Mechanization of Inertial Navigation System in Inertially Stable Frame of Reference

Inertial Navigation Systems

Muhammad Ushaq

Mechanization of Inertial Navigation System in Inertially Stable Frame of Reference

Institute of Space TechnologyIslamabad, [email protected]

Muhammad Ushaq 2

c

P

Xi

Zi & Ze

Xe

Yi

yil

yg0 zil, zg0

xil

xg0

R

Oi

O1,Og

ψo

Do Ro

Local

Meridian

Earth Frame

Meridian Xg

Yg Zg

Greenwich

Meridian

Inertial Ref

Meridian Ye

Equatorial Plane

ωieΔt

λ

λo

λi

Earth Rotation

North

h

Muhammad Ushaq 3

Earth frame meridian

Greenwich meridian

Inertial reference

meridian

0

i

iet

ix

ex

eziz

ie

Local meridian

iy

gx

gy

N

Equatorial plane

iO0

0cilx

ilz

gz

ilygO

ilO

Space Stable Mechanization of SINS

Muhammad Ushaq 4

In this navigation scheme inertial reference frame is used for navigation

computations.

The essential difference between a local-vertical and space-stabilized

platform configuration is the absence of platform torquing on the space-

stabilized case.

The error equations for a space-stabilized configuration are simpler than

those of the local-vertical system.

For a space stabilized platform mounted in a vehicle, the

accelerometers attached to the platform will measure accelerations in

a coordinate system fixed in inertial space.

The accelerometer triad is held in a non-rotating inertial frame by a

three-axis gyroscopic stabilized platform


Muhammad Ushaq 5

An inertial rather than a geographic reference frame is used for the

navigation computations

As in every "all attitude" gimbaled inertial navigation system, four

gimbals are normally required to isolate the inertial platform from vehicle

angular motion.

In the navigate mode, the gyroscopes are un-torqued, or at most they

are torqued at a very low level in order to compensate for the known

gyroscope drift rates

The inertial platform is un-commanded, so within the limits imposed by

the gyroscope drift, the platform will remain inertially non-rotating


Muhammad Ushaq 6

The commanded platform inertial angular velocity is equal to the desired

platform angular velocity, which is equal to zero ( 0p

ip )

The gyroscope and platform axes are non-orthogonal and are related

by a small angle transformation. Therefore, since the platform rotation

is very small, the angular velocity of the gyroscope frame (G

iG ) is equal

to the angular velocity of the platform, p

ip . Thus, p G

ip iG

The platform axes could be aligned with the axes of any inertially non-

rotating frame, it will be assumed here that the li (Launch Site Inertial

Frame) is instrumented

Space-stabilized systems are mostly used in spacecraft and missile

platform mechanizations, since in these applications no geographic

navigation information (online) is needed. Also, space-stable navigation

systems are used in some terrestrial aircraft and marine navigation

applications.


Muhammad Ushaq 7

Coordinate Fames Employed in Space Stable SINS Scheme

Earth Centered Inertial Frame ( i i iX Y Z )

Earth Centered Earth Fixed Frame ( e e eX Y Z )

The Local Geographic Frame ( g g gX Y Z ) East, North, Up

Launch Site Inertial Frame (l l li i iX Y Z )

This is the reference coordinate system for SINS. It is considered as

inertial frame. Origin of this frame is the starting point of the vehicle.

liX and

liY axes are in the local horizontal plane at starting point, and

liY points in the direction of target.

liZ is perpendicular to the local

horizontal plane and points up.

Muhammad Ushaq 8

Launch Site Inertial Frame (l l li i iX Y Z )

This frame is North Referenced Tangent Plane. This frame does not

move or rotate with the moving vehicle. The angle o (the angle between

liY and north at starting point) gives the initial reference from north


At starting point latitude ( o ), longitude ( o ), and ( o ) i.e. the angle

between li

Y and north gives us the position and orientation of this

particular frame with respect to the earth centered inertial frame

Muhammad Ushaq 9


Initial Local Geographic Frame (0 0 0g g gX Y Z )

This particular frame is used for defining initial orientation of inertial

reference frame as well as body frame.

Body Frame ( b b bX Y Z )

Origin of body frame is located at the center of mass of vehicle. Axes of

body frame are oriented as follows

bX : Along right wing

bY : Along Longitudinal axis (forward)

bZ : Lies in longitudinal Symmetrical Plane and points up.

