CN103198187A - Track design method of deep space probe and based on differential modification - Google Patents

Abstract

A track design method of a deep space probe and based on differential modification relates to the field of track design of the deep space probe. The track design method includes the following steps: calculating an initial value according to track parameter of the deep space probe and determined by a genetic algorithm or a Pork Chop Plots method, conducting track numerical integration operation under an accurate kinetic model according to the initial value of control parameter to obtain a terminal parameter value, comparing the parameter value obtained by calculation with a standard parameter to obtain a parameter deviation value to calculate a new control parameter according, utilizing the new control parameter to conduct track integral operation on the kinetic model again to obtain a new terminal parameter value deviation value and repeating the above process till the terminal parameter meets the requirement for accuracy. The track design method breaks up a partial derivative array into three portions, and provides a specific expression form. The new track parameter obtained by calculation meets any requirement for accuracy. The partial derivative array is suitable for different requirements for navigation calculation and error analysis and the like.

Description

Rail design method based on the deep space probe of differential correction

Technical field

The present invention relates to deep space probe track design field, be specifically related to the accurate rail design method based on the deep space probe of differential correction.

Background technology

The accurate track design of deep space probe is the initial value that obtains according to primary design, utilizes accurate kinetic model and numerical method to find the solution the process of nominal track.It is a two-point boundary value problem, i.e. Lambert problem, and great majority carry out the differential modification method based on intelligent optimization algorithm and partial derivative information and calculate both at home and abroad at present.Wherein the accurate track design based on the differential revised law is widely used, and in business softwares such as STK, be applied, wherein partial derivative matrix finds the solution the methods that great majority adopt state transitions, its fast convergence rate, but high to the initial value accuracy requirement.Therefore says, design and a kind ofly have the method for finding the solution partial derivative matrix that precision is new preferably and be applicable to that navigate calculating and error analysis are imperative.Disome model and approximation method are only adopted in the preliminary track design of deep space probe, and its precision can not satisfy the requirement of aerial mission, and accurately the track design is to utilize the track initial value to find the solution the nominal track under the standard power model.A large amount of scholar have proposed big metering method one after another both at home and abroad, wherein are used widely based on the rapid solving method of differential correction, and are comparatively complicated based on finding the solution of the partial derivative matrix in the differential correction.

Summary of the invention

That the accurate rail design method that the invention solves existing deep space probe exists is high to the initial value accuracy requirement, cause the bigger problem of track design accuracy error, and then a kind of rail design method of the deep space probe based on the differential correction is provided.Utilized a kind of new partial derivative matrix method for solving in determining based on the accurate track of the deep space probe of differential correction, the partial derivative matrix method for solving is a kind of differential correction.

The present invention solves the problems of the technologies described above the technical scheme of taking to be:

A kind of rail design method of the deep space probe based on the differential correction, the detailed process of described method is:

The orbit parameter of step 1, the deep space probe determined by genetic algorithm or Pork Chop Plots method is calculated initial value: mainly comprise earth escape orbit parameter, canonical parameter (B _T0, B _R0, T _Tof0), choose orbit inclination i, the orbit radius r of earth escape orbit _p, track initial velocity v _p, rise focus right ascension Ω and argument of perigee ω and be the control parameter; Subscript " ₀" the expression value;

Step 2, according to control parameter be that initial value carries out the integral operation of track numerical value under kinetic model, try to achieve the terminal parameter value, terminal parameter is chosen B plane parameter (B _R, B _T) and time of arrival T _Tof

The B plane is defined as the barycenter of the celestial body that overdoes and enters asymptotic plane perpendicular to Mars probes, be that the B plane is perpendicular to the infinite distance velocity reversal (as shown in Figure 1) of Mars probes, the target component of mars exploration task middle orbit adopts the B plane parameter in the B plane coordinate system, the initial point of its B plane coordinate system is selected in the Mars center, the unit vector of note detector injection asymptotic line direction is the S axle, chooses the normal direction of mars equatorial

,

With

Multiplication cross be the T axle, R axle and S axle and T axle constitute right-handed coordinate system, namely

$S=\frac{v_{\∞}}{\left|\right|v_{\∞}\left|\right|},T=\frac{S\×N}{\left|\right|S\×N\left|\right|},R=S\×T$

Usually vector B is defined as Mars barycenter directed towards detector track and B plane point of intersection, and its parameter is generally elected it as at the component of T axle and R axle, namely

B _T＝B·T

B _R＝B·R

Step 3, the parameter value that calculates and canonical parameter compare, and obtain parameter error amount (Δ B _T, Δ B _R, Δ T _Tof), thereby try to achieve new control parameter value;

Step 4, utilize new control parameter value again kinetic model to be carried out the orbit integration computing, obtain new terminal parameter value deviation;

Step 5, judge whether the departure reduce gradually satisfies accuracy requirement, if finish computing, otherwise execution in step two is to step 5, is reduced to up to departure and satisfies accuracy requirement.

In step 5, described accuracy requirement refers to that the periareon height error is in ± 50m scope.

In step 1, the position of selection perigee and periareon and speed are used P as the initial time parameter of deep space probe ₀Expression, P ₀Calculate by five parameters of earth escape orbit; The terminal parameter of deep space probe is chosen as the inclination angle of target track, nearly heart distance and B plane parameter; P ₀Five parameter: orbit inclination i, the orbit radius r of earthing ball escape orbit _p, track initial velocity v _p, rise focus right ascension Ω and argument of perigee ω and calculate; Deep space probe arrives at the terminal parameter note of target interval and is Q, then between deep space probe original state and the end of a period state with representing as minor function, namely

Q＝f(P)

Actual track is only kept linear term after carrying out Taylor expansion near the nominal track,

ΔQ＝KΔP

Wherein, Δ P is controlled variable, partial derivative matrix

Detailed process to the numerical solution of partial derivative matrix is:

If deep space probe is respectively X at the state matrix that speed and the position of perigee and periareon constitute _{6 * 1}(0), X _{6 * 1}(t), determine as if the set out moment and flight time t, then P, X _{6 * 1}(0), X _{6 * 1}(t) and the pass that exists of Q be

Q＝f _q(X _6×1(t))

