Research Article Study on Zero-Doppler Centroid Control ...downloads.hindawi.com/journals/ijap/2014/549269.pdf · Study on Zero-Doppler Centroid Control for GEO SAR Ground Observation

Research ArticleStudy on Zero-Doppler Centroid Control forGEO SAR Ground Observation

Yicheng Jiang Bin Hu Yun Zhang Meng Lian and Zhuoqun Wang

Research Institute of Electronic Engineering Technology Harbin Institute of Technology No 714 Harbin 150001 China

Correspondence should be addressed to Bin Hu hubin hitqqcom

Received 7 March 2014 Revised 11 June 2014 Accepted 8 July 2014 Published 26 November 2014

Academic Editor Bing Liu

Copyright copy 2014 Yicheng Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In geosynchronous Earth orbit SAR (GEO SAR) Doppler centroid compensation is a key step for imaging process which could beperformed by the attitude steering of a satellite platform However this zero-Doppler centroid control method does not work wellwhen the look angle of radar is out of an expected range This paper primarily analyzes the Doppler properties of GEO SAR in theEarth rectangular coordinate Then according to the actual conditions of the GEO SAR ground observation the effective range ispresented by the minimum and maximum possible look angles which are directly related to the orbital parameters Based on thevector analysis a new approach for zero-Doppler centroid control in GEO SAR performing the attitude steering by a combinationof pitch and roll rotation is put forward This approach considering the Earthrsquos rotation and elliptical orbit effects can accuratelyreduce the residual Doppler centroid All the simulation results verify the correctness of the range of look angle and the proposedsteering method

1 Introduction

There has been an increasing interest in GeosynchronousEarth orbit SAR (GEO SAR) for the purpose of surveillancewith large observation In fact this broadside-looking SARhas a non-zero-Doppler centroid due to the Earthrsquos rotationand elliptical orbit effects [1] Compensating the residualDoppler centroid is an important step to solve the range-azimuth coupling which decreases the quality of SAR imag-ing The zero-Doppler attitude steering method was first putforward by Raney in the literature [2] which used a 1D yawsteering method to compensate the Doppler shift induced bythe Earthrsquos rotation This method is useful in circular orbitsbut it causes larger Doppler shifts in elliptical satellite orbitslike GEO SAR Later a method (the TerraSAR-X method) of2D attitude steering was carried out in the low Earth ellipticalorbits (LEO) SAR in literatures [3 4] yet this 2D attitudesteering method does not work well particularly with thelarge residual Doppler centroid exited inGEOSARThen thetotal zero-Doppler steering (TZDS) method was proposed inthe literatures [5 6] taking the Earthrsquos rotation and ellipticalorbit effects into account The method in [5] can minimizethe Doppler residuals in the LEO SAR system but it isnot suitable for the GEO SAR system Subsequently a 2D

phase scanmethod was introduced in literatures [7 8] whichcarried out a highly accurate compensation of the Dopplercentroid in GEO SAR By using the 2D phased scan instead ofthe attitude steering the approach in [7] can avoid the satelliteplatform rotation and stabilization

However these methods mentioned above are invalid orinaccurate when the look angle of radar is out of an effectiverange which is elaborated as follows

(i) The Earthrsquos surface could not be observed by the radarwhen the look angle is greater than the maximum ofthe range

(ii) The coverage of radar cannot provide valid echoeswith a zero-Doppler centroid when the look angle isless than the minimum of the range

For the above issues this paper introduces a generalapproach to compensate the Doppler centroid frequency forthe spaceborne SAR which works with a look angle within aneffective range The paper is organized as follows Section 2describes the GEO SAR geometry and notation and theDoppler vector of the satellite-borne SAR is analyzed inthe Earth rectangular coordinate (ERC) In Section 3 thederivation of the effective range of the look angle is firstly

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 549269 7 pageshttpdxdoiorg1011552014549269

2 International Journal of Antennas and Propagation

S

xo

yo

zo

z4

y4

x4

120579

120579

120593

P Beam center point

S

Q

R119956

R119956

Rst

R119957

Oe

Figure 1 Geometry of the earth rectangular coordinates andsatellite local coordinates

presented in detail The range is determined by the satelliteorbit parameters such as the orbital eccentricity the argumentof perigee Then based on a reasonable look angle a newmethod of zero-Doppler centroid control performed by thepitch and roll attitude steering is introduced In Section 4simulation results validate the correctness of derivation in thepaper Conclusions appear in Section 5

2 Vector Analysis in GEO SAR Model

The geometry of the Earth rectangular coordinate (ERC)119874119890-119909411991041199114is shown in Figure 1 The ERC can be defined as

the coordinate origin is the Earthrsquos center 119874119890 the 119874

1198901199094-axis

is along the Greenwich meridian direction in the equatorialplane the axis of 119874

1198901199114-axis is along the Earth angular

momentum direction and the 1198741198901199104-axis obeys a right-hand

rule In Figure 1 119875 is the intersection of the beam-pointingdirection and the Earthrsquos surface which is called the ldquobeamcenter pointrdquo (BCP) in this paper and 119876 is the nadir point ofthe satelliteRt andRs are the position vectors of the BCP andthe satellite respectively Rs and Rt are the first-order timederivative of Rs and Rt The relative position vector betweenthe satellite and the BCP is expressed as Rst = Rs minus Rt Theslant range 119877st =

1003816100381610038161003816Rs minus Rt1003816100381610038161003816 varies along the orbit which can

be calculated by

119877st (119905) = 119877119904 (119905) cos (120593) minus radic1198772119905 minus 1198772119904 (119905) sin (120593) (1)

where 120593 is the look angle of radar (shown in Figure 1) 119877119904(119905)

and 119877119905are the magnitudes of Rs(119905) and Rt(119905) respectively

Note that 119877119905is a constant that is the length of the Earthrsquos

radius The formula of 119877119904(119905) is given by the literature [9]

The Doppler centroid 119891dc of the beam center point alongthe orbit is given by the literature [7]

119891dc (119905) = minus2

120582

Rst (119905) sdot Rst (119905)1003816100381610038161003816Rst (119905)

1003816100381610038161003816

= minus2

120582

(Rs (119905) minus Rt (119905)) sdot (Rs (119905) minus Rt (119905))1003816100381610038161003816Rs (119905) minus Rt (119905)

1003816100381610038161003816

(2)

where 120582 is the radar wavelength the symbol || indicates thevector magnitude Noting that Rt(119905) = 0 in the ERC theformula of the Doppler centroid can be rewritten as

