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Efficient Informative Sensing using Multiple Robots Amarjeet Singh, Andreas Krause, Carlos Guestrin and William J. Kaiser (Presented by Arvind Pereira for CS-599 Sequential Decision Making in Robotics)

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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

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3 Challenges for informative path planning Use robots to monitor environment Not just select best k locations A for given F(A). Need to take into account cost of traveling between locations cope with environments that change over time need to efficiently coordinate multiple agents Want to scale to very large problems and have guarantees

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4 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

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5 Y1Y1 Y2Y2 Y3Y3 Y4Y4 Y5Y5 Selection B = {Y 1,, Y 5 } Key observation: Diminishing returns Y1Y1 Y2Y2 Selection A = {Y 1, Y 2 } Adding Y will help a lot! Adding Y doesnt help much Y New observation Y Y B A + + Large improvement Small improvement For A B, F(A [ {Y}) F(A) F(B [ {Y}) F(B) Submodularity: Many sensing quality functions are submodular*: Information gain [Krause & Guestrin 05] Expected Mean Squared Error [Das & Kempe 08] Detection time / likelihood [Krause et al. 08] *See paper for details

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6 Selecting the sensing locations Lake Boundary G1G1 G2G2 G3G3 G4G4 Greedy selection of sampling locations is (1-1/e) ~ 63% optimal [Guestrin et. al, ICML05] 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

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7 Greedy - reward/cost maximization Available Budget = B s Reward = B Cost = B reward cost = 2 reward cost = 1

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

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

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10 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, FOCS05]

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11 Recursive path planning algorithm [Chekuri et.al, FOCS05] Start (s) Finish (t) vmvm Recursively search middle node v m P1P1 P2P2 Solve for smaller subproblems P 1 and P 2

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12 v m2 Recursive path planning algorithm [Chekuri et.al, FOCS05] Start (s) Finish (t) P1P1 v m1 v m3 Maximum reward Recursively search v m C(P 1 ) B 1 Lake boundary vmvm

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13 Recursive path planning algorithm [Chekuri et.al, FOCS05] 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!

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14 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 6080100120140160 Cost of output path (meters) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Execution Time (Seconds) OOPS ! Small problem with 23 sensing locations Recursive path planning algorithm [Chekuri et.al, FOCS05]

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15 Almost a day!! Recursive path planning algorithm [Chekuri et.al, FOCS05] 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

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Recursive-Greedy Algorithm (RG)

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17 Selecting sensing locations Given: finite set V of locations Want: A * V such that Typically NP-hard! Greedy algorithm: Start with A = ; For i = 1 to k s* := argmax s F(A [ {s}) A := A [ {s*} G1G1 G2G2 G3G3 G4G4 How well does the greedy algorithm do?

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18 Selecting sensing locations Given: finite set V of locations Want: A * V such that Typically NP-hard! Greedy algorithm: Start with A = ; For i = 1 to k s* := argmax s F(A [ {s}) A := A [ {s*} G1G1 G2G2 G3G3 G4G4 Theorem [Nemhauser et al. 78] : F(A G ) (1-1/e) F(OPT) Greedy near-optimal!

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Sequential Allocation

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Sequential Allocation Example

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Spatial Decomposition in recursive-eSIP

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recursive-eSIP Algorithm

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SD-MIPP

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eMIP

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Branch and Bound eSIP

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Experimental Results

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Experimental Results : Merced

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Comparison of eMIP and RG

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Comparison of Linear and Exponential Budget Splits

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Computation Effort w.r.t Grid size for Spatial Decomposition

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Collected Reward for Multiple Robots with same starting location

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Collected Reward for Multiple Robots with different start locations

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Paths selected using MIPP

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Running Time Analysis Worst-case running time for eSIP for linearly spaced splits is: Worst-case running time for eSIP for exponentially spaced splits is: Recall that Recursive Greedy had:

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Approximation guarantee on Optimality

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Conclusions eSIP builds on RG to near-optimally solve max collected information with upper bound on path-cost SD-MIPP allows multiple robot paths to be planned while providing a provably strong approximation gurantee Preserves RG approx gurantee while overcoming computational intractability through SD and branch & bound techniques Did extensive experimental evaluations