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

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 1 / 19 Lecture Topics

1 Coordinate Frames

2 -Centered Inertial (ECI) Frame

3 Earth-Centered Earth-Fixed (ECEF) Frame

4 Local Navigation (Nav) Frame

5 Body Frame

6 Other Frames

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 2 / 19 Coordinate Frames

Right-hand coordinate z frame α has α

1 origin oα at which frame is located, and 2 orthonormal vectors oα xα, yα, zα that serve as axes and indicate y xα α positive directions.

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 3 / 19 Coordinate Frames

This definition implies zα xα · xα = yα · yα = zα · zα = 1

xα · yα = yα · zα = zα · xα = 0 oα xα × yα = zα

yα × zα = xα y xα α zα × xα = yα

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 4 / 19 Coordinate Frames

Coordinate frames used as means to describe position and orientation/attitude of one frame with respect to another.

zβ

zα yβ

oβ

oα xβ

yα xα

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 5 / 19 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

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 6 / 19 The ECI coordinate frame does not rotate with the earth

ECI Frame

origin oi of ECI is located near the center of mass (center of ellipsoidal representation) of the earth zi -axis points along the nominal axis of rotation of the earth true north not magnetic north! xi -axis lies in the equatorial plane and points from the earth to the at the vernal (spring) equinox defined by the intersection of the equatorial plane and the earth-sun orbital plane ◦ yi -axis chosen to complete right hand coordinate system (90 ahead of xi in direction of earth’s rotation)

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 7 / 19 ECI Frame

origin oi of ECI is located near the center of mass (center of ellipsoidal representation) of the earth zi -axis points along the nominal axis of rotation of the earth true north not magnetic north! xi -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 ◦ yi -axis chosen to complete right hand coordinate system (90 ahead of xi in direction of earth’s rotation)

The ECI coordinate frame does not rotate with the earth

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 7 / 19 oi at earth’s center

zi -axis points along the earth’s axis of rotation

xi -axis points towards sun at vernal (spring) equinox

yi -axis completes a right hand coordinate system

ECI Frame

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 8 / 19 zi -axis points along the earth’s axis of rotation

xi -axis points towards sun at vernal (spring) equinox

yi -axis completes a right hand coordinate system

ECI Frame

oi at earth’s center

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 8 / 19 xi -axis points towards sun at vernal (spring) equinox

yi -axis completes a right hand coordinate system

ECI Frame

oi at earth’s center

zi -axis points along the earth’s axis of rotation

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 8 / 19 yi -axis completes a right hand coordinate system

ECI Frame

oi at earth’s center

zi -axis points along the earth’s axis of rotation

xi -axis points towards sun at vernal (spring) equinox

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 8 / 19 ECI Frame

oi at earth’s center

zi -axis points along the earth’s axis of rotation

xi -axis points towards sun at vernal (spring) equinox

yi -axis completes a right hand coordinate system

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 8 / 19 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

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 9 / 19 ECEF Frame

origin oe is located (nearly) at the center of the mass of the earth (co-located with ECI’s oi )

ze -axis points along the nominal axis of earth’s rotation (same as ECI’s zi ) xe -axis lies at the intersection of the equatorial plane and the reference meridian plane (i.e., Greenwich/) tied to concept of and xe points from oe towards 0◦ longitude and 0◦ latitude (a little west of central Africa)

ye -axis is chosen to complete right hand coordinate system

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 10 / 19 ze -axis points along axis of earth’s rotation

xe -axis points towards zero latitude and longitude

ye -axis completes right hand coordinate system NMT’s (lat, long) ≈ (34.07◦, −106.9◦) = (34.07◦, 253.1◦)

ECEF Frame

ze ≡ zi

ωe N

ye NMT

G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 11 / 19 xe -axis points towards zero latitude and longitude

ye -axis completes right hand coordinate system NMT’s (lat, long) ≈ (34.07◦, −106.9◦) = (34.07◦, 253.1◦)

ECEF Frame

ze ≡ zi

ze -axis points along axis of earth’s rotation ωe N

ye NMT

G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 11 / 19 ye -axis completes right hand coordinate system NMT’s (lat, long) ≈ (34.07◦, −106.9◦) = (34.07◦, 253.1◦)

