Presentation SolvingFKPparallelRobots MUN RoSe 2014

8/12/2019 Presentation SolvingFKPparallelRobots MUN RoSe 2014

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Solving the forward kinematics ofparallel robots, a review ofavailable methodsMemorial University of Newfounland

Solving the forward kinematics ofparallel robots, a review ofavailable methodsMemorial University of Newfounland

# "$orial % niv"r&ity


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IntroductionKinematics formulationForward Kinematics roblem

Solving the system!esults and "nalysisSummary

IntroductionKinematics formulationForward Kinematics roblem

Solving the system!esults and "nalysisSummary

Outline




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#he truly parallel manipulator' (ou)h *latfor$ +St",art *latfor$-

' on" fi "d a&"' on" $o il" *lat"for$' 6 in"$atic& chain&ach in"$atic& chain' ,ith on" *ri&$atic actuator' throu)h univ"r&al or all oint&

#he truly parallel manipulator' (ou)h *latfor$ +St",art *latfor$-

' on" fi "d a&"' on" $o il" *lat"for$' 6 in"$atic& chain&ach in"$atic& chain' ,ith on" *ri&$atic actuator' throu)h univ"r&al or all oint&

Introduction




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Inverse kinematics problem

: (iv"n th" )"n"rali "d coordinat"& of th" $ani*ulator "nd'"ff"ctor 7 find th" oint *o&ition& L.

*licit &olution.R"al &olution 9 2

Inverse kinematics problem

: (iv"n th" )"n"rali "d coordinat"& of th" $ani*ulator "nd'

"ff"ctor 7 find th" oint *o&ition& L.

*licit &olution.R"al &olution 9 2

Introduction




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Forward kinematics problem

: (iv"n th" oint *o&ition& L find th" )"n"rali "d coordinat"&7 of th" $ani*ulator "nd'"ff"ctor.

a difficult *ro l"$ +Roth-Proven:40 co$*l" &olution& +La ard-R"al &olution 9 ; co$*l" 9

Forward kinematics problem

: (iv"n th" oint *o&ition& L find th" )"n"rali "d coordinat"&

7 of th" $ani*ulator "nd'"ff"ctor.

a difficult *ro l"$ +Roth-Proven:40 co$*l" &olution& +La ard-R"al &olution 9 ; co$*l" 9

Introduction




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Position based equationsin" varia l"& : = th" fir&t 3 $o il" *latfor$ oint& = A 1 y1 1 2 y2 2 3 y3 3B

ro$ th" CD4 F>5 and F>6 ar" ,ritt"n in

t"r$& of varia l"&

= or$ "t,""n Ei and >i

KinematicsFormulation




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Displacement based equations

#od"l&



Position based equations

#od"l&thr"" *oint $od"l ,ith *latfor$

di$"n&ional con&traint&thr"" *oint $od"l ,ith *latfor$con&traint& ,ith *ointin) a i&

th" thr"" *oint $od"l ,ith con&traint&and function r"co$ ination

th" &i *oint $od"l

Forwards Kinematics

Problem





Transformation into anoptimization problem

or o*ti$i ation t"chniqu"& Hn" o "ctiv" function

"riv"d fro$ th" CD L"t l )i " th" l") l"n)th of

in"$atic& chain i +in*ut ofth" *ro l"$-.

au)$"nt"d y on" con&traint &"t : th" *latfor$ fi "d

di&tanc"& "t,""n th" thr""&"l"ct"d oint *oint& : >1I>2and >3 di&tinct *oint&

,h"r"

,h"r"

# iddl" a&t



Numeric Methods

S"cant #"thod 'J on" &olution

",ton $"thod 'J on" &olution

Fontinuation $"thod ,ithho$otho*y 'J &"v"ral &olution&

yallitic li$ination 'J &"v"ral&olution&

Cnt"rval analy&i& 'J all &olution& orno an&,"r +c"rtifi"d-

("o$"tric Ct"rativ" #"thod 'J on"&olution

Solving the system





Algebraic Methods

yallitic li$ination 'J &"v"ral&olution&R"&ultant& $"thod 'J &"v"ral&olution&(ro" n"r a&"& 'J all " act&olution& +c"rtifi"d-

