HAL Id: hal-00668921https://hal.inria.fr/hal-00668921

Submitted on 10 Feb 2012

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Calibration of a fully-constrained parallel cable-drivenrobot

Julien Alexandre Dit Sandretto, David Daney, Marc Gouttefarde, CédricBaradat

To cite this version:Julien Alexandre Dit Sandretto, David Daney, Marc Gouttefarde, Cédric Baradat. Calibration of afully-constrained parallel cable-driven robot. [Research Report] RR-7879, INRIA. 2012, pp.21. <hal-00668921>

ISS

N0

24

9-6

39

9IS

RN

INR

IA/R

R--

78

79

--F

R+

EN

G

RESEARCH

REPORT

N° 7879Janvier 2012

Project-Teams COPRIN

Calibration of a

fully-constrained parallel

cable-driven robot

Julien Alexandre dit Sandretto

David Daney

Marc Gouttefarde

Cedric Baradat

RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93

06902 Sophia Antipolis Cedex

rt♦♥ ♦ ②♦♥str♥ ♣r

r♥ r♦♦t

♥ ①♥r t ♥rtt♦∗

♥②†

r ♦ttr‡

r rt§

Pr♦t♠s P

sr ♣♦rt ♥ ♥r ♣s

strt ♥ ♥tt♦♥ ♦ t ♠♦ ♣r♠trs ♦r ♣r r♥ r♦♦t s ♣r♦r♠ ② s♥ ♦t rt♦♥ ♥ srt♦♥ ♣♣r♦ ♠♥♣t♦r st ss ♦♥ ♣r rttr ♥ s t♦ ♦♥tr♦ t rs ♦ r♦♠ ♦ ts ♠♦♣t♦r♠ s♦ tt t ♠♦ ♣t♦r♠ s ② ♦♥str♥ ② t s ❯♥r s♦♠ ②♣♦tss♦♥ ♣r♦♣rts t ♥trst ♦ r♥♥② ♥ tt♦♥ s ①♣♦t t♦ srt ② s♥♣r♦♣r♦♣t s♥s♦rs s ♣♣r♦ s ♦♠♣r t♦ t ts t♦ ♠♣♠♥t rt♦♥♣r♦ss t♦♥② ♥ t♦♦s ♥ ♦rt♠ ♠♣r♦♠♥ts r ♣rs♥t t♦ ♣r♦r♠ t♣r♠tr ♥tt♦♥ ♦♠♣t ①♣r♠♥tt♦♥ ts t r♦♦t r② ♠♣r♦♠♥ttr rt♦♥ ♦r srt♦♥ ❲ s♦ tt t s ②♣♦tss ♦♥ ♣r♦♣rts r r ♦r♦r t ♥st♠♥t ♥ tr♠s ♦ t♠ ♥ ♦st t♦ ♦t♥ t ①tr♥ ♠sr♠♥ts ♦rrt♦♥ ♣r♦ss ♦s ♥♦t r♥ ttr rsts ♥ ♦s ♥♦t ♥ t s♠♣t② ♥ ♥②♦ t s rt♦♥ ♣r♦ss

②♦rs rt♦♥ r♥ ♦♦t ♥♠ts

∗ ♥r† ♥r‡ r♠♠§ ♥

t♦♥♥ ♥ r♦♦t ♣rè à âs

sr♦♥tr♥t

r♣♣♦rt rr

♥r

és♠é ♦♠♥t ♣rés♥t s réstts ♥ ét♦♥♥ t ♥ t♦ét♦♥♥ ♥ r♦♦t ♣rè à âs ♣s rt♥s ♠ét♦s ♥tt♦♥s♦♥t ♦♠♣rés t ♥ ♥♦ ♣♣r♦ t ♦ st ért

♦tsés t♦♥♥ ♦♦t à âs ♥é♠tq

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

♦♥t♥ts

♥tr♦t♦♥

r♥ r♦♦t ♦ ♦ t r♦♦t

♥♠t rt♦♥s♣ tt qr♠ ♦rr ♥♠ts sss♦♥ ♦♥ ♥rs ♥♠t ♦♥

♥♥② ♥ tt♦♥

rt♦♥ ♥r

❲t st qrs rt♦♦♥ st♥ rss♦♥ ❳ ♦♥ ♥r st qrs s♦♥ ♦t♥s

r♥ r♦♦t s rt♦♥ t ①tr♥ ♠srs rt♦♥ t♦t ①tr♥ ♠srs

①♣r♠♥ts sr♣t♦♥ ♦ t ♣r♦t♦t②♣

s ♦ ♥ t♦♦s

♦ ♦♦s

sr♠♥t qst♦♥ ♣r♦ss t②

sts Pr♠tr ♥tt② rt♦♥ t ①tr♥ ♠srs rt♦♥ t♦t ①tr♥ ♠srs ❱t♦♥

sss♦♥

♦♥s♦♥s ♥ tr ♦rs ♦♥s♦♥s tr ♦rs

♥♦♠♥ts

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

♥tr♦t♦♥

♣ts ♦ ♣r r♦♦ts ①♣♥ tr s ♥ t ♥str ♣♣t♦♥sr♦♠ s♣ ♣♥♣ t♦ ♣rs srr② ❬❪

s ♦ t♦r♥s ♥ ♠♥tr♥ ♦r ss♠② t ♦♠tr② ♦ t t ♠♥♣t♦r ♦s ♥♦t ♦rrs♣♦♥ t♦ t sr s♥ ♥ ts t♦rt♥♠t ♠♦ ♦♥sq♥t② t ♣r♦r♠♥s ♦ t ♠♥♣t♦r s sts r② r r ♥♦t ♦st s ♣r♦♠ ♦ ②♣ss ② ♠♣r♦♥ t t♦rt ♥♠t ♠♦ ② ♥♥ t t s ♦ t♥♠t ♣r♠trs s ♣r♠trs ♥♥ t ♦♠tr② ♦ t r♦♦tr♠ ♥ ♣t♦r♠ ♥ tt♦r ♣rts r ♣r♦ ② ♥♠t rt♦♥ ♣r♦r rt♦♥ ♦♥ssts ♥ ♥t②♥ ♠♦ ♣r♠trs tr♦r♥♥t ♥♦r♠t♦♥ ♦♥ t stt ♦ t r♦♦t ♣r♦ ② ♠sr♠♥ts ts ♦ rt♦♥ r ♦t ♣rt ♥ ♦♠♣tt♦♥ ♥ ①♣r♠♥t ♣♥ tr♠♥s t t②♣ ♦ r♥♥t ♥♦r♠t♦♥ ♥ t ② t♦♦t♥ t♠ ♥ ♦rr t♦ ♠♣r♦ t ♥♠r ♥tt♦♥ ♦ t ♥♠t♣r♠trs ♦♠♣t ♥ ♥①t st♣

