772 50 3MB
Pages 117 Page size 399.863 x 617.346 pts Year 2011
Springer Tracts in Advanced Robotics Volume 12 Editors: Bruno Siciliano · Oussama Khatib · Frans Groen
Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo
K. Iagnemma S. Dubowsky
Mobile Robots in Rough Terrain Estimation, Motion Planning, and Control with Application to Planetary Rovers With 58 Figures
Professor Bruno Siciliano, Dipartimento di Informatica e Sistemistica, Universit`a degli Studi di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy, email: [email protected] Professor Oussama Khatib, Robotics Laboratory, Department of Computer Science, Stanford University, Stanford, CA 94305-9010, USA, email: [email protected] Professor Frans Groen, Department of Computer Science, Universiteit van Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands, email: [email protected] STAR (Springer Tracts inAdvanced Robotics) has been promoted under the auspices of EURON (European Robotics Research Network)
Authors Dr. Karl Iagnemma Prof. Dr. Steven Dubowsky Massachusetts Institute of Technology (MIT) Department of Mechanical Engineering 77 Massachusetts Avenue Cambridge, MA 02139-4307 USA
ISSN 1610-7438 ISBN 3-540-21968-4 Springer Berlin Heidelberg New York Library of Congress Control Number: 2004106986 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by authors. Data-conversion and production: PTP-Berlin Protago-TeX-Production GmbH, Germany Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 62/3020Yu - 5 4 3 2 1 0
Editorial Advisory Board EUROPE Herman Bruyninckx, KU Leuven, Belgium Raja Chatila, LAAS, France Henrik Christensen, KTH, Sweden Paolo Dario, Scuola Superiore Sant’Anna Pisa, Italy R¨udiger Dillmann, Universit¨at Karlsruhe, Germany AMERICA Ken Goldberg, UC Berkeley, USA John Hollerbach, University of Utah, USA Lydia Kavraki, Rice University, USA Tim Salcudean, University of British Columbia, Canada Sebastian Thrun, Carnegie Mellon University, USA ASIA/OCEANIA Peter Corke, CSIRO, Australia Makoto Kaneko, Hiroshima University, Japan Sukhan Lee, Sungkyunkwan University, Korea Yangsheng Xu, Chinese University of Hong Kong, PRC Shin’ichi Yuta, Tsukuba University, Japan
Foreword
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¶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aples, Italy March 2004
Bruno Siciliano STAR Editor
3UHIDFH
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³WUDGLWLRQDO´URERWLFVUHVHDUFK DUHDV
X
3UHIDFH
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¶V!FXUUHQW SRVLWLRQ!WR!D!GLVWDQW!JRDO!SRVLWLRQ! 7KH!JRDO!RI!WKH!VHFRQG!SODQQLQJ PHWKRG!LV!WR!GHWHUPLQH!WKH!RSWLPDO!FRQILJXUDWLRQ!RI!DQ!DFWLYHO\!DUWLFXODWHG VXVSHQVLRQ!URYHU!WR!LPSURYH!WLSRYHU!VWDELOLW\!GXULQJ!WUDYHO!LQ!URXJK!WHU UDLQ!&KDS ! DGGUHVVHV WKH SUREOHP RI!URXJK WHUUDLQ FRQWURO!E\!SUHVHQWLQJ D!VHUYROHYHO!FRQWURO!PHWKRG!IRU!LPSURYHG!ZKHHO!WUDFWLRQ!RU!UHGXFHG SRZHU!FRQVXPSWLRQ!LQ!URXJK!WHUUDLQ! &KDS!!VXPPDUL]HV!WKH!FRQWULEX WLRQV!RI!WKLV!PRQRJUDSK!DQG!SUHVHQWV!VXJJHVWLRQV!IRU!IXWXUH!ZRUN!)RU!DOO RI WKH ZRUN VLPXODWLRQ DQG H[SHULPHQWDO UHVXOWV DUH SUHVHQWHG IRU!ZKHHOHG PRELOH!URERWV!RSHUDWLQJ!LQ!URXJK!RXWGRRU!HQYLURQPHQWV! 7KHVH!UHVXOWV GHPRQVWUDWH WKH HIIHFWLYHQHVV RI WKH SURSRVHG PHWKRGV 7KH!FRQFHSWV!SUHVHQWHG!LQ!WKLV!PRQRJUDSK!ZHUH!LQLWLDOO\!GHYHORSHG!LQ WKH!ILUVW!DXWKRU¶V!!3K'!WKHVLV!DW!WKH!0DVVDFKXVHWWV!,QVWLWXWH!RI!7HFK QRORJ\!0XFK!RI!WKH!ZRUN!KRZHYHU!KDV!EHHQ!XSGDWHG!DQG!H[SDQGHG!IRU WKLV!YROXPH!7KLV ZRUN ZDV VXSSRUWHG E\!WKH 1$6$!-HW 3URSXOVLRQ /DER UDWRU\!-3/ ! 7KH!DXWKRUV!ZRXOG!OLNH!WR!DFNQRZOHGJH!WKH!DVVLVWDQFH!DQG HQFRXUDJHPHQW!RI!'UV!3DXO!6FKHQNHU!6DPDDG!+D\DWL!DQG!5LFK!9ROSH!RI -3/ ,W LV WKH DXWKRUV¶!KRSH!WKDW!WKLV!PRQRJUDSK ZLOO!EH!RI!LQWHUHVW WR!URERWLFV UHVHDUFKHUV DQG HQJLQHHUV ZKR VWXG\ DQG GHYHORS URERWLF V\VWHPV GHVLJQHG IRU!WKH!RXWGRRU!ZRUOG!$V!PRELOH!URERWV!EHFRPH!PRUH!DGYDQFHG!DQG!WKHLU XVH!PRUH!ZLGHVSUHDG!ZH!IHHO!WKDW!WKLV!VXEMHFW!DUHD!FDQ!RQO\!JURZ!LQ!LP SRUWDQFH Cambridge, March 2004
