PaperlayerView exact revision on arXiv

Source-linked AI summary

Efficient Informative Sensing using Multiple Robots

Amarjeet Singh, Andreas Krause, Carlos Guestrin, William J. Kaiser

arXiv:1401.3462v1cs.ROcs.AI

TL;DR

Environmental monitoring needs informative robot paths despite bounded resources and an NP-hard planning problem. The paper develops eSIP and sequential allocation to plan near-optimal single- and multi-robot paths, evaluates them in lake and river deployments and simulations, and reports practical effectiveness. The approach’s scope includes computational and modeling qualifications described by the authors.

  • Problem

    Environmental robots must coordinate resource-constrained paths to collect information about spatio-temporal phenomena, although optimal informative path planning is NP-hard.

  • Method

    The paper combines Gaussian-Process mutual-information modeling with eSIP for single robots and sequential allocation to extend single-robot planners to multiple robots.

  • Results

    The approach was evaluated in field experiments for lake and river monitoring and in simulations using real-world sensor-network data sets.

  • Takeaways & Limitations

    The paper presents efficient informative path planning intended to make provably near-optimal sensing practical for real-world environmental monitoring.

  • Takeaways & Limitations

    Recursive-greedy has quasi-polynomial running time, with complexity O((MB)O(log2 M)), making it impractical for several real-world sensing problems.

Abstract

from arXiv · show

The need for efficient monitoring of spatio-temporal dynamics in large environmental applications, such as the water quality monitoring in rivers and lakes, motivates the use of robotic sensors in order to achieve sufficient spatial coverage. Typically, these robots have bounded resources, such as limited battery or limited amounts of time to obtain measurements. Thus, careful coordination of their paths is required in order to maximize the amount of information collected, while respecting the resource constraints. In this paper, we present an efficient approach for near-optimally solving the NP-hard optimization problem of planning such informative paths. In particular, we first develop eSIP (efficient Single-robot Informative Path planning), an approximation algorithm for optimizing the path of a single robot. Hereby, we use a Gaussian Process to model the underlying phenomenon, and use the mutual information between the visited locations and remainder of the space to quantify the amount of information collected. We prove that the mutual information collected using paths obtained by using eSIP is close to the information obtained by an optimal solution. We then provide a general technique, sequential allocation, which can be used to extend any single robot planning algorithm, such as eSIP, for the multi-robot problem. This procedure approximately generalizes any guarantees for the single-robot problem to the multi-robot case. We extensively evaluate the effectiveness of our approach on several experiments performed in-field for two important environmental sensing applications, lake and river monitoring, and simulation experiments performed using several real world sensor network data sets.

1. Introduction

Environmental monitoring requires robots to observe phenomena that vary across space and time, despite limited resources. The paper develops efficient single- and multi-robot planning methods that seek near-optimal information collection.

  • Motivation: Environmental sensing applications require high-fidelity observation of spatio-temporal phenomena such as algal blooms and river salinity.These applications motivate robotic sensors for collecting measurements across environmental domains.
  • Motivation: Robots face resource constraints that limit travel distance or measurements, requiring coordinated paths that maximize information within available budgets.The constraints can include storage battery energy and limited time before the phenomenon changes significantly.
  • Approach: The paper uses Gaussian Processes and mutual information to quantify how observations reduce uncertainty at unvisited locations.Mutual information provides an information-based objective for selecting informative sensing paths.
  • Contributions: eSIP improves the practicality of recursive-greedy single-robot planning through spatial decomposition and branch-and-bound search.The techniques address recursive-greedy’s quasi-polynomial running time for environmental sensing problems with many observation locations.
  • Contributions: Sequential allocation extends single-robot planning algorithms to multiple robots while preserving approximation guarantees with only a minimal reduction.Combining sequential allocation with eSIP yields an efficient multi-robot path-planning approach.
  • Evaluation: The approach is evaluated in field campaigns for lake and river monitoring using the NIMS robotic system.The experiments include deployments at a California river confluence and a lake at UC Merced.

