2458-14

Workshop on GNSS Data Application to Low Latitude Ionospheric Research

6 - 17 May 2013

Inertial Navigation Systems

VAN GRAAS Frank Ohio University School of Electrical Engineering and Computer Science 345 Stocker Center Athens OH 45701-2979 U.S.A.

Inertial Navigation Systems

Applications of GNSS

Workshop on GNSS Data Application to Low Latitude Ionospheric Research

Trieste – Italy, 06-17 May 2013

Prof. Frank van Graas Ohio University

Inertial Navigation Systems p.1 Course Overview

• Introduction to inertial navigation • Accelerometer operation and error sources • Gyroscope operation and error sources • Coordinate frames • Inertial navigation mechanization • Dynamics • Initialization • Strapdown terminology • Movement over ellipsoid • Attitude, velocity, and position updating • Future trends

Inertial Navigation Systems p.2 Two Inertial Measurement Systems

Both systems use inertial sensors and know about the orientation of the device, but only the system on the left can be used to navigate using inertial measurements only.

An inertial navigation system has two components: • Aviation Ring Laser Gyro Standalone Navigation Grade Inertial Reference Unit 1) Sensor package • 1 nautical mile per hour performance 2) Navigation computer • Outputs: – 50Hz body rates and accelerations, pitch, roll Autonomous navigation system – 25Hz heading – 20Hz velocities

Inertial Navigation Systems p.3 What Makes Inertial Navigation Work?

• In order to move, we first need to accelerate • If we keep track of our acceleration over time, then we can determine how much we move » Use accelerometers to measure acceleration • If we could also keep track of the direction in which we accelerate, then we can determine how much we move in a particular direction » Use gyroscopes to measure changes in direction over time • If we knew where we started and in which direction we were pointing, then we can determine our position, velocity and orientation as time goes on » Use initial conditions

Inertial Navigation Systems p.4 Inertial Application Areas

• Navigation Grade » Spacecraft, sub-marine, military aircraft, commercial aircraft, ship • Tactical Grade » Attitude and heading reference system (artificial horizon), short-term tactical guidance, land vehicles, aircraft autopilot, guidance stabilization, GPS integration • Short-term (Commercial Grade) » Small unmanned aerial systems (UAS), camera stabilization,,g airbag sensors, car electronic stability control, video games, smart phones, athletic shoes, motion sensors (e.g. shipping crates, electronics), medical

Inertial Navigation Systems p.5 What About the Ionosphere? • Primary application: » Mitigate the effects of the ionosphere by aiding the receiver tracking loop with inertial measurements » First need to understand the ionosphere … • Secondary applications: » Earthquakes and nuclear detonations cause disturbances in the ionosphere • Earthquake detection networks are starting to use inertial measurements to obtain the high-frequency content of the quake along with GNSS measurements for the lower frequencies » Tsunami prediction networks are starting to use accelerometers along with GNSS receivers

Inertial Navigation Systems p.6 Short-Term, Tactical, Navigation Grade

Navigation Inertial Grade Short-Term Tactical Grade Flight Control Example Missiles Aircraft Consumer Applications AHRS Navigation System Electronics 0.1 – 10 deg/hr 0.001 – 0.1 deg/hr > 10 deg/hr Gyroscopes Fiber Optic, Fiber Optic, Silicon MEMS Silicon MEMS Ring Laser 100 Pg – 1 mg 10 – 100 Pg > 1 mg Accelerometers Quartz Resonant, Quartz Resonant, Silicon MEMS Silicon MEMS Silicon MEMS

AHRS = Attitude and Heading Reference System (Artificial Horizon) MEMS = Micro-Electro-Mechanical Systems All numbers are root-mean-square (rms) values Note: Navigation grade INS is integrated with airdata computer for vertical channel stabilization

Inertial Navigation Systems p.7 Accelerometer

• Consider a proof-mass glued to a scale that can measure both positive (push) and negative (pull) weights Proof mass Scale

