introduction to inertial sensors - ushaq - oct 2015

An Introduction

To

Inertial Sensors

Muhammad Ushaq

Institute of Space Technology

Islamabad, Pakistan

[email protected]

0092-322-2992772

Oct-15 Introduction to Inertial Sensors (Muhammad Ushaq) 1

mailto:[email protected]

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)


Inertial Navigation Systems (INS)


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.


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.

http://www.gyroscopes.org/showfull.asp?imagename=1.jpg


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).


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

2 Axis Platform.avi


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.


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


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


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


SDOF Gyroscope

Single Degree of Freedom Gyro


SDOF Gyroscope


2-DOF Gyroscope

Basic Parts of 2-DOF Gyroscope


http://en.wikipedia.org/wiki/Image:Gyroscope_wheel-text.png

http://en.wikipedia.org/wiki/Image:Gyroscope_wheel-text.png



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




Mechanical Gyro Principle




Mechanical Gyro – How it Works?

When torque is applied about the inner gimbal axis, the gyro will

precess about the outer gimbal axis




Force

Plane of rotation Plane of force

Plane of precession

Mechanical Gyro Principle

Oct-15

Introduction to Inertial Sensors (Muhammad

Ushaq) 17


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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.


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


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


Input Axis

Ring Laser Gyro


Fibre Optic Rate Sensor

• When gyro is rotated the rate of rotation is proportional to the phaseshift between the beams (Sagnac phase shift)


• 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


Typical performance characteristics



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


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


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



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


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

http://www.ktjmyq.com/en/shiying/JB_01JiaSuDuJi/

https://aerospace.honeywell.com/~/media/Brochures/Q-Flex QA-2000 Accelerometer.ashx


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


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


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.


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.


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.



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.


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.


Photo-elastic accelerometer



Piezoelectric Accelerometers


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


Mach-Zehnder Accelerometer


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.



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.



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.


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


http://www.animationlibrary.com/animation/28828/Digger_thinks/

http://www.animationlibrary.com/animation/28828/Digger_thinks/

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.



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.



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.





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





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




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.


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


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


Testing of Inertial Sensors


Index Head

Precision CentrifugeCourtesy: www.ktjmyq.com/ceshi/4_2_JingMiLiXinJi/inJi/

Turn Tables (1,2,3-axis)

Equipment Used for Testing & Calibration


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


Next Lecture

MEMS & NEMSGyroscopes and Accelerometers

introduction to inertial sensors - ushaq - oct 2015

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