Passive Bilateral Teleoperation with Constant Time Delays

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1 Proceengs of the 6 IEEE Internatonal Conference on Robotcs an Automaton Orlano, Flora - May 6 Passve Blateral eleoperaton wth Constant me Delays Dongjun Lee an Mark W Spong Coornate Scence Laboratory Unversty of Illnos at Urbana-Champagn 138 W Man St Urbana, IL 6181 USA E-mal: -lee@controlcsluuceu, mspong@uuceu Abstract We propose a novel control framework for blateral teleoperaton of a par of mult-egree-of-freeom (DOF) nonlnear robotc systems uner constant communcaton elays he propose framework utlzes the smple proportonal-ervatve (PD) control, e the master an slave robots are rectly connecte va sprng an amper over the elaye communcaton channels Usng the controller passvty concept, the Lyapunov- Krasovsk technque, an Parseval s entty, we can passfy the combnaton of the elaye communcaton an control blocks altogether robustly, as long as the elays are fnte constants an an upper-boun for the roun-trp elay s known Havng explct poston feeback through the elaye P-acton, the propose framework enforces master-slave poston coornaton whch s often compromse n the velocty-base schemes (eg conventonal scatterng-base teleoperaton) he propose control framework proves humans wth extene physologcal proprocepton so that s/he can affect an sense the remote slave envronments manly relyng on her/hs musculoskeletal systems Experments are performe to valate the propose control framework I INRODUCION Energetcally, as llustrate n Fg 1, a close-loop teleoperator s a two-port system wth the master an slave ports beng couple wth the human operators an slave envronments, respectvely herefore, the foremost an prmary goal of the control (an communcaton) esgn for the teleoperaton shoul be to ensure nteracton safety an/or couple stablty [1] when mechancally couple wth a broa class of slave envronments an humans o ensure such nteracton safety an stablty, energetc passvty (e mechancal power as the supply rate []) of the close-loop teleoperator has been wely utlze as the control objectve [3], [4], [5] In [6], energetc passvty of the elaye teleoperaton s acheve by passfyng the communcaton block wth (possbly unknown) fnte constant tme-elays hs passfcaton was mae possble by applyng scatterng theory In [7], ths scatterng-base result s further extene an the noton of the wave varables was ntrouce Snce these two semnal works, scatterng-base (or wave-base) teleoperaton has been vrtually the only way to enforce energetc passvty of the tme-elaye blateral teleoperaton (eg [3], [8], [9]) In ths paper, we propose a novel control framework for blateral teleoperaton of a par of mult-dof nonlnear robotc systems wth fnte constant communcaton elays he propose framework s base on the smple proportonalervatve (PD) control, e rectly connectng the master an slave robots va sprng an amper over the elaye Human Operator F 1 1 τ F Master 1 Master Slave Comm+ Slave Robot Delay Comm+ Control Robot Control q 1 Fg 1 Sensng q 1 Delay τ Control an Communcaton Close-loop eleoperator q Sensng Close-loop teleoperator as a two-port system q Slave Envron communcaton channels hen, utlzng the controller passvty concept [5], [1], the Lyapunov-Krasovsk technque for elaye systems [11], an Parseval s entty [1], we can enforce energetc passvty of the close-loop teleoperator as long as the