New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels: Stabilization and Performance under Variable Time Delays

Transcription

1 New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels: Stabilization and Performance under Variable Time Delays Bo Zhang LAGIS, CNRS UMR 8219, Laboratoire d Automatique, Génie Informatique et Signal - Ecole Centrale de Lille - France Thesis supervisor: Prof. Jean-Pierre Richard - EC Lille Thesis co-advisor: Dr. Alexandre Kruszewski - EC Lille 10/07/2012 1/102

2 PLAN 1. PRELIMINARIES 2. NOVEL CONTROL SCHEMES 3. ROBUSTNESS ASPECTS 4. EXPERIMENTATIONS 5. CONCLUSIONS & PERSPECTIVES 2/102

3 PRELIMINARIES Background & Challenges Background & Challenges; Delayed Teleoperation; Positioning; Modeling of Time Delays; Stability of Time Delay Systems ; 1. Preliminaries 3/102

4 PRELIMINARIES Background & Challenges Cooperative System : Haptic interface cooperative system, LAGIS Performance Objectives : Stability; Synchronization (the position tracking, from the master to the slave); Transparency (the force tracking, from the slave to the master); 4/102

5 PRELIMINARIES Background & Challenges Cooperative System : Haptic interface cooperative system, LAGIS Background & Challenges : Wireless or long-distance; Complex environment; Modeling uncertainties; Collaborative sensing; Path planning and trajectory tracking; 5/102

6 PRELIMINARIES Delayed Teleoperation Background & Challenges; Delayed Teleoperation; Positioning; Modeling of Time Delays; Stability of Time Delay Systems ; 1. Preliminaries 6/102

7 PRELIMINARIES Delayed Teleoperation Delayed Teleoperation - Two Cases : Unilateral master/slave teleoperation Bilateral master/slave teleoperation 7/102

8 PRELIMINARIES Delayed Teleoperation Delayed Teleoperation - Two Cases : Unilateral master/slave teleoperation Bilateral master/slave teleoperation 7/102

9 PRELIMINARIES Delayed Teleoperation Bilateral Teleoperation : 8/102

10 PRELIMINARIES Delayed Teleoperation Bilateral Teleoperation : 9/102

11 PRELIMINARIES Delayed Teleoperation General Teleoperation Structure : F = force ; x = position/velocity; τ = delay; m, s, h, e = master, slave, human operator, environment ; 10 /102

12 PRELIMINARIES Delayed Teleoperation General Teleoperation Structure : F = force ; x = position/velocity; τ = delay; m, s, h, e = master, slave, human operator, environment ; 10 /102

13 PRELIMINARIES Delayed Teleoperation General Teleoperation Structure : F = force ; x = position/velocity; τ = delay; m, s, h, e = master, slave, human operator, environment ; 10 /102

14 PRELIMINARIES Delayed Teleoperation General Teleoperation Structure : F = force ; x = position/velocity; τ = delay; m, s, h, e = master, slave, human operator, environment ; 10 /102

15 PRELIMINARIES Delayed Teleoperation General Teleoperation Structure : F = force ; x = position/velocity; τ = delay; m, s, h, e = master, slave, human operator, environment ; 10 /102

16 PRELIMINARIES Delayed Teleoperation General Teleoperation Structure : Performance Objectives : Stability; Synchronization (the position tracking, from the master to the slave); Transparency (the force tracking, from the slave to the master) ; Properties : 1. Linear master/slave systems ; 2. Internet/Ethernet/Wifi...; 3. Time-stamped data packets; 11 /102

17 PRELIMINARIES Positioning Background & Challenges; Delayed Teleoperation; Positioning; Modeling of Time Delays; Stability of Time Delay Systems ; 1. Preliminaries 12 /102

18 PRELIMINARIES Positioning Main Capabilities of Recent Control Strategies : Control Strategy Passivity Robust Freq. Predict. SMC Adapt. Lyapunov Time Delays Position Tracking Constant Time-varying Force Tracking P. Arcara et al., Robotics and Autonomous Systems, 2002 ; S. Zampieri, IFAC, 2008 ; E. Nuño et al., Automatica, 2011 ; Control Strategy Goals of this thesis Time Delays Position Tracking Force Tracking Constant Time-varying + Performance 13 /102

19 PRELIMINARIES Positioning Main Capabilities of Recent Control Strategies : Control Strategy Passivity Robust Freq. Predict. SMC Adapt. Lyapunov Time Delays Position Tracking Constant Time-varying Force Tracking P. Arcara et al., Robotics and Autonomous Systems, 2002 ; S. Zampieri, IFAC, 2008 ; E. Nuño et al., Automatica, 2011 ; Control Strategy Goals of this thesis Time Delays Position Tracking Force Tracking Constant Time-varying + Performance 13 /102

20 PRELIMINARIES Modeling of Time Delays Background & Challenges; Delayed Teleoperation; Positioning; Modeling of Time Delays; Stability of Time Delay Systems ; 1. Preliminaries 14 /102

21 PRELIMINARIES Modeling of Time Delays Network Induced Delays : Communication delay τ c (t) Asynchronous sampling delay τ s (t) Data loss delay 15 /102

22 PRELIMINARIES Modeling of Time Delays Network Induced Delays : Notations and Constraints for Delays : τ(t) = h + η(t). (2) τ(t) = τ c (t) + τ s (t) + NT. (1) Constant delay : η(t) = 0, h 0; Non-small time-varying delay : η(t) µ < h, so τ(t) [h µ, h + µ] ; Interval time-varying delay : 0 η(t) µ, so τ(t) [h, h + µ] ; Time delay with the constraint on the derivative : τ(t) d < 1, d > 0, or τ(t) 1 ; 16 /102

23 PRELIMINARIES Modeling of Time Delays Network Induced Delays : Notations and Constraints for Delays : τ(t) = h + η(t). (2) τ(t) = τ c (t) + τ s (t) + NT. (1) Constant delay : η(t) = 0, h 0; Non-small time-varying delay : η(t) µ < h, so τ(t) [h µ, h + µ] ; Interval time-varying delay : 0 η(t) µ, so τ(t) [h, h + µ] ; Time delay with the constraint on the derivative : τ(t) d < 1, d > 0, or τ(t) 1 ; 16 /102

