a stochastic processes approach to studying robotic swarm … · 2005-06-11 · a stochastic...

Analyzing Swarms A Stochastic Processes Approach to Studying Robotic Swarm Behavior

Kristina LermanUSC Information Sciences Institute

Tad HoggHP Labs

Thanks to Aram Galstyan Alcherio Martinoli Maja Matarić Chris Jones Brian Gerkey Bernardo Huberman

ScheduleScheduleMotivation How to design swarmsAnalysis based on stochastic processesndash basic theoryndash applications amp comparing to experiments

BREAK

Generalizationsndash adaptive robotsndash reconfigurable and microscopic robots

Summary and future directions

June 8 2005 Analyzing Swarms Tutorial 2118

What is a Robot SwarmWhat is a Robot SwarmLarge number of simple robotsndash limited hardware capabilities

power CPU communications

ndash limited view of environmentlocal neighborhood noisy sensors

ndash simple control methods

Tasks involve overall behavior of groupndash generally average not extreme behaviorndash less interest in specific individual robots


How to Design a SwarmHow to Design a SwarmWhat hardware capabilitiesWhat local control program

Does resulting behavior solve my taskndash reliably (eg even with some failures)ndash within reasonable time and resource use

What if task changes during operation


Local Control amp Collective BehaviorLocal Control amp Collective BehaviorLarge design space for swarmsHow to find good designs

key issuerelate designed robot capabilities to resulting swarm behavior


Poor Design for One Poor Design for One Ok for ManyOk for Many

Example search an areaControl gives a robot 1 chance to find targetndash bad for one robotndash good with swarm of 1000 robots

swarm likely to find target if behaviors more-or-less independent


Good Design for One Good Design for One Bad for ManyBad for Many

Example search an areaControl gives a robot 100 chance to find targetndash good for one robotndash bad for swarm of 1000 robots

all may pile up at targetmay interfere with each other


Swarm Sensitivity to Robot BehaviorsSwarm Sensitivity to Robot BehaviorsRandom variation averages out

But correlated changes can have large effectsndash eg save power by increasing time before communicating

interesting eventsndash gradually increasing delay can lead to abrupt system change

from steady-state to oscillatory or chaotic system

ndash such abrupt changes not evident with just a few robotsndash analogous to phase transitions in physical systems

ndash for design identify parameter values giving abrupt changes


Coordination within SwarmsCoordination within SwarmsRobots mostly act independentlyndash occasional coordination with nearby robotsndash behavior based on their local environment

maintaining no history of past interactions

Complicationsndash robots modify behavior based on historyndash robots interact continuously with neighborsndash robots also have changing global constraints

eg broadcast instructions to whole swarm


System Design ApproachesSystem Design ApproachesSimulation Experiments Deployment


Simulation Simulation Examine behavior of large swarms prior to building robots

Can be computationally intensive to explore design spaceAccuracy depends on knowledge of robots and task environment


ExperimentExperimentBuild a few robots test in task environment

Expensive especially to explore different hardware capabilitiesMay not have enough robots to see large-scale behaviors


DeploymentDeploymentBuild many robots try swarm on full task

Very expensive


Another Choice AnalysisAnother Choice AnalysisStochastic models to capture key issuendash relating local control to swarm behavior

Readily explore design spacendash eliminate bad designs

Validate designs with simulation amp experimentAn option pick designs with simple analysisndash Ie deliberately satisfy simplifying assumptions


System Design ApproachesSystem Design ApproachesAnalysis with stochastic modelsSimulation Experiments Deployment


OutlineOutline1 Stochastic processes

1 Historical perspective2 As a framework for studying robot swarms

2 Classes of stochastic processes with applications1 Ordinary Markov Processes

Basic theoryApplications Reactive robots

BREAK2 Generalized Markov Processes

Basic theoryApplications Dynamic task allocation

3 Stochastic processes with spatial dependenceApplication Medical nano-robots

3 Summary


Historical PerspectiveHistorical PerspectiveTheory of stochastic processes grew out of thermodynamics

and probability theoryThermodynamics 1700rsquos ndash Early 1800rsquosEmpirical laws derived from large body of experimental data

Kinetic theory of gases Mid-1800rsquosBulk properties of matter (eg thermodynamic laws for gases) arose

from the dynamics of its constituent elements (gas molecules)

Probabilistic methods first used to calculate gas properties

Statistical mechanicsLate 1800rsquos ndash early 1900rsquosStochastic processes theory - Mathematical foundation (along with

quantum theory) of modern physics


ldquoThere is no hope to compute [irregular position of a Brownian particle] in detail but hellip certain averaged features vary in a regular way which can be described by simple lawsrdquo

Van Kampen 1992


Stochastic ProcessStochastic ProcessStochastic Process Y(t) is a random variable (or a

function of a random variable) that changes with time

Defined by joint probability density

p(y1 t1 y2 t2 hellip yn tn)

probability that Y(t) takes values y1 y2 hellip at times t1 t2 hellip


Some ExamplesSome ExamplesCoin flipping amp random walk (probability theory)Bacterial chemotaxis (biology)Chemical reactions (chemistry)Stock market prices (finance)Spread of disease through a population (epidemiology)Decay of nuclear particles (physics)Network traffic (computer science)