communcaton elays are fnte constants an an upper-boun of the roun-trp elay (e sum of the forwar an backwar elays) can be known, even f the elays are asymmetrc (e τ 1 τ n Fg 1) or ther exact estmates are not avalable Relyng on the controller passvty concept, ths passvty s ensure even n the presence of moel parametrc uncertanty (e robust passvty [5], [1]) In [13], we propose a smlar PD-base teleoperaton scheme, whch also enforces passvty However, t requres that the communcaton elays be exactly known an symmetrc (e τ 1 = τ ) In ths paper, we completely remove these two requrements, whch are ffcult to satsfy n practce Compare to the velocty-base schemes (eg conventonal scatterng-base teleoperaton), the man avantage of the propose framework s the explct poston feeback through the elaye P-control acton (e sprng term wth elaye set poston) he lack of such explct poston feeback has been known as the major rawback of the conventonal scatterngbase teleoperaton, n whch, roughly speakng, the velocty nformaton s extracte from the communcate scatterng varables, an then, ntegrate to recover the set poston nformaton herefore, f ths ntegraton becomes naccurate (eg slave robot makes a har contact wth a rg wall or communcaton s blacke out shortly), the master an slave postons may start rftng away from each other (see [14] for example) Havng the explct poston feeback, our propose framework prevents such a poston rftng hs explct poston feeback also enables us to guarantee (e theoretcally prove) asymptotc master-slave poston coornaton In contrast to the scatterng-teleoperaton where the elaye communcaton block s passfe so that the close /6/$ 6 IEEE 9

2 loop teleoperator becomes an nterconnecton of passve submoules (e passfe communcaton, passve control, an passve master/slave robots), the propose framework passfes the combnaton of the communcaton an control blocks altogether Whle the scatterng-base teleoperaton can be use wthout any knowlege on the (fnte constant) elays, our propose framework requres to know an upper-boun of the roun-trp elay However, we thnk that ths s a ml restrcton, snce, n many applcatons, such a roun-trp elay s relatvely easy to measure/estmate [15] Although t acheves at least a level of eal transparency [3], the man goal of the propose control framework s to prove humans wth extene physologcal proprocepton (EPP) [16], e the close-loop teleoperator as a tool, by whch the human operator can affect an sense the remote slave envronments manly relyng on her/hs musculoskeletal systems In ths sense, the propose framework s along the lne of such research works as common passve mechancal tool [4] an vrtual tool for wave-base teleoperaton [9] he rest of ths paper s organze as follows he control problem s formulate n secton II, an the control law s esgne an ts propertes are etale n secton III Expermental results are presente n secton IV an secton V contans some conclung remarks hs paper s a short verson of our recent journal paper [17], to whch we refer reaers for more etals II PROBLEM FORMULAION A Moelng of eleoperators uner Constant me Delay Let us conser a teleoperator consstng of a par of n- egree-of-freeom (DOF) nonlnear robotc systems: M 1 (q 1 ) q 1 (t)+c 1 (q 1, q 1 ) q 1 = 1 (t)+f 1 (t), (1) M (q ) q (t)+c (q, q ) q = (t)+f (t), () where q (t),f (t), (t) R n are the confguratons, human/envronmental force, an controls, an M (q ), C (q, q ) R n n are symmetrc an postve-efnte nerta matrces an Corols matrces, respectvely, st Ṁ (q ) C (q, q ) are skew-symmetrc ( =1, ) Here, we assume that the gravty effects are ether nclue n F 1 (t),f (t) or pre-compensate by the local controls In ths work, we assume the communcaton structure as shown n Fg 1, where the forwar an backwar communcatons are elaye by fnte constant tme-elays τ 1 an τ, respectvely hen, the controls 1 (t), (t) n (1)-() can be efne as functons of the current local nformaton an the elaye remote nformaton (rectly receve from the communcaton lne), e 1 (t) := 1 (q 1 (t), q 1 (t),q (t τ ), q (t τ )) R n, (3) (t) := (q (t), q (t),q 1 (t τ 1 ), q 1 (t τ 1 )) R n (4) B Control Objectves We woul lke to esgn 1 (t), (t) n (3)-(4) to acheve master-slave poston coornaton: f(f 1 (t),f (t)) =, q E (t) :=q 1 (t) q (t), t ; (5) an statc force reflecton: wth ( q 1 (t), q (t), q 1 (t), q 1 (t)), F 1 (t) F (t) (6) For safe nteracton an couple stablty, we woul also lke to enforce the followng energetc passvty of the close-loop teleoperator (1)-(): a fnte constant Rst [ F 1 (θ) q 1 (θ)+f (θ) q (θ) ] θ, t, (7) e maxmum extractable energy from the two-port closeloop teleoperator s boune (see Fg 1) Let us also efne controller passvty [4], [5], [1]: a fnte constant c Rst [ 1 (θ) q 1 (θ)+ (θ) q (θ) ] θ c, t, (8) e the two-port controller n Fg 1 generates only boune amount of energy Lemma 1 [5], [1] For the mechancal teleoperator (1)-(), controller passvty (8) mples energetc passvty (7) Proof: Let us efne the total knetc energy κ f (t) := 1 q 1 (t)m 1 (q 1 ) q 1 (t)+ 1 q (t)m (q ) q (t), (9) then, usng (1)-() wth ts skew-symmetrc property, we have t κ f (t) =[ 1 (t)+f 1 (t)] q 1 (t)+[ (t)+f (t)] q (t) (1) hus, by ntegratng (1) wth the controller passvty conton (8) an the fact that κ f (t), wehave, t, [F 1 (θ) q 1 (θ)+f (θ) q (θ)]θ κ f () c =: Lemma 1 s smple but powerful n the sense that t enables us to analyze energetc passvty (7) of the close-loop teleoperator by examnng only the controller structure whch s often much smpler than that of the close-loop ynamcs Furthermore, by enforcng controller passvty (8), energetc passvty (7) wll be guarantee robustly (robust passvty [5], [1]), because controller passvty (8) oes not epen on the possbly uncertan open-loop ynamcs (1)-() III CONROL DESIGN o acheve the master-slave coornaton (5), blateral force reflecton (6), an energetc passvty (7), we esgn the master an slave controls 1 (t), (t) n (3)-(4) to be 1 (t) := K v ( q 1 (t) q (t τ )) (K + P ɛ ) q 1 (t) K p (q 1 (t) q (t τ )), (11) (t) := K v ( q (t) q 1 (t τ 1 )) (K + P ɛ ) q (t) K p (q (t) q 1 (t τ 1 )), (1) where τ 1,τ are the forwar an backwar fnte constant elays, K v,k p R n n are the symmetrc an postveefnte proportonal (P) an ervatve (D) control gans, K R n n s the sspaton to passfy the elaye P-acton (e wth K p ) n (11)-(1) (to be esgne below), an P ɛ R n n s an atonal ampng ensurng master-slave coornaton 93