♠♦r s♠♣ ♥ ♦♠♠♦♥ ♣♣r♦ t♦ rt ♣r r♦♦t s t♥rs ♠t♦ s ♣rs♥t ♥ ❬ ❪ ss t ♠sr♠♥ts ♦ t ♣♦s♦♦r♥ts ♦t♥ ② t♦♦t ❬❪ ♦r ♠r ❬❪ s r♥♥t♥♦r♠t♦♥ ♥ ♦t♥ ② ♦tr ①tr s♥s♦rs s s ♥♥♦♠trs ❬❪ ♦r♥② t②♣ ♦ ♦♠tr ♦♥str♥ts ❬❪ ♥ t srt♦♥ s t ♥ssr②t r ♣r♦ ② t♦♥ ♥tr♥ s♥s♦rs ♥② s♦t♦♥s ♥♣r♦♣♦s ♦r ♣r ♠♥♣t♦rs ♥ s♦♠ ♦ t♠ ❬ ❪ ♠② s②♣t t♦ t s t t ♥ t ♣rs♥t ♣♣r ♥ ♥ ❬ ❪ ♥t♠sr♥ s s ♥ t♦♥ s♠♥t tt ♥s t s ♥ t♠♦ ♦ ♦ ♣t♦r♠ s ♦♥sr

s ♣♣r ♦ss ♦♥ ②♦♥str♥ ♣r r♥ r♦♦t r♥ r♦♦ts sr ♥trst♥ ♣r♦♣rts r ♠ss ♦ ♠♦♥♣rts ♦r s ♦ ♥ ♠ss s ♦ r♦♥rt♦♥ ♥ s♣② ♣♦t♥t② r② r ♦rs♣ ② r ♥♦t② s ♦r ②♥ ♠rs②st♠ ❬❪ ♥ ♥ ♣r♦♣♦s ♦r ② ♦s tr♥s♣♦rtt♦♥ ♦r ♦r♥t♥ ② s ♥ ♦r ♦♥t♦r rt♥ ❬❪ ♦st ♦ ts ♥ts s ♦♠♣① ♥♠t ♥ ②♥♠ ♦r t♦ t ①t② ♠ss ♥ stt② ♦ t s ♥ ♦rr t♦ ② ♦♥str♥ t ♠♦ ♣t♦r♠ t ♥♠r♦ s ♠st rtr t♥ t ♥♠r ♦ ❬❪ ♥ s r♥♥t②tt r♥ r♦♦ts st♥ss s ♥r② ♥ ♥ ♦r st r♦♦t s tt ② t s ♦r s① r sts ♥ r♥♦♥ r♥ r♦♦t ♥♠ts ❬ ❪ t ♦♥r♥♥ tr rt♦♥ ♥♠t rt♦♥ s ♦♥ sr t♦♥ ♦r t ♦♥t s♥s♦rs ♥ r♣♦rt ♥ ❬ ❪ ♥ ♥ ❬❪ rs♣t② rs srt♦♥♣r♦r ♦r ♣♥r r♦♦t s ♥tr♦ ♥ ❬❪

s ♠♥t♦♥ ♦r ♦t ♥♦r♠t♦♥ ♣r♦ r♥♥t qt♦♥ss ♦r ♣r♠tr ♥tt♦♥ ♦t ♦ ♦♠♣tt♦♥ ♠t♦s ♥♦♣ s ♦♥ s t ♥♦♥ ♥r st sqrs ♣♣r♦ ♦♠♣ts t ♣r♠trs s♦ s t♦ ♠t ♠♦ st♠t♦♥s t ♠srs ♠r♠t♦s r ♦rt♦♦♥ st♥ rrss♦♥ ❬❪ ♥ χ2 s ② Pt ♥❬❪ r♥t ♣♣r♦s ♥ ♣r♦♣♦s tr♥ ♣t ② ❲♠♣r♥ ❬❪ ♦r ♥ ♦r♥ ♥tr ♣♣r♦ ♣r♦♣♦s ② ♥② ♥ ♥ ❬❪

♥ ts ♣♣r ♥♠ts ♥ r♥♥② ♦ t r♥ r♦♦ts rst ❲ ♣r♦♣♦s s♠♣ ♥ r♦st ♠t♦ ♦r srt♦♥ ♥♦♠♣r t t ♠♦r st♥r ♥rs rt♦♥ ♥ tst tr♦①♣r♠♥ts t r♥t ♣r♠trs ♥tt♦♥ ♠t♦s s ♦♥ ♥♦♥ ♥r

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

st sqrs ♣♣r♦ ♥② t rsts ♦t♥ ♥ t s ♦ t str♥♥t r♥ ♠♥♣t♦r r t♥ sss

r♥ r♦♦t

s st② s ♣rt ♦ ♣r♦t ♦♦ ♦♥tr♦ ♦ ♥t ♦♦t ♥♦t② ♠s t s♥♥ ♣r r♦♦t ♥ n = 6 rs ♦ r♦♠♥ r② r ♦rs♣ ♥ ts ♣♣r s♠ s ♣r♦t♦t②♣ ♦ ts ♣r r♦♦t s st t ss m = 8 s ♦♥tr♦♥ t ♠♦t♦♥ ♦ ts♠♦ ♣t♦r♠ ♥ ts ♦♠tr② s ♥ ♦s♥ s♦ tt t ♣t♦r♠ s ②♦♥str♥ ② t s

♦ ♦ t r♦♦t

r r♥ r♦♦t

♠♦♥ ♣t♦r♠ ♦r ♥t♦r ♠♦ rr♥ r♠ ΩC s ♦♥♥t t♦ t s ① rr♥ r♠ ΩO ② m = 8 s m > n t♦ ② ♦♥tr♦ ❬❪ ith ♦♥♥ts t ♣♦♥t Ai ♦ t s ♦♦r♥t ai ♥ ΩO t♦ t ♣♦♥t Bi ♦♥ t ♠♦ ♣t♦r♠ ♦♦r♥t bi ♥ ΩC ♣♦s ♦ t ♠♦ ♥ ② t ♣♦st♦♥ P ♥ t ♦r♥tt♦♥ ♦ ΩC

①♣rss ♥ ΩO s rt② ♦♥tr♦ ② t ♥t ♥ t t♥s♦♥ ♦

♥♠t rt♦♥s♣

♠♣t ♥♠t s②st♠ ♦ qt♦♥s s ♥ ②

||P + Rbi − ai||2 − L2

i = 0, i = [1...m]

r Li s t st♥ AiBi

tt qr♠

stt qr♠ s ♥ ② ♠♥s♦♥ qt♦♥ s②st♠

W · τ = F

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

r τ = [τ1, ..., τm] s t t♦r ♦ t♥s♦♥s s t t♦r ♦r♥s ♦♠♥t♦♥ ♦ ♦rs ♥ ♠♦♠♥ts ♣♣ ♦♥ t ♣t♦r♠ ♥ W =−J−T J−1 ♥ t s♦ ♥rs ♥♠ts ♦♥ ♠tr① ♥t♦♥ ♦t ♣♦s ♥ t ♦♠tr ♣r♠trs ♦ t r♦♦t ❬❪

♦rr ♥♠ts

♦rr ♥♠ts ♦♥ssts ♥ t s♦♥ ♦ q s♦ s t♦ tr♠♥t ♣♦st♦♥ ♥ ♦r♥tt♦♥ ♦r ♥ st ♦ ♥ts Li ♥ t s rm = n tr r s♦♠ ♦rt s♦t♦♥s ❬❪ ♣r♦ ①t s♦t♦♥s ♦r ♦r r♥ r♦♦t r m > n ♥ ①t ♦rr ♥♠ts♦♥ssts ♥ s♦♥ ♥ ♦r♦♥str♥ s②st♠ ♦ qt♦♥s s② s♥t s♦t♦♥ ♦r r♥♦♠ st ♦ ♥ts ♥ t tr r s♦♠ ♦♥str♥tst ♠♥s♦♥ q t♦ m−n ♥♥ t ♦♠tr② ♦ t r♦♦t t t st♦ ♦♥t ♦♦r♥ts ♥♦r♠t♦♥ tt ♥ s t♦ srt t r♦♦t

