Contact instability. integral-action motion controller coupling to more mass evokes instability

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1 Conac insabiliy Probl: Conac and inracion wih objcs coupls hir dynaics ino h anipulaor conrol sys This chang ay caus insabiliy Exapl: ingral-acion oion conrollr coupling o or ass voks insabiliy Ipdanc conrol affords a soluion: Mak h anipulaor ipdanc bhav lik a passiv physical sys Hogan, N. (1988) On h Sabiliy of Manipulaors Prforing Conac Tasks, IEEE Journal of Roboics and Auoaion, 4: Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 1

2 Exapl: Ingral-acion oion conrollr Sys: (s 2 + bs + k) x = cu - f Mass rsraind by linar spring & x c = dapr, drivn by conrol acuaor & u s 2 + bs + k xrnal forc Conrollr: g u = (r x) Ingral of rajcory rror s Sys + conrollr: (s 3 + bs 2 + ks + cg) x = cgr - s f x cg = 3 r s + bs 2 + ks + cg s: Laplac variabl Isolad sabiliy: x: displacn variabl bk f: xrnal forc variabl Sabiliy rquirs uppr bound on > g u: conrol inpu variabl conrollr gain c r: rfrnc inpu variabl : ass consan b: daping consan k: siffnss consan c: acuaor forc consan g: conrollr gain consan Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 2

3 Exapl (coninud) Objc ass: f = s 2 x 3 Coupld sys: [( + )s + bs 2 + ks + cg] x = cgr x cg = r ( + )s 3 + bs 2 + ks + cg Coupld sabiliy: bk > cg( + ) : objc ass consan Choos any posiiv conrollr gain bk > g ha will nsur isolad sabiliy: c Tha conrolld sys is dsabilizd by coupling o a sufficinly larg ass > bk cg Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 3

4 Probl & approach Probl: Find condiions o avoid insabiliy du o conac & inracion Approach: Dsign h anipulaor conrollr o ipos a dsird inracion-por bhavior Dscrib h anipulaor and is conrollr as an quivaln physical sys Find an (quivaln) physical bhavior ha will avoid conac/coupld insabiliy Us our knowldg of physical sys bhavior and how i is consraind Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 4

5 Gnral objc dynaics Assu: L( q, q ) = E k (, q Lagrangian dynaics d L d q L Passiv = P D ( q, q ) q Sabl in isolaion p = L q = E * k q Lgndr ransfor: * E k ( p, q ) = p q E k ( q, q ) Kinic co-nrgy o kinic H ( p q ) = p q q, ) * q ) E p ( q ) nrgy, L( q Lagrangian for o Hailonian for Hailonian = oal sys nrgy H ( p, q ) = E k (, q q = H p p = H q D + P q : (gnralizd) coordinas L: Lagrangian E * k : kinic co-nrgy p ) + E p ( q ) E p : ponial nrgy D : dissipaiv (gnralizd) forcs P : xognous (gnralizd) forcs H : Hailonian Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 5

6 Sir Willia Rowan Hailon Willia Rowan Hailon Born 1805, Dublin, Irland Knighd 1835 Firs Forign Associa lcd o U.S. Naional Acady of Scincs Did 1865 Accoplishns Opics Dynaics Quarnions Linar opraors Graph hory and or hp:// HisMah/Popl/Hailon/ Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 6

7 Passiviy Basic ida: sys canno supply powr indfinily Many alrnaiv dfiniions, h bs ar nrgy-basd Wya al. (1981) Wya, J. L., Chua, L. O., Gann, J. W., Göknar, I. C. and Grn, D. N. (1981) Enrgy Concps in h Sa-Spac Thory Passiv: oal sys nrgy is lowr-boundd of Nonlinar n-pors: Par I Passiviy. Mor prcisly, availabl nrgy is lowr-boundd IEEE Transacions on Circuis and Syss, Vol. CAS-28, No. 1, pp Powr flux ay b posiiv or ngaiv Convnion: powr posiiv in Powr in (posiiv) no lii Powr ou (ngaiv) only unil sord nrgy xhausd You can sor as uch nrgy as you wan bu you can wihdraw only wha was iniially sord (a fini aoun) Passiviy sabiliy Exapl: Inracion bwn opposily chargd bads, on fixd, on fr o ov on a wir Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 7

