Control of industrial robots. Robot dynamics
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- Ashley Watts
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1 Coto of dut oot Root dy of. oo Roo oteo d Mo Dteto d Eetto, Ifozoe e Bogege
2 Itoduto Wth thee de we w deve the dy ode of the uto he dy ode out fo the eto etwee the oue of oto (foe d oet) d the eutg oto (oto d veote) Sytet ethod ext to deve the dy ode of the uto, whh w e evewed hee, og wth the oete of uh ode u(t) (t) Coto of dut oot Root dy oo Roo
3 Ket eegy Code ot wth, whoe oto deed y veto wth eet to xyz fe. z We defe et eegy of the ot the utty: Sy, fo yte of ot: Code ow gd ody, wth, voue V d dety ρ. he et eegy defed wth the teg: V ρdv x x O z O dv y y Coto of dut oot Root dy oo Roo
4 otet eegy A yte of oto foe (.e. deedg oy o the oto of the ot of to) d to e oevtve the wo de y eh foe doe ot deed o the tetoy foowed y the ot of to, ut oy o the t d f oto. I th e the eeety wo ode wth the dffeet, wth hged g, of futo ed otet eegy: dw du A exe of yte of oevtve foe the gvtto foe. Fo ot we hve the otet eegy: U g whee g the gvty eeto veto. Fo gd ody: U ρdv g g V whee the oto of the ete of. Coto of dut oot Root dy oo Roo
5 Syte of gd ode Let u ode yte of gd ode ( fo exe, the of oot). If thee ode e fee to ove e, the oto of the yte, t eh te tt, deed y e of 6 oodte x. Suoe ow tht tto ext the oto of the ode of the yte ( fo exe the eee of ot, whh ete fve out of the x etve degee of feedo etwee two oeutve ). A ott thu ext o the oto of the ode, whh we w exe wth the eto: h( x ) Suh ott d to e hooo (t deed oy o oto oodte, ot veote) d ttoy (t doe ot hge wth te). Coto of dut oot Root dy oo Roo
6 Fee oodte h( x ) If the ott h e ooed of euto d they e otuouy dffeete, t oe, y e of the ott, to ete oodte fo the yte euto. he eg 6 oodte e ed fee, o Lgg, o tu, o geezed oodte. the ue of degee of feedo of the eh yte. Fo exe oot wth 6 ot, out of the 36 og oodte, 3 e eted y vtue of the ott oed y the 6 ot. he eg 6 e the Lgg oodte (tyy the ot ve ued the et ode). Coto of dut oot Root dy oo Roo
7 Lgge euto Gve yte of gd ode, whoe oto d oetto e exeed y e of geezed oodte, we defe Lgg of the eh yte the utty: L (, ) (, ) U whee d U e the et d the otet eege, eetvey. Let the ξ e the geezed foe oted wth the geezed oodte. he eeety wo efoed y the foe tg o the yte e exeed : dw ξ d It e ove tht the dy of the yte of ode exeed y the foowg Lgge euto: d dt L L ξ,,, Coto of dut oot Root dy oo Roo
8 A exe Let u ode yte ooed of oto gdy oeted to od, ueted to the gvtto foe. Let: I d I the oet of et of the oto d the od wth eet to the oto g x the of the od the dte of the ete of of the od fo the x of the oto. τ y C g x Ket eegy of the oto: I Ket eegy of the od: I Coto of dut oot Root dy oo Roo
9 A exe τ g Gvtto otet eegy: o U g g g [ ] L U I I g Lgg: Lgge euto: d dt L L τ he: d dt ( I I ) ) g o τ ( I I ) g o τ h euto e ey teeted the euu of oet oud the otto x. Coto of dut oot Root dy oo Roo
10 Ket eegy of he otuto of et eegy of ge e outed wth the foowg teg: * * ρdv V * gee ot og the * oto of the ete of : ρ dv V Veoty of the gee ot: ( ) whee: S z y x * z S y x (ew-yet tx) Coto of dut oot Root dy oo Roo
11 Coto of dut oot Root dy oo Roo to otuto: V dv ρ Mutu otuto: * ρ ρ V V dv dv S S Rotto otuto: V V dv dv I S S S S ρ ρ S S Note: Ket eegy of he: I veoty of ete of gu veoty of the Kög heoe
12 Iet teo I We defe et teo the tx: ( y z ) R * * ρdv xyρ ( x z ) * dv ρdv (yet tx) xzρdv Ixx Ixy Ixz yzρdv * I yy Iyz ρ Izz x y dv * * he et teo, f exeed the e fe, deed o the oot ofguto. If the gu veoty exeed wth efeee to fe gdy tthed to the (fo exe the DH fe): the et teo efeed to th fe ott tx. Moeove: I R I R Coto of dut oot Root dy oo Roo