24 PRELIMINARIES Stability of Time Delay Systems Background & Challenges; Delayed Teleoperation; Positioning; Modeling of Time Delays; Stability of Time Delay Systems ; 1. Preliminaries 17 /102

25 PRELIMINARIES Stability of Time Delay Systems Delay-Independent Stability : If a time-delay system is asymptotically stable for any delay values belonging to R +, the system is said to be delay-independent asymptotically stable. Delay-Dependent Stability : If a time-delay system is asymptotically stable for all delay values belonging to a compact subset D of R +, the system is said to be delay-dependent asymptotically stable. Rate-Independent Stability : For a delay-dependent asymptotically stable time delay system, if the stability does not depend on the variation rate of delays or on the time derivative of delays, the system is said to be rate-independent asymptotically stable. Rate-Dependent Stability : For a delay-dependent asymptotically stable time delay system, if the stability depends on the variation rate of delays or on the time derivative of delays, the system is said to be rate-dependent asymptotically stable. 18 /102

26 PRELIMINARIES Stability of Time Delay Systems Delay-Independent Stability : If a time-delay system is asymptotically stable for any delay values belonging to R +, the system is said to be delay-independent asymptotically stable. Delay-Dependent Stability : If a time-delay system is asymptotically stable for all delay values belonging to a compact subset D of R +, the system is said to be delay-dependent asymptotically stable. Rate-Independent Stability : For a delay-dependent asymptotically stable time delay system, if the stability does not depend on the variation rate of delays or on the time derivative of delays, the system is said to be rate-independent asymptotically stable. Rate-Dependent Stability : For a delay-dependent asymptotically stable time delay system, if the stability depends on the variation rate of delays or on the time derivative of delays, the system is said to be rate-dependent asymptotically stable. 18 /102

27 PRELIMINARIES Stability of Time Delay Systems General System with Piecewise Time-Varying Delays : ẋ(t) = f(x(t), x(t τ 1 (t)), u(t), u(t τ 2 (t))), x(t 0 + θ) = φ(θ), ẋ(t 0 + θ) = φ(θ), u(t 0 + θ) = ζ(θ), θ [ h, 0]. (3) Linear Case : { ẋ(t) = A 0 x(t) + n i=1 A ix(t τ 1i (t)) + B 0 u(t) + m j=1 B ju(t τ 2j (t)), x(t 0 + θ) = φ(θ), ẋ(t 0 + θ) = φ(θ), θ [ h, 0]. (4) 19 /102

28 PRELIMINARIES Stability of Time Delay Systems Lyapunov-Krasovskii Functionals (LKF) - Linear Case : Search of suitable V (x(t), ẋ(t)) [E. Fridman, IMA Journal of Mathematical Control and Information, 2006] : t V (x(t), ẋ(t)) = x(t) T Px(t) + x(s) T S a x(s)ds + t h 2 + h h 1 t q (h 2 h 1 ) i=1 ẋ(s) T Rẋ(s)dsdθ t+θ h1 t h 2 t+θ ẋ(s) T R ai ẋ(s)dsdθ. t t h 1 x(s) T Sx(s)ds (5) Asymptotical stability condition depending of the derivative of V (x(t), ẋ(t)) along the system trajectories : V (x(t), ẋ(t)) > 0, V (x(t), ẋ(t)) < 0, for any xt 0, and V (x(0), ẋ(0)) = 0, V (x(0), ẋ(0)) = 0. (6) 20 /102

29 CONTROL SCHEMES System Description & Assumptions 2. Novel Control Schemes System Description & Assumptions; Robust Stability Conditions; Control Scheme 1 - Bilateral State Feedback Control Scheme [B. Zhang et al., CCDC, 2011] ; Control Scheme 2 - Force-Reflecting Control Scheme ; Control Scheme 3 - Force-Reflecting Proxy Control Scheme [B. Zhang et al., ETFA, 2011] ; Results and Analysis; 21 /102

30 CONTROL SCHEMES System Description & Assumptions General Teleoperation Structure : Our Master/Slave Controller Solution : Lyapunov-Krasovskii functional (LKF); H control; Linear Matrix Inequality (LMI) optimization; 22 /102

31 CONTROL SCHEMES System Description & Assumptions General Teleoperation Structure : From Property 1 - Assumption 1 : Linear master/slave systems = ẋ m (t) = (A m B m K m 0 )x m (t) + B m (F m (t) + F h (t)), ẋ s (t) = (A s B s K s 0 )x s(t) + B s (F s (t) + F e (t)), (7) x m (t) = θ m (t) R n, x s (t) = θ s (t) R n ; K m 0 & K s 0 : supposed to be known; 23 /102

32 CONTROL SCHEMES System Description & Assumptions General Teleoperation Structure : From Property 2 - Assumption 2 : Internet/Ethernet/Wifi... = τ 1 (t), τ 2 (t) [h 1,h 2 ], h 1 0 ; 24 /102

33 CONTROL SCHEMES System Description & Assumptions General Teleoperation Structure : From Property 3 - Assumption 3 : Time-stamped data packets = ˆτ 1 (t) = τ 1 (t), ˆτ 2 (t) = τ 2 (t) ; 25 /102

34 CONTROL SCHEMES Robust Stability Conditions 2. Novel Control Schemes System Description & Assumptions; Robust Stability Conditions; Control Scheme 1 - Bilateral State Feedback Control Scheme ; [B. Zhang et al., CCDC, 2011] ; Control Scheme 2 - Force-Reflecting Control Scheme ; Control Scheme 3 - Force-Reflecting Proxy Control Scheme [B. Zhang et al., ETFA, 2011] ; Results and Analysis; 26 /102

35 CONTROL SCHEMES Robust Stability Conditions Linear Time Delay System : { ẋ(t) = A 0 x(t) + q i=1 A ix(t τ i (t)), x(t 0 + θ) = φ(θ), ẋ(t 0 + θ) = φ(θ), θ [ h 2, 0]. (8) Asymptotical Stability Theorem [E. Fridman, IMA Journal of Mathematical Control and Information, 2006] : P > 0, R > 0, S > 0, S a > 0, R ai > 0, and P 2, P 3, Y 1, Y 2, i = 1, 2,..., q ; LMI condition is feasible; Rate-independent asymptotically stable for time-varying delays τ i (t) [h 1, h 2 ], i = 1, 2,..., q ; 27 /102

