review of essential mathematics and basic concepts -navigation systems - ushaq

Inertial Navigation Systems

Muhammad Ushaq

Institute of Space [email protected]

0092-322-2992772

Review of Essential MathematicsFor Inertial Navigation Systems

mailto:[email protected]

Navigation Algorithm

Navigation algorithms involve various coordinate frames and the

mutual transformation of coordinates.

Inertial sensors measure translational acceleration and angular

rotation of the body (or platform) with respect to an inertial frame

which is resolved in the host platform’s body frame.

This information is further transformed to a navigation frame.

Review of Essential Mathematics for INS (Muhammad Ushaq)2

01-Oct-15

Vector Representation

Representation of a Vector in a Coordinate System

i

j

k

X

r

O rx ry

Z

Y

rz

Cos

Cos

Cos

x

y

z

r r

r r

r r

x y zr i r j r kr

If 1r x y zr r r r Cos Cos Cos


01-Oct-15

Vectors Dot Product

Let we have two vector a and b defined in same frame as follows:

x y za i a j a ka

x y zb i b j b kb

x x

x y z x y z x y z y x y z y

z z

x x y y z z

b a

a b a i a j a k b i b j b k a a a b b b b a

b a

a b b a a b a b a b

1, 1, 1, 0, 0, 0i i j j k k i j j k i k

2 2 2

i j k i j k

1

r r Cos Cos Cos Cos Cos Cos

Cos Cos Cos

Inner Product or Dot Product


01-Oct-15

Vectors Cross Product

0, 0, 0

, , , , ,

i i j j k k

i j k j k i k i j j i k k j i i k j

kajaiaa zyx

kbjbibb zyx

( ) ( )

( ) ( ) ( )

x y z x y z

x y x z y x y z z x z y

y z z y z x x z x y y x

a b a i a j a k b i b j b k

a b k a b j a b k a b i a b j a b i

a b a b i a b a b j a b a b k

Cross Product


01-Oct-15

Vectors Cross Product (Cont)

0

0

0

y z z y z y x

z x x z z x y

x y y x y x z

a b a b a a b

a b a b a b a a b

a b a b a a b

0

0

0

z y x

z x y

y x z

a a b

a b a a b Ab

a a b

ABCddcba Similarly

0

0 is the skew symmetric matrix for the vector a=

0

z y x

z x y

y x z

a a a

A a a a

a a a


01-Oct-15

Synthesis of Motion

Let a point P is moving with respect to a frame “m” which is moving with

respect to a fixed frame “ f ”.

Frame “m” has two types of velocities i.e. translational and

rotational as well as angular velocity

( ) ( )fp f fp f fp f fm f fm f fm f mp m mp m mp mx i y j z k x i y j z k x i y j z k

Absolute Position = Relative Position + Following Position

fm


01-Oct-15

Synthesis of Motion (Cont)

d d d ( ) ( ) ( )

dt dt dt

m m mfp f fp f fp f fm f fm f fm f mp m mp m mp m mp mp mp

i j kV i V j V k V i V j V k V i V j V k x y z

( ) ( ( )fm fmfp f fp f fp f fm f f f mp m mp m mp m mp m mp m mp mV i V j V k V i V j V k V i V j V k x i y j z k

( ) ( ) ( )fp f fp f fp f fm f fm f fm f mp m mp m mp m fm mp m mp m mp mV i V j V k V i V j V k V i V j V k x i y j z k

a fm mp mpV V V r mp mp m mp m mp mr x i y j z k

Taking derivative on both sides of prev eq we get velocity

Derivative of unit vectors in fixed frame is zero. But derivative of unit

vectors for moving frame exists

We can take out from last bracket

Whereas


01-Oct-15

Synthesis of Motion (Cont)

fp fm mp mpV V V r

( )fp fm mp mp mp mp mpV V V V r V r

2fp fm mp mp mp mpV V V V r r

fmV

mpV

2 mpV

mpr

mpr

Taking derivative

: Translational Acceleration

: Following Translational Acceleration

: Coriolis Acceleration

: Tangential Acceleration

: Centripetal/centrifugal Acceleration


01-Oct-15

Introduction to Geodetic Datums

Geodetic datums define the size and shape of the earth and

the origin and orientation of the coordinate systems used to

map the earth.