119891dc (119905) = minus2

120582

(Rs (119905) minus Rt (119905)) sdot Rs (119905)1003816100381610038161003816Rs (119905) minus Rt (119905)

1003816100381610038161003816

(3)

It is apparent that the Doppler centroid 119891dc(119905) is not onlydetermined by the satellite position but also dependent onthe position of the BCP Both are important for the analyticalderivation of the 2D attitude steering angles which will beshown in Section 3

To obtain the analytic formula for the position vectorof the BCP it is needed to find the relationship betweenthe satellite and the BCP in the satellite local coordinatesystem (SCS) 119878-119909

119900119910119900119911119900(see Figure 1) The SCS is defined

as the coordinate origin is the center of the satellite massthe 119878119911

119900-axis is along the satellite position vector towards the

Earthrsquos center the axis of 119878119909119900is perpendicular to the 119878119911

119900-axis

in the orbital plane (the angle between the 119878119909119900axis and the

satellite velocity vector is less than 90∘) and the 119878119910119900-axis obeys

the right-hand rule The formula for the slant range vector ofthe BCP in the SCS can be expressed as

Rst (119905) = [119877st (119905) sin (120593) cos (120579 (119905))

minus119877st (119905) sin (120593) sin (120579 (119905)) 119877st (119905) cos (120593)]119879

(4)

where 120579(119905) is the angle between the axis 119878119909119900and the projection

of the beam pointing vector in the 119909119900-119910119900plane (as shown in

Figure 1) and can be presented as

120579 (119905) = arccos(sin (120579119904)

cos (120595 (119905))) (5)

where 120579119904is the squint angle of radar and 120595(119905) is the angle

between the 119878119909119900-axis and the satellite velocity vector

120595 (119905) = arccos( Rs (119905) sdot Rs (119905)1003816100381610038161003816Rs (119905)

1003816100381610038161003816 sdot10038161003816100381610038161003816Rs (119905)

10038161003816100381610038161003816

) minus120587

2 (6)

The position equation of the BCP in the Earth rectangularcoordinate is given by

Rt (119905) = [

[

119883t (119905)119884t (119905)119885t (119905)

]

]

=WA (119905) timesWB timesWc (119905) times Rsp (119905) (7)

International Journal of Antennas and Propagation 3

where

WA (119905) = [

[

cos (119908119890sdot 119905) sin (119908

119890sdot 119905) 0

minus sin (119908119890sdot 119905) cos (119908

119890sdot 119905) 0

0 0 1

]

]

WB = [

[

cos (Ω0) minus sin (Ω

0) 0

sin (Ω0) cos (Ω

0) 0

0 0 1

]

]

times [

[

1 0 0

0 cos (119894) sin (119894)0 minus sin (119894) cos (119894)

]

]

times [

[

cos (1205960) minus sin (120596

0) 0

sin (1205960) cos (120596

0) 0

0 0 1

]

]

Wc (119905) = [

[

cos (119891 (119905)) minus sin (119891 (119905)) 0sin (119891 (119905)) cos (119891 (119905)) 0

0 0 1

]

]

Rsp (119905) = [

[

0 0 minus1

1 0 0

0 minus1 0

]

]

times Rst (119905) + [

[

119877119904(119905)

0

0

]

]

(8)

where the symbol times represents cross product119908119890is the Earthrsquos

angular velocity 119894 indicates the orbit inclination Ω0is the

longitude of ascending node 1205960represents the argument of

perigee and 119891(119905) indicates the argument of latitude (ie thesum of the true anomaly and the perigee argument)

3 Zero-Doppler Centroid Control

As we known when the angle 120579 = 90∘ the beam-pointing

vector is perpendicular to the velocity vector of satellite andRst(119905) sdot Rst(119905) = 0 in (2) However owing to the influences ofearth rotation and elliptical orbit sometimesRst(119905)sdotRst(119905) = 0especially in GEO SAR and it affects the quality of image bythe residual Doppler centroid Fortunately it can be resolvedby steering the satellitersquos attitude angles to reduce the offset ofthe Doppler centroid

The geometry of the spaceborne radar after zero-Dopplercentroid control is illustrated in Figure 2 in which the radarobserves the Earthrsquos surface with the look angle 120593 120572 is theangle of instance 120573 is the geocentric angle which is given by

120573 (119905) = arcsin [119877119904(119905)

119877119905

sin (120593)] minus 120593 (9)

The distance between the radar and the BCP over theorbital period can be calculated by

1003816100381610038161003816Rst (119905)1003816100381610038161003816

2

=1003816100381610038161003816Rs (119905)

1003816100381610038161003816

2

+1003816100381610038161003816Rt (119905)

1003816100381610038161003816

2

minus 21003816100381610038161003816Rs (119905)

10038161003816100381610038161003816100381610038161003816Rt (119905)

1003816100381610038161003816 cos (120573 (119905)) (10)

After the attitude steering the relative velocity vectorbetween the satellite and the BCP is perpendicular to therelative position vector According to the properties of thevector inner product we have

Rst (119905) sdot Rs (119905) = 0 (11)

Earth

Equator

Beam centerpoint

O

120572

120573

120593

R119956

R119956

Rst

R119957

Figure 2 Geometry of the satellite after zero-Doppler centroidcontrol

Then combining the aforementioned expressions theposition equation of the new beam center point (after the atti-tude steering) in ERCwhich is expressed as [119909 (119905) 119910 (119905) 119911(119905)]for the convenience of clarity can be calculated by thefollowing equations

1199092

(119905) + 1199102

(119905) + 1199112

(119905) = 1198772

119905

(119909 (119905) minus 119909119904(119905))2

+ (119910 (119905) minus 119910119904(119905))2

+ (119911 (119905) minus 119911119904(119905))2

= 1198772

119904(119905) + 119877

2

119905minus 2119877119904(119905) 119877119905cos (120573 (119905))

(119909 (119905) minus 119909119904(119905)) 119904(119905) + (119910 (119905) minus 119910

119904(119905)) 119910119904(119905)

+ (119911 (119905) minus 119911119904(119905)) 119904(119905) = 0

(12)

where the position vector of the satellite [119909119904(119905) 119910119904(119905) 119911119904(119905)]

and its velocity vector [119904(119905) 119910119904(119905) 119904(119905)] are given by the

literature [10] 119909(119905) 119910(119905) and 119911(119905) are solved as follows

119909 (119905) =119861 (119905) 119909

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119909

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119910119904(119905) 119904(119905) minus 119910

119904(119905) 119911119904(119905)

1198772119904(119905)

