Objectives - Penn Engineeringesteager/Dynamic_Vehicle... · 2010-02-23 · 1 motion coordination:...

Overview of MURI Activities:Coverage control, partitioning policies, pursuit

algorithms, and dynamic vehicle routing

Francesco Bullo

Center for Control,Dynamical Systems & Computation

University of California at Santa Barbara

http://motion.me.ucsb.edu

Review Meeting, ARO SWARMS MURI W911NF-05-1-0219University of Pennsylvania, February 23, 2010

Francesco Bullo (UCSB) SWARMS MURI Overview 23feb10 @ Penn 1 / 53

Objectives

Develop broad set of coordination algorithms for autonomous networkedvehicles:

1 Coverage and surveillanceminimal gossip asynchronous comm

2 Pursuit strategies

3 Task allocation and dynamic vehicle routing


Technical Approach

Distributed algorithms

Graph theory and combinatorics

Computational geometry and geometric optimization

Nonlinear control theory

Queueing theory and stochastic processes


Major Breakthroughs

Coverage control

gossip communicationadaptive coverage and explorationboundary monitoring via synchronization

Cooperative pursuit strategies

pursuit with sensing limitations (limited range and range-only)pursuit with agility limitations

Task allocation and dynamic vehicle routingstatic and dynamic problemsgeometric heuristics and approximation algorithmsgrowing range of models and scenarios


DoD Benefits

Coverage and surveillance strategies

minimal communicationautomatic synchronization

Novel bio-inspired approaches to pursuit/evasion

strategies incorporating sensing/mobility constraints

Geometric algorithms for target assignment and vehicle routingprovably optimal or constant-factor under simplified assumptionsgrowing range of models and scenarios


SWARMS Collaborations

1 David Skelly: bio-inspired cooperative pursuit strategies

2 Daniela Rus and David Skelly: adaptive coverage and explorationStephen Smith (from UCSB PhD to postdoc with CSAIL @

MIT)

3 Ali Jadbabaie: discrete coverage control and network flow

4 Brian Anderson and Stephen Morse: network localization

5 Dan Koditschek: topology of the space of partitions

6 Emilio Frazzoli: dynamic vehicle routing


Success Stories

Plenary speaker: NESC 05, BMSC 06, HSCC 06, Int Conf Applied Math andComputing 08, IEEE MSC ’09, IFAC NECSYS ’10

Promoted to Full Professor & ME vicechair at UCSB

Elevated IEEE Fellow, 2010

Article selection for inclusion in SIGEST section of SIAM Review ’09

Outstanding Paper Award, IEEE Control Systems Magazine, 2008

Students recognition: Best student paper award at ACC06; finalist atCDC05, CDC07, and ACC’10. UCSB CCDC Best PhD Thesis. DoDSMART fellowship

6 PhD thesis on cooperative control: Sara Susca, Ketan Savla, AnuragGanguli, Giuseppe Notarstefano, Stephen Smith, Shaunak Bopardikar

Special Issue in SIAM Journal of Control and Optimization on Control andOptimization in Cooperative Networks, Jan 2009

Workshops: at Centro De Giorgi in Dec 2007, CDC in Dec 08, minitutorialat 2009 SIAM Conf on Dynamical Systems, BMSC 2009, ACC 2010

Open source software library for visibility computation, http://visilibity.org


Publications: Journal articles

1 S. Susca, S. Martınez, and F. Bullo. Monitoring environmental boundaries with a robotic sensor network. IEEETransactions on Control Systems Technology, 16(2):288–296, 2008

2 C. Gao, J. Cortes, and F. Bullo. Notes on averaging over acyclic digraphs and discrete coverage control. Automatica,44(8):2120–2127, 2008

3 K. Plarre and F. Bullo. On Kalman filtering for detectable systems with intermittent observations. IEEE Transactions onAutomatic Control, 54(2):386–390, 2009

4 S. L. Smith and F. Bullo. Monotonic target assignment for robotic networks. IEEE Transactions on Automatic Control,54(9):2042–2057, 2009

5 S. D. Bopardikar, F. Bullo, and J. P. Hespanha. On discrete-time pursuit-evasion games with sensing limitations. IEEETransactions on Robotics, 24(6):1429–1439, 2008

