Simulation dynamique de perte d'équilibre : Application aux passagers debout de transport en commun

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Simulation dynamique de perte d’équilibre : Application

aux passagers debout de transport en commun

Zohaib Aftab

To cite this version:

Zohaib Aftab. Simulation dynamique de perte d’équilibre : Application aux passagers debout de transport en commun. Médecine humaine et pathologie. Université Claude Bernard - Lyon I, 2012. Français. �NNT : 2012LYO10243�. �tel-00982026�

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Thèse

Dynamic Simulation of Balance

Recovery: Application to the

standing passengers of

public transport

Présentée devant

L’UNIVERSITE CLAUDE BERNARD LYON I

Pour l'obtention

du DIPLÔME DE DOCTORAT

Formation doctorale : Biomécanique École doctorale : École doctorale MEGA

Par

M. Zohaib AFTAB

Soutenue le 21 Novembre 2012 devant la Commission d’examen

Jury

Rapporteur M. Patrick LACOUTURE Professeur (Université de Poitiers) Rapporteur M. Philippe FRAISSE Professeur (Université Montpellier 2) Examinateur Mme. Laurence CHÈZE Professeur (Université Lyon 1)

Directeur de thèse M. Bernard BROGLIATO Directeur de recherche (INRIA Grenoble) Co-directeur M. Thomas ROBERT Chargé de recherche (IFSTTAR, Bron) Co-directeur M. Pierre-Brice WIEBER Chargé de recherche (INRIA Grenoble)

Laboratoire de Biomécanique et Mécanique des Chocs,

Ifsttar, 25 av. Francois Mitterrand, case 24, 69 675 BRON Cedex

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ʹǤͳ š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”› X X X X X X X X X X X R9 ʹǤͳǤͳ š’‡”‹‡–ƒŽ ‡…Šƒ‹•• X X X X X X X X X X X X X X X X X X R9 ʹǤͳǤʹ ›–Š‡•‹• ‘ˆ š’‡”‹‡–ƒŽ ‹†‹‰• X X X X X X X X X X X X X R8 ʹǤͳǤ͵ …‘•‹•–‡…› ‹ š’‡”‹‡–ƒŽ ‡•—Ž–• X X X X X X X X X X X RN ʹǤͳǤͶ ‡ƒ”• X X X X X X X X X X X X X X X X X X X X X X X X X X X ky ʹǤʹ ‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡ X X X X X X X X X X X X X X X X X X X X ky ʹǤʹǤͳ ‹’Ž‹ϐ‹‡† ”‡’”‡•‡–ƒ–‹‘• ‘ˆ Š—ƒ „‘†› X X X X X X X X X kR ʹǤʹǤʹ ƒŽŽ ’”‡†‹…–‘”• X X X X X X X X X X X X X X X X X X X X X X X X kk ʹǤʹǤ͵ ‘–”‘Ž •…Š‡‡• X X X X X X X X X X X X X X X X X X X X X X X k9 ʹǤʹǤͶ ‡ƒ”• X X X X X X X X X X X X X X X X X X X X X X X X X X X k8 ʹǤ͵ ›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–› X X X X X X X X X X X X X X X X ke ʹǤ͵Ǥͳ “—‹Ž‹„”‹— ‘ˆ ‹’‡† ›•–‡• X X X X X X X X X X X X X X X X kd ʹǤ͵Ǥʹ ‡ϐ‹‹‰ –ƒ„‹Ž‹–› ˆ‘” –Š‡ „‹’‡† •›•–‡• X X X X X X X X X X X k3 ʹǤ͵Ǥ͵ Š‡ ‘ ’‘‹–ǡ ƒ •–‡’ ˆ‘”™ƒ”† –‘™ƒ”†• ˆƒŽŽ ‡•–‹ƒ–‹‘ X X jy ʹǤ͵ǤͶ ‹ƒ„‹Ž‹–› ‡”‡Ž X X X X X X X X X X X X X X X X X X X X X X X X jR ʹǤ͵Ǥͷ ‘…Ž—†‹‰ ”‡ƒ”• X X X X X X X X X X X X X X X X X X X X X jj ʹǤͶ ‹’‡† …‘–”‘Ž •…Š‡‡• X X X X X X X X X X X X X X X X X X X X X X X jj ʹǤͶǤͳ  ƒ• ƒ ‘–‹‘ †”‹˜‡” ƒ† ƒ†ƒ’–‹˜‡ •–‡’’‹‰ X X X X X X X j9 ʹǤͷ ‘…Ž—•‹‘X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X j9

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

ʹǤͳ š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

h#H2 kXR,  b2H2+iBQM Q7 BKTQ`iMi THi7Q`K /Bbim`#M+2 bim/B2b M/ i?2B` +?`+i2`BbiB+bX

hQ #2 +Q?2`2Mi #2ir22M bim/B2b- i?2 /Bbim`#M+2 Bb `2TQ`i2/ BM i2`Kb Q7 T2F THi7Q`K ++2H2`iBQM BM HH +b2b

aim/v .Bbim`#M+2 .B@

`2+iBQM .Bbim`#M+2 S`Q}H2 _M;2 Q7 T2F THi@7Q`K ++2H2`iBQM ai2TTBM;HHQr2/ \ P#b2`p2/LmK#2` Q7 ai2Tb

6Q+mb Q7 aim/v /2 :`7 M/ pM

q2T2`2M URNNdV

6Q`r`/- "+F@

r`/- aB/2rvb ZmbB@a[m`2 ++2H2`@iBQM T`Q}H2 yXj@RXe KXb

−27Q` k b LQ y GBKBi Q7 #HM+2 rBi?@

Qmi bi2TTBM; _mM;2 2i HX URNNNV "+Fr`/ h`T2xQB/H p2HQ+Biv<sub>T`Q}H2</sub> eX8 @ jk KXb<sub>yXje @ 9 b</sub> −2 7Q` LQ y MFH2 M/ >BT bi`i2@<sub>;B2b</sub> axim`K M/ 6HHM;

URNN3V

6Q`r`/- "+F@

r`/- aB/2rvb ZmbB@b2KBb[m`2 p2@HQ+Biv T`Q}H2 k @ 9 KXb

−27Q` yX9 @

yX3 b LQ y JQp2K2Mi Sii2`Mb-JQiQ` +QMi`QH J+AH`Qv M/ JFB

URNNeV

6Q`r`/- "+F@

r`/ a[m`2 rp2 ++2H2`@iBQM T`Q}H2 RX8@k KXb

−27Q` yXe b u2b >R ;2@`2Hi2/ bi2TTBM; +?`+i2`BbiB+b o2``B2bi 2i HX UkyRyV 6Q`r`/- "+F@_{r`/- aB/2rvb</sub> ZmbB@i`T2xQB/H +@_{+2H2`iBQM T`Q}H2} 9 KXb−27Q` 8 b u2b >k EBM2KiB+b Q7 #H@<sub>M+2 `2+Qp2`v BM}

2K2`;2M+v #`FBM; +QM/BiBQMb _Q#2`i UkyyeV "+Fr`/ a2KB@bBMQbQB/H_{++2H2`iBQM T`Q}H2} k@Ry KXb−27Q` yX9 b u2b R@9 EBM2KiB+b M/ EB@_{M2iB+b Q7 KmHiBTH2 bi2T}

#HM+2 `2+Qp2`v

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

6B;m`2 kXR, .Bbim`#M+2 bim/B2b mM/2` +QMbB/2`iBQM BM i?Bb i?2bBb, h2i?2`@`2H2b2 K2+?@

MBbK UG27iV M/ h`MbHiBM; THi7Q`K T`/B;K U_B;?iV ƒŽƒ…‡ ‡…‘˜‡”› –”ƒ–‡‰‹‡•

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

6B;m`2 kXk, "bB+ KQp2K2Mi bi`i2;B2b 2KTHQv2/ #v ?mKMb iQ `2iBM i?2B` #HM+2, MFH2

UH27iV- >BT UKB//H2V M/ ai2TTBM; U`B;?iV UEMKBv 2i HX- kyRyV

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

6B;m`2 kXj, h?2 i2KTQ`H T?b2b Q7 #HM+2 `2+Qp2`v BMpQHpBM; bi2TTBM;

Š‡ –‡’’‹‰ ƒ”‹ƒ„Ž‡•

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

ʹǤͳǤ͵ …‘•‹•–‡…› ‹ š’‡”‹‡–ƒŽ ‡•—Ž–•

‹•–—”„ƒ…‡ …Šƒ”ƒ…–‡”‹•–‹…• ƒ”‡ ‘– ™‡ŽŽǦ†‡ϔ‹‡†

š’‡”‹‡–ƒŽ –—†‹‡• ‘ˆ ƒŽƒ…‡ ‡…‘˜‡”›

‹ˆˆ‡”‡– ’”‹‘” ‹•–”—…–‹‘• ’”‘†—…‡ †‹ˆˆ‡”‡– ”‡•—Ž–•

Š‡ ”‡’‘”–‡† ”‡•—Ž–• ƒ› ‘– …ƒ””› ƒŽŽ –Š‡ ‹ˆ‘”ƒ–‹‘

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

ʹǤʹ ‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

6B;m`2 kX9, aQK2 +QKKQM bBKTHB}2/ KQ/2Hb mb2/ iQ TT`QtBKi2 KmHiB@`iB+mHi2/ ?mKM

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

6B;m`2 kX8, h?2 bi#BHBiv Q7 MFH2 bi`i2;v +H+mHi2/ BM SB M/ SiiQM URNNdV Ub?/2/

`2;BQMV M/ TT`QtBKi2/ #v >Q7 2i HX Ukyy8V U/Qii2/ HBM2V mbBM;  bBKTH2 BMp2`i2/ T2M/mHmK KQ/2HX U6B;m`2 iF2M 7`QK >Q7 2i HX Ukyy8VV

