Inverse Manipulator Kinematics (1/3)
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- Johnathan Gibson
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1 Invee Mniulto Kinemti /
2 Diet Veu Invee Kinemti Diet Fow Kinemti Given: Joint ngle n lin geomety Comute: Poition n oienttion o the en eeto eltive to the be me B Invee Kinemti Given: Poition n oienttion o the en eeto eltive to the be me Comute: All oible et o joint ngle n lin geomety whih oul be ue to ttin the given oition n oienttion o the en eeto N B N
3 Centl oi - Invee Mniulto Kinemti - Exmle Geometi Solution - Conet Deomoe til geomety into evel lne geomety Exmle - Pln RRR R mniulto - Geometi Solution Algebi Solution - Conet N N N x y z Diet Kinemti Gol Numei vlue Exmle - PUMA - Algebi Solution
4 Solvbility - PUMA Given : PUMA - DOF, Solve: x y z otl Numbe o Eqution: Ineenent Eqution: - Rottion Mtix ye o Eqution: Non-line - Poition Veto
5 Solvbility Exitene o Solution Multile Solution Metho o olution Cloe om olution Algebi olution Geometi olution Numeil olution
6 Solvbility - Exitene o Solution N Fo olution to exit, mut be in the woe o the mniulto Woe - Deinition Dexteou Woe DW: he ubet o e in whih the obot en eeto n eh ll oienttion. Rehble Woe RW: he ubet o e in whih the obot en eeto n eh in t let oienttion he Dexteou Woe i ubet o the Rehble Woe DW RW
7 Solvbility - Exitene o Solution - Woe - R Exmle - L L Rehble Woe Dexteou Woe
8 Solvbility - Exitene o Solution - Woe - R Exmle - L L Rehble Woe NO Dexteou Woe
9 Solvbility - Exitene o Solution - Woe - R Exmle - L L En Eeto Rottion Rehble Woe & Dexteou Woe
10 Solvbility - Multile Solution Multile olution e ommon oblem tht n ou when olving invee inemti beue the ytem h to be ble to hoe one he numbe o olution een on the numbe o joint in the mniulto but i lo untion o the lin mete i i i i Exmle: he PUMA n eh etin gol with 8 ieent olution m onigution Fou olution e eite Fou olution e elte to lie wit 8 8 o o
11 Solvbility - Multile Solution Poblem: he t tht mniulto h multile multile olution my ue oblem beue the ytem h to be ble to hooe one Solution: Deiion itei he loet geometilly - minimizing the mount tht eh joint i equie to move Note : inut gument - eent oition o the mniulto Note : Joint Weight - Moving mll joint wit inte o moving lge joint Shoule & Elbow Obtle exit in the woe - voiing olliion
12 Solvbility - Multile Solution - Numbe o Solution Deinition - Poition the en eeto in eii oint in the lne D No. o DOF No. o DOF o the t Numbe o olution: elbow u/own No. o DOF > No. o DOF o the t Numbe o olution: Sel Motion - he obot n be move without moving the the en eeto om the gol
13 Solvbility - Metho o Solution Solution Invee Kinemti- A olution i the et o joint vible oite with n en eeto eie oition n oienttion. No genel lgoithm tht le to the olution o invee inemti eqution. Solution Sttegie Cloe om Solution - An nlyti exeion inlue ll olution et. Algebi Solution - igonometi Nonline eqution Geometi Solution - Reue the lge oblem to eie o lne geomety oblem. Numeil Solution - Itetive olution will not be oniee in thi oue.