36 CONTROL SCHEMES Robust Stability Conditions Asymptotical Stability Theorem : Γ 1 11 Γ1 12 R+ q i=1 PT 2 A i qy 1 T qy 1 T PT 2 A 1 +Y 1 T... P 2 T Aq+Y 1 T Y 1 T... Y T 1 Γ 1 q 22 i=1 PT 3 A i qy 2 T qy 2 T PT 3 A 1 +Y 2 T... P 3 T Aq+Y 2 T Y 2 T... Y T 2 S R Γ 1 Sa = R a < 0, Raq R a Raq (9) Γ 1 11 = S + S a R + A T 0 P 2 + P2 T A 0, Γ 1 12 = P P2 T + A T 0 P 3, q Γ 1 22 = P 3 P3 T + h 2 1R + (h 2 h 1 ) 2 R ai. i=1 (10) 28 /102

37 CONTROL SCHEMES Robust Stability Conditions H Performance - Delay-Free with Perturbation : { ẋ(t) = Ax(t) + Bw(t), z(t) = Cx(t). (11) J(w) = 0 (z(t) T z(t) γ 2 w(t) T w(t))dt < 0, (sup w ( z(t) 2 w(t) 2 ) < γ). V (x(t)) + z(t) T z(t) γ 2 w(t) T w(t) < 0, V (x(t)) = x(t) T Px(t). Robust Stability Theorem : P > 0, P 2, P 3, and a positive scalar γ > 0; LMI condition is feasible (see manuscript); Asymptotically stable with H performance J(w) < 0; (12) 29 /102

38 CONTROL SCHEMES Robust Stability Conditions H Performance - Delay-Free with Perturbation : { ẋ(t) = Ax(t) + Bw(t), z(t) = Cx(t). (11) J(w) = 0 (z(t) T z(t) γ 2 w(t) T w(t))dt < 0, (sup w ( z(t) 2 w(t) 2 ) < γ). V (x(t)) + z(t) T z(t) γ 2 w(t) T w(t) < 0, V (x(t)) = x(t) T Px(t). Robust Stability Theorem : P > 0, P 2, P 3, and a positive scalar γ > 0; LMI condition is feasible (see manuscript); Asymptotically stable with H performance J(w) < 0; (12) 29 /102

39 CONTROL SCHEMES Robust Stability Conditions H Performance - Delay-Free with Perturbation : { ẋ(t) = Ax(t) + Bw(t), z(t) = Cx(t). (11) J(w) = 0 (z(t) T z(t) γ 2 w(t) T w(t))dt < 0, (sup w ( z(t) 2 w(t) 2 ) < γ). V (x(t)) + z(t) T z(t) γ 2 w(t) T w(t) < 0, V (x(t)) = x(t) T Px(t). Robust Stability Theorem : P > 0, P 2, P 3, and a positive scalar γ > 0; LMI condition is feasible (see manuscript); Asymptotically stable with H performance J(w) < 0; (12) 29 /102

40 CONTROL SCHEMES Robust Stability Conditions Robust Stability Condition - Teleoperation Case : LKF asymptotical stability condition with several time-varying delays + H performance improvement condition without time-varying delays but with the perturbation = Robust stability condition with several time-varying delays and the perturbation : ẋ(t) = A 0 x(t) + q i=1 A ix(t τ i (t)) + Bw(t), z(t) = Cx(t), x(t 0 + θ) = φ(θ), ẋ(t 0 + θ) = φ(θ), θ [ h 2, 0], (13) V (x(t), ẋ(t)) + z(t) T z(t) γ 2 w(t) T w(t) < /102

41 CONTROL SCHEMES Robust Stability Conditions Robust Stability Condition - Teleoperation Case : LKF asymptotical stability condition with several time-varying delays + H performance improvement condition without time-varying delays but with the perturbation = Robust stability condition with several time-varying delays and the perturbation : ẋ(t) = A 0 x(t) + q i=1 A ix(t τ i (t)) + Bw(t), z(t) = Cx(t), x(t 0 + θ) = φ(θ), ẋ(t 0 + θ) = φ(θ), θ [ h 2, 0], (13) V (x(t), ẋ(t)) + z(t) T z(t) γ 2 w(t) T w(t) < /102

42 CONTROL SCHEMES Robust Stability Conditions Robust Stability Condition - Teleoperation Case : LKF asymptotical stability condition with several time-varying delays + H performance improvement condition without time-varying delays but with the perturbation = Robust stability condition with several time-varying delays and the perturbation : ẋ(t) = A 0 x(t) + q i=1 A ix(t τ i (t)) + Bw(t), z(t) = Cx(t), x(t 0 + θ) = φ(θ), ẋ(t 0 + θ) = φ(θ), θ [ h 2, 0], (13) V (x(t), ẋ(t)) + z(t) T z(t) γ 2 w(t) T w(t) < /102

43 CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme 2. Novel Control Schemes System Description & Assumptions; Robust Stability Conditions; Control Scheme 1 - Bilateral State Feedback Control Scheme ; [B. Zhang et al., CCDC, 2011] ; Control Scheme 2 - Force-Reflecting Control Scheme ; Control Scheme 3 - Force-Reflecting Proxy Control Scheme [B. Zhang et al., ETFA, 2011] ; Results and Analysis; 31 /102

44 CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme Control Scheme 1 - Bilateral State Feedback Control Scheme : Delayed state feedback C 1 & C 2 : C 1 : C 2 : F s (t) = K1 1 θ s (t ˆτ 1 (t)) K1 2 θ m (t τ 1 (t)) K1 3 (θ s(t ˆτ 1 (t) θ m (t τ 1 (t))), F m (t) = K2 1 θ s (t τ 2 (t)) K2 2 θ m (t ˆτ 2 (t)) K2 3 (θ s(t τ 2 (t) θ m (t ˆτ 2 (t))). (14) ˆτ 1 (t) = τ 1 (t) and ˆτ 2 (t) = τ 2 (t) from Assumption 3; 32 /102

