position estimation 4 - field robotics estimation 4: inertial navigation ... solutions to these...

Alon

Position Estimation 4:1

use of inertial principles

in German Peenemunde

ited to Charles Draper et

d reckoning available.ursion (outdoor) missions.here gravity is known.- requ i re no ex te rna l

exhibit perfect stealth.

elerations of unpowered

ibit Schuler oscillation

zo Kelly February 21, 2005

• Radiate nothing - 1.3 Disadvantages

• Cannot sense accspace flight.• Most errors exh(advantage?).

Mobile Robot Systems 1.1History

Position Estimation 4:Inertial Navigation Systems

1 Introduction1.1 History

• Name comes from(Newton’s Laws).• Historical roots Group.• Modern form credal. @MIT.

1.2 Advantages• Most accurate dea• Useful in wide exc• Work anywhere w• Are jamproof information.

Alon


••

1.4•o•

nertial Navigation

ee accelerometers to theintegrating their outputs:

rules of math and physics:e te rs measure wrong

re it in wrong reference

nt it in wrong coordinate

elerometers


• Acceleromquantity.• They measuframe.• They represesystem.

Mobile Robot Systems 2 Principles of Inertial Navigation1.4Fundamental Idea

Most errors are time dependent. Requires input of initial conditions. Fundamental Idea Mount three accelerometers along threerthogonal axes. Integrate twice to determine position.

2 Principles of I2.1 Naive Concept

• Try strapping thrvehicle and double

• This breaks many

acc

Alon


Force to Acceleration

•sr

2.2

•bfo

•im

he quantity . We canl acceleration in terms ofgravitational forces as

gravity is known), it ist the specific force intot ion, so that i t can be

licit knowledge of therength is required at everyicle. Inertial navigation is this field is known (orficant).ove Apparent Forcesaw twice.y scary math - but its just put into code to use aMU.ference:(inertial frame).

T m⁄

Wm-----+ t w+= (1)


Fundamental equation of inertial navigations Newton’s second law applied to the proofass:

W

F∑ T W+ mai= =

2.3 Second Fix: Rem• Use the Coriolis L• This is moderatelwhat you need tomodern strapdown I• Three frames of re

• i: geocentric

Mobile Robot Systems 2 Principles of Inertial Navigation2.2First Fix: Convert Specific

The quest for ever better engineeringolutions to these problems is the primaryeason for the complexity of the modern INS. First Fix: Convert Specific Force to

Acceleration Accelerometer is a specific force transducerecause the calibrated restraint responds toorce in the spring, (which is not the net forcen the mass).

T

Accelerometer

• Specific force is texpress the inertiathe specific and follows:

• At this point, (ifpossible to converinert ial acceleraintegrated. • Notice that expgravitational field stposition of the vehonly viable whenknown to be insigni

aiTm----=

Alon


rent Forces

•r•r•b•

•vm•tot

ocity of the vehicle wrt the wrt the earth plus the parttion of the earth wrt the

erentiating the above onertial frame is:

ters inherently provideg the axes of the vehicleost convenient to express

s of this frame so that thectly integrated. The firstnd side can be referred toy another application of

) [5] and then (2) [3] into

te⎠⎞i

Ω tddre⎝ ⎠⎛ ⎞

i×+

(4)

ω ve× (5)


he inertial and earth frames (the i and erigins are coincident), the inertial velocity ofhe vehicle can be expressed as:

(3)vi td

dri⎝ ⎠⎛ ⎞

i tddre⎝ ⎠⎛ ⎞

i

tddre⎝ ⎠⎛ ⎞

eΩ re×+ ve Ω re×+=

= = = • By substituting (2(2) [4]:

tddve⎝ ⎠⎛ ⎞

i tddve⎝ ⎠⎛ ⎞

v+=

Mobile Robot Systems 2 Principles of Inertial Navigation2.3Second Fix: Remove Appa

• e: earth frame (rotating).• v: vehicle frame (fixed to accels).

Let the constant rotation of the earth withespect to inertial space be given by . Let the rotation of the vehicle frame withespect to the earth be given by . Let the inertial rotation of the vehicle framee given by . Clearly:

Let , , and be the position vector,elocity, and acceleration of the vehicleeasured in the frame x.

