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!"# $% # !" & ' ( $ !"'' ) & *++ 7 +E-mailaddress: F++) + Collisionsarebiglosses5B &( ! ( + ( ( C +/2 ! 5$ !& ( : $ $ & ! (" ! !'(+6 (; &$ % ' "( $ $!"! ! $++G: (!! ! ! ( ' ! ! # -&' ! # +@( $& % (!+1.1.Theplanofthepaper3! '(- !( $ $+ $ ( !% ( !"( + ' & $(& % + $ ' & & !" &! C +4 & ( $ ( + 3! $(+2.1.Mechanicallyconsistentmodelsoflocomotion/$$ ! !(+/ (( ( & ( & $!' C#$ + - $! &( $ " (-! ' & $+.; % & ( & (' + ( $ '( #+B &! +B " "(& &! # ;+B & %&# "+B C( !$ & $!"+H $!" +2.2.Positiveandnegativemusclework $ !"( !"+ ! !" $ # ; %+B'? 2! & ( !" "+B& " $ 4 ++ & G$ ,( 299 +. ! &( &+ (! ! ++ % ,( (,90* ! $ + ( +A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 &(!" !+B? 2! ? 2 $ $( (( !"musc "&! !&! & (( !" $ +2.3.Muscleworkandmetaboliccost/ C (((% C & & C ( +.!( + (! $ & C&% & % !& & $ $ ! + '( $ $# $ +* $'; ( (I ' # ++, %&& $ &( J &+3' % '; , &A1%&+ ! +3( ! $+ , &A+, $*#+(% !" ! +,*#+(% (( % $ ' ' ; ! ; $+,( '! +/&&!"& &&+3 $! &! !" - & ( ($ ! ( ! !" ! &+2.4.Mechanicalenergybalance % & ( (# ' $$ $!" # +!" &( $' (( +/$$ (2" + ; ; &$ & $ !"+ &!" " +3 $!" $;% & &$& $!" muscles ! ? ; 2 +@ - & ' & ( $&! " $!" + " $!"1((( ' % +2.5.Energybalanceandmetaboliccostoflocomotion3! (!( ! (# ( +7( *#+ ( $ &A! *#+;+B & $! $!"& %3 *#+ $!" ! ; !" A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 (K&! $!" ; K ' 'K; +. & $!"K# $!"+I!$& $ $!"( (# $!"+7$& $ $!" ((# $!"+ : (!*#+A+ % & $ /$ &% $!"(((#% $!"&$ $& $!" ( ( # $!"+ ( $ C(( C( +B!&*#+ & % !"& ( % ! ! ( 09K' $!"' ( 0K +4 ' &! (&!$' B& ( !(&&9K; (' $ 2.6.EnergeticcostoftransportB $ $ # + ''(; ; & ! $ $@ ( ' D"+! % ( ' *#+& &! (& $ & $! & $! !" +. ;!"'( ! !( B $$ +3 $ ! ((+ +B & ( ! 2( ! + ! ( + $ (!! (- ! ( ' +B &(- ( $(+ ( ' !'' +/ ! ( +3.1.Passive-dynamicwalking7 passive-dynamic!";$ ,8&(+ ( ( &! + & ' &!"! ! ' " '7 +& ($ !" +*% & % & $' !" $ +* & ' ""' "& ! &' ( (&8 +& ' $' !" 8 +9+ ! ($ ! + 8 +9 &9! +8 + ! *#+9 ! ($ % $' ! "& & (+ A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 ' !"( ( $(' ( +4 & !' % ! '+3.2.Othercollisionalmodelsoflocomotion) $"00&09 $(; + '! ($ !+ ! & ++""&A1, &A1)($ 4" &9 ! (! !" ( +4 ( ( C ! %& ' ( +! J&&3.3.Recenthumanwalkingresearchthatusescollisionalaccounting) G+&&(J &1J&& $ !(' ( ?? '' 22 !" +6 " & '' ! ( $' '(( (+ !" ?? '' 22 % ??!'' 22 ! + !" C C !"+3.4.Thelackofcollisionsinbrachiation:BouncingwithoutspringsB ( $'&( ! ( ( (( ??