X _6×1(t)＝f _t(X _6×1(0))

X _6×1(0)＝f _p(P)

F wherein _q, f _t, f _pBe multivariate function group, by multivariate function rule, consider the single order item,

ΔQ＝KqΔX _6×1(t)

ΔX _6×1(t)＝K _tΔX _6×1(0)

ΔX _6×1(0)＝K _pΔP

Namely

ΔQ＝(K _q·K _t·K _p)ΔP

K wherein _q, K _t, K _pBe the matrix that the single order partial derivative constitutes, then the partial derivative matrix of the relative P of Q is:

K＝K _q·K _t·K _p

K _q, K _t, K _pConcrete solution procedure is as follows: K wherein _pBe about earth escape orbit inclination angle i, orbit radius r _p, the earth's core transfer orbit initial velocity v _p, rise the function of focus right ascension Ω and argument of perigee ω, K _qBe about terminal B plane parameter (B _R, B _t) and the function of time of arrival:

$K_p=\left[\begin{array}{ccccc}\frac{{\∂r}_0}{{\∂r}_p}& \frac{{\∂r}_0}{{\∂v}_p}& \frac{{\∂r}_0}{\∂i}& \frac{{\∂r}_0}{\∂\mathrm{\Ω}}& \frac{{\∂r}_0}{\∂\mathrm{\ω}}\\ \frac{{\∂v}_0}{{\∂r}_p}& \frac{{\∂v}_0}{{\∂v}_p}& \frac{{\∂v}_0}{\∂i}& \frac{{\∂v}_0}{\∂\mathrm{\Ω}}& \frac{{\∂v}_0}{\∂\mathrm{\ω}}\end{array}\right]$

K _tBe state-transition matrix, its each element value can adopt numerical integration to try to achieve:

$K_t=\left[\begin{array}{cc}\frac{{\∂r}_t}{{\∂r}_0}& \frac{{\∂r}_t}{{\∂v}_0}\\ \frac{{\∂v}_t}{{\∂r}_0}& \frac{{\∂v}_t}{{\∂v}_0}\end{array}\right]=\left[\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]$

$\stackrel{\·}{K_t}=\left[\begin{array}{cc}\frac{{\∂v}_t}{{\∂r}_0}& \frac{{\∂v}_t}{{\∂v}_0}\\ \frac{{\∂a}_t}{{\∂r}_0}& \frac{{\∂a}_t}{{\∂v}_0}\end{array}\right]=\left[\begin{array}{cc}\frac{{\∂v}_t}{{\∂r}_0}& \frac{{\∂v}_t}{{\∂v}_0}\\ \frac{{\∂a}_t}{{\∂r}_t}\·\frac{{\∂r}_t}{{\∂r}_0}& \frac{{\∂a}_t}{{\∂r}_t}\·\frac{{\∂r}_t}{{\∂v}_0}\end{array}\right]=\left[\begin{array}{cc}{K}_{21}& K_{22}\\ \frac{{\∂a}_t}{{\∂r}_t}{K}_{11}& \frac{{\∂a}_t}{{\∂r}_t}{K}_{12}\end{array}\right]$

K _qBy descending or trying to achieve:

$K_q=\left[\begin{array}{cc}\frac{{\∂B}_T}{{\∂r}_t}& \frac{{\∂B}_T}{{\∂v}_t}\\ \frac{{\∂B}_R}{{\∂r}_t}& \frac{{\∂B}_R}{{\∂v}_t}\\ \frac{{\∂T}_{\mathrm{tof}}}{{\∂r}_t}& \frac{{\∂T}_{\mathrm{tof}}}{{\∂v}_t}\end{array}\right]$

If be respectively r with respect to the position of target star and the component of speed during deep space probe arrival flight time terminal point _t=[r _xr _yr _z], v _t=[v _xv _yv _z], then

$v_{\∞}=\sqrt{v_t^2-\frac{{2\mathrm{\μ}}_t}{r_t}}$

By h=B * v _∞=r _t* v _t, as can be known

$v_{\∞}\×(B\×v_{\∞})=v_{\∞}^{2}B=v_{\∞}\×(r_t\×v_t)=(v_{\∞}\·v_t)r_t-(v_{\∞}\·r_t)v_t$

obtain

$B=\frac{(v_{\∞}\·v_t)r_t-(v_{\∞}\·r_t)v_t}{v_{\∞}^2}$

Formula $B=\frac{(v_{\∞}\·v_t)r_t-(v_{\∞}\·r_t)v_t}{v_{\∞}^2}$ To r _t, v _tAsk local derviation,

$\frac{\∂B}{{\∂r}_m}=\frac{1}{v_{\∞}^2}\left[\begin{array}{ccc}-v_{y\∞}{r}_y-v_{z\∞}{r}_z& v_{y\∞}{r}_x& v_{z\∞}{r}_x\\ v_{x\∞}{r}_y& -v_{x\∞}{r}_x-v_{z\∞}{r}_z& v_{z\∞}{r}_y\\ v_{x\∞}{r}_z& v_{y\∞}{r}_z& -v_{x\∞}{r}_x-v_{y\∞}{r}_y\end{array}\right]$

$\frac{\∂B}{{\∂v}_m}=\frac{1}{v_{\∞}^2}\left[\begin{array}{ccc}{v}_{y\∞}{v}_y+v_{z\∞}{v}_z& {-v}_{y\∞}{v}_x& {-v}_{z\∞}{v}_x\\ -v_{x\∞}{v}_y& v_{x\∞}{v}_x+v_{z\∞}{v}_z& {-v}_{z\∞}{v}_y\\ -v_{x\∞}{v}_z& {-v}_{y\∞}{v}_z& v_{x\∞}{v}_x+v_{y\∞}{v}_y\end{array}\right]$

By

$\begin{array}{cc}\frac{\∂B\·T}{{\∂r}_t}=\frac{\∂B}{{\∂r}_t}\·T+\frac{\∂T}{{\∂r}_t}\·B& \frac{\∂B\·T}{{\∂v}_t}=\frac{\∂B}{{\∂v}_t}\·T+\frac{\∂T}{{\∂v}_t}\·B\end{array}$

