the sphere with two colors (black and white). Fixed ... ios. They are illustrated in figure 1 and include (1) robotic wrist, elbow, and shoulder actuators which.
Proceedings of the 2001IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
7KHRU\ 'HVLJQ DQG ,PSOHPHQWDWLRQ RI D 6SKHULFDO (QFRGHU 'DYLG 6WHLQ EUDZOHU#MKXHGX *UHJRU\ 6 &KLULNMLDQ JUHJF#MKXHGX (GZDUG 5 6FKHLQHUPDQ HUV#MKXHGX
'HSDUWPHQW RI 0HFKDQLFDO (QJLQHHULQJ 'HSDUWPHQW RI 0DWKHPDWLFDO 6FLHQFHV 7KH -RKQV +RSNLQV 8QLYHUVLW\ %DOWLPRUH 0' 86$
$EVWUDFW :H GHYHORS D PHWKRGRORJ\ IRU DEVROXWH HQFRGLQJ RI VSKHULFDO PRWLRQ 7KLV LV DFFRPSOLVKHG E\ SDLQWLQJ WKH VXUIDFH RI D PRYLQJ VSKHUH LQ WZR FRORUV DQG VHQVLQJ WKH FRORU DW D nQLWH VHW RI SRLQWV :H VKRZ KRZ SRLQW PHDVXUHPHQWV RQ WKH VSKHUH FDQ UHVROYH DQ DUELWUDU\ URWDWLRQ WR ZLWKLQ D UDQJH GHSHQGLQJ RQ WKH QXPEHU RI VHQVRUV DQG WKH SDLQWLQJ RI WKH VSKHUH 7KHVH WHFK QLTXHV DUH DSSOLHG WR D SURWRW\SH 7KH LPSOHPHQWDWLRQ RI WKLV SURWRW\SH LV GLVFXVVHG
EHFRPH D UHDOLW\ ERWK UREXVW DQG LQH[SHQVLYH VSKHUL FDO PRWRUV DQG HQFRGHUV PXVW EHFRPH DYDLODEOH
,QWURGXFWLRQ
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nJXUH DQG LQFOXGH URERWLF ZULVW HOERZ DQG VKRXOGHU DFWXDWRUV ZKLFK XVH VSKHULFDO PRWRUV FDPHUD DFWXDWRUV IRU FRP SXWHU YLVLRQ >@ RPQLGLUHFWLRQDO ZKHHOV IRU PRELOH URERWV DFWXDWRU DUUD\V FDSDEOH RI WUDQVSRUWLQJ RE MHFWV LQ DQ\ GLUHFWLRQ > @ )RU WKHVH DSSOLFDWLRQV WR 0-7803-6475-9/01/$10.00© 2001 IEEE
)LJXUH $SSOLFDWLRQV RI 6SKHULFDO $FWXDWLRQ
3UHYLRXV :RUN
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
1773
'HVSLWH WKH UDWKHU ODUJH OLWHUDWXUH RQ VSKHULFDO PRWRUV VHH HJ> @ WKH VWXG\ RI RSWLFDO HQFRGHUV IRU VSKHUL FDO PRWLRQ DSSHDUV QRW WR KDYH EHHQ DSSURDFKHG IURP D PDWKHPDWLFDO RU DOJRULWKPLF SHUVSHFWLYH ,Q WKLV SD SHU ZH nOO WKLV JDS E\ SURYLGLQJ D JHQHUDO WHFKQLTXH IRU VSKHULFDO HQFRGLQJ 2XU V\VWHP LV DQDORJRXV WR WKH RSWLFDO HQFRGHU IRU D[LDO URWDWLRQV VKRZQ LQ nJXUH 7KH PDLQ GLmHUHQFH LV WKDW LQ VSKHULFDO PRWLRQ ZH FDQQRW WDNH DGYDQWDJH RI D GLPHQVLRQ SHUSHQGLFXODU WR PRWLRQ 7KDW LV WKH SDLQWLQJ DQG VHQVRU PHDVXUH PHQWV PXVW EH SHUIRUPHG RQ WKH VXUIDFH RI WKH VSKHUH 7KLV LV DQDORJRXV WR RQO\ FLUFXPIHUHQWLDO SDLQWLQJ DQG VHQVRU PHDVXUHPHQWV LQ WKH D[LDO FDVH