6 R. Carli and F. Bullo. Quantized coordination algorithms for rendezvous and deployment. SIAM Journal on Control andOptimization, 48(3):1251–1274, 2009

7 S. L. Smith and F. Bullo. The dynamic team forming problem: Throughput and delay for unbiased policies. Systems &Control Letters, 58(10-11):709–715, 2009

8 S. D. Bopardikar, F. Bullo, and J. P. Hespanha. A cooperative Homicidal Chauffeur game. Automatica, 45(7):1771–1777,2009

9 S. D. Bopardikar, S. L. Smith, F. Bullo, and J. P. Hespanha. Dynamic vehicle routing for translating demands: Stabilityanalysis and receding-horizon policies. IEEE Transactions on Automatic Control, January 2010. (Submitted, Mar 2009) toappear

10 M. Pavone, E. Frazzoli, and F. Bullo. Distributed and adaptive algorithms for vehicle routing in a stochastic and dynamicenvironment. IEEE Transactions on Automatic Control, August 2009. (Submitted, Apr 2009) to appear

11 G. Notarstefano and F. Bullo. Distributed abstract optimization via constraints consensus: Theory and applications. IEEETransactions on Automatic Control, October 2009. Submitted

12 F. Bullo, R. Carli, and P. Frasca. Gossip coverage control for robotic networks: Dynamical systems on the the space ofpartitions. SIAM Review, January 2010. Submitted

plus 25 conference articles and book chapters


Publications: Book on Robotic Networks

1 intro to distributed algorithms (graphtheory, synchronous networks, andaveraging algos)

2 geometric models and geometricoptimization problems

3 model for robotic, relative sensingnetworks, and complexity

4 algorithms for rendezvous,deployment, boundary estimation

Manuscript by F. Bullo, J. Cortes, and

S. Martınez. Princeton Univ Press, 2009,

ISBN 978-0-691-14195-4. Freely downloadable

at http://coordinationbook.info with

tutorial slides and software libraries.


Link to other efforts

ARO Institute for Collaborative Biotechnology, “Bio-inspiredStochastic Search and Decision Making for Robotic Networks”

NSF “Distributed Illumination Problems for Visually-guided Agents”

AFOSR MURI on “Decision Dynamics in Mixed Networks”

NSF “Minimalist Mapping and Monitoring”


Technology Transition

Established connections (conversations, mutual visits, planning):1 UtopiaCompression, Dr Jacob Yadegar2 Northrop Grumman, Dr Han Park3 PARC, Dr Haitham Hindi4 Mayachitra, Dr Marco Zuliani5 Los Alamost National Lab, Dr Misha Chertkov

PhD students in DoD labs1 Current PhD candidate Karl Obermeyer, SMART fellow at

UCSB/AFRL Wright Patterson

Graduate students in industry:1 Anurag Ganguli, UtopiaCompression2 Sara Susca, Honeywell Research Labs3 Nathan Owen, Boeing Space & Intelligence Systems

PhD students in academia:1 Ketan Savla, Research Scientist at MIT2 Stephen Smith, Postdoc at MIT with Daniela Rus3 Shaunak Bopardikar, Postdoc at UCSB4 Giuseppe Notarstefano, Professor at University of Lecce


Plans for Years 4-5

1 Research planned in proposal:

Design of motion coordination algorithms, e.g., deployment,connectivity maintenance, target assignmentCoordination with minimal (gossip, quantized, intermittent)communicationBioinspired pursuit behaviors

2 New unanticipated directions:

Distributed linear programming with applications to formation controland target localizationSynchronization for surveillanceQueueing theory for robotic networks


Dynamic Vehicle Routing for Robotic Networks:Models, Fundamental Limits and Algorithms

Francesco Bullo

Center for Control,Dynamical Systems & Computation

University of California at Santa Barbara

http://motion.me.ucsb.edu

Review Meeting, ARO SWARMS MURI W911NF-05-1-0219University of Pennsylvania, February 23, 2010

Francesco Bullo (UCSB) Dynamic Vehicle Routing 23feb10 @ Penn 13 / 53

Today’s Outline

1 IntroductionSummaryFrom coordination and static routing to Dynamic Vehicle RoutingLiterature review

2 Prototypical Dynamic Vehicle Routing ProblemBasic modelPreliminary attemptsAlgorithm design and analysis

3 DVR Extensions, Variations, Applications, . . .Vehicle dynamics: Dubins TSP problemVehicle coordination: Gossip partitioningTargets: Moving targetsTargets: Heterogeneous targets requiring teamsTargets: Priority levels


