页面提取自-星载SAR用到的坐标系关系_Thesis

1、AppendixA. Coordinate Frames and TransformationsSeveral different coordinate frames are used in this work. Each of them has some advantages concerning the calculations that are done on its basis. Most of the frames are Cartesian, because they allow rather simple vector algebra. And beyond it, a set

2、of simple matrices can be used to perform vector rotation around each axis Appendix A.2.The final results are mostly given in ellipsoidal WGS84 coordinates to relate them to the surface of the rotating earth Appendix A.1.2.A.1 Definition of the Coordinate FramesA.1.1 Earth Centered Inertial System (

3、IN) vsatPsatFig A. 1: Earth Centered Inertial SystemAppendix3The Earth Centered Inertial System is a right handed Cartesian coordinate frame. All vectors related to an orbital platform (e.g. position, velocity) can be calculated independently from the earths rotation in this system.Definition:The po

4、int of origin is the geocenter. The z-axis is parallel to the spin axis of the earth, the x-axis lies in the equatorial plane and points towards the vernal equinox (). Orthogonal to the x-z-plane, the y-axis completes the right handed coordinate system.A.1.2 Earth Centered Earth Fixed System (EF)Lik

5、e the Earth Centered Inertial System, the Earth Centered Earth Fixed System is right handed and Cartesian. In this coordinate frame all vectors can be expressed related to the rotating earth.Fig. A. 2: Earth Centered Earth Fixed System with ellipsoidal coordinatesDefinition:Again the point of origin

6、 is the geocenter. The z-axis is aligned with the z-axis of the EarthCentered Inertial System. The x-axis lies in the equatorial plane and points towards the intersection of equator and zero meridian. The y-axis completes the right handed system. Relative to the Earth Centered Inertial System, the e

7、arth fixed system rotates around the z-axiswith the angular velocity of .All vectors in the earth fixed system can also be expressed by ellipsoidal coordinates. Theseare related to the WGS84 ellipsoid. The parameters are:aearth6378137 msemi major axis of the WGS84 ellipsoideearth0.081819190eccentric

8、ity of the WGS84 ellipsoid0.72921151467 E-04 rad/sangular velocityTab. A. 1: Parameters of the WGS84 EllipsoidThe ellipsoidal coordinates are:latitudelongitudehheight above the WGS84 ellipsoidTab. A. 2: Ellipsoidal CoordinatesNote that the point where all ellipsoidal coordinates (, , h) are zero is

9、not the center of the EF frame but the point of intersection between zero meridian and equator on the surface of the WGS84 ellipsoid.The earth fixed Cartesian coordinates can be derived from the ellipsoidal coordinates by using the Helmert projection NI06:xEF = ( RN + h) cos ( ) cos ()yEF = (RN + h)

10、 cos ( ) sin ()(A.1.1)z= R (1 e2)+ h sin ()where:EFNearthRN =1 e2aearth sin2 ( )(A.1.2)earthThe ellipsoidal coordinates are obtained from the earth fixed Cartesian coordinates by inverse Helmert projection: y = a tan EF xEF 22h = xEF + yEFcos ( )1 e2aearth sin () = a tan earthEFz+ e2 b sin3 ()2223wh

11、ere:xEF+ yEF eearth aearth cos (A.1.3) = a tan aearthzEF b 22+ yxEFEF b = a 1 e2earthearth22eearth =aearth baearthe =22a bearth=aearth eearth=eearthba1 e21 e2earthearthearthAppendix5For calculations concerning the gravitation model and its gradient, the earth fixed Cartesian coordinates must be expr

12、essed in spherical coordinates. In literature, usually the polar angle is used, which is the angle between the Cartesian z-axis and the vector to the point P whichshall be expressed in spherical coordinates. Here, the geocentric latitude is used, which is thernorm of the Cartesian vector to the poin

