An Analysis of Stability of Systems with Impulse Effects: Application to Biped Robots

Transcription

1 A Aalyss of Stablty of Systems wth Impulse Effects: Applcato to Bped Robots L me Lu,, Ya tao a, Pe je Zhag, Zhe ze Lu College of Commucato Egeerg, J l Uversty, Chag chu, 3005, Cha Departmet of Mathematcs of Chag chu axato College, Chag chu, 307, Cha dalu37@sacomc, tayt@maljlueduc Abstract he approxmate Jacoba matrx of the Pocaré retur map at the fxed pot s preseted for the olear system wth mpulse effects Ad the suffcet codto to the exstece of ths approxmate Jacoba matrx s gve wth the dsturbace theory ad learzato method Sce ths approxmate expresso depeds oly o the cofgurato of the system wth mpulse effects, the the uqueess of ths approxmate expresso ca be guarateed ad ths approxmate expresso ca be obtaed precsely I addto, ths approxmate Jacoba matrx ca be used as a tool to study the asymptotcal stablty of the system wth mpulse effects Sce the bped robot gats ca be descrbed by the olear system wth mpulse effects, the the stablty of the bped robot walkg cycle ca be studed wth ths tool I order to study the stablty, ths approxmate Jacoba matrx s appled to the compass-lke passve bped robot gats It s show that the approxmate Jacoba matrx proposed ths paper s as useful as the oes proposed the umercal methods I the ed ths result s cofrmed by smulatos Keywords stablty, Pocaré map, Jacoba matrx, lmt cycle, olear system wth mpulse effects, bped robot Ⅰ INRODUCION he olear system wth mpulse effects s a hybrd dyamc system cosstg of the ordary dfferetal equatos ad the algebrac mappg dyamc systems If the olear system wth mpulse effects s stable, the the system has a stable lmt cycle he stablty of the olear system wth mpulse effects has bee studed usg the theory of the Lyapuov fucto method But the most prmary method s usg the theory of the Pocaré retur map Sce a asymptotcally stable fxed pot of Pocaré retur map correspods to a asymptotcally stable perodc trajectory of the system It s well kow how to use the umercal methods to compute the Pocaré retur map McGeer, Goswam ad Garca have studed the stablty of passve gat va the methods of the Pocaré retur map [,, 3, 4, 5, 6], but they all use the umercal methods Sce Jacoba matrx of the Pocaré retur map at the fxed pot has ucertaty through the umercal method, so the result of the stablty o bped robot gat s rough Grzzle has proposed the restrcted Pocaré secto theory to study the system wth the mpulse he correspodg author: tayt@maljlueduc hs work was supported by the Natoal Hgh echology Research ad Developmet Program of Cha (863Program ( Grat No: 006AA04Z5 effects, ad appled t to the bped locomoto[7,8,9] But themethod proposed by Grzzle s ot coveet to the hyperspace I ths paper, a approxmate expresso of the Jacoba matrx of the Pocaré retur map at the fxed pot s proposed for the system wth mpulse effects Ad ths expresso s obtaed o the bass of the dsturbace theory ad the learzato method Furthermore, the suffcet codto to the exstece of ths expresso s proposed hs approxmate expresso bases oly o the cofgurato of the system; however the Jacoba matrx the umercal method bases o the data that are chose So the result of the approxmate expresso proposed ths paper s uque, but the results umercal methods are varous Sce the stablty of the system wth mpulse effects ca be determed o the bass of Jacoba matrx of the Pocaré retur map, the usg the approxmate expresso proposed ths paper ca acheve the stablty of the olear system wth mpulse effects exactly Specally, ether plaar bped robot gats or o-plaar bped robot gats ca be all expressed by the system wth mpulse effects, so the stablty of bped robot