking error are to location influence.

Summary of the invention

The object of the present invention is to provide a kind of pulsar/CNS Combinated navigation method, effectively eliminate pulsar site error and clocking error, improved bearing accuracy.

A kind of pulsar/CNS Combinated navigation method, be specially: at t epoch, starlight elevation angle observed reading during t is as the input of Unscented kalman filtering device with satellite position estimated value that epoch, t-1 obtained and epoch, and the Unscented kalman filtering device is output as intermediate, satellite location estimation value; Judge epoch, whether t equaled N _pK, N _pBe the sampling period ratio of pulsar navigation and celestial navigation, k is a natural number, if be not equal to, then intermediate, satellite location estimation value is final satellite position estimated value, otherwise, with intermediate, satellite location estimation value with at N epoch _pThe pulse arrival time difference of pulsar navigation observation is as H during k _∞The input of wave filter, H _∞The output of wave filter is final satellite position estimated value;

At N epoch _pThe pulse arrival time of pulsar navigation observation is poor during k ${\stackrel{\‾}{Z}}_2\left(N_{p}k\right)={\tilde{Z}}_2(N_{p}k-N_p)+Z_2\left(N_{p}k\right)-Z_2(N_{p}k-N_p),$ Epoch N _pK-N _pThe time pulsar navigation the measurement reference value ${\tilde{Z}}_2(N_{p}k-N_p)=h_2(\tilde{X}(N_{p}k-N_p),N_{p}k-N_p),$ h ₂() is the pulsar navigation measurement equation, For to (N epoch _pK-N _p+ 1), (N _pK-N _p+ 2) ..., (N _pK-1) value that obtains after the smoothing processing, Z are made in the starlight elevation angle that obtains the time ₂(N _pK) and Z ₂(N _pK-N _p) be respectively pulsar navigation at N epoch _pK and N _pK-N _pThe actual pulse arrival time that measures is poor.

The present invention's advantage compared with prior art is:

(1) the present invention is when obtaining the pulse arrival time difference of pulsar navigation observation, utilized the actual measurement pulse of current epoch and last epoch to make calculus of differences, difference result makes clocking error and pulsar site error offset separately, both eliminated the clocking error influence, make pulsar navigation not be subjected to the restriction of spaceborne clock accuracy, reduced the cost of spaceborne clock; Eliminated the influence of pulsar site error again, made pulsar navigation not be subjected to the restriction of measurement technology means, made it under the prior art level, just can reach higher positioning accuracy.

(2) the present invention adopts UKF/H _∞The locating information that wave filter fusion celestial navigation and pulsar navigation provide is issued to pinpoint effect in the prerequisite that guarantees system robustness, and it is little to have calculated amount, real-time, and simplicity of design is easy to advantages such as Project Realization.

Description of drawings

Fig. 1 is pulsar of the present invention/CNS navigation flowcharts.

Embodiment

Be that the invention will be further described for example below with the satellite.

The satellite orbit kinetic model is expressed as

$\stackrel{\·}{X}\left(t\right)=f(X\left(t\right),t)+w\left(t\right)---\left(1\right)$

Wherein, t is epoch, state vector X (t)=[x y z v _xv _yv _z] ^T, v _xSpeed at x place, the position of X-direction, v _ySpeed at y place, the position of Y direction, v _zSpeed at z place, the position of Z-direction. $w\left(t\right)={\left[\begin{array}{cccccc}{w}_x& w_y& w_z& w_{v_x}& w_{v_y}& w_{v_z}\end{array}\right]}^T$ Be X (t)=[x y z v _xv _yv _z] ^TCorresponding state processing noise, its covariance matrix is Q, T represents the vector transposition.It is as follows that this model embodies formula:

