From: AAAI Technical Report SS-02-04. Compilation copyright © 2002, AAAI (www.aaai.org). All rights reserved.
Distributed
Adaptive Constrained Optimization for Smart Matter Systems
MarkusP.J. Fromherz,Lara S. Crawford,Christophe Guettier, and Yi Shang
Palo Alto Research Center
3333 Coyote Hill Road
Palo Alto, CA94304
{ fromherz,lcrawford,cguettier, yshang) @parc.xerox.com
Abstract
Theremarkableincrease in computing
powertogether with a
similar increasein sensorandactuatorcapabilitiesuowunder
wayis enablinga significant changein howsystemscan sense
and manipulatetheir environment.Thesechangesrequire
control algorithmscapableof operatinga multitudeof interconnectedcomponents.
In particular, novel "smart matter"
systemswill eventually use thousandsof embedded,
microsize sensors,actuatorsandprocessors.
In this paper,weproposea newframework
for a on-line, adaptive constrainedoptimizationfor distributedembedded
applications. In this approach,on-line optimizationproblemsare
decomposed
and distributed across the network,and solvers
are conm311ed
by an adaptivefeedbackmechanism
that guarantees timely solutions. Wealso present examplesfromour
experiencein implementing
smartmattersystemsto motivate
ourideas.
a)
b)
Introduction
The remarkable increase in computing power together with
a similar increase in sensor and actuator capabilities now
under wayis enabling a significant change in howsystems
can sense and manipulate their environment. These changes
require control algorithms capable of operating a multitude
of interconnected components.In particular, novel "smart
matter" systems will eventually use thousands of embedded,
micro-size sensors, actuators and processors. As an example, consider an air-jet paper mover,consisting of 576 actuators and 30Ksensor pixels embeddedinto an active surface that movesa sheet of paper on an air bed by controlling each jet valve individually (Jackson eta/. 2001).
another example, consider a hyper-redundant modularrobot
that consists of hundreds of interchangeable robotic modules connected into complexconfigurations, each module
with its ownactuated joint, sensors, processor, and communication (Yim1994). As yet another example, consider materiais with embedded,co-located force actuators and sensors for active vibration control (Chase, Yim,&Berlin 1999;
Hogg& Huberman1998) or damageidentification (Wang
Chang2000). Suchsystems will require control, sensing and
diagnostic algorithmsthat are robust, scalable, andreconfigurable.
Constrained optimization is at the core of manyplanning, control, reconfiguration, and fault diagnosis applica-
34
c)
Legend:
Q Controllers
u Sensors
& Actuators
I seining &control communication
Figure 1: Sketchesof a) centralized, b) decentralized, and
hierarchical control organization.
tions (Bondarenko,Bortz, & Mort 1998). In fact, the developmentof optimal controllers and diagnostic routines for
smart matter systems often starts with the formulation of a
model of the system dynamicsand an objective function to
be optimizedby the algorithm (e.g., the cost function of an
LQRcontroller). For somesystems, such as modularreconfigurable robots, only an on-line, model-basedcontrol approach is able to robustly handlethe variety of non-standard
objectives and constraints required (Fromherz,Hoeberechts,
& Jackson 1999; Fromherzet al. 2001).
Today,solutions to these control problemsare usually at
either one of two extremes: they arc either completelycentralized or completelydecentralized. Centralized algorithms
(Fig. la) provide optimal performance(taking into account
the complete system behavior whencontrolling each actuator), but clearly do not scale to the large numbersof elements
in smart matter systems. Completelydecentralized (i.e., local) algorithms (Fig. lb), on the other hand, scale well
makeuse of a system’s distributed processing capabilities,
but they typically result in poor performance(e.g., introducing large errors or resulting in inordinateactuation energies).