Robot as a Stochastic ProcessRobot as a Stochastic Process

Robot can be viewed as a stochastic process

An individual robotrsquos behavior subject tondash External forces

cannot be anticipated

ndash Noise fluctuations and random events

ndash Other robots with complex trajectoriesCanrsquot predict which robots will interact

ndash Randomness in individualrsquos behavior rules eg collision avoidance procedures in robot controllers


Stochastic Approach to Studying SwarmsStochastic Approach to Studying Swarmsunpredictable individuals predictable swarms

Stochastic processes framework provides simple model of behavior of ensemble of individually unpredictable robots

Details of robotsrsquo trajectories who interacts with whom and when are not importantSingle robot characteristics determine collective behavior of the swarm









3 Summary


Ordinary Markov Process DefinitionOrdinary Markov Process DefinitionStochastic process takes values n1 n2hellip at times

t0 t1 hellip is a Markov process if it has a

Markov property value at time ti+1 depends only on its value at time ti and no other times

Dynamics of the process is fully determined byndash Initial state p(nt0)ndash Transition probability p(n ti+1|nrsquo ti)

Or p(n t+∆t|nrsquo t)


Random WalkRandom Walk

Random walk is a Markov process

Probability distribution of positionndash p(n t)

Transition probabilitiesndash p(n+1 t+∆t|n t)=12ndash p(n-1 t+∆t|n t)=12

n n+1n-1n-2 n+2


Dynamics of a Random WalkDynamics of a Random Walk

n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=


Stochastic Master EquationStochastic Master Equation

sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(

Describes dynamics of a Markov processTransition rates w

ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0


From One to ManyFrom One to ManyCollective state of an ensemble of stochastic

processesEach process ndash Independent and indistinguishablendash Can take exactly one value from n1hellip nn at time t

Collective state = occupation vector N1 hellip Nmndash Ni is number of processes that have a value ni

ndash P(N t) is probability system is in configuration N at time t


Collective State of Ensemble of Random Collective State of Ensemble of Random WalkersWalkers

n n+1n-1n-2 n+2

ρ(n t) ndash Number (density) of particles at position x

n

t0

t1


Collective DynamicsCollective DynamicsCollective Master Equationsndash Describes how probability density of the collective configuration

changes in timendash Can be derived from the probability distribution of the

ensemble and the transition rates

Butndash It is often difficult to specify the correct probability

distribution of the collective statendash Instead average over the ensemble to get the Rate

Equation


Rate EquationRate Equation

sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd

ndash Describes how Nn average number of processes with value nchanges in time

ndash No need to know exact probability distributionsRegardless of underlying probability distribution the mean evolves according to the Rate Equation

ndash Transition rates w are individual transition ratescan depend on the N values (ldquodensity dependentrdquo)

ndash Usually phenomenologicalCan be written down simply by assessing what important characteristics of the problem are


Analysis of Rate EquationsAnalysis of Rate EquationsSolve dNndtndash subject to initial conditions values of Nn at t=0ndash To obtain Nn vs t

Analyze solutionsndash Does a steady state (s s) exist

dNndt=0ndash If not what is dynamics

oscillation chaosndash If so

How long to converge to s sHow does s s depend on critical parameters

ndash How quickly does system respond to changes


Roadmap to Modeling Robot SwarmsRoadmap to Modeling Robot SwarmsApply stochastic processes framework to

study swarms of reactive robotsStarting with an individual robotndash Described by its controllerndash Compute transition rates w from physical parameters

Make transition to a multi-robot swarmndash Practical ldquoreciperdquo for constructing the Rate Equation from the

robot controllerndash Solve equations for a given set of parameters


Reactive RobotsReactive RobotsReactive robot is a minimalist robot in which

perception and action are tightly coupled

Reactive robot makes decision about what action to take based on the action it is currently executing and input from sensors


Reactive Robot as Markov ProcessReactive Robot as Markov ProcessReactive robot as a Markov Processndash Robotrsquos action at time t+∆t depends only on the action it is

executing at time tndash Inputs from sensors trigger transitions between actions

Individual descriptionndash p(n t) is probability robot is executing action n at time tndash Stochastic Master Equation describes dynamics of p(n t)

Collective descriptionndash Homogeneous group of reactive robots executing the same

controllerndash Rate Equation describes dynamics of the average number of

robots executing action n at time t


Representation of a Reactive RobotRepresentation of a Reactive RobotReactive robot controller as a finite state automaton (FSA)ndash State = action robot is executingndash Transitions between states triggered by sensory inputs

Example simplified foraging scenario

searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home


MultiMulti--robot System Representationrobot System Representation

Homogeneous swarm is represented by the same FSAndash State = (average) number of robots executing an actionndash Transitions between states triggered by sensory inputs

searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home


CoarseCoarse--graininggraining

searchDetectobject

Avoidobstacle

search Avoidobstacle

search

bull Coarse-graining reduces the complexity of the modelbull Helps construct a minimal model that explains experiments