Karl Iagnemma
7DEOHRI&RQWHQWV
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
XII
7DEOH RI &RQWHQWV
,QFRUSRUDWLQJ8QFHUWDLQW\ LQWKH5DSLG3DWK6HDUFK ,QFRUSRUDWLQJ8QFHUWDLQW\LQWKH0RGHO%DVHG(YDOXDWLRQ 6LPXODWLRQ5HVXOWV²5RXJK7HUUDLQ3ODQQLQJ 5RXJK7HUUDLQ$UWLFXODWHG6XVSHQVLRQ&RQILJXUDWLRQ3ODQQLQJ $UWLFXODWHG6XVSHQVLRQ&RQILJXUDWLRQ3ODQQLQJ3UREOHP 'HVFULSWLRQ 0RELOLW\ $QDO\VLV $UWLFXODWHG6XVSHQVLRQ&RQILJXUDWLRQ3ODQQLQJ IRU(QKDQFHG7LSRYHU6WDELOLW\ 5HVXOWV²5RXJK7HUUDLQ$UWLFXODWHG6XVSHQVLRQ&RQILJXUDWLRQ 3ODQQLQJ 6LPXODWLRQ5HVXOWV ([SHULPHQWDO5HVXOWV 6XPPDU\DQG&RQFOXVLRQV &KDSWHU5RXJK7HUUDLQ&RQWURO ,QWURGXFWLRQ 0RELOH5RERW5RXJK7HUUDLQ&RQWURO57& :KHHO7HUUDLQ&RQWDFW)RUFH2SWLPL]DWLRQ 2SWLPL]DWLRQ&ULWHULD 3UREOHP&RQVWUDLQWV 5HVXOWV²5RXJK7HUUDLQ&RQWURO 6LPXODWLRQ5HVXOWV ([SHULPHQWDO5HVXOWV 6XPPDU\DQG&RQFOXVLRQV &KDSWHU&RQFOXVLRQVDQG6XJJHVWLRQVIRU)XWXUH:RUN &RQWULEXWLRQVRI7KLV0RQRJUDSK 6XJJHVWLRQVIRU)XUWKHU:RUN 5HIHUHQFHV ,QGH[
&KDSWHU ,QWURGXFWLRQ
3UREOHP6WDWHPHQWDQG0RWLYDWLRQ 0RELOH URERWV DUH LQFUHDVLQJO\ EHLQJ HPSOR\HG LQ URXJK RXWGRRU WHUUDLQ IRUDSSOLFDWLRQV VXFKDVIRUHVWU\PLQLQJVHDUFKDQGUHVFXHDQG KD]DUGRXV VLWHLQVSHFWLRQ2VERUQ*RQWKLHUDQG3DSDGRSRXORV&XQQLQJ KDP HWDO0DHHWDO 7KHVHDSSOLFDWLRQVRIWHQUHTXLUHURERWV WRWUDYHODFURVVXQSUHSDUHGUXJJHGWHUUDLQWR LQVSHFWDORFDWLRQRUWUDQVSRUW PDWHULDO,Q RXWGRRUVHWWLQJV DURERW¶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³URYHUV´ ODQGHGRQWKHVXUIDFHRI0DUV WR FRQGXFW VFLHQWLILF H[SHULPHQWV IRFXVHG RQ XQGHUVWDQGLQJ WKH SODQHW¶V FOLPDWH KLVWRU\ VXUIDFH JHRORJ\ DQG SRWHQWLDO IRU SDVW RU SUHVHQW OLIH 7KHVH 1$6$ -3/ URYHUV ZHUH H[SHFWHG WR QHJRWLDWH ORRVH VORSHG DQG URFNILOOHG WHUUDLQ WR DFFHVV VFLHQWLILFDOO\ LQWHUHVWLQJ VLWHV 7KH 6R MRXUQHU URYHU OLPLWHG LWV WUDYHUVHV WR UHODWLYHO\ VKRUW GLVWDQFHV LH D IHZ PHWHUV WKURXJKIODWURFN\WHUUDLQ XQGHUFORVHKXPDQVXSHUYLVLRQVHH)LJ 7KH 0DUV ([SORUDWLRQ 5RYHUV DUH DV WKLV PRQRJUDSK JRHV WR
K. Iagnemma and S. Dubovsky: Mobile Robots in Rough Terrain, STAR 12, pp. 1–15, 2004. © Springer-Verlag Berlin Heidelberg 2004
&KDSWHU
,QWURGXFWLRQ
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³KXQJ XS´RQDODUJHURFN ZKLFKFRXOGOHDG WRHQWUDSPHQW
)LJ 6RMRXUQHU URYHU RQ WKH VXUIDFH RI 0DUV 1$6$ -HW 3URSXOVLRQ /DERUDWRU\
7R DFFRPSOLVK WKHVH FKDOOHQJLQJ REMHFWLYHV IXWXUH URERW GHVLJQV PD\ HYROYHIURPWUDGLWLRQDO³IL[HGFRQILJXUDWLRQ´GHVLJQVWRWKRVHZLWKDFWLYHO\ DUWLFXODWHG VXVSHQVLRQV 6FKHQNHU HW DO 5RERWV ZLWK DUWLFXODWHG VXVSHQVLRQV FDQ LPSURYH URXJK WHUUDLQ PRELOLW\ E\ PRGLI\LQJ WKHLU VXV
3XUSRVH RI7KLV0RQRJUDSK
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³WUD GLWLRQDO´GHVLJQVRUDFWLYHO\ DUWLFXODWHGVXVSHQVLRQV7KH\PXVWEHDEOH WR DFFRPSOLVKSODQQHGWDVNV ZLWKVRPH GHJUHH RI DXWRQRP\ ZKLOHHQVXULQJ URERWVDIHW\
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
&KDSWHU
,QWURGXFWLRQ
WKH!PHWKRGV!DQG!DOJRULWKPV!SUHVHQWHG!KHUH!FDQ!EH!DSSOLHG!WR!D!ZLGH!UDQJH RI URXJK WHUUDLQ VLWXDWLRQV (VWLPDWLRQ!DOJRULWKPV!DUH!SUHVHQWHG!WKDW!IRFXV!RQ!URERWWHUUDLQ!LQWHUDF WLRQ! 6SHFLILFDOO\!PHWKRGV!DUH!SUHVHQWHG!IRU!HVWLPDWLQJ!WHUUDLQ!FRKHVLRQ DQG!LQWHUQDO!IULFWLRQ!DQJOH!WZR!LPSRUWDQW!SK\VLFDO!SDUDPHWHUV!UHODWHG!WR WHUUDLQ WUDYHUVDELOLW\ DQG ZKHHOWHUUDLQ FRQWDFW DQJOHV!$ PRGHO RI URERW WHUUDLQ!LQWHUDFWLRQ!PHFKDQLFV!LV!DOVR!SUHVHQWHG!ZKLFK!XQLWHV!WKH!HVWLPDWHG WHUUDLQ!TXDQWLWLHV!ZLWK!³FODVVLFDO´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
%DFNJURXQGDQG/LWHUDWXUH5HYLHZ
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¶VOHJ DQGLVQRWDSSOLFDEOHWRZKHHOHG V\VWHPV 7HUUDLQSDUDPHWHUHVWLPDWLRQIRUWUDFNHG YHKLFOHVKDVEHHQSUR SRVHGLQ/HHWDO 7KLVDSSURDFKDVVXPHVDVLPSOLILHG³IRUFHFRHI ILFLHQW´ PRGHO RI WUDFNWHUUDLQ LQWHUDFWLRQ WKDW LV QRW YDOLG LQ GHIRUPDEOH WHUUDLQ 2WKHUUHVHDUFKHUVKDYHDWWHPSWHG WRFLUFXPYHQWH[SOLFLWSDUDPHWHU HVWLPDWLRQ E\ XVLQJ VLPSOLILHG PRGHOV WR HVWLPDWH ZKHHO VOLS RU GUDZEDU SXOO/LQGJUHQHWDOT T L VLQ T VLQ T @ · ¨ ¸ N F V T WDQ I ¨ H ¸ © ¹