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nJXUH DOO Q ELW SDWWHUQV DSSHDU H[DFWO\ RQFH DQG ZH FDQ UHVROYH Q GLmHUHQW URWDWLRQV RI WKH GLVN 6HH > @ 1RZ ZH ZLOO SUHVHQW D WHFKQLTXH WR XVH RSWLFDO VHQVRUV WR HVWDEOLVK WKH RUL HQWDWLRQ RI WKH VSKHUH
6SKHUH 3DLQWLQJ
)LJXUH 7KH DQJXODU RULHQWDWLRQ RI D GLVN FDQ EH GH WHUPLQHG XVLQJ RSWLFDO VHQVRUV $ERYH ZH VHH D VWDQGDUG HQFRGLQJ LQ ZKLFK WKH IRXU VHQVRUV DUH SRVLWLRQHG UDGLDOO\ %HORZ ZH VHH DQ HQFRGHU EDVHG RQ D GH %UXLMQ VHTXHQFH WKH FRORUV DQG VHQVRUV DUH SODFHG RQ WKH HGJH RI WKH GLVN
7KH JHQHUDO FRQFHSW EHKLQG VSKHUH SDLQWLQJ LV WKDW WKH ELFRORU SDWWHUQ ZH DSSO\ WR WKH VSKHUH PXVW EH XQLTXH IURP HYHU\ YLHZSRLQW 8QIRUWXQDWHO\ D VLPSOH UHJX ODU SDLQWLQJ RI WKH VSKHUH LV XQDFFHSWDEOH VXFK SDLQW LQJV RIWHQ KDYH V\PPHWULHV DQG FRQVHTXHQWO\ GLmHU HQW RULHQWDWLRQV RI WKH VSKHUH SURGXFH LGHQWLFDO VHQVRU UHDGLQJV 2Q WKH RWKHU KDQG D KLJKO\ LUUHJXODU FRO RULQJ PLJKW EH GLpFXOW WR GHVFULEH DQG EH LQHpFLHQW WR KDQGOH FRPSXWDWLRQDOO\ ,Q RXU SUHYLRXV ZRUN LQ WKLV VXEMHFW >@ ZH GHVFULEH D PHWKRG FDOOHG UDQGRP 9RURQRL FRORULQJ 7KLV WHFKQLTXH SURGXFHV D FRORULQJ RI WKH VSKHUH WKDW LV UDQGRP EXW LV VWLOO FRPSXWDWLRQ DOO\ HpFLHQW 7KH SDLQWLQJ XVHG LQ RXU SURWRW\SH LV GLVFXVVHG LQ VHFWLRQ
7KHRU\ RI DQ (QFRGHU XVLQJ 'LVFUHWH 6HQVRUV
2XU PHWKRG RI GHWHUPLQLQJ WKH RULHQWDWLRQ RI D VSKHUH LV LQVSLUHG E\ WKH IROORZLQJ WHFKQLTXH IRU GHWHUPLQLQJ WKH RULHQWDWLRQ RI D GLVN WKDW LV IUHH WR URWDWH DERXW D VLQJOH D[LV :H SDLQW WKH GLVN DQG ORFDWH RSWLFDO VHQ VRUV WR GHWHUPLQH WKH DQJOH RI URWDWLRQ )RU H[DPSOH WKH WRS RI nJXUH VKRZV D VWDQGDUG RSWLFDO HQFRGLQJ +RZHYHU IRU WKLV HQFRGHU WKH VHQVRUV PXVW EH DEOH WR VHH WKH LQWHULRU RI WKH GLVN $Q DOWHUQDWLYH PHWKRG VKRZQ RQ WKH ERWWRP RI nJXUH LV EDVHG RQ GH %UXLMQ VHTXHQFH ,W DOORZV XV WR GHWHUPLQH WKH DQJOH RI WKH
6HQVRU 2SHUDWLRQ
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nHG E\ D URWDWLRQ PDWUL[ $ 62
1774
,I WKH VSKHUH KDV EHHQ URWDWHG IURP WKH KRPH SRVL WLRQ YLD D URWDWLRQ $ WKHQ WKH FRORU GHWHFWHG E\ VHQVRU L LV F$7 VL :H GHnQH WKH FRORU YHFWRU RI D URWDWLRQ $ WR EH WKH IROORZLQJ YHFWRU ZKRVH HQWULHV DUH h F$
d F$7 V F$7 V
ccc
F$7 VQ
e7