Solving the system





Optimization Techniques

("n"tic El)orith$ 'J &"v"ral&olution&

Si$ulat"d Enn"alin) 'J &o$"&olution&

Ky rid ("n"tic El)orith$ andSi$ulat"d Enn"alin) 'J all

&olution&(3' F7 'J all &olution&

Solving the system





Newton%s Method' " hav" on" &olution

' i"udonn" in 1?M2

&bservations ' Nuadratic conv"r)"nc"' S$all co$*utation ti$"&' #ay not conv"r)"' Oaco ian inv"r&ion' u$"ric in&ta iliti"&'(ample' P"ry fa&t $"thod for control' on &in)ularity fr"" SS#: 5Q failur"&

' ""d& conv"r)"nc" t"&t a& th" Dantorovich th"or"$

Newton%s Method' " hav" on" &olution

' i"udonn" in 1?M2

&bservations ' Nuadratic conv"r)"nc"' S$all co$*utation ti$"&' #ay not conv"r)"' Oaco ian inv"r&ion' u$"ric in&ta iliti"&'(ample' P"ry fa&t $"thod for control' on &in)ularity fr"" SS#: 5Q failur"&' ""d& conv"r)"nc" t"&t a& th" Dantorovich th"or"$

Solving Methods





Interval "nalysis' Ell &olution&

' #"rl"t in 2005

&bservations ' Nuadratic conv"r)"nc"' Lon) co$*utation ti$"&' #ay not conv"r)"' Oaco ian inv"r&ion' Eccount& for i$*r"ci&ion'(ample' ""d& ",ton & $"thod' Hn &in)ularity fr"" SS#: 5Q failur"&

' ""d& "nclo&ur" t"&t a& ,ith th" Dantorovich th"or"$

Interval "nalysis' Ell &olution&

' #"rl"t in 2005

&bservations ' Nuadratic conv"r)"nc"' Lon) co$*utation ti$"&' #ay not conv"r)"' Oaco ian inv"r&ion' Eccount& for i$*r"ci&ion'(ample' ""d& ",ton & $"thod' Hn &in)ularity fr"" SS#: 5Q failur"&' ""d& "nclo&ur" t"&t a& ,ith th" Dantorovich th"or"$

Solving Methods





$ontinuation method with homothopy

' Ra)havan in 1??3

' " hav" &olution& for a &i$*l" "quation &y&t"$ +7- 0' " ,i&h &olution& for &i$ilar (+7- 0' Fontinuation: K+7 - (+7-T + +7-U(+7--' A0 V 1B&bservations ' #ay $i&& &olution&

' #ay add &olution&' Fro&&in) &olution&' ""d& it"rativ" $"thod'(ample' ro l"$ )oin) fro$ th" SS# to th" 6'6

$ontinuation method with homothopy' Ra)havan in 1??3

' " hav" &olution& for a &i$*l" "quation &y&t"$ +7- 0' " ,i&h &olution& for &i$ilar (+7- 0' Fontinuation: K+7 - (+7-T + +7-U(+7--' A0 V 1B&bservations ' #ay $i&& &olution&

' #ay add &olution&' Fro&&in) &olution&' ""d& it"rativ" $"thod'(ample' ro l"$ )oin) fro$ th" SS# to th" 6'6

Solving Methods





)yallitic 'limination * Numeric

' C&olation to a univariat" "quation

' Ku&ty in 1??4&bservations' "rha*& all &olution&' Fo$*l" &olution& $ay "co$" r"al &olution&' S*uriou& &olution& ar" add"d'(ample' Si$*l"r *arall"l ro ot&: HD' ro l"$: 40 &olution& for th" SS#

)yallitic 'limination * Numeric


' Ku&ty in 1??4&bservations' "rha*& all &olution&' Fo$*l" &olution& $ay "co$" r"al &olution&' S*uriou& &olution& ar" add"d

'(ample' Si$*l"r *arall"l ro ot&: HD' ro l"$: 40 &olution& for th" SS#

Solving Methods





!esultants * "lgebraic


' Ku&ty in 1??4&bservations' "rha*& all &olution&' S*uriou& &olution& ar" add"d' R"quir"& "li$ination &t"* ,ith CD'(ample' Si$*l"r *arall"l ro ot&: HD' ro l"$: 40 &olution& for th" SS#



' Ku&ty in 1??4&bservations' "rha*& all &olution&' S*uriou& &olution& ar" add"d' R"quir"& "li$ination &t"* ,ith CD

'(ample' Si$*l"r *arall"l ro ot&: HD' ro l"$: 40 &olution& for th" SS#

Solving Methods






' Solvin) for R"&+f ) 1- 0 "quival"nt to d"t+#- 0

' Cn c"rtain in&tanc"& th" h"ad t"r$& of th" *olyno$ial& canc"lW th" canc"llation of th" d"t"r$inant

W it add& on" " tran"ou& root.