Clearly, hierarchical approaches(Fig. lc) promisean intermediate solution: at various levels of the hierarchy, the
algorithm takes into account groupings of system elements
and their behaviors, while also being able to distribute the
computation amonga numberof processing nodes. Consider
for examplethe force allocation problemin the air-jet paper
mover(Fig. 2) (Jackson et al. 2001; Fromherz& Jackson
2001), wherethe goal is to generate an optimal allocation for
the air jets such that together they producethe desired forces
and torques that will movethe sheet of paper (see belowfor
details). Ourimplementedapproachis a self-similar, hierarchical decompositionof this task, wherethe actuators are
partitioned into smaller and smaller groups (Fig. 3): at every
level in the hierarchy, jets are combinedinto groups that act
like "super jets" (each delivering forces that act on the paper), and no level has to knowwhetherthe next-lower level
consists of individual jets or jet groups. Similar considerations have been madefor other smart matter systems (Hngg
& Huberman1998).
From a model-based computing perspective, however,
there are at least two majorbarriers to employingthis hierarchical, distributed approachin real-world systems: (1) the
decompositionof a centralized formulation into a hierarchical structure is non-trivial, as it often involvescomplexproblem reformulations and the balancing of trade-offs between
performance and resource uses (computing, communication)
that have to be modeledexplicitly; and (2) today’s optimization / constraint solving algorithms are not ready for realtime, resource-constrainedapplications.
Wehave studied and developed smart matter applications
for a numberof years and found several suitable modelbased control solutions (Fromherz, Hoeberechts, & Jackson 1999; Jackson et al. 2001). From this experience,
we are proposing a newframeworkfor an (on-line) adaptive constrained optimization service for distributed embedded applications. In this approach, model-basedcontrol and
thus on-line optimization problemsare decomposed
and distributed across the network,and solvers are controlled by an
adaptive feedback mechanismthat guarantees timely solutions. The frameworkaddresses the two barriers described
above, i.e., problemscale and real-time constrained optimization. This paper discusses the two main componentsof
our approach, decompositionand adaptive control of solving, in the followingtwo sections. Wealso describe illustrative examplesto motivate our ideas and end with conclusions
and future work.
35
a)
olnl
32.000~.S~mr~a~Valvosw~576
$emmrmmdout
Vslw drivvn
I
r.,rjl+~.+
o~,t,lr.+r~r.,rjir.,o-.+
’~ +"I"" °~r.,l~.,~r.,~,lr.+~.0
o4. *.,. [#.,. t.,. °.,. °4. |°4. #.,. : °.,. t.,. |°.. +.,.
Airje~
r.,r.,F.,r., r.,~or., o~,~,1°~.,
°.~°,. I+.~°~*~°gI°. *,.
b)
Figure 2: Air-jet system: a) Photograph of 35 cm x 35
cmair-jet paper movermodule: 16-element arrays are flap
valves and associated jets, black bars are sensor arrays, b)
Systemarchitecture and layout of the board.
Problem Decomposition for
Distributed Solving
Becauseof the large numbersof sensors, actuators, and computing elements in smart matter systems, we suggest that
problemdecompositionand distribution techniques are core
to enabling and delivering a scalable and robust solver service for such systems. In particular, decompositionallows
us to balance control requirements with communicationand
resource realities. For problemswith thousandsof variables
and constraints, representing the large numberof distributed
but connected physical elements in large-scale smart matter systems, it is often both infeasible and unnecessaryto
solve the constrained optimization problemas a single problem on a single processor. Instead, our proposedapproachis
based on the idea that systemswith tens or hundredsof thousands of sensors andactuators, whether discrete or continuous at the individual element level, approacha continuum
Decomposition
andhierarchical
allocation
Figure3: Decomposition
of a force allocation probleminto
sub-problems
on an active surface
a)
b)
Figure
4:Aggregation
heuristics
applied
totheair-jet
table(excerpts):
a)neighborhoods
ingeometric
space
(tiles);
b)neighborhoods
ingeometric
andforce
space
(rows
and
columns)
inthelimit
andthuscanbeapproximated
bymuchlowerdimensional
continuous
models
athigher
levels.
