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78

Inertial Navigation Systems

Centrale SEXTANT SIRAL

Integrated INS/GPS miniature navigation system, ELEKTROPRIBOR, Russia

120 mm

Sigma 40DX

Sigma 95

MARINS

EPSILON 10

KN-4072A

LN120G LASEREF VI

QUASAR 3000

ADIRU

79

Inertial Technology Manufacturers

1 mg

Gyrometers accuracy

AccelerometerAccuracy

150°/h 1°/h 0.1°/h 0.01°/h 0.001°/h

10 mg

100 µg

10 µg

30°/h

HWL

HWLHWL

Litton

Litef

Sagem

Sagem

Sagem

AISAIS

L3Com

Sagem

Kearfott

Kearfott

Litef

Litton

Astrium

Thales

ThalesThales

Litef

Thales

MEMSDomaine

RLG - FOG RLG - FOG HPMEMS

Driven by price and technology breakdown

KVHHWL

Sagem

HWL

LITEF

LITEF

iXBlue

80

Inertial navigation systems

Principles Basic ideas Compensation of Attitude reference

Architectures : strap-down and gimbaled

Gimbaled systems

Strap-down systems

Key issues

gr

81

Inertial navigation systems

Global design Basic idea : double integrate inertial forces in order to compute

velocity and position

∫ ∫V P

Reference frame

Acceleros.

Gyros.

Inertial force

And then, how would the attitude be measured ?

(to be continued…)

82

Aboard the vehicle, the pilot is subject to all the forces exerted by the seat.

These forces are such that the pilot remains fixed in the cockpit.

F = m ( γ − g)

What an accelerometer does measure

Thrust

= true vehicle’s acceleration m γγγγ

R2 = mγγγγThrust transmission

Reactive force to gravity mg

R1 = −−−− mg

Resulting force exerted by the seat to the pilot

mγ γ γ γ −−−− mg

83

An angular reference frame is necessary !

Let a vehicle turn clockwise at a constant e rotational speed ω rad.s−1

and constant tangential velocity v.

The accelerometers provide :

• Nul acceleration on the X-axis • Constant centripetal acceleration Rω2 on the Y-axis

γ γ γ γ −−−− g−−−− g

γγγγvehicle

The orientation of the measured acceleration is needed.

If the INS system assumed constant orientation, it would compute a parabolic trajectory along the Y-axis !

84

Inertial navigation systems : two architectures

Strap-down INS

Gimbaled INS

Cre

dits

:In

ertia

l Nav

igat

ion

–F

orty

Yea

rs o

f Evo

lutio

n, b

y A

. D. K

ING

, B

.Sc.

, F.R

.I.N

., M

arco

ni E

lect

roni

c S

yste

ms

Ltd.

85

Design of inertial navigation systems : two archite ctures

The angular reference is used to compensate for . Then The linear acceleration may be integrated. The position is the result of numerical integration of the speed.

The angular reference is also the reference frame, with respect to which the attitude is defined: it may then be measured or computed.

Gimbaled Strap-down

The attitude is measured on the gimbals : the reference frame is

indeed a piece of hardware.

The attitude of accelerometers is constantly integrated : the attitude of the vehicle is

computed.

gr

86

Strap-down inertial navigation systems

The inertial attitude of the accelerometer cluster is constantly integrated, so that the inertial force may be properly oriented in inertial space.

Other definitions of attitude are obtained from computations.

Elektropribor THALESTOTEM 3k

87

Gimbaled inertial navigation systems

The angular position of the vehicle is constantly mechanically compensated, so that the accelerometer platform keeps steady. Implies the use of gimbals.The vehicle attitude is measured on gimbal pickoffs, which provide the movement of the outer body around the steady platform.

Elektropribor

88


89

Accéléromètres et gyromètres

Accels

Gyros

axe d’entrée

fr

efrr

⋅er

Projection de l’accélération apparente sur l’axe d’entrée →

mesure

Principe de mesure : étudier et/ou rendre immobile une masse d’épreuve interne m par

rapport au boîtier

charnière

masse d’épreuve m

Impossibilité de séparer les effets de la gravité locale et des mouvements

propres du véhicule

Mesure d’une « accélération apparente » dite force spécifique

gfrrr

−= γ

axe d’entréeer

ωr err

⋅ωProjection de la vitesse de

rotation instantanée sur l’axe d’entrée → mesure Impossibilité de mesurer l’orientation

absolue du véhicule

Principe de mesure : effets Coriolis sur une masse interne m ou étude d’effets relativistes

Mesure d’une vitesse de rotation instantanée (ou débattement angulaire)

90


Change of coordinate frame

x

y

z

Angular integration of

gyros

x

y

z

Accels

Compensation of gravity

∫∫

Attitude of the sensor unitrelative to the reference frame

Initial attitude of the accelerometers relative to the reference frame

Initial position and velocity of the vehicle

Gyros

( ) ( ) frame refacc ffrr

→rvrr

,

rr

Vehicle attitude relative to local geodetic frame

( ) ( )( ) frame ref

frame refframe ref

g

fr

rr

+=γ

( )( ) frame refrgrr

Local gravity model

rr

Examples of reference frame : Earth Fixed Earth Centered, Copernic, …

Inertial navigation systemDouble integration of an apparent, measured acceleration • reconstructed angular position of the accel cluster, through gyros• compensated by local gravity, modeled as a function of the estimated position

Inertial navigation systemDouble integration of an apparent, measured acceleration • reconstructed angular position of the accel cluster, through gyros• compensated by local gravity, modeled as a function of the estimated position