♥ rt♦♥ ♣r♦ss t ♥♠t ♣r♠trs r ♥♦t st♠t♥ t ♥ts r ♥♦♥ ♣ t♦ ♠sr♠♥t rr♦rs ♥ ts s t s♦ ♥ ♥rt rr♦rs ♥ ①t s♦t♦♥ ♦ ♦♥str♥t ♦ ♦♥ ♥ ♦♠♣① ♥ s♦t♦♥ s t♦ s s ♦♥ t♦♥♣s♦♥s♠ FKNR t m = n t ts ♦♥r♥ t r♦♥st♠t ts ♥♦t rt♥ m > n ♥ ♣♣r♦①♠t ♥♠ FKLS s s ♦♥ ♥♦♥♥r st sqrs ♦rt♠ t rs rr♦r ♣r♦s ♥ ♥① s♦♥ t qt② ♦ ♥♦ ♦ ♥♠t ♣r♠trs ♦r ts s♠♦ ♦♥r t♦ ♦ ♠♥♠♠ tr♦r ♣r♦♥ r♦♥ ♥♦r♠t♦♥ ♥t ♥tt♦♥ ♣r♦ss

sss♦♥ ♦♥ ♥rs ♥♠t ♦♥

r♥ ①♣r♠♥tt♦♥ r♠r r♥ t♥ t ♥rs ♥♠t♦♥ ♦♠♣t t ♥t r♥s ♥ t ♥rs ♥♠t ❬❪ ♥st ①♣tt♦♥s t ♥t r♥s ♦♥ s ttr rsts ♥ tr♠s ♦♥♠r ♦ trt♦♥s ♥ ♣rs♦♥ r♥ t t♦♥♣s♦♥ s♠ s♦r s♦♥ t rt ♥♠t ♣r♦♠ ❲ s t ♦♦♥ t♦r ♥ ♦rrt♦ r♣rs♥t t ♣♦st♦♥ ♥ t ♦r♥tt♦♥s X = [x, y, z, αv] t ♦r♥tt♦♥t♦r δ = αv s ♦♥ t t ①s t♦r ♥ α t r♦tt♦♥ ♥ ❲ ♥♦tt

Ω = αv + sinαv + (1− cosα)v × v

❲ s② ♠ ♦♥ ♦rr ②♦r ♣♣r♦①♠t♦♥ t α s♠

Ω ≈ αv + αv = δ

Pr♦♣♦st♦♥ rst ♦rr ②♦r ♣♣r♦①♠t♦♥ s t♦♦ str♦♥ ♦r t ♦s

♥s ♣r♦♣♦s s♦♥ ♦rr cosα ≈ 1− α2

2♥ sinα ≈ α

♣♣r♦①♠t♦♥ ♦ Ω ♦♠s

Ω2 = αv + αv + (1− (1− α2

2))v × v

♦♣

Ω2 = δ + α2

2v × v = δ + 1

2· αv × αv

❲ δ = αv + αv ♥ s♦ αv = δ − αvΩ2 = δ + 1

2· αv × (δ − αv) = δ + α

2· v × δ t♦ v × v = 0

Pr♦♣♦st♦♥ ♥ ♣♣r♦①♠t♦♥ ♥ ♦rr ♦r ♠♦r ♥s ♥ st♠t♦♥ ♦t rt ♦ ♥ ♥ t ♥ s ♥♦t s♥t

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

♦♥ ♥t r♥s rst ♦rr ♦♥ ♦rr♠r ♦ trt♦♥s

Prs♦♥ ♠ ♣s ♠s ♠s ♠s

r sts ♠♠ ♥ s♦♠ rs♦t♦♥ ♦ rt ♥♠ts

Pr♦♣♦st♦♥ ss ♥rs ♥♠t ♦♥ Jinv = [ni, ni × BiC]♦ rrtt♥ s

λi = [ni, ni ×BiC] · X + [ni, ni×BiC] · (0,α

2v × δ)

= [ni, ni ×BiC] · X + [ni, (ni×BiC)×α

2v] · (0, δ)

= [ni, ni ×BiC + (ni ×BiC)×α

2v] · X

♥ s♦ Jinv ♦♠s

Jinv = [ni, ni ×BiC + (ni ×BiC)×α

2v]

r ♣rr t♦ ♣ t ss Pr ♥ ♦♦r♥ts ♦♥ ♦ rt

Jinv2 = Jinv · (I + M)

r s t ♣r r♦ss ♣r♦t ♦ α2v

sts ♥ s♠t♦♥ ♦rrt tr♠ α2· v × (ni × BiC) s tt♥

s♦♠t♠s r♥ ♥♦ t♦ ♥ trt♦♥s ♥ t♦r ♥ t ♣rs♦♥ ♦ t

sts ♥ ♥♠r ♦ trt♦♥s t♠ ♥ ♣rs♦♥ ♥ ♦rr ♥♠ts s♠ r ♣rs♥t ♥ ❲ s♦ ♦♦ rsts ♥ srt♦♥ ♥ ts s t ♣♦st♦♥ r ♥♦t ♥♦♥ ♥ t rr♦r ♦ ♠♣♦rt♥t ts ♥ ♦♥ s s♦ r② s

♦ ♦♥ ♦rr ♥♠t s♠ s♥ ts ♦rrt tr♠ sttr ♥ str s♦t♦♥ srt♦♥ ♣r♦r ♥s ♦t ♦ t♠ ♥♥ s♦♠ s ♦♥r ♥ t ss ♦♥ ♥t ♦ t

♥♥② ♥ tt♦♥

s ♥ ♥ ❬ ❪ t r♦♦t st ♥ ts ♣♣r s r♥♥t ♥tr♠s ♦ tt♦♥ s t ♥t♦r ♠♦t♦♥ s ♦r♦♥str♥ ② ttt♦rs ♥ qst♦♥ s t♦ ♥ ♦ ts r♥♥② ♥ s t♦ ♣r♦r♥♥t ♥♦r♠t♦♥ ♦♥ t stt ♦ t r♦♦t

♦♥sr t stt② ♥ t ♠ss ♦ t s t ♥ts ♣♥♦♥ t t♥s♦♥s ♥ ♦♥ t ♣♦s ♦ t r♦♦t s s ♠♦ ② mt♦♥ qt♦♥s Li = L(ρi, τi) t ρi t rtr ♦♦r♥t ♥t ♦t i ♥♦♥ ② r♠ s ♦♥sq♥ t ♥♠t ♥ t stt♠♦ qs ♥ r ♥ t ♥♠r ♦ qt♦♥s ♦r ♦♥ ♦♥rt♦♥♦ t r♦♦t s q t♦ m ♥ q ♥ ♥ q srt♦♥ ♣r♦♠ s♦♥sr t ♣♦s ♦♦r♥ts P ♥ R r ♥♥♦♥ ♦r ♥♦♥