13 Coto of dut oot Root dy oo Roo Su of the otuto Let u u the tto d otto otuto: R R I Le veoty: J [ ] J Agu veoty: J O O O [ ] O O O J
14 Coto of dut oot Root dy oo Roo Coutto of the Jo [ ] J [ ] O O O J ot otto t ot O z z z Cou of the Jo:
15 Coto of dut oot Root dy oo Roo Iet tx Suttutg exeo fo the e d gu veote: J R R I J J J O O By ug the otuto of the we ot the et eegy of the whoe : B whee: O O J R R I J J J B the et tx of the uto. yet otve defte deed o
16 otet eegy he otet eegy of gd eted ut to the gvtto foe: U V * g ρdv g whee g the gvty eeto veto exeed the e fe. he otet eegy of the whoe uto the the u of the ge otuto: U U g Coto of dut oot Root dy oo Roo
17 Coto of dut oot Root dy oo Roo Euto of oto he Lgg of the uto : U L,, g If we dffeette the Lgg: Futheoe: g g g U dt d dt d dt d L dt d
18 Euto of oto Fo Lgge euto we ot: h g ξ,, whee: h gvtto te, deed oy o the ot oto Aeeto te: Cetfug d Coo te: : et oet ee fo the x of ot : effet of the eeto of ot o the ot. h : etfug effet dued t ot y the veoty of ot.. h : Coo effet dued t ot y the ve. of ot e h Coto of dut oot Root dy oo Roo
19 No oevtve foe Bede the gvtto oevtve foe, othe foe t o the uto: tuto toue vou fto toue tt fto toue f τ F v (, ) F v f (, ) dgo tx of vou fto oeffet futo tht ode the tt fto t the ot Coto of dut oot Root dy oo Roo
20 Coete dy ode I veto fo the dy ode e exeed foow: B C( ) F f (, ) g τ, v whee C ute tx, whoe eeet tfy the euto: h C ot yet gee Coto of dut oot Root dy oo Roo
21 Coto of dut oot Root dy oo Roo Coutto of the eeet of C he hoe of tx C ot uue. Oe oe hoe the foowg oe: h he gee eeet of C : Chtoffe yo of the ft d whee:
22 Coto of dut oot Root dy oo Roo Sew-yety of tx BC he evou hoe of tx C ow to ove ott oety of the dy ode of the uto. Mtx:. C B N,, ew-yet: w w N w,, I ft: (ew-yet)
23 Eegy oevto he euto: (, ) N (tu e of the evou oe) vd whteve the hoe of tx C. Fo the eegy oevto e, the devtve of the et eegy eu the owe geeted y the foe tg t the ot of the uto: d dt ( B ) ( τ F f (, ) g ) v g the devtve t the eft hd de d ug the euto of the ode: d ( B ) B B ( B C (, )) dt ( τ F f (, ) g ) v fo whh the euto foow. Coto of dut oot Root dy oo Roo
24 Lety the dy ete If we ue fed exeo fo the tt fto futo: f ( ) F g( ), t oe to ove tht the dy ode of the uto e wth eet to ute et of dy ete (e, oet of et) We the wte: τ Y (,, )π π: veto of ott ete Y: tx, futo of ot oto, veote d eeto (egeo tx) Coto of dut oot Root dy oo Roo
25 wo- Cte uto d z Code two- Cte uto, htezed y e e. he veto of geezed oodte : d z x d d he Jo eeded fo the outto of the et tx e: ( J ) [ ] [ z ] [ ] [ z z ] ( ) ( ) ( J ) whe thee e o otuto to the gu veote. Coto of dut oot Root dy oo Roo
26 Coto of dut oot Root dy oo Roo Coutg the et tx wth the gee fou, we ot: A B ott, C,.e. thee e o etfug d Coo te. he, e: the veto of the gvtto te : g g J J J J B g g wo- Cte uto
27 wo- Cte uto If thee e o fto toue d o foe t the ed-effeto: ( ) d ( ) d f g f f e f : foe whh t og the geezed oodte Coto of dut oot Root dy oo Roo
28 wo- uto y, I, I x Let u ode two- uto: e: d egth: d dte of the ete of fo the ot xe: d oet of et oud xe g though the ete of d e to z : I d I Geezed oodte: he Jo eeded fo the outto of the et tx deed o the foowg veto:,,, z z Coto of dut oot Root dy oo Roo
29 Coto of dut oot Root dy oo Roo wo- uto L [ ] [ ] z J [ ] [ ] z J O O L [ ] [ ] z z J [ ] [ ] z z J O O O
30 Coto of dut oot Root dy oo Roo g to out tht the gu veoty veto d e ged wth z t ot eey to oute otto te R, o tht the outto of the et tx gve: I I I I I I O O O O J J J J J J J J B wo- uto deed o deed o ott!
