CN103076607A - Method for realizing sliding spotlight mode based on SAR (Synthetic Aperture Radar) satellite attitude control - Google Patents

Abstract

The invention discloses a method for realizing a sliding spotlight mode based on SAR (Synthetic Aperture Radar) satellite attitude control, which belongs to the technical field of signal processing. According to the method, a satellite attitude is regulated and controlled, so that the direction angle of a satellite antenna is changed continuously, and the problem of angular quantification is solved; and a complete space relation geometrical model is established, so that a sliding spotlight mode can be realized more accurately, and the SAR azimuth resolution is increased. Moreover, the antenna is simple in structure, the satellite cost can be lowered, the size and the weight of a satellite are reduced, and the satellite has a wide application prospect. Compared with the conventional method for realizing the sliding spotlight mode through a phase control array, the method disclosed by the invention has the advantages that the direction of the antenna is changed continuously, and the accuracy is higher; and the structure of the satellite antenna can be greatly simplified, the satellite cost is reduced, the satellite weight is reduced, and the satellite antenna has a very important application value.

Description

A kind of method based on SAR attitude of satellite control realization slip beam bunching mode

Technical field

The present invention proposes a kind of method based on SAR attitude of satellite control realization slip beam bunching mode, belongs to the signal processing technology field.

Background technology

The slip beam bunching mode is a kind of synthetic aperture radar (SAR) imaging pattern of novelty, it comes control azimuth resolution by the control antenna irradiated site in the translational speed on ground, the Area Ratio beam bunching mode of its imaging is large, and its resolution can be higher than same antenna size band pattern, and it can make good balance in high resolving power and large tracts of land imaging.And phased array antenna is to realize at present the Main Means of slip beam bunching mode, on the radar antenna scan mode, phased array antenna is pointed to the angle and affected by the digital phase shifter quantization, there is the angular quantification error, it is impossible realizing changing continuously, this quantification problem causes echo master energy discontinuous, affects the azimuth resolution of SAR.And phased array antenna structure is complicated, and cost is high, needs complicated central processing unit to control.

Summary of the invention

The present invention proposes a kind of method based on SAR attitude of satellite control realization slip beam bunching mode, the method is passed through satellite roll angle θ _r, crab angle θ _yAnd pitching angle theta _pControl, adjust satellite aerial directing, make it to keep pointing to underground point of rotation C, thereby realize the slip beam bunching mode, the present invention is a kind of new method that the high maneuverability satellite is realized the slip beam bunching mode that is applicable to.

A kind of method based on SAR attitude of satellite control realization slip beam bunching mode comprises following step:

Step 1: Calculation of Satellite is rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s);

Step 2: the positive side-looking beam center of Calculation of Satellite points to the moment T of observed object regional center ₀, antenna coordinate is E _aMiddle unit vector (0,1,0) is being rotated geocentric coordinate system E _gIn coordinate;

Step 3: calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is to the distance R on ground ₀

Step 4: calculate T ₀Point of rotation C points to oblique distance Δ R a little to ground constantly ₀

Step 5: calculate point of rotation C and rotating geocentric coordinate system E _gIn coordinate

Step 6: calculate complete sliding poly-(slip pack) process Satellite and point of rotation C and rotating geocentric coordinate system E _gIn relative vector (R _x, R _y, R _z);

Step 7: antenna direction is rotating geocentric coordinate system E in the complete sliding collecting process of calculating _gIn coordinate (x ' ₁, y ' ₁, z ' ₁);

Step 8: antenna direction is E at antenna coordinate in the complete sliding collecting process of calculating _aIn coordinate (x " ₁, y " ₁, z " ₁);

Step 9: calculating antenna coordinate is E _aLower unit vector (0,1,0), the coordinate (x after attitude control _ZT, y _ZT, z _ZT);

Step 10: calculate complete sliding collecting process Satellite attitude pilot angle;

Step 11: calculate complete sliding collecting process Satellite attitude control law, and each axis angular rate control curve.

The present invention proposes a kind of method based on SAR attitude of satellite control realization slip beam bunching mode, the method is by regulating and control the attitude of satellite, realize that the satellite aerial directing angle changes continuously, there is not the angular quantification problem, by setting up more perfect spatial relationship geometric model, can realize more accurately the beam bunching mode that slides, improve the SAR azimuth resolution.And antenna structure is simple, can reduce satellite cost, reduces the satellite volume and weight, has broad application prospects.

The invention has the advantages that:

(1) method of the present invention's proposition is than the method that realizes at present the slip beam bunching mode by phased array, and antenna direction changes continuously, and accuracy is higher;

(2) method of the present invention's proposition realizes in very accurate celestial body motion model, and degree of accuracy is very high;

(3) method of the present invention's proposition can be simplified the satellite antenna structure greatly, reduces satellite cost and alleviates satellite weight, has very important using value.

Description of drawings

Fig. 1 is method flow diagram of the present invention;

Fig. 2 be among the embodiment angle of pitch with change curve sweep time;

Fig. 3 be among the embodiment crab angle with change curve sweep time;

Fig. 4 is that embodiment Satellite x axis angular rate is with change curve sweep time;

Fig. 5 is that embodiment Satellite y axis angular rate is with change curve sweep time;

Fig. 6 is that embodiment Satellite z axis angular rate is with change curve sweep time;

Fig. 7 is the sliding poly mode imaging results of phased array antenna enlarged drawing among the embodiment;

Fig. 8 is the sliding poly mode imaging results of attitude regulation and control enlarged drawing among the embodiment;

Fig. 9 is the sliding poly mode point target of phased array antenna analysis result among the embodiment;

Figure 10 is the sliding poly mode point target of attitude regulation and control analysis result among the embodiment.

Embodiment

The present invention is described in further detail below in conjunction with drawings and Examples.