Summary: Dynamic Vehicle Routing for Robotic Networks

Problem Setup:DVR = planning policies for vehicles tovisit targets arriving in real time

1 vehicle: dynamics, comm models

2 coordination: partitions/teaming

3 targets: priority, mobile

Technical Approach

models: dimensional analysis,intrinsic regimes, phase transitions

fundamental limits onperformance (delay vs throughput)

algorithm design: optimal orconstant-factor, adaptive,distributed

Publications and Dissemination

3 PhD Theses @ UCSB: K. Savla,S. L. Smith and S. D. Bopardikar

9 journal articles (6 in TAC, 1SICON, 1 SCL, 1 AIAA JGCD)

plenary at ’09 IEEE Conf ControlApps, tutorial at ’09 SIAM DynamSystems Conf, ACC’10 workshop


Acknowledgements

Ketan Savla(PhD @ UCSB, nowresearch scientist @

MIT)

Stephen Smith(PhD @ UCSB, now

postdoc @ MIT)

Marco Pavone(graduating PhD @

MIT)

Emilio Frazzoli (MIT)

Shaunak D. Bopardikar (UCSB) and Joao P. Hespanha (UCSB):translating targets

Ruggero Carli (UCSB), Joey W. Durham (UCSB), andPaolo Frasca (Universita di Roma): gossip coordination


From coordination and static routing toDynamic Vehicle Routing

Simple coordination problems arise in static environments

1 motion coordination: rendezvous, deployment, flocking

2 task allocation, target assignment3 static vehicle routing (P. Toth and D. Vigo ’01)

Routing policies vs planning algorithms

dynamic, stochastic and adversarial events take place

1 design policies (in contrast to pre-planned routes or motion planningalgorithms) to specify how to react to events

2 dynamic demands add queueing phenomena to the combinatorialnature of vehicle routing


Literature review

Shortest path through randomly-generated and worst-case points(Beardwood, Halton and Hammersly, 1959 — Steele, 1990)

Traveling salesman problem solvers (Lin, Kernighan, 1973)

DVR formulation on a graph (Psaraftis, 1988)

DVR on Euclidean plane (Bertsimas and Van Ryzin, 1990–1993)

Unified receding-horizon policy (Papastavrou, 1996)

Recent developments in DVR for robotic networks:

Adaptation and decentralization (Pavone, Frazzoli, FB: TAC, in press)

Nonholonomic / Dubins UAVs (Savla, Frazzoli, FB: TAC 2008)

Pickup delivery tasks (Waisanen, Shah, and Dahleh: TAC 2008)

Heterogeneous vehicles and team forming (Smith and Bullo: SCL 2009)

Distinct-priority targets (Smith, Pavone, FB, Frazzoli: SICON, in press)

Moving targets (Bopardikar, Smith, Hespanha, FB: TAC, in press)


Today’s Outline





Prototypical Dynamic Vehicle Routing Problem

Given:

a group of vehicles, and

a set of service demands

Objective:provide service in minimum timeservice = take a picture at location

Vehicle routing (All info known ahead of time, Dantzig ’59)

Determine a set of paths that allow vehicles to service the demands

Dynamic vehicle routing (New info in real time, Psaraftis ’88)

New demands arise in real-time

Existing demands evolve over time



Given:











Given:











Given:











Given:










Today’s Outline





Plain-vanilla re-optimization?

Example: DVR on segment

Objective: minimize averagewaiting time

Strategy: re-optimize at eachevent

10 0.5

1 For adversarial target generation, vehicle travels forever without everservicing any request =⇒ unstable queue of outstanding requests

2 Even if queue remains bounded, what about performance? how farfrom the optimal?






10 0.5








10 0.5








10 0.5








10 0.5








10 0.5








10 0.5




Online algorithms?

Online algorithms (Jaillet and M. R. Wagner ’06)

online algorithm operates based on input information up to thecurrent time

online algorithm is r -competitive if

Costonline ≤ rCostoptimal offline(I ), ∀ problem instances I .