$\left\{\begin{array}{c}\frac{\mathrm{dx}}{\mathrm{dt}}=v_x+w_x\\ \frac{\mathrm{dy}}{\mathrm{dt}}=v_y+w_y\\ \frac{\mathrm{dz}}{\mathrm{dt}}=v_z+w_z\\ \frac{{\mathrm{dv}}_x}{\mathrm{dt}}=-\mathrm{\μ}\frac{x}{{\left|r\right|}^3}[1-J_2{\left(\frac{R_e}{\left|r\right|}\right)}^2(7.5\frac{z^2}{{\left|r\right|}^2}-1.5)]+{\mathrm{\ΔF}}_x+w_{v_x}\\ \frac{{\mathrm{dv}}_t}{\mathrm{dt}}=-\mathrm{\μ}\frac{y}{{\left|r\right|}^3}[1-J_2{\left(\frac{R_e}{\left|r\right|}\right)}^2(7.5\frac{z^2}{{\left|r\right|}^2}-1.5)]+{\mathrm{\ΔF}}_y+w_{v_y}\\ \frac{{\mathrm{dv}}_z}{\mathrm{dt}}=-\mathrm{\μ}\frac{z}{{\left|r\right|}^3}[1-J_2{\left(\frac{R_e}{\left|r\right|}\right)}^2(7.5\frac{z^2}{{\left|r\right|}^2}-4.5)]+{\mathrm{\ΔF}}_z+w_{v_z}\end{array}\right.---\left(2\right)$

Wherein, r=[x, y, z] ^TBe the position vector of satellite with respect to the earth, the then distance between spacecraft and the earth centroid $\left|r\right|=\sqrt{x^2+y^2+z^2},$ μ is the terrestrial gravitation constant, J ₂Be humorous coefficient of second order band, R _eBe earth mean radius, Δ F _x, Δ F _y, Δ F _zBe respectively the influence of perturbative force on X, Y, three directions of Z axle, perturbative force include perturbation of earths gravitational field the high-order perturbing term, solar-lunar perturbate, solar radiation pressure perturbation and atmospheric perturbation.

Integrated navigation step to satellite is specially:

1) makes up celestial navigation and measure model

It is as follows that celestial navigation measures model:

$Z_1=h_1(X\left(t\right),t)+v_1\left(t\right)=\arccos (-\frac{r\·s}{\left|r\right|})-\arcsin \left(\frac{R_e}{\left|r\right|}\right)+v_1\left(t\right)---\left(3\right)$

Wherein, z ₁Be the observed quantity of the starlight elevation angle, s is the direction vector of fixed star, v ₁(t) be celestial navigation measurement noise, celestial navigation measurement equation h ₁(X (t), t) expression formula is as follows:

$h_1(X\left(t\right),t)=\arccos (-\frac{r\·s}{\left|r\right|})-\arcsin \left(\frac{R_e}{\left|r\right|}\right)---\left(4\right)$

2) make up pulsar navigation and measure model

(21) make up traditional pulsar navigation and measure model

Select Crab (Crab Nebula) pulsar as nautical star.The pulse TOA that satellite measures is t _SC, the corresponding time t that it is located at SSB (Solar System Barycentre, solar system barycenter) _bPulse arrival time difference t _b-t _SCWith the pass of satellite position be:

$t_b-t_{\mathrm{SC}}=\frac{1}{c}n\·r_{\mathrm{SC}}+\frac{1}{2c{D}_0}[-r_{\mathrm{SC}}^2+{(n\·r_{\mathrm{SC}})}^2-2b\·r_{\mathrm{SC}}+2(n\·b)(n\·r_{\mathrm{SC}})]+\frac{2{\mathrm{\μ}}_{\mathrm{Sum}}}{c^3}\ln |\frac{n\·r_{\mathrm{SC}}+r_{\mathrm{SC}}}{n\·b+b}+1|---\left(5\right)$

Wherein, c is the light velocity, D ₀Be the distance of pulsar to solar system barycenter, b is the position vector of SSB with respect to the sun, μ _SumBe the solar gravitation constant, n is the actual direction vector of pulsar, r _SCBe the position vector of satellite with respect to SSB, r _EBe the position vector of the earth with respect to SSB.