13、t Pgeographic longitudegeocentric latitudeangle between the vector to the point P and the Cartesian x-y-plane. The modified spherical coordinates are:Fig. A. 3: Modified Spherical CoordinatesThe Cartesian coordinates expressed in modified spherical coordinates are:xEF = r cos ( ) cos ( ) yEF = r cos

14、 ( ) sin ( ) zEF = r sin ( )(A.1.4)The transformation matrix is given by the relation between the unit vectors. From geometry we obtain: er = cos ( ) cos ( ) ex,EF + cos ( ) sin ( ) ey ,EF + sin ( ) ez,EF e = sin ( ) ex,EF + cos ( ) ey.EF e = sin ( ) cos ( ) ex,EF sin ( ) sin ( ) ey ,EF + cos ( ) ez

15、,EF(A.1.5)with:, of the vector the modified spherical coordinate frame is related toThe vectors in modified spherical coordinates and in earth fixed coordinates are:vr ,MS vx,EF vMS= v ,MS ;v ,MS vEF= v y ,EF vz ,EF (A.1.6)Now the coordinate transformation is done by writing equation (A.1.5) as matr

16、ix equation: v= T EF vMSMSEFvr ,MS cos ( ) cos ( )cos ( ) sin ( )sin ( )vx,EF (A.1.7) v ,MS = sin ( )cos ( )0 vy ,EF v ,MS sin ( ) cos ( ) sin ( ) sin ( )cos ( )vz ,EF It is important to note that the unit vectors of the modified spherical coordinate frame depend on a Cartesian reference vector that

17、 corresponds to the location of the point of origin of the modified spherical coordinates in the Cartesian frame.A.1.3 Orbital Plane System (OS)Fig. A. 3: Orbital Plane SystemStrictly speaking, an orbital plane can only be defined if the orbit is an ideal Kepler ellipse. Because of gravitational inf

18、luences, this is not the case in reality. Nevertheless the Orbital Plane System is used to calculate a Kepler orbit by a simple set of equations.Definition:The point of origin is the geocenter. The z-axis is orthogonal to the orbital plane and the z-direction is chosen so that the satellite is movin

19、g right handed around this axis. The x-axisAppendix111points towards the perigee and the y-axis completes the right handed Cartesian frame. If the orbit is ideal Kepler, all z-values are zero. The (time dependent) position vectors of the satellite are calculated using the eccentric anomaly E.A.1.4 T

20、rajectory System (TS)Fig. A. 4: Trajectory SystemIt is useful to do all calculations that are directly related to the spacecrafts trajectory in the Trajectory System. E.g. the attitude angles pitch, roll and yaw of a satellite are defined in this coordinate frame.Definition:The point of origin is th

21、e mass center of the satellite. The z-axis points to the inverse nadirdirection (zenith), the x-axis is orthogonal to the plane defined by the satellites velocity vector and the z-axis ant the y-axis completes the right handed Cartesian system.The unit vectors are given by:zTSe= n0,IN (inverse nadir

22、 direction)xTSTSe= ev ,IN ez(A.1.8)yzxe= e eTSTSTSA.1.5 Satellite System (SS)In the Satellite System, each axis is fixed to the satellite. The point of origin is the mass center of the spacecraft. If the attitude angles are zero, the Satellite System is aligned with Trajectory System.Even though the

23、re is only a small difference between the Trajectory System and the Satellite System, nevertheless it is useful to distinguish between the two. One reason is the fact that the Antenna Coordinate System (see) is only related to the spacecraft and not to its attitude angles. So the antenna pointing ve

24、ctor, especially in the case of antenna steering, can bedescribed independent from the spacecrafts attitude. Once the antenna pointing vector is calculated and transformed into the Satellite System, the effect oft the attitude angles can be described by simple rotations around the corresponding axes

25、.Fig. A. 5: Satellite SystemA.1.6 Antenna Coordinate System (AC)tFig. A. 6: Antenna Coordinate SystemThe antenna pointing vector, including antenna steering in azimuth and elevation, is described in the Antenna Coordinate System. This frame is shifted and distorted relative to the satellite System b

