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

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?