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

♦ t ①tr♥ ♦r F t st ♦ m rs τi t♦ t♥ ♥ ♦♥t srt♦♥ s ♣♦ss ♦♥② ♥ ♥♦r♠t♦♥ ♥ ♣rt r ♥ ♦tt t♥s♦♥s τi

s♣♣♦s tt t stt② ♥ t ♠ss ♦ t s r ♥ t ♥♠t ♥ t stt rt♦♥s♣s r t♥ ♥♣♥♥t ♥ ♥ s♦ s♣rt② ♥ ♦tr ♦rs t 6 ♣♦s ♦♦r♥ts ♣♦st♦♥ ♥♦r♥tt♦♥ ♦ t ♠♦ ♣t♦r♠ s♦ r♦♠ t m = 8 ♥♠t rt♦♥s ♦ q ② r s♠♣② ♥t♦♥s ♦ ♦♠tr ♣r♠trsξ = [a1..m, b1..m,∆l1..m] ♥ ♦♥tr♦ ♥ts ρ1..m ❲ rtrt♦♥ Li = ρi + ∆li r ∆li s t ♦st ♥t ♥ t ith i = 1..ms s②st♠ s ♥♦ ♦r♦♥str♥ ♥ s t♦ qt♦♥s tr ♣♦s ♦♦r♥t ♠♥t♦♥ tt ♥ t ♥tr♥ ♠sr♠♥t ρi ♥ t ♥♠t♣r♠trs ξ s ②♣♦tss ♦♥ ♣r♦♣rts s♠♣s t rt♦♥ t② ♦ ts ②♣♦tss ♥ r t ♥ r♥ ♠♦ s♠t♦♥♦r ♦r tst ❬ ❪ ❲ s ♥ t ♦♦♥ tt t ♥t r♥♣r♦t♦t②♣ s ♥ ts s

rt♦♥

rt♦♥ ♦ s t♦ ♥♥ t r♦♦t ♣r♦r♠♥s ② ♠♣r♦♠♥t♦ ♠♦ ♥♦ ttr ♦♥ssts ♥ ♥t②♥ t ♠♦ ♣r♠trstr♦ r♥♥t ♥♦r♠t♦♥ ♦♥ t stt ♦ t r♦♦t ♣r♦ ② ♠sr♠♥ts

❲ s tt rt♦♥ ♥ ♦♥sr s ♥r ♣r♦ss ❬❪❲ ♠ r♥ t♥ t s r t♦♥ ①tr♥♠srs ♦♥ t stt ♦ t r♦♦t ♥ t s r t ♣r♦♣r♦♣t s♥s♦rt ♦ t r♦♦t r s♥t ♦r rt♦♥ s♦ srt♦♥

r♦♦ts st m > n r r♥♥t ♥ tr♠s ♦ ♠sr♠♥t ♠ t ②♣♦tss ♦ ♥♦♥ st ♥ ♠ssss s s st♦♥

♥r

s ♦♥ ❬❪ ♦r ♦ t NC ♠sr ♦♥rt♦♥ t rt♦♥ qt♦♥s ♥s tr t②♣s ♦ rs ♠sr♠♥ts Mk k = 1..NC t ♣r♠trs ξ ♥t t♦ ♥t② ♦♠tr ♣r♠trs ♥ ♥♥♦♥s rs Υ rqr t♦ ♠♦ ♦r qt♦♥s s rs Υ = [Υ, Υ1..NC

] s♦

❼ ♦♥st♥t Υ tr s ♦ ♥♦t ♥ r♥ t rt♦♥ ♣r♦ss

❼ ❱r s ♥t♦♥ ♦ t r♦♦t ♦♥rt♦♥s Υk=1..NC

❲ ♦♥sr s②st♠ ♦ qt♦♥s ♥♥ st ♦ ♠srs M ♥ t♥♥♦♥s V = [ξ,Υ] ♥ t rt♦♥ qt♦♥s

fk(Mk, V ) ≃ 0, k = [1...NC ]

s♦t♦♥ ♦ t s②st♠ ♦ ♦♠♣t ② r♥t ♠t♦s ♠♦st ♦t♠ ♥♦♥ ♥r st sqrs s♦t♦♥ ♠♥♠③s t rtr FT .Ft F = [f1, . . . , fNc

]T s ♦ ♦t♥ t ss ♥rrqrt ♦rt♠ t s♦♠ ♠♣r♦♠♥t ♦♥ rtr♦♥ ♥t♦♥ r ♣♦ss

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

❲t st qrs

❲ t♥q ♥tr♦s t t② t♦ ♣r♦rt③ ♠sr ② ♦♥sr♥t rtr♦♥ FT ΣF F t ♠tr① ΣF s t s ♥t♦♥ ♦ ♥♦♦♥ s♦♠ ♥rt♥ts ♥ t t ♠sr♠♥t ♠♦ tr♦ ♦r♥ ♠tr① ΣM ♥r ♣♣r♦①♠t♦♥ ♦ ΣF s ♦t♥ s ΣF = JT

MΣMJM

t JM = ∂F∂M

rt♦♦♥ st♥ rss♦♥

ts ♥t♦ ♦♥t t ♣♦ss rr♦rs ♥ ♠sr♠♥ts ❬❪ ♥ ♦♥srst rtr FT ΣF F + MT ΣMM ❲ ♣t M s t r♥ t♥ trr♥t M ♥ t ♥t M

❳

χ2 ♣r♠ts ♦♥tr♦ ♦ t♦♥ ♥ ♥tt♦♥ ♦ t ♥♥♦♥s ❬❪ ♦♥sr rtr s ♥♦ FT ΣF F + MT ΣMM + V T ΣV V ❲ ♣t V s tr♥ t♥ t rr♥t V ♥ t ♥t V

♦♥ ♥r st qrs s♦♥ ♦t♥s

ss ♥♦♥ ♥r st sqrs ♣r♦♠s s♦♥ s s ♦♥ trt ♥r♣♣r♦①♠t♦♥ s♥ ②♦rs ♥r③t♦♥ Prt r♥tt♦♥ s t♦♦♥ ♦r♦♥str♥ ♥r s②st♠

JV ·∆V = ∆F

t t ♦♥ JV = ∂f∂V

♥ t ♦♥t♦♥ t♦ ♦t♥ ♥ ♦r♦♥str♥s②st♠ ♦ qt♦♥s ♥ ②

NC × dimfk > dimξ + dimΥ + NC × dimΥk

s②st♠ ♦ q ♥ s♦ t ♥ ❱ s♥r ♦♠♣♦st♦♥ ♦ JV

♥ ♦rr t♦ ♣r♠trs r s♦ t ♦♦♥ ♦♥t♦♥ s t♦ r ❬❪

rank(JV ) = dim(V )

= dimξ + dimΥ + NC × dimΥk

r♥ r♦♦t s

♦r t ♣r r♦♦t rt♦♥ t qt♦♥s s r rt② t ♥♠t rt♦♥s♣s ♣r♦ tt t ②♣♦tss ♦ ♥ stt②♥ ♠ss s st♦♥ s ♣t

fk,i(Mk, V ) = ||Pk + Rkbi − ai||2 − (ρk,i + ∆li)

2

= 0

♦r k = 1...NC ♥ i = 1...m♦ ts sss t♦ r♥t rt♦♥ ♣♣r♦s t ♥ t♦t

①tr♥ ♠sr♠♥ts

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

rt♦♥ t ①tr♥ ♠srs

♥ t♦♥ t♦ t rtr ♦♦r♥ts ♥ ② t ♣r♦♣r♦♣t s♥s♦rst ♠sr♠♥t ♦ t r♦♦t ♣♦s ♣♦st♦♥ ♥ ♦r♥tt♦♥ ♣r♦ ② ♥①tr♥ ♠r ♦r sr trr s ss♠ t♦ rt♦♥ s②st♠ t♦ s♦ s ♠ ♦ t ♥t♦♥s fk,i(Mk, V ) t t♦♦♥ t