14 Solvbility Robot - DOF Single Seie Chin Revolute & Pimti Joint Rel-ime Anlyti Solution Numei Solution Non Rel-ime Inutil Robot Cloe Fom Solution Suiient Conition hee jent xe oty o imti mut inteet Itetion
15 Mthemtil Eqution Lw o Sinu / Coine - Fo genel tingle A B C b in A in B b in C b bo A Sum o Angle in o
16 Invee Kinemti - Pln RRR R - Algebi Solution - / L
17 L i i i i i i i i i i i i i i i i i i L Invee Kinemti - Pln RRR R - Algebi Solution - /
18 Invee Kinemti - Pln RRR R - Algebi Solution - / Uing tigonometi ientitie to imliy, the olution to the ow inemti i: wee L L L L L L B W B W L L L L B W o in
19 Invee Kinemti - Pln RRR R - Algebi Solution - / Given: Diet Kinemti: he homogenou tnomtion om the be to the B wit W Gol Point Deinition: Fo ln mniulto, eiying the gol n be omlihe by eiying thee mete: he oition o the wit in e x, y n the oienttion o lin in the lne eltive to the Xˆ xi
20 Invee Kinemti - Pln RRR R - Algebi Solution - / Poblem: Wht e the joint ngle oienttion x, y,,, untion o the wit oition n Solution: he gol in tem o oition n oienttion o the wit exee in tem o the homogeneou tnomtion i eine ollow B W Gol x y B W L L L L
21 Invee Kinemti - Pln RRR R - Algebi Solution - / B W Gol A et o ou nonline eqution whih mut be olve o,, Solving o I we que n them while ming ue o ; we obtin x y x l y l l l x y l l l l
22 Invee Kinemti - Pln RRR R - Algebi Solution - / Continue l l y l l x l l l l y x
23 Invee Kinemti - Pln RRR R - Algebi Solution - 7/ Solving o we obtin x y l l l l Note: In oe o olution to exit, the ight hn ie mut hve vlue between - n. Phyilly i thi ontint i not tiie, then the gol oint i too wy o the mniulto to eh. Auming the gol i in the woe, n ming ue o we wite n exeion o Note: he hoe o the ign oeon to the multile olution in whih we n hooe the elbow-u o the elbow-own olution
24 Invee Kinemti - Pln RRR R - Algebi Solution - 8/ Finlly, we omute uing the two gument tngent untion, tn, Atn l l l l y x A
25 Invee Kinemti - Pln RRR R - Algebi Solution - 8/ Solving o Fo olving we ewite the oiginl nonline eqution uing hnge o vible ollow whee l l y l l x y x l l l l
26 Geometil Inteettion Invee Kinemti - Pln RRR R - Algebi Solution - 8/ ontinue
27 Invee Kinemti - Pln RRR R - Algebi Solution - 9/ Chnging the wy in whih we wite the ontnt n Atn, hen o in l
28 Invee Kinemti - Pln RRR R - Algebi Solution - / Be on the eviou two tnomtion the eqution n be ewitten o o x y o in x y x y x o o in in y o in in o o o in in o in in o o in
29 Invee Kinemti - Pln RRR R - Algebi Solution - / Uing the two gument tngent we inlly get olution o Note: By Deinition y x Atn, Atn y, x A tn, Atn y, x Atn, l l l x y l l When hoie o ign i me in the olution o Atn, Atn,, it will ue ign hnge in thu eting x y I then the olution beome uneine - in thi e i bity l l
30 Invee Kinemti - Pln RRR R - Algebi Solution - / Solving o Be on the oiginl eqution We n olve o the um o,, Atn, Note: It i tyil with mniulto tht hve two o moe lin moving in lne tht in the oue o olution, exeion o um o joint ngle ie
31 Given: Mniulto Geomety Invee Kinemti - Pln RRR R - Geometi Solution - / Gol Point Deinition: he oition n oienttion o the wit in e Poblem: Wht e the joint ngle oienttion,, x, y untion o the gol wit oition n
32 Invee Kinemti - Pln RRR R - Geometi Solution - / Solution: We n ly the lw o oine to olve o x y l l ll o8 Sine o 8 o We hve x y l l l l
33 Invee Kinemti - Pln RRR R - Geometi Solution - / x y l l l l Atn,