Satellite is with respect to the position vector r of SSB _SCSatisfy equation (6) with satellite with respect to the position vector r of the earth:

r＝r _SC-r _E (6)

Obtain traditional pulsar navigation measurement model in conjunction with equation (5) and (6), be expressed as:

z ₂＝h ₂(X(t)，t)+V ₂(t)＝t _b-t _SC+v ₂(t) (7)

Wherein, Z ₂Be pulse arrival time difference t-t _SC, v ₂(t) be the pulsar navigation measurement noise, pulsar navigation measurement equation h ₂(X (t), t) expression formula is as follows:

$h_2(X\left(t\right),t)=\frac{1}{c}\hat{n}\·r_{\mathrm{SC}}+\frac{1}{2c{D}_0}[-r_{\mathrm{SC}}^2+{(\hat{n}\·r_{\mathrm{SC}})}^2-2b\·r_{\mathrm{SC}}+2(\hat{n}\·b)(\hat{n}\·r_{\mathrm{SC}})]+\frac{2{\mathrm{\μ}}_{\mathrm{Sum}}}{c^3}\ln |\frac{\hat{n}\·r_{\mathrm{SC}}+r_{\mathrm{SC}}}{\hat{n}\·b+b}+1|---\left(8\right)$

Be the pulsar direction vector that measures.

(22) eliminate pulsar site error and the clocking error that pulsar navigation measures model.

Pulsar navigation measurement noise v ₂(t) can be expressed as:

v ₂(t)＝v _c(t)+v _p(t)+v _m(t) (9)

Wherein, v _c(t) be clock correction, v _m(t) be the TOA measurement noise, its variance is σ _R ²Variance can be calculated by Taylor (Taylor) method.v _p(t) system deviation that causes for the pulsar site error, but approximate representation is

$v_p\left(t\right)=(\hat{n}-n)\·r_{\mathrm{SC}}/c---\left(10\right)$

From formula (9) as can be seen, the pulsar navigation measurement noise is made up of three parts.Wherein, v _m(t) be white noise, available filtering method is handled, v _c(t) and v _p(t) can regard normal value deviation at short notice as, can not eliminate its influence with the method for filtering, core of the present invention place proposes between epoch method of difference exactly to eliminate v _c(t) and v _p(t).

For ease of explanation, introduce the implication of sampling period ratio earlier.Sampling period is compared N _pBe calculated as follows:

N _p＝P _p/P _CNS (11)

Wherein, P _pAnd P _CNSBe respectively the sampling period of pulsar navigation and CNS.Select suitable P _pAnd P _CNS, make N _pBe integer.That is to say that each epoch is the may observe starlight elevation angle all, only at N epoch _pIt is poor that k (k is a natural number) just can observe pulse arrival time.

Between epoch in the method for difference, the estimated value that once measures before at first calculating is added this poor with preceding actual amount survey, can obtain one and not contain v _c(t) and v _p(t) new measurement, specific as follows:

Step 221: the starlight elevation angle measurement information that provides according to the celestial navigation mode, (document S.Sarkka sees reference to adopt Unscented (no mark) Rauch-Tung-Striebel smoother, " Unscented Rauch-Tung-Striebel Smoother ", IEEE Transaction on AutomaticControl, Vol.53, No.3, pp845-849,2008.) processing (N epoch _pK-N _p+ 1)～(N _pK-1) state estimation the time can obtain the reference state of satellite position

With the reference state variance

Thereby obtain N epoch _pK-N _pThe time pulsar navigation the measurement reference value

${\tilde{Z}}_2(N_{p}k-N_p)=h_2(\tilde{X}(N_{p}k-N_p),N_{p}k-N_p)$

$=h_2(X(N_{p}k-N_p),N_{p}k-N_p)+v_{\mathrm{process}}(N_{p}k-N_p)---\left(12\right)$

Wherein, v _ProcessBe that average is 0, variance is R _EstNoise.