119862 (119905)


119910 (119905) =119861 (119905) 119910

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 1199102

119904(119905) 119860 (119905) minus 119910

119904(119905) 1198602

(119905) minus 119910119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119911119904(119905) 119904(119905) minus

119904(119905) 119909119904(119905)

1198772119904(119905)

119862 (119905)

119911 (119905) =119861 (119905) 119911

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119911

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119909119904(119905) 119910119904(119905) minus 119910

119904(119905) 119904(119905)

1198772119904(119905)

119862 (119905)

(13)

where

119860 (119905) = 119909119904(119905) 119904(119905) + 119910

119904(119905) 119910119904(119905) + 119911

119904(119905) 119904(119905)

119861 (119905) = 119877119904(119905) 119877119905cos (120573 (119905))

119862 (119905)

= radic1198772119904(119905) 1198772

119905sin2 (120573 (119905)) minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

(14)

In (13) 119896 = +1 denotes the radar looks from the left-sidewhile 119896 = minus1means the radar is right looking

Under the condition of circular orbit 119860(119905) = 0 and (13)can have solutions For an elliptical orbit SAR like GEOSAR the necessary and sufficient condition when (13) hassolutions is that the square root terms in 119862(119905) need to meetthe following requirements

1198772

119904(119905) 1198772

119905sin2 (120573 (119905))

minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

ge 0

(15)

Meanwhile to ensure the observation of a target fixedon the Earth by a space-borne SAR the following inequalityneed to be satisfied

minus arcsin(119877119905

119877119904(119905)) le 120601 (119905) le arcsin(

119877119905

119877119904(119905)) (16)

Combining (15) and (16) without the loss of generalitythe range of the look angle is expressed as

arcsin( |119860 (119905)|

119877119904(119905)10038161003816100381610038161003816Rs (119905)

10038161003816100381610038161003816

) le1003816100381610038161003816120593 (119905)

1003816100381610038161003816 le arcsin(119877119905

119877119904(119905)) (17)

0 50 100 150 200 250 300 350minus1500

minus1000

minus500

0

500

1000

1500

Argument of Latitude (deg)

fdc (

Hz)

No attitude steering2D attitude steering

Perigee Apogee

Figure 3 Comparison of the Doppler residuals before and after theattitude steering

Through the transformation matrix between the SCS andthe ERC the track of the new beam center point in the SCScan be calculated by

[

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

= [

[

0 1 0

0 0 minus1

minus1 0 0

]

]

times UA (119905) times UB times UC (119905)

times [

[

119909 (119905)

119910 (119905)

119911 (119905)

]

]

+ [

[

0

0

119903 (119905)

]

]

(18)

where

UA (119905) = [

[

cos (119891 (119905)) sin (119891 (119905)) 0

minus sin (119891 (119905)) cos (119891 (119905)) 00 0 1

]

]

UB =[[

[

cos (1205960) sin (120596

0) 0

minus sin (1205960) cos (120596

0) 0

0 0 1

]]

]

times[[

[

1 0 0


]]

]

times[[

[

cos (Ω0) sin (Ω

0) 0

minus sin (Ω0) cos (Ω

0) 0

0 0 1

]]

]

UC (119905) =[[

[

cos (119908119890sdot 119905) minus sin (119908

119890sdot 119905) 0

sin (119908119890sdot 119905) cos (119908

119890sdot 119905) 0

0 0 1

]]

]

(19)


minus4

minus3

minus2

minus1

0

1

2

3

4


Pitc

h an

gle (

deg)

Perigee Apogee

0 50 100 150 200 250 300 350

(a)

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0


Roll

angl

e (de

g)

Perigee Apogee

(b)

Figure 4 The pitch angles and the roll angles of the 2D method along the orbit

0

20

40

60

80

minus60

minus40

minus20

0 20 40 60 80

Longitude (deg)

Latit

ude (

deg)

Track of the nadir pointTrack of the ldquobeam centerrdquo before the attitude steeringTrack of the ldquobeam centerrdquo

Perigeepoint

Startingpoint

after the attitude steering

Figure 5 Comparisons of the track of the nadir point and two tracksof the BCP before and after the attitude steering

Then combining (4) the transformation matrixesbetween the track of the new BCP and the track of theoriginal one in the SCS can be presented as

[

[

1 0 0

0 cos (120579119883(119905)) minus sin (120579

119883(119905))

0 sin (120579119883(119905)) cos (120579

119883(119905))

]

]

times [

[

cos (120579119884(119905)) 0 sin (120579

119884(119905))

0 1 0

minus sin (120579119884(119905)) 0 cos (120579

119884(119905))

]

]

times [

[

119877st sin (120593) cos (120579)minus119877st sin (120593) sin (120579)

119877st cos (120593)]

]

= [

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

(20)

where 120579119884(119905) is the angle that the satellite rotates around the

axis of 119878119910119900and 120579

119883(119905) is the angle that the satellite rotates

around the axis of 119878119909119900 Based on the relationship between the

attitude steering angles and the axis angles (120579119883(119905) and 120579

119884(119905))

in (20) the pitch steering angle 120579119875(119905) and the roll steering

angle 120579119877(119905) can be expressed as

120579119875(119905)=arcsin(

1199090(119905)

119877st (119905)) minus arcsin (sin (120593) cos (120579 (119905))) minus 120595 (119905)

120579119877(119905) = minus arcsin(

1199100(119905)

119877st (119905) sdot cos (120579119875 (119905)))

minus arcsin(sin (120593) sin (120579 (119905))

radic1 minus sin2 (120593) cos2 (120579 (119905)))

(21)

4 Simulation and Analysis

For investigations of the numerically applied 2D steeringmethod the relevant orbit parameters of GEO SAR as listedin Table 1 are used

Before the 2D attitude steering the Doppler centroid 119891dcvaries along the orbit as the blue line shown in Figure 3 Thenon-zero-Doppler centroid is caused by Earthrsquos rotation andthe elliptical orbit The red line in Figure 3 shows that 119891dccan be perfectly compensated to zero by using the approachpresented in this paper

In simulations the GEO satellite starts at the ascendingnode and the radar is left looking during the orbital periodFigure 4 depicts the two attitude angles of the proposedmethod which can minimize the Doppler centroid shifts InFigure 4 the positivenegative values of the angle indicate theangles that the satellite rotates around its axis anticlockwiseclockwise It is apparent that both the attitude angles varywithin a small range of degreesThemaximum pitch steering


0 50 100 150 200 250 300 3500

02

04

06

08

1


Min

angl

e (de

g)