45 CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme Control Scheme 1 - Master/Slave Controller Design : ẋ ms (t) u ms (t) z ms (t) = (A ms B ms K 0 )x ms (t) + B ms u ms (t) + B ms w ms (t), = Kmsx 1 ms (t τ 1 (t)) Kmsx 2 ms (t τ 2 (t)), = C ms x ms (t), (15) ( θs(t) ) x ms(t) = θm(t) θs(t) θm(t) ( ), u ms(t) = z ms(t) = θs(t) θm(t), ( As 0 0 ) A ms = 0 Am ( K s ) K 0 = K 0 m 0, K 1 ( ms = K 1 1 K 1 2 K ( ) C ms = ( ) Fs(t), w Fm(t) ms(t) = ( ) Bs 0 ( ), B ms = 0 Bm = B ms 1 B2 ms, 0 0 ( ) Fe(t), F h (t) ), K 2 ( ) ms = K 2 1 K2 2 K3, 2 (16) (17) 33 /102

46 CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme Control Scheme 1 - Master/Slave Controller Design : Transformation so to apply robust control condition : ẋ ms (t) z ms (t) = A 0 ms x ms(t) + A 1 ms x ms(t τ 1 (t)) + A 2 ms x ms(t τ 2 (t)) +B ms w ms (t), = C ms x ms (t), (18) A 0 ms = A ms B ms K 0, A 1 ms = B mskms 1 = B1 ms K 1, A 2 ms = B mskms 2 = B2 ms K 2, (19) ( ) ( ) K 1 = K1 1 K1 2 K1 3, K 2 = K2 1 K2 2 K2 3. (20) 34 /102

47 CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme Control Scheme 1 - Control Objectives : LKF : the system stability under the time-varying delays τ 1 (t), τ 2 (t) [h 1, h 2 ] ; H control : the impact γ of disturbances w ms (t) on z ms (t) (θ s (t) θ m (t)) ; Control Scheme 1 - Master/Slave Controller Design Theorem : P > 0, R > 0, S > 0, S a > 0, R a1 > 0, R a2 > 0, P 2, W 1, W 2, Y 1, Y 2, and positive scalars γ and ξ ; LMI condition is feasible (see manuscript); Rate-independent asymptotically stable with H performance J(w) < 0 for time-varying delays τ 1 (t), τ 2 (t) [h 1, h 2 ] : K 1 = W 1 P 1 2, K 2 = W 2 P 1 2. (21) 35 /102

48 CONTROL SCHEMES Control Scheme 2 - Force-Reflecting Control Scheme 2. Novel Control Schemes System Description & Assumptions; Robust Stability Conditions; Control Scheme 1 - Bilateral State Feedback Control Scheme [B. Zhang et al., CCDC, 2011] ; Control Scheme 2 - Force-Reflecting Control Scheme ; Control Scheme 3 - Force-Reflecting Proxy Control Scheme [B. Zhang et al., ETFA, 2011] ; Results and Analysis; 36 /102

49 CONTROL SCHEMES Control Scheme 2 - Force-Reflecting Control Scheme Control Scheme 2 - Force-Reflecting Control Scheme : ˆF e (t) : F m (t) = ˆF e (t τ 2 (t)); C with controller gain K i, i = 1, 2, 3 : C : F s (t) = K 1 θs (t ˆτ 1 (t)) K 2 θm (t τ 1 (t)) K 3 (θ s (t ˆτ 1 (t) θ m (t τ 1 (t))). (22) 37 /102

50 CONTROL SCHEMES Control Scheme 2 - Force-Reflecting Control Scheme Control Scheme 2 - Slave Controller Design : { ẋ ms (t) = (A ms B ms K 0 )x ms (t) + B ms u ms (t) + B ms w ms (t), u ms (t) = K ms x ms (t τ 1 (t)), (23) ( ) K ms = K 1 K 2 K 3. (24) { ẋ ms (t) = A 0 ms x ms(t) + A 1 ms x ms(t τ 1 (t)) + B ms w ms (t), (25) z ms (t) = C ms x ms (t), ( ) A 1 ms = B msk ms = Bms 1 K, K = K 1 K 2 K 3. (26) 38 /102

51 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme 2. Novel Control Schemes System Description & Assumptions; Robust Stability Conditions; Control Scheme 1 - Bilateral State Feedback Control Scheme ; [B. Zhang et al., CCDC, 2011] ; Control Scheme 2 - Force-Reflecting Control Scheme ; Control Scheme 3 - Force-Reflecting Proxy Control Scheme [B. Zhang et al., ETFA, 2011] ; Results and Analysis; 39 /102

52 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Force-Reflecting Proxy Control Scheme : From master to slave : θ m (t)/θ m (t)/ ˆF h (t), the position tracking; From slave to master : F m (t) = ˆF e (t τ 2 (t)), the force tracking; 40 /102

53 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Slave Controller Description : P : ẋ p (t) = (A m B m K m 0 )x p(t) B m F p (t) + B m ( ˆF e (t ˆτ 1 (t)) + ˆF h (t τ 1 (t))), (27) x p (t) = θ p (t) R n. 41 /102

54 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Slave Controller Description : ) L = (L 1 L 2 L 3 = θ p (t) θ m (t) : ( F p (t) = L θ p(t ˆτ 1(t)) θ m(t τ 1(t)) θ p(t ˆτ 1(t)) θ m(t τ 1(t)) ). (27) 41 /102

55 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Slave Controller Description : ) K = (K 1 K 2 K 3 of the controller C = θ s (t) θ p (t) : ) F s (t) = K ( θs(t) θ p(t) θ s(t) θ p(t). (28) 41 /102

56 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Master-Proxy Synchronization : { ẋ mp (t) = A 0 mpx mp (t) + A 1 mpx mp (t τ 1 (t)) + B mp w mp (t), z mp (t) = C mp x mp (t). θp(t) ( ) x mp(t) = θm(t) ˆFe(t ˆτ, w mp(t) = 1 (t))+ ˆF h (t τ 1 (t)), Fm(t)+F θp(t) θm(t) h (t) ( ) z mp(t) = θp(t) θm(t). (29) (30) 42 /102

57 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Master-Proxy Synchronization : ( ) Am B mk m A 0 mp = A m B mk m 0 0, B mp = ( ) C mp = 0 0 1, ( A 1 mp = BmL ) 1 B ml 2 B ml ( Bm ) ( ) 0 0 B m = B 1 mp B2 mp, 0 0 = B 1 mp L. (31) 43 /102