ʹǤʹǤ͵ ‘–”‘Ž •…Š‡‡•

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

+iBp2

6B;m`2 kXe,  HBM2` 722/#+F +QMi`QH KQ/2H- r?2`2 K Bb  Ki`Bt Q7 722/#+F ;BMb T`Q/m+@

BM; DQBMi iQ`[m2 +QKKM/b u b  7mM+iBQM Q7 #Q/v KQp2K2Mi x US`F 2i HX-kyy9VX

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡

6B;m`2 kXd, 6HQr /B;`K Q7 i?2 ?mKM biM+2 +QMi`QH KQ/2H T`QTQb2/ #v pM /2` EQQBD

2i HX URNNNVX h?2 mMFMQrM /Bbim`#M+2b wst `2 bm++2bb7mHHv Qzb2i #v i?2 Kmb+H2 +iBQMb ustX AM //BiBQM iQ b2MbQ`v BMTmi- i?2 b2MbQ`v +QMi`QH +2Mi2` HbQ BMTmib Kmb+H2 +iBQMb 7Q` #2bi 2biBKi2 Q7 i?2 #Q/v brv

BMi2`KBii2Mi bHB/BM;@KQ/2

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–›

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–› biiB+ 2[mBHB#`BmK /vMKB+ 2[mBHB#`BmK bi#BHBiv

ʹǤ͵Ǥͳ “—‹Ž‹„”‹— ‘ˆ ‹’‡† ›•–‡•

Wi = Wg+ Wc Wi _Wg _Wc :Bp2M i?i i?2 ;`p@ BiiBQMH 7Q`+2 Uq;_{V Bb +QMbiMi M/ +QMi+i 7Q`+2b Uq}+_{V #2ir22M 722i M/}

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–›

‡”‘ ‘‡– ‘‹–

ʹǤ͵Ǥʹ ‡ϐ‹‹‰ –ƒ„‹Ž‹–› ˆ‘” –Š‡ „‹’‡† •›•–‡•

≃ /vMKB+ 2[mBHB#`BmK bi#H2

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–›

:Bp2M i?i i?2 ;`pBiiBQMH 7Q`+2 Bb +QMbiMi- i?2 QMHv rv iQ +?B2p2  /2bB`2/ KQiBQM Bb iQ bQK2?Qr #2 #H2 iQ +?M;2 i?2 +QMi+i 7Q`+2b

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–›

ʹǤ͵Ǥ͵ Š‡ ‘ ’‘‹–ǡ ƒ •–‡’ ˆ‘”™ƒ”† –‘™ƒ”†• ˆƒŽŽ ‡•–‹ƒ–‹‘

m l j α ¨ cx px cx mg(px− cx) = jα ≈ −ml2 ¨ cx l ¨ cx = ω20(cx− px) ω0 =g/l cx(t) = px+ (cx(0) − px)cosh(ω0t) + ˙cx(0) ω0 sinh(ω0t) cx(t) ≤ px cx+ ˙cx ω0 ≤ px cx + _ω˙cx0

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–›

6B;m`2 kX3, G27i, h?2 bBKTH2 AMp2`i2/ S2M/mHmK UASV Y 6QQi KQ/2H bbmK2b TQBMi Kbb

+QM+2Mi`i2/ i *QJX h?2 ;`QmM/ `2+iBQM 7Q`+2 U:_6V +ib i i?2 *QS r?B+? Bb +QMbi`BM2/ iQ `2KBM rBi?BM i?2 7QQiX _B;?i, h?2 HBKBib Q7 i?2 /vMKB+ bi#BHBiv `2;BQM U;`2v `2V 7Q` biM/BM; #HM+2X Mv bii2 i?i HB2b rBi?BM i?Bb `2;BQM rQmH/ bm++2bb7mHHv i2`KBMi2 Bib KQiBQM Qp2` i?2 "Qa U7QQiVX

≃

›ƒ‹… ‡“—‹Ž‹„”‹— ƒ† •–ƒ„‹Ž‹–›

6B;m`2 kXN, h?2 pB#BHBiv F2`M2H ;i?2`b HH i?2 bii2b 7`QK r?B+? Bi Bb TQbbB#H2 iQ pQB/ iQ

7HHX G2pBM; Bi BKK2/Bi2Hv BKTHB2b M mMpQB/#H2 7HH UqB2#2`- kyykV

 ’”ƒ…–‹…ƒŽ •‘Ž—–‹‘ ≃ tk+T tk c (n) x 2dt

‹’‡† …‘–”‘Ž •…Š‡‡•

ʹǤ͵Ǥͷ ‘…Ž—†‹‰ ”‡ƒ”•

‹’‡† …‘–”‘Ž •…Š‡‡•

‘…Ž—•‹‘

*?Ti2` j

*QMi2Mib

͵Ǥͳ ”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–• X X X X X X X X X X X X jN ͵ǤͳǤͳ ƒ’–—”‡ ‘‹– Ž‰‘”‹–Š X X X X X X X X X X X X X X X X X X X jN ͵ǤͳǤʹ –‡’’‹‰ ‘†‡Ž ‘ˆ ‘ˆƒ ȋʹͲͲ͸Ȍ ™‹–Š ’ƒ…– X X X X X X 9R ͵ǤͳǤ͵ ‹‹ƒŽ •–‡’ Ž‡‰–Š ‘†‡Ž „› — ‡– ƒŽǤ ȋʹͲͲ͹Ȍ X X X X X X X 9j ͵ǤͳǤͶ ‘’Ž‹ƒ– Ž‡‰ ‘†‡Ž ‘ˆ •‹ƒ‘ ƒ† ‘„‹‘˜‹–…Š ȋͳͻͻͻȌ X X X 98 ͵Ǥʹ ˜ƒŽ—ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–• X X X X X X X X X X X X X 9d ͵Ǥ͵ ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡…‘˜‡”› X X X X 9N ͵Ǥ͵Ǥͳ ‘’ƒ”‹•‘ ™‹–Š –Š‡ ƒ’–—”‡ ‘‹– X X X X X X X X X X X X X 8y ͵Ǥ͵Ǥʹ ‡•—Ž–• X X X X X X X X X X X X X X X X X X X X X X X X X X X X 8y ͵Ǥ͵Ǥ͵ ‹•…—••‹‘ X X X X X X X X X X X X X X X X X X X X X X X X X X 8R ͵ǤͶ ‘†‡Ž ”‡†‹…–‹˜‡ ‘–”‘Ž X X X X X X X X X X X X X X X X X X X X X 8j ͵ǤͶǤͳ ‹’‡†  •…Š‡‡• X X X X X X X X X X X X X X X X X X X X X 89 ͵ǤͶǤʹ  …‘–”‘ŽŽ‡” ™‹–Š ƒ”‹ƒ„Ž‡ –‡’’‹‰ X X X X X X X X X X X 88 ͵Ǥͷ  ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘• X X X X X X X X X X 8d ͵ǤͷǤͳ ‡…Šƒ‹…ƒŽ ƒ† –‡”ƒŽ ‘†‡Ž• X X X X X X X X X X X X X X X 83 ͵ǤͷǤʹ ‘•– —…–‹‘ X X X X X X X X X X X X X X X X X X X X X X X X X 8N ͵ǤͷǤ͵ –‡’ ‹‹‰• X X X X X X X X X X X X X X X X X X X X X X X X X ey ͵ǤͷǤͶ ‡Ž‡…–‹‘ ‘ˆ –Š‡ ‘†‡Ž ƒ”ƒ‡–‡”•X X X X X X X X X X X X X X ey ͵ǤͷǤͷ ’Ž‡‡–ƒ–‹‘ ‘ˆ –Š‡ ˆ‡‡†„ƒ… Ž‘‘’ X X X X X X X X X X X X ek ͵ǤͷǤ͸ ‡•—Ž–• X X X X X X X X X X X X X X X X X X X X X X X X X X X X ek ͵ǤͷǤ͹ ‹•…—••‹‘ X X X X X X X X X X X X X X X X X X X X X X X X X X ej ͵Ǥ͸ ‘…Ž—•‹‘X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X e8

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

͵Ǥͳ ”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

cx+ ˙cx ω₀ ≤ px px cx + _ω˙cx₀

͵ǤͳǤͳ ƒ’–—”‡ ‘‹– Ž‰‘”‹–Š

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

6B;m`2 jXR, h?2 GBM2` AMp2`i2/ S2M/mHmK UGASV Y 6Hvr?22H KQ/2H rBi? +QMbiMi ?2B;?i

*QJ M/ i2H2b+QTB+ H2;b ¨ cx = ω2(cx− px) ω =g/h ELIP = 1 2˙c 2 x− g 2h(cx− px) 2 px = xcapt = cx+ ˙cx ω

