Fundamentals of Strapdown
Inertial and GPS-Aided
Navigation
By
Professor Dominick Andrisani
Purdue University, West Lafayette, IN 47907-1282
andrisan@ecn.purdue.edu 765-494-5135
Tactical Imagery Geopositioning Workshop
March 12, 2002
Chantilly, VA
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Purposes of this talk
•To provide a tutorial overview of inertial
navigation systems (INS).
•Illustrate ideas with a 2-D navigator.
•Discuss inertial sensors (simple rate gyros and linear
accelerometers).
•Discuss characteristic errors in the INS.
•To demonstrate the need by the INS for altitude
aiding.
•To show how GPS aids the INS and leads to far
superior navigation accuracy.
3/12/02 - 2
Types of Inertial Navigation Systems (INS)
Inertial Platform based INS
Strapdown INS <<(emphasized here)
Aided Navigators of either type
Altitude-aided
Altitude and X-aided
Heading-aided
GPS aided
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Inertial Platform
Ref 4.
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Mechanization of Inertial Platform
Ref 4.
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Strapdown INS
Ref 4.
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Mechanization of Strapdown INS
Ref 4.
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Properties of Platforms
Advantages
Simpler gyros (platform rotates at small rates,
lower dynamic range).
High accuracy (North and East accelerometers do
not see a component of gravity).
Self alignment by gyro compassing.
Sensor calibration by platform rotations.
Disadvantages
Complexity and cost.
Gimbal magnetics (torquers must not leak
magnetic flux).
Reliability (bearings and slip rings tend to wear).
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Properties of Strapdown Systems
Advantages
Simple structure, low cost.
More rugged and lighter.
Reliability (no gimbal magnetics, no slip rings, no
bearings, electronics more reliable then
machinery).
Disadvantages
More difficult to align.
More difficult to calibrate.
Motion induced errors which can only be partly
compensated for.
Accelerometer errors (each accelerometer may feel
1 g from gravity).
Requires a computer that can perform coordinate
rotations in <.01 sec).
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Simple Example: Two Dimensional Motion
Xb, Yb, and Zb
are body fixed
XNED is northerly
YNED is easterly
ZNED is down
Yb=YNED
.
q
ZNED
Xb
Pitch angle
XNED
North Pole
Zb
South Pole
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Equations of Motion of the Aircraft
dQ/dt=vx/(Ro+h) + wy
dvx/dt =vxvy/(Ro+h) + fx
dvz/dt =-vz2/(Ro+h) + fz + g(h)
dx/dt =vx
dh/dt =-vz
where
fx=fxbcos(Q)+fzbsin(Q)
fz=-fxbsin(Q) + fzbcos(Q)
Q is pitch angle
vx is velocity in northerly direction
vz is velocity in down direction
x is northerly position
h is altitude (positive up)
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Inertial Sensors
Rate gyros measure the components of inertial angular
rate of the aircraft in the sensitive direction of the
instrument.
Linear accelerometers are used to measure the
components of aircraft linear acceleration minus the
components of gravity in its sensitive direction.
Newton’s Law for the aircraft is
F=ma=Faero+Fthrust+mg
Accelerometer measures
a-g=(Faero+Fthrust)/m=specific force
In this simple two-dimensional example, two linear
accelerometers and one rate gyro are used.
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A Single Axis Angular Rate Gyro
This device measures inertial angular rate about its
sensitive direction. Three of these arranged
orthogonally measure the components of the angular
velocity vector.
Ref 3.
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A Simple “Open Loop” Accelerometer
This device measures specific force= a-g=(Faero+Fthrust)/m.
They cannot distinguish between acceleration and gravity.
g
Ref 3.
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Simulation of Aircraft and INS
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Aircraft Simulation
Coriolis acceleration
Coriolis acceleration
Transport rate
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INS Simulation
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INS Simulation (Free integrator)
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INS Free Integrator
The free integrator will create the following types of
errors.
•For initial condition errors on x, the resulting xposition error will neither decay or grow.
•For initial condition errors on Vx, the resulting xposition errors will grow linearly with time.
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INS Simulation (Unstable Altitude Loop)
Unstable Altitude Loop
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INS Unstable Altitude Loop
The unstable altitude loop results because errors in altitude
means that there will be errors in the determination of the
acceleration of gravity.