58 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Proxy-Slave Synchronization : { ẋ ps (t) = A ps x ps (t) + B ps w ps (t), z ps (t) = C ps x ps (t), θs(t) ( ) x ps(t) = θp(t), z ps(t) = θs(t) θp(t), θs(t) θp(t) ( ) F e(t) w ps(t) = ˆF e(t τˆ 1 (t)) + ˆF. h (t τ 1 (t)) F p(t) (32) (33) 44 /102

59 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Proxy-Slave Synchronization :. ( ) Bs 0 B ps = 0 Bm 0 0 ( ) ( ) = B ps 1 B2 ps, C ps = 0 0 1, ( As BsK s 0 BsK 1 BsK 2 ) BsK 3 A ps = 0 Am BmK 0 m ( As BsK s 0 0 ) 0 = 0 Am BmK 0 m = (A 0 ps B1 ps K). ( BsK1 BsK 2 BsK ) (34) 45 /102

60 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Global Performance Analysis : { ẋ mps (t) z mps (t) x mps(t) = = A 0 mpsx mps (t) + A 1 mpsx mps (t τ 1 (t)) + B mps w mps (t), = C mps x mps (t), (35) θs(t) θp(t) θm(t) θs(t) θp(t) θp(t) θm(t) (, wmps(t) = Fe(t) ) ˆFe(t ˆτ 1 (t))+ ˆF h (t τ 1 (t)) Fm(t)+F h (t), z mps(t) = ( ) θs(t) θp(t). θp(t) θm(t) (36) As BsK s 0 BsK 1 BsK 2 0 BsK 3 0 A 0 mps = 0 Am BmK 0 m Am BmK 0 m 0 0, Bs 0 0 A 1 mps = 0 BmL 1 BmL 2 0 BmL Bm 0 ( ), B mps = 0 0 Bm, C mps = (37) 46 /102

61 CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Conclusions : Control Scheme 1 - Bilateral state feedback control scheme; Control Scheme 2 - Force-reflecting control scheme; Control Scheme 3 - Force-reflecting proxy control scheme; Control Schema Time Delays Position Tracking Constant Time-varying Force Tracking Force estimation/measure : 2 & 3; Better performance, but additional computation load : 3; 47 /102

62 CONTROL SCHEMES Results and Analysis 2. Novel Control Schemes System Description & Assumptions; Robust Stability Conditions; Control Scheme 1 - Bilateral State Feedback Control Scheme [B. Zhang et al., CCDC, 2011]; [B. Zhang et al., CCDC, 2011] ; Control Scheme 2 - Force-Reflecting Control Scheme ; Control Scheme 3 - Force-Reflecting Proxy Control Scheme [B. Zhang et al., ETFA, 2011] ; Results and Analysis; 48 /102

63 CONTROL SCHEMES Results and Analysis Simulation Conditions : Maximum amplitude of time-varying delays : 0.2s; Master, proxy and slave models : 1/s, 1/s and 2/s; Poles : [ 100.0]. Km 0 = 100, K0 s = 50; ( ) K 1 = K 2 = , ( ) , γ C1/C2 min = (38) ( ) L = , γmin L = , ( ) K = , γmin K = (39) Global stability of the system is verified with γ g min = ; 49 /102

64 CONTROL SCHEMES Results and Analysis Abrupt Tracking Motion : 50 /102

65 CONTROL SCHEMES Results and Analysis Abrupt Tracking Motion : 51 /102

66 CONTROL SCHEMES Results and Analysis Wall Contact Motion : Hard wall : the stiffness K e = 30kN/m, the position x = 1.0m; Our aims : 1. When the slave robot reaches the wall, the master robot can stop as quickly as possible; 2. When the slave robot returns after hitting the wall (F e (t) = 0), the system must restore the position tracking between the master and the slave ; 3. When the slave contacts the wall, the force tracking from the slave to the master can be assured ; 52 /102

67 CONTROL SCHEMES Results and Analysis Wall Contact Motion : 53 /102

68 CONTROL SCHEMES Results and Analysis Wall Contact Motion : 54 /102

69 CONTROL SCHEMES Results and Analysis Wall Contact Motion : Force response in wall contact motion (F m(t) ; ˆFe(t)). 55 /102

70 CONTROL SCHEMES Results and Analysis Wall Contact Motion under Large Delays h 2 = 1.0s : 56 /102

71 ROBUSTNESS Discrete-Time Approach 3. Robustness Aspects Discrete-Time System [B. Zhang et al., TDS, 2012] : Discrete-Time Approach; Slave Controller Design; Results and Analysis ; Linear Parameter-Varying System (LPV) : Polytopic-type uncertainties [B. Zhang et al., TDS, 2012] ; Norm-bounded uncertainties [B. Zhang et al., ROCOND, 2012] ; Results and Analysis ; 57 /102

72 ROBUSTNESS Discrete-Time Approach Discrete-Time System : { x(k + 1) = q i=0 A ix(k τ i (k)) + Bw(k), z(k) = Cx(k). (40) Delay-Free Case : { x(k + 1) = A 0 x(k) + Bw(k), z(k) = Cx(k). (41) 58 /102

73 ROBUSTNESS Discrete-Time Approach Robust Stability Condition : Discrete LKF, y(k) = x(k + 1) x(k) : V (x(k)) = x(k) T Px(k) h 1 k 1 i= h 1 j=k+i k 1 y(j) T Ry(j) + i=k h 2 x(i) T S a x(i) + q (h 2 h 1 ) i=1 H control : J(w) = i=0 [z(k) T z(k) γ 2 w(k) T w(k)] < 0 V (x(k)) = V (x(k + 1)) V (x(k)) : h 1 1 k 1 i=k h 1 x(i) T Sx(i) k 1 j= h 2 l=k+j y(l) T R ai y(l). (42) V (x(k)) + z(k) T z(k) γ 2 w(k) T w(k) < 0. (43) LMI condition is feasible (see manuscript) ; Rate-independent asymptotically stable and H performance J(w) < 0 for time-varying delays τ i (k) [h 1, h 2 ], h 2 h 1 0, i = 1, 2,..., q ; 59 /102

74 ROBUSTNESS Discrete-Time Approach 3. Robustness Aspects Discrete-Time System [B. Zhang et al., TDS, 2012] : Discrete-Time Approach; Slave Controller Design; Results and Analysis ; Linear Parameter-Varying System (LPV) : Polytopic-type uncertainties [B. Zhang et al., TDS, 2012] ; Norm-bounded uncertainties [B. Zhang et al., ROCOND, 2012] ; Results and Analysis ; 60 /102