Disadvantages

1 cumulative cost

2 worst-case analysis

3 not possible to include a-priori information (e.g., arrival rate)

4 not as clear what competitive ratio means

5 so far, only few simple DVR problems admit online algorithms


Today’s Outline





Algorithm design

1 Distinct light load and high load behaviors

2 For light load, optimal assets placement for fastest response3 For high load,

1 solve combinatorial optimization over available info2 repeat in receding-horizon fashion

4 For multi-agent problems,1 (often) optimal load balancing via territory partitioning2 (other times) teaming


Algorithm design






Algorithm design






Algo #1: Receding-Horizon Shortest-Path policy

Receding-Horizon Shortest-Path (RH-SP)

For η ∈ (0, 1], single agent performs:

1: while no customers, move to center2: while customers waiting

1 compute shortest path through current targets

2 service η-fraction of path


Algo #1: Receding-Horizon Shortest-Path policy

Receding-Horizon Shortest-Path (RH-SP)

For η ∈ (0, 1], single agent performs:

1: while no customers, move to center2: while customers waiting

1 compute shortest path through current targets

2 service η-fraction of path

M. Pavone, E. Frazzoli, and F. Bullo. Distributed and adaptive algorithms for vehiclerouting in a stochastic and dynamic environment. IEEE Transactions on AutomaticControl, August 2009. (Submitted, Apr 2009) to appear


RH-SP analysis

Implementation:

NP-hard computation, but effective heuristics

Stability:1 queue is stable if service time < interarrival time

2 service time =length shortest path(n)

n(n = # customers)

3 queue is stable if (length of shortest path(n)) = sublinear f(n)


RH-SP analysis

Implementation:




n(n = # customers)



RH-SP analysis

Implementation:




n(n = # customers)



RH-SP analysis

Implementation:




n(n = # customers)



RH-SP analysis

Implementation:


Stability:

1 queue is stable if service time < interarrival time


n(n = # customers)


Combinatorics in Euclidean space (Steel ’90)

Worst-case and expected bounds

length shortest path(n) ≤ βworst

√n

limn→+∞

length shortest path(n) = βexpected

√n


RH-SP analysis: continued

Adaptation: the policy does not require knowledge of

1 vehicle velocity v , environment Q

2 arrival rate λ and spatial density function f

3 expected on-site service s

Performance:

1 in light load, delay is optimal

2 in high load, delay is within a multiplicative factor from optimal

3 multiplicative factor depends upon f and is conjectured to equal 2

no known adaptive algo with better performancevery little known outside of asymptotic regimes


Today’s Outline





Euclidean TSP and Dubins TSP

Euclidean TSP (ETSP)

NP-hard

effective heuristics available

length(ETSP) ∈ O(√

n)

Dubins TSP (DTSP)Given a set of points find the shortest tour with bounded curvature

not a finite dimensional problem

no prior algorithms or results (asof 2006)

length(DTSP) sub-linear in n ?

K. Savla, E. Frazzoli, and F. Bullo. Traveling Salesperson Problems for the Dubinsvehicle. IEEE Transactions on Automatic Control, 53(6):1378–1391, 2008


Stochastic DTSP

Problem Statement Given a set of n independently and uniformlydistributed points, design polynomial-time algorithm with smallestexpected DTSP tour length

Theorem: For n iid uniformly distributed points:

E[length of DTSP(n)] ∼ n2/3

Lower bound proof based on “area of reachable set”

1 area of reachable set in time t by Dubins with radius ρ is t3

3ρ

2 expected distance to nearest target (n iid uniform targets) is 34(3ρ

n )1/3

3 lenght of tour cannot be less than n times this distance


Constructive upper boundbased on environment tiling tuned to vehicle dynamics

10

!

p! p+B!(!)

!

Fig. 2. Construction of the “bead” B!(!). The figure shows how the upper half of the boundary is constructed, the bottom half is symmetric.

Next, we study the probability of targets belonging to a given bead. Consider a bead B entirely contained in Q

and assume n points are uniformly randomly generated in Q. The probability that the ith point is sampled in B is

µ(!) =Area(B!(!))

Area(Q).

Furthermore, the probability that exactly k out of the n points are sampled in B has a binomial distribution, i.e.,

indicating with nB the total number of points sampled in B,

Pr[nB = k| n samples] =!

n

k

"µk(1! µ)n!k.