( ) frame refγr

Angular position

91

x

y

zBody coordinate frame

x

yz

Local geodetic / navigation coordinate frame

Lon

Latxy

z

Earth Fixed Earth Centered coordinate frame

nn

bb

ee

True north pole

Common coordinate frames

tt

Inertialframe

ii

x y

z

Fixed relative to stars

92

Computation of position and velocity (1)

Velocities

So, for terrestrial applications, need to integrate :

Hence for t = EFEC :

t

rvrr ∆

= for terrestrial applications

t : EFEC or North-East-Down

for spatial vesselsi

rvrr ∆

=

( )( )rvrv

rgfr

ititit

it

ii

rrrrrrr

rrrr

×+×+×+=

+=

/// 2 ωωω

( ) ⊥Ω+×Ω−+= rvrgfv EE

t rrrrrrr 22

Time derivatives wrt rotary frames

93

Plumb bob vs Newton gravity

Earth is rotating in the inertial frame, thus inducing an inertial force that adds to the gravity field.

By the definition, we denote

And will omit the « Plumb bob » subscript in the following. Therefore :

( ) ( ) ( )rrgrgrrrrrrr ×Ω×Ω−=bob Plumb

( ) vrgfv E

e rrrrrr ×Ω−+= 2

94


Actual computation should be able to add measured in accelerometer axes to

usually available in some global frame (eg. ECEF)

Therefore : a coordinate transform between accelerometer axes and global axes is absolutly needed !

some data is measured : modeled : computed : perfectly known :

fr

( )rgrr

gyros

measdatamoddata

data

cdata

95


How do gyros provide proper coordinate transform between global and accelerometer axes (body) ?

They measure : in body axes.

and the transition matrix from i to b obeys

ib /ωr

( )

=

z

y

x

bib

ωωω

ω /

r

( )[ ]

ib

xy

xz

yz

ibbibib

T

TTdt

d

/

///

0

0

0

⋅

−−

−=

⋅×−=

ωωωωωω

ωr

96


(accelero compensation)

Henceforth :

Then

Similarly :

( ) ( )

⋅⋅=

⋅==

z

y

x

biie

bbeECEFECEF

f

f

f

TT

fTff

//

/

rr

( )( ) ( )e

z

y

x

biieECEF g

f

f

f

TTrgfrrrr

+

⋅⋅=+ //

( ) ( )

⋅⋅⋅=

⋅==

z

y

x

biieen

bbnnn

f

f

f

TTT

fTff

///

/

rr

97


Actual computation ; let’s summarize :

is measured in accelerometer axes

is a model of the true gravitational field, computed at , which is the best estimate of the position we have so far.

is a mathematical function of time

z

y

x

f

f

f

( )ECEFgr cr

r

ECEFiT /

98


So the actual computation of position and velocity will involve the following in the ECEF axes :

Nota bene : need initial estimates !

( ) ( )

⋅

−−

−=

Ω+×

Ω−

+

⋅⋅=

=

⊥

cib

cib

cE

c

E

c

z

y

x

ECEFic

ibc

cc

TTdt

d

rvr

g

g

g

tTTvdt

d

vrdt

d

//

2// 0

0

2

mod

meas

0

0

0

meas

z

y

x

xy

xz

yz

ωω

ωω

ωω

f

f

f

cib

cc Tvr /,, ( ) ( ) ( )000 cb/i

cc T,v,r

99

Computation of position and velocity in n (7)

Le cas de n = Trièdre Géographique Local (TGL) ou North-East-Down

avec

( ) ( ) vrgfv E

n rrrrrrr ×Ω+−+= 2ρ

en /ωρ rr ∆=

100

( )

( ) ( ) ( ) ( ) ( )

⋅

−−

−=

×

Ω⋅⋅+−

+

⋅⋅⋅=

×−=

cib

cib

c

E

cen

cc

z

y

xcbiie

cen

c

cccc

TTdt

d

vrTvr

g

g

g

TtTrTvdt

d

rvvrdt

d

//

//// 0

0

2

mod

meas

0

0

0

meas

z

y

x

xy

xz

yz

ωω

ωω

ωω

f

f

f

ρ

ρ

Computation of position and velocity in n (8)

101

Hardware design of computation

102

Strap-down INS : two time scales

Schematic of strap-down calculus

Sen

sor

acqu

isiti

on

Acquisition and processing of sensor data

Thermal compensations

Virtual platform Interfaceacceleros +

acceleros -

dither

gyros

velocity increments

angular increments

attitudes

quaternion

angular velocity

linear velocity

acceleros sums

temperatures

Compensation parameters

High frequency

Low frequency

103

Strap-down INS : two time scales, why ?

ConingSculling Centrifugal

accelerations

aktz −=Ω∫ π2d(Goodman theorem)

ΩΩΩΩ

Ωz

YX

Gyro input axis

Z

Sinusoidal acceleration input γ = γο cosωt

Simultaneous rotation of input axis at ωrad/s

ψ = ψ0 cos (ωt + φ)then the accelerometer X will be biased by

γγγγ

Rotational perturbation

φψγ cos2

100sculling =B

γγγγcentrifugal

L

ω=ϕ&Rotation of IMU

2centrifuge ωγ L=

104


High frequency process

acquisition gyros

acquisition capteurs de

vitesse d’activation

filtrage d’activation

mise à l’échelle

correction mésalignements

correction mouvements

coniques

correction de bras de levier

correction de sculling

(mvt. de godille)

correction mésalignements

mise à l’échelle

acquisition accéléros +

acquisition accéléros

−correction

de non linéarité

facteurs d’échelle mésalignements

mésalignementsfacteurs d’échelle

dérives

incréments angulaires

incréments de vitesse

biais

cumuls gyros

cumuls accéléros

+

cumuls accéléros

−

+

+

+

+

+

−

−

vitesses d’activation

Acquisition

Correction

105


Low frequency process

intégration quaternion

normalisation quaternion

matrice de cosinus directeurs

changement de repère angulaire

intégration

transformation

incréments de vitesse

incréments angulaires

attitudes

vitesses angulaires

quaternion

vitesses

accélérations

ωq2

1q ⋅=&

Intégration

106

Hardware design of system

107

Strap-down INS

Strap-down INS TOTEM 3000 (THALES)