Parameters of formula statement part of the present invention:

T ₀Point to the moment of observed object regional center for the positive side-looking beam center of satellite;

A _NumFor satisfying observation band orientation to length requirement, the transponder pulse number;

f _PRFBe pulse repetition rate PRF;

θ _LBe the radar antenna visual angle;

ρ _aBe azimuth resolution;

K _aFor the orientation to the beam-broadening factor;

L _sBe antenna eliminator length;

For satellite at rail motion mean angular velocity;

Be the average spin velocity of the earth;

A is semimajor axis of ellipsoid length;

B is semiminor axis of ellipsoid length;

M is average nearly heart angle;

E is eccentric angle;

θ is very near heart angle;

E is excentricity;

R is radius vector;

P is half positive focal length;

H _GGreenwich hour angle (GHA) for the first point of Aries;

Ω is the ascending node of orbit right ascension;

I is orbit inclination;

ω is the perigee of orbit argument;

γ is the flight-path angle of satellite;

θ _rBe the roll angle of satellite around the x axle;

θ _yBe the crab angle of satellite around the y axle;

θ _pBe the angle of pitch of satellite around the z axle;

θ _{Y_c}Be the satellite crab angle after the driftage control;

ω ₃₂Be the equivalent rotary angular velocity vector;

R ₂Rotation matrix when going off course for satellite;

R ₃Rotation matrix during for the satellite pitching.

Conversion Matrix of Coordinate statement of the present invention (for convenient expression, represent cosine cos with c, represent sinusoidal sin with s):

$A_{\mathrm{go}}=\left(\begin{array}{ccc}{c}_{H_G}& s_{H_G}& 0\\ {-s}_{H_G}& {\mathrm{<figure-callout\; id="0"\; label="c\; H\; G"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">c</figure-callout>}}_{H_G}& 0\\ 0& 0& 1\end{array}\right)$ For not rotating geocentric coordinate system E _oTo rotating geocentric coordinate system E _gTransition matrix;

$A_{\mathrm{ov}}=\left(\begin{array}{ccc}{c}_{\mathrm{\&Omega;}}& s_{\mathrm{\&Omega;}}& 0\\ {-s}_{\mathrm{\&Omega;}}& c_{\mathrm{\&Omega;}}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}1& 0& 0\\ 0& c_i& s_i\\ 0& {-s}_i& c_i\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}{c}_{\mathrm{\&omega;}}& s_{\mathrm{\&omega;}}& 0\\ {-s}_{\mathrm{\&omega;}}& c_{\mathrm{\&omega;}}& 0\\ 0& 0& 1\end{array}\right)$ Be orbit plane coordinate system E _vTo not rotating geocentric coordinate system E _oTransition matrix;

$A_{\mathrm{vr}}=\left(\begin{array}{ccc}{-s}_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& {-c}_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& 0\\ c_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& {-s}_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& 0\\ 0& 0& 1\end{array}\right)$ Be satellite platform coordinate system E _rTo orbit plane coordinate system E _vTransition matrix;

$A_{\mathrm{re}}=\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_r}& 0& {-s}_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">r</figure-callout>}}}\\ 0& 1& 0\\ s_{{\mathrm{\&theta;}}_r}& 0& c_{{\mathrm{\&theta;}}_r}\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_p}& {-s}_{{\mathrm{\&theta;}}_p}& 0\\ s_{{\mathrm{\&theta;}}_p}& c_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">p</figure-callout>}}}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{{\mathrm{\&theta;}}_y}& {-s}_{{\mathrm{\&theta;}}_y}\\ 0& s_{{\mathrm{\&theta;}}_y}& c_{{\mathrm{\&theta;}}_y}\end{array}\right)$ Be satellite celestial body coordinate system E _eTo satellite platform coordinate system E _rTransition matrix;

$A_{\mathrm{ea}}=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{{\mathrm{\&theta;}}_L}& s_{{\mathrm{\&theta;}}_L}\\ 0& {-s}_{{\mathrm{\&theta;}}_L}& c_{{\mathrm{\&theta;}}_L}\end{array}\right)$ For antenna coordinate is E _aTo satellite celestial body coordinate system E _eTransition matrix;

Opposite transition can realize by matrix inversion.

Via satellite attitude control of the present invention realizes the slip beam bunching mode, and method flow specifically comprises following step as shown in Figure 1.

Step 1: Calculation of Satellite is rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s);

When launching m pulse, average nearly heart angle M is:

$M={\stackrel{\&OverBar;}{\mathrm{\&omega;}}}_{\mathrm{sat}}\&CenterDot;({\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">T</figure-callout>}}_0+\frac{m}{f_{\mathrm{PRF}}}-\frac{{\mathrm{<figure-callout\; id="2"\; label="A\; num"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">A</figure-callout>}}_{\mathrm{num}}}{2\&CenterDot;f_{\mathrm{PRF}}})---\left(1\right)$

And then calculating eccentric angle E is:

$E=M+e\&CenterDot;(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}{8}+\frac{{\mathrm{<figure-callout\; id="4"\; label="e"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">e</figure-callout>}}^4}{192})\&CenterDot;\sin M+e^2\&CenterDot;(\frac{1}{2}-\frac{{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}{6})\&CenterDot;\sin \left(2M\right)+$ $\left(2\right)$

$e^3\&CenterDot;(\frac{3}{8}-\frac{{27\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}{128})\&CenterDot;\sin \left(3M\right)+\frac{1}{3}{\mathrm{<figure-callout\; id="4"\; label="e"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">e</figure-callout>}}^4\&CenterDot;\sin \left(4M\right)+\frac{125}{384}{\mathrm{<figure-callout\; id="5"\; label="e"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">e</figure-callout>}}^5\&CenterDot;\sin \left(5M\right)$

Calculating very near heart angle θ is:

$\mathrm{\&theta;}=2\&CenterDot;\arctan (\sqrt{\frac{1+e}{1-e}}\&CenterDot;\tan \frac{E}{2})---\left(3\right)$

Calculating radius vector r is:

$r=\frac{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">P</figure-callout>}}{1+e\&CenterDot;\cos \mathrm{\&theta;}}---\left(4\right)$

By on can get satellite at orbit plane coordinate system E _vIn coordinate (x _Vs, y _Vs, z _Vs) be:

$\left(\begin{array}{c}{x}_{\mathrm{vs}}\\ y_{\mathrm{vs}}\\ z_{\mathrm{vs}}\end{array}\right)=\left(\begin{array}{c}r\&CenterDot;\cos \mathrm{\&theta;}\\ r\&CenterDot;\sin \mathrm{\&theta;}\\ 0\end{array}\right)---\left(5\right)$

By the coordinate system conversion, be converted to and rotate geocentric coordinate system E _g, obtain satellite and rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s) be:

$\left(\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right)=A_{\mathrm{go}}\&CenterDot;A_{\mathrm{ov}}\&CenterDot;\left(\begin{array}{c}{x}_{\mathrm{vs}}\\ y_{\mathrm{vs}}\\ z_{\mathrm{vs}}\end{array}\right)---\left(6\right)$

Therefrom get T ₀Satellite is rotating geocentric coordinate system E constantly _gCoordinate be

Step 2: calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is being rotated geocentric coordinate system E _gIn coordinate (x ₁, y ₁, z ₁);

By the coordinate system conversion, be converted to satellite celestial body coordinate system E _e, be converted to satellite platform coordinate system E _r, be converted to orbit plane coordinate system E _v, be converted to the geocentric orbital reference system E that does not rotate _o, be converted to again and rotate geocentric coordinate system E _g, obtaining antenna coordinate is E _aMiddle unit vector (0,1,0) is being rotated geocentric coordinate system E _gIn coordinate (x ₁, y ₁, z ₁) be:

$\left(\begin{array}{c}{x}_1\\ y_1\\ {\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1\end{array}\right)=A_{\mathrm{go}}\&CenterDot;A_{\mathrm{ov}}\&CenterDot;A_{\mathrm{vr}}\&CenterDot;A_{\mathrm{re}}\&CenterDot;A_{\mathrm{ea}}\&CenterDot;\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)---\left(7\right)$

Attention: do not consider attitude adjustment control this moment, at A _ReMiddle attitude angle all is zero.From satellite celestial body coordinate system E _eBe converted to satellite platform coordinate system E _rIn the process, consider driftage control.Through driftage control, satellite crab angle θ _{Y_c}For:

${\mathrm{\&theta;}}_{y\_c}={\mathrm{\&theta;}}_y+\arctan \left(\frac{\cos (\mathrm{\&theta;}+\mathrm{\&omega;})-\sin i}{{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}_{\mathrm{sat}}/{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}_{\mathrm{earth}}-\cos i}\right)---\left(8\right)$

Step 3: calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is to the distance R on ground ₀

${\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">r</figure-callout>}}_1=b^2{x}_1^2+b^2{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_1^2+a^2{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1^2---\left(9\right)$

${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=2(b^2{x}_1{\mathrm{<figure-callout\; id="0"\; label="x\; s"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">x</figure-callout>}}_{s_{}}+b^2{y}_1{y}_{s_0}+a^2{z}_1{z}_{s_0})---\left(10\right)$

${\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">r</figure-callout>}}_3=b^2{\mathrm{<figure-callout\; id="0"\; label="x\; s"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">x</figure-callout>}}_{s_{}}^{{}_0}+b^2{y}_{s_0}^2+a^2{\mathrm{<figure-callout\; id="0"\; label="z\; s"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_{s_{}}^{{}_0}-a^2{b}^2---\left(11\right)$

Wherein, r ₁, r ₂, r ₃Be respectively intermediate variable.

Antenna coordinate is E _aMiddle unit vector (0,1,0) is to the distance R on ground ₀:

${\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0=\frac{-\sqrt{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2-{4r}_1{r}_3}-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}{{2r}_1}---\left(12\right)$

Step 4: calculate T ₀Point of rotation C points to oblique distance Δ R a little to ground constantly ₀

Point of rotation C is under the slip beam bunching mode, and the antenna fixed directional earth is more underground, is antenna beam centre focus point.By slip pack azimuth resolution computing formula

:

${\mathrm{\&Delta;<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0=\frac{{2\mathrm{\&rho;}}_a{R}_0}{K_a{L}_s-2{\mathrm{\&rho;}}_a}---\left(13\right)$

Step 5: calculate point of rotation C and rotating geocentric coordinate system E _gIn coordinate

$\left\{\begin{array}{c}{x}_{{\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}=x_1\&CenterDot;({\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+{\mathrm{\&Delta;R}}_0)+x_{s_0}\\ {\mathrm{<figure-callout\; id="0"\; label="y\; c"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_{c_{}}={\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_1\&CenterDot;({\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+{\mathrm{\&Delta;R}}_0)+y_{s_0}\\ {\mathrm{<figure-callout\; id="0"\; label="z\; c"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_{c_{}}={\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1\&CenterDot;({\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+{\mathrm{\&Delta;R}}_0)+z_{s_0}\end{array}\right.---\left(14\right)$

Because under the slip beam bunching mode, point of rotation C invariant position can be selected constantly T ₀Lower, this moment, attitude of satellite angle all was zero.

Step 6: calculate complete sliding collecting process Satellite and point of rotation C and rotating geocentric coordinate system E _gIn relative vector (R _x, R _y, R _z);

$\left\{\begin{array}{c}{R}_x=x_s-x_{c_0}\\ R_y=y_s-y_{c_0}\\ R_z=z_s-z_{c_0}\end{array}\right.---\left(15\right)$

Step 7: antenna direction is rotating geocentric coordinate system E in the complete sliding collecting process of calculating _gIn coordinate (x ' ₁, y ' ₁, z ' ₁);

$\left\{\begin{array}{c}{x}_1^{\&prime;}=\frac{x_{c_0}-x_s}{\sqrt{R_x^2+{\mathrm{<figure-callout\; id="2"\; label="R\; y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_y^{}+R_z^2}}\\ {\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_1^{\&prime;}=\frac{y_{c_0}-y_s}{\sqrt{R_x^2+{\mathrm{<figure-callout\; id="2"\; label="R\; y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_y^{}+R_z^2}}\\ {\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1^{\&prime;}=\frac{z_{c_0}-z_s}{\sqrt{R_x^2+{\mathrm{<figure-callout\; id="2"\; label="R\; y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_y^{}+R_z^2}}\end{array}\right.---\left(16\right)$

