introduction to inertial sensors - ushaq - oct 2015
TRANSCRIPT
An Introduction
To
Inertial Sensors
Muhammad Ushaq
Institute of Space Technology
Islamabad, Pakistan
0092-322-2992772
Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 1
Navigation
• The estimation of the state (position, velocity, and attitude) of
moving body in real time, with respect to some known reference
• A navigation system may be completely self-contained aboard the
navigating body e.g. Inertial Navigation System
Or
• It may require an external infrastructure as well as user segments,
such as radio navigation systems (GPS, GLONASS, Galileo, Beiduo,
etc)
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Inertial Navigation Systems (INS)
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A complete 3-D Navigation Solution (Position, Velocity, Attitude)
IMU (Accelerometers, Gyroscopes, Electronics)
INS relies on knowing initial position,
velocity, and attitude and thereafter
measuring accelerations and attitude
rates.
Accelerometer measures the
acceleration and gyroscope measure
the angular rotation.
Gyroscopes provide the info on where
the accelerations are directed/oriented.
Inertial Navigation Sensors
Gyroscopes and accelerometers have been in use for last 7~8 decades.
INS have been on the market since forties and fifties of the last century.
In the past 50 years, INS technology has developed rapidly, and the
precision/accuracy has been greatly enhanced.
In 1944, the "V-2" rocket made the first use of an INS. V-2’s range
was 320 kilometers and its deviation from target was approx 1.6
kilometers.
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Over the years INS has also been widely used in other applications
including the navigation of aircraft, tactical and strategic missiles,
spacecraft, submarines, ships, land vehicle, tunnels etc
Advances in MEMS → miniaturized INS. MEMS have widened the
range of possible applications to include areas such as human and
animal motion capture.
Inertial Measurement Unit (IMU)
The INS is made from a navigation computer and a set of
gyroscopes and accelerometers.
The group of inertial sensors is commonly called an inertial
measurement unit (IMU) or an inertial reference unit (IRU).
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The sensors are fastened to the vehicle.
Measurements from Gyros and
Accelerometers are in vehicle frame.
These measurements are mathematically
transformed in reference frame and used
for computation of position, velocity and
attitude.
Strapdown
The inertial sensors are mounted
on a platform isolated from angular
rotations of host body employing
gimbals so the sensors stay
oriented in a desired frame no
matter how the vehicle moves.
Platform or Gimbaled
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Gyroscopes
Gyroscopes
Gyroscope measures change in angularorientation either directly (displacementgyroscopes) , or through integrating ameasured rotational rate (rategyroscopes).
Conventional gyroscopes make use ofthe inertial properties of a wheel or rotorspinning at high speed.
A spinning wheel tends to maintain theorientation of its spin axis in space byvirtue of its angular momentumvector (𝐻 = 𝐼𝜔), the product of its inertiaand spin speed, and so defines areference direction.
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Angular Momentum (H) is along the
spin-axis which tend to remain
stabilized in inertial space.
Newton 2nd Law: Angular Momentum
𝑯 = 𝑰𝝎𝒔 of a body will remain
unchanged unless it is acted upon by a
torque T and that the rate of change of
angular momentum𝑑𝐻
𝑑𝑡is equal to the
magnitude of applied torque T given
as:
Law of Gyroscope
If the applied torque acts about the
spin-axis its effect is to increase only
spin velocity
𝑇 =𝑑𝐻
𝑑𝑡
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s s s
s s s
d I ddHT I I
dt dt dt
If T is ┴ to H, the T will change the direction
of H, casing angular rate 𝜔𝑝 (precession)
about an axis ┴ to both H and T.
and we have
Therefore
p
p p
p
dHdH Hd T
dt
Hd ddHT H H
dt dt dt
Law of Gyroscope
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There are two basic classes of rotation sensing gyros:
Rate gyros
The output is relative to the angular speed
Rate integrating gyros
Indicate the actual turn angle or heading
The angle is relative => must be initially referenced to a
known orientation
Angle is anyway integrated from angular speed
The primary measuring magnitude of a gyro is always
angular speed!!
Two Broader Types of Gyros
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SDOF Gyroscope
Single Degree of Freedom Gyro
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SDOF Gyroscope
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2-DOF Gyroscope
Basic Parts of 2-DOF Gyroscope
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2-DOF Gyroscope - Rigidity
The base surface turns around the outer gimbal axis or around
the inner gimbal axis, but the spin axis is stabilized in space
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Mechanical Gyro Principle
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Mechanical Gyro – How it Works?