108

Strap-down INS

Inertial measurement unit

109

Strap-down INS

FIN3110 instrument cluster

Marconi FIN3110 ring laser gyro inertial navigation unit

Credits : Inertial Navigation – Forty Years of Evolution, by A. D. KING, B.Sc., F.R.I.N., Marconi Electronic Systems Ltd.

110

Strap-down INS

MIMU (Space applications)

HG9900 IMU (from datasheet online)

111

Inertial Navigation SystemsKey issues

112

Key elements of inertial systems

Supporting mechanics

Inertial sensorsGyroscopes and rate gyros

AccelerometersPerformance under

environment, reliability

Stiffness, EM shielding, precision,

optical alignment

SuspensionsShock damage avoidance

Vibration isolation

Control electronicsDedicated compensations

Mechanical housingStiffness, EM shielding, precision, optical alignment, thermal draining, handling

Navigation computerOptimized and modular algorithms,

safety design objectives

Input/Output boardVersatility, performance, safety design objectives

Power supplyStability, power consumption

113

INS : thermal issues

Thermal stabilization

The performances of inertial sensors may significantly depend on temperature. Therefore it might be necessary to control the effects of thermal drifts :

control the temperature of the sensors and/or environment : needs thermal insulation materials and thermostating devicesGimbaled INS are more concerned.

correct thermal drifts using models : needs calibrationModels of order 2, 3 or 4 are used on strap-down systems, whose computation power is appropriate.

114

INS : vibratory environment issues

Suspension The main purpose is to reject vibratory perturbances due to the

environment or dithers. It might be crucial for the inertial sensors to survive the mechanical environment ! performances (coning, sculling, …)

The design of the vibration dampers is a compromise between the passband the efficiency of correction algorithms (coning, sculling) the need to activate laser gyros (dithers).

The design of a strap-down system usually requires the cancellation of assymetric effects and angular disturbances.

Popular vibration damping : pods rings

115

INS : design issues

TechnologyChallenges Gimbaled Strap-down

DesignHard & Complex

Servoing of gyrosTransmitting signals through gimbals requires slip rings

Easy…but HF computations and balancing of IMU

Manufacturing& servicing/maintenance

ComplexMany piecesResolvers

EasyVery few pieces

Calibration Easy Hard

Computation time Small Large

Range of vehicle orientationsRestricted

Gimbals may lock when reaching some positions

FullIf gyro integration algorithm appropriate (quaternion)

IntegrityIntrinsic property

of the « inertial » platformHard

fault in electronics may induce loss of all computations !

116

Inertial navigation systems : take home ideas

Discrete, independent Non-jammableAvailable, reliableUniversal coverage

Very expensiveAccuracy is also a function of volume and mass

Divergent accuracyOperational constraints

117

Structure of INS errors

118

Computation of position and velocity (recall)

The actual computation of position and velocity will involve the following in the ECEF axes :

Nota bene : need initial estimates !

( ) ( )

⋅

−−

−=

Ω+×

Ω−

+

⋅⋅=

=

⊥

cib

cib

cE

c

E

c

z

y

x

ECEFic

ibc

cc

TTdt

d

rvr

g

g

g

tTTvdt

d

vrdt

d

//

2// 0

0

2

mod

meas

0

0

0

meas

z

y

x

xy

xz

yz

ωω

ωω

ωω

f

f

f

cib

cc Tvr /,, ( ) ( ) ( )000 cb/i

cc T,v,r

119

Error analysis : coordinate transform error (1)

Coordinate transform errorsLet’s denote the (small) vector such that

We could show that satisfies the following :

where is defined in the accelerometer axes as

( )( ) ibic

ib TψT // Id ⋅×−≅ r

ψr

ψr

gyroεψ rr−=

i

gyroεψ rr−=

i

gyroεr

( ) ( ) ( )bibbibb /meas

/gyro ωωε rrr−=

« The speed of coordinate transform error as seen from the stars is exactly the gyro error. »

120

Position & Velocity errors

121

Error analysis : position and velocity errors (1)

Velocity errorLet’s denote the amount of specific force measurement error :

One can show that

accεr

( )

−

=

z

y

x

z

y

x

b

f

f

f

f

f

fmeas

accεr

( )

( ) ( ) ( )

( ) ( )⊥−⋅Ω+−×

Ω−

−+−⋅∂∂+

×

⋅+⋅=−

rrvv

rgrgrrgr

f

f

f

TTvv

cE

c

E

ECEFECEFc

ECEF

ECEF

z

y

x

ECEFbc

ECEFbc

2

modmod

/acc/

0

0

r

r&& ψε

122

Error analysis : position and velocity errors (2)

(velocity error continued)

With the assumption that , we end up with the following

« grand model » : ( ) ( )ECEFECEF rgrg ≈mod

⊥⋅Ω+⋅∂∂++×=×Ω+ rr

r

gfvv EE

t rrr

rrvrrrr δδεψδδ 2

acc2 ⊥⋅Ω+⋅∂∂++×=×Ω+ rr

r

gfvv EE

t rrr

rrvrrrr δδεψδδ 2

acc2

t here : ECEF

123

Error analysis : 3D mid-term behaviour (1)

For short to mid term applications (a few hours at most), the effects of are negligible if and are small enough.