Since the position vector is the same in both

Ω

ρ

ω

ω Ω ρ+= (2)

rx vx ax

• Intuitively, the velinertial frame is thatcaused by the rotainertial frame.• The result of diffmore time in the ine

• The acceleromemeasurements alonframe. Hence, it is mderivatives in termoutputs can be direterm on the right hathe vehicle frame bthe Coriolis law:

ai tddvi⎝ ⎠⎛ ⎞

i ddv⎝⎛= =

Alon


rent Forces

•(

•tu

a

a

a

a

on the right hand side arenly the position and are

ether and called gravity.itation and part centrifugal

tial guidance system, you m/s/s for . You have aalue everywhere on earth.e quantities of interest (byhicle frame):

s can be solved for the, and velocity, given theon:

Ω re×( )× (8)

g

(9)

Ω) ve× g+ ]dt ve0+

0


This is one of the most convenient forms ofhe equation of motion of the vehicle. We willse it as THE equation of inertial navigation.

tddve⎝ ⎠⎜ ⎟⎛ ⎞

vt ω Ω+( ) ve×– w Ω Ω re×( )×–+= (7)

• These equationunknown positionfollowing informati

re vedt0∫ re+=

Mobile Robot Systems 2 Principles of Inertial Navigation2.3Second Fix: Remove Appa

Substituting the specific force equationequation (1)) and rearranging gives:


vω Ω+( ) ve× Ω Ω re×( )×+ +=

(6)

i tddvi⎝ ⎠⎛ ⎞


iΩ td

dre⎝ ⎠⎛ ⎞

i×+= =


vω ve× Ω td

dre⎝ ⎠⎛ ⎞

i×+ +=


vω ve× Ω ve Ω re×+[ ]×+ +=

ai t w+ tddve⎝ ⎠⎜ ⎟⎛ ⎞

vω Ω+( ) ve× Ω Ω re×( )×+ += =

• The last two termsboth functions of ooften grouped togGravity is part gravforce:

• In a precision inerdo not just use 9.8map of its precise v• Solving (2) for thintegrating in the ve

g w Ω–=

ve t ω +(–[0

t

∫=

t

Alon


rent Forces

ravitation as the force per unit massa test mass in the samehe earth. force that’s proportional totons law of gravitation:

easuring and modellingdesy. o the surface of the earthtrifugal force as viewedframe spinning with the

e surface of the earth doesits center but is ratherward the equator.l be distinguished from.

GMeR3-----------R–=


• A plumb bob at thnot point toward displaced slightly to• Gravitation wilgravity from now on

Mobile Robot Systems 3 Gravity and Gravitation2.3Second Fix: Remove Appa

• A model of the earth’s acceleration dueto gravitation as a function ofposition.• The earth’s sidereal rate of rotation .• The spec i f i c fo rces f rom theaccelerometers.• The initial position .• The initial velocity .

w m⁄

Ωt

re0ve0

3 Gravity and G• Gravity is definedrequired to keep position relative to t• Gravitation is the the masses. By New

• The practice of mgravity is called geo• An object fixed texperiences a cenfrom a reference earth.

W wm----=

Alon


E

anization of an

der the problem in i tsdependent form.

ionn equations amount to a

gs:on to specific force .rifugal force icle offset from the center

iolis force ehicle’s motion on thearth.integrations, incorporatingns.Propagationll these correction terms

s are only tiny anyway?uted position is actuallyto them.

w tΩ Ω re×( )×

ω Ω+( ) ve×


• Perform two initial conditio

4.2 Temporal Error • Why mess with awhose magnitudeBecause the compextremely sensitive

Mobile Robot Systems 4 Generic Mechanization of an INS4.1Vector Formulation

quator

Ω Ω Ω R×⎝ ⎠⎛ ⎞×–

circleof constantlatitude

RG g

4 Generic MechINS

• For now, consicoordinate system in

4.1 Vector Formulat• The mechanizationeed to do four thin

• Add gravitati• Remove centdue to the vehof the earth.• Remove cordue to the vsurface of the e

Alon


on

•esft

Sp

G

Co

gra

integration multipliessquare of time, and 1 hours squared.glecting the centrifugalunts for over 3 kilometersr. Point one milli-g equalsn seconds is 3600*3600 =

or occurs even when the for that time period - so itent.

