Strapdown Inertial Navigation Systems
(INS) Sensors and UAVs Avionic
By
Dr. Hamid Bolandhemmat
CONTENTS
•Overview of Inertial Navigation Systems (INS) and their application
•General navigation equations in spaced fixed and rotating reference frames
• Inertial sensors
•Gyroscopes
•Underlying theory
•Rebalance loop
•Sources of errors
•calibration
•Accelerometers
•Underlying theory
•Rebalance loop
•Sources of errors
•Calibration
•Compass
•Air data systems
•Airspeed
•Pressure Altitude
•Global Positioning Systems (GPS)
•Underlying concept
•Pseudo ranges
•Clock drift
INERTIAL NAVIGATION SYSTEMS OVERVIEW)
• In order to autonomously travel from one point to another,
information regarding position and attitude vector of the UAV
must be determined:
Guidance
Control laws
Sensor Fusion
Filter
Aiding Sensors
Servos
Navigation
Mechanization
Equations
UAV
Inertial Sensors
INS
• These states are also required to calculated other flight parameters
such as wind vector, crab angle, etc
2D NAVIGATION W.R.T. SPACED FIXED FRAME:
Yb=E
.
Xb
q
Zb
N
earth
D
2D NAVIGATION W.R.T. SPACED FIXED FRAME:
Yb=E
.
Xb
q
Zb
N
D
UAV
2D NAVIGATION W.R.T. ROTATING FRAME:
q
Yb=E
.
Xb
N
Zb
ZE
D
XE
2D NAVIGATION W.R.T. ROTATING FRAME:
q
Yb=E
.
Xb
Zb
N
ZE
D
XE
3D STRAPDOWN INERTIAL NAVIGATION SYSTEMS:
Onboard Navigation
processor
3 accelerometers
N
3 Gyros
UAV
INERTIAL SENSORS - GYROSCOPES
Angular momentum vector
INERTIAL SENSORS - GYROSCOPES
• Angular velocity measurement (the application of the precession
principle):
• The spinning wheel or rotor is put on a gimbal inside an instrument
case to be held isolated in space; any changes in the angles of the
gimbal then will represent changes in the orientation of the case
with respect to the reference direction maintained by the rotor
angular momentum
INERTIAL SENSORS - GYROSCOPES
Spin axis
output axis
Torque generator
or springs
Input axis
GYROSCOPES – REBALANCE LOOP
• With the springs:
• With the torque generator:
GYROSCOPES – REBALANCE LOOP
• With the torque generator:
• PI controller instead of a gain to guarantee zero steady state error :
GYROSCOPES – DYNAMICS EQUATIONS
Gimbal
case
GYROSCOPES – FIBER OPTIC GYROS (FOG):
• Underlying principle (Sagnac effect):
• The transit time for two counter-propagating beams of light travelling in a
fiber optic ring is not the same when the ring is spinning.
CCW wave
CW wave
GYROSCOPES – FIBER OPTIC GYROS (FOG):
• two counter-propagating beams of light are travelling through a looped fiber
optic coils (in a closed path)
• The transmit time difference or the phase shift between the two beams are
used to compute the rotational speed
• The time difference:
• The phase shift:
Light source
Light detector
GYROSCOPES – FIBER OPTIC GYROS (FOG):
• Advantages:
• High reliability and low maintenance cost (no mechanical and spinning
parts)
• Wide dynamic range
• Insensitivity to acceleration, shock and vibration
• Digital output
• Instant start-up time
Light source
Light detector
GYROSCOPES – RING LASER GYROS (RLG):
• Underlying principle (Sagnac effect):
• Frequency and wave length of two counter-propagating beams of
laser travelling within an optical cavity (a triangular closed path with
reflecting mirrors at each corner) are different when the cavity is
spinning.