CI’his
puts
theconventional
idea
ofhybrid
systems
onitshead,
where
discrete
models
areusually
atthetopandcontinuous
models Jackson
2001):
atthebottom.)
A second
keyinsight
isthat
such
large-scale
I N ]~+
1 N ,tf
a,+
problems
areoften
self-similar
atmultiple
levels,
asalready
illustrated
above.
Finally,
wehaveobserved
that
different
N
S.t. Fx = ~’~iz~l fxi
groupings
ofelements
canlead
towidely
different
computa(l)
tional
complexities
atlower
levels.
Even
ifthehierarchical
abstraction
isfixed
ahead
oftime,
wecanstill
decide
dynamT. ---- E,+=IZJy,- ~"~N~_1 Y,fxi
ically
which
concrete
actuators
togroup
together,
e.g.,
based
whereforces Fx and Fy and z-torqueTz are the desiredtoonrun-time
information
about
available
actuators.
tal forcesto be appliedto the sheet, f~ = (fz~ fy~) are the
allocatedx andy forcesfor jet i, (zi,~i) is a jet’s position,
Constraint-directed problem decomposition
andwi = (wx~wy~)are the weightingfactors for eachjet’s
contributionof x andy forces.
This problemcan be decomposed
recursively into sets of
Weare investigating twoschemesfor automaticallydecom(smaller)optimizationproblems
to be solvedbysubgroups
posing tasks on many-element
systems. In a constrainlthe next-lowerlevels of the hierarchy.Exceptfor the bottom
directedapproach,weare developingstructure analysis and
level, a "jet" i in Eq.(1) refersto the "virtualjet" that arises
abstraction techniquesfor decomposing
and approximating
out of a groupof jets. Weare free to choosehowto aggrelarge-scale constraint problems.Thegoal is to discover
gate jets into virtual jets. Oneobviousheuristicis to follow
structure in constraint problemsthat can be exploited by
the geometric
layout,e.g., eachtile of jets becomes
a virtual
solvers for the subproblems.For instance, we havefound
jet(Fig.
4a). Theadvantage
ofthis
heuristic
isthat
a moduthat maximizingsymmetriesin subproblemsby grouping
larstructure
ofthesystem
leads
toa corresponding
modular
actuatorsthat are interchangeable
withrespect to the constructureof the allocationalgorithm.
stralnts can makea dramaticdifference. Asshownin an examplebelow,by usingthe problem’sconstraintsto guidethe
Anotherheuristic is to aggregatejets in neighborhoods
of
a different parameterspace,namelythat of force directions
grouping,solving subproblems
at lowerlevels of the hierandlocationalongjust oneof the axes. Thisheuristic is inarchy can become
trivial. In certain systems,it mayeven
spired by the linear superpositionmodelin the constraints
be possibleto precompute
the lower-levelsolving step and
of Eq. (I). Weobserve,for example,that all x-jets withthe
replaceit by a constant-timetable lookup.Problemcharacteristics such as symmetriesseemto be common
in systems
samey positionare interchangeable
withrespectto the constraints in Eq.(I). In fact, if our systemconsistedonlyof
whereconstraintsarise out of (often uniform)physicalcharx-jets withthe samey position,the instantiationof the alloacteristics. Thisproposed
approachof structureanalysisand
cation functionswouldsimplybe
abstractioncanlead to considerableincreasesin efficiency
withreasonableerror boundsfor problems
in large-scalesysFx
tems (promhcrz&Jackson 2001).