Muhammad Ushaq 10

Definition of Longitudes

At starting point ( 0t ), the inertially fixed meridian (which is assumed

non-rotating with respect to the fixed stars), earth frame meridian, and

local meridians are all coincident

Following relation holds true at any arbitrary time onwards

: Terrestrial Longitude (from Greenwich meridian to local meridian)

o : Initial Terrestrial Longitude (between Xe and Greenwich meridian)

i : Celestial Longitude (between local meridian and Xi)

o i iet

ie : Earth sidereal Rate (7.292115 x 10-5

rad/sec)

t : Time elapsed

Muhammad Ushaq 11

Computation of Acceleration

In case of real platform the output of accelerometers are directly

measured in the reference frame i.e. li

l

l

l li iiiif r G

l

l

l li iiiir f G

Muhammad Ushaq 12


For a strapdown system, the out puts of accelerometers are directly

measured in body frame and are transformed into reference inertial frame

(l

i ) then the inertially referenced acceleration is calculated from following

relation

l l li i ibb ibr C f G

lir : Computed acceleration in reference inertial frame (l

i )

li

bC : Transformation matrix form body frame to reference frame (l

i )

bibf : Inertial specific force referenced in body frame

Muhammad Ushaq 13


liG : Acceleration due to Gravitational field, computed in reference

inertial frame (l

i )

l li i iiG C G

Muhammad Ushaq 14

.Computation of Strapdown DCM (l

biC ) or li

bC

l

biC is the transformation matrix from reference inertial frame to body

frame

/ /1 1 1l l l

i i il l l

i i ii i i b b bX Z Y

X Y Z X Y Z

Hereil

,il

and il

are vehicle’s pitch, yaw and roll angles with respect to

reference inertial frame ( li )

Muhammad Ushaq 15

.Computation of Strapdown DCM (l

biC ) or li

bC

l l l l l l l l l l l l

l l l lil

i i i i i il l l l l il l l l l ll

l

i i i i i i i i i i i i

i i i i

i i i i i

bi

Cos Cos Cos Sin Cos Sin Sin Cos Sin Sin Sin Cos

Sin Cos Cos Cos Sin

Sin Cos Sin Sin Cos Cos Sin Sin Sin Sin Cos Cos

C

=( )l

l

i b TibC C

Muhammad Ushaq 16

.Computation of Initial li

bC

Initial li

bC is calculated from initial alignment

0

0

l li i ggb bC C C

og is the initial local geographic north pointing frame (E N U)

Muhammad Ushaq 17

.Computation of Initial li

bC

0

ligC is the transformation matrix from initial local geographic frame to

the reference inertial frame (l

i ) At start time ( 0t )

0

( ) ( ) 0

( ) ( ) 0

0 0 1

l

o o

o o

ig

Cos Sin

Sin CosC

o is the angle between li

Y and north (0gY ) at launch site

Muhammad Ushaq 18

.Updating Attitude DCM li

bC

1

2Q Q

The quaternion is obtained as

1

0b

i b

1 1

b b b

i b ib ii

As 1

0b

ii as frame i and li are inertially fixed with respect to each

other

1

b b

i b ib (Readings of gyroscopes)

Muhammad Ushaq 19

Computation of Gravitational Acceleration

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

Ti i i i

x y zG g g g

Muhammad Ushaq 20

Computation of Gravitational Acceleration

is the product of the Earth’s mass and universal gravitational constant

7 14 3 2(3.9860305 3 10 ) 10 [ / sec ]m

2 2 2

i i i

x y zr r r r is geocentric position vector magnitude

3

2 (1.08230 0.0002) 10J is a constant coefficient

6378137e mr is the Earth’s equatorial radius

Muhammad Ushaq 21

Geocentric Position Vector in Geographic Coordinates

l

l l

l

l

iixxi ii i

y i y o

i iz z

rr

r C r r

r r

l l o

o

i

o

i gg or C r

0oT

o o o o

go r SinD r CosDr

oD is the angle between Geodetic and

Geocentric vertical as shown in Figure

0D

er

prh

P

N

D

r

0r

io

lo

0c

c

Muhammad Ushaq 22

Geocentric Position Vector in Geographic Coordinates

(2 )oD eSin

e is the eccentricity of earth

21 ( )eor R eSin

ogor is the geocentric vector

in geographic frame

0D

er

prh

P

N

D

r

0r

io

lo

0c

c

Muhammad Ushaq 23

.Computation of l

iiC

The reference inertial coordinate frame ( li ) has following relation

with local geographic frame (E N U) at the launch site.