• Most accelerometers (like the one above) measure “specific force,” or force per unit mass (in m/s2), which consists of: » Acceleration specific force = acceleration = a (m/s2) » Gravitation specific force ≈ 9.8 (m/s2) • If the scale is at rest on the floor, then the output of the accelerometer will be +g ≈ 9.8 (m/s2) • If the accelerometer is in “free fall,” then there is no gravity force on the scale, such that the output = 0 (m/s2)

Inertial Navigation Systems p.8 Accelerometer Technologies • Several sensing technologies are in common use: » Force-feedback pendulum » Vibrating beam (or dual vibrating beam) » Micro-machined with electrostatic control of proof mass • In general, the force is measured indirectly by measuring how much signal is required to keep the proof mass from moving Pendulum pivot Electromagnetic coil Permanent magnet Force Measured specific force Proof mass Feedback Capacitive displacement sensor

Sensitive axis Differential amplifier

Inertial Navigation Systems p.9 Delta Velocity instead of Acceleration

• Most accelerometers output 'V instead of acceleration

t2 ' ta )(V dt (m/s) ,tt 12 ³ t1 • In words: 'V is the change in velocity over the time interval 't measured from t1 to t2 » For most applications: 't = 0.01 (s) is sufficient » High-dynamic applications use 't as small as 0.00025 (s) • If an accelerometer measures gravity, then a ≈ 9.8 (m/s2), such that for 't = 0.01 (s), 'V = 0.098 (m/s)

• Convenient format as it simplifies subsequent processing; some sensors already directly measure change; reduces data rate and dynamic range compared to implementation of the single integration outside the sensor

Inertial Navigation Systems p.10 Example Inertial Measurement Unit

Accelerometers Miniature MEMS Quartz IMU Range ±10 g TM Systron Donner MMQ 50 Bias turn-on to turn-on stability ≤ 2.5 mg, 1V Bias in-run stability ≤ 3 mg, 1V Velocity random walk 0.5 mg/√Hz Scale factor error ≤ 0.5% Alignment ≤ 5 mrad Bandwidth 50 Hz 6.5 cm Gyroscopes Range 200°/sec Bias turn-on to turn-on stability ≤ 100°/hr, 1V Bias instability ≤ 4-15°/hr Angle random walk 0.3°/√hr Scale factor error ≤ 0.5% 5 cm Alignment ≤ 5 mrad From: www.systron.com Bandwidth (-90° phase shift) 50 Hz

Inertial Navigation Systems p.11 Hardware Integration Research Platform

Inertial Navigation Systems p.12 Inertial Navigation in the Vertical Direction • Accelerometer sensitive axis in the Z direction z (perpendicular to local level defined by gravity) • Scenario: » t = 0: Start at height = 0 (h = 0) » t = 1 s to 2 s: accelerate upwards, reach h= 0 highest point, stay for 2 seconds and return to h = 0

Inertial Navigation Systems p.13 MEMS Accelerometer: Vertical Acceleration

• Assume » Accelerometer does not have any errors except for noise » Gravity does not vary as a function of height • Use first second of data to estimate gravity and subtract it from the measured 'V ("zero-velocity update")

Inertial Navigation Systems p.14 Vertical Velocity and Displacement

• To obtain velocity and displacement from 'V measurements:

' tt )(V ' vv ' V tt 12 12, 2 1 ,tttt 12 2

v2 ' tt 12, )(V tt 12 'vt t,  ttvrr )(  v 2 1 ttt 112 12 2 1

t1 t2 • Note that acceleration is time-varying, not constant 1 2 » In general, cannot directly use: 00 tvr)t(r  at (because it assumes constant acceleration) 2 » However, can use this by integrating in small steps and assuming that acceleration is constant during each step

Inertial Navigation Systems p.15 Measured Vertical Velocity and Displacement

z

Very good h= 0 short-term performance for tracking loop aiding Drift = -0.6 m

Inertial Navigation Systems p.16 What Happened in the X and Y Directions?