3 (5) Inherent evce vscous ampng can substtute ths P ɛ Note that the control laws (11)-(1) contan the explct poston feeback (va the P-acton), whch s generally absent n the conventonal scatterng-base teleoperaton o enforce energetc passvty (7), we esgn sspaton K an P-gan K p n (11)-(1) to satsfy the followng conton: K [ ] sn w(τ 1+τ ) K p K 1 w K p, w R, (13) where, for square matrces A, B, A B mples that A B s postve-semefnte As to be shown below, wth ths conton (13), the elaye P-control acton (e wth K p )s sn wτ passfe by the sspaton K From the fact that τ w, w R, one possble soluton for the conton (13) s K = τ rt K p, (14) where τ rt s an upper-boun of the roun-trp elay τ rt := τ 1 + τ st τ rt τ rt heorem 1 Conser the mechancal teleoperator (1)-() wth the controls (11)-(1) uner the conton (13) 1) (Robust Passvty) he close-loop teleoperator s energetcally passve (e satsfes (7)) regarless of parametrc uncertanty n the open-loop ynamcs (1)-(); ) (Couple Stablty) Suppose that the human operator an slave envronment n Fg 1 are energetcally passve: fnte constants 1, Rst t, F (θ) q (θ) }{{} θ, = {1, }, (15) power nflow to human/envron e the maxmum extractable energy from them are boune hen, q 1 (t), q (t) L hus, f the human an slave envronment are L -stable nput-output mpeance maps, we wll also have F 1 (t),f (t) L ; 3) (Poston Coornaton) Suppose that the human operator an slave envronment are passve n the sense of (15) Suppose further that M jk M (q ), jk (q ) an M jk m (q ) are ml all boune wrt q, where M jk (q ) an q m are the jkth an the m-th components of M (q ) an q, respectvely hen, q E (t) =q 1 (t) q (t) s boune t Moreover, f (F 1 (t),f (t)) = t, (q E (t), q E (t)) ; 4) (Statc Force Reflecton) If ( q 1 (t), q (t), q 1 (t), q (t)), then, F 1 (t) F (t) K p (q 1 q ) Proof: 1) Let us enote the mechancal power generate by the controls (11)-(1) by s c (t) :=1 (t) q 1 (t)+ (t) q (t) = s (t)+s p (t) P (t), (16) where s (t) an s p (t) are the supply rates assocate to the elaye D-acton, an elaye P-acton (+ sspaton K ), respectvely efne by s (t) := q 1 (t)k v q 1 (t)+ q 1 (t)k v q (t τ ) q (t)k v q (t)+ q (t)k v q 1 (t τ 1 ), (17) s p (t) := q 1 (t)k q 1 (t) q 1 (t)k p (q 1 (t) q (t τ )) q (t)k q (t) q (t)k p (q (t) q 1 (t τ 1 )), (18) an P (t) s the followng quaratc form: ( ) [ ) q1 (t) Pɛ ]( q1 (t) P (t) := (19) q (t) P ɛ q (t) We want the total controller supply rate s c (t) n (16) to satsfy the controller passvty (8) Let us frst conser the elaye D-acton supply rate s (t) n (17) hen, usng the fact that, for (, k) ={(1, ), (, 1)}, q (t)k v q k (t τ k ) q (t)k v q (t)+ q k (t τ k )K v q k (t τ k ), () we can show that s (t) =1 1 [ q (t)k v q (t) q (t τ )K v q (t τ ) ] = t V v(t), (1) where V v (t) s a Lyapunov-Krasovsk functonal for elaye systems [11] efne by 1 V v (t) := q (t + θ)k v q (t + θ)θ () =1 τ hen, by ntegratng the nequalty (1), we have s (θ)θ V v (t)+v v (), (3) e energy generaton by the elaye D-acton s always boune by the energy store n V v (t) (e V v (t) s the storage functon for the supply rate s (t)) Now, let us conser the supply rate s p (t) n (18) hen, we can rewrte s p (t) n (18) st: wth N := {(1, ), (, 1)}, s p (t) = q 1 (t)k q 1 (t) q (t)k q (t) q 1 (t)k p (q 1 (t) q (t)) q 1 (t)k p (q (t) q (t τ )) q (t)k p (q (t) q 1 (t)) q (t)k p (q 1 (t) q 1 (t τ 1 )) = q 1 (t)k q 1 (t) q (t)k q (t) t V p(t) q (t)k p (q k (t) q k (t τ k )) (4) (,k) N where V p (t) s the P-acton sprng energy efne as V p (t) := 1 q E(t)K p q E (t), (5) wth q E (t) =q 1 (t) q (t) gven n (5) Let us efne the truncate sgnal q t(θ) of q (t) st { q (θ) t q (θ) f θ [,t] :=, (6) otherwse 94