If the bead length ! is chosen as a function of n in such a way that " = n · µ(!(n)) is a constant, then the limit

for large n of the binomial distribution is [31] the Poisson distribution of mean ", that is,

limn"+#

Pr[nB = k| n samples] ="k

k!e!" .

C. The Recursive Bead-Tiling Algorithm

In this section, we design a novel algorithm that computes a Dubins path through a point set in Q. The proposed

algorithm consists of a sequence of phases; during each of these phases, a Dubins tour (i.e., a closed path with

bounded curvature) will be constructed that “sweeps” the set Q. We begin by considering a tiling of the plane such

June 30, 2006 DRAFT

Q

Key properties of the bead

1 Beads tile the plane

2 Approaching and leaving a bead horizontally, Dubins can service a target

first analysis of joint combinatorics, dynamics and stochasticextensions to STLC systems by Itani, Dahleh and Frazzoli

extensions to multi-vehicle Dubins


Today’s Outline





Algo #2: Load balancing via territory partitioning

RH-SP + Partitioning

Each agent i:

1: computes own cell vi in optimal partition2: applies RH-SP policy on vi


Territory partitioning akin to animal territory dynamics

Tilapia mossambica, “Hexagonal

Territories,” Barlow et al, ’74

Red harvester ants, “Optimization, Conflict, and

Nonoverlapping Foraging Ranges,” Adler et al, ’03

Sage sparrows, “Territory dynamics in a sage sparrows

population,” Petersen et al ’87


Optimal partitioning cost functions

Expected wait time (light load problem)

H(p, v) =

∫v1

‖q − p1‖dq + · · ·+∫vn

‖q − pn‖dq

n robots at p = {p1, . . . , pn}environment is partitioned into v = {v1, . . . , vn}

H(p, v) =n∑

i=1

∫vi

f (‖q − pi‖)φ(q)dq

φ : R2 → R≥0 density

f : R≥0 → R penalty function


From optimality conditions to algorithms

H(p, v) =n∑

i=1

∫vi

f (‖q − pi‖)φ(q)dq

Theorem (Alternating Algorithm, Lloyd ’57)

1 at fixed positions, optimal partition is Voronoi

2 at fixed partition, optimal positions are “generalized centers”

3 alternate v-p optimization=⇒ local optimum = center Voronoi partition


Gossip partitioning policy

1 Random communication between two regions2 Compute two centers3 Compute bisector of centers4 Partition two regions by bisector

F. Bullo, R. Carli, and P. Frasca. Gossip coverage control for robotic networks: Dynam-ical systems on the the space of partitions. SIAM Review, January 2010. Submitted


Gossip partitioning policy: sample implementation

Player/Stage platform

realistic robot models in discretized environments

integrated wireless network model & obstacle-avoidance planner

J. W. Durham, R. Carli, P. Frasca, and F. Bullo. Discrete partitioning and cover-age control with gossip communication. In ASME Dynamic Systems and ControlConference, Hollywood, CA, October 2009


Gossip partitioning policy: analysis results

1 class of dynamical systems on space of partitionsi.e., study evolution of the regions rather of the agents

2 convergence to centroidal Voronoi partitions (under mild conditions)

3 novel results in topology, analysis and geometry:1 compactness of space of finitely-convex partitions with respect to the

symmetric difference metric2 continuity of various geometric maps (Voronoi as function of

generators, centroid location as function of set, multicenter functions)3 LaSalle convergence theorems for dynamical systems on metric

spaces with deterministic and stochastic switches

conjectures about topology of space of partitionsasymmetric gossip algorithms, akin to stigmergy

tolerance to failures, arrivals, and dynamic environments


Asymmetric gossip partitioning policy

Asymmetric communication akin to animal stigmergy

1 unidirectional delayed randomized links

2 use covering instead of partitions

3 doubly-greedy strategy:improve performance & diminish overlap

Asymmetric gossip partitioning algorithmRandom directed communication from region vi to region vj :

1 Compute two centers

2 Compute bisector of centers3 Perform two operations on region j

1 add every point in region vi that is closer to jth center2 del every point in region vj ∪ vi that is closer to ith center