ZKHUH N LV WKH VKHDU GHIRUPDWLRQ PRGXOXV U LV WKH ZKHHO UDGLXV DQG L LV WKHZKHHOVOLSGHILQHGDV L 9 UZ IRUZ !:RQJ 7KHQRUPDOVWUHVVDWWKHZKHHOWHUUDLQLQWHUIDFHLVJLYHQE\ § NF · NI ¸ ] Q E © ¹
V ] ¨
ZKHUH NFDQGNIDUHSUHVVXUHVLQNDJH PRGXOLDQGQLV WKHVLQNDJHH[SRQHQW %HNNHU 7KLVHTXDWLRQFDQ EHH[SUHVVHGDVDIXQFWLRQRIWKHZKHHO DQJXODUORFDWLRQ T E\QRWLQJWKDWVLQNDJHLVUHODWHG WRTDV ] T U FRV T FRV T
6XEVWLWXWLQJ (T LQWR(T \LHOGVH[SUHVVLRQVIRUWKHQRUPDO VWUHVVGLVWULEXWLRQDORQJWKHZKHHOWHUUDLQLQWHUIDFHDV § NF · NI ¸U FRV T FRV T Q E © ¹
V T ¨
Q
V T
T T P ·¸ FRVT ·¸ ·¸ ·§ § § § NF ¨ NI ¸¨ U ¨¨ FRV¨¨T T ¸ ¸ T P ¸¹ ¹¨© © © ©E ¹¹
7RGHYHORSDSDUDPHWHUHVWLPDWLRQDOJRULWKP DQDO\WLFDOH[SUHVVLRQVIRU WKH IRUFH EDODQFH HTXDWLRQV (TV DUH UHTXLUHG VLQFH WKHVH HTXDWLRQVUHODWHSK\VLFDOO\ PHDVXUDEOH TXDQWLWLHV:7] Z9 WRWKHSD UDPHWHUV RILQWHUHVWF I +RZHYHU (TV DUHQRWDPHQDEOHWR FORVHGIRUPLQWHJUDWLRQ GXHWRWKHLUFRPSOH[LW\7KLVPRWLYDWHVWKHXVHRI DQ DSSUR[LPDWHIRUPRIWKHVWUHVVHTXDWLRQV(TV
7HUUDLQ&KDUDFWHUL]DWLRQDQG ,GHQWLILFDWLRQ
!(TXDWLRQ 6LPSOLILFDWLRQ )LJVKRZVW\SLFDOVLPXODWHGSORWVRIVKHDUDQGQRUPDOVWUHVVGLVWULEX WLRQVDV GHILQHG E\ (TV DQG UHVSHFWLYHO\ DURXQGWKH ULPRIDGULYHQULJLGZKHHORQYDULRXVWHUUDLQVDWPRGHUDWHZKHHOVOLS 3D UDPHWHUVXVHGLQWKHVHSORWVDUHOLVWHGLQ7DEOHDQGUHSUHVHQWDUDQJHRI WHUUDLQ W\SHV GU\ VDQG VDQG\ ORDP FOD\H\ VRLO DQG VQRZ :RQJ ,WFDQEHVHHQLQ)LJWKDWWKHVKHDUDQGQRUPDOVWUHVVGLVWULEX WLRQV DUH DSSUR[LPDWHO\ OLQHDU IRU D GLYHUVH UDQJH RI WHUUDLQV 9LQFHQW
1RUPDO6WUHVV
1RUPDO6WUHVV
6KHDU6WUHVV
6KHDU6WUHVV
6WUHVVN3D
6WUHVVN3D
$QJOHGHJ
'U\ 6DQG
$QJOHGHJ
6DQG\ /RDP
1RUPDO6WUHVV
1RUPDO6WUHVV
6WUHVVN3D
6WUHVVN3D
6KHDU6WUHVV
6KHDU6WUHVV
$QJOHGHJ
&OD\H\6RLO
$QJOHGHJ
6QRZ
)LJ1RUPDODQGVKHDU VWUHVVGLVWULEXWLRQIRUYDULRXV WHUUDLQ W\SHVDWPRGHUDWH ZKHHOVOLS
%DVHG RQ WKLV REVHUYDWLRQ OLQHDU DSSUR[LPDWLRQV RI WKH VKHDU DQG QRU PDOVWUHVVHTXDWLRQVFDQEHZULWWHQDV
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
T T V P T T P T V / T V P TP T T W P W / T T T P T W / T F W P F TP V / T
ZKHUH VPDQG WPDUHWKHPD[LPXPYDOXHVRIWKHQRUPDODQGVKHDUVWUHVVUHVSHF WLYHO\(T FRQWDLQVDQRIIVHWWHUPIRUFRKHVLYHVRLOV
7DEOH3DUDPHWHUVIRUYDULRXVWHUUDLQW\SHV Q F>N3D@ I >GHJ@ NF>N3DPQ@ NI>N3DPQ@ N>P@
'U\VDQG
6DQG\ORDP
&OD\H\VRLO
6QRZ
6LPSOLILHGIRUPVRIWKHIRUFHEDODQFHHTXDWLRQVFDQEHZULWWHQE\ FRP ELQLQJ(TV ZLWK(TV T · § TP ¸ ¨ / / ¨ V T FRV T GT V T FRV T GT ¸ ¸ ¨ TP ¸ UE¨ T ¸ ¨ TP ¸ ¨ / / ¨ W T VLQ T GT W T VLQ T GT ¸ ¸ ¨ TP ¹ © T · §TP ¸ ¨ / / ¨ W T FRV T GT W T FRV T GT ¸ ¨ ¸ TP ¸ UE¨ T ¨ TP ¸ ¨ ¸ / / V T VLQ T G T V T VLQ T G T ¨ ¸ ¨ ¸ T P ¹ © T §TP · ¨ ¸ 7 U E¨ W / T GT W / T GT ¸ ¨ ¸ TP © ¹
³
:
³
³
³
³
'3
³
³
³
³
³
7HUUDLQ&KDUDFWHUL]DWLRQDQG,GHQWLILFDWLRQ
(YDOXDWLRQ RI(TV DQG \LHOGV WKHIROORZLQJH[SUHVVLRQVIRU WKHQRUPDOORDGDQGWRUTXH § V P T P FRVT T FRVT P T ¨ UE W P T P VLQ T T VLQ T P T P T T P ¨¨ © F T VLQ T P T P VLQ T P T PT T P
:
· ¸ ¸ ¸ ¹
U E W PT FT P
7
7ZR DVVXPSWLRQVDUHPDGHLQVROYLQJ(TV DQG IRUFDQGI 7KHILUVWLVWKDWWKHORFDWLRQRIWKH PD[LPXPVKHDU DQGQRUPDOVWUHVVRF FXUV DW WKH VDPH ORFDWLRQ TP $QDO\VLV DQG VLPXODWLRQ KDYH VKRZQ WKDW WKLVDVVXPSWLRQLVUHDVRQDEOHIRUDZLGHUDQJHRIVRLOW\SHV:LWKWKLVDV VXPSWLRQDQ DGGLWLRQDOUHODWLRQFDQEHZULWWHQEDVHGRQ(T WP
§
F V P WDQ I ¨¨ H
U >T T P L VLQ T VLQ T P @ · ¸ N
¸ ¹
©
7KHVHFRQGDVVXPSWLRQLVWKDWWKHDQJXODUORFDWLRQRI PD[LPXPVWUHVV TPRFFXUV PLGZD\ EHWZHHQ TDQGTLH T T TP
7KLVDVVXPSWLRQLVUHDVRQDEOHIRUDZLGHUDQJHRIVRLOVDWPRGHUDWHVOLS UDWLRV .DQJ 7KLV FDQ EH MXVWLILHG E\ QRWLQJ WKDW TP FDQ EH HVWL PDWHGIURPWKH UHODWLRQ TP F LF TZKHUH F DQG F DUHWHUUDLQ SD UDPHWHUV 7KH UDQJH RI F DQG F LV JHQHUDOO\ F § DQG F :RQJDQG5HHFH 7KXVIRU DZLGHUDQJHRI VOLSUDWLRV TPZLOOEH QHDU 7KHV\VWHPRI(TV FDQEHFRPELQHGLQWRD VLQJOHHTXDWLRQ UHODWLQJ FRKHVLRQDQGLQWHUQDOIULFWLRQ DQJOHDV N WDQ I N F N WDQ I N ZKHUH T · § N $¨T:U 7 VLQ T 7 VLQ ¸ ¹
©
N
T § · 7 ¨ FRV T FRV ¸ © ¹
N
T § · $TU E¨ VLQ T VLQ T ¸ © ¹
N
TU E¨ FRV T FRV
§
T
©
$ FRV T $ FRV
T
· $ ¸ ¹
&KDSWHU
DQG $ H
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ § U ªT § T · ·º « L ¨¨ VLQ T VLQ ¨ ¸ ¸¸ » N ¬« © ¹ ¹ ¼» ©
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
ZLWK .
