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

  1. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  2. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  3. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels PRELIMINARIES Stability of Time Delay Systems General System with Piecewise Time-Varying Delays :  x ( t ) = f ( x ( t ) , x ( t − τ 1 ( t )) , u ( t ) , u ( t − τ 2 ( t ))) , ˙     x ( t 0 + θ ) = ˙ x ( t 0 + θ ) = φ ( θ ) , ˙ φ ( θ ) , (3)  u ( t 0 + θ ) = ζ ( θ ) ,    θ ∈ [ − h, 0] . Linear Case : � x ( t ) = A 0 x ( t ) + � n i =1 A i x ( t − τ 1 i ( t )) + B 0 u ( t ) + � m ˙ j =1 B j u ( t − τ 2 j ( t )) , x ( t 0 + θ ) = ˙ x ( t 0 + θ ) = φ ( θ ) , ˙ φ ( θ ) , θ ∈ [ − h, 0] . (4) 19 / 102

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

  5. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  6. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  7. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES System Description & Assumptions General Teleoperation Structure : From Property 1 - Assumption 1 : Linear master/slave systems = ⇒ x m ( t ) = ( A m − B m K m ˙ 0 ) x m ( t ) + B m ( F m ( t ) + F h ( t )) , (7) x s ( t ) = ( A s − B s K s ˙ 0 ) x s ( t ) + B s ( F s ( t ) + F e ( t )) , θ s ( t ) ∈ R n ; K m x m ( t ) = ˙ θ m ( t ) ∈ R n , x s ( t ) = ˙ 0 & K s 0 : supposed to be known; 23 / 102

  8. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  9. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES System Description & Assumptions General Teleoperation Structure : From Property 3 - Assumption 3 : Time-stamped data packets = ⇒ ˆ τ 1 ( t ) = τ 1 ( t ) , ˆ τ 2 ( t ) = τ 2 ( t ) ; 25 / 102

  10. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  11. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Robust Stability Conditions Linear Time Delay System : � x ( t ) = A 0 x ( t ) + � q ˙ i =1 A i x ( t − τ i ( t )) , (8) x ( t 0 + θ ) = ˙ x ( t 0 + θ ) = φ ( θ ) , ˙ φ ( θ ) , θ ∈ [ − h 2 , 0] . 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

  12. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Robust Stability Conditions Asymptotical Stability Theorem :   12 R + � q Γ1 11 Γ1 i =1 P T 2 Ai − qY T qY T − P T 2 A 1+ Y T ... − P T 2 Aq + Y T Y T Y T ... 1 1 1 1 1 1 � q   Γ1 i =1 P T 3 Ai − qY T qY T − P T 3 A 1+ Y T ... − P T 3 Aq + Y T Y T Y T  � ...  22 2 2 2 2 2 2    − S − R 0 0 0 0 0 0 0  � �   − Sa 0 0 0 0 0 0 � � � Γ 1 =    − Ra 1 0 0 0 0 0  < 0 , � � � �   � � � � � ... 0 0 0 0    − Raq 0 0 0  � � � � � �    � � � � � � � − Ra 1 0 0  0 � � � � � � � � ... � − Raq � � � � � � � � (9) Γ 1 11 = S + S a − R + A T 0 P 2 + P T Γ 1 12 = P − P T 2 + A T 2 A 0 , 0 P 3 , q � (10) Γ 1 22 = − P 3 − P T 3 + h 2 1 R + ( h 2 − h 1 ) 2 R ai . i =1 28 / 102

  13. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Robust Stability Conditions H ∞ Performance - Delay-Free with Perturbation : � x ( t ) ˙ = Ax ( t ) + Bw ( t ) , (11) z ( t ) = Cx ( t ) . � ∞ ( z ( t ) T z ( t ) − γ 2 w ( t ) T w ( t )) dt < 0 , J ( w ) = 0 (12) ( sup w ( � z ( t ) � 2 ) < γ ) . � w ( t ) � 2 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 ; 29 / 102

  14. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Robust Stability Conditions H ∞ Performance - Delay-Free with Perturbation : � x ( t ) ˙ = Ax ( t ) + Bw ( t ) , (11) z ( t ) = Cx ( t ) . � ∞ ( z ( t ) T z ( t ) − γ 2 w ( t ) T w ( t )) dt < 0 , J ( w ) = 0 (12) ( sup w ( � z ( t ) � 2 ) < γ ) . � w ( t ) � 2 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 ; 29 / 102