1.5x10-4 g

rm Magnitudes

xpression Nominal Value

Ω re××


Table 1: Term Magnitudes

Term Name Expression Nominal Value

ecific Force 0.1 g

ravitational 1.0 g

riolis 0.03 g

tg2Ω ve×

Mobile Robot Systems 4 Generic Mechanization of an INS4.2Temporal Error Propagati

For a vehicle at the equator, movingastward at a velocity of 10 meters perecond, and accelerat ing at 0 .1 g, theollowing table gives the magnitude of eacherm:

+

-

+ +

- -

t

ai

ve0ve re0

re

Ω Ω _××

ω Ω+( ) _×

dt

0

t

∫dt

0

t

∫

GMe_( )3

------------

from the accelerometers

vitation

coriolis

centrifugal

• The process ofacceleration by the is 13 million second• After 1 hour, ne(smallest) term accoof accumulated erro3 Km! because i10 million.• Note that this errsystem is stationaryis truly time depend

Centrifugal

Table 1: Te

Term Name E

Ω

t2

Alon


rticular Coordinate System

5 I5.1

•as•

•mu•kal•c

hniquesf systems exist on this

temsknown as gimballed ors, employ a stabilized

actively servoed to the.m was the first practical

oped historically sincermation was availablebal angles.

require only minimalcity. have been replaced by

tems, called semi-analytic. Systemsed systems which control a platform in order toation frame only.

nsform the sensor outputsme. Computations are also


We either actively control or passivelyeeping track of the orientation of theccelerometers as the vehicle moves. Theatter is the modern strapdown approach. A trade-off exists between mechanicalomplexity and computational complexity

another class of sys 5.2.2 Semi-Analytic

• These are gimballthe orientation ofinstrument the navig• Computations trafrom the inertial fra

Mobile Robot Systems 5 Implementation5.1Third Fix: Mechanize a Pa

mplementation Third Fix: Mechanize a Particular

Coordinate System You can’t just add up abstract vectors like nd . You have to establish some coordinateystem to actually do the computing. All inertial navigation systems use:

• Gyroscopes to measure vehicle rotationin inertial space.• Accelerometers to measure inertialspecific force.

This package is often called the inertialeasurement unit (IMU) or inertial reference

nit (IRU).

tw

5.2 Stabilization Tec• Three classes ospectrum:

5.2.1 Gimballed Sys• The first class, geometric systemplatform which isrequired orientation• This type of systeclass to be develnavigational infodirectly from the gim• These systemscomputational capa• In practice, they

Alon


tems

ut•w•isrl•c

5.2•ac•f

liminating the platformf causing the components

ic rotation of the vehicle.hanization places morements on the sensory

and on the computationale systems have appearedt l y i n t he h i s t o ry o ftial navigation.omise to replace semi-

most applications.dinate Systemst important way to classify

ed.ed.d Systemstially fixed coordinate

tions are simple (trivial


rate..3 Strapdown Systems

The third class, known as strapdown orn a l y t i c sy s t ems , a r e s a id t o beomputationally stabilized. Strapdown systems get their name from theact that they strap the sensors directly to the

• Earth stabiliz 5.3.1 Space Stabilize

• Employ an inersystem.• Navigation equaeven).

Mobile Robot Systems 5 Implementation5.3Navigation Coordinate Sys

sed to determine the latitude and longitude ofhe vehicle. Gimballed systems employ gyro torquershich drive the platform to rotate as required.

Untorqued gyros naturally implement annertial reference but this is only useful inpace. Instrumenting an earth referenceequires gyro torquing to keep the platformevel. The total torquer signal may includeompensation for:

• The gyro drift rate.• The sidereal rotation of the earth, calledearth rate.• The angular velocity of the vehicle withrespect to the earth, called the vehicle

vehicle chassis, egimbals at the cost oto suffer the dynam• Strapdown mecstr ingent requirecomponents used, throughput, so thesr e l a t i ve ly r ecendevelopment of iner• Such systems pranalytic systems in

5.3 Navigation Coor• A second and mossystems:

• Space stabiliz

Alon


igation Systems

•a

5.3•w•s(•ecd•s•c•arf

rtial Navigation

gation Systems

earthstabilized

earthfixed(basepoint)

freeazimuth

wanderazimuth

above can be nalytic, or geometric


oordinate axis pointed north at all times. Free azimuth and wander azimuth systemsllow the level coordinate axes to rotate withespect to north about the vertical axis as aunction of latitude: Note: any system

analytic, semi-a

Mobile Robot Systems 5 Implementation5.4Taxonomy Of Inertial Nav

Very inconvenient for use in terrestrialpplications..2 Earth Stabilized Systems

Maintain orientation referenced in variousays to the earth.

