Complete Identification of Isotropic Configurations of a Caster Wheeled Mobile Robot with Nonredundant/Redundant Actuation

Transcription

1 486 Ieraoal Joural Sugbok of Corol Km Auomao ad Byugkwo ad Sysems Moo vol 4 o 4 pp Augus 006 Complee Idefcao of Isoropc Cofguraos of a Caser Wheeled Moble Robo wh Noreduda/Reduda Acuao Sugbok Km ad Byugkwo Moo Absrac: I hs paper we prese he complee soropy aalyss of a caser wheeled omdrecoal moble robo (COMR) wh oreduda/reduda acuao I s desrable for robus moo corol o keep a COMR close o he soropy bu away from he sgulary as much as possble Frs wh he characersc legh roduced he kemac model of a COMR s obaed based o he orhogoal decomposo of he wheel veloces Secod a geeral form of he soropy codos of a COMR s gve erms of physcally meagful vecor quaes whch specfy he wheel cofgurao Thrd for all possble oreduda ad reduda acuao ses he algebrac expressos of he soropy codos are derved so as o defy he soropc cofguraos of a COMR Fourh he umber of he soropc cofguraos he soropc characersc legh ad he opmal al cofgurao are dscussed Keyword: Caser wheeled moble robo characersc legh soropc cofgurao reduda acuao INTRODUCTION The omdrecoal mobly of a moble robo s requred o avgae daly lfe evrome whch s resrced space ad cluered wh obsacles Several omdrecoal wheel mechasms have bee proposed cludg uversal wheels Swedsh wheels orhogoal wheels ball wheels ad so o [] Recely caser wheels were employed as a praccal ad effecve meas o develop a omdrecoal moble robo a Saford Uversy whch was laer commercalzed by Nomadc Techologes as XR4000 [] Sce caser wheels operae whou addoal perpheral rollers or suppor srucure a caser wheeled omdrecoal moble robo or a COMR ca maa good performace eve uder varyg payload or groud codo There have bee several works o he kemac ssues of a COMR For a geeral form of wheeled moble robos a sysemac kemac modelg procedure was developed [] Regardg he mmal admssble acuao was show ha a leas four jos ou of wo caser wheels should be Mauscrp receved March 8 005; revsed Jauary 006; acceped Aprl 006 Recommeded by Edoral Board member Sooyog Lee uder he dreco of Edor Jae- Bok Sog Ths work was suppored by Hakuk Uversy of Foreg Sudes Research Fud of 006 Sugbok Km ad Byugwo Moo are wh he School of Elecrocs ad Iformao Egeerg Hakuk Uversy of Foreg Sudes Yog-s Kyugk-do Korea (emals: {sbkm kwo-e}@hufsackr) acuaed o avod he sgulary [4] For some specfc acuao ses he global soropc characerscs over he ere wheel cofguraos was cosdered o oba he opmal desg parameers of he mechasm [5] For represeave acuao ses he algebrac codos for he (local) soropy were derved o defy he soropc cofguraos [6] O he oher had for a omdrecoal moble robo employg Swedsh wheels he soropy aalyss was made bu he resuls are complee ad eed furher elaborao [7] For a COMR havg ( ) acuaed jos he relaoshp bewee he jo velocy Θ R ad he ask velocy x R ca be expressed he form of Ax = BΘ where A R ad B R are he Jacoba marces [6] Noe ha A s a fuco of a gve wheel cofgurao whle B s always osgular depedely of he wheel cofgurao Whe he rak of A s less ha hree a COMR falls o he sgular cofguraos whch he ask veloces wh he ullspace of A ca be produced eve wh all he acuaed jos locked [8] O he oher had whe hree sgular values of B A become decal a COMR reaches he soropc cofguraos whch he jo veloces requred for a uy ask velocy all drecos are uform magude [9] Obvously s desrable for robus moo corol o keep a COMR away from he sgular cofguraos bu close o he soropc cofguraos as much as possble [7]