  15. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Robust Stability Conditions H ∞ Performance - Delay-Free with Perturbation : � x ( t ) ˙ = Ax ( t ) + Bw ( t ) , (11) z ( t ) = Cx ( t ) . � ∞ ( z ( t ) T z ( t ) − γ 2 w ( t ) T w ( t )) dt < 0 , J ( w ) = 0 (12) ( sup w ( � z ( t ) � 2 ) < γ ) . � w ( t ) � 2 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 ; 29 / 102

  16. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 :  x ( t ) = A 0 x ( t ) + � q  ˙ i =1 A i x ( t − τ i ( t )) + Bw ( t ) ,  z ( t ) = Cx ( t ) , (13)   x ( t 0 + θ ) = ˙ x ( t 0 + θ ) = φ ( θ ) , ˙ φ ( θ ) , θ ∈ [ − h 2 , 0] , x ( t )) + z ( t ) T z ( t ) − γ 2 w ( t ) T w ( t ) < 0 . ˙ V ( x ( t ) , ˙ 30 / 102

  17. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 :  x ( t ) = A 0 x ( t ) + � q  ˙ i =1 A i x ( t − τ i ( t )) + Bw ( t ) ,  z ( t ) = Cx ( t ) , (13)   x ( t 0 + θ ) = ˙ x ( t 0 + θ ) = φ ( θ ) , ˙ φ ( θ ) , θ ∈ [ − h 2 , 0] , x ( t )) + z ( t ) T z ( t ) − γ 2 w ( t ) T w ( t ) < 0 . ˙ V ( x ( t ) , ˙ 30 / 102

  18. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 :  x ( t ) = A 0 x ( t ) + � q  ˙ i =1 A i x ( t − τ i ( t )) + Bw ( t ) ,  z ( t ) = Cx ( t ) , (13)   x ( t 0 + θ ) = ˙ x ( t 0 + θ ) = φ ( θ ) , ˙ φ ( θ ) , θ ∈ [ − h 2 , 0] , x ( t )) + z ( t ) T z ( t ) − γ 2 w ( t ) T w ( t ) < 0 . ˙ V ( x ( t ) , ˙ 30 / 102

  19. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  20. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 : 1 ˙ 1 ˙ F s ( t ) = − K 1 τ 1 ( t )) − K 2 C 1 : θ s ( t − ˆ θ m ( t − τ 1 ( t )) − K 3 1 ( θ s ( t − ˆ τ 1 ( t ) − θ m ( t − τ 1 ( t ))) , (14) 2 ˙ 2 ˙ F m ( t ) = − K 1 θ s ( t − τ 2 ( t )) − K 2 C 2 : θ m ( t − ˆ τ 2 ( t )) − K 3 2 ( θ s ( t − τ 2 ( t ) − θ m ( t − ˆ τ 2 ( t ))) . � ˆ τ 1 ( t ) = τ 1 ( t ) and ˆ τ 2 ( t ) = τ 2 ( t ) from Assumption 3 ; 32 / 102

  21. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme Control Scheme 1 - Master/Slave Controller Design :   x ms ( t ) ˙ = ( A ms − B ms K 0 ) x ms ( t ) + B ms u ms ( t ) + B ms w ms ( t ) ,  = − K 1 ms x ms ( t − τ 1 ( t )) − K 2 u ms ( t ) ms x ms ( t − τ 2 ( t )) ,   z ms ( t ) = C ms x ms ( t ) , (15) � � ˙ θs ( t ) � Fs ( t ) � � Fe ( t ) � x ms ( t ) = ˙ , u ms ( t ) = , w ms ( t ) = , θm ( t ) Fh ( t ) Fm ( t ) θs ( t ) − θm ( t ) (16) � � z ms ( t ) = , θs ( t ) − θm ( t ) � As � � Bs � � � 0 0 0 B 1 ms B 2 A ms = 0 Am 0 , B ms = = , 0 Bm ms 1 − 1 0 0 0 � � � � � � Ks 0 0 K 1 1 K 2 1 K 3 0 0 0 K 1 K 2 (17) 0 K 0 = ms = ms = , 1 , K 1 2 K 2 2 K 3 , Km 0 0 0 0 0 0 2 � � C ms = 0 0 1 . 33 / 102

