1. Inertial Navigation Systems and GPS Juan Jacobo Van der Dys April 20 th, 2005

2. 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.

3. Introduction The 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.

4. 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.

5. Determining Position Courtesy of C.F.Savant Jr et. al. acceleration ∫∫ position Newton’s Law

6. Determining Attitude Based on Euler Angles Kinematic differential equations Courtesy of Wie, Bong

7. 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

8. Inertial Platforms An 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 rotations Cons: Complexity and cost. Gimbal Mechanics. Reliability.

9. Strapdown Systems The 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.

10. ERRORS (and consequences) From Anthony Lawrence “Modern Inertial Technology”

11. 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”

12. Precision/Accuracy is $$$ The better the sensor, the more expensive it is. –Most of the time this also means size and weight. From Avidyne CorporationFrom Raytheon Company

13. Accumulated error correction Airspeed/Baro-altitude, –Δv ~ ∫a bias 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.

14. Pseudo Attitude from GPS Pseudo roll: Pseudo pitch: Pseudo yaw: Images from the Queensland University of Technology

15. 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

16. The Kalman filter Sensors: 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.

17. 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.

18. References Mohinder 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.com Lawrence, Anthony “Modern Inertial Technology” Springer- Verlag (1993) Shephard, William et al. “Inertial Navigation” D Van Nostrand Co. (1962) www.raytheon.com www.bee.qut.edu.au http://psas.pdx.edu www.avidyne.com

19. Questions?