Perigee Apogee

t1 t2 t3 t4

(a)

0 50 100 150 200 250 300 35084

85

86

87

88

89


Max

angl

e (de

g)

Perigee

Apogee

(b)

Figure 6The effective look angles of the zero-Doppler centroid control along the orbit (a) Minimum value of the look angle (b) maximumvalue of the look angle

0 0025 005 0075 01 0125 0150

1

2

3

4

5

6

7

8

9

10

Orbital eccentricity

Min

imum

dow

n lo

ok an

gle (

deg)

120596 = 30∘

120596 = 40∘

120596 = 50∘

120596 = 60∘

120596 = 70∘

120596 = 80∘

120596 = 90∘

(a)

74

76

78

8

82

84

86

88

9

Max

imum

dow

n lo

ok an

gle (

deg)

0 0025 005 0075 01 0125 015


(b)

Figure 7 Relationship between the range of the look angle and the orbit parameters (a) Minimum look angle variation with the eccentricityat different arguments of perigee (b) Maximum look angle variation with the eccentricity

angle is about 37∘ and the maximum roll steering angleis nearly 24∘ As can be seen there is no demand on theattitude steering when the satellite platform lies in the perigeeor apogee because the relative velocity of the satellite isperpendicular to the beam direction at that moment

Figure 5 depicts the tracks of the BCP before and afterthe attitude steering and the track of the nadir point in theEarthrsquos Latitude and Longitude Coordinate represented bythe red blue and green dotted lines The red blue and greenlines describe these three tracks (from the starting point to

the perigee point) during the time that the satellite runs fromthe ascending node to the perigee

As can be seen from Figure 5 there is a great differencebetween the red and blue lines at the starting point wherethe residual Doppler centroid is great After that the red andblue lines come together at the perigee point which meansthat there is no demand on the zero-Doppler centroid controlwhen the satellite reaches its absolute perigee

Figure 6 shows the effective range of the look angle 120593along the orbit Specifically the look angle should not be less


Table 1 System parameters of GEO SAR

Specification ValueSemimajor axis 42220000mOrbit inclination 60 degreesOrbit eccentricity 002Argument of perigee 90 degreesRight ascension of ascending node 40 degreesLook angle 25 degreesBeam width 048 degreesSignal wavelength 025mAntenna diameter 30mPulse repetition frequency 250Hz

than the minimum angle (the blue line shown in Figure 6(a))and not be greater than the maximum angle (shown inFigure 6(b)) If the look angle is 06∘ after attitude steeringas the green line shown in Figure 6(a) the zero-Dopplercentroid control is effective only during the time [119905

1 1199052] and

[1199053 1199054] Therefore it is needed to take the effective range

into account when the zero-Doppler centroid method isperformed

According to (17) the orbit parameters have a bearing onthe range of look angle Figure 7(a) illustrates the relationshipbetween the minimum look angle and the orbital eccentricityat different arguments of perigee There is an upward trendof the minimum look angle with the increase of eccentricityat all arguments of perigee As the argument of perigee is90∘ the minimum value grows at the lowest speed and ithas the fastest growth rate when the argument of perigee is60∘ According to (16) the maximum value of the look angleonly relates to 119877

119905and 119877

119904(119905) and it decreases with increasing

the orbital eccentricity (shown in Figure 7(b)) Generally theminimum value is supposed to be less than the maximumone Otherwise the radar beammay radiate the space outsidethe Earth after the zero-Doppler centroid control as in thecase that the eccentricity is about 014 and 120596

0= 90∘

5 Conclusion

The contribution of this paper is that it introduces theeffective range of the look angle in which the zero-Dopplercentroid control is feasible for GEO SAR system Meanwhilea new calculation approach is proposed for the attitudesteering anglesThe 2Dmethod combining the pitch steeringwith roll steering provides a new implementation for thesatellite attitude control This approach is deduced fromthe aspect of the Earth observation rather than the satellitemotion modelTherefore it is suitable for general Earth orbitSAR system Simulations validate the expressions derived andthe proposed zero-Doppler centroid control method Lastlythe relationship between the range of the look angle and theorbital parameters is illustrated which would be helpful forthe design of the satellite orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the National NaturalScience Foundation of China (61201308) and the sponsoringof the Aerospace Science and Technology Innovation Fundwas acknowledged

References

[1] X Dong Y Gao C Hu T Zeng and D Chao ldquoEffects of Earthrotation on GEO SAR characteristics analysisrdquo in Proceedingsof the 6th International Conference on Radar (RADAR rsquo11) vol1 pp 34ndash37 October 2011

[2] R K Raney ldquoDoppler properties of radars in circular orbitrdquoInternational Journal of Remote Sensing vol 7 no 9 pp 1153ndash1162 1986

[3] E Boerner H Fiedler G Krieger and J Mittermayer ldquoA newmethod for total zero doppler steeringrdquo in Proceedings of theIEEE International Geoscience and Remote Sensing Symposium(IGARSS rsquo04) pp 1526ndash1529 September 2004

[4] H Fiedler E Boerner J Mittermayer and G Krieger ldquoTotalZero Doppler steering a new method for minimizing theDoppler centroidrdquo IEEE Geoscience and Remote Sensing Lettersvol 2 no 2 pp 141ndash145 2005

[5] Y Ze Z Yinqing C Jie et al ldquoA new satellite attitude steeringapproach for zero Doppler centroidrdquo in Proceedings of the IETInternational Radar Conference pp 1ndash4 April 2009

[6] Y J Zhang Y S ZhangH F Huang and ZDong ldquoResearch onelliptic orbit total zero Doppler steeringrdquo in Proceedings of the8th European Conference on Synthetic Aperture Radar (EUSARrsquo10) pp 1ndash4 Aachen Germany 2010

[7] X Dong Z Ding and Z Zhao ldquoA method of zero Dopplercentroid controlrdquo in Proceedings of the 8th European Conferenceon Synthetic Aperture Radar (EUSAR rsquo10) pp 1ndash4 VDE 2010

[8] T Long X Dong C Hu and T Zeng ldquoA new method of zero-doppler centroid control in GEO SARrdquo IEEE Geoscience andRemote Sensing Letters vol 8 no 3 pp 512ndash516 2011

[9] L J Cantafio Space-Based Radar Handbook chapter 2 ArtechHouse 1989

[10] T Zeng W Yin L Zhao and Z Ding ldquoAn improved velocitycalculation model in GEO SAR systemrdquo in Proceedings of theIET International Radar Conference pp 1ndash5 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



