Team-Triggered Coordination of Robotic Networks for Optimal Deployment

Transcription

1 Team-Triggered Coordination of Robotic Networks for Optimal Deployment Cameron Nowzari 1, Jorge Cortés 2, and George J. Pappas 1 Electrical and Systems Engineering 1 University of Pennsylvania Mechanical and Aerospace Engineering 2 University of California, San Diego American Control Conference Chicago, Illinois July 3, 2015

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 Fri. July 3rd 2 / 22

3 -Optimal deployment- of robotic sensor networks Objective: optimal task allocation and space partitioning optimal 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 Fri. July 3rd 3 / 22

4 Time-triggered (periodic) coordination Agents take actions at some fixed period T C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 4 / 22

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 Self-triggered coordination Agents decide when to take actions based on available information C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 5 / 22

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 -Team-triggered coordination- Cooperative agents share more information with each other C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 6 / 22

25 -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

26 -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

27 -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

28 -Team-triggered coordination- Cooperative agents share more information with each other C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 6 / 22

29 -Team-triggered coordination- Cooperative agents share more information with each other C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 6 / 22

30 -Team-triggered coordination- Cooperative agents share more information with each other C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 6 / 22

31 -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

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 7 / 22

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

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

35 Expected-value multicenter function Objective: Given sensors/nodes/robots/sites (p 1,...,p n ) moving in environment S achieve optimal coverage φ : R d R 0 density agent performance decreases with distance [ ] minimize H(p 1,...,p n ) = E φ min q p i 2 i {1,...,n} C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 9 / 22

36 Voronoi partitions Let (p 1,...,p n ) S n denote the positions of n points The Voronoi partition V(P) = {V 1,...,V n } generated by (p 1,...,p n ) V i = {q S q p i q p j, j i} = S j HP(p i,p j ) where HP(p i,p j ) is half plane (p i,p j ) 3 generators 5 generators 50 generators C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 10 / 22

37 Optimal configurations of H Alternative expression in terms of Voronoi partition, H(p 1,...,p n ) = n i=1 H as a function of agent positions and partition, H(p 1,...,p n,w 1,...,W n ) = V i q p i 2 2φ(q)dq n f( q p i 2 )φ(q)dq W i n f( q p i 2 )φ(q)dq V i For fixed positions, Voronoi partition is optimal For fixed partition, centroid configurations are optimal i=1 i=1 C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 11 / 22

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

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

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

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 D i = ((p i 1,ri 1 ),...,(pi n,ri n )), p i j rj i : last known location of agent j : maximum distance traveled by agent j since last info p i i = p i and ri i = 0 Agents move at max speed v max C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 14 / 22

42 Guaranteed Voronoi diagram Guaranteed Voronoi diagram gv(d 1,...,D n ) = {gv 1,...,gV n } of S generated by D 1,...,D n S, gv i = {q S max x D i q x 2 min y D j q y 2 for all j i} gv i contains points guaranteed to be closer to any point in D i than to any other point in D j, j i In general, for p i D i, gv i V i C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 15 / 22

43 Dual guaranteed Voronoi diagram Dual guaranteed Voronoi diagram dgv(d 1,...,D n ) = {dgv 1,...,dgV n } of S generated by D 1,...,D n S, dgv i = {q S min x D i q x 2 max y D j q y 2 for all j i} Points outside dgv i are guaranteed to be closer to any point of D j than to any point of D i In general, for p i D i, V i dgv i C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 16 / 22

44 When is motion good? [Nowzari, Cortés 12] With outdated info, agent i cannot calculate C Vi Proposition Let L V U. Then, for any density function φ, ( C V C L 2 bound(l,u) = 2cr(U) 1 mass(l) ) mass(u) C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 17 / 22

45 When is motion good? [Nowzari, Cortés 12] With outdated info, agent i cannot calculate C Vi Proposition Let L V U. Then, for any density function φ, ( C V C L 2 bound(l,u) = 2cr(U) 1 mass(l) ) mass(u) Agent i moves from p i to p i making sure that C Vi p i C gvi 2 bound i = bound(gv i,dgv i ) C Vi C gvi 2 C gvi move towards C gvi as much as possible in one time step until it is within distance bound i of it. As time elapses without new info, bound grows p i p i C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 17 / 22

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

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

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

49 Simulations Periodic Self-triggered Team-triggered C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 19 / 22

50 Simulations Communication cost and performance dbmw power units: n P i = 10log 10 i,j comm pi pj Periodic Team-trigger (λ=.25) Team-trigger (λ=.5) Self-trigger Periodic Team-trigger (λ=.25) Team-trigger (λ=.5) Self-trigger H N comm Timestep Timestep C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 20 / 22

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) P avg Team-triggered Periodic T con Team-triggered Periodic λ λ C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 21 / 22

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

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

54 Thank you! C. Nowzari (Penn) Team-triggered deployment Fri. July 3rd 22 / 22