Inertial Navigation SystemsandGPS
Juan Jacobo Van der Dys
April 20th, 2005
Outline• Introduction• Inertial Navigation Systems
– Definition– Determining Position– Determining Attitude– Inertial Platforms– Strapdown Systems– Gyro and Accelerometer errors and their
consequences.
• The money issue• Pseudo-attitude from GPS• The Kalman filter.
IntroductionThe five basic forms of navigation are:- Pilotage: Recognizing landmarks to know where you
are. Older than human kind.- Dead reckoning: Knowing where you started from,
some sort of heading and some estimate of speed.- Celestial Navigation: Knowing time and angle between
vertical and celestial objects.- Radio Navigation: rely on radio-frequency sources with
known locations.- Inertial Navigation: Knowing initial position, velocity,
attitude and measuring accelerations and attitude rates to determine position and attitude. No external references.
Inertial Navigation Systems
INS (or INU) = Navigation Computer
+ gyroscopes + acceleration
IMU (or IRU) = Inertial Sensors.
Once aligned to a set of reference axes (North, East, Down) the computer carries out continuous dead-reckoning calculations.
Determining Position
Courtesy of C.F.Savant Jr et. al.
acceleration ∫ ∫ position Newton’s Law
Determining AttitudeBased on Euler Angles
Kinematic differential equations
Courtesy of Wie, Bong
Courtesy of Wie, Bong
Why do we care about attitude?Instrument Flight Rules (IFR): Flying below
minimum weather requirements.
Guidance: Auto-pilots, cruise missiles, precision approaches.
Courtesy of Cessna Aircraft Company
Inertial PlatformsAn inertial platform uses gyros to maintain
accelerometers in a fixed attitude.
Courtesy of Dr. Walter Haussermann,Marshal Space Flight Center
Pros:• Simpler Gyros• Higher Accuracy• Self-alignment by
gyrocompassing• Sensor Calibration by
platform rotationsCons:• Complexity and cost.• Gimbal Mechanics.• Reliability.
Strapdown SystemsThe gyroscopes and accelerometers are rigidly
mounted to the vehicle structure so that they move with the vehicle.
Courtesy of Minneapolis-Honeywell
Pros:• Simpler structure, low cost,
lighter weight.• Ruggedness.• Reliability.Cons:• Alignment.• Sensor calibration.• Motion induced errors.• Strapdown computer.
ERRORS (and consequences)
From Anthony Lawrence “Modern Inertial Technology”
Errors• Scale Factor• Non-linearity• Asymmetry• Bias
– Change over time/temperature
– Tilt misalignment
• Random Drift/Walk• Dead Band, Threshold,
and Resolution• Hysteresis• Gyro acceleration
sensitivity.
From Anthony Lawrence “Modern Inertial Technology”
Precision/Accuracy is $$$• The better the sensor, the more
expensive it is.– Most of the time this also means
size and weight.
From Avidyne Corporation From Raytheon Company
Accumulated error correction• Airspeed/Baro-altitude,
– Δv ~ ∫abias
• Magnetic Field, is constant for short range flights.
• GPS:– Derive position (twice)
to compute acceleration
– Trajectory angle ~ velocity (heading - beta (side slip angle) +
airspeed) + wind.
– Flight path angle ~ pitch angle + alpha (angle of attack)
– In coordinated flight constant heading means zero roll angle.
aXt
2
2
Pseudo Attitude from GPS• Pseudo roll:
• Pseudo pitch:
• Pseudo yaw:Images from the QueenslandUniversity of Technology
The Kalman filter
From Portland State Aerospace Society
The state vector• Θ pitch• Φ roll• Ψ yaw• p• q• r• Xaccel bias• Yaccel bias• Zaccel bias• Xgyro bias• Ygyro bias• Zgyro bias• α angle of attack• β sideslip angle• Airspeed
The Kalman filterSensors:• Accelerometers• Gyros• GPS• Magnetometer• Airdata
Compute the results:– Read the sensors– Estimate signal covariance– Calculate gains– Compute state
Tuning the Kalman filter may be very challenging. Initializing can be tricky.
Conclusion
• GPS is a great aid for INS
• GPS allows engineers use inexpensive sensors and have good accuracy.
• System susceptible to GPS jamming (not liked by FAA).
• Requires lots of computer power.
ReferencesMohinder S. Grewal, "Global Positioning Systems, Inertial
Navigation, and Integration" Wiley-Interscience; Book&Disk edition (December 15, 2000)
Savant, C.F. et al. “Principles of Inertial Navigation” McGraw-Hill (1961)
Wie, Bong “Space Vehicle Dynamics and Control” AIAA Education Series (1998)
www.cessna.comLawrence, Anthony “Modern Inertial Technology” Springer-
Verlag (1993)Shephard, William et al. “Inertial Navigation” D Van
Nostrand Co. (1962)www.raytheon.comwww.bee.qut.edu.auhttp://psas.pdx.eduwww.avidyne.com
Questions?