Earth fixed systems employ a coordinateystem fixed with respect to the rotating earth.aka. base point systems) Local vertical systems are the most commonar th s t ab i l i zed sys tems and employoordinates referenced to the local verticalirection with the other two axes horizontal. There are three variants of local verticalystems: Nor th-s laved sys tems main ta in one

5.4 Taxonomy Of IneSystems

Inertial Navi

spacestabilized

localvertical(locallylevel)

northslaved

Alon


rdinate Systems

5.5

•mp•l

5.5

•c••

c

ω

ω

ω

xyz

cruise systems.oles .

Locally Level SystemtddΛ

λ

Ω

equator

prim

e m

erid

ian

n

ezR

cα tddλsα–

sα tddλcα–

vehicle

Λ

αyx

ith northith east


Like all locally level systems, avoids gravityomputations. Gyro torquers required. Permits easy calculation of lat, long.

equator axis horizontal, north axis horizontal, east axis vertical, down

center at present locationx axis horizontal, angle α wy axis horizontal, angle α wz axis vertical, down

Mobile Robot Systems 5 Implementation5.5Common Mechanized Coo

Common Mechanized Coordinate Systems

The three most common terrestrial INSechanizations differ only in the way that the

latform rotates with respect to the earth ( ). Let the vehicle latitude be denoted and itsongitude ..1 North Slaved, Locally Level System

ωλ

Λ

λ

Λ

prim

e m

erid

ian

x

yzR

enter at present location

x ΩtddΛ+⎝ ⎠

⎛ ⎞ cλ=

y tddλ–=

z Ω tddΛ+⎝ ⎠

⎛ ⎞ sλ–=vehicle

Ω

• Used in long term• Singularity at the p

5.5.2 Free Azimuth,

ωx Ω tddΛ+⎝ ⎠

⎛ ⎞ cλ=

ωy Ω tddΛ+⎝ ⎠

⎛ ⎞ cλ=

ωz 0=

ωz Ωsλ–=For Wander Azimuth:

Alon


•bt•tt•en

5.5

cex y z

o

moves with the vehicle,ith respect to the earthinitial value. of the fact that constantbe generated much moreable ones. d excursions only.

mechanical stabilization.s used to regulate the table prevent the precession of

follows:, tries to rotate gyro about

precesses about its output

ses the output, signal

rives motor to rotate thehere it started.


λ0

equator

prim

e m Λ0

basepointnter at present location

axis horizontal, initial northaxis horizontal, initial eastaxis vertical, initial down

• Gyro actuallyaxis.• Pickoff senamplified.• Servo loop dtable back to w

Mobile Robot Systems 5 Implementation5.6Stable Table

Solves the high polar platform rate problemy providing no inertial platform rate abouthe vertical axis. Azimuth is allowed to wander with respecto north and computations keep track of therue direction of north. In the wander azimuth variation, only thearth rate is compensated for - vehicle rate isot..3 Tangent Plane, Base Point System

Ω

erid

ian

xyzR

ωx Ωcλ0=ωy 0=ωz Ω– cλ0=

vehicle

x

yz

ne should be sin?

• While the platformits orientation wremains fixed at its • Takes advantageplatform rates can accurately than vari• Suitable for limite

5.6 Stable Table• This is the basis of• A feedback loop iin a such a way as tothe gyroscopes. • The loop works as

• Vehicle yawsits input axis.

Alon

Position Estimation 4:ation 14

•n

ics of Inertial

hich navigate close to therizontal oscillatory errorsus Schuler period of 84

error, being oscillatory, is and this is contrary to our expect a quadratic growth acceleration is integrated

the tuning of a device toquency the same as that oflength is the radius of thee, for example, makes theune to the motions of a

ed to a vehicle moving on earth will be deflected


earth. This techniqugyrocompass immship. • A pendulum attachthe surface of the

1. Invented by Max Schuler.

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.1Schuler Tuning

Although the vehicle yaws, the table doesot.

