Exponential Stability Analysis of a System Comprised of a Robot and its Associated Safety Mechanism

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1 rongs of nnul onfrn of hn nsu of ommunons Eponnl Sbl nlss of Ssm omprs of obo n s sso Sf Mhnsm Whu GUO ng YNG prmn of Mhms n nforms sn Zhngzhou Unvrs of lgh nusr Zhngzhou hn; E-ml: whguosr@hooomn; ngp66@hoon bsr: Th ssm omprs of robo n s sso sf mhnsm s su n h ppr h mho of srong onnuous sm-group h ppr nlzs h rsron of ssnl sprl growh boun of h ssm opror Th ssnl sprl rus of h ssm opror s lso suss bfor n fr prurbon Th rsuls show h h nm soluon of h ssm s ponnl sbl n ns o h s soluon of h ssm K wors: robo; srl omnn gnvlu; ssnl sprum; surbn; ponnl sbl nrouon Wh rp vlopmn of sn n hnolog h pplon of robos hs nrs n mprssv r n h r of nusrl sor onsruon fr prvnon unrwr ploron our sp ploron n mn Unoubl robo wll b houshol goo s norml h nrs fuur Hn robo sf n rlbl hs bom n mporn for Th uhors sblsh h mhm mols of h ssm onssng of obo sso sf mhnsm s ssm n su h s soluon of h ssm b h mho of plrnsform [] usng h lnr opror hor n nh sp h uhors prov h h soluon of h ssm s smpo sbl n s s soluon s h gnvor whh s orrsponng o h gnvlu of h ssm opror [] n hs ppr unr mor norml ssumpons w wll prov h sn of rgorous omnn gnvlu n nlz h rsron of ssnl sprl growh boun of h ssm opror n h hng of h ssnl sprl rus fr prurbon Th rsuls show h h nm soluon of h ssm s ponnl sbl n ns o h s soluon of h ssm Th s ssumpons n Mhm Mol of h Ssm Th s ssumpons Th followng ssumpons r sso wh h nlss prsn n hs rl: Th ssm s ompos of wo ms: robo n s sso sf mhnsm ssm Flurs r ssll npnn Nur Sn Fun of Hnn rovnl Thnolog prmn 97 Tms o flur ohr hn h of ommon-us flurs r ponnll srbu Ssm fls whn h robo fls Th rpr robo or s sso sf mhnsm s s goo s nw 6 Th fl ssm rpr ms r rbrrl srbu 7 Th prll fl ssm rpr ms r ponnll srbu 8 Th pplon of h v of sgs mho m nvolv som ppromon 9 ommon-us flur ms r gmm srbu Th followng smbols r sso wh Fgur or s rl nlss: m h s of h ssm: = obo n s sso sf mhnsm worng normll =l obo worng normll sf mhnsm fl = obo fl wh n nn = obo fl sfl = obo fl sf ssm oprng normll = obo fl u o ommon-us flur = umm s onsn flur r of h sf mhnsm / ssm onsn flur r of h robo flng wh n nn onsn flur r of h robo flng sfl onsn flur r of h robo rmrs sso wh ommon-us flurs onsn rpr r from s ; = Tm-pnn rpr r whn h fl Ss

2 ssm s n s n hs n lps rpr m of ; for = robbl h h robo s n s m ; = p [ Th probbl h m h fl ssm s n s n h lps rpr m ls n h nrvl ; for = ] Th s-sp grm of h ssm s shown n Fgur Fgur S sp grm of h ssm onnng robo n s sso sf mhnsm Th Mhml Mol n h ssumpons s bfor b supplmnr vrbls w n g h followng ngrl ffrnl quon group whh srbs h ssm Th rpr funon s ssum o b boun [] So whn w suss h ssm som prl ul mnng n som goo proprs wll b los n f n h prl pplon s unboun gnrll n orr o prf h ssm w ssum ; sup nf ssum s sp s followng For h norm of s fn Obvousl s nh sp no n w fn opror g h fn rgon of b } { fn opror n : T ; Th ssm n b srb s n bsr uh problm: Th S Soluon of h Ssm Frsl w su h nonzro soluon s sn bou whr Th quon n b prss b solvng w n g Ss rongs of nnul onfrn of hn nsu of ommunons

