Inverse Kinematics From Position to Angles

Transcription

1 Invere Knemat From Poton to Ange

2 Invere Knemat Gven a dered poton P & orentaton R o the end-eetor Y,, z, O, A, T z q,, n Fnd the jont varabe whh an brng the robot to the dered onguraton.

3 Sovabt 0 Gven the numera vaue o we attempt to nd vaue o,,, n. The PUMA 560: T Gven 0 a 6 numera vaue, 6T ove or6 jont ange,,,, 6. equaton and 6 unnown 6 equaton and 6 unnown nonnear, tranendenta equaton

4 Invere Knemat More dut. The equaton to ove are nonnear thu temat oed-orm outon not awa avaabe. Souton not unque. Redundant robot. Ebow-up/ebow-down onguraton. Robot dependent. outon!

5 The Worpae Worpae: voume o pae whh an be reahed b the end eetor Detrou worpae: voume o pae where the end eetor an be arbtrar orented Reahabe worpae: voume o pae whh the robot an reah n at eat one orentaton

6 Two-n manpuator I = reahabe wor pae a d o radu. The detrou Worpae a pont: orgn I there no detrou worpae. The reahabe wor pae a rng o outer radu + and nner radu -

7 Etene o Souton A outon to the IKP et the target beong to the worpae. Worpae omputaton ma be hard. In prate t made ea b pea degn o the robot.

8 Method o Souton A manpuator ovabe the jont varabe an be determned b an agorthm. The agorthm houd nd a pobe outon. Souton oed orm outon numera outon

9 umera Souton Reut n a numera, teratve outon to tem o equaton, or eampe ewton/raphon tehnque. Unnown number o operaton to ove. On return a nge outon. Aura dtated b uer. Beaue o thee reaon, th muh e derabe than a oed-orm outon. Can be apped to a robot.

10 Coed-orm outon Anata outon to tem o equaton Can be oved n a ed number o operaton thereore, omputatona at/nown peed Reut n a pobe outon to the manpuator nemat Oten dut or mpobe to nd Mot derabe or rea-tme ontro Mot derabe overa

11 Invere Knemat Probem Gven: Poton & Orentaton Fnd: jont oordnate o ED-EFFECTOR 0 T q, q, q 3,, q eed to ove at mot ndependent equaton n unnown.

12 Invere Knemat Probem ISSUES Etene o outon Worpae Detrou Worpae Le than 6 jont Jont mt prata Mutpe outon Crtera Agebra Sovabt oed orm numera Geometr number o outon = 6 d, r 0 or pont

13 Souton To Invere Knemat 0 T = 0 T T T 3 - T = A A A 3 A Gven: 0 T n n nz o o o z a a a z p p A p z θ -θ θ d θ θ θ - θ d θ r Fnd: q = q, q, q 3,, q jont oordnate

14 Souton To Invere Knemat 3 z z z z...a A A A p a o n p a o n p a o n Equaton 6 ndependent 6 redundant unnown LHS,j = RHS,j row =,, 3 oumn j =,, 3, 4

15 Souton To Invere Knemat Genera Approah: Ioate one jont varabe at a tme A - 0 T = A A 3 A = T unton o q unton o q,, q Loo or ontant eement n T Equate LHS,j = RHS,j Sove or q

16 Souton To Invere Knemat A - A -0 T = A 3 A = T unton o q 3,, q unton o q, q on one unnown q ne q ha been oved or Loo or ontant eement o T Equate LHS,j = RHS,j Sove or q Mabe an nd equaton nvovng q on ote: There no agorthm approah that 00% eetve Geometr ntuton requred

17 A Smpe Eampe Revoute and Prmat Jont Combned, Fndng : θ artan More Spea: θ artan artan pee that t n the rt quadrant Y S Fndng S: X S

18 Invere Knemat o a Two Ln Manpuator, Gven:,,, Fnd:, Redundan: A unque outon to th probem doe not et. ote, that ung the gven two outon are pobe. Sometme no outon pobe.,

19 The Geometr Souton,

20 The Geometr Souton, Ung the Law o Cone: aro θ oθ oθ θ o80 θ o80 o C ab b a Ung the Law o Cone: artan α α θ θ nθ θ n80 nθ n n C b B artan nθ arn θ Redundant ne oud be n the rt or ourth quadrant. Redundan aued ne ha two pobe vaue

21 The Agebra Souton,

22 aro θ n n n The Agebra Souton, θ θ θ 3 n θ o θ o θ On Unnown n o o n n n n o o o : b a b a b a b a b a b a ote

23 n o n o n n n o o o : a b b a b a b a b a b a ote n We now what rom the prevou de. We need to ove or. ow we have two equaton and two unnown n and o Subttutng or and mpng man tme ote th the aw o one and an be repaed b + arn θ

24 Invere Knemat Anata outon on wor or a ar mpe truture umera/teratve outon needed or a ompe truture

25 umera Approahe Invere nemat an be ormuated a an optmzaton probem

26 Funton Optmzaton Fndng the mnmum or nonnear unton

27 Formuaton So how to onvert the IK proe n an optmzaton unton? arg mn F θ θ θ C=C,C Bae 0,0

28 Iteratve Approahe Fnd the jont ange θ that mnmze the dtane between the hpothezed harater poton and uer peed poton arg mn C hpothezed poton peed poton θ θ θ C=C,C Bae 0,0

29 Iteratve Approahe Fnd the jont ange θ that mnmze the dtane between the hpothezed harater poton and uer peed poton arg mn, o o n n θ θ θ C=, Bae 0,0

30 Iteratve Approahe Mathemata, we an ormuate th a an optmzaton probem: arg mn The above probem an be oved b man nonnear optmzaton agorthm: - Steepet deent - Gau-newton - Levenberg-marquardt, et

31 Gradent-baed Optmzaton

32 Gau-ewton Approah Step : ntaze the jont ange wth 0 Step : update the jont ange:

33 Gau-ewton Approah Step : ntaze the jont ange wth 0 Step : update the jont ange: How an we dede the amount o update?

34 Gau-ewton Approah Step : ntaze the jont ange wth 0 Step : update the jont ange: arg mn

35 Gau-ewton Approah Step : ntaze the jont ange wth 0 Step : update the jont ange: arg mn Known!

36 Gau-ewton Approah Step : ntaze the jont ange wth 0 Step : update the jont ange: arg mn Taor ere epanon

37 Gau-ewton Approah Step : ntaze the jont ange wth Step : update the jont ange: mn arg mn arg 0 Taor ere epanon rearrange

38 Gau-ewton Approah Step : ntaze the jont ange wth Step : update the jont ange: mn arg mn arg 0 Taor ere epanon rearrange Can ou ove th optmzaton probem?

39 Gau-ewton Approah Step : ntaze the jont ange wth Step : update the jont ange: mn arg mn arg mn arg 0 Taor ere epanon rearrange Th a quadrat unton o

40 Gau-ewton Approah Optmzng an quadrat unton ea It ha an optma vaue when the gradent zero,...,,...,,..., M M M mn arg

41 Gau-ewton Approah Optmzng an quadrat unton ea It ha an optma vaue when the gradent zero,...,,...,,..., M M M mn arg b J Δθ Lnear equaton!

42 Gau-ewton Approah Optmzng an quadrat unton ea It ha an optma vaue when the gradent zero,...,,...,,..., M M M mn arg b J Δθ

43 Gau-ewton Approah Optmzng an quadrat unton ea It ha an optma vaue when the gradent zero,...,,...,,..., M M M mn arg b J Δθ b J J J T T