Coordinate Transformations

Transcription

1 Global Positioning Systems, Inertial Navigation, and Integration, Mohinder S. Grewal, Lawrence R. Weill, Angus P. Andrews Copyright # 001 John Wiley & Sons, Inc. Print ISBN X Electronic ISBN Appendix C Coordinate Transformations C.1 NOTATION We use the notation C from to to denote a coordinate transformation matrix from one coordinate frame designated by ``from'') to another coordinated frame designated by ``to''). For example, C ECI ENU denotes the coordinate transformation matrix from earth-centered inertial ECI) coordinates to earth- xed east±north±up ENU) local coordinates and C RPY NED denotes the coordinate transformation matrix from vehicle body- xed roll± pitch±yaw RPY) coordinates to earth- xed north±east±down NED) coordinates. Coordinate transformation matrices satisfy the composition rule C B CC A B ˆ C A C; where A, B, and C represent different coordinate frames. What we mean by a coordinate transformation matrix is that if a vector v has the representation v x v ˆ v y 5 C:1 v z

2 C.1 NOTATION 5 in XYZ coordinates and the same vector v has the alternative representation v u v ˆ v v 5 C: v w in UVW coordinates, then v x v y v z 5 ˆ C UVW XYW v u v v v w 5; C: where ``XYZ'' and ``UVW '' stand for any two Cartesian coordinate systems in threedimensional space. The components of a vector in either coordinate system can be expressed in terms of the vector components along unit vectors parallel to the respective coordinate axes. For example, if one set of coordinate axes is labeled X, Y and Z, and the other set of coordinate axes are labeled U, V, and W, then the same vector v can be expressed in either coordinate frame as v ˆ v x 1 x v y 1 y v z 1 z ˆ v u 1 u v v 1 v v w 1 w ; C: C:5 where the unit vectors 1 x, 1 y, and 1 z are along the XYZ axes; the scalars v x, v y, and v z are the respective components of v along the XYZ axes; the unit vectors 1 u, 1 v,and1 w are along the UVW axes; and the scalars v u, v v, and v w are the respective components of v along the UVW axes. The respective components can also be represented in terms of dot products of v with the various unit vectors, v x ˆ 1 T x v ˆ v u 1 T x 1 u v v 1 T x 1 v v w 1 T x 1 w ; v y ˆ 1 T y v ˆ v u 1 T y 1 u v v 1 T y 1 v v w 1 T y 1 w ; v z ˆ 1 T z v ˆ v u 1 T z 1 u v v 1 T z 1 v v w 1 T z 1 w ; C: C: C:8

3 COORDINATE TRANSFORMATIONS which can be represented in matrix form as v x 1 T x 1 u 1 T x 1 v 1 T x 1 w v y 5 ˆ 1 T y 1 u 1 T y 1 v 1 T y 1 w 5 v u v v 5 C:9 v z ˆ v w 1 T z 1 u 1 T z 1 v 1 T z 1 w v u XYZ v v 5 ; C:10 def C UVW v w which de nes the coordinate transformation matrix C UVW XYZ from UVW to XYZ coordinates in terms of the dot products of unit vectors. However, dot products of unit vectors also satisfy the cosine rule de ned in Section B.1..5) 1 T a 1 b ˆ cos y ab ; C:11 where y ab is the angle between the unit vectors 1 a and 1 b. As a consequence, the coordinate transformation matrix can also be written in the form C UVW XYZ cos y xu cos y xv cos y xw ˆ cos y yu cos y yv cos y yw 5 ; C:1 cos y zu cos y zv cos y zw which is why coordinate transformation matrices are also called ``direction cosines matrices.'' Navigation makes use of coordinates that are natural to the problem at hand: inertial coordinates for inertial navigation, orbital coordinates for GPS navigation, and earth- xed coordinates for representing locations on the earth. The principal coordinate systems used in navigation, and the transformations between these different coordinate systems, are summarized in this appendix. These are primarily Cartesian orthogonal) coordinates, and the transformations between them can be represented by orthogonal matrices. However, the coordinate transformations can also be represented by rotation vectors or quaternions, and all representations are used in the derivations and implementation of GPS=INS integration. C. INERTIAL REFERENCE DIRECTIONS C..1 Vernal Equinox The equinoxes are those times of year when the length of the day equals the length of the night the meaning of ``equinox''), which only happens when the sun is over the

4 C. INERTIAL REFERENCE DIRECTIONS equator. This happens twice a year: when the sun is passing from the Southern Hemisphere to the Northern Hemisphere vernal equinox) and again when it is passing from the Northern Hemisphere to the Southern Hemisphere autumnal equinox). The time of the vernal equinox de nes the beginning of spring the meaning of ``vernal'') in the Northern Hemisphere, which usually occurs around March 1±. The direction from the earth to the sun at the instant of the vernal equinox is used as a ``quasi-inertial'' direction in some navigation coordinates. This direction is de ned by the intersection of the equatorial plane of the earth with the ecliptic earth±sun plane). These two planes are inclined at about :5, as illustrated in Fig. C.1. The inertial direction of the vernal equinox is changing ever so slowly, on the order of 5 arc seconds per year, but the departure from truly inertial directions is neglible over the time periods of most navigation problems. The vernal equinox was in the constellation Pisces in the year 000. It was in the constellation Aries at the time of Hipparchus 190±10 BCE) and is sometimes still called ``the rst point of Aries.'' C.. Polar Axis of Earth The one inertial reference direction that remains invariant in earth- xed coordinates as the earth rotates is its polar axis, and that direction is used as a reference direction in inertial coordinates. Because the polar axis is by de nition) orthogonal to the earth's equatorial plane and the vernal equinox is by de nition) in the earth's equatorial plane, the earth's polar axis will always be orthogonal to the vernal equinox. A third orthogonal axis can then be de ned by their cross-product) such that the three axes de ne a right-handed de ned in Section B..11) orthogonal coordinate system. Fig. C.1 Direction of Vernal Equinox.

5 8 COORDINATE TRANSFORMATIONS C.COORDINATE SYSTEMS Although we are concerned exclusively with coordinate systems in the three dimensions of the observable world, there are many ways of representing a location in that world by a set of coordinates. The coordinates presented here are those used in navigation with GPS and=or INS. C..1 Cartesian and Polar Coordinates Rene Descartes 159±150) introduced the idea of representing points in threedimensional space by a triplet of coordinates, called ``Cartesian coordinates'' in his honor. They are also called ``Euclidean coordinates,'' but not because Euclid discovered them rst. The Cartesian coordinates x; y; z and polar coordinates y; f; r of a common reference point, as illustrated in Fig. C., are related by the equations x ˆ r cos y cos f ; C:1 y ˆ r sin y cos f ; C:1 z ˆ r sin f ; C:15 p r ˆ x y z ; C:1 f ˆ arcsin z 1 r p f 1 p ; C:1 y ˆ arctan y p < y p ; C:18 x with the angle y in radians) unde ned if f ˆ 1 p. C.. Celestial Coordinates The ``celestial sphere'' is a system for inertial directions referenced to the polar axis of the earth and the vernal equinox. The polar axis of these celestial coordinates is Fig. C. Cartesian andpolar coordinates.