8 Sabiliy Sabiliy: Convrgnc o quilibriu Us Lyapunov s scond hod A gnralizaion of nrgy-basd analysis Lyapunov funcion: posiiv-dfini non-dcrasing sa funcion Sufficin condiion for asypoic sabiliy: Ngaiv si-dfiniiv ra of chang of Lyapunov funcion For physical syss oal nrgy ay b a usful candida Lyapunov funcion Equilibria ar a an nrgy inia Dissipaion nrgy rducion convrgnc o quilibriu Hailonian for dscribs dynaics in rs of oal nrgy Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 8

9 Sady sa & quilibriu Sady sa: Kinic nrgy is a posiiv-dfini non-dcrasing funcion of gnralizd onu Assu: Dissipaiv (inrnal) forcs vanish in sady-sa Ruls ou saic (Coulob) fricion Ponial nrgy is a posiivdfini non-dcrasing funcion of gnralizd displacn Sady-sa is a uniqu quilibriu configuraion Sady sa is quilibriu a h origin of h sa spac {p,q } q = 0 = H p = E k p E p = 0 p = 0 k p = 0 = H q D + P Assu D ( 0, q ) = 0 Isolad P = 0 H E k E + p = q q q p = 0 p = 0 E k H E = p = 0 q q q p = 0 p = 0 E q = 0 q = 0 p Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 9

10 Noaion Rprsn parial drivaivs using H H subscrips q = q H is a scalar H h Hailonian sa funcion H p = p H q is a vcor Parial drivaivs of h Hailonian q = H p ( p, q ) w.r.. ach ln of q H is a vcor p Parial drivaivs of h Hailonian w.r.. ach ln of p p, D ( p, )+ P = H q ( p q ) q Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 10

11 Isolad sabiliy q q H pp Us h Hailonian as a Lyapunov dh d = H + funcion dh d = HqH p + H p ( H q D + P ) Posiiv-dfini non-dcrasing funcion of sa dh d = q P q D Ra of chang of sord nrgy = powr in powr dissipad Isolad P Sufficin condiion for asypoic = 0 sabiliy: dh d = q D Dissipaiv gnralizd forcs ar a posiiv-dfini funcion of q D > 0 dh d < 0 p 0 gnralizd onu Dissipaion ay vanish if p = 0 and sys is no a quilibriu Bu p = 0 dos no dscrib any sys rajcory LaSall-Lfshz hor Enrgy dcrass on all non- quilibriu sys rajcoris Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 11

12 Physical sys inracion Inracion of gnral dynaic Inracion of physical syss syss If u i and y i ar powr conjugas Many possibiliis: cascad, G i ar ipdancs or adiancs paralll, fdback Powr-coninuous conncion: y 1 = G 1 ( s ) u Two linar syss: y = G 2 () 1 Powr ino coupld sys 2 s u 2 us qual n powr ino coponn syss Cascad coupling y 3 = y 2 quaions: u y u 2 = y 3 3 = u 1y 1 + u 2 y 2 1 u 1 = u 3 Cobinaion: y 3 = G 3 ( s ) u 3 G 3 ()= s G 2() s G 1() s No powr-coninuous y u y u 2 + y u Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 12

13 Inracion por Assu coupling occurs a a s of poins on h objc X This dfins an inracion por X is as a funcion of gnralizd X = L ( q ) coordinas q Gnralizd vlociy drins V = J ( q ) q por vlociy Por forc drins gnralizd P = J ( q ) F forc Ths rlaions ar always wlldfind Guarand by h dfiniion of gnralizd coordinas Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 13

