Μια προσπαθεια για την επιτευξη ανθρωπινης επιδοσης σε ρομποτικές εργασίες με νέες μεθόδους ελέγχου

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1 Μια προσπαθεια για την επιτευξη ανθρωπινης επιδοσης σε ρομποτικές εργασίες με νέες μεθόδους ελέγχου Towards Achieving Human like Robotic Tasks via Novel Control Methods Zoe Doulgeri Department of Electrical and Computer Engineering Aristotle University of Thessaloniki 54124, Thessaloniki, GREECE

2 Current Trends in Robotic System Development Robotics Research Goal Build robots to assist humans, navigate around human spaces, deal with human tools, do bimanual manipulation, deal with deformable objects, manipulate objects blindly etc. with human like dexterity. Implementation of soft components at various levels materials sensing actuation

3 Soft actuation Automation and Robotics Lab. Low stiffness Low gear ratios High back drivability Associated with uncertainties Robot Inertias, couplings, nonlinearities Load and external force disturbances adversely affect performance (speed and accuracy) Associated with increased model complexity increased control complexity Variable stiffness actuators VIA, antagonistic muscles or tendons Can we guarantee prescribed, time dependant performance bounds in robot motion through simple control algorithms??

4 Control objective Automation and Robotics Lab. Design a simple controller that guarantees Prescribed Performance In terms of: Maximum overshoot Minimum speed of convergence Maximum steady state error Irrespective of (not affected by): Model uncertainties Bounded disturbances So far robot control solutions guarantee stability and convergence but not prescribed transient performance Joint or end-effector position error? yes Contact force error? Actuator stiffness? yes I do not know yet

5 0 Automation and Robotics Lab. r() t lt ( r r ) e - = - + r max steady state Max initial error ρ 0 > e(0) Prescribed Performance min speed of convergence lim r( t) = r > 0 t error Overshoot 0 M 1 - Mr() t < et () < r() t Performance bound functions ( ), ρ( ) Mρ t t - r() t < et () < Mr() t

6 p ( ) e = Q p p d Contact Maintenance Automation and Robotics Lab. e = f f f d f = f ( χ ) > 0 χ > 0 0 χ = 0 - lt r() t = ( r - r ) e + r 0 f() t > 0 T T T = t t p p φ ( ) ( ) ( ) M ρ t < f t f t f f d - Mr() t < et () < r() t e f ( ) 0 0 ( ) ( ) M ρ t < f t f f d OK for M f = 0 e f ( ) 0 0 f ( t) f ( t) f ( t) ρ < d - r() t < et () < Mr() t f ( t) f ( t) ρ < d

7 Basic Idea: Error Transformation Modulate the position error using the performance function Define a bijective map of the performance region: ( M,1) or ( 1, M ) to (, ) e et ( ) r ( t) ζet ( ) φ ( t) = T ηθr ( t) χψ - et ( ) M < r ( t) < 1 - ( ) 1 < et r ( t) < M ζ e φ M ζeφ + r T ln in case e a = ( 0) ³ 0 ηr θ χψ e η 1- ηθ r χψ ζ e φ 1 ζeφ + r T ln in case e a = ( 0) 0 ηr θ χψ e ηm - ηθ r χψ The inverse transformation exists and it is bounded - T 1 () e < 1 IEEE TAC 2008

8 or a shifted transformation such that T (0) = 0 b ζ e φ M ζeφ + r ln η in case e( 0) 0 r χ= ³ ηθ χψ ζ eφ M 1 - η r θ ηθ χ χψ e( 0) ³ 0 ( ) e 0 0 b ζ e ζ φ M 1 + ζeφ r ηθ ψχ χ ln η in case e( 0) 0 r χ= ηθ χψ e M η - r ηθ χψ Boundedness of ε(t) achieves PP for e(t). ( ) ( t) et - M < < r e > 4 ( M + ) 1 1 r () t e ( ) ( t) et - 1 < < M r ICRA10