>N
N N M N M
@ . 7
.ª F º «¬ WDQ I »¼
7
ª º « N N N M N M » ¬ ¼
7DEOH 3DUDPHWHUVSDFHIRUDOJRULWKPDQDO\VLV 0LQLPXPYDOXH
3DUDPHWHU Q Iҏ>GHJ@ F>N3D@ NF>N3DPQ@ NI>N3DPQ@ N>P@
0D[LPXP YDOXH
7HUUDLQ&KDUDFWHUL]DWLRQDQG,GHQWLILFDWLRQ
,Q SUDFWLFHPRUHWKDQWZRHTXDWLRQVDUHXVHG WR IRUP DQHVWLPDWHRIF DQG IWR GHFUHDVHVHQVLWLYLW\WRVHQVRUQRLVH,QWKLVFDVH.LV QRQVTXDUH DQG(T FDQ EH VROYHGLQDOHDVWVTXDUHVVHQVHXVLQJ WKHSVHXGRLQ YHUVHRI. ª F º «¬ WDQ I »¼
.
7 7 . . .
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
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW0RGHOLQJDQG(VWLPDWLRQ
XUHPHQWVFDQEHKLJKO\GHJUDGHG E\QRLVH,QWKLVFDVHYLVXDORGRPHWU\ FDQ \LHOGPRUH DFFXUDWHUHVXOWV2OVRQ HWDO 7KH VHQVRUV GHVFULEHG DERYH LH URERW FRQILJXUDWLRQ VHQVRUV PRWRU FXUUHQWVHQVRUZKHHOWDFKRPHWHU,08DQGYLVLRQV\VWHP ZRXOGEHSDUW RI PRVWDXWRQRPRXVURERWV\VWHPV7KXVDOOUHTXLUHGLQSXWVFDQEH PHDV XUHG RU HVWLPDWHGZLWKRQERDUG URERW VHQVRUV $Q LPSRUWDQW LPSOHPHQWDWLRQ LVVXH LV PLQLPL]LQJ VHQVRU QRLVH 0RVW URERWVHQVRUVFDQEH PRGHOHG DVD³WUXH´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ª F º «¬ WDQ I »¼
.
7 .
G ,
. 7 .
ZKHUH G LV D VPDOO SRVLWLYH FRQVWDQW WKDW FDQ EH RSWLPL]HG E\ WHFKQLTXHV VXFKDVFURVVYDOLGDWLRQ*ROXE DQG YDQ /RDQ ,Q SUDFWLFHDQGLQ WKH UHVXOWV SUHVHQWHG EHORZ QDWXUDO WHUUDLQ YDULDWLRQ ZLOO XVXDOO\ OHDG WR DFFHSWDEOH HTXDWLRQ FRQGLWLRQLQJ 'HOLEHUDWHO\ LQGXFLQJ YDULDEOH ZKHHO VOLSDOVRLPSURYHVSDUDPHWHUHVWLPDWHVRQIODWWHUUDLQ
5HVXOWV7HUUDLQ ,GHQWLILFDWLRQ
!5HVXOWV 7HUUDLQ ,GHQWLILFDWLRQ
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qIUDQJHVIURPqq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q $V H[SHFWHG LQFUHDVLQJ WKH VHQVRU QRLVH OHYHOV OHDGVWRLQFUHDVHGSDUDPHWHU HVWLPDWLRQHUURU 6LPXODWLRQVZHUHWKHQUXQ
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
ZLWKWKHQXPEHURIVDPSOHGGDWDSRLQWVLQFUHDVHGWR 7KH506HUURU EHWZHHQWKHHVWLPDWHGDQG DFWXDOF ZDVUHGXFHGWR N3DDQGIRU IWR q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stimated Cohesion−−Noise−free 6
Estimated Cohesion−−Noisy, 15 data pts. Estimated Cohesion−−Noisy, 30 data pts.
Cohesion (kPa)
5
4
3
2
1 True Cohesion
0
−1 0
5
10
15
Time (sec)
)LJ 6LPXODWHGHVWLPDWLRQRIFRKHVLRQRIGU\VDQGIRU QRLVHIUHHFDVH QRLV\ FDVHXVLQJ GDWDSRLQWVDQGQRLV\ FDVHXVLQJGDWDSRLQWV
5HVXOWV7HUUDLQ ,GHQWLILFDWLRQ
35 Estimated Friction Angle−−Noise−free Estimated Friction Angle−−Noisy, 15 pts.
Internal Friction Angle (deg)
Estimated Friction Angle−−Noisy, 30 pts.