Hundreds of different datums have been used to frame

position descriptions since the first estimates of the earth's size

were made by Aristotle.

Datums have evolved from those describing a spherical earth

to ellipsoidal models derived from years of satellite

measurements


01-Oct-15

History of World Geodetic Datums


01-Oct-15

Ellipsoid Semi-major axis 1/flattening

Airy 1830, 6377563.396 299.3249646

Modified Airy 6377340.189 299.3249646

Australian National 6378160 298.25

Bessel 1841 (Namibia) 6377483.865 299.1528128

Bessel 1841 6377397.155 299.1528128

Clarke 1866, 6378206.4 294.9786982

Clarke 1880, 6378249.145 293.465

Everest (India 1830)" 6377276.345 300.8017

Everest (Sabah Sarawak) 6377298.556 300.8017

Everest (India 1956) 6377301.243 300.8017

Everest (Malaysia 1969) 6377295.664 300.8017

Everest (Malay. & Sing) 6377304.063 300.8017

Everest (Pakistan) 6377309.613 300.8017

Modified Fischer 1960 6378155 298.3

Helmert 1906 6378200 298.3

Hough 1960 6378270 297

Indonesian 1974 6378160 298.247

International 1924 6378388 297

Krassovsky 1940 6378245 298.3

GRS 80 6378137 298.257222101

South American 1969 6378160 298.25

WGS 72 6378135 298.26

WGS 84 6378137 298.257223563


WGS-84 and Shape of Earth

01-Oct-15

WGS-84 and Shape of Earth

WGS84 is an Earth-centered, Earth-fixed terrestrial reference

system and geodetic datum. It is based on a consistent set of

constants and model parameters that describe the Earth's size,

shape, gravity and geomagnetic fields.

WGS84 is the standard U.S. Department of Defense definition of a

global reference system for geospatial information


01-Oct-15

Earth Parameters (WGS-84)


Semi-major axis (Equatorial Radius) a 6,378,137.0m

Reciprocal flattening 1/f 298.257223563

Earth’s rotation rate ωe 7.292115 x 10-5

Gravitation Constant GM 3.986004418 x 1014m3/s2

Flatness 0.00335281

Semi-minor axis b=a(1-f) 6356752.3142m

Eccentricity

Mass of earth (including atmosphere M 5.9733328 x 1024 kg

Theoretical (normal) gravity at equator γe 9.7803267714 m/s2

Theoretical (normal) gravity at poles γp 9.8321863685 m/s2

Mean Value of Theoretical (normal) gravity γ 9.7976446561 m/s2

a bf

a

2 2

22 0.0818191908426

a be f f

a

01-Oct-15

Shape of Earth

Geometric Figure of Earth—Geoid:

The equipotential surface (surface of constant

gravity) best fitting the average sea level. It can be

thought of as the idealized mean sea level

extended over the land portion of the globe.

Reference Ellipsoid—Ellipsoid

The mathematically defined surface approximates

the geoid by an ellipsoid that is made by rotating

an ellipse about its minor axis, which is coincident

with the mean rotational axis of the Earth. The

center of the ellipsoid is coincident with the

Earth’s center of mass. Its shape is defined by

two geometric parameters called the semi major

axis (a) and the semi minor axis (b).


01-Oct-15

Local Radius of Curvatures

The normal radius RN is defined for the east-west direction or

radius of curvature of the prime vertical. RN governs the rate at which the longitude changes as a navigating platform moves on or near the surface of the Earth.

The meridian radius of curvature RM is defined for the north-south direction and is the radius of the ellipse. RM governs the rate at which the latitude changes as a navigating platform moves on or near the surface of the Earth.


01-Oct-15

Rectangular and Geodetic Coordinates

Rectangular Coordinates in ECEF

T

e e ex y z

Geodetic Coordinates in the ECEF Frame h

Latitude is the angle in the meridian plane

from the equatorial plane to the ellipsoidal normal at the point of interest

Longitude is the angle in the equatorial plane from the prime meridian to the projection of the point of interest onto the equatorial plane

Altitude h is the distance along the ellipsoidal normal, between the surface of the ellipsoid and the point of interest


01-Oct-15

Geodetic Coordinates Rectangular Coordinates

1 1

2 2 2

2

tan tan

1sin

e N e

e e eN

eN

y R h z

x b x yR h

a

zh R


01-Oct-15

Earth Sidereal and Solar Day


01-Oct-15

The duration of a solar day is 24h, the time taken between successive

rotations for an Earth-fixed object to point directly at the Sun. The Sidereal day

represents the time taken for the Earth to rotate to the same orientation ins

space and is slightly shorter duration than the solar day, 23h, 56m, 4.1s. The

Earth rotates thought one geometric revolution each Sidereal day, not in 24h,

which accounts for the slightly strange value of Earth’s rate.