6 C. COORDINATE SYSTEMS 9 Fig. C. Celestial coordinates. parallel to the polar axis of the earth and its prime meridian is xed to the vernal equinox. Polar celestial coordinates are right ascension the celestial analog of longitude, measured eastward from the vernal equinox) and declination the celestial analog of latitude), as illustrated in Fig. C.. Because the celestial sphere is used primarily as a reference for direction, no origin need be speci ed. Right ascension is zero at the vernal equinox and increases eastward in the direction the earth turns). The units of right ascension RA) can be radians, degrees, or hours with 15 deg=h as the conversion factor). By convention, declination is zero in the equatorial plane and increases toward the north pole, with the result that celestial objects in the Northern Hemisphere have positive declinations. Its units can be degrees or radians. C.. Satellite Orbit Coordinates Johannes Kepler 151±10) discovered the geometric shapes of the orbits of planets and the minimum number of parameters necessary to specify an orbit called ``Keplerian'' parameters). Keplerian parameters used to specify GPS satellite orbits in terms of their orientations relative to the equatorial plane and the vernal equinox de ned in Section C..1 and illustrated in Fig. C.1) include the following: Right ascension of the ascending node and orbit inclination, specifying the orientation of the orbital plane with respect to the vernal equinox and equatorial plane, is illustrated in Fig. C.. a) Right ascension is de ned in the previous section and is shown in Fig. C.. b) The intersection of the orbital plane of a satellite with the equatorial plane is called its ``line of nodes,'' where the ``nodes'' are the two intersections of the satellite orbit with this line. The two nodes are dubbed ``ascending'' 1 1 The astronomical symbol for the ascending node is, often read as ``earphones.''

7 0 COORDINATE TRANSFORMATIONS Fig. C. Keplerian parameters for satellite orbit. i.e., ascending from the Southern Hemisphere to the Northern Hemisphere) and ``descending''. The right ascension of the ascending node RAAN) is the angle in the equatorial plane from the vernal equinox to the ascending node, measured counterclockwise as seen looking down from the north pole direction. c) Orbital inclination is the dihedral angle between the orbital plane and the equatorial plane. It ranges from zero orbit in equatorial plane) to 180. Semimajor axis a and semiminor axis b de ned in Section C..5. and illustrated in Fig. C.) specify the size and shape of the elliptical orbit within the orbital plane. Orientation of the ellipse within its orbital plane, speci ed in terms of the ``argument of perigee,'' the angle between the ascending node and the perigee of the orbit closest approach to earth), is illustrated in Fig. C.. Position of the satellite relative to perigee of the elliptical orbit, speci ed in terms of the angle from perigee, called the ``argument of latitude'' or ``true anomaly,'' is illustrated in Fig. C.. For computer simulation demonstrations, GPS satellite orbits can usually be assumed to be circular with radius a ˆ b ˆ R ˆ ;50 km and inclined at 55 to the equatorial plane. This eliminates the need to specify the orientation of the elliptical orbit within the orbital plane. The argument of perigee becomes overly sensitive to orbit perturbations when eccentricity is close to zero.)

8 C. COORDINATE SYSTEMS 1 Fig. C.5 ECI andecef Coordinates. C.. ECI Coordinates Earth-centered inertial ECI) coordinates are the favored inertial coordinates in the near-earth environment. The origin of ECI coordinates is at the center of gravity of the earth, with Fig. C.5) 1. axis in the direction of the vernal equinox,. axis direction parallel to the rotation axis north polar axis) of the earth, and. an additional axis to make this a right-handed orthogonal coordinate system, with the polar axis as the third axis hence the numbering). The equatorial plane of the earth is also the equatorial plane of ECI coordinates, but the earth itself is rotating relative to the vernal equinox at its sidereal rotation rate of about ;9;115; rad=s, or about deg=h, as illustrated in Fig. C.5. C..5 ECEF Coordinates Earth-centered, earth- xed ECEF) coordinates have the same origin earth center) and third polar) axis as ECI coordinates but rotate with the earth, as shown in Fig. C.5. As a consequence, ECI and ECEF longitudes differ only by a linear function of time. Longitude in ECEF coordinates is measured east ) and west ) from the prime meridian passing through the principal transit instrument at the observatory at Greenwich, UK, a convention adopted by 1 representatives of 5 nations at the International Meridian Conference, held in Washington, DC, in October of 188.

9 COORDINATE TRANSFORMATIONS Latitudes are measured with respect to the equatorial plane, but there is more than one kind of ``latitude.'' Geocentric latitude would be measured as the angle between the equatorial plane and a line from the reference point to the center of the earth, but this angle could not be determined accurately before GPS) without running a transit survey over vast distances. The angle between the pole star and the local vertical direction could be measured more readily, and that angle is more closely approximated as geodetic latitude. There is yet a third latitude parametric latitude) that is useful in analysis. The latter two latitudes are de ned in the following subsections. C..5.1 Ellipsoidal Earth Models Geodesy is the study of the size and shape of the earth and the establishment of physical control points de ning the origin and orientation of coordinate systems for mapping the earth. Earth shape models are very important for navigation using either GPS or INS, or both. INS alignment is with respect to the local vertical, which does not generally pass through the center of the earth. That is because the earth is not spherical. At different times in history, the earth has been regarded as being at rst-order approximation), spherical second-order), and ellipsoidal third-order). The thirdorder model is an ellipsoid of revolution, with its shorter radius at the poles and its longer radius at the equator. C..5. Parametric Latitude For geoids based on ellipsoids of revolution, every meridian is an ellipse with equatorial radius a also called ``semimajor axis'') and polar radius b also called ``semiminor axis''). If we let z be the Cartesian coordinate in the polar direction and x meridional be the equatorial coordinate in the meridional plane, as illustrated in Fig. C., then the equation for this ellipse will be x meridional a z b ˆ 1 ˆ cos f parametric sin f parametric ˆ a cos f parametric a b sin f parametric b C:19 C:0 C:1 That is, a parametric solution for the ellipse is ˆ a cos f parametric Š a b sin f parametric Š b : C: x meridional ˆ a cos f parametric ; z ˆ b sin f parametric ; C: C: as illustrated in Fig. C.. Although the parametric latitude f parametric has no physical signi cance, it is quite useful for relating geocentric and geodetic latitude, which do have physical signi cance.