When torque is applied about the inner gimbal axis, the gyro will
precess about the outer gimbal axis
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Force
Plane of rotation Plane of force
Plane of precession
Mechanical Gyro Principle
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Mechanical Gyro – How it Works?
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Mechanical Gyro – How it Works?
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Mechanical Gyro – How it Works?
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Errors in Gyroscope
Error Definitions Causes
Fixed Drifts
The sensor output which is present even in the absence of an applied
input rotation. The size of the bias is independent
of any motion to which the gyroscope may be. It is usually expressed
in units of degrees per hour (°/h.
Residual torques from flexible leads
spurious magnetic fields and
temperature gradients.
g-Dependent Drifts
Proportional to the magnitude of the applied acceleration. The
relationship between these components of bias and the applied
acceleration can be expressed by means of coefficients having units of
o/h/g.
Mass unbalance in the rotor suspension, that
is, non-coincidence of the rotor center of
gravity and the center of the suspension
mechanism.
An-isoelastic drifts
(g2-dependent
drifts)
Biases which are proportional to the product of acceleration along
orthogonal pairs of axes. The anisoelastic coefficients have units of
o/h/g2
Gyroscope rotor suspension structure,
particularly the bearings, has finite
compliances which are unequal in different
directions.
Anisoinertia
errors:
Inequalities in gyroscope moments of inertia about different axes.
The resulting biases are proportional to the product of angular rates
applied about pairs of orthogonal axes. The anisoinertia coefficients
may be expressed in units of°/h/(rad/s)2.
These are consequence of the elastic coupling
between the magnetic ring on the rotor and
the rotating magnetic field.
Scale-factor
errors:
Scale-factor non-linearity relates to thermal
changes that result in changes of the
magnetic flux
Cross-coupling
errors:
Erroneous gyroscope outputs resulting from gyroscope sensitivity to
turn rates about axes normal to the input axis. Expressed as parts per
million or a percentage of the applied angular rate
Due to non-orthogonality of the sensor axes.
Angular
acceleration
sensitivity:
This error increases with increasing frequency of input motion.
mechanical gyroscopes are sensitive to
angular acceleration owing to the inertia of
the rotor.
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Components of Errors in Gyroscope
Each of the errors will, in general, include some or all of the following
components:
• Fixed or repeatable terms
• Temperature induced variations
• Switch-on to switch-on variations
• In-run variations.
ˆ 1x fx x x y x z z gx x gz z axz x z xB S M M B a B a B a a
Typical Mathematical Equation for Spinning Wheel Gyros
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𝜔𝑥 𝜔𝑥, 𝜔𝒚, 𝜔𝒛 𝐵𝑓𝑥 𝑆𝑥 𝑀𝑦 , 𝑀𝑧 𝐵𝑔𝑥, 𝐵𝑔𝑧 𝐵𝑎𝑥𝑧 𝜂𝑥
Indicated
rate about
x-axis
Acceleration applied
along x,y,z axis
Fixed Drift
along x-axis of
gyro
Scale
Factor
Error
Cross-coupling
coefficients
g-sensitive bias
coefficients,
Anisoelastic
drift coefficient
zero-mean
random drift
Ring Laser Gyro
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Input Axis
Ring Laser Gyro
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Fibre Optic Rate Sensor
• When gyro is rotated the rate of rotation is proportional to the phaseshift between the beams (Sagnac phase shift)
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• Performance Specifications Measurement range
Number of sensing axes SDOF or 2DOF
Nonlinearity
Bandwidth
Angular Random Walk (ARW) [for optical gyros]
Drift
Drift Instability
Cost
Working temperature range
Shock survivability
Temperature range
Size/Mass/dimensions
• Specification Guidehttp://www.globalspec.com/SpecSearch/SearchForm/sensors_transducers_detectors/tilt_sensing/gyroscopes
Common Gyroscope Criteria
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Typical performance characteristics
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Accelerometers
Inertial navigation depend upon the measurement and integration of linear
acceleration (in a reference frame) to compute velocity and position. It is the
function of accelerometer to provide measurement of acceleration in a
known reference e frame.
Choice of Accelerometer
Cost
Overall accuracy requirement of the INS.
Size
Weight
Power consumption
Reliability
Temperature range
Accelerometer
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Accelerometer operate by measuring the inertial force generated when
a mass accelerates.