Photo Diodes
Laser beams
• The frequency difference resulting
from the difference in “effective
path lengths” can then be calculated
as:
Helium-Neon Laser
GYROSCOPES – RING LASER GYROS (RLG):
• The frequency difference will be then:
Photo Diodes
Laser beams
Helium-Neon Laser
GYROSCOPES – RING LASER GYROS (RLG):
• Lock-in effect: due to imperfections in the lasing cavity
(mirrors), below an input rate thresholds, there would
be no output frequency difference.
• The lock-in dead zone is in the order of 100deg/hr
(earth rotation rate is 15deg/hr)
• Mechanical dither the laser block at 400500Hz, with peak amplitude of approximately
1 arc-sec to remove the dead zone.
Dead
zone
GYROSCOPES – MEMS GYROS:
• In 2D, polar position coordinate of the object is:
• Double differentiating with respect to time,
gives the acceleration:
GYROSCOPES – MEMS GYROS:
Driven
vibrations
Coriolis
forces
Tuning fork
Torsional
vibration
• Size and shape of the tuning fork is designed such as the
torsional vibration frequency is identical to the flexural
frequency of the tuning fork.
Courtesy of Ref. 3
GYROSCOPES – MEMS GYROS:
• Usually the sensing forks are coupled to a similar fork which produces the
rate output signal:
Courtesy of Ref. 2
Coriolis
forces
• The piezo-electric drive tines are oscillated at precise amplitudes. In the
presence of the angular velocity, the tines of the pick up fork move up and
down in an out of the plane of the fork assembly. An electrical output signal
is then produced by the pick up amplifier which is proportional to the input
angular rate.
GYROSCOPES – ERROR TERMS
• Constant bias [4]:
• Bias is the sensor average output when zero output is expected
(no rotation). A constant bias error 𝜀 causes ∆𝜃 = 𝜀. 𝑡
• Thermo-mechanical noise:
• Bandwidth of The thermo-mechanical noise on the (MEMS)
gyroscopes are much larger than the sampling frequency. As a
result, the noise due to the thermo-mechanical fluctuations
behaves similar to white noise.
• Assume that 𝑁𝑖 is the 𝑖 𝑡ℎ sample of the sensor white noise
sequence, then
𝑛
𝑡
𝜀 𝑡 𝑑𝑡 = ∆𝑡
0
𝑁𝑖
𝑖=1
GYROSCOPES – ERROR TERMS
• Hence, the accumulated angle random walk error has a mean and covariance
of [4]:
• Hence the output angle random walk has zero mean and a variance
which grows proportional to square root of time:
GYROSCOPES – ERROR TERMS
Ref. 4
GYROSCOPES – ERROR TERMS
• Flicker noise / Bias stability [4]
• Bias of the MEMS gyro changes due to flicker noise (low
frequency noise with a 1/f spectrum)
• Bias fluctuations due to flicker noise is modelled as random
walk (not accurate as the bias variance doesn’t grow with time).
• Bias stability parameter (1𝜎 value) is defined to show how the
sensor bias changes over a specified period of time (usually 100sec
– constant temperature)
• for example, if the bias stability (based on 100 seconds time
𝑑𝑒𝑔
period) is calculated to be 0.01 ℎ𝑟 , it means that the sensor bias
after 100sec, would have a mean equal to the original sensor bias
𝑑𝑒𝑔
and an standard deviation of 0.01 ℎ𝑟 .
• If using the random walk model for the bias variations, then the
variance change is expressed to be proportional to square root of
time:
𝑑𝑒𝑔
𝐵𝑆(
)
𝑑𝑒𝑔
ℎ𝑟
𝐵𝑅𝑊 2
= 2
ℎ𝑟
𝑡(ℎ)
GYROSCOPES – ERROR TERMS
• Flicker noise / Bias stability [4]
• The attitude error due to the bias fluctuations (with bias
random walk model), would then be a second order angle
random walk model.
• Temperature variations:
• Changes in the sensor bias due to the temperature changes
(could be also caused due to the electronics self heating)
• The effect is nonlinear usually of order 3
• Either the sensors must be calibrated with temperature or the
sensors unit must be temperature controlled.