0

0

0

0 0 1

l

o oig o o

Cos Sin

C Sin Cos

Keeping this relation in mind we can deduce that ( )l

l

i i Ti iC C is

obtained by following sequence of angular rotations

(0) 90 90o oi o o

o o o l l li i goi i i g g g i i iZ X Z

X Y Z X Y Z X Y Z

Muhammad Ushaq 24

.Computation of l

iiC

(0) 90 90o oi o o

o o o l l li i goi i i g g g i i iZ X Z

X Y Z X Y Z X Y Z

l l o

o

i i gi g iC C C

(0) (0)

(0) (0)

0 1 0 0 0

) 0 0 0

0 0 1 0 0 0 1

l

o o i i

o o o o i i

o o

ii

Cos Sin Sin Cos

Sin Cos Sin Cos Cos Sin

Cos Sin

C

(0) (0) (0) (0)

(0) (0) (0) (0)

(0) (0)

l

o i o o i o i o o i o o


o i o i o

ii

Cos Sin Sin Sin Cos Cos Cos Sin Sin Sin Sin Cos

Sin Sin Cos Sin Cos Sin Cos Cos Sin Sin Cos Cos

Cos Cos Cos Sin Sin

C

Muhammad Ushaq 25

.Computation of l

iiC

(0) (0) (0) (0)

(0) (0) (0) (0)

(0) (0)

l



o i o i o

ii

Cos Sin Sin Sin Cos Cos Cos Sin Sin Sin Sin Cos

Sin Sin Cos Sin Cos Sin Cos Cos Sin Sin Cos Cos

Cos Cos Cos Sin Sin

C

o is defined positive from north line to west

o is latitude at launching site

(0)( )i is the celestial longitude at launching site

Muhammad Ushaq 26

Velocity Update

The inertially referenced acceleration is given by

l l li i ibb ibr C f G

0

(0)il

til ilr r dt r

( ) ( )

l l l

t ti i ix x x

t

t t tV V r dt

( ) ( )

l l l

t ti i iy y y

t

t t tV V r dt

( ) ( )

l l l

t ti i iz z z

t

t t tV V r dt

Muhammad Ushaq 27

Position Update

( ) ( )

( ) ( )

( ) ( )

l l l

l l l

l l l

t ti i ix x x

t

t ti i iy y y

t

t ti i iz z z

t

t t t

t t t

r r V dt

r t t r t V dt

r r V dt

Velocity in i-frame isi i il

ilr C r

Now position in i-frame is calculated as follows

0

(0)i

ti ir r dt r

0

0(0)(0) gi i il

gilr C C r

Muhammad Ushaq 28

Latitude, Longitude and Altitude

The geocentric latitude is related to the polar component of position by

the expression sini

zc

r

r in accordance with the figure

0D

er

prh

P

N

D

r

0r

io

lo

0c

c

Muhammad Ushaq 29

Latitude Update

The geographic latitude, , is related to the

geocentric latitude, c , through the deviation of

the normal, D. Thus the computed geographic

latitude is given by

1sin ( )i

zc

rD

rD

sin 2D e

1sin ( ) sin 2i

zr er

0D

er

prh

P

N

D

r

0r

io

lo

0c

c

Muhammad Ushaq 30

The celestial longitude i is related to the equatorial position

components by the expression

1tan ( )

i

y

i i

x

r

r

But the celestial longitude is related to the terrestrial longitude as

o i iet

Thus the terrestrial longitude is

1

0 tan ( / ) i i

y x ier r t

Longitude Update

Muhammad Ushaq 31

Altitude Update

oh r r

2 2 2( ) ( ) ( )i i ix y zr r r r

21 eor r eSin

Muhammad Ushaq 32

Attitude Computation

l l l l l l l l l l l l

l l l lil

i i i i i il l l l l il l l l l ll

l

i i i i i i i i i i i i

i i i i

i i i i i

bi

Cos Cos Cos Sin Cos Sin Sin Cos Sin Sin Sin Cos

Sin Cos Cos Cos Sin

Sin Cos Sin Sin Cos Cos Sin Sin Sin Sin Cos Cos

C

=( )l

l

i b TibC C

1(1,2)( )li

m bSin C

(1,3)1

(1,1)

( )l

l

i

b

m i

b

CTan

C

(3,2)1

(2,2)