( 224 +0 +&+) ( ' & '! & : (! +B $ ( ': ( ( ! & !! ! + $(& $ ( %! + & ( % $ &( ( ! +/ ??( 22 ( ' "% +) & $ ('$ ( ! ! +' "'( 8) & ! '" ' ' ( ++ ( : (!+ !( ! +) (( ' 5 (!! ! ! ' & " ! 'C $$ & ' C +4 ' ! ( &(( $ +3.5.Canananimalswingagainsttheground?B CC ((?2&( '!&! $ &( ' ' + ' $;-7.8;' ' / ($ ! &?? 22++%&+( ( $ $! !"+ $ ! ; & " '$ ( ((& + ! !( (&!; + ! % ! &! $ ! & ' $+ ( $' ( $ +3 % ( % & ! # +4' !& ! & $ ( (+3 %&$ : ($ ( $+ A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 / ?? ' 22 ' ("(! 0 +I! $ & ! &+I&!" 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( ( (# $ & ( $ (' 4.4.Inferredenergeticcostofagaitcyclewithexactlyonecollision $$ $ $!" ((! + $(# ++*#+ !"+B $!" ' +/ & $!"( ' +3" inferredenergeticcostofcollisionoveragaitcycle( (( $!" &" $!" $((! $+ $!" $!"( $&C $+3 & $!" $!" +4($ ! & $!"$ ! ( (+ &!( ( $!" +H % ! *#+0& ( A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 EmbEarEarEa fvmr . &! & ( ! $!" +. ( EmbEgrEa vmfrf $ ( *#+A+I!$& " !" $+4.5.MultiplesequencedcollisionspergaitcycleB!'' $ (' +* (' % ( # ($& ! !+4 ($ !'' + $(! % (% " *#+A! ( +4& $% % % # + !"# $B ! ! + ($ ( $+ ' ! ( $ $ % +5.1.Aglancingperfectlyplasticcollisioncansuperciallyappearelastic.( $ !? 2? 2 ( ! % +7 (( ( 4 ++ $ ( $( + (&$ &! : + ' I!$&$ ! !! ! &(+/ ; ! $ (- ! + $ $( (1 ; !$ (# !!$ + $ ! ! % $(( ' $; &( + &++7 -) 9&! $ & ! & (! ; (! (-+& ! $&& $ + ' '$ % ( ( +5.2.Apseudo-elasticcollisionwithnorecoveryhasonequarterthecostofafullyabsorbingcollision ! % ! ' ' $ C $+$ : +7 ! +4 ' ! $1 !" $ ( ( $ A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 $ +4 ! $(*#+(*#+A9+9@% (( +B + *#+ ! ! *#+9&$! $4 ' $!"# $ ! ! : +/ & $!" " ! ! + # $$ & ' &! & '# +5.3.Pseudo-elasticcollisionsminimizeenergeticcost! $ ! (! ; ( + $+4! C &*#+ C(*#+A C( + &$!; C ( + ($ #5$! $& (; "$ ' $ (( C $!"$ # (( : + #( ' +/ & (( % ( $ ' $!" ' +5.4.Dividingonecollisionintotwosmallercollisionshalvesthecost7 (( ' ! : + " ! $ 4 ++4 +@! ' : : ! : + (! 1'# + ( ! ': 0 + " $ $ # : +I$ # +.! ' $ ( ' + # +5.5.Dividingonecollisionintonsub-collisionsreducesthecostbyafactorofnI! C +7 ! $ :(! ; $@! ! #(' ! : + ( + *#+&A+ EBmEAm +&(*#+&! $ + & $ ( !" " ( $ % &! $ ( ! ! : #+" " (' $ ! +. ! +5.6.Aninnitesequenceofplasticsub-collisionscanproduceoneelasticcollision@!!" $ +H *#+! ( C! ( ! $ A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 + & &# ' ! ((' +5.7.Thecollisionalcostoftransportforapointmass ( (($ $ + % $ ""A) $"09+7 !