True Friction Angle
30
25
20 0
5
10
15
Time (sec)
)LJ6LPXODWHGHVWLPDWLRQRILQWHUQDOIULFWLRQDQJOHRIGU\VDQGIRUQRLVHIUHH FDVH QRLV\ FDVHXVLQJGDWDSRLQWVDQGQRLV\ FDVHXVLQJGDWDSRLQWV
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
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
ORDG:DQGGUDZEDUSXOO '37KHWHVWEHGLVFRQWUROOHGE\D0+]3& 7KHVRLO ELQLV FP ORQJFPZLGHDQG FP GHHS 9HUWLFDO$[LV
&DUULDJH
:KHHO
)LJ7HUUDLQ FKDUDFWHUL]DWLRQWHVWEHG
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
5HVXOWV7HUUDLQ ,GHQWLILFDWLRQ
:RQJ N3D@
I>GHJ@
7DEOH5HVXOWVIURP WHUUDLQ SDUDPHWHUHVWLPDWLRQ H[SHULPHQWV :DVKHG6DQG 'ULHG%HQWRQLWH&OD\ &RPSDFWHG7RSVRLO
F>N3D@
I>GHJ@
7KHVH UHVXOWV VKRZ WKDW WKH HVWLPDWLRQ DOJRULWKP SURGXFHV UHDVRQDEO\ DFFXUDWH RQOLQH SDUDPHWHU HVWLPDWHV RQ DQ H[SHULPHQWDO V\VWHP ZLWK QRLV\VHQVRUVIRUDYDULHW\ RIWHUUDLQW\SHV 7KHOHYHORIDFFXUDF\ VKRZQ E\WKHDOJRULWKPZRXOGDOORZLWWR GLVWLQJXLVKEHWZHHQGLVWLQFWVRLOW\SHV VXFKDVFUXVW\PDWHULDO GULIWPDWHULDOFOD\HWF7KHFRPSXWDWLRQWLPHIRU HDFK HVWLPDWLRQ F\FOH ZDV DSSUR[LPDWHO\ PVHF RQ D 0+] GHVNWRS 3&7KXVWKHDSSURDFKLVVXLWDEOHIRUV\VWHPVZLWKOLPLWHGRQERDUGFRP SXWDWLRQVXFKDVSODQHWDU\ URYHUV
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
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
WHUUDLQWDQJHQWGXHWRWKHULJLGWHUUDLQDVVXPSWLRQ7KHGLVWDQFHEHWZHHQ WKHZKHHOFHQWHUVLVGHILQHGDV O
n2
l
n1
a
g2
g1
)LJ3ODQDUWZRZKHHOHGV\VWHP
)RUWKLVV\VWHPWKHIROORZLQJNLQHPDWLFHTXDWLRQVFDQEHZULWWHQ Q FRVJ D Q FRVJ D
Q VLQ J D Q VLQ J D O D
(T UHSUHVHQWV WKH NLQHPDWLF FRQVWUDLQW WKDW WKH ZKHHO FHQWHU OHQJWKOGRHVQRWFKDQJH1RWHWKDWWKLV FRQVWUDLQWUHPDLQV YDOLGLQFDVHV ZKHUH FKDQJHVLQWKHYHKLFOHVXVSHQVLRQFRQILJXUDWLRQ FDXVHFKDQJHVLQO DVORQJ DVO YDULHVVORZO\(T LV DULJLGERG\ NLQHPDWLFUHODWLRQEH WZHHQWKHYHORFLWLHVRIWKHZKHHOFHQWHUVDQGWKHYHKLFOHSLWFKUDWH D &RPELQLQJ(TV DQG \LHOGV O D VLQ J D J D FRVJ D Q :LWKWKHGHILQLWLRQV T { J D E { D J D { O D Q E { Q Q (TV DQG EHFRPH
E VLQ T VLQ E FRV T FRV E
D FRV T
E FRV T
:KHHO7HUUDLQ&RQWDFW$QJOH (VWLPDWLRQ
6ROYLQJ (TV DQG IRU WKH ZKHHOWHUUDLQ FRQWDFW DQJOHV J DQGJ \LHOGV
ZKHUH K {
J
D FRV K
J
FRV K D E
D E D E D E D
7KHUH DUH WZR VSHFLDO FDVHV WKDW PXVW EH FRQVLGHUHG LQ WKLV DQDO\VLV 7KHILUVWVSHFLDOFDVHRFFXUVZKHQWKHURERWLVVWDWLRQDU\,QWKLVFDVH(TV DQG GRQRW\LHOGDVROXWLRQVLQFHLI D Q Q ERWK D DQGE DUHXQGHILQHG3K\VLFDOO\WKHODFNRIDVROXWLRQUHVXOWVIURPWKHIDFW WKDWDVWDWLRQDU\URERWFDQKDYHDQLQILQLWHVHWRISRVVLEOHFRQWDFWDQJOHVDW HDFKZKHHO 7KHVHFRQGVSHFLDOFDVHRFFXUVZKHQFRVT HTXDOV ]HUR,QWKLVFDVH J rS DIURPWKHGHILQLWLRQRI TDQG(T \LHOGVWKHVROXWLRQ J rS D3K\VLFDOO\ WKLVFRUUHVSRQGVWRWZRSRVVLEOH FDVHVWKHURERWXQ GHUJRLQJSXUHWUDQVODWLRQ RUSXUHURWDWLRQDVVKRZQLQ)LJ
n2
n2
n1
n1
3XUH7UDQVODWLRQ
3XUH5RWDWLRQ
)LJ3K\VLFDOLQWHUSUHWDWLRQVRIFRVT
:KLOH WKHVH FDVHV DUH XQOLNHO\ WR RFFXU LQ SUDFWLFH WKH\ DUH HDVLO\ GH WHFWDEOH)RUWKHFDVH RISXUHURWDWLRQQ Q7KHVROXWLRQVIRUJDQG J FDQ EH ZULWWHQE\LQVSHFWLRQDV S J D VJQ D
J
D
S
VJQ D
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
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
ª º [ Z N «¬ »¼ N
ZKHUH ZN LV D[YHFWRUUHSUHVHQWLQJWKHSURFHVVQRLVH(T LP SOLHV WKDW JURXQG FRQWDFW DQJOH HYROXWLRQ LV D :LHQHU SURFHVV 7KLV LV SK\VLFDOO\ UHDVRQDEOH VLQFH WHUUDLQ YDULDWLRQ LV LQKHUHQWO\ XQSUHGLFWDEOH 7KHSURFHVVQRLVHFRYDULDQFHFDQEHDVVLJQHGDVWKHH[SHFWHGWHUUDLQYDULD WLRQ
:KHHO7HUUDLQ&RQWDFW$QJOH (VWLPDWLRQ
( [ N [ N
5N
7KLV LQIRUPDWLRQ FRXOG EH HVWLPDWHG IURP NQRZOHGJH RI ORFDO WHUUDLQ URXJKQHVVRUFRPSXWHGIURPIRUZDUGORRNLQJUDQJHGDWD 7KH(.)PHDVXUHPHQWHTXDWLRQFDQEHZULWWHQDV ª º [ Q N «¬ »¼ N
\N