9 in a multi dof system e i ( t) ( ) ( t) ζe t φ i = T η ηθr χψ J Ti ( e r) i 1 > r 0 T = ι ω κ K 1 n ϊ e() t e e λ ϋ - lt r() t = ( r - r ) e + r 0 ( ei ) ρ 0 > max (0) & e= J () t ιe+ a() teω T κ& λ ϊϋ J () t = diag ιj J T κ K ω λ T 1 Tn ϊ ϋ a() t & r () r () t 0 < a() t < l lim a( t) = 0 t

10 v Automation and Robotics Lab. Robot Joint Position Regulation H() qq& + C(, qqq & )& + Cq& + gq () + F() t = ut () Ft () Any bounded model error or disturbance qd = const Typical regulator structure: Setting: u=-ke- Kq& - v v=- gq () or v=-gq ( d ) e= q- q d p Positive Definite Diagonal Matrices v Conventional PD controller v = K e( t ) dt I ς v = K y ( t ) dt 0 I ς 0 0 t 0 y = & q+ ase () t PID controller with local asymptotic stability PID controller with global asymptotic stability Saturation function U( q) - U( q) - e gq ( ) ³ - c e T d d g 2 T e ( gq ()- gq ( ) d ) ³ - c e g 2

11 Prescribed Performance Regulator TP-PID t =- - &- - p v T Iς0 u Ke Kq KJ ()() tet K y( t ) dt e K, K, K, K k, k, k, k p v e I p v e I Diagonal positive definite gain matrices Minimum diagonal entries TP - term - 1 yt () = & q+ k() te= J () t & e k() t = a() t + b b b 0 T ζeφ in case of T η ηθr χψ ζeφ in case of T η ηθr χψ = a η χ > 0 b η χ SYROCO 09 ICRA10

12 PP Regulation Stability Result Using TP-PID control law with a choice of gains satisfying k v p It is proved that > 2c g ( ) H H 0 0 k > ( l + b) l + r nc + 2l 4 l l C(, qab ) c a b 0 ( ) = max ι κλ l ( ) ω ϊϋ H q M Hq (a) all signal in the closed loop are bounded (b) Position error remains in the performance region at all times without even approaching its boundary, hence prescribed performance is guaranteed - Mr() t < et () < r() t - r() t < et () < Mr() t (c ) the joint velocity asymptotically converge to zero & q 0 e 0 (d) the error asymptotically converge to zero Provided transformation T b (.) is used in all joints and disturbance F is constant

13 Remarks on Prescribed Performance Regulator TP-PID t =- - &- - p v T Iς0 u Ke Kq KJ ()() tet K y( t ) dt e Structure Gain tuning TP - term I - term Omitting the I - term Simple PID-type regulator with minimum robot information required to satisfy theorem conditions Significantly simplified as it is not related to performance Choose values that lead to reasonable input torques Is responsible for the prescribed performance stability result Compensates for any bounded constant disturbance and hence Is responsible for the asymptotic convergence of the joint velocity and error to zero Prescribed performance of joint error is guaranteed as well as the uub of joint velocity Omitting the K p - term Possible when using the shifted transformation T b, replacing the lower bound of K p with a K ε lower bound.

14 Simulation Example Automation and Robotics Lab. masses lengths Inertias Initial joint positions

15 PID Simulation Results Regulation task Set-point q = d T [ ] deg T q (0) = [ ] deg Overshoot Index M = 0.1 M = 0 PID gains K = 50I p 3 K = 5 I, K = di ag[2, 20, 2] v 3 I Performance function r( t) = ( r - 10 ) e t p 2e 0 0 i 3 r = =

16 PID state and gain sensitivity Regulation task PID Reduce K p from Change initial configuration 0 q(0) = q(0) = 60 40

17 PID robustness Constant disturbance

18 PID robustness Time varying disturbance

19 TP-PID Regulation Task Automation and Robotics Lab. T () Χ a TP-PID t =- - &- - p v T Iς0 u Ke Kq KJ ()() tet K y( t ) dt e Ke = 0.01I b = 1 With the original transformation zero overshoot can be prescribed 3 T () Χ b With the shifted transformation convergence to zero allows a wider steady state performance band to cater for noisy measurements