The full model then reads :

In North-East-Down axes (NED),

EΩr

−=

⋅∂∂++×=

gyro

acc

εψ

δεψδrv

rr

rrvrr

NED

t

rr

gfv

−=

⋅∂∂++×=

gyro

acc

εψ

δεψδrv

rr

rrvrr

NED

t

rr

gfv

R

g

r

gs

s

s

s

NED

02

2

2

2

200

00

00

=

−−

≅

∂∂ ω

ωω

ωwithr

r

(spherical Earth with radius R)

rrδ v

rδ

Mid-Term Error Model (MTERM)

124

Error analysis : vertical mid-term behaviour (2)

Vertical divergenceOn the local vertical axis (D in NED), MTERM may be further developed as :

This is an exponentially unstable dynamical system, with divergence time

Consequently, when a stable navigation system is needed on the vertical axis, it may not be inertial purely !

( ) ( ) zfv sDDD δωεψδ 2acc 2++×= rvr

& ( ) ( ) zfv sDDD δωεψδ 2acc 2++×= rvr

&

s5662

1 ≅=sω

τ

125

Error analysis : horizontal mid-term behaviour (1)

Horizontal errorsOn the local horizontal plane of NED, equation MTERM now reads

( ) ( ) ( )( ) ( )

( )

−=−=

−+×+×=

DD

NENE

NEsNENEDDNENE rffv

gyro

gyro

2acc

εψεψ

δωεψψδ

r&r

r&r

rrvrvr&r( ) ( ) ( )

( ) ( )( )

−=−=

−+×+×=

DD

NENE

NEsNENEDDNENE rffv

gyro

gyro

2acc

εψεψ

δωεψψδ

r&r

r&r

rrvrvr&r

126

Horizontal errors : simplified modelWhen assuming horizontal accelerations only,

So :

The gravitational field is responsible for errors on the horizontal plane when gyros show significant horizontal reference error ( )

Significant errors due to azimut misalignment occur only if significant horizontal accelerations occur.

Error analysis : horizontal mid-term behaviour (2)

( ) ( ) ( )( ) ( )

( )

−=−=

−+×−×=

DD

NENE

NEsNENEDNENE rgv

gyro

gyro

2acc

εψεψ

δωεψψγδ

r&r

r&r

rrvrvr&r( ) ( ) ( )( ) ( )

( )

−=−=

−+×−×=

DD

NENE

NEsNENEDNENE rgv

gyro

gyro

2acc

εψεψ

δωεψψγδ

r&r

r&r

rrvrvr&r

0rr

≠NEψ

NEγrDψr

127

Error analysis : solving for horizontal errors in mi d-term (1)

Now assume constant sensor errors, as seen from the EFEC coordinate frame

Then :

and

stationary=accεrstationary=gyroεr

0 gyro, NENENE t ψεψ rrr +⋅= 0

gyro, DDD t ψεψ rrr +⋅=

in EFEC

128

Error analysis : solving for horizontal errors in mi d-term (2)

(constant sensor errors continued)

short term contributions of simply follow

short term contributions of are linear (!) :

global, mid-term oscillatory behaviour at fundamental Schüler’s frequency

NEg ψrr×−

!2

0

gyro,0

2

gyro,

apple sNewton' of speed

apple sNewton' of position

×+

×=⋅×+⋅×

NE

NENENE tgt

g

ψ

εψεr

rrrrr

accεr t⋅accεr

129

Error analysis : solving for horizontal errors in s hort-term (more graphically !)

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

Vel

ocity

err

or (

m/s

)

Time (min)

1 mg

0,1 mg

0,01 mg

10 mg

0,001 °/h

0,01 °/h

0,1 °/h

1 °/h

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (min)

Pos

ition

err

or (

m)

0,001 °/h

0,01 °/h

0,1 °/h

1 °/h

0,01 mg

0,1 mg

1 mg

10 mg

accelerometer bias or g ×××× initial alignment error rate gyro bias

130

Error analysis : solving for horizontal errors in m id-term (more graphically !)

Mid-term Schüler damping effect

Initial position error

Rate gyro bias

Accelero bias

0

0

0

positionsvelocities

84 min

131

P & V mid-term errors : orders of magnitude

0 0.5 1 1.5 2 2.5 3-8

-6

-4

-2

0

2

4

6

8

Time in h

Vel

ocity

err

or (

m/s

)

0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Time in h

Pos

ition

err

or (

m)

1 mg

0.1 mg

1 mg

0.1 mg

0.1 °/h

0.01 °/h0.01 °/h

0.1 °/h

0.001 °/h 0.001 °/h

accelerometer bias or g ×××× initial alignment error rate gyro bias

132

Analytical expressions of mid-term velocity errors

It can be shown that velocity errors have a response to gyro drift as follows

Maximum errors are thus

( )( )tDR

tDR

v

v

SN

SW

W

N

ωω

δδ

cos1

cos1

Terre

Terre

−⋅⋅−⋅⋅−

==

Application: consider 0,01°/h on each input axis N, W, U⇒ 0,01×60’ /h = 0,60 ’/h = 0,60 Nm/h = 0,60 ktssince 1°= 60’ = 60 Nm on the Earth (otherwise stated: RTerre x 1° = 60 Nm)

NWMax DRv ,Terre2 ⋅⋅=δ

133

Schüler stabilizing effect

Medium term error propagation : Schüler’s term starts beeing more and more sensitiver

r

g rr

r

∆∂∂

The difference between the computedgravitational horizontal component and thetrue component has an opposite sign to thehorizontal position error.