$\begin{array}{cc}\frac{\∂B\·R}{{\∂r}_t}=\frac{\∂B}{{\∂r}_t}\·R+\frac{\∂R}{{\∂r}_t}\·B& \frac{\∂B\·R}{{\∂v}_t}=\frac{\∂B}{{\∂v}_t}\·R+\frac{\∂R}{{\∂v}_t}\·B\end{array}$

Because the B plane coordinate system is with r _t, v _tChange lessly, think

So following formula can turn to

$\begin{array}{cc}\frac{\∂B\·T}{{\∂r}_t}=\frac{\∂B}{{\∂r}_t}\·T& \frac{\∂B\·T}{{\∂v}_t}=\frac{\∂B}{{\∂v}_t}\·T\end{array}$

$\begin{array}{cc}\frac{\∂B\·R}{{\∂r}_t}=\frac{\∂B}{{\∂r}_t}\·R& \frac{\∂B\·R}{{\∂v}_t}=\frac{\∂B}{{\∂v}_t}\·R\end{array}$

By $T_L=\sqrt{\frac{a^3}{{\mathrm{\μ}}_m}}(\frac{R}{a}-\ln \frac{2R}{a})$

$\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;r}_t}+\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\&CenterDot;\frac{{\&PartialD;v}_{\&infin;}}{{\&PartialD;r}_t}=\left[\begin{array}{c}\frac{r_x}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_x}{v_{\&infin;}^3{\mathrm{<figure-callout\; id="2"\; label="r\; t"\; filenames="HDA00003003299500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_t^{}}+\frac{{\&PartialD;T}_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_x}{v_{\&infin;}{r}_t^3}\\ \frac{r_y}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_y}{v_{\&infin;}^3{\mathrm{<figure-callout\; id="2"\; label="r\; t"\; filenames="HDA00003003299500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_t^{}}+\frac{{\&PartialD;T}_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_y}{v_{\&infin;}{r}_t^3}\\ \frac{r_z}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_z}{v_{\&infin;}^3{\mathrm{<figure-callout\; id="2"\; label="r\; t"\; filenames="HDA00003003299500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_t^{}}+\frac{\&PartialD;T_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_z}{v_{\&infin;}{r}_t^3}\end{array}\right]$

$\frac{\&PartialD;T_{\mathrm{tof}}}{{\&PartialD;v}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\&CenterDot;\frac{{\&PartialD;v}_{\&infin;}}{{\&PartialD;v}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\left[\begin{array}{c}\frac{{\mathrm{\&mu;}}_s{v}_x}{v_{\&infin;}{v}_t}\\ \frac{{\mathrm{\&mu;}}_s{v}_y}{v_{\&infin;}{v}_t}\\ v_{\&infin;}{v}_t\end{array}\right]$

$\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}=-\frac{r_t}{v_{\&infin;}^2}+\frac{{3\mathrm{\&mu;}}_s}{v_{\&infin;}^4}\ln \frac{{2r}_m{v}_{\&infin;}^2}{{\mathrm{\&mu;}}_s}-\frac{{2\mathrm{\&mu;}}_m}{v_{\&infin;}^4}$

So far obtain K _q

r _t, v _tThe expression deep space probe arrives flight time position and speed with respect to target star during terminal point.

In step 2, the expression formula of described kinetic model is:

$-\frac{{\mathrm{\&mu;}}_e}{r^2}r-{\mathrm{\&mu;}}_a(\frac{r_{\mathrm{ad}}}{r_{\mathrm{ad}}^3}+\frac{r_a}{r_a^3})-{\mathrm{\&mu;}}_s(\frac{r_{\mathrm{sd}}}{r_s^3}+\frac{r_s}{r_s^3})$

Wherein subscript e represents the earth, and a represents the sun, and s represents Mars, and a represents the acceleration of deep space probe.

The invention has the beneficial effects as follows:

The inventive method resolves into three parts with partial derivative matrix, and has provided the form of embodying, and satisfies mission requirements by calculating new orbit parameter precision, and the partial derivative matrix of this invention is applicable to that navigation calculates and different demand such as error analysis.Adopt the track design of the deep space probe of the inventive method, make track design accuracy error little, precision has satisfied the requirement of aerial mission.The inventive method can be applicable to different demands such as navigation calculating, and requires not high enough to initial value.The present invention is applicable to survey of deep space task in the solar system.

Description of drawings

Fig. 1 is the B floor map, among the figure: 1 expression objective plane (B plane), 2 expression target celestial bodies, 3 expression detector flight paths, 4 expression error ellipses, 5 expressions enter the asymptotic line direction; Fig. 2 is the process flow diagram of accurate rail design method of the present invention.

Embodiment

As depicted in figs. 1 and 2, the implementation procedure of the rail design method of the described deep space probe based on the differential correction of present embodiment (based on the accurate rail design method of the deep space probe of B plane parameter) is:

The orbit parameter of step 1, the deep space probe determined by genetic algorithm or Pork Chop Plots method is calculated initial value: mainly comprise earth escape orbit parameter, canonical parameter (B _T0, B _R0, T _Tof0), choose orbit inclination i, the orbit radius r of earth escape orbit _p, track initial velocity v _p, rise focus right ascension Ω and argument of perigee ω and be the control parameter; Subscript " ₀" the expression value;

Step 2, according to control parameter be that initial value carries out the integral operation of track numerical value under kinetic model, try to achieve the terminal parameter value, terminal parameter is chosen B plane parameter (B _R, B _T) and time of arrival T _Tof

The B plane is defined as the barycenter of the celestial body that overdoes and enters asymptotic plane perpendicular to Mars probes, be that the B plane is perpendicular to the infinite distance velocity reversal (as shown in Figure 1) of Mars probes, the target component of mars exploration task middle orbit adopts the B plane parameter in the B plane coordinate system, the initial point of its B plane coordinate system is selected in the Mars center, the unit vector of note detector injection asymptotic line direction is the S axle, and the unit vector of getting certain reference direction is

, its in theory direction be arbitrarily, but generally elect the normal direction of mars equatorial as,

With

Multiplication cross be the T axle, R axle and S axle and T axle constitute right-handed coordinate system, namely

$S=\frac{v_{\&infin;}}{\left|\right|v_{\&infin;}\left|\right|},T=\frac{S\&times;N}{\left|\right|S\&times;N\left|\right|},R=S\&times;T$

Usually vector B is defined as Mars barycenter directed towards detector track and B plane point of intersection, and its parameter is generally elected it as at the component of T axle and R axle, namely

B _T＝B·T

B _R＝B·R

Step 3, the parameter value that calculates and canonical parameter compare, and obtain parameter error amount (Δ B _T, Δ B _R, Δ T _Tof), thereby try to achieve new control parameter value;

Step 4, utilize new control parameter value again kinetic model to be carried out the orbit integration computing, obtain new terminal parameter value deviation; This departure reduces gradually;

Step 5, judge whether the departure reduce gradually satisfies accuracy requirement (periareon height error ± 50m scope in), if the end computing, otherwise execution in step two is to step 5, is reduced to up to departure and satisfies accuracy requirement.