$R_{\mathrm{est}}=E\left[v_p(N_{p}k-N_p)v_p(N_{p}k-N_p)\right]=H^T(N_{p}k-N_p)\tilde{P}(N_{p}k-N_p)H(N_{p}k-N_p)---\left(13\right)$

Wherein, H () is for measuring matrix, E[] the expression mathematical expectation.

$H(N_{p}k-N_p)=\frac{{\∂h}_2(X\left(t\right),t)}{\∂X\left(t\right)}{|}_{X\left(t\right)=\tilde{X}(N_{p}k-N_p)}---\left(14\right)$

Step 222: the new measurement model of structure pulsar navigation ${\stackrel{\‾}{Z}}_2\left(N_{p}k\right)={\tilde{Z}}_2(N_{p}k-N_p)+Z_2\left(N_{p}k\right)-Z_2(N_{p}k-N_p)$

Wherein, z ₂(N _pK-N _p) and Z ₂(N _pK) respectively the expression before once with this actual amount measured value.Further derive, obtain

${\stackrel{\‾}{Z}}_2\left(N_{p}k\right)={\tilde{Z}}_2(N_{p}k-N_p)+Z_2\left(N_{p}k\right)-Z_2(N_{p}k-N_p)$

$=h_2(X(N_{p}k-N_p),N_{p}k-N_p)+(t_b\left(N_{p}k\right)-t_{\mathrm{SC}}\left(N_{p}k\right))-(t_b(N_{p}k-N_p)-t_{\mathrm{SC}}(N_{p}k-N_p))+{\stackrel{\‾}{v}}_2\left(N_{p}k\right)$

$=t_b(N_{p}k-N_p)-t_{\mathrm{SC}}(N_{p}k-N_p)+t_b\left(N_{p}k\right)-t_{\mathrm{SC}}\left(N_{p}k\right)-(t_b(N_{p}k-N_p)-t_{\mathrm{SC}}(N_{p}k-N_p))+{\stackrel{\‾}{v}}_2\left(N_{p}k\right)$

$=h_2(X\left(N_{p}k\right),N_{p}k)+{\stackrel{\‾}{v}}_2\left(N_{p}k\right)---\left(15\right)$

New measurement noise is a coloured noise, and its expression formula is as follows:

${\stackrel{\‾}{v}}_2\left(N_{p}k\right)=v_{\mathrm{process}}(N_{p}k-N_p)+v_2\left(N_{p}k\right)-v_2(N_{p}k-N_p)$

${=v}_{\mathrm{process}}(N_{p}k-N_p)+v_m\left(N_{p}k\right)-v_m(N_{p}k-N_p)$

As can be seen, new measurement noise

With clock correction v _c(t) and pulsar site error v _p(t) irrelevant, overcome the influence of clock correction and pulsar site error to whole navigation.

The pulsar navigation that the present invention adopts measures

The pulsar navigation noise variance $R_2={2\mathrm{\σ}}_R^2+R_{\mathrm{est}}.$

(4) adopt UKF/H _∞Wave filter merges navigation information

The present invention proposes UKF/H _∞Wave filter merges navigation data.This wave filter is by UKF and H _∞Wave filter is in series, as shown in Figure 1.At UKF/H _∞In the wave filter, H _∞The state processing noise of wave filter and state-transition matrix are respectively 0 and unit matrix.

CNS and pulsar navigation subsystem adopt UKF and H respectively _∞Wave filter, promptly the starlight elevation angle and based on the pulse arrival time difference of method of difference between epoch not as UKF and H _∞The measurement of wave filter.Concrete filtering flow process is in two kinds of situation: (1) no pulse is poor time of arrival.During pulse observation, only CNS moves and the output state estimated value.(2) there is pulse arrival time poor.In case it is poor to have produced a pulse arrival time, utilizes the pulse arrival time difference that the state estimation value of CNS output is upgraded, thereby obtain high-precision state estimation value.From above flow process as can be seen, the navigation information that celestial navigation and pulsar navigation provide is all revised state value, has realized information fusion.