Henceforth the horizontal Schüler componenthas a stabilizing effect on horizontal error.

Mtrue

Mcomputed

∆∆∆∆g

Mtru

e

Mcomputed

∆∆∆∆g

The difference between the computedgravitational vertical component and the truecomponent has the same sign as the verticalposition error.

Henceforth the vertical Schüler componenthas a destabilizing effect on vertical error.

g(Mcomputed )

g(Mtrue )

134

Mid-term response of velocity to laser gyro Random Walk

2.5 10−3 °/√h

7.5 10−3 °/√h

25 10−3 °/√h

Time (s)

Vel

ocity

err

or (

m/s

)

0 2000 4000 6000 8000 10000 12000-6

-4

-2

0

2

4

6

135

Altitude error

Time in min

Altitude error in m

Mean time of divergence : 9 min 30 s

Exponentially divergent altitude errors

Such a divergent nature

• precludes using of INS as a sole reference for altitude

• calls upon vertical hybridization, with GPS and/or pressure sensor

Initial errors :altitude = 10 mspeed = 0,2 m.s−1

No sensor error !

136

Attitude & Heading errors

137

Error analysis : attitude error (1)

The attitude is defined as the transition matrix from the angular reference frame to the body frame :

So for instance, when reference = NED (which is a local frame):

This leads to the actual computation of attitude :

reference/bT

NEDiib

NEDbb

TT

TT

//

/reference/

⋅==

cc NEDb

cb TT

/reference/ =

function of computed position

138


With the following conventional notations

one can show that

Therefore the computed attitude error reads :

( ) ×−=⇒⋅×−≅ ψψ IdId/// bbib

cib cTTT

( ) ×−=⇒⋅×−≅ θθ IdId/// NEDNEDiNED

ciNED cTTT

( ) NEDbNEDbTT cc //

Id ⋅×−≅ ϕ

ϕθψ rrv +=

ψθϕ vrr −=

139


The main result to keep in mind about attitude errors wrt the local geodetic frame (NED) is that horizontal components are pretty much bounded !

The divergent horizontal position errors are compensating for the inertial reference error : horizontal errors are reflecting inertial reference errors.

Horizontal components of attitude errors are damped by gravity through Schüler loops (accelerometers) : since the gravitation field is essentially vertical, the local horizontal plane is the orthogonal to accelerometers’ signals. This information is biased when the vehicle is accelerating along the horizontal plane – but it shan’t be long term.

Note : due to Earth’s rotation around the North-South axis (so-called « World axis »), is roughly spir aling around this axis : hence the equatorial components are bou nded, the world component is linearly diverging in time. This property leads to longitude divergence of horizonta l errors, that is to say North component of .

θr

ψv

ψv

θr

140

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1,0 °/h

10 °/h

0,1 °/h

10 mg

1,0 mg

0,10 mg

Mid-term vertical errors (roll & pitch)

Time in seconds

Degrees

141

Mid and long term true heading errors

0 2 4 6 8 10 12 14 16 18 20

0

5

10

15

20

25

Time in hours

1,0 °/h

10 °/h

0,1 °/h 10 mg

Degrees

142

Inertia Market Segmentation (Precision)

1 mg

Gyrometers accuracy

AccelerometerAccuracy

150°/h 1°/h 0.1°/h 0.01°/h 0.001°/h

10 mg

100 µg

10 µg

Launchers

Military Aero

Military Land Navigation

30°/h

Civil Aero

Ammunitions

Short range& medium range

Military Helos

Gyro stabilization

Air Defense

AHRSMilitary Aerial

Transportation

Cruise Missiles

G-IRS

Stand-By&

Flight Controls

Rocket

Industrie

0.05°/hSubmarines

Navigation

ICBM

HRG - RLG - FOG RLG - FOG HPMEMS

ME

MS

PO

L -

PC

L

143

Hybridization

144

Hybrid technologies (inertial system/ x)

The hybridization of an inertial navigation system with a non-inertial is necessary to prevent from the vertical divergence (when mid-term navigation is a requirement). altitude sensor : GNSS, baro-altimeters, vision, … altitude blocking

The hybridization is also a mean to preclude horizontal (velocity) oscillations, using : odometers GNSS zero velocity updates (the user or some sensor raises a flag as

soon as the vehicle stops in the environment)

The most popular technique of hybridization is the Kalman filter (and brother filters)

145

Stochastic hybridization techniques (1)

Kalman filter Optimal choice when the dynamics and measurement are

linear with gaussian noise, and perfectly known If the linear dynamics and noises are not perfectly known, a

robust tuning of the filter might fit : nevertheless a robust set of parameters may provide ultra-conservative results !

Extended Kalman filter Used when dynamics and/or measurement are analytically

non-linear Poor performance if non-linearities are too strong No theoretical guarantee of optimality

146

Stochastic hybridization techniques (2)

Particle filtering Used when dynamics and/or measurement are strongly non-linear Particle = (roughly speaking) one simulation of the system at a given

initial state. Hence particle filtering makes massively parallel simulations of the real system → the particle which best explains measurements is selected probabilistically

No theoretical guarantee of optimality, unless the number of particles is infinite !

Unscented (or Sigma points) Kalman filter Used when dynamics and/or measurement are analytically non-linear Tries to inherit nice properties of particle filtering in presence of strong

non-linearities : the idea is to extend the exploration space at most, without increasing the number of particles. Hence UKF chooses deterministic values in state space to explore (so-called « Sigma points »).

