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Wave-based control of flexible mechanical systems William O ’ Connor University College Dublin National University of Ireland ICINCO 2006 Setúbal, Portugal.

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A powerful, new, robust, generic solution Control of flexible mechanical systems

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Growing importance Robotics Cranes Disk drives Long reach manipulators Cleaning, construction equipment Medical & human assistance devices Space structures …

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Traditional robots Heavy Short arms Dynamically sub-optimal Expensive But rigid

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Lighter robots Dynamically more responsive Faster Cheaper Less energy/force Sometimes necessary

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Cranes Cable is inherently flexible The “ Gantry Crane problem ”

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“Reconciling” position control and active vibration damping

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Rigid System x0x0 x1x1 x 0 = x 1 Control Easy!

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Flexible System Actuator c c c kk k m m x0x0 x1x1 x2x2 x3x3 What actuator “input”, x 0 (t), to control “output”, x 3 (t) ? m Tip mass

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“ Second order ” systems load actuator load “actuator” Distributed and/or Lumped

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“ Fourth order ” systems nn y1y1 ynyn y2y2 y3y3 Actuator y k1k1 k2k2 k3k3 knkn Load n y n

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A difficult problem?

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“ to date a general solution to the control problem [of flexible structures] has yet to be found. ” “ One important reason is that computationally efficient (real-time) mathematical methods do not exist for solving the extremely complex sets of partial differential equations and incorporating the associated boundary conditions that most accurately model flexible structures. ” p.165 of “Flexible robot dynamics and controls” Kluwer Academic/Plenum, New York, 2002

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Approaches to date Classical & state-space control Modal control Input / command shaping Bang-bang control Sliding mode control FOC Wave & Virtual system ideas …

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Generally, at best get asymptotic position & vibration control. Major focus on system identification and modelling (cf: “… extremely complex PDEs & BCs …” ) “ no general solution to date ”

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“ Wave-based ” techniques

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Experimental Crane Model

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Space structures simulated

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Flexing rig, no control Lab rig in University of Castilla-La Mancha, Ciudad Real, Spain

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Flexing rig, w-b control

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“ Wave-based ” techniques Control of complex system in a simple, natural way

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Simple idea … The “ interface ” is key a) Understand, b) Measure, & c) Manage the interface All using Wave concepts

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Generic flexible system Actuator Load Arbitrary flexible system “load” / “end point”

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Same control strategy Load Actuator Flexible system

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Also for laterally flexing systems Actuator y Load n y n Arbitrary flexible system

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What must controller do? ? ? ? Computer controller? Assume actuator / trolley can move themselves

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Wave view of flexible system (Assuming displacement waves)

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Wave-control ideas When actuator moves, it launches “wave”—x. “Wave” pass though each of the masses. “Wave” reaches load mass. It deflects 2x. If actuator absorbs this returned “wave”, system will comes to rest at displacement 2x! system will comes to rest at displacement 2x! Returning “wave” moves each mass by x again.

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Actuator does 2 jobs x 0 (t) = launch + absorb x 0 (t) = a 0 (t) + b 0 (t) x 0 (t)

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Set launch component, a 0 (t) to reach ½(Target_x) while simultaneously measuring and adding b 0 (t) to “absorb” returning motion. (Guarantees success) Key idea 1 “Push”=“Pull” Absorbing action causes system to move the other half-target distance. (For how? and why? See below.) Newton III?

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Response tt Launch_x = ½ Trgt absorb_x Actuator posn End mass posn

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Works very well, but … Assymptotic approach to target

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Key idea 2 ½Target) “Launch_x” time profile is arbitrary provided the final value is correct (½Target)

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“Best” way for Launch-x to arrive at ½Target position is a time-reversed and inverted “re-play” of absorbed wave from start-up Best launch profile? “Wave-Echo” control

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Wave echo control, single mass Response tt 0.5 Mirror of absorb Absorb 1.0

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Response tt Wave echo control, 3 mass-springs

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Magic! Many other outstanding features (see later) Works wonderfully! Load stops dead!

