Team-Triggered Coordination of Robotic Networks for Optimal - - PowerPoint PPT Presentation

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Team-Triggered Coordination of Robotic Networks for Optimal Deployment Cameron Nowzari 1 , Jorge Cort es 2 , and George J. Pappas 1 Electrical and Systems Engineering 1 University of Pennsylvania Mechanical and Aerospace Engineering 2

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SLIDE 1

Team-Triggered Coordination of Robotic Networks for Optimal Deployment

Cameron Nowzari1, Jorge Cort´ es2, and George J. Pappas1

Electrical and Systems Engineering1 University of Pennsylvania Mechanical and Aerospace Engineering2 University of California, San Diego American Control Conference Chicago, Illinois July 3, 2015

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SLIDE 2
  • Coordination of robotic networks-

Each individual senses immediate environment communicates with others processes information gathered takes action in response Multiple agents provide inherent robustness adaptive behavior enable tasks beyond individuals’ capabilities

  • C. Nowzari (Penn)

Team-triggered deployment

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2 / 22

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SLIDE 3
  • Optimal deployment- of robotic sensor networks

Objective: optimal task allocation and space partitioning

  • ptimal placement and tuning of sensors

Why? servicing resource allocation environmental monitoring data collection force protection surveillance search and rescue

  • C. Nowzari (Penn)

Team-triggered deployment

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SLIDE 4

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 5

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T Eve, where are you?!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 6

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T Eve, where are you?! I’m here!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 7

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T Now where are you?

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 8

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T Now where are you? Still here!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 9

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T How about now?

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 10

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T How about now? Still here...

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 11

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T And now??

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 12

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T And now?? Leave me alone.

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 13

Real-time implementations of -optimal deployment-

Time-triggered (periodic) coordination

Agents take actions at some fixed period T And now?? Leave me alone. Simple, but Wasteful

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

4 / 22

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SLIDE 14

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 15

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information Eve, where are you?!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 16

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information Eve, where are you?! I’m here!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 17

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information (She probably hasn’t moved too far...)

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 18

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information (She probably hasn’t moved too far...)

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 19

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information (I’ll ask her again in a few seconds)

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 20

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information (I’ll ask her again in a few seconds)

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 21

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information Now where are you?

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 22

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information Now where are you? Still here!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 23

Real-time implementations of -optimal deployment-

Self-triggered coordination

Agents decide when to take actions based on available information Now where are you? Still here! Actions taken only when necessary, but potentially still conservative!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

5 / 22

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SLIDE 24

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 25

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other Eve, where are you?!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 26

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other Eve, where are you?! I’m here!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 27

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other But I’m going this way!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 28

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 29

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 30

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 31

Real-time implementations of -optimal deployment-

  • Team-triggered coordination-

Cooperative agents share more information with each other Higher quality information allows for less communication!

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

6 / 22

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SLIDE 32
  • Team-triggered- control

Objective: Combine best properties of event- and self-triggered strategies into a unified, implementable approach How?

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

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SLIDE 33
  • Team-triggered- control

Objective: Combine best properties of event- and self-triggered strategies into a unified, implementable approach How? Agents make promises to neighbors about their future states Agents warn each other when promises need to be broken

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

7 / 22

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SLIDE 34

Outline

1 Motivation 2 Problem Formulation

aggregate objective optimization Voronoi partition

3 Triggered Deployment Algorithms

self-triggered deployment algorithm team-triggered deployment algorithm simulations

4 Conclusions

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

8 / 22

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SLIDE 35

Expected-value multicenter function

Objective: Given sensors/nodes/robots/sites (p1, . . . , pn) moving in environment S achieve optimal coverage φ : Rd → R≥0 density agent performance decreases with distance minimize H(p1, . . . , pn) = Eφ

  • min

i∈{1,...,n} q − pi2

  • C. Nowzari (Penn)

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  • Fri. July 3rd

9 / 22

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SLIDE 36

Voronoi partitions

Let (p1, . . . , pn) ∈ Sn denote the positions of n points The Voronoi partition V(P) = {V1, . . . , Vn} generated by (p1, . . . , pn) Vi = {q ∈ S| q − pi ≤ q − pj , ∀j = i} = S ∩j HP(pi, pj) where HP(pi, pj) is half plane (pi, pj)

3 generators 5 generators 50 generators

  • C. Nowzari (Penn)

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  • Fri. July 3rd

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SLIDE 37

Optimal configurations of H

Alternative expression in terms of Voronoi partition, H(p1, . . . , pn) =

n

  • i=1
  • Vi

q − pi2

2φ(q)dq

H as a function of agent positions and partition, H(p1, . . . , pn, W1, . . . , Wn) =

n

  • i=1
  • Wi

f(q − pi2)φ(q)dq ≤

n

  • i=1
  • Vi

f(q − pi2)φ(q)dq For fixed positions, Voronoi partition is optimal For fixed partition, centroid configurations are optimal

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

11 / 22

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SLIDE 38

Centroid algorithm

[Cortes, Martinez, Karatas, Bullo ’04]

At each round, agents synchronously execute: transmit position and receive neighbors’ positions; compute centroid of own cell determined according to some notion of partition of the environment Between communication rounds, each robot moves toward centroid

initial configuration gradient descent final configuration

Properties: provably correct, adaptive, distributed over Voronoi graph

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

12 / 22

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SLIDE 39

Outline

1 Motivation 2 Problem Formulation

aggregate objective optimization Voronoi partition

3 Triggered Deployment Algorithms

self-triggered deployment algorithm team-triggered deployment algorithm simulations