❼ Mk = [ρi,k, Pk, Rk]

❼ ξ = [ai, bi,∆li]

❼ V = [Υ, ξ] = [∅, ξ] = ξ

♥ t ①♣r♠♥tt♦♥ st♦♥ s♦ ts s②st♠ t tst sqrs ♥ ♥ ♠t♦

rt♦♥ t♦t ①tr♥ ♠srs

♦♥t ♥② ①tr♥ ♠sr♠♥t ♥ rt t r♦♦t t t♣r♦♣r♦♣t s♥s♦rs ♦♥②

rt♦♥ s②st♠ t♦ s♦ s st ♠ ♦ t ♥t♦♥s fk,i(Mk, V )t t t ♦♦♥ t

❼ Mk = [ρi,k]

❼ ξ = [ai, bi,∆li]

❼ V = [Υ, ξ] = [Pk, Rk, ξ]

❲t ts t t ♦♥ JV ♦ q s ♦♠♣♦s ♦ t ♦♥ ♦♥♠ts ♣r♠trs s ♥ rt♦♥ s Jξ ♥ ♦ t ♥rs ♥♠ts♦♥ JΥ

♥ t② ♦ rt♦♥ s t♦ ♠♥t t Υk = [Pk, Rk] rs ❬❪♥ t ♥tt♦♥ t♦r V = [Υk, ξ] ♥ ❬❪ t s ♦♥ ♥rt② t ♥trt ♦rr ♥♠ts ♥ ♦rr t♦ tr♠♥ Υ ♥ trt♦♥ ♦ t♥tt♦♥ ♦rt♠ ❲ ♣r♦♣♦s ♦♠♣t ♥tt♦♥ ♦♦s ♦rΥ t♦tr t ξ s ♦s s t♦ ♦ t ♣r♦♠ ♦ t ♦♥r♥

rtr ♦♠♣r ♠♥t♦♥ ♥ t ♦rr ♥♠ts ♥ t ♦♠♣t ♣♣r♦

♦rt♠ ♦rr ♥♠t

∆F > ǫ ♦[P,R]j+1 ← FK(ξj , [P , R]j , M)

ξj+1 ← IS(ξj , [P , R]j+1, M)♥

♥ ♦rt♠ s ♥ ♥tt♦♥ s♠ t② ♦ ♥♦♥ ♥r st sqr s♠ s♦♥ t s②st♠

♦rr ♥♠ts st♣ ss ♥ ♣♣r♦①♠t♦♥ ♦ t ♣r♠trsξ t♦ ♥ t ♣♦s ♦ t r♦♦t [P,R] ♦r ♦♥rt♦♥s t s♦t♦♥ s t♥♥♦t sts ♥ ♥ r r♦♠ t ①♣t ♣♦st♦♥ st♦♥ ♦♥sq♥t② ♥ trt♦♥ ♠② ♥♦t ♠♣r♦ t rt♦♥ t ♥st tr♦rtst s ♣♥♦♠♥♦♥ s ♦sr ♥ ♦r s s s ② s t ♦♦♥♠t♦ rrr t♦ s ♦♠♣t ♥tt♦♥ ♣r♦ss

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

♦rt♠ ♦♠♣t ♥tt♦♥

∆F > ǫ ♦Vj+1 ← IS(Vj , M)

♥

♦♠♣t ♥tt♦♥ ss ♦♥② ♥ ♥tt♦♥ s♠ t ♣♦s ♦♦r♥ts ♥ t ♦♠tr ♣r♠trs r ♦♠♣t t t s♠ t♠ ♥tr♦r ♥rt♥t② ♦♥ t ♣♦s tr♠♥t♦♥ ♦♥r♥ s ♠♥t

♥ t ①♣r♠♥tt♦♥ st♦♥ s♦ ts s②st♠ t t stsqrs ♥ χ2 ♠t♦

①♣r♠♥ts

sr♣t♦♥ ♦ t ♣r♦t♦t②♣

♣r♦t♦t②♣ s♦♥ ♥ s t ② ♥ ♦♦rt♦♥ tt ♥ s s s s♠ s ♣r♦t♦t②♣ ♦ t tr r ♣r r♦♦t t♦ t ♥ ♦♦ ♣r♦t r♠♦r

r Prt ♦ t ♣r♦t♦t②♣

① s r♦♥r r♥ r♦♦t t s ♦♥ r♦♥♥s r tt t♦ t t ♦r♥rs ♦ s♣ ♣t♦r♠ ♦♦t ♥t♠trs ② ♠♥s ♦ s♣r ♦♥ts ♥s r ① ② ♦♥ ♦r ♣♦sts ♣ t♦ tr ♠trs rr♥ t t ♦r ♦r♥rs ♦ tr ②♦r ♠trs rt♥

s

s r ♦ ♥③ st t s♥str♥ ① strtr ♥ ♠tr♦ ♠♠

s♥ ♦ t tt♦♥ s②st♠ s s tt t ♣t ♥t ♦ t str♦ t r♥t ♣②s s ♦♥st♥t r♦♠ t r♠ t♦ t ②t r♠

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

s ♥ t♦ t ♠♦t♦r stt♦r t ♣rs♠t ♦♥t ♥r r♥ ♥ r♦s♥ sr♥t s②st♠ tr♥sts t r♠ tt ② t ②s♥s ♣ t t s♠ ♣ ♦♥ t r♠ ②t s stt ♥ rt♦♥ ♦t s ♠♥♠③ ② ♣r♦r♥t♥ t ②t ①s t♦r t ♥tr ♦ t♦rs♣ tt♦rs r rt rs r♦♠ t ♦t♦♥

tt♦rs ♥ ♦♣t ♥r♠♥t ♥♦rs t ♥s ♦ ♠sr t t♥s♦♥ ♥ t s ♦♥ ♣② ♦♥ r♦♦t ①s s q♣♣ t♦r s♥s♦rs r♦♠ ♥s sr♠♥t

♦♥tr♦ s②st♠ s s ♦♥ ①P rt r♦♠ t♦rs ♥ ♦rr t♦♠♥♠③ t r♥ t rs r ♥tr③ ♥ ♣ ♥r t ♠♦t♦rs♦r s♥s♦rs ♥ ♥♦rs ② r ♦♥♥t t♦ t ♦♥tr♦ s②st♠ ② ♠♥s♦ rt♠ s

♦ ♥ t♦♦s

♥ t ♦♦♥ ♣rr♣ sr t ♠♦ ♥ t t♦♦s ♠♣♠♥t♦r t ♥tt♦♥ ♦ ① ♣r♦t♦t②♣ ♥♠t ♣r♠trs

♦

❲ ♦♦s t ss ♦♠tr ♠♦ ♦ q t♦t t stt qr♠rt♦♥s♣ ♦ q ♥ t ♥ ♥t t stt② ♥ ♠ss ♦ t ss ss♠♣t♦♥ s ♥ r ② ♦♠♣t♥ t ♣♦ss strt♥ ♦♦♥ ♠♠tr ♥ ② r♥ ♠♦ ♦♠♣rs♦♥ ❬ ❪

♦♦s

❲ s t♦♦♦① s ♦♥ ❯ ♥t rr② ♦♥t♥s ♥t♦♥s ♦r r♦♦t ♠♦♥ r♥t ♠t♦s ♦r ♣r♠trs ♥tt♦♥ χ2 ♥ ♥♦♥ ♥r s♦r s ♦♥ ♥rrqrt