gats ca be acheved by the approxmate expresso proposed ths paper I ths paper, the stablty of a compass-lke passve bped robot gats s argued wth the approxmate Jacoba matrx both proposed ths paper ad umercal method It s demostrated that the approxmate Jacoba matrx proposed ths paper has uqueess ad accuracy of judgmet o the stablty of the system wth mpulse effects Ⅱ SYSEM WIH IMPULSE EFFECS Cosder a ordary dfferetal equato ( x t x ( t = f ( g x( t u( t ( where x( t Χ, ad Χ s a coected ope subset of R, m ut (, ad f ad the colums of g are cotuous vector felds o Χ Let S : = { xt Χ r( xt = 0}, where r : Χ R, ad r s called the mpulse effect detecto fucto Let H : S Χs called the mpulse effect trasto fucto A system wth mpulse effects s preseted as follows: xt ( = f xt ( g xt ( ut ( x( t : S x ( t = H( x ( t x ( t S ( /08 /$ IEEE RAM 008

2 ad x (: t = lm x( τ ad x (: t = lm x( τ deote respectvely τ t τ t the left ad the rght lmts of the trajectory x( τ atτ = t I smple words, a trajectory of the above model (s specfed by the dfferetal equato(utl ts state mpacts the hyper surfaces At ths pot, the mpulse mode H compresses the mpact evet to a stataeous momet of tme, resultg a dscotuty the state trajectory he ultmate result of the mpact model s a ew tal codto, from whch the dfferetal equato model evolves utl the ext mpact wths I order to avod the state havg to take o two values at the mpact tme, the mpact evet s descrbed terms of the values of the state just pror to mpact at tme " t " ad just after mpact at tme" t " hese values are represeted by the left lmt x ad the rght lmt x of equato (respectvely Usg the theory of the Pocaré retur map s the prmary method to study the stablty of the system wth mpulse effects Let x = P( x be the Pocaré retur map of k k the model( he stat secto after mpact s chose as the Pocaré secto Let ϕ ( t,0, xk be a soluto of equato(, so P( xk : = H ϕ ( t,0, xk he relatoshp betwee modules of the egevalues λ of the Pocar é retur map P ad the stablty of the model(s as follows: λ < λ λ > asymptotcally stable stable ustable Ⅲ MAIN RESUL he Pocar é retur map s deoted as x P( x =, k k where x k represets the state after the kthmpulse effect he stat after the mpulse effect just happeed s chose as the Pocaré secto Sce x s a fxed pot of the mappg, that s Px = x, the learzato of the mappg P at the fxed pot s P( x x P( x x = x x he P( x x x x J : = s the Jacoba matrx of P at the fxed pot x I partcular,, where = J ( x x x x P x P x, so x J x [6,0,] I ths secto, the moto betwee mpulse effects s expressed as a ordary dfferetal equato x ( t = f ( x( t Let x ( t be a lmt cycle trajectory wth the tal pot x at 0 x t = f x( t ad letτ deote the lmt cycle tme t = of ( betwee the mpulse effects x H( x * effect trasto fucto, such that x H[ x ( τ ] = deotes the mpulse = he mpulse effect detecto fucto s defed as r( x herefore a mpulse effect occurs whe r( x = 0, that s, rx [ ( τ ] = 0 Uder the above hypotheses, the system wth mpulse effects ths secto s deoted as xt ( = f xt ( x( t : S, (3 x ( t = H( x ( t x ( t S { } where xt R r( xt S : = Χ = 0 Lemma : Cosder the system wth mpulse effects(3 satsfyg the followg hypotheses: ( f ( x( t s cotuous ad dfferetable o Χ ; ( a soluto of x ( t = f ( x( t from a gve tal codto s uque ad depeds cotuously o the tal codto; (3 the model(3exts a lmt cycle trajectory x ( t wth the tal pot x at t = 0 ; (4 a perturbato xˆ(0 s gve at the tal pot x he the perturbato trajectory xˆ( t of the lmt cycle trajectory x ( t ca be obtaed as ( * Df x ( t ˆ t f x ( t x ( t e xˆ(0, where Df ( x ( t : = Proof: If a perturbato xˆ(0 s gve at the tal pot x, the lmt cycle trajectory has chaged mmedately Sce xˆ( t deotes the perturbato trajectory of the lmt cycle trajectory x ( t, so x ( t xˆ ( t deotes the lmt cycle of model (3wth the tal pot x xˆ(0 at t = 0 Ad Forε s suffcetly small, the x ( t ε x ˆ( t * * ( x ( t εx ˆ ( t f ( x ( t εxˆ( t f ( x ( t Sce x ( t = f ( x ( t where Df ( x t = f x ( t ( : = f x ( t ε xˆ( t f x ( t, so x ˆ( t xˆ( t, Df ( x ( t t, so xˆ( t e xˆ(0 Lemma : Supppose that rx ( s a cotuous ad dfferetable fucto o Χ, the the secod dervatve of rx ( s also cotuous