31 Coto of dut oot Root dy oo Roo Fo the et tx t e to oute the Chtoffe yo: h h h h wo- uto
32 Coto of dut oot Root dy oo Roo he exeo of tx C the:, h h h C We vefy tht tx N ew-yet: N C B N,,, h h h h h h h h h h Futheoe, e g [ g ], the veto of the gvtto te : g g g g wo- uto
33 Coto of dut oot Root dy oo Roo τ : toue ed t the ot τ g g I I I τ g I I Wthout fto t the ot, the euto of oto e: wo- uto
34 Coto of dut oot Root dy oo Roo By e eto, we ot the dy ete wth eet to whh the ode e,.e. thoe ete fo whh we wte: π τ Y,, We hve: [ ] π π π π π π I I π π π π π wo- uto
35 wo- uto Y y y y y y y y g y g y (,, ) y g y y 4 y y 5 g he oeffet of Y deed o,, the ft d eod devtve, g d. Coto of dut oot Root dy oo Roo
36 Idetfto of dy ete he ety of the dy ode wth eet to the dy ete: τ Y (,, )π ow to etu oedue fo the exeet detfto of the e ete, whh e uuy uow o uet. Sute oto tetoe. ut e exeuted, og whh the ot oto e eoded, the veote e eued o oted y ue dffeetto,.. d the eeto e oted wth fteed (o o-u) dffeetto. Ao the toue τ e eued, dety (wth ute eo) o dety, fo the eueet of uet the oto. Suoe to hve the eueet (det o det oe) of the ve fo the te tt t,, t N. Coto of dut oot Root dy oo Roo
37 Idetfto of dy ete Wth N eueet et: τ τ τ ( t ) ( t ) ( t ) Y Y N t N π Yπ Sovg wth et-ue tehue: π ( Y Y ) Y τ Left eudo-vee tx of Y Oy the eeet of π fo whh the oeodg ou dffeet fo zeo e detfed. Soe ete e detfe oy oto wth othe oe. he tetoe to e ued ut e uffety h (good odtog of tx Y Y): they eed to vove ooet the dy ode Coto of dut oot Root dy oo Roo
38 Newto-Eue fouto A tetve wy to foute the dy ode of the uto the Newto-Eue ethod. It ed o e of foe d oet tg o the ge, due to the teto wth the ey the et h. We ot yte of euto tht ght e oved euve wy, ogtg the veote d eeto fo the e to the ed effeto, whe the foe d oet the oote wy: veote eeto foe oet Reuo e Newto-Eue goth outtoy effet. Coto of dut oot Root dy oo Roo
39 Defto of the ete Let u ode the gee of the et h: h tue te fo the textoo: B. So, L. Svo, L. V, G. Ooo: Root: Modeg, g d Coto, 3d Ed. Sge, 9 We defe the foowg ete: d I -,C,C -, d et teo of the veto fo the og of fe (-) to the ete of C veto fo the og of fe to the ete of C veto fo the og of fe (-) to the og of fe Coto of dut oot Root dy oo Roo
40 Defto of the ve. C e veoty of the ete of C. e veoty of the og of fe gu veoty of the.. C e eeto of the ete of C.. e eeto of the og of fe. gu eeto of the g gvty eeto f f foe exeted y - o foe exeted y o µ oet exeted y - o wth eet to the og of fe - µ oet exeted y o wth eet to the og of fe A veto e exeed the e fe. Coto of dut oot Root dy oo Roo
41 Coto of dut oot Root dy oo Roo Newto-Eue fouto Newto euto (tto oto of the ete of ) C g f f Eue euto (otto oto) C C dt d µ µ I I I f f,, Geezed foe t ot : t e ove gyoo effet τ ot otto t ot z z f µ
42 Coto of dut oot Root dy oo Roo Aeeto of ogto of the veote: ot otto t ot z ot otto t ot,, d z ogto of the eeto: ot otto t ot z z ot otto t ot,,,, d d z z of ete,, C C C Note: devtve of veto tthed to the ovg fe t t t t S t dt d t dt d R R