2 Complee Idefcao of Isoropc Cofguraos of a Caser Wheeled Moble Robo wh Noreduda 487 The purpose of hs paper s o compleely defy he soropc cofguraos of a COMR wh oreduda/reduda acuao Ths paper s orgazed as follows Wh he characersc legh roduced [7] Seco preses he kemac model based o he orhogoal decomposo of he wheel veloces I Seco a geeral form of he soropy codos s gve erms of physcally meagful vecor quaes specfyg he wheel cofgurao For all possble oreduda ad reduda acuao ses Secos 4 ad 5 derve he algebrac expressos of he soropy codos o defy he soropc cofguraos Seco 6 dscusses he umber of soropc cofguraos he soropc characersc legh ad he opmal al cofgurao Fally he cocluso s made Seco 7 KINEMATIC MODEL Cosder a COMR wh hree caser wheels aached o a regular ragular plaform movg o he xy-plae as show Fg Le l be he sde legh of he plaform wh he ceer O b ad hree verces O = For h he caser wheel wh he ceer P = we defe he followg Le d ad r be he legh of he seerg lk ad he radus of he wheel respecvely Le θ ad ϕ be he agles of he roag ad he seerg jos respecvely Le u ad v be wo orhogoal u vecors alog he seerg lk ad he wheel axs respecvely such ha cosϕ u = sϕ Noe ha sϕ v = () cosϕ uu + vv = I () v q P θ r ϕ P v s ϕ O ω y u O O P Fg A caser wheeled omdrecoal moble robo O b v v x d l ϕ u = 0 v = 0 () where I s he dey marx ad 0 s he zero vecor Le p be he vecor from O b o P ad q be he roao of p by 90 couerclockwse Noe ha q = 0 p = 0 (4) p = 0 u = 0 (5) Le ad v be ω he lear ad he agular veloces a O of he plaform respecvely For he h b caser wheel = he lear velocy a he po of coac wh he groud ca be expressed by v+ ωq = r θ u + d ϕ v = (6) Premulpled by + ω = rθ + ω = dϕ u ad v we have u v u q = (7) v v v q = (8) Assume ha ( 6) jos of a COMR are acuaed Wh he characersc legh L ( > 0) roduced [6] he kemacs of a COMR ca be wre as Ax = BΘ (9) where x= [ v L ω] R ad s he ask velocy vecor Θ R s he jo velocy vecor ad g g h L A = R g g h L (0) c 0 B= R 0 c () are he Jacoba marces I (0) gk k = correspods o eher u or v = whle h k k = correspods o q = I () c k k = correspods o eher r or d = I should be meoed ha he roduco of he characersc legh L makes all hree colums of A o be cosse physcal u The expresso of g h k = ca be k k

3 488 Sugbok Km ad Byugkwo Moo smplfed as follows I he case of he roag jo for whch gk = u ad h k = q = k k = = g h u q v p () Ad he case of he seerg jo for whch gk = v ad h k = q = k k = = g h v q u p () I s worhwhle o meo ha our kemac modelg of a COMR does o volve marx verso ulke he rasfer mehod proposed [5] For a gve ask velocy he saaeous moo of he wheel s decomposed o wo orhogoal compoes: he saaeous moo of he roag jo ad he saaeous moo of he seerg jo The resulg kemac model allows us o perform a geomerc ad uve aalyss o he soropy of a COMR ISOTROPY CONDITIONS Based o (9) he ecessary ad suffce codo for he soropy of a COMR s gve by ( B A) ( B A) I (4) I [5] s foud o be opmal for global soropc characerscs ha hree caser wheels are decal o have he seerg lk legh equal o he wheel radus ha s c = d > 0 k = B I (5) k Uder he assumpo of (5) (4) becomes A A I (6) Noe ha fac (5) ad (6) are he suffce codos for he soropy of a COMR From (0) ad (6) he soropy codo o A s obaed by A A= I (7) whch leads o he followg hree codos: C : gk gk = I R k k k C : ( g h ) g = 0 R C : ( gk hk) = R L 6 (8) I geeral C ad C correspod o hree ad wo scalar cosras respecvely whch are mposed o hree seerg jo agles ϕ k k = Thus he soropy of a COMR ca occur oly a specfc values of ϕ k k = called soropc cofguraos for whch C ad C are sasfed smulaeously For a gve soropc cofgurao C correspodg o oe scalar cosra deermes he characersc legh requred for he soropy deoed by L so I wha follows s assumed ha a COMR has hree decal caser wheels havg he seerg lk legh equal o he wheel radus 4 ISOTROPY ANALYSIS FOR NONREDUNDANT ACTUATION 4 Noreduda acuao ses A COMR wh oreduda acuao ca have hree acuaed jos ( = ) each of whch ca be eher roag or seerg oe Accordg o he umber of acve wheels ad he combao of acuaed jos all possble oreduda acuao ses Θ ca be dvded o hree groups deoed by NAG I II ad III as lsed Table 4 Isoropy aalyss for NAG I Cosder Θ = { θ θ θ } where hree roag jos of hree caser wheels are acuaed for whch [ g g g] = [ u u ] Uder he codo of C we have whch s uu k k = 5 I (9) k= c + c + c = 5 cs + cs + cs = 00 Table Three oreduda acuao groups Number Number of of acve acuaed wheels jos = Acuao se Θ = { θ θ θ } Θ = { ϕ ϕ ϕ } Θ = { ϕ θ θ } Θ = { ϕ ϕ θ } Θ = { θ ϕ θ } Θ = { θ ϕ ϕ } (0) Noreduda acuao group NAG I NAG II NAG III

4 Complee Idefcao of Isoropc Cofguraos of a Caser Wheeled Moble Robo wh Noreduda 489 where ck = cos( ϕk) ad sk = s ( ϕk) k = There are egh dffere dsrbuos of { u k k = } sasfyg (0) whch ca be dvded o wo dscve groups characerzed respecvely by π π ϕ = ϕ+ π ϕ ϕ = ϕ+ ϕ π () π π ϕ = ϕ+ ϕ π ϕ = ϕ+ π ϕ () as show Fg The frs group of four dsrbuos characerzed by () s commo ha u ad le o hree sdes of a regular ragle couerclockwse order as show Fg (a) O he oher had he secod group of four dsrbuos characerzed by () s commo ha u u ad le o hree sdes of a regular ragle clockwse order as show Fg (b) Uder he codo of we have ( ) ( ) k k k k k k u q u = v p u = 0 () whch s equvale o ( ) k k k ak k v q v = v = 0 (4) where αk = vk p k k = s he projeco of p k oo v k For he frs group of four dsrbuos characerzed by () ca be show ha [5] α = α = α = α (5) α v + α v + α v = 0 (6) Whle C places wo scalar cosras gve by (0) o hree varables ϕ ϕ ad ϕ C does o place addoal cosra As a resul here are - v u p v (v p )v (v p )v (v p )v fely may soropc cofguraos geeral Fg (a) llusraes a soropc cofgurao of a COMR where hree seerg lks form a regular ragle ceered a he plaform ceer scrbed by a crcle of radus α O he oher had ca be show ha he secod group of four dsrbuos characerzed by () cao sasfy (4) so ha he soropy of A cao be acheved Smlar aalyss o he above ca be made for Θ = { ϕ ϕ ϕ} where hree seerg jos of hree caser wheels are acuaed Fg (b) llusraes a soropc cofgurao of a COMR where hree wheel axes form a regular ragle ceered a he plaform ceer whch s scrbed by a crcle of β ( = u p = u p = u p ) radus 4 Isoropy aalyss for NAG II Cosder Θ = { ϕ θ θ} where oe seerg ad wo roag jos of hree caser wheels are acuaed for whch [ g g g] = [ v u ] Frs uder C we have + + = 5 vv uu uu I (7) Nex uder C we have p u v v p ( p ) v (u p (u p ) )u (a) (b) Fg Isoropc cofguraos for NAG I : (a) Θ = Θ = ϕ ϕ ϕ { θ θ θ } ad (b) { } p u v u u u u or ( v q ) v + ( u q ) u + ( u q ) u = 0 (8) (a) ( u p ) u + ( v p ) v + ( v p ) v = 0 (9) u u (b) Fg Two dscve groups of he dsrbuos of { u k k = } : (a) couerclockwse order ad (b) clockwse order u u Wh (7) beg held ca be show ha (9) cao be sasfed uless d s equal o zero [5] Ths ells ha he soropy of A ca be acheved oly whe caser wheels reduce o coveoal wheels whou seerg lk Fg 4(a) llusraes a soropc cofgurao of a coveoal wheeled moble robo Smlar aalyss o he above ca be made for Θ = ϕ ϕ θ where wo seerg ad oe roag { }

5 490 Sugbok Km ad Byugkwo Moo Fg 4 Wh d = 0 soropc cofguraos for NAG II: (a) Θ = { ϕ θ θ} ad (b) Θ = { ϕ ϕ θ } jos of hree caser wheels are acuaed Fg 4(b) llusraes a soropc cofgurao of a coveoal wheeled moble robo 44 Isoropy aalyss for NAG III Cosder Θ = { θ ϕ θ} where boh roag ad seerg jos of oe caser wheel ad he roag jo of aoher caser wheel are acuaed for whch [ g g g] = [ u v u ] Frs uder C we have whch s + + = 5 uu vv u u I (0) c v v (u p ) p (v p )v (a) (v p )v p u cs v = 05 = 00 () There does o exs ϕ sasfyg () ad he soropy of A cao be acheved a all Smlar aalyss o he above ca be made for Θ = { θ ϕ ϕ} where boh roag ad seerg jos of oe caser wheel ad he seerg jo of aoher caser wheel are acuaed 5 ISOTROPY ANALYSIS FOR REDUNDANT ACTUATION 5 Reduda acuao ses A COMR wh reduda acuao ca have four fve ad sx acuaed jos ( = ) each of whch ca be eher roag or seerg oe Accordg o he umber ad combao of acuaed jos ad he umber of acve wheels all possble reduda acuao ses Θ ca be dvded o fve groups deoed by RAG I II III IV ad V as lsed Table 5 Isoropy aalyss for RAG I Cosder Θ = { θ ϕ θ ϕ} where boh roag ad seerg jos of wo caser wheels are acuaed v v (u p ) p (u p )u (v p (b) )v p v u Table Fve reduda acuao groups Number of acuaed jos for whch [ ] = [ ] g g g g4 u v u v Frs uder C we have ( uu k k + uu k k ) = I () whch always holds Nex uder C we have or k k k k k k k Number of acve wheels {( u q ) u + ( v q ) v } = q = 0 () p+ p = 0 (4) whch yelds l ϕ arcs = ϕ = π ϕ (5) d Sce C places o cosra ad C places wo scalar cosras gve by (4) o wo varables ϕ ad ϕ geeral here are mulple soropc cofguraos depedely of ϕ Fg 5 llusraes a soropc cofgurao where he seerg lks of wo caser wheels are symmerc wh respec o y-axs Acuao se = { θ ϕ θ ϕ} = 4 Θ = { θ ϕ θ θ} Θ = { θ ϕ ϕ ϕ} = { θ ϕ θ ϕ} = 5 Θ = { θ ϕ θ ϕ θ} Θ = { θ ϕ θ ϕ ϕ} = 6 Θ = { θ ϕ θ ϕ θ ϕ } u u Fg 5 A soropc cofguraos for Θ = { θ ϕ θ ϕ } belogg o RAG I p Reduda acuao group Θ RAG I RAG II Θ RAG III RAG IV RAG V

6 Complee Idefcao of Isoropc Cofguraos of a Caser Wheeled Moble Robo wh Noreduda 49 wh he ceers of wo caser wheels ad he ceer of l he plaform lyg o he le of y = Noe ha he soropc cofgurao does o exs f he l seerg lk legh s less ha 5 Isoropy aalyss for RAG II Cosder Θ = { θ ϕ θ θ} where boh roag ad seerg jos of oe caser wheel ad wo roag jos of wo caser wheels are acuaed for whch [ g g g g4] = [ u v u ] Frs uder C we have whch s hece + = uu uu I (6) cs + = (7) cs 00 π ϕ = ϕ ± (8) Noe ha u ad are perpedcular o each oher ad so are v ad v Nex uder C we have or ( ) ( ) q + u q u + u q u = 0 (9) ( ) ( ) p + v p v + v p v = 0 (40) Sce C places oe scalar cosra gve by (7) ad C places wo scalar cosras gve