Drift = 4.5 m

Drift = -5 m

Inertial Navigation Systems p.17 Leveling Error

Z’ Z Z’: g cos(D) | g X’: g sin(D) |Dg X’ D If D = 2 degrees, then X X’ acceleration: 0.34 m/s2 After 5 s: 'X=4.28 m g

Inertial Navigation Systems p.18 Other Complications • Accelerometers have many errors in addition to noise • Gravity is not constant (anomalies), doesn't always point to the center of the Earth (deflections), and varies as a function of height.

Gravity anomaly map

1 mGal = 10-5 m/s2

From: http://mrdata.usgs.gov/ geophysics/gravity.html

For the WGS 84 Earth Gravitational Model: http://earth-info.nga.mil/GandG/wgs84/gravitymod/ and http://www.ngs.noaa.gov/GEOID/GEOID96/ngdc-indx.html

Inertial Navigation Systems p.19 Gyroscope • Gyroscopes are used to measure orientation or changes in orientation and are often based on the principle of conservation of angular momentum.

Spinning wheel

Angle measurements (resolvers)

Gimbals Note: Earth's rotation rate ≈ 15° per hour (7.292115×10-5 rad/sec)

Inertial Navigation Systems p.20 Some Gyroscope Technologies

• Mechanical (see previous slide) – measures actual angles relative to some orientation • Micro Electro-Mechanical System (MEMS) Gyroscopes use the Coriolis effect: uZ vmF )(2 on a vibrating proof mass » Single beam oscillator, balanced or tuning fork oscillators, shell or wine glass or cylindrical oscillators F v

Z Oscillation axis • Ring Laser Gyroscopes (RLG) and Fiber Optic Gyroscopes (FOG) use the Sagnac effect to measure angular rate » RLG: laser beam around a closed path with mirrors » FOG: lasers through optical fiber

Inertial Navigation Systems p.21 Fiber Optic Gyroscope

Fiber Optic Coil • Path for the CW beam is longer due to the rotation CW Beam • Path for the CCW beam is shorter due to the rotation Laser Z Measure the interference CCW Beam pattern between the CW and CCW beam to determine 'T

• Benefits include reliability and high bandwidth (small mass enables strapdown without vibration isolation)

Inertial Navigation Systems p.22 MEMS Gyroscope Example

• Stationary, followed by a clock-wise (CW) rotation of 90° » t = 0: Start at angle = 0° 90° » t = 1.5 s to 3.5 s: rotate CW up to 90° • Use stationary portion to estimate the gyro bias • Integrate (accumulate) 'T to determine the change in angle

Inertial Navigation Systems p.23 Measured Angle (around Z Axis)

Inertial Navigation Systems p.24 Gimballed and Strapdown

• Early days: gimballed » Platform with accelerometers is held level and north- pointing such that position is calculated "directly" by integrating the 'V's

• Today: mostly strapdown » Sensors are "rigidly" mounted to the vehicle/device and leveling and north-pointing is implemented in software.

Inertial Navigation Systems p.25 Strapdown INS Terminology

Ref: Proposed IEEE Inertial Systems Terminology Standard and Other Inertial Sensor Standards, Randall K. Curey, Michael E. Ash, Leroy O. Thielman, and Cleon H. Barker, IEEE PLANS 2004 (IEEE Std 1559).

Inertial Navigation Systems p.26 IMU Accelerometer Parameters of Interest

• Input dynamic range (g) • Bias stability during operation (mg) • Bias error at turn-on at room temperature (mg) • Bias error variation over temperature (mg) • Noise (mg, 1-V in a x-Hz bandwidth) • Velocity random walk (m/s/√hr) • Bias vibration sensitivity (mg/g2) • Scale factor error (% of full scale) • Scale factor linearity (%) • Frequency response (-3-dB bandwidth in Hz) • Cross-axis sensitivity (%)