43 Reuve goth A fowd euo of veote.. d. eeto de: t odto o, -g,..... outto of,,, C veote eeto A wd euo of foe d oet de: te odto o f d µ outto: f f C f (,, C ) µ f C I ( ), µ I foe oet he geezed foe t ot outed: fto otuto τ f µ z z F v F v d f f t ot otto ot Coto of dut oot Root dy oo Roo
44 Lo efeee fe U to ow we hve uoed tht the veto e efeed to the e fe. It oe oveet to exe the veto wth eet to the uet fe o. I th wy, veto -, e,c d the et teo I e ott, whh e the goth outtoy oe effet. he euto e odfed oe te (we eed to uty the veto y ute otto te) ut othg hge the tue of the ethod. Coto of dut oot Root dy oo Roo
45 wo- uto y, I, I x Let u ode g two- uto wth otto ot, whoe ode h ee edy deved wth the Eue-Lgge ethod: e: d egth: d dte of the ete of fo the ot xe: d oet of et oud xe g though the ete of d e to z : I d I It odto fo the fowd euo of veote d eeto; [ g ] g, It odto fo the wd euo of foe d oet: f µ 3, Coto of dut oot Root dy oo Roo
46 Coto of dut oot Root dy oo Roo Defto of veto d te Let u efe the utte to the uet fe o the. We deve thee ott veto: he otto te e:,,,,,,, C C,, 3 R R R
47 Coto of dut oot Root dy oo Roo Fowd euo: z R z z R,, g g R,, g g C C C
48 Coto of dut oot Root dy oo Roo z R z z R,, g g R,, g g C C C Fowd euo:
49 Coto of dut oot Root dy oo Roo g g C C f R f, ,, g I C C µ µ I I f R R f g I I τ z R µ Bwd euo: (det to the euto oted wth Eue- Lgge ethod)
50 Coto of dut oot Root dy oo Roo g g C f R f,,, g g I I C C µ µ I I f R R f g g I I I τ z R µ Bwd euo: (det to the euto oted wth Eue- Lgge ethod)
51 Eue-Lgge v. Newto-Eue Eue-Lgge fouto t ytet d ey to udetd t etu the euto of oto yt d ot fo, etg the et tx, the Coo d etfug te, the gvtto te. A thee eeet e uefu fo the deg of ode ed otoe t ed tef to the toduto to the ode of oe oex effet (e ot o defoto) Newto-Eue fouto t outtoy effet euve ethod Coto of dut oot Root dy oo Roo
52 Det d vee dy B C( ) g F f (, ) τ, v Det dy.. Fo gve ot toue τ(t), detee. the ot eeto (t) d, f t oto (t ) d veote (t ) e ow, the oto (t) d the veote. (t). oe whoe outo uefu ode to oute the uto ode of the oot uto It e oved oth wth Eue-Lgge d wth Newto-Eue ohe Ivee dy... Fo gve eeto (t), veote (t) d oto (t) detee the ot toue τ(t) eeded fo oto geeto. oe whoe outo uefu fo tetoy g d ode ed oto It e effety oved wth the Newto-Eue fouto Coto of dut oot Root dy oo Roo
53 Coutto of det d vee dy Coutto of the vee dy e ey doe oth wth the Eue- Lgge ethod d wth the Newto-Eue oe. A fo the outto of the det dy, et u ewte the dy ode of the uto thee te: B (, ) τ whee: ( ) C(, ) g F f (, ), v We thu hve to uey tegte the ext yte of dffeet euto: ( τ ( )) B, whee the eeet eeded to ud the yte e dety outed y the Eue-Lgge ethod. Coto of dut oot Root dy oo Roo
54 Coutto of det d vee dy How to oute the det dy wth the Newto-Eue ethod? Newto-Eue t (Mt, C, ): τ ΝΕ (,, ). Wth the.. uet vue of d, ft teto of the t efoed, ettg. I th wy the toue τ outed y the ethod dety etu the veto. he. we et g de the t ( ode to ete the gvtto effet) d ( ode to ete Coo, etfug.. d.. fto effet). teto of the t e efoed, wth d,. h wy tx B foed ou y ou d eeet to fo the yte of euto e ve. Coto of dut oot Root dy oo Roo
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