2I FRXUVH WKH FRORU YHFWRU GHSHQGV QRW RQO\ RQ WKH URWDWLRQ $ EXW DOVR RQ WKH SDLQWLQJ RI WKH VSKHUH F DQG WKH SODFHPHQW RI WKH VHQVRUV IVL J +RZHYHU RQO\ WKH RULHQWDWLRQ $ FKDQJHV WKH SDLQWLQJ DQG VHQVRU SODFHPHQW DUH n[HG 7KH SUREOHP ZH ZDQW WR VROYH LV *LYHQ D FRORU YHFWRU F$ GHWHUPLQH $ 7KH SUREOHP GRHV QRW KDYH D XQLTXH VROXWLRQ %HFDXVH WKHUH DUH RQO\ Q VHQ VRUV WKHUH DUH DW PRVW Q SRVVLEOH FRORU YHFWRUV EXW WKHUH DUH LQnQLWHO\ PDQ\ GLmHUHQW URWDWLRQ PDWULFHV $ 7KHUHIRUH WKH SUREOHP ZH DFWXDOO\ VROYH LV JLYHQ D FRORU YHFWRU F$ nQG D URWDWLRQ PDWUL[ % VR WKDW F$ F% DQG % LV FORVH WR $ :H FDQ PHDVXUH WKH GLVFUHSDQF\ EHWZHHQ $ DQG % DV DQ DQJOH 6LQFH $% 7 62 ZH FDQ FRQVLGHU $% 7 DV D URWDWLRQ DERXW DQ D[LV WKH DQJOH RI WKDW URWDWLRQ LV RXU PHD VXUH RI FORVHQHVV 7KXV r s WKH ?GLVWDQFH EHWZHHQ $ DQG 7 b WU$% b % LV FRV +RZ PDQ\ VHQVRUV VKRXOG ZH XVH DQG ZKHUH VKRXOG WKH\ EH SODFHG" 2XU H[SHULPHQWV LQGLFDWH WKDW WKH QXPEHU RI VHQVRUV LV PRUH LPSRUWDQW WKDQ WKHLU SODFH PHQW $V ORQJ DV WKH VHQVRUV DUH UHDVRQDEO\ GLV WULEXWHG DURXQG WKH VXUIDFH RI WKH VSKHUH ZH DFKLHYH QHDUO\ LGHQWLFDO SHUIRUPDQFH RI RXU GHFRGLQJ PHWKRG 7KH PRUH VHQVRUV ZH XVH WKH PRUH DFFXUDWHO\ FDQ ZH GHWHUPLQH $ IURP F$ +RZHYHU WKH FRPSXWDWLRQDO EXUGHQ ULVHV DFFRUGLQJO\ :LWK VHQVRUV ZH XVXDOO\ nQG $ WR ZLWKLQ RU GHJUHHV :LWK VHQVRUV ZH W\SLFDOO\ nQG $ WR ZLWKLQ KDOI D GHJUHH
/RFDO VROXWLRQ
2XU SUREOHP LV *LYHQ F$ nQG D % 62 VR WKDW F% F$ 7R WKLV HQG ZH GHnQH D IXQFWLRQ I 62 5 E\ I %
S NF% b F$ N Q
S 7KH IDFWRU Q LV QRW VWULFWO\ QHFHVVDU\ EXW VLPSO\ UHVFDOHV WKH LPDJH RI WKH IXQFWLRQ VR WKDW LW DOZD\V OLHV LQ > @ UHJDUGOHVV RI WKH VL]H RI Q VLQFH WKH HQWULHV RI F$ DQG S F% DUH h WKH YDOXH RI NF% b F$ N LV DW PRVW Q 7KH SUREOHP FDQ WKHQ EH UHVWDWHG nQG % 62 VR WKDW I% (TXLYDOHQWO\ VLQFH I LV QRQQHJDWLYH
ZH FDQ WKLQN RI WKLV SUREOHP DV VHDUFKLQJ IRU D JOREDO PLQLPL]HU RI I ,Q WKLV VHFWLRQ ZH SUHVHQW DQ LWHUDWLYH PHWKRG WKDW OHDGV WR D VROXWLRQ SURYLGHG ZH DUH JLYHQ DQ LQLWLDO % WKDW LV FORVH VD\ ZLWKLQ GHJUHHV WR WKH GHVLUHG VR OXWLRQ 7KLV DVVXPSWLRQ LV MXVWLnDEOH IRU WZR UHDVRQV )LUVW DV WKH VSKHUH WXUQV LQ LWV KRXVLQJ WKH VHQVRUV FRQWLQXRXVO\ WUDFN LWV SURJUHVV :H FDQ XVH WKH SUHYL RXV RULHQWDWLRQ RI WKH VSKHUH DV DQ LQLWLDO JXHVV IRU WKH FXUUHQW RULHQWDWLRQ 6HFRQG WKH PHWKRG GHVFULEHG LQ WKH QH[W VHFWLRQ FDQ SURYLGH XV ZLWK DQ LQLWLDO JXHVV % UHDVRQDEO\ FORVH WR WKH VROXWLRQ YDOXH :H EHJLQ E\ H[DPLQLQJ D SORW RI I% ZKHUH % LV UHVWULFWHG WR D VLQJOH D[LV RI URWDWLRQ VXFK D SORW LV SUHVHQWHG LQ nJXUH