6 Error DynamNavigation

6.1 Schuler Tuning• All INS systems wearth experience horeflecting the famominutes. • Hence, horizontalinherently boundedintuition. We wouldwith time (becausetwice).• Schuler tuning1 isexhibit a natural frea pendulum whose

Alon


bb

•aatmcse

6.2•cveo

erent minus sign in theed above using a simpleccelerometer:

compensate

g

θ 3) Which is apparentmotion this way

4) Whichcauses theplatform to

this way tostay level

v rdt∫

1R---

dt∫

ωR dt∫


Schuler Loop Consider a stable platform servo, which isalled a Schuler loop. It computes the angularelocity of the vehicle with respect to thearth and rotates the table appropriately inrder to keep it level.

spring model of an a

+-

aAccelerometer

gθgθ

a gθ–

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.2Schuler Loop

ecause of the vehicle acceleration as shownelow.

The hypothetical Schuler pendulum, having length equal to the earth’s radius, can beccelerated arbitrarily around the surface ofhe earth without being disturbed by theotion of the vehicle. In practice, one cannot

onstruct such a pendulum, but any dynamicystem with the correct natural frequency willxhibit this same stability property.

g

θ

a

• There is an inhprocess as illustrat

2) Spring

1) Platformtilts off-levelthis way

deflectsthis way

Alon


•rftie•w

•u

is twice to obtain thential equation of the loop:

orcing function whatever,lly oscillate at a period of:

d of a pendulum whoseof the earth!ntal oscilation happensven if there is no stable

tyation is much worse:e posit ion estimate is altitude.s e s g r av i t y t o be.

(12)

gR--- a gθ–( )

aR---=

(13)4minutes


a gθ– aR--- a gθ–[ ]∫ dt∫–=

6.3 Vertical Instabili• Vertically, the situ

• Suppose thslightly high in• Th i s c auunderestimated

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.3Vertical Instability

The feedback path is constructed byecognizing that the accelerometer specificorce includes any component of gravity dueo an off-level condition. This feedback isnherent in the accelerometer - there is nolectrical connection required. If the table is off level, the accelerometerill measure the gravity component:

We can invest igate the dynamics bynwinding the loop:

(10)a g θ gθ≈sin= (for small angles)

(11)

a gθ– a g ωdt∫– a gR--- vdt∫–= =

g

• Differentiate thcharacteristic differe

• Thus, given any fthe table will natura

which is the periolength is the radius • The same horizocomputationally etable.

gθ·· =

θ·· gR---θ+

T 2π Rg--- 8= =

Alon

Position Estimation 4:ation 17nertial Navigation

6.4

•Sb•a•vta

•a

ed specific force and the the sys tem output i s

specific force include an and cause errors in thend gravitation denoted by

accomplished through the

cripts i and t representuantities. Substituting thisal equation and cancellingation yields.

rential equation whichgation of errors from the

the position and gravity

(15)δaδrδg

(16)δa δg+


erms of the specific force indicated by theccelerometers , and gravitation :

We will use a technique called perturbativenalysis. A hypothetical perturbative error is

a g

(14)ainertialt2

2

dd r a g+= = • This is the diffe

describes the propaaccelerometer to computations.

t22

dd δr =

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.4Simple Error Analysis of I

• Which means the actual acceleromeerreadings will be interpreted partly as anacceleration upward.• Which cause s an even h ighe roverestimate of altitude.• Yadda yadda!

Simple Error Analysis of Inertial Navigation

Most errors in an INS oscillate with thechuler period. Lets compute the aboveehaviors from first principles. Cons ide r a s i ng l e e r ro r sou rce -ccelerometer bias Consider the basic navigation equation inector form - inertial acceleration expressed in

applied to the senseffect of th is oninvestigated.• Let the indicatederror denoted ,computed position a

and . This issubstitutions:

• Where the subsindicated and true qback into the originout the original equ

δa

∆r ∆g

ai at +=ri rt +=gi gt +=

Alon

Position Estimation 4:ation 18nertial Navigation

•pagt

•

•

•totTi

δ

d

lution to these equations of the trajectory followedt the start point for the

e z axis on the surface of vehicle trajectory remain

t so that andthis assumption, the crossthe equations cancel and

be the g rav i t a t iona lradius to the local region,

ssumed t ra jec tory aretant. If the accelerometer

(19)

yδy zδz+ + )x δax=

yδy zδz+ + )y δay=

yδy zδz+ + )z δaz=

x y 0= =

(20)

