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Robertson3d:\Corel\Suite8\Programs\Masters\Color\DEFAULT.MSTAdmin68$D:\COREL\SUITE8\PROGRAMS\PRWIN8.EXEd 5Background 6 TitleBulleted ListTextOrg ChartData ChartCombinationWPCT}C:P\\TIN-3'XXXXKKKK5/ @1 'Q; ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?WPCd ~ t(uAO,{QoWohono>|"XXXX'3XXXX3' >!,Q} %C \ yb%&&'T3) **+,-_%/ ;dB 2PTU \b2 nmr"WPC 6,B-  62  I  .3  InverseDynamics....2  1  .3  D.GordonE.Robertson,Ph.D. SchoolofHumanKineticsUniversityofOttawa #-#$BCD-2  I  .3  InverseDynamics....2  1  .3  Definition..2  A  .3  Theprocessofderivingthe #kinetics #(i.e.,forcesandmomentsofforce)necessarytoproducethe #kinematics #(observedmotion)ofbodieswithknown #inertialproperties #(i.e.,massandmomentofinertia)...2  A  .3  Typicallytheprocessisusedtocompute #internalforcesandmoments #when #externalforces #areknownandthereare #no #closedkinematicchains.-# %HIJ-2  I  .3  Two-dimensionalDerivation....2  1  .3  Step-by-stepexplanationofderivation..2  A  .3  Thefollowingslidesoutlinethederivationoftheequationsfordetermining # @netforcesandmomentsofforce # @forthetwo-dimensionalcase...2  A  .3  Thethree-dimensionalcasefollowsthesameprocedure.2(./022  I  .3  InverseDynamics....2  1  .3  KinematicChains,Segment&Assumptions..2  A  .3  Firstdividebodyinto # @kinematicchains # @..2  A  .3  Nextdividechainsinto # @segments # @..2  A  .3  Assumethateachsegmentisa  # @rigidbody # @..2  A  .3  Assumethateachjointisrotationally # @frictionless # @,"!0!" ,2  I  .3  2(, 22  I  .3  InverseDynamics....2  1  .3  OrderingofSegments..2  A  .3  Startwiththe # @terminal  @segmentofakinematicchain..2  A  .3  The # @groundreactionforces  @oftheterminalsegmentmustbeknown(i.e.,measured)orzero(i.e.,free-ended)..2  A  .3  Ifnot,usetheotherendofthechain..2  A  .3  If # @externalforces  @areunknown,measurethem,otherwise, # youcannotanalyze # @2(#$+#22  I  .3  InverseDynamics....2  1  .3  Free-bodyDiagram..2  A  .3  Makea # @free-bodydiagram  @(FBD)oftheterminalsegment..2  A  .3  # @Rules: # @....2  1  .3  Addallknownforcesthatdirectlyinfluencethefree-body....2  1  .3  Whereverfree-bodycontactstheenvironmentoranotherbodyaddunknownforceandmoment....2  1  .3  Simplifyunknownforceswhenpossible(i.e.,doesaforcehaveaknowndirection,canforcebeassumedtobezero,issurfacefrictionless) # @1'12  12  I  .3  1.....2  1  .3  Drawfree-bodydiagramoffoot(aterminalsegment)1';< 12  I  .3  2.....2  1  .3  Addweightvectortofree-bodydiagramatcentreofgravity1'  12  I  .3  3.....2  1  .3  Addgroundreactionforceatcentreofpressure # @1'  12  I  .3  4.....2  1  .3  Addallmuscleforcesattheirpointsofapplication1' 12  I  .3  5.....2  1  .3  Addbone-on-bone&ligamentforces&jointmomentofforce2( @A?22  I  .3  EquationsareIndeterminant....2  1  .3  ToomanyUnknowns,ToofewEquations..2  A  .3  In # @twodimensions # @therearethreeequationsofmotion,buttherearemorethanthreeunknownforces(twoormoremusclesperjoint,severalligaments,skin,jointcapsule,bone-on-bone(actuallycartilage)forces,etc.)..2  A  .3  Thus,equationsofmotionareindeterminantandcannotbesolved # @...2  A  .3  In # @threedimensions # @therearesixequationsbutmanymorethansixunknowns.(FGE(2  I  .3  Solution....2  1  .3  Reducenumberofunknownstothree(2D)orsix(3D)..2  A  .3  Thesolutionistoreducethenumberofunknownstothree(orsixfor3D)..2  A  .3  Thesearecalledthe # @netforce # @( # @Fx,Fy # @)andthe # @netmomentofforce # @( # @Mz # @)for2D̀or( # @Fx, # @ # Fy,Fz # @)and( # @` # @Mx,My,Mz # @)for3D1'  12  I  .3  5a.....2  1  .3  Considerasinglemuscleforce(F)onthefoot1'   12  I  .3  5b.....2  1  .3  Movemuscleforcetojointcentre(F*)1' 12  I  .3  5c.....2  1  .3  Addbalancingforce(!F*)1' 12  I  .3  5d.....2  1  .3  Forcecouple(F,!F*)isequaltofreemomentofforce(MFk) # @1' 12  I  .3  5e.....2  1  .3  Replacecouplewithfreemoment1'  12  I  .3  5.....2  1  .3  Showallforcesagain1'   12  I  .3  6.....2  1  .3  Replaceeachmuscleforcewithequivalentforceandfreemomentofforce1'   12  I  .3  7.....2  1  .3  Addallankleforcesandmomentstoobtainnetankleforceandmomentofforce1'   12  I  .3  8.....2  1  .3  Showcompletefree-bodydiagram1'=> 12  I  .3  9.....2  1  .3  Showpositionvectors(rankle,rground # @ # @)-#-2  I  .3  ThreeEquationsofMotionfor # @Foot `L @....2  1  .3  NewtonsSecondLaw..2  A  .3  c&d9 Z6Times New Roman Regularc@@@ # @$Fx=max:Q?