10 C. COORDINATE SYSTEMS Fig. C. Geocentric, parametric, and geodetic latitudes in meridional plane. C..5. Geodetic Latitude Geodetic latitude is de ned as the elevation angle above ) or below ) the equatorial plane of the normal to the ellipsoidal surface. This direction can be de ned in terms of the parametric latitude, because it is orthogonal to the meridional tangential direction. The vector tangential to the meridian will be in the direction of the derivative to the elliptical equation solution with respect to parametric latitude: a cos f v tangential / # parametric parametric b sin f parametric " ˆ a sin f parametric # ; C: b cos f parametric and the meridional normal direction will be orthogonal to it, or " v normal / b cos f parametric # ; C: a sin f parametric as illustrated in Fig. C.. The tangent of geodetic latitude is then the ratio of the z- and x-components of the surface normal vector, or tan f geodetic ˆasin f parametric C:8 b cos f parametric ˆ a b tan f parametric ; C:9

11 COORDINATE TRANSFORMATIONS from which, using some standard trigonometric identities, tan f geodetic sin f geodetic ˆq 1 tan f geodetic C:0 a sin f parametric ˆ q ; C:1 a sin f parametric b cos f parametric 1 cos f geodetic ˆq 1 tan f geodetic The inverse relationship is C: b cos f parametric ˆ q : C: a sin f parametric b cos f parametric tan f parametric ˆb a tan f geodetic ; from which, using the same trigonometric identities as before, tan f parametric sin f parametric ˆq 1 tan f parametric C: C:5 b sin f geodetic ˆ q ; C: a cos f geodetic b sin f geodetic 1 cos f parametric ˆq 1 tan f parametric C: a cos f geodetic ˆ q ; C:8 a cos f geodetic b sin f geodetic and the two-dimensional X -Z Cartesian coordinates in the meridional plane of a point on the geoid surface will in terms of geodetic latitude. x meridional ˆ a cos f parametric C:9 a cos f geodetic ˆ q ; C:0 a cos f geodetic b sin f geodetic z ˆ b sin f parametric C:1 b sin f geodetic ˆ q C: a cos f geodetic b sin f geodetic

12 C. COORDINATE SYSTEMS 5 Equations C.0 and C. apply only to points on the geoid surface. Orthometric height h above ) or below ) the geoid surface is measured along the surface normal, so that the X -Z coordinates for a point with altitude h will be x meridional ˆ cos f geodetic 0 1 a B h qa; C: a cos f geodetic b sin f geodetic z ˆ sin f geodetic 0 1 b B h qa: C: a cos f geodetic b sin f geodetic In three-dimensional ECEF coordinates, with the X -axis passing through the equator at the prime meridian at which longitude y ˆ 0), x ECEF ˆ cos y x meridional C:5 ˆ cos y cos f geodetic 0 1 a B h qa; C: a cos f geodetic b sin f geodetic y ECEF ˆ sin y x meridional C: ˆ sin y cos f geodetic 0 1 a B h qa; C:8 a cos f geodetic b sin f geodetic z ECEF ˆ sin f geodetic 0 1 b B h qa; C:9 a cos f geodetic b sin f geodetic in terms of geodetic latitude f geodetic, longitude y, and orthometric altitude h with respect to the reference geoid.

13 COORDINATE TRANSFORMATIONS The inverse transformation, from ECEF XYZ to geodetic longitude±latitude± altitude coordinates, is y ˆ atan y ECEF ; x ECEF ; C:50 f geodetic ˆ atan z ECEF e a sin z ; x e a cos z ; C:51 b h ˆ x cos f r T; C:5 where atan is the four-quadrant arctangent function in MATLAB and z ˆ atan az ECEF ; bx ; C:5 x ˆ q x ECEF y ECEF; C:5 a r T ˆ p ; C:55 1 e sin f where r T is the transverse radius of curvature on the ellipsoid, a is the equatorial radius, b is the polar radius, and e is elliptical eccentricity. C..5. Geocentric Latitude For points on the geoid surface, the tangent of geocentric latitude is the ratio of distance above ) or below ) the equator [z ˆ b sin f parametric ] to the distance from the polar axis [x meridional ˆ a cos f parametric ], or tan f GEOCENTRIC ˆb sin f parametric a cos f parametric ˆ b a tan f parametric ˆ b a tan f geodetic ; C:5 C:5 C:58

14 C. COORDINATE SYSTEMS from which, using the same trigonometric identities as were used for geodetic latitude, tan f geocentric sin f geocentric ˆq C:59 1 tan f geocentric b sin f parametric ˆ q a cos f parametric b sin f parametric C:0 b sin f geodetic ˆ q ; C:1 a cos f geodetic b sin f geodetic 1 cos f geocentric ˆq 1 tan f geocentric a cos f parametric ˆ q a cos f parametric b sin f parametric C: C: a cos f geodetic ˆ q : C: a cos f geodetic b sin f geodetic The inverse relationships are tan f parametric ˆa b tan f geocentric ; tan f ˆa geodetic b tan f geocentric ; from which, using the same trigonometric identities again, tan f parametric sin f parametric ˆq 1 tan f parametric C:5 C: C: a sin f geocentric ˆ q ; C:8 a sin f geocentric b cos f geocentric a sin f geocentric sin f geodetic ˆq ; C:9 a sin f geocentric b cos f geocentric 1 cos f parametric ˆq 1 tan f parametric C:0 b cos f geocentric ˆ q ; C:1 a sin f geocentric b cos f geocentric b cos f geocentric cos f geodetic ˆq : C: a sin f geocentric b cos f geocentric

15 8 COORDINATE TRANSFORMATIONS C.. LTP Coordinates Local tangent plane LTP) coordinates, also called ``locally level coordinates,'' are a return to the rst-order model of the earth as being at, where they serve as local reference directions for representing vehicle attitude and velocity for operation on or near the surface of the earth. A common orientation for LTP coordinates has one horizontal axis the north axis) in the direction of increasing latitude and the other horizontal axis the east axis) in the direction of increasing longitude, as illustrated in Fig. C.. Horizontal location components in this local coordinate frame are called ``relative northing'' and ``relative easting.'' C...1 Alpha Wander Coordinates Maintaining east±north orientation was a problem for some INSs at the poles, where north and east directions change by 180. Early gimbaled inertial systems could not slew the platform axes fast enough for near-polar operation. This problem was solved by letting the platform axes ``wander'' from north but keeping track of the angle a between north and a reference platfrom axis, as shown in Fig. C.8. This LTP orientation came to be called ``alpha wander.''' C... ENU=NED Coordinates East±north±up ENU) and north±east±down NED) are two common right-handed LTP coordinate systems. ENU coordinates may be preferred to NED coordinates because altitude increases in the upward direction. But NED coordinates may also be preferred over ENU coordinates because the direction of a right clockwise) turn is in the positive direction with respect to a downward axis, and NED coordinate axes coincide with vehicle- xed roll±pitch±yaw RPY) coordinates Section C..) when the vehicle is level and headed north. The coordinate transformation matrix C ENU NED from ENU to NED coordinates and the transformation matrix C NED ENU from NED to ENU coordinates are one and the same: C ENU NED ˆ C NED ENU ˆ : C: Fig. C. ENU coordinates.