A ldquoReciperdquo for Rate EquationsA ldquoReciperdquo for Rate Equationssearching

Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt

dNssps Nw rarrminus hsh Nw rarr+

=dt

dN psps Nw rarr+ php Nw rarrminus psh NNNN ++=

Initial conditions Ns(t=0)=N Nh(0)=0 Np(0)=0June 8 2005 Analyzing Swarms Tutorial 39118

Transition Rates Transition Rates wwjj kkTransition between states is triggered

ndash By a stimulus Obstacle another robot in a particular state location (eg home) communication

ndash By a timerTurn in a random direction for x seconds

Computing transition ratesndash Calculated from theory

Microscopic theoryMust know exact probability distributions

Phenomenological ldquoscattering cross-sectionrdquo approachTriggers are uniformly distributed in spaceRobots encounter triggers randomly

ndash Estimated from data by Calibration Fitting model to the data


ldquoldquoScattering CrossScattering Cross--sectionrdquo Approachsectionrdquo ApproachUseful idealization for roughly estimating model

parameters

A is arena area v is robotrsquos speeddi is robotrsquos detection widthMi is number of objects i

v∆t

di

AMvdw ii=


Calibration ApproachCalibration ApproachIn simulation or experiment measure single robot

interactions To calculate rate robots encounter each otherndash Run experiment or simulation in an empty arena with two robotsndash Keep track of the number of collisions

To calculate rate robots encounter objectsndash Run experiment or simulation with a single robot and objects

scattered around the arenandash Keep track of the rate robot encounters them (eg picks up

pucks)






2 Generalized Markov ProcessesBasic theoryApplications Dynamic task allocation


3 Summary


Robot ForagingRobot ForagingCollect objects scattered in

the arena and assemblethem at a ldquohomerdquo locationndash Can be accomplished by a

single robot or a group of robots

Foraging model validated using PlayerStage grounded simulations

Goldberg amp Matarić


Single Single vsvs Group of RobotsGroup of RobotsBenefits of group foragingndash Group is robust to individualrsquos failurendash Group can speed up collection by working in parallel

Disadvantages of a groupndash Increased interference due to collision avoidance

Optimal group sizendash beyond some group size interference outweighs the benefits of

the grouprsquos increased robustness and parallelism


Robot Controller DiagramRobot Controller Diagram

start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle


CoarseCoarse--grained Macroscopic Diagramgrained Macroscopic Diagram

searching homing

avoidingavoiding

startstart searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle


Model VariablesModel VariablesMacroscopic dynamic variables Ns(t) = number of searching robots at time t

Nh(t) = number of homing robots at time t

Nsav(t) Nh

av(t) = number of avoiding robots at time t

Mu(t) = number of undelivered pucks at time t

searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av


Model ParametersModel ParametersParameters ndash connect the model to experimentsαr αrsquor = rate of encountering a robot while searching homing

αp = rate of encountering a puck

τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh


Mathematical Model of ForagingMathematical Model of Foraging

=dt

dNsusp MNαminus [ ]0NNN ssr +minusα h

hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1

minususp MNα [ ]0NNN hhr +primeminusα avhN

τ1

+



( )0NNN hhr +primeα

searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs

avs NNNNN minusminusminus= 0

Initial conditions Ns(t=0)=N0



Uncollected puckshavhu NNMM minusminus= 0

Number of pucks in the arena changes when robots deliver them home

hh

Ndt

dMτ1

minus=

Initial conditions M(t=0)=M0


Searching Robots and Pucks Searching Robots and Pucks vsvs TimeTime

robots

pucks


Pucks Pucks vsvs Time for Different Group SizesTime for Different Group Sizes

N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)

5 avoid while homing4 avoid3 reverse home2 home1 collect0 search

PlayerStage SimulationsPlayerStage SimulationsAverage time spent in each behavior


Linking Model to SimulationsLinking Model to SimulationsGet transition rates from experimental parameters

To calculate robot encounter rate αrndash Run experiments in an empty arena with two robotsndash Keep track of the number of collision avoidance maneuvers

During searching αr=008 (includes wall avoidance)During homing αrsquor=005

To calculate puck encounter rate αpndash Run experiments with a single robot and pucks scattered around

the arenandash Keep track of the rate robot picks them up

αp =002


Average Homing Time Average Homing Time ττhh

0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing

[ ]Nrhh ταττ prime+= 10Empirical results suggest


ldquoldquoInterference Strengthrdquo Interference Strengthrdquo ττ

260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)

avoid whilesearchingavoid whilehoming

Time per collision avoidance maneuver τ=τ0[1+βN]τ0 measures ldquointerference strengthrdquo ndash studied τ0=3s τ0=15s


Comparison with SimulationsComparison with Simulations

0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)

avoid time = 3savoid time = 1smodel (3 s)model (1 s)