(2)
A~= "~-, fy+ = 0 (i = 1,..., N)
Fordiscrete actuators, one cansimplycalculate the number
An example for problem decomposition
of necessaryjets as Fz/fm~roundedto the nearest integer
and then openthat manyjets (fins= is the maximum
force
Weillustrate the potentialof constraint-directeddecomposi- availablefroma singlejet).
tion on the air-jet papermoverintroducedabove.ForexamGenerally,rowsof x-jets with the samey position and
ple, the air-jet force allocationproblem
canbe formulated
as
columnsof y-jets with the samex position lead to paran optimizationproblem,attemptingto achievethe desired
ticularly simpleallocation functionswithin a rowor coltotal forces whileminimizing
actuation energy(Fromherz
unto. Therefore,wehaveimplemented
an aggregationof jets
36
wherethe N virtual jets are dividedinto "x modules,"each
withonly x-directed jets with a common
y position Yi, and
"y modules,"eachwith onlyy-directedjets witha common
x
positionzi (Fig. 4b). Now,allocationto thesevirtual jets has
to compute
onlyeither fx~or fyi for the x andy modules,
respectively,andallocationwithina virtual jet followsEq.(2)
for x modulesand its equivalentfor y modules,instead of
solving the morecomplexEq. (1). Thus, the decomposition
at onelevel greatlysimplifiesthe allocationat anotherlevel.
E
Nested-loop partitioning
In a secondapproachto decomposition,we are developing a nested-looppartitioningapproachthat takes as input
a nested-loopformulationof the task, as well as modelsof
the systemand the application(e.g., computing
and memory
limits, performance
objectives)anda rangeof suggestedhierarchicalstructures. In a nested-loopformulation,eachvertex of the iteration spacerepresentsa givennodeor agent
(e.g., sensoror actuator) (Aocourtet al. 1997).Examples
are sensors and actuators embedded
on a two-dimensional
surface, suchas the sensorsand jets in the papermoverand
the vibration sensors and actuators embedded
on a surface
grid for vibrationcontrol.There,a controlcyclecanbe modeled as nested,two-dimensional
loopsiterating overtheseelementsfor the sensing,control, and actuationtasks. Partitioning groupsandeffectivelyparallelizes these loops such
that the resourceconstraintsare satisfied. Forhierarchical
applications,partitioningis appliedrecursivelyin orderto
determinean appropriatehierarchicalstructure andparameters, suchas the branching
factorandheightof the hierarchy.
~
Adaptive Control of Problem Solving
Thesecondelementof our approachis the on-line tuningof
solvers for the environmentand problemsof embedded
applications. In general, the designof a problemsolver for a
particular problemdependson the problemtype, the system
resources, andthe applicationrequirements,as well as the
specific probleminstance. Oneapproachto this designissue is the use of adaptivecontrol of problemsolving. We
define adaptivecontrol of solvingas modifyingthe solver
or the problemrepresentation in responseto environment
or problemchanges. Adaptivesolving control makesuse
of a modelor set of rules concerningthe relationship betweenvarious problem,system,and application parameters
and choicesfor solvers, heuristics, andproblemtransformations. Thesechoicesdefine the control parametersof the
solver. This modelor rule basealso enablesthe prediction
of the behaviorof the solver definedby these controlparameters. Thepredicted behaviorcan be used to monitorthe
actual on-line behaviorof the solver andupdatethe control
parametersaccordingly.If the predictedbehavioris consistently at oddswiththe actual behavior,the accumulated
performancefeedbackcan be used to update the modelor rule
baseitself.
A frameworkfor adaptive solver control
Theseideas are directly analogousto on-line adaptationin
control systems(Astr6m&Wittenmark
1995;Levine1996),
37
l
Control Module [[ u.
=
T
$olv~Module
Y
Figure 5: Adaptivesolving systemframework
withthe solver takingthe place of the plant. Basedon this
viewpoint,we proposea genericframework
for the adaptive
control of solving(Crawford
et al. 2001),shownin Fig. 5.
Thisframework
includesthree levels of increasinglysophisticated control: open-loop,feedback,andadaptivecontrol.
In a basic control system,the controllertakes the reference
input andoutputsa control signal to the solver withoutany
feedback.For such a feed-forward,open-loopcontrol systemto producethe desired behavior,the controller mustbe
designedandtuned off-line using a very accuratemodelof
the solver. Anyinaccuraciesin the modelor variancesto the
system(e.g., the problems
to be solved)will not be takeninto
accountat run-time.