Variation of gravitational field over the Earth


In order to extract the precise estimates of true acceleration needed for very accurate navigation in the vicinity of the Earth, it is necessary to model accurately the Earth's gravitational field.

Gravity Anomalies:

Variation between the mass attractions of the Earth Variation in gravity vector the centrifugal acceleration being a function

of latitude. Variation with position on the Earth because of the in-homogenous

mass distribution of the Earth The deflection of the local gravity vector from the vertical are expressed

as angular deviations about the north and east axes of the local geographic frame as

[ , , ]T

lg g g g

Where is the meridian deflection and is the deflection perpendicular to

the meridian

01-Oct-15


The magnitude of the gravity vector with latitude at sea level (h = 0)

and its rate of change with altitude above ground is given as:

3 2 6 2(0) 9.780318 (1 5.3024 10 sin 5.9 10 sin 2 )g

3 2(0) 0.0000030877 (1 1.39 10 sin )d

gdh

For many applications, it’s sufficient to assume that the variation of gravity

with latitude is as follows:

2

0( ) (0) / (1 / )g h g h R

Variation of gravitational field over the Earth

01-Oct-15

Reference Frames

A frame of reference consists of an abstract coordinate system and the set of

physical reference points that uniquely fix (locate and orient) the coordinate

system and standardize measurements.

Acceleration, velocity, position and attitude are expressed as vectors.

These vectors need to be expressed with respect to some preselected & predefined

reference coordinate system.

The definition of a suitable coordinate system employed in INS requires:

Knowledge of the motion of the earth.

The initial orientation of the reference coordinate frame

Initial position, initial velocity and orientation (attitude)

These coordinate frames are orthogonal, right-handed Cartesian frames and differ

in the location of the origin, the relative orientation of the axes, and the relative

motion between the frames.


01-Oct-15

Reference Frames

Greenwich meridian

Inertial reference

meridian

iet

ix

ex

eziz

ie

Local meridian

iy

gx

gy

wxwy

N

0( 90 )ey E

S

00

00

Equatorial plane

gzwz

c

Orthogonal, right handed, co-ordinate frame or axis set

In many inertial navigation systems latitude , longitude , and altitude hare the desired outputs, and consequently the system should bemechanized to yield these outputs directly.

There are generally fol fundamental coordinate frames of interest fornavigation:

i. True inertial frameii. Earth-centered inertial frameiii. Earth-centered earth-fixed frameiv. Local Level Framev. Body framevi. Wander azimuth framevii. Navigation frameviii. Platform Frameix. Computational Framex. True Frame


01-Oct-15

The Earth Centered inertial frame(i-frame)

Origin : the centre of the Earth.

Axes :non – rotating with respect to the fixed stars

Oxi, Oyi, Ozi

Ozi :Coincident with the Earth’s polar axis


XiOi

Zi

Yi

01-Oct-15

The Earth frame(e-frame)

origin : the centre of the Earth.

axes: fixed with respect to the Earth.

Oxe, Oye, Oze

Oxe: along the intersection of the plane of the Greenwich meridian with the Earth’s equatorial plane.

Oze: along the Earth’s polar axis.

e-frame rotates with respect to i-frame at a rate Ώie about the axis Ozi


Xi

Xe

Oe

Zi Ze

Yi

Ye

01-Oct-15

The Local geographic frame(n-frame)

A local geographic frame

origin: the location of navigation system

axes:

Oxn : east

Oyn : north

Ozn : local vertical up


01-Oct-15

Greenwich Meridian

eX

eZie

Local Meridian

gx

gyN

( )eY E

S

0o

0o

gz

:E

:U:N

ie

The wander azimuth frame(w-frame)

Used to avoid the singularities inthe computation which occur atthe poles of the navigationframe.

