Probabilistic Analysis of a Robot System with Redundant Safety Units and Common-Cause Failures
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1 Illg Io Mg do:.436/.9.3 ublhd Ol Dcb 9 hp:// obblc Aly of Robo Sy wh Rdud Sfy U d Coo-Cu Flu B. S. DHION Zh I Dp of Mchcl Egg Uvy of Ow Oo Cd El: dhllo@g.uow.c Abc: Th pp p lbly d vlbly ly of odl pg y hvg o obo d -dud fy u wh coo-cu flu. A l fy u u fuco uccfully h obo y ucc. Th obo d oh flu d h plly fld y p ud co d h fld obo-fy y p ud bly dbud. Mov d upply vbl hod w ud o p hcl ly of h odl. Glzd po pobbl y vlbl lbly o flu d vc of o flu dvlopd. lo of o ulg po how. Kywod: obo fy vlbly lbly coo-cu flu flu p dudcy. Ioduco Robo copl d ophcd ch. pc dc h obo c cou ouc of g dg o hu. Fo pl ov h y ub of ou ccd d oh fy-ld pobl volvg obo hv occud [. Th dc h fy u p coc h dg llo opo d c of obo. Ndl o y obo o oly h o b lbl bu lo f. Thu h fy u po l of h obo y. Mo pcfclly obo y d up of obo d ocd fy u. Th ffcv obo lbly ly h couplg bw lbly d fy u b udd d h occuc of coo-cu flu codd. A coo-cu flu y b dfd y c ulpl u o l fl du o gl cu [. Th cocp of dudcy wdly ud o c h fy d lbly of y. I c lo b ppld o h obo y pcul o fy u. Thu h pp p lbly d vlbly ly of obo y hvg o obo d -dud fy u ubc o coo-cu flu. A l fy u u fuco olly h uccful opo of h obo y. Th bloc dg of h obo-fy y how Fgu d copodg pc dg gv Fgu. Th ul d l d h bo d llp of Fgu do y. A = h obo d ll fy u opg. Th obo-fy y c fl h du o h flu of h obo lf h lfuco of h -+ h fy u o h occuc of coo-cu flu. Nohl h obo-fy y wll fuco uccfully ul l fy u d h obo opg olly. Th y go hough -+ dc opg. A coo-cu flu c occu oly f l fy u d h obo fucog uccfully. Th obo-fy y h ol of -+4 dc. I h y of ul pg y y b dcouou. Fo pl -ou-of-4 fy u h y of ul pg y Mo pcfclly h y of ul ul 3 d 4 g. Th dgdd o fully fld obo-fy y pd. Th followg upo ocd wh h odl: Th obo-fy y copod of o obo d dcl fy u. Th obo d dud fy u opg ulouly. 3 All flu clly dpd. 4 All flu d h plly fld y p co. 5 Th fld obo-fy y p c b co o o-co. 6 Th pd obo o fy u good w. 7 Th ovll obo-fy y fl wh h cv Copygh 9 ScR
2 B. S. DHION ET A. 5 Fgu. Th bloc dg of h obo-fy y wh coo-cu flu Fgu. Th pc dg of h obo-fy y wh coo-cu flu. Th ul d l d qu cgl d llp do y d f =- = - obo fl coo-cu flu occu o h -+ h fy u fl.. Noo Th followg ybol ocd wh h odl: h of h ovll obo-fy y: = obo d ll fy u pfc wog codo; = obo d - fy u opg olly whl o fy u h fld; = =3 -- d = - h obo d - fy u opg olly whl fy u hv fld; =- = obo d fy u opg olly whl - fy u hv fld. h of h fld obo-fy y: =+ obo-fy y fld du o h lfuco of h -+ h fy u ; =+ obo-fy y fld du o h flu of h obo lf; =+3 obo-fy y fld du o coo-cu flu. 3 : Co flu of h fy u. : Co flu of h obo. c : Co coo-cu flu of h obofy y ; = -. : Co p of h fy u ; = -. : F p vl. : T-dpd p wh h fld obo-fy y d h lpd p of ; = p : Th pobbly h h fld obo-fy y d h lpd p l h vl [ +; = df: obbly dy fuco. z : pdf of p wh h fld obo-fy y d h lpd of ; = : obbly h h obo-fy y ; = -. : obbly h h obo-fy y ; = : Sdy- pobbly h h obo-fy y ; = -. : Sdy- pobbly