inertial navigation systems - indico...
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2458-14
Workshop on GNSS Data Application to Low Latitude Ionospheric Research
VAN GRAAS Frank
6 - 17 May 2013
Ohio University School of Electrical Engineering and Computer Science
345 Stocker Center Athens OH 45701-2979
U.S.A.
Inertial Navigation Systems
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 GraasOhio 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
• Aviation Ring Laser Gyro Standalone Navigation Grade Inertial Reference Unit
• 1 nautical mile per hour performance• Outputs:
– 50Hz body rates and accelerations, pitch, roll– 25Hz heading– 20Hz velocities
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:1) Sensor package2) Navigation computer
Autonomous navigation system
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 whichwe accelerate, then we can determine howmuch 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 tacticalguidance, land vehicles, aircraft autopilot,guidance stabilization, GPS integration
• Short-term (Commercial Grade)» Small unmanned aerial systems (UAS), camera stabilization, airbag
sensors, car electronicstability control, video games, smart phones,athletic shoes, motion sensors (e.g. shippingcrates, electronics), medical
, g
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
Inertial Grade Short-Term Tactical Navigation Grade
Example Applications
Flight ControlConsumer Electronics
MissilesAHRS
Aircraft Navigation System
Gyroscopes> 10 deg/hrSilicon MEMS
0.1 – 10 deg/hrFiber Optic, Silicon MEMS
0.001 – 0.1 deg/hrFiber Optic,Ring Laser
Accelerometers> 1 mgSilicon MEMS
100 g – 1 mgQuartz Resonant,Silicon MEMS
10 – 100 gQuartz Resonant,Silicon MEMS
AHRS = Attitude and Heading Reference System (Artificial Horizon)MEMS = Micro-Electro-Mechanical SystemsAll numbers are root-mean-square (rms) valuesNote: 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
• 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 theoutput of the accelerometer will be +g ≈ 9.8 (m/s2)
• If the accelerometer is in “free fall,” then there is noforce on the scale, such that the output = 0 (m/s2)
Proof massScale
gravity
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
ForceFeedback
Permanent magnet
Capacitive displacement sensor
Differentialamplifier
Proof mass
Electromagnetic coil
Measured specific force
Pendulum pivot
Sensitive axis
Inertial Navigation Systems p.9
Delta Velocity instead of Acceleration• Most accelerometers output V instead of acceleration
• 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
(m/s) )(V2
1
12 ,
t
ttt dtta
Inertial Navigation Systems p.10
Example Inertial Measurement Unit
Miniature MEMS Quartz IMUSystron Donner MMQTM 50
Accelerometers
Range ±10 g
Bias turn-on to turn-on stability ≤ 2.5 mg, 1
Bias in-run stability ≤ 3 mg, 1
Velocity random walk 0.5 mg/√Hz
Scale factor error ≤ 0.5%
Alignment ≤ 5 mrad
Bandwidth 50 Hz
Gyroscopes
Range 200°/sec
Bias turn-on to turn-on stability ≤ 100°/hr, 1
Bias instability ≤ 4-15°/hr
Angle random walk 0.3°/√hr
Scale factor error ≤ 0.5%
Alignment ≤ 5 mrad
Bandwidth (-90° phase shift) 50 HzFrom: www.systron.com
6.5 cm
5 cm
Inertial Navigation Systems p.11
Inertial Navigation in the Vertical Direction• Accelerometer sensitive axis in the Z direction
(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
highest point, stay for 2 seconds and return to h = 0
z
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
• To obtain velocity and displacement from V measurements:
• Note that acceleration is time-varying, not constant» In general, cannot directly use:
(because it assumes constant acceleration)» However, can use this by integrating in small steps and
assuming that acceleration is constant during each step
Vertical Velocity and Displacement
221
00 attvr)t(r
2)(V
)(
V
12,12
,
12
112
1212
ttttvrr
vv
ttttt
tttt
t1 t2
v1
v2
2)(V 12, 12
tttt
1t,2tv
Inertial Navigation Systems p.15
Measured Vertical Velocity and Displacement
z
h = 0
Drift = -0.6 m
Inertial Navigation Systems p.16
Very goodshort-termperformancefortrackingloop aiding
What Happened in the X and Y Directions?
