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

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

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

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

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

Cranes Cable is inherently flexible The “ Gantry Crane problem ”

“Reconciling” position control and active vibration damping

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

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

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

“ Fourth order ” systems    nn y1y1 ynyn y2y2 y3y3 Actuator    y  k1k1 k2k2 k3k3 knkn Load  n   y n

A difficult problem?

“ 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

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

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 ”

“ Wave-based ” techniques

Experimental Crane Model

Space structures simulated

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

Flexing rig, w-b control

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

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

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

Same control strategy Load Actuator Flexible system

Also for laterally flexing systems Actuator    y  Load  n   y n Arbitrary flexible system

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

Wave view of flexible system (Assuming displacement waves)

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.

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

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?

Response tt Launch_x = ½ Trgt absorb_x Actuator posn End mass posn

Works very well, but … Assymptotic approach to target

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

“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

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

Response tt Wave echo control, 3 mass-springs

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

“ Waves ” ??...

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

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

“ Waves ” “ Waves ” in lumped systems?

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”

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

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)

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

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

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

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

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

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

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 ”

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

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

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

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

The End Thank you

Space structures simulated

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

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

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? ? ? ?

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

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

“ 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