)LJXUH 3ORW RI I % DV % YDULHV WKURXJK D VLQJOH D[LV RI URWDWLRQ 2Q WKH KRUL]RQWDO D[LV p FRUUHVSRQGV WR WKH KRPH RULHQWDWLRQ )RU WKLV nJXUH ZH XVHG D VSKHUH ZLWK EODFN DQG ZKLWH UHJLRQV DQG VHQVRUV
1RWLFH WKDW WKH JUDSK RI I LV SLHFHZLVH oDW 7KLV KDSSHQV EHFDXVH DV WKH VSKHUH URWDWHV VHQVRUV FURVV WKH ERXQGDULHV EHWZHHQ WKH FRORUHG UHJLRQV DQG VXG GHQO\ FKDQJH VWDWH 7KLV FDXVHV D VWHSMXPS LQ WKH YDOXH RI I $OVR QRWLFH WKDW ZLWKLQ p RI WKH DFWXDO SRVLWLRQ WKH JUDSK RI I GHFUHDVHV VWHSZLVH WR WKH PLQLPXP YDOXH 7KXV D VHQVLEOH DSSURDFK WR PLQLPL]LQJ I% ZRXOG EH WR IROORZ D VWHHSHVWGHVFHQW WUDMHFWRU\ +RZ HYHU WR PDNH WKLV SUHFLVH ZH QHHG WR EH FOHDU RQ ZKDW ZH PHDQ E\ WKH JUDGLHQW RI I DQG WKHQ GHDO ZLWK WKH IDFW WKDW ZKHUHYHU WKH JUDGLHQW LV GHnQHG LWV YDOXH LV ]HUR %HFDXVH WKH GRPDLQ RI I LV 62 ZH XVH FRQFHSWV IURP /LH JURXSVDOJHEUDV > @ WR VSHDN FDUH IXOO\ RI WKH JUDGLHQW RI I VHH >@ /HW ; EH D d
1775
VNHZV\PPHWULF PDWUL[ :H VHW b W; c nn GI %H n ; 5 I % n n GW
W 5 :H FDQ WKLQN RI ; DV D ULJKW GLUHFWLRQDO GHULYDWLYH
RI I LQ WKH GLUHFWLRQ ; 'HnQH WKH ULJKW JUDGLHQW RI I DW % WR EH WKH YHFWRU e d 5 (5 I (5 I ( I ZKHUH (
(
(
b b b
*OREDO VROXWLRQ
,Q WKH SUHYLRXV VHFWLRQ ZH SUHVHQWHG D WHFKQLTXH WR nQG D % 62 WKDW VROYHV WKH HTXDWLRQ F% F$ SURYLGHG ZH DUH JLYHQ D VWDUWLQJ % VXpFLHQWO\ QHDU $ ZLWKLQ p +RZHYHU D VWHHSHVWGHVFHQW PHWKRG IDLOV LI ZH VWDUW WRR IDU IURP WKH VROXWLRQ EH FDXVH RI WKH SUHVHQFH RI PDQ\ ORFDO PLQLPD DQG WKH KLJKO\ QRQVPRRWK QDWXUH RI WKH IXQFWLRQ )LJXUH LO OXVWUDWHV I $ DV $ YDULHV DORQJ WZR RUWKRJRQDO D[HV
