DYNAMIC PROPERTIES OF ELECTRONIC GYROSCOPES FOR INERTIAL MEASUREMENT UNITS

VSB – Technical University of Ostrava Faculty of Mechanical Engineering Ostrava, Czech Republic

Jiří Tůma & Jiří Kulhánek

© VSB-TU Ostrava, 2007 2

Outline

MotivationKalman filter – Signal fusionPrinciple of iMEMS gyroscopes and accelerometersTest stand for electronic gyroscopes Frequency response function measurementsKalman filter for drift-rate biasConclusion

© VSB-TU Ostrava, 2007 3

Motivation

Global position systemInertial navigation system

© VSB-TU Ostrava, 2007 4

Angular velocity

Inertial Navigation System (1 DOF)

Kalman filter

Accelerometer

Gyroscope

Angle

Angle TrueΘ

mω

Error

Drift-rate bias

mΘ

Trueb

Θ

Θn

Error rn

TrueΘ&

Measured quantities

Estimation

bDrift-rate bias

Unknown (hidden)quantities

© VSB-TU Ostrava, 2007 5

Kalman Filter – Gyro Noise Model

rTruemTrue nb ++= ωΘ& wTrue nb =&

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

w

rm

True

True

True

True

nn

bbdtd ω

ΘΘ01

0010

nxFdtdx

+=

[ ] Θ

ΘΘ n

bTrue

Truem +⎥

⎦

⎤⎢⎣

⎡= 01

ΘΘ nxHm +=

Continuous time t , angle , angular velocity , drift-rate bias

mm ωΘ ,Measured quantities TrueTrue b,ΘUnknown quantities

White noise error

Model

Θ ω b

Matrix form

( )( ) ( )( ) ( )( ) ΘΘ NtnENtnENtnE wwrr === 222 ,,

⇒

⇒

© VSB-TU Ostrava, 2007 6

Kalman Filter – Signal Fusion

HPRPHQPFFPP TT 1−−++=&

( )⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +==⇒= −

2

110kk

NNNNNNRPHKP

w

wrT

Θ

ΘΘ&

( ) ( ) ( ) ( )21

2

2

212

21

212

2

,ˆksks

ssFsksks

ksks

sksks

ss mm

++=

+++

+++

= ΘΩΘ

[ ]ΘNRN

NQ

w

r =⎥⎦

⎤⎢⎣

⎡= ,

00

The continuous Kalman filter equation for the state covariance matrix P is:

The Kalman gain at the steady state operation:

where

The Laplace transform of the angle estimate:

© VSB-TU Ostrava, 2007 7

Kalman Filter – Frequency Response Function

( ) ( ) ( ) ( )( ) ( )ssFs

ssFs mm ΘΩΘ −+= 1ˆ

The Laplace transform of the angle estimate

integration

Roumeliotis, Sukhatme, Bekey: Circumventing Dynamic Modeling: Evaluation of the Error-State Kalman filter applied to Mobile Robot Localization

dB

0 Log ω

|F(jω)| |1-F(jω)|+40 dB/dec|F(jω)/jω|+20 dB/dec

2kRoll-off:

s = jω

© VSB-TU Ostrava, 2007 8

Common principle of iMEMSgyroscopes

Direction of rotation

iMEMS: Integrated Micro–Electro Mechanical System

Inner frame

Resonating mass

Mass drive direction

Springs

Coriolis sense fingers

Analog DevicesADXRS300

© VSB-TU Ostrava, 2007 9

Application Circuit

© VSB-TU Ostrava, 2007 10

Common principle of iMEMSaccelerometers

iMEMS: Integrated Micro–Electro Mechanical System

Acceleration

Suspensions

Substrate

Anchor

Sense fingers

Analog DevicesADXL105

© VSB-TU Ostrava, 2007 11

Test Stand

shaker

spring

gyroscope

spindle

© VSB-TU Ostrava, 2007 12

Instrumentation

TIRA shakerTIRA power amplifierLabShop PULSESignal Analyzer software

© VSB-TU Ostrava, 2007 13

Response to Swept Sine Input SignalTachometer : Sec Swept Sine : Expanded Time(Acc)

01020304050

0 5 10 15 20 25 30 35 40

Hz

Time History : Sec Swept Sine : Expanded Time(Acc)

-20-10

01020

0 5 10 15 20 25 30 35

rad/

s

Time History : Sec Swept Sine : Expanded Time(Gyroskop)

-20-10

01020

0 5 10 15 20 25 30 35Time [s]

rad/

s

© VSB-TU Ostrava, 2007 14

Gyroscope Frequency Response Function FRF : Time(Gyroskop) / RE: Time(Acc)

-40-30-20-10

01020

1 10 100Frequency [Hz]

FRF

[dB

]

FRF: Time(Gyroskop) / RE:Time(Acc)

0,00,20,40,60,81,01,2

1 10 100Frequency [Hz]

FRF

Coh

eren

ce

Autospectrum : Time(Acc) ; Time(Gyroskop)

0,0000,0010,0020,0030,0040,0050,0060,007

1 10 100

Frequency [Hz]

RM

S m

/s

Acc Gyroskop

© VSB-TU Ostrava, 2007 15

Gyroscope Drift-Rate Bias Spectrum

Autospectrum : Signal

0,01

0,10

1,00

0 1 2 3 4 5 6 7

Frequency [Hz]

PS

D (d

eg/s

)^2/

Hz

© VSB-TU Ostrava, 2007 16

Kalman filter for Drift-Rate Bias

Time History : Signal & Kalman Filter (Signal)

475476477478479480481482

0 1000 2000 3000

Time [s]U

nit

Signal

Kalman

Statistics : Signal

1

10

100

1000

10000

100000

476

477

478

479

480

481

Unit

Num

ber

wnn nbb += −1ModelHistogram

© VSB-TU Ostrava, 2007 17

ConclusionThe paper is focused at the gyroscope frequency response function and thermal noise measurementsAs a paper motivation, it is to design the Kalman filter for inertial navigation systemsThe Kalman filter is way how to fuse the acceleration and gyroscope signalThe test stand for frequency response function was designedThe theory is illustrated by result of measurements.

© VSB-TU Ostrava, 2007 18

Thank you for your attention