16 C. COORDINATE SYSTEMS 9 C... ENU=ECEF Coordinates The unit vectors in local east, north, and up directions, as expressed in ECEF Cartesian coordinates, will be sin y 1 E ˆ cos y 5; C: 0 cos y sin f geodetic 1 N ˆ sin y sin f geodetic 5; C:5 cos f geodetic cos y cos f geodetic 1 U ˆ sin y cos f geodetic 5; C: sin f geodetic and the unit vectors in the ECEF X, Y, and Z directions, as expressed in ENU coordinates, will be sin y 1 X ˆ cos y sin f geodetic 5; C: cos y cos f geodetic Fig. C.8 Alpha wander.

17 0 COORDINATE TRANSFORMATIONS cos y 1 Y ˆ sin y sin f geodetic 5; C:8 sin y cos f geodetic 0 1 Z ˆ cos f geodetic 5: C:9 sin f geodetic C... NED=ECEF Coordinates It is more natural in some applications to use NED directions for locally level coordinates. This coordinate system coincides with vehicle-body- xed RPY coordinates shown in Fig. C.9) when the vehicle is level headed north. The unit vectors in local north, east, and down directions, as expressed in ECEF Cartesian coordinates, will be cos y sin f geodetic 1 N ˆ sin y sin f geodetic 5; C:80 cos f geodetic sin y 1 E ˆ cos y 5; C:81 0 cos y cos f geodetic 1 D ˆ sin y cos f geodetic 5; C:8 sin f geodetic Fig. C.9 Roll-pitch-yaw axes.

18 C. COORDINATE SYSTEMS 1 and the unit vectors in the ECEF X, Y, and Z directions, as expressed in NED coordinates, will be cos y sin f geodetic 1 X ˆ sin y 5; C:8 cos y cos f geodetic sin y sin f geodetic 1 Y ˆ cos y 5; C:8 sin y cos f geodetic cos f geodetic 1 Z ˆ 0 sin f geodetic 5; C:85 C.. RPY Coordinates The RPY coordinates are vehicle xed, with the roll axis in the nominal direction of motion of the vehicle, the pitch axis out the right-hand side, and the yaw axis such that turning to the right is positive, as illustrated in Fig.C.9. The same orientations of vehicle- xed coordinates are used for surface ships and ground vehicles. They are also called ``SAE coordinates,'' because they are the standard body- xed coordinates used by the Society of Automotive Engineers. For rocket boosters with their roll axes vertical at lift-off, the pitch axis is typically de ned to be orthogonal to the plane of the boost trajectory also called the ``pitch plane'' or ``ascent plane''). C..8 Vehicle Attitude Euler Angles The attitude of the vehicle body with respect to local coordinates can be speci ed in terms of rotations about the vehicle roll, pitch, and yaw axes, starting with these axes aligned with NED coordinates. The angles of rotation about each of these axes are called Euler angles, named for the Swiss mathematician Leonard Euler 10± 18). It is always necessary to specify the order of rotations when specifying Euler pronounced ``oiler'') angles. A fairly common convention for vehicle attitude Euler angles is illustrated in Fig. C.10, where, starting with the vehicle level with roll axis pointed north: 1. Yaw=Heading. Rotate through the yaw angle Y ) about the vehicle yaw axis to the intended azimuth heading) of the vehicle roll axis. Azimuth is measured clockwise east) from north.. Pitch. Rotate through the pitch angle P) about the vehicle pitch axis to bring the vehicle roll axis to its intended elevation. Elevation is measured positive upward from the local horizontal plane.

19 COORDINATE TRANSFORMATIONS Fig. C.10 Vehicle Euler angles.. Roll. Rotate through the roll angle R) about the vehicle roll axis to bring the vehicle attitude to the speci ed orientation. Euler angles are redundant for vehicle attitudes with 90 pitch, in which case the roll axis is vertical. In that attitude, heading changes also rotate the vehicle about the roll axis. This is the attitude of most rocket boosters at lift-off. Some boosters can be seen making a roll maneuver immediately after lift-off to align their yaw axes with the launch azimuth in the ascent plane. This maneuver may be required to correct for launch delays on missions for which launch azimuth is a function of launch time. C..8.1 RPY/ENU Coordinates With vehicle attitude speci ed by yaw angle Y ), pitch angle P), and roll angle R) as speci ed above, the resulting unit vectors of the roll, pitch, and yaw axes in ENU coordinates will be sin Y cos P 1 R ˆ cos Y cos P 5 ; C:8 sin P cos R cos Y sin R sin Y sin P 1 P ˆ cos R sin Y sin R cos Y sin P 5 ; C:8 sin R cos P sin R cos Y cos R sin Y sin P 1 Y ˆ sin R sin Y cos R cos Y sin P 5 ; C:88 cos R cos P

20 C. COORDINATE SYSTEMS the unit vectors of the east, north, and up axes in RPY coordinates will be sin Y cos P 1 E ˆ cos R cos Y sin R sin Y sin P 5 ; C:89 sin R cos Y cos R sin Y sin P cos Y cos P 1 N ˆ cos R sin Y sin R cos Y sin P 5 ; C:90 sin R sin Y cos R cos Y sin P sin P 1 U ˆ sin R cos P 5 ; C:91 cos R cos P and the coordinate transformation matrix from RPY coordinates to ENU coordinates will be C RPY ENU ˆ 1 R 1 P 1 Y Šˆ 1 T E 1 T N 1 T U 5 C:9 S Y C P C R C Y S R S Y S P S R C Y C R S Y S P ˆ C Y C P C R S Y S R C Y S P S R S Y C R C Y S P 5 ; C:9 S P S R C P C R C P where S R ˆ sin R ; C R ˆ cos R ; S P ˆ sin P ; C P cos P ; S Y ˆ sin Y ; C Y ˆ cos Y : C:9 C:95 C:9 C:9 C:98 C:99

21 COORDINATE TRANSFORMATIONS C..9 GPS Coordinates The parameter O in Fig. C.1 is the RAAN, which is the ECI longitude where the orbital plane intersects the equatorial plane as the satellite crosses from the Southern Hemisphere to the Northern Hemisphere. The orbital plane is speci ed by O and a, the inclination of the orbit plane with respect to the equatorial plane a 55 for GPS satellite orbits). The y parameter represents the location of the satellite within the orbit plane, as the angular phase in the circular orbit with respect to ascending node. For GPS satellite orbits, the angle y changes at a nearly constant rate of about 1:58 10 rad=s and a period of about,08 s half a day). The nominal satellite position in ECEF coordinates is then given as x ˆ R cos y cos Osin y sin O cos aš; C:100 y ˆ R cos y cos O sin y sin O cos aš; C:101 z ˆ R sin y sin a; C:10 0 y ˆ y 0 t t 0 deg; ;08 C:10 0 O ˆ O 0 t t 0 deg; C:10 8;1 R ˆ ;50;000 m. C:105 GPS satellite positions in the transmitted navigation message are speci ed in the ECEF coordinate system of WGS 8. A locally level x 1 ; y 1 ; z 1 reference coordinate system described in Section C..) is used by an observer location on the earth, where the x 1 y 1 plane is tangential to the surface of the earth, x 1 pointing east, y 1 pointing north, and z 1 normal to the plane. See Fig. C.11. Here, X ENU ˆ C ECEF ENU X ECEF S; C ECEF ENU C ECEF ENU ˆ coordinate transformation matrix from ECEF to ENU; S ˆ coordinate origin shift vector from ECEF to local reference; sin y cos y 0 ˆ sin f cos y sin f sin y cos f 5; cos f cos y cos f sin y sin f X U sin y Y U cos y S ˆ X U sin f cos y Y U sin f sin y Z U cos f 5; X U cos f cos y Y U cos f sin y Z U sin f X U ; Y U ; Z U ˆ user's position; y ˆ local reference longitude; f ˆ local geometric latitude:

22 C. COORDINATE SYSTEMS 5 Fig. C.11 Pseudorange. Fig. C.1 Satellite coordinates.

23 COORDINATE TRANSFORMATIONS C. COORDINATE TRANSFORMATION MODELS Coordinate transformations are methods for transforming a vector represented in one coordinate system into the appropriate representation in another coordinate system. These coordinate transformations can be represented in a number of different ways, each with its advantages and disadvantages. These transformations generally involve translations for coordinate systems with different origins) and rotations for Cartesian coordinate systems with different axis directions) or transcendental transformations between Cartesian and polar or geodetic coordinates). The transformations between Cartesian and polar coordinates have already been discussed in Section C..1 and translations are rather obvious, so we will concentrate on the rotations. C..1 Euler Angles Euler angles were used for de ning vehicle attitude in Section C..8, and vehicle attitude representation is a common use of Euler angles in navigation. Euler angles are used to de ne a coordinate transformation in terms of a set of three angular rotations, performed in a speci ed sequence about three speci ed orthogonal axes, to bring one coordinate frame to coincide with another. The coordinate transformation from RPY coordinates to NED coordinates, for example, can be composed from three Euler rotation matrices: Yaw Pitch Roll z } { z } { z } { C Y S Y 0 C P 0 S P CNED RPY ˆ S Y C Y C R S R 5 C: S P 0 C P 0 S R C R C Y P P S Y C R C Y S P S R S Y S R C Y S P C R S Y C P C Y C R S Y S P S R C Y S R S Y S P C R ˆ ; C:10 S P C P S R C P C R 5 roll axis pitch axis yaw axis {z } in NED coordinates where the matrix elements are de ned in Eqs. C.9±C.99. This matrix also rotates the NED coordinate axes to coincide with RPY coordinate axes. Compare this with the transformation from RPY to ENU coordinates in Eq. C.9.) For example, the coordinate transformation for nominal booster rocket launch attitude roll axis straight up) would be given by Eq. with pitch angle P ˆ 1 p C P ˆ 0; S P ˆ 1), which becomes C RPY NED ˆ 0 sin R Y cos R Y 0 cos R Y sin R Y 5: 1 0 0

24 C. COORDINATE TRANSFORMATION MODELS That is, the coordinate transformation in this attitude depends only on the difference between roll angle R) and yaw angle Y ). Euler angles are a concise representation for vehicle attitude. They are handy for driving cockpit displays such as compass cards using Y ) and arti cial horizon indicators using R and P), but they are not particularly handy for representing vehicle attitude dynamics. The reasons for the latter include the following: Euler angles have discontinuities analogous to ``gimbal lock'' Section..1.) when the vehicle roll axis is pointed upward, as it is for launch of many rockets. In that orientation, tiny changes in vehicle pitch or yaw cause 180 changes in heading angle. For aircraft, this creates a slewing rate problem for electromechanical compass card displays. The relationships between sensed body rates and Euler angle rates are mathematically complicated. C.. Rotation Vectors All right-handed orthogonal coordinate systems with the same origins in three dimensions can be transformed one onto another by single rotations about xed axes. The corresponding rotation vectors relating two coordinate systems are de ned by the direction rotation axis) and magnitude rotation angle) of that transformation. For example, the rotation vector for rotating ENU coordinates to NED coordinates and vice versa) is r ENU NED ˆ p p= p p= 5; C:108 0 which has magnitude jr ENU NED jˆp 180 ) and direction north±east, as illustrated in Fig. C.1. For illustrative purposes only, NED coordinates are shown as being translated from ENU coordinates in Fig. C.1. In practice, rotation vectors represent pure rotations, without any translation.) The rotation vector is another minimal representation of a coordinate transformation, along with Euler angles. Like Euler angles, rotation vectors are concise but also have some drawbacks: 1. It is not a unique representation, in that adding multiples of p to the magnitude of a rotation vector has no effect on the transformation it represents.. It is a nonlinear and rather complicated representation, in that the result of one rotation followed by another is a third rotation, the rotation vector for which is a fairly complicated function of the rst two rotation vectors. But, unlike Euler angles, rotation vector models do not exhibit ``gimbal lock.''

25 8 COORDINATE TRANSFORMATIONS Fig. C.1 Rotation from ENU to NED coordinates. C...1 Rotation Vector to Matrix The rotation represented by a rotation vector r 1 r ˆ r 5 C:109 r can be implemented as multiplication by the matrix C r ḓef exp r C: r r def ˆ exp B r 0 r 1 5A C:111 r r r r ˆ cos jrj I 1cos jrj jrj rr T sin jrj r jrj 0 r 1 5 C:11 r r u u ˆ cos y I 1cos y 1 r 1 T r sin y u 0 u 1 5 ; C:11 u u 1 0 y ḓef jrj; ḓef r 1 r rj ; C:11 C:115 which was derived in Eq. B.1. That is, for any three-rowed column vector v, C r v rotates it through an angle of jrj radians about the vector r.

26 C. COORDINATE TRANSFORMATION MODELS 9 The form of the matrix in Eq. C.11 is better suited for computation when y 0, but the form of the matrix in Eq. C.11 is useful for computing sensitivities using partial derivatives used in Chapter 8). For example, the rotation vector r ENU NED in Eq. C.108 transforming between ENU and NED has magnitude and direction y ˆ p sin y ˆ0; cos y ˆ1Š; p 1= p 1 r ˆ 1= 5; 0 respectively, and the corresponding rotation matrix 0 u u NED ˆ cos p I 1cos p Š1 r 1 r sin p u 0 u 1 5 u u 1 0 C ENU ˆI 1 r 1 T r ˆ ˆ transforms from ENU to NED coordinates. Compare this result to Eq. C..) Because coordinate transformation matrices are orthogonal matrices and the matrix C ENU NED is also symmetric, C ENU NED is its own inverse. That is, C ENU NED ˆ C NED ENU: C:11 C... Matrix to Rotation Vector Although there is a unique coordinate transformation matrix for each rotation vector, the converse is not true. Adding multiples of p to the magnitude of a rotation vector has no effect on the resulting coordinate transformation matrix. The following approach yields a unique rotation vector with magnitude jrj p. The trace tr C of a square matrix M is the sum of its diagonal values. For the coordinate transformation matrix of Eq. C.11, tr C r Š ˆ 1 cos y ; C:11 Linear combinations of the sort a 1 I a 1 r a 1 r 1 T r, where 1 is a unit vector, form a subalgebra of matrices with relatively simple rules for multiplication, inversion, etc.