@  Z$Symbol RegularQ@@@  @....2  1  .3  # @Fx(ankle)  @+Fx(ground)=max(foot)..2  A  .3  c&d9 Z6Times New Roman Regularc@@@ # @$Fy=may:Q?@  Z$Symbol RegularQ@@@  @....2  1  .3   # @Fy(ankle)  @+Fy(ground)!mg=may(foot)..2  A  .3  c&d9 Z6Times New Roman Regularc@@@ # @$Mz=IQ?@  Z$Symbol RegularQ@@@Ԁac&d9 Z6Times New Roman Regularc@@@:Q?@  Z$Symbol RegularQ@@@  @....2  1  .3  # @Mz(ankle)  @+[r # @ankle'Fankle]z̀+[rground'Fground]z=IfootQ?@  Z$Symbol RegularQxxxac&d9 Z6Times New Roman Regularcxxx(foot)2(89:22  I  .3  MomentofForceasCrossProduct....2  1  .3  AlsocalledVectorProduct..2  A  .3  the # @momentofaforce # @( # @M # @)isdefinedasthecross-product( # @x # @)ofapositionvector( # @r # @ނ)anditsforce( # @F # @ނ).I.e.,M=r # @ނx # @ނF..2  A  .3  Mz=[r'F]z=rxFy # @ނ!  @ނryFx..2  A  .3  rankle=(xankle!xfoot,yankle!yfoot)..2  A  .3  [...]zmeanstakethescalarportioninthez-direction  @ނ7- %&'72  I  .3  EquationsofMotionfor # @ނFoot # @ނ....2  1  .3  SolvefortheUnknowns..2  A  .3  # @ނ$Fx=max:  @#....2  1  .3   # @ނFx(ankle) # @ނ # @=max(foot)!Fx(ground) # @ނ..2  A  .3  # @ނ$Fy=may:  @#....2  1  .3  # @ނFy(ankle)  @#  @=may(foot)!Fy(ground)+mg  @#..2  A  .3  # @ނ$Mz=IQ?@  Z$Symbol RegularQ@@@ac&d9 Z6Times New Roman Regularc@@@:  @#....2  1  .3  # @ނMz(ankle)  @#  @=IfootQ?@  Z$Symbol RegularQxxxac&d9 Z6Times New Roman Regularcxxx(foot)![rankle'Fankle]z̀![rground'Fground]z # @ނ....2  1  .3  # @ނNote, # @ނmomentofinertia # @ނ( # @ނIfoot # @ނ)isaboutcentreofmassoffoot,butproximalordistalends) # @ނ2(45322  I  .3  ApplyNewtonsThirdLawtoLeg....2  1  .3  Reaction=!Action..2  A  .3  Netforceandmomentofforceatproximalendof # @ނankle # @ނcausesreactionforceandmomentofforceatdistalendof # @ނleg  @ނ..2  A  .3  Reactionsareoppositeindirectiontoactions..2  A  .3  I.e.,reactionforce=  @ނ!  @ނactionforcereactionmoment= # @ނ!  @ނactionmoment1'67 12  I  .3  10.....2  1  .3  Drawfree-bodydiagramofleg7- 72  I  .3  EquationsofMotionfor # @ނLeg # @ނ....2  1  .3  NewtonsSecondandThirdLaws..2  A  .3  # @ނ$Fx=max:  @#....2  1  .3  # @ނFx(knee)  @# # @ނ!  @#Fx(ankle)=max(leg)..2  A  .3  # @ނ$Fy=may:  @#....2  1  .3  # @ނFy(knee)  @# # @ނ!  @#Fy(ankle)!mg=may(leg)..2  A  .3  # @ނ$Mz=IQ?@  Z$Symbol RegularQ@@@Ԁac&d9 Z6Times New Roman Regularc@@@:  @#....2  1  .3   # @ނMz(knee)  @#+[r # @ނknee'Fknee]z # @ނ!  @#Mz(ankle)  @ނ̀+[rankle # @ނ' # @ނ! # @ނFankle]z=IlegQ?@  Z$Symbol RegularQxxxac&d9 Z6Times New Roman Regularcxxx(leg)7- ()*72  I  .3  EquationsofMotionfor # @ނThigh # @ނ....2  1  .3  NewtonsSecondandThirdLaws..2  A  .3  # @ނ$Fx=max:  @#....2  1  .3  # @ނFx(hip)  @# # @ނ!  @#Fx(knee)=max(thigh)..2  A  .3  # @ނ$Fy=may:  @#....2  1  .3  # @ނFy(hip)  @# # @ނ!  @#Fy(knee)!mg=may(thigh)..2  A  .3  # @ނ$Mz=IQ?@  Z$Symbol RegularQ@@@ac&d9 Z6Times New Roman Regularc@@@:  @#....2  1  .3  # @ނMz(hip)  @#+[r # @ނhip'Fhip]z # @ނ!  @#Mz(knee)̀+[rknee' # @ނ! # @ނFknee]z=IthighQ?@  Z$Symbol RegularQxxxac&d9 Z6Times New Roman Regularcxxx(thigh)-#&KLM-2  I  .3  Interpretation....2  1  .3  Mathematicalconceptsnotanatomicalkinetics..2  A  .3  Theseforcesandmomentsare # @ނmathematicalconstructs # 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 9׿aQ 4 @ @ @ @ @ @ f" dddddd$   M .+%#13# L - XO, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxAddgroundreactionforceatcentreofpressure     #   @#WPCXX81%XX81%S3'XXXXKKKK9"R$ O2 OC xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R O.+%#13# Q O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxAddallmuscleforcesattheirpointsofapplicationWPCXX81%XX81%&3'XXXXKKKK9 %& 2& C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrtuvwxyz{|}~##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   %& .+%#13# $8& O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxAddbone-on-bone&ligamentforces&jointmomentofforceWPC XX81%XX81%f3'XXXXKKKK9"e : 2 : C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   e : .+%#13# d O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxConsiderasinglemuscleforce(F)onthefootWPCXX81%XX81%73'XXXXKKKK9"6# 2 C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   6 .