34 Invee Kinemti - Pln RRR R - Geometi Solution - / Note : Conition - Shoul be hee by the omuttionl lgoithm to veiy exitene o olution. l l x y Auming tht the olution exit it lie in the nge o o o 8 he othe oible olution my oun by ymmety to be '
35 Invee Kinemti - Pln RRR R - Geometi Solution - / By einition Deining untion o x,y Atn y, x Alying the lw o oine to in Note: o x l y x l o 8 l y Atn y, x Atn o,o
36 Invee Kinemti - Pln RRR R - Geometi Solution - / Angle in the lne u to eine the oienttion o the lt lin
37 Invee Mniulto Kinemti /
38 Centl oi - Invee Mniulto Kinemti - Exmle Geometi Solution - Conet Deomoe til geomety into evel lne geomety Exmle - Pln RRR R mniulto - Geometi Solution Algebi Solution - Conet N K N N x y z Diet Kinemti Gol Numei vlue Exmle - PUMA - Algebi Solution
39 Given: Mniulto Geomety Invee Kinemti - Pln RRR R - Geometi Solution - / x, y Gol Point Deinition: he oition n oienttion o the wit in e Poblem:, Wht e the joint ngle untion o the gol wit oition n oienttion, φ φ
40 Invee Kinemti - Pln RRR R - Geometi Solution - / Solution: We n ly the lw o oine to olve o x y l l l l o8 Sine o 8 o We hve x y l l l l
41 Invee Kinemti - Pln RRR R - Geometi Solution - / x y l l l l ± Atn,
42 Invee Kinemti - Pln RRR R - Geometi Solution - / Note : Conition - Shoul be hee by the omuttionl lgoithm to veiy exitene o olution. l l x y Auming tht the olution exit it lie in the nge o o o 8 he othe oible olution my oun by ymmetytobe to '
43 Invee Kinemti - Pln RRR R - Geometi Solution - / By einition β ± ψ < > β Deining untion o x,y β A tn y, x Alying the lw o oine to in oψ x l y x l l y ψ Note: 8 o
44 Invee Kinemti - Pln RRR R - Geometi Solution - / Angle in the lne u to eine the oienttion o the lt lin φ φ
45 Invee Kinemti - PUMA - Algebi Solution - / Given: Diet Kinemti: he homogenou tnomtion om the be to the B wit W Gol Point Deinition: he oition n oienttion o the wit in e
46 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / Poblem: Wht e the joint ngle untion o the wit oition n L oienttion o when i given numei vlue x z y { Gol Diet Kinemti
47 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / Solution Genel ehnique: Multilying eh ie o the iet inemti eqution by n invee tnomtion mtix o eting out vible in eh o olvble eqution Put the eenene on on the let hn ie o the eqution by multilying the iet inemti eq. with give ] [ q g ] [ ] [ ] [ I ] [ ] [ ] [ ],, [ ],,, [ ],,, [ ],,,, [ ],,,, [
48 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / Put the eenene on on the let hn ie o the eqution by multilying the iet inemti eq. with give ] [ ] [ ] [ { I ] [ BORG A A B A B B A P R R ] [ B A A B ] [
49 Invee Kinemti - PUMA - Algebi Solution - / x y z
50 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / ' ' ' ' ' ' ' ' ' ' ' ' y x z
51 Invee Kinemti - PUMA - Algebi Solution - 7/ y z x Equting the, element om both ie o the eqution we hve x y o olve the eqution o thi om we me the tigonometi ubtitution x y ρ oφ ρ inφφ
52 Invee Kinemti - PUMA - Algebi Solution - 8/ ρ x y φ Atn x, y, ρ, φ Subtituting with we obtin x y φ φ ρ Uing the ieene o ngle omul in φ ρ
53 Invee Kinemti - PUMA - Algebi Solution - 9/ Be on φ o φ in n o o φ ± ρ φ A tn, ± ρ ρ he olution o my be witten A tn, ± ρ ρ A tn y, x Note: we hve oun two oible olution o oeoning to the /- ign
54 Invee Kinemti - PUMA - Algebi Solution - / Equting the, element n, element x y z We obtin x y z
55 Invee Kinemti - PUMA - Algebi Solution - / I we que the ollowing eqution n the eulting eqution x y x y z
