Mechanization of General Navigation Equations

Inertial Navigation Systems

Muhammad Ushaq

[email protected]

0092-322-2992772

Mechanization of General Navigation Equations

mailto:[email protected]

Review of Accelerometer

Muhammad Ushaq 2

A force F acting on a body of mass m causes the body to accelerate with

respect to inertial space. This translational acceleration (a) is given by:

F ma

Impractical to determine the acceleration of a vehicle by measuring

the total force (F) on the navigating body!

Solution?

Measure the small force (f) acting on a small mass contained within the

vehicle which is constrained to move with the vehicle. The small mass,

known as a proof mass, forms a part of an accelerometer.


Muhammad Ushaq 3

ProofMass

Under steady state conditions, the force acting on

the mass will be balanced by the tension (or

compression) in the spring, the net extension of the

spring providing a measure of the applied force and

is proportional to the applied acceleration.

The total force (F) acting on a mass (m) in space:

F ma mf mg

f : Acceleration due to inertial forces

g : Acceleration due to gravitational force

mg : Force contribution due to gravitation force

F

m: Force per unit

Fa f g

m

The acceleration (a) experienced is the total force

per unit mass. f a g


Muhammad Ushaq 4

Accelerometer provides the measure of specific fore (difference between

the total acceleration and the acceleration due to gravitational force. This

quantity is the non-gravitational force per unit mass exerted on the

instrument.

Output of Accelerometer under free-fall condition, near the surface of earth?

Output of Accelerometer placed on the surface of earth?

As f a g , the knowledge of the gravitational field is essential to

enable the measurement provided by the accelerometer to be related to

the inertial acceleration.

The measurements provided by the accelerometers must be combined

with knowledge of the gravitational field to determine the acceleration of

the vehicle with respect to inertial space. Using this information, vehicle

acceleration relative to the body may be derived.

One-dimensional example of navigation

Muhammad Ushaq 5

To determine the instantaneous speed

of the train and the distance it has

travelled from a known starting point we

need to take measurements of its

acceleration along the track by using a

single accelerometer.

The time integral of the acceleration measurement provides a continuous

estimate of the instantaneous speed of the train, provided its initial speed

was known. A 2nd

integration yields the distance travelled with respect to

a known starting point.

Muhammad Ushaq 6

The accelerometer together with a computer, or other suitable device

capable of integration, constitutes a one-dimensional navigation system.

One-dimensional example of navigation

Output of Accelerometer is f (1)

a V f g

X V

(0)X X Vdt

Y-C

oo

rdin

ate

of

Po

siti

on

X-Coordinate of Position

FixedHeading

Muhammad Ushaq 7

Two-dimensional Navigation

Y-C

oo

rdin

ate

of

Po

siti

on

X-Coordinate of Position

Detect continuously the translational

motion of the moving body in two directions

and changes in its direction of travel, that

is, to detect the rotations of the train about

the perpendicular to the plane of motion as

the train moves along the track.

Two accelerometers are required to detect the translational motion in

perpendicular directions along and perpendicular to the track, one

gyroscope will also be required for measurement of the rotational motion.

Here, it is possible to construct a simple, two-dimensional, navigation

system using one gyroscope, two accelerometers and a computer.

Muhammad Ushaq 8

Realization of Two-dimensional Navigation

Gyro

Y-axisAccel

X-axis

Acc

elZ-

axis

R

ibZf

ibXf

Xb

ipY

Yb

Zb

ixg

iyg

Vz

Vz X

ZIt is assumed that a system is required to

navigate a vehicle which is constrained to

move in a single plane. The system contains

two accelerometers and a single axis rate

gyroscope, all of which are attached rigidly

to the body of the vehicle.