  22. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 1 - Bilateral State Feedback Control Scheme Control Scheme 1 - Master/Slave Controller Design : � Transformation so to apply robust control condition :  = A 0 ms x ms ( t ) + A 1 ms x ms ( t − τ 1 ( t )) + A 2  x ms ( t ) ˙ ms x ms ( t − τ 2 ( t ))  + B ms w ms ( t ) ,   z ms ( t ) = C ms x ms ( t ) , (18) A 0 ms = A ms − B ms K 0 , A 1 ms = − B ms K 1 ms = − B 1 A 2 ms = − B ms K 2 ms = − B 2 ms K 1 , ms K 2 , (19) � � � � K 1 K 2 K 3 K 1 K 2 K 3 K 1 = , K 2 = . (20) 1 1 1 2 2 2 34 / 102

  23. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 a 1 > 0 , R a 2 > 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 K 2 = W 2 P − 1 , . (21) 2 2 35 / 102

  24. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  25. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 2 - Force-Reflecting Control Scheme Control Scheme 2 - Force-Reflecting Control Scheme : � ˆ F e ( t ) : F m ( t ) = ˆ F e ( t − τ 2 ( t )) ; � C with controller gain K i , i = 1 , 2 , 3 : F s ( t ) = − K 1 ˙ τ 1 ( t )) − K 2 ˙ C : θ s ( t − ˆ θ m ( t − τ 1 ( t )) (22) − K 3 ( θ s ( t − ˆ τ 1 ( t ) − θ m ( t − τ 1 ( t ))) . 37 / 102

  26. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 2 - Force-Reflecting Control Scheme Control Scheme 2 - Slave Controller Design : � x 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 1 K 2 K 3 K ms = . (24) 0 0 0 ⇓ � = A 0 ms x ms ( t ) + A 1 x ms ( t ) ˙ 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 ms K ms = − B 1 ms K, K = . (26) K 1 K 2 K 3 38 / 102

  27. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  28. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 ) / ˆ F h ( t ) , the position tracking ; � From slave to master : F m ( t ) = ˆ F e ( t − τ 2 ( t )) , the force tracking ; 40 / 102

  29. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Slave Controller Description : � ’P’ : x p ( t ) = ( A m − B m K m ˙ 0 ) x p ( t ) − B m F p ( t ) (27) + B m ( ˆ τ 1 ( t )) + ˆ F e ( t − ˆ F h ( t − τ 1 ( t ))) , x p ( t ) = ˙ θ p ( t ) ∈ R n . 41 / 102

  30. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Slave Controller Description : � � � L = = ⇒ θ p ( t ) → θ m ( t ) : L 1 L 2 L 3 � � ˙ θ p ( t − ˆ τ 1 ( t )) F p ( t ) = L . (27) ˙ θ m ( t − τ 1 ( t )) θ p ( t − ˆ τ 1 ( t )) − θ m ( t − τ 1 ( t )) 41 / 102

  31. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Slave Controller Description : � � of the controller ¯ � K = K 1 K 2 K 3 C = ⇒ θ s ( t ) → θ p ( t ) : � � ˙ θ s ( t ) F s ( t ) = − K . (28) ˙ θ p ( t ) θ s ( t ) − θ p ( t ) 41 / 102

  32. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Master-Proxy Synchronization : � = A 0 mp x mp ( t ) + A 1 x mp ( t ) ˙ mp x mp ( t − τ 1 ( t )) + B mp w mp ( t ) , (29) z mp ( t ) = C mp x mp ( t ) .   ˙ � ˆ � θp ( t ) τ 1( t ))+ ˆ Fe ( t − ˆ Fh ( t − τ 1( t ))   , x mp ( t ) = ˙ w mp ( t ) = , θm ( t ) Fm ( t )+ Fh ( t ) θp ( t ) − θm ( t ) (30) � � z mp ( t ) = . θp ( t ) − θm ( t ) 42 / 102

  33. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Master-Proxy Synchronization : � A m − B m K m � � B m � � � 0 0 0 0 A 0 mp = , B mp = = , A m − B m K m B 1 mp B 2 0 0 0 B m 0 mp 0 0 1 − 1 0 � � C mp = , 0 0 1 � − B m L 1 − B m L 2 − B m L 3 � A 1 = − B 1 mp = mp L. 0 0 0 0 0 0 (31) 43 / 102