)δx δax=

)δy δay=3)δz δaz=


Further analysis of this equation requireshat a coordinate system be adopted. Let therigin be placed at the center of the earth, andhe three cartesian axes be oriented arbitrarily.hen the above equation in component form

s:

t

• Le t and acceleration of and which for our aapproximately cons

δy·· GM R3⁄(+

δz·· 2GM R⁄(–

g0 R0

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.4Simple Error Analysis of I

The gravitational force is a function only ofosition. By a Taylor series expansionssuming a spherical homogeneous earth, theravitation error can be rewritten in terms ofhe error in the position as follows:

By the product rule of differentiation:

Substituting this into (2) yields:

(17)g GMr3

---------r– GMr r⋅( )3 2/---------------------r–= =

gr∂∂ g δr GM

r3---------– δr 3GM

r5--------- r δr⋅( )r+= =

(18)2

2d δr δa GMr3

---------δr 3GMr5

--------- r δr⋅( )r+–=

• Any particular sorequires knowledgeby the vehicle. Lesystem be along ththe earth, and let theclose to this poin

. Under coupling terms in they reduce to:

δx·· GMr3

---------δx 3GMr5

--------- xδx(–+

δy·· GMr3

---------δy 3GMr5

--------- xδx(–+

δz·· GMr3

---------δz 3GMr5

--------- xδx(–+

z r R= =

δx·· GM R3⁄(+

Alon


esc

•if((•qa•i

error in any direction andor.

amicsan be conducted for manyf error. Some interestingzed below.chuler frequency, bearth’s radius, be the

26 39 52Time, min

δv

δh

gVc


sinusoidal) at the cost of a divergentexponential) vertical channel. Without such a field, the errors all growuadra t i ca l ly wi th t ime fo r cons tan tccelerometer bias. The development of position error over times expected to resemble the following, where

• Similiar analyses cdifferent sources oresults are summari• Let be the Sgravity, be the e

ω0R

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.5Other Error Dynamics

rrors are assumed to be constant biases, theolutions to these equations for zero initialonditions are:

Hence, the accelerometer feedback that isnherent when operating in a gravitationalield bounds the horizontal error channels

(21)

δxδaxg0 R0⁄--------------- 1

g0R0------t

⎝ ⎠⎜ ⎟⎛ ⎞

cos–=

δyδayg0 R0⁄--------------- 1

g0R0------t

⎝ ⎠⎜ ⎟⎛ ⎞

sin–=

δzδaz

2g0 R0⁄-------------------

2g0R0--------t

⎝ ⎠⎜ ⎟⎛ ⎞

cosh=

is the horizontal is the vertical err

6.5 Other Error Dyn

δhδv

Erro

r, ft

X 1

03

0

1

2

3

4

5

6

13

Alon


ib

ABi

InEr

VeD

InA

InA

, it would be nice to knowformulas for and INS withMS or “vehicle motion

instrument

ation of INS Errors

osition Error Physical Source

t2

2----ω0tcos 1–

ω02---------------------------+


lignment ditioncalibrationinstrument

itial Azimuthal lignment

initial con-ditioncalibrationinstrument

θ0 0 0

ψ0ψ0VIct

Mobile Robot Systems 6 Error Dynamics of Inertial Navig6.5Other Error Dynamics

nertial velocity in the crosstrack direction, e the present latitude, and be time. Then:

Table 2: Propagation of INS Errors

Error Source Position Error Physical Source

cceleromoeter as

instrument

itial Velocity ror

initial con-dition

rtical Gyro rift

instrument

itial Vertical initial con-

λt

εaεa

ω02------- 1 ω0tcos–( )

εv εvω0sin tω0

-----------------

ωv Rωv tω0sin tω0

-----------------–⎝ ⎠⎜ ⎟⎛ ⎞

Rθ 1 ω tcos–( )

• For mobile robotsthe error dynamics odometry (aka Vsensor”) inputs.

Azimuth Gyro Drift

Table 2: Propag

Error Source P

ωz ωzVIc

Alon


7 R7.1

•esapw•aoss•

rn to same spot (survey

update

ding

started, a sequence ofseveral minutes, is oftene system is operational. involves spinning up ther the components to reachature (often, precisionironmentally stabilized).omple ted , the sys temce which effects the initialccelerometers along the

on coordinate system. nt, the local vertical is the gravity vector and thes a reference for north. driving the platform untilccelerometers read zero.