UKF/H _∞The detailed process of Filtering Processing is as follows:

Given system model:

X(t+1)＝f(X(t)，t)+w(t) (16)

Z ₁(t)＝h ₁(X(t)，t)+v ₁(t) (17)

${\stackrel{\‾}{Z}}_2\left(t\right)=h_2(X\left(t\right),t)+{\stackrel{\‾}{v}}_t\left(t\right)---\left(18\right)$

T is t epoch.

Step 41:UKF filtering (celestial navigation)

The navigation information that utilizes celestial navigation to provide according to the operation of the UKF filtering shown in formula (19)～(34), can obtain the state estimation of satellite.During pulse TOA observation, pulsar navigation can't provide navigation information, need not execution in step 42.

(a) initialization

$\hat{X}\left(0\right)=E\left[X\left(0\right)\right]---\left(19\right)$

$P\left(0\right)=E\left[{(X\left(0\right)-\hat{X}\left(0\right))\left(X\left(0\right)\hat{X}\left(0\right)\right))}^T\right]---\left(20\right)$

Wherein, E[] the expression mathematical expectation.

(b) calculate SIGMA (sigma) point

$\mathrm{\χ}(t-1)=\left[\begin{array}{ccc}\hat{X}(t-1)& \hat{X}(t-1)+\sqrt{m+\mathrm{\τ}}{\left(\sqrt{P(t-1)}\right)}_i& \hat{X}(t-1)-\sqrt{m+\mathrm{\τ}}(\sqrt{P(t-1)}{)}_i\end{array}\right]---\left(21\right)$

W ₀＝τ/(m+τ) (22)

W _i＝1/[2(m+τ)] (23)

W _i+m＝1/[2(m+τ)] (24)

Wherein, τ ∈ R, It is matrix square root

I row, i=1,2 ... m, m are state vector Dimension.

(c) time upgrades

χ(t/t-1)＝f(χ(t-1)，t-1) (25)

Wherein, χ (t/t-1) expression t-1 one-step prediction SIGMA point constantly.

$\hat{X}\left(t\right)=\underset{i=0}{\overset{2m}{\mathrm{\Σ}}}{W}_i\mathrm{\χ}(t/t-1)---\left(26\right)$

$P\left(t\right)=\mathrm{\Σ}{W}_i{[{\mathrm{\χ}}_i(t/t-1)-\hat{X}\left(t\right)][{\mathrm{\χ}}_i(t/t-1)-\hat{X}\left(t\right)]}^T+Q\left(t\right)---\left(27\right)$

Wherein, χ _i(t/t-1) i of expression χ (t/t-1) row.

Z ₁(t/t-1)＝h(χ(t/t-1)，t)

Wherein, Z ₁(t/t-1) expression t-1 one-step prediction measuring value constantly.

${\hat{Z}}_1\left(t\right)=\underset{i=0}{\overset{2m}{\mathrm{\Σ}}}{W}_i{Z}_1^i(t/t-1)---\left(29\right)$

Wherein, Z ₁ ⁱ(t/t-1) expression Z ₁(t/t-1) i row.

(d) measure renewal

$P_{\hat{z}\hat{z}}\left(t\right)=\underset{i=0}{\overset{2m}{\mathrm{\Σ}}}{W}_i[Z_1^i(t/t-1)-{\hat{Z}}_1\left(t\right)]{[Z_1^i(t/t-1)-{\hat{Z}}_1\left(t\right)]}^T+R_1\left(t\right)---\left(30\right)$

$P_{\mathrm{xz}}\left(t\right)=\underset{i=0}{\overset{2m}{\mathrm{\Σ}}}{W}_i{[{\mathrm{\χ}}_i(t/t-1)-\hat{X}\left(t\right)][Z_1^i(t/t-1)-{\hat{Z}}_1\left(t\right)]}^T---\left(31\right)$