7KH JUDGLHQW SRLQWV LQ WKH GLUHFWLRQ RI VWHHSHVW DV FHQW VR E\ IROORZLQJ D SDWK LQ WKH RSSRVLWH GLUHFWLRQ ZH DUH OHG WR D ORFDO PLQLPXP RI WKH IXQFWLRQ +RZ HYHU IRU WKH IXQFWLRQ ZH DUH FRQVLGHULQJ WKH JUDGLHQW RI I DW % LV IRU DOPRVW DOO % 62 DQG LV XQGH nQHG RWKHUZLVH 2XU VROXWLRQ LV WR XVH DSSUR[LPDWH JUDGLHQWV :H UHSODFH WKH ULJKW GHULYDWLYHV (M5 I ZLWK M ZLWK nQLWH GLmHUHQFH DSSUR[LPDWLRQV I %HW(M b I %HbW(M W :H EHJLQ ZLWK D VSHFLnF VWHS VL]H IRU W HJ W ,I WKH DSSUR[LPDWH JUDGLHQW ZH FDOFXODWH LV ZH LQFUHDVH W E\ D FRQVWDQW IDFWRU :H GR WKLV XQWLO WKH DSSUR[LPDWH JUDGLHQW LV VRPH QRQ]HUR YHFWRU [ :H WKHQ XSGDWH % E\ D VWHS RI VL]H W LQ WKH QHJDWLYH JUDGLHQW GLUHFWLRQ LH [ [ [ % % H[S bW b[ b[ b[
ZKHUH [M LV WKH nQLWH GLmHUHQFH DSSUR[LPDWLRQ WR (M5 I % :H WKHQ LWHUDWH WKLV SURFHGXUH $W HDFK VXFFHVVLYH VWHS ZH GHFUHDVH W E\ D FRQVWDQW IDFWRU VR ZH GR QRW RYHUUXQ WKH PLQLPXP ,GHDOO\ ZH FRQWLQXH WKLV SURFHGXUH XQWLO ZH UHDFK D % DW ZKLFK I % 2I FRXUVH ZH FRXOG JHW WUDSSHG DW D ORFDO PLQLPXP ,Q WKLV FDVH LI I % LV IDLUO\ VPDOO VD\ I% ZH DUH DFWXDOO\ TXLWH FORVH WR WKH PLQLPXP DQG PD\ VWRS WKHUH
)LJXUH 3ORW RI I $ DV $ YDULHV DORQJ WZR RUWKRJRQDO D[HV
2XU VWUDWHJ\ LV VLPSOH 6LQFH 62 LV FRPSDFW DQG VLQFH WKH GHVFHQW PHWKRG MXVW QHHGV D VWDUWLQJ YDOXH WKDW LV UHDVRQDEO\ FORVH WR WKH FRUUHFW YDOXH ZH FDQ HYDOXDWH I% RYHU D GLVFUHWH VXEVHW RI 62 7KDW LV ZH FDQ FKRRVH D WKUHVKROG VD\ p DQG nQG D nQLWH VXEVHW = | 62 VR WKDW IRU DOO $ 62 WKHUH LV D % = VR WKDW WKH URWDWLRQDO GLmHUHQFH EHWZHHQ $ DQG % LV EHORZ WKH WKUHVKROG 2XU PHWKRG LV HTXDOO\ VLPSOH :H SUHFRPSXWH WKH VHW = | 62 :H UHFRUG LQ D WDEOH WKH YHFWRU F% IRU DOO % = 6XSSRVH = I% % %N J :H FUHDWH D N d Q PDWUL[ 0 ZKRVH L MHQWU\ LV F%L7 VM LH WKLV HQWU\ LV WKH FRORU REVHUYHG E\ WKH M WK VHQVRU LI WKH VSKHUH LV URWDWHG E\ %L IURP LWV KRPH SRVLWLRQ /HW $ 62 EH DUELWUDU\ 7KHQ ZH FKRRVH %L LQ 62 VR WKDW F$ c F%L LV DV ODUJH DV SRVVLEOH EHVW SRVVLEOH PDWFK 7KLV FDQ EH GRQH E\ D VLQJOH PDWUL[YHFWRU PXOWLSO\ 0 F$ DQG nQGLQJ WKH FRRU GLQDWH ZLWK ODUJHVW LQGH[ 7KLV FDOFXODWLRQ LV UHDVRQDEO\ IDVW EXW WKH PDWUL[ 0 LV TXLWH ODUJH IRU VHQVRUV DQG VDYHG RUL HQWDWLRQV ZH QHHG RQH PLOOLRQ HQWULHV 2QH LGHD WKDW VDYHV VRPH PHPRU\ DQG VSHHGV XS WKH FDOFXODWLRQ LV WR UHSODFH 0 E\ D UHGXFHGUDQN DSSUR[LPDWLRQ
1776