( ( : !( &+3" ! += 4 + : 4 ++4*#+& &*#+! $ vm9Tregfvmgreg9dgmreg9v&cmb EmTmgvb 4' "( ( ! " + 4( +@!&! ! ! ! +5.8.Passiverunningwithnospringsmaybelikehip-poweredrunning &! $! "( $ + 4 +( $' !"+. + ($&( ! &! $'' # : +4 ' $&+4*#+ EmTmgvb 0+* # $( $ !"! +4!!!!"% + ( % & ! 4 ++. # $# % $# && +3 % '&(% ! !% & !"! +&(,8 %!" ,8&& ,8&(&& ' ( ! %+4 & ( % ++ ! 7/,& '+@% & & 7/, + ; ! ( # ( # % $( $ + &*#+ & % & #4 +& $ 4 +(&" ((+ downhillImpulse from "leg torque"Active bouncing.Flight phase duration = Path of CoM trajectory.(a)(b)legd -V+ plastic legpseudo-elastic leg BC (c)iii 4 ++7 ' $ ? ' "2' +(G! $ ! & 0 &$ $&6$' ( #& (+I ! - $ $& & $+ ! !5 ' & ' ! +A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 5.9.Runningwithpseudo-elasticsprings '(( ' $ ' & . '6B$.6B& ' "+B: ! (+G (' & $ &: ! +B $$ + '! ( (() (&++) (&9A&+ $! ,8(+ ! $ $ ( ' -511;.300;B$ $ ' ! $ $+4 !( (" +4( !$ $ !"+ (*#+ ( EmTmgvb A+07 *#+0! 0 ! ' ! $!"( !+ &% & ; #" +5.10.Caveatontheneglectofleg-swingcosts C +) $( ' $& ! ! ' ' +! ' & & (!! % ;%+4% & $ C C+ ' (!!+ ! & C (;% ( +/ ! C' ! C# ; +% &8 ' # +H( C ( $& ( ( $ C +7 $ $ +I!$&( $ + swingswingflightflight'arm' (b) (c)(d)surfaceoval rolling& skippingSpherical horse:eccentric sphere 'le g s'G transition 4 +0+ ' ( +&+ ! (-+ (!: $( ( ' ! + : - ! +( $ + ( ( ; (! #+* ; +@& $$ +(' "((! !! C$ -(+ '' ! +B ! & ! '' (&$! A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 6.1.Gallopingasslidingonabumpysurface3 -(" ' $ ( 4 +04 +0( + ' " $ +4( ': +@!! + $ ((*#+ ; ( ; &$ & # $ + 4 +0( ! ( ( ; ( ' $ + &#$'! ( % + ; # ; ' ( +6.2.Assumeasphericalhorse*$! ; ($ ' &$' ( $( :! 4 +0+ ('$'( ( ( +B: !' ( ! $ +3 % !'! !! ! C!!$ +B C ! ! C $$ +3 $ ' +( & ! + 'inelastic ! $ ( ( ''& I+ " ( " & C ( C C$ + $ (' " ! ' + $ & ( $ ' : + $5 ' (! ; ; 1; C ;++B ( ' ($ '($+@ ' $( & +) & - + '( ' ( : "3! ((' "' ' ! ; ( +6.3.Thepoint-masscollisionalmodelofgalloping3 ( ! ' (' 4 ++ ( &; (' ' ! $ ;+4*#+&9 ! + & *#+&! $ rf0b rg0vfb r0 7 *#+0! ' ( &% *#++ $ ! " ! + $ & ' ! $& $ "+' ()I! '' !" ' ' ' $+ $ $ ++J&( $ C #?? 224 ++B $ $ '' &! & $+ $ '-$ +4 + ! $ ( ! - ! +B ( !& $+G $ $ +. !$ $ ' - & ! ! +*$ !" $ - & $ A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 ( +B! ( C(! '5 ! && 1 ! +G & C+ $ +!! $ + $ +( +L& $# $ ( +7.1.Therimlesswheelandpassive-dynamicwalking ! ""&A1, &A! (4 +A ,8+ . 3"8 +&9+B !