The inertial force might deflect a spring, it might change the tension in a
string and its vibrating frequency, or it might generate a torque that will
precess a gyro.
Parts of Accelerometer
1. Proof Mass
2. Suspension mechanism for locating
the proof mass.
3. Pickoff mechanism which puts out a
signal proportional to the applied
acceleration
4. Electro-magnetic force generator to
oppose the inertial force.
5. Electronic servomechanism
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2
2 x
d x dxF m c K x
dt dt
For steady acceleration and mass displacement is steady (transient oscillation died
away)2
2 x x
x
d x x mm K x ma K x
dt a K
Inertia force is balanced by the opposing spring force and x is a measure of applied
acceleration. The scale factor is m/Kx
xn
K
m
Pendulous Accelerometer
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Working Principle of Pendulous Accelerometer
On experiencing acceleration, a torque is produced about output axis
given as:
T FlNewton Law: F ma
( )T mal a ml aP For close-loop, T is proportional to rebalance feedback current i (T=Kfi).
The same current is taken as output of the accelerometer.
f
f f
KT K i aP K i a i
P
Scale Factor
1
( )
( )
f
PaK
f
Output Signal i P PendulosityK
input acceleration g a K Forcer Scale Factor
Quartz Flexure Accelerometer
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Quartz Flexure Accelerometer
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S/N Parameters Required Value1 Measuring Range ±50g
2 Threshold & Resolution 5μg
3 Bias K0/K1 ≤(±4 mg)
4 Scale Factor Kl 1.3±0.2 mA/g
5 2nd Order nonlinearity coefficient K2/Kl ≤±20μg /g2
6 0g, 4-hours short time stability ≤15 μg
7 1g, 4-hours short time stability ≤15 ppm
8 Bias Standard deviation σK0( 1σ,one month) ≤20 μg
9 Std Deviation of scale factor σK1/K1( 1σ,1month) ≤20ppm
10 Std Deviation of 2nd Order nonlinearity Coeff σK2/K1( 1σ,1 month) ≤±15 μg /g2
11 Bias thermal coefficient ≤±20 μg /℃12 Scale factor thermal coefficient ≤±30 ppm /℃13 Noise (across sample resistance 840Ω) ≤5mV
14 Natural Frequency 400~800 Hz
15 Bandwidth 800~2500 Hz
16 Vibration 5g(20-2000Hz)
17 Shock 100g,5ms,1/2sin
18 Temperature range(Operating) -40-+85℃19 Temperature range -60-+120℃20 Operating Voltage ±12~±15V
21 Current ≤±20mA
22 Temp. sensor Optional
23 Size Ф25.4X30mm
24 Weight ≤80garm
Typical Performance Specification of Q-Flex Accelerometer
Ref:
http://www.ktjmyq.com/en/shiying/JB_01JiaSuDuJi/
https://aerospace.honeywell.com/~/media/Brochures/Q-Flex%20QA-2000%20Accelerometer.ashxOct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 35
Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 36
2 3
2 3
1
ˆi o i i i io p ip i p ip o io i o
Ea K a K a K a d a K a a d a K a a
K
𝑎𝑖 : Indicated Acceleration – output of accelerometer (g)
𝐸 : Output in the sensor units (V, I or Hz)
𝐾1: Scale Factor, (V or I or Hz)/g
Ko: Fixed Bias, g
ai: Acceleration along Input Axis
K2: 2nd Order Nonlinearity Coefficient (g/g2)
K3 : 3rd Order Nonlinearity Coefficient (g/g3)
dio: Misalignment of input axis about output axis
ap : Acceleration along Pendulous Axis (g)
Kip : Cross Coupling input axis along pendulous axis (g/g2)
dip : Misalignment of Input about Pendulous axis
ao : Acceleration along output axis (g)
Kio : Cross Coupling input axis along output axis (g/g2)
Typical Mathematical Equation for Accelerometer
Some other types of
Accelerometers
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2-dimensional Accelerometer
On application of acceleration optical fiber is deflected and displacement is
sensed by a laser beam passing through the optical fiber and being focused on
a two dimensional photo-sensitive array.
Optical Fiber Accelerometer
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Pneumostatic Accelerometer
The deflection of mass along the longitudinal axis will be sensed by the pickoff and an
output voltage will be developed, proportional to the displacement. After amplifying,
demodulating, and filtering this signal, it is fed back negatively to the force generator to
develop a force fd equal and opposite to the reaction force f. In steady state, the two
forces will be equal and the current input if to the forcers will be a measure of the applied
acceleration ai.
