Navigation Mathematics Tuesday 15 Jan 2013 NMT EE 570 Location and Navigation Theory amp Practice Slide 1 of 14 Navigation Mathematics Coordinate Frames ECI The EarthCentered Inertial ECI Coordinate Frame ID: 289631
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Slide1
EE 570: Location and Navigation: Theory & Practice
Navigation Mathematics
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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14Slide2
Navigation Mathematics :
Coordinate Frames - ECI
The Earth-Centered Inertial (ECI) Coordinate Frame
An Inertial coordinate frame is one that does NOT accelerate (rectilinearly) or change its orientation (
wrt
the “stars”)
All inertial sensors measure “inertial” motionGyroscopes measure rate of change of inertial orientationAccelerometers measure inertial accelerationThe ECI frame will be referred to as the i-frame
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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Navigation Mathematics :
Coordinate Frames - ECI
The
Earth-Centered Inertial (ECI) Coordinate FrameThe origin of the ECI frame is located at the center of mass of the Earth
The
z
-axis points along the nominal axis of rotation of the earthTrue North NOT magnetic North!!The x-axis lies in the equatorial plane and points from the Earth to the Sun at the vernal equinoxDefined by the intersection of the equatorial plane and Earth-Sun orbital plane
The y-axis is simply chosen to conform to a right hand coordinate system
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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The ECI coordinate frame does NOT rotate with the EarthSlide4
Navigation Mathematics :
Coordinate Frames - ECI
The Earth-Centered
Inertial (ECI) Coordinate Frame
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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4 of 14
Equatorial Plane
Vernal
Equinox
z
-axis points along the nominal E
arth axis
of rotation
x
-axis
points
towards vernal (spring) equinox
y
-axis completes a right hand
coord
sys
hudsonvalleygeologist.blogspot.com/Slide5
Navigation Mathematics :
Coordinate Frames - ECEF
The Earth-Centered Earth-Fixed (ECEF) Coordinate Frame
The ECEF coordinate frame is NOT an inertial frameThe ECEF coordinate frame
is fixed with respect to the Earth
The ECEF frame will be referred to as the
e-frameTuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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The ECEF coordinate frame is fixed with respect to the EarthSlide6
Navigation Mathematics :
Coordinate Frames - ECEF
The
Earth-Centered Earth-Fixed (ECEF) Coordinate FrameThe origin of the ECEF frame is located at the center of mass of the Earth (same as ECI)
The
z
-axis points along the nominal axis of rotation of the earth (same as ECI)The x-axis lies at the intersection of the equatorial plane and the reference meridian plane (i.e. Greenwich meridian)Concept of latitude and longitudeThe y-axis is simply chosen to conform to a right hand coordinate system
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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Navigation Mathematics :
Coordinate Frames - ECEF
The Earth-Centered Earth-Fixed (ECEF) Coordinate Frame
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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z-axis points along the nominal
Earth axis
of rotation
x
-axis
points
towards zero longitude
y
-axis completes a right hand
coord
sysSlide8
Navigation Mathematics :
Coordinate Frames - Nav
The
Local Navigation (Nav) Coordinate Frame
The
Nav
coordinate frame is typically NOT fixed with respect to the EarthThe x/y axes lie in a plane which is locally-level or tangential to the Earth’s surfaceThe Nav frame is sometimes called the goedetic
, geographic, locally-level, or tangential frameThe Nav frame will be referred to as the
n-frame
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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The
Nav
coordinate frame moves with the vehicle of interestSlide9
Navigation Mathematics :
Coordinate Frames - Nav
The Local Navigation (
Nav) Coordinate FrameThe origin of the
Nav
frame is located at the center of mass of the vehicle
The z-axis points “down” along the normal to the Earth’s surfaceApproximately towards the center of the EarthThe x-axis points to the North poleThe y-axis is simply chosen to conform to a right hand coordinate system
This configuration is often referred to as a NED framex North,
y East, z
Down
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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Navigation Mathematics :
Coordinate Frames - Nav
The Local Navigation (
Nav) Coordinate Frame
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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z
-axis points
“down”
x
-axis
points
North
y
-axis points EastSlide11
Navigation Mathematics :
Coordinate Frames - Body
The
Body Coordinate FrameThe body coordinate frame is fixed with respect to the vehicle
The body frame will be referred to as the
b
-frameTuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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The Body coordinate frame is attached to the vehicle of interest
www.grc.nasa.gov/WWW/K-12/airplane/rotationsSlide12
Navigation Mathematics :
Coordinate Frames - Body
The Body Coordinate Frame
The origin of the body frame is located at the center of mass of the vehicle (same as the Nav frame)
The
x
-axis points “forward” wrt the moving vehicleThe z-axis points loosely “down”Changes with the roll/pitch of the vehicleThe y-axis is simply chosen to conform to a right hand coordinate system
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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Navigation Mathematics :
Coordinate Frames - Body
The Body Coordinate Frame
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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z-axis points
“down”
x
-axis
points
forward
y
-axis points to the rightSlide14
Navigation Mathematics :
Coordinate Frames – Other Frames
Other Coordinate 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 FrameTypically, refers to another type of ECEF frame fixed to the Earth’s surface (not moving like the n-frame)Computer FrameVirtual coordinate frame that represents where we think that we are
Tuesday 15 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
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