Step 8: antenna direction is E at antenna coordinate in the complete sliding collecting process of calculating _aIn coordinate (x " ₁, y " ₁, z " ₁);

$\left(\begin{array}{c}{x}_1^{\&prime;\&prime;}\\ {\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_1^{\&prime;\&prime;}\\ {\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1^{\&prime;\&prime;}\end{array}\right)=A_{\mathrm{ae}}\&CenterDot;A_{\mathrm{er}}\&CenterDot;A_{\mathrm{rv}}\&CenterDot;A_{\mathrm{vo}}\&CenterDot;A_{\mathrm{og}}\&CenterDot;\left(\begin{array}{c}{x}_1^{\&prime;}\\ {\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_1^{\&prime;}\\ {\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1^{\&prime;}\end{array}\right)---\left(17\right)$

Step 9: calculating antenna coordinate is E _aLower unit vector (0,1,0), the coordinate (x after attitude control _ZT, y _ZT, z _ZT);

In theory, the attitude angle that changes any two dimensions or all three dimensions can be finished the control of slip pack attitude, gets the change pitching angle theta _pWith crab angle θ _yBe example, control first the angle of pitch and control again coordinate (x after crab angle obtains attitude control _ZT, y _ZT, z _ZT) be:

$\left(\begin{array}{c}{x}_{\mathrm{ZT}}\\ y_{\mathrm{ZT}}\\ z_{\mathrm{ZT}}\end{array}\right)=A_{\mathrm{ae}}\&CenterDot;\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_y}& 0& s_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}}\\ 0& 1& 0\\ -s_{{\mathrm{\&theta;}}_y}& 0& c_{{\mathrm{\&theta;}}_y}\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_p}& s_{{\mathrm{\&theta;}}_p}& 0\\ {-s}_{{\mathrm{\&theta;}}_p}& c_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">p</figure-callout>}}}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;A_{\mathrm{ea}}\&CenterDot;\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)---\left(18\right)$

Abbreviation gets:

$\left\{\begin{array}{c}{x}_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_L}-s_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_L}\\ y_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">L</figure-callout>}}}^2+c_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}+s_{{\mathrm{\&theta;}}_L}^2{c}_{{\mathrm{\&theta;}}_y}\\ z_{\mathrm{ZT}}=s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_p}-c_{{\mathrm{\&theta;}}_L}^2{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}-s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_y}\end{array}\right.---\left(19\right)$

Step 10: calculate complete sliding collecting process Satellite attitude pilot angle: pitching angle theta _pWith crab angle θ _y

$\left\{\begin{array}{c}{x}_1^{\&prime;\&prime;}=x_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_L}-s_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="1"\; label="L\; y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">L</figure-callout>}}}& \\ y_{}^{}{}_1^{\&prime;\&prime;}=y_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">L</figure-callout>}}}^2+c_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}+s_{{\mathrm{\&theta;}}_L}^2{c}_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="1"\; label="y\; z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}}& \\ z_{}^{}{}_1^{\&prime;\&prime;}=z_{\mathrm{ZT}}=s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_p}-c_{{\mathrm{\&theta;}}_L}^2{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}-s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_y}\end{array}\right.---\left(20\right)$

Solve an equation (20) can get:

$\left\{\begin{array}{c}{s}_{{\mathrm{\&theta;}}_y}=\frac{-A\&CenterDot;B-\sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">A</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">B</figure-callout>}}^2}}{1+B^2}\\ c_{{\mathrm{\&theta;}}_y}=A+B\&CenterDot;s_{{\mathrm{\&theta;}}_y}\\ s_{{\mathrm{\&theta;}}_p}=\frac{x_1^{\&prime;\&prime;}+s_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_L}}{c_{{\mathrm{\&theta;}}_y}{c}_{{\mathrm{\&theta;}}_L}}\\ c_{{\mathrm{\&theta;}}_p}=\frac{{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1^{\&prime;\&prime;}}{c_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}}+\frac{1}{c_{{\mathrm{\&theta;}}_y}}+\frac{s_{{\mathrm{\&theta;}}_y}\&CenterDot;x_1^{\&prime;\&prime;}}{c_{{\mathrm{\&theta;}}_y}\&CenterDot;s_{{\mathrm{\&theta;}}_L}}\end{array}\right.---\left(21\right)$

Wherein, $\left\{\begin{array}{c}A=\frac{s_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="1"\; label="L\; y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">L</figure-callout>}}}}{y_{}^{}}\\ B=\frac{x_1^{\&prime;\&prime;}}{{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}_1^{\&prime;\&prime;}{s}_{{\mathrm{\&theta;}}_L}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">z</figure-callout>}}_1^{\&prime;\&prime;}{c}_{{\mathrm{\&theta;}}_L}}\end{array}\right..$

Step 11: calculate complete sliding collecting process Satellite attitude control law, and each axis angular rate control curve;

Even because controlled quentity controlled variable is certain, the sequencing difference of control also can cause different control results, need to use Euler's quaternary element formula to represent the variation of control angle, obtain the Changing Pattern of attitude Eulerian angle.