Inertial Navigation Systems p.27 IMU Gyroscope Parameters of Interest

• Input dynamic range (deg/s) • Bias stability during operation (deg/s) • Bias error at turn-on at room temperature (deg/s) • Bias error variation over temperature (deg/s) • Noise (deg/s, 1-V in a x-Hz bandwidth) • Angular random walk (deg/s/√hr) • Bias vibration sensitivity (deg/s/g2) • g-sensitivity (deg/s/g) • Scale factor error (% of full scale) • Scale factor linearity (%) • Frequency response (-3-dB bandwidth in Hz) • Cross-axis sensitivity (%)

Inertial Navigation Systems p.28 Gyroscope Drift & Angle Random Walk • Gyroscope drift is expressed in terms of degrees per hour and represents the long-term (average) angular drift of the gyro • Gyroscope noise is the short-term variation in the output of the gyro »Can be measured in deg/sec »Can be expressed as a Power Spectral Density (PSD) with units of deg/sec/√Hz or (deg/sec)2/Hz to express the output noise as a function of bandwidth »To track angular changes, output is integrated over time to find the angle as a function of time. Angular changes exhibit Angle Random Walk (ARW) in units of deg/√hr • If ARW = 1 deg/√s, then »After 1 s, V of the angular change will be 1 deg »After 100 s, V of the angular change will be 10 deg »After 1000 s, V of the angular change will be 31.6 deg

Inertial Navigation Systems p.29 Random Walk Process

• “Take successive steps in random directions” • In continuous time, this process can be generated by feeding an integrator with white noise. w(t) t x(t) uw )( du ³o t uwtx )()( du ³0 t ^` ^`uwEtxE du 0)()( ³0 t t tt ^`2 ^`uwEtxE du vw )()()( dv ^`vwuwE )()( dudv 0 0 ³³³³ 00 t tt ^`2 txE G  vu )()( dvdudv t ³0 ³³00 • Mean is zero, but variance increases linearly with time.

Inertial Navigation Systems p.30 Gyro Random Walk

Gaussian noise 5

0

Angular noise in deg/s -5 0 200 400 600 800 1000 Integrated Gaussian noise 50

0 V of angular

Angle in degrees change after -50 0 200 400 600 800 1000 1000 s Time in seconds

Inertial Navigation Systems p.31 Gyroscope and Accelerometer Time-Varying Biases

• Gyroscope and accelerometer biases are often modeled using a first-order Gauss-Markov process: a stationary Gaussian process with an exponential autocorrelation function.

w(t) t x(t) )( + ³ uw du - o

E

• Continuous time:  E  wxx time constant = 1/E E' t 2 • Discrete time: k  1  wxex kk Var(x) = V /(2E)

where: wk is an uncorrelated zero-mean Gaussian sequence

Inertial Navigation Systems p.32 1st Order Gauss-Markov Example

N = 7200*10; % 20 hours beta = 1/7200; % time constant is 2 hours dt = 1; % time interval is 1 second w = randn(N,1); % generate unity Gaussian t= zeros(N,1); % pre-allocate memory for speed x = zeros(N,1); % pre-allocate memory for speed x(1,1) = 0; % initial condition t(1,1) = 0; % initial condition for i = 2 : 1 : N t(i,1) = i; x(i,1) = exp(-beta*dt) * x(i-1,1) + w(i,1); end

Var(x) = V2/(2E)=3600 Std(x) = 60

1st Order Gauss-Markov processes are often used in engineering problems Mean and variance statistics are “stable”, but as can be seen from the above example, the process is not always “well-behaved.”

Inertial Navigation Systems p.33 INS Functional Components

Calibration of inertial sensors; realized post- processing or through bench test Generic INS Generic IMU computational process hardware

~ ' A tV )( Pre-processing IMU Conversion (pre-integration Coordinate ~ and instrument 'T transformation A t)( compensation) ˆ 'TA t)(

Attitude ˆ P/A tT )( ' ˆ )( computation N tV (DCM)

ˆ tR )( Compensation of N gravity and non- Integration ˆ N tV )( inertial components

Inertial Navigation Systems p.34 Conversion of Measured Data to State Estimates • Gyro increments are converted into angular orientation changes. • Accelerometer outputs are combined with gravity effect and transformed into velocity changes in navigation axes. • Result is combined with known kinematical adjustments and propagated into position increments. • Current position estimate at each measurement time is used to predict observations. • Residuals (innovations) are formed by comparing predictions versus measurements. • Residuals (innovations) are weighted to produce updated states. • Position, velocity, and misorientation state updates are assimilated into current estimates, resetting their a posteriori estimates to zero.