26、y a translation vector t and rotation angles, which depend on the localization and orientation of the antenna on the carrier platform.Definition:The point of origin is the antenna phase center. The y-axis is the normal to the antenna. If no antenna steering is applied, the direction of radiation is

27、aligned with the y-axis. The x-axis is fixed lengthwise on the antenna array and the z-axis completes the right handed Cartesian frame.A.1.7 Scene Centered Tangential System (SCT)Fig. A. 7: Scene Centered Tangential SystemThe Scene Centered Tangential System was introduced to relate the airplanes na

28、vigation to the scene that is illuminated by the spacecraft.Definition:The point of origin is the center of the illuminated scene on the WGS84 ellipsoid. The z-axisis aligned with the zenith direction of this point, the x-axis is orthogonal to the zenith and points to the east direction and the y-ax

29、is is orthogonal to the z-x-plane, pointing to the north direction.A.1.8 North East Down Coordinates (NED)Navigation parameters of an aircraft are often given in NED coordinates. This is an internal coordinate frame which is a basis for the inertial navigation system (INS). The attitude angles of th

30、e airplane can also be defined in NED coordinates.Definition:The point of origin is the mass center of the airplane. To obtain the direction of the z-axis(down component), a perpendicular is dropped from the point of origin to the earths surface. The x-axis is perpendicular to the z-axis and points

31、towards the north direction and the y-axis is perpendicular to the z-x-plane, pointing towards the east direction.A.2 Coordinate TransformationsA.2.1 Transformation Matrices TAll coordinate transformations in this work are rotational or translational. While a translational coordinate transformation

32、is performed by simply adding the translation vector, the rotational transformations are done by using rotation matrices. They are defined to rotate a vector right handed around the corresponding coordinate axis, if the rotation angle is positive.Letting be the rotation angle andfollows: Rotation ar

33、ound the x-axis:x = x1x2x3 , the rotation matrices are defined as100D1 () = 00 x = D1 () xcos ()sin () sin ()cos () (A.1.9) Rotation around the y-axis:= x1 100 x1 x = 0c 2 os () sin () x2 x3 0sin ()cos () x3 cos ()0sin () D2 () = 010 sin ()0 x = D2 () xcos ()(A.1.10) x1 cos ()0sin () x1 = x2 = 010 x

34、2 Rotation around the z-axis: x3 sin ()0cos () x3 cos () sin ()0D3 () = sin ()cos ()0001 x = D3 () x(A.1.11) x1 cos () sin ()0 x1 = x2 = sin ()cos ()0 x2 x3 001 x3 A.2.2 Transformation IN EFAccording to the definition in A.1.2, the earth fixed coordinate frame rotates relative to the inertial frame

35、with the angular velocity of around the common z-axis. Hence thetransformation of position vectors between the EF and the IN is a rotation using D3 (A.1.11): xIN = D3 ( t ) xEF x1 cos ( t ) sin ( t )0 x1 (A.1.12)= x2 = sin ( t )cos ( t )0 x2 x3 IN001 x3 EFThe opposite transformation direction is exp

36、ressed by the inverse of D3:3IN xEF= D1 ( t ) xwith333D1 ( t ) = DT ( t ) = D( t )(A.1.13)It is important to note that in the case of transforming velocity or acceleration the time dependency of the transformation matrix must be taken into account: vIN = xIN = D3 ( t ) xEF + D3 ( t ) xEFwith vEF = x

37、EFand(A.1.14) gIN = vIN = D3 ( t ) xEF + 2 D3 ( t ) xEF + D3 ( t ) xEFwith gEF = xEFIn case the calculations have to be done subject to date, a starting angle has to be added to the angular argument in the rotation matrix. This is the angle between the x-axis in the EarthCentered Inertial System and