14 Sipl ipdanc Targ (idal) bhavior of anipulaor F z = K (X z X o ) + B ( V z ) Elasic and viscous bhavior In Hailonian for: Hailonian = ponial nrgy q Assu V = 0 for sabiliy analysis o p = H zq ( q )+ B ( V ) q z = X z X o z z z z = V z V o H z ( q )= K ( q )dq F z = p z z z z Isolad: V z = 0 or F z = 0 V o = V z = 0 q = consan F z = consan z Sufficin condiion for isolad asypoic sabiliy: F z = 0 H zq = B dh z d = H zq q z = B q z B q z > 0 0 Unconsraind ass in Hailonian q = H p ( p ) H (p ) = 1 p M for p = F Hailonian = kinic nrgy Arbirarily sall ass V = q V z 1 p 2 Coupl hs wih coon vlociy V = V z F V + z F V z = 0 Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 14

15 Mass coupld o sipl ipdanc Hailonian for H ( p, q z ) = H (p ) + H z(q z ) Toal nrgy = su of coponns p = H q ( q z ) B ( H p (p )) q z = H p ( p ) Assu posiiv-dfini, nondcrasing ponial nrgy Equilibriu a (p,q z ) = (0,0 ) dh d = H p p + H Ra of chang of Hailonian: dh d = H H q H B + H H p = q B Sufficin condiion for asypoic z B sabiliy And bcaus ass is unconsraind, sabiliy is global q > 0 p 0 q q z p p q z Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 15

16 Gnral objc coupld o sipl ipdanc Toal Hailonian (nrgy) is su H ( p, q ) = H (, q z z of coponns H ( p, q ) = E k ( p, q ) + E p (q ) + H z( ( q o Assu Boh syss a quilibriu p ) + H (q ) Inracion por posiions coincid a coupling Toal nrgy is a posiiv-dfini, dh d = H J H + H H H H non-dcrasing sa funcion H p D p J H zq p J B Ra of chang of nrgy: dh d = q D q B zq p q p p q z Th prvious condiions sufficin for sabiliy of Objc in isolaion Sipl ipdanc coupld o arbirarily sall ass nsur global asypoic coupld sabiliy Enrgy dcrass on all non-quilibriu sa rajcoris Tru for objcs of arbirary dynaic ordr L ) X ) Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 16

17 Sipl ipdanc conrollr iplnaion 1 p I 1 ( q ) Robo odl: q = H p H = 2 p J Inrial chanis, saically p = H q D + P a + F balancd (or zro graviy), ffor- V = J q conrolld acuaors Hailonian = kinic nrgy X = L ( q ) Conrollr: = J { K ( L (q ) X ) B ( J q )} Transfor sipl ipdanc o anipulaor configuraion spac P a o q Conrollr coupld o robo: = H cp Sa srucur as a physical p = H cq D J B + J F sys wih Hailonian H c V = J q H c = H + H q : gnralizd coordinas (configuraion variabls) z X = L ( q ) p : gnralizd ona H : Hailonian I : inria D : dissipaiv (gnralizd) forcs P a : acuaor (gnralizd) forcs X,V,F : inracion por posiion, vlociy, forc L,J : kinaic quaions, Jacobian Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 17

18 Sipl ipdanc conrollr isolad sabiliy cq cp cp H cp Ra of chang of Hailonian: dh c d = H H H H cq D Enrgy dcrass on all non- H cp J B + H cpj F quilibriu rajcoris if dh d = q D V B + V F Sys is isolad F = 0 c Dissipaiv forcs ar posiiv- F = 0 dh c d = q D V B dfini q D > 0, V B > 0 p 0 Miniu nrgy is a q z = 0, X = X o Assu: Bu his ay no dfin a uniqu Non-rdundan chanis anipulaor configuraion Hailonian is a posiiv-dfini Non-singular Jacobian non-dcrasing funcion of q z bu Thn usually no of configuraion q Hailonian is posiiv-dfini & Inracion-por ipdanc ay no non-dcrasing in a rgion abou conrol inrnal dgrs of frdo q = L 1 ( X ) Could add rs o conrollr bu for sipliciy Local asypoic sabiliy o Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 18

