EE 570: Location and Navigation: Theory & PracticeNavigation Sensors and INS Mechanization Navigation Equations – Inertial Navigation •Inertial Navigation The process of “integrating”

EE 570: Location and Navigation: Theory & Practice

Navigation Sensors and INS Mechanization

Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 14

Navigation Sensors and INS Mechanization Navigation Equations – The Fundamental Problem

Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice

• The Fundamental Inertial Navigation Problem:

Using inertial sensors (accels & gyros) and an initial position and orientation, determine the vehicle’s (i.e. body frame) current position, velocity, and attitude (PVA)

Assumptions:

1. Know where we started (initial PVA: 𝑟 ?𝑏? , 𝑣 ?𝑏

? , & 𝐶𝑏? )

2. Inertial sensors (𝜔𝑖𝑏𝑏 and 𝑓 𝑖𝑏

𝑏) are error free (relax later)

3. Have a gravity (𝑔 𝑏? ) and/or gravitational (𝛾 ?𝑏

? ) model

Where am I ? – Current PVA ?

o With respect to which frame?

Slide 2 of 14

Navigation Sensors and INS Mechanization Navigation Equations – Inertial Navigation

• Inertial Navigation The process of “integrating” angular velocity & acceleration

to determine one’s position, velocity, and attitude (PVA) o Effectively “dead reckoning”

To measure the acceleration and angular velocity vectors we need at least 3-gyros and 3-accels o Typically configured in an orthogonal

triad

The “mechanization” can be performed wrt: o The ECI frame,

o The ECEF frame, or

o The Nav frame.


Navigation Sensors and INS Mechanization Navigation Equations – ISA, IMU, & INS

• An Inertial Navigation System (INS)

ISA – Inertial Sensor Assembly

o Typically, 3-gyros + 3-accels + basic electronics (power, …)

IMU – Inertial Measurement Unit

o ISA + Compensation algorithms (i.e. basic processing)

INS – Inertial Navigation System

o IMU + gravity model + “mechanization” algorithms

IMU

- Basic Processing

ISA Comp Algs

Raw sensor outputs

b

b

if

b

b

i

Mechanization Equations

INS

Initialization

Position, Velocity,

and Attitude (PVA)


Gravity Model

Slide 4 of 14

Mechanization Equations

Navigation Sensors and INS Mechanization Navigation Equations – Mechanization Process

Gravity / Gravitational

Model

b

ib b

ibf

Prior Attitude

Prior Velocity

Prior Position

Updated Attitude

Updated Velocity

Updated Position

Pri

or

PV

A

Updated PVA

IMU Measurements


Navigation Sensors and INS Mechanization Navigation Equations – A Four Step Mechanization

• Can be generically performed in four steps:

1. Attitude Update

o Update the prior attitude (rotation matrix) using the current

angular velocity measurement (𝐶 10 = 𝐶1

0 𝛺011 = 𝛺01

0 𝐶10 )

2. Transform the specific force measurement (𝑓 𝑖𝑏? = 𝐶𝑏

?𝑓 𝑖𝑏𝑏 )

o Typically, using the attitude computed in step 1.

3. Update the velocity

o Essentially integrate the result from step 2. with the use of

a gravity/gravitation model (𝑓 𝑖𝑏 = 𝑎 𝑖𝑏 − 𝛾 𝑖𝑏)

4. Update the Position

o Essentially integrate the result from step 3.


Navigation Sensors and INS Mechanization Navigation Equations – A Four Step Mechanization

2. SF Transform

b

ib

Prior Attitude

b

ibf

3. Position Update

Grav

Model Prior Velocity

Prior Position

3. Velocity Update

1. Attitude Update

Updated Attitude

Updated Velocity

Updated Position

Pri

or

PV

A

Updated PVA

IMU Measurements


Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization


• CASE 1: ECI Frame Mechanization

Determine the Position, Velocity, and Attitude of the Body frame with respect to the Inertial Frame

• Determine our PVA wrt the ECI frame

Position: Vector from the origin of the inertial frame to the

origin of the body frame resolved in the inertial frame: 𝑟 𝑖𝑏𝑖

Velocity: Velocity of the body frame wrt the inertial frame

resolved in the inertial frame: 𝑣 𝑖𝑏𝑖

Attitude: Orientation of the body frame wrt the inertial

frame 𝐶𝑏𝑖

Slide 8 of 14


1. Attitude Update: Method A Body orientation frame at time “k” wrt time “k-1”

o t = Timek – Timek-1

( 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”

i i b

b b ibC C

b

ib


lim ( ) ( 1)

0

i ii b bb

C k C kC

t t

( 1)i b

b ibC k

( ) ( ) ( )i i i b

b b b ibC C C t

( ) ( )i i b

b b ibC C I t

Slide 9 of 14


1. Attitude Update: Method B

Body orientation frame at time “k” wrt time “k-1”

o t = Timek – Timek-1

( 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 e

( ) ( )bib ti i

b bC C e

b

ib


( )i b

b ibC I t

3 2

3 1

2 1

ˆ ˆ0

ˆ ˆ0

ˆ ˆ 0

ˆ( , )

k k

k k

k k

kR e e

K

2 2 3 3

2! 3!I

K KK

2sin( ) 1 cos( )I K K

e K

ˆb

ib t k

Slide 10 of 14


1. Attitude Update:

High Fidelity

Lower Fidelity

2( ) ( ) sin( ) 1 cos( )i i

b bC C I K K


( ) ( )i i b

b b ibC C I t

ˆb

ib t k

Slide 11 of 14

ˆSk kK

b b

ib ibSk


2. Specific Force Transformation

Simply coordinatize the specific force

3. Velocity Update

Assuming that we are in space (i.e. no centrifugal component)

Thus, by simple numerical integration

4. Position Update

By simple numerical integration

( )i i

ib i

b

b bf C f

a ib i

i i

ib

i

bf

( ) ( ) aib ib

i

ib

i iv v t

2

( ) ( ) ( ) a2

ib ib ib i

i

b

i i i tr r v t


aib i

i i

ib

i

bf


b

ib

1. Attitude Update

( )i

bC

( )i

bC

Grav

Model

3. Velocity Update

i

ib( )ib

iv

( )ib

iv

3. Position Update

( )ib

ir

( )ib

ir

2. SF Transform

b

ibf

i

ibf

( ) ( )i i b

b b ibC C I t

( )i i

ib i

b

b bf C f

( ) ( ) 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




• In continuous time notation:

Attitude: 𝐶 𝑏𝑖 = 𝐶𝑏

𝑖 𝛺𝑖𝑏𝑏

Velocity: 𝑣 𝑖𝑏𝑖 = 𝐶𝑏

𝑖 𝑓 𝑖𝑏𝑏 + 𝛾 𝑖𝑏

𝑖

Position: 𝑟 𝑖𝑏𝑖 = 𝑣 𝑖𝑏

𝑖

• Combining into a state-space equation:

a ib i

i i

ib

i

bf

i i b

bib ibf C f

ii

i i b i

b

i i b

b

ibib

ib ib ib

b ib

vr

v C f

C C

ib

iv

Slide 14 of 14

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EE 570: Location and Navigation: Theory & PracticeNavigation Sensors and INS Mechanization Navigation Equations – Inertial Navigation •Inertial Navigation The process of “integrating”