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Inertial NavigationInertial Navigation• Inertial Navigation
The process of “integrating” angular velocity & acceleration to determine One’s position, velocity, and attitude (PVA)
• Effectively “dead reckoning”
To measure the acceleration and angular velocity vectors we need at least 3-gyros and 3-accels
• Typically configured in an orthogonaltriad
The “mechanization” can be performed wrt:
• The ECI frame,
• The ECEF frame, or
• The Nav frame.
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 1
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial Navigation• An Inertial Navigation System (INS)
ISA – Inertial Sensor Assembly
• Typically, 3-gyros + 3-accels + basic electronics (power, …)
IMU – Inertial Measurement Unit
• ISA + Compensation algorithms (i.e. basic processing)
INS – Inertial Navigation System INS – Inertial Navigation System
• IMU + gravity model + “mechanization” algorithms
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 2
IMUIMU
- Basic Processing
ISAISAComp
Algs
Raw sensor
outputs
b
ibf
b
ibω
Mechanization
Equations
Gravity
Model
INSINS
- More Sophisticated Processing
Initialization
Position,
Velocity, and
Attitude (PVA)
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationA Four Step MechanizationA Four Step Mechanization
• Can be generically performed in four steps:1. Attitude Update
• Update the prior attitude (a DCM, say) using the current
angular velocity measurement
2. Transform the specific force measurement2. Transform the specific force measurement
• Typically, using the attitude computed in step 1.
3. Update the velocity
• Essentially integrate the result from step 2. with the use of
a gravity model
4. Update the Position
• Essentially integrate the result from step 3.
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 3
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationA Four Step MechanizationA Four Step Mechanization
2. SF
Transform
b
ibω
Prior
Attitude
b
ibf
GravPrior 3. Velocity
1. Attitude
Update
Pri
or
PV
AIMU Measurements
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 4
3. Position
Update
Grav
Model
Prior
Velocity
Prior
Position
3. Velocity
Update
Updated
Attitude
Updated
Velocity
Updated
Position
Pri
or
PV
A
Updated PVA
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 1: ECI Frame MechanizationCase 1: ECI Frame Mechanization
• Determine our PVA wrt the ECI frame Namely: Position , Velocity, , and attitude
1. Attitude Update: Body orientation frame at time “k” wrt time “k-1”
• ∆t = Time k – Time k-1bω
i
ibv
i
bC
i
ibr
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 5
( 1)b kx
−
( 1)b ky
−
( 1)b kz
−
( )b kx
( )b ky
( )b kz
Body Frame
at time “k-1”
Body Frame
at time “k”
( 1)
( ) ( 1) ( )
i i b k
b k b k b kC C C−
−=
( 1)
( )
bib tb k
b kC eΩ ∆− =
( )( ) ( )
( )
bib ti i
b b
i b
b ib
C C e
C I tω
Ω ∆+ = −
− + × ∆
b
ibω
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 1: ECI Frame MechanizationCase 1: ECI Frame Mechanization
2. Specific Force Transformation Simply coordinatize the specific force
3. Velocity Update Assuming that we are in space (i.e. no centrifugal component)
( )i i
ib i
b
b bf C f= +
Assuming that we are in space (i.e. no centrifugal component)
Thus, by simple numerical integration
4. Position Update By simple numerical integration
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 6
aib i
i i
ib
i
bf γ= −
( ) ( ) aib ib
i
ib
i iv v t+ = − + ∆
2
( ) ( ) ( ) a2
ib ib ib i
i
b
i i i tr r v t
∆+ = − + − ∆ +
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 1: ECI Frame MechanizationCase 1: ECI Frame Mechanization
2. SF
Transform
b
ibω b
ibf
Grav3. Velocity
1. Attitude
Update( )
i
bC −
iγ
i
ibf
( )( ) ( )i i b
b b ibC C I tω + = − + × ∆
( )i i
ib i
b
b bf C f= +
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 7
3. Position
Update
Grav
Model
3. Velocity
Update
( )i
bC +
i
ibγ
( )ib
iv −
( )ib
iv +
( )ib
ir +
( )ib
ir −
( ) ( ) aib ib
i
ib
i iv v t+ = − + ∆
2
( ) ( ) ( ) a2
ib ib ib i
i
b
i i i tr r v t
∆+ = − + − ∆ +
a ib i
i i
ib
i
bf γ= −
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 2: ECEF Frame MechanizationCase 2: ECEF Frame Mechanization
• Determine our PVA wrt the ECI frame Namely: Position , Velocity, , and attitude