20 TP-PID Gain sensitivity K p = 20I 3 TP-PID Gain tuning is significantly simplified T () Χ a T () Χ b

21 TP-PID Constant disturbances T () Χ a T () Χ b

22 TP-PID Time-varying disturbances T () Χ a T () Χ b With the shifted transformation choice we still get practical convergence despite time varying disturbances Is such a performance and robustness achieved with increased control effort? no

23 Simulation Input torques Automation and Robotics Lab. TP-PID Comparable control effort with PID T () Χ T () Χ b a PID

24 Can we achieve the same results in a trajectory tracking task keeping such a simple control structure? yes

25 Prescribed Performance Tracking q ()Ξ t C 2 q, & q, q& Ξ L d d d d Ft () TP-PD u =-K e- K & e- K J ()() t e t p v e T using the shifted transformation e () t i ζ e φ T χ η ηθr () t ψχ i = b χ Using TP-PD control law with a choice of gains satisfying ( ) 2 1 k > ( l + b) l + r nc + l l + ( c v c 2 v 2 ) v H 0 0 H 0 d 0 d 4 2 It is proved that (a) Position error remains in the performance region at all times without even approaching its boundary, hence prescribed performance is guaranteed - Mr() t < et () < r() t - r() t < et () < Mr() t (b) e( t) e& are uniformly ultimately bounded with respect to a set involving control gains and system constants

26 Simulation TP-PD Tracking (1) Desired and output trajectories Desired positions (dotted lines) ι5 ω ι30ω q = (1 cos(2 pt )) deg d + - κ15ϊ κ40 λ ϋ λ ϊϋ Gains K = 5 I, K = 50I, K = 0.1I v 3 p 3 e 3 Tracking errors Overshoot Index M = 0.1 r( t) = ( r - 10 ) e + 10 r Performance function 0 p = 2e = 0 0i t - 2

27 Simulation TP-PD Tracking (2) Velocity errors Input torques Disturbances

28 Can a model based controller be endowed with prescribed performance guarantees? yes

29 Model based control structures endowed with prescribed performance guarantees Reference Velocity Model based control structure (Slotine&Li) Parameter update laws Ft () F & q = & q - ae r d u= Z(, qqq &,&, q& )() q t - kv- Ds r & q () t =-G r r p Z(,, qqq & &, q& ) q = H() qq& + C(, qqq & )& + gq () r r r r s = & q- & q = & e+ ae T Z s Γ>0 - - b - J 1 T K q e + b - J 1 T e Conventional Controller 0 < α const v = 0 Prescribed Performance Controller ρ() α = t ρ() t v = J T ε MED09

30 Reference Velocity Model based control structure Adaptive laws for Conventional Controller 0 < α, β const v v p f = e f Automation and Robotics Lab. Model based control structures endowed with prescribed performance guarantees ( α ) ˆ χ β( χ ˆ χ ) ( ) p = Q p e + n r d p d d T T u = Mqr + Cqr + Fq + g + J nfd + J QF Ds k J nv k J Qv T T q f f p p ( d β d)( f f f ) hˆ = γ f + f e + kv Prescribed Performance Controller p = ep vp = JTpε p 1 ( f = k, ) uncertainties sχ h= ks Parametric uncertainty Structural uncertainty ( ) ρ p () t α = ρ f () t β = ρ () t ρ () t v = J ε f Tf f f ICRA09 f( χ) Z χ θ MSC09 f ( χ) = T f f = f ( ) χ χ ( ) T f( χ) = θ Z χ + w ( χ) f f f IEEE Trans. NN 2010

31 Is it possible to design a prescribed performance guaranteeing controller that is completely model knowledge free? Yes First results in MED10 More in upcoming publications

32 One degree of freedom Experimental setup Maxon Motor's analogue ADS Servoamplifier 50/5 in current mode (internal current loop) link length: 15 cm total mass: 145 g Dc motor (Faulhaber CR) equipped with an incremental encoder and a low rate reduction gear box (1:14),

