EE 570: Location and Navigation
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1 EE 570: Location and Navigation Navigation Mathematics: Coordinate Frames Kevin Wedeward Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA In Collaboration with Stephen Bruder Electrical and Computer Engineering Department Embry-Riddle Aeronautical Univesity Prescott, Arizona, USA January 25, 2016 Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
2 Lecture Topics 1 Coordinate Frames 2 Earth-Centered Inertial (ECI) Frame 3 Earth-Centered Earth-Fixed (ECEF) Frame 4 Local Navigation (Nav) Frame 5 Body Frame 6 Other Frames Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
3 Coordinate Frames Right-hand coordinate frame α has z α 1 origin o α at which frame is located, and 2 orthonormal vectors x α, y α, z α that serve as axes and indicate positive directions. x α o α y α Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
4 Coordinate Frames This definition implies x α x α = y α y α = z α z α = 1 x α y α = y α z α = z α x α = 0 x α y α = z α z α o α y α z α = x α z α x α = y α x α y α Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
5 Coordinate Frames Coordinate frames used as means to describe position and orientation/attitude of one frame with respect to another. z β z α o β y β o α x β x α y α Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
6 Earth-Centered Inertial (ECI) Frame ECI Frame defined as an inertial frame, i.e., it is assumed not to accelerate or rotate with respect to the universe ECI will be attached to earth, but won t spin with earth inertial sensors measure inertial motion relative to ECI frame Gyroscopes measure rate of change of orientation Accelerometers measure linear acceleration referred to as i-frame Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
7 ECI Frame origin o i of ECI is located near the center of mass (center of ellipsoidal representation) of the earth z i -axis points along the nominal axis of rotation of the earth true north not magnetic north! x i -axis lies in the equatorial plane and points from the earth to the sun at the vernal (spring) equinox defined by the intersection of the equatorial plane and the earth-sun orbital plane y i -axis chosen to complete right hand coordinate system (90 ahead of x i in direction of earth s rotation) Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
8 ECI Frame origin o i of ECI is located near the center of mass (center of ellipsoidal representation) of the earth z i -axis points along the nominal axis of rotation of the earth true north not magnetic north! x i -axis lies in the equatorial plane and points from the earth to the sun at the vernal (spring) equinox defined by the intersection of the equatorial plane and the earth-sun orbital plane y i -axis chosen to complete right hand coordinate system (90 ahead of x i in direction of earth s rotation) The ECI coordinate frame does not rotate with the earth Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
9 ECI Frame Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
10 ECI Frame o i at earth s center Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
11 ECI Frame o i at earth s center z i -axis points along the earth s axis of rotation Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
12 ECI Frame o i at earth s center z i -axis points along the earth s axis of rotation x i -axis points towards sun at vernal (spring) equinox Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
13 ECI Frame o i at earth s center z i -axis points along the earth s axis of rotation x i -axis points towards sun at vernal (spring) equinox y i -axis completes a right hand coordinate system Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
14 Earth-Centered Earth-Fixed (ECEF) Frame ECEF Frame not an inertial frame fixed with respect to the earth, i.e., attached to the earth and spins with earth referred to as e-frame Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
15 ECEF Frame origin o e is located (nearly) at the center of the mass of the earth (co-located with ECI s o i ) z e -axis points along the nominal axis of earth s rotation (same as ECI s z i ) x e -axis lies at the intersection of the equatorial plane and the reference meridian plane (i.e., Greenwich/Prime Meridian) tied to concept of latitude and longitude x e points from o e towards 0 longitude and 0 latitude (a little west of central Africa) y e -axis is chosen to complete right hand coordinate system Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
16 ECEF Frame z e z i N ω e NMT y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
17 ECEF Frame z e z i z e -axis points along axis of earth s rotation N ω e NMT y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
18 ECEF Frame z e z i z e -axis points along axis of earth s rotation x e -axis points towards zero latitude and longitude N ω e NMT y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
19 ECEF Frame z e z i z e -axis points along axis of earth s rotation x e -axis points towards zero latitude and longitude y e -axis completes right hand coordinate system NMT N ω e y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
20 ECEF Frame z e z i z e -axis points along axis of earth s rotation x e -axis points towards zero latitude and longitude y e -axis completes right hand coordinate system NMT s (lat, long) (34.07, ) = (34.07, ) x i NMT θ lat N ω e θlon Greenwichh y e x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
21 Local Navigation (Nav) Frame Nav Frame typically not fixed with respect to the earth, i.e., free to move, but has specified orientation also called geodetic, geographic, locally level, or tangential frame referred to as n-frame Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
22 Nav Frame origin o n is located at the center of mass of the body (e.g., air, land or sea vehicle) of interest z n -axis points down normal to the earth s surface (approximately towards the center of the earth) x n - y n axes then constrained to lie in plane locally-level (tangential) to the earth s surface x n -axis points to the north pole y n -axis is chosen to complete right hand coordinate system frame s configuration is often referred to as the NED frame x n North, y n East, and z n Down Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
23 Nav Frame z e z i N ω e xn NMT yn zn y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
24 Nav Frame z e z i o n on (potentially moving) body xn N ω e NMT yn zn y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
25 Nav Frame z e z i o n on (potentially moving) body x n-axis points north NMT xn yn zn N ω e y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
26 Nav Frame z e z i o n on (potentially moving) body x n-axis points north y n-axis points east NMT xn yn zn N ω e y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
27 Nav Frame z e z i o n on (potentially moving) body x n-axis points north y n-axis points east z n-axis points down x i xn NMT θ lat yn zn N ω e θlon Greenwichh y e x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
28 Body Frame Body Frame attached to moving body (e.g., land, air or sea vehicle) and moves (position and orientation/attitute) with body origin o b located at the center of mass of the body (co-located with Nav frame s o n ) x b -axis points forward wrt moving body z b -axis points loosely down varies with the roll/pitch of the vehicle y b -axis chosen to complete right hand coordinate system referred to as b frame Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
29 Body Frame y b pitch Center of Gravity roll x b yaw z b Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
30 Body Frame y b pitch Center of Gravity body frame is fixed with respect to the vehicle roll x b yaw z b Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
31 Body Frame y b pitch Center of Gravity body frame is fixed with respect to the vehicle x b forward roll x b yaw z b Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
32 Body Frame y b pitch Center of Gravity body frame is fixed with respect to the vehicle x b forward z b down x b roll yaw z b Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
33 Body Frame y b pitch Center of Gravity body frame is fixed with respect to the vehicle x b forward z b down y b completes right hand coordinate system ( right ) x b roll yaw z b Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
34 Body Frame z e z i xb yb zb N ω e xn NMT yn zn y e x i θ lat θlon Greenwichh x e Vernal Equinox y i Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
35 Other Frames Wander Azimuth Frame (alternative to the Nav frame) does not always point north to avoid numerical stability problems near the poles Other locally level frames Tangential Frame typically, refers to another type of the ECEF frame fixed to the Earth s surface (not moving like the n-frame) Computer Frame virtual coordinate frame that represents where we think that we are Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
36 The End Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, / 19
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