by (40) o hree varables ϕ ϕ ad ϕ here are mulple soropc cofguraos geeral Fg 6(a) llusraes a soropc cofgurao where he seerg lks of wo caser wheels wh acuaed roag jo are perpedcular o each oher ad he ceer of he oher caser wheel wh acuaed roag ad seerg jos s locaed uder he cosra of (40) Smlar aalyss o he above ca be made for Θ = { θ ϕ ϕ ϕ} where boh roag ad seerg jos of oe caser wheel ad wo seerg jos of wo caser wheels are acuaed Fg 6(b) llusraes a soropc cofgurao where he seerg lks of wo caser wheels wh acuaed seerg jo are perpedcular o each oher ad he ceer of he oher caser wheel wh acuaed roag ad seerg jos s locaed uder he followg cosra: ( ) ( ) p + u p u + u p u = 0 (4) 54 Isoropy aalyss for RAG III Cosder Θ = { θ ϕ θ ϕ} where boh roag ad seerg jos of oe caser wheel ad oe roag ad oe seerg jos of wo caser wheels are acuaed for whch [ gggg4] = [ uvu v ] Frs uder C we have whch s hece + = u u v v I (4) cs cs = 00 (4) ϕ = ϕ (44) Nex uder C we have or ( ) ( ) q + u q u + v q v = 0 (45) ( ) ( ) q + u q u + v q v = 0 (46) Sce C places oe scalar cosra gve by (4) ad C places wo scalar cosras gve by (46) o hree varables ϕ ϕ ad ϕ here are mulple soropc cofguraos geeral Fg 7 llusraes a soropc cofgurao where he seerg lks of oe caser wheel wh acuaed v v v p ( v p ) v (a) p v (v p )v u ( u p ) u p (b) (u p ) u Fg 6 Isoropc cofguraos for RAG II: (a) Θ = θ θ θ Θ = θ ϕ ϕ ϕ { ϕ } ad (b) { } p u u (v p )v u p (u p )u Fg 7 A soropc cofgurao for Θ = { θ ϕ θ ϕ } belogg o RAG IV p v

7 49 Sugbok Km ad Byugkwo Moo roag jo ad aoher caser wheel wh acuaed seerg jo are parallel o each oher ad he ceer of he oher caser wheel wh acuaed roag ad seerg jos s locaed uder he cosra of (46) 55 Isoropy aalyss for RAG IV Cosder Θ = { θ ϕ θ ϕ θ} where boh roag ad seerg jos of wo caser wheels ad he roag jo of oe caser wheel are acuaed for whch [ g g g g4 g5] = [ u v u v ] Frs uder C we have u = 05I (47) whch s cs c = 05 = 00 (48) There does o exs ϕ sasfyg (48) ad he soropy of A cao be acheved a all Smlar aalyss o he above ca be made for Θ = { θ ϕ θ ϕ ϕ} where boh roag ad seerg jos of wo caser wheels ad he seerg jo of oe caser wheel are acuaed 56 Isoropy aalyss for RAG V Cosder Θ = { θ ϕ θ ϕ θ ϕ} where boh roag ad seerg jos of hree caser wheels are g g g g g g fully acuaed for whch [ ] = [ u v u v u v ] Frs C holds always Nex uder C we have pk = 0 (49) whch yelds p π π ϕ = ϕ± ϕ = ϕ (50) Sce C places o cosras ad C places wo scalar cosras gve by (49) o hree varables ϕ ϕ ad ϕ here are fely may soropc cofguraos geeral Fg 8 llusraes a soropc cofgurao of a COMR where he ceers of hree caser wheels are symmerc wh respec o he ceer of he plaform 6 SOME DISCUSSIONS 6 Number of soropc cofguraos Depedg o he seleco of acuaed jos he umber of soropc cofguraos whch sasfy C ad C ca be eher oe mulple(fe) or fe Table lss he oreduda ad he reduda acuao ses resulg more ha oe soropc cofgurao From Table he followg observaos ca be made Whe he acuao of hree caser wheels are homogeeous cludg Θ = { θ θ θ} { ϕ ϕ ϕ } ad { θ ϕ θ ϕ θ ϕ } here are fely may soropc cofgureos Whe he umber of acuaed jos are reduda cludg Θ = { θ ϕ θ θ} { θ ϕ ϕ ϕ } { θ ϕ θ ϕ } ad { θ ϕ θ ϕ } here are mulple soropc cofguraos The oly Θ = θ ϕ θ ϕ θ ad wo excepos are { } { θ θ } ϕ ϕ ϕ I should be meoed ha boh homogeey wheel acuao