1 Efficient planning of informative paths for multiple robots Amarjeet Singh *, Andreas Krause +, Carlos Guestrin +, William J. Kaiser *, Maxim Batalin * * Center for Embedded Networked Sensing, University of California, Los Angeles + Machine Learning Department, Carnegie Mellon University

2 Predicting spatial phenomena in large environments Constraint: Limited fuel for making observations Fundamental Problem: Where should we observe to maximize the collected information? Biomass in lakes Salt concentration in rivers

3 How to quantify collected information? Mutual Information (MI): reduction in uncertainty (entropy) at unobserved locations [Caselton & Zidek, 1984] MI = 4 Path length = 10 MI = 10 Path length = 40

4 Selecting the sensing locations Lake Boundary G1G1 G2G2 G3G3 G4G4 Greedy selection of sampling locations is (1-1/e) ~ 63% optimal [Guestrin et. al, ICML’05] Result due to Submodularity of MI:  Diminishing returns Greedy may lead to longer paths! Greedily select the locations that provide the most amount of information

5 Greedy - reward/cost maximization Available Budget = B s  Reward = B  Cost = B reward cost = 2 reward cost = 1

6 Greedy - reward/cost maximization Available Budget = B-  s  B  B B Too far! Greedy Reward = 2 

7 Greedy - reward/cost maximization Available Budget = 0 s  B  B Greedy Reward = 2  Optimal Reward = B Greedy can be arbitrarily poor! 

8 Informative path planning problem max p MI(P) MI – submodular function Lake Boundary Start- s Finish- t P C(P) · B Informative path planning – special case of Submodular Orienteering Best known approximation algorithm – Recursive path planning algorithm [ Chekuri et. Al, FOCS’05]

9 Recursive path planning algorithm [Chekuri et.al, FOCS’05] Start (s) Finish (t) vmvm Recursively search middle node v m P1P1 P2P2 Solve for smaller subproblems P 1 and P 2

10 v m2 Recursive path planning algorithm [Chekuri et.al, FOCS’05] Start (s) Finish (t) P1P1 v m1 v m3 Maximum reward Recursively search v m C(P 1 ) · B 1 Lake boundary vmvm

11 Recursive path planning algorithm [Chekuri et.al, FOCS’05] Start (s) Finish (t) P1P1 vmvm Recursively search v m C(P 1 ) · B 1 Commit to the nodes visited in P 1 Recursively optimize P 2 C(P 2 ) · B-B 1 P2P2 Maximum reward Committing to nodes in P 1 before optimizing P 2 makes the algorithm greedy!

12 Quasi-polynomial running time O (B*M) log(B*M) B: Budget  Reward Chekuri ¸ Reward Optimal log(M) M: Total number of nodes in the graph Cost of output path (meters) Execution Time (Seconds) OOPS ! Small problem with 23 sensing locations Recursive path planning algorithm [Chekuri et.al, FOCS’05]

13 Almost a day!! Recursive path planning algorithm [Chekuri et.al, FOCS’05] Quasi-polynomial running time O (B*M) log(B* M) B: Budget  Reward Chekuri ¸ Reward Optimal log(M) M: Total number of nodes in the graph Small problem with 23 sensing locations

14 Our contributions Algorithm with significantly improved running time exploiting recursive path planning Spatial decomposition of sensing region Branch and bound - Calculating bounds using submodularity and other heuristics to prune search space Extended single robot path planning to multiple robots with strong approximation guarantee Extensive empirical evaluation on several real world sensing datasets Including data collected using robotic boat at Lake Fulmor, California

15 Lake Boundary Spatial decomposition into cells Ending node t Starting node s Ending Cell C t Starting Cell C s Search for middle Cell C m Perform recursive path planning on cells P1P1 P2P2

16 Ending Cell C t Starting Cell C s Middle Cell C m P1P1 P2P2 Greedily select locations without path cost constraint: 1-1/e optimal Node selection inside the cell Incoming pathExiting path G1G1 G2G2 G4G4 G3G3 Tradeoff: Larger cell size ) Faster Execution, Increased additional traveling cost Smaller cell size ) Slower Execution, Reduced additional traveling cost Small cells: Traveling cost inside cell can be ignored Additional cost for traveling to the sensing locations

17 Recursive Path Planning Approximation guarantees Running time: O ((B*N) log B*N ) Required budget: O (B)  Collected Reward ¸ (1-1/e) Reward Optimal log(N) N: Total number of cells in the graph Cost of output path (meters) Execution Time (seconds) Efficient Path Planning Approx. a day Approx. 2 min. Small problem – 23 sensing locations Too slow for larger problems! 

18 Further improvement in running time Search space represented as SUM-MAX tree (similar to AND-OR tree) Execution Time (seconds) Cost of output path (meters) Upto 400 meters calculated within approx. 15 min. Pruned search space using branch and bound Upper bound exploiting submodularity Lower bound exploiting known heuristic [Chao et. al’ 96] Several other tricks – see paper Larger problem – 109 sensing locations

19 Multi robot informative path planning problem max P 1,P 2,P 3 MI(P 1 U P 2 U P 3 ) MI – submodular function s t C(P 1 ) · B; C(P 2 ) · B; C(P 3 ) · B P2P2 P3P3 P1P1

20 Multi robot path planning – Simple sequential allocation approach s t P2P2 P3P3 P1P1 Use algorithm for single robot instance to find path P 1 for the first robot Optimize for second robot (P 2 ) committing to nodes in P 1 Optimize for third robot (P 3 ) committing to nodes in P 1 and P 2

21 Performance evaluation Works for any single robot path planning algorithm Independent of number of robots used Reward MR ¸ Reward Optimal 1 +  Greedy selection of nodes with no path cost constraint Arbitrarily Poor Recursive Greedy path planning Reward RG ¸ Reward Opt  (=log(M)) Sequential allocation for multiple robots – Greedy over paths ??

22 Efficient multi-robot information path planning Spatial Decomposition Obtain cell path exploiting submodularity, branch and bound A B C D Greedy node selection within visited cells to get node path Sequential Allocation for multi-robot path planning

23 Robotic boat measuring surface temperature and chlorophyll at Lake Fulmor, California Empirical evaluation 52 sensor motes used to monitor temperature at Intel Research laboratory, Berkeley Precipitation data collected from 167 regions in Pacific NW, during the years

Execution Time (seconds) Cost of output path (meters) Empirical evaluation – varying the cell size Precipitation Dataset No. of cells = 36 No. of cells = 25No. of cells = 16 Lower is better Higher information quality Cost of output path (meters)

Higher information quality Cost of output path (meters) Chekuri Algorithm Proposed Efficient Algorithm Intel Laboratory Temperature Dataset Empirical evaluation – reward comparison Reduced execution time by several factors Similar collected reward

26 Empirical evaluation – heuristic comparison Lake Temperature Dataset Higher information quality Cost of output path (meters) Efficient informative path planning algorithm Known heuristic [Chao et. al’ 96]

Total RMS Error Cost of output path per robot (meters) Empirical evaluation – multi robot Robot-1 Robot-3 Robot-2 1 Robot 2 Robots 3 Robots Lower is better

28 Conclusions First efficient multi robot informative path planning algorithm with performance guarantee Exploits spatial decomposition Exploits submodularity and other heuristics for branch and bound Near optimal extension of single robot path planning algorithm to multiple robots Extensive empirical evaluation on several real world sensor network datasets Including data collected using robotic boat in real lake Planning on a deployment at Lake Merced, California with robotic boat in February