Whenthe feedbackloop is addedto the control system,
the controllergainsthe ability to react to the qualityof the
solver’s output. Thecontroller still mustbe designedand
tunedoff-line, but the systemhas a muchbetter chanceof
performingas desired.
In an adaptivesystem,the controlleris able to adapt to
compensate
for systematicperformance
errors causedby inaccurate solver modeling.The controller is designedand
tunedoff-line, but canbe formulatedparametricallyin terms
of solver parametersthat maybe unknown
or imprecisely
known.The modelbecomesmore accurate over time and
can also adapt to changesin the problemsand solver’s environment.
Thisfunctionalframework
is usefulfor structuringa principled discussionof adaptivesolvingapproaches.Thefollowingdescription
oudines
the proposed
functionality.
Examples
ofpossible
implementations
andinstantiations
ofits
elements
areprovided
in(Crawford
etal.2001).
Referring
toFig.
5,theSolver
Module
represents
software
forproblem
transformation
andsolving,
possibly
containing
multiple
algorithms
andheuristics
among
which
toselect,
as
wellastunable
parameters.
(Byproblem
transformation,
we
meanchanges
totheproblem
formulation,
suchaschanging
thegranularity
ofvariable
domains
toimprove
efficiency.)
Thealgorithmsandparametersare selected with the input
u, providedby the ControlModule.This modulemakesthe
solver algorithmselectionand tuningdecisionsbasedon the
environment
specification, E, the problemto be solved, P,
the on-line (during a run) behaviorof the solver, y, and some
internal tuning parameters, a. The environment,E, includes
modelsof both the systemspecifications and the application
requirements. Including the on-line behavior of the solver
as an input to the Control Moduleallows the solver control
to have a feedback component.The role of the Adaptation
Moduleis to monitor the behavior y of the Control Module/Solver Modulesystem and adapt the parameters, a, of
the Control Moduleto improveperformance. Its outputs, ~,
are the current estimates of the optimal Control Moduleparameters. The Adaptation Modulewould typically act at a
muchslower time scale than the feedback loop. As it acreally modifies parameters of the Control Module,its actions
wouldgenerally not be based on system behavior for a single problemor subproblem,as the feedback control would.
Both the Control Moduleand the Adaptation Modulecontain, either explicitly or implicitly, information about the
solver and its expected behavior. A predictive modelof expected solver behavior wouldprovide a basis for the Control
Moduleto makeits solver parameter choices. The Adaptation Modulewoulduse the predicted behavior as a yardstick
for measuringthe true behavior, y. Thecontroller parameters
a adjusted by the Adaptation Modulemight then be parameters of the solver model.
Previous approaches to adaptive solving for different
classes of problemshave involved the first of the levels of
control enumerated above and sometimes the second. In
contrast, mostof the workon adaptive solving to date has in
fact not beenadaptive,at least not in the control sense (Crawford et al. 2001). In other words, these approachesare not
adaptive to the run-timeenvironmentof the solver, a crucial
prerequisite for deployingsolvers in embedded
applications.
a)
I~ °.o
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am~ mllo 1 IS~
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38
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,I
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iiiilT[]]],!i!TiT
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10
An example for feedback and adaptation in solving
Wepresent the following examplein order to illustrate the
value of feedbackand adaptation in on-line solving. It has
been shownthat the combinationof global and local solvers
can be particularly effective for complex,realistic problems (Shangetal. 2001b). Consider, for instance, a cooperative global/local solver for continuousconstraint satisfaction
problems that uses the Nelder-Meadalgorithm (Lagarias et
al. 1998)for global search followedby sequential quadratic
programming
for local search(fminsearchandfmincon,respectively, in the Matlab Optimization Toolbox(Matlab Optimization Toolbox2001)). Such a solver has several tuning parametersthat need to be set. Oneof these is the size
of the initial simplex for the Nelder-Meadalgorithm, which
determines the search scale. Animportant indicator of complexity for this type of problemseemsto be the constraintto-variableratio (constraint ratio for short). Thus,a first step
might be to analyze the performanceof various initial simplex sizes for various constraint ratios, as is shownin Fig. 6a
for a 25-variable problem.Suchan analysis provides the basis for a solver modeland as such the foundationforan openloop control scheme.This schemespecifies that the simplex
size should be chosenas the one yielding the lowest median
complexitygiven the problem’sconstraint ratio.