+%#13# 5 O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxMovemuscleforcetojointcentre(F*)WPC!XX81%XX81%c3'XXXXKKKK9<Ѐ| " 2C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   -.)*+%#'153# bM O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxAddbalancingforce(!F*)WPCXX81%XX81%k3'XXXXKKKK9"j!h 2!h C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)? 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O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxForcecouple(F,!F*)isequaltofreemomentofforce(MFk)     #   @#WPCXX81%XX81%p3'XXXXKKKK9"o# 2 C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   o .+%#13# n O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxReplacecouplewithfreemomentWPCXX81%XX81%t3'XXXXKKKK9"s% 2 C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   s .+%#13# r O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxShowallforcesagainWPCXX81%XX81%-3'XXXXKKKK9",#(2(C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   ,(.+%#13# +(hO, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxReplaceeachmuscleforcewithequivalentforceandfreemomentofforceWPC+XX81%XX81%23'XXXXKKKK9"1#02#0C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   1#0.+%#13# 0#z O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxAddallankleforcesandmomentstoobtainnetankleforceandmomentofforceWPCXX81%XX81%3'XXXXKKKK9""J 2J C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   J .+%#13#  O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxShowcompletefree-bodydiagramWPCXX81%XX81%w3'XXXXKKKK9"q&u/"u/"E @@?@  Z$Symbol Regular +#W xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#5!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   qu/"'13# p+:/1c&d9 Z6Times New Roman Regularc        @#] !!!!!!]Q?@  Z$Symbol RegularQ@@@0M..c&d9 Z6Times New Roman Regularc@@@     #   @#$Fx=max:Q?@  Z$Symbol RegularQ@@@        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @#Fx(ankle)        @#+Fx(ground)=max(foot)fQ?@  Z$Symbol RegularQ@@@0M..c&d9 Z6Times New Roman Regularc@@@     #   @#$Fy=may:Q?@  Z$Symbol RegularQ@@@        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @#Fy(ankle)        @#+Fy(ground)!mg=may(foot)fQ?@  Z$Symbol RegularQ@@@0M..c&d9 Z6Times New Roman Regularc@@@     #   @#$Mz=IQ?@  Z$Symbol RegularQ@@@Ԁac&d9 Z6Times New Roman Regularc@@@:Q?@  Z$Symbol RegularQ@@@        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @#Mz(ankle)        @#+[r     #   @#ankle'Fankle]z̀+[rground'Fground]z=IfootQ?@  Z$Symbol RegularQxxxa..c&d9 Z6Times New Roman Regularcxxx(foot)WPC%XX81%XX81%w3'XXXXKKKK9"t2R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  tR/#%`L13Y r/#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO ThreeEquationsofMotionfor     #   @YFoot     `L   @Y.+%19׿3aQ `sR//SRSWPCXX81%XX81%w3'XXXXKKKK9"v=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   v/.+%#13# u5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularONewtonsSecondLawWPCXX81%XX81%/3'XXXXKKKK9",2R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  ,R/#%`L13Y *C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO EquationsofMotionfor     #   @ނLeg     `L   @Y.+%19׿3aQ `+R// R WPCXX81%XX81%/3'XXXXKKKK9".=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   ./.+%#13# -5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularONewtonsSecondandThirdLawsWPCRXX81%XX81%/3'XXXXKKKK9")@u/"u/"W @@&d9 Z6Times New Roman Regular +#W xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#5!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   )u/"'13# (+:/c&d9 Z6Times New Roman Regularc        @#] !!!!!!]c&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Fx=max:        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @ނFx(knee)        @#     #   @ނ!        @#Fx(ankle)=max(leg)fc&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Fy=may:        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @ނFy(knee)        @#     #   @ނ!        @#Fy(ankle)!mg=may(leg)fc&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Mz=IQ?@  Z$Symbol RegularQ@@@Ԁac&d9 Z6Times New Roman Regularc@@@:        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @ނMz(knee)        @#+[r     #   @ނknee'Fknee]z     #   @ނ!        @#Mz(ankle)        @ނ̀+[rankle     #   @ނ'     #   @ނ!     #   @ނFankle]z=IlegQ?@  Z$Symbol RegularQxxxaQ..Qc&d9 Z6Times New Roman Regularcxxx(leg)WPCXX81%XX81%&3'XXXX,,,,9"#2,R/# R/# #R/#%`L'13Y !6 /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO InverseDynamics.+%19׿3aQ `"R// R WPCXX81%XX81%=3'XXXXKKKK9":2+R/# R/# :R/#.%`L13Y 8C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO InverseDynamics.+%19׿3aQ `9R// R WPC@XX81%XX81%=3'XXXXKKKK9"7&ue/"Bue/"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   7ue/".'13 6+e:/O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Startwiththe     #   @ނterminal        @ނsegmentofakinematicchainfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..The     #   @ނgroundreactionforces        @ނoftheterminalsegmentmustbeknown(i.e.,measured)orzero(i.e.,free-ended)fO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Ifnot,usetheotherendofthechainfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..If     #   @ނexternalforces        @ނareunknown,measurethem,otherwise,     #   ނyoucannotanalyze     #   @ނWPC&XX81%XX81%3'XXXXKKKK9"+.0 2.0 C ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;Y)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  .0 %O;'13Y / O, \AZ"Arial RegularO` ` `      O;   @Y  Q QO, \AZ"Arial RegularO` ` `  ComputerizetheProcess.+%19׿3aQ `.0.0VVWPCSXX81%XX81%g3'XXXXKKKK9"20220C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   20-.)*+%#'153# f|/O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxE.g.,2D:BiomechMotionAnalysisSystem;3D:Visual3D,Polygon,KinTools,KinTrak,Kwon3DWPCXX81%XX81%D3'XXXXKKKK9">Su/"Bu/"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   >u/".'13 =+:/O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Makea     #   @ނfree-bodydiagram        @ނ(FBD)oftheterminalsegmentfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..     #   @ނRules:     #   @ނO, \AZ"Arial RegularOxxx        @#M..M        @    O, \AZ"Arial RegularOxxx        @#0..Addallknownforcesthatdirectlyinfluencethefree-bodyM..MO, \AZ"Arial RegularOxxx        @    O, \AZ"Arial RegularOxxx        @#0..Whereverfree-bodycontactstheenvironmentoranotherbodyaddunknownforceandmomentM..MO, \AZ"Arial RegularOxxx        @    O, \AZ"Arial RegularOxxx        @#0..Simplifyunknownforceswhenpossible(i.e.,doesaforcehaveaknowndirection,canforcebeassumedtobezero,issurfacefrictionless)M..M     #   @#WPCXX81%XX81%D3'XXXXKKKK9"A2%R/# R/# AR/#.%`L13Y ?C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO InverseDynamics.+%19׿3aQ `@R// R WPCXX81%XX81%(3'XXXXKKKK9"%2R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  %R/#%`L13Y #C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO EquationsofMotionfor     #   @ނFoot     `L   @Y.+%19׿3aQ `$R// R WPCXX81%XX81%(3'XXXXKKKK9"'=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   '/.+%#13# &5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOSolvefortheUnknownsWPCXX81%XX81%(3'XXXXKKKK9""&u/"u/"W @@&d9 Z6Times New Roman Regular +#W xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#5!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   "u/"'13# !+:/ec&d9 Z6Times New Roman Regularc        @#] !!!!!!]c&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Fx=max:        @#c&d9 Z6Times New Roman Regularcxxx        @#M..M0..     #   @ނFx(ankle)     #   @ނ     #   @=max(foot)!Fx(ground)     #   @ނfc&d9 Z6Times New Roman Regularc@@@        @#0M..     #   @ނ$Fy=may:        @#c&d9 Z6Times New Roman Regularcxxx        @#M..M0..     #   @ނFy(ankle)        @#        @=may(foot)!Fy(ground)+mg        @#fc&d9 Z6Times New Roman Regularc@@@        @#0M..     #   @ނ$Mz=IQ?@  Z$Symbol RegularQ@@@ac&d9 Z6Times New Roman Regularc@@@:        @#c&d9 Z6Times New Roman Regularcxxx        @#M..M0..     #   @ނMz(ankle)        @#        @=IfootQ?@  Z$Symbol RegularQxxxac&d9 Z6Times New Roman Regularcxxx(foot)![rankle'Fankle]zM..M![rground'Fground]z     #   @ނc&d9 Z6Times New Roman Regularcxxx        @#M..M0..     #   @ނNote,     #   @ނmomentofinertia     #   @ނ(     #   @ނIfoot     #   @ނ)isaboutcentreofmassoffoot,butproximalordistalends)     #   @ނWPC XX81%XX81%3'XXXXKKKK9"2R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  R/#%`L13Y C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO EquationsofMotionfor     #   @ނThigh     `L   @Y.+%19׿3aQ `R// R WPCXX81%XX81%3'XXXXKKKK9"=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   /.+%#13# 5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularONewtonsSecondandThirdLawsWPC\XX81%XX81%3'XXXXKKKK9"@u/"u/"W @@&d9 Z6Times New Roman Regular +#W xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#5!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   u/"'13# +:/c&d9 Z6Times New Roman Regularc        @#] !!!!!!]c&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Fx=max:        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @ނFx(hip)        @#     #   @ނ!        @#Fx(knee)=max(thigh)fc&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Fy=may:        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @ނFy(hip)        @#     #   @ނ!        @#Fy(knee)!mg=may(thigh)fc&d9 Z6Times New Roman Regularc@@@0M..     #   @ނ$Mz=IQ?@  Z$Symbol RegularQ@@@ac&d9 Z6Times New Roman Regularc@@@:        @#M..Mc&d9 Z6Times New Roman Regularcxxx0..     #   @ނMz(hip)        @#+[r     #   @ނhip'Fhip]z     #   @ނ!        @#Mz(knee)̀+[rknee'     #   @ނ!     #   @ނFknee]z=IthighQ?@  Z$Symbol RegularQxxxa..c&d9 Z6Times New Roman Regularcxxx(thigh)WPCXX81%XX81%D3'XXXXKKKK9"C=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   C/.+%#13# B5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOFree-bodyDiagramWPCXX81%XX81%=3'XXXXKKKK9"<=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   </.+%#13# ;5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOOrderingofSegmentsWPCXX81%XX81%&3'XXXX,,,,9"%=l/2l/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   %l/.+%#13# $5lF/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOD.GordonE.Robertson,Ph.D.        @#SchoolofHumanKineticsUniversityofOttawa     #   @#WPCXX81%XX81%63'XXXXKKKK9"32)R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  3R/#.%`L13Y 1C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO InverseDynamics.+%19׿3aQ `2R// R WPCXX81%XX81%63'XXXXKKKK9"5=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   5/.+%#13# 45F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOKinematicChains,Segment&AssumptionsWPCXX81%XX81%63'XXXXKKKK9"0&u /" u /"  0u /".'13 /+ :/O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Firstdividebodyinto     #   @ނkinematicchains        @ނfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Nextdividechainsinto     #   @ނsegments        @ނfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Assumethateachsegmentisa      #   @ނrigidbody        @ނfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Assumethateachjointisrotationally     #   @ނfrictionless     #   @ނWPCXX81%XX81%I3'XXXXKKKK9"FJ.0 2.0 C ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;Y)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  F.0 %O;'13Y D/ O, \AZ"Arial RegularO` ` `      O;   @Y  Q QO, \AZ"Arial RegularO` ` `  1..+%19׿3aQ `E.0.0VVWPCXX81%XX81%I3'XXXXKKKK9"H 2 C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   H .+%#13# G - O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxDrawfree-bodydiagramoffoot(aterminalsegment)WPCXX81%XX81%3'XXXXKKKK9"&u/"Bu/"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   u/".'13 +:/ O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Netforceandmomentofforceatproximalendof     #   @ނankle     #   @ނcausesreactionforceandmomentofforceatdistalendof     #   @ނleg        @ނfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..ReactionsareoppositeindirectiontoactionsfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..I.e.,reactionforce=        @ނ!        @ނactionforceπreactionmoment=     #   @ނ!        @ނactionmomentWPC.XX81%XX81%3'XXXXKKKK9"2,R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  R/#.%`L13Y C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO ApplyNewtonsThirdLawtoLeg.+%19׿3aQ `R// R WPCXX81%XX81%3'XXXXKKKK9"=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   /.