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Inertial Navigation Systems
Muhammad Ushaq
0092-322-2992772
Mechanization of General Navigation Equations
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.
Review of 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
Review of Accelerometer
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
Realization of Two-dimensional Navigation
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
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
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
( ) ( )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
General Navigation Equations
14
32
3.9
860306
10
(/se
c)
m
22
2(
)(
)(
)i
ii
xy
zR
rr
r
Muhammad Ushaq 20
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations
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
General Navigation Equations