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–• xcapt xcapt xcapt ƒ’–—”‡ ‡‰‹‘ xcapt = cx+ ˙cx ω ± τmax mg [ eωTR2− 2eω(TR2−TR1)+ 1 eωTR2 ] τmax TR1 TR2

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

6B;m`2 jXk, h?2 bi2TTBM; KQ/2H T`QTQb2/ #v >Q7KMM UkyyeV M/ 2ti2M/2/ #v ai2T?2Mb

UkyydVX h?2 bvbi2K #2?p2b HBF2  bBKTH2 BMp2`i2/ T2M/mHmK mMiBH BKT+i rBi? +QMbiMi H2; H2M;i? lX 7i2` BKT+i- i?2 H2; H2M;i?b `2 /Dmbi2/ iQ KQp2 *QJ i  +QMbiMi ?2B;?i 7`QK ;`QmM/X

͵ǤͳǤʹ –‡’’‹‰ ‘†‡Ž ‘ˆ ‘ˆƒ ȋʹͲͲ͸Ȍ ™‹–Š ’ƒ…–

l

"27Q`2 BKT+i

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–• i BKT+i ˙c+ x − ˙c−x ˙c+ z − ˙c−z = tanθ ˙c− x ˙c+x ˙c− z ˙c+z 7i2` BKT+i ˙c+ z = 0 ˙c+x = ˙c−x − ˙c−z tanθ cx+ ˙c+ x ω0 = 0 ˙θ = ωsinθcos1/2θ cos2θ θ 2θ

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

6B;m`2 jXj, h?2 i`D2+iQ`B2b mM/2` BMp2`i2/ T2M/mHmK KQ/2H bbmKTiBQM 7`QK /Bz2`2Mi H2p@

2Hb Q7 BMBiBH /Bbim`#M+2 U#Hm2 +m`p2bV BMi2`b2+i i?2 QTiBKH bi2T +m`p2 U#H+FV i QTiBKH bi2T /BbiM+2 b?QrM BM 6B;m`2 jX9

6B;m`2 jX9, h?2 QTiBKmK bi2T H2M;i?b U#H+FV M/ +Q``2bTQM/BM; +QMi+i iBK2b U#Hm2V 7Q`

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

mMB[m2 bi2T H2M;i?@bi2T /m`iBQM +QK#BMiBQM

͵ǤͳǤ͵ ‹‹ƒŽ •–‡’ Ž‡‰–Š ‘†‡Ž „› — ‡– ƒŽǤ ȋʹͲͲ͹Ȍ

͵ǤͳǤͶ ‘’Ž‹ƒ– Ž‡‰ ‘†‡Ž ‘ˆ •‹ƒ‘ ƒ† ‘„‹‘˜‹–…Š ȋͳͻͻͻȌ

θc θ0 tcont θc = θ0cosh(  3g 2l − 3ka ml2tcont) θc Fmax α = 2θc tcont

”‡•‡–ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

6B;m`2 jX8, JBMBKH bi2T H2M;i? UMQ`KHBx2/ rBi? i?2 H2M;i? Q7 7QQiV M22/2/ 7Q` #HM+2

`2+Qp2`v Q7 p`BQmb BMBiBH /BbTH+2K2Mib UMQ`KHBx2/ #v i?2 H2M;i? Q7 7QQiV M/ p2HQ+BiB2b UMQ`KHBx2/ #v i?2 √gH - r?2`2 > Bb ?2B;?i Q7 #Q/v M/ ; Bb i?2 ++2H2`iBQM /m2 iQ ;`pBivV Q7 *PJ #v Uqm 2i HX- kyydV

6B;m`2 jXe, h?2 bi2TTBM; KQ/2H Q7 >bBQ M/ _Q#BMQpBi+? URNNNV rBi? iQ`bBQMH M/ HBM2`

˜ƒŽ—ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

͵Ǥʹ ˜ƒŽ—ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

–‡’ †‡Žƒ›

bi2T /m`iBQM bvbi2K bii2 i i?2 iBK2 Q7 BKT+i

˜ƒŽ—ƒ–‹‘ ‘ˆ •–‡’’‹‰ ”‡ˆ‡”‡…‡ ’‘‹–•

ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡…‘˜‡”›

—Ž–‹’Ž‡ •–‡’•

‘•‹†‡”ƒ–‹‘ ‘ˆ —Ž–‹’Ž‡ …”‹–‡”‹ƒ

͵Ǥ͵ ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡Ǧ

…‘˜‡”›

ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡…‘˜‡”› st nd st 6B;m`2 jXd, h?2 THi7Q`K /Bbim`#M+2 T`Q}H2

͵Ǥ͵Ǥͳ ‘’ƒ”‹•‘ ™‹–Š –Š‡ ƒ’–—”‡ ‘‹–

τmax = 190N.m ◦

ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡…‘˜‡”›

͵Ǥ͵Ǥʹ ‡•—Ž–•

6B;m`2 jX3, h?2 /BbiM+2 . #2ir22M i?2 bi2TTBM; 7QQi M/ i?2 +Tim`2 TQBMi i BKT+iX

1st b2`B2b bm#D2+ib r?Q bi2TT2/ 7`i?2` 7`QK i?2 +Tim`2 TQBMi Ur?Bi2 #`V iQQF b2p2`H bi2Tb iQ `2+Qp2` #HM+2X

st

ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡…‘˜‡”›

nd _{13±4.6 cm}

͵Ǥ͵Ǥ͵ ‹•…—••‹‘

ƒ’–—”‡ ‘‹– ‡•–‹ƒ–‹‘ ˆ‘” —ƒ „ƒŽƒ…‡ ”‡…‘˜‡”›

6B;m`2 jXN, h?2 2pQHmiBQM Q7 i?2 *QJ U#H+FV M/ *Tim`2 SQBMi U#Hm2V BM iBK2 7Q` QM2

bm#D2+i r?Q bi2TT2/ p2`v +HQb2 iQ i?2 +Tim`2 TQBMi- M/ QM i?2 +Tim`2 `2;BQM-M/ `2+Qp2`2/ #HM+2 BM 2t+iHv QM2 bi2TX h?2 ;`2v `2 `2T`2b2Mib i?2 +Tim`2 `2;BQM +H+mHi2/ mbBM; ~vr?22H `QiiBQM rBi? Mi?`QTQKQ`T?B+ T`K2i2`b BM #Qi? /B`2+iBQMbX

6B;m`2 jXRy, 1tKTH2 Q7  bm#D2+i r?Q /B/ MQi bi2T QM i?2 +Tim`2 TQBMi f `2;BQM BM i?2

‘†‡Ž ”‡†‹…–‹˜‡ ‘–”‘Ž

͵ǤͶ ‘†‡Ž ”‡†‹…–‹˜‡ ‘–”‘Ž

tk L T  tk+T tk L(q(t), ˙q(t), ¨q(t), u(t), λ(t))dt u(t) t_k+1

͵ǤͶǤͳ ‹’‡†  •…Š‡‡•

‘†‡Ž ”‡†‹…–‹˜‡ ‘–”‘Ž

͵ǤͶǤʹ  …‘–”‘ŽŽ‡” ™‹–Š ƒ”‹ƒ„Ž‡ –‡’’‹‰

N T c ˆ ck = ⎡ ⎢ ⎢ ⎣ ck ˙ck ¨ ck ⎤ ⎥ ⎥ ⎦ , tk C_k+1 = ⎡ ⎢ ⎢ ⎣ ck+1 ck+N ⎤ ⎥ ⎥ ⎦ , ˙C_k+1= ⎡ ⎢ ⎢ ⎣ ˙ck+1 ˙ck+N ⎤ ⎥ ⎥ ⎦ , ¨C_k+1 = ⎡ ⎢ ⎢ ⎣ ¨ ck+1 ¨ ck+N ⎤ ⎥ ⎥ ⎦

‘†‡Ž ”‡†‹…–‹˜‡ ‘–”‘Ž Ck+1 = ⎡ ⎢ ⎢ ⎣ c_k+1 c_k+N ⎤ ⎥ ⎥ ⎦ Ck+1 = Spcˆk+ UpCk, ˙ Ck+1 = Svˆck+ UvCk, ¨ C_k+1 = Saˆck+ UaCk. zx = cx− h g c¨x px zx Zk+1 = ⎡ ⎢ ⎢ ⎣ zk+1 zk+N ⎤ ⎥ ⎥ ⎦ Ck Z_k+1 = Szˆck + UzCk,

‘†‡Ž ”‡†‹…–‹˜‡ ‘–”‘Ž Sz = Sp− h gSa, Uz = Up− h gUa. ‘•–”ƒ‹–• ƒ† ‘–”‘ŽŽ‡” ‡•‹‰ f′ i  ˙f′ i ≤ ˙fKt′ . zx D(zi− fi) ≤ b, D b fi ˙ Ck+1 ˙ C_k+1ref Ck Zk+1 Fk+1

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘• 1 c2 1 ˙ Ck+1 − C˙k+1ref2 + 1 c2 2C k2 + 1 c2 3Z k+1 − Fk+12, c1 c2 c3 u =  Ck ¯ Fk+1 Ck ¯ F_k+1