75 ROBUSTNESS Discrete-Time Approach System Description : (Σ d m ) x m(k + 1) = (A md B md Kmd 0 )x m(k) + B md (F m (k) + F h (k)), (Σ d s ) x s(k + 1) = (A sd B sd Ksd 0 )x s(k) + B sd (F s (k) + F e (k)), (Σ d p) x p (k + 1) = (A md B md Kmd)x 0 p (k) B md F p (k) + B md ( ˆF e (k ˆτ 1 (k)) + ˆF h (k τ 1 (k))). (44) 61 /102

76 ROBUSTNESS Discrete-Time Approach Slave Controller Design) - Master-Proxy Synchronization : L d = (L d1 L d2 L d3 : F p (k) = L d ( θ p(k ˆτ 1(k)) θ m(k τ 1(k)) θ p(k ˆτ 1(k)) θ m(k τ 1(k)) ). (45) x mp (k + 1) = A 0 mpd x mp(k) + A 1 mpd x mp(k τ 1 (k)) (Σ d mp ) +B mpd w mp (k), z mp (k) = C mpd x mp (k), (46) A 1 mpd = L d ; 62 /102

77 ROBUSTNESS Discrete-Time Approach Slave Controller Design) - Proxy-Slave Synchronization : K d = (K d1 K d2 K d3 : F s (k) = K d ( θs(k) θ p(k) θ s(k) θ p(k) ). (47) (Σ d ps ) { ( Σ d ps ) { x ps (k + 1) = A psd x ps (k) + B K F s (k) + B psd w ps (k), z ps (k) = C psd x ps (k), x ps (k + 1) = (A psd B K K d )x ps (k) + B psd w ps (k), z ps (k) = C psd x ps (k). (48) (49) 63 /102

78 ROBUSTNESS Discrete-Time Approach Slave Controller Design - Global Performance Analysis : { (Σ d x(k + 1) = Ampsd mps ) x(k) + Bmpsd K Fs(k) BL mpsd Fp(k) + B mpsdw(k), z(k) = C mpsd x(k), F s (k) = K ( ) d x(k) = K d1 K d2 0 K d3 0 x(k), F p (k) = L ( ) d x(k τ 1 (k)) = 0 L d1 L d2 0 L d3 x(k τ 1 (k)), ( Bsd Bmpsd K = ( Σ d mps ) { ) x(k + 1) z(k) ( 0, Bmpsd L = B md ), = A d 0 x(k) + Ad 1 x(k τ 1(k)) + B mpsd w(k), = C mpsd x(k), (50) (51) (52) A d 0 = A mpsd B K mpsd K d, A d 1 = BL mpsd L d. 64 /102

79 ROBUSTNESS Discrete-Time Approach 3. Robustness Aspects Discrete-Time System [B. Zhang et al., TDS, 2012] : Discrete-Time Approach; Slave Controller Design; Results and Analysis ; Linear Parameter-Varying System (LPV) : Polytopic-type uncertainties [B. Zhang et al., TDS, 2012] ; Norm-bounded uncertainties [B. Zhang et al., ROCOND, 2012] ; Results and Analysis ; 65 /102

80 ROBUSTNESS Discrete-Time Approach Simulation Conditions : T = 0.001s, h 1 = 1, h 2 = 100 (in continuous-time domain, h 1 = 0.001s, h 2 = 0.1s); ( ) L d = ( ) K d = γ g min = , γ L d min = ,, γ K d min = , (53) 66 /102

81 ROBUSTNESS Discrete-Time Approach Abrupt Tracking Motion : 67 /102

82 ROBUSTNESS Discrete-Time Approach Abrupt Tracking Motion : 67 /102

83 ROBUSTNESS Discrete-Time Approach Abrupt Tracking Motion : 68 /102

84 ROBUSTNESS Discrete-Time Approach Wall Contact Motion : 69 /102

85 ROBUSTNESS Discrete-Time Approach Wall Contact Motion : 70 /102

86 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties 3. Robustness Aspects Discrete-Time System [B. Zhang et al., TDS, 2012] : Discrete-Time Approach; Slave Controller Design; Results and Analysis ; Linear Parameter-Varying System (LPV) : Polytopic-type uncertainties [B. Zhang et al., TDS, 2012] ; Norm-bounded uncertainties [B. Zhang et al., ROCOND, 2012] ; Results and Analysis ; 71 /102

87 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties System Description : (Σ m)ẋ m(t) = (A m(ρ m(t)) B m(ρ m(t))k 0 m )xm(t) + Bm(ρm(t))(Fm(t) + F h(t)), (Σ s)ẋ s(t) = (A s(ρ s(t)) B s(ρ s(t))k 0 s )xs(t) + Bs(ρs(t))(Fs(t) + Fe(t)), (Σ p)ẋ p(t) = (A m(ρ p(t)) B m(ρ p(t))k 0 m )xp(t) + Bm(ρp(t))( ˆF e(t ˆτ 1 (t)) + ˆF h (t τ 1 (t)) F p(t)), N N [A m(ρ m(t)), B m(ρ m(t))] = ρ mj (t)[a mj, B mj ], [A s(ρ s(t)), B s(ρ s(t))] = ρ sj (t)[a sj, B sj ], j=1 j=1 N [A m(ρ p(t)), B m(ρ p(t))] = ρ pj (t)[a mj, B mj ]. j=1 (54) 72 /102

88 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties Controller Design - Problem 1 : Km 0 & K0 s : robust stability w.r.t polytopic-type uncertainties; Controller Design - Problem 2 : L & K : stability & position/force tracking w.r.t time-varying delays & polytopic-type uncertainties ; Robust Stability Theorem [E. Fridman, IMA Journal of Mathematical Control and Information, 2006] : P > 0, R > 0, S > 0, S a > 0, R ai > 0, P 2, P 3, Y 1, Y 2, i = 1,2,...,q, and a positive scalar γ > 0; N LMI conditions are feasible (see manuscript); Rate-independent asymptotically stable with H performance J(w) < 0 for time-varying delays τ i (t) [h 1,h 2 ], i = 1,2,...,q ; 73 /102