ZKHUH\N LVDV\QWKHWLF³PHDVXUHPHQW´RIWKHJURXQG FRQWDFWDQJOHVFRP SXWHG DQDO\WLFDOO\ IURP (TV DQG UDZ VHQVRU GDWD LH \ N I ] N ZKHUH ] >D D Q Q @7 ,WLVDVVXPHGWKDWWKH YHKLFOHSLWFK D DQG SLWFK UDWH D DUH GLUHFWO\ VHQVHG DQG VSHHGV Q DQG Q FDQ EH DS SUR[LPDWHGIURP NQRZOHGJHRIWKHZKHHODQJXODUYHORFLWLHVDQGUDGLL 7KHIROORZLQJLVDGHVFULSWLRQRIWKH(.)LPSOHPHQWDWLRQSURFHGXUH ,QLWLDOL]DWLRQRIWKHVWDWHHVWLPDWH [Ö DQGWKHHVWLPDWHGHUURUFRYDULDQFH PDWUL[ 3 +HUH [Ö \ DQG 3 5 [ ZKHUH 5 [ (^Z Z7 ` 3URSDJDWLRQ RI WKH FXUUHQW VWDWH HVWLPDWH DQG FRYDULDQFH PDWUL[ 7KH VWDWH HVWLPDWH LV JHQHUDOO\ FRPSXWHG IURP D VWDWH WUDQVLWLRQ PDWUL[ ZKLFKKHUHLVWKHLGHQWLW\ PDWUL[7KXV [Ö N
[Ö N
7KH DSULRUL FRYDULDQFH PDWUL[LVFRPSXWHG DV 3N
3N 5 [
&RPSXWDWLRQ RI WKH .DOPDQ JDLQ DQG XSGDWLQJ WKH D SULRUL VWDWH HVWL PDWH DQG D SULRUL FRYDULDQFH PDWUL[ 7KH .DOPDQ JDLQ PDWUL[ . LV JLYHQE\ .N
3N 3N 5 \N
7KHVHQVRUQRLVHFRYDULDQFHPDWUL[ 5 \N FDQEHFRPSXWHGDV 5 \N
^
( Q N Q N7
`
§ w\ N ¨¨ © w]
7
§ w\ · ¸¸ 5 ] ¨¨ N © w] ¹
· ¸¸ ¹
ZKHUH 5]LVD[GLDJRQDOPDWUL[ RINQRZQ QRLVHFRYDULDQFHVDVVRFL DWHG ZLWK ] 5 ] GLDJ V D V D V Q V Q 1RWH WKDW DQ HVWLPDWH RI 5 \N
FDQDOVREHIRUPHGE\ FRPSXWLQJWKHXQVFHQWHG WUDQVIRUPRI(TV -XOLHUDQG8KOPDQQ 7KHVWDWHHVWLPDWHLVXSGDWHGDV
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
[Ö N . N \ N [Ö N
[Ö N
DQGWKHFRYDULDQFHPDWUL[LVXSGDWHG DV 3N
, . N 3N
7KHVSHFLDOFDVHVGLVFXVVHGLQ6HFWFDQOHDGWRDODFN RI REVHUYDELO LW\LQWKHILOWHU+RZHYHUDVGHVFULEHGDERYHWKHVHVLWXDWLRQVDUH GHWHFW DEOH 7KXV QHZ PHDVXUHPHQW XSGDWHV IRU WKH ILOWHU DUH QRW WDNHQ ZKHQ WKHVHVSHFLDOFDVHVDUHGHWHFWHG $SLFWRULDO GLDJUDPRIWKH (.)HVWLPDWLRQSURFHVVLV VKRZQLQ)LJ DGDSWHGIURP:HOFKDQG%LVKRS Initial estimates for x^0 and P0
Measurement y k = f (z k )
Compute Kalman gain: Project state forward: k
xˆ = xˆ
k -1
Project covariance forward: Pk- = Pk -1 + R x
(
)
-1
K k = Pk- Pk- + R yk Update state estimate with measurement: xˆ k = xˆ -k + K k y k - xˆ -k Update error covariance: Pk = (I - K k )Pk-
(
)
)LJ 'LDJUDPRI(.)HVWLPDWLRQSURFHVVIURP:HOFKDQG%LVKRS
!5HVXOWV :KHHO7HUUDLQ &RQWDFW $QJOH ,GHQWLILFDWLRQ
!6LPXODWLRQ 5HVXOWV 7KHZKHHOWHUUDLQFRQWDFWDQJOHHVWLPDWLRQDOJRULWKPZDVLPSOHPHQWHGLQ VLPXODWLRQRQDSODQDUYHKLFOHVLPLODUWRWKHRQHLQ)LJZLWKO P 7KHSLWFK DZDVFRUUXSWHGZLWKZKLWHQRLVHRIVWDQGDUGGHYLDWLRQq 7KH UHDUDQGIURQWZKHHOYHORFLWLHV Q!DQG QZHUHFRUUXSWHGZLWKZKLWHQRLVH
5HVXOWV:KHHO7HUUDLQ&RQWDFW$QJOH,GHQWLILFDWLRQ
RI VWDQGDUG GHYLDWLRQ FPVHF 7KLV PRGHOV HUURU GXH WR HIIHFWV RI ZKHHOVOLSDQGWDFKRPHWHUQRLVH )LJVKRZVWKHUHVXOWVRIDUHSUHVHQWDWLYHVLPXODWLRQWULDORQXQGX ODWLQJWHUUDLQ +HUHWKHDFWXDODQGHVWLPDWHGZKHHOWHUUDLQFRQWDFWDQJOHV DUHFRPSDUHG ,WFDQ EH VHHQWKDW DIWHUDQLQLWLDOWUDQVLHQWWKH(.) HVWL PDWHVRIWKHWHUUDLQFRQWDFWDQJOHVDUHTXLWHDFFXUDWHZLWK506HUURUVRI qDQGqIRUWKHIURQWDQGUHDUDQJOHVUHVSHFWLYHO\ (UURULQFUHDVHV DWIODWWHUUDLQUHJLRQVLHZKHUHWKHYDOXHVRIIURQWDQGUHDUFRQWDFWDQJOHV DUH LGHQWLFDO VLQFH WKH DQJOH HVWLPDWLRQ HTXDWLRQV EHFRPH SRRUO\ FRQGL WLRQHGGXHWRUHDVRQVGLVFXVVHGSUHYLRXVO\ +RZHYHUWKHHUURUFRYDULDQFH PDWUL[UHPDLQHGVPDOOGXULQJWKHVLPXODWLRQ ,QJHQHUDOWKH(.)GRHVDQ H[FHOOHQW MRE LQ VLPXODWLRQ RI HVWLPDWLQJ ZKHHOWHUUDLQ FRQWDFW DQJOHV LQ WKHSUHVHQFHRIQRLVH !([SHULPHQWDO 5HVXOWV 7KHZKHHOWHUUDLQFRQWDFWDQJOHHVWLPDWLRQDOJRULWKPZDVLPSOHPHQWHGRQ WKH VL[ZKHHOHG H[SHULPHQWDO URYHU V\VWHP VKRZQ LQ )LJ %XUQ 7KLVURYHU SRVVHVVHVDURFNHUERJLHVXVSHQVLRQVLPLODUWRWKH6R MRXUQHUURYHU%LFNOHU 7KHVL[ZKHHOVDUHGULYHQE\JHDUHG'&PR WRUVZLWKDSHDNWRUTXHRIR]LQDQGPD[LPXPDQJXODUYHORFLW\RI USP 7KH UHVXOWLQJ PD[LPXP YHORFLW\ RI WKH URYHU LV DSSUR[LPDWHO\ FPVHF 7KHURYHULVVWHHUHGZLWKVNLGVWHHULQJ $ P HFKDQLFDOGLIIHUHQWLDO LQ WKH URYHU IUDPH DOORZV WKH ERG\ WR ³VSOLW WKH GLIIHUHQFH´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¶V DUWLFXODWHG VXVSHQVLRQ +RZHYHU LW GRHV PDLQWDLQDIL[HGGLVWDQFHWRWKHERJH\IUHHSLYRWMRLQWDQGLWVFRQWDFWDQJOH FDQEHHVWLPDWHGIURP(TXDWLRQV DQG ZLWKO WDNHQDVWKHGLV WDQFHIURPWKHUHDUZKHHOFHQWHUWRWKHFHQWHURIWKHERJH\IUHHSLYRWMRLQW