h h obo-fy y ; = : plc vbl. : plc of h pobbly h h obo-fy y ; = -. : plc of h pobbly h h obo-fy y ; = AV : plc of h obo-fy y vlbly wh h obo wog wh l fy u. AV : Robo-fy y -dpd vlbly wh h obo wog wh l fy u. SSAV : Robo-fy y dy vlbly wh h obo wog wh l fy u. R : plc of h obo-fy y lbly wh h obo wog wh l fy u. R : Robo-fy y lbly wh h obo wog wh l fy u. MTTF : Robo-fy y o flu wh h obo wog wh l fy u. : Robo-fy y vc of o flu wh h obo wog wh l fy u. 3. Aly Ug h upply hod [ 3 h y of Equo ocd wh Fgu c b pd follow: 3 d d d Copygh 9 ScR
3 B. S. DHION ET A d d d d c... c c Th ocd boudy codo follow: 5 6 c 3 7 A = = d ll oh l codo pobbl qul o zo. 3.. T Dpd Avlbly Aly Ug h plc T chqu d h l codo Equo 7 w g 3 d c 3 4 Solvg dffl Equo w g h followg po: d p[ 5 3 Sc 3 d 6 d ogh wh Equo 5 w g 3 Z 7 p[ d d Z d z Z 9 p[ d z z h fld obo-fy y p pobbly dy fuco. Ug Equo 9 d 7 ogh wh 4 w g h followg plc T of pobbly oluo:... M N 3 M N... Copygh 9 ScR
4 B. S. DHION ET A. 53 [ 3 c c 3 M Z 3 N 4 N N 5... [ Z N 3 6 Thu h plc of h obo-fy y vlbly wh l wog fy u N M AV 7 Subug h plc of z dff p dbuo Equo 7 d g h v plc of h ulg quo w c g h -dpd obo-fy y vlbly AV. 3.. Sdy S Avlbly Aly A ppoch fy pobbl ch h dy. Thu Equo 7 duc o Equo 8 34 pcvly. 3 d d d Solvg Equo 3 w g 34 p[ c 3 d 35 Th dy codo of h pobbly h du o flu h obo-fy y ud p d 3 36 Subug Equo 35 o Equo 36 yld E [ 3 37 E p[ z d d d 38 whch h o obo-fy y p wh h fld obo-fy y d h lpd p of. Subug Equo 3 34 o Equo 37 w g: E[ 39 E[ 4 3 c E3[ 4 Solvg Equo 9 3 d 39-4 ogh wh 4 4 yld h followg dy pobbl: 3 E [ 43 G Copygh 9 ScR
5 54 B. S. DHION ET A. E [ c 3 c G E [ 47 Th dy vlbly of h obo-fy y wh l wog fy u SSAV G 48 Fo dff fld y p dbuo h vlu of G obd follow:. Wh h fld obo-fy y p polly dbud h h pobbly dy fuco of h p z 3 49 h p d h co p of. Thu h o obo-fy y p E [ h pol dbuo E [ z d 3 5 Subug Equo 5 o Equo 47 w g 3 G G 5. Wh h fld obo-fy y p g dbud h h pobbly dy fuco of h p z 3 5 h p h g fuco d d h hp d cl p pcvly. Thu h o obo-fy y p E [ h g dbuo E [ z d 3 53 Subug Equo 53 o Equo 47 w g G G 3 g Wh h fld obo-fy y p Wbull dbud h h pobbly dy fuco of h p pd by z 3 55 h p d d h hp d cl p of h Wbull dbuo pcvly. Thu h o obo-fy y p E [ h Wbull dbuo gv by / E [ z d 56 3 Subug Equo 56 o Equo 47 w g G G 3 w [ / Wh h fld obo-fy y p Rylgh dbud h h pobbly dy fuco of h Rylgh dbuo pd by z / 3 58 h p d h cl p. Thu h o obo-fy y p E [ h Rylgh dbuo E [ z d Subug Equo 59 o Equo 47 w g G G Wh h obo-fy y p log- Copygh 9 ScR
6 B. S. DHION ET A. 55 ol dbud h h pobbly dy fuco of h p z y l y [ y 3 6 h p d l h ul logh of wh d vc d pcvly. Th codo o p follow: y y l 4 l 6 3 Hc h fld obo-fy y o p E [ h logol dbuo E [ y y 3 Subug Equo 63 o Equo 47 w g G G l 3 [ y y Robo-Sfy Sy Rlbly MTTF d Vc of o flu Sg + = + = +3 = Fgu d pplyg h Mov hod w g h followg dffl quo: d 65 d d 66 d... d 67 d d 68 d d d 69 d 3 c d 7 A = = d ll oh l codo pobbl qul o zo. Tg h plc of Equo 65 7 d olvg h ulg of quo w ob h followg plc of pobbl: 3 [ Th plc of h obo-fy y lbly wh l wog fy u R 74 Ug Equo 74 h obo-fy y o h flu obd follow [4: MTTF l R 3 75 Th -dpd obo-fy y lbly R c b obd by g h v plc of Equo 74. Th obo-fy y vc of o flu pd by lr d 3 ' MTTF MTTF 3 3 d 76 R do h dvv of R wh pc o. Copygh 9 ScR