33 Experiment Regulation Automation and Robotics Lab. PID controller t u k ι ω =- T( & q+ ke) + ( q+ kedt ) PID v v p 0 p κ ς & λ ϊϋ k = 9.5, k = , T = p v v TP-PID controller k = 0.02, k = 0.9, k = 10, k = 1 e v p I b = 1 Performance bounds M = 0.15 for T b ( ) r - 14t ( t) = (2-0.07) e Convergence in less than 0.4 s Less than 5% of the setpoint error

34 Desired trajectory q d =- 30 cospt deg Control gains k 0.5, k v p 2, k e Performance bounds r () t = ( - ) e + M = 1 p p - 4t p Experiment Tracking Symmetric performance envelope Automation and Robotics Lab.

35 Conclusions & Future Application Prescribed performance controllers incorporate performance quality constraints via an error transformation Prescribed performance controllers guarantee transient and steady state in complex nonlinear uncertain robotic systems Prescribed performance controllers can be model free Prescribed performance controllers have been applied in robot position regulation and tracking and in robot force/position tracking guaranteeing contact maintenance Future Applications Stiffness performance guarantee Rolling motion guarantee

36 Publications Automation and Robotics Lab. C. P. Bechlioulis and G. A. Rovithakis, Robust Adaptive Control of Feedback Linearizable MIMO Nonlinear Systems with Prescribed Performance, IEEE Transaction on Automatic Control, vol. 53, no. 9, pp , Z. Doulgeri, O. Zoidi, Prescribed Performance Regulation for robot manipulators, 9th IFAC Symposium on Robot Control (SYROCO 09), Gifu, Japan, September 9-12, 2009, pp Z. Doulgeri, Y. Karayiannidis, PID Type Robot Joint Position Regulation with Prescribed Performance Guaranties, 2010 IEEE International Conference on Robotics and Automation, ICRA 2010, May 3-8, 2010, Anchorage Alaska, USA, pp Zoe Doulgeri and Leonidas Droukas, Robot Task Space PID type regulation with prescribed performance guaranties, The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2010), Taipei, Taiwan, October 18-22, 2010, pp Z. Doulgeri, Y. Karayiannidis, O. Zoidi, Prescribed Performance control for robot joint trajectory tracking under parametric and model uncertainties, 17th Mediterranean Conference on Control and Automation (MED09), Thessaloniki, Greece, June 24-26, 2009, pp Ch. Bechlioulis, Z. Doulgeri and G. Rovithakis, Robot Force/Position Tracking with guaranteed Prescribed Performance, IEEE International Conference on Robotics and Automation, ICRA 2009, May 12-17, 2009, Kobe, Japan, pp Ch. Bechlioulis, Z. Doulgeri and G. Rovithakis, Prescribed Performance Adaptive Control for Robot Force/Position Tracking, 2009 IEEE International Symposium on Intelligent Control, Part of 2009 IEEE Multi-conference on Systems and Control, MSC09, July 8-10, 2009, Saint Petersburg, Russia, pp Ch. Bechlioulis, Z. Doulgeri and G. Rovithakis, Neuro-Adaptive Force/Position Control with Prescribed Performance and Guaranteed Contact Maintenance, IEEE Transactions on Neural Networks, 2010 Ch. Bechlioulis, Z. Doulgeri and G. Rovithakis, Model Free Force/Position Robot Control with Prescribed Performance, 18th Mediterranean Conference on Control and Automation (MED10), Marrakesh, Morocco, June 23-25, 2010, pp

37 THANK YOU FOR YOUR ATTENTION Zoe Doulgeri Department of Electrical and Computer Engineering Aristotle University of Thessaloniki 54124, Thessaloniki, GREECE

38 Talk Outline Automation and Robotics Lab. Current Trends in Robotic System Development-Soft Robotics Prescribed Performance-Basic Idea PP Model free Joint Position Regulation & Tracking Model based control structures endowed with prescribed performance guarantees Experimental Results Conclusions and Future Work