7/24/01 - 1

Aircraft Simulation StudyIncluding Inertial Navigation System

(INS) Model with Errors

Navigation Team:Takayuki Hoshizaki1 and Dominick Andrisani2

July 24, 2001

School of Aeronautics and AstronauticsPurdue University

West Lafayette, IN 47907-1282

1Ph.D. Candidate2Professor.

7/24/01 - 2

Model and Parameters to Drive Simulation

Aircraft Motion

Aircraft Model

Trajectory Input

Time Input

Turbulence Input

ErrorsGPS

Satellite Constellation

Processing Mode

AntennasNumber, Location

Errors

INS Position, Attitude, Rates Position, Attitude, Rates

Filter

Aircraft Position & Attitude Estimate and Uncertainty

Transformation to Sensor Position, Attitude, and Uncertainty

Errors

ErrorsSensor Parameters

Image AcquisitionParameters

Site Model

Imaging System

Target CoordinatesUncertainty, CE90

Graphic Animation

Multi-ImageIntersection

Synthetic Image GenerationErrors

Target Tracking

7/24/01 - 3

Objectives

1. Development of mathematical INS model Including errors

2. Analysis of error impact on the target position accuracy

7/24/01 - 4

Inertial Navigation System (INS)

Inertial Sensor Cluster

Sensor Support

Electronics

Attitude Computation

Navigation Computation

Inertial Measurement Unit(IMU)

Attitude & Heading reference System(AHRS)

Inertial Navigation System(INS)

7/24/01 - 5

Major Error Sources

1. Inertial sensor errors

2. Initial misalignments

(significant but assumed to be 0 in the simulation)

3. Computational errors

e.g. - Higher order truncation

- Ignorance of high frequency motion

(assumed to be small relative to 1 & 2 in the simulation)

7/24/01 - 6

SIMULINK Diagram

Turbulence Input

Control Input R

Aircraft Model

INS Model

uIC(3)

deltaRic1

uIC(1)

deltaEic1

uIC(2)

deltaAic1

uIC(4)

bhpIC1

t

To Workspace8

ynav

To Workspace1

y

To Workspace

In1

In2

In3

In4

In5

In6

Out1

Strapdown INS

Mux

Mux1

Mux

Mux

Out1

Out2

Out3

Gust Subystem

Demux

Demux

Clock1

In1 Out1

Aircraft Subsystem Fbx/m

Fby /m

P

Q

Fbz/m

Aircraft Angular Rates

Aircraft Accelerations

True Aircraft Simulation

INS Simulation

7/24/01 - 7

INS Model

Accelerometers with Errors

Rate Gyroswith Errors

1

Out1

In7 Out2

RLGyros2w

Mux

PQR

In1 Out1

Navigation Equation

Mux

Mux2

Mux

FB/m

In7 Out2

A4Accelerometers

6

In6

5

In5

4

In4

3

In3

2

In2

1

In1Aircraft Accelerations

Aircraft Angular Rates

• Un-aided Strapdown INS

IMUUn-aided INS

Navigation Computation

Initial Conditions

7/24/01 - 8

ii

i

jiij

i

ds with noise hiteGaussian wDparameter noise Random

)(1s BiasB)(1sparameter nality NonorthogoMM

)(1sfactor ScaleSrun)each in different run, a during(Constant constants Random

==•

==−==

•

Accelerometer Model With Errors

errortMeasuremenonaccelerati(actual)Input

onacceleratiMeasured~whereDDD

BBB

aaa

dadada

~

z

y

x

z

y

x

z

y

x

z

y

x

===

+

+

=

=

+=

daaa

da

daaa

zzyzx

yzyyx

xzxyx

SMMMSMMMS

7/24/01 - 9

Rate Gyro Model With Errors

errortMeasuremen:rateangular(actual)Input:

rateangularMeasured:~whereDDD

BBB

???

d?d?d?

~

z

y

x

z

y

x

z

y

x

z

y

x

d???

d?

d???