( )l

l

i

b

m i

b

CTan

C

Muhammad Ushaq 33

Attitude Computation

Yaw

m

Roll

m if 1

(1,1)

i

bC > 0

0180m if 1

(1,1)

i

bC < 0 and 0m

0180m if 1

(1,1) m0 0i

bC

Pitch

m if 1

(2,2)

i

bC > 0

0180m if 1

(2,2)

i

bC < 0 and 0m

0180m if 1

(2,2) 0i

bC and 0m

Muhammad Ushaq 34

Calculation of Attitude in Geographic Frame

For computation of attitude in geographical frame first we have to

compute the attitude DCM from body frame to local geographical

frame. It can be evaluated that the required transformation matrix has

the following relation with the other DCMs involved in the scheme:

o l

o l

igg g ii g ib bC C C C C

giC is obtained by the following sequence of rotation

90 90o oi

i ii i i g g gZ XX Y Z X Y Z

Muhammad Ushaq 35


90 90o oi

i ii i i g g gZ XX Y Z X Y Z

1 0 0 0

0 0

0 0 0 1

i i

g

i i i

Sin Cos

C Sin Cos Cos Sin

Cos Sin

0i

g

i i i

i i

Sin Cos

C Sin Cos Sin Sin Cos

Cos Cos Cos Sin Sin

Muhammad Ushaq 36


b

g

Cos Cos Sin Sin Sin Cos Sin Sin Sin Cos Sin Cos

C Cos Sin Cos Cos Sin

Sin Cos Cos Sin Sin Sin Sin Cos Sin Cos Cos Cos

T

g b

b gC C

1 21

22

( )g

Ctan

C 1 13

33

( )g

Ctan

C

1

23( )g Sin C

Muhammad Ushaq 37

Heading

If 22C >0 and g >0 then = g

Else if 22C >0 and g <0 then = 2 g

Else if 22C <0 then = g Roll

If 33C >0 then g

Else if 33C <0 and g <0 then g

Else if 33C <0 and g >0 then g

Pitch

g


Muhammad Ushaq 38

.Calculation of Initial Velocity in Reference Inertial Frame ( li )

Just before launch ( 0t ) although vehicle has no relative velocity with

respect to earth, but it is not stationary with respect to reference inertial

frame ( li )

The contributing factor for initial velocity of vehicle with respect to

li -frame is the earth rate.

This initial velocity will cause the celestial longitude to change with

following rate

1(0 )( )

X

og

i b

i

oN

V

R h Cos

Muhammad Ushaq 39


This rate of change of celestial longitude will be exactly equal to earth

rate at 0t

( )X

o

l

g

i b

N

ie

oR h

V

Cos

(0) ( )o

l X

g

i b ie N oV hR Cos

It is clear that there will be no component of initial velocity in north and

up direction i.e.

(0)

(0)

0

0

o

l

o

l

gyi b

gzi b

V

V

Muhammad Ushaq 40


( )

(0) 0

0

o

l

N

g

i b

ie oR h

V

Cos

(0) (0)o

l

l l

o

g

i b

i igib VV C

Muhammad Ushaq 41

Calculation of Earth Referenced Velocity

Applying the Coriolis law, the earth referenced velocity is given by

following relation

( )i ig g ieg i ier rV C

Here iie is the skew symmetric matrix of

iie given by

0 0

0 0

0 0 0

ie

i

ie ie

Muhammad Ushaq 42

Calculation of Earth Referenced Velocity

0 0Ti

ie ie

ir is updated by solving following equation

l

l l

i iii i br C V

Ti i i i

x y zr r r r

Muhammad Ushaq 43

Calculate (0) (0) l lli ii

f GV

Measure (1) (2) (1), b b b

ib ib iband f i.e. the outputs of gyros and accelerometers

Update Q, Compute (1), l l li i i b

b b ibC f C f

Compute ( i

xr , i

yr and i

zr ), iG , and l li i i

iG C G

Calculate (1) (1) l l li i i

V f G

Update Velocity (1)li

V and position vector components , ,l l li i i

x y zr r r & , ,i i i

x y zr r r

Update h, , , , ,

(0) (2), b b

ib ib (0) (1) (0) (1), l l l li i i i

V V V V

Measure initial readings of Gyros and accelerometers i.e. (0)b

ib and (0)b

ibf

Input 0 0 0 0, , and h

Set the values ofie , Re, e, , J2, Tf, t

Initial Alignment: 1 (0)i

bC , Q(0), 1 (0)i

V

Calculate initial , , l l li i ii i

i iC G G C G , ,i i i

x y zr r r

Muhammad Ushaq 44

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Mechanization of Inertial Navigation System in Inertially Stable Frame of Reference