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Vibrating string accelerometer
At zero input acceleration, each wire will, theoretically, oscillate at the same frequency
since the tensions T1 and T2 of the strings are equal. However, when and input
acceleration ai exists, the tension T1 in one string increases while the tension T2 in the
2nd wire decreases. As a result of the change in string tensions, each string will have
new natural frequency and they will no longer be equal. The difference between two
frequencies is a measure of the applied acceleration.
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Vibrating Beam Accelerometer
Each beam is made to vibrate at its
own resonant frequency. In the
absence of any acceleration along the
axis sensitive to acceleration, both
beams vibrate at the same resonant
frequency. When an acceleration is
applied along the sensitive axis, one
beam experiences compression
whilst the other is stretched, or under
tension, owing to the inertial reaction
of the proof mass. The result is that
the beam in compression experiences
a decrease in frequency, whereas the
beam in tension has an increase in
frequency. The difference in
frequency is measured and this is
directly proportional to the applied
acceleration.
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Vibrating String 3-Axis Accelerometer
Surface acoustic wave accelerometer
When an acceleration is applied
normal to the plane containing the
beam, the inertial reaction of the
assembly causes the beam to
bend. When the surface of the
beam is subjected to an applied
strain, as occurs when the beam
bends, the frequency of the
surface acoustic wave changes in
proportion to the applied strain.
Comparison of this change with
the reference frequency provides a
direct measure of the acceleration
applied along the sensitive axis.
Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 43
Resonant silicon accelerometer
frequency sensitive resonant tie bars
integrally attached to a silicon
seismic mass. These tie bars are
maintained at mechanical resonance,
typically vibrating at frequencies
between 40 and 100 kHz depending
on the configuration. When an
acceleration is applied along the
sensitive axis, movement of the
seismic mass induces a strain in the
tie bars resulting in a change in
frequency of the order of tens of hertz
for each applied unit g. This change
in frequency is reasonably
detectable.
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Photo-elastic accelerometer
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Piezoelectric Accelerometers
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It has a chamber of gas with a heating element in the center and four
temperature sensors around its edge.· Just as hot air rises and cooler air
sinks, the same applies to hot and cool gasses.· If you hold the accelerometer
still, all it senses is gravity.· When you hold the accelerometer level, the hot
gas pocket is rises to the top-center of the accelerometer’s chamber, and all
the temperature sensors measure the same temperature.· Depending on how
you tilt the accelerometer, the hot gas will collect closer to one or maybe two
of the temperature sensors.·
Heated Gas Accelerometer
Capacitive Accelerometers (MEMS)
Stationary Polysilicon fingers
Based on ADXL accelerometers, Analog Devices, Inc.
Spring Inertial Mass
Anchor to substrate
Displacement
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Mach-Zehnder Accelerometer
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GeneralDefinitions
&
Specifications
Sensor Specifications - Definitions
The bias of a sensor is the signal it gives when
there is no input
Bias:
Scale Factor: The scale factor is the ratio between a change in
the output signal and the change in input.
Scale factor Asymmetry:Instrument have a different scale factor for
positive and negative inputs, known as scale
factor asymmetry.Input Axis:
Input axis is the axis along which (accelerometer) or about
which (gyro) an input causes a maximum output.
Residuals:
Residuals are the differences between the actual outputs
and the value that would be predicted using the calculated
scale factor.
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Sensor Specifications - Definitions
The composite error is the ratio of the largest residual to the full scale range.
Composite Error:
Gyro Drift:
A gyro drift is the change in misalignment angle over time, this time
varying misalignment causes cross-coupling into the accelerometer channels.
Random Drift:
If the sensor is allowed to run on a stable base, its output will wander
some small amount due to disturbances inside the sensor, called random
drift. It is characterized by the standard deviation of the output measured
periodically for some specified time.
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Sensor Specifications - Definitions
Sensors have a lower limit below which they can not
detect small changes in input, which can be regarded
as a dead band around null.
Threshold:
Dead Band:
Threshold is defined as the largest value of the
minimum input that produces an output of at least
half the expected value.
Resolution: Resolution is defined as the largest value of the
minimum change in input that produces an output of
a specified proportion of the value expected using the
scale factor.