4 =1, hen, the energy generaton by the supply rate s p (t) n (4) can be wrtten as: s p (θ)θ = V p (t)+v p () [ + θ q 1 t (θ)k p [ + θ q t (θ)k p =1 q t (ξ)ξ θ τ q t (θ)k q (θ)θ t q t (ξ)ξ }{{} q t 1(ξ)ξ :=g t(θ) θ τ1 q t 1(ξ)ξ } {{ } :=g1 t(θ) ] ] θ θ (7) Let us enote the Fourer transform of q t ( = 1, ) by V t(w) := q t (θ)e θ θ = q (θ)e θ θ, where j = 1 hen, the Fourer transform of g t (θ) n (7) (e G t (w) := gt (θ)e θ θ) s gven by G t (w) = 1 e τ V t (w) Let us enote the complex conjugate transpose of a complex vector C n by (e = ) hen usng the Parseval s entty [1], (7) can be rewrtten as s p (θ)θ = V p (t)+v p () (8) =1 (,k) N 1 π 1 π V t (w)k V t (w)w V t = V p (t)+v p () 1 π where N = {(1, ), (, 1)} an [ H(w) = K p K e τ e τ 1 1 e τ k (w)k p Vk t (w)w ( ) V t ( ) 1 (w) V t V t H(w) 1 (w) (w) V t (w) K p e τ 1 e τ K w, ] C n n For more etals on these ervatons, see [17] Snce H(w) s Hermtan (e H (w) = H (w) =H(w)) wth postveefnte block agonal matrces K, followng [18, pp473], H(w) (e H(w) s postve-semefnte) f an only f K eτ1 e τ e τ e τ1 K p K 1 K p, (9) whch s nothng but the conton (13) hus, wth the conton (13) st H(w), we can show from (8) that s p (θ)θ V p (t)+v p (), (3) e energy generaton by the supply rate s p (t) s always boune by the P-acton sprng energy V p (t) n (5) hus, by summng up (3) an (3) wth (16) an the fact that V v (t) an V p (t) t, we can prove controller passvty (8) st for all t, [ 1 (θ) q 1 (θ)+ (θ) q (θ)]θ V v (t)+v v () V p (t)+v p () P (θ)θ V v () + V p () =: c, (31) where the term V v () wll be zero f we start from zero veloctes (e ( q 1 (t), q (t)) = t (, ]), whle the term V p () woul be small, f the ntal coornaton error q E () = q 1 () q () s small Fnally, from Lemma 1, energetc passvty (7) of the close-loop teleoperator follows ) By ntegratng (1) wth (31) an (15), we have, t, κ f (t)+v v (t)+v p (t) (3) κ f () + V v () + V p () P (θ)θ + 1 +, where P (t) n (19) Here, snce V v (),V p (), 1, an κ f () are all boune, κ f (t) s boune hus, q 1 (t), q (t) are also boune t (e q 1 (t), q (t) L ) herefore, f the human operator an the slave envronment are L -stable mpeance maps, F 1 (t),f (t) L 3) Bouneness of q E (t) =q 1 (t) q (t) s a rect consequence of (3) wth the efnton of V p (t) n (5) Frst step of the poston-coornaton proof s to show that ( q 1 (t), q (t)) Suppose that F 1 (t),f (t) =, t hen, from (3) wth 1 = =an the bouneness of P ɛ,m 1 (q 1 ),M (q ),wehave: t, κ f (t) κ f () + c κ f () + c γ P (θ)θ κ f (θ)θ, (33) where c = V v () + V p () as gven n (31) an γ>s a constant scalar Here, snce κ f (t), the term κ f (θ)θ s monotoncally ncreasng an upper boune, thus, t converges to a lmt herefore, followng Barbalat s lemma, f κ f (t) s unformly contnuous, κ f (t) wll also converge to (e ( q 1 (t), q (t)) ) o show ths, let us conser t κ f (t) hen, from (1) wth F 1 (t) =F (t) =,wehave t κ f (t) = 1 (t) q 1 (t)+ (t) q (t), where q 1 (t), q (t) are boune from tem of ths theorem Also, 1 (t), (t) are boune, snce, n ther efntons (11)-(1), for (,k) ={(1, ), (, 1)}, 1) q (t) q k (t τ k ) s boune (wth boune q (t)); an ) q (t) q k (t τ k )=q E (t)+ τ k q k (t + θ)θ s also boune (wth boune q (t), q E (t) an τ k ) hus, t κ f (t) s boune, an κ f (t) s unformly contnuous herefore, κ f (t) an ( q 1 (t), q (t)) he next step of the proof s to show that ( q 1 (t), q (t)) Let us conser the ynamcs (1)-() wth F 1 (t) =F (t) =, where, as shown n the above paragraph, the controls 1 (t), (t) n (11)-(1) are boune Also, from the bouneness assumpton of Mjk (q ), the Corols terms C m (q, q ) q ( = 1, ) n (1)-() are boune hus, the acceleratons 95