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“ Waves ” ??...

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x 0 (t) = launch + absorb x0(t)x0(t) a0(t)a0(t) b0(t)b0(t)

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\/\/ For rest to rest manoeuvres … \/\/ Implications (for any value of Z)

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“ Waves ” “ Waves ” in lumped systems?

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G n-1 G n G 1 G 2 H n-1 H n H 1 H 2 F A1A1 A2…A2… A n-1 AnAn A0A0 B1B1 B2…B2… B n-1 BnBn B0B0 X0X0 - + Ref x 0 k1k1 k2k2 knkn m2m2 mnmn m1m1 x2x2 x1x1 xnxn x0x0 k n-1 m n-1 x n-1 X0X0 X1X1 X2X2 X n-1 XnXn X i = A i + B i “Wave transfer functions”

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Have neither poles nor zeros Steady-state gain of unity Close to second order Zero instantaneous response Dominated by local dynamics Lagging phase

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x 0 (t) = launch + absorb x 0 (t) X 0 (s) a 0 (t) A 0 (s) b 0 (t) B 0 (s) X 1 (s)

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X0X0 G 1 (s) H 1 (s) X1X1 B0b0B0b0 A0a0A0a0 + – – +

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X0X0 G 1 (s) H 1 (s) X1X1 B0b0B0b0 A0a0A0a0 + – – + ½ + + X trgt CcCc Absorb wave b 0 added to launch wave set to half ref

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X0X0 G 1 (s) H 1 (s) X1X1 b0b0 a0a0 + – – + X trgt CcCc

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x 0 x 1 c= (km) m k Ĝ(s) = X 1 (s)/X 0 (s) = n 2 s 2 +s n + n 2 G 1 (S), H 1 (s) approximated

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Absorbing motion Ensures stability Dampens vibrations Moves system second half of motion Gives real-time system identification It “ opens the loop ”, cancelling poles

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G n-1 G n G 1 G 2 H n-1 H n H 1 H 2 F A1A1 A2…A2… A n-1 AnAn A0A0 B1B1 B2…B2… B n-1 BnBn B0B0 X0X0 - + Ref x 0 k1k1 k2k2 knkn m2m2 mnmn m1m1 x2x2 x1x1 xnxn x0x0 k n-1 m n-1 x n-1

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Features of control system Works for n masses Uniform or non-uniform With or without internal damping No system model needed Or: System itself is “ model ” and “ computer ”

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continued … Sensing is minimal, and local “ Real ” Actuator OK Zero steady-state error Minimal vibration in transit

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Continued … Very rapid Very energy efficient Does deliberately & naturally what other approaches do, perhaps unconsciously, with difficulty. Generic

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“ Take care of the interface ” “… And the system will take care of itself ”

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Other applications Stabilized platforms “ Power assisted ” motion Multiple actuator systems A complex problem may have a simple solution!

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The End Thank you

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Space structures simulated

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Actuator-System interface Two-way motion flow in all flexible system types: distributed & lumped 2nd order systems distributed & lumped 4th order systems uniform or non-uniform mixed lumped and distributed Energy, momentum, dynamics, sensing, control … all via the interface

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Wave-based models A new way to model lumped mechanical systems Two-way motion revealed, made “ legitimate ” (defined), made measurable, and thereby allowing control.

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Problem definition How should you move the actuator (or trolley) a) to move the end-point from A to B, and b) to control the vibrations? ? ? ?

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Magic! Zero vibration. Zero steady state error No system model needed, nor modal info Arbitrary order Adapts automatically to system changes Works fine with real actuator System can be non-linear, non-uniform Very energy efficient, very rapid Computationally very simple No jerk, nor chatter, nor precise switching … etc

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System modelling New method of modelling lumped flexible dynamic systems Already proving itself very powerful and adaptable to new demands

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“ Point to point ” control Typical of robots, some cranes But sometimes the end point is not known initially. Typical of manual operation of cranes. “ Open ended ” control

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