Inertial Navigation Systems p.35 Aircraft Euler Angles

ª 001 º ªcos 0 sin TT º « » « » >@I 0 cos sin II >@T 010 I x « » y « » Roll « II » « T T » ¬0 sin cos ¼ ¬sin 0 cos ¼ T ª cos sin \\ 0º ­ I Roll Pitch ° >@\ « sin cos \\ 0» T Pitch \ z « » ® ° ¬« 100 ¼» ¯ \ Heading b

A/P CT n >@>@>x y \TI z @ Heading

b Transformation from Platform to Aircraft / CT nPA (Platform = Navigation Frame)

Inertial Navigation Systems p.36 Small Angles

ª coscos sin coscos sinsinsinsin cos sin cos IT\I\IT\I\T\ º «sin coscoscos sinsinsin cos sinsinsin cos IT\I\IT\I\T\ » T / AP « » ¬« sin T cos sin IT cos cos IT ¼» • For small angles: sin , sin , sin , cos 1, cos 1, cos |I|T|\I|IT|T\|\ 1

ª 1 T\ º ª 0 T\ º « 1 I\ »  « 0 I\ » T / AP « » I « » ¬« IT 1 ¼» ¬« IT 0 ¼» ª 1 T\ º ª 0 T\ º ' « 1 I\ »  « 0 I\ » TT // APPA « » I « » ¬« IT 1 ¼» ¬« IT 0 ¼»

Inertial Navigation Systems p.37 Gimbal Lock

ª coscos sin coscos sinsinsinsin cos sin cos IT\I\IT\I\T\ º «sin coscoscos sinsinsin cos sinsinsin cos IT\I\IT\I\T\ » T / AP « » ¬« sin T cos sin IT cos cos IT ¼»

ª 0 sin( ) cos( \I\I ) º S « » WhenT , then T 0 cos( ) sin( \I\I ) 2 / AP « » ¬« 01 0 ¼» • Now only the difference between the roll and heading angle determines the transformation from Aircraft to Platform. • The roll and heading angles are not unique. • Same occurs when the pitch angle = -90°, then the sum of the roll and heading angles determines the transformation.

• For aircraft, both roll and heading undergo a discontinuity when the pitch angle passes through ±90° Æ gimbal lock.

Inertial Navigation Systems p.38 Four-Parameter Set (Quaternions)

• A quaternion is described by (nonsingular, unambiguous): ª § T ·º E1 sin¨ ¸ q 1 « 2 » « © ¹» ªq1 º § T · « » T / EE « » E2 sin¨ ¸ PA q2 « © 2 ¹» q « » « » 2 2 2 «q » § T · 1 2  EEE 3 1 3 «E sin¨ ¸» « » « 3 2 » ¬q4 ¼ © ¹ § TTT 332211  1· « § T · » T arccos¨ ¸ « cos¨ ¸ » © 2 ¹ ¬ © 2 ¹ ¼ • Example: 2 2 2 2 ª 4 1 2  3 2  3421 2  qqqqqqqqqqqq 2431 º « »  2 2 2  2  T / PA « 2 3421 4 1 2 3 2 qqqqqqqqqqqq 1432 » «   2 2 2  2 » ¬ 2 2431 2 1432 4 1 2 qqqqqqqqqqqq 3 ¼

1 ,1 EEE 32 ,0 then q1 sin(T 2/ ), qq 32 0 ª 001 º T TT 22 sin( )2/ cos(TT )2/ sin(T ) «0 cos( ) sin(TT )» qq 41 T / PA « » 2 2 2 2 «0 sin( ) cos(TT )» 4 qq 1 cos )2/( sin TT )2/( cos(T ) ¬ ¼