3 So uder the hypotheses of lemma, the perturbato ˆτ of the lmt cycle tme betwee mpulse effectsτ almost s Dr x ˆ x( τ ˆ τ, where rx Dr x : Dr x x = ( τ Proof: From Lemma, t s ecessary to chage the tme betwee mpulse effects he perturbato of the lmt cycle tme betwee mpulse effectsτ s expressed by ˆ τ Forε s suffcetly small, 0 = r x ( τ ετˆ εxˆ( τ ετˆ rx ( τ ετˆ rx ( ˆ τ ετ εxˆ( τ ετˆ o( ε rx r x x rx rx x ˆ ε xˆ( τ ετˆ o( ε t rx r x x rx ( τ xˆ( τ ε o ε rx rx x ( τ ετˆ xˆ( τ ε Dr x xˆ So ˆ τ τ τ Dr x x, ( τ where Dr x ( τ : = rx s cotuous ad dfferetable o Χ Moreover uder the hypotheses of lemma Lemma 3: It s assumed that H ( x ( t ad lemma, the perturbato ˆx of the result of the mpulse effect s expressed as x ( τ Drx xˆ DHx I xˆ( τ, Dr x x ( τ x r where DH x = : ; Dr x x : = Proof:Sce a perturbato s added to the tal pot x of the lmt cycle trajectory x ( t, terms of lemma ad lemma, the result of the mpulse effect s chaged too Let ˆx be the perturbato of the result of the mpulse effect he x εxˆ = H x ( τ ετˆ εxˆ( τ ετˆ Forε s suffcetly small, x εxˆ = H x ( τ ετˆ εxˆ( τ ετˆ H x ( τ ετˆ H x ( ˆ τ ετ εxˆ( τ ετˆ o( ε x ( τ H x x H x ( τ ετˆ ε xˆ( τ x ˆ o( ε x ( τ x x x εxˆ( τ o( ε x x ( τ Drx xˆ I xˆ τ, Dr x x ( τ So where I deotes the detty matrx ad Dr x : = r x heorem : Cosder the olear system wth mpulse effects xt ( = f xt ( x( t : S x ( t = H( x ( t x ( t S { ( ( 0} where S xt R r( xt = Χ = followg codtos:, whch satsfes the ( f ( x( t s cotuous ad dfferetable o Χ ; ( a soluto of x ( t = f ( x( t from a gve tal codto s uque ad depeds cotuously o the tal codto; (3 the system exts a lmt cycle trajectory x ( t wth the tal pot x at t = 0, ad the lmt cycle tme

4 betwee the mpulse effectsτ s kow ; (4 x s the fxed pot of the Pocaré retur map P that correspods to ths system; (5 rx ( s a cotuous ad dfferetable fucto o Χ ; (6 H ( x ( t s cotuous ad dfferetable o Χ he Jacoba matrx J of the Pocaré retur map P at the fxed pot x for ths system wth mpulse effects s expressed approxmately as x ( τ Drx DH x I e Dr x x ( τ x where DH x : = ; Dr x : = rx Proof: Wth the whole hypotheses, Df ( x ( τ x ( τ Drx ( τ xˆ DHx I Dr x x ( τ Df ( x ( τ τ e xˆ(0 x where DH x : = s obtaed mmedately from lemma ad lemma 3 I terms of lemma 3, f xˆ deotes x ˆ(0, ad x ˆ deotes ˆx, the xˆ I the ed, f( x J : = x x x ( τ Drx I Drx x ( τ Df ( x ( τ τ e xˆ x ( τ Drx DH x ( τ I e Dr x x ( τ ( ( τ τ Df x τ, hs expresso oly eeds to fd a lmt cycle trajectory x ( t ad the lmt cycle tme betwee the mpulse effectsτ Ad the lmt cycle ad the lmt cycle tme betwee mpulse effects ca be obtaed through the umercal method So the stablty of the olear system wth mpulse effects ca be studed by the Jacoba matrx of the Pocaré retur map at the fxed pot Ⅳ BIPED ROBO AND SABILIY I ths secto, the goal s to study the stablty of a walkg cycle for the compass-lke passve bped robot usg the Jacoba matrx of the Pocaré retur map at the fxed pot So the smplest passve bped robot s troduced he smplest passve robot cossts of a hp ad two legs of equal legth wth o akles ad o kees [] Fgure presets the sketch of the smplest passve robot he walkg cycle s composed of the swg phase of moto (the sgle support-oly oe leg s