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EE 570: Location and Navigation: Theory & Practice

Navigation Mathematics

Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 15

Navigation Mathematics : Coordinate Frame Transformations

Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice

• Determine the detailed kinematic relationships between the 4 major frames of interest

The Earth-Centered Inertial (ECI) Coordinate Frame (i-frame)

The Earth-Centered Earth-Fixed (ECEF) Coordinate Frame (e-frame)

The Local Navigation (Nav) Coordinate Frame (n-frame)

The Body Coordinate Frame (b-frame)

Slide 2 of 15

Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF


• Relationship Between the ECI and ECEF Frames

ECI & ECEF have co-located origins

o 𝑟 𝑖𝑒 = 𝑟 𝑖𝑒 = 𝑟 𝑖𝑒 = 0

The x, y, and z axes of the ECI & ECEF frames are coincident at time t0

The ECEF frame rotates about the common z-axis at a fixed rate (𝜔𝑖𝑒)

o Ignoring minor speed variations (precession & nutation) 𝜔𝑖𝑒 = 72.921151467 µ𝑟𝑎𝑑/𝑠 (WGS84) which is 15/hr

Slide 3 of 15

Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF


• The angular velocity and acceleration are:

• The angle of rotation is

• The orientation of frame {e} wrt frame {i} becomes

0ie ie t t

ie GMSTt

GMST: Greenwich Mean Sidereal time

,

Cos( ) Sin( )

Sin( ) Cos( )

0

0

0

0 1ie

ie ie

ie

i

e z ieC R

0

0i

ie

ie

0

0

0

i

ie

NOTE: 𝜔𝑖𝑒𝑖 = 𝜔𝑖𝑒

𝑒

Slide 4 of 15

0

b b b b b

b b b b b

b b

L L

L L

L L

s c s c c

s s c c s

c s

Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav


• Descriptions wrt the Navigation frame

Orientation of the n-frame 𝑤𝑟𝑡 the e-frame

, , 90b b

e

n z y LC R R

b

cos sin 0 sin 0 cos

sin cos 0 0 1 0

0 0 1 cos 0 sin

b b b b

b b

b b

L L

L L

sin cos sin cos cos

sin sin cos cos sin

cos 0 sin

b b b b b

b b b b b

b b

L L

L L

L L

Compact Notation: cos 𝜃 = 𝑐𝜃 & sin 𝜃 = 𝑠𝜃

bL

Geodetic Lat = 𝐿𝑏 and Geodetic Lon = 𝑏

Slide 5 of 15



Angular velocity of the n-frame wrt the e-frame resolved in the e-frame as a Skew-Symmetric matrix

The angular velocity vector

Te e e

en n nC C

0 0

b b b b b b b b b b b b b b

b b b b b b b b b b b b b b

b b b b

L b L b b L b L b L L

L b L b b L b L b L L

L b L b L

T

L

s s c c L c c s L c s c s s c c

c s L c s s s s L c c s s c c s

s L c L c s

0 cos

0 sin

cos sin 0

b b b

b b b

b b b b

L

L

L L

sin

cos

b

b

e

en

b

b

b

L

L

cos

sin

b b

b

b

Tn e e

en n en

b

L

L

L

C

e e n e e

n n en en nC C C

3 2

3 1

2 1

0

0

0

Slide 6 of 15

Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF/Nav


Hence the orientation of the n-frame wrt the i-frame becomes

c s

s c

0

0

0 10 0

ie ie b b b b b

ie ie b b b b b

b b

i i e

n e

L L

n L L

L L

s c s c c

s s c cC C C s

c s

ie ie ie

ie ie ie

sin cos sin cos cos

sin sin cos cos sin

cos 0 sin

b b b b b

b b b b b

b b

L L

L L

L L

Slide 7 of 15

? ? ?

in ie en

Navigation Mathematics : Coordinate Frame Transformations - ECI/ECEF/Nav


The angular velocity of the n-frame wrt the i-frame resolved in the i-frame is:

i i i e

in ie e enC

Cos( ) Sin(0 0

0 0

) sin

Sin( )

0

Cos( ) cos

10

ie ie b b

i

ie ie b

e

b

b

L

L

ie

ie

ie

sin

cos

b b

b b

b

L

L

Slide 8 of 15



• The vector from origin of the e-frame to the n-frame org resolved in the e-frame (from the last lecture)

• The velocity of the n-frame wrt the e-frame resolved in the e-frame (see handout#1 for proof)

2

( ) ( )

( ) ( )

(1 ) ( )

E b b b

e

E b b b

E

b

b

e

b

R h Cos L Cos

r R h Cos L Sin

R e h Sin L

Cos Sin Sin Cos Cos

Sin Sin Cos Cos Sin Cos

Cos 0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )Sin

b b b b b N b b

b b b b b b E b b

b b b

L L R h L

L L L R h

L L h

e

en en

edv r

dt

n

e

erOrigins of the {n} and

{b} frames are the same

e e

en en enb b

b b b

e

b

r r rL h

L h

Slide 9 of 15

../handouts/handout_1.pdf



• Recalling the form of 𝐶𝑛𝑒 suggests that

• and hence,

Cos( )

N b b

e

en n b E b b

b

ev

R h L

C L R h

h

?e

nCne

n enC v

Cos( )

N b b

n

en b E b b

b

R h L

L Rv h

h

Slide 10 of 15



• Restating 𝑣 𝑒𝑛𝑛 as

• and recalling that

• Suggests that

,

,

,

Cos( )

N b b en N

en b E b b en E

b e

n

n n

n

n D

R h L

L R h

v

v

h

v

v

cos

sin

b b

b

b b

n

en

L

L

L

,

,

,tan

en E E b

e

n

n n

e n N N b

b en E E b

n

n

R h

R h

v

vL h

v

R

Slide 11 of 15

Navigation Mathematics : Coordinate Frame Transformations – Body Frame


• Descriptions wrt the Body frame

Orientation of the b-frame 𝑤𝑟𝑡 the n-frame in terms of relative yaw(), pitch(), then roll() angles

, , ,

n

b z y xR R RC

0 0 1 0 0

0 0 1 0 0

0 0 1 0 0

c s c s

s c c s

s c s c

c c c s s c s c c s s s

c s c c s s s c s s c s

s c s c c

www.grc.nasa.gov/WWW/K-12/airplane/rotationsbx bz

by

Slide 12 of 15



• The angular velocity of the b-frame wrt the i-frame resolved/coordinatized in the i-frame

i i i n

ib in n nbC

i i e i n

ie e en n nbC C

Slide 13 of 15



• Position vectors to the origin of the body frame

The origins of the body and Nav frames are co-incident

The origins of the ECI and ECEF frames are co-incident

Velocity of the b-frame wrt the i-frame resolved in the i-frame

o Case #2: A moving point in a rotating frame

0nbr

eb ib en inr r r r

i

ib ib

idv r

dt

eb

i e e i e

e ie e ebC r C v

i e e e

e ie eb ebC r v

i e

e eb

dC r

dt

Slide 14 of 15



Acceleration of the b-frame wrt the i-frame resolved in the i-frame

o Case #2: A moving point in a rotating frame

i

ib ib

ida v

dt

e

i e e e i e

e ie eb b ee iC r v C e e

eb eb eb

e e

ier r v

2eb

i e e e e e e

e eb ebie ie ieC r v a

i e e e

e ie eb eb

dC r v

dt

Slide 15 of 15

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