♦r t ♦trs tr♥ ♥ ♠♣♦rt♥t st♣ sr ♥ ❬❪ s tr s ♦♥ s ss♥ ♦r t♦ ③r♦ ♥r ♣r♠ts t♦♠♣tt♦♥ ♦ ♦♦ t ♦r ♠sr

♥ t♦♥ ♥t♦♥ ♦♥ t rt♦♥ ♦ t ♣r♠tr ♦srt② s ♦♥ ♦♠♣♦st♦♥ ♦ t ♥tt♦♥ ♠tr① s ♥ s

sr♠♥t

♠sr♠♥ts r ♠ ② ♠♥s ♦ sr trr s②st♠ ♥ ♣♦rt ♠sr♥ r♠

qst♦♥ ♣r♦ss

qst♦♥ ♦ t ♠sr♠♥ts r ♠ ♥ r♥t st♣srst ♦♥ t st♠t♦♥ ♦ ♦♠tr ♣r♠trs ❲ ♠sr t

②t ♣♦st♦♥s ♦♥ t r♠ t t sr trr ♥ t tt♠♥t ♣♦♥ts♦♥ t ♠♦ ♣t♦r♠ t ♣♦rt ♠sr♥ r♠

♣♦s ♠sr♠♥t st♣ ♦ t♥ strt ❲ ♣ t ♠♦ ♣t♦r♠♥ r♥t ♣♦ss ♥ t♦♦ ♠srs ♦ tr t②♣s

❼ Pr♦♣r♦♣t s♥s♦rs s ♥ts

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

❼ r♦♦t ♦r s♥s♦rs s t t♥s♦♥ ♥ t s ♦♥t sts t ♥ ts ♣♣r

❼ P♦st♦♥s ♦ tr ♣♦♥ts ♠sr t t sr s t ♣♦st♦♥ ♥♦r♥tt♦♥ ♦ t ♠♦ ♣t♦r♠

♦♠♠♥t t♦♦ t ♠srs ♦♥ t ♦rrs ♦ t ♦rs♣ ❬❪ tt ♣r♦t♦t②♣ t♦ ♥♦♥♠tr ♦♥tr♦ ♥t ♣r♦ r ♦rs♣♦♠♣r t ai ♥ t t ♣t♦r♠ ♦ ♠♦ ♦t ♠ r♦♥ t♥tr ♥ r♦tt rs ♥ tr rt♦♥s

♦ r ♦♥ t♦ s tt ts ♦ ♥ ♣r♦♠ ♦r t ♥tt♦♥ ♦ ♣r♠trs

t②

♠sr s s ts ♣rs♦♥ s ♥♦♥ ♥ ts ♥ ♠♥ st♠tt ①♣t rr♦r ♦r s rr♦r s s ♥ t rt♦♥st♣ ♥ ♥ t ♦♠♣tt♦♥ ♦ t ts ♦ t ♥tt♦♥ ♣r♦ss ♥♦r s ♦♥sr ♠① t♥ t ♣rs♦♥ ♦ ♠sr♠♥t ♥ t ② t qst♦♥ s ♦♥♠①♠③ ♦♥ ♣r♣♦s σposes = 5mmσlength = 5mm σA = 20mm σB = 10mm σ∆l = 100mm

sts

r♦♠ t ♠srs ♦♥ s ♠srs ♦r t ♥tt♦♥ ♥ ♦r t♦♥ ♦trs r ♠♥t

♥ ♦r rt♦♥ st② ♥ t t ♦ ♦ ♣t♥ t ♦♥ ♥t r♦♦t ♠ ♥ ♠♣♦rt♥t ♦r ♥st♠♥t t♦ ♦t♥ st♠t ♥♠t ♣r♠trs ② sr ♥ ♠sr♠♥ts ♥ ♦rr t♦ t r♦st♥ss♦ t ♦rt♠s s

rst s ♥ tr♠s ♦ ♦♥r♥ ♥t ♥ ♥ rr♦rs ♦♥ ♥ts ♥ ♠♣r♦♠♥t ♦♥ t t♦♥ ♠srs

Pr♠tr ♥tt②

♥t② ♥t t [ai, bi,∆li]i=1..8 t ♦♥ str♦♥ ♣♥♥t♥ t ai ♥ t bi s ♠ r♦♠ t s♠ r♦tt♦♥s ♦ ② t♣r♦t♦t②♣ ♣r♠trs ai ♥ bi r ♥ ② t rt♦♥ Rbi − ai ♥ t♥tt♦♥ qt♦♥s ts ♥♦t ♥ ♠♣♦rt♥t ♣r♦♠ ♥ t ♣r♠trs♦ t ♣t♦r♠ r ♥♦♥ s② t♦ ♠sr ♥ ♦♥t ♥ ♥ ∆li ♥ai ♥s t ♥ ♦♥rt♦♥ rstrt♥ ♦ ♠♦t♦rs t ♥ t♣rtr s ♦ srt♦♥ ts ♥ssr② t♦ ♦♦s t rr♥ r♠ ♦t r♦♦t s ♦♦s ❬❪ a1x = a1y = a1z = a2y = a2z = a3z = 0

rt♦♥ t ①tr♥ ♠srs

rsts ♦ t rt♦♥ q rs rr♦r ♦♥ t ♥ts r♦t ♥ t rst ♣rt ♦ t

tr trt♦♥s r ♦rrt ♠♥♠♠ t ♠♠ rr♦r t ♠♠♦r ❲ ♥ ♠♠ ♦r ♥ tr ♦t trt♦♥s t s♦r st♦♣s tt ①♣t ♣rs♦♥ ∆F < 10−8 ♦♥r♥ r♦♥ s ♥♦tt ♥t ♣r♠trs r ♥♦t ♥ssr②

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

❲ ♥tt♦♥ ♦♥rt♦♥s

♥t rr♦r ♠♥st ♠♥st♠♠ ♠♠ ♠♠ ♠♠

♥ rr♦r ♠♥st ♠♥st♠♠ ♠♠ ♠♠ ♠♠

❱t♦♥ ♦♥rt♦♥srr♦r ♦♥ rt♦♥ qt♦♥

♥t ♥ ♥t ♥

♠♠ ♠♠ ♠♠ ♠♠rr♦r ♦♥ ♣♦st♦♥♥

♥t ♥ ♥t ♥

rr♦r ♦♥ ♦r♥t♥

♥t ♥ ♥t ♥

sts ♦r t rt♦♥

rt♦♥ t♦t ①tr♥ ♠srs

rsts ♦ t rt♦♥ q rs rr♦r ♦♥ t ♥ts r♦t ♥ t rst ♣rt ♦ t

♥♥♦♥ ♣r♠trs Υk = [P,R]k r ♥t③ t FKLS ♣r♦ss ♦♥r♥ s ♦r t ♠sr♠♥t ♦♥rt♦♥s ♦t tt t ♥♠t ♣r♠tr st♠t♦♥ s t♦♦ r♦ FKLS ♦s♥t ♥ ♣ts♦t♦♥ ♥ t srt♦♥ ♥♥♦t strt

s ♦r t rt♦♥ tr trt♦♥s r ♦rrt ♠♥♠♠ t ♠♠ rr♦r t ♠♠ ♦r ❲ ♥ ♠♠ ♦r χ2 ♥ tr ♦t trt♦♥st s♦r st♦♣s t t ①♣t ♣rs♦♥ ∆F < 10−8

