inertial navigationelosery/spring_2011/ee570/lectures/... · 11 march 2011 ee 570: location and...

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


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.


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


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


( 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


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


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

∆+ = − + − ∆ +


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= +


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= +


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× = ×


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


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 +


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 +


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 + +


4. Position Update By simple numerical integration

2

( ) ( ) ( ) a2

eb eb eb e

e

b

e e e tr r v t

∆+ = − + − ∆ +



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= +


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


n

ebv

n

bC

ex

ey

ez

Inertial Frame

ECEF Framei

x

iy

iz

?

ieω

nx

ny

nz

?

enω

Nav Frame

bx

by?

nbω


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

= = − −

= − −


( )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


( )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


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× = × ⇒ × = ×


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


,( ) ( ) ( )

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+ = − +


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

++ = − + ∆

−

inertial navigationelosery/spring_2011/ee570/lectures/... · 11 march 2011 ee 570: location and...

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