Optimal group size is smaller for higher interferenceOptimal group size is smaller for higher interference


CaveatsCaveatsParametersndash αr=008 (calibration) vs 004 (best fit)ndash αrsquor=005 vs 008ndash αp= 002 vs 002ndash τh

1=15s+-1s vs 16s

Despite these differences the minimal model shows good quantitative agreement with the dataModel does not take into account reverse homing or collecting behaviors of simulated robots


StickStick--Pulling Experiments in RobotsPulling Experiments in Robots

Collaboration in a group of reactive robotsndash Task completed only through collaborationndash Experiments with 2 ndash 6 Khepera robotsndash Minimalist robot controller

(Ijspeert Martinoli amp Billard 2001)


Experimental ResultsExperimental ResultsKey observationsbull Different dynamics for

different ratio of robots to sticks

bull Optimal gripping time parameter


Flowchart of the Robot ControllerFlowchart of the Robot Controllerstart look for sticks

object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u

start look for sticks

object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al


Model VariablesModel VariablesMacroscopic dynamic variables Ns(t) = number of robots in search state at time t

Ng(t) = number of robots gripping state at time t

M(t) = number of uncollected sticks at time t M(t)=M0-Ng(t)

search

grip

s u

Ns

Ng


Model ParametersModel Parametersconnect the model to the real systemα = rate of encountering a stick

αRG = rate of encountering a gripping robot

γ = stick release rate (corresponds to experimental 1τ)

search

grip

s uα

αRG γ


Mathematical Model of CollaborationMathematical Model of Collaborationsuccessful collaboration

find amp grip sticks

unsuccessful collaboration0NNN gs =+

( ) ggsGgss NNNRNMN

dtdN γαα ++minusminus= 0

consttM =)( for static environment

Initial conditions 00 )0(0)0()0( MMNNN gs ===


Dimensional AnalysisDimensional AnalysisRewrite equations in dimensionless form by making the following transformations

ndash only the parameters β and γ appear in the eqns and determine the behavior of solutions

Collaboration ratendash rate at which robots pull sticks out

βββ G

s

RMN

NNn

==

rarr~

00

0

( )nnR minus= 1~)( ββγβ


Searching Robots Searching Robots vsvs TimeTimeSe

arch

ing

robo

ts f

ract

ion

time


SteadySteady--state Searching Robots state Searching Robots vsvs γγ

β=10

β=15

β=05


Collaboration Rate Collaboration Rate vsvs ττ

β=10

β=05

Analytic resultscritical β

optimal stick release rate

Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15


Comparison with ExperimentsComparison with Experiments

theory experiment + simulation

4 robots

6 robots

2 robots

4 robots

6 robots

2 robots

Experiments+simulationsIjspeert et al 2001

Minimal 2-state model with γ



Minimal 2-state modelLerman et al 2001

Complete model+simulationsMartinoli et al 2004


BreakBreak

Adaptive RobotsAdaptive RobotsDynamic task allocation in a multi-foraging scenario

Dynamically achieve an appropriate division of laborFraction of Red robots = fraction of Red pucks

even as the distribution of Red pucks changes

Jones 2004








3 Limitations of the approach


Generalized Markov Process DefinitionGeneralized Markov Process DefinitionStochastic process takes values n1 n2hellip at times t0

t1 hellip is a generalized Markov process of order m if Value at time ti+1 depends on its values at times tiand ti-1 ti-2 hellip ti-m

Dynamics of the process is fully determined byndash Initial states p(n1t1) hellip p(nmtm)ndash Transition probability p(ni+1 ti+1|ni tihellip ni-m ti-m)

Continuous time p(n t+∆t|n1 t-∆thellip nm t-m∆t) - p(n t|h)History h=(t-∆thellip nm t-m∆t)


Rate Equation for Rate Equation for gMPgMPDerive a macroscopic equation for rate of change of

Nn average number of processes with value n at time t

Same as before except transition rates are averaged over histories

sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(


Dynamic Task AllocationDynamic Task AllocationMulti-foraging task

Robots are assigned to forage for Red or Green pucksGoal

Dynamically achieve an appropriate division of laborbull Fraction of Red robots = fraction of Red pucks


Dynamic Task Allocation MechanismDynamic Task Allocation MechanismRobot makes local observations and adds them to memory (length m)

Each robot estimates the proportion of pucks and robots in the environment (from memory) and switches its foraging state accordingly

Robot can be modeled as a generalized Markov process of order m


Model VariablesModel VariablesNR(t) number of robots in Red foraging stateNG(t) number of robots in Green foraging stateMR(t) number of Red pucksMG(t) number of Green pucks

SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG


Model ParametersModel ParametersεfR G rate robots switch from Red to Green states εfG R rate robots switch from Green to Red statesαR rate robots collect Red pucksαG rate robots collect Green pucksmicro rate at which new pucks are added