Strapdown INS with body frame: b b bX Y Z

Reference Frame: i i iX Y Z

: Angular displacement between the body and reference frames

Muhammad Ushaq 9


Xi

Xb

ZiZb

Attitude is updated as: 0 ibydt .

is used to resolve the measurements of specific

force, ibxf and ibzf , into the reference frame. A gravity model, stored in the computer provide estimates of the gravity components in the

reference frame, ixg and izg . These quantities are combined with the resolved measurements of

specific force ixf and izf to determine true

accelerations, denoted by ixV and izV

These derivatives are subsequently integrated twice to obtain estimates of vehicle velocity and position.

Muhammad Ushaq 10


Gyro

Y-axisAccel

X-axis

Acc

elZ-

axis

R

ibZf

ibXf

Xb

ipY

Yb

Zb

ixg

iyg

Vz

Vz X

Z

iby

Cos Sinix ibx ibzf f f

Sin Cosiz ibx ibzf f f

ix ix ix ixa V f g

iz iz iz iza V f g

i ixX V

i izZ V

Xi

Xb

ZiZb

(0) i i ixX X V dt

(0) i i izZ Z V dt

Muhammad Ushaq 11

General Navigation Equations

The acceleration of a point P with respect to a space-fixed axis set,

termed as i-frame is defined as:

2

2i

ii

d r dVa

dt dt

Navigation Mechanization are the equations and procedures used with

inertial navigation system to generate position and velocity information.

A triad of perfect accelerometers will provide a measure of the specific

force (if ) acting at point P where

2

2

( )

( ) , ( )

ii i

m

i

i i i i

m i m

i

dVf g R

dt

d rf g R f a g R

dt

Muhammad Ushaq 12


Differential equation of motion of inertial navigation of a vehicle relative to

an inertial frame can be written in vector form as

i iR V

( )i

i i

m

i

dVf g R

dt

iR = geocentric position vector

Ti

x y zV V V V = velocity of the vehicle relative to the i frame

if = the acceleration sensed or measured by an accelerometer

( )i

mg R =gravitational acceleration due to mass attraction

Muhammad Ushaq 13


Motion of the navigating-body in inertial coordinates (ECI) frame

( )i

i i

m

i

dVf g R

dt

2

2( )

ii i

m

i

d Rf g R

dt

2

2

i

i

d R

dt

is the inertial acceleration wrt the center of non-rotating earth.

A Triade of Accelerometers is aligned and mounted on platform (P).

The output of accelerometerspf coordinatized (referenced) in platform

frame will be given as:

2

2( )p p i

i m

i

d Rf C g R

dt

OR ( )p p i i

i mf C R g R

Muhammad Ushaq 14


Rearranging ( )p p i i

i mf C R g R we get

( )i i p i

p mR C f g R

This is the basic “Inertial Navigation Equation” and INS is based on solving

this equation for velocity and position by the onboard computer

Accelerometers

p

if i

pC

iR+

+

Initial Velocity

Initial Position

( )i

mg R

iRp

if

NavigationComputer

Latitude

Longitude

Altitude

iR

Muhammad Ushaq 15


For navigation at or near the surface of the earth, we need to refer the

position and velocity of the vehicle to an earth-fixed coordinate system,

which rotates with the earth.

ie ie

i e

dR dRR V R

dt dt

Coriolis acceleration comes into effect when a vehicle is moving with some

velocitydR

Vdt

with respect to the rotating coordinate frame.

2

2 ie

i ii

d R dV dR

dt dt dt

As 0ied

dt

earth spin rate is fixed in space

Muhammad Ushaq 16


By putting value of

i

idR

dt

2

2

i

ie ie

ii

d R dVV R

dt dt

2

2( )ie ie ie

ii

d R dVV R

dt dt

The output of the accelerometers gives quantities which are measured

along the platform. Differentiation or integration of these components

should be carried out with respect to platform axes.