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Error Specifications
In general, errors fall into either of the two categories:
In dealing with errors and error analysis, it is necessary to describe errors
mathematically in order to make possible the several computations
necessary for studying the propagation of errors, as well as to place a
quantitative limit or tolerance on the value of error.
Predictable
Unpredictable
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Error Specifications (Cont’…)
The predictable errors are usually simple in form
and easy to describe mathematically, such as with
constant coefficients.
The unpredictable errors are usually random in
nature and statistical techniques are generally
used for their description.
An unpredictable error is one in which the
measurement history does not provide means for
accurately knowing what will happen at any
arbitrary future time.
Errors in INS which are predictable are generally
compensated.
Those errors which are unpredictable are treated
statistically to obtain a mathematical specification
of the error.
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Error Specifications (Cont’…)
Experiences have shown that the random errors associated with INS
components have a Gaussian or normal distribution.
When two or more errors are combined with one another, it is generally
assumed the errors are independent.
In view of the Gaussian distribution of random errors in INS, the
arithmetic mean and standard deviation serve as a complete description
of the random variation.
The standard deviation is the root mean square (RMS) deviation of the
values from their arithmetic mean. For example, in the population {4, 8},
the mean is 6 and the standard deviation is 2.
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Error Specifications (Cont’…)
Standard deviation is the most common measure of statistical
dispersion, measuring how widely spread the values in a data set are.
If the data points are all close to the mean, then the standard
deviation is close to zero.
If many data points are far from the mean, then the standard
deviation is far from zero.
If all the data values are equal, then the standard deviation is
zero.
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Error Specifications (Cont’…)
The standard deviation of a discrete uniform random variable X
can be calculated as follows:
For each value xi calculate the difference between xi and the
average value
Calculate the squares of these differences.
Find the average of the squared differences. This quantity is the
variance σ2.
Take the square root of the variance
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Error Specifications (Cont’…)
The value P refers to the part of the area that is enclosed by the
curve and the base line between the values + and -1 sigma,
(light blue area) and + and -2 sigma (light blue + medium blue
area), respectively, or + and -3 sigma (light blue + medium blue
+ dark blue areas).Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 59
Error Specifications (Cont’…)
This means that 68.26 percent of all readings of an ideal distribution scatter
with 1 sigma, 95.46 percent with 2 sigma and 99.73 percent with 3 sigma
around the mean.
From probability viewpoint, the 1-sigma value means that the function x(t)
will be less than + 1-sigma value 68% of the time.
It is 68% probable that the value of x(t) will not exceed + 1-σ.
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INS Error Sources - Accelerometer
Error of random
nature added to the
measurement
Accelerometer measurement noise
A fixed bias in the
measured valueAccelerometer bias
Errors in the calibrated
accelerometer scale
factors
Accelerometer scale factors
Error in the alignment of
the accelerometer axes
from the platform axesAccelerometer alignment
Deviation from the
defined linear
input/output relationship
Accelerometer non-linearity
Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 61
INS Error Sources - Gyroscope
random additive error on the
measurement
Gyro measurement noise
a standard bias in the measured
angular rateGyro drift (bias)
Gyro scale factorerror in the calibrated scale
factor of the gyro
error in the alignment of the gyro
axes from the orthogonal
platform axes
Gyro alignment
Gyro g-sensitivity sensitivity of the instrument output to force applied along or
perpendicular to the sensitive axis of the gyro
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Testing of Inertial Sensors
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Index Head
Precision CentrifugeCourtesy: www.ktjmyq.com/ceshi/4_2_JingMiLiXinJi/inJi/
Turn Tables (1,2,3-axis)
Equipment Used for Testing & Calibration
Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 64
1. Titterton, David, and John L. Weston. Strapdown inertial navigation
technology. Vol. 17. IET, 2004.
2. Lawrence, Anthony. Modern inertial technology: navigation, guidance, and
control. Springer Science & Business Media
3. Chen, Zhe. "Strapdown inertial navigation system." (1986).
4. Inertial Technology, Class Notes, Professor Zhang Chang Yun, Beihang
University, 2012
5. Inertial Navigation Systems, Class Notes, Professor Yu Wen Bo, Beihang
University, 2012
6. Research on SINS Based Navigation Techniques, Master Thesis,
Muhammad Ushaq, Beihang University, 2003
7. https://en.wikipedia.org/wiki/Gyroscope
Salient of References
Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 65
Next Lecture
MEMS & NEMSGyroscopes and Accelerometers
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