Locally level frame

Rotated through the wanderangle α about the local vertical

with respect to the n-frame


Xe

Oe

Ot 、Ow

Ze

Yt

Ye

Yw

Xi

Yi

01-Oct-15

The body frame(b-frame)

An orthogonal axis set which is aligned

with the roll, pitch and yaw axes of the

vehicle in which the navigation system

is installed.


ForwardRight

Down

Xy

z

01-Oct-15

Frame Transformation

The techniques for transforming a vector from one coordinate frameinto another.

Various mathematical representations can be used to define the attitudeof a body with respect to a co-ordinate reference frame Or the attitudeof one frame with respect to another frame frame.

The parameters associated with each method may be stored within acomputer and updated as the vehicle rotates using the measurementsof turn rate provided by the strapdown gyroscopes. Three attituderepresentations are described here, namely:


01-Oct-15

Representation of Attitude

Direction Cosine Matrix is a 3 x 3 matrix, the columns of whichrepresent unit vectors in one axes projected along the referenceaxes.

Euler Angles: A transformation from one co-ordinate frame toanother is defined by three successive rotations about different axestaken in sequence. The three angles correspond to the angles whichwould be measured between a set of mechanical gimbals, whichsupports a stable platform, where the axes of the stable platformrepresent the reference frame, and with the body being attached viaa bearing to the outer gimbal.

Quaternion attitude representation allows a transformation fromone co-ordinate frame to another to be effected by a single rotationabout a vector defined in the reference frame. The quaternion is afour-element vector representation, the elements of which arefunctions of the orientation of this vector and the magnitude of therotation.


01-Oct-15

The Direction Cosine Matrix (DCM)

11 12 13

21 22 23

31 32 33

b

a

C C C

C C C C

C C C

Direction Cosine Matrix, maps the three

components of a vector resolved in one frame into

the same vector's components resolved into the

other frame.

Achieved by the computation of direction cosines

between each axis of one frame and every axis of

another one or vector dot products between the

axes.

cosij i j ijC i i i j

Each element Cij of the DCM represents the cosine of the angle or a projection

between the ith axis of the a-frame and the jth axis of the b-frame.

a b

a

b

x X

y C Y

z Z

a a b

bR C R


01-Oct-15

DCM Differential Equation

Let us denote transformation matrix from frame a to b at time t as

( )b

aC t and that at time ( )t t as ( )b

aC t t . Let the b frame at time t is denoted

by ( )b b bX Y Z t and that at time ( )t t it is denoted by ( )b b bX Y Z t t . Let during

this very small span of time following rotations take place in body frame.

' ''

b b b

( ) ( ) X Y Z

yx zb b b b b bX Y Z t X Y Z t t

about about about

Hence the corresponding transformation matrix at time ( )t t will be given

as follows

( ) ( ) 0 ( ) 0 ( ) 1 0 0

( ) ( ) 0 0 1 0 0 ( ) ( )

0 0 1 ( ) 0 ( ) 0 ( ) ( )

( ) ( )

z z y y

z z x x

y y x x

b b

a g

Cos Sin Cos Sin

Sin Cos Cos Sin

Sin Cos Sin Cos

C t t C t


01-Oct-15


As x , y

z are very small (because Δt is assumed to be very short

time) so Cosines of these angles are equal to unity and Sines are equal

to the angles themselves. Using this trigonometric identity we have

following

1 0 1 0 1 0 0

( ) 1 0 0 1 0 0 1 ( )

0 0 1 0 1 0 1

z y

b b

a z x a

y x

C t t C t

0

( ) ( ) 0 ( )

0

z y

b b b

a a z x a

y x

C t t C t C t

Therefore

0

( ) ( ) 0 ( )

0

z y

b b b

a a z x a

y x

C t t C t C t


01-Oct-15

Dividing Previous equation by t and taking limit 0t , we have following

0 0

0 0 0

0 0

0 lim lim

( ) ( )lim lim 0 lim ( )

lim lim 0

yzt t

b bba a xz

t t t a

y xt t

t t

C t t C tC t

t t t

t t

We know that 0

( ) ( )lim ( )

b bba a

t a

C t t C tC t

t

Hence we have

0 0

0 0

0 0

0 lim lim

( ) lim 0 lim ( )

lim lim 0

yz

t t

b bxz

a t t a

y x

t t

t t

C t C tt t

t t



01-Oct-15


0 0 0lim , lim and lim

yx z

t t t

t t t

are the components of angular rate of b frame

with respect to a frame during the time from t to t t

So we can write Equation as

0

( ) 0 ( )