❱t♦♥

rst s♠♣ t♦♥ s ♦♥ ② t ♥ ♦ ♠♦ ♠♣r♦♠♥t ♦♥t t♦♥ ♠srs rsts rs rr♦r ♦♥ t ♥tsr t♥ ♣rs♥t ♥ t s♦♥ ♣rt ♦ t ♦r rt♦♥ ♥ t ♦rsrt♦♥

♦♥ t t ♠♥♣t♦r r② ② ♦♠♣t♥ t ♠♣r♦♠♥t ♦ ♥r ♣♦st♦♥♥ ❲ ♠sr ♠♦♥ r♥ t♥ t♦t♦♥ ♦♥rt♦♥s ♦♥ ♣♦st♦♥ ∆Pmeas ♥ ♦♥ ♦r♥tt♦♥ ∆Rmeas ♥r ♥s ❲ ♦♠♣t t t♦rt ♠♦♥ ∆Pλ ♥ t r♦tt♦♥ ∆Rλ

t ♣r♦ss r λ ♠♥s t ♥ ♦ ♥♠t ♣r♠trs s ♥t ♥t ♣♦st ❲ rt♦♥ ♣♦st ♣♦st ❲ srt♦♥♦r ♣♦st χ2 ♥tt♦♥ rsts r ♥ s rt rr♦r ♥ ♣r♥ts♥ ♥ ② 100 ∗ (∆Pλ−∆Pmeas

∆Pmeas)

r♣ rst ♦r ♦♥ s♣♠♥t s s♦♥ ♥ t ♥ ♦r t♣♦st♦♥♥ ♥ ♦r t ♦r♥t♥

♦♠♣t rsts r ①♣rss t ♠①♠ rr♦r ♦♥ P ♥ R ♥♥ ♣r♥ts ② r s♦♥ ♥ t ♦r ♣rt ♦ t ♦r rt♦♥ ♥t ♦r srt♦♥

❲ s tt t ♦r♥t♥ s ♥♦t ♣rt② ♦rrt ①♣t ♦r t χ2

♠t♦ ♣r♦② s ♦ ♥♦t ♦t♥ ♠sr♠♥ts ♥ ♦rs♣

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rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

❲ χ2

♥tt♦♥ ♦♥rt♦♥s♥t rr♦r ♠♥st ♠♥st

♠♠ ♠♠ ♠♠ ♠♠♥ rr♦r ♠♥st ♠♥st

♠♠ ♠♠ ♠♠ ♠♠❱t♦♥ ♦♥rt♦♥s

rr♦r ♦♥ rt♦♥ qt♦♥

♥t ♥ ♥t ♥

♠♠ ♠♠ ♠♠ ♠♠rr♦r ♦♥ ♣♦st♦♥♥

♥t ♥ ♥t ♥

rr♦r ♦♥ ♦r♥t♥

♥t ♥ ♥t ♥

sts ♦r t srt♦♥

r rr♦r ♦♥ ♣♦st♦♥♥ ♦r ♥t ♣r♠trs ♥ tr r♥t♥tt♦♥ ♠t♦s

t r♥t ♦r♥tt♦♥s rr t♥ rs

sss♦♥

rstrt ♦rs♣ t s♠ ♣♦ss ♠♦s ♥ rt♦♥ t t r♠♠♥s♦♥ ♥ s♠ r♦tt♦♥ ♠♣ts ss t♥ rs s♠ t♦ ♥t ♦♠tr② s ♦r t ♣r♦t♦t②♣ tr ♣r r♦♦t rttr r♥t ♥ ♦r♥ t ts r♠r

s ts s t♦ rt ♦♥② t ♣♦st♦♥ ♦ t ai ♥ t ∆lit② t ♦♠tr② ♦ t ♣t♦r♠ ♥ ♥♦♥ ♥ s② t♦ t ts♣r♦♠ s ♥tr② ②♣ss

♦rt♠s s ♦r t ♥tt♦♥ r ♣♦r ♥ tr♠ ♦ ♦♥r♥ ♥ rs rr♦r sss♦♥s ♦s ♦♥ t rsts ♦t ♥♣♥♥t♠srs ♦ ♠t♦ ♥ ♠♥② ♦♥ t ♦♠♣rs♦♥ t♥ rt♦♥♥ srt♦♥

♦♠♣rs♦♥ t♥ rt♦♥ t st sqrs ♥ s ♥♦ts♠♣ s t ♣r♠tr ♥tt♦♥ rsts r ♦s t ♠♥s tt t♠srs t sr trr ♦ rr♦rs

♥ t s ♦ srt♦♥ ♥t ♣r♠tr st♠t♦♥ s ss♥t s t st② rs ♦♥ t ♥ s t♦ ♦♦s s♣ ♣r♦ss ♥ ♦rs ♦♦ rst st♠t♦♥ ♥ tr♦r ♥ t♦♥ t♦ ♦♦ ♦♥r

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rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

r rr♦r ♦♥ ♦r♥t♥ ♦r ♥t ♣r♠trs ♥ tr r♥t♥tt♦♥ ♠t♦s

♥ t χ2 ♣r♦s t♦♥ ♦♥tr♦ s♦♠ ♣r♠trs r ♥♦t ♣rt②♥t t♦ t r♦♦t ♦♠tr② ♦r ♠sr♠♥t s ♦sr ♥ t♦r♥tt♦♥ r② st sqrs s♦ ♦♦ rsts

r ♦♠♣rs♦♥ s ts t♥ rt♦♥ ♥ srt♦♥ ♥ ts①♣r♠♥ts ♥ ♦♥ tt srt♦♥ s ♦♦ ♥♦ rsts

♦r♦r t srt♦♥ s qt ♣ t ♥ts r ♥ ②t ♦♥tr♦ t s♥t r②

♥ t ♦tr ♥ t st♠t♦♥ ♦ ♣r♠trs t rt♦♥♣r♦ss s ♠♦r r♦st t♥ srt♦♥ ♥ s② r rst♥♥ ♥tt♦♥ ♦ t r♦♦t ♣r♠trs

♦♥s♦♥s ♥ tr ♦rs

♦♥s♦♥s

♥ ts ♣♣r r ①♣r♠♥t② t ②♣♦tss ♦ srt♦♥♣t② ♦r ♣rtr ♣r r♥ r♦♦t ♦ t♦s ♦ s t♦ tst s♠♣ ♥ ♣♣r♦ ♦r t ♠♥t♦♥ ♦ ♣♦s rs ♥ t srt♦♥♣r♦ss ❲ tr② tr r♥t ♠t♦s r r♦♠ t st sqrs ♣♣r♦♦r t ♣r♠tr ♥tt♦♥ ♥ ♠ s♦♠ ♣r♦♣♦ss ♦♥ tr s ♦♦♥ ♦r r♦♦t ♥ tr rt ♦♥t rt ♥♠ts♣r♠tr st♠t♦♥ ♦r srt ♦t t r♦st ♦rt♠s

tr ♦rs

r♦♦t ♥r ♦♥strt♦♥ ♦r t ♦♦ ♣r♦t r♥t ♦♠tr② ♥ r♥t s t♦ ♥ ② ♦s ♦r tt ♥ ♠♦ s♥ ♣r♦rss t ♠ss ♥ stt② ♦♥srt♦♥ tr ♦rs ♥rtt♦♥ ♦ t ♥tt♦♥ rsts ♥ t ♣r♦t ♣♥ t♦ s ♠r♦r ♣♦s s♥s♥ ♥ r ♦♣♥ rt♦♥ ♠t♦ s ♦♥ ts♦♥ rst