• Radar altimeters• Doppler radar velocity• GPS• Landmarks• Map matching

alignment of the aaxes of the navigati• In self alignmedetermined throughearth’s spin provide

• Levelling is the horizontal a

Mobile Robot Systems 7 Real Inertial Systems7.1Aided Inertial Systems

eal Inertial Systems Aided Inertial Systems The use of external measurements to reducerror is referred to as damping, and theystems employing the technique are calledided or augmented inertial systems. Inractice, all INS systems are aided in someay.

The damping of the vertical channel can beccomplished in many ways. Measurementsf position, velocity, and attitude can be usedingly or in any combination. The particularolution adopted is application dependent. Some alternatives are:

• Barometric altitude

• Periodic retupoints)• Zero velocity• Odometry• Magnetic hea

7.2 Alignment• When an INS isoperations, taking performed before th• Mechanical setupgyros and waiting fooperating tempercomponents are env• Af ter th is i s cundergoes a sequen

Alon


•brv

7.3•aoahabv•0

ed to 0.1° and heading to


oth normal directions (crosstrack andertical) are distinguished. Land vehicle navigation systems achieve.2% to 2% of distance travelled. Pitch and

1. Although the term is used commonly, it does not mean that the gyro operates according to the principle of the gyrocompass. It means that it does the same thing as the gyrocompass, namely, finds north.

Mobile Robot Systems 7 Real Inertial Systems7.3Accuracy

• North alignment or gyrocompassing1isaccomplished by rotating the now levelplatform about the vertical until one gyrosenses no component of the earth’srotation.

Modern GPS aided systems do “movingase alignment” where the difference in GPSeadings over time can be used to determineehicle heading. Accuracy Commercial cruise systems can achieveccuracies on the order of 0.2 nautical milesf error per hour of operation. Pitch and rollre often accurate to 0.05° and true geographiceading to 0.5°. In some cases, positionccuracy along the trajectory (alongtrack) and

roll can be measur0.5°.

Alon


8 S•lk•r•ako•Ta

•cpt

ystems deliberately do noth. Hence, they have nog near the poles where

instantaneously.rizontal position error in

are oscillatory with the4 minutes and hence are

is unstable in free inertialother aiding devices are


Law• in the wrong Coord System <-> gyrostrack attitude

Mode rn “ s t r apdown” sy s t ems a r eomputationally stabilized. The stablelatform is no longer necessary in an INS

hough it has other uses in pointing systems.

Mobile Robot Systems 8 Summary7.3Accuracy

ummary Inertial navigation is based on Newton’saws and works everywhere that gravity isnown. It is stealthy and jamproof. The basic process is accelerometer deadeckoning (so-called “free inertial” operation). Errors of 1 part in 10,000 in measuringcceleration of predicting gravity causeilometers of position error after 1 hour ofperation regardless of any motion. Naive approaches are seriously flawed.hree problems and their solutions are thatccelerometers measure:

• the wrong quantity <-> model gravity• wrt the wrong ref frame <-> Coriolis

• Wander azimuth stry to point Nortproblems operatin“North” can change• Some forms of hofree inertial modeSchuler period of 8bounded.• Vertical position mode and various used to damp it.

Alon


9 N- m

10[1[2[3P[4[5H[6[7IE[8S[9N[1[1Y[1In[11[1o[1W

rtial systems which make use ofments of navigation quantities.rallel to the direction of motion.ystems which compute rather thaname quantities.for system which maintain theitial orientation with respect to the

perpendicular to the direction oflly.of filtering redundant externalS output in order to reduce the

the matrix which describes thexes into the navigation frame.

n for systems which maintain aefined with respect to the earth in

in UTM coordinates. for systems providing no platform

systems which perform geometricle platform.


1] Pittman, G. R. Jr (ed.), “Inertial Guidance”, John Wiley Z& Sons Inc, Nework 1962.2] Sutherland, A. A. Jr, and Gelb, A., “Application of the Kalman Filter to Aidedertial Systems”, Analytical Sciences Corporation, Winchester, Mass.3] Savage, P. G., “Strapdown Systems Algorithms”, AGARD Lecture Series No.

33, Strapdown Associates Inc., Minnetonka, Minnesota.4] Savant C. J. Jr. , R. C. Howard, C. B. Solloway, and C. A. Savant, “Principles

f Inertial Navigation”, McGraw Hill Book Company, Inc, New York, 19615] Unknown, “An Introduction to Inertial Navigation”, Litton Systems Inc,oodland Hills, Ca,

navigation frme which is dsome way.easting - eastern coordinate free azimuth - a designationrate about the azimuth.geometric - designation for stabilization utilising a stab

Mobile Robot Systems 9 Notes7.3Accuracy

otesine perceptOR INS unit survey for info.