Drift = 4.5 m
Drift = -5 m
Inertial Navigation Systems p.17
Leveling Error
X
X’
ZZ’
g
Z’: g cos( ) g
X’: g sin( ) g
If = 2 degrees, then
X’ acceleration: 0.34 m/s2
After 5 s: X=4.28 m
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.
For the WGS 84 Earth Gravitational Model: http://earth-info.nga.mil/GandG/wgs84/gravitymod/ andhttp://www.ngs.noaa.gov/GEOID/GEOID96/ngdc-indx.html
From: http://mrdata.usgs.gov/geophysics/gravity.html
1 mGal = 10-5 m/s2
Gravity anomaly map
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 rotationrate ≈ 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: on a vibrating proof mass
» Single beam oscillator, balanced or tuning fork oscillators, shell or wine glass or cylindrical oscillators
• 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
)(2 vmF
Oscillation axis
vF
Inertial Navigation Systems p.21
Fiber Optic Gyroscope
Fiber Optic Coil
Laser
CW Beam
CCW Beam
• Path for the CW beam is longer due to the rotation
• Path for the CCW beam is shorter due to the rotation
Measure the interference pattern between the CW and CCW beam to determine
• 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°» t = 1.5 s to 3.5 s: rotate CW up to 90°
• Use stationary portion to estimate the gyro bias• Integrate (accumulate) to determine the change in angle
90°
Inertial Navigation Systems p.23
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- 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- 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, of the angular change will be 1 deg»After 100 s, of the angular change will be 10 deg»After 1000 s, 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.
• Mean is zero, but variance increases linearly with time.
t
oduuw )(
x(t)w(t)
t tt
tttt
t
t
tdvdudvvutxE
dudvvwuwEdvvwduuwEtxE
duuwEtxE
duuwtx
0 00
2
0000
2
0
0
)()(
)()()()()(
0)()(
)()(
Inertial Navigation Systems p.30
Gyro Random Walk
0 200 400 600 800 1000-5
0
5Gaussian noise
Ang
ular
noi
se in
deg
/s
0 200 400 600 800 1000-50
0
50Integrated Gaussian noise
Time in seconds
Ang
le in
deg
rees
of angular change after
1000 s
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.
• Continuous time: time constant = 1/• Discrete time: Var(x) = 2/(2 )
where: wk is an uncorrelated zero-mean Gaussian sequence
t
oduuw )(
x(t)w(t)
+-
wxx
kkt
k wxex 1
Inertial Navigation Systems p.32
1st Order Gauss-Markov Example
N = 7200*10; % 20 hoursbeta = 1/7200; % time constant is 2 hoursdt = 1; % time interval is 1 secondw = randn(N,1); % generate unity Gaussian t= zeros(N,1); % pre-allocate memory for speedx = zeros(N,1); % pre-allocate memory for speedx(1,1) = 0; % initial conditiont(1,1) = 0; % initial conditionfor i = 2 : 1 : N
t(i,1) = i;x(i,1) = exp(-beta*dt) * x(i-1,1) + w(i,1);
end
Var(x) = 2/(2 )=3600Std(x) = 60
1st Order Gauss-Markov processes are often used in engineering problemsMean 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
Generic INScomputational process
)(P/A tT
Compensation ofgravity and non-
inertial componentsIntegration
)(ˆN tR
)(N tV
)(N tV
)(~ tVA
)(~ tA
IMU Conversion
Attitudecomputation
(DCM)
Coordinatetransformation
Calibration of inertial sensors; realized post-processing or
through bench test
Pre-processing(pre-integrationand instrumentcompensation)
)(A t
Generic IMUhardware
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 posterioriestimates to zero.