89 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties Problem 2 Slave Controller Design Master-proxy synchronization : { ẋmp(t) = A 0 mp (ρmp(t))xmp(t) + A1 mp (ρmp(t))xmp(t τ 1(t)) + B mp(ρ mp(t))w mp(t), z mp(t) = C mpx mp(t), K 0 m & K0 s = A0 mp (ρ mp(t)) ; A 1 mp (ρ mp(t)) = L ; Proxy-slave synchronization : (55) { ẋ ps(t) z ps(t) = A ps(ρ ps(t))x ps(t) + B ps(ρ ps(t))w ps(t), = C psx ps(t), (56) A ps (ρ ps (t)) = K ; Global performance analysis with K 0 m & K0 s, L & K : ẋ mps(t) z mps(t) = A 0 mps (ρmps(t))xmps(t) + A1 mps (ρmps(t))xmps(t τ 1(t)) +B mps(ρ mps(t))w mps(t), = C mpsx mps(t). (57) 74 /102

90 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties Remark - Nonlinear Systems : (Σ m) M m(θ m) θ m(t) + C m(θ m, θ m) θ m(t) + g m(θ m) = F h (t) + F m(t), (Σ p) M m(θ p) θ p(t) + C m(θ p, θ p) θ p(t) + g m(θ p) = ˆF e(t ˆτ 1 (t)) + ˆF h (t τ 1 (t)) F p(t), (Σ s) M s(θ s) θ s(t) + C s(θ s, θ s) θ s(t) + g s(θ s) = F e(t) + F s(t), (58) ( Σ m) θm(t) = A m(θ m, θ m) θ m(t) + B m(θ m)(f h (t) + F m(t) g m(θ m)), ( Σ p) θp(t) = A m(θ p, θ p) θ p(t) + B m(θ p)( ˆF e(t ˆτ 1 (t)) + ˆF h (t τ 1 (t)) F p(t) g m(θ p)), ( Σ s) θs(t) = A s(θ s, θ s) θ s(t) + B s(θ s)(f e(t) + F s(t) g s(θ s)), A m(θ m, θ m) = M 1 m (θm)cm(θm, θ m), A m(θ p, θ p) = M 1 m (θp)cm(θp, θ p), B m(θ m) = M 1 m (θm), B m(θ p) = M 1 m (θp), A s(θ s, θ s) = M 1 s (θs)cs(θs, θ s), B s(θ s) = M 1 s (θs). (59) (60) 75 /102

91 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties 3. Robustness Aspects Discrete-Time System [B. Zhang et al., TDS, 2012] : Discrete-Time Approach; Slave Controller Design; Results and Analysis ; Linear Parameter-Varying System (LPV) : Polytopic-type uncertainties [B. Zhang et al., TDS, 2012] ; Norm-bounded uncertainties [B. Zhang et al., ROCOND, 2012] ; Results and Analysis ; 76 /102

92 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties System Description : (Σ m) ẋ m(t) = ((A m + A m(t)) (B m + B m(t))k 0 m )xm(t) + (B m + B m(t))(f m(t) + F h (t)), (Σ s) ẋ s(t) = ((A s + A s(t)) (B s + B s(t))k 0 s )xs(t) + (B s + B s(t))(f s(t) + F e(t)), (Σ p) ẋ p(t) = ((A m + A p(t)) (B m + B p(t))k 0 m )xp(t) (B m + B p(t))f p(t) + (B m + B p(t))( ˆF e(t ˆτ 1 (t)) + ˆF h (t τ 1 (t))), (61) i = {m, s, p} : A i (t) = G i (t)d i, B i (t) = H i (t)e i, (t) T (t) I. (62) 77 /102

93 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties System Description : (Σ m) ẋ m(t) = ((A m + A m(t)) (B m + B m(t))k 0 m )xm(t) + (B m + B m(t))(f m(t) + F h (t)), (Σ s) ẋ s(t) = ((A s + A s(t)) (B s + B s(t))k 0 s )xs(t) + (B s + B s(t))(f s(t) + F e(t)), (Σ p) ẋ p(t) = ((A m + A p(t)) (B m + B p(t))km 0 )xp(t) (B m + B p(t))f p(t) + (B m + B p(t))( ˆF e(t ˆτ 1 (t)) + ˆF h (t τ 1 (t))), (63) i = {m, s, p} : A i (t) = G i (t)d i, B i (t) = H i (t)e i, (t) T (t) I. (64) Slave Controller Design Solution : Transformation from norm-bounded uncertain system to linear time-delay system; 78 /102

94 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Problem 2 Slave Controller Design - Master-Proxy Synchronization : ẋ mp(t) z mp(t) = (A 0 mp + A0 mp (t))xmp(t) + (A1 mp + A1 mp (t))xmp(t τ 1(t)) +(B mp + B mp(t))w mp(t), = C mpx mp(t), (65) K 0 m & K 0 s = A 0 mp + A 0 mp(t); ϕ p(t) = ( A p(t) B p(t)k 0 m ) θ p(t) + B p(t)( ˆF e(t ˆτ 1 (t)) + ˆF h (t τ 1 (t))), ϕ m(t) = ( A m(t) B m(t)k 0 m ) θ m(t) + B m(t)(f m(t) + F h (t)), µ p(t) = B p(t)lx mp(t τ 1 (t)). { ẋ mp(t) z mp(t) = A 0 mp xmp(t) + A1 mp xmp(t τ 1(t)) + B mp w mp(t), = C mpx mp(t), ( Bm ˆFe(t ˆτ w(t) = 1 (t))+bm ˆF ) h (t τ 1 (t))+ϕp(t)+µp(t), Bmp = BmFm(t)+BmF h (t)+ϕm(t) (66) (67) ( ) , (68) 0 0 A 1 mp = L; 79 /102

95 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Slave Controller Design - Proxy-Slave Synchronization : { ẋ ps(t) = (A ps + A ps(t))x ps(t) + (B ps + B ps(t))w ps(t), z ps(t) = C psx ps(t), A ps + A ps (t) = K ; (69) ϕ s(t) = ( A s(t) B s(t)k 0 s ) θ s(t) + B s(t)f e(t), µ s(t) = B 1 ps (t)kxps(t). (70) { ẋ ps(t) z ps(t) = A psx ps(t) + B ps w ps(t), = C psx ps(t), ( ) BsFe(t)+ϕs(t)+µs(t) w ps(t) = Bm ˆFe(t ˆτ 1 (t))+bm ˆF h (t τ 1 (t)) BmFp(t)+ϕp(t)+µp(t) A ps = K ;, Bps = ( ) , 0 0 (71) (72) 80 /102