56 Invee Kinemti - PUMA - Algebi Solution - / Continue
57 Invee Kinemti - PUMA - Algebi Solution - / Continue
58 Invee Kinemti - PUMA - Algebi Solution - / we obtin K whee K x y z Note tht the eenene on h be emove. Moeove the eq. o i o the me om the eq. o n o my be olve by the me in o tigonometi ubtitution to yiel olution o
59 Invee Kinemti - PUMA - Algebi Solution - / A tn, A tn, K ± K Note tht the /- ign le to two ieent olution o
60 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / ] [ ] [ I ] [ ],, [ z y x Equting the, element n, element we obtin z y x hee eqution my be olve imultneouly o n eulting in z y x
61 Invee Kinemti - PUMA - Algebi Solution - / Continue
62 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / y x z y x z y x z Sine the enominto e equl n oitive, we olve o the um o n y x z ], tn [ y x z y x z A he eqution omute ou vlue o oing to the ou oible ombintion o olution o n
63 Invee Kinemti - PUMA - Algebi Solution - / Algebi Solution / hen, ou oible olution o e omute Equting the, n the, element y x we get z y
64 A long we n olve o Invee Kinemti - PUMA - Algebi Solution - 7/ At tn, When the mniulto i in ingul onigution in whih joint xe n line u n ue the me motion o the lt lin o the obot. In thi e ll tht n be olve o i the um o ieene o n. hi itution i etete by heing whethe both gument o Atn e ne zeo. I o i hoen bity uully hoen to be equl to the eent vlue o joint, n i omute lte, it will be omute oingly
65 Invee Kinemti - PUMA - Algebi Solution - 8/ Algebi Solution 8/ ],,, [ ],,, [ x z y Equting the, n the, element we get We n olve o tn A, tn A
66 Invee Kinemti - PUMA - Algebi Solution - 9/ Algebi Solution 9/ ],,,, [ ],,,, [ z y x z Equting the, n the, element we get We n olve o, tn A, tn A
67 Invee Kinemti - PUMA - Algebi Solution - / Summy - Numbe o Solution Fou olution Atn y, x Atn, ± ρ ρ A tn, A tn, K ± K Fo eh o the ou olution the wit n be lie ' ' ' 8 8 o o
68 Invee Kinemti - PUMA - Algebi Solution - / Ate ll eight olution hve been omute, ome o ll o them my hve to be ie beue o joint limit violtion. O the emining vli olution, uully the one loet to the eent mniulto onigution i hoen.
69 Invee Mniulto Kinemti /
70 Centl oi - Invee Mniulto Kinemti - Exmle Geometi Solution - Conet Deomoe til geomety into evel lne geomety Exmle - D - RRR R mniulto - Geometi Solution Algebi Solution loe om - Piee Metho - Lt thee oneutive xe inteet t one oint Exmle - Pum
71 Given: Mniulto Geomety Invee Kinemti - D RRR R - Geometi Solution - / x, y, z Gol Point Deinition: he oition o the wit in e Poblem:,, Wht e the joint ngle untion o the gol wit oition n oienttion
72 Invee Kinemti - D RRR R - Geometi Solution - /
73 Invee Kinemti - D RRR R - Geometi Solution - / he ln geomety - to view o the obot y L x y x L z x x z A tn y, x x y
74 Invee Kinemti - D RRR R - Geometi Solution - /
75 Invee Kinemti - D RRR R - Geometi Solution - / he ln geomety - ie view o the obot: L L ẑ LL z z ˆ ˆ z L L z x y z x y whee zˆ z L L
76 Invee Kinemti - D RRR R - Geometi Solution - / By Aly the lw o oine we get L L LL o8 L L LL o Renging give n Solving o we get L L L L Atn ±, Whee i eine bove in tem o nown mete L, L, x, y,n z
77 Invee Kinemti - D RRR R - Geometi Solution - 8/ L β L ẑ Finlly we nee to olve o β whee Atn zˆ, x y z z L L ˆ
78 Invee Kinemti - D RRR R - Geometi Solution - 9/ Geometi Solution 9/ Be on the lw o oine we n olve o β o β L L L L L L β, tn β β β A ±, tn, tn β β A y x L L z A ±,, β β y
