Passive Control of Mechanical Systems

Transcription

1 1 Perry Y. Li Deprtment of Mechnicl Engineering University of Minnesot Minnepolis MN

2 2 Outline Introduction / Motivtion Pssive Velocity Field Control (PVFC) problem PVFC design methodology Properties of PVFC Appliction to contour following problems Conclusions

3 3 Introduction Control for mechnicl systems (e.g. robots) trditionlly consists of: trnslting tsk into desired timed desired trjectory qd: R + G designing of feedbck control to trck the desired trjectory q( t) q ( t) d trjectory plnner q d (t) Torque T q, dq/dt time, t controller

4 4 Timed trjectory pproch my not be the most nturl for mny pplictions: Single minded Does not tke into ccount current stte of the system trjectory plnner q d (t) Torque T q, dq/dt controller

5 5 Contour following Importnt in mchining (cutting, deburring, etc) Timed trjectory convenient wy to prmeterize contour Exct timing is not very importnt Stying on contour is more criticl thn keeping pce with trjectory trcking errors do not reflect contouring errors trjectory trcking controllers cn cuse system to devite more from desired contour => Rdil Reduction Plnning should tke into ccount current configurtion Desired position trcking error Actul pos x x contour error Contour following Coordintion

6 6 Interction with Environments Mny pplictions involve interctions with the physicl environments, environments properties nd geometries my not be well chrcterized environment cn be delicte New pplictions involve humn in the loop e.g. robotic surgery, humn extenders, hptic devices, exercise mchines Mchine should not injure humn opertor or dmge environment stble nd sfe interction Timed trjectory control pproch will ttempt to single mindedly trck trjectory potentilly unsfe

7 Pssive Control System 7 Definition: A system is pssive w.r.t. the mechnicl power if for ny environment force, nd t ny time t, c 2 t 0 T F ( τ) q ( τ) dτ c e F e ( ) where is the vlue of (energy) storge t t = 0. 2 F T e q F e T Controller Mnipultor q Environment Direction of NET energy flow

8 8 To ensure sfe interction with environment: Requires feedbck system to be pssive!! F e Controller Mnipultor Environment q Direction of NET energy flow energy in system is limited by energy put in by environment previously most environments re themselves strictly pssive (dissiptive) Smll gin theorem (L-2 norm) gurntees stbility interconnection between pssive nd strictly pssive system is lwys stble

9 9 Open loop mechnicl systems hve well known pssivity! property! F e! t! T 2! Fe ( τ) q ( τ) dτ c Mnipultor Environment 0!! q! Closed loop mechnicl systems re generlly NOT pssive timed trjectory control pproches re NOT pssive ctively genertes energy to overcome obstcles F e Controller Mnipultor Environment q

10 10 A new control prdigm for mechnicl systems: design controllers such tht closed loop system is pssive use behvioril pproch to encode tsk so tht current stte of the system is reflected encode tsk using velocity field...

11 Tsk Encoding Vi Velocity Fields 11 Use specifiction more tuned to coordintion problems A velocity field V :G T G d ssigns for ech position, velocity vector Tsk is ccomplished by trcing the rrows: Time is not used explicitly ( ) V : q V ( q) = (q, q) d q ( t) = β V ( q( t)) d d

12 12 Plnt: M( q) q + C( q, q ) q = T + F e T : Control (generlized) force F e : environment (generlized) force C( q, q ) q : Coriolis forces M( q) : inerti mtrix T F e Controller Mnipultor ( q, q ) Environment

13 13 Pssive Velocity Field Control Problem (PVFC) Control Problem: Given desired velocity field V :G T G d, find feedbck controller for s.t. F e ( ) The feedbck system is PASSIVE : i.e. for ny nd t, T t 0 T F ( τ) q ( τ) dτ c e 2 If environment force, velocity converges to scled multiple of : F ( e t ) = 0 t 0 V d q β Vd ( q( t)) β depends on current energy in the system

14 14 Motivtions for P.V.F.C. Explicitly ccount for environment interction Enhnce stbility nd sfety importnt for interction with frgile objects: e.g. humn useful for rehbilittion nd exercise mchines, surgicl robots etc. Mke desired motion dependent on current stte of system Decouples tsk specifiction (e.g. trcing out contour) from speed of trversl (determined by energy level in the system)

15 P.V.F. Control Design Kinetic energy of robot vries when q ( t) = α V ( q( t)) T K. E.( t) = α 2 Vd ( q( t)) M( q( t)) Vd ( q( t)) 2 d 15 K.E.(t) Control injects energy here t Control hs to inject energy into system to trck desired velocity field environment my extrct n infinite mount of energy from robot hence possibly NOT pssive

16 16 P.V.F. Control Design Min ide: Mke controller look like n energy storge element (e.g. flywheel) controller must be dynmic Mkes sure tht if controller INJECTS energy to mechnicl system, controller energy storge LOSES equl mount. Flywheel F e controller Mnipultor q Environment Pssive feedbck system