Bibliography] Broxmeyer, C., “Inertial Navigation Systems”, McGraw Hill, 1964] Britting, K., “Inertial Navigation System Analysis”, Wiley, 1971.] Draper, C. S., W. Wrigley, and J Hovorka: “Inertial Guidance”, Pergammon

ress, New York 1960] Farrell, J., “Integrated Aircraft Navigation”, Academic Press, 1976.] Fernandez M., and Macomber G. R., “Inertial Guidance Engineering”, Prenticeall, Englewood Cliffs, NJ, 1962] Ivey, D. G., “Physics”, University of Toronto Press, 1982] Kuritsky M. M. , and Goldstein M. S. , “Inertial Navigation”, Proceedings of theEE, vol 71, no 10, 1156-1176] McGreevy, J. “Fundamentals of Strapdown Inertial Navigation”, Litton

ystems Inc, Moorpark Ca, Rev C, May 1986] Odonnell, C. F. (ed.), “Inertial Navigation Analysis & Design”, McGraw Hill,ew York, 19640] Parvin, R., “Inertial Navigation”, Van Nostrand, 1962

11 Glossaryaided inertial systems - ineredundant external measurealongtrack - the direction paanalytic - a designation for sinstrument the navigation frbase point - designation platform orientation at the inearth.crosstrack - the direction motion, measured horizontadamping - the process measurements with the INmagnitude of system error.direction cosine matrix - transformation of the body aearth fixed - see base point.earth-stabilized - designatio

Alon


gimplatgimgravgridlatitspagyrthe inergyrinerlatitequlevealiglocaplatlocalongprimmecare mernornorthe

atitude.ch measures a gimbal angle.on of performance experienced byroach the earth’s poles.l loop which actively prohibitsnsducer.

alized mathematical model of the

ich employs gravity feedback in

d of oscillation of the Schuler

s of designing a system to exhibity.of aligning an INS without the aid

on for systems which instrument

ff.ion for systems which maintainientation.al acceleration minus gravitational

whose orientation is stable in a


itude - angle of a point on the earth measured from thee meridian.hanization - the process by which the navigation equationsimplemented in hardware and software.idian - a line of constant longitude.thing - northern coordinate in UTM coordinates.th-slaved - designation for systems which physically trackdirection of north by rotating the platform.

space-stabilized - designatconstant inertial platform orspecific force - the net inertiattractions.stable platform - a platformparticular reference frame.strapdown - see analytic.

Mobile Robot Systems 11 Glossary7.3Accuracy

balled - designation for systems incorporating a stableform.bal lock - singularity of the platform gimbal mechanism.ity - vector sum of gravitation and centrifugal force. - a regular coordinate grid placed on a map. Unlikeude and longitude, grid lines are parallel and equallyced.ocompassing - the process of using the earth’s spin to finddirection of north.tial measurement unit - the sensor suite consisting of theos and accelerometers.tial reference unit - see inertial measurement unit.ude - angle of a point on the earth measured from theatorial plane.lling - the process of levelling a stable platform duringnment.lly level - designation for systems which maintain theform level.l vertical - see locally level.

parallel - a line of constant lpickoff - the transducer whipole problem - the degradatiinertial systems as they apprebalance loops - a controdeflection of a compliant trareference ellipsoid - an ideearth.Schuler loop - the loop whorder to level a table.Schuler period - the periopendulum, or 84 minutes.Schuler tuning - the procesthe Schuler natural frequencself alignment - the process of external measurements.semi-analytic - a designationly the navigation frame.signal generator - see picko

Alon


starstartorqvehthe wanto nwanazim


Mobile Robot Systems 11 Glossary7.3Accuracy

-tracker - a device which tracks ad reports the bearing of a/uer - device for applying torque to a gyro or gimbal.icle rate - the angular velocity of the vehicle with respect tocenter of the earth.der angle - the rotation of the coordinate axes with respectorth.der azimuth - a designation for a system which torques theuth to follow the earth rate only.

position estimation 4 - field robotics estimation 4: inertial navigation ... solutions to these...

Documents