review of essential mathematics and basic concepts -navigation systems - ushaq
TRANSCRIPT
Inertial Navigation Systems
Muhammad Ushaq
Institute of Space [email protected]
0092-322-2992772
Review of Essential MathematicsFor Inertial Navigation Systems
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.
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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
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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
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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
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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
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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
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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
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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
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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
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History of World Geodetic Datums
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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
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WGS-84 and Shape of Earth
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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
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Earth Parameters (WGS-84)
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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
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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).
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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.
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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
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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
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Earth Sidereal and Solar Day
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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
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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
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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.
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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
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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
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XiOi
Zi
Yi
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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
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Xi
Xe
Oe
Zi Ze
Yi
Ye
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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
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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
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Xe
Oe
Ot 、Ow
Ze
Yt
Ye
Yw
Xi
Yi
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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.
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ForwardRight
Down
Xy
z
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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:
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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.
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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
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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
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DCM Differential Equation
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
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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
DCM Differential Equation
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DCM Differential Equation
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 .
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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
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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
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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.
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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
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Euler Angles Transformation
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
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Euler Angles Transformation
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)
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Euler Angles Transformation
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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.
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Euler Angles Transformation
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Body Frame to Navigation Frame
Orientation:
Origin: Aircraft center of mass
Roll ( ) Pitch ( ) Yaw ( )
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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
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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:
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n
n
n
x N
y E
z D
b n
b
b n n
b n
x x
y C y
z z
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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)
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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
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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
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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
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1 13
2
13
tan ( )1
C
C
1 12
11
tan ( )C
C
1 23
33
tan ( )C
C
Review of Essential Mathematics for INS (Muhammad Ushaq)51
01-Oct-15
Earth-fixed to Navigation
Origin: System Location (INS)
Orientation
xn : up
yn : east
zn : north.
Review of Essential Mathematics for INS (Muhammad Ushaq)52
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
Review of Essential Mathematics for INS (Muhammad Ushaq)53
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
Review of Essential Mathematics for INS (Muhammad Ushaq)54
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
Review of Essential Mathematics for INS (Muhammad Ushaq)55
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
Review of Essential Mathematics for INS (Muhammad Ushaq)56
01-Oct-15
or
n e
n
n e e
n e
x x
y C y
z z
Review of Essential Mathematics for INS (Muhammad Ushaq)57
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)
Review of Essential Mathematics for INS (Muhammad Ushaq)58
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
Review of Essential Mathematics for INS (Muhammad Ushaq)59
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
Review of Essential Mathematics for INS (Muhammad Ushaq)60
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
Review of Essential Mathematics for INS (Muhammad Ushaq)61
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
Review of Essential Mathematics for INS (Muhammad Ushaq)62
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
Review of Essential Mathematics for INS (Muhammad Ushaq)63
Transformation between Frame
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
Review of Essential Mathematics for INS (Muhammad Ushaq)64
Transformation between Frame
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
Review of Essential Mathematics for INS (Muhammad Ushaq)65
Transformation between Frame
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
Review of Essential Mathematics for INS (Muhammad Ushaq)66
Transformation between Frame
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
Review of Essential Mathematics for INS (Muhammad Ushaq)68
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
Review of Essential Mathematics for INS (Muhammad Ushaq)69
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
Review of Essential Mathematics for INS (Muhammad Ushaq)70
Quaternions
01-Oct-15
pq
33221100 ,,, pqpqpqpq
pq
2)
kpqjpqipqpq
)()()( 33221100
Review of Essential Mathematics for INS (Muhammad Ushaq)71
Quaternions
1) Equality
Addition/Subtraction
01-Oct-15
3)
qa
kaqjaqiaqaq
3210
q
4)
kqjqiqq
3210
0q
5)
kjiq
0000
Review of Essential Mathematics for INS (Muhammad Ushaq)72
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
=
Review of Essential Mathematics for INS (Muhammad Ushaq)73
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
Review of Essential Mathematics for INS (Muhammad Ushaq)74
01-Oct-15
( )q M p qM qp
( )q M p qp Mp
qp pq
Review of Essential Mathematics for INS (Muhammad Ushaq)75
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
Review of Essential Mathematics for INS (Muhammad Ushaq)76
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
Review of Essential Mathematics for INS (Muhammad Ushaq)77
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
Review of Essential Mathematics for INS (Muhammad Ushaq)78
01-Oct-15
kzjyixr bbbb
kzjyixR bbbb
0
QQRR bn
wherekqjqiqqQ
3210
kqjqiqqQ
3210
kzjyixR nnnn
0
Review of Essential Mathematics for INS (Muhammad Ushaq)79
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
Review of Essential Mathematics for INS (Muhammad Ushaq)80
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
Review of Essential Mathematics for INS (Muhammad Ushaq)81
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
Review of Essential Mathematics for INS (Muhammad Ushaq)82
Inter-conversion b/w direction cosines ,Euler angles , quaternions
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
Review of Essential Mathematics for INS (Muhammad Ushaq)83
Inter-conversion b/w direction cosines ,Euler angles , quaternions
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
Review of Essential Mathematics for INS (Muhammad Ushaq)84
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
Review of Essential Mathematics for INS (Muhammad Ushaq)85
Inter-conversion b/w direction cosines ,Euler angles , quaternions
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
Review of Essential Mathematics for INS (Muhammad Ushaq)86
Inter-conversion b/w direction cosines ,Euler angles , quaternions
01-Oct-15
Review of Essential Mathematics for INS (Muhammad Ushaq)87
01-Oct-15