  34. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Proxy-Slave Synchronization : � x ps ( t ) ˙ = A ps x ps ( t ) + B ps w ps ( t ) , (32) z ps ( t ) = C ps x ps ( t ) ,   ˙ θs ( t ) � �   , x ps ( t ) = ˙ z ps ( t ) = θs ( t ) − θp ( t ) , θp ( t ) θs ( t ) − θp ( t ) (33) � � F e ( t ) w ps ( t ) = . ˆ τ 1 ( t )) + ˆ F e ( t − ˆ F h ( t − τ 1 ( t )) − F p ( t ) 44 / 102

  35. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Proxy-Slave Synchronization : � Bs � � � � � 0 B 1 ps B 2 B ps = = , C ps = , 0 Bm 0 0 1 ps 0 0 � As − BsKs � 0 − BsK 1 − BsK 2 − BsK 3 A ps = Am − BmKm 0 0 0 1 − 1 0 (34) � As − BsKs � � − BsK 1 − BsK 2 − BsK 3 � 0 0 0 Am − BmKm = + 0 0 0 0 0 0 0 0 0 1 − 1 0 = ( A 0 ps − B 1 ps K ) . . 45 / 102

  36. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Control Scheme 3 - Force-Reflecting Proxy Control Scheme Control Scheme 3 - Global Performance Analysis : � = A 0 mps x mps ( t ) + A 1 x mps ( t ) ˙ mps x mps ( t − τ 1 ( t )) + B mps w mps ( t ) , z mps ( t ) = C mps x mps ( t ) , (35)   ˙ θs ( t ) � � � � ˙  θp ( t )  Fe ( t )   θs ( t ) − θp ( t ) x mps ( t ) = w mps ( t ) = ˆ τ 1( t ))+ ˆ z mps ( t ) =  ˙  ,  Fe ( t − ˆ Fh ( t − τ 1( t )) , . θm ( t ) θp ( t ) − θm ( t )  θs ( t ) − θp ( t ) Fm ( t )+ Fh ( t ) θp ( t ) − θm ( t ) (36)   As − BsKs 0 − BsK 1 − BsK 2 0 − BsK 3 0  Am − BmKm  0 0 0 0  0  A 0 mps =  Am − BmKm  ,  0 0 0 0  0 1 − 1 0 0 0 0 1 − 1 0 0     0 0 0 0 0 Bs 0 0 � � 0 − BmL 1 − BmL 2 0 − BmL 3 0 0  Bm  A 1   , 0 0 0 1 0 mps = B mps =   , C mps = . 0 0 0 0 0 0 0 Bm 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (37) 46 / 102

  37. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 ; Time Delays Control Schema Position Tracking Force Tracking Constant Time-varying √ √ √ 1 √ √ √ √ 2 √ √ √ √ 3 � Force estimation/measure : 2 & 3 ; � Better performance, but additional computation load : 3 ; 47 / 102

  38. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  39. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Simulation Conditions : � Maximum amplitude of time-varying delays : 0 . 2 s ; � Master, proxy and slave models : 1 /s , 1 /s and 2 /s ; Poles : [ − 100 . 0] . K 0 m = 100 , K 0 s = 50 ; � � � K 1 = − 0 . 1870 − 0 . 0368 65 . 0846 , � � (38) γ C 1 /C 2 K 2 = 0 . 4419 0 . 0813 − 153 . 8704 , = 0 . 0123 . min � � � γ L L = − 1 . 4566 0 . 1420 282 . 482 , min = 0 . 0081 , � � (39) γ K K = − 29 . 9635 − 3 . 6393 618 . 536 , min = 0 . 0075 . Global stability of the system is verified with γ g min = 0 . 0062 ; 49 / 102

  40. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Abrupt Tracking Motion : 50 / 102

  41. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Abrupt Tracking Motion : 51 / 102

  42. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Wall Contact Motion : � Hard wall : the stiffness K e = 30 kN/m , the position x = 1 . 0 m ; � 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

  43. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Wall Contact Motion : 53 / 102

  44. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Wall Contact Motion : 54 / 102