'



+roebner ases * "lgebraic

' calculation of (ro" n"r a&i&: canonical for$ of id"al

' conv"r&ion to a Rational %niv"ariat" R"*r"&"ntation' La ard au)"r" and Rouilli"r in 1??6 = 2000 *"riod&bservations' Ell " act &olution&

' Rational or int")"r co"ffici"nt&' R"quir"& &olvin) th" %nivariat" "quation'(ample' 36 &olution& for th" SS#' 6'6 co$*utation ti$"&: 1 $in in #a*l"

+roebner ases * "lgebraic

' calculation of (ro" n"r a&i&: canonical for$ of id"al

' conv"r&ion to a Rational %niv"ariat" R"*r"&"ntation

' La ard au)"r" and Rouilli"r in 1??6 = 2000 *"riod&bservations' Ell " act &olution&' Rational or int")"r co"ffici"nt&' R"quir"& &olvin) th" %nivariat" "quation'(ample' 36 &olution& for th" SS#' 6'6 co$*utation ti$"&: 1 $in in #a*l"

Solving Methods





+enetic "lgorithms' " hav" on" &olution

' >oudr"au in 1??6

&bservations ' K"uri&tic co$*utation ti$"&' #ay not conv"r)"' #od"lin) i&&u"' Startin) &olution d"*"ndant'(ample' #ay find $any &olution&throu)h r"*"at"d trial&' S$all"r ro ot&

+enetic "lgorithms' " hav" on" &olution

' >oudr"au in 1??6

&bservations ' K"uri&tic co$*utation ti$"&' #ay not conv"r)"' #od"lin) i&&u"' Startin) &olution d"*"ndant'(ample' #ay find $any &olution&throu)h r"*"at"d trial&' S$all"r ro ot&

Solving Methods





Ooint varia l"& L : X1250I1250I 1250I 1250I 1250I1250YFa&" ,ith 16 r"al r"&ult&

confir$"d y al)" raic$"thod

FKP OOT !" T#F#"D "$%&T$!ONF#'% AT#ON TA(&"

# iddl" a&t # co$*ati l" F ,ith 1.M4 (K dual cor" *roc"&&or& ,ith Linu

Results nalysis





Results nalysis

(ro" n"r a&i& T Rational %nivariat" R"*r"&"ntation




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Results nalysis

(ro" n"r a&i& T Rational %nivariat" R"*r"&"ntation




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Succ"&& rat"&: SE i& 52 Q oth"r& 100 Q Solvin): (3' F7 o tain"d all 16 &olution& (3' F7 out*"rfor$"d th" oth"r& on all account& o*ulation &i " of 200 : "tt"r r"&*on&" ti$"&

# iddl" a&t


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Succ"&& rat"&: SE i& 52 Q oth"r& 100 Q Solvin): (3' F7 o tain"d all 16 &olution& (3' F7 out*"rfor$"d th" oth"r& on all account& o*ulation &i " of 200 : "tt"r r"&*on&" ti$"&

# iddl" a&t


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E&&"$ ly #od"&

Results nalysis




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E&&"$ ly #od"&

Results nalysis




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$olving methods ",tonZ& $"thod ith Dantorovich P"ry fa&t calculation&

Interval Analysis

= Certified solutions = But long computation times

Algebraic methods (Groebner Ell " act &olution&

>ut lon) co$*utation ti$"& or ch"c in) *ur*o&"&

(3' F7 ("n"tic El)orith$

= Ell &olution& = ot v"ry *r"ci&"

Summary RoSe, March 14, 2014

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Presentation SolvingFKPparallelRobots MUN RoSe 2014