NA4CA Appliation Note Datum ranformation of NA4 Referene Frame Giri Baleri, Sr. Appliation Engineer Crobow ehnology, In. http://www.xbow.om hi appliation note explain how to onvert variou referene frame for Crobow Inertial Sytem in general and NA4CA in partiular. he Crobow Navigation Sytem or NA4CA ue a 3-axi aelerometer and a 3-axi rate enor to make a omplete meaurement of the dynami of the ytem. he addition of a 3-axi magnetometer inide the Crobow AHRS allow it to make a true meaurement of magneti heading without an external flux valve. When GPS reeiver i added to the ytem, the ombined ytem beome a low-ot INS that an output loation, veloity and aeleration. Inertial Coordinate Frame he NA4CA ha a label on one fae illutrating the NA4CA oordinate ytem a hown in Figure. With the onnetor faing you, and the mounting plate down, the axe are defined a: roll X yaw pith Y X-axi from fae with onnetor through the NA4CA Y-axi along the fae with onnetor from left to right Z-axi along the fae with the onnetor from top to bottom Z Figure : NA4CA Coordinate Sytem he axe form an orthogonal right-handed oordinate ytem. It origin i nominally loated at the vehile CG. In Figure, the body frame of the ytem i hown relative to the tangent frame or loal level horizontal frame. X b -North (X tan ) γ α β LC eloity in angent Frame λ Y b -Eat (Y tan ) Z b -Down Along Altitude (Z tan ) Figure : Coordinate Frame Angle Definition Page
NA4CA Appliation Note In thi formulation, the body axi ha been hoen to point toward north along it x-axi when there i no yaw angle. herefore when the vehile attitude i zero, or when the Euler angle roll, pith, and yaw are zero, the tranformation from body to tangent frame i imply: Here the matrix repreent the Coine rotation matrix, whih take you from Body () to angent frame. hi erve a the tarting point for further rotation of the body in the tangent frame due to vehile attitude hange. Several angle defined in the Figure are of importane for the vehile. he bold line repreent the vehile' enterline, and the dahed line repreent the vehile' veloity vetor in the tangent frame. he angle, whih the veloity vetor make with repet to the vehile enterline, are the typial aerodynami ontrol angle, angle of attak α, and angle of idelip β. he angle that the veloity vetor make with repet to the tangential plane are the typial air veloity angle, flight path angle λ and heading angle γ. he Euler body angle, whih the enterline of the vehile body make with repet to the tangential frame, are the pith and yaw angle. he vehile body roll angle i rotated along it enterline. he Euler angle deribe the vehile attitude and form a 3-- rotation of the body in the tangent frame. In expliit term the rotation matrix i: From thi rotation matrix, whih will tranform a vetor from the body frame into the tangent frame, the attitude Euler angle an be derived a follow: ( ) (,) (,) atan (3,) ain (3,3) (3,) atan ) ( ) ( ) ( yaw pith roll GPS Coordinate Frame Coordinate repreenting poition on the earth an be given in two format, Spherial or Carteian. Spherial or Geodeti oordinate are three dimenional with the omponent of latitude (), longitude ( and height above ellipoid (h). With two of the omponent being non-linear with angular unit, omputation are more omplex for oordinate geometry problem. Alternatively, Carteian oordinate are entirely linear and provide for an exellent platform for mathemati. he origin and orientation of the oordinate frame are dependant on the uer appliation and many well defined ytem already exit. For global appliation the ytem known a earth entered - earth fixed (ECEF) i preferred. Figure 3 illutrate the relationhip between Spherial oordinate (, λ, h) and Carteian ECEF oordinate (X, Y, Z) with repet to a referene ellipoid. Page
NA4CA Appliation Note Figure 3: Relationhip between Carteian and Spherial Coordinate Sytem LLA Coordinate Sytem he mot ommonly ued oordinate ytem today i the latitude, longitude, and altitude (LLA) ytem. he origin of the ytem i at the ma enter of the earth. he Prime Meridian and the Equator are the referene plane ued to define latitude and longitude. he geodeti latitude (there are many other defined latitude) of a point i the angle from the equatorial plane to the vertial diretion of a line normal to the referene ellipoid. he geodeti longitude of a point i the angle between a referene plane and a plane paing through the point, both plane being perpendiular to the equatorial plane. he geodeti altitude at a point i the ditane from the referene ellipoid to the point in a diretion normal to the ellipoid. ECEF Coordinate Sytem Earth Centered, Earth Fixed (ECEF) Carteian oordinate are alo ued to define three-dimenional poition. Earth entered, earth-fixed, X, Y, and Z, Carteian oordinate (XYZ) define three-dimenional poition with repet to the enter of ma of the referene ellipoid and follow rotation of the earth. he origin of the ytem i at the ma enter of the earth. he Z-axi i along the axi of rotation and point toward the North Pole. he X-axi i defined by the interetion of the plane define by the prime meridian and the equatorial plane. he Y-axi omplete a right-handed orthogonal ytem by a plane 9 eat of the X-axi and it interetion with the equator. Page 3