27 50 COORDINATE TRANSFORMATIONS from which the rotation angle jrj ˆy C:118 tr C r Š 1 ˆ arcos ; C:119 a formula that will yield a result in the range 0 < y < p, but with poor delity near where the derivative of the cosine equals zero at y ˆ 0 and y ˆ p. The values of y near y ˆ 0 and y ˆ p can be better estimated using the sine of y, which can be recovered using the antisymmetric part of C r, A ˆ 0 a 1 a 1 a 1 0 a 5 C:10 a 1 a 0 def ˆ 1 C r CT r Š ˆ sin y y C:11 0 r r r 0 r 1 5 ; C:1 r r 1 0 from which the vector will have magnitude a a 1 a 1 5 ˆ sin y 1 jrj r q a a 1 a 1 ˆ sin y C:1 C:1 and the same direction as r. As a consequence, one can recover the magnitude y of r from q y ˆ atan a a 1 a 1; tr C r Š1 C:15 using the MATLAB function atan, and then the rotation vector r as when 0 < y < p. r ˆ y a a sin y 1 5 C:1 a 1

28 C. COORDINATE TRANSFORMATION MODELS 51 C... Special Cases for sin u 0 For y 0, r 0, although Eq. C.1 may still work adequately for y > 10,say. For y p, the symmetric part of C r, s 11 s 1 s 1 S ˆ s 1 s s 5 C:1 s 1 s s def 1 ˆ C r CT r Š C:18 ˆ cos y I 1cos y rr T y C:19 I y rrt C:10 and the unit vector 1 r ḓef 1 y r C:11 satis es u 1 1 u 1u u 1 u S u 1 u u 1 u u 5; C:1 u 1 u u u u 1 which can be solved for a unique u by assigning u k > 0 for 0 B k ˆ argmax@ s 11 s 1 C 5A; C:1 s q 1 u k ˆ s kk 1 then, depending on whether k ˆ 1, k ˆ, or k ˆ, 9 k ˆ 1 k ˆ k ˆ r s u s 1 s 1 u u r >= s u 1 s 1 s u 1 u r s u 1 s s 11 1 >; u 1 u C:1 C:15

29 5 COORDINATE TRANSFORMATIONS and u 1 r ˆ y u 5: C:1 u C... Time Derivatives of Rotation Vectors The mathematical relationships between rotation rates o k and the time derivatives of the corresponding rotation vector r are fairly complicated, but they can be derived from Eq. C.1 for the dynamics of coordinate transformation matrices. Let r ENU be the rotation vector represented in earth- xed ENU coordinates that rotates earth- xed ENU coordinate axes into vehicle body- xed RPY axes, and let C r be the corresponding rotation matrix, so that, in ENU coordinates, 1 E ˆ 1 0 0Š T ; 1 N ˆ 0 1 0Š T ; 1 U ˆ 0 0 1Š T ; C r ENU 1 E ˆ 1 R ; C r ENU I N ˆ 1 P ; C r ENU 1 U ˆ 1 Y ; C RPY ENU ˆ 1 R 1 P 1 Y Š; C:1 ˆ C r ENU 1 E C r ENU 1 N C r ENU 1 U Š ˆ C r ENU 1 E 1 N 1 U Š ˆ C r ENU C RPY ENU ˆ C r ENU : C:18 That is, C r ENU is the coordinate transformation matrix from RPY coordinates to ENU coordinates. As a consequence, from Eq. C.1, d dt C r ˆd ENU dt CRPY ENU ˆ 0 o U o N o U 0 o E o N o E 0 C:19 0 o Y o P ENU C RPY ENU o Y 0 o R 5; o P o R 0 5C RPY C:10 0 o U o N 0 o Y o P d dt C r ENU ˆ o U 0 o E 5C r ENU C r ENU o Y 0 o R 5; o N o E 0 o P o R 0 C:11

30 C. COORDINATE TRANSFORMATION MODELS 5 where o R v RPY ˆ o P 5 C:1 o Y is the vector of inertial rotation rates of the vehicle body, expressed in RPY coordinates, and o E v ENU ˆ o N 5 C:1 o U is the vector of inertial rotation rates of the ENU coordinate frame, expressed in ENU coordinates. The matrix equation C.11 is equivalent to nine scalar E _r N _r U _r U ˆc 1; o P c 1; o Y c ;1 o N c ;1 o U E _r N _r U _r U ˆ c 1; o R c 1;1 o Y c ; o N c ; o U E _r N _r U _r U ˆc 1; o R c 1;1 o P c ; o N c ; o U E _r N _r U _r U ˆc ; o P c ; o Y c ;1 o E c 1;1 o E N U _r U ˆ c ; o R c ;1 o Y c ; o E c 1; o E N U _r U ˆc ; o R c ;1 o P c ; o E c 1; o U E _r N _r U _r U ˆc ; o P c ; o Y c ;1 o E c 1;1 o E N U _r U ˆ c ; o R c ;1 o Y c ; o E c 1; o E N U _r U ˆc ; o R c ;1 o P c ; o E c 1; o N ; where c 11 c 1 c 1 c 1 c c 5 ḓef C renu c 1 c c and the partial derivatives

31 5 COORDINATE 11 ˆ ue 1 u E 1 cos y Šy sin y E 11 ˆ un u E 1 cos y Šy sin y 1 u E N 11 ˆ uu u E 1 cos y Šy sin y 1 u E U 1 ˆ un 1 u E 1 cos y Š ue u U sin y yu E u U cos y yu N u E sin y E 1 ˆ ue 1 u N 1 cos y Š uu u N sin y yu N u U cos y yu E u N sin y N 1 ˆ u Eu N u U 1 cos y Š 1 u U sin y yu U cos y yu U u N u E sin y U 1 ˆ uu 1 u E 1 cos y ŠuE u N sin y yu E u N cos y yu U u E sin y E 1 ˆ u Eu N u U 1 cos y Š 1 u N sin y yu N cos y yu U u N u E sin y N 1 ˆ ue 1 u U 1 cos y ŠuU u N sin y yu N u U cos y yu E u U sin y U 1 ˆ un 1 u E 1 cos y ŠuE u U sin y yu E u U cos y yu N u E sin y E 1 ˆ ue 1 u N 1 cos y ŠuU u N sin y yu N u U cos y yu E u N sin y N 1 ˆ u Eu N u U 1 cos y Š sin y 1 u U yu U cos y yu U u N u E sin y U ˆ ue u N 1 cos y Šy 1 u N sin y E ˆ un 1 u N 1 cos y Šy sin y N ˆ uu u N 1 cos y Šy 1 u N sin y U ˆ u Eu N u U 1 cos y Š 1 u E sin y yu E cos y yu E u N u U sin y E ˆ uu 1 u N 1 cos y Š ue u N sin y yu E u N cos y yu N u U sin y N ˆ un 1 u U 1 cos y Š ue u U sin y yu E u U cos y yu U u N sin y ; y

