Fundamentals of Strapdown Inertial and GPS-Aided Navigation By Professor Dominick Andrisani Purdue University, West Lafayette, IN 47907-1282 firstname.lastname@example.org 765-494-5135 Tactical Imagery Geopositioning Workshop March 12, 2002 Chantilly, VA. Purposes of this talk.
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Fundamentals of Strapdown Inertial and GPS-Aided Navigation
Professor Dominick Andrisani
Purdue University, West Lafayette, IN 47907-1282
Tactical Imagery Geopositioning Workshop
March 12, 2002
Inertial Platform based INS
Strapdown INS <<(emphasized here)
Aided Navigators of either type
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.
Complexity and cost.
Gimbal magnetics (torquers must not leak magnetic flux).
Reliability (bearings and slip rings tend to wear).
Simple structure, low cost.
More rugged and lighter.
Reliability (no gimbal magnetics, no slip rings, no bearings, electronics more reliable then machinery).
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).
Xb, Yb, and Zb
are body fixed
XNED is northerly
YNED is easterly
ZNED is down
vx is velocity in northerly direction
vz is velocity in down direction
x is northerly position
h is altitude (positive up)
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
In this simple two-dimensional example, two linear accelerometers and one rate gyro are used.
This device measures inertial angular rate about its sensitive direction. Three of these arranged orthogonally measure the components of the angular velocity vector.
This device measures specific force= a-g=(Faero+Fthrust)/m.
They cannot distinguish between acceleration and gravity.
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.
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).
Nonlinear navigator equations of motion
Linear error equations of motion
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)
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).
Given three inputs, we can find all outputs including errors.
Examine navigation errors when the IC on X is in error.
Errors remain constant.
Examine navigation errors when the IC on H is in error.
Errors are dominated by unstable altitude mode.
Examine navigation errors when IC on Vx is in error.
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.
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
Includes a steady state (constant gain) Kalman filter with
gains on (Hmeasured-Hestimated).
Includes a steady state (constant gain) Kalman filter with gains on (Hmeasured-Hestimated) and (Hmeasured-Hestimated).
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.
: Satellite Position
: Platform Position
: Pseudorange equvalent
Clock Bias (Random Walk)
: Pseudorange rate equivalent
Clock Drift (Random Walk)
: Normally Distributed Random Numbers
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.
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
Poles of H and X aided observer
-0.0078 + 0.0076i
-0.0078 - 0.0076i
-0.0715 + 0.1238i
-0.0715 - 0.1238i
Poles of H aided observer
-0.0057 + 0.0116i
-0.0057 - 0.0116i
-0.0116 + 0.0057i
-0.0116 - 0.0057i
Poles of various systems
0.00002389235751 + 0.00123800034072i
0.00002389235751 - 0.00123800034072i