SearchRed

SearchGreen

o1 o2 hellip omfR G fG R

start


Modeling Adaptive Task AllocationModeling Adaptive Task Allocation

=dt

dNR

NNN RG =+

Initial conditions NR(t=0)=N

SearchRed

SearchGreen

o1 o2 hellip om

start

RGR Nhf )(rarrminus εGRG Nhf )(rarrε


Mathematical Form of Mathematical Form of ffTransition rates depend on observed densities of robots and pucks

nR observed density of Red robotsmR observed density of Red pucks

Mathematical form of transition ratesfR G=(1- mR)g(nR-mR)fG R=mRg(mR-nR)

ndash Guarantees nR=mR in steady state

g(mR-nR)


Observed DensitiesObserved Densitiest-∆t-2∆hellipt-|h|∆

Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

helliphellip hellip hellip

)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr


Physical ParametersPhysical Parameters

Simulations performed in 3D physically realistic world with full dynamics50 pucks 20 robots N=20 M=50Robot speed = 02 msDistance between observations = 2m

10s between observations ε=01 s-1

Robotrsquos view area Avis= 131 m2

Arena area A = 315 m2 αM=AvisMA=21


Dynamics of Red Robots Dynamics of Red Robots ndashndash Linear Linear ggSolutions show oscillations characteristic of delay equationsSolutions eventually relax to puck distributionMagnitude of oscillations and relaxation time depend on history length

Dynamics of Red Robots Dynamics of Red Robots ndashndash Power Power ggSolutions relax to correct puck distributionOscillation appear for much larger history length values

Lessons LearnedLessons LearnedStochastic processes theory provides a fruitful

framework for analyzing robot swarms

Created a models of collective swarm dynamics based on theory of stochastic processes ndash No need to know exact trajectoriesndash Allow quantitative analysis of collective behavior

Application to distributed robotic swarmsndash Reactive robotsndash Adaptive robots

Resultsndash Theoretical predictions agree with experimental resultsndash Analytic results not obtainable by other methodsndash Insights into robot design


LimitationsLimitationsSimplifying assumptions made by the stochastic

processes approachDilute limit ndash Robotsrsquo actions are independent of one anotherndash Robotsrsquo detection footprints do not overlap significantlyndash Valid for low densities of robots in the arena

Homogeneous robot systemsndash Robots are largely similar to one another (sensing actuation

control program)ndash Collective behavior is well characterized by mean parameter

values not distributionsndash These parameters can be calculated from individual robotrsquos

behavior


LimitationsLimitationsRate Equations describe dynamics of average behaviorndash Swarm behavior is well characterized by its average behavior

Fluctuations are not significantBetter description of larger systems

ndash Should be compared to results averaged over many experiments not a single trial

Spatially uniform environmentndash Transition rates well represented by average valuesndash Probability densities independent of position robotrsquos trajectories








3 Summary

Swarms with Significant Spatial DependenceSwarms with Significant Spatial DependenceReconfigurable robotsndash robots made of many smaller robots (ldquomodulesrdquo)ndash tight physical constraints

analogous to solids not dilute gases

ndash spatial fields defined by modules used for local controleg to adjust shape to move through pipes

Microscopic robotsndash interesting applications involve environments with spatial

gradients


Microscopic RobotsMicroscopic RobotsRobots the size of bacteriandash eg programmable bacteriandash eg via molecular engineered devices

Caveat canrsquot build yetndash design studies indicate plausible trade-offsndash analysis and simulation are the only options currently

Swarm motivationndash need many to have large effectndash limited hardware capabilities


Some Design StudiesSome Design Studies

Artificial blood cells (ldquorespirocytesrdquo)ndash hold ~100x O2 as red cells in same volume

R Freitas Artif Cells Blood Substitutes and Immobilization Biotechnology 26411 (1998)

Artificial immune cellsndash fast detection of infectious microbe

Casal et al Proc of Stanford Biomedical Computation Symposium 2003

Microsurgical repair of damaged nervesndash before nerve degenerates

Hogg amp Sretavan Proc of AAAI-2005


Task Respond to InjuryTask Respond to InjuryMonitor for chemical signalFollow gradient to sourcendash coordinate avoid too many responders

ie donrsquot block vessels

Identify infectious microbendash chemical contact sensing

fast and accurate at small distances

Pass info to attending physicianndash which immune cells canrsquot do


a robotrsquos microenvironmentschematic of one robot in ~20 micron blood vessel

cf artist conceptions oftenshow much more open space

vessels lt01mm diameter~10 total blood volume

~95 of ~500m2 surface areagt99 of ~5x104 km length




Scenario Scenario Find Chemical SourceFind Chemical Source

1012 robots in 5-liter blood volumendash use about 10-5 of blood volume

compared to ~40 used by red cells

ndash total mass of all robots ~02 g

Power to move ~10-12 watt at 1mmsndash so if all move at once 1 wattndash vs a person at rest using ~100 watts


task nerve repairtask nerve repairApproaches ndash regeneration via appropriate chemicalsndash repair via replacement with graft tissue