͵Ǥͷ  ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

͵ǤͷǤͳ ‡…Šƒ‹…ƒŽ ƒ† –‡”ƒŽ ‘†‡Ž•

͵ǤͷǤʹ ‘•– —…–‹‘

˙ C_k+1`27 = 0 T<sub>?Q`BxQM</sub>= 1 TbKTHBM;= 25

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

6B;m`2 jXRR, h?2 `2T`2b2MiiBQM Q7 i?2 ?mKM #Q/v mb2/ 7Q` i?2 bQ@+HH2/ K2+?MB+H

KQ/2HX Ai +QMbBbib Q7  bBKTH2 BMp2`i2/ T2M/mHmK Y 7QQi KQ/2H- r?2`2 i?2 *QJ 7QHHQrb  +B`+mH` `+ `QmM/ i?2 MFH2 DQBMiX

m

͵ǤͷǤ͵ –‡’ ‹‹‰•

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘• T_T`2T T_bi2T

͵ǤͷǤͶ ‡Ž‡…–‹‘ ‘ˆ –Š‡ ‘†‡Ž ƒ”ƒ‡–‡”•

T_HM/ = T`2++ TT`2T+ Tbi2T T`2+ T<sub>T`2T</sub> T_bi2T T_HM/ T<sub>`2+</sub>+ TT`2T

h#H2 jXR, Mi?`QTQKQ`T?B+ T`QTQ`iBQMb mb2/ BM i?2 bBKmHiBQM Ub22 HbQ 6B;m`2 jXRRVX

l _×

lf ×

a _{× l}f

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

h#H2 jXk, aBKmHiBQM T`K2i2`b mb2/ b BMTmi 7Q` i?2 }p2 bBKmHi2/ b+2M`BQb, H2M M;H2b

UθV- #Q/v ?2B;?i UHV `2+iBQM iBK2b UTreacV- bi2T T`2T`iBQM iBK2b UTprepV M/ H2; brBM; iBK2b UTstepVX

H θ Treac Tprep Tstep

m.s−1

c1 c3

c₁ = 1 m.s−1

c₂ c₃ 100

h#H2 jXj, h?2 +QMi`QHH2` T`K2i2`b `2Hi2/ iQ +Qbi 7mM+iBQM M/ +QMbi`BMib

× c1 −1 c2 −3 c₂ lf vmax −1

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

͵ǤͷǤͷ ’Ž‡‡–ƒ–‹‘ ‘ˆ –Š‡ ˆ‡‡†„ƒ… Ž‘‘’

T_bKTHBM;

6B;m`2 jXRk, h?2 722/#+F HQQT BKTH2K2Mi2/ iQ bBKmHi2 i?2 i2i?2`@`2H2b2 +QM/BiBQM mbBM;

i?2 JS* +QMi`QHH2`

͵ǤͷǤ͸ ‡•—Ž–•

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

6B;m`2 jXRj, ai2T H2M;i?b 7Q` bBM;H2 bi2T `2+Qp2`v b+2M`BQb 7`QK >bBQ@q2+FbH2` M/ _Q#B@

MQpBi+? UkyydV, 2tT2`BK2MiH Ur?Bi2 #`b- p2`;2/ +`Qbb bm#D2+ib ± QM2 biM/`/ /2pBiBQMV p2`bmb bBKmHi2/ U#H+F #`bV `2bmHibX

 ‡•–‹ƒ–‹‘• ˆ‘” š’‡”‹‡–ƒŽ ‹–—ƒ–‹‘•

6B;m`2 jXR9, ai`B/2 H2M;i? 7Q` KmHiBTH2 bi2T `2+Qp2`v b+2M`BQ 7`QK *v` M/ aK22bi2`b

UkyyNVX 1tT2`BK2MiH Ur?Bi2 #`b- p2`;2/ +`Qbb bm#D2+ib ± QM2 biM/`/ /2pBiBQMV p2`bmb bBKmHi2/ U#H+F #`bV `2bmHibX

‘…Ž—•‹‘

6B;m`2 jXR8, 1pQHmiBQM Q7 i?2 K2+?MB+H KQ/2H /m`BM; i?2 T`2/B+i2/ `2+Qp2`v 7Q` i?2 b+2@

M`BQ Q7 *v` M/ aK22bi2`b UkyyNV UbMTb?Qib 2p2`v kyy KbVX

]Pm` 2tT2`B2M+2 ?Bi?2`iQ DmbiB}2b mb BM i`mbiBM; i?i Mim`2 Bb i?2 `2HBxiBQM Q7 i?2 bBKTH2bi i?i Bb Ki?2KiB+HHv +QM+2Bp#H2X] H#2`i 1BMbi2BM- RNjj

*?Ti2` 9

 —Ž–‹’Ž‡Ǧ–”ƒ–‡‰› ƒŽƒ…‡

‡…‘˜‡”› ‘†‡Ž

*QMi2Mib

ͶǤͳ š‹•–‹‰ —Ž–‹’Ž‡Ǧ•–”ƒ–‡‰› …‘–”‘ŽŽ‡”• X X X X X X X X X X X X X X dy ͶǤʹ ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡ X X X X X X X X X X X X X X X X X X dR ͶǤʹǤͳ ›•–‡ ›ƒ‹…• X X X X X X X X X X X X X X X X X X X X X X X dR ͶǤʹǤʹ ‹‡ƒ–‹… ƒ† †›ƒ‹… …‘•–”ƒ‹–• X X X X X X X X X X X X X dj ͶǤʹǤ͵ ‘–”‘ŽŽ‡” †‡•‹‰ X X X X X X X X X X X X X X X X X X X X X X X d8 ͶǤ͵ ‘•– —…–‹‘ ƒŽ›•‹• X X X X X X X X X X X X X X X X X X X X X X X d8 ͶǤ͵Ǥͳ –ƒ„‹Ž‹–› —•‹‰ ‹š‡†Ǧ•—’’‘”– •–”ƒ–‡‰‹‡• X X X X X X X X X X X de ͶǤ͵Ǥʹ ‘‘–Š‹‰ ‘ˆ ‘–‹‘ –”ƒŒ‡…–‘”‹‡• X X X X X X X X X X X X X X dN ͶǤ͵Ǥ͵ –‡’’‹‰ ƒ† ‘–ƒ…– –‹‡ ‘’–‹‹œƒ–‹‘ X X X X X X X X X X X dN ͶǤ͵ǤͶ  ‘–‡ ‘ –Š‡ ”‡Žƒ–‹˜‡ …‘–”‘Ž ™‡‹‰Š–• X X X X X X X X X X X X 3k ͶǤͶ ‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡• X X X X X X X X X X X X X X X X X X X X X X 39 ͶǤͶǤͳ Ž‡ ƒ† ‹’ •–”ƒ–‡‰‹‡• X X X X X X X X X X X X X X X X X X X 39 ͶǤͶǤʹ ‡‰—Žƒ–‹‘ ‘ˆ ͵ •–”ƒ–‡‰‹‡• X X X X X X X X X X X X X X X X X X 3N ͶǤͷ ‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ X X X X X X X X X X X X X X X Ny ͶǤͷǤͳ ‘†‡Ž ƒ”ƒ‡–‡”• X X X X X X X X X X X X X X X X X X X X X X NR ͶǤͷǤʹ ‡•—Ž–• X X X X X X X X X X X X X X X X X X X X X X X X X X X X NR ͶǤͷǤ͵ –‡’’‹‰ ’”‡†‹…–‹‘• ™‹–Š —’’‡”Ǧ„‘†› ‹‡”–‹ƒ X X X X X X X Nj ͶǤͷǤͶ –‡’’‹‰ ’”‡†‹…–‹‘• ™‹–Š •–‡’ –‹‡ ‘’–‹‹œƒ–‹‘ X X X X X Ne ͶǤ͸ ‘…Ž—•‹‘X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X NN

š‹•–‹‰ —Ž–‹’Ž‡Ǧ•–”ƒ–‡‰› …‘–”‘ŽŽ‡”•

ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡

ͶǤʹ ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡

ͶǤʹǤͳ ›•–‡ ›ƒ‹…•

mhc¨x+ j ¨θ= mg(cx− zx) cx zx θ j g N T c θ ˆ θk= ⎡ ⎢ ⎢ ⎣ θk ˙θk ¨ θk ⎤ ⎥ ⎥ ⎦ tk Θk+1 = ⎡ ⎢ ⎢ ⎣ θk+1 θk+N ⎤ ⎥ ⎥ ⎦ , ˙Θk+1 = ⎡ ⎢ ⎢ ⎣ ˙θk+1 ˙θk+N ⎤ ⎥ ⎥ ⎦ , ¨Θk+1 = ⎡ ⎢ ⎢ ⎣ ¨ θk+1 ¨ θk+N ⎤ ⎥ ⎥ ⎦ Θk+1 = ⎡ ⎢ ⎢ ⎣ θk+1 θk+N ⎤ ⎥ ⎥ ⎦

ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡ Θk+1 = Spθˆk+ UpΘk, ˙ Θk+1 = Svθˆk+ UvΘk, ¨ Θk+1 = Saθˆk+ UaΘk. Sp, Up z = cx− h g ¨cx− j mgθ.¨ Zk+1 = ⎡ ⎢ ⎢ ⎣ z_k+1 z_k+N ⎤ ⎥ ⎥ ⎦ Ck Θk Z_k+1 = Sz  ˆ ck ˆ θk + Uz  Ck Θk , Sz =  Sp− h_gSa −_mgj Sa  , Uz =  Up− h_gUa −_mgj Ua  .

ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡

ͶǤʹǤʹ ‹‡ƒ–‹… ƒ† †›ƒ‹… …‘•–”ƒ‹–•

i_{∈ [k + 1, . . . k + N]} θ_{KBM≤ θ}i ≤ θKt. j_|¨θi| ≤ τKt fi ti ci x− fi ≤ lKt. f′ i  ¨f′ i ≤ ¨fKt′ zx D(zi− fi) ≤ b,

ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡ D b fi Fk+1 = ⎡ ⎢ ⎢ ⎣ f_k+1 f_k+N ⎤ ⎥ ⎥ ⎦ fk ¯ F_k+1 V_k+1 ¯ Vk+1 ti F_k+1 = Vk+1fk+ ¯Vk+1F¯k+1. Vk+1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ¯Vk+1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 1 0 1 0 0 1 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ƒŽƒ…‡ ‡…‘˜‡”›  •…Š‡‡

ͶǤʹǤ͵ ‘–”‘ŽŽ‡” †‡•‹‰

˙ C_k+1 ˙ Θk+1 ¨ F′ k+1 1 c2₁ ˙Ck+1 2₊ 1 c2₂ ˙Θk+1 2₊ 1 c2₃ ¨F ′ k+12. Ck Θk Z_k+1 F_k+1 1 c2₁ ˙Ck+1 2₊ 1 c2₂ ˙Θk+1 2₊ 1 c2₃ ¨F ′ k+12 + 1 c2 4C k2+ 1 c2 5Θ k2+ 1 c2 6Z k+1− Fk+12, u=Ck Θk F¯k+1 T c₁ c₆

‘•– —…–‹‘ ƒŽ›•‹•

h#H2 9XR, Mi?`QTQKQ`T?B+ T`QTQ`iBQMb mb2/ BM i?2 bBKmHiBQM

H m h_{= 0.575 × H} lf = 0.152 × H 0.81 × lf 0.19 × lf θ_Kt π/2 θ_{KBM −π/2} j 2 τmax

ͶǤ͵ ‘•– —…–‹‘ ƒŽ›•‹•

ͶǤ͵Ǥͳ –ƒ„‹Ž‹–› —•‹‰ ‹š‡†Ǧ•—’’‘”– •–”ƒ–‡‰‹‡•

1 c2 1 ˙ Ck+12+ 1 c2 2 ˙Θ k+12. u = Ck Θk T

‘•– —…–‹‘ ƒŽ›•‹•

6B;m`2 9XR, 1z2+i Q7 BM2`iB r?22H iQ`[m2 HBKBi QM i?2 KtBKmK /Bbim`#M+2 i?i +M #2

bmbiBM2/ rBi?Qmi bi2TTBM;X h?2 KtBKmK `QiiBQM M;H2 Bb }t2/ iQ π 2

6B;m`2 9Xk, 1z2+i Q7 BM2`iB r?22H M;H2 HBKBi QM i?2 KtBKmK /Bbim`#M+2 i?i +M #2

‘•– —…–‹‘ ƒŽ›•‹• τmax θmax −1 −1 τmax = 190N.m θmax = π₂ −1 −1

6B;m`2 9Xj, h?2 }t2/@bmTTQ`i bi#BHBiv `2;BQM BM *QJ bii2 bT+2 rBi? Qm` JS* +QMi`QHH2`

U9XkjV mbBM; Mi?`QTQK2i`B+ T`K2i2`bX w2`Q BM/B+i2b i?2 TQbBiBQM Q7 MFH2 BM b;BiiH THM2X Ai Bb 2pB/2Mi i?i i?2 /BbTQbBiBQM Q7 mTT2`@#Q/v HHQrb `2+Qp2`BM; 7`QK Km+? H`;2` /Bbim`#M+2b i?M rBi? MFH2 bi`i2;v HQM2 rBi?Qmi bi2TTBM;

‘•– —…–‹‘ ƒŽ›•‹•

ͶǤ͵Ǥʹ ‘‘–Š‹‰ ‘ˆ ‘–‹‘ –”ƒŒ‡…–‘”‹‡•

−1 Ck2 −1 c₄ c₅ c₄ c₅ −3 −3

‘•– —…–‹‘ ƒŽ›•‹•

6B;m`2 9X9, hBK2 T`Q}H2b Q7 *QJ M/ BM2`iB r?22H ++2H2`iBQMb 7QHHQrBM;  ;Bp2M /Bbim`@

#M+2 Q#iBM2/ #v KBMBKBxBM; +Qbi 7mM+iBQM U9XkjVX LQi2 i?2 `2bmHiBM; ~m+im@ iBQMb /m2 iQ i?2 #b2M+2 Q7 i?2B` i?B`/ /2`BpiBp2 T2MHBxiBQM

6B;m`2 9X8, hBK2 T`Q}H2b Q7 *QJ M/ BM2`iB r?22H ++2H2`iBQMb r?2M QMHv *QJ D2`F UH27iV

M/ #Qi? *QJ M/ BM2`iB r?22H D2`Fb U`B;?iV `2 T2MHBx2/X

‘•– —…–‹‘ ƒŽ›•‹•

6B;m`2 9Xe, aBM;H2 `2+Qp2`v bi2T H2M;i?b 7Q`  ;Bp2M T2`im`#iBQM b 7mM+iBQM Q7 bi2T /m`@

iBQMb −1  ˙Ck+12 ¨ F′ k+1  ¨F′ k+12

‘•– —…–‹‘ ƒŽ›•‹•

6B;m`2 9Xd, h?2 *QJ p2HQ+Biv +Qbi UH27iV M/ brBM; 7QQi ++2H2`iBQM +Qbi U`B;?iV ;BMbi

p`vBM; bi2T /m`iBQMb M/ 7Q` bi2T H2M;i?b THQii2/ BM 6B;m`2 9XeX LQi2 i?i #Qi? i?2 +Qbib b?Qr +QKTH2i2Hv QTTQbBi2 i`2M/ ;BMbi bi2T /m`iBQM

1 c2 1 ˙ C_k+12 + 1 c2 3 ¨ F′ k+12 c3 c1 −2

ͶǤ͵ǤͶ  ‘–‡ ‘ –Š‡ ”‡Žƒ–‹˜‡ …‘–”‘Ž ™‡‹‰Š–•

‘•– —…–‹‘ ƒŽ›•‹•

6B;m`2 9X3, JBMBKH pHm2b Q7 i?2 +Qbi 7mM+iBQM 7Q` /Bz2`2Mi T2MHBxiBQM +Q2{+B2Mib c3

c₄ c₅ c₁ −1 _c 4 −3 c₅ −3 c₂ c₃ c₂ c2 −1 c3  ¨F′ k+12 103 104  ˙Ck+12

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

h#H2 9Xk,  ;2M2`H ;mB/2HBM2 7Q` b2H2+iBM; +QMi`QH r2B;?i pHm2b

c₁ −1 c2 −1 c₃ −2 c4 −3 c5 −3 c₆ c₁ c_{3 ≈} −2

ͶǤͶ ‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

ͶǤͶǤͳ Ž‡ ƒ† ‹’ •–”ƒ–‡‰‹‡•

1 c2 1 ˙ Ck+12+ 1 c2 2 ˙Θ k+12+ 1 c2 3 ¨ F′ k+12 + 1 c2₄Ck 2 ₊ 1 c2₅Θk 2₊ 1 c2₆Zk+1− Fk+1 2

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

6B;m`2 9XN, hBK2 THQib Q7 `2bmHiBM; ?BT iQ`[m2 7i2` i?2 bvbi2K Bb 2tTQb2/ iQ /Bbim`#M+2b

Q7 BM+`2bBM; K;MBim/2b iBHH i?2 HBKBi Q7 i?2 +QK#BM2/ MFH2Y?BT bi`i2;B2b

6B;m`2 9XRy, hBK2 THQib Q7 `2bmHiBM; MFH2 iQ`[m2 7i2` i?2 bvbi2K Bb 2tTQb2/ iQ /Bbim`#M+2b

Q7 BM+`2bBM; K;MBim/2b iBHH i?2 HBKBi Q7 i?2 +QK#BM2/ MFH2Y?BT bi`i2;B2b

c₂ −1 _c

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡• −1 c₂ c₅ c2 −1 c5 −3 c2 c5 c₂ −1 −1 ◦

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

6B;m`2 9XRR, h?2 `2HiBp2 H2p2H Q7 mb2 Q7 MFH2 M/ ?BT bi`i2;B2bX S2F MFH2 U#H+FV M/ ?BT

U#Hm2V iQ`[m2b `2 THQii2/ ;BMbi p`vBM; T2`im`#iBQMbX q?Bi2 /Qi BM/B+i2b i?2 TQBMi r?2`2 mTT2`@#Q/v M;H2 HBKBi Q7 Ny◦ Bb `2+?2/

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

c2 c5 c2

6B;m`2 9XRk, 1z2+i Q7 p`vBM; *QJ D2`F T2MHBxiBQM c4 QM `2HiBp2 +QMi`B#miBQM Q7 MFH2 M/ ?BT iQ`[m2bX

c₆ c₄

c4 = 1000 m.s−2

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

6B;m`2 9XRj, 1z2+i Q7 p`vBM; *QS /Bp2`;2M+2 +Q2{+B2Mi c6QM `2HiBp2 +QMi`B#miBQM Q7 MFH2 M/ ?BT iQ`[m2bX ‡ƒ”• c2 c6