+

+

=

=

+=

zzyzx

yzyyx

xzxyx

SMMMSMMMS

1s with noise hiteGaussian wwparameterwalkRandomr

w)/r(Dparameter noise Random

)(1s BiasB)(1sparameter nality NonorthogoMM

)(1sfactor ScaleSrun)each in different run, a during(Constant constants Random

i

i

iii

i

jiij

i

===×=

•==−==

•

t

7/24/01 - 10

Simulated INS Model

(rms) deg 0.015 : Roll) (Pitch, (rms) deg 0.05 : (Azimuth) Attitude(rms) ft/sec 2.5 : xisVelocity/A

nmile/hr 0.8 : Position

)(ft/s5d)(ft/s25B

(2arcsec) rad 6e7.9M )10(ppm ppm50S

values1 ter"Accelerome 4-A"

2i

2i

ij

6-i

gg

µµ

σ

==

−===

• Litton LN-100L Launch Vehicle Inertial Guidance System

)hr/(0.0015deg srad/736.4rhr)(0.003deg/rad/s845.1B

rad 0M )10(ppm ppm1S

values1 gyrolaser Ring

i

i

ij

6-i

−=−=

===

ee

σ

Try to reproduce Performance Data using given Sensor Accuracy

- Performance Data

- Sensor Accuracy

7/24/01 - 11

Reference Frames

,iz

ix

iy

ex

ey

ez

O

λcλ

φψ

(North)

(East)

(Down)

nx

ny

nz

bx

by

bz

h

P

p

• Nomenclaturei : Earth Centered Inertial Frame(ECI)e : Earth Centered Earth Fixed Frame (ECEF)n : North East Down Frame (NED)b : Aircraft Body Fixed Frame (B)w : Wind Frame (W)

: Geocentric Latitude: Geodetic Latitude: Geodetic Longitude: Celestial Longitude: Altitude

ψφλ

cλh

7/24/01 - 12

Navigation Equation

)1,1()1,2(

tan

)3,3(1

)1,3(tan

)3,3()2,3(

tan

)cos()(

)cos()2(~)cos()2()sin()2(~

)sin()2(~

13

2

12

11

bn

bn

bn

bn

bn

bn

bb

nbnbn

TT

T

TTT

OTT

−

−

−

=

−

−=

=

⋅=−=

+=

+=

+−+−=++++=

++−=

E

E

E

vhhRv

?

hRv

gvv?avv?v?av

vv?av

D

E

E

N

N

nNEeDD

DeNeEE

DEeNN

&&

&

&

&&&&&&

&&&

φ

φ

φφωφωφω

φφω

[ ]

[ ]frameBodyinoutputs

sgyro'rateis~where),(~,,

00

0NEDtoBodyfrommatrixtionTransforma:)E(T)E(T)E(T

frameBodyinoutputsteracceleromeis~where,~a~a~a~~

lyrespective curvature,ofradiuswest East&Meridian:R,Rlyrespectiveanglesrollpitch,yaw,Euler:E,E,E

frame NEDin described vector,onacceleratiincluded)Gravity(:]g00[)(

frameNEDtoBodyfrommatrixtionTransforma:EarthofrateAngular:

Tzyx

xy

xz

yz

123

TDEN

EN

123

Tn

e

bb

inn

een

inbbb

ibb

n

bb

n

bn

bbbn

n

ieiein

bn

???T??

O

T

aaTa

Gp??GgT

+−==ωωω

ωω−ω−ω

ωω−=

=

==−

−=××−=

ωNomenclature

7/24/01 - 13

Simulated Trajectory

Nominal Trajectory Conditions

Velocity : 300 ft/s

Altitude : 15000 ft

Direction : West on the Equator

Flight time : 3600 s

ex

ey

ez

7/24/01 - 14

Numerical Simulation : Un-aided INS

0 500 1000 1500 2000 2500 3000 3500 4000-1

0

1

VN (ft/s)

Actual & Measured Velocities

0 500 1000 1500 2000 2500 3000 3500 4000-320

-310

-300

-290

VE (ft/s) Actual velocity

Measured velocity

0 500 1000 1500 2000 2500 3000 3500 4000-400

-200

0

200

VD (ft/s)

time (s)

0 500 1000 1500 2000 2500 3000 3500 4000-1

-0.5

0

0.5

δ VN (ft/s)