7 56 B. S. DHION ET A. d l d l ' d d d ' l ' '... 3 l ' 3 d d 3 l c ' do h dvv of wh pc o. do h dvv of wh pc o. Th ub of fy u copod wh h obo-fy y h of dd lvl of fy. Mo fy u w u h b y fy lbly d MTTF w c chv. 4. Spcl C Modl: = =3 Fo = d =3 Fgu d h odl bco y hvg o obo d h dud fy u. Howv l wo fy u u fuco uccfully h obo-fy y ucc. Th copodg y of Equo c b obd fo Equo 7 by g = d =3. Fuho obo-fy y pobbl [ vlbl [AV SSAV lbly [R o flu [MTTF d vc of o flu [ h pcl c odl c lo b obd by g = d =3 o h copodg glzd Equo. 4.. T Dpd Avlbly lo = d =3 Sg: =.6 =.6 c =. c =. =.9 4 =. 5 =. 6 =.6 Equo d 7 d g dbud fld y p ug Mpl copu pog [5 h -dpd plo of obo-fy y pobbl d vlbly how Fgu 3 d 4 pcvly. 4.. Sdy S Avlbly lo = d =3 Sg: =.6 =.6 c =. =.9 4 =. 5 =. 6 =.6 Equo 48 d g d Wbull dbud fld y p ug Mpl copu pog d [5 plo SSAV how Fgu 5 d 6 pcvly. Fgu 3. T-dpd pobbly plo obofy y wh g dbud = fld y p Fgu 4. T-dpd vlbly plo obofy y wh g dbud = fld y p Fgu 5. Robo-fy y dy vlbly vu coo-cu flu c plo wh g dbud =.5.5 fld y p. Copygh 9 ScR
8 B. S. DHION ET A. 57 fy y how Fgu 7. Slly plo of h obo-fy y o flu ug Equo 75 fuco of coo-cu flu c how Fgu Cocluo Fgu 6. Robo-fy y dy vlbly vu coo-cu flu c plo wh Wbull dbud =...6 fld y p. Th pp pd lbly ly of y hvg o obo d -dud fy u wh coo-cu flu. Th ul of h ly dc h dud fy u hlp o pov obo y lbly d h occuc of coo-cu flu dc h obo y lbly. I codd h h ul of h udy wll b uful o g d gg pofol o vou obo y lbly vlbly d fy-ld dco. REFERENCES Fgu 7. Rlbly plo of h obo-fy y Fgu 8. M o flu plo of h obo-fy y fuco of coo-cu flu c 4.3. Rlbly d MTTF lo = d =3 Sg: =.6 =.6 c =. c =. 4 = 5 = 6 = Equo 74 d ug Mpl copu pog [5 h -dpd lbly plo of h obo- [. Ncol Sfy pobl ld o obo Roboc Vol. 3 pp [ M. Ngch T fl ccd du o obo Jp Egooc of Hybd Auod Sy I d. H. R. Kwow d M. R. Elv Ad pp [3 B. S. Dhllo Robo lbly d fy Spg- Vlg Nw Yo 99. [4 J. Fy Fuu pco ol obo fy Roboc Wold Vol. 4 No. pp [5 S. Nl Ipovg obo fy gg uoo Vol. 8 No. pp [6 D. Kulc d E. Cof -collo fy g hu-obo co Auooou Robo Vol. No. pp [7 E. J. Vdp d S. S. Mhov Ovll vlbly of obo wh l fy dvc Copu d Idul Egg Vol. 56 No. pp [8 S. Hddd S. A. Albu-SuchCucy d G. Hzg Rqu f obo: u ly d w gh Iol Joul of Roboc Vol. 8 No. pp [9 J.. Ml Ivl ly d lbly oboc Iol Joul of Rlbly d Sfy Vol. 3 No. 3 pp [ B. S. Dhllo d S. Chg obblc ly of pbl obo-fy y copod of - dby obo A Sfy U d Swch Joul of Quly Mc Egg Vol. 4 No. 3 pp [ B. S. Dhllo Rlbly gg y dg d opo V Nod Rhold Nw Yo 983. [ D.. Gv T o flu d vlbly of pllld Copygh 9 ScR
9 58 B. S. DHION ET A. y wh p IEEE Tco o Rlbly Vol. pp [3 R. C. Gg Dpdbly of copl y hvg wo yp of copo IEEE Tco o Rlbly Vol. pp [4 B. S. Dhllo Dg lbly: fudl d pplco CRC Boc Ro Flod 999. [5 R. M. Col El MAE: A oduco o cfc pog Spg Vlg Nw Yo 995. Copygh 9 ScR
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