ad redudacy jo acuao play a sgfca role for ehacg he soropy of a COMR 6 Isoropc characersc legh As descrbed Seco he soropy of a COMR ca be acheved uder hree codos C C ad C Oce a soropc cofgurao s defed uder C ad C he characersc legh requred for he soropy L so ca be deermed uder C As a example le us cosder he case of Θ = { θ θ θ } Uder C we have p k k k k L ( u q ) = ( v p ) = 5 (5) L Wh () beg held from (5) he characersc legh of a soropc COMR s obaed by L = ( v p = v p = v p ) (5) so Fg 8 A soropc cofgurao for Θ = { θ ϕ } θ θ } ϕ ϕ belogg o RAG V For all acuao ses wh more ha oe soropc cofgurao Table also lss he resulg soropc characersc legh L so Noe ha he soropy of a COMR cao be acheved uless L = L so

8 Complee Idefcao of Isoropc Cofguraos of a Caser Wheeled Moble Robo wh Noreduda 49 Acuao group NAG I Table All acuao ses resulg more ha oe soropc cofgurao Number of soropc Acuao se Θ Isoropc characersc legh L cofguraos so { θ θ θ } Ife ( v p = v p = v p ) { ϕ ϕ ϕ } Ife ( u p = u p = u p ) RAG I { θ ϕ θ ϕ } Mulple ( p = p ) RAG II θ ϕ θ θ Mulple ( p = v p) + ( v p ) { } { } θ ϕ ϕ ϕ Mulple ( p = u p) + ( u p ) θ ϕ θ ϕ Mulple ( p = v p) + ( u p ) RAG III { } RAG V { θ ϕ θ ϕ θ ϕ } Ife ( p = p = p ) 6 Opmal al cofgurao For a gve ask velocy rajecory he wheel cofguraos are subjec o udergo dffere chages depedg o he al cofgurao chose Uless he al cofgurao s se carefully a COMR may suffer from poor soropc characerscs durg ask execuo whch s udesrable for robus moo corol To fd he opmal al cofgurao wh mmal compuao a smple bu effecve measure should be devsed whch ca evaluae he soropc characerscs of a gve wheel cofgurao As a example le us cosder he case of Θ = { θ ϕ θ ϕ θ ϕ} Suppose ha a ask velocy rajecory over some me erval x () [ 0T] wh he prespecfed al poso x (0) s gve o a COMR Usg (8) he seerg jo agle a me ϕ () of he h caser wheel s obaed by 0 ϕ () = ϕ (0) + a ( ϕ ( τ)) x ( τ) dτ = (5) where a ( ϕ) = [ v ( ϕ) vk ( ϕ) q k( ϕ)] Noe ha ϕ () = s sesve o s al codo ϕ (0) The cofgurao rajecory durg ask execuo Π ca be descrbed as Π = {( ϕ( ) ϕ( ) ϕ( )) [0 T]} = Π( ϕ (0) ϕ (0) ϕ (0)) (54) Noe ha Π s a fuco of he al cofgurao ( ϕ(0) ϕ(0) ϕ (0)) Wh he pror kowledge of he soropc cofguraos we propose o evaluae he soropc characerscs of a gve wheel cofgurao based o he dsace from he soropc cofguraos Wh ϕ = ϕ ϕ = he soropc cofguraos gve by (50) ca be expressed as wo pos π π π π ( ϕ ϕ) -space ( ) ad ( ) For a gve wheel cofgurao a me ( ϕ ( ) ϕ ( ) ϕ ()) ds( ) he local soropy measure deoed by ca be defed as he smaller value of he weghed dsaces from ( ϕ ( ) ϕ ( )) o he wo soropc pos A proper weghg would be a fuco whch reurs zero whe ( ϕ( ) π π ϕ ()) = ( ± ) ad reurs larger value as ϕ () or ϕ () approaches o 0 or ± π or ( ϕ( ) π π ϕ ()) = ( ± ± ) Now for a gve cofgurao rajecory Π ( ϕ(0) ϕ(0) ϕ (0)) he global soropy measure deoed by DIST( Π ) ca be defed as DIST( Π) = max ds( ) (55) [0 T ] Geomercally DIST( Π) ca be erpreed as he radus of he smalles crcle ( ϕ ϕ) -space whch coas all he devaos from he soropc cofguraos alog Π Fally amog all possble cofgurao rajecores resulg from dffere al cofguraos he opmal al cofgurao op op op ( ϕ (0) ϕ (0) ϕ (0)) ca be deermed hrough he opmzao gve below: m DIST( Π) (56) = m [max ds( )] ( ϕ(0) ϕ(0) ϕ(0) ) ( ϕ(0) ϕ(0) ϕ(0) ) [ 0 T ]