5 q 1 (t), q (t) are also boune t Now, let us conser the acceleraton q (t) n (1)-() (wth F (t) =): q = M 1 (q )C (q, q ) q + M 1 (q ) (t), =1,, (34) where the tme-ervatves of the terms n the RHS are all boune, ue to the bouneness of q (t), q (t), q E (t), t M 1 (q )= M 1 (q ) t M (q )M 1 (q ) (from the bouneness of Mjk (q ) ), an m t [C (q, q ) q ] (from the bouneness of M jk (q ), q, an q ) hs mples that the RHS ml of (34) s unformly contnuous hus, q 1 (t), q (t) are also unformly contnuous herefore, followng Barbalat s lemma, ( q 1 (t), q (t)) as ( q 1 (t), q (t)) Now, let us conser the ynamcs (1)-() wth the controls 1 (t), (t) n (11)-(1) an F 1 (t) = F (t) = hen, snce ( q 1 (t), q (t), q 1 (t), q (t)), we have K p (q (t) q k (t τ k )), (, k) = {(1, ), (, 1)} hs conton can be rewrtten as K p (q E (t) + τ k q k (t + θ)θ), where the secon term n the parenthess goes to zero, because q k (t) an τ k s fnte herefore, snce K p s postveefnte, q E (t) (e q 1 (t) q (t)) 4) Suppose that ( q 1 (t), q (t), q 1 (t), q (t)) hen, from the ynamcs (1)-() an ther controls (11)-(1), we have: F 1 (t) K p (q 1 q ), F (t) K p (q q 1 ), (35) wth q (t τ) q (t) q as q (t), q (t) In heorem 1, the negatve sgns n the passvty conton for the human an slave envronment (15) come from the fact that the power nflows to those systems are gven by F (t) q (t), e the prouct of the reacton force F (t) an the nteracton velocty q (t) he bouneness assumpton on M jk (q ), Mjk (q ) an M jk m (q ) n heorem 1 s guarantee, ml f the robot s confguraton space s compact an ts nerta matrx s smooth Such compact confguraton space an smooth nerta are possesse by many practcal robotc systems (eg revolute jont robots) he conton (14) (or (13)) enables us to passfy the elaye P-acton n (11)-(1) by the sspaton K hs elaye P-acton contans an explct poston feeback nformaton, the lack of whch s recognze as the man cause of the master-slave poston rft n the conventonal scatterng-base teleoperaton In contrast, the elaye D-acton n (11)-(1) s tself passve wth the Lyapunov-Krasovsk functon V v (t) n () as ts storage functon (see (3)) Snce the conton (14) can be acheve as long as the elays τ 1,τ are fnte constants an ther roun-trp elay (e τ 1 + τ ) s upperboune, energetc passvty (7) can also be ensure wth such fnte constant elays, even f they are not exactly known or asymmetrc (e τ 1 τ ) In [13], we propose a smlar PD-base scheme, whch also enforces passvty However, t requres the elays to be symmetrc an exactly known he conton (14) mposes the followng mplcatons on the system performance: 1) wth the same statc force reflecton performance (e same K p, see tem 4 of heorem 1), the moton aglty (e less K ) woul be compromse as the elays becomes longer, snce, n the conton (14), the requre sspaton K s proportonal to the roun-trp elay τ rt ; an ) wth the elays fxe, there s a trae-off between the statc force reflecton performance (e K p ) an moton aglty (e K ), snce, uner the conton (14), a large K p (e sharp force reflecton) requres a large K (e poor moton aglty), or a small K (e agle free-moton) permts only a small K p (e poor force reflecton) he key step n the proof of heorem 1 s the use of the Parseval s entty n (8) whch we assume to be true A suffcent conton for the Parseval s entty to hol s that q 1 (t), q (t) L [1] As the followng Lemma shows, ths suffcent conton