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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
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
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
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
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
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
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
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Navigation Mathematics : Coordinate Frame Transformations - ECEF/Nav
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Navigation Mathematics : Coordinate Frame Transformations – Body Frame
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Navigation Mathematics : Coordinate Frame Transformations – Body Frame
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• 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
Navigation Mathematics : Coordinate Frame Transformations – Body Frame
Thursay 7 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
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