ͶǤͶǤʹ ‡‰—Žƒ–‹‘ ‘ˆ ͵ •–”ƒ–‡‰‹‡•

¨ F′′ k+1

‡‰—Žƒ–‹‘ ‘ˆ •–”ƒ–‡‰‹‡•

c3 = −2

c₃ = −2

6B;m`2 9XR9, ai2T H2M;i? r?2M `2+iBM; iQ p`vBM; TQbi@BKT+i *QJ p2HQ+BiB2b 7Q` /Bz2`2Mi

T2MHBxiBQM +Q2{+B2Mib c3- Q` 7Q` MQ T2MHBxiBQM i HH Q7 i?2 brBM; 7QQi ++2H2`iBQM ¨F′

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

ͶǤͷ ‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

Tsampling

ͶǤͷǤͳ ‘†‡Ž ƒ”ƒ‡–‡”•

Treac Tprep Tstep

ͶǤͷǤʹ ‡•—Ž–•

◦

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

h#H2 9Xj, h?2 +QMi`QHH2` T`K2i2`b `2Hi2/ iQ bBKmHiBQM M/ +QMbi`BMib

× lf ¨ f_′Kt −2 θKt π/2 τmax c1 −1 c2 −1 c₃ −2 c4 −3 c5 −3 c₆ Tland

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9XR8, .Bz2`2M+2 #2ir22M  bi2T M/  bi`B/2 H2M;i?X 6Q` i?2 722i BMBiBHHv HB;M2/ i?2

b;BiiH THM2- #Qi? bi2T M/ bi`B/2 H2M;i?b `2 i?2 bK2 7Q` i?2 }`bi bi2T

Tstep

ͶǤͷǤ͵ –‡’’‹‰ ’”‡†‹…–‹‘• ™‹–Š —’’‡”Ǧ„‘†› ‹‡”–‹ƒ

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9XRe, ai2T H2M;i?b 7Q` bBM;H2 bi2T `2+Qp2`v b+2M`BQb 7`QK >bBQ@q2+FbH2` M/ _Q#B@

MQpBi+? UkyydV, 2tT2`BK2MiH Ur?Bi2 #`b- p2`;2/ +`Qbb bm#D2+ib ± QM2 biM/`/ /2pBiBQMV p2`bmb bBKmHi2/ UrBi? M/ rBi?Qmi l"AV `2bmHibX h?2 bi2T iBKBM;b `2 }t2/ BM /pM+2 ++Q`/BM; iQ i?2 `2TQ`i2/ pHm2b BM i?2 bim/vX

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9XRd, h?2 T2F ?BT iQ`[m2 pHm2b +?B2p2/ /m`BM; i?2 9 `2+Qp2`v b+2M`BQb 7`QK

>bBQ@q2+FbH2` M/ _Q#BMQpBi+? UkyydV M/ +Q``2bTQM/BM; }MH `2+Qp2`v TQb@ im`2b Q7 i?2 K2+?MB+H KQ/2HX LQi2 i?i i?2 T2F ?BT iQ`[m2 M/ mTT2`@#Q/v `QiiBQM M;H2 `2 TQbBiBp2Hv b+H2/ rBi? i?2 T2`im`#iBQM H2p2HX

6B;m`2 9XR3, ai`B/2 H2M;i? 7Q` KmHiBTH2 bi2T `2+Qp2`v b+2M`BQ 7`QK *v` M/ aK22bi2`b

UkyyNVX 1tT2`BK2MiH Ur?Bi2 #`b- p2`;2/ +`Qbb bm#D2+ib ± QM2 biM@ /`/ /2pBiBQMV p2`bmb bBKmHi2/ UrBi? M/ rBi?Qmi l"AV `2bmHibX ai`B/2 iBK2b `2 }t2/ BM /pM+2X

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9XRN, 1pQHmiBQM Q7 i?2 K2+?MB+H KQ/2H /m`BM; i?2 T`2/B+i2/ `2+Qp2`v 7Q` i?2 b+2@

M`BQ Q7 *v` M/ aK22bi2`b UkyyNVX LQi2 i?i i?2 FM22 DQBMi ?b #22M //2/ 7Q` #2ii2` pBbmH +QKT`BbQM M/ Bb MQi i?2 T`i Q7 i?2 KQ/2HX

ͶǤͷǤͶ –‡’’‹‰ ’”‡†‹…–‹‘• ™‹–Š •–‡’ –‹‡ ‘’–‹‹œƒ–‹‘

Tstep

c3 −2

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9Xky, ai2T H2M;i?b 7Q` bBM;H2 bi2T `2+Qp2`v b+2M`BQb 7`QK >bBQ@q2+FbH2` M/ _Q#B@

MQpBi+? UkyydV rBi? QTiBKBxiBQM Q7 bi2T HM/BM; iBK2b, 2tT2`BK2MiH Ur?Bi2 #`b- p2`;2/ +`Qbb bm#D2+ib ± QM2 biM/`/ /2pBiBQMV p2`bmb bBKmHi2/ UrBi? M/ rBi?Qmi l"AV `2bmHibX

6B;m`2 9XkR, h?2 T`2/B+i2/ bi2T HM/BM; iBK2b Tland 7Q` i?2 9 T2`im`#iBQM b+2M`BQb Q7 >bBQ@q2+FbH2` M/ _Q#BMQpBi+? UkyydV- rBi? M/ rBi?Qmi l"A +QMbB/2`iBQMb +QKT`2/ ;BMbi 2tT2`BK2MiH pHm2b Ur?Bi2 #`bV

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9Xkk, h?2 T2F ?BT iQ`[m2 pHm2b +?B2p2/ /m`BM; i?2 9 `2+Qp2`v b+2M`BQb 7`QK

>bBQ@q2+FbH2` M/ _Q#BMQpBi+? UkyydV r?2M Tstep Bb HbQ QTiBKBx2/X

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9Xkj, ai`B/2 H2M;i? `2bmHib 7Q` KmHiBTH2 bi2T `2+Qp2`v b+2M`BQ 7`QK *v` M/ aK22bi2`b

UkyyNV r?2`2 bi`B/2 iBK2b `2 HbQ QTiBKBx2/X 1tT2`BK2MiH Ur?Bi2 #`b-p2`;2/ +`Qbb bm#D2+ib ± QM2 biM/`/ /2pBiBQMV p2`bmb bBKmHi2/ UrBi? M/ rBi?Qmi l"AV `2bmHibX

6B;m`2 9Xk9, h?2 QTiBKBx2/ bi`B/2 iBKBM;b- rBi? M/ rBi?Qmi l"A- +QKT`2/ ;BMbi 2t@

T2`BK2MiH pHm2b 7Q` KmHiBTH2 bi2T `2+Qp2`v b+2M`BQ 7`QK *v` M/ aK22bi2`b UkyyNVX

‘’ƒ”‹•‘ ™‹–Š š’‡”‹‡–ƒŽ ƒ–ƒ

6B;m`2 9Xk8, h?2 bBM;H2 bi2T `2+Qp2`v T`2/B+iBQMb 7Q` k `2+Qp2`v b+2M`BQb 7`QK JQ;HQ M/

aK22bi2`b Ukyy8V- rBi? M/ rBi?Qmi l"A +QKT`2/ ;BMbi 2tT2`BK2MiH `2@ bmHib Ur?Bi2 #`bVX

6B;m`2 9Xke, PTiBKBx2/ bi2T /m`iBQMb 7Q` k `2+Qp2`v b+2M`BQb 7`QK JQ;HQ M/ aK22bi2`b

‘…Ž—•‹‘

]A i?BMF i?i QMHv /`BM; bT2+mHiBQM +M H2/ mb 7m`i?2` M/ MQi ++mKmHiBQM Q7 7+ibX] H#2`i 1BMbi2BM- RN8k

*?Ti2` 8

*QMi2Mib

ͷǤͳ ‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡• X X X X X X X X X X X X X X X X Ry8 ͷǤͳǤͳ š’‡”‹‡–ƒŽ †ƒ–ƒ X X X X X X X X X X X X X X X X X X X X X X Ry8 ͷǤͳǤʹ ‹—Žƒ–‹‘• X X X X X X X X X X X X X X X X X X X X X X X X X X RyN ͷǤͳǤ͵ ‘…Ž—•‹‘ X X X X X X X X X X X X X X X X X X X X X X X X X X RRj ͷǤʹ ‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰X X X X X X X X RR8 ͷǤʹǤͳ ‹•–—”„ƒ…‡ ˆ‘”‡…ƒ•–‹‰ X X X X X X X X X X X X X X X X X X X RR8 ͷǤʹǤʹ ’Ž‡‡–ƒ–‹‘ X X X X X X X X X X X X X X X X X X X X X X X RRd ͷǤʹǤ͵ ‡•—Ž–• X X X X X X X X X X X X X X X X X X X X X X X X X X X X RR3 ͷǤʹǤͶ ‡ƒ”• X X X X X X X X X X X X X X X X X X X X X X X X X X X RkR ͷǤ͵ ‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘ X X X X X X X X X X Rkk ͷǤ͵Ǥͳ Šƒ”ƒ…–‡”‹•–‹…• ‘ˆ ƒ ‡Ž†‡”Ž› ”‡•’‘•‡ X X X X X X X X X X X X Rkk ͷǤ͵Ǥʹ ‹—Žƒ–‹‘ ‘ˆ ƒ‰‡ ‡ˆˆ‡…–• X X X X X X X X X X X X X X X X X X X Rkj ͷǤ͵Ǥ͵ ‡•—Ž–• ƒ† †‹•…—••‹‘ X X X X X X X X X X X X X X X X X X X X Rk9 ͷǤͶ ‘…Ž—•‹‘X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Rk3