+%#13# 5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOReaction=!ActionWPCXX81%XX81%3'XXXXKKKK9"L.0 2.0 C ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;Y)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  .0 %O;'13Y / O, \AZ"Arial RegularO` ` `      O;   @Y  Q QO, \AZ"Arial RegularO` ` `  10..+%19׿3aQ `.0.0MMWPCXX81%XX81%3'XXXXKKKK9"020C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   0-.)*+%#'153# z O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxDrawfree-bodydiagramoflegWPC-XX81%XX81%~3'XXXXKKKK9"{2R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  {R/#.%`L13Y y/#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO MomentofForceasCrossProduct.+%19׿3aQ `zR//SRSWPCXX81%XX81%~3'XXXXKKKK9"}=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   }/.+%#13# |5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOAlsocalledVectorProductWPC7XX81%XX81%~3'XXXXKKKK9"x&ue/"Bue/"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   xue/".'13 w+e:/O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..the     #   @ނmomentofaforce     #   @ނ(     #   @ނM     #   @ނ)isdefinedasthecross-product(     #   @ނx     #   @ނ)ofapositionvector(     #   @r     #   @ނ)anditsforce(     #   @F     #   @ނ).I.e.,M=r     #   @ނx     #   @ނFfO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Mz=[r'F]z=rxFy     #   @ނ!        @ނryFxfO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..rankle=(xankle!xfoot,yankle!yfoot)fO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..[...]zmeanstakethescalarportioninthez-direction        @ނWPCXX81%XX81%@3'XXXXKKKK9"=L.0 2.0 C ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;Y)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  =.0 %O;'13Y ;/ O, \AZ"Arial RegularO` ` `      O;   @Y  Q QO, \AZ"Arial RegularO` ` `  2..+%19׿3aQ `<.0.0MMWPC XX81%XX81%@3'XXXXKKKK9"? 2 C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   ? .+%#13# > - O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxAddweightvectortofree-bodydiagramatcentreofgravityWPCXX81%XX81%73'XXXXKKKK9"4H.0 2.0 C ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;YC ` ` , \AZ"Arial Regular +O;Y)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  4.0 %O;'13Y 2/ O, \AZ"Arial RegularO` ` `      O;   @Y  Q QO, \AZ"Arial RegularO` ` `  9..+%19׿3aQ `3.0.0KKWPCXX81%XX81%73'XXXXKKKK9"6 2 C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##C xx, \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   6 .+%#13# 5 O, \AZ"Arial RegularOxxx     #   @#  Q QO, \AZ"Arial RegularOxxxShowpositionvectors(rankle,rground     #   @#     #   @#)WPCXX81%XX81%Z3'XXXXKKKK9"T8Qu/)Bu/)C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   Tu/).'13 S+9/ O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..In     #   @ނtwodimensions     #   @ނtherearethreeequationsofmotion,buttherearemorethanthreeunknownforces(twoormoremusclesperjoint,severalligaments,skin,jointcapsule,bone-on-bone(actuallycartilage)forces,etc.)fO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Thus,equationsofmotionareindeterminantandcannotbesolved     #   @ނ.fO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..In     #   @ނthreedimensions     #   @ނtherearesixequationsbutmanymorethansixunknowns.WPC'XX81%XX81%Z3'XXXXKKKK9"W2&R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  WR/#.%`L13Y UC /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO EquationsareIndeterminant.+%19׿3aQ `VR// R WPCXX81%XX81      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{|}~%Z3'XXXXKKKK9"Y=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   Y/.+%#13# X5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOToomanyUnknowns,ToofewEquationsWPCXX81%XX81%(3'XXXXKKKK9"%2,R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  %R/#.%`L13Y #C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO InverseDynamics.+%19׿3aQ `$R// R WPCXX81%XX81%(3'XXXXKKKK9"'=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   '/.+%#13# &5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularODefinitionWPCXX81%XX81%(3'XXXXKKKK9""&u/"Bu/"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   "u/".'13 !