Challengendash disconnected axons degrade in 1-2 days

no longer receive proteins from cell body


Nerve RepairNerve Repair

graft ~1cm

undamagedhost

undamagedhost

MEMS device1mm3surgery area

(10rsquos of microns long and wide)showing a few exposed axons

in fluid lower than body temp reduces tissue injury

in vitro repair demonstrated for single axons with MEMSin vivo must measure and manipulate ~1000 axons in nerve

D Sretavan et al Neurosurgery to appear 2005June 8 2005 Analyzing Swarms Tutorial 100118

MEMS + Swarm of Microscopic RobotsMEMS + Swarm of Microscopic Robotseg 104 microscopic robotsExamine individual axons for viabilityMove and connect axonsLong-range coordination via MEMS devices

spatial variation due to chemicals and locations of other robots


Application for Stochastic AnalysisApplication for Stochastic AnalysisBrownian motion adds randomnessextremely large numbers of robots

Stochastic assumptions are reasonablendash but must include spatial variation


Stochastic Processes in External FieldsStochastic Processes in External FieldsPrevious theory does not describe systems with spatial correlationsndash Swarms of agents interacting through chemical fields

Ant colonies interacting through pheromonesRobots monitoring chemicals released into fluid

Generalizationsndash Probability a stochastic process takes value k at location x at

time tp(k x t)

ndash Density of processes taking value k at location x at time tNk(x t)

ndash External field ρ(x) with which the processes are interacting


Generalized Rate EquationGeneralized Rate Equation

int sum

int sum

primeprimeminus

primeprimeprime=part

part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ

Transition rates wnnrsquo depend not only on values n and nrsquo but also on spatial coordinates and field valuesndash Contains kinematics


SimplificationSimplificationMotion (transitions in space) is decoupled from state transitions

))(()())()(( xwxxxxxxWw jkkjkjk ρδρρδ primeminus+primeprime=

Change in position while in state k

Change in state while at position x


Simplified Rate EquationSimplified Rate Equation

sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ

Kinematics and state transitions decoupledndash Motion operator describes kinematics of a process

may depend on state of robot

ndash Transition rates describe state transitions at some locationbased only on value of field


Motion OperatorMotion Operator

2nabla=image kk D Diffusive motion with diffusion constant D

nablasdotminusnabla=image vDkk2 Diffusion in a fluid

moving with velocity v

Chemotaxis diffusion and drift in direction of the gradient of ρ with velocity VD

)(2 ρDkk VD sdotnablaminusnabla=image


Applying Stochastic Analysis to Applying Stochastic Analysis to Swarms with Spatial VariationSwarms with Spatial Variation

External field may also be dynamicndash eg time-dependent diffusion

Robots could modify the fieldndash eg releasing chemicals when in certain states

Need more design studies amp simulationsndash compare predictions with stochastic analysis


Example Chemical Diffusion in FluidExample Chemical Diffusion in Fluid

flow ~1mms

10 micro

m

chemical source

30 microm

fluid flow pushing objects of size comparable to cellschemical diffusion coef ~300microm2s

robot attempts to detect chemical follow gradient to sourcefield can be complex and dynamic

One example of analysis approach ndash Galstyan et al at this conferencendash start with simple scenario


LimitationsLimitationsContinuous field approximationndash relevant scale of variation larger than underlying

discrete systemseg size of modules (for reconfigurable robots)eg mean-free path of molecules

ndash no abrupt changes to swarm due to small motionseg reconfigurable robot modules lose power if disconnected








3 Summary

System Design ApproachesSystem Design ApproachesAnalysis with stochastic modelsSimulationExperimentsDeployment


Stochastic Models amp SwarmsStochastic Models amp SwarmsSwarms use many robotsndash each robot fairly simple

limited view of the worldsimple control

ndash focus on tasks where overall behavior is keynot behavior of individual robots

each robot has little effect on swarm as a whole

These features stochastic models

Some examples comparing analysis with experimentsndash good match even for fairly small number of robots


ExamplesExamplesFinite-state machinesndash foragingndash collaborative stick pulling

Machines with memoryndash dynamic task allocation

Spatial variationndash shape changing (reconfigurable robots) ndash find chemical source (microscopic robots)


Some Future Uses for Stochastic AnalysisSome Future Uses for Stochastic AnalysisSwarms of different types of robots egndash a few with special hardware

eg long-range communication

ndash a few large robots + many smaller ones

Accuracy of stochastic assumptionsndash eg pick designs for which analysis works well

Dynamic overall controlndash can analysis suggest how user could alter behaviors in response

to changing global task requirementseg with broadcast communication to the swarm


Benefits of Stochastic ModelsBenefits of Stochastic ModelsEvaluate average behavior resulting from individual robot controlsndash identify thresholds for drastic changes in collective behavior

Explore wide range of design choicesndash for robot hardware capabilitiesndash for control methodsndash for swarms that cannot yet be built

helping guide future developmenteg identifying trade-offs among hardware capabilities control and task performance


Limitations of Stochastic ModelsLimitations of Stochastic ModelsNo info on extreme individual robot behaviorsndash eg as relevant for safety guarantees