0

z y

z x

y x

b b

ab ab

b b b b

a ab ab a

b b

ab ab

C t C t

0

( ) 0 ( )

0

b b

gbz gby

b b b b

g gbz gbx g

b b

gby gbx

C t C t

( ) ( )b

a

b b

aabC t C t

Where b

ab is the skew symmetric matrix corresponding to

b

ab .


01-Oct-15

DCM Differential Equations

11 12 13

21 22 23

31 32 33

21 31 22 32 23 33

31 11 32 12 33 1

0

( ) 0

0

( )

b b

gbz gby

b b b

g gbz gbx

b b

gby gbx

b b b b b b

gbz gby gbz gby gbz gby

b b b b b b b

g gbx gbz gbx gbz gbx gbz

C C C

C t C C C

C C C

C C C C C C

C t C C C C C C

3

11 21 12 22 13 23

11 21 31 12 22 32 13 23 33

21 31

In component form (we have 9 differential equations)

, ,

b b b b b b

gby gbx gby gbx gby gbx

b b b b b b

gbz gby gbz gby gbz gby

b

gbx

C C C C C C

C C C C C C C C C

C C

11 22 32 12 23 33 13

31 11 21 32 12 22 33 13 23

, ,

, ,

b b b b b

gbz gbx gbz gbx gbz

b b b b b b

gby gbx gby gbx gby gbx

C C C C C C C

C C C C C C C C C


01-Oct-15

Euler Angles

Three ordered right-handed rotations

Determine the orientation of the body

Euler angles ( , , ) correspond to the conventional roll-

pitch-yaw angles

Not uniquely defined

The rotation order once defined must be used consistently


01-Oct-15

Euler Angles Transformation

A transformation from one co-ordinate frame to another can be carried

out as three successive rotations about different axes. For instance, a

transformation from reference axes to a new co-ordinate frame may be

expressed as follows:

Rotate through angle about reference z-axis

Rotate through angle about new y-axis

Rotate through angle about new x-axis

Where , and are referred to as the Euler rotation angles. Euler

angles correspond to the angles which would be measured by angular

pick-offs between a set of three gimbals in a stable platform inertial

navigation system.


01-Oct-15

z

y,

z,

x,

x

yo

Rotation about the Z axis of XYZ through

an angle results in a new set of axes

(X’,Y’,Z’)

cos sin 0

sin cos 0

0 0 1

X X

Y A Y

Z Z

X

Y

Z



01-Oct-15

o

x’

y’

z’

x’’

y’’

z’’

Rotating the (X’,Y’,Z’) about the y’

axis through and angle results

in a new (X’’,Y’’,Z’’) frame

cos 0 sin

0 1 0

sin 0 cos

X X

Y B Y

Z Z

X

Y

Z



01-Oct-15

1 0 0

0 cos sin

0 sin cos

x X

y D Y

z Z

X

Y

Z

Rotation of (X’’,Y’’,Z’’)

about X’’ axis through an

angle

results in to the

final frame (x,y,z)



01-Oct-

15

2 2

x X X

y D B A Y C Y

z Z Z

cos cos cos sin sin

sin cos sin sin cos sin sin sin cos cos cos sin

sin sin cos sin cos cos sin sin sin cos cos cos

X

Y

Z

where the matrix [C] is the product of [D]. [B],

and [A] in that order in terms of the angles Φ, θ, Ψ.

[C] represents the Euler angle transformation

matrix.