Inertial Navigation Systems p.39 Inertial Navigation on a Flat Non-Rotating Earth

Zz y

x

Z \ Z Z YI YA FF ,, AXIX cos( ) F ,AY sin(\\ )

,IY FF ,AX sin( ) F ,AY cos(\\ )

vxFv X I   ,,,, IXIXIXIX \   vxFv ,,,, IYIYIYIY X A

Inertial Navigation Systems p.40 Rotation around an Axis in Inertial Space

ω

uZuZ R ω angular velocity in rad/s v ωu R R a ωu v a ωω uu R

Inertial Navigation Systems p.41 Inertial Space & Earth-Centered-Earth-Fixed

As derived in: Goldstein, H., Classical Mechanics, 2nd Ed., 1980, pp. 174-176.

Inertial Navigation Systems p.42 Apply to radius and inertial space velocity vectors

Inertial Navigation Systems p.43 Force in Inertial Space (measured by accelerometer)

I amF I with : eeEeEI uZuZuZ )r()v(2aa or : I EI eeEe uZuZuZ )r(m)v(m2amamF specific E eeEe uZuZuZ )r(m)v(m2amF

Centrifugal force 2 Coriolis force Note : 1 cm/s (points outwards) 5.1v2 u Z 104 v after 5 min e 2 4.3r cm/s2 if v 300 m/s e |Z displacement : 2 Sum of earth’s gravitational e Z 5.4v2 cm/s 1 .0( 01)(300)2 450 m field and centrifugal force = 2 gravity field Based on: Goldstein, H., Classical Mechanics, 2nd Ed., 1980, pp. 177-179.

Inertial Navigation Systems p.44 Movement over Ellipsoid

Radius of curvature in a meridian

2 a (1 EE )e RM 22 2/3 >@ e1 E sin (Lat) Radius of curvature in planes toparallel the equator

a R E P 22  e1 E sin (Lat) a 6378137 m; e2 (2  f;f) f 1 E E 298.257223563

Inertial Navigation Systems p.45 Basis of Schuler Effect Estimated True Position Position R u Rˆ Computed u gravity vector gˆ g has a backward gˆ directional component

horizontal position error :  g r r R Position error g frequency R Acceleration error period | 84 minutes

Inertial Navigation Systems p.46 INS Functional Components

Calibration of inertial sensors; realized post- processing or through bench test Generic INS Generic IMU computational process hardware

~ ' A tV )( Pre-processing IMU Conversion (pre-integration Coordinate ~ and instrument 'T transformation A t)( compensation) ˆ 'TA t)(

Attitude ˆ P/A tT )( ' ˆ )( computation N tV (DCM)

ˆ tR )( Compensation of N gravity and non- Integration ˆ N tV )( inertial components

Inertial Navigation Systems p.47 Initialization

• Position is initialized with » Estimated longitude » Estimated geodetic latitude » Estimated altitude above ellipsoid (baro-altimeter) » Zero wander angle • Pitch and roll angles are calculated from the accelerometers (gravity defines the apparent vertical) • Heading is obtained from gyrocompassing » Need high-quality gyroscopes » If not available Æ integration with other sensors

• Form corresponding matrix TP/E from Earth to nav-reference (local level, wander-azimuth) coordinates

Inertial Navigation Systems p.48 Short Term INS Position Error

Position Error in Cruise – Short Term (<< 84 min)

Position Error at time t = Initial Position Error + Velocity Error u t + Leveling Error u g u ½ t2 + Accelerometer Bias u ½ t2 + Drift Rate u g u 1/6 t3

Inertial Navigation Systems p.49 Future Trends

• Cost of inertial sensors continues to drop, while performance continues to increase » Non-tactical grade » Automotive, consumer applications • Low-cost navigation grade IMUs are not on the horizon • Performance improvements feasible through advanced calibration techniques • Continued developments in the area of integration with GNSS and other navigation sensors » Ultra-tight integration with carrier phase • Long signal integration and tracking loop design » Integration with software-defined radios and batch processing (open-loop tracking)

Inertial Navigation Systems p.50