GYROSCOPES – ERROR TERMS – ALLAN VARIANCE
• Allan variance method [Ref. 4]:
• It is a time domain technique to characterize the noise
• Is a function of “averaging period”
• For an averaging period 𝑇, the Allan variance is calculated as:
 Create 𝑛 bins of data out of a long sequence of the sensor
readings where each bin contains data with the length of
the averaging period 𝑇 (at least 9 bins are required).
 Take average of the data for each bin
𝑎(𝑇)1 , 𝑎(𝑇)2 , … , 𝑎(𝑇)𝑛 , where 𝑛 is total number of bins.
 Calculate the Allan variance by:
1
𝐴𝑉𝐴𝑅 𝑇 =
2(𝑛 − 1)
𝑛
(𝑎 𝑇
𝑖+1
− 𝑎 𝑇 𝑖 )2
𝑖=1
 Also, Allan deviation (equivalent to standard deviation)
would be the square root of Allan variance:
𝐴𝐷 𝑇 =
𝐴𝑉𝐴𝑅(𝑇)
GYROSCOPES – ERROR TERMS – ALLAN VARIANCE
• Allan variance method is plotted as a function of the averaging period T
on a log-log scale
• Different random process usually appear in different region of T:
Courtesy of Ref. 5
T
GYROSCOPES – CALIBRATION
• Calibration of the sensor suite is accomplished on temperaturecontrolled precise turn-tables (1 arcsec tilt accuracy) to determine the
sensors bias, scale factor, misalignment factors, g-sensitivity factors, etc:
• For gyroscopes:
• For accelerometer:
• Kalman Filter or Least Square method to determine the unknown calibration
parameters.
ACCELEROMETERS - CONCEPT
• Accelerometers measure specific force
1
𝑓 = 𝑎 − 𝑔 = 𝑚 (𝐹𝐴𝑒𝑟𝑜 + 𝐹𝑇ℎ𝑟𝑢𝑠𝑡 )
𝑎
Acceleration
with respect to
inertial space
case
Displacement
pick-off
− +
Signal proportional
to specific force 𝑓
Proof mass
g
ACCELEROMETERS – REBALANCE LOOP
• Accelerometers measure specific force
1
𝑓 = 𝑎 − 𝑔 = 𝑚 (𝐹𝐴𝑒𝑟𝑜 + 𝐹𝑇ℎ𝑟𝑢𝑠𝑡 )
Proof mass
𝑎
𝜃
𝑧
Torque
generator
Angle
pick-off
controller
𝑦 Input
axis
𝑥
ACCELEROMETERS – REBALANCE LOOP
• Euler’s law for the proof mass pendulous in the instrument case:
𝐼 𝜃 + 𝜑 + 𝑐 𝜃 = 𝐹𝑦 𝑏 + 𝐹𝑧 𝜃𝑏 − 𝑇𝑇𝑔
𝜑
𝐹𝑦
𝑓𝑧
𝑚𝑏
𝐼
𝑚𝑏
+
Σ
−
−
Σ
1
𝐼𝑠 2 + 𝑐𝑠
𝑘 𝑇𝑔
𝜃
𝐾(𝑠)
𝑖
ACCELEROMETERS – MEMS SENSORS
• Spring and mass from Silicon
• Change in the displacement causes an output voltage due to the
change in capacitance
Courtesy of Ref. 6
REFERENCES
1. A. Lawrence, Modern Inertial Technology, Springer, 1998
2. R.P.G. Collinson, Introduction to Avionic Systems, third
edition, Springer.
3. D.H. Titterton and J.L. Weston, Strapdown Inertial
Navigation Technology, Peter Peregrinus, Ltd., 1997.
4. O.J. Woodman, An Introduction to inertial navigation.
Technical Report, University of Cambridge, 2007.
5. IEEE Std 962-1997 (R2003) Standard Specification Format
Guide and Test Procedure fro Single-Axis Interferometric
Fiber Optic Gyros, Annex C. IEEE, 2003.
6. www.ett.bme.hu/memsedu