32 @c 1 ˆ uu 1 u E 1 cos y Š ue u N sin y yu E u N cos y yu U u E sin y E 1 ˆ u Eu N u U 1 cos y Š 1 u N sin y yu N cos y yu U u N u E sin y N 1 ˆ ue 1 u U 1 cos y Š uu u N sin y yu N u U cos y yu E u U sin y U ˆ u Eu N u U 1 cos y Š 1 u E sin y yu E cos y yu U u N u E sin y E ˆ uu 1 u N 1 cos y ŠuE u N sin y yu E u N cos y yu N u U sin y N ˆ un 1 u U 1 cos y ŠuE u U sin y yu E u U cos y yu U u N sin y U ˆ ue u U 1 cos y Šy sin y 1 u U E ˆ un u U 1 cos y Šy sin y 1 u U N ˆ uu 1 u U 1cos y Šy sin U y for y ḓef jrenu j; ḓef r u E E y ; u ḓef r N N y ; u ḓef r U u y : These nine scalar linear equations can be put into matrix form and solved in least squares fashion as _r E L _r N 5 ˆ R _r U C. COORDINATE TRANSFORMATION MODELS 55 o R o P o Y o E o N o U _r E _r N 5 ˆ L T L n L T R {z } _r ; C:1 5 v RPY v ENU : C:15

33 5 COORDINATE TRANSFORMATIONS The matrix product L T L will always be invertible because its determinant det L T LŠˆ 1cos y Š y ; C:1 lim y!0 det LT LŠˆ8; C:1 and the resulting equation for r ENU can be put into the form _r ENU v ENU : C:18 The can be partitioned @v ENU C:19 with ˆ RPY jrj sin jrj rr T jrj sin jrj jrj 1cos jrj Š 1cos jrj Š I 1 rš ˆ 1 r 1 T r y sin y 1cos y Š I 1 r1r T Š y 1 rš; lim r P@_r ˆ I; ˆ ENU jrj sin jrj rr T jrj 1cos jrj Š jrj sin jrj 1cos jrj Š I 1 rš ˆ1 r 1 T r y sin y 1cos y Š I 1 r1 T rš y 1 lim ˆI: ENU C:150 C:151 C:15 C:15 C:15 C:155 For locally leveled gimbaled systems, v RPH ˆ 0. That is, the gimbals normally keep the accelerometer axes aligned to the ENU or NED coordinate axes, a process modeled by v ENU alone.

34 C. COORDINATE TRANSFORMATION MODELS 5 C...5 Time Derivatives of Matrix Expressions The Kalman lter implementation for integrating GPS with a strapdown INS in Chapter 8 will require derivatives with respect to time of the RPH Eq: C:150 ENU Eq: C:15 : We derive here a general-purpose formula for taking such derivatives and then apply it to these two cases. General Formulas There is a general-purpose formula for taking the time derivatives d=dt M r of matrix expressions of the sort M r ˆ M s 1 r ; s r ; s r C:15 ˆ s 1 r I s r rš s r rr T ; C:15 that is, as linear combinations of I, r, and rr T with scalar functions of r as the coef cients. The derivation uses the time derivatives of the basis matrices, where the vector d dt I ˆ 0 ; C:158 d dt rš ˆ _r Š; C:159 d dt rrt ˆ _rr T r_r T ; C:10 _r ˆ d dt r; C:11 and then uses the chain rule for differentiation to obtain the general formula d dt M r r _rr Š s r _rš; _rrrt s r _rr T r_r T ; C:1 where the i r =@r are to be computed as row vectors and the inner i r =@rš_r will be scalars.

35 58 COORDINATE TRANSFORMATIONS Equation C.1 is the general-purpose formula for the matrix forms of interest, which differ only in their scalar functions s i r. These scalar functions s i r are generally rational functions of the following scalar functions shown in terms of jrjp ˆ pjrj p r T ; sin ˆ cos jrj jrj1 r T ; cos ˆ sin jrj jrj1 r T C:15 Time Derivative ENU =@v RPY In this case Eq. C.150). s 1 r ˆ jrj sin jrj 1cos jrj Š ; 1 ˆ1 jrj1 sin jrj r T ; 1cos jrj Š s r ˆ 1 ; ˆ 0 1; C:19 s r ˆ 1 jrj sin jrj ; jrj 1cos jrj Š ENU 1 r ˆ 1 jrj1 sin jrj jrj 1cos jrj Š jrj r T ; 1cos jrj Š _rr Š s r _rš _rrrt s r _rr T r_r T ; C:1 ˆ 1 jrj1 sin jrj r T _r I 1 1cos jrj Š _r Š; 1 jrj1 sin jrj jrj 1cos jrj Š jrj r T _r rr T ; 1cos jrj Š 1 jrj sin jrj _rr T r_r T : C:1 jrj 1cos jrj Š

36 C. COORDINATE TRANSFORMATION MODELS 59 Time Derivative ENU ENU In this case Eq. C.15), jrj sin jrj s 1 r ˆ 1cos jrj Š ; 1 ˆ 1 jrj1 sin jrj r T ; C:15 1cos jrj Š s r ˆ 1 ˆ 0 1; C:1 s r ˆ 1 jrj sin jrj ; C:18 jrj 1cos jrj ˆ1 jrj1 sin jrj jrj 1cos jrj Š jrj r T ; C:19 1cos jrj Š ENU 1 r _rr Š s r _r _rrrt s r _rr T r_r T C:180 ˆ 1 jrj1 sin jrj r T _r I 1 1cos jrj Š r _rš; 1 jrj1 sin jrj jrj 1cos jrj Š jrj r T _r rr T ; 1cos jrj Š 1 jrj sin jrj _rr T r_r T : C:181 jrj 1cos jrj Š C... Partial Derivatives with Respect to Rotation Vectors Calculation of the dynamic coef cient matrices F and measurement sensitivity matrices H in linearized or extended Kalman ltering with rotation vectors r ENU as part of the system model state vector requires taking derivatives with respect to r ENU of associated vector-valued f-orh-functions, as F r ENU; v ; ENU H r ENU ; v ; ENU where the vector-valued functions will have the general form f r ENU ; v or h renu ; v ˆ s 0 r ENU I s 1 r ENU renu s r ENU renu r T ENU v; C:18