  45. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Wall Contact Motion : � Force response in wall contact motion ( F m ( t ) ; ˆ F e ( t ) ). 55 / 102

  46. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels CONTROL SCHEMES Results and Analysis Wall Contact Motion under Large Delays h 2 = 1 . 0 s : 56 / 102

  47. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  48. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Discrete-Time System : � x ( k + 1) = � q i =0 A i x ( k − τ i ( k )) + Bw ( k ) , (40) z ( k ) = Cx ( k ) . Delay-Free Case : � x ( k + 1) = A 0 x ( k ) + Bw ( k ) , (41) z ( k ) = Cx ( k ) . 58 / 102

  49. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Robust Stability Condition : � Discrete LKF, y ( k ) = x ( k + 1) − x ( k ) : k − 1 k − 1 � � V ( x ( k )) = x ( k ) T Px ( k ) + x ( i ) T S a x ( i ) + x ( i ) T Sx ( i ) i = k − h 2 i = k − h 1 q − 1 k − 1 − h 1 − 1 k − 1 � � � � � y ( j ) T Ry ( j ) + y ( l ) T R ai y ( l ) . + h 1 ( h 2 − h 1 ) i = − h 1 j = k + i i =1 j = − h 2 l = k + j (42) � 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 )) : △ 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

  50. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  51. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach System Description : (Σ d x m ( k + 1) = ( A md − B md K 0 m ) md ) x m ( k ) + B md ( F m ( k ) + F h ( k )) , (Σ d x s ( k + 1) = ( A sd − B sd K 0 s ) sd ) x s ( k ) + B sd ( F s ( k ) + F e ( k )) , (Σ d x p ( k + 1) = ( A md − B md K 0 p ) md ) x p ( k ) − B md F p ( k ) + B md ( ˆ τ 1 ( k )) + ˆ F e ( k − ˆ F h ( k − τ 1 ( k ))) . (44) 61 / 102

  52. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Slave Controller Design - Master-Proxy Synchronization : � � � L d = L d 1 L d 2 L d 3 : � � ˙ θ p ( k − ˆ τ 1 ( k )) F p ( k ) = L d . (45) ˙ θ m ( k − τ 1 ( k )) θ p ( k − ˆ τ 1 ( k )) − θ m ( k − τ 1 ( k )) �  = A 0 mpd x mp ( k ) + A 1  x mp ( k + 1) mpd x mp ( k − τ 1 ( k ))  (Σ d mp ) (46) + B mpd w mp ( k ) ,   z mp ( k ) = C mpd x mp ( k ) , A 1 mpd = ⇒ L d ; 62 / 102

  53. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Slave Controller Design - Proxy-Slave Synchronization : � � � K d = K d 1 K d 2 K d 3 : � � ˙ θ s ( k ) F s ( k ) = − K d ˙ . (47) θ p ( k ) θ s ( k ) − θ p ( k ) � � x ps ( k + 1) = A psd x ps ( k ) + B K F s ( k ) + B psd w ps ( k ) , (Σ d ps ) (48) 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 ) , (¯ Σ d ps ) (49) z ps ( k ) = C psd x ps ( k ) . 63 / 102

  54. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Slave Controller Design - Global Performance Analysis : � = A mpsd x ( k ) + B K mpsd F s ( k ) − B L x ( k + 1) mpsd F p ( k ) + B mpsd w ( k ) , (Σ d mps ) (50) = C mpsd x ( k ) , z ( k ) � � F s ( k ) = − ¯ K d x ( k ) = − x ( k ) , K d 1 K d 2 0 K d 3 0 � � F p ( k ) = ¯ L d x ( k − τ 1 ( k )) = x ( k − τ 1 ( k )) , 0 L d 1 L d 2 0 L d 3 (51) � B sd � � � 0 B md 0 B K B L mpsd = , mpsd = , 0 0 0 0 0 0 ⇓ � = A d 0 x ( k ) + A d x ( k + 1) 1 x ( k − τ 1 ( k )) + B mpsd w ( k ) , (¯ Σ d mps ) (52) = C mpsd x ( k ) , z ( k ) mpsd ¯ mpsd ¯ A d 0 = A mpsd − B K K d , A d 1 = − B L L d . 64 / 102

