ee 570: location and navigation: theory &...
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EE 570: Location and Navigation: Theory & Practice
Navigation Sensors and INS Mechanization
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
• CASE 3: Nav Frame Mechanization
Determine the Position, Velocity, and Attitude of the Body frame with respect to the Navigation Frame
• Determine our PVA wrt the Nav frame
Position: Typically described in curvilinear coordinates: 𝐿𝑏 , 𝜆𝑏 , ℎ𝑏
𝑇
Velocity: Typically the velocity of the body wrt the earth frame resolved in navigation frame coords: 𝑣 𝑒𝑏
𝑛
Attitude: Typically the orientation of the body described in the nav frame: 𝐶𝑏
𝑛
Slide 2 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
• Determine our PVA wrt the Nav frame
ex
ey
ez
Inertial Frame
ECEF Frame
ix
iy
iz
?
ie
nx
ny
nz
?
en
Nav Frame
bx
by
bzBody Frame
?
nb
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
Body & Nav frame have the same origin
Inertial & Earth frame have the same origin
Slide 3 of 13
? ? ? ?
ib ie en nb
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
1. Attitude Update: Method A
Note that:
Now
nb ib ie
b
n
b
e
b b
n n b
b b nbC C
Measured by the gyro
cos( )
cos( ) 0
* * * 0
* * * 0 0
(in in) ( )
n
ie e ie
b
b b b
n e
T
ie
ie
C
L
L s L s L
See next slide
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
ib i
b b
e en nb
b b
n b n b n b
b ib b bie enC C C
ib i
n b n
e
n n
b benC C
ib
n b b b
eb ie nC
TSk C C Sk C
C Sk Sk C C
Slide 4 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
e e n
n n enC C
Last term: 𝛺𝑒𝑛𝑛 = 𝑆𝑘 𝜔𝑒𝑛
𝑛
0 sin( ) -
-sin( ) 0 -cos( )
cos( ) 0
b b b
b b b b
b b b
L L
L L
L L
Courtesy of Mathematica
,
n
en x
,
n
en y
,
n
en z
cos( )
-
-sin( )
b b
en b
b
n
b
L
L
L
,
,
,
Cos( )
n
n
eb N
N b
b
eb E
b
b E b
b
e
n
b D
R hL
L R hh
v
v
v
From handout #2
,
,
,
-
tan( )-
eb E
E b
eb N
en
N b
b e
n
b
n
E
n
E
n
b
v
R h
R h
L
R h
v
v
( ) ( )n n n
b b bC C tC
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
( ) ( )ib ie en
n b n n n
b bC I t C t
T
n e e
nen nC C
Slide 5 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
1. Attitude Update: Method B
Note that: 𝛺𝑛𝑏𝑏 = 𝛺𝑖𝑏
𝑏 − 𝛺𝑖𝑒𝑏 − 𝛺𝑒𝑛
𝑏
Hence, 𝐶𝑏𝑛(+) = 𝐶𝑏
𝑛(−)𝑒𝛺𝑛𝑏𝑏 𝛥𝑡
nb i
b b
b ie en
b b
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
2( ) ( ) sin( ) 1 cos( )n n
b bC C I K K
bnb t
e e K
b b n n b n
ib ie en
n
n b n bC C C C
TSk C C Sk C
ie
b
e
b n
n iSk Sk C i
b
n be
n nC Sk C
Slide 6 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
1. Attitude Update:
High Fidelity
Lower Fidelity
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
ˆb
nb t k
( ) ( ) ( )ib ie
n n b n n n
b b n beC C I t C t
2( ) ( ) sin( ) 1 cos( )n n
b bC C I K K
nb i
b b
b ie en
b b ib ie
b b n
nn e
b n
nC C
Slide 7 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
2. Specific Force Transformation:
Simply coordinatize the specific force
( )n n
ib i
b
b bf C f
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 8 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
3. Velocity Update
Note that: 𝑣 𝑒𝑏𝑛 = 𝐶𝑒
𝑛 𝑣 𝑒𝑏𝑒
Finally,
TSk C C Sk C
2n n n n n
ib b e ebn ief g v
eb eb
n n e
eb
n e
e ev C v C v
2n n e n e e e e
ne e e i ib ebb b eeC v C f g v
2n n n n n e e
ib b en e ebieebf g v C v
2n n n n n n e
ib b en ie ebeebf g v C v
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice
( ) ( ) 2 ( )n n n n n n n
ib beb e ieeb ebnv v t f g v
C Sk Sk C C
a 2e e e e e e
eb eb ib b ie ebv f g v
Slide 9 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
4. Position Update
Recalling the relationship between 𝑣 𝑒𝑏𝑛 and the curvilinear
coordinates (see handout #2)
,( ) ( ) ( )n
b b eb Dh h t v
, ( )( ) ( )
eb
n
b b
N
N
b
vL L t
R h
, ( )
( ) ( )cos( )
eb E
n
b
E
b
b b
vt
R h L
,
,
,
Cos( )
n
n
eb N
N b
b
eb E
b
b E b
b
e
n
b D
R hL
L R hh
v
v
v
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 10 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
( )eb
nv
( )n
bC
b
ib
1. Attitude Update
( )
( )
( )b
b
b
h
L
( ) ( ) ( )ib ie
n n b n n n
b b n beC C I t C t
2. SF Transform
b
ibf
n
ibf
( )n n
ib i
b
b bf C f
Grav
Model
3. Velocity Update
n
bg
( )eb
nv
( ) ( )
2 ( )
n n
n n n n n
ib b
eb
e ie
eb
ebn
v v
t f g v
3. Position Update
( )n
bC ( ), ( ), ( )b b bh L
,( ) ( ) ( )n
b b eb Dh h t v
, ( )( ) ( )
eb
n
b b
N
N
b
vL L t
R h
, ( )
( ) ( )cos( )
eb E
n
b
E
b
b b
vt
R h L
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 11 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
• In continuous time notation:
Attitude: 𝐶 𝑏𝑛 = 𝐶𝑏
𝑛 𝛺𝑖𝑏𝑏 − 𝛺𝑖𝑒
𝑛 + 𝛺𝑒𝑛𝑛 𝐶𝑏
𝑛
Velocity: 𝑣 𝑒𝑏𝑛 = 𝑓 𝑖𝑏
𝑛 + 𝑔 𝑏𝑛 − 𝛺𝑒𝑛
𝑛 + 2𝛺𝑖𝑒𝑛 𝑣 𝑒𝑏
𝑛
Position:
𝐿 𝑏𝜆 𝑏ℎ 𝑏
=
𝑣𝑒𝑏,𝑁𝑛
𝑅𝑁+ℎ𝑏
𝑣𝑒𝑏,𝐸𝑛
Cos(𝐿𝑏) 𝑅𝐸+ℎ𝑏−𝑣 𝑒𝑏,𝐷
𝑛
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 12 of 13
Navigation Sensors and INS Mechanization Navigation Equations – Case 3: Nav Mechanization
• Combining into a state-space equation:
,
,
,
C
2
os( )
n
n
nn
n n n n nn
ib b
eb N
N bb
eb Eb
b E bb
eb Deb
eb
ib i
en ieb
n b n n n
b be en
v
v
vv
R hL
L R h
vC
C C
h
f g
Tuesday 5 March 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 13 of 13