In the survey of deep space task, deep space probe from earth escape orbit till meet with the target astrology, angle from fuel consumption, survey of deep space is revised midway and is not track is modified to the nominal track, but apply a proper speed increment in the position that error is arranged, thereby make deep space probe satisfy requirement to the end of a period state along a new orbit maneuver.The terminal parameter of deep space probe is typically chosen in the inclination angle of target track, nearly heart distance and B plane parameter.Choose terminal parameter, the note initial time is P ₀(selecting perigee and position, periareon and speed is variable, can try to achieve by earth escape orbit parameter).Deep space probe arrives at the terminal parameter note of target interval and is Q, then can represent with certain function between deep space probe original state and the end of a period state, namely

Q＝f(P) (0.1)

Actual track is only kept linear term after carrying out Taylor expansion near the nominal track,

ΔQ＝KΔP (0.2)

Wherein, Δ P is controlled variable, partial derivative matrix

Numerical solution to partial derivative matrix is elaborated below:

If deep space probe is respectively X at the state matrix that speed and the position of perigee and periareon constitute _{6 * 1}(0), X _{6 * 1}(t), determine as if the set out moment and flight time t, then P, X _{6 * 1}(0), X _{6 * 1}(t) and the pass that exists of Q be

Q＝f _q(X _6×1(t))

X _6×1(t)＝f _t(X _6×1(0)) (0.3)

X _6×1(0)＝f _p(P)

F wherein _q, f _t, f _pBe multivariate function group, by multivariate function rule, consider the single order item,

ΔQ＝K _qΔX _6×1(t)

ΔX _6×1(t)＝K _tΔX _6×1(0) (0.4)

ΔX _6×1(0)＝K _pΔP

Namely

ΔQ＝(K _q·K _t·K _p)ΔP

K wherein _q, K _t, K _pBe the matrix that the single order partial derivative constitutes, then the partial derivative matrix of the relative P of Q

K＝K _q·K _t·K _p (0.5)

Provide K below _q, K _t, K _pConcrete solution formula, wherein P is about earth escape orbit inclination angle i, orbit radius r _p, the earth's core transfer orbit initial velocity v _p, rise the function of focus right ascension Ω and argument of perigee ω, Q is about terminal B plane parameter (B _R, B _T) and the function of time of arrival.

$K_p=\left[\begin{array}{ccccc}\frac{{\&PartialD;r}_0}{{\&PartialD;r}_p}& \frac{{\&PartialD;r}_0}{{\&PartialD;v}_p}& \frac{{\&PartialD;r}_0}{\&PartialD;i}& \frac{{\&PartialD;r}_0}{\&PartialD;\mathrm{\&Omega;}}& \frac{{\&PartialD;r}_0}{\&PartialD;\mathrm{\&omega;}}\\ \frac{{\&PartialD;v}_0}{{\&PartialD;r}_p}& \frac{{\&PartialD;v}_0}{{\&PartialD;v}_p}& \frac{{\&PartialD;v}_0}{\&PartialD;i}& \frac{{\&PartialD;v}_0}{\&PartialD;\mathrm{\&Omega;}}& \frac{{\&PartialD;v}_0}{\&PartialD;\mathrm{\&omega;}}\end{array}\right]---\left(0.6\right)$

Concrete computation process is:

$\frac{{\&PartialD;r}_0}{{\&PartialD;r}_p}={\left[\begin{array}{ccc}\cos \mathrm{\&Omega;}\cos \mathrm{\&omega;}-\sin \mathrm{\&Omega;}\cos i\sin \mathrm{\&omega;}& \sin \mathrm{\&Omega;}\cos \mathrm{\&omega;}+\cos \mathrm{\&Omega;}\cos i\sin \mathrm{\&omega;}& \sin i\cos \mathrm{\&omega;}\end{array}\right]}^T$

$\frac{{\&PartialD;r}_0}{{\&PartialD;v}_p}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$

$\frac{\&PartialD;r_0}{\&PartialD;i}=r_p{\left[\begin{array}{ccc}\sin \mathrm{\&Omega;}\sin i\sin \mathrm{\&omega;}& -\cos \mathrm{\&Omega;}\sin i\sin \mathrm{\&omega;}& \cos i\cos \mathrm{\&omega;}\end{array}\right]}^T$

${\frac{{\&PartialD;r}_0}{\&PartialD;\mathrm{\&Omega;}}=r_p\left[\begin{array}{ccc}-\sin \mathrm{\&Omega;}\cos \mathrm{\&omega;}-\cos \mathrm{\&Omega;}\cos i\sin \mathrm{\&omega;}& \cos \mathrm{\&Omega;}\cos \mathrm{\&omega;}-\sin \mathrm{\&Omega;}\cos i\sin & 0\end{array}\right]}^T$

$\frac{\&PartialD;r_0}{\&PartialD;\mathrm{\&omega;}}=r_p{\left[\begin{array}{ccc}-\cos \mathrm{\&Omega;}\sin \mathrm{\&omega;}-\sin \mathrm{\&Omega;}\cos i\cos \mathrm{\&omega;}& -\sin \mathrm{\&Omega;}\sin \mathrm{\&omega;}+\cos \mathrm{\&Omega;}\cos i\cos \mathrm{\&omega;}& \sin i\sin \mathrm{\&omega;}\end{array}\right]}^T$