  55. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  56. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Simulation Conditions : � T = 0 . 001 s , h 1 = 1 , h 2 = 100 (in continuous-time domain, h 1 = 0 . 001 s , h 2 = 0 . 1 s ) ; � � � γ L d L d = 4 . 6815 − 5 . 1390 540 . 7828 , min = 0 . 0051 , � � (53) γ K d min = 2 . 9568 × 10 − 4 , K d = 273 − 127 10961 , γ g min = 0 . 0327 . 66 / 102

  57. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Abrupt Tracking Motion : ♥ ♥ 67 / 102

  58. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Abrupt Tracking Motion : ♥ ♥ 67 / 102

  59. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Abrupt Tracking Motion : 68 / 102

  60. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Wall Contact Motion : 69 / 102

  61. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS Discrete-Time Approach Wall Contact Motion : 70 / 102

  62. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  63. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties System Description : x m ( t ) = ( A m ( ρ m ( t )) − B m ( ρ m ( t )) K 0 (Σ m ) ˙ m ) x m ( t ) + B m ( ρ m ( t ))( F m ( t ) + F h ( t )) , x s ( t ) = ( A s ( ρ s ( t )) − B s ( ρ s ( t )) K 0 (Σ s ) ˙ s ) x s ( t ) + B s ( ρ s ( t ))( F s ( t ) + F e ( t )) , x p ( t ) = ( A m ( ρ p ( t )) − B m ( ρ p ( t )) K 0 m ) x p ( t ) + B m ( ρ p ( t ))( ˆ τ 1 ( t )) + ˆ (Σ p ) ˙ F e ( t − ˆ F 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

  64. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties Controller Design - Problem 1 : K 0 m & K 0 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

  65. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties Problem 2 Slave Controller Design � Master-proxy synchronization : � = A 0 mp ( ρ mp ( t )) x mp ( t ) + A 1 x mp ( t ) ˙ mp ( ρ mp ( t )) x mp ( t − τ 1 ( t )) + B mp ( ρ mp ( t )) w mp ( t ) , z mp ( t ) = C mp x mp ( t ) , (55) K 0 m & K 0 ⇒ A 0 A 1 s = mp ( ρ mp ( t )) ; mp ( ρ mp ( t )) = ⇒ L ; � Proxy-slave synchronization : � x ps ( t ) ˙ = A ps ( ρ ps ( t )) x ps ( t ) + B ps ( ρ ps ( t )) w ps ( t ) , (56) z ps ( t ) = C ps x ps ( t ) , A ps ( ρ ps ( t )) = ⇒ K ; � Global performance analysis with K 0 m & K 0 s , L & K :  = A 0 mps ( ρ mps ( t )) x mps ( t ) + A 1  x mps ( t ) ˙ mps ( ρ mps ( t )) x mps ( t − τ 1 ( t ))  (57) + B mps ( ρ mps ( t )) w mps ( t ) ,   z mps ( t ) = C mps x mps ( t ) . 74 / 102

  66. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Polytopic-Type Uncertainties Remark - Nonlinear Systems : M m ( θ m )¨ θ m ( t ) + C m ( θ m , ˙ θ m ) ˙ (Σ m ) θ m ( t ) + g m ( θ m ) = F h ( t ) + F m ( t ) , M m ( θ p )¨ θ p ( t ) + C m ( θ p , ˙ θ p ) ˙ θ p ( t ) + g m ( θ p ) = ˆ τ 1 ( t )) + ˆ (Σ p ) F e ( t − ˆ F h ( t − τ 1 ( t )) − F p ( t ) , M s ( θ s )¨ θ s ( t ) + C s ( θ s , ˙ θ s ) ˙ (Σ s ) θ s ( t ) + g s ( θ s ) = F e ( t ) + F s ( t ) , (58) ⇓ (¯ ¨ θ m ( t ) = A m ( θ m , ˙ θ m ) ˙ Σ m ) θ m ( t ) + B m ( θ m )( F h ( t ) + F m ( t ) − g m ( θ m )) , (¯ θ p ( t ) = A m ( θ p , ˙ ¨ θ p ) ˙ θ p ( t ) + B m ( θ p )( ˆ τ 1 ( t )) + ˆ Σ p ) F e ( t − ˆ F h ( t − τ 1 ( t )) − F p ( t ) − g m ( θ p )) , (¯ θ s ( t ) = A s ( θ s , ˙ ¨ θ s ) ˙ Σ s ) θ s ( t ) + B s ( θ s )( F e ( t ) + F s ( t ) − g s ( θ s )) , (59) θ m ) = − M − 1 B m ( θ m ) = M − 1 A m ( θ m , ˙ m ( θ m ) C m ( θ m , ˙ θ m ) , m ( θ m ) , θ p ) = − M − 1 B m ( θ p ) = M − 1 A m ( θ p , ˙ m ( θ p ) C m ( θ p , ˙ (60) θ p ) , m ( θ p ) , A s ( θ s , ˙ θ s ) = − M − 1 ( θ s ) C s ( θ s , ˙ B s ( θ s ) = M − 1 θ s ) , ( θ s ) . s s 75 / 102

