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16.333: Lecture#15
InertialSensorsComplementaryfilteringSimpleKalmanfiltering
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Image removed due to copyright considerations.
Image removed due to copyright considerations.
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Fibre Coil
Lens
Spatial
Filter
LaserSource
Splitters
Sagnac0
Phase Shift
Lens
Fringe Pattern
Detector
Polariser Beam
Optical Intensity
Phase Shift
Proportional to
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Fall2004 16.3331511
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Examples of Estimation Filters
from Recent Aircraft Projects at MIT
November 2004
Sanghyuk Park and Jonathan How
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Complementary Filter(CF)Often, there are cases where you havetwodifferent measurement sourcesforestimatingonevariable and the noise properties of the two measurementsare such that one source gives good information only in low frequency regionwhile the other is good only in high frequency region.
You can use a complementary filter !
accelerometer rate gyro
g
outputaccel.sin 1
- not good in long termdue to integration
- only good in long term
- not proper during fast motionLow Pass Filter
High Pass Filter
dtrate)(angular
est
Example: Tilt angle estimation using accelerometer and rate gyro
+
=1
1
s
+
= examplefor,1s
s
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Complementary Filter(CF) Examples
CF1. Roll Angle Estimation
CF2. Pitch Angle EstimationCF3. Altitude Estimation
CF4. Altitude Rate Estimation
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CF1. Roll Angle Estimation
High freq. : integrating roll rate (p) gyro output
Low freq. : using aircraft kinematics
- Assuming steady state turn dynamics,
roll angle is related with turning rate, which is close to yawrate (r)
=
sin
sin
r
mgL
mVL
r
g
V
CF setup 1s
HPF
LPFVg
RollRate
Gyro
YawRate
Gyro
+
+
Roll
angle
estimat
p
r
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CF2. Pitch Angle Estimation
High freq. : integrating pitch rate (q) gyro output
Low freq. : using the sensitivity of accelerometers to gravity direction- gravity aiding
In steady state
cos
sin
gA
gA
Z
X
=
=
=
z
x
A
A1tan
outputsteraccelerome, ZX AA
Roll angle compensation is needed
CF setup
estmeasq cos
= est
z
xss
AA costan 1
measqest
xA
zAest
s1
LPF
HPF
++ est
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CF3. Altitude Estimation
Motivation : GPS receiver gives altitude output, but it has ~0.4 seconds of delay.In order of overcome this, pressure sensor was added.
Low freq. : from GPS receiver
High freq. : from pressure sensor
CF setup & flight data
KFGPSfromh LPF
HPF
++ esth
SensorPressurefromh
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CF4. Altitude Rate Estimation
Motivation : GPS receiver gives altitude rate, but it has ~0.4 seconds of delay.In order of overcome this, inertial sensor outputs were added.
Low freq. : from GPS receiver
High freq. : integrating acceleration estimate in altitude directionfrom inertial sensors
CF setup
KFGPSfromh
za
ya
xa
Angular Transform
estest , s1
LPF
HPF
++
ha
esth
outputsteraccelerome, zx AA
[ ][ ]
=
gA
A
A
a
a
a
estest
z
y
x
z
y
x
0
0
:note
[ ] [ ] matricestiontransformaangular:, estest
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Kalman Filter(KF) Examples
KF1. Manipulation of GPS Outputs
KF2. Removing Rate Gyro Bias Effect
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KF 1. Manipulation of GPS Outputs
Background & Motivation
Stand-alone GPS receiver gives position and velocity
position pseudo-ranges velocity Doppler effect
These are obtained by independent methods :
and are certainly related )( vx=
Kalman filter can be used to combine them !
Motivation :
Position ~30 m
Velocity ~0.15 m/s
Typical Accuracies
Many GPS receivers provide high quality velocity information
Use high quality velocity measurement to improve position estimate
KF 1 K l Filt S t
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KF 1. Kalman Filter Setup
2+=vvmeas
avdt
d=
1=jdt
d
Measurements Filter Dynamics
measx 1+= xxmeas
measv jadt
d=
vx
dt
d=
est
x
estv
esta
NorthEast
Down
a
v
jii ,
x : velocity
: acceleration : jerk
: position
: white noises
:esta
noisy, but not biased combined with rate gyros in removing the gyro biases (KF2)
KF 2. Removing Rate Gyro Bias Effect
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KF 2. Removing Rate Gyro Bias Effect
Background & Motivation
In aircraft control, roll anglecontrol is commonly used in inner-loop to create requiredlateral accelerationwhich is commanded from guidance outer-loop
Biased roll angle estimate can cause steady-state error in cross-track
1s
HPF
LPFVg
Roll
RateGyro
Yaw
RateGyro
+
+
Rollangle
estimat
p
r
Drawback : biased estimate
Complementary filter with roll & raw gyros (CF1)Single-Antenna GPS Based
Aircraft Attitude Determination- Richard Kornfeld, Ph.D. (1999)
Drawback : sampling rate limit (GPS),typical filter time constant ~0.5 sec.
rVgas p
KF 2 Kalman Filter Setup
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KF 2. Kalman Filter Setup
sa
V
ii ,r
: velocity
: acceleration in sideways direction
: roll rate : yaw rate
: bank angle
: white noises
2++= pmeas biaspp
2=pdt
d
3=pbiasdtd
Measurement Equations Filter Dynamics
measp
estpbias( )estsa 1+=gas
3 ++= rmeas biasV
grmeasr
4=rbias
dt
d
1 += pdt
d
estp
( )estrbias
estfrom Rate Gyros
from GPS Kalman Filter
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KF 2. Simulation Result
Simulation for 10 degree bank angle holdRoll rate gyro bias=0.03 rad/s, yaw rate gyro bias = 0.02 rad/swere used in simulation
References
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References
Applied Optimal EstimationEdited by Arthur Gelb, MIT Press, 1974
Fundamentals of Kalman Filtering A Practical Approach
Paul Zarchan& Howard Musoff, Progress in Astronautics and Aeronautics Vol. 190
Avionics and Control System Development for Mid-Air Rendezvous of Two Unmanned Aerial VehiclesSanghyuk Park, Ph.D. Thesis, MIT, Feb. 2004
Fundamentals of High Accuracy Inertial Navigation
Averil Chatfield, Progress in Astronautics and Aeronautics Vol. 174
Applied Mathematics in Integrated Navigation SystemsR. Rogers, AIAA Education Series, 2000
The Impact of GPS Velocity Based Flight Control on Flight Instrumentation Architecture
Richard Kornfeld, Ph.D. Thesis, MIT, Jun. 1999
Autonomous Aerobatic Maneuvering of Miniature HelicoptersValdislavGavrilets, Ph.D. Thesis, MIT, May 2003