$\frac{{\&PartialD;r}_0}{{\&PartialD;v}_p}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$

$\frac{\&PartialD;v_0}{{\&PartialD;v}_p}={\left[\begin{array}{ccc}-\cos \mathrm{\&Omega;}\sin \mathrm{\&omega;}-\sin \mathrm{\&Omega;}\cos i\cos \mathrm{\&omega;}& -\sin \mathrm{\&Omega;}\sin \mathrm{\&omega;}+\cos \mathrm{\&Omega;}\cos i\cos \mathrm{\&omega;}& \sin i\sin \mathrm{\&omega;}\end{array}\right]}^T$

$\frac{\&PartialD;r_0}{\&PartialD;i}=v_p{\left[\begin{array}{ccc}\sin \mathrm{\&Omega;}\sin i\sin \mathrm{\&omega;}& -\cos \mathrm{\&Omega;}\sin i\sin \mathrm{\&omega;}& \cos i\cos \mathrm{\&omega;}\end{array}\right]}^T$

${\frac{{\&PartialD;v}_0}{\&PartialD;\mathrm{\&Omega;}}=v_p\left[\begin{array}{ccc}\sin \mathrm{\&Omega;\&omega;}\sin -\cos \mathrm{\&Omega;}\cos i\cos \mathrm{\&omega;}& -\cos \mathrm{\&Omega;\&omega;}\sin -\sin \mathrm{\&Omega;}\cos i\cos \mathrm{\&omega;}& 0\end{array}\right]}^T$

$\frac{{\&PartialD;v}_0}{\&PartialD;\mathrm{\&omega;}}=v_p{\left[\begin{array}{ccc}-\cos \mathrm{\&Omega;\&omega;}\cos +\sin \mathrm{\&Omega;}\cos i\sin \mathrm{\&omega;}& -\sin \mathrm{\&Omega;}\sin \mathrm{\&omega;}-\cos \mathrm{\&Omega;}\cos i\sin \mathrm{\&omega;}& -\sin i\sin \mathrm{\&omega;}\end{array}\right]}^T$

K _tBe state-transition matrix, its each element value can adopt numerical integration to try to achieve.

$K_t=\left[\begin{array}{cc}\frac{{\&PartialD;r}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;r}_t}{{\&PartialD;v}_0}\\ \frac{{\&PartialD;v}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;v}_t}{{\&PartialD;v}_0}\end{array}\right]=\left[\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]$ (0.7)

$\stackrel{\&CenterDot;}{K_t}=\left[\begin{array}{cc}\frac{{\&PartialD;v}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;v}_t}{{\&PartialD;v}_0}\\ \frac{{\&PartialD;a}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;a}_t}{{\&PartialD;v}_0}\end{array}\right]=\left[\begin{array}{cc}\frac{{\&PartialD;v}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;v}_t}{{\&PartialD;v}_0}\\ \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;v}_0}\end{array}\right]=\left[\begin{array}{cc}{K}_{21}& K_{22}\\ \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}{K}_{11}& \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}{K}_{12}\end{array}\right]$

Consider the sun, the earth, Mars and deep space probe limbs kinetic model, the expression formula of a is

$\frac{{\mathrm{\&mu;}}_e}{r^3}r-{\mathrm{\&mu;}}_a(\frac{r_{\mathrm{ad}}}{r_{\mathrm{ad}}^3}+\frac{r_a}{r_a^3})-{\mathrm{\&mu;}}_s(\frac{r_{\mathrm{sd}}}{r_s^3}+\frac{r_s}{r_s^3})---\left(0.8\right)$

Wherein subscript e represents the earth, and a represents the sun, and s represents Mars, and initial value is respectively

$K_{21}^0=0_{3\&times;3},$ $K_{22}^0=I_{3\&times;3}.$

Consider impact point not on actual track, can only influence at the target star and find the solution two-body problem in the ball and solve so seek on B plane parameter and the track relation of some states arbitrarily, then

$K_q=\left[\begin{array}{cc}\frac{{\&PartialD;B}_T}{{\&PartialD;r}_t}& \frac{{\&PartialD;B}_T}{{\&PartialD;v}_t}\\ \frac{{\&PartialD;B}_R}{{\&PartialD;r}_t}& \frac{{\&PartialD;B}_R}{{\&PartialD;v}_t}\\ \frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}& \frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_t}\end{array}\right]---\left(0.9\right)$

If position and speed with respect to the target star during deep space probe arrival flight time terminal point are respectively $r_t=\left[\begin{array}{ccc}{r}_x& r_y& r_z\end{array}\right],v_t=\left[\begin{array}{ccc}{v}_x& v_y& v_z\end{array}\right],$ Then

$v_{\&infin;}=\sqrt{v_t^2-\frac{{2\mathrm{\&mu;}}_t}{r_t}}$

By h=B * v _∞=r _t* v _t, as can be known

$v_{\&infin;}\&times;(B\&times;v_{\&infin;})=v_{\&infin;}^{2}B=v_{\&infin;}\&times;(r_t\&times;v_t)=(v_{\&infin;}\&CenterDot;v_t)r_t-(v_{\&infin;}\&CenterDot;r_t)v_t$

obtain

$B=\frac{(v_{\&infin;}\&CenterDot;v_t)r_t-(v_{\&infin;}\&CenterDot;r_t)v_t}{v_{\&infin;}^2}---\left(0.10\right)$

Formula (0.10) is to r _t, v _tAsk local derviation,

$\frac{\&PartialD;B}{{\&PartialD;r}_m}=\frac{1}{v_{\&infin;}^2}\left[\begin{array}{ccc}-v_{y\&infin;}{r}_y-v_{z\&infin;}{r}_z& v_{y\&infin;}{r}_x& v_{z\&infin;}{r}_x\\ v_{x\&infin;}{r}_y& -v_{x\&infin;}{r}_x-v_{z\&infin;}{r}_z& v_{z\&infin;}{r}_y\\ v_{x\&infin;}{r}_z& v_{y\&infin;}{r}_z& -v_{x\&infin;}{r}_x-v_{y\&infin;}{r}_y\end{array}\right]$ (0.11)