Today’s Outline





DVR for moving targets

Very little is know about moving targets:

1 no polynomial time algorithms for shortest path

2 no length estimates

3 no efficient DVR algorithms

S. D. Bopardikar, S. L. Smith, F. Bullo, and J. P. Hespanha. Dynamic vehicle routing fortranslating demands: Stability analysis and receding-horizon policies. IEEE Transactions onAutomatic Control, January 2010. (Submitted, Mar 2009) to appear


Translating targets

Problem parameters:

speed ratio v :

v =target speed

vehicle speed

arrival rate λ

segment width W

deadline distance L

W

L

L = +∞ L is finiteStabilize queue Maximize capture fraction

v < 1 translational path policy translational path policy

v ≥ 1 Not possible for any λ > 0 longest path policy


Translating targets

Problem parameters:

speed ratio v :

v =target speed

vehicle speed

arrival rate λ

segment width W

deadline distance L

W

L


v < 1 translational path policy translational path policy

v ≥ 1 Not possible for any λ > 0 longest path policy


Shortest Translational Path Fraction (v < 1)

Shortest translational path fraction policy

1 Compute shortest translational path through all demands higherthan L/2.

2 Service demands on path for min of: L/(2v) time units, and time totravel entire path

3 Repeat

Translational path computation(Hammar and Nilsson, 2002)

Order: scaled shortest static pathMotion: intercept on straight line

L/2


Maximizing capture fraction for v ≥ 1

For v ≥ 1, it is optimal to remain on deadline

Reachable targets

Reachability graph is directed and acyclic

Longest path can be computed in polynomial time


Longest path policy (v ≥ 1)

Longest path (LP) policy

1 Compute reachability graph ofunserviced demands

2 Compute longest path in graph

3 Capture first demand on path byintercepting on deadline

4 Repeat


Summary of boundary guarding


v < 1

v ≥ 1 Not possible for any λ > 0


DVR for moving targets

Results (limits and constant-factor analysis removed for brevity):

problem structure: four regimes

fundamental limits on stability/capture fraction

efficient algorithms in all four regimes


v < 1 Translational path policy Translational path policy

v ≥ 1 Not possible for any λ > 0 Longest path policy


Studying more general scenarios

Relaxed assumptions:

Non-Poisson

Non-uniform

Different speeds

Different directions

Finite capture radius

More general setup:

Higher dimensions

Advance information

very little known about generic target motion models


Targets: Heterogeneous targets requiring teams

Problem setup:

Heterogeneous vehicles

Tasks require vehicle teams

Goal: Minimize task delay

Consider only unbiased policies:Equal expected delay to all tasks

Provably efficient policies in certain scenariosVery rich problem

S. L. Smith and F. Bullo. The dynamic team forming problem: Throughput anddelay for unbiased policies. Systems & Control Letters, 58(10-11):709–715, 2009


Targets: Priority levels

Problem setup:

n vehicles

Two classes of tasks α, β

α – high priorityβ – low priority

Goal: minimize cDα + (1− c)Dβ

c ∈ (0, 1) gives bias toward α

c = 0.80p = 0.82

Provably efficient policy

Extends to m classes

S. L. Smith, M. Pavone, F. Bullo, and E. Frazzoli. Dynamic vehicle routing withpriority classes of stochastic demands. SIAM Journal on Control and Optimization,48(5):3224–3245, 2010


Today: Dynamic Vehicle Routing for Robotic Networks

Problem Setup:DVR = planning policies for vehicles tovisit targets arriving in real time

1 vehicle: dynamics, comm models

2 coordination: partitions/teaming

3 targets: priority, mobile

Technical Approach

models: dimensional analysis,intrinsic regimes, phase transitions

fundamental limits onperformance (delay vs throughput)

algorithm design: optimal orconstant-factor, adaptive,distributed

Publications and Dissemination

3 PhD Theses @ UCSB: K. Savla,S. L. Smith and S. D. Bopardikar

9 journal articles (6 in TAC, 1SICON, 1 SCL, 1 AIAA JGCD)

plenary at ’09 IEEE Conf ControlApps, tutorial at ’09 SIAM DynamSystems Conf, ACC’10 workshop


Objectives - Penn Engineeringesteager/Dynamic_Vehicle... · 2010-02-23 · 1 motion coordination:...

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