  67. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  68. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties System Description : x m ( t ) = (( A m + ∆ A m ( t )) − ( B m + ∆ B m ( t )) K 0 (Σ m ) ˙ m ) x m ( t ) + ( B m + ∆ B m ( t ))( F m ( t ) + F h ( t )) , x s ( t ) = (( A s + ∆ A s ( t )) − ( B s + ∆ B s ( t )) K 0 (Σ s ) ˙ s ) x s ( t ) + ( B s + ∆ B s ( t ))( F s ( t ) + F e ( t )) , (61) x p ( t ) = (( A m + ∆ A p ( t )) − ( B m + ∆ B p ( t )) K 0 (Σ p ) ˙ m ) x p ( t ) − ( B m + ∆ B p ( t )) F p ( t ) + ( B m + ∆ B p ( t ))( ˆ τ 1 ( t )) + ˆ F e ( t − ˆ F h ( t − τ 1 ( t ))) , ∆( t ) T ∆( t ) � I. (62) i = { m, s, p } : ∆ A i ( t ) = G i ∆( t ) D i , ∆ B i ( t ) = H i ∆( t ) E i , 77 / 102

  69. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties System Description : x m ( t ) = (( A m + ∆ A m ( t )) − ( B m + ∆ B m ( t )) K 0 (Σ m ) ˙ m ) x m ( t ) + ( B m + ∆ B m ( t ))( F m ( t ) + F h ( t )) , x s ( t ) = (( A s + ∆ A s ( t )) − ( B s + ∆ B s ( t )) K 0 (Σ s ) ˙ s ) x s ( t ) + ( B s + ∆ B s ( t ))( F s ( t ) + F e ( t )) , (63) x p ( t ) = (( A m + ∆ A p ( t )) − ( B m + ∆ B p ( t )) K 0 (Σ p ) ˙ m ) x p ( t ) − ( B m + ∆ B p ( t )) F p ( t ) + ( B m + ∆ B p ( t ))( ˆ τ 1 ( t )) + ˆ F e ( t − ˆ F h ( t − τ 1 ( t ))) , ∆( t ) T ∆( t ) � I. (64) i = { m, s, p } : ∆ A i ( t ) = G i ∆( t ) D i , ∆ B i ( t ) = H i ∆( t ) E i , Slave Controller Design Solution : Transformation from norm-bounded uncertain system to linear time-delay system ; 78 / 102

  70. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Problem 2 Slave Controller Design - Master-Proxy Synchronization :  = ( A 0 mp + ∆ A 0 mp ( t )) x mp ( t ) + ( A 1 mp + ∆ A 1  x mp ( t ) ˙ mp ( t )) x mp ( t − τ 1 ( t ))  +( B mp + ∆ B mp ( t )) w mp ( t ) , (65)   z mp ( t ) = C mp x mp ( t ) , K 0 m & K 0 ⇒ A 0 mp + ∆ A 0 s = mp ( t ) ; ⇓ ϕ p ( t ) = (∆ A p ( t ) − ∆ B p ( t ) K 0 m ) ˙ θ p ( t ) + ∆ B p ( t )( ˆ τ 1 ( t )) + ˆ F e ( t − ˆ F h ( t − τ 1 ( t ))) , ϕ m ( t ) = (∆ A m ( t ) − ∆ B m ( t ) K 0 m ) ˙ (66) θ m ( t ) + ∆ B m ( t )( F m ( t ) + F h ( t )) , µ p ( t ) = − ∆ B p ( t ) Lx mp ( t − τ 1 ( t )) . � = A 0 mp x mp ( t ) + A 1 mp x mp ( t − τ 1 ( t )) + � x mp ( t ) ˙ B mp � w mp ( t ) , (67) z mp ( t ) = C mp x mp ( t ) , � � � 1 0 � Bm ˆ τ 1( t ))+ Bm ˆ Fe ( t − ˆ Fh ( t − τ 1( t ))+ ϕp ( t )+ µp ( t ) � w ( t ) = � , B mp = , (68) 0 1 BmFm ( t )+ BmFh ( t )+ ϕm ( t ) 0 0 A 1 mp = ⇒ L ; 79 / 102