Inertial Navigation Systems p.35
Heading Pitch
Roll
1000cossin0sincos
cos0sin010
sin0cos
cossin0sincos0
001
zyxA/P
z
yx
bnCT
Roll
Pitch
Heading
Aircraft Euler Angles
Transformation from Platform to Aircraft(Platform = Navigation Frame)
bnPA CT /
Inertial Navigation Systems p.36
Small Angles
• For small angles:
coscossincossincossinsinsincossinsinsincoscoscossin
cossincossinsinsinsincoscossincoscos
/ APT
1cos 1,cos 1,cos ,sin ,sin ,sin
00
0
11
1
00
0
11
1
'//
/
ITT
IT
APPA
AP
Inertial Navigation Systems p.37
Gimbal Lock
coscossincossincossinsinsincossinsinsincoscoscossin
cossincossinsinsinsincoscossincoscos
/ APT
When , then
• 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° gimbal lock.
2 001)sin()cos(0
)cos()sin(0
/ APT
Inertial Navigation Systems p.38
Four-Parameter Set (Quaternions)• A quaternion is described by (nonsingular, unambiguous):
• Example:2
cos
2sinE
2sinE
2sinE
3
2
1
4
3
2
1
qqqq
q
23
22
21
2414322431
143223
22
21
243421
2431342123
22
21
24
/
222222
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
T PA
21arccos
1
1
332211
23
22
21
/
TTTEEE
T PA EE
q
)cos()2/(sin)2/(cos
)sin()2/cos()2/sin(220 ),2/sin( then ,0 ,1
2221
24
41
321321
qqqq
qqqEEE
)cos()sin(0)sin()cos(0
001
/ PAT
Inertial Navigation Systems p.39
Inertial Navigation on a Flat Non-Rotating Earth
x
yz
IYIYIYIY
IXIXIXIX
AYAXIY
AYAXIX
Z
vxFvvxFv
FFFFFF
,,,,
,,,,
,,,
,,,
)cos()sin(
)sin()cos(
IX
AX
AYIY
Inertial Navigation Systems p.40
Rotation around an Axis in Inertial Space
R
ω
RωωavωaRωv
rad/s invelocity angular ωR
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
Force in Inertial Space (measured by accelerometer)
Based on: Goldstein, H., Classical Mechanics, 2nd Ed., 1980, pp. 177-179.
)r(m)v(m2amF)r(m)v(m2amamF :or
)r()v(2aa :withamF
eeEeEspecific
eeEeEII
eeEeEI
II
Coriolis force Centrifugal force(points outwards)
Sum of earth’s gravitational field and centrifugal force =
gravity field
22e cm/s 4.3r
2e
4e
cm/s 5.4v2
m/s 300 vifv105.1v2
m 450)300)(01.0(
:ntdisplacememin 5after
cm/s 1 :Note
221
2
Inertial Navigation Systems p.44
Movement over Ellipsoid
563298.2572231 m; 6378137
equator the toparallel planesin curvature of Radius
meridian ain curvature of Radius
f f;f)(2ea
(Lat) sine1aR
(Lat)sine1
)e(1aR
2EE
22E
EP
2/322E
2EE
M
Inertial Navigation Systems p.45
Basis of Schuler Effect
g
TruePosition
EstimatedPosition
Acceleration error
Position error
Computedgravity vectorhas a backwarddirectional component
g
minutes 84 period
gfrequency
rgr
:error position horizontal
R
R
RR
g
Inertial Navigation Systems p.46
INS Functional Components
Generic INScomputational process
)(P/A tT
Compensation ofgravity and non-
inertial componentsIntegration
)(ˆN tR
)(N tV
)(N tV
)(~ tVA
)(~ tA
IMU Conversion
Attitudecomputation
(DCM)
Coordinatetransformation
Calibration of inertial sensors; realized post-processing or
through bench test
Pre-processing(pre-integrationand instrumentcompensation)
)(A t
Generic IMUhardware
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 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 t +Leveling Error g ½ t2 +Accelerometer Bias ½ t2 +Drift Rate g 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