Velocity Measurement Errors

0 500 1000 1500 2000 2500 3000 3500 4000-20

-10

0

10

δ V

E (ft/s)

0 500 1000 1500 2000 2500 3000 3500 4000-300

-200

-100

0

δ V

D (ft/s)

time (s)

Velocities in NED frame (ft/s) )/( sftVVV actualmeasured −=δ

NV

EV

DV

NVδ

EVδ

DVδ

unbounded. isDVδ

unbounded.also isVEδ

7/24/01 - 15

Numerical Simulation : Un-aided INS

0 500 1000 1500 2000 2500 3000 3500 4000-5

0

5x 10

-5

φ (rad)

Actual & Measured Geodetic Positions

0 500 1000 1500 2000 2500 3000 3500 4000-0.06

-0.04

-0.02

0

λ (rad)

Actual geodetic position Measured geodetic position

0 500 1000 1500 2000 2500 3000 3500 40000

5

10

15x 10

4

h (ft)

time (s)

0 500 1000 1500 2000 2500 3000 3500 4000-10

-5

0

5x 10

-5

δ φ (rad)

Position Measurement Errors

0 500 1000 1500 2000 2500 3000 3500 4000-4

-2

0

2x 10

-4

δ λ (rad)

0 500 1000 1500 2000 2500 3000 3500 40000

1

2x 10

5

δ h (ft)

time (s)

Geodetic Positions

)(radφ

)(radλ

)( fth

)(radδφ

)(radδλ

)( fthδ

actualmeasured XXX −=δ

2t toproportionin increaseshδ Vertical channel error is

dominant

7/24/01 - 16

Error Equation

D

2E

E

E

EE

E

2N

NN

N

v-dhd

dhcos)hR(

vd

cos)hR(tanv

dvcos)hR(

1d

dh)hR(

vdv

)hR(1

d

~)2()2(~~

=φ+

−φφ+

φ+φ+

=λ

+−

+=φ

++×+−×+−×=−+×=−

&

&

&

&&

bbn

nnnn

eeb

in

nn

eeb

inn

bb

ibnnn

inn

i

adTdgv?d?dvd??eavd?dT?de?e

T

majorsemi

levelsea

TDEN

T

]hR

g2,0,0[

]v,v,v[anglesntmisalignme:],,[

where

δ−

=

δδδ=δγδβδα=

n

n

dg

dve

Attitude :

Velocity :

Position :

.iablesvarallerioratesdetandrapidlyincreacesh.errorsvelocityaffectsdirectly)size~(tersacceleromeofAccuracy

.errorsattitudeaffectsdirectly)size~(gyrosrateofAccuracy

δ⋅⋅⋅

bb

i

bb

i

?d?d

7/24/01 - 17

INS Aided With Barometric Pressure Altitude

tmeasuremenpressurebarometrichh =

Instead of (Un-aided INS)

∫−= dtvh D

added)is10ftswithnoise(white =

7/24/01 - 18

Numerical Simulation : INS Aided with Barometric Altitude

0 1000 2000 3000 4000 5000 6000-1

0

1

VN (ft/s)

Actual & Measured Velocities

0 1000 2000 3000 4000 5000 6000-305

-300

-295

VE (ft/s)

Actual velocity Measured velocity

0 1000 2000 3000 4000 5000 6000-20

-10

0

10

VD (ft/s)

time (s)

0 1000 2000 3000 4000 5000 6000-1

-0.5

0

0.5

δ VN (ft/s)

Velocity Measurement Errors

0 1000 2000 3000 4000 5000 6000-0.5

0

0.5

δ V

E (ft/s)

0 1000 2000 3000 4000 5000 6000-10

-5

0

δ VD (ft/s)

time (s)

Velocities in NED frame (ft/s)

NV

EV

DV

NVδ

EVδ

DVδ

)/( sftVVV actualmeasured −=δ

Horizontal velocity errors are bounded

7/24/01 - 19

Numerical Simulation : INS Aided with Barometric Altitude

0 1000 2000 3000 4000 5000 6000-5

0

5

10x 10

-5

φ (rad)

Actual & Measured Geodetic Positions

0 1000 2000 3000 4000 5000 6000-0.1

-0.05

0

λ (rad)