147

Hybrid technologies (inertial system/ x)

Various architectures are practiced: from « loose » to « ultra-tight » « loose » = GPS position and/or velocity updates « ultra-tight » = « radio-inertial navigation system »

148

Principle of a navigation filter

1 2

p v a ω v p

−

δp0 δv0 Ψ εb εg

δp δv δa δω

Deviation

model

δp0

δv0

Ψ

εb

εgNavigation filter

1 2

INS Other equipment

p v a ωFiltered

149

Advantages and drawbacks… hybridized !

The hybridization often allows for better accuracy and/or cheaper sensors mutual calibration « in run calibration » fault detection and analysis

sensor redundancy

Be always aware ! Appropriate modeling of causality (e.g.dynamical models) is the fundamental condition to these positive effects of hybridization.

An hybridized system also inherits drawbacks of subsystems less availability dependancy on American DoD, when hybridizing GPS

150

Hybridization of INS and GPS provides better equilibri um in cost, accuracy, accuracy, dependency, vulnerability

INS and GPS : two navigation means thatcomplement each other

Discrete, independent

Non-jammable

Available, reliable, highpass

Universal coverage

Provides whole navigation data

Very accurate over the short term

Fully autonomous

Very expensive

Accuracy is a function of volume and mass

Poor, divergent accuracy over the long term

Operational constraints

Very accurate anytime

Free service very cheap function (civil)

No operational constraint in civil services

Very compact form factor

Provides position and velocity only (no angles)

Availability not compatible with high integrity reqs.

Low/no availability in certain zones

Operational constraints in military services (key management)

Noisy over the short term

Highly jammable

Dependent on the USA

151

Hybridization example : INS/GPS (2)

Specific terminology closed loop : the filter estimates are backwarded to the INS

loops (velocity for instance) tight coupling : GPS pseudo-ranges are provided by the

receiver, as updates to the hybridization filter Ultra-tight coupling : updates are INS increments and GPS

pseudo-ranges (no separate computation of velocity or position)

Inertial aid : the GPS receiver stage is provided with inertial informations

152


Loose coupling Advantage :

easy to integrate

Drawbacks : Limited precision when jammed or antenna hidden Cascade of two locally optimal filters → suboptimal No GPS navigation with less than 4 satellites low frequency of GPS

Navigation filterPosition, speed, attitudes

position, speed, time

GPS Navigation computer

IMU

GPS RF stage

INS Navigation computer

153


Loose INS / GPS with filter loopbackAdvantage :

INS errors maintained small → linear error models

Drawbacks : possible biased GPS measurements loop back to INS short term noise pollution of the INS

corrections Navigation filterPosition, speed, attitudes


INS Navigation computerIMU

GPS RF stage position, speed, time

154


Tight INS / GPS coupling with INS or filter aid of GPSAdvantages :

reduced lock-on time of the GPS receiver stage more robust to jamming higher robustness against signal losses due to dynamics higher GPS frequency

Drawbacks : IMU is necessary robustness against jammers can be deceiptful

pseudo-ranges, doppleror GPS solutions

Navigation filterPosition, speed, attitudes


IMU

GPS RF stage

INS Navigation computer

155


Ultra-tight INS / GPS couplingAdvantages :

very compact system auto-adaptive GPS signal processing navigation under limit of 4 satellites integrity checks of satellite lines of sight high anti-jamming capacity

Drawbacks : high technological skills required in the field of GNSS and INS design

Navigation filterattitude and velocity increments

pseudo-ranges, dopplerPhase lock

loops

IMU

GPS RF stage

156

Appendices

157

French/English terminology

Centrale inertielle de navigation Inertial navigatio n system

Cardan Gimbal

Plate-forme à cardans Gimbaled platform

Contacts tournants Slip rings

Thermostat Thermostat

Thermostatisation Thermostating

Suspension Suspension

Suspension antivibratoire Vibration damping element

Mécanisation Mechanization

Spire Coil

Fibre optique Optical fiber

Odomètre Odometer

Centrifugeuse Centrifuge

Calage angulaire Setting angle

Panne Fault

Intégrité Fault tolerance, integrity

Guidage Guidance

Pilotage Control

Navigation Navigation

Localisation Localization

Missile balistique Ballistic missile

Spécification Requirement

Prédiction Predication phase

Recalage Update phase

Moindres carrés Least squares

Hybridation, couplage Coupling

Couplage lâche, serré, très serré (GPS) Loose, tight , ultra-tight coupling (speaking of GPS)

158

Gravitation field ϕ

MA

MB)( BM MFA

x

B

x

A

Aϕ

Aϕ

Aϕ

R+ h

2

2

AA h)(R

R(R) . h) (R

+=+ ϕϕ

Mearth = 5,98 .1024 kg Rearth = 6 370 km ϕearth(M at R) ≈ -9.83 N/kg (or -9.83 m/s²)

Variation LRBA - Vernon (down town) ≈≈≈≈ 30 µg

)R

2h(R) . (1-h)(R AA ϕϕ ≈+

LRBA (h = 130 m) : 1 g ≈ 9.809370 m/s² Vernon (h = 30 m) : 1g ≈ 9.809670 m/s²

Accelerometers - Gravity

159

Gyros : what is deg.h -1 ?