Requires estimates for transition ratesndash especially difficult if robots have complicated control program

Large fluctuationsndash eg through global broadcast instructions


Further ReadingFurther ReadingLerman Martinoli amp Galstyan in Swarm Robotics Workshop (LNCS 3342) 2005ndash Mathematical models amp comparison with experimentsndash And references therein

Huberman amp Hogg in The Ecology of Computation North-Holland 1988ndash dynamics due to delays and uncertainty (steady-state

oscillations chaos)



Schedule

What is a Robot Swarm

How to Design a Swarm

Local Control amp Collective Behavior

Poor Design for One Ok for Many

Good Design for One Bad for Many

Swarm Sensitivity to Robot Behaviors

Coordination within Swarms

System Design Approaches

Simulation

Experiment

Deployment

Another Choice Analysis


Outline

Historical Perspective

Stochastic Process

Some Examples

Robot as a Stochastic Process

Stochastic Approach to Studying Swarms

Outline

Ordinary Markov Process Definition

Random Walk

Dynamics of a Random Walk

Stochastic Master Equation

From One to Many

Collective State of Ensemble of Random Walkers

Collective Dynamics

Rate Equation

Analysis of Rate Equations

Roadmap to Modeling Robot Swarms

Reactive Robots

Reactive Robot as Markov Process

Representation of a Reactive Robot

Multi-robot System Representation

Coarse-graining

A ldquoReciperdquo for Rate Equations

Transition Rates wjk

ldquoScattering Cross-sectionrdquo Approach

Calibration Approach

Outline

Robot Foraging

Single vs Group of Robots

Robot Controller Diagram

Coarse-grained Macroscopic Diagram

Model Variables

Model Parameters

Mathematical Model of Foraging



Searching Robots and Pucks vs Time

Pucks vs Time for Different Group Sizes

PlayerStage Simulations

Linking Model to Simulations

Average Homing Time th

ldquoInterference Strengthrdquo t

Comparison with Simulations

Caveats

Stick-Pulling Experiments in Robots

Experimental Results

Flowchart of the Robot Controller


Model Variables

Model Parameters

Mathematical Model of Collaboration

Dimensional Analysis

Searching Robots vs Time

Steady-state Searching Robots vs g

Collaboration Rate vs t

Comparison with Experiments


Break

Adaptive Robots

Outline

Generalized Markov Process Definition

Rate Equation for gMP

Dynamic Task Allocation

Dynamic Task Allocation Mechanism

Model Variables

Model Parameters

Modeling Adaptive Task Allocation

Mathematical Form of f

Observed Densities

Physical Parameters

Dynamics of Red Robots ndash Linear g

Dynamics of Red Robots ndash Power g

Lessons Learned

Limitations

Limitations

Outline

Swarms with Significant Spatial Dependence

Microscopic Robots

Some Design Studies

Task Respond to Injury

Scenario Find Chemical Source

task nerve repair

Nerve Repair

MEMS + Swarm of Microscopic Robots

Application for Stochastic Analysis

Stochastic Processes in External Fields

Generalized Rate Equation

Simplification

Simplified Rate Equation

Motion Operator

Applying Stochastic Analysis to Swarms with Spatial Variation

Example Chemical Diffusion in Fluid

Limitations

Outline


Stochastic Models amp Swarms

Examples

Some Future Uses for Stochastic Analysis

Benefits of Stochastic Models

Limitations of Stochastic Models

Further Reading

ScheduleScheduleMotivation How to design swarmsAnalysis based on stochastic processesndash basic theoryndash applications amp comparing to experiments

BREAK

Generalizationsndash adaptive robotsndash reconfigurable and microscopic robots

Summary and future directions












































Very expensive















3 Summary











Van Kampen 1992





























3 Summary












n n+1n-1n-2 n+2



n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading











































Very expensive















3 Summary











Van Kampen 1992





























3 Summary












n n+1n-1n-2 n+2



n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading














3 Summary











Van Kampen 1992





























3 Summary












n n+1n-1n-2 n+2



n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading










Van Kampen 1992





























3 Summary












n n+1n-1n-2 n+2



n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




























3 Summary












n n+1n-1n-2 n+2



n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading











n n+1n-1n-2 n+2



n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


n n+1n-1n-2 n+2

)()(

)1()1()(

)1()1(

)1()1(

tnpwtnpw

tnpwtnpwdt

tndp

nnnn

nnnn

minusrarr+rarr

rarr+rarrminus

minusminus

++minus=

[ ] )()1()1(21)( tnptnptnp

dttndp

minus++minus=



sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


sumsumprime

primeprime

prime minusprime=n

nnn

nn tnpwtnpwdt

tndp )()()(


ttnttnpw

tnn ∆

prime∆+=

rarr∆prime

)|(lim0








n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading







n n+1n-1n-2 n+2


n

t0

t1







Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading






Equation



sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


sumsumprime

primeprime

primeprime minus=n

nnnn

nnnn NwNw

dtNd































searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


























searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home




searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading



searching

pickuphoming

start

Gripperclosed

PuckdetectedReach

home



searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


searchDetectobject

Avoidobstacle


search




Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


Ns

pickupNp

homingNh

startws p

wp h

wh s

=dt


=dt












parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading










parameters


v∆t

di

AMvdw ii=






pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading





pucks)