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
25 Actual Angle
Front Wheel Contact Angle (deg)
Estimated Angle
20
15
10
5
0
0
5
10 15 20 Travel Distance (m)
25
30
25
30
)URQWZKHHO 25 Actual Angle
Rear Wheel Contact Angle (deg)
Estimated Angle
20
15
10
5
0 0
5
10 15 20 Travel Distance (m)
5HDU:KHHO )LJ6LPXODWHGZKHHOWHUUDLQFRQWDFWDQJOHVDQGHVWLPDWHVIRUIURQWDQGUHDU ZKHHOV
5HVXOWV:KHHO7HUUDLQ&RQWDFW$QJOH,GHQWLILFDWLRQ
)LJ ([SHULPHQWDOURYHU WHVWEHG
34.0 cm Y
Differential joint Z
F
Y
Free pivot joint pc X
Q
14.0 cm 11.4 cm 7.0 cm 12.1 cm
20.0 cm
34.8 cm 9.1 cm
)LJ ([SHULPHQWDO URYHUWHVWEHGNLQHPDWLFGHVFULSWLRQ
0HDVXUHPHQWQRLVHFRYDULDQFHVIRUWKH(.)ZHUHHVWLPDWHGRIIOLQHE\ DQDO\]LQJWKHVWDQGDUGGHYLDWLRQ RIVHQVRUUHDGLQJVIRUWKHZKHHOWDFKRPH
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
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q LQFOLQH DQG WKH IURQW DQG PLGGOH ZKHHOWHUUDLQ FRQWDFW DQJOHV ZHUH HVWL PDWHG 5HVXOWV RIWKLVH[SHULPHQW FDQEHVHHQ LQ )LJXUH 7KH506 HUURU RI WKH IURQW ZKHHO FRQWDFW DQJOH HVWLPDWH LV q ZKLOH WKH PLGGOH ZKHHO 506 HUURU LV q $YHUDJH FRPSXWDWLRQ WLPH IRU ERWK H[SHUL PHQWVZDVPVHFF\FOHZKLFKLVUHDVRQDEOHIRURQERDUGLPSOHPHQWD WLRQ7KXV LWFDQEHFRQFOXGHGWKDWZKHHOWHUUDLQFRQWDFWDQJOHVFDQEHDF FXUDWHO\HVWLPDWHGE\ DURYHUZLWKQRLV\VHQVRUVDQGOLPLWHGFRPSXWDWLRQDO UHVRXUFHV 20 Front Wheel Middle Wheel Rear Wheel
Wheel-Terrain Contact Angle (deg)
15 10 5 0 -5 -10 -15 -2 0 -2 5
0
2
4
6
8
10 12 Time (s)
14
16
18
20
22
)LJ(VWLPDWHGZKHHOWHUUDLQFRQWDFWDQJOHRIIURQWPLGGOHDQG UHDUULJKW VLGHZKHHOV RIH[SHULPHQWDOURYHUWUDYHUVLQJD URFN
5HVXOWV:KHHO7HUUDLQ&RQWDFW$QJOH,GHQWLILFDWLRQ
30 Actual Angle
Front Wheel Contact Angle (deg)
25
Estimated Angle
20 15 10 5 0 -5 -10
0
5
10
15
20
25
20
25
Time (s)
)URQW:KHHO 30
Actual Angle
Middle Wheel Contact Angle (deg)
25
Estimated Angle
20 15 10 5 0 -5 -10
0
5
10
15
Time (s)
0LGGOH:KHHO )LJ (VWLPDWHGDQGDFWXDOZKHHOWHUUDLQ FRQWDFWDQJOHV IRUIURQWDQGPLGGOH ZKHHOVRIH[SHULPHQWDOURYHUWUDYHUVLQJ qLQFOLQH
&KDSWHU
5RXJK 7HUUDLQ0RELOH5RERW 0RGHOLQJDQG(VWLPDWLRQ
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
&KDSWHU 5RXJK 7HUUDLQ 0RWLRQ 3ODQQLQJ
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¶V VROXWLRQ OLHV LQ WKH VSDFH RI WKH URERW¶V SRVLWLRQ DQG KHDGLQJ DQG WKH VHFRQG DOJRULWKP¶V VROXWLRQ OLHV LQ D VSDFH RI WKH URERW¶V VXVSHQVLRQ GHJUHHVRIIUHHGRP %RWK DOJRULWKP DOVR FRQVLGHU LPSRUWDQW WHUUDLQ HI IHFWV
K. Iagnemma and S. Dubovsky: Mobile Robots in Rough Terrain, STAR 12, pp. 51–79, 2004. © Springer-Verlag Berlin Heidelberg 2004
&KDSWHU
5RXJK7HUUDLQ0RWLRQ3ODQQLQJ
7KHURXJKWHUUDLQPRWLRQ SODQQLQJWHFKQLTXHSUHVHQWHGLQ6HFWFRQ VLGHUV WHUUDLQ FKDUDFWHULVWLFV DQG UHOHYDQW UHDOZRUOG LPSOHPHQWDWLRQ LV VXHV6LPXODWLRQUHVXOWVIRUWKLVDOJRULWKPDUHSUHVHQWHGLQ6HFW,WLV VKRZQWKDWWKDWWKHPHWKRGLVDEOHWRHIIHFWLYHO\SODQ VDIHURXWHVWKURXJK URXJK WHUUDLQ $ FRQILJXUDWLRQ SODQQLQJ PHWKRG IRU URERWV ZLWK DFWLYHO\ DUWLFXODWHG VXVSHQVLRQV LV SUHVHQWHG LQ 6HFW 6LPXODWLRQ DQG H[SHUL PHQWDOUHVXOWVDUHVKRZQLQ6HFW,W LV VKRZQ WKDW WKHDOJRULWKP FDQ JUHDWO\LPSURYHURERWVWDELOLW\LQ URXJKWHUUDLQ
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
5RXJK 7HUUDLQ0RWLRQ 3ODQQLQJ
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³FRQYHQWLRQDO´PRWLRQSODQQLQJ PHWKRGDQGLWLVVKRZQWKDWWKH XVH RI SK\VLFDO PRGHOV RI WKH URERW DQG WHUUDLQ DOORZV WKH URXJK WHUUDLQ SODQQHU WR DYRLG KD]DUGV WKDW WKH FRQYHQWLRQDO SODQQLQJ PHWKRGV FDQQRW GHWHFW INPUT: Range Map
Rapid Path Search (Sec. 3.2.1)