3 So h frs quon of bom 7 hn w hv quon group s followng 8 no h offn rmnn of 8 b n hn w hv Whn n s n gnvlu of hn n urn f n ssfs hn h quon group 8 hs nonzro soluon hn n s soluon of Spll whn hn So s n gnvlu of h opror n s ln vor hs h followng omponns: 9 hn w hv For n w hv hn s n smpl gnvlu of h opror * Hn w n g h s posv soluon of h ssm ˆ whr p p r prss s n 9 Th Eponnl Sbl of h Soluon of Ssm Th uhors hv prov h h soluon of ssm s h mol s progrssvl sn s smpl gnvlu of h opror of h ssm [] n hs son w wll llumn h wh h mor srongl onons h ssm hs mor wll sbl Thorm [] s smpl gnvlu of h opror ; { } : Thorm Suppos h s fn s bfor n hr s posv numbr ssf } mn{ Thn whn w hv n roof Whn for n gvn w onsr h rsolvn quon whr Ss rongs of nnul onfrn of hn nsu of ommunons

4 rongs of nnul onfrn of hn nsu of ommunons Th nl prsson of h rsolvn quon s s followng Whn w hv solvng w hv Snorng o h nfrn [] w hv p o llumns h whn : s boun hn n orng o h umr-hlps horm [] w n onlu h followng nfrn nfrn Suppos h n s fn s bfor hn h omprsson sm-group S spnn b opror s ponnll grssv nml for n hn S Sn h opror s fn rn opror n hn s omp opror orng o h opror sm-group horm n omp prurbon of sm-groupw hv h followng onluson Thorm Suppos h n s fn s bfor hn h omprsson sm-group T whh s spnn b + posssss h proprs s follow Whn w hv ; Suppos h for n { } whr N hn w hv s rgorous omnn gnvlu Suppos h ˆ s h s soluon of h ssm n hn for n hr ss onsn M suh h T ˆ whr roof Whn orng o h horm w hv Sn s hr orr rn opror hn s omp opror hn f n onl f s no n gnvlu of So whn w hv Sn s n nl funon hn mos hr r fn zro pons n hr s no umulon pon n fn rgon Suppos h hn w hv { } Whr N orng o h srnss of gnvlu n horm w hv Sn h gnfunon h orrsponng o s posv hn w hv s rgorous omnn gnvlu Fnll b h prurbon horm of sm-group omp prurbon os no hng h ssnl sprum boun of h sm-grouphn h sm-group T spnn b + n sm-group S spnn b hv h sm h ssnl sprum boun [-6] Hn for h ssnl sprum boun of T w hv Suppos h ˆ s fn s n n orng o h fn pnson Ss 8

5 rongs of nnul onfrn of hn nsu of ommunons horm of sm-groupw hv h followng onluson For n hr ss onsn M suh h T ˆ M whr Th bov onlusons show h unr som fn onons h nm soluon of h ssm s ponnl sbl n ns o h s soluon of h ssm frns [] HON S FSHN M obo Ssms robbls nlss [J] Mrolron lb 997 7: - [] Guo Whu u Gnq Sbl nlss of h Ssm onssng of obo n s sso Sf Mhnsm [J] Mhms n r n Thor 9: 6- [] GUO WhuEponnl Sbl nlss of rll prbl Ssm wh Two Non-nlUn[J] Mhms n rn Thor 99: 8- [] z Smgroup of nr Oprors n pplons o rl ffrnl Equons [M] Sprngr Nw Yor 98 [] u Gnq rurbon Thorm for us of Srong onnuous Sm-group Essnl Sprum [J] T mhm Sn99 6:77-76 [6] u Gnq Th Esmon of Srongl onnuous _ Sm-group rurbon Essnl Sprum [J] T Mhm Sn99 6: Ss

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