With

Difference order of representation rotating shaft vector $\hat{1}={\left(\begin{array}{ccc}1& 0& 0\end{array}\right)}^T,$ $\hat{2}={\left(\begin{array}{ccc}0& 1& 0\end{array}\right)}^T,$ $\hat{3}={\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^T,$ Then have:

${\mathrm{\&omega;}}_{32}=R_2\left({\mathrm{\&theta;}}_y\right)\&CenterDot;[{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\&CenterDot;\hat{2}+R_3\left({\mathrm{\&theta;}}_p\right)\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\&CenterDot;\hat{3}]$

$=\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_y}& 0& s_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}}\\ 0& 1& 0\\ {-s}_{{\mathrm{\&theta;}}_y}& 0& c_{{\mathrm{\&theta;}}_y}\end{array}\right)\&CenterDot;\left[\begin{array}{c}{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">y</figure-callout>}}\end{array}\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)+\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_p}& s_{{\mathrm{\&theta;}}_p}& 0\\ {-s}_{{\mathrm{\&theta;}}_p}& c_{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">p</figure-callout>}}}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">p</figure-callout>}}\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\right]---\left(22\right)$

$=\left(\begin{array}{c}{s}_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\\ {\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\\ c_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\end{array}\right)$

Can get attitude control law thus, x, y, the angular velocity of rotation ω of z three axles _x, ω _y, ω _zBe respectively:

$\left(\begin{array}{c}{\mathrm{\&omega;}}_x\\ {\mathrm{\&omega;}}_y\\ {\mathrm{\&omega;}}_z\end{array}\right)=\left(\begin{array}{c}{s}_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\\ {\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\\ c_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\end{array}\right)---\left(23\right)$

Through above 11 steps, finished the calculating to slip beam bunching mode attitude control law.

Embodiment:

The method that the present invention proposes has been carried out the emulation experiment checking.Simulating, verifying is divided into two parts, and first programmes by C Plus Plus, calculates the attitude control law that the present invention proposes; Second portion is programmed by C Plus Plus, according to the attitude control law that first calculates the attitude of satellite is controlled, emulation generates echo data, and echo data carried out imaging and assessment, by with the artificial echo Data Comparison of phased array antenna mode, verified correctness of the present invention and superiority.

First: calculate the attitude control law that the present invention proposes;

Given parameters in the emulation experiment: semimajor axis of ellipsoid 6378140.0m, semiminor axis of ellipsoid 6356755.0m, radius of a ball 6371140.0m fifty-fifty, rotational-angular velocity of the earth 0.000073reg/s, semi-major axis of orbit 7003819.0m, orbit inclination 97.889, argument of perigee 90.0, right ascension of ascending node 121.0, excentricity 0.0011, operation wavelength 0.03m, center of antenna visual angle 20.0deg, antenna length 3.3m, signal sampling rate 350MHz, signal bandwidth 300MHz, pulse repetition rate 5000Hz, azimuth resolution 0.5m, wave beam unrolling times 1.2, by in point target of scene center arrangement, concrete calculation procedure is:

1.1 Calculation of Satellite is rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s);

Obtain satellite at orbit plane coordinate system E according to formula (1)-(4) _vIn coordinate (x _Vs, y _Vs, z _Vs), suc as formula (5), by the coordinate system conversion, be converted to and rotate geocentric coordinate system E _g, obtain satellite and rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s), suc as formula (6), and therefrom get T ₀Satellite is rotating geocentric coordinate system E constantly _gCoordinate be

1.2 calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is being rotated geocentric coordinate system E _gIn coordinate (x ₁, y ₁, z ₁);

Consider driftage control, by the coordinate system conversion, be converted to and rotate geocentric coordinate system E _g, suc as formula (7).

1.3 calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is to the distance R on ground ₀

According to formula (9)-(11), calculate the result suc as formula (12).

1.4 calculate T ₀Point of rotation C points to oblique distance Δ R a little to ground constantly ₀

Because $\frac{\mathrm{\&Delta;}{R}_0}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+{\mathrm{\&Delta;<figure-callout\; id="0"\; label="R"\; filenames="HDA00002698049800021.png,HDA00002698049800022.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}\&CenterDot;K_a\frac{L_s}{2}={\mathrm{\&rho;}}_a,$ Can get Δ R ₀Suc as formula (13).

Rotating geocentric coordinate system E 1.5 calculate point of rotation C _gIn coordinate

Because under the slip beam bunching mode, point of rotation invariant position can be selected constantly T ₀Lower, this moment, attitude of satellite angle all was zero, calculated the result suc as formula (14).

Rotating geocentric coordinate system E 1.6 calculate complete sliding collecting process Satellite and point of rotation C _gIn relative vector (R _x, R _y, R _z);

Rotate geocentric coordinate system E _gLower, in the sliding collecting process, satellite position changes entirely, and point of rotation C is constant, can calculate relative vector suc as formula (15).

Antenna direction is rotating geocentric coordinate system E in the complete sliding collecting process 1.7 calculate _gIn coordinate (x ' ₁, y ' ₁, z ' ₁);

Result of calculation is suc as formula (16).

Antenna direction is E at antenna coordinate in the complete sliding collecting process 1.8 calculate _aIn coordinate (x " ₁, y " ₁, z " ₁);

Do the coordinate system conversion, suc as formula (17).

1.9 calculating antenna coordinate is E _aLower unit vector (0,1,0), the coordinate (x after attitude control _ZT, y _ZT, z _ZT);

Through coordinate system conversion and attitude control, suc as formula (18), obtain the result suc as formula (19).

1.10 calculate complete sliding collecting process Satellite attitude pilot angle: pitching angle theta _pWith crab angle θ _y

According to formula (20), solvable equation obtains the result suc as formula (21).

1.11 calculate complete sliding collecting process Satellite attitude control law, and each axis angular rate control curve;

According to formula (22), can get attitude control law, x, y, the angular velocity of rotation of z three axles is suc as formula (23).

During emulation, the orientation gets 65536 to sampling number, this moment, maximum scan angle had ± 2.808 degree, different maximum scan angle can be by changing the orientation to the employing realization of counting, in order to make things convenient for simulated program to write, 2 integer power is got in the orientation to sampling number in this example, and actual treatment is only got the scan angle scope and spent to+223 for-2.23 degree, survey and draw strip length 10km this moment, calculates crab angle θ _pAnd pitching angle theta _yChanging Pattern is presented at respectively among Fig. 2 and Fig. 3, and then the controlled attitude of satellite shows in Fig. 4, Fig. 5 and Fig. 6 respectively with the three axle angular velocity of rotations of realizing the slip beam bunching mode.

Second portion: emulation generates echo data, and echo data is carried out imaging and assessment.