touchg the groud ad the collso stage (the double support-two legs are all touchg the groud Sce the swg phase of moto ca be descrbed by the olear dfferetal equato, ad the double support stage ca be descrbed by the olear algebrac mappg dyamc system So the model of the walkg cycle of ths bped robot ca be descrbed by the olear system wth mpulse effects he dyamc model of the robot betwee successve mpacts s derved from the theory of Lagrage, ad the expresso a stadard secod order system terms of the ormalzed parameters µ ad β s where θ ( θ, θ M ( θ θ N ( θ, θ θ g ( θ = 0, a = are the agles made respectvely by the s s swg ad the support leg wth the vertcal(couterclockwse postve ( θ, ( θ, θ, ( θ M N g deped o µ ad β, ad they are deduced from the Lagrage,where M β ( β βcos α ( θ = ; ( β βcos α ( β ( µ 0 ( ββθ ss( θs θ s N (, θθ = ; ( ββθ s s( θs θs 0 g ( θ g β s ( θs ( = µ β g s( θs m H b Ad β = ; µ = a m So the olear dfferetal equato of the sgle support s expressed by θ dt ω a = he state space for the system s take as d x = = M ( θ N ( θ, ω ω g ( θ : f ( x Χ= : { x : = ( θ, ω } ' θ π, π ; ω R he collso stage s descrbed as ( α x = W x, ad

5 W ( α deotes the collso trasto fucto, where W K 0 ( α = 0 H ( α, that s, x = H( x = W( α x, where Q Q 0 K = ; 0 H( α = ( Q ( α Q ( α ; β β ( µ β β cosα ; α = 0 β β( β ( β cosα ( β (( β βcosα ( α = µ ( β β β( β cosα he collso surface s {( θ, ω } ' θ ;, ; s θs φ θ π π ω S = = R he mpact detecto fucto s r( x θ θ φ 0 s s = = he premses uderlyg ths model are that: ( the mpact takes place stataeously; ( the mpact of the swg leg wth the groud s assumed to be elastc ad wthout sldg; (3 the tp of the support leg s assumed ot to be slp, ad the robot behaves as a ballstc double-pedulum; (4 a kee-less robot wth a rgd leg ca ot clear the groud We choose the parameters as follows: β = ; µ = ; α = 070; φ = 3 ; a = b = 05 hrough the umercal method, a fxed pot s x ( 0 = [ 0334, 086, 03779, 09] he lmt cycle tme betwee the collsos sτ = [ ] x ( τ = 086, 0334, 804, 4936 By meas of calculato wth theorem, the Jacoba matrx J of the Pocaré retur map at fxed pot x (0 s J = he egevalues of the Jacoba matrx J of the Pocaré retur map at fxed pot x (0 are , , 0, 0958 he modules of these egevalues are 057, 057,0, ad 0958 Sce all of the modules of egevalues are strctly less tha oe, the bped gats are asymptotcally stable he exstece ad the characterstcs of steady passve compass gats are cofrmed by smulatos (fg Uder the same parameters codtos, the Jacoba matrx of Pocaré retur map at the fxed pot x (0 obtaed by the umercal method whch s troduced [] s J = he egevalues of the Jacoba matrx J of Pocar é retur map at the fxed pot x (0 obtaed by the umercal method are , , 0, 039 he modules of the egevalues of J are 0763, 0763, 0 ad 039 hey are all less tha However uder the same hypotheses, the Jacoba matrx of the Pocaré retur map at the fxed pot x (0 obtaed by the same umercal method[] s J = he egevalues of the Jacoba matrx J of the Pocaré retur map at fxed pot x (0 are 05 05, 05 05, 0, 004 he modules of the egevalues of J are 033, 033, 004 ad hey are less tha, too Sce J s obtaed o bass of the cofgurato of the model, the J s uque However J ; J are acheved o the data that are chose the umercal method, so J ; J are dfferet I the followg, the learty errors of J ad J are compared the same codtos he results are show able It s show that the learty error of J s close to the learty error of J So the accuracy of judgmet o the stablty of the system wth mpulse effects s guarateed usg the approxmate Jacoba matrx proposed ths paper I a word, the Jacoba matrx of the Pocaré retur map at the fxed pot the umercal method has ucertaty