2. The Multi-robot Informative Path Planning Problem

MIPP plans bounded-cost paths for multiple robots so their visited locations maximize a sensing-quality function. The formulation supports mutual information and other normalized, monotonic submodular objectives.

  • Problem formulation: MIPP seeks k robot paths with specified starts and finishes whose sensing locations maximize information under individual path-cost budgets.Each path cost includes sensing and travel costs, and the collection’s quality depends on the union of visited locations.
  • Problem formulation: A path is a sequence from a starting location to a finishing location, with cost determined by sensing and travel costs along consecutive locations.The formulation treats a path both as an ordered sequence and as its subset of sensing locations.
  • Sensing quality: Gaussian Processes model uncertainty at unobserved locations, while mutual information measures the uncertainty reduction produced by sensing selected locations.For Gaussian Processes, mutual information can be computed efficiently and analytically.
  • Computational challenge: Even without path-length constraints, maximizing mutual information over selected locations is NP-hard, so the goal is to find efficiently computed near-optimal solutions.The paper therefore targets sensing quality provably close to the optimum rather than exact optimization in general.
  • Objective properties: The objective is modeled as normalized, monotonic, and submodular, capturing diminishing returns as more locations are sensed.This formulation allows objectives beyond mutual information when they satisfy the same properties.
  • Objective properties: Informative path planning balances exploration of uncertain environments against exploitation of the current belief under sensing constraints.The paper frames this trade-off as relevant to applications including search and rescue.

3. Approximation Algorithm for MIPP

The paper extends single-robot informative path planning to multiple robots through sequential allocation, repeatedly planning paths using residual information from earlier selections. Its approximation guarantee remains close to that of the underlying single-robot algorithm.

  • Assumptions: Mutual information is only approximately monotonic, although the paper states this suffices for its purposes.This qualification applies to the mutual-information objective rather than to the general monotonicity assumption in the formulation.
  • Sequential allocation: The algorithm accepts budgets, robot start and finish locations, and candidate sensing locations, then returns one informative path per robot.Its pseudocode maintains the cumulative set of selected locations across planning stages.
  • Sequential allocation: Sequential allocation applies a single-robot planner successively to each robot and commits previously selected locations before planning the next path.The next planner call optimizes residual information after accounting for locations already visited.
  • Guarantees: Theorem 1 gives sequential allocation an approximation guarantee of (1 + η) for MIPP when the single-robot planner has guarantee η.With common starting and finishing locations, the guarantee improves to 1/(1 −exp (−1/η)) ≤1 + η.
  • Guarantees: The sequential-allocation result extends earlier guarantees from additive sensing functions to general submodular functions.This broadens the result beyond the modular case considered by Blum et al.
  • Guarantees: Using recursive-greedy as the single-robot planner yields an MIPP performance guarantee of O(log2 |P∗|).The guarantee follows from recursive-greedy’s single-robot approximation guarantee through sequential allocation.

4. A Note on Greedy Path Planning

The paper shows that intuitive greedy path-planning rules can perform arbitrarily poorly when travel costs constrain which informative locations can be reached. This motivates more sophisticated algorithms for informative path planning.

  • Greedy selection based on mutual-information benefit without travel costs is near-optimal only in the unconstrained sensor-placement setting.The cited prior work selects k locations to maximize mutual information under submodularity and excludes path costs.
  • A natural path-planning extension chooses the next location by maximizing the benefit-cost ratio relative to the current selected set.The incremental cost is defined as CAi(v) = C(Ai ∪{v}) −C(Ai), where C(A) is the cheapest path connecting A.
  • 1(B/ϵ −1) + ϵ reward is achieved by an optimal route that first visits o1 and then covers the remaining clustered locations within budget.The route pays B/2 to reach o1, then visits the remaining B/ϵ−1 locations and returns to s with total travel cost 2B.
  • 2B reward is obtained by the reward-cost greedy rule, while reward-only greedy obtains 1, compared with 1(B/ϵ −1) + ϵ for the optimum.Because B/ϵ can grow arbitrarily as ϵ →0, both intuitive greedy rules can be arbitrarily poor relative to the optimal route.
  • The example uses a modular reward, yet the authors state that the analogous submodular optimal reward can also be arbitrarily larger than simple greedy rewards.The difference between modular and submodular reward depends on correlations among selected observation locations.