79 Invee Kinemti - D RRR R - Geometi Solution - / Geometi Solution / Summy, tn y x A, tn, tn L L L z y x L L L L z y x L L L z y x L L L L z y x A y x L L z A ± L L L z y x L L L z y x, tn L L L L z y x L L L L z y x A ±, tn L L L L A ±
80 Invee Kinemti - Genelize Algebi Anlytil l Solution Ce -7
81 Invee Kinemti - Genelize Algebi Anlytil Solution Ce Eqution in o b [-,] b [] [-,] Solution Unique Atn, b
82 Invee Kinemti - Genelize Algebi Anlytil Solution Ce Eqution in [-,] o b [-,] o ± in ± b Solution Atn, ± Atn ± b, b wo Solution 8 Singulity t the Bouny o When ±9, When o,8, b
83 Invee Kinemti - Genelize Algebi Anlytil Solution Ce Eqution o bin in o b Solution 8 wo Solution t Atn, b Atn, b Singulity b
84 Invee Kinemti - Genelize Algebi Anlytil Solution Ce Eqution o bin, b, Solution wo Solution A tn ± b, Atn b, Fo olution to exit b > No olution outie o the woe b < One olution ingulity b
85 Invee Kinemti - Genelize Algebi Anlytil Solution Ce
86 Invee Kinemti - Genelize Algebi Anlytil Solution Ce
87 Invee Kinemti - Genelize Algebi Anlytil Solution Ce Eqution in inφ o inφφ b Solution Atn, b i in φ i oitive Atn, b i inφ i negtive
88 Invee Kinemti - Genelize Algebi Anlytil Solution Ce Eqution o in e o in g Solution Atn g e, g Fo n exiting olution the eteminnt mut be oitive e >
89 Algebi Solution by Reution to Polynomil nenentl eqution e iiult to olve beue they e untion o,, Ming the ollowing ubtitution yiel n exeion in tem o ingle veitble u, Uing thi ubtitution, tnenentl eqution e onvete into olynomil eqution u tn u o u u in u
90 Algebi Solution by Reution to Polynomil - Exmle nenentl eqution b, Subtitute with the ollowing eqution yiel u u o u u in u bu u u bu
91 Algebi Solution by Reution to Polynomil - Exmle Whih i olve by the quti omul to be u b ± b tn b ± b Note I u i omlex thee i no el olution to the oiginl tnenentl eqution I then o 8
92 Solvbility Robot - DOF Single Seie Chin Revolute & Pimti Joint Rel-ime Anlyti Solution Numei Solution Non Rel-ime Inutil Robot Cloe Fom Solution Suiient Conition hee jent xe oty o imti mut inteet
93 Piee Solution - hee oneutive Axe Inteet Piee Solution - Cloe om olution o eil DOF in whih thee oneutive xe inteet t oint inluing obot with thee oneutive llel xe, ine they meet t oint t ininity Piee metho lie to the mjoity o ommeilly vilble inutil obot Exmle: Pum All joint e evolute joint he lt joint e inteeting
94 Piee Solution - hee oneutive Axe Inteet Given: Mniulto Geomety: DOF & DH mete All joint e evolute joint he lt joint e inteeting Gol Point Deinition: he oition n oienttion o the wit in e Poblem: x y z,,,,, Wht e the joint ngle untion o the gol wit oition n oienttion
95 Piee Solution - hee oneutive Axe Inteet When the lt thee xe o DOF obot inteet, the oigin o lin me {}, {}, n {} e ll lote t the oint o inteetion. hi oint i given in the be oointe ytem og P og P Fom the genel ow inemti metho o etemining homogeneou tnom uing DH mete, we now: i i i R i i iog i P i i i i i i i i i i i i i i i i
96 Piee Solution - hee oneutive Axe Inteet Fo i P og R Uing the outh olumn n ubtituting o we in P Uing the outh olumn n ubtituting o, we in P P P og P P P P og og P P og og
97 Piee Solution - hee oneutive Axe Inteet whee
98 Piee Solution - hee oneutive Axe Inteet Reeting the me oe gin P P og og g g g g g g g
99 Piee Solution - hee oneutive Axe Inteet g g Reeting the me oe o the lt time g g g g g P P og og g g g g P og