This in turn will propagate into an error in vertical
acceleration which will be in the direction to drive the
altitude error further from the correct value. This is an
unstable mechanism since altitude error leads to greater
altitude error.
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INS Simulation (Schuler Pendulum)
Schuler Pendulum Loop
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Schuler Pendulum Loop
The Schuler pendulum loop creates dynamic errors that
oscillate with an 84 minute period.
The Schuler pendulum loop, while creating persistent
oscillations, does limit the growth of errors in velocity (Vx).
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The Schuler Pendulum
•Imagine we have a pendulum to provide a vertical reference.
•As we accelerate horizontally, the pendulum tilts, giving a
false vertical indication.
•Schuler showed that this would not occur with a pendulum
having a period of 84 minutes (a ball on a string with length
equal to the radius of the Earth has this period).
•Correcting an inertial system so that it does not tilt when
accelerated is known as Schuler tuning.
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Error Analysis Via Linearization
Nonlinear navigator equations of motion
dX/dt=f(X,U)
Error model
e=XINS-Xsimulation
u=UINS-Usimulation
Linear error equations of motion
de/dt=Ae+Bu
where
A=df/dX evaluated at the a reference
state X and input U)
B=df/dU evaluated at the a reference
state X and input U)
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Error Analysis Via Linearization, continued
Linear error equations of motion
de/dt=Ae+Bu with initial condition e(0)
System matrix A will have 5 poles (eigenvalues), two
complex poles for the Schuler Pendulum, two real poles
for the altitude modes (one unstable, one stable, equal
magnitude), and one pole at zero (X-pole).
The error system provides a useful way to study INS
error propagation using linear methods and as the basis
for designing Kalman filters to implement the various
aiding techniques (e.g. altitude aiding).
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Nonlinear Simulation of Aircraft and INS
Given three inputs, we can find all outputs including errors.
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Results of Typical Simulation
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Error Analysis Using Nonlinear Simulation
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Error Analysis Using Nonlinear Simulation
Examine navigation errors when the IC on X is in error.
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Error Due to X Initial Condition
Errors remain constant.
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Simulation of Aircraft and INS
Examine navigation errors when the IC on H is in error.
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Errors Due to H initial Condition
Errors are dominated by unstable altitude mode.
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Simulation of Aircraft and INS
Examine navigation errors when IC on Vx is in error.
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Errors Due to Vx Initial Condition
In the flat earth navigator the X-error would go to infinity.
The Schuler pendulum mode limits the X-error.
Note both Schuler oscillation and unstable altitude mode.
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INS Aiding
Altitude Aiding
Velocity and position errors in the vertical channel are
not bounded and can quickly become quite large.
Barometric altitude provides a measure of height
above sea level, typically to an accuracy of 0.1%.
Most airborne INS operate with barometric aiding in
order to bound the growth of vertical channel errors.
Many other types of aiding are typically used
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Simulation of Several Aided INS
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Altitude-aided INS
Includes a steady state (constant gain) Kalman filter with
gains on (Hmeasured-Hestimated).
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Stabilized altitude errors in altitude-aided INS
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Altitude and X-aided INS
Includes a steady state (constant gain) Kalman filter with gains
on (Hmeasured-Hestimated) and (Hmeasured-Hestimated).
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Stabilized errors in Alt. and H-aided INS
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GPS Aiding of INS
GPS can provide aiding to an INS by providing an
independent measurement of x, y, and z (altitude).
Furthermore, certain GPS implementation can
provide velocity aiding by providing independent
measurements of Vx, Vy and Vz.
A Kalman filter is often used to help blend the GPS
measurements with the INS outputs in an optimal
way.
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Integrated INS/GPS Block Diagram
Velocity
accelerations
angular rates
Position
Orientation
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Measurements from the GPS Receiver Model
Pseudorange
ρGPS  (X  x)2  (Y  y)2  (Z  z)2  c Δt  w1
Pseudorange Rate
dρ
dρ
d

 c Δt  w 2
dt GPS dt
dt
X, Y, Z
: Satellite Position
x, y, z
: Platform Position
c Δt
: Pseudorange equvalent
d
c Δt
dt
Clock Bias (Random Walk)
: Pseudorange rate equivalent
Clock Drift (Random Walk)
w1 , w 2
: Normally Distributed Random Numbers
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Benefits of Integrated INS/GPS Systems
•INS gives accurate estimates of aircraft
orientation.