01-Oct-15

Body Frame to Navigation Frame

Orientation:

Origin: Aircraft center of mass

Roll ( ) Pitch ( ) Yaw ( )


xb： longitudinal direction

yb ： right wing

zb ： down

n

n

n

x N

y E

z D

b n

b

b n n

b n

x x

y C y

z z

01-Oct-15


Local Geographic

Navigation axes

Inertial axes

Greenwich

meridian

N

ED

exey

ez iz

ix

iy

L

o

Equatorial

plane

Local

meridian

plane

Earth

axes

01-Oct-15

Angle b/w the project of longitudinal axis of body frame on horizontal plane and

north

Positive : the aircraft nose is rotating from north to east

Angle b/w lateral axis and its projection on horizontal plane

positive： right wing dips below the horizontal plane.

negative ：bring yb into the horizontal plane

Angle between longitudinal axis of body frame and its projection on the

horizontal plane

positive ：the nose of the aircraft is elevated above the horizontal plane

roll angle：

pitch angle:

Yaw angle:


01-Oct-15

n

n

n

x N

y E

z D

b n

b

b n n

b n

x x

y C y

z z


01-Oct-15

The order of rotation :

(1) through about the down or zn-axis

(2) through θ about yn’ axis

(3) through about xn” axis (xb axis)


01-Oct-15

For platform INS : Cnb = Cp

b.

11 12 13

21 22 23

31 32 33

b

n

C C C

C C C C

C C C


01-Oct-15

The direction cosine elements are as follows

11

12

13

cos cos

sin cos

sin

C

C

C

21

22

23

cos sin sin sin cos

sin sin sin cos cos

cos sin

C

C

C

31

32

33

cos sin cos sin cos

sin sin cos cos sin

cos cos

C

C

C


01-Oct-15

23

33

sin cos sintan

cos cos cos

C

C

12

11

sin cos sintan

cos cos cos

C

C

13

2 2

13

sin sintan

cos1 1 sin

C

C


01-Oct-15

1 13

2

13

tan ( )1

C

C

1 12

11

tan ( )C

C

1 23

33

tan ( )C

C


01-Oct-15

Earth-fixed to Navigation

Origin: System Location (INS)

Orientation

xn : up

yn : east

zn : north.


01-Oct-15

is realized by two rotations

(1). Through the angle λ about ze

(2). Through the angle about ye ’

(3). Rotation about the ye” (yn) axis through the angle -90o

n

eC


01-Oct-15

(I)

cos sin 0

sin cos 0

0 0 1

e e

e e

e e

x x

y y

z z


01-Oct-15

(II)

cos 0 sin

0 1 0

sin 0 cos

e e

e e

e e

x xL L

y y

L Lz z


01-Oct-15

cos cos cos sin sin

sin cos 0

sin cos sin sin cos

e

e

e

xL L L

y

L L L z

cos 0 sin cos sin 0

0 1 0 sin cos 0

sin 0 cos 0 0 1

n e

n e

n e

x xL L

y y

L Lz z


01-Oct-15

or

n e

n

n e e

n e

x x

y C y

z z


01-Oct-15

Rotation about the ye’’ (yn) axis through the

angle -900

0 0 1

0 1 0

1 0 0

n e

n e

n e

x x

y y

z z

(III)


01-Oct-15

0 0 1 cos cos cos sin sin

0 1 0 sin cos 0

1 0 0 sin cos sin sin cos

n e

n e

n e

x xL L L

y y

L L Lz z

sin cos sin sin cos

sin cos 0

cos cos cos sin sin

e

e

e

xL L L

y

L L L z


01-Oct-15

xb : right wing

yb : longitudinal direction

zb :up of aircraft

Navigation frame

xn :east yn : north zn : up

''' about ,about , about ,' ' ' '' '' '' pn nYZ X

n n n n n n n n n b b bx y z x y z x y z x y z

Chinese Conventions


01-Oct-15

Inertial FrameECI

(i-Frame)

Earth CenteredEarth-fixed Frame

(e-Frame)

Wander-azimuthFrame

(c-Frame)

NavigationalFrame

(n-Frame)

PlatformFrame

(p-Frame)

Body Frame(b-Frame)

, ,i i ix y z

, ,e e ex y z

, ,n n nx y z

, ,c c cx y z

, ,p p px y z

e

iCn

eC

c

nC

p

cC

b

pC

,

, ,

b e n c p b

i i e n c pC C C C C C


01-Oct-15

Transformation between Frame

i i iX Y Z To e e eX Y Z

e e eX Y Z Frame is related with i i iX Y Z by a single positive rotation about the