Given parameter constant in the emulation experiment, programme by C++, emulation produces the echo data under phased array antenna mode and the attitude control mode of the present invention respectively, again by imaging, obtain respectively the result and be presented among Fig. 7 and Fig. 8, imaging data is assessed again, the result is presented at respectively among Fig. 9 and Figure 10, can find out from analysis result, attitude control realizes that the slip pack promotes to some extent than phased array antenna on azimuthal resolution.

The method that above-mentioned two parts emulation experiment explanation the present invention proposes is a kind of very accurate method.

Claims (1)

1. the method based on SAR attitude of satellite control realization slip beam bunching mode is characterized in that, comprises following step:

Step 1: Calculation of Satellite is rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s);

When launching m pulse, average nearly heart angle M is:

$M={\stackrel{\&OverBar;}{\mathrm{\&omega;}}}_{\mathrm{sat}}\&CenterDot;(T_0+\frac{m}{f_{\mathrm{PRF}}}-\frac{A_{\mathrm{num}}}{2\&CenterDot;f_{\mathrm{PRF}}})---\left(1\right)$

And then calculating eccentric angle E is:

$E=M+e\&CenterDot;(1-\frac{e^2}{8}+\frac{e^4}{192})\&CenterDot;\sin M+e^2\&CenterDot;(\frac{1}{2}-\frac{e^2}{6})\&CenterDot;\sin \left(2M\right)+$ $\left(2\right)$

$e^3\&CenterDot;(\frac{3}{8}-\frac{{27e}^2}{128})\&CenterDot;\sin \left(3M\right)+\frac{1}{3}{e}^4\&CenterDot;\sin \left(4M\right)+\frac{125}{384}{e}^5\&CenterDot;\sin \left(5M\right)$

Calculating very near heart angle θ is:

$\mathrm{\&theta;}=2\&CenterDot;\arctan (\sqrt{\frac{1+e}{1-e}}\&CenterDot;\tan \frac{E}{2})---\left(3\right)$

Calculating radius vector r is:

$r=\frac{P}{1+e\&CenterDot;\cos \mathrm{\&theta;}}---\left(4\right)$

By on can get satellite at orbit plane coordinate system E _vIn coordinate (x _Vs, y _Vs, z _Vs) be:

$\left(\begin{array}{c}{x}_{\mathrm{vs}}\\ y_{\mathrm{vs}}\\ z_{\mathrm{vs}}\end{array}\right)=\left(\begin{array}{c}r\&CenterDot;\cos \mathrm{\&theta;}\\ r\&CenterDot;\sin \mathrm{\&theta;}\\ 0\end{array}\right)---\left(5\right)$

By the coordinate system conversion, be converted to and rotate geocentric coordinate system E _g, obtain satellite and rotating geocentric coordinate system E _gIn coordinate (x _s, y _s, z _s) be:

$\left(\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right)=A_{\mathrm{go}}\&CenterDot;A_{\mathrm{ov}}\&CenterDot;\left(\begin{array}{c}{x}_{\mathrm{vs}}\\ y_{\mathrm{vs}}\\ z_{\mathrm{vs}}\end{array}\right)---\left(6\right)$

Therefrom get T ₀Satellite is rotating geocentric coordinate system E constantly _gCoordinate be

Step 2: calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is being rotated geocentric coordinate system E _gIn coordinate (x ₁, y ₁, z ₁);

By the coordinate system conversion, be converted to satellite celestial body coordinate system E _e, be converted to satellite platform coordinate system E _r, be converted to orbit plane coordinate system E _v, be converted to the geocentric orbital reference system E that does not rotate _o, be converted to again and rotate geocentric coordinate system E _g, obtaining antenna coordinate is E _aMiddle unit vector (0,1,0) is being rotated geocentric coordinate system E _gIn coordinate (x ₁, y ₁, z ₁) be:

$\left(\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right)=A_{\mathrm{go}}\&CenterDot;A_{\mathrm{ov}}\&CenterDot;A_{\mathrm{vr}}\&CenterDot;A_{\mathrm{re}}\&CenterDot;A_{\mathrm{ea}}\&CenterDot;\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)---\left(7\right)$

Step 3: calculate T ₀Antenna coordinate is E constantly _aMiddle unit vector (0,1,0) is to the distance R on ground ₀

$r_1=b^2{x}_1^2+b^2{y}_1^2+a^2{z}_1^2---\left(9\right)$

$r_2=2(b^2{x}_1{x}_{s_0}+b^2{y}_1{y}_{s_0}+a^2{z}_1{z}_{s_0})---\left(10\right)$

$r_3=b^2{x}_{s_0}^2+b^2{y}_{s_0}^2+a^2{z}_{s_0}^2-a^2{b}^2---\left(11\right)$

Wherein, r ₁, r ₂, r ₃Be respectively intermediate variable;

Antenna coordinate is E _aMiddle unit vector (0,1,0) is to the distance R on ground ₀:

$R_0=\frac{-\sqrt{r_2^2-{4r}_1{r}_3}-r_2}{{2r}_1}---\left(12\right);$

Step 4: calculate T ₀Point of rotation C points to oblique distance Δ R a little to ground constantly ₀

By slip pack azimuth resolution computing formula

:

${\mathrm{\&Delta;R}}_0=\frac{{2\mathrm{\&rho;}}_a{R}_0}{K_a{L}_s-2{\mathrm{\&rho;}}_a}---\left(13\right);$

Step 5: calculate point of rotation C and rotating geocentric coordinate system E _gIn coordinate

$\left\{\begin{array}{c}{x}_{c_0}=x_1\&CenterDot;(R_0+{\mathrm{\&Delta;R}}_0)+x_{s_0}\\ y_{c_0}=y_1\&CenterDot;(R_0+{\mathrm{\&Delta;R}}_0)+y_{s_0}\\ z_{c_0}=z_1\&CenterDot;(R_0+{\mathrm{\&Delta;R}}_0)+z_{s_0}\end{array}\right.---\left(14\right);$