If, for instance, for a problemwith constraint ratio 6.6,
the simplex size were chosenbased on Fig. 6a, a size of 0.3
DI
::T
ml
I
,~zl
I
I
I
15
!0
i
~ni~
;*
I
I
I
I I ,I,I
. l I
I
I
1i!i
aO Im
I
¯
I
4.- a,,Itll
o o~.m
÷ atolL!
¯ ih!
4O aS
60
h)
Figure 6: Continuousconstraint satisfaction problemswith
different initial simplex sizes: This figure showsdata on
sum-of-sines type continuous constraint satisfaction problems with 25 variables. The top diagram shows the median
(over 50 instances) of the numberof function evaluations
required for problemswith different constraint ratios, using
initial simplexsizes s between0.03 and I0. Thebottom diagramshows,for a single constraint ratio, the variation among
all instances and all simplexsizes
wouldbe selected (the lowest point in the second-to-last set
of data). Since this is the best choice only on average for
this particular constraint ratio, it will not be the best choice
for every problem instance. Fig. 6b shows the complexity
for 50 different instances for this constraint ratio. Theinstances have been ordered by the complexity for the openloop simplex size, 0.3. Clearly, there are a numberof instances where different choices of the simplex wouldyield
muchbetter results. Therefore, a feedbacklaw wouldbe useful here in order to fine tune the systemfor the particular instance. Whenmonitoringthe on-line solver behavior, if the
numberof function evaluations were growingtoo large, the
simplex size could be changed. This type of feedback could
improvethe solving time in cases like those on the right-hand
side of Fig. 6b.
Finally, if, over the course of manyproblems,the initial
open-loopestimate of the best simplexsize is in error more
often than not, the adaptation portion of the solver control
comesinto play. The AdaptationModulecan monitor the behavior of the solver and compareit to that predicted by the
solver model (here, the median complexity given the constraint ratio and the choice of simplexsize). If this prediction appearsto be incorrect, such as if the simplexsize of 0.3
is yielding consistently worseresults (and the feedbacklaw
described above has to act frequently), adaptation can adjust the solver modelaccordingly. Therefore, it mayhappen
that, as moreexamplesare gathered on-line, the estimate of
the complexityfor the 0.3 solver increases enoughthat 0.3 is
no longer the preferred simplex size for problemswith constraint ratio 6.6.
Thus, open-loop control implements parameter choices
based on a solver modelor a set of rules, either of which
can be generatedoff-line basedon analysis or statistical sampling. The feedback control componentperforms on-line
corrections to compensatefor individual instances deviating from the norm. Finally, the adaptive control component
serves to correct persistent errors in the solver modelor rule
base, whichcan affect both the open-loop and feedback components.
Conclusion
Constrained optimization is at the core of manyembedded
applications. Weare proposing a newframeworkfor an (online) adaptive constrained optimization service for such applications that addresses two central issues: problemdistribution and representation as well as adaptive algorithm selection and tuning.
The adaptation frameworkpresented here allows for, as
far as weare aware,the first principled classification of different techniques for adaptive control of problemsolving.
Our current focus is on generating suitable problemensembles for complexityanalysis, on a better understandingof the
relevant problemparameters that control or at least predict
complexity, and on cooperative solvers (Shang et al. 2001a;
Fromherzet al. 2001).
Webelieve that this workis relevant for a wide variety of
applications andis particularly importantfor large-scale, dynamic, embeddedproblems.
Acknowledgment This work is supported
DARPA
under contract F33615-0I-C- 1904.
in part
by
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