+:/ O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Theprocessofderivingthe     #   @ނkinetics     #   @ނ(i.e.,forcesandmomentsofforce)necessarytoproducethe     #   @ނkinematics     #   @ނ(observedmotion)ofbodieswithknown     #   @ނinertialproperties     #   @ނ(i.e.,massandmomentofinertia).fO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Typicallytheprocessisusedtocompute     #   @ނinternalforcesandmoments     #   @ނwhen     #   @ނexternalforces     #   @ނareknownandthereare     #   @ނno     #   @ނclosedkinematicchains.WPCXX81%XX81%a3'XXXXKKKK9"[&u /"Bu /"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   [u /".'13 Z+ :/aO, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Thesolutionistoreducethenumberofunknownstothree(orsixfor3D)fO, \AZ"Arial RegularO@@@        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Thesearecalledthe     #   @ނnetforce     #   @ނ(     #   @ނFx,Fy     #   @ނ)andthe     #   @ނnetmomentofforce     #   @ނ(     #   @ނMz     #   @ނ)for2D̀or(     #   @ނFx,     #   @ނ     #   ނFy,Fz     #   @ނ)and(     #   @`ނ     #   @ނMx,My,Mz     #   @ނ)for3DWPCXX81%XX81%a3'XXXXKKKK9"^2+R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  ^R/#.%`L13Y \C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO Solution.+%19׿3aQ `]R// R WPCXX81%XX81%a3'XXXXKKKK9"`=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   `/.+%#13# _5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOReducenumberofunknownstothree(2D)orsix(3D)WPCXX81%XX81%/3'XXXXKKKK9",2+R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   ,R/#%`L *C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO Two-dimensionalDerivation.+%19׿3aQ `+R// R WPCXX81%XX81%/3'XXXXKKKK9".=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   ./.+%#13# -5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOStep-by-stepexplanationofderivationWPC XX81%XX81%/3'XXXXKKKK9")&u /"2u /"C @@, \AZ"Arial Regular +YC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#)?  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   )u /"%'13Y (+ :/uO, \AZ"Arial RegularO        @Y  Q QO, \AZ"Arial RegularO@@@Thefollowingslidesoutlinethederivationoftheequationsfordetermining     #   @Ynetforcesandmomentsofforce     #   @Yforthetwo-dimensionalcase.f  O, \AZ"Arial RegularO@@@        @YThethree-dimensionalcasefollowsthesameprocedure.WPCXX81%XX81%3'XXXXKKKK9"2+R/#2R/#C , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LYC , \AZ"Arial Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$  R/#.%`L13Y C /#O, \AZ"Arial RegularO     `L   @Y  Q QO, \AZ"Arial RegularO Interpretation.+%19׿3aQ `R// R WPCXX81%XX81%3'XXXXKKKK9"=/2/C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##C , \AZ"Arial Regular +##)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   /.+%#13# 5F/O, \AZ"Arial RegularO     #   @#  Q QO, \AZ"Arial RegularOMathematicalconceptsnotanatomicalkineticsWPCVXX81%XX81%3'XXXXKKKK9"&u/"Bu/"C @@, \AZ"Arial Regular +ނC xx, \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#C , \AZ"Arial Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   u/".'13 +:/' O, \AZ"Arial RegularO        @a /!!!!!!aO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Theseforcesandmomentsare     #   @ނmathematicalconstructs     #   @ނNOTactualforcesandmoments.fO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Theactualforcesinsidejointsandthemomentsacrossjointsare     #   @ނhigher     #   @ނbecauseofthe     #   @ނcocontractionsofantagonists     #   @ނ.fO, \AZ"Arial RegularO@@@        @ނ        @  /  O, \AZ"Arial RegularO@@@        @ނ0M..Furthermore,thereisnocertainmethodtoapportionthenetforcesandmomentstotheindividualanatomicalstructures.op}q w#Y)nLl4J=s nlv\<x>-kjX]nw v2WPCoXX81%G >"(b%a3'XXXX____ ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9`@ # sG >"(b%$*eF 2'<(`` @@@ ---;;;???@@@EEEGGGQQQRRRTTTVVVWWWYYY[[[\\\]]]^^^___```aaabbbdddeeeggghhhjjjkkklllmmmqqqrrrsssvvvwww{{{PP`` @`p(P0`@P`p @`pP` @@``pp(P0`@P`p @`pȐРذPP`` @`pP(`0@P`p @`pp@ P0`0`@`@p@pPP`pఀа`````````p````p```````~8}4~xoTox~xo$w[w$oximS[`w ~xo>S[w$Toxgm'('%S[`w~x:'9WOW'O7'S[w$Tox~K=WIQZWG=('S[w$ox}ewW(_n6nfn6nfn7pgp/'S`w~x O BhPPPPQQIW9'Sw$oxy~x O5QPPPPPPPIW9'Sw$ToxubwWZn6nfn6nfn6nfn6nfn6nfp/'S`w~K';pPPPPPPPPPPIW9''('Sw$oxq~x:+=VLPPPPPPPPPPPIW'WOW9'Sw$Toxn~1K>tPPPPPPPPPPPPPZQIW9'S[w xm~xo>( BhPPPPPPPPPPPPPPPPIW9'O$oxk~:xo$wS'0QPPPPPPPPPPPPPPPPPIW9'Sw$Toxh\wW/'/pfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfp+S`wZw`S/ofn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nZS[`wYwW.h6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6ndS%S`wXwW-*ofn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn('(/m+S[`wVw`S.'