The derivative of the velocity V with respect to the platform axes can be

related to the derivative wrt inertial reference velocity as follows

ip

i p

dV dVV

dt dt

Muhammad Ushaq 17


By combining last two equations we have:

2

2( )ip ie ie ie

pi

d R dVV V R

dt dt

2

2( ) ( )ip ie ie ie

pi

d R dVV R

dt dt

By putting this value of

2

2

i

d R

dt in Equation

2

2( )

ii i

m

i

d Rf g R

dt

( ) ( ) ( )i i

ip ie ie ie m

p

dVf V R g R

dt

The term ( )ie ie R represent the centripetal acceleration caused by

rotation of earth and is a function of position on the earth only (as ie is

constant), it can be combined with the mass gravity term as

( ) ( ) ( )i i

m ie ieg R g R R

Muhammad Ushaq 18


( ) ( )i i

ip ie

p

dVf V g R

dt

( ) ( )i i

ip ie

p

dVf V g R

dt

This equation is the generalized mechanization equation

For the locally level coordinate frame:

Spatial rate ( ip ) is the sum of the earth rate and the vehicle angular rate,

or transport rate with respect to the earth fixed frame given as:

+p p p

ip ie ep

By substitution we have:

( ) ( )i i

ep ie ie

p

dVf V g R

dt

( 2 ) ( )i i

ep ie

p

dVf V g R

dt

Muhammad Ushaq 19

( 2 ) ( 2 ) i

x x epy ie y z epz iez y xV f V V g

( 2 ) ( 2 ) i

y y epz iez x epx iex z yV f V V g

( 2 ) ( 2 ) i

z z epx iex y epy iey x zV f V V g

Ti i i i

x y zg g g g

2 23223

Re1 1 5( ) ( )i

i zx

ix r

R Rgr

JR

2 23223

Re1 1 5( ) ( )i

i zy

iy r

R Rgr

JR

2 23223

Re1 3 5( ) ( )i

i zz

iz r

R Rgr

JR


14

32

3.9

860306

10

(/se

c)

m

22

2(

)(

)(

)i

ii

xy

zR

rr

r

Muhammad Ushaq 20


For the east-north-up (ENU) coordinate frame we have

0

Cos

Sin

X E

Y N

Z U

p p

ie ie

p p p

ie ie ie ie

p pieie ie

0 0T

p p

zg g

2 -69.783 0.051799 0.94 10p

zg Sin h

o -536015.04106874 /h=7.2921159 10 rad/s

23 [56 (4.9 / 600] / 60ie

( 2 ) ( 2 )p p p p p

x x ep y ie y z ep z ie z yV f V V

( 2 )p p p p

y y ep z ie z x ep x zV f V V

( 2 )p p p p

z z ep x y ep y ie y x zV f V V g

Muhammad Ushaq 21


Velocity can be update by integration

( ) ( )

t t

t

V t t V t Vdt

Latitude, longitude and altitude can be updated with following equations

( ) ( )

t t

t

t t t dt

( ) ( )

t t

t

t t t dt

( ) ( )

t t

z

t

h t t h t V dt

Muhammad Ushaq 22


Local Radii of Curvatures of Earth

2

3 22 2

(1 )

1 sin

eM

R eR h

e L

122 21 sin

eN

RR h

e L

The radii of curvature of the reference ellipsoid can be approximated with sufficient accuracy as

21 2 3( )M eR R e Sine

2R 1( )N eR Sine

Muhammad Ushaq 23

The transport rate (the rate caused by movement of vehicle w.r.t. earth) is given by following relation

p

y

M

pp x

ep

N

p

x

M

V

R h

V

R h

VTan

R h

and +

p

y

M

pp p p x

ip ie ep

N

p

x

M

ie

ie

V

R h

VCos

R h

VSin Tan

R h


Muhammad Ushaq 24

The coordinates in (ECEF) system can be computed by a transformation from geodetic to earth-fixed coordinates.

2

Cos Cos

Cos Sin

1 Sin

e N

e N

eN

X R h

Y R h

Z R e h

The inverse transformation:

1tan e

e

y

x

21

sin

eN

zh R

1

2 2 2 tan N e

e eN

R h z

b x yR h

a


Date post:	18-Jan-2017
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Mechanization of General Navigation Equations

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