$\frac{\&PartialD;B}{{\&PartialD;v}_m}=\frac{1}{v_{\&infin;}^2}\left[\begin{array}{ccc}{v}_{y\&infin;}{v}_y+v_{z\&infin;}{v}_z& {-v}_{y\&infin;}{v}_x& {-v}_{z\&infin;}{v}_x\\ {-v}_{x\&infin;}{v}_y& v_{x\&infin;}{v}_x+v_{z\&infin;}{v}_z& {-v}_{z\&infin;}{v}_y\\ {-v}_{x\&infin;}{v}_z& {-v}_{y\&infin;}{v}_z& v_{x\&infin;}{v}_x+v_{y\&infin;}{v}_y\end{array}\right]$

By

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;T+\frac{\&PartialD;T}{{\&PartialD;r}_t}\&CenterDot;B& \frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;T+\frac{\&PartialD;T}{{\&PartialD;v}_t}\&CenterDot;B\end{array}$

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;R+\frac{\&PartialD;R}{{\&PartialD;r}_t}\&CenterDot;B& \frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;R+\frac{\&PartialD;R}{{\&PartialD;v}_t}\&CenterDot;B\end{array}$

Consider that the B plane coordinate system is with r _m, v _mChange lessly, can think So formula can turn to

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;T& \frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;T\end{array}$ (0.12)

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;R& \frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;R\end{array}$

By $T_L=\sqrt{\frac{a^3}{{\mathrm{\&mu;}}_m}}(\frac{R}{a}-1n\frac{2R}{a})$

$\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;r}_t}+\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\&CenterDot;\frac{{\&PartialD;v}_{\&infin;}}{{\&PartialD;r}_t}=\left[\begin{array}{c}\frac{r_x}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_x}{v_{\&infin;}^3{\mathrm{<figure-callout\; id="2"\; label="r\; t"\; filenames="HDA00003003299500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_t^{}}+\frac{{\&PartialD;T}_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_x}{v_{\&infin;}{r}_t^3}\\ \frac{t_y}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_y}{v_{\&infin;}^3{\mathrm{<figure-callout\; id="2"\; label="r\; t"\; filenames="HDA00003003299500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_t^{}}+\frac{{\&PartialD;T}_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_y}{v_{\&infin;}{r}_t^3}\\ \frac{r_z}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_z}{v_{\&infin;}^3{\mathrm{<figure-callout\; id="2"\; label="r\; t"\; filenames="HDA00003003299500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_t^{}}+\frac{\&PartialD;T_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_z}{v_{\&infin;}{r}_t^3}\end{array}\right]$

$\frac{\&PartialD;T_{\mathrm{tof}}}{{\&PartialD;v}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\&CenterDot;\frac{{\&PartialD;v}_{\&infin;}}{{\&PartialD;v}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\left[\begin{array}{c}\frac{{\mathrm{\&mu;}}_s{v}_x}{v_{\&infin;}{v}_t}\\ \frac{{\mathrm{\&mu;}}_s{v}_y}{v_{\&infin;}{v}_t}\\ \frac{{\mathrm{\&mu;}}_s{v}_s}{v_{\&infin;}{v}_t}\end{array}\right].$

$\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}=-\frac{r_t}{v_{\&infin;}^2}+\frac{{3\mathrm{\&mu;}}_s}{v_{\&infin;}^4}\ln \frac{{2r}_m{v}_{\&infin;}^2}{{\mathrm{\&mu;}}_s}-\frac{{2\mathrm{\&mu;}}_m}{v_{\&infin;}^4}$

Embodiment:

The present invention determines for the accurate track that solves deep space probe, the preliminary orbit parameter of the deep space probe of determining according to genetic algorithm or Pork Chop Plots method, method based on the B plane parameter is optimized, thereby obtains the accurate orbit parameter, is example with the mars exploration task below.

Be example with the mars exploration, at first set up the contact between initial control parameter and the terminal parameter, wherein initially control parameter and be the function about earth escape orbit inclination angle, initial velocity, right ascension of ascending node and argument of perigee, terminal parameter is about the B plane parameter and arrives the function of Mars time.Then partial derivative matrix is resolved into 3 parts, i.e. original state matrix, state-transition matrix and SOT state of termination matrix, wherein the original state matrix is that the matrix that constitutes of position, perigee and velocity is to the partial derivative of initial control parameter; The partial derivative of the state-transition matrix matrix that to be the matrix that constitutes of position, periareon and velocity constitute position, perigee and velocity, and consider the limbs model; The partial derivative of the SOT state of termination matrix matrix that to be terminal parameter constitute position, periareon and velocity because B plane impact point is on the practical flight track, so in Mars influence ball solution disome and problem solves.So obtain the form that embodies of three matrixes, obtained based on partial derivative matrix in the accurate track design of the Mars probes of differential correction, it can be applicable to different demands such as navigation calculating, and requires not high enough to initial value.

Claims (4)

1. rail design method based on the deep space probe of differential correction, it is characterized in that: the detailed process of described method is:

The orbit parameter of step 1, the deep space probe determined by genetic algorithm or Pork Chop Plots method is calculated initial value: mainly comprise earth escape orbit parameter, canonical parameter (B _T0, B _R0, T _Tof0), choose orbit inclination i, the orbit radius r of earth escape orbit _p, track initial velocity v _p, rise focus right ascension Ω and argument of perigee ω and be the control parameter; Subscript " ₀" the expression value;

Step 2, according to control parameter be that initial value carries out the integral operation of track numerical value under kinetic model, try to achieve the terminal parameter value, terminal parameter is chosen B plane parameter (B _R, B _T) and time of arrival T _Tof

The B plane is defined as the barycenter of the celestial body that overdoes and enters asymptotic plane perpendicular to Mars probes, be that the B plane is perpendicular to the infinite distance velocity reversal of Mars probes, the target component of mars exploration task middle orbit adopts the B plane parameter in the B plane coordinate system, the initial point of its B plane coordinate system is selected in the Mars center, the unit vector of note detector injection asymptotic line direction is the S axle, chooses the normal direction of mars equatorial

With

Multiplication cross be the T axle, R axle and S axle and T axle constitute right-handed coordinate system, namely

$S=\frac{v_{\&infin;}}{\left|\right|v_{\&infin;}\left|\right|},T=\frac{S\&times;N}{\left|\right|S\&times;N\left|\right|},R=S\&times;T$

Vector B is defined as Mars barycenter directed towards detector track and B plane point of intersection, and its parameter is elected it as at the component of T axle and R axle, namely

B _T＝B·T

B _R＝B·R

Step 3, the parameter value that calculates and canonical parameter compare, and obtain parameter error amount (Δ B _T, Δ B _R, Δ T _Tof), thereby try to achieve new control parameter value;

Step 4, utilize new control parameter value again kinetic model to be carried out the orbit integration computing, obtain new terminal parameter value deviation;

Step 5, judge whether the departure reduce gradually satisfies accuracy requirement, if finish computing, otherwise execution in step two is to step 5, is reduced to up to departure and satisfies accuracy requirement.