s guarantee n many practcal stuatons where the human an slave envronment are passve n the sense of (15), the nerta matrces of the master an slave robots (1)-() have boune partal-ervatve wrt q (e M jk (q ) are boune), an the master an slave veloctes m an the coornaton error are ntally boune Due to the space lmtaton, we omt the proof here an refer reaers to [17] he Parseval s entty was also use n [19] to ensure the energetc passvty of haptc-nterfaces uner zero-orer-hol Lemma Suppose that the human an slave envronment are passve n the sense of (15) an efne L -stable mpeance maps (e f q 1 (t), q (t) L, F 1 (t),f (t) L ) Suppose further that Mjk (q ) are boune hen, f q m 1 (), q (),q E () are boune, q 1 (t), q (t) L, thus, the Parseval s entty n (8) hols an the tems 1-3 of heorem 1 are ensure IV EXPERIMEN For the experment, we use a par of rect-rve planar -DOF seral-lnks revolute-jont robots More etals on the harware/software constructon of ths system can be foun n [17] along wth more experment an smulaton results he control laws (11)-(1) are erve for the Cartesanspace ynamcs so that, followng tem 4 of heorem 1, the Cartesan (statc) force reflecton can be acheve We nstall an alumnum wall n the slave envronment, an 5[ms] samplng rate s obtane We set the elays st (τ 1,τ ) = (1, 18)[sec] (e τ rt = 3[sec]) he P-gan K p an sspaton K n (11)-(1) are esgne accorng to the conton (14) Atonal ampng P ɛ s omtte n the control mplementaton, as we leave the evce frcton uncompensate We conser the followng scenaro: 1) ntally, a human stablzes the teleoperator; ) then, wthout seeng, moves the slave close to the alumnum wall an keeps pushng the master untl s/he perceves the wall; 3) makes a har contact; an 4) fnally, retracts the slave from the wall Expermental results are gven n Fg As shown by the force profle n Fg, the human operator can perceve the alumnum wall through the force reflecton Also, when the contact s remove (e free-moton), the master an slave postons become coornate wth each other hese freemoton an contact behavor are all stable, as we enforce passvty (7) through the conton (14) After the contact, 96

6 the human operates the master so that the slave returns to ts startng poston hs leas nto bumps n the human force n Fg after aroun 6[sec] In Fg, there are some errors n both the force reflecton an poston coornaton (eg aroun 8[sec]) hese errors are ue to the substantal (bearng) Coulomb frcton of the robots, whch we foun can go up to 1[N] For nstance, suppose n (1)-() that F 1 = F h μ h an F = F e μ e, where F h,f e are the human/contact force, an μ h,μ e are the frctons Durng the contact, the frctons always oppose the human/contact forces hen, from tem 4 of heorem 1, F h F e (μ e + μ h ), e (μ e + μ h ) causes force reflecton error (eg aroun 4[sec]) hese errors were not observe when we performe a smulaton wthout such frctons [17] Although we not observe any substantal problems urng the experment, the requre sspaton K (from (14)) may cause sluggsh system behavor an lmt the framework s usablty, especally when the task requres agle operaton but the elays are large How to further mnmze ths requre K whle enforcng passvty s a topc for future work V CONCLUSIONS he man avantage of the propose framework s n that we can explot the benefts of explct poston feeback whle stll retanng passvty Also, ue to ts smple PD-base structure whch are pervasve n