100 Piee Solution - hee oneutive Axe Inteet Fme {} - he me tthe to the be o the obot o lin lle me {} hi me oe not move n o the oblem o m inemti n be oniee the eeene me. Aign {} to mth {} when the it joint veitble i zeo g g P g P og g g g g g g g P og
101 Piee Solution - hee oneutive Axe Inteet hough lgebi mniultion o thee eqution, we n olve o the eie joint ngle.,, he it te i to que the mgnitue o the itne om the me {} oigin to me {} oigin. g g g P P P ogz ogy ogx Uing the eviouly eine untion o we hve g i
102 Piee Solution - hee oneutive Axe Inteet Alying ubtitution o temoy vible, we n wite the mgnitue que tem long with the z omonent o the {} me oigin to the {} me g P Z ogz que tem long with the z-omonent o the {} me oigin to the {} me oigin itne. h ti l b h b li i t Z hee eqution e ueul beue eenene on h been eliminte, n eenene on te imle om
103 Piee Solution - hee oneutive Axe Inteet Conie e while olving o : Ce - Solution Methoology - Reution to Ploynomil > Quti Eqution in o tn u u u u u u u
104 Piee Solution - hee oneutive Axe Inteet Ce - Z Z Solution Methoology - Reution to Ploynomil > Quti Eqution in o tn u u u u u
105 Piee Solution - hee oneutive Axe Inteet Ce Genel e : We n in though the ollowing lgebi mniultion: Z quing both ie, we in Z q g, Z
106 Piee Solution - hee oneutive Axe Inteet Aing thee two eqution togethe n imliying uing the tigonomety ientity Reution to Ploynomil, we in outh oe eqution o Z
107 Piee Solution - hee oneutive Axe Inteet With olve, ubtitute into to in Z, Z
108 Piee Solution - hee oneutive Axe Inteet With olve, ubtitute into to in, og P g g g g g P og g g P ogx g g P ogy g g Solve o uing the eution to olynomil metho g g
109 Piee Solution - hee oneutive Axe Inteet, o omlete ou olution we nee to olve o, Sine the lt thee xe inteet thee joint ngle et the oienttion o only the lt lin. We n omute them be only uon the ottion otion o the eiie gol R R R R R R R R - he oienttion o lin me {} eltive to the be me {} when,, R e the Eule ngle lie to
110 Invee Mniulto Kinemti /
111 Centl oi - Invee Mniulto Kinemti - Exmle Algebi Solution loe om - Piee Metho Continue - Lt thee oneutive xe inteet t one oint Conie DOF wit non-ln obot whoe xe ll inteet t oint.
112 Ming - Rotte Fme - Z-Y-Z Eule Angle Stt with me {}. β} Rotte me {} bout Ẑ by n ngle Rotte me {} bout Yˆ by n ngle Eule Angle B Rotte me {} bout Ẑ by n ngle Note - Eh ottion i eome bout n xi o the moving eeene me {}, the then ixe eeene.
113 Ming - Rotte Fme - X-Y-Z Eule Angle β β,, ' ' ' β β β β β β R R R R Z Y Z Z Y Z β β β β β β β R Z Y Z,, ' ' ' β β β β β β β Z Y Z,, ' ' ',, β R R Z Y X A B
114 hee oneutive Axe Inteet - wit β β β β β β β β β { β β β Gol Diet Kinemti
115 hee oneutive Axe Inteet - wit β,, Solve o uing element β ββ β ± Uing the Atn untion, we in β Atn ±,
116 hee oneutive Axe Inteet - wit Solve o uing element, β ββ / β, / β Atn
117 hee oneutive Axe Inteet - wit Solve o uing element, β β / β, / β Atn
118 hee oneutive Axe Inteet - wit β Note: wo nwe exit o ngle whih will eult in two nwe eh o ngle n. β Atn ±, / β, / β Atn / β, / β Atn I o o the olution egenete β, β 8 β
119 hee oneutive Axe Inteet - wit β β β β β β β β β β β β o β We e let with o evey e. hi men we n t olve o eithe, jut thei ieene.
120 hee oneutive Axe Inteet - wit One oible onvention i to hooe o he olution n be lulte to be β β 8, Atn,, Atn, Atn Atn
121 hee oneutive Axe Inteet - wit Fo thi exmle, the ingul e eult in the bility o el-ottion. ht i, the mile lin n otte while the en eeto oienttion neve hnge.
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