Gyro’s main performance expressed in deg.h-1

Earth rotation = 1 revolution / day = 360 deg / 24 hours~ 15 deg.h-1

Think of 1 deg.h-1 as roughly 1/15 of the earth rotation during 1 hour

Earth rotation rate

15.041 deg.h -1

EΩr

160

Sensor performance modeling

Main concepts

161

Output repeatability

Property lying underneath the performance : ability of the sensor to provide identical output when submitted to identical input and environment

Identical input

Identical environmentIdentical output

+ Should be a very simple function of the input and the environment

« Ideal sensor » =• output is purely proportional to input, with perman ent scale factor• output is insensitive to environment

162

Unitary sensor model

K1

K2

K3

K3

K2

K2+K3

Output (as)

True input

2K

3K

second order sensitivity

third order sensitivity

ε residuals

Input

•

•

•

Output

Measurements

Error / model

+⋅+⋅+⋅

∆++⋅= ε33

22

1

101 1 EKEKE

K

KKKS

E input

first order sensitivity (scale factor)1K

offset0K

163

Thermal sensitivity

Repeatability of the parameters K0, K1, K2, K3, … as functions of temperature is modeled with polynomials

-600

-400

-200

0

200

400

600

800

-50 -30 -10 10 30 50 70 90

Temperature in °C

Bias residual (µg)

-6000

-4000

-2000

0

2000

4000

6000

8000Bias variation (µg)

Bias residual (2nd order)Bias (uncompensated)

Bias = f(θ)

Uncompensated bias : 100 mm/s²/°C

(10 µg/°C)

Hysteresis

Unmodeled Biais error after compensation

(2nd order)

( )( )( )3

ref03

2ref02

ref01000

TTK

TTK

TTKKK

−⋅+

−⋅+

−⋅+=

( )( )( )3

ref13

2ref12

ref11101

TTK

TTK

TTKKK

−⋅+

−⋅+

−⋅+=

( )ref21202 TTKKK −⋅+=

( )ref31303 TTKKK −⋅+=

Offset/bias

2nd order NL

Scale factor

3rd order NL

164

Sensor mounting errors (outside of the packaging)

AX

AY

RY

RZ

RX

AZ

Accelero or gyro

AX

AY

RY

RX

2 axes - seen from above

3 axes

AXRX

Reference X-axis of the sensor cluster

Input X-axis defined on the packaging of the X-gyro/accelero-meter

165

3D sensor errors

zzyyxx efefeffrrrr

++=Measurement equation as seen from the computer,

inducing false 3D measurement : ff

rr⋅= Km

=1

1

1

K

zyzx

yzyx

xzxy

εεεεεε

fz

fr

xer

fx

fy

True measurement

yer

zer

fzm

fr

xe′r

fxm

ye′r

ze′rfy

mPerfectly aligned sensorsTheoretical

Misaligned (=real) sensors

zzyyxx efefeff ′+′+′= rrrrm

fr

mfr

zyx eeerrr

,,

zyx eee ′′′ rrr,,

False projection of the computer

166

Stability, accuracy, precision

Accuracy, stability, temporal driftEnvironment : temperature, in run, vibrations, radiations, pressure, acceleration,…

167

Vocabulary of sensor stability

« In run » stability

timeSensor off Sensor on…Short term

Self heatingPower on electrical

transients

« Run to run » stability

« Power on » / « Warm up » repeatability

Short term

Sensor output

« Warm up » time

Note : on IESI, run-to-run does not include power-on & warm-up.

168

Useful formulae

Transform of apparent velocity

Reference frame T2

1/2ωr

xxxrrrr ×−= 1/2

12

ω

( )xxxx

xxxx

xxx

rrrrrr&rr

rrrrrrr

rrrr

××−×−×−=

×−×−

×−=

×−=

1/21/2

1

1/21/2

11

1/2

1

1/2

1

1/2

1

2

1/2

122

ωωω2ω

ωωω

ω

First derivative

Second derivativehence 1/21/2

1

1/2

2

ωωω &rrr==

Reference frame T1

xr

169

Quaternions

Spectral Representation of a Rotation

αααα

When α tends to 0, the exact development simplifies to the first order as:

e : rotation axis

U

∆∆∆∆U

For any angle α,

( ) [ ] ( ) [ ]2eαcos1eαsinIαe,Rot ⋅×−+⋅×+=

( ) [ ] ( )2αeαIαe,Rot O+⋅×+=µ

170

Quaternions

Rotations vs. pass matrices

Let (a) and (b) be two orthonormal, clockwise bases.The pass-matrix from (a) to (b) is denoted Ta/b. Then the pass-matrix

from (a) to (b) is Ta/b−1 = Tb/a.

In this context, the rotation operator R=Rot(e,α) transforming (a) into (b), that is to say such that

Is represented in the base (a) by the matrix:

a/b

a

TR ≡

( ) ( )baR =

171

Strap-down inertial navigation unitFull attitude integration (1)

How is the angular orientation computed from the rate gyros signals ? Using a mathematical tool known as « attitude quaternion »

instead of the more classical rotation matrix. The quaternion associated with the rotation with angle θ

and unit vector u, that is such that Tfixed is transformed into Tmobile reads:

)2θsin()2θcos(q ur+=

Tfixed Tmobile

u

θ

172

Strap-down inertial navigation unitFull attitude integration (2)

Then The attitude integration in terms of quaternion space reads :

with ω : the (« quaternionized ») rotation vector from the rate gyros measurement.

This is the theoretical ordinary differential equation, that needs to be numerically integrated, using specific techniques.