6 ad the judgmet o the stablty s cosstet the judgmet through the Jacoba matrx proposed ths paper So the method ths paper s pror to the umercal method, whe the stablty of the system wth mpulse effects s studed ABLE RESULS OF COMPARISON BEWEEN J AND J tal pot of lmt cycle learty error of J learty error of J [ 034, 086, 03779, 09] 03% 07% [ 0334, 096, 03779, 09] 03% 04% [ 0334, 086, 03779, 09] 05% 0% [ 0333, 087, 03778, 09] 0% 8657e-004 [ 0334, 086, 03779, 09] 03% 06% [ 0334, 096, 03779, 093] 09% 037% [ 0334, 086, 03879, 09] [ 0334, 086, 0779, 09] [ 03334, 086, 03779, 09] a b= l a θ s b m 0% 04% 5% 58% 09% 48% θ s α φ m H Fgure Model of a compass-lke passve bped robot walkg dow a slope (Swg stage m Fgure A lmt cycle of stable bped gats Ⅴ CONCLUSION Wth the dsturbace theory ad the learzato method, a approxmate Jacoba matrx of the Pocaré retur map at the fxed pot s preseted for the olear system wth mpulse effects he suffcet codto to the exstece of ths approxmate Jacoba matrx s gve Ad ths approxmate Jacoba matrx s compared wth the oes obtaed umercal method I coclusos, frstly ths approxmate Jacoba matrx depeds oly o the cofgurato of the model; secodly ths approxmate Jacoba matrx has uqueess that does ot exst the umercal method; thrdly the stablty of the system wth mpulse effects s the same usg ths approxmate Jacoba matrx ad the oes obtaed umercal method So the approxmate Jacoba matrx ths paper ca be used to study the stablty of the system accurately ad coveetly Sce the bped robot gats ca be preseted by a system wth mpulse effects, so the method proposed ths paper ca be used to study the stabltes of the bped robot gats But ths approxmate expresso eeds to fd the lmt cycle x ( t ad the lmt cycle tme betwee mpulse effectsτ So the ext job s to fd the fxed pots of the Pocaré map wth the theory of mathematcs REFERENCES [] McGeer, Passve dyamc walkg, Iteratol Joural of Robotcs Research, vol9, No, pp 6-8, Aprl, 990 [] A Goswam, B hulot, ad BEspau, Compass-lke bped robot Part: Stablty ad bfurcato of passve gats, Rapport de recherche INRIA, RhoeALPES, 996, ute de Rapport stmart Frace, october 996 [3] Bhulot, AGoswam, ad BEspau, Bfurcato ad chaos a smple passve bpedal gat, IEEE Iteratoal Coferece o Robotcs ad Automato,vol, Issue, 0-5, pp79 798, Aprl,997 [4] A Goswam, B hulot, ad B Espau, A study of the passve gat of a compass-lke bped robot:symmetry ad chaos, Iteratoal Joural of Robotcs Research,vol7, No, pp 8 30, December,998 [5] AGoswam, BEspau, ad AKeramae, Lmt cycles a passve compass gat bped ad passvty-mmckg cotrol laws, Joural of Autoomous Robots, vol 4, No3, pp 73-86, 997 [6] MGarca, ARua, AChatterjee, ad MColema, he Smplest Walkg Model: Stablty, Complexty ad Scalg, ASME Joural of Bomechacal Egeerg, vol 0, No, pp 8-88, Aprl, 998 [7] J WGrzzle, F Plesta, ad GAbba, Pocaré maps s method for systems wth mpulse effects: applcato tomechacal bped locomoto, Decso ad Cotrol, 999 Proceedgs of the 38th IEEE Coferece,vol 4, pp , 999 [8] JWGzzle, GAbba,ad FPlesta, Asymptotcally Stable Walkg for Bped Robots: Aalyss va Systems wth Impulse Effects, IEEE ras Autom Cotrol, vol46, No, pp5-64, Jauary, 00 [9] EPlesta, JWGzzle, ERWestervelt, ad Gabba, Stable walkg of a 7-DOF bped robot, IEEE rasactos o Robotcs ad Automato,vol9, No4, pp , August, 003 [0] MColema, AChatterjee, ad ARua, Moto of a rmless spoked wheel: A smple 3D system wth mpact, Dyamc ad Stablty of Systems, vol, No 3, pp 39-69, Spetember, 997 [] MWSpog ad FBullo, Cotrolled symmetres ad passve walkg, IEEE rasactos o Automato ad Cotrol,vol50, No7, pp 05-03, July, 005 [] Zheze Lu, O Passve-Dyamc Model of Huma Gat ad Some Useful Cotrol Strateges, Doctor thess, October 007, upublshed