NA4CA Appliation Note Figure 4: ECEF Coordinate Referene Frame he Global Poitioning Sytem (GPS) i baed on the World Geodeti Sytem of 984 (WGS84) datum. WGS84 i a geoentri ytem, whih provide an exellent mathematial repreentation in relation to the orbiting atellite ontellation. Upon the introdution of atellite navigation, everal national geodeti organization immediately graped the tehnology to update their datum with modernized geoentri ellipoid and to redue exiting ditortion. A referene ellipoid an be deribed by a erie of parameter that define it hape and whih inlude a emi-major axi (a), a emi-minor axi (b) and it firt eentriity (e) and it eond eentriity (e ) a hown in Figure 5. Depending on the formulation ued, ellipoid flattening (f) may be required. hi ellipoid ha it origin oinident with the ECEF origin. he X-axi piere the Greenwih meridian (where longitude degree) and the XY-plane make up the equatorial plane (latitude degree). Altitude i deribed a the perpendiular ditane above the ellipoid. WGS84 Parameter: a 637837 b a( f 635675.34458 e e' f ) 98.573563 a b a a b b Figure 5: Referene Ellipoid Parameter b a Page 4
NA4CA Appliation Note Converion between ECEF and Loal angential Plane GPS oordinate frame onverion are aomplihed by variou method. Complete datum onverion i baed on even parameter tranformation that inlude three tranlation parameter, three rotation parameter and a ale parameter. Simple three parameter onverion between latitude, longitude, and height in different datum an be aomplihed by onverion through ECEF X, Y, Z Carteian oordinate in one referene datum and three origin offet that approximate differene in rotation, tranlation and ale. LLA to ECEF he onverion from LLA to ECEF i hown below. where, X ( N h)oϕ oλ Y ( N h)oϕ in λ b Z a λ longitude ϕ latitude N N h inϕ h height above ellipoid Radiu of Curvature, defined a : a e in ϕ ertial Datum Knowledge of the geoidal undulation of a partiular poition, allow the orthometri height to be derived from GPS meaured ellipoidal height. ertial datum an refer to either a urfae uh a a geoid or the urfae of a referene ellipoid. he height determined by GPS meaurement relate to the perpendiular ditane above the referene ellipoid and hould not be onfued with the more well-known height datum Mean Sea Level (MSL). he datum that define the MSL (alo alled the geoid) i a omplex urfae that require dene and aurate gravity data to define it hape. he WGS84 ellipoid approximate the geoid on a worldwide bai with deviation between the two datum never exeeding meter. he relationhip between the geoid and ellipoid i hown in Figure 6 and the algebrai differene between the two i known a the geoidal undulation (N). he onverion between the two referene datum i hown by: where, h H N h ellipoidal height (Geodeti), H orthometri height (MSL), N geoid eparation (undulation) Page 5
NA4CA Appliation Note Figure 6: Ellipoid Height and Geoid Height Relationhip Repreenting geoidal undulation for a relatively large area with a mathematial model beome diffiult due to high frequeny pherial harmoni. o overome the problem of irregular geoidal undulation, an idential approah to the horizontal datum tranformation an be done with the ue of two dimenional data grid. Eah node on the grid ha a geoidal undulation (N) value and intermediate loation are interpolated. Converting ECEF eloitie to Loal angent Plane eloitie GPS alo reolve the peed and diretion of travel in the ECEF XYZ referene frame. o onvert thee value to a loal tangent plane (LP), the veloity vetor mut be rotated to reflet diretion in term more uable to the uer. he LP ue the orientation of North, Eat, and Down, (NED) whih i onitent with the geodeti oordinate LLA. o tranform the veloity vetor, you ue the following diretion oine matrix (North, Eat, Down) and olving for eah omponent reult in the following matrix tranformation: north eat down in( ϕ)o( in( o( ϕ)o( in( ϕ)in( o( o( ϕ)in( o( ϕ) in( ϕ) x y z he peed and heading data an be derived from the veloity information uing the following relationhip. Speed north Heading artan Referene eat eat north Crobow ehnology, In. http://www.xbow.om Global Poitioning Sytem Overview, Peter H. Dana http://www.olorado.edu/geography/graft/note/gp/gp_f.html Uer Guide, SiRF Star Evaluation Kit http://www.linkwave.o.uk/aet/gps.g-e--c.pdf Page 6