+ - ( 4 ++ ; ( ($ !" ! $+H & &*#+& +A ! 4 +& !" +7.2.Activecollisionreduction:push-offpreceedingheel-strike " ,8& !" (( (( ' " ! ' % ! ! $ + $ ((0 !J&H !"#! ! + ; $ ' ! (( ! +H *#+9 -V+ g collision reductionfacto r 1/4= J4 ++= '' !" + ! - $ $& & $+!" ! !+.$ !+ B $' ! !" & !+ !" $ ! +$ 7 % $(( + !"# "' 7 ($ + # $ ! ( !+ B ' ' "!" $ ! &!( (( ' " !+ H ! $ # $ + $ $( ! & ' " ( '+$B ! 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' " +4% ((( ! 4 +(&+ ( ( #+ ! &$ $& ! $ ' " &$! $ $$+. %!( (; &( + & ' ?? 22M + $( & (& $+3 (# $ ' ! 3 3*#+0! dg9vAr ! $ ; ( $09K& ! '+8.2.Horsegalloping3 ) (( '(& ; ' (;" "& $ ! !"(! ++) (&9A1) (+&9A '++ " 4& ! !") (2( !" !+ $ ! + $ "+ + ( !' "& (" '( + $$ !"( '( + ( +I ! ! !'' 5 ;' ' ' + 4 ++ ' % +B& ' ' & ' !$ + ' ' " +) & 2 " !" & ( # "& "+ 4 ++ ' & ! 4 +0 %'( " (b)(c) vxof CoM (m/s)Hei g hth of CoM (m) 0.60.8 4 ++ ' ' ' +( ( ! ' (;& & ( & ( ': &" +! ,"+&$ ( + ! ! +7'' C$ : A+9D!, +&+ ! ! % 4 ++ ; : ! )) (+I C$ &( ( $ (2 +,! +7 6) )4 ( +G +4 64?2 ("&( ! ! +G ' ( $ %: +G ( C ! ( ! +@&! ( $!( : +A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 +B &( ; !& ( '' +. ! ' + : & ! 4 +& : !+ ' ' ' ! (( ! ( $ ( ( + ' $ $+/$& & ( ! +/ ( ! ! # +/ & " ' + ( ( $! I &9*+&1+&916+&1+&13+&91, +&, + (! #(A+, + % (!D"+H $ ; (!++5(+04*#+ ! brg0vf & $ (AKK +I!$& "53 $$1 $ $! $ 1 C& ( '(+($ (( ( ) 2$+ 'C$! + & &(( +9+8.2.1.Sequencedversusevenlyspacedcollisions $ $( ! ( '(! ba-da-dump+/ $ + 0( +7 & $$0'( B ??"22 ( $' !"+8.3.Humanwalking3 ( # $# $ ! !"' +. 58.3.1.ThemetaboliccostofwalkingG 4 +9+# !(-! '% C( " '% + % + ;*#+! !&0+ (; (( !" (- #+ ( !" + & $& ; +8.3.2.Coordinationpatterns, ;&% ( ( ",8& m / E d /2 (non-dimensional step-length) = 1.732.102.40Stridefrequency(Hz) 00.20.40.60.81.01.2 4 bJ(1-r) = 5/4 4 +9+I!" +G +' ! ( ( ( !" +A.Ruinaetal./JournalofTheoreticalBiology237(2005)170192 $( ' !!" % +4 & $( & ' ( & $ & % ( "& $+ $ ( (' ' " !J+&+ !" !& & '' ! ! !&( & !! 05 ' $ & ' & ' & ' $' $ + $ '! ( ' "+8.3.3.Thewalktoruntransition!! &! % !"'' +B& !$&! $ $#! ! ! +H &&*#+! !" ! $+H *#+! ! $+ +@ ( #+B& !" % ! ( ( !" + % (! % ' ( ' !" +8.4.Childrensskipping ,90, 9 2" &! !(!' $: & ( +/$ " ? 2( + ! , &9 $ ! ! +B ! % ! %: ! ! ! + ( +& & " " ( !" % (' 4 +(, &9& &! & ; % ! "" +I!$& $$ & ' C # ' " ( ! ( +/# ; - " +8.5.Classicationofgaits8 ( ;(!++( &% : &'& $ & &! % '&+4 $! ! 8 # $ !'' 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