Model-Based Evaluation (Sec. 3.2.2)
Modify A* Cost Function
NO
Is path safe? YES END
)LJ +LJKOHYHOIORZFKDUWRIWKHSODQQLQJDOJRULWKP
!6WHS 2QH 5DSLG 3DWK 6HDUFK 7KH!SXUSRVH!RI!WKH!UDSLG!SDWK!VHDUFK!LV!WR!TXLFNO\!ILQG!D!GLUHFW!UHDVRQDEOH SDWK!IURP!WKH!FXUUHQW!URERW!SRVLWLRQ!WR!WKH!JRDO!SRVLWLRQ!,Q!WKH!LQWHUHVW!RI UHGXFLQJ!FRPSXWDWLRQ!WLPH!IRU!RQERDUG!LPSOHPHQWDWLRQ!WKH!UDSLG!VHDUFK
&KDSWHU
5RXJK7HUUDLQ0RWLRQ3ODQQLQJ
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
10
5
0 10 8 6 Z
4
z(x,y) 2
Y X
0
0
2
4
6
)LJ ([DPSOHRI WHUUDLQGDWDLQSXWWRUDSLGSDWKVHDUFK
8
10
5RXJK 7HUUDLQ0RWLRQ 3ODQQLQJ
7HUUDLQURXJKQHVVFDQEHGHILQHGLQ QXPHURXVZD\V+HUHDURXJKQHVV GHILQLWLRQLV SURSRVHGEDVHG RQWKHYDULDQFH RIWKHWHUUDLQHOHYDWLRQRYHUD URERWVL]HGSDWFK&RQVLGHU DURERW FHQWHUHGDWDSRLQW[\ DQGRULHQWHG LQWKHGLUHFWLRQRIPRWLRQ DORQJDSURSRVHGSDWK /HW EHWKHVHWRI WHU UDLQ HOHYDWLRQ SRLQWV LQVLGH WKH FRQYH[ KXOO GHILQHG E\ WKH ZKHHOWHUUDLQ FRQWDFWSRLQWV RIWKHURERWRQ IODWJURXQG DVVKRZQLQ )LJ 7KHWHUUDLQ URXJKQHVV UDW[\ LVGHILQHGDVWKHVTXDUHURRWRIWKHYDULDQFHRIDOOHOH YDWLRQSRLQWVLQ U [ \
YDU]
G
ZKHUHG LVWKHURERWZKHHOGLDPHWHUDQGLVLQFOXGHGWRQRQGLPHQVLRQDOL]H DQGVFDOHWKHPHWULFWRWKHURERWVL]H1RWHWKDWLWLVLPSRUWDQWWR LQFOXGH ³LQWHULRU´WHUUDLQ SRLQWVLHLQVLGHWKHURERWIRRWSULQW LQRUGHUWR SURYLGH DQ HVWLPDWH RI SRWHQWLDO URERW KDQJXS IDLOXUH +DQJXS IDLOXUH RFFXUV ZKHQ WKH URERW ERG\ EHFRPHV ORGJHG DWRS DQ REVWDFOH FDXVLQJ ORVV RI ZKHHOWUDFWLRQDQG URERWHQWUDSPHQW 7HUUDLQDUHDVZLWKODUJHURXJKQHVVUHODWLYHWRWKHURERWZKHHOGLDPHWHU DUHOLNHO\XQWUDYHUVDEOH,WLVGHVLUDEOHWRDVVLJQKLJKFRVWWRWKHVHUHJLRQV 7RDFFRPSOLVKWKLVDPRGLILHGURXJKQHVVPHDVXUHUc LVGHILQHG DV § YDU] · ¸ U c [ \ ¨ ¨ ¸ G © ¹
D
ZKHUHDLVDFRQVWDQWJUHDWHUWKDQRQH7KLVKDVWKHHIIHFWRILQFUHDVLQJO\ SHQDOL]LQJWHUUDLQUHJLRQVWKDWDUHFOHDUO\XQWUDYHUVDEOH ZKLOHPDLQWDLQLQJ DFRQWLQXRXV DVRSSRVHGWRELQDU\ REVWDFOHUHSUHVHQWDWLRQ
)LJ7HUUDLQ URXJKQHVVUHJLRQGHILQLWLRQ
&KDSWHU
5RXJK7HUUDLQ0RWLRQ3ODQQLQJ
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
§ YDU] · ¸ ¨ ¸ ¨ G ¹ ©
D
ZKHUH D LV D FRQVWDQW JUHDWHU WKDQ RQH DQG ƍ LV GHILQHG DV WKH VHW RI SRLQWV LQVLGH D FLUFOH FHQWHUHG DW [\ ZLWK UDGLXV HTXDO WR WKH GLVWDQFH IURPWKHURERWFHQWHUWR WKHPRVWGLVWDQWZKHHOWHUUDLQFRQWDFWSRLQW 7KXV ƍ FDQEH YLHZHGDVDVXSHUVHWRI DXJPHQWHGWRLQFOXGHWKHDUHDVZHSW GXULQJWXUQLQJ7KHZKHHOGLDPHWHULV DJDLQXWLOL]HGIRUVFDOLQJDQGQRQ GLPHQVLRQDOL]LQJWKHPHWULF 7KHFRVW O DVVRFLDWHGZLWKSDWKOHQJWKLV VLPSO\WDNHQDVWKHOHQJWK /RI WKHFDQGLGDWHSDWK GLYLGHGE\ DURERWZKHHOGLDPHWHU O [ \
/ G
7KH SHUIRUPDQFH LQGH[ ) LV IRUPHG DV D ZHLJKWHG VXP RI WKH PHWULFV FRQVLGHULQJ WHUUDLQ XQHYHQQHVV WXUQLQJ DQG SDWK OHQJWK DQG D SHQDOW\ WHUP S ĭ
N U c N W N O S
ZLWKFRQVWDQWV NNDQGN FKRVHQ WRDGMXVWWKHUHODWLYHYDOXHVRIUc WDQG OWR DGGUHVV PLVVLRQVSHFLILFFRQVWUDLQWV7KHSHQDOW\WHUP SLV]HURIRUDOO SRLQWVH[FHSWWKRVHLGHQWLILHGDVKD]DUGRXVE\WKHPRGHOEDVHGHYDOXDWLRQ GHVFULEHG EHORZ)RUWKRVHSRLQWVLWLVDODUJHSRVLWLYHQXPEHU $OHDVWFRVWSDWK 3LVIRXQG IURPWKHFXUUHQWURERWSRVLWLRQWRWKHJRDO SRVLWLRQXVLQJWKH$ DOJRULWKP7KHSDWK3LVFRPSRVHGRIQQHLJKERULQJ [\ SDLUV 3 ^[ \«[Q \Q` DQG FRPELQHV WHUUDLQ VPRRWKQHVV PLQL PXPWXUQLQJDQGVKRUWGLVWDQFH7KHSDWKLVQRWJXDUDQWHHGWREHKD]DUG IUHHGXHWRWKHVLPSOHKHXULVWLFQDWXUHRIWKHSHUIRUPDQFHLQGH[$JDLQ WKHSHUIRUPDQFHLQGH[ ZDVVHOHFWHG GXHWRLWVHDVHRIHYDOXDWLRQDQGLQWXL WLYHUHODWLRQVKLSWRURERWPRELOLW\ ,W LVLQWHQGHGWRUDSLGO\ OHDGWRDUHD VRQDEOHSDWKWKURXJKWKHWHUUDLQ
5RXJK 7HUUDLQ0RWLRQ 3ODQQLQJ
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¶V PZKHHOWHUUDLQFRQWDFWSRLQWVSLL ^«P` DUHQXP EHUHGLQ DVFHQGLQJ RUGHULQ DFORFNZLVHPDQQHUZKHQYLHZHGIURPDERYH DVVKRZQLQ )LJ1RWHWKDWDOOYHFWRUVLQWKLVDQDO\VLVDUH H[SUHVVHGLQ WKHLQHUWLDOIUDPH ^;