9 494 Sugbok Km ad Byugkwo Moo Noe ha for a gve Π compug ds( ) ca be sopped mmedaely afer he value of ds() becomes greaer ha or equal o he smalles amog he global soropy measures already compued By choosg op op op ( ϕ (0) ϕ (0) ϕ (0)) he maxmal devao from he soropc cofguraos durg ask execuo ca be kep small as much as possble whch s desrable for robus moo corol Sce he al cofgurao opmzao for mproved soropc characerscs requres esed mulple loops he assocaed compuaoal cos depeds heavly o how o devse a local soropy measure wh he ermos loop The local soropy measure ds() whch s devsed usg he pror kowledge of he soropc cofguraos s o oly physcally meagful bu also smple compuao A alerave measure whou such kowledge would be he codo umber of he Jacoba marx A whch requres far more expesve compuao ha ds() 7 CONCLUSION Ths paper preseed he complee soropy aalyss of a caser wheeled omdrecoal moble robo (COMR) wh oreduda ad reduda acuao All possble acuao ses wh dffere umber ad combao of roag ad seerg jos were cosdered Frs wh he characersc legh roduced he kemac model was obaed Secod a geeral form of he soropy codos was gve erms of physcally meagful vecor quaes Thrd for all possble oreduda ad reduda acuao ses he algebrac expressos of he soropy codos were derved so as o defy he soropc cofguraos compleely Fourh he umber of he soropc cofguraos he soropc characersc legh ad he opmal al cofgurao were dscussed We hoped ha he soropy aalyss made hs paper ca serve for beer desg ad corol of a COMR wh mproved soropc characerscs REFERENCES [] P F Mur ad C P Neuma Kemac modelg of wheeled moble robos Joural of Roboc Sysems vol 4 o pp [] W Km B Y ad D Lm Kemac modelg of moble robos by rasfer mehod of augmeed geeralzed coordaes Joural of Roboc Sysems vol o 6 pp [] R Holmberg Desg ad Developme for Powered-caser Holoomc Moble Robo PhD Thess Dep of Mechacal Eg Saford Uversy 000 [4] G Campo G Bas ad B D`Adrea Novel Srucural properes ad classfcao of kemac ad dyamc models of wheeled moble robos IEEE Tras o Robos ad Auomaos vol o pp [5] W Km D Km B Y B You ad S Yag Desg of a om-drecoal moble robo wh caser wheels Tras o Corol Auomao ad Sysems Egeerg ( Korea) vol o 4 p [6] S Km ad H Km Isoropy aalyss of caser wheeled omdrecoal moble robo Proc of IEEE I Cof o Robocs ad Auomao pp [7] S K Saha J Ageles ad J Darcovch The desg of kemacally soropc rollg robos wh omdrecoal wheels Mechasm ad Mache Theory vol 0 o 8 p [8] C Gossel ad J Ageles Sgulary aayss of closed-loop kemac chas IEEE Tras o Robos ad Auomaos vol 6 o pp [9] T Yoshkawa Aalyss ad corol of robo mapulaors wh redudacy Robocs Research pp MIT Press 984 Sugbok Km receved he BS degree Elecrocs Egeerg from Seoul Naoal Uversy Korea 980 he MS degree Elecrcal ad Elecrocs Egeerg from KAIST Korea 98 ad he PhD degree Elecrcal Egeerg from he Uversy of Souher Calfora USA 99 Sce 994 he has bee wh he School of Elecrocs ad Iformao Egeerg Hakuk Uversy of Foreg Sudes Korea where he s currely a Professor Hs rece research eress clude he aalyss desg ad corol of moble robos ad humaods Byugkwo Moo receved he BS degrees Dgal ad Iformao Egeerg from Hakuk Uversy of Foreg Sudes Korea 005 He s ow workg oward a MS degree Elecrocs ad Iformao Egeerg a Hakuk Uversy of Foreg Sudes Korea Hs research eress clude he desg ad corol of omdrecoal moble robos