17 17 STEP I: Incorporte energy storge in controller dynmics Fictitious flywheel dynmics: Augmented system dynmics: M( q) 0 q C( q, q ) T Fe n 0 M 1 2 q + 0 = T + n q Let q = n S 1 q +1 G : = G Kinetic energy of ugmented system: n 1 M2q + = Tn +1 M ( q ) q + C ( q, q ) q = T + F 1 2 e T T n+ 1 2 κ ( q ) = q M ( q ) q = q M( q) q + M2( q ) mnipultor flywheel

18 18 STEP II: Define desired velocity field for ugmented system V V :G TG d d Vd = n T V d +1 G Augmented desired V.F. is followed implies originl V.F. is followed: q α V q α V d d V d Kinetic energy of ugmented system is constnt when is followed: 1 T κ ( V d ) = V d M ( q ) Vd = E = constnt > 0 2 E Choose lrge enough nd solve for V d n+1 q E energy

19 19 STEP III: Define coupling torque T M ( q ) q + C( q, q ) q = T + F e in If T = Ω( q, q ) q nd Ω( q, q ) R n n ( ) ( ) is skew symmetric, i.e. T Ω( q, q ) = Ω ( q, q ) Then, the closed loop system would be pssive coupling torque re-distributes energy energy will in fct be conserved

20 Find suitble Ω( q, q ) Differentil forms re skew symmetric, so choose to be differentil 2-form Ω( q, q ) 20 Define desired momentum field ctul momentum p Covrint derivtive of the desired momentum field P = M V = M q V w P M q q q C q q V = q = ( ) + (, ) this turns out to be the inverse dynmics function for the ugmented system trcking the desired ugmented velocity field d

21 Define Ω( q, q ) = P w E γ 2 { P p } 21 In coordintes, where Ω( q, q ) = G( q, q ) + γ R( q, q ) 1 G( q q E w P, ) = 2 is positive gin constnt. [ T P w T ] [ T T ] R( q, q ) = p P P p γ T = q Ω( q, q ) Control: T = G( q q ) q γ R( q q ) q,,

22 Convergence Properties 22 F e q Closed loop system ( ) is pssive ; F ( e t ) = 0 t 0 If, velocity converges exponentilly to scled multiple of the desired velocity field: d dt K E F T.. q = e q sgn( γ ) β V ( q ( t)) d nd q sgn( γ ) β V ( q( t)) d where β 2 = K. E. 2E nd γ = feedbck gin Remrks: control lw distributes energy between robot nd flywheel so tht multiple of the desired velocity field is followed { } Pth q ( t): t for free response is invrint to scling of 0 q ( t = 0)

23 Pth Invrince Property 23 Closed loop dynmics: M ( q ) q + Y( q, q ) q = Y ( q, q ) = C ( q, q ) Ω( q, q ) where is liner in F e q So, if initil condition ( q ( 0), q ( 0)) ( q ( 0), α q ( 0)) Then, t q ( t) q α

24 Pth Invrince 24 Hence, the pth trced out is invrint to scling in initil velocity decomposes the execution of tsk (pth), from the speed of execution; perhps more nturl for humn to interct with This is not so for trjectory bsed pproches.

25 Robustness Properties 25 Decompose environment force into components prllel nd norml to the desired momentum: where P = M( q ) Vd( q ) F ( t) = δ( t) P + F ( t) e e is the desired momemtum, nd F e ( t), V = 0 d Let e ( t) = q β( t) V ( q ( t)) β d where β( t) = sgn( γ ) K. E.( t) E 1 2

26 Robustness Properties (2) 26 Theorem: Assume tht for some bounds, b, c, d > 0, i) δ( t) > β( t) 2 (decelertive forcing) ii) iii) d 2 3 β (t) > -2b β(t) (rte of energy dissiption) dt 2 ( ) F ( t) 2E min d β(t), c β( t) (norml disturbnce) e Then 1) When 2) In generl, given F ( ) = 0 nd F, q is finite e e β = 0 eβ ( t) e is semi-globlly exponentilly stble ε > 0 is semi-globlly ultimtely bounded by ε

27 27 Interprettion of Robustness Properties Prllel disturbnce ccelertive disturbnces (pushing) do not hurt decelertive disturbnces (brking) cn be compensted by high gin Norml disturbnce error is ultimtely bounded with high gin Disturbnce rejection IMPROVES s energy in the system increses Worst disturbnces norml, dissiptive, nd occurs when energy is low Consequence of feedbck being qudrtic in velocity