S

xo

yo

zo

z4

y4

x4

120579

120579

120593

P Beam center point

S

Q

R119956

R119956

Rst

R119957

Oe

Figure 1 Geometry of the earth rectangular coordinates andsatellite local coordinates

presented in detail The range is determined by the satelliteorbit parameters such as the orbital eccentricity the argumentof perigee Then based on a reasonable look angle a newmethod of zero-Doppler centroid control performed by thepitch and roll attitude steering is introduced In Section 4simulation results validate the correctness of derivation in thepaper Conclusions appear in Section 5

2 Vector Analysis in GEO SAR Model

The geometry of the Earth rectangular coordinate (ERC)119874119890-119909411991041199114is shown in Figure 1 The ERC can be defined as

the coordinate origin is the Earthrsquos center 119874119890 the 119874

1198901199094-axis

is along the Greenwich meridian direction in the equatorialplane the axis of 119874

1198901199114-axis is along the Earth angular

momentum direction and the 1198741198901199104-axis obeys a right-hand

rule In Figure 1 119875 is the intersection of the beam-pointingdirection and the Earthrsquos surface which is called the ldquobeamcenter pointrdquo (BCP) in this paper and 119876 is the nadir point ofthe satelliteRt andRs are the position vectors of the BCP andthe satellite respectively Rs and Rt are the first-order timederivative of Rs and Rt The relative position vector betweenthe satellite and the BCP is expressed as Rst = Rs minus Rt Theslant range 119877st =

1003816100381610038161003816Rs minus Rt1003816100381610038161003816 varies along the orbit which can

be calculated by

119877st (119905) = 119877119904 (119905) cos (120593) minus radic1198772119905 minus 1198772119904 (119905) sin (120593) (1)

where 120593 is the look angle of radar (shown in Figure 1) 119877119904(119905)

and 119877119905are the magnitudes of Rs(119905) and Rt(119905) respectively

Note that 119877119905is a constant that is the length of the Earthrsquos

radius The formula of 119877119904(119905) is given by the literature [9]

The Doppler centroid 119891dc of the beam center point alongthe orbit is given by the literature [7]

119891dc (119905) = minus2

120582

Rst (119905) sdot Rst (119905)1003816100381610038161003816Rst (119905)

1003816100381610038161003816

= minus2

120582

(Rs (119905) minus Rt (119905)) sdot (Rs (119905) minus Rt (119905))1003816100381610038161003816Rs (119905) minus Rt (119905)

1003816100381610038161003816

(2)

where 120582 is the radar wavelength the symbol || indicates thevector magnitude Noting that Rt(119905) = 0 in the ERC theformula of the Doppler centroid can be rewritten as

119891dc (119905) = minus2

120582

(Rs (119905) minus Rt (119905)) sdot Rs (119905)1003816100381610038161003816Rs (119905) minus Rt (119905)

1003816100381610038161003816

(3)

It is apparent that the Doppler centroid 119891dc(119905) is not onlydetermined by the satellite position but also dependent onthe position of the BCP Both are important for the analyticalderivation of the 2D attitude steering angles which will beshown in Section 3

To obtain the analytic formula for the position vectorof the BCP it is needed to find the relationship betweenthe satellite and the BCP in the satellite local coordinatesystem (SCS) 119878-119909

119900119910119900119911119900(see Figure 1) The SCS is defined

as the coordinate origin is the center of the satellite massthe 119878119911

119900-axis is along the satellite position vector towards the

Earthrsquos center the axis of 119878119909119900is perpendicular to the 119878119911

119900-axis

in the orbital plane (the angle between the 119878119909119900axis and the

satellite velocity vector is less than 90∘) and the 119878119910119900-axis obeys

the right-hand rule The formula for the slant range vector ofthe BCP in the SCS can be expressed as

Rst (119905) = [119877st (119905) sin (120593) cos (120579 (119905))

minus119877st (119905) sin (120593) sin (120579 (119905)) 119877st (119905) cos (120593)]119879

(4)

where 120579(119905) is the angle between the axis 119878119909119900and the projection

of the beam pointing vector in the 119909119900-119910119900plane (as shown in

Figure 1) and can be presented as

120579 (119905) = arccos(sin (120579119904)

cos (120595 (119905))) (5)

where 120579119904is the squint angle of radar and 120595(119905) is the angle

between the 119878119909119900-axis and the satellite velocity vector

120595 (119905) = arccos( Rs (119905) sdot Rs (119905)1003816100381610038161003816Rs (119905)

1003816100381610038161003816 sdot10038161003816100381610038161003816Rs (119905)

10038161003816100381610038161003816

) minus120587

2 (6)

The position equation of the BCP in the Earth rectangularcoordinate is given by

Rt (119905) = [

[

119883t (119905)119884t (119905)119885t (119905)

]

]

=WA (119905) timesWB timesWc (119905) times Rsp (119905) (7)


where

WA (119905) = [

[

cos (119908119890sdot 119905) sin (119908

119890sdot 119905) 0


119890sdot 119905) 0

0 0 1

]

]

WB = [

[


0) 0

sin (Ω0) cos (Ω

0) 0

0 0 1

]

]

times [

[

1 0 0


]

]

times [

[

cos (1205960) minus sin (120596

0) 0

sin (1205960) cos (120596

0) 0

0 0 1

]

]

Wc (119905) = [

[


0 0 1

]

]

Rsp (119905) = [

[

0 0 minus1

1 0 0

0 minus1 0

]

]

times Rst (119905) + [

[

119877119904(119905)

0

0

]

]

(8)









120573 (119905) = arcsin [119877119904(119905)

119877119905

sin (120593)] minus 120593 (9)


1003816100381610038161003816Rst (119905)1003816100381610038161003816

2

=1003816100381610038161003816Rs (119905)

1003816100381610038161003816

2

+1003816100381610038161003816Rt (119905)

1003816100381610038161003816

2

minus 21003816100381610038161003816Rs (119905)

10038161003816100381610038161003816100381610038161003816Rt (119905)

1003816100381610038161003816 cos (120573 (119905)) (10)


Rst (119905) sdot Rs (119905) = 0 (11)

Earth

Equator

Beam centerpoint

O

120572

120573

120593

R119956

R119956

Rst

R119957



1199092

(119905) + 1199102

(119905) + 1199112

(119905) = 1198772

119905

(119909 (119905) minus 119909119904(119905))2

+ (119910 (119905) minus 119910119904(119905))2

+ (119911 (119905) minus 119911119904(119905))2

= 1198772

119904(119905) + 119877

2

119905minus 2119877119904(119905) 119877119905cos (120573 (119905))