/HW 0 8 h9 7 EH 0 V VLQJXODU YDOXH GHFRPSRVL WLRQ >@ +HUH 8 LV D N d N XQLWDU\ PDWUL[ 9 LV DQ Q d Q XQLWDU\ PDWUL[ DQG h LV D N d Q GLDJRQDO PDWUL[ ZKRVH GLDJRQDO HQWULHV DUH UHDO QRQQHJDWLYH DQG LQ GHFUHDVLQJ RUGHU /HW U EH D PRGHVW SRVLWLYH LQWHJHU HJ U DQG OHW A EH WKH U d U XSSHU OHIW FRUQHU RI h q h A EH WKH N d U PDWUL[ IRUPHG E\ FKRRVLQJ MXVW q 8 WKH nUVW U FROXPQV RI 8 DQG q 9A EH WKH Q d U PDWUL[ IRUPHG E\ FKRRVLQJ MXVW WKH nUVW U FROXPQV RI 9 Ah A 9A 7 7KLV DSSUR[LPDWH GHFRPSRVL 7KHQ 0 { 8 WLRQ FDQ EH FRPSXWHG ZLWKRXW nQGLQJ WKH IXOO VLQJX ODU YDOXH GHFRPSRVLWLRQ ,W LV D RQHWLPH FRPSXWDWLRQ WKDW LV UHDVRQDEO\ IDVW 1RWLFH WKDW WKHVH WKUHH PDWULFHV FRQVXPH D WRWDO RI Q DQG UN U UQ VWRUDJH )RU N U WKLV LV DERXW ZKLFK LV VLJQLnFDQWO\ OHVV WKDQ RQH PLOOLRQ :H FDQ WKHQ DSSUR[LPDWH 0F$ Ah A 9A 7 F$ DQG VHOHFW WKH ODUJHVW FRPSRQHQWV WR E\ 8 JLYH UHDVRQDEOH VWDUWLQJ YDOXHV WR WKH GHVFHQW PHWKRG 2XU H[SHULHQFH KDV EHHQ WKDW ZH QHHG WR WU\ D IHZ RI WKH ODUJHVW VWDUWLQJ YDOXHV RI WKH DSSUR[LPDWLRQ WR 0 F$ WR JHW D JRRG VWDUWLQJ YDOXH
)LJXUH 3DLQWHG 6SKHUH VSOLW LQ KDOI
7KH VSKHUH XVHG LQ WKH SURWRW\SH LQ D $%6 SODVWLF VSKHUH WKDW VSOLWV LQ KDOI )XOO VL]H GUDZLQJV ZHUH PDGH VR FRQVWUXFWLRQ OLQHV FRXOG EH WUDQVIHUUHG WR HDFK KHPLVSKHUH 7KH DUHDV WKDW UHPDLQHG ZKLWH ZHUH PDVNHG Rm DQG WKH UHPDLQLQJ DUHDV ZHUH SDLQWHG ZLWK oDW EODFN HSR[\ HQDPHO :H H[SHULPHQWHG ZLWK PDQ\ GLmHUHQW EUDQGV RI SDLQW XQWLO ZH IRXQG RQH WKDW LV GXUDEOH ZLOO FRYHU LQ RQH FRDW DQG LV FRPSDWLEOH ZLWK WKH VHQVRUV ZH FKRVH )LJXUHV DQG VKRZ RXU SURWRW\SH VSKHUH
&RQVWUXFWLRQ RI D 6SKHULFDO (QFRGHU
7KLV VHFWLRQ GHVFULEHV WKH SURWRW\SH VSKHULFDO HQFRGHU WKDW ZH FRQVWUXFWHG :H RXWOLQH WKH FRQVWUXFWLRQ DQG WDON DERXW WKH LPSOHPHQWDWLRQ RI WKH RULHQWDWLRQ DO JRULWKP
3URWRW\SH 6SKHUH 3DLQWLQJ
)LJXUH 3DLQWHG 6SKHUH LQ +RXVLQJ
$V ZH VWDWH SUHYLRXVO\ WKH VSKHUH SDLQWLQJ KDV WR EH LUUHJXODU DQG FRQWDLQ QR V\PPHWULHV 7KLV JXDUDQWHHV WKDW WKH SLFWXUH WKH VHQVRUV VHH RQ WKH VXUIDFH RI WKH VSKHUH LV XQLTXH IRU DQ\ RULHQWDWLRQ RI WKH VSKHUH $ UDQGRP 9RURQRL FRORULQJ LV D YHU\ JRRG FKRLFH EXW LW LV GLpFXOW WR SDLQW E\ KDQG RQ D VSKHUH )RU RXU SURWRW\SH ZH FKRVH D VSKHUH SDLQWLQJ WKDW LV VLPSOH EXW VWLOO JLYHV WKH VHQVRUV D XQLTXH SLFWXUH IRU HDFK RUL HQWDWLRQ 7KH GRZQVLGH WR XVLQJ WKLV VLPSOH SDLQWLQJ LV WKDW LW UHGXFHV WKH HpFLHQF\ DQG DFFXUDF\ RI RXU WHFKQLTXH
3URWRW\SH 6HQVRU $UUD\