ωq2

1q ⋅=&

173

Quaternions

Quaternion representationUsing the following notations,

We have :

=

=

=

=

2

αcos

2

αsine

2

αsine

2

αsine

4

33

22

11

ρ

ρ

ρ

ρ

( ) ( )( ) ( )( ) ( )

++−+−−+−+++−+−−

=24

23

22

2141324213

413224

23

22

214321

4213432124

23

22

21

a/b

22

22

22

T

ρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρρ

174

Quaternions

Quaternions’ algebra HSet of real linear combinations of 1, i, j and k, where

Isomorphism between H and the direct rotation groupWith Ta/b written as the previous slide, we uniquely associate

denoting

kkjjii

jikkiikjjkkjiij

kji

−=−=−==−==−==−=

−===

∗∗ *

222 1

2

αsin

2

αcos

4321a/b

u

kjiQ

+=

+++= ρρρρ

321 eee kjiu ++=

175

Quaternions

Base (a) to base (b) change of coordinates, matrix and quaternion form

∗=⇔= a/ba/bab

a/ba

T QxQxrrb

176

Quaternions

Attitude integration

With the « quaternionized » rotation vector:

( )[ ] ( )aa/ba/ba/ba/baa/ba/b 2

1TωT Ω=⇔⋅×−= QQ&&

( ) ( )44 344 21

! quaternionimaginary pure

aa/baa/b ωωω

ω

ω

ω

ω kji zyx

z

y

x

++=Ω⇔

=

( ) b/aaa/bb/a2

1QQ Ω−=&

Since Qa/b is a unit quaternion, and Ωa/b a pure imaginary quaternion, the integration formula is equivalent to its quaternion conjugate :

177

Quaternion numerical integration (1)

We need to numerically integrate the following attitude quaternion differentialequation :

with, as previously stated :

A convenient way to see this equation is to consider that a quaternion is a 4-dimensional vector in IR4 :

Then (1) is equivalent to the vector integration :

with

( )abababa QQ /// 2

1 Ω⋅=&

( ) ( )44 344 21

! quaternionimaginary pure

aa/baa/b ωωω

ω

ω

ω

ω kji zyx

z

y

x

++=Ω⇔

=

=

3

2

1

0

/

q

q

q

q

Q ba

(1)

baba QMQ // 2

1 ⋅⋅=&

−−

−−−−

=

0

0

0

0

xyz

xzy

yzx

zyx

M

ωωωωωωωωωωωω

q0 q1, q2, q3real part imaginary parts

178

We can solve this last equation between tk−1 and tk as:

provided that the vector is constant between tk-1 and tk

When the time step is small enough, angular increments are verysmall, and so is the argument of the exp. Then:

BTW, note that


∫∫−−

+≈

k

k

k

k

t

t

t

tMM

11 2

1Id

2

1exp 4

−−

−−−−

== ∫−

0

0

0

0

1

xyz

xzy

yzx

zyx

t

t

k

k

MA

αααααααααααα

∫−

= k

k

t

t zyxzyx1

,,,, ωαwhere are the (debiased) angular increments

( ) ( )1//12

1exp −⋅

= ∫−

kba

t

tkba tQMtQk

k

[ ]

×−

=

−−

−−−−

=.

0

0

0

0

0

ααα


T

xyz

xzy

yzx

zyx

A where

=

z

y

x

ααα

α

ba /ωr

179


Typical numerical schemes in practice, more accurate approximations of the previous

discrete equation may be derived a popular one :

Check [Titterton] pages 319-324 for more details

4444 34444 2144 344 21scorrection

112

2

npropagatio

11 6

1

8

1

2

1

⋅+⋅⋅−⋅+= −−−− kkkkk QAQQAQQ α

where

23

22

21

2

2vvvv ++=

[ ]

×−

=

−−

−−−−

=.

0

0

0

0

0

ααα


T

xyz

xzy

yzx

zyx

A

∫−

= k

k

t

t zyxzyx1

,,,, ωα are the (debiased) angular increments

( ) ( )1/1/ −− == kbakkbak tQQtQQ

180

Wilcox integration of quaternion

Second order

Third order

Forth order

⋅+⋅⋅−⋅+= −−−− 11

2

11 61

81

21

kkkkk QAQQAQQ α

⋅+⋅⋅+

⋅+⋅⋅−⋅+= −−−−−− 11

4

11

2

11 101

3841

61

81

21

kkkkkkk QAQQAQQAQQ αα

1

2

11 81

21

−−− ⋅⋅−⋅+= kkkk QQAQQ α

181

Strategies to reduce computation load whenunder coning motion

Any of the previous numerical schemes can lead to apparent drift errors (when coning motion exists) : finite angular resolution of the RLG superactivation high frequency vibrations vs frequency of digitial computation digital signal processing stages inside the angular integration

loop

When integrated at very high frequency.Classes of numerical schemes (Miller, Bortz) are intended to allow (rather) low frequency computation while combining high frequency « correction » terms, to circumvent virtual drifts.

182

Elements of proof

Recall that for a square matrix X,

Observe that for we have andso A² simplifies to

further leading to an exact expression of any power of A :

The remarkable consequence for the exp is that it has the following closed form:

Taylor expansions of cos and sine to the 3rd order give the numerical scheme of previous slide.

[ ]

×−

=.

0

ααα T

A

( ) ∑+∞

==

0 !

1exp

p

pXp

X

[ ][ ]

××+⋅−−=

×

×

..0

0

13

31

22

ααααα

TA [ ][ ] 3

2.. IT ⋅−⋅=×× ααααα

4

22 IA ⋅−= α

( ) 4

22 1 IAppp ⋅−= α ( ) AA

ppp ⋅−=+ 212 1 α0≥∀ p

AIA ⋅+⋅=

α

αα 2

sin

2cos

2exp 4

11 2sin

2cos −− ⋅⋅+⋅= kkk Q

AQQ

ααα

inertial navigation adiru systems - ensta bretagnemems domaine mems rlg - fog rlg - fog hp driven by...

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