3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading







3 Summary














start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading













start searching

avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle



searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


searching homing

avoidingavoiding


avoid obstacle

collect

homing

detect object

reverse homing

detect object

avoid obstacle




Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading



Nsav(t) Nh



searching homing

avoidingavoiding

start

Ns Nh

Nsav Nh

av




τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading



τ = avoiding time

τh = homing time

searching homing

avoidingavoiding

start

αr αrsquor

αp

1τ1τ1τh



=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


=dt


hN

τ1

+ avsN

τ1

+

searching homing

avoidingavoiding

start

=dt

dNhh

h

Nτ1


τ1

+




searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading



searching homing

avoidingavoiding

start

=dt

dN avh av

hNτ1

minus

avhhs







hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




hh

Ndt

dMτ1

minus=




robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


robots

pucks



N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


N=5

N=15

N=20


000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading

000

20000

40000

60000

80000

100000

120000

1 2 3 4 5 6 7 8 9 10

number of robots

time

(s)









αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading






αp =002



0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

number of robots

ti

me

hom

ing




260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


260

300

340

380

420

460

0 2 4 6 8 10

number of robots

time

colli

sion

(s)





0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10

number of robots

time

(s)





1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


1=15s+-1s vs 16s












object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading










object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al



search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


search

grip

s u


object detected

obstacle

gripped

grip amp wait

time out

teammatehelp

release

obstacleavoidance

success

Y

N

Y

N

N

NN

Y

Y

YIjspeert et al





search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




search

grip

s u

Ns

Ng





search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




search

grip

s uα

αRG γ





( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




( ) ggsGgss NNNRNMN








βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




βββ G

s

RMN

NNn

==

rarr~

00

0




arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


arch

ing

robo

ts f

ract

ion

time



β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


β=10

β=15

β=05



β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


β=10

β=05



Gc R+

=1

2β

)1(2

1 Gopt R+minus=βγ

β=15




4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading



4 robots

6 robots

2 robots

4 robots

6 robots

2 robots








BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading





BreakBreak




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




Jones 2004


















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading

















sumsumprime

primeprime

primeprime minus=n

nhnnn

nhnnn NtwNtw

dttdN )()()(











SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading










SearchRed

SearchGreen

o1 o2 hellip om

NR

start

NG



SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


SearchRed

SearchGreen


start



=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


=dt

dNR

NNN RG =+


SearchRed

SearchGreen

o1 o2 hellip om

start







g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading





g(mR-nR)



Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


Robotrsquosmemory

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn

hobsR

hobsR

m

n

1

1

obsR

obsR

mn

0

0

obsR

obsR

mn


)()(

∆minus

∆minus

htmhtn

R

R

)2()2(

∆minus

∆minus

tmtn

R

R

)()(

∆minus

∆minus

tmtn

R

RAllrobots

hobsR

hobsR

mn

2

2

obsR

obsR

mn

1

1

obsR

obsR

mn

sum ∆minus=

sum ∆minus=

=

=

h

iRR

h

iRR

itmh

m

itnh

n

1

1

)(1

)(1

)()()1(

RRRRG

RRRGRnmgmf

mngmfminus=

minusminus=

rarr

rarr





















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading




















behavior













3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading












3 Summary





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading





gradients






































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading





































graft ~1cm

undamagedhost

undamagedhost















time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading










time tp(k x t)





int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading


int sum

int sum

primeprimeminus


part

)()(

)()()(

txNtxxwxd

txNtxxwxdt

txN

kj

kj

jjjk

k

ρ

ρ









sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading







sum

sum

minus

+image=part

part

jkkj

jjjkkk

k

txNw

txNwtxNt

txN

)()(

)()()()(

ρ

ρ


















flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading














flow ~1mms

10 micro

m

chemical source

30 microm















3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading











3 Summary































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading































oscillations chaos)



Schedule









Simulation

Experiment

Deployment



Outline


Stochastic Process

Some Examples



Outline


Random Walk



From One to Many


Collective Dynamics

Rate Equation



Reactive Robots




Coarse-graining





Outline

Robot Foraging




Model Variables

Model Parameters











Caveats





Model Variables

Model Parameters








Break

Adaptive Robots

Outline





Model Variables

Model Parameters



Observed Densities

Physical Parameters



Lessons Learned

Limitations

Limitations

Outline


Microscopic Robots

Some Design Studies



task nerve repair

Nerve Repair





Simplification


Motion Operator



Limitations

Outline



Examples




Further Reading

a stochastic processes approach to studying robotic swarm … · 2005-06-11 · a stochastic...

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