2. the rail design method of a kind of deep space probe based on the differential correction according to claim 1 is characterized in that: in step 5, described accuracy requirement refers to that the periareon height error is in ± 50m scope.

3. the rail design method of a kind of deep space probe based on the differential correction according to claim 1 and 2 is characterized in that: in step 1, select the position of perigee and periareon and speed as the initial time parameter of deep space probe, use P ₀Expression, P ₀Calculate by five parameters of earth escape orbit; The terminal parameter of deep space probe is chosen as the inclination angle of target track, nearly heart distance and B plane parameter; P ₀Five parameter: orbit inclination i, the orbit radius r of earthing ball escape orbit _p, track initial velocity v _p, rise focus right ascension Ω and argument of perigee ω and calculate; Deep space probe arrives at the terminal parameter note of target interval and is Q, then between deep space probe original state and the end of a period state with representing as minor function, namely

Q＝f(P)

Actual track is only kept linear term after carrying out Taylor expansion near the nominal track,

ΔQ＝KΔP

Wherein, Δ P is controlled variable, partial derivative matrix

Detailed process to the numerical solution of partial derivative matrix is:

If deep space probe is respectively X at the state matrix that speed and the position of perigee and periareon constitute _{6 * 1}(0), X _{6 * 1}(t), determine as if the set out moment and flight time t, then P, X _{6 * 1}(0), X _{6 * 1}(t) and the pass that exists of Q be

Q＝f _q(X _6×1(t))

X _6×1(t)＝ft(X _6×1(0))

X _6×1(0)＝f _p(P)

F wherein _q, f _t, f _pBe multivariate function group, by multivariate function rule, consider the single order item,

ΔQ＝K _qΔX _6×1(t)

ΔX _6×1(t)＝K _tΔX _6×1(0)

ΔX _6×1(0)＝K _pΔP

Namely

ΔQ＝(K _q·K _t·K _p)ΔP

K wherein _q, K _t, K _pBe the matrix that the single order partial derivative constitutes, then the partial derivative matrix of the relative P of Q is:

K＝K _q·K _t·K _p

K _q, K _t, K _pConcrete solution procedure is as follows: K wherein _pBe about earth escape orbit inclination angle i, orbit radius r _p, the earth's core transfer orbit initial velocity v _p, rise the function of focus right ascension Ω and argument of perigee ω, K _qBe about terminal B plane parameter (B _R, B _T) and the function of time of arrival:

$K_p=\left[\begin{array}{ccccc}\frac{{\&PartialD;r}_0}{{\&PartialD;r}_p}& \frac{{\&PartialD;r}_0}{{\&PartialD;v}_p}& \frac{{\&PartialD;r}_0}{\&PartialD;i}& \frac{{\&PartialD;r}_0}{\&PartialD;\mathrm{\&Omega;}}& \frac{{\&PartialD;r}_0}{\&PartialD;\mathrm{\&omega;}}\\ \frac{{\&PartialD;v}_0}{{\&PartialD;r}_p}& \frac{{\&PartialD;v}_0}{{\&PartialD;v}_p}& \frac{{\&PartialD;v}_0}{\&PartialD;i}& \frac{{\&PartialD;v}_0}{\&PartialD;\mathrm{\&Omega;}}& \frac{{\&PartialD;v}_0}{\&PartialD;\mathrm{\&omega;}}\end{array}\right]$

K _tBe state-transition matrix, its each element value can adopt numerical integration to try to achieve:

$K_t=\left[\begin{array}{cc}\frac{{\&PartialD;r}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;r}_t}{{\&PartialD;v}_0}\\ \frac{{\&PartialD;v}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;v}_t}{{\&PartialD;v}_0}\end{array}\right]=\left[\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]$

$\stackrel{\&CenterDot;}{K_t}=\left[\begin{array}{cc}\frac{{\&PartialD;v}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;v}_t}{{\&PartialD;v}_0}\\ \frac{{\&PartialD;a}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;a}_t}{{\&PartialD;v}_0}\end{array}\right]=\left[\begin{array}{cc}\frac{{\&PartialD;v}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;v}_t}{{\&PartialD;v}_0}\\ \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;r}_0}& \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;v}_0}\end{array}\right]=\left[\begin{array}{cc}{K}_{21}& K_{22}\\ \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}{K}_{11}& \frac{{\&PartialD;a}_t}{{\&PartialD;r}_t}{K}_{12}\end{array}\right]$

K _qBy descending or trying to achieve:

$K_q=\left[\begin{array}{cc}\frac{{\&PartialD;B}_T}{{\&PartialD;r}_t}& \frac{{\&PartialD;B}_T}{{\&PartialD;v}_t}\\ \frac{{\&PartialD;B}_R}{{\&PartialD;r}_t}& \frac{{\&PartialD;B}_R}{{\&PartialD;v}_t}\\ \frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}& \frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_t}\end{array}\right]$

If be respectively with respect to the position of target star and the component of speed during deep space probe arrival flight time terminal point $r_t=\left[\begin{array}{ccc}{r}_x& r_y& r_z\end{array}\right],v_t=\left[\begin{array}{ccc}{v}_x& v_y& v_z\end{array}\right],$ Then