7KH VHQVRUV XVHG WR GHWHFW WKH FRORU RI WKH VSKHUH DUH 47 2SWRHOHFWURQLFV PRGHO 45' 7KH 45' FRQVLVWV RI DQ LQIUDUHG HPPLWLQJ GLRGH DQG D SKRWRUH VLVWRU PRXQWHG VLGH E\ VLGH LQ D FRPSDFW SDFNDJH 7KH GHWHFWRU OHQVH LV XQIRFXVHG ZLWK D GD\OLJKW nOWHU VR LW FDQ EH XVHG RQ FXUYHG RU LUUHJXODU VXUIDFHV 7KH HQFRGHU SURWRW\SH ZDV EXLOW WR EH SDUW RI D VSKHULFDO VWHSSHU PRWRU > "@ WKDW ZH DUH SUHVHQWO\ ZRUNLQJ RQ 7KH PRVW ORJLFDO SODFH IRU WKH VHQVRUV WR EH ORFDWHG LV LQ WKH VWDWRU KHPLVSKHUH RI WKH VSKHU
1777
LFDO PRWRU 7KH GLpFXOW\ LQ ORFDWLQJ WKH VHQVRUV LQ WKH VWDWRU LV WKDW PXFK RI WKH VXUIDFH RI WKH VWDWRU KHPLVSKHUH LV FRYHUHG E\ VWDWRU HOHFWURPDJQHWV :H DFWXDOO\ XVH WKH VWDWRU HOHFWURPDJQHWV WR RXU DGYDQ WDJH E\ XVLQJ WKHP DV D PRXQWLQJ SRLQWV IRU VHQVRU DUUD\
)LJXUH ,5 6HQVRU 0RGXOH
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nJXUH 6LQFH WKH VHQVRUV XVH GLJLWDO ORJLF WKH\ PDNH LW UHODWLYHO\ HDV\ WR SROO WKH VWDWH RI WKH VHQVRU DUUD\ E\ XVLQJ D GLJLWDO ,2 ERDUG :H XVH WKH $UERU 7HFKQRORJ\ 3&, ,2 ERDUG ZKLFK KDV ,2 OLQHV
,PSOHPHQWDWLRQ RI WKH 2ULHQ WDWLRQ $OJRULWKP
8VLQJ WKH DOJRULWKPV SUHVHQWHG LQ SUHYLRXV VHFWLRQ ZH EXLOW D PRGHO RI WKH HQFRGHU LQ 0DWODE 7KLV DOORZHG XV WR WHVW RXU VSKHUH SDLQWLQJ DQG VHQVRU SODFHPHQW 0DWODE KDV D JUHDW IURQW HQG DQG EXLOW LQ IXQFWLRQV ZKLFK PDNH LW YHU\ HDV\ IRU XV WR YLVXDOL]H SDLQWLQJV DQG PDNH TXLFN FKDQJHV WR WKH PRGHO SDUDPHWHUV 0DWODE FDQ EH PDGH WR H[HFXWH GRV SURJUDPV VR LW ZDV UHODWLYHO\ HDV\ WR LQWHJUDWH RXU DFWXDO SURWRW\SH VHQVRU LQSXWV LQWR WKH 0DWODE PRGHO 2XU H[SHULPHQWV RQ WKH SURWRW\SH VKRZ WKDW WKH SURWRW\SHV UHVROXWLRQ PDWFKHV ZKDW WKH WKHRU\ SUH GLFWHG 7KH nJXUH RQ WKH OHIW LQ nJXUH VKRZV D WRS YLHZ RI WKH SDLQWHG VSKHUH LQ LWV KRXVLQJ ZKLOH WKH ULJKW VLGH VKRZV WKH RULHQWDWLRQ RI WKH VSKHUH WKH PRGHO FDOFXODWHG IURP DFWXDO VHQVRU GDWD :H SHU IRUPHG UHSHDWHG WHVWV ZLWK WKH VSKHUH LQ UDQGRP RUL HQWDWLRQV DQG WKH PRGHO SUHGLFWHG WKH DFWXDO VSKHUH RULHQWDWLRQ ZLWKLQ p :LWK RQO\ VHQVRUV DQG VXE RSWLPDO VSKHUH SDLQWLQJ ZH IHHO WKDW p RI HUURU LV H[ WUHPHO\ JRRG IRU HQFRGLQJ VSKHULFDO PRWLRQV
)LJXUH $FFXUDF\ RI WKH HQFRGHU 7KH nJXUH RQ WKH OHIW LV D WRS YLHZ RI WKH DFWXDO VSKHUH ZKLOH WKH nJXUH RQ WKH ULJKW LV WKH RULHQWDWLRQ RI WKH VSKHUH FDOFXODWHG IURP DFWXDO VHQVRU GDWD