28 Circulr Contour Experiments 28

29 Humn intercting with mchine 29

30 Energy vrition in experiment 30

31 31 Comprison with Trjectory Bsed Control Timed trjectory control lw: Desired Compenstion Control Lw (DCCL) (Sdegh nd Horowitz) model bsed, pssivity bsed, feedforwrd + PD feedbck Whitcomb nd Koditschek showed tht dptive version of DCCL outperforms other trjectory bsed control Compre contour trcking performnce t different nominl speeds Compre robustness to computer shutdown

32 32 Experimentl Results Speed Control Friction Compenstion Mnipultor q, q P.V.C.F. Friction compenstion (signum functions) in some experiements Regulte speed using n dditionl dmping loop

33 33 1 rd /s At low speed, Pssive Velocity Field Control performnce is dominted by friction / stiction system hs little energy to mnipulte trjectory bsed control (DCCL) performs better

34 Moderte Speed (2 rd/s) 34

35 35 High Speed (3 rd/s) Performnce of trjectory bsed control degrdes rpidly t moderte speeds, but PVFC performnce improves t high speeds.

36 36 Recovery from Power Cut-off (DCCL) Turn off power t middle of experiment turn bck on fter few seconds

37 37 Recovery from power cut-off (PVFC) After power cut-off, Timed trjectory bsed DCCL re-pproches contour in wkwrd mnner PVFC recovers from power cut-off by returning to the contour directly

38 38 At low speed, Pssive Velocity Field Control performnce is dominted by friction / stiction system hs little energy to mnipulte trjectory bsed control (DCCL) performs better Performnce of trjectory bsed control degrdes rpidly t moderte speeds, but PVFC performnce improves t high speeds. After power cut-off, Timed trjectory bsed DCCL re-pproches contour in wkwrd mnner PVFC recovers from power cut-off by returning to the contour directly

39 39 Velocity field design for generl contours Need to design desired velocity field,, so tht rrows converge to desired contour V d Difficult or my not exist (e.g. tngent is not unique) τ = 0 τ = 0. 2 Prmeterized trjectory: prmeterized the contour q : I G q : τ q ( τ) d d d τ progression prmeter is similr to time in ordinry trjectory, however, τ is not necessrily 1.

40 Suspended Mechnicl System Suspend dynmics of Progression prmeter, M( q) 0 q C( q, q ) q T F e = + τ τ 0 τ 40 Design desired velocity field for suspended mechnicl system : Progression term moves long the prmeterized trjectory t speed determined by progression rte, Grdient field forces position to converged to q d ( τ ) / q grd U d Vd ( q, τ) = λ1( q, τ) + λ2 ( q, τ) 1 0 Design P.V.F.C. for suspended system to follow its desired V.F.

41 41 Design of Desired V.F. : Self-pcing / q grd U d Vd ( q, τ) = λ1( q, τ) + λ2 ( q, τ) 1 0 Choose nd intelligently, so tht s position devites from prmeterized trjectory, Rte of progression decreses Increses strength of the grdient field Exmple: λ 1 λ 2 q d / λ 1 λ 2 Error Error

42 Free Response 42

43 Effects of Self Pcing 43 no self-pcing with self-pcing Strt

44 Effect of Self-Pcing 44

45 Without Friction Compenstion 45

46 Effect of Contouring Speed without self-pcing 46

47 47 Conclusions New Pssive velocity field control is proposed ensures pssive interction with environment tsk is encoded using velocity fields does not use timed trjectory Robust to environment forces best disturbnce is ccelertive nd occur t high speed robustness improves s energy increses P.V.F. control pproch to contour following time is not explicitly used decouples the tsk of following the contour nd the speed of movement Velocity field design for contours described by prmeterized trj. Experimentl results verify theoreticl findings

48 48 Applictions Robotic mchining (cutting, deburring... ) Mchines tht interct with humn robotic surgicl tools intelligent exercise mchines

49 49 Cn you tell? F e F e Mnipultor q Environment Blck Box q Environment F e P.V.C.F. controller Mnipultor q Environment

50 50 References P.Y. Li nd R. Horowitz, "Control of Smrt Exercise Mchines: Prt I, Problem Formultion nd Non Adptive Control", IEEE / ASME Trnsctions on Mechtronics, Specil Issue on Humn Friendly Mechtronics, Dec, 1997 (To pper) P.Y. Li nd R. Horowitz, "Pssive Velocity Field Control of Mechnicl Mnipultors" IEEE Trnsction on Robotics nd Automtion (To pper). P. Y. Li, Pssive Dynmic Approch to the Control of Bilterl Teleopertors, Americn Control Conference, Phildelphi, June 1998 (submitted) P.Y. Li nd R. Horowitz, "Pssive Velocity Field Control. Prt 1: Geometry nd Robustness", IEEE Trnsctions on Automtic Control, (submitted). July, 1997 P.Y. Li nd R. Horowitz, "Pssive Velocity Field Control. Prt 2: Appliction to Robot Contour Following Problems ", IEEE Trnsctions on Automtic Control, (submitted). July,1997