96 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Slave Controller Design - Global Performance Analysis : ẋ mps(t) z mps(t) = (A 0 mps + A0 mps (t))xmps(t) + (A1 mps + A1 mps (t))xmps(t τ 1(t)) +(B mps + B mps(t))w mps(t), = C mpsx mps(t). (73) { ẋ mps(t) z mps(t) = A 0 mps xmps(t) + A1 mps xmps(t τ 1(t)) + B mps w mps(t), = C mpsx mps(t), ( ) BsFe(t)+ϕs(t) w mps(t) = Bm ˆFe(t ˆτ 1 (t))+bm ˆF h (t τ 1 (t))+ϕp(t), Bmps = (75) BmFm(t)+BmF h (t)+ϕm(t) (74) 81 /102

97 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties 3. Robustness Aspects Discrete-Time System [B. Zhang et al., TDS, 2012] : Discrete-Time Approach; Slave Controller Design; Results and Analysis ; Linear Parameter-Varying System (LPV) : Polytopic-type uncertainties [B. Zhang et al., TDS, 2012] ; Norm-bounded uncertainties [B. Zhang et al., ROCOND, 2012] ; Results and Analysis ; 82 /102

98 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Simulation Conditions : polytopic type : A m(ρ m(t)) = A s(ρ s(t)) = 0, B m(ρ m(t)) = B s(ρ s(t)) =, ρ(t) [0.5, 1], ρ(t) 1 norm bounded type : A m = A s = 0, G i = D i = 0, B m = B s = 1.5, H i = 0.5, E i = 1, i = {m, p, s}. (76) polytopic type : K 0 m = , K0 s = , (77) norm bounded type : K 0 m = , K0 s = ( ) polytopic type : L = , γmin L = , ( ) K = , γ K min = , γ g min = , ( ) norm bounded type : L = , γmin L = 0.05, ( ) K = , γmin K = , (78) γ g min = /102

99 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Abrupt Tracking Motion : 84 /102

100 ROBUSTNESS H Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Wall Contact Motion : 85 /102

101 EXPERIMENTATIONS Experimental Test-Bench Experimental Test-Bench; Force Estimation; Results and Analysis; 4. Experimentations 86 /102

102 EXPERIMENTATIONS Experimental Test-Bench Experimental Test-Bench : Master, Phantom Premium 1.0A : θ m (t) = θ m (t) τ m τ m = 0.448s, K m = s/kg ; + K m τ m u m (t), (79) 87 /102

103 EXPERIMENTATIONS Experimental Test-Bench Experimental Test-Bench : Slave, Mitsubishi Robot + CRIO card : θ s (t) = θ s (t) τ s τ s = 0.32s, K s = 1.85s/kg, F s = 0.30m/s 2 ; + K s τ s u s (t) F s sign( θ s (t)), (80) 88 /102

104 EXPERIMENTATIONS Experimental Test-Bench Experimental Test-Bench : Networks : UDP ; 89 /102

105 EXPERIMENTATIONS Force Estimation Experimental Test-Bench; Force Estimation; Results and Analysis; 4. Experimentations 90 /102

106 EXPERIMENTATIONS Force Estimation Force Estimation F h (t) : ẋ m (t) = (A m B m K 0 m )x m(t) + B m (F m (t) + F h (t)). (81) F (n+1) (t) = 0 for some n : h ε h (t) = x m(t) F h (t) Ḟ h (t). F (n) h (t) ε h (t) = A h ε h (t) + B h F m (t), (82), A h = y h (t) = x m(t) = C h ε h (t) : A m B mkm 0 Bm I I , B h =. B m (83) ˆε h (t) = A hˆε h (t) + B h F m (t) + L h (y h (t) ŷ h (t)). (84) e h (t) = ε h (t) ˆε h (t) : ė h (t) = (A h L h C h )e h (t). (85) 91 /102

107 EXPERIMENTATIONS Force Estimation Force Estimation F h (t) : n = 1, the eigenvalues of Luenberger observer as [ 11, 10, 9] ; 92 /102

108 EXPERIMENTATIONS Results and Analysis Experimental Test-Bench; Force Estimation; Results and Analysis; 4. Experimentations 93 /102

109 EXPERIMENTATIONS Results and Analysis Abrupt Tracking Motion : h 2 = 0.3s ; 94 /102

110 EXPERIMENTATIONS Results and Analysis Wall Contact Motion : 95 /102

111 EXPERIMENTATIONS Results and Analysis Wall Contact Motion under Constant F h (t) : 96 /102

112 CONCL. & PERSP. 5. Conclusions & Perspectives 97 /102

113 CONCL. & PERSP. Conclusions Conclusions : 3 novel control schemes (3 2 1) : 1. Bilateral state feedback control scheme (Position/Velocity-Position/Velocity); 2. Force-reflecting control scheme (Position/Velocity-Force); 3. Force-reflecting proxy control scheme (Position/Velocity/Force-Force); Formal proofs : 1. Stability (LKF); 2. Performance (H control) ; 3. Time-varying (large) delays; 4. Uncertainties (polytopic or norm bounded) ; 5. Continuous or discrete time; LMI design of the controllers; Simulations and experimentations : 1. Abrupt tracking motion ; 2. Wall contact motion ; 98 /102

114 CONCL. & PERSP. Perspectives Perspectives : Nonlinear systems e.g. master : M(θ) θ(t) + C(θ, θ) θ(t) + g(θ) = F h (t) + F m (t). (86) One master/multiple slaves teleoperation system ; 99 /102

115 CONCL. & PERSP. Perspectives Perspectives : Delay-scheduled state-feedback controller design : K(τ(t)) = K 0 + K 1 τ(t) + K 2 τ 2 (t) + + K h τ h (t) = h K i τ i (t). (87) Switch control strategy between the passivity-based approach and the approach proposed in this thesis; i=0 100 /102

116 CONCL. & PERSP. Perspectives Perspectives : Global stability analysis with perturbation observers ; Dynamics of the human operator and the environment; 101 /102

117 CONCL. & PERSP. Questions Thank you for your attention! 102 /102