$G_1\left(t\right)=P_{\mathrm{xz}}\left(t\right)P_{\hat{z}\hat{z}}^{-1}\left(t\right)---\left(32\right)$

$\hat{X}\left(t\right)=\hat{X}\left(t\right)+G_1\left(t\right)(Z_1\left(t\right)-{\hat{Z}}_1\left(t\right))---\left(33\right)$

$P\left(t\right)=P\left(t\right)-G_1\left(t\right)P_{\hat{z}\hat{z}}\left(t\right)G^T\left(t\right)---\left(34\right)$

Wherein, Q (t) and R ₁(t) be respectively processing noise and celestial navigation measurement noise covariance matrix.When X (t) is Gaussian distribution, get m+ τ=3 usually, T representing matrix transposition.

Step 42:H _∞Filtering (pulsar navigation)

In case it is poor to have obtained a pulse arrival time, then carries out this step.Utilize the pulse arrival time difference that the corrected state estimation of celestial navigation is upgraded, promptly handle according to formula (35)～(38).Like this, the information that celestial navigation and pulsar navigation provide is all upgraded the state estimation of satellite, has also just realized information fusion.

Estimation procedure is as follows:

$G_2\left(t\right)=P\left(t\right){[I-\mathrm{\θP}\left(t\right)+H^T\left(t\right)R_2^{-1}\left(t\right)H\left(t\right)P\left(t\right)]}^{-1}{H}^T\left(t\right)R_2^{-1}\left(t\right)---\left(35\right)$

$P\left(t\right)=P\left(t\right){[I-\mathrm{\θP}\left(t\right)+H^T\left(t\right)R_2^{-1}\left(t\right)H\left(t\right)P\left(t\right)]}^{-1}---\left(36\right)$

$\hat{X}\left(t\right)=\hat{X}\left(t\right)+G_2\left(t\right)({\stackrel{\‾}{Z}}_2\left(t\right)-h_2(\hat{X}\left(t\right),t))---\left(37\right)$

Wherein, θ is user's preset threshold, and span is 10 ^-1～10 ^-6R ₂(t) be pulsar navigation measurement noise covariance matrix.

For this scheme is separated, must meet the following conditions.

$P^{-1}\left(t\right)-\mathrm{\θI}+H^Y\left(t\right)R_2^{-1}\left(t\right)H\left(t\right)>0---\left(38\right)$

Wherein, I is a unit matrix.

The content that is not described in detail in the instructions of the present invention belongs to this area professional and technical personnel's known prior art.

Claims (1)

1. pulsar/celestial navigation system Combinated navigation method, be specially: at t epoch, starlight elevation angle observed reading during t is as the input of Unscented kalman filtering device with satellite position estimated value that epoch, t-1 obtained and epoch, and the Unscented kalman filtering device is output as intermediate, satellite location estimation value; Judge epoch, whether t equaled N _pK, N _pBe the sampling period ratio of pulsar navigation and celestial navigation, k is a natural number, if be not equal to, then intermediate, satellite location estimation value is final satellite position estimated value, otherwise, with intermediate, satellite location estimation value with at N epoch _pThe pulse arrival time difference of pulsar navigation observation is as H during k _∞The input of wave filter, H _∞The output of wave filter is final satellite position estimated value;

At N epoch _pThe pulse arrival time of pulsar navigation observation is poor during k

Epoch N _pK-N _pThe time pulsar navigation the measurement reference value

h ₂() measures expression formula for pulsar navigation,

For to (N epoch _pK-N _p+ 1), (N _pK-N _p+ 2) ..., (N _pK-1) value that obtains after the smoothing processing, Z are made in the starlight elevation angle that obtains the time ₂(N _pK) and Z ₂(N _pK-N _p) be respectively pulsar navigation at N epoch _pK and N _pK-N _pThe actual pulse arrival time that measures is poor.

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