Actual geodetic position Measured geodetic position

0 1000 2000 3000 4000 5000 60001.48

1.49

1.5

1.51x 10

4

h (ft)

time (s)

0 1000 2000 3000 4000 5000 6000-10

-5

0

5x 10

-5

δ φ (rad)

Position Measurement Errors

0 1000 2000 3000 4000 5000 6000-2

0

2x 10

-5

δ λ (rad)

0 1000 2000 3000 4000 5000 6000-50

0

50

δ h (ft)

time (s)

Geodetic Positions actualmeasured XXX −=δ

)(radφ

)(radλ

)( fth

)(radδφ

)(radδλ

)( fthδ

Horizontal position errors are reduced

7/24/01 - 20

Status

• Simulation: Completed Components– Aircraft model– Ellipsoidal earth model– Turbulence– INS including errors

• Under Development– Kalman filter– Integration of external navigation aids

To do• Accuracy comparison changing

navigation aids / sensor performance

7/24/01 - 21

Conclusions

• Simulation of aircraft and INS is nearly completed.

• Error impact analysis and navigation technique comparison remain to be completed.

7/24/01 - 22

References

[1] Titterton, D. H. and Weston, J. L. (1997). “Strapdown Inertial Navigation Technology”. Peter Peregrinus Ltd, Herts, U. K.

[2] Rogers, R. M. (2000). “Applied Mathematics In Integrated Navigation Systems”. AIAA Education Series.

[3] Chatfield, A. B. (1997). “Fundamentals of High Accuracy Inertial Navigation”. Volume 174, Progress in Astronautics and Aeronautics. AIAA.

[4] Britting, K. R. (1971). “Inertial Navigation Systems Analysis”. Wiley Interscience.

[5] Brown, R. G. and Hwang, P. Y. C. (1985). “Introduction to Random Signals and Applied Kalman Filtering”. John Wiley & Sons.

7/24/01 - 23

Numerical Simulation : Un-aided INS

0 500 1000 1500 2000 2500 3000 3500 4000-5

0

5x 10

-3

E1 (rad)

Actual & Measured Euler Angles

0 500 1000 1500 2000 2500 3000 3500 4000-0.04

-0.03

-0.02

-0.01

E2 (rad)

0 500 1000 1500 2000 2500 3000 3500 4000-1.6

-1.55

-1.5

E3 (rad)

time (s)

Actual Euler angles Measured Euler angles

0 500 1000 1500 2000 2500 3000 3500 4000-4

-2

0

2x 10

-5

δ E1 (rad)

Euler Angle Measurement Errors

0 500 1000 1500 2000 2500 3000 3500 40000

2

4x 10

-4

δ E2 (rad)

0 500 1000 1500 2000 2500 3000 3500 4000-6

-4

-2

0x 10

-5

δ E3 (rad)

time (s)

Euler Angles (rad)

)(1

rollE

)(2

pitchE

)(3

yawE

)(1

rollEδ

)(2

pitchEδ

)(3

yawEδ

)(radEEE actualmeasured −=δ

7/24/01 - 24

Numerical Simulation : INS Aided with Barometric Altitude

0 1000 2000 3000 4000 5000 6000-5

0

5x 10

-3

E1 (rad)

Actual & Measured Euler Angles

0 1000 2000 3000 4000 5000 6000-0.04

-0.02

0

E2 (rad)

0 1000 2000 3000 4000 5000 6000-1.6

-1.55

-1.5

E3 (rad)

time (s)

Actual Euler angles Measured Euler angles

0 1000 2000 3000 4000 5000 6000-4

-2

0

2x 10

-5

δ E1 (rad)

Euler Angle Measurement Errors

0 1000 2000 3000 4000 5000 60000

2

4x 10

-5

δ E2 (rad)

0 1000 2000 3000 4000 5000 6000-1

-0.5

0x 10

-4

δ E3 (rad)

time (s)

Euler Angles (rad)actualmeasured EEE −=δ

)(1

rollEδ

)(2

pitchEδ

)(3

yawEδ

)(1

rollE

)(2

pitchE

)(3

yawE