Zi, axis through an angle ie t

Whereas o -5360

15.04106874 /h=7.2921159 10 rad/s23 [56 (4.9 / 600] / 60

ie

the vector ie is expressed with respect to the ECEF frame as

0

0ie

ie

cos sin 0

sin cos 0

0 0 1

cos sin 0

sin cos 0

0 0 1

e ie ie i

e ie ie i

e i

i ie ie e

i ie ie e

i e

x t t x

y t t y

z z

x t t x

y t t y

z z


01-Oct-15

From Earth fixed frame to Geographic Frame

/

e

90 90( )

Z axis X axis

o o

e e e g g gX Y Z X Y Z ENUabout about

1 0 0 ( 90) ( 90) 0

0 (90 ) (90 ) ( 90) ( 90) 0

0 (90 ) (90 ) 0 0 1

g

e

Cos Sin

C Cos Sin Sin Cos

Sin Cos

By using following trigonometric relations

(90 ) ( ) , (90 ) ( ), ( 90) ( ) , ( 90) ( )Cos Sin Sin Cos Cos Sin Sin Cos

We can simplify geC as follows

1 0 0 0 0

0 0

0 0 0 1

g g

e e

Sin Cos Sin Cos

C Sin Cos Cos Sin C Sin Cos Sin Sin Cos

Cos Sin Cos Cos Cos Sin Sin



01-Oct-15

ECEF to Wander Azimuth Frame.

/

e g

90 90( )

Z axis Z (U) X axis

o o

e e e g g g w w wX Y Z X Y Z ENU X Y Zabout aboutabout

According to this sequence of rotation weC

will be formed as follows

( ) ( ) 0 1 0 0 ( 90) ( 90) 0

( ) ( ) 0 0 (90 ) (90 ) ( 90) ( 90) 0

0 0 1 0 (90 ) (90 ) 0 0 1

we

Cos Sin Cos Sin

Sin Cos Cos Sin Sin Cos

Sin Cos

C

0 0

0

0 0 1

w

e

Cos Sin Sin Cos

C Sin Cos Sin Cos Sin Sin Cos

Cos Cos Cos Sin Sin



01-Oct-15

we

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

0

0

0 0 1

w g

w g

w g

w g

w

w g g

w g

x Cos Sin x

y Sin Cos y

z z

x x

y C y

z z



01-Oct-15

From navigation frame (geographic) to body frame

g Z axis axis axisg g g g g g g g g b b b

g g

X Y Z X Y Z X Y Z X Y Zabout about X about Y

0 1 0 0 0

0 1 0 0 0

0 0 0 0 1

b

g

Cos Sin Cos Sin

C Cos Sin Sin Cos

Sin Cos Sin Cos

( )

bg

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

Cos Sin Cos Cos Sin

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

C



01-Oct-15

Propagation of Euler Angles with time

, and are the the gimble angles read from the gimbal pick-offs and

, and the gimbal rates. The gimbal rates are related to the body

rates as fol:

3 3 2

0 0

0 0

0 0

n

x

n

y

n

z

C C C

By putting respective values we ca get fol

1( )( sin cos )cos

cos sin

tan ( sin cos )

y z

y z

x y z

The eqs can be solved in a strapdown system to update the Euler rotations of the body with respect to the reference frame. However, their

use is limited since the solution of the , and become indeterminate

when 90o Review of Essential Mathematics for INS (Muhammad Ushaq)67

Euler Angles Differential Equation

01-Oct-15

Quaternions

The quaternion attitude representation is afour-parameter representation based on theidea that a transformation from one co-ordinate frame to another may be acheivedby a single rotation about a vector definedin the reference frame.

The quaternion is a four-element vectorrepresentation, the elements of which arefunctions of the orientation of this vectorand the magnitude of the rotation


01-Oct-15

)2/sin()/(

)2/sin()/(

)2/sin()/(

)2/cos(

3

2

1

0

z

y

x

q

q

q

q

q＝

= ComponentS of the angle vector, ,x y z

= magnitude of or magnitude of rotation


Quaternions

01-Oct-15

kqjqiqqq

3210

kqjqiqqq

3210

kpjpippp

3210

. 1, . 1, . 1

. , . , .