1. Attitude Update: Start with the angular velocity ? ? ?
ib ie ebω ω ω= +
?ω
e
ebv
e
bC
e
ebr
by
bz
ib ie eb
e e eω ω ω= +
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 8
ex
ey
ez
Inertial Frame
ECEF Frame
( )( ) ( )
( )
( ) ( )
eeb te e
b b
e b b i e
b e bib ie
ib i
e b
b be
i e
C e C
I C C t t C
C I t C t
Ω ∆+ = −
+ ∆ − ∆ −
= − + ∆ − − ∆
Ω Ω
Ω Ω
ix
iy
iz
ieω
bx
by
?
ebω
Body Frame
ib ie eb
eb i
e e
b ie
eb ib ie
b i
b
e e b b i
b e
C
C C
ω ω ω
ω ω ω
= +
= −
= Ω Ω−Ω
( ) [ ] TC C Cω ω× = ×
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 2: ECEF Frame MechanizationCase 2: ECEF Frame Mechanization
2. Specific Force Transformation: Simply coordinatize the specific force
3. Velocity Update ECEF & ECI have the same origin
( )e e
ib i
b
b bf C f= +
by
bz
e e
eb eb
e i e i
i ib i ib
e e i e i
ei i ib i ib
v r
C r C r
C r C v
=
= +
= Ω +
ECEF & ECI have the same origin
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 9
ex
ey
ez
Inertial Frame
ECEF Frameix
iy
iz
bx
by
e
ebr
Body Frame
i i i e
ib ie e ebr r C r= +
0e e i
eb i ibr C r⇒ =
e e e i
ie eb i ibr C v= −Ω +
( )a
a
e e e e e i
eb eb ie eb i ib
e e e i e i
ie eb i ib i ib
e e e e i e i
ie eb ei i ib i ib
dv r C v
dt
r C r C v
v C v C
= = −Ω +
= −Ω + +
= −Ω + Ω +
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 2: ECEF Frame MechanizationCase 2: ECEF Frame Mechanization
e i e e e
i ib ie eb ebC v r v= Ω +
? ? ?a
ib ib ibf γ= −
ae e e
ib ib ibf γ= +
ae e e e e e e i
ie eb ie eb ie eb i ibv v r C = −Ω − Ω + Ω +
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 10
e e e e e
b ib ie ie ebg rγ= − Ω Ω
( ) ( ) a
( ) 2 ( )
e e e
eb eb eb
e e e i e
eb ib b ie eb
v v t
v f g v t
+ = − + ∆
= − + + − Ω − ∆
ae e e e e e
ib ib b ie ie ebf g r= + + Ω Ω
2 a
ie eb ie eb ie eb i ib
e e e e e e
ie eb ie ie eb ibv r
= − Ω − Ω Ω +
2i e e e
ie eb ib bv f g= − Ω + +
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 2: ECEF Frame MechanizationCase 2: ECEF Frame Mechanization
4. Position Update By simple numerical integration
2
( ) ( ) ( ) a2
eb eb eb e
e
b
e e e tr r v t
∆+ = − + − ∆ +
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 11
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 2: ECEF Frame MechanizationCase 2: ECEF Frame Mechanization
2. SF
Transform
b
ibω b
ibf
Grav3. Velocity
1. Attitude
Update( )
e
bC −
( )b e
e e
bg r
e
ibf
( ) ( ) ( )ib i
e e b i e
b b e bC C I t C t + = − + ∆ Ω− − ∆ Ω
( )e e
ib i
b
b bf C f= +
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 12
3. Position
Update
Grav
Model
3. Velocity
Update
( )e
bC +
( )b eb
g r( )
eb
ev −
( )eb
ev +
( )eb
er +
( )eb
er −
( )ib ib bf C f= +
( ) ( ) ae e e
eb eb ebv v t+ = − + ∆
2
( ) ( ) ( ) a2
eb eb eb e
e
b
e e e tr r v t
∆+ = − + − ∆ +
a 2 ( )e e e i e
eb ib b ie ebf g v= + − Ω −
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 3: Navigation Frame MechanizationCase 3: Navigation Frame Mechanization
• Determine our PVA wrt the Nav frame Position: Typically described in curvilinear coordinates
Velocity: Typically the velocity of the body wrt
the earth frame described in the navigation
frame coords
[ ], ,b b b
TL hλ
by
bz Body Frame
?ω
frame coords
Attitude: Typically the orientation
of the body described in the nav
frame
This unusual choice facilitates
the guidance system function
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 13
n
ebv
n
bC
ex
ey
ez
Inertial Frame
ECEF Framei
x
iy
iz
?