ECEF Frame

ze ≡ zi

ze -axis points along axis of earth’s rotation ωe N xe -axis points towards zero latitude and longitude ye NMT

G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 11 / 19 NMT’s (lat, long) ≈ (34.07◦, −106.9◦) = (34.07◦, 253.1◦)

ECEF Frame

ze ≡ zi

ze -axis points along axis of earth’s rotation ωe N xe -axis points towards zero latitude and longitude y ye e -axis completes right NMT hand coordinate system G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 11 / 19 ECEF Frame

ze ≡ zi

ze -axis points along axis of earth’s rotation ωe N xe -axis points towards zero latitude and longitude y ye e -axis completes right NMT hand coordinate system G

r NMT’s (lat, long) ≈ e θ e ◦ ◦ lat θlon n

(34.07 , −106.9 ) = w ◦ ◦ i

(34.07 , 253.1 ) c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 11 / 19 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

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 12 / 19 Nav Frame

origin on is located at the center of mass of the body (e.g., air, land or sea vehicle) of interest

zn-axis points “down” normal to the earth’s surface (approximately towards the center of the earth) xn - yn axes then constrained to lie in plane locally-level (tangential) to the earth’s surface

xn-axis points to the north pole yn-axis is chosen to complete right hand coordinate system frame’s configuration is often referred to as the NED frame

xn → North, yn → East, and zn → Down

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 13 / 19 on on (potentially moving) body

xn-axis points north yn-axis points east

zn-axis points ”down”

Nav Frame

ze ≡ zi

ωe N

xn

yn ye NMT zn G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 14 / 19 xn-axis points north yn-axis points east

zn-axis points ”down”

Nav Frame

ze ≡ zi

ωe N on on (potentially moving) body xn

yn ye NMT zn G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 14 / 19 yn-axis points east

zn-axis points ”down”

Nav Frame

ze ≡ zi

ωe N on on (potentially moving) body xn

xn-axis points north yn ye NMT zn G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 14 / 19 zn-axis points ”down”

Nav Frame

ze ≡ zi

ωe N on on (potentially moving) body xn

xn-axis points north yn ye yn-axis points east NMT zn G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 14 / 19 Nav Frame

ze ≡ zi

ωe N on on (potentially moving) body xn

xn-axis points north yn ye yn-axis points east NMT zn G zn-axis points ”down” r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 14 / 19 Body Frame

Body Frame attached to moving body (e.g., land, air or sea vehicle) and moves (position and orientation/attitute) with body

origin ob located at the center of mass of the body (co-located with Nav frame’s on)

xb-axis points “forward” wrt moving body zb-axis points loosely “down” varies with the roll/pitch of the vehicle

yb-axis chosen to complete right hand coordinate system referred to as b−frame

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 15 / 19 body frame is fixed with respect to the vehicle

xb “forward”

zb “down” yb completes right hand coordinate system (“right”)

Body Frame

yb pitch Center of Gravity

roll

xb yaw

zb

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 16 / 19 xb “forward”

zb “down” yb completes right hand coordinate system (“right”)

Body Frame

yb pitch Center of Gravity

body frame is fixed with respect to the vehicle roll

xb yaw

zb

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 16 / 19 zb “down” yb completes right hand coordinate system (“right”)

Body Frame

yb pitch Center of Gravity

body frame is fixed with respect to the vehicle xb “forward” roll

xb yaw

zb

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 16 / 19 yb completes right hand coordinate system (“right”)

Body Frame

yb pitch Center of Gravity

body frame is fixed with respect to the vehicle xb “forward” roll zb “down” xb yaw

zb

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 16 / 19 Body Frame

yb pitch Center of Gravity

body frame is fixed with respect to the vehicle xb “forward” roll zb “down” yb completes right hand x coordinate system (“right”) b yaw

zb

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 16 / 19 Body Frame

ze ≡ zi

xb

yb ωe zb N

xn

yn ye NMT zn G

r e θlat θlon e

n

w

i c

h xe h xi y Vernal Equinox i

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 17 / 19 Other Frames

Wander 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

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 18 / 19 The End

Frames ECI ECEF Nav Body Other Kevin Wedeward, Aly El-Osery (NMT) EE 570: Location and Navigation January 25, 2016 19 / 19