.i , . , .

i i j j k k

i j k j k i k i j

j k k j i i k j


Quaternions

01-Oct-15

pq

33221100 ,,, pqpqpqpq

pq

2)

kpqjpqipqpq

)()()( 33221100


Quaternions

1) Equality

Addition/Subtraction

01-Oct-15

3)

qa

kaqjaqiaqaq

3210

q

4)

kqjqiqq

3210

0q

5)

kjiq

0000


Quaternions

Multiplication by scalar

Negative Quaternion

Zero Quaternion

01-Oct-15

pq

6)

0 1 2 3 0

1 0 3 2 1

2 3 0 1 2

3 2 1 0 3

q q q q p

q q q q p

q q q q p

q q q q p

＝


Quaternions

Multiplication of two quaternions

01-Oct-15

q p p q

( ) ( )q p M q p M

aq qa ( ) ( )ab q a bq

( )a b q aq bq

( )a q p aq ap

( ) ( )qM p q Mp


01-Oct-15

( )q M p qM qp

( )q M p qp Mp

qp pq


Quaternions

01-Oct-15

Conjugate number and Norm

qqQ

0 q complex part

0Q q q

1 2 1 2( ) ( )Q Q Q Q

1 2 2 1( )Q Q Q Q


Quaternions

01-Oct-15

The norm of quaternion is defined as N

2

3

2

2

2

1

2

0 qqqqQQQQN

If N=1,

Q

is defined as unit quaternion

( )( )QMN QM QM QMM Q

M M Q MQN Q QQ N N N


Quaternions

01-Oct-15

Inverse and division

Q-1=Q*/N

NQ-1=1/NQ

only for those N≠0, Q-1 exist.

For unit quaternion,

•Q-1= Q*/N= Q*

• Q-1Q=1


01-Oct-15

kzjyixr bbbb

kzjyixR bbbb

0

QQRR bn

wherekqjqiqqQ

3210

kqjqiqqQ

3210

kzjyixR nnnn

0


Transformation of Vector b to n Frame

01-Oct-15

Propagation of a quaternion with time

1

2

b

nbQ QP b

nbz

b

nby

b

nbx

b

nbP ,,,0


0 0 1 2 3

1 1 0 3 2

2 2 3 0 1

3 3 2 1 0

0

1

2

b

nbx

b

nby

b

nbz

q q q q q

q q q q q

q q q q q

q q q q q

0

1

2

3

0

01

02

0

b b b

nbx nby nbz

b b b

nbx nbz nby

b b b

nby nbz nbx

b b b

nbz nby nbx

q

q

q

q

01-Oct-15

Inter-conversion b/w direction cosines ,Euler angles , quaternions

2

3

2

2

2

1

2

011 coscos qqqqC

)(2sincos 302112 qqqqC

)(2sin 203113 qqqqC


01-Oct-15

21

1 2 0 3

sin sin cos cos sin

2( )

C

q q q q

22

2 2 2 2

0 1 2 3

sin sin sin cos cosC

q q q q

)(2cossin 013223 qqqqC



01-Oct-15

31

1 3 0 2

cos sin cos sin sin

2( )

C

q q q q

32

2 3 0 1

cos sin sin sin cos

2( )

C

q q q q

2

3

2

2

2

1

2

033 coscos qqqqC



01-Oct-15

Quaternions expressed in terms of direction cosines

2/1

2322110 )1(2

1CCCq

)(4

12332

0

1 CCq

q

)(4

13113

0

2 CCq

q

)(4

11221

0

3 CCq

q


01-Oct-15

In terms of Eular angles Ψ, θand Φ

2sin

2sin

2sin

2cos

2cos

2cos0

q

2cos

2cos

2sin

2cos

2sin

2sin1

q

2cos

2sin

2sin

2cos

2cos

2sin2

q

2cos

2sin

2sin

2cos

2cos

2sin3

q



01-Oct-15

2 2 2 2

0 1 2 3 1 2 0 3 1 3 0 2

2 2 2 2

1 2 0 3 0 1 2 3 1 3 0 1

2 2 2 2

1 3 0 2 2 3 0 1 0 1 2 3

2( ) 2( )

2( ) 2( )

2( ) 2( )

b

a

q q q q q q q q q q q q

C q q q q q q q q q q q q

q q q q q q q q q q q q



01-Oct-15


01-Oct-15

review of essential mathematics and basic concepts -navigation systems - ushaq

Documents