  71. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Slave Controller Design - Proxy-Slave Synchronization : � x ps ( t ) ˙ = ( A ps + ∆ A ps ( t )) x ps ( t ) + ( B ps + ∆ B ps ( t )) w ps ( t ) , (69) z ps ( t ) = C ps x ps ( t ) , A ps + ∆ A ps ( t ) = ⇒ K ; ⇓ ϕ s ( t ) = (∆ A s ( t ) − ∆ B s ( t ) K 0 s ) ˙ θ s ( t ) + ∆ B s ( t ) F e ( t ) , (70) µ s ( t ) = − ∆ B 1 ps ( t ) Kx ps ( t ) . � = A ps x ps ( t ) + � x ps ( t ) ˙ B ps � w ps ( t ) , (71) z ps ( t ) = C ps x ps ( t ) , � � � 1 0 � BsFe ( t )+ ϕs ( t )+ µs ( t ) � w ps ( t ) = � , B ps = , 0 1 Bm ˆ τ 1( t ))+ Bm ˆ Fe ( t − ˆ Fh ( t − τ 1( t )) − BmFp ( t )+ ϕp ( t )+ µp ( t ) 0 0 (72) A ps = ⇒ K ; 80 / 102

  72. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Slave Controller Design - Global Performance Analysis :  = ( A 0 mps + ∆ A 0 mps ( t )) x mps ( t ) + ( A 1 mps + ∆ A 1  x mps ( t ) ˙ mps ( t )) x mps ( t − τ 1 ( t ))  (73) +( B mps + ∆ B mps ( t )) w mps ( t ) ,   z mps ( t ) = C mps x mps ( t ) . ⇓ � = A 0 mps x mps ( t ) + A 1 mps x mps ( t − τ 1 ( t )) + � x mps ( t ) ˙ B mps � w mps ( t ) , (74) z mps ( t ) = C mps x mps ( t ) ,   � � 1 0 0 BsFe ( t )+ ϕs ( t ) 0 1 0 Bm ˆ τ 1( t ))+ Bm ˆ �   . w mps ( t ) = � Fe ( t − ˆ Fh ( t − τ 1( t ))+ ϕp ( t ) , B mps = (75) 0 0 1 0 0 0 BmFm ( t )+ BmFh ( t )+ ϕm ( t ) 0 0 0 81 / 102

  73. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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

  74. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels 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 , 1 B m ( ρ m ( t )) = B s ( ρ s ( t )) = , ρ ( t ) ∈ [0 . 5 , 1] , ρ ( t ) (76) 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 } . � K 0 K 0 polytopic − type : m = 2 . 6585 , s = 2 . 6585 , (77) K 0 K 0 norm − bounded − type : m = 9 . 5117 , s = 3 . 3812 . � � � γ L polytopic − type : L = 1 . 3218 − 1 . 3219 6 . 3602 , min = 0 . 4436 , � � γ K K = min = 0 . 0164 , 20 . 4799 − 21 . 2537 575 . 2051 , γ g min = 0 . 4595 , (78) � � γ L norm − bounded − type : L = 1 . 6218 − 1 . 6264 29 . 5012 , min = 0 . 05 , � � γ K K = 16 . 4993 − 12 . 0059 434 . 4988 , min = 0 . 0072 , γ g min = 0 . 0376 . 83 / 102

  75. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Abrupt Tracking Motion : 84 / 102

  76. New Control Schemes for Bilateral Teleoperation under Asymmetric Communication Channels ROBUSTNESS H ∞ Robust Teleoperation under Time-Varying Model Uncertainties - Norm-Bounded Model Uncertainties Wall Contact Motion : 85 / 102

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