(+_n6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn1W+vwZ'S[`wVm3ogp7nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nfn6nZ+uwW+v/('S`w~xoT>5%>QQPPPPPPPPPPPPPPPPPPPPF [zK=^y 'Q9('Sw xUSm4k>oen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9h-pn.vw5(N\vRvT5(4oen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nbPqvTqv\X]w.Tv~O7("YQ\\\\\\\\\\\\\\\\\\\\\\Ov 2O2y"JX KA*_#(C"yQQvT7.qen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nbUqnqvT Obq.N\v~>yS8C=s\\\\\\\\\\\\\\\\\\\\\\\[!tU 22y"C Ot%_8NJ8WSyLOm4&!j9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nbOqvTqvT nqN\vNvT2k>oen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nbOnp)nmTn-TvN5m(4oen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nek&TvqnvTTq.Tvv\X\vM6vToen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen4Jk.)pvTnmnv.NXN NX\vM5m&!h9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nX(4qqmTvnnv5(NX\v~5y9 i\\\\\\\\\\\\\\\\\\\\\\[!O("_;~+ }y:_"(NXJ8Sy!L3m')\n9nen9nen9nen9nen9nen9nen9nen9nen9nen9nen9nbOqp-XwnnY'N\v~4y9 ~Xe\\\\\\\\\\\\\\\\\\\\\yO2 u Cyn[N()(&&&()((NJ"y~2O :t\\\\\\\\\\\\\\\\\\\\\Q!\ q 63 y"7C&9_&_"H"(9M>M"YLY"Yu9)(()(OJ2vToen9nen9nen9nen4Tbk!ken9nen9nen9nen9nen9nen9nXTvwXXwqONk>vw5(*]o>oen9nen9qeq>qeq.TvJ4mh9nen9nen9nbOOTG!G!ken9nen9nen9nen9nen9nbO&m m)pp--qw5(N5nen9nen9nen9nen9nXmIvT-oen9nen9nekG!ken9nen9nen9nen9nbO.n5(O\vnqv.qqXn9nen9nen9nen9nek-p~O =p\\\[!{,&G:,D\\\\\Q:[yO8Xa:YOG yX922q\\\\\\\\\~yS" @d\\[!tOG()"H"(G:!Q\\\\lCt, {+y2"SyyX9\u q\\\\\\\\3%yG m-oen9nek()5wqT4nek!O]ON\wv5(N\vn-pv5(-pp))nen9nen9nen9nen9nXmEv\O*]n9nek!G.wnT-J &-pw.N\vp-Xww5)(XwqXn9nen9nen9nen9nen)nDvTj9nen4J.vvT J&()(!qv.N\vwXmq Tvn)nen9nen9nen9nen9nXTvDm-oen9nX.vm()5n>ww>pv5( N\vmnv\XO TvnXn9nen9nen9nen9nen)mCvTXn9nek(5vp4( &(5ww5(mnTvm& mvT)nen9nen9nen9nen9n]*nCm'-nen4Jqq()5n>ww5(nnTvn.n.TvmXn9nen9nen9nen9nen9jTvAvYKodnvYKWr=ndj3w x3Wr8) KUKEPj)KUKKUK'!im (1ndnp23rdnbKM&y y'MwyVDMK)'xD$ )-/H$'/EXXXXXXXXXX}&Vy;vPk=ndn'E&y4vYQ2odnnan-ryCo=,,2=ysnx7 6qTE9nan>nan>nan>iLu/m i>nan>nT6qy=2,,2=yq6Lux7 rp5+Sn>nan>nan>nao<,m/n kan>nan.ryCyrLur Lusjan>nan>nan>naoLu~ ___z ;Yz[vLv X9z)V #,/X______>(X9z ~z" ___NI{s ",X9z~N/!z ~3_Pj______bDV .p.i>nan>ir y7HVvx7Luq<,I[n>nan>nan>n[JLu.vT kan>nao6nrHQM ms-nan>nan>nan5(Lu.m i>nan>nTLuq<,(Luq<,ML9nan>nan>nas7Lu.n kan>nan.my=2,!l7LuTSn>nan>nan>nX0m.q6 i>nan>nTnyCoT Lup5+jan>nan>nan?lIVu~~N/___z vLz  V :R%,_____^#,IO~c.X__v ,"z~~V*,!zv2x#e_____]C!z~z Pj__gExX9z~w2U/X9zbNJB______X9z0x7.nan>nan5+muVur LuuL!jan>nan>nao<,Lu1rTn>nan>nas7IVup-mq6 muLi>nan>nan>nas7m~@'e___o( IE=^Yztu^~N/!zzU`Uzz9X(#e_____o($Yz1n -nan>nan?lHVuuL!qx7Lum!HQH,=o=,-nan>nan>nan?lm1q<,Tn>nan>nao<,HVuuVH6qr HQI(0(ssML9nan>nan>nao6Lu3sjan>nan>nas=,(LuuVHrq6 ssnq<,Sn>nan>nan>nX0Lu~)/X____\_%,9D*,X9ziOH;Yz~N/ ,")~3_G'e____g\X9z4rSn>nan>nan>nasCoaoHVuuLHVuvT sx=,Luq6I[n>nan>nao<,m4nI[n>nan>nan>nan>i!(LuuZoZumLum LuTLux=,-nan>nan>nas=,IVu~",IH'e______bD6(X=Yz$zU{hD=^Yz1zY$X9`Uz! Va7Vw2_y_____\_/IO6s-nan>nan>nan>naoClHVuuZoVQHHVoVum H Lusnq<,jan>nan>nan>nTIVu}6nTn>nan>nan>nan>nao<,HVuuL LuuVI,2,Luq6nsi>nan>nan>nan5+m}6q6.nan>nan>nan>nan>nas=,HQVuuZoVHHVum,2=ysLux7LunJ[n>nan>nan>naoIVu{7uJ0;WsBn]nAn]nAn]nAn]nAnXF0@yyyyo@o@0*H38pq:muRECEKE sp30?quRFlo]nAn]nAn]j!C0?qy8uJ0;WsBn]nAn]nAn]nAn]nAnXFC/70070** JFFSF5QYsFRuuWoRKRuuJ ;yuJnm*!iAn]nAn]nAn]j!Csy9uRF0;VnAn]nAn]nAn]nAn]nAn]j!C!CCFSJ=n]nAnXJ6J07@yp3Fnn :quJsuJj]nAn]nAn]nAn]j:qx~z]+0#l|))JFJF;)C($`$k$wYJ07XIOvc +("%z'Hvz0@t^nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAnXF0@o@ysJuuJ5H i!Jum Pvm/;QJ5VnAn]nAn]nAn]nAinnNuRE>0@t^nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAnXF*;o@7@o@7@yy;muJ5H*!nuRF;yvRE/;H;t^nAn]nAn]nAn]jtnPuRE>0@t^nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAnXF**070FRuuJ5H sm;yuRE5H0?o]nAn]nAn]nAnYK0@ylRuRE>0@t^oCo]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAn]nAnXFmvP* 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