♥♦♠♥ts

s ♦r s s♣♣♦rt ♦r ♦♥ ♣rt ② t r♥ t♦♥ sr ♥② ♥r r♥t ♦♦ ♣r♦t ♥ ♦r t ♦tr♣rt ② t é♦♥ ♥♦♦ss♦♥ ♥r r♥t ♥s t♦ s r♦♠tt♦♥ ♦r rt sss♦♥s

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

r♥s

❬❪ ②♠

❬❪ st♥ s♥r ♥ ❲s♠ ♥t ♣r♠trs ♦r ♣rr♦♦ts ♥♠t rt♦♥ ♥ ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♦♦ts ♥t♦♠t♦♥ ♦♠ ♣s ♦ ♦r ♠②

❬❪ st♥ s♥r ♥ ❲s♠ rt♦♥ ♦ ♣r r♦♦tst t♦ ♥♥♦♠trs ♥ ♦♦ts ♥ t♦♠t♦♥ ♦♠ ♣s

❬❪ P ♦s r ②r ♥ ♦rt ♥ st ♥ ♥t ♦rt♠ ♦r ♥♦♥♥r ♦rt♦♦♥ st♥ rrss♦♥ ♦r♥♦♥ ♥t ♥ ttst ♦♠♣t♥

❬❪ P ♦rstr♦♠ ♦r♥ ♦rstr♦♠ t② t♠ t♥ ♥ ❲ sr ♠s♣ r♥r♦♦t t srt♦♥ ♣ts ♦♦ts r♥st♦♥s ♦♥ ♦t

❬❪ P ♦ssr ♥ ss♣♥ r♦♦t ♦♥t♦r rt♥ s②st♠t♦♠t♦♥ ♥ ♦♥strt♦♥

❬❪ ♠ ♦r ♦♠tr s r♦♦ts ♣rs ♥tr♥s ♣r s sP tss ❯♥rst

❬❪ ♥② rt♦♥ ♦ ♦ ♣t♦r♠ s♥ ♠♦t② ♦♥str♥ts♥ ♥ Pr♦ t ❲♦r ♦♥rss ♦r② ♦ ♥ ♥ ♥s♠s ♥♥ ♣s

❬❪ ♥② t♦♥♥ ♦♠trq s r♦♦ts ♣rs P tss ♦♣ ♥t♣♦s

❬❪ ♥② ♦s ♥r s rt ♥ ❨s P♣② ♥tr ♠t♦ ♦r rt♦♥ ♦ ♣r r♦♦ts ❱s♦♥ s ①♣r♠♥ts♥s♠ ♥ ♥ ♦r②

❬❪ Prr♦t r♥♦s Pr ♠♥s♠s ♥ r♥♥② ♥ st ♥t♦♦q♠ ♦♦rt sr ♥tr ♣s

❬❪ ♦ttr P rt ♥ ♥② tr♠♥t♦♥ ♦ t r♥♦sr ♦rs♣ ♦ ♦ ♣r r♥ ♠♥s♠s ♥ r♥♥♥r ♥ ♦t t♦rs ♥s ♥ ♦♦t ♥♠ts ♣s ♣r♥r tr♥s

❬❪ r♥ ♥ ② ♥r t♦r② ♦ r rt♦♥s ♦ ss♣♥ Pr♦♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥ rs t♠t ♥ P②s ♥s ♣♣

❬❪ ♥Prr rt Pr ♦♦ts ♥ t♦♥ ♦♠ ♣r♥r

❬❪ ♥ ♥ t② ♥ t♣ rr♦♠ P♦st♦♥♥ ♥s♠ ❯s♥ ❲rs ♦♣♠♥t P♥r ♦♠♣t② str♥ P♦st♦♥♥ ♥s♠♥tr♥t♦♥ ♦r♥ ♣♥ ♦t② ♦r Prs♦♥ ♥♥r♥ ♣

♥

rt♦♥ ♦ ②♦♥str♥ ♣r r♥ r♦♦t

❬❪ ♠t Pt ♥ ♦r♥ ♠♥♥ rt♦♥ ♦ ①♣♦ ♠♥t♦♦ s♥ r♥♥t ♥tr♥t♦♥ ♦r♥ ♦ ♥ ♦♦s ♥♥tr ♠r

❬❪ P ♥ ♥r P rt♥t ♥ ♦ ♥♠t rt♦♥♦ ♣r ♠♥s♠s ♥♦ ♣♣r♦ s♥ s ♦srt♦♥ ♦♦ts r♥st♦♥s ♦♥

❬❪ ♦♦r♦ ❱r♦♥ r ♥ ♠♦r ♣♦rt ♣r ♠♥♣t♦r ♦r sr ♥ rs t rs r♥ rtqs ♥♥ ♥tt♦♥ ♦rt♠ ♦r t ♥stt♦♥ ♥ ♥strtr ♥r♦♥♠♥ts ♥ ♥t♥t ♦♦ts ♥ ②st♠s Pr♦♥s ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♦♠ ♣s ♦

❬❪ ❨♦ ♥ ♥ ♥ r♦ ♥s s ♥♠trt♦♥ ♠t♦ ♦r ♥Pr tt ♠♥s♠s s♥ ♦rr srs ♦r♥ ♦ ♥ s♥

❬❪ ❱r③r ♥ ♦ts ♥♠t rt♦♥ ♦ rtt♣r r♦♦t ♥s♠ ♥ ♥ ♦r②

❬❪ ❲♠♣r ♥ r rt♦♥ ♦ r♦♦ts ♥ ♥♠t ♦s♦♦♣s s♥ ♥♦♥♥r stsqrs st♠t♦♥ Pr♦ ♦ ♥t②♠♣ ♦r② ♣s

❬❪ rs ❲ ❲♠♣r ♦♥ ♦r ♥ ts♦ r ♥ ♠♣t♦♦♣ ♠t♦ ♦r ♥♠t rt♦♥ ♥ ts ♣♣t♦♥ t♦ ♦s♥♠♥s♠s r♥st♦♥s ♦♥ ♦♦ts ♥ t♦♠t♦♥ ♦t

❬❪ ♥q ❩♥ s♦r② ♥ ❨♥ ♥♠t rt♦♥ ♦ strt ♣t♦r♠ s♥ ♣♦s ♠sr♠♥ts ♦t♥ ② s♥ t♦♦t ♥ ♥t♥t ♦♦ts ♥ ②st♠s ♠♥ ♦♦t ♥trt♦♥♥ ♦♦♣rt ♦♦ts Pr♦♥s ♥tr♥t♦♥♦♥r♥ ♦♥ ♦♠ ♣s ♦

❬❪ ♥q ❩♥ ❨♥ ♥ r♥ s♦r② rt♦♥ ♦ strt ♣t♦r♠s ♥ ♦tr ♣r ♠♥♣t♦rs ② ♠♥♠③♥ ♥rs ♥♠trss ♦r♥ ♦ ♦♦t ②st♠s

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RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93

06902 Sophia Antipolis Cedex

PublisherInriaDomaine de Voluceau - RocquencourtBP 105 - 78153 Le Chesnay Cedexinria.fr

ISSN 0249-6399