Step 6: calculate complete sliding collecting process Satellite and point of rotation C and rotating geocentric coordinate system E _gIn relative vector (R _x, R _y, R _z);

$\left\{\begin{array}{c}{R}_x=x_s-x_{c_0}\\ R_y=y_s-y_{c_0}\\ R_z=z_s-z_{c_0}\end{array}\right.---\left(15\right);$

Step 7: antenna direction is rotating geocentric coordinate system E in the complete sliding collecting process of calculating _gIn coordinate (x ' ₁, y ' ₁, z ' ₁);

$\left\{\begin{array}{c}{x}_1^{\&prime;}=\frac{x_{c_0}-x_s}{\sqrt{R_x^2+R_y^2+R_z^2}}\\ y_1^{\&prime;}=\frac{y_{c_0}-y_s}{\sqrt{R_x^2+R_y^2+R_z^2}}\\ z_1^{\&prime;}=\frac{z_{c_0}-z_s}{\sqrt{R_x^2+R_y^2+R_z^2}}\end{array}\right.---\left(16\right);$

Step 8: antenna direction is E at antenna coordinate in the complete sliding collecting process of calculating _aIn coordinate (x " ₁, y " ₁, z " ₁);

$\left(\begin{array}{c}{x}_1^{\&prime;\&prime;}\\ y_1^{\&prime;\&prime;}\\ z_1^{\&prime;\&prime;}\end{array}\right)=A_{\mathrm{ae}}\&CenterDot;A_{\mathrm{er}}\&CenterDot;A_{\mathrm{rv}}\&CenterDot;A_{\mathrm{vo}}\&CenterDot;A_{\mathrm{og}}\&CenterDot;\left(\begin{array}{c}{x}_1^{\&prime;}\\ y_1^{\&prime;}\\ z_1^{\&prime;}\end{array}\right)---\left(17\right);$

Step 9: calculating antenna coordinate is E _aLower unit vector (0,1,0), the coordinate (x after attitude control _ZT, y _ZT, z _ZT);

Control first the angle of pitch and control again coordinate (x after crab angle obtains attitude control _ZT, y _ZT, z _ZT) be:

$\left(\begin{array}{c}{x}_{\mathrm{ZT}}\\ y_{\mathrm{ZT}}\\ z_{\mathrm{ZT}}\end{array}\right)=A_{\mathrm{ae}}\&CenterDot;\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_y}& 0& s_{{\mathrm{\&theta;}}_y}\\ 0& 1& 0\\ -s_{{\mathrm{\&theta;}}_y}& 0& c_{{\mathrm{\&theta;}}_y}\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_p}& s_{{\mathrm{\&theta;}}_p}& 0\\ {-s}_{{\mathrm{\&theta;}}_p}& c_{{\mathrm{\&theta;}}_p}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;A_{\mathrm{ea}}\&CenterDot;\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)---\left(18\right);$

Abbreviation gets:

$\left\{\begin{array}{c}{x}_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_L}-s_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_L}\\ y_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_L}^2+c_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}+s_{{\mathrm{\&theta;}}_L}^2{c}_{{\mathrm{\&theta;}}_y}\\ z_{\mathrm{ZT}}=s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_p}-c_{{\mathrm{\&theta;}}_L}^2{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}-s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_y}\end{array}\right.---\left(19\right);$

Step 10: calculate complete sliding collecting process Satellite attitude pilot angle: pitching angle theta _pWith crab angle θ _y

$\left\{\begin{array}{c}{x}_1^{\&prime;\&prime;}=x_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_L}-s_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_L}\\ y_1^{\&prime;\&prime;}=y_{\mathrm{ZT}}=c_{{\mathrm{\&theta;}}_p}{c}_{{\mathrm{\&theta;}}_L}^2+c_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_L}{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}+s_{{\mathrm{\&theta;}}_L}^2{c}_{{\mathrm{\&theta;}}_y}\\ z_1^{\&prime;\&prime;}=z_{\mathrm{ZT}}=s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_p}-c_{{\mathrm{\&theta;}}_L}^2{s}_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_p}-s_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_y}\end{array}\right.---\left(20\right)$

Solve an equation (20):

$\left\{\begin{array}{c}{s}_{{\mathrm{\&theta;}}_y}=\frac{-A\&CenterDot;B-\sqrt{1-A^2+B^2}}{1+B^2}\\ c_{{\mathrm{\&theta;}}_y}=A+B\&CenterDot;s_{{\mathrm{\&theta;}}_y}\\ s_{{\mathrm{\&theta;}}_p}=\frac{x_1^{\&prime;\&prime;}+s_{{\mathrm{\&theta;}}_y}{s}_{{\mathrm{\&theta;}}_L}}{c_{{\mathrm{\&theta;}}_y}{c}_{{\mathrm{\&theta;}}_L}}\\ c_{{\mathrm{\&theta;}}_p}=\frac{z_1^{\&prime;\&prime;}}{c_{{\mathrm{\&theta;}}_L}{c}_{{\mathrm{\&theta;}}_L}}+\frac{1}{c_{{\mathrm{\&theta;}}_y}}+\frac{s_{{\mathrm{\&theta;}}_y}\&CenterDot;x_1^{\&prime;\&prime;}}{c_{{\mathrm{\&theta;}}_y}\&CenterDot;s_{{\mathrm{\&theta;}}_L}}\end{array}\right.---\left(21\right)$

Wherein, $\left\{\begin{array}{c}A=\frac{s_{{\mathrm{\&theta;}}_L}}{y_1^{\&prime;\&prime;}{s}_{{\mathrm{\&theta;}}_L}-z_1^{\&prime;\&prime;}{c}_{{\mathrm{\&theta;}}_L}}\\ B=\frac{x_1^{\&prime;\&prime;}}{y_1^{\&prime;\&prime;}{s}_{{\mathrm{\&theta;}}_L}-z_1^{\&prime;\&prime;}{c}_{{\mathrm{\&theta;}}_L}}\end{array}\right.;$