ieω
nx
ny
nz
?
enω
Nav Frame
bx
by?
nbω
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 3: Navigation Frame MechanizationCase 3: Navigation Frame Mechanization
1. Attitude Update: Note that
Now
? ? ? ?
ib ie en nbω ω ω ω= + +
nb ib ie
b
n
b
e
b bω ω ω ω= − −⇒
( )n n b n b b b
b b b
n b
nb ib ie en
ib i
n b n b
b b be en
C C C
C C C
Ω Ω Ω Ω
Ω Ω Ω
= = − −
= − −
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 14
( )ib ib b b
n b n
e en
ib ie be
n n
nb
C C C
C C
Ω Ω Ω
Ω Ω Ω
= − −
= − +
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
ω
ω
ω
ω
= =
=
− −
See next slide
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 3: Navigation Frame MechanizationCase 3: Navigation Frame Mechanization
( )en
Te e n n e e
n n n nenC C C CΩ Ω= ⇒ =
en e
n n
nω⇔Ω
Last term:
0 sin( ) -
-sin( ) 0 -cos( )
cos( ) 0
b b b
b b b b
b b b
L L
L L
L L
λ
λ λ
λ
=
Courtesy of Mathematica
,
n
en xω
,
n
en yω
,
n
en zω
cos( )b bL λ
n
v
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 15
cos( )
-
-sin( )
b b
en b
b
n
b
L
L
L
λ
λ
ω
=
( )
( )
,
,
,
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
λ
+ +
=
−
From Eqn. 16 (Wedeward handout) ⇒
( )
( )
( )
,
,
,
-
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
ω
∴ =
+
+
+
( ) ( )( ) ( )
( ) ( )
n n n
b b b
n b n n n
b ib e en bi
C C tC
C I t C t
+ − + ∆
= − + ∆ Ω +Ω− Ω − ∆
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase Case 3: 3: NavNav Frame Frame MechanizationMechanization
2. Specific Force Transformation: Simply coordinatize the specific force
3. Velocity Update Recall that
( )n n
ib i
b
b bf C f= +
( ) [ ] [ ] ( )TC C C C C Cω ω ω ω× = × ⇒ × = ×
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 16
n n e
eeb ebv C v=
( )2n n n n n
ib b e ebn ief g v= + − Ω + Ω
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= Ω + + − Ω
2n n n n n e e
ib b en e ebieebf g v C v= + − Ω − Ω
2n n n n n n e
ib b en ie ebeebf g v C v= + − Ω − Ω
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase Case 3: 3: NavNav Frame Frame MechanizationMechanization
Finally
4. Position Update Recalling the relationship between and the curvilinear
coordinates
( )( ) ( ) 2 ( )n n n n n n n
ib beb e ieeb ebnv v t f g v + = − + ∆ + − Ω + Ω −
n
ebv
coordinates
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 17
,( ) ( ) ( )
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 Lλ λ
++ = − + ∆
−
( )
( )
,
,
,
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
λ
+ +
=
−
Inertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationInertial NavigationCase 3: Case 3: NavNav Frame MechanizationFrame Mechanization
2. SF
Transform
b
ibω b
ibf
1. Attitude
Update( )
n
bC −
n
ibf
( ) ( )( ) ( ) ( )ib ie
n n b n n n
b b n beC C I t C t+ = − + ∆ − + Ω −Ω ∆Ω
( )n n
ib i
b
b bf C f= +
( )
( ) ( )n n
eb ebv v+ = − +
11 March 2011 EE 570: Location and Navigation: Theory & Practice Lecture 7: Slide 18
3. Position
Update
Grav
Model
3. Velocity
Update
( )n
bC +
n
bg
( )eb
nv −
( )eb
nv +
ibf
( )
( )
( )b
b
b
h
L
λ
−
−
−
( ), ( ), ( )b b bh L λ+ + +
( )2 ( )n n n n n
ib b e ie ebnt f g v ∆ + − Ω + Ω −
,( ) ( ) ( )
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 Lλ λ
++ = − + ∆
−
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