EE 495 Modern Navigation Systems

EE 495 Modern Navigation Systems

Inertial Navigation in the ECI Frame

Wed, Feb 18 EE 495 Modern Navigation Systems Slide 1 of 14

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Inertial Navigation in the ECI Frame Background– The Fundamental Problem

Wed, Feb 18 EE 495 Modern Navigation Systems

• 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: , , & )2. Inertial sensors ( and ) are error free (relax later)3. Have a gravity () and/or gravitational () model

• QUESTION: Where am I ? – Current PVA ?o With respect to which frame?

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Inertial Navigation in the ECI Frame Background – 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-accelso Typically configured in an orthogonal

triad The “mechanization” can be

performed wrt:o The ECI frame,o The ECEF frame, oro The Nav frame.



INS

Mec

hani

zatio

nEq

uatio

ns

Initialization

GravityModel

Position

Velocity

Attitude

Inertial Navigation in the ECI FrameBackground – ISA, IMU, & INS

• An Inertial Navigation System (INS) ISA – Inertial Sensor Assembly

o Typically, 3-gyros + 3-accels + basic electronics (power, ADCs, …) IMU – Inertial Measurement Unit

o ISA + Compensation algorithms (i.e., basic processing) INS – Inertial Navigation System

o IMU + gravity model + “mechanization” algorithms


IMUCo

mpe

nsati

on

Algo

rithm

s

bbif

bbi

ISA

Accels

Gyros Raw

sen

sor

sign

als


MechanizationEquations

Inertial Navigation in the ECI FrameBackground – The Mechanization Process

Gravity / Gravitational

Model

bib

bibf

PriorAttitude

PriorVelocity

PriorPosition

UpdatedAttitude

UpdatedVelocity

UpdatedPosition

Prio

r PVA

Updated PVA

Current IMU Measurements


gyro accel


Inertial Navigation in the ECI FrameBackground – A Four Step Mechanization Process

• Can be generically performed in four steps:1. Attitude Update

o Update the prior attitude (rotation matrix) using the current angular velocity measurement ()

2. Transform the specific force measurement ()o Typically, using the attitude computed in step 1.

3. Update the velocityo Essentially, integrate the result from step 2. with the use of

a gravity/gravitation model ()

4. Update the Positiono Essentially, integrate the result from step 3.



Inertial Navigation in the ECI FrameBackground – A Four Step Mechanization

2. SFTransform

bib

PriorAttitude

bibf

4. PositionUpdate

GravModel

PriorVelocity

PriorPosition

3. VelocityUpdate

1. Attitude Update

UpdatedAttitude

UpdatedVelocity

UpdatedPosition

Prio

r PVA

Updated PVA

IMU Measurements



Inertial Navigation in the ECI FrameCase 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: Velocity: Velocity of the body frame wrt the inertial frame

resolved in the inertial frame: Attitude: Orientation of the body frame wrt the inertial

frame

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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 Frameat time “k-1”

Body Frameat time “k”

i i bb b ibC C

bib


lim ( ) ( 1)0

i ii b bb

C k C kCt t

( 1)i bb ibC k

( ) ( ) ( )i i i bb b b ibC C C t

( ) ( )i i bb b ibC C I t

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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 Frameat time “k-1”

Body Frameat time “k”

( 1)( ) ( 1) ( )i i b kb k b k b kC C C

( 1)( )

bib tb k

b kC e

( ) ( )bib ti i

b bC C e

bib


( )i bb ibC I t

3 2

3 1

2 1

ˆ ˆ0ˆ ˆ0ˆ 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 Kˆb

ib t k

Slide 10 of 14



1. Attitude Update: High Fidelity

Lower Fidelity

2( ) ( ) sin( ) 1 cos( )i ib bC C I K K



ˆbib t k

Slide 11 of 14

ˆSk k

K

b bib 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 iib i

bb bf C f

a ib ii i

ibi

bf

( ) ( ) aib ibiib

i iv v t

2

( ) ( ) ( ) a2ib ib ib i

ib

i i i tr r v t


aib ii i

ibi

bf



GravModel

3. VelocityUpdate

iib( )ib

iv

( )ibiv

4. PositionUpdate

( )ibir

( )ibir

2. SFTransform

bibf

iibf


( )i iib i

bb bf C f

( ) ( ) aib ibiib

i iv v t

2

( ) ( ) ( ) a2ib ib ib i

ib

i i i tr r v t

a ib ii i

ibi

bf


2( ) ( ) sin( ) 1 cos( )i ib bC C I K K

or

bib

( )ibC

( )ibC

1. Attitude Update




• In continuous time notation: Attitude: Velocity: Position:

• Combining into a state-space equation:

a ib ii i

ibi

bf

i i bbib ibf C f

ii

i i b ib

i i bb

ibib

ib ib ib

b ib

vrv C fC C

ibiv

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EE 495 Modern Navigation Systems

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