Step 11: calculate complete sliding collecting process Satellite attitude control law, and each axis angular rate control curve;

With

Difference order of representation rotating shaft vector $\hat{1}={\left(\begin{array}{ccc}1& 0& 0\end{array}\right)}^T,$ $\hat{2}={\left(\begin{array}{ccc}0& 1& 0\end{array}\right)}^T,$ $\hat{3}={\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^T,$ Then have:

${\mathrm{\&omega;}}_{32}=R_2\left({\mathrm{\&theta;}}_y\right)\&CenterDot;[{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\&CenterDot;\hat{2}+R_3\left({\mathrm{\&theta;}}_p\right)\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\&CenterDot;\hat{3}]$

$=\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_y}& 0& s_{{\mathrm{\&theta;}}_y}\\ 0& 1& 0\\ {-s}_{{\mathrm{\&theta;}}_y}& 0& c_{{\mathrm{\&theta;}}_y}\end{array}\right)\&CenterDot;\left[\begin{array}{c}{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\end{array}\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)+\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_p}& s_{{\mathrm{\&theta;}}_p}& 0\\ {-s}_{{\mathrm{\&theta;}}_p}& c_{{\mathrm{\&theta;}}_p}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\right]---\left(22\right)$

$=\left(\begin{array}{c}{s}_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\\ {\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\\ c_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\end{array}\right)$

Get thus attitude control law, x, y, the angular velocity of rotation ω of z three axles _x, ω _y, ω _zBe respectively:

$\left(\begin{array}{c}{\mathrm{\&omega;}}_x\\ {\mathrm{\&omega;}}_y\\ {\mathrm{\&omega;}}_z\end{array}\right)=\left(\begin{array}{c}{s}_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\\ {\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_y\\ c_{{\mathrm{\&theta;}}_y}\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_p\end{array}\right)---\left(23\right)$

Through above 11 steps, finished the calculating to slip beam bunching mode attitude control law;

In the above-mentioned steps, alphabetical parameter is defined as follows:

T ₀Point to the moment of observed object regional center for the positive side-looking beam center of satellite; A _NumFor satisfying observation band orientation to length requirement, the transponder pulse number; f _PRFBe pulse repetition rate PRF; θ _LBe the radar antenna visual angle; ρ _aBe azimuth resolution; K _aFor the orientation to the beam-broadening factor; L _sBe antenna eliminator length;

For satellite at rail motion mean angular velocity;

Be the average spin velocity of the earth; A is semimajor axis of ellipsoid length; B is semiminor axis of ellipsoid length; M is average nearly heart angle; E is eccentric angle; θ is very near heart angle; E is excentricity; R is radius vector; P is half positive focal length; H _GGreenwich hour angle (GHA) for the first point of Aries; Ω is the ascending node of orbit right ascension; I is orbit inclination; ω is the perigee of orbit argument; γ is the flight-path angle of satellite; θ _rBe the roll angle of satellite around the x axle; θ _yBe the crab angle of satellite around the y axle; θ _pBe the angle of pitch of satellite around the z axle; θ _{Y_c}Be the satellite crab angle after the driftage control; ω ₃₂Be the equivalent rotary angular velocity vector; R ₂Rotation matrix when going off course for satellite; R ₃Rotation matrix during for the satellite pitching;

$A_{\mathrm{go}}=\left(\begin{array}{ccc}{c}_{H_G}& s_{H_G}& 0\\ {-s}_{H_G}& c_{H_G}& 0\\ 0& 0& 1\end{array}\right)$ For not rotating geocentric coordinate system E _oTo rotating geocentric coordinate system E _gTransition matrix;

$A_{\mathrm{ov}}=\left(\begin{array}{ccc}{c}_{\mathrm{\&Omega;}}& s_{\mathrm{\&Omega;}}& 0\\ {-s}_{\mathrm{\&Omega;}}& c_{\mathrm{\&Omega;}}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}1& 0& 0\\ 0& c_i& s_i\\ 0& {-s}_i& c_i\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}{c}_{\mathrm{\&omega;}}& s_{\mathrm{\&omega;}}& 0\\ {-s}_{\mathrm{\&omega;}}& c_{\mathrm{\&omega;}}& 0\\ 0& 0& 1\end{array}\right)$ Be orbit plane coordinate system E _vTo not rotating geocentric coordinate system E _oTransition matrix;

$A_{\mathrm{vr}}=\left(\begin{array}{ccc}{-s}_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& {-c}_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& 0\\ c_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& {-s}_{(\mathrm{\&theta;}-\mathrm{\&gamma;})}& 0\\ 0& 0& 1\end{array}\right)$ Be satellite platform coordinate system E _rTo orbit plane coordinate system E _vTransition matrix;

$A_{\mathrm{re}}=\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_r}& 0& {-s}_{{\mathrm{\&theta;}}_r}\\ 0& 1& 0\\ s_{{\mathrm{\&theta;}}_r}& 0& c_{{\mathrm{\&theta;}}_r}\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}{c}_{{\mathrm{\&theta;}}_p}& {-s}_{{\mathrm{\&theta;}}_p}& 0\\ s_{{\mathrm{\&theta;}}_p}& c_{{\mathrm{\&theta;}}_p}& 0\\ 0& 0& 1\end{array}\right)\&CenterDot;\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{{\mathrm{\&theta;}}_y}& {-s}_{{\mathrm{\&theta;}}_y}\\ 0& s_{{\mathrm{\&theta;}}_y}& c_{{\mathrm{\&theta;}}_y}\end{array}\right)$ Be satellite celestial body coordinate system E _eTo satellite platform coordinate system E _rTransition matrix;

$A_{\mathrm{ea}}=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{{\mathrm{\&theta;}}_L}& s_{{\mathrm{\&theta;}}_L}\\ 0& {-s}_{{\mathrm{\&theta;}}_L}& c_{{\mathrm{\&theta;}}_L}\end{array}\right)$ For antenna coordinate is E _aTo satellite celestial body coordinate system E _eTransition matrix;

Opposite transition realizes by matrix inversion.

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