(119909 (119905) minus 119909119904(119905)) 119904(119905) + (119910 (119905) minus 119910

119904(119905)) 119910119904(119905)

+ (119911 (119905) minus 119911119904(119905)) 119904(119905) = 0

(12)




119909 (119905) =119861 (119905) 119909

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119909

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119910119904(119905) 119904(119905) minus 119910

119904(119905) 119911119904(119905)

1198772119904(119905)

119862 (119905)


119910 (119905) =119861 (119905) 119910

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 1199102

119904(119905) 119860 (119905) minus 119910

119904(119905) 1198602

(119905) minus 119910119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119911119904(119905) 119904(119905) minus

119904(119905) 119909119904(119905)

1198772119904(119905)

119862 (119905)

119911 (119905) =119861 (119905) 119911

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119911

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119909119904(119905) 119910119904(119905) minus 119910

119904(119905) 119904(119905)

1198772119904(119905)

119862 (119905)

(13)

where

119860 (119905) = 119909119904(119905) 119904(119905) + 119910

119904(119905) 119910119904(119905) + 119911

119904(119905) 119904(119905)

119861 (119905) = 119877119904(119905) 119877119905cos (120573 (119905))

119862 (119905)

= radic1198772119904(119905) 1198772

119905sin2 (120573 (119905)) minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

(14)



1198772

119904(119905) 1198772

119905sin2 (120573 (119905))

minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

ge 0

(15)



119877119904(119905)) le 120601 (119905) le arcsin(

119877119905

119877119904(119905)) (16)


arcsin( |119860 (119905)|

119877119904(119905)10038161003816100381610038161003816Rs (119905)

10038161003816100381610038161003816

) le1003816100381610038161003816120593 (119905)

1003816100381610038161003816 le arcsin(119877119905

119877119904(119905)) (17)

0 50 100 150 200 250 300 350minus1500

minus1000

minus500

0

500

1000

1500


fdc (

Hz)


Perigee Apogee



[

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

= [

[

0 1 0

0 0 minus1

minus1 0 0

]

]


times [

[

119909 (119905)

119910 (119905)

119911 (119905)

]

]

+ [

[

0

0

119903 (119905)

]

]

(18)

where

UA (119905) = [

[

cos (119891 (119905)) sin (119891 (119905)) 0

minus sin (119891 (119905)) cos (119891 (119905)) 00 0 1

]

]

UB =[[

[

cos (1205960) sin (120596

0) 0

minus sin (1205960) cos (120596

0) 0

0 0 1

]]

]

times[[

[

1 0 0


]]

]

times[[

[

cos (Ω0) sin (Ω

0) 0


0) 0

0 0 1

]]

]

UC (119905) =[[

[


119890sdot 119905) 0

sin (119908119890sdot 119905) cos (119908

119890sdot 119905) 0

0 0 1

]]

]

(19)


minus4

minus3

minus2

minus1

0

1

2

3

4


Pitc

h an

gle (

deg)

Perigee Apogee

0 50 100 150 200 250 300 350

(a)

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0


Roll

angl

e (de

g)

Perigee Apogee

(b)


0

20

40

60

80

minus60

minus40

minus20

0 20 40 60 80

Longitude (deg)

Latit

ude (

deg)


Perigeepoint

Startingpoint




[

[

1 0 0

0 cos (120579119883(119905)) minus sin (120579

119883(119905))

0 sin (120579119883(119905)) cos (120579

119883(119905))

]

]

times [

[

cos (120579119884(119905)) 0 sin (120579

119884(119905))

0 1 0

minus sin (120579119884(119905)) 0 cos (120579

119884(119905))

]

]

times [

[


119877st cos (120593)]

]

= [

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

(20)


axis of 119878119910119900and 120579




119884(119905))



120579119875(119905)=arcsin(

1199090(119905)


120579119877(119905) = minus arcsin(

1199100(119905)

119877st (119905) sdot cos (120579119875 (119905)))



(21)






0 50 100 150 200 250 300 3500

02

04

06

08

1


Min

angl

e (de

g)

Perigee Apogee

t1 t2 t3 t4

(a)

0 50 100 150 200 250 300 35084

85

86

87

88

89


Max

angl

e (de

g)

Perigee

Apogee

(b)


0 0025 005 0075 01 0125 0150

1

2

3

4

5

6

7

8

9

10


Min

imum

dow

n lo

ok an

gle (

deg)

120596 = 30∘

120596 = 40∘

120596 = 50∘

120596 = 60∘

120596 = 70∘

120596 = 80∘

120596 = 90∘

(a)

74

76

78

8

82

84

86

88

9

Max

imum

dow

n lo

ok an

gle (

deg)

0 0025 005 0075 01 0125 015


(b)











1 1199052] and




119905and 119877



0= 90∘

5 Conclusion




Acknowledgments


References













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

WA (119905) = [

[

cos (119908119890sdot 119905) sin (119908

119890sdot 119905) 0


119890sdot 119905) 0

0 0 1

]

]

WB = [

[


0) 0

sin (Ω0) cos (Ω

0) 0

0 0 1

]

]

times [

[

1 0 0


]

]

times [

[

cos (1205960) minus sin (120596

0) 0

sin (1205960) cos (120596

0) 0

0 0 1

]

]

Wc (119905) = [

[


0 0 1

]

]

Rsp (119905) = [

[

0 0 minus1

1 0 0

0 minus1 0

]

]

times Rst (119905) + [

[

119877119904(119905)

0

0

]

]

(8)









120573 (119905) = arcsin [119877119904(119905)

119877119905

sin (120593)] minus 120593 (9)


1003816100381610038161003816Rst (119905)1003816100381610038161003816

2

=1003816100381610038161003816Rs (119905)

1003816100381610038161003816

2

+1003816100381610038161003816Rt (119905)

1003816100381610038161003816

2

minus 21003816100381610038161003816Rs (119905)

10038161003816100381610038161003816100381610038161003816Rt (119905)

1003816100381610038161003816 cos (120573 (119905)) (10)


Rst (119905) sdot Rs (119905) = 0 (11)

Earth

Equator

Beam centerpoint

O

120572

120573

120593

R119956

R119956

Rst

R119957



1199092

(119905) + 1199102

(119905) + 1199112

(119905) = 1198772

119905

(119909 (119905) minus 119909119904(119905))2

+ (119910 (119905) minus 119910119904(119905))2

+ (119911 (119905) minus 119911119904(119905))2

= 1198772

119904(119905) + 119877

2

119905minus 2119877119904(119905) 119877119905cos (120573 (119905))