38、 the x-axis of the Earth Centered Earth Fixed System. It can be obtained using the MJD (Modified Julian Date).A.2.3 Transformation OS INTo obtain the inertial coordinates of a vector given in the Orbital Plane System, a cascade of three rotations is done. The angular arguments of the rotation matric

39、es are the Kepler elementsinclination (i), angular argument of ascending node () and angular argument of the perigee (p). They define the orientation of the Kepler ellipse in the Earth Centered Inertial System: xIN = D3 () D1 (i ) D3 () xOSand: xOS = D3 () D1 (i ) D3 ( ) xIN(A.1.15)Note that the seq

40、uence of rotations must not be changed, because related to the “source” frame the coordinate axes change after each rotation. The inverse transformation is done by cancelling out the transformation matrices by left multiplying the equation with the inverse transformation matrices according to the ro

41、tation sequence.A.2.4 Transformation TS INBefore doing the transformation it is necessary to calculate the unit vectors of the Trajectory System in inertial coordinates (A.1.8). It is useful to first calculate the nadir direction in earth fixed ellipsoidal coordinates (WGS84) and then transform it i

42、nto inertial coordinates:cos (s ) cos (s )n0,EF = cos (s ) sin (s ) sin (s )with s ,s : longitude and latitude of the satellite n0,IN = D3 ( t ) n0,EF(A.1.16)Furthermore, the unit vector of the satellites velocity in the inertial system is needed. If the velocity vector is already given in the inert

43、ial system, the corresponding unit vector is calculated by dividing the velocity vector by its norm. If it is given in the Orbital Plane System, a coordinate transformation must be done: ev ,IN =vIN vIN (A.1.17)= D D i Dv3 ()1 ( )3 ( per )aearthOS (1 + eearth cos (E )Now the transformation matrix ca

44、n be calculated: D = ex ,TS , ey ,TS , ez ,TS IN (A.1.18)= n0,IN ev ,IN , (n0,IN ev ,IN ) n0,IN , n0,IN The transformation itself is done by: xIN = D xTS 1xTS = D T xIN = D (A.1.19) xINA.2.5 Transformation TS SSTransformations between the Trajectory System and the Satellite System are performed by r

45、otation with the attitude angles: xTS = D3 ( yaw) D2 (roll ) D1 ( pitch) xSSinverse direction: xSS = D1 ( pitch) D2 (roll ) D3 ( yaw) xTS(A.1.20)A.2.6 Transformation SS ACSince the orientation of the Antenna Coordinate System depends on how the antenna is mounted on the carrier platform, also the tr

46、ansformation rules depend on the particular situation. The angles for the rotation as well as length and direction of the translation vector tare defined by the geometry of the platform. In this work, the simplified caset = 0was assumed. That means that the centers of bothcoordinate frames are ident

47、ical. So the transformation can be done by just using rotations. The first rotation angle is /2 around the z-axis. After that, a rotation with around the x-axis is performed. Assuming that the roll angle is zero, the last rotation angle is /2- around the y- axis for a right looking system or /2+ for

48、 a left looking system respectively. is the off-nadir angle. xSS = D2 2 D1 () D3 2 xACinverse direction:(A.1.21) xAC = D3 D1 () D2 + xSS 2 2It is important to note that is the off-nadir angle when the roll angle is zero. If the roll angle is not zero, this effects the transformation from the Satelli

49、te System to the Trajectory System. The result will be that the real angle between the antenna pointing vector and the nadir vectorwill not be equal to any more.A.2.7 Transformation EF SCTA vector in the Earth Centered Earth Fixed coordinate frame is defined as: xEF = xEFyEFzEF = xEF ex ,EF + yEF ey ,EF + zEF ez ,EF(A.1.22)In the same way a vector in the Scene Cent red Tangential System can be expressed: xSCT = xSCTySCTzSCT (A.1.23)= xSCT ex ,SCT + ySCT ey ,SCT + zSCT ez ,SCTEquat