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

ͷǤͳ ‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

ͷǤͳǤͳ š’‡”‹‡–ƒŽ †ƒ–ƒ

st nd š’‡”‹‡–ƒŽ ‡•—Ž–• st nd

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

6B;m`2 8XR, h?2 THi7Q`K /Bbim`#M+2 T`Q}H2 mb2/ iQ /2bi#BHBx2 i?2 bm#D2+ib #v

U_Q#2`i-kyyeVX

6B;m`2 8Xk, h?2 irQ `2+Qp2`v ibFb ;Bp2M iQ i?2 bm#D2+ib ;BMbi i?2 bK2 H2p2H Q7 /Bbim`@

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

h#H2 8XR, *QKTH2i2 bi2T FBM2KiB+b- T`2pBQmbHv mMTm#HBb?2/- 7Q` i?2 irQ #HM+2 `2+Qp2`v

b+2M`BQb +QMbB/2`2/ BM _Q#2`i UkyyeV Treac Tprep Tstep ± ± ± ± ± ± ± ± ± ± ± ± ± ± ‡ƒ”•

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

6B;m`2 8Xj, hBK2 T`Q}H2b Q7 *QJ p2HQ+Biv Q7 M 2tKTH2 bm#D2+i BM #Qi? `2+Qp2`v ibFbc

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

ͷǤͳǤʹ ‹—Žƒ–‹‘•

1 c2₁ ˙Ck+1 2₊ 1 c2₂ ˙Θk+1 2₊ 1 c2₃ ¨F ′ k+12+ 1 c2₄Ck 2₊ 1 c2₅Θk 2₊ 1 c2₆Zk+1−Fk+1 2 c₁ c₆

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

6B;m`2 8X9, h?2 BKTH2K2Mi2/ 722/#+F HQQT rBi? JS* +QMi`QHH2` BM i?2 HQQTX h#H2 8Xk, *QMi`QHH2` T`K2i2`b Treac Tprep st nd Tstep × ¨ f_′Kt −2 θ_Kt π/2 τmax c₁ −1 c2 −1 c3 −2 c₄ −3 c5 −3 c6

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

6B;m`2 8X8, h?2 bBKmHiBQM Q7 irQ #HM+2 `2+Qp2`v ibFb UH`;2 bT+2 pbX HBKBi2/ bT+2V

mbBM; i?2 bK2 +QMi`QHH2` T`K2i2`b U+X7X h#H2 8XkXV h?2 T`2/B+i2/ bi2TTBM; #2?pBQ` Bb +HQb2` iQ i?2 HBKBi2/ bT+2 2tT2`BK2MiH b+2M`BQ U`B;?iV i?M rBi? i?2 H`;2 bT+2X ‹—Žƒ–‹‰ –Š‡ •—„Ǧƒš‹ƒŽ „‡Šƒ˜‹‘”•  ¨F′ k+12 Ck2 c3 c4 c₃ c₄ c₃ −2 _c 4 −3

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

6B;m`2 8Xe, h?2 H2M;i?b Q7 T`2/B+i2/ }`bi bi2T ;BMbi p`vBM; pHm2b Q7 +QMi`QH r2B;?ib

c3 M/ c4- r?B+? T2MHBx2 i?2 *QJ D2`F M/ brBM; 7QQi ++2H2`iBQM i2`Kb `2bT2+iBp2HvX h?2 MmK#2` i i?2 #QiiQK Q7 2+? THQi `2T`2b2Mi i?2 iQiH MmK#2` Q7 T`2/B+i2/ 7Q`r`/ bi2TbX Ai +M #2 b22M i?i i?2 bKHH2` +QMi`QH r2B;?i pHm2b UBKTHvBM; bi`QM;2` T2MHBxiBQM Q7 +Q``2bTQM/BM; +Qbi 7mM+iBQM i2`KbV `2bmHi BM b?Q`i2` bi2T H2M;i?b M/ KQ`2 bi2TbX

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

ͷǤͳǤ͵ ‘…Ž—•‹‘

‹—Žƒ–‹‰ ’Žƒ–ˆ‘” †‹•–—”„ƒ…‡•

6B;m`2 8Xd, *QKT`BbQM Q7 p2`;2 2tT2`BK2MiH bi2T H2M;i?b rBi? T`2/B+i2/ H2M;i?b rBi?

c3 4 kyy KXb−2 M/ c4 4 jy KXb−3X

6B;m`2 8X3, *QKT`BbQM #2ir22M i?2 *QJ p2HQ+Biv 7Q`  `2H bm#D2+i U/Qii2/ THQi- H`;2

bT+2 b+2M`BQV M/ i?2 bBKmHiBQM U+QMiBMmQmb THQiV rBi? c3 4 kyy KXb−2- c4 4 jy KXb−3X

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

ͷǤʹ ‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

−3 −2

6B;m`2 8XN, h?2 ivTB+H ++2H2`iBQM T`Q}H2 Q7 i?2 2K2`;2M+v #`FBM; +QM/BiBQM BM m`#M

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

6B;m`2 8XRy, h?2 7Q`2+bi2/ THi7Q`K /Bbim`#M+2 T`Q}H2b U/Qii2/ #Hm2V `2 b?QrM rBi?

`2bT2+i iQ i?2 +imH T`Q}H2 U#H+F THQiV i 7Qm` bKTH2 BMbiMib Ui 4 yX8- RXy-RX8- k bV 7i2` i?2 /Bbim`#M+2 QMb2iX i 2+? BMbiMi- i?2 +QMi`QHH2` 7Q`2+bib i?2 /Bbim`#M+2 iQ +QMiBMm2 7Q` i?2 M2ti Rb, G27i, qBi? M ++2H2`iBQM 2[mH iQ i?2 p2`;2 Q7 Tbi Rb Ub+2M`BQ #V- Q` _B;?i, qBi? i?2 bK2 BMbiMiM2Qmb ++2H2`iBQM Ub+2M`BQ +VX q?2M i?2 +imH /Bbim`#M+2 /BbTT2`b i i4 kb-i?2 7Q`2+bi Bb HbQ K/2 iQ x2`QX

ͷǤʹǤʹ ’Ž‡‡–ƒ–‹‘

¨ cx = g h(cx− zx) − j mhθ¨

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰ ¨ cx= g h(cx− zx) − j mhθ¨ ± ¨x pf F cast ¨ xpf_{F cast}

ͷǤʹǤ͵ ‡•—Ž–•

−2

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

6B;m`2 8XRR, h?2 T`2/B+i2/ `2+Qp2`v #2?pBQ` mM/2` i?2 2K2`;2M+v #`FBM; b+2M`BQ U+X7X

6B;m`2 8XNV rBi? MQ Q` p`vBM; H2p2Hb Q7 /Bbim`#M+2 7Q`2+biBM;X hBK2 T`Q}H2b Q7 *QJ TQbBiBQM UH27i- #H+F THQiV M/ p2HQ+Biv U`B;?iV `2 THQii2/X h?2 bi2T TQbBiBQMb `2 BM/B+i2/ rBi? #Hm2 /QibX

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

nd

‹—Žƒ–‹‰ Ž‘‰‡” †‹•–—”„ƒ…‡• ™‹–Š ‘”‡…ƒ•–‹‰

6B;m`2 8XRk, h?2 iQiH *QJ i`p2H b 7mM+iBQM Q7 +QMi`QHH2` bKTHBM; iBK2 pHm2b 7Q` /Bz2`2Mi

H2p2Hb Q7 /Bbim`#M+2 7Q`2+biBM;- mM/2` i?2 THi7Q`K /Bbim`#M+2 T`Q}H2 ;Bp2M BM 6B;m`2 8XNX

‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

ͷǤ͵ ‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

ͷǤ͵Ǥʹ ‹—Žƒ–‹‘ ‘ˆ ƒ‰‡ ‡ˆˆ‡…–•

− −2 −2 π/2 π/3

‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

h#H2 8Xj, *QMi`QHH2` T`K2i2`b 7Q` i?2 2H/2`Hv `2+iBQM

lf × Tprep Treac Tstep ¨ f_′Kt −2 θKt π/3 τmax − − c1 c6 Tstep

ͷǤ͵Ǥ͵ ‡•—Ž–• ƒ† †‹•…—••‹‘

‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

◦

‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

6B;m`2 8XRj, PTiBKBx2/ bi2T H2M;i?b M/ iBKBM;b ;BMbi 7Q`r`/ BM+HBMiBQMb b+2M`BQ 7Q`

2H/2`Hv 7`QK >bBQ@q2+FbH2` M/ _Q#BMQpBi+? UkyydV ;BMbi KtBKmK 7QQi ++2H2`iBQM pHm2b Up`B2/ HQM; t@tBbV 7Q` j /Bz2`2Mi `2+iBQM iBK2 pHm2b Ud8- Ryy M/ R8y KbV M/ k /Bz2`2Mi 2z2+iBp2 "Qa H2M;i?b, UH27i `2/ THQib 7Q` `2/m+2/ "Qa pbX `B;?i #Hm2 THQib 7Q` 7mHH "QaVX