many nustral control systems, we thnk that the propose framework woul make a goo mpact n the real practce of teleoperaton applcatons We beleve that the propose framework s promsng for the Internet teleoperaton, where ts explct poston feeback woul o an mportant role to recover the poston coornaton n the presence of packet-loss an tme-varyng elays ACKNOWLEDGMEN Research partally supporte by the Offce of Naval Research uner grants N an N , the Natonal Scence Founaton uner grants IIS , CCR -9 an ECS-1-41, an the College of Engneerng at the Unversty of Illnos REFERENCES [1] J E Colgate Couple stablty of multport systems - theory an experments ransactons of the ASME, Journal of Dynamc Systems, Measurement an Control, 116(3):419 48, 1994 [] J C Wllems Dsspatve ynamcal systems part1: general theory Arch Ratonal Mech Anal, 45():31 351, 197 [3] D A Lawrence Stablty an transparency n blateral teleoperaton IEEE ransactons on Robotcs an Automaton, 9(5):64 637, 1993 [4] D J Lee an P Y L Passve blateral feeforwar control of lnear ynamcally smlar teleoperate manpulators IEEE ransactons on Robotcs an Automaton, 19(3): , 3 [5] D J Lee an P Y L Passve blateral control an tool ynamcs renerng for nonlnear mechancal teleoperators IEEE ransactons on Robotcs, 1(5): , 5 [6] R J Anerson an M W Spong Blateral control of tele-operators wth tme elay IEEE ransactons on Automatc Control, 34(5):494 51, 1989 [7] G Nemeyer an J J E Slotne Stable aaptve teleoperaton IEEE Journal of Oceanc Engneerng, 16(1):15 16, 1991 [8] N Chopra, M W Spong, S Hrche, an M Buss Blateral teleoperaton over the nternet: the tme varyng elay problem In Proceengs of Amercan Control Conference, pages , 3 Fg x Force [N] y Force [N] q1 q [m] Master an Slave x Forcng: τ rt =3sec, Half Kp Master an Slave y Forcng: τ rt =3sec, Half Kp F1x Fx 1 F1y Fy tme [sec] Master Slave Poston Error: τ rt =3sec, Half Kp tme [sec] Expermental results wth the roun-trp elay τ rt =3[sec] [9] G Nemeyer an J J E Slotne elemanpulaton wth tme elays Internatonal Journal of Robotcs Research, 3(9):873 89, 4 [1] Dongjun Lee Passve Decomposton an Control of Interactve Mechancal Systems uner Coornaton Requrements Doctoral Dssertaton, Unversty of Mnnesota, 4 [11] K Gu an S Nculescu Survey on recent results n the stablty an control of tme-elay systems ASME Journal of Dynamc Systems, Measurements, an Control, 15: , 3 [1] R R Golberg Fourer ransforms he Cambrge Unversty Press, New York, NY, 1961 [13] D J Lee an M W Spong Passve blateral control of teleoperators uner constant tme-elay In Proceengs of the IFAC Worl Congress, 5 [14] S Hrche an M Buss Packet loss effects n passve telepresence systems In Proceengs of the IEEE Conference on Decson an Control, pages , 4 [15] O Gurewtz an M S Estmatng one-way elays from cyclc-path elay measurements In Proceengs of IEEE INFOCOM, pages , 1 [16] D S Chlress Control strategy for upper-lmb prostheses Proceengs of the th Annual Internatonal Conf of the IEEE Engneerng n Mecne an Bology Socety, (5):73 75, 1998 [17] D J Lee an M W Spong Passve blateral teleoperaton wth constant tme-elay IEEE ransactons on Robotcs, 6 o appear Preprnt avalable at lee/leespongdelayro4pf [18] R A Horn an C R Johnson Matrx analyss Cambrge Unversty Press, Cambrge, UK, 1985 [19] J E Colgate an G Schenkel Passvty of a class of sample-ata systems: applcaton to haptc nterfaces Journal of Robotc Systems, 14(1):37 47,