)LJXUH &RPSOHWH 6WDWRU ZLWK 6HQVRU 0RGXOHV
:H DOVR SHUIRUPHG H[SHULPHQWV WR WHVW WKH VSHHG RI RXU DOJRULWKPV :H FKRVH UDQGRP RULHQWDWLRQV DQG JDYH WKH PRGHO JXHVVHV ZLWK HUURUV UDQJLQJ IURP p WR p 7KH )/236 UHTXLUHG WR nQG WKH DFWXDO RULHQ WDWLRQ ZDV UHFRUGHG IRU HDFK JXHVV DQG ZDV UHSHDWHG IRU WZHQW\ UDQGRP RULHQWDWLRQV $ VFDWWHU SORW RI WKH UHVXOWV LV VKRZQ LQ nJXUH 7KH DYHUDJH WLPH LW WRRN WKH 0DWODE PRGHO WR FRQYHUJH RQ WKH DFWXDO RULHQWD WLRQ LV DSSUR[LPDWHO\ VHFRQGV RQ D 3HQWLXP 7KLV PLJKW VHHP OLNH D ORQJ WLPH EXW LW UHoHFWV WKH GRZQVLGH RI 0DWODE 7KH VDPH DOJRULWKP FRGHG LQ & ZRXOG EH VLJQLnFDQWO\ IDVWHU DSSUR[LPDWHO\ WLPHV
1778
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pFH
)LJXUH )/236 UHTXLUHG IRU WKH 0DWODE 0RGHO
&RQFOXVLRQ
:H KDYH SUHVHQWHG DQ HQFRGHU IRU VSKHULFDO PRWLRQ XVLQJ GLVFUHWH RSWLFDO VHQVRUV 7KH SURWRW\SH ZH FRQVWUXFWHG ZDV SUHVHQWHG DQG LWV SHUIRUPDQFH ZDV GHPRQVWUDWHG 2XU H[SHULPHQWV VKRZ WKDW RXU HQ FRGHU LV QRW MXVW D QRYHO LGHD EXW ZLWK VRPH n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
5HIHUHQFHV >@ &KLULNMLDQ *6 6WHLQ ' ?.LQHPDWLF 'HVLJQ DQG &RPPXWDWLRQ RI D 6SKHULFDO 6WHSSHU 0R WRU $60(,((( 7UDQV RQ 0HFKDWURQLFV 9RO 1XPEHU SS 'HF >@ /HH .0 .ZDQ &. 'HVLJQ &RQFHSW 'HYHO RSPHQW RI D 6SKHULFDO 6WHSSHU IRU 5RERWLF $SSOL FDWLRQV ,((( 7UDQV RQ 5RERWLFV DQG $XWRPDWLRQ 9RO 1R )HE SS
>@ .0 /HH 2ULHQWDWLRQ VHQVLQJ V\VWHP DQG PHWKRG IRU D VSKHULFDO ERG\ 86 3DWHQW 2pFH >@ %RQG\ -$ DQG 0XUW\ 865 *UDSK 7KHRU\ ZLWK $SSOLFDWLRQV 1RUWK+ROODQG >@ :HVW '% ,QWURGXFWLRQ WR *UDSK 7KHRU\ 3UHQWLFH +DOO >@ 6FKHLQHUPDQ (5 &KLULNMLDQ *6 6WHLQ ' ?(QFRGHUV IRU VSKHULFDO PRWLRQ XVLQJ GLVFUHWH VHQ VRUV :RUNVKRS RQ WKH $OJRULWKPLF )RXQGDWLRQV RI 5RERWLFV +DQRYHU 1+ 0DUFK >@ &KLULNMLDQ *6 .\DWNLQ $% (QJLQHHULQJ $S SOLFDWLRQV RI 1RQFRPPXWDWLYH +DUPRQLF $QDO\VLV &5& 3UHVV >@ *HO IDQG ,0 0LQORV 5$ 6KDSLUR =@ +HOJDVRQ 6 'LmHUHQWLDO JHRPHWU\ /LH JURXSV DQG V\PPHWULF VSDFHV $FDGHPLF 3UHVV >@ 0LOOHU -U : /LH WKHRU\ DQG VSHFLDO IXQFWLRQV $FDGHPLF 3UHVV >@ 7D\ORU &- .ULHJPDQ '- 0LQLPL]DWLRQV RQ WKH /LH JURXS 62 DQG UHODWHG PDQLIROGV 7HFKQL FDO 5HSRUW @ +RUQ 5$ -RKQVRQ &5 0DWUL[ $QDO\VLV &DPEULGJH 8QLYHUVLW\ 3UHVV
1779