4 Conclusions

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

13 / 22

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SLIDE 40

Trading computation for communication/sensing

Balance cost of up-to-date information with limited resources what can agents do with outdated information about each other? Agents have uncertainty regions on other agents how up-to-date information must be to positively contribute to task when information must be updated

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

14 / 22

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SLIDE 41

Trading computation for communication/sensing

Balance cost of up-to-date information with limited resources what can agents do with outdated information about each other? Agents have uncertainty regions on other agents how up-to-date information must be to positively contribute to task when information must be updated Each agent i stores Di = ((pi

1, ri 1), . . . , (pi n, ri n)),

pi

j : last known location of agent j

ri

j : maximum distance traveled by agent j since last info

pi

i = pi and ri i = 0

Agents move at max speed vmax

  • C. Nowzari (Penn)

Team-triggered deployment

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14 / 22

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SLIDE 42

Guaranteed Voronoi diagram

Guaranteed Voronoi diagram gV(D1, . . . , Dn) = {gV1, . . . , gVn} of S generated by D1, . . . , Dn ⊂ S, gVi = {q ∈ S | max

x∈Di q − x2 ≤ min y∈Dj q − y2 for all j = i}

gVi contains points guaranteed to be closer to any point in Di than to any

  • ther point in Dj, j = i

In general, for pi ∈ Di, gVi ⊂ Vi

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

15 / 22

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SLIDE 43

Dual guaranteed Voronoi diagram

Dual guaranteed Voronoi diagram dgV(D1, . . . , Dn) = {dgV1, . . . , dgVn}

  • f S generated by D1, . . . , Dn ⊂ S,

dgVi = {q ∈ S | min

x∈Di q − x2 ≤ max y∈Dj q − y2 for all j = i}

Points outside dgVi are guaranteed to be closer to any point of Dj than to any point of Di In general, for pi ∈ Di, Vi ⊂ dgVi

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

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SLIDE 44

When is motion good?

[Nowzari, Cort´ es ’12]

With outdated info, agent i cannot calculate CVi

Proposition

Let L ⊂ V ⊂ U. Then, for any density function φ, CV − CL2 ≤ bound(L, U) = 2 cr(U)

  • 1 − mass(L)

mass(U)

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

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SLIDE 45

When is motion good?

[Nowzari, Cort´ es ’12]

With outdated info, agent i cannot calculate CVi

Proposition

Let L ⊂ V ⊂ U. Then, for any density function φ, CV − CL2 ≤ bound(L, U) = 2 cr(U)

  • 1 − mass(L)

mass(U)

  • Agent i moves from pi to p′

i making sure that

p′

i − CgVi2 ≥ boundi = bound(gVi, dgVi)

≥ CVi − CgVi2 move towards CgVi as much as possible in one time step until it is within distance boundi of it. As time elapses without new info, bound grows pi p′

i

CVi CgVi

  • C. Nowzari (Penn)

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17 / 22

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SLIDE 46

Triggered coordination algorithms

Reachable sets self-triggered centroid algorithm combines motion law self-triggered update policy (requesting information)

Proposition

Set of Centroidal Voronoi Configurations is globally asymptotically stable

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

18 / 22

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SLIDE 47

Triggered coordination algorithms

Reachable sets Promise sets self-triggered centroid algorithm combines motion law self-triggered update policy (requesting information)

Proposition

Set of Centroidal Voronoi Configurations is globally asymptotically stable

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

18 / 22

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SLIDE 48

Triggered coordination algorithms

Reachable sets Promise sets team-triggered centroid algorithm combines motion law self-triggered update policy (requesting information) event-triggered update policy (broken promises)

Proposition

Set of Centroidal Voronoi Configurations is globally asymptotically stable

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

18 / 22

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SLIDE 49

Simulations

Periodic Self-triggered Team-triggered

  • C. Nowzari (Penn)

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SLIDE 50

Simulations

Communication cost and performance

dBmW power units: Pi = 10 log10  

n

  • i,j comm

100.1+pi−pj2  

10 20 30 40 50 60 70 80 2 4 6 8 10 12 14 16 18

Periodic Self-trigger Team-trigger (λ = .25) Team-trigger (λ = .5)

Timestep H

10 20 30 40 50 60 70 80 50 100 150 200 250 300

Periodic Self-trigger Team-trigger (λ = .25) Team-trigger (λ = .5)

Timestep Ncomm

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

20 / 22

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SLIDE 51

Simulation

Effect of varying promise sizes

λ captures the tightness of promises λ = 0 corresponds to exact trajectories for promises λ = 1 corresponds to no promises (recovers self-triggered case)

0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140

Team-triggered Periodic

λ Pavg

0.2 0.4 0.6 0.8 1 10 20 30 40 50 60 70 80

Team-triggered Periodic

λ Tcon

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

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SLIDE 52

Conclusions

Team-triggered deployment of robotic networks for optimal deployment team-triggered centroid algorithm correct, adaptive, distributed, asynchronous same convergence guarantees as synchronous algorithm with perfect information at all times reduced communication efforts throughout the network

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

22 / 22

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SLIDE 53

Conclusions

Team-triggered deployment of robotic networks for optimal deployment team-triggered centroid algorithm correct, adaptive, distributed, asynchronous same convergence guarantees as synchronous algorithm with perfect information at all times reduced communication efforts throughout the network Things I skipped: how agents update information guaranteeing no Zeno behavior maximum times without communication

  • C. Nowzari (Penn)

Team-triggered deployment

  • Fri. July 3rd

22 / 22

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SLIDE 54

Thank you!

  • C. Nowzari (Penn)

Team-triggered deployment

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