19 Sipl ipdanc conrollr coupld sabiliy q ( q, ) Coupling kinaics q = q Coupling rlas q o q bu no nd o solv xplicily Toal Hailonian (nrgy) is su H = H ( p, q ) + H c ( p, q ) of coponns Ra of chang of Hailonian q p p q dh d = H H + H ( H D + J F ) dh d = q D + q J F q ( D + J B )+ q J F cq cp cp J B + J ) dh d = q D + V F q D V B + V F + H H + H ( H cq D F Coupling canno gnra powr V F + V F = 0 Th prvious condiions sufficin for sabiliy of Objc in isolaion Sipl ipdanc conrolld robo nsur local asypoic coupld sabiliy V dh d = q D q D B Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 19

20 Kinaic rrors ( ( o B ( )} Assu conrollr and inracion ~ ~ ~ P a = J { K L q ) X ) J q por kinaics diffr Conrollr kinaics aps ~ ~ ~ X = L ( q ) L ( q ) configuraion o a poin X Corrsponding ponial funcion ~ ~ ~ H z ( q ) = H z ( q z ) = H z ( L ( q ) X o ) is posiiv-dfini, non-dcrasing ~ ~ in a rgion abou q 1 = L ( X o ) Assu slf-consisn conrollr ~ ~ L q = J kinaics ~ ~ dx d = V = ~ J ( q ) q h (rronous) Jacobian is h corrc drivaiv of h ~ (rronous) kinaics ~ ~ ~ dh z d = H L zq q = H zq J q = Hzq V q Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 20

21 Kinaic rrors (coninud) ~ ~ Hailonian of his conrollr H c ( p, q ) = H ( p, q ) + H z ( q z ) ~ ~ coupld o h robo H c ( p, q ) = H ( p, q ) + H z ( ( q L ) X o ) q = H p Hailonian sa quaions ~ ~ p = H q D J H zq J B + F J ~ ~ dh c d = H zq J H p + HqHp Ra of chang of h Hailonian ~ ~ + H ( H D p q J H zq J B + J F ) ~ ~ dh c d = q D V B + J F In isolaion ~ ~ F = 0 dh c d = q D V B Prvious condiions on D & B ar sufficin for isolad local asypoic sabiliy Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 21

22 Insnsiiviy o kinaic rrors Th sa condiions ar also sufficin o nsur local ~ asypoic coupld sabiliy H, = E k ( p q ) + E ~ p (q ) + Coupld sys Hailonian and H ( p, q ) + H z ( L ( q ) X o ) is ra of chang: ~ ~ dh d = q D q D V B Sabiliy propris ar insnsiiv o kinaic rrors Providd hy ar slf-consisn No ha hs rsuls do no rquir sall kinaic rrors Could aris whn conac occurs a unxpcd locaions.g., on h robo links rahr han h nd-poin Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 22

23 Paralll & fdback conncions Powr coninuiy y 3 u 3 = y 2 u 2 + y 1 u 1 Paralll conncion quaions Powr balanc OK Fdback conncion quaions Powr balanc OK y 3 = ± y 2 ± y 1 u 3 = u 2 = y u u 1 = ± y u ± y u y 3 = y 1 = u 2 u 1 = u 3 y 2 u 1y 1 = u 3 y 3 y 2 u 2 Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 23

24 Suary rarks Inracion sabiliy Srucur ars Th abov rsuls can b xndd Dynaics of physical syss ar Nurally sabl objcs consraind in usful ways Kinaic consrains I ay b bnficial o ipos no dynaics physical sys srucur on a Inrfac dynaics gnral dynaic sys.g., du o snsors.g. a robo conrollr A sipl ipdanc can provid a robus soluion o h conac insabiliy probl Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 24

25 So ohr Irishn of no Bishop Gorg Brkly Robr Boyl John Boyd Dunlop Gorg Francis Fizgrald Willia Rowan Hailon Willia Thoson (Lord Klvin) Josph Laror Charls Parsons Osborn Rynolds Gorg Gabril Soks Mod. Si. Dyn. Sys. Inracion Sabiliy Nvill Hogan pag 25