•GPS provides accurate estimates of aircraft
position.
•INS solutions are generally computed 100 times
per second.
•GPS solutions are computed once per second.
•GPS in subject to jamming, INS is not.
•Combining GPS and INS provides accurate and
robust determination of both translational and
rotational motion of the aircraft.
•Both translational and rotational motion are
required to locate targets on the ground from the
aircraft.
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Conclusions
Unaided INS have troublesome errors that grow with
time or oscillate with an 84 minute period.
Various aiding schemes are often implemented to
stabilize the INS errors.
GPS aiding of INS is an effective means to stabilize INS
position and velocity errors.
Integrated INS and GPS systems are useful for
determining both the position and orientation of an
aircraft. Such systems are therefore helpful in locating
of targets on the ground.
3/12/02 - 46
Additional Purdue Resources
Presented at the The Motion Imagery Geolocation Workshop, SAIC Signal Hill Complex, 10/31/01
1. Dominick Andrisani, Simultaneous Estimation of Aircraft and Target Position With a Control
Point
2. Ade Mulyana, Takayuki Hoshizaki, Simulation of Tightly Coupled INS/GPS Navigator
3. James Bethel, Error Propagation in Photogrammetric Geopositioning
4. Aaron Braun, Estimation Models and Precision of Target Determination
Presented at the The Motion Imagery Geopositioning Review and Workshop, Purdue University,
24/25 July, 2001
1. Dominick Andrisani, Simultaneous Estimation of Aircraft and Target Position
2. Jim Bethel, Motion Imagery Modeling Study Overview
3. Jim Bethel, Data Hiding in Imagery
4. Aaron Braun, Estimation and Target Accuracy
5. Takayuki Hoshizaki and Dominick Andrisani, Aircraft Simulation Study Including Inertial
Navigation System (INS) Model with Errors
6. Ade Mulyana, Platform Position Accuracy from GPS
3/12/02 - 47
References
1. B.H. Hafskjold, B. Jalving, P.E. Hagen, K. Grade, Integrated
Camera-Based Navigation, Journal of Navigation, Volume
53, No. 2, pp. 237-245.
2. Daniel J. Biezad, Integrated Navigation and Guidance
Systems, AIAA Education Series, 1999.
3. D.H. Titterton and J.L. Weston, Strapdown Inertial
Navigation Technology, Peter Peregrinus, Ltd., 1997.
4. A. Lawrence, Modern Inertial Technology, Springer, 1998.
5. B. Stietler and H. Winter, Gyroscopic Instruments and
Their Application to Flight Testing, AGARDograph No.
160, Vol. 15,1982.
6. A.K. Brown, High Accuracy Targeting Using a GPSAided Inertial Measurement Unit, ION 54th Annual
Meeting, June 1998, Denver, CO.
3/12/02 - 48
Errors due to Theta Initial Condition
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Errors Due to Vz Initial Condition
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Errors Due to wy Measurement Bias
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Errors Due to Fxb Measurement Bias
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Errors Due to Fzb Measurement Bias
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Poles of Various systems
Poles of various systems
OLpoles =
0
0.00002389235751 + 0.00123800034072i
0.00002389235751 - 0.00123800034072i
-0.00175119624036
0.00175119562360
Poles of H aided observer
Poles of H and X aided observer
Wn41 =
0.0129
0.0129
0.0130
0.0130
Wn52 =
0.0110
0.0110
0.1430
0.1430
0.1430
Z41 =
0.4427
0.4427
0.8973
0.8973
Z52 =
0.7161
0.7161
1.0000
0.5000
0.5000
P41 =
-0.0057 + 0.0116i
-0.0057 - 0.0116i
-0.0116 + 0.0057i
-0.0116 - 0.0057i
P52 =
-0.0078 + 0.0076i
-0.0078 - 0.0076i
-0.1430
-0.0715 + 0.1238i
-0.0715 - 0.1238i
3/12/02 - 54