(119909 (119905) minus 119909119904(119905)) 119904(119905) + (119910 (119905) minus 119910

119904(119905)) 119910119904(119905)

+ (119911 (119905) minus 119911119904(119905)) 119904(119905) = 0

(12)




119909 (119905) =119861 (119905) 119909

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119909

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119910119904(119905) 119904(119905) minus 119910

119904(119905) 119911119904(119905)

1198772119904(119905)

119862 (119905)


119910 (119905) =119861 (119905) 119910

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 1199102

119904(119905) 119860 (119905) minus 119910

119904(119905) 1198602

(119905) minus 119910119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119911119904(119905) 119904(119905) minus

119904(119905) 119909119904(119905)

1198772119904(119905)

119862 (119905)

119911 (119905) =119861 (119905) 119911

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119911

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119909119904(119905) 119910119904(119905) minus 119910

119904(119905) 119904(119905)

1198772119904(119905)

119862 (119905)

(13)

where

119860 (119905) = 119909119904(119905) 119904(119905) + 119910

119904(119905) 119910119904(119905) + 119911

119904(119905) 119904(119905)

119861 (119905) = 119877119904(119905) 119877119905cos (120573 (119905))

119862 (119905)

= radic1198772119904(119905) 1198772

119905sin2 (120573 (119905)) minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

(14)



1198772

119904(119905) 1198772

119905sin2 (120573 (119905))

minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

ge 0

(15)



119877119904(119905)) le 120601 (119905) le arcsin(

119877119905

119877119904(119905)) (16)


arcsin( |119860 (119905)|

119877119904(119905)10038161003816100381610038161003816Rs (119905)

10038161003816100381610038161003816

) le1003816100381610038161003816120593 (119905)

1003816100381610038161003816 le arcsin(119877119905

119877119904(119905)) (17)

0 50 100 150 200 250 300 350minus1500

minus1000

minus500

0

500

1000

1500


fdc (

Hz)


Perigee Apogee



[

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

= [

[

0 1 0

0 0 minus1

minus1 0 0

]

]


times [

[

119909 (119905)

119910 (119905)

119911 (119905)

]

]

+ [

[

0

0

119903 (119905)

]

]

(18)

where

UA (119905) = [

[

cos (119891 (119905)) sin (119891 (119905)) 0

minus sin (119891 (119905)) cos (119891 (119905)) 00 0 1

]

]

UB =[[

[

cos (1205960) sin (120596

0) 0

minus sin (1205960) cos (120596

0) 0

0 0 1

]]

]

times[[

[

1 0 0


]]

]

times[[

[

cos (Ω0) sin (Ω

0) 0


0) 0

0 0 1

]]

]

UC (119905) =[[

[


119890sdot 119905) 0

sin (119908119890sdot 119905) cos (119908

119890sdot 119905) 0

0 0 1

]]

]

(19)


minus4

minus3

minus2

minus1

0

1

2

3

4


Pitc

h an

gle (

deg)

Perigee Apogee

0 50 100 150 200 250 300 350

(a)

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0


Roll

angl

e (de

g)

Perigee Apogee

(b)


0

20

40

60

80

minus60

minus40

minus20

0 20 40 60 80

Longitude (deg)

Latit

ude (

deg)


Perigeepoint

Startingpoint




[

[

1 0 0

0 cos (120579119883(119905)) minus sin (120579

119883(119905))

0 sin (120579119883(119905)) cos (120579

119883(119905))

]

]

times [

[

cos (120579119884(119905)) 0 sin (120579

119884(119905))

0 1 0

minus sin (120579119884(119905)) 0 cos (120579

119884(119905))

]

]

times [

[


119877st cos (120593)]

]

= [

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

(20)


axis of 119878119910119900and 120579




119884(119905))



120579119875(119905)=arcsin(

1199090(119905)


120579119877(119905) = minus arcsin(

1199100(119905)

119877st (119905) sdot cos (120579119875 (119905)))



(21)






0 50 100 150 200 250 300 3500

02

04

06

08

1


Min

angl

e (de

g)

Perigee Apogee

t1 t2 t3 t4

(a)

0 50 100 150 200 250 300 35084

85

86

87

88

89


Max

angl

e (de

g)

Perigee

Apogee

(b)


0 0025 005 0075 01 0125 0150

1

2

3

4

5

6

7

8

9

10


Min

imum

dow

n lo

ok an

gle (

deg)

120596 = 30∘

120596 = 40∘

120596 = 50∘

120596 = 60∘

120596 = 70∘

120596 = 80∘

120596 = 90∘

(a)

74

76

78

8

82

84

86

88

9

Max

imum

dow

n lo

ok an

gle (

deg)

0 0025 005 0075 01 0125 015


(b)











1 1199052] and




119905and 119877



0= 90∘

5 Conclusion




Acknowledgments


References













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119910 (119905) =119861 (119905) 119910

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 1199102

119904(119905) 119860 (119905) minus 119910

119904(119905) 1198602

(119905) minus 119910119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119911119904(119905) 119904(119905) minus

119904(119905) 119909119904(119905)

1198772119904(119905)

119862 (119905)

119911 (119905) =119861 (119905) 119911

119904(119905)

1198772119904(119905)

+1198772

119904(119905) 2

119904(119905) 119860 (119905) minus 119911

119904(119905) 1198602

(119905) minus 119904(119905) 119860 (119905) 119861 (119905)

1198772119904(119905) (2119904(119905) + 1199102

119904(119905) + 2

119904(119905))

+ 119896119909119904(119905) 119910119904(119905) minus 119910

119904(119905) 119904(119905)

1198772119904(119905)

119862 (119905)

(13)

where

119860 (119905) = 119909119904(119905) 119904(119905) + 119910

119904(119905) 119910119904(119905) + 119911

119904(119905) 119904(119905)

119861 (119905) = 119877119904(119905) 119877119905cos (120573 (119905))

119862 (119905)

= radic1198772119904(119905) 1198772

119905sin2 (120573 (119905)) minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

(14)



1198772

119904(119905) 1198772

119905sin2 (120573 (119905))

minus

1198602

(119905) (1198772

119905+ 119877119904(119905)2

minus 2119861 (119905))

(2119904(119905) + 1199102

119904(119905) + 2

119904(119905))2

ge 0

(15)



119877119904(119905)) le 120601 (119905) le arcsin(

119877119905

119877119904(119905)) (16)


arcsin( |119860 (119905)|

119877119904(119905)10038161003816100381610038161003816Rs (119905)