$v_{\&infin;}=\sqrt{v_t^2-\frac{{2\mathrm{\&mu;}}_t}{r_t}}$

By h=B * v _∞=r _t* v _t, as can be known

$v_{\&infin;}\&times;(B\&times;v_{\&infin;})=v_{\&infin;}^{2}B=v_{\&infin;}\&times;(r_t\&times;v_t)=(v_{\&infin;}\&CenterDot;v_t)r_t-(v_{\&infin;}\&CenterDot;r_t)v_t$

obtain

$B=\frac{(v_{\&infin;}\&CenterDot;v_t)r_t-(v_{\&infin;}\&CenterDot;r_t)v_t}{v_{\&infin;}^2}$

Formula $B=\frac{(v_{\&infin;}\&CenterDot;v_t)r_t-(v_{\&infin;}\&CenterDot;r_t)v_t}{v_{\&infin;}^2}$ To r _t, v _tAsk local derviation,

$\frac{\&PartialD;B}{{\&PartialD;r}_m}=\frac{1}{v_{\&infin;}^2}\left[\begin{array}{ccc}-v_{y\&infin;}{r}_y-v_{z\&infin;}{r}_z& v_{y\&infin;}{r}_x& v_{z\&infin;}{r}_x\\ v_{x\&infin;}{r}_y& -v_{x\&infin;}{r}_x-v_{z\&infin;}{r}_z& v_{z\&infin;}{r}_y\\ v_{x\&infin;}{r}_z& v_{y\&infin;}{r}_z& -v_{x\&infin;}{r}_x-v_{y\&infin;}{r}_y\end{array}\right]$

$\frac{\&PartialD;B}{{\&PartialD;v}_m}=\frac{1}{v_{\&infin;}^2}\left[\begin{array}{ccc}{v}_{y\&infin;}{v}_y+v_{z\&infin;}{v}_z& -v_{y\&infin;}{v}_x& {-v}_{z\&infin;}{v}_x\\ {-v}_{x\&infin;}{v}_y& v_{x\&infin;}{v}_x+v_{z\&infin;}{v}_z& {-v}_{z\&infin;}{v}_y\\ {-v}_{x\&infin;}{v}_z& -v_{y\&infin;}{v}_z& v_{x\&infin;}{v}_x+v_{y\&infin;}{v}_y\end{array}\right]$

By

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;T+\frac{\&PartialD;T}{{\&PartialD;r}_t}\&CenterDot;B& \frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;T+\frac{\&PartialD;T}{{\&PartialD;v}_t}\&CenterDot;B\end{array}$

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;R+\frac{\&PartialD;R}{{\&PartialD;r}_t}\&CenterDot;B& \frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;R+\frac{\&PartialD;R}{{\&PartialD;v}_t}\&CenterDot;B\end{array}$

Because the B plane coordinate system is with r _t, v _tChange lessly, think

So following formula can turn to

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;T& \frac{\&PartialD;B\&CenterDot;T}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;T\end{array}$

$\begin{array}{cc}\frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;r}_t}=\frac{\&PartialD;B}{{\&PartialD;r}_t}\&CenterDot;R& \frac{\&PartialD;B\&CenterDot;R}{{\&PartialD;v}_t}=\frac{\&PartialD;B}{{\&PartialD;v}_t}\&CenterDot;R\end{array}$

By $T_L=\sqrt{\frac{a^3}{{\mathrm{\&mu;}}_m}}(\frac{R}{a}-1n\frac{2R}{a})$

$\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;r}_t}\&CenterDot;\frac{{\&PartialD;r}_t}{{\&PartialD;r}_t}+\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\&CenterDot;\frac{{\&PartialD;v}_{\&infin;}}{{\&PartialD;r}_t}=\left[\begin{array}{c}\frac{r_x}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_x}{v_{\&infin;}^3{r}_t^2}+\frac{{\&PartialD;T}_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_x}{v_{\&infin;}{r}_t^3}\\ \frac{r_y}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_y}{v_{\&infin;}^3{r}_t^2}+\frac{{\&PartialD;T}_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_y}{v_{\&infin;}{r}_t^3}\\ \frac{r_z}{v_{\&infin;}{r}_t}-\frac{{\mathrm{\&mu;}}_a{r}_z}{v_{\&infin;}^3{r}_t^2}+\frac{\&PartialD;T_L}{{\&PartialD;v}_{\&infin;}}\frac{{\mathrm{\&mu;}}_a{r}_z}{v_{\&infin;}{r}_t^3}\end{array}\right]$

$\frac{\&PartialD;T_{\mathrm{tof}}}{{\&PartialD;v}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\&CenterDot;\frac{{\&PartialD;v}_{\&infin;}}{{\&PartialD;v}_t}=\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}\left[\begin{array}{c}\frac{{\mathrm{\&mu;}}_s{v}_x}{v_{\&infin;}{v}_t}\\ \frac{{\mathrm{\&mu;}}_s{v}_y}{v_{\&infin;}{v}_t}\\ \frac{{\mathrm{\&mu;}}_s{v}_z}{v_{\&infin;}{v}_t}\end{array}\right]$

$\frac{{\&PartialD;T}_{\mathrm{tof}}}{{\&PartialD;v}_{\&infin;}}=-\frac{r_t}{v_{\&infin;}^2}+\frac{{3\mathrm{\&mu;}}_s}{v_{\&infin;}^4}\ln \frac{{2r}_m{v}_{\&infin;}^2}{{\mathrm{\&mu;}}_s}-\frac{{2\mathrm{\&mu;}}_m}{v_{\&infin;}^4}$

So far obtain K _q

r _t, v _tThe expression deep space probe arrives flight time position and speed with respect to target star during terminal point.

4. the rail design method of a kind of deep space probe based on the differential correction according to claim 3, it is characterized in that: in step 2, the expression formula of described kinetic model is:

$\frac{{\mathrm{\&mu;}}_e}{r^2}r-{\mathrm{\&mu;}}_a(\frac{r_{\mathrm{ad}}}{r_{\mathrm{ad}}^3}+\frac{r_a}{r_a^3})-{\mathrm{\&mu;}}_s(\frac{r_{\mathrm{sd}}}{r_s^3}+\frac{r_s}{r_s^3})$

Wherein subscript e represents the earth, and a represents the sun, and s represents Mars, and a represents the acceleration of deep space probe.

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