‹—Žƒ–‹‰ –Š‡ •‹–—ƒ–‹‘ ‘ˆ ‡Ž†‡” ’‘’—Žƒ–‹‘

6B;m`2 8XR9, PTiBKBx2/ bi2T H2M;i?b M/ iBKBM;b ;BMbi 7Q`r`/ BM+HBMiBQMb b+2M`BQ 7Q`

2H/2`Hv 7`QK >bBQ@q2+FbH2` M/ _Q#BMQpBi+? UkyydV ;BMbi KtBKmK 7QQi ++2H2`iBQM pHm2b Up`B2/ HQM; t@tBbV 7Q` j /Bz2`2Mi `2+iBQM iBK2 pHm2b Ud8- Ryy M/ R8y KbV M/ k /Bz2`2Mi 2z2+iBp2 "Qa H2M;i?b, UH27i `2/ THQib 7Q` `2/m+2/ "Qa pbX `B;?i #Hm2 THQib 7Q` 7mHH "QaVX ai``2/ pHm2 U⋆V `2T`2b2Mib  7HHX

‘…Ž—•‹‘

]_2K2K#2` i?i HH KQ/2Hb `2 r`QM;c i?2 T`+iB+H [m2biBQM Bb ?Qr r`QM; /Q i?2v ?p2 iQ #2 iQ MQi #2 mb27mHX] 1KTB`B+H KQ/2H@#mBH/BM; M/ `2bTQMb2 bm`7+2b #v :2Q`;2 1X SX "Qt- qBH2v- RN3d

*?Ti2` e

Š‡ ƒ•– ‘”†

—ƒ”› ƒ† …‘–”‹„—–‹‘•

‘†‡Ž‹‰ ‘ˆ Š—ƒ „ƒŽƒ…‡ —†‡” •–”‘‰ †‹•–—”„ƒ…‡•

+Q@2tBbi2M+2 M/ `2;mHiBQM

 —Ž–‹Ǧ•–”ƒ–‡‰›  •…Š‡‡ ˆ‘” „‹’‡† ”‘„‘–•

l"A M/ M 2z2+iBp2 bi2T iBK2 QTiBKBxiBQM BM M JS* b+?2K2

’’Ž‹…ƒ–‹‘ –‘ ˜‡Š‹…Ž‡ †‡•‹‰ ƒ† ‘’‡”ƒ–‹‰ ’”ƒ…–‹…‡•

b@ b2bbK2Mi Q7 `BbFb

‡”•’‡…–‹˜‡•

”‘ ’”‡†‹…–‹‘ –‘ ‡™ ”‡•‡ƒ”…Š “—‡•–‹‘•

͸Ǥʹ ‡”•’‡…–‹˜‡•

‡”•’‡…–‹˜‡•

‡”•’‡…–‹˜‡•

6B;m`2 eXR, h?2 +m``2Mi ?mKM@#Q/v `2T`2b2MiiBQM UiQTV +M #2 `2TH+2/ #v KQ`2 `2HBbiB+

KQ/2Hb bm+? b  /Qm#H2@BMp2`i2/ T2M/mHmK U.ASV KQ/2H /m`BM; i?2 bBM;H2@ bmTTQ`i T?b2 M/  bT`BM; KQ/2H /m`BM; i?2 /Qm#H2@bmTTQ`i T?b2X

‡”•’‡…–‹˜‡•

6B;m`2 eXk,  TQbbB#H2 BMi2;`iBQM Q7 ?mKM b2MbQ`v bvbi2Kb BM i?2 "_ iQQH rQmH/ BMpQHp2

 b2T`i2 b2MbQ` /vMKB+ KQ/2H Q7 2+? b2MbQ`v KQ/HBiv UU2X;X "Q`? 2i HX-RN33VV M/  b2MbQ`v BMi2;`iBQM +2Mi2`X

‡”•’‡…–‹˜‡•

_ûbmKû 2M 6`MÏBb

‘–‡š–‡ ‡– ‘–‹˜ƒ–‹‘

‘–‡š–‡ ǣ Ž‡ ’”‘„Ž°‡ †‡ Žƒ •–ƒ„‹Ž‹–± †‡• ’ƒ••ƒ‰‡”• †‡„‘—– †‡

–”ƒ•’‘”–• ‡ …‘—

‡”•’‡…–‹˜‡•

6B;m`2 eXj, G T`QTQ`iBQM /2b T2`bQMM2b ;û2b /Mb H TQTmHiBQM 7`MÏBb2 /2TmBb RN8y

2i T`QD2iû Dmb[m^¨ ky8y T` H^AMbiBimi LiBQMH /2 H aiiBbiB[m2 2i /2b 1im/2b 1+QMQKB[m2b UAMb22- kyyeVX

‡”•’‡…–‹˜‡•

‡• ±–—†‡• ‡š’±”‹‡–ƒŽ‡• ǣ —–‹Ž‡• ƒ‹• ’ƒ• ’”±†‹…–‹˜‡•

6B;m`2 eX9, G2 KQv2M /^2bbB +Qm`KK2Mi miBHBbû TQm` ;ûMû`2` /2b T2`im`#iBQMb /2 ivT2

‡”•’‡…–‹˜‡•

6B;m`2 eX8, G2b bi`iû;B2b 7QM/K2MiH2b miBHBbû2b T` H2b ?mKBMb TQm` `ii`T2` H2m` û[mBHB@

#`2, G bi`iû;B2 /Bi2 /2 +?2pBHH2 U;m+?2V- /2 ?M+?2 U+2Mi`2V 2i /2 Tb /2 `ii`T;2 U/`QBi2V UEMKBv 2i HX- kyRyV

‡”•’‡…–‹˜‡•

Wi = Wg_{+ W}c

‡”•’‡…–‹˜‡•

6B;m`2 eXe, G2 MQvm /2 pB#BHBiû `bb2K#H2 iQmb H2b ûiib ¨ T`iB` /2b[m2Hb BH 2bi TQbbB#H2

‡”•’‡…–‹˜‡•

‡”•’‡…–‹˜‡•

’Ž±‡–ƒ–‹‘

1 c2₁ ˙Ck+1 − C˙ ref k+12 + 1 c2₂Ck 2 ₊ 1 c2₃Zk+1 − Fk+1 2_, c₁ c₂ c₃ u =  Ck ¯ Fk+1 Ck F¯k+1

‘’ƒ”ƒ‹•‘ ƒ˜‡… Ž‡• ”±•—Ž–ƒ–• ‡š’±”‹‡–ƒ—š

‡”•’‡…–‹˜‡•

6B;m`2 eXd, GQM;m2m`b /2b Tb /2 `ii`T;2 TQm` H2 b+ûM`BQb /2 >bBQ@q2+FbH2` M/ _Q#B@

MQpBi+? UkyydV, 2tTû`BK2MiH2b U2M #HM+- KQv2MM2 2i û+`i@ivT2V pb bBKmHû2b T` H T`2KBĕ`2 p2`bBQM /m KQ/ĕH2UMQB`V

‡”•’‡…–‹˜‡•

6B;m`2 eX3, G2b TQbBiBQMb /2b i`QBb Tb /2 `ii`T;2 TQm` H2 b+ûM`BQ /2 *v` M/ aK22bi2`b

UkyyNV, 1tTû`BK2MiH2b U2M #HM+- KQv2MM2 2i û+`i@ivT2V pb bBKmHû2b T` H T`2KBĕ`2 p2`bBQM /m KQ/ĕH2UMQB`V mhc¨x+ j ¨θ= mg(cx− zx) z = cx− h g ¨cx− j mgθ.¨ j θ ˙ Ck+1 ˙ Θk+1 Q#i2MB` mM2 pû`Bi#H2 QTiBKBbiBQM /2 H /m`û2 /2 Tb ¨ F′ k+1 1 c2 1 ˙ Ck+12+ 1 c2 2 ˙Θ k+12+ 1 c2 3 ¨ F′ k+12.

‡”•’‡…–‹˜‡•

6B;m`2 eXN, G2 KQ/ĕH2 /2 T2M/mH2 BMp2`bû HBMûB`2 p2+ mM pQHMi /^BM2`iB2 TQm` T`2M/`2 2M

+QKTi2 H2b 2z2ib BM2`iB2Hb /m ?mi /m +Q`Tb

Ck Θk Z_k+1 1 c2₁ ˙Ck+1 2₊ 1 c2₂ ˙Θk+1 2₊ 1 c2₃ ¨F ′ k+12 + 1 c2₄Ck 2₊ 1 c2₅Θk 2₊ 1 c2₆Zk+1− Fk+1 2_, u =Ck Θk F¯k+1 T c₁ c₆

‡”•’‡…–‹˜‡•

6B;m`2 eXRy, G2b HQM;m2m`b /2 Tb /2 `ii`T;2 TQm` H2 b+ûM`BQ /2 >bBQ@q2+FbH2` M/

_Q#BMQpBi+? UkyydV p2+ H^QTiBKBbiBQM /2 H /m`û2 /2 Tb, 1tTû`BK2MiH2 U2M #HM+- KQv2MM2 2i û+`i@ivT2V +QMi`2 bBKmHû2 p2+ UMQB`V 2i bMb U;`BbV +QMbB/û`iBQM /2 H^BM2`iB2 /m ?mi /m +Q`TbX