37 0 COORDINATE TRANSFORMATIONS and s 0 ; s 1 ; s are scalar-valued functions of r ENU and v is a vector that does not depend on r ENU. We will derive here the general formulas that can be used for taking the partial r ENU ; v =@renu r ENU ; v =@renu. These formulas can all be derived by calculating the derivatives of the different factors in the functional forms and then using the chain rule for differentiation to obtain the nal result. Derivatives of Scalars The derivatives of the scalar factors s 0, s 1, i r i r ENU ˆ ENU i r i r U ; C:185 a row vector. Consequently, for any vector-valuedfunction g r ENU by the chain rule, the derivatives of the vector-valued product s i r ENU g renu i r ENU g r ENU g ˆ g i r ENU s i r r ENU ; C:18 {z } {z } ENU matrix matrix the result of which will be the Jacobian matrix of that subexpression in f or h. Derivatives of Vectors in Eq. C.18 are The three potential forms of the vector-valued function g g r ENU ˆ 8 < Iv ˆ v; r ENU v; : r ENU r T ENU v; C:18 each of which is considered ˆ ; ENU ENU r T ENU ENU C:188 v r ENU Š ; ENU ˆv Š; C:190 0 v v ˆ v 0 v 1 v v 1 0 5; C:191 ˆ r T ENU T ENU r ENU ; ENU C:19 ˆ r T ENU v I r ENU v T : C:19

38 C. COORDINATE TRANSFORMATION MODELS 1 General Formula Combining the above formulas for the different parts, one can obtain the following ENU s 0 r ENU I s 1 r ENU renu s r ENU renu r T ENU v ˆ 0 r 0 r 0 r 1 r r ENU v 1 r N s 1 r ENU 1 r U r T ENU r r r ENU U s r ENU r T ENU v I r ENU v T ; C:19 applicable for any differentiable scalar functions s 0, s 1, s. C..Direction Cosines Matrix We have demonstrated in Eq.C.1 that the coordinate transformation matrix between one orthogonal coordinate system and another is a matrix of direction cosines between the unit axis vectors of the two coordinate systems, C UVW XYZ cos y XU ˆ cos y YU cos y ZU cos y XV cos y YV cos y ZV cos y XW cos y YW 5: C:195 cos y ZW Because the angles do not depend on the order of the direction vectors i.e., y ab ˆ y ba ), the inverse transformation matrix C XYZ UVW ˆ ˆ cos y UX cos y VX cos y WX cos y XU cos y YU cos y ZU cos y UY cos y VY cos y WY cos y XV cos y YV cos y ZV cos y UZ cos y VZ 5 cos y WX C:19 T cos y XW cos y YW 5 ; cos y ZW C:19 ˆ C UVW T: XYZ C:198

39 COORDINATE TRANSFORMATIONS That is, the inverse coordinate transformation matrix is the transpose of the forward coordinate transformation matrix. This implies that the coordinate transformation matrices are orthogonal matrices. C...1 Rotating Coordinates Let ``rot'' denote a set of rotating coordinates, with axes X rot,y rot,z rot, and let ``non'' represent a set of non-rotating i.e., inertial) coordinates, with axes X non,y non,z non, as illustrated in Fig. C.1. Any vector v rot in rotating coordinates can be represented in terms of its nonrotating components and unit vectors parallel to the nonrotating axes, as v rot ˆ v x;non 1 x;non v y;non 1 y;non v z;non 1 z;non C:199 v x;non ˆ 1 x;non 1 y;non 1 z;non Š 5 C:00 v y;non ˆ C non rot v non ; v z;non C:01 where v x;non ; v y;non ; v z;non are nonrotating components of the vector, 1 x;non ; 1 y;non, 1 z;non ˆ unit vectors along X non ; Y non ; Z non axes, as expressed in rotating coordinates v rot ˆ vector v expressed in RPY coordinates v non ˆ vector v expressed in ECI coordinates, C non rot ˆ coordinate transformation matrix from nonrotating coordinates to rotating coordinates and C non rot ˆ 1 x;non 1 y;non 1 z;non Š: C:0 Fig. C.1 Rotating coordinates.

40 C. COORDINATE TRANSFORMATION MODELS The time derivative of C non rot, as viewed from the non-rotating coordinate frame, can be derived in terms of the dynamics of the unit vectors 1 x;non, 1 y;non and 1 z;non in rotating coordinates. As seen by an observer xed with respect to the nonrotating coordinates, the nonrotating coordinate directions will appear to remain xed, but the external inertial reference directions will appear to be changing, as illustrated in Fig. C.1. Gyroscopes xed in the rotating coordinates would measure three components of the inertial rotation rate vector v rot ˆ o x;rot o y;rot o z;rot 5 C:0 in rotating coordinates, but the non-rotating unit vectors, as viewed in rotating coordinates, appear to be changing in the opposite sense, as d dt 1 x;non ˆv rot 1 x;non ; d dt 1 y;non ˆv rot 1 y;non ; d dt 1 z;non ˆv rot 1 z;non ; C:0 C:05 C:05 as illustrated in Fig. C.1. The time-derivative of the coordinate transformation represented in Eq. C.0 will then be d dt Cnon rot ˆ d dt 1 x;non d dt 1 y;non d dt 1 z;non C:0 ˆ v rot 1 x;non v rot 1 y;non v rot 1 z;non Š ˆ v rot Š 1 x;non Š 1 y;non 1 z;non Š ˆ v rot ŠC non rot ; C:08 0 o z;rot o y;rot v rot Š ḓef o z;rot 0 o x;rot 5: C:09 o y;rot o x;rot 0 The inverse coordinate transformation non ˆ C non 1 rot C:10 ˆ C non T; rot C:11 C rot

41 COORDINATE TRANSFORMATIONS the transpose of C non rot, and its derivative d dt Crot non ˆ d C non T rot C:1 dt ˆ d T C:1 dt Cnon rot ˆ v rot C non T rot C:1 ˆC non T T; rot vrot C:15 ˆ C rot non v rot : C:1 In the case that ``rot'' is ``RPY'' roll-pitch-yaw coordinates) and ``non'' is ``ECI'' earth centered inertial coordinates), Eq. C.1 becomes d dt CRPY ECI ˆ C RPY ECI v RPY ; C:1 and in the case that ``rot'' is ``ENU'' east-north-up coordinates) and ``non'' is ``ECI'' earth centered inertial coordinates), Eq. C.08 becomes d dt CECI ENU ˆv ENU C ECI ENU; C:18 and the derivative of their product C RPY ENU ˆ C ECI ENUC RPY ECI ; d dt CRPY ENU ˆ d dt CECI ENU C RPY ECI C ECI d ENU dt CRPY ECI ˆ v ENU ŠC ECI ENU C RPY ECI C ECI ENU C RPY ECI v RPY Š ˆ v ENU Š C ECI ENUC RPY {z } ECI C ECI ENUC RPY {z } ECI v RPY Š; C RPY ENU C RPY ENU d dt CRPY ENU ˆ v ENU ŠC RPY ENU C RPY ENU v RPY Š: C:19 C:0 C:1