10038161003816100381610038161003816

) le1003816100381610038161003816120593 (119905)

1003816100381610038161003816 le arcsin(119877119905

119877119904(119905)) (17)

0 50 100 150 200 250 300 350minus1500

minus1000

minus500

0

500

1000

1500


fdc (

Hz)


Perigee Apogee



[

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

= [

[

0 1 0

0 0 minus1

minus1 0 0

]

]


times [

[

119909 (119905)

119910 (119905)

119911 (119905)

]

]

+ [

[

0

0

119903 (119905)

]

]

(18)

where

UA (119905) = [

[

cos (119891 (119905)) sin (119891 (119905)) 0

minus sin (119891 (119905)) cos (119891 (119905)) 00 0 1

]

]

UB =[[

[

cos (1205960) sin (120596

0) 0

minus sin (1205960) cos (120596

0) 0

0 0 1

]]

]

times[[

[

1 0 0


]]

]

times[[

[

cos (Ω0) sin (Ω

0) 0


0) 0

0 0 1

]]

]

UC (119905) =[[

[


119890sdot 119905) 0

sin (119908119890sdot 119905) cos (119908

119890sdot 119905) 0

0 0 1

]]

]

(19)


minus4

minus3

minus2

minus1

0

1

2

3

4


Pitc

h an

gle (

deg)

Perigee Apogee

0 50 100 150 200 250 300 350

(a)

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0


Roll

angl

e (de

g)

Perigee Apogee

(b)


0

20

40

60

80

minus60

minus40

minus20

0 20 40 60 80

Longitude (deg)

Latit

ude (

deg)


Perigeepoint

Startingpoint




[

[

1 0 0

0 cos (120579119883(119905)) minus sin (120579

119883(119905))

0 sin (120579119883(119905)) cos (120579

119883(119905))

]

]

times [

[

cos (120579119884(119905)) 0 sin (120579

119884(119905))

0 1 0

minus sin (120579119884(119905)) 0 cos (120579

119884(119905))

]

]

times [

[


119877st cos (120593)]

]

= [

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

(20)


axis of 119878119910119900and 120579




119884(119905))



120579119875(119905)=arcsin(

1199090(119905)


120579119877(119905) = minus arcsin(

1199100(119905)

119877st (119905) sdot cos (120579119875 (119905)))



(21)






0 50 100 150 200 250 300 3500

02

04

06

08

1


Min

angl

e (de

g)

Perigee Apogee

t1 t2 t3 t4

(a)

0 50 100 150 200 250 300 35084

85

86

87

88

89


Max

angl

e (de

g)

Perigee

Apogee

(b)


0 0025 005 0075 01 0125 0150

1

2

3

4

5

6

7

8

9

10


Min

imum

dow

n lo

ok an

gle (

deg)

120596 = 30∘

120596 = 40∘

120596 = 50∘

120596 = 60∘

120596 = 70∘

120596 = 80∘

120596 = 90∘

(a)

74

76

78

8

82

84

86

88

9

Max

imum

dow

n lo

ok an

gle (

deg)

0 0025 005 0075 01 0125 015


(b)











1 1199052] and




119905and 119877



0= 90∘

5 Conclusion




Acknowledgments


References













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus4

minus3

minus2

minus1

0

1

2

3

4


Pitc

h an

gle (

deg)

Perigee Apogee

0 50 100 150 200 250 300 350

(a)

0 50 100 150 200 250 300 350minus25

minus2

minus15

minus1

minus05

0


Roll

angl

e (de

g)

Perigee Apogee

(b)


0

20

40

60

80

minus60

minus40

minus20

0 20 40 60 80

Longitude (deg)

Latit

ude (

deg)


Perigeepoint

Startingpoint




[

[

1 0 0

0 cos (120579119883(119905)) minus sin (120579

119883(119905))

0 sin (120579119883(119905)) cos (120579

119883(119905))

]

]

times [

[

cos (120579119884(119905)) 0 sin (120579

119884(119905))

0 1 0

minus sin (120579119884(119905)) 0 cos (120579

119884(119905))

]

]

times [

[


119877st cos (120593)]

]

= [

[

1199090(119905)

1199100(119905)

1199110(119905)

]

]

(20)


axis of 119878119910119900and 120579




119884(119905))



120579119875(119905)=arcsin(

1199090(119905)


120579119877(119905) = minus arcsin(

1199100(119905)

119877st (119905) sdot cos (120579119875 (119905)))



(21)






0 50 100 150 200 250 300 3500

02

04

06

08

1


Min

angl

e (de

g)

Perigee Apogee

t1 t2 t3 t4

(a)

0 50 100 150 200 250 300 35084

85

86

87

88

89


Max

angl

e (de

g)

Perigee

Apogee

(b)


0 0025 005 0075 01 0125 0150

1

2

3

4

5

6

7

8

9

10


Min

imum

dow

n lo

ok an

gle (

deg)

120596 = 30∘

120596 = 40∘

120596 = 50∘

120596 = 60∘

120596 = 70∘

120596 = 80∘

120596 = 90∘

(a)

74

76

78

8

82

84

86

88

9

Max

imum

dow

n lo

ok an

gle (

deg)

0 0025 005 0075 01 0125 015


(b)











1 1199052] and




119905and 119877



0= 90∘

5 Conclusion




Acknowledgments


References













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 50 100 150 200 250 300 3500

02

04

06

08

1


Min

angl

e (de

g)

Perigee Apogee

t1 t2 t3 t4

(a)

0 50 100 150 200 250 300 35084

85

86

87

88

89


Max

angl

e (de

g)

Perigee

Apogee

(b)


0 0025 005 0075 01 0125 0150

1

2

3

4

5

6

7

8

9

10


Min

imum

dow

n lo

ok an

gle (

deg)

120596 = 30∘

120596 = 40∘

120596 = 50∘

120596 = 60∘

120596 = 70∘

120596 = 80∘

120596 = 90∘

(a)

74

76

78

8

82

84

86

88

9

Max

imum

dow

n lo

ok an

gle (

deg)

0 0025 005 0075 01 0125 015


(b)











1 1199052] and




119905and 119877



0= 90∘

5 Conclusion




Acknowledgments


References













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1 1199052] and




119905and 119877



0= 90∘

5 Conclusion




Acknowledgments


References













RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Study on Zero-Doppler Centroid Control ...downloads.hindawi.com/journals/ijap/2014/549269.pdf · Study on Zero-Doppler Centroid Control for GEO SAR Ground Observation