5. The Recursive-greedy Algorithm

Recursive-greedy searches informative paths by dividing each route at a middle location, allocating budget across two subpaths, and recursively optimizing them. Although this commitment-based search has a logarithmic approximation guarantee, its quasi-polynomial growth makes it impractical at realistic problem sizes.

  • The Recursive-greedy Algorithm: Recursive-greedy divides an s−t path at a middle location into two subpaths with approximately equal numbers of locations.It uses a divide-and-conquer strategy to recursively search for informative subpaths.
  • The Recursive-greedy Algorithm: fP1(A) = f(A ∪P1) −f(P1) measures the incremental reward of a second-subpath location set A after selecting first-subpath locations P1.The residual scores add exactly: f(P1) + fP1(P2) = f(P1 ∪P2).
  • The Recursive-greedy Algorithm: For every reachable middle location and feasible budget split, recursive-greedy plans one subpath, commits to it, and optimizes the other using the remaining budget.The second subpath accounts for locations already selected in the first through a residual reward function.
  • The Recursive-greedy Algorithm: IX(P) ≥ IX(P∗)/⌈1 + log k⌉ when recursion depth i ≥⌈1 + log k⌉, giving recursive-greedy a logarithmic approximation guarantee.Here P∗ is an optimal s−t path and k is the number of locations in that optimal path.
  • Running Time: O((MB)O(log2 M)) running time makes recursive-greedy impractical even for M = 64 observation locations.The computation grows quasi-polynomially with M and the budget B.
  • Running Time: recursive-eSIP decomposes the sensing domain into cells, jointly optimizes cell-paths and within-cell experiment allocations, then concatenates and smooths paths for eMIP.Within cells, sensing locations connect to cell centers before between-cell and within-cell paths are combined.
  • Running Time: Spatial decomposition reduces branching by clustering sensing locations, while branch and bound prunes subtrees using lower and upper reward bounds.The two techniques target the search tree’s large branching factor and unnecessary recursion.

6. Spatial Decomposition – Approximating MIPP as SD-MIPP

Spatial decomposition replaces location-level path planning with cell-level planning, then selects sensing locations within visited cells. Combined with budget splitting, smoothing, and sequential allocation, this yields efficient approximate solutions for single- and multi-robot informative path planning.

  • Spatial decomposition: The sensing region is partitioned into non-overlapping cells containing clusters of sensing locations, with inter-cell distance measured between centroids.This reframes path planning as choosing sub-regions and then selecting sensing locations within them.
  • Budget allocation: The decomposed problem separates the budget into travel between cells and experiments at locations within visited cells.Travel within cells is treated as zero in the decomposed model, while sensing costs remain bounded by the experimental budget.
  • Approximation: An optimal MIPP path can be represented by an SD-MIPP path that collects the same information with additional budget bounded by 2B + 4L.The bound accounts for the spatial decomposition and the cell width L.
  • Algorithm construction: eMIP solves SD-MIPP and smooths paths over selected observation locations, while recursive-eSIP, eSIP, and sequential allocation provide its component procedures.eSIP varies the travel and experimental budget split, and eMIP applies eSIP sequentially across robots.
  • Running time: Spatial decomposition changes recursive-greedy’s complexity from O((MB)O(log2 M)) to a form using log2 B and log2 N, making the approach more viable.The improvement reduces both the budget-dependent base and the logarithm of the sensing-location count in the exponent.

7. Branch and Bound

Branch-and-bound makes recursive-eSIP tractable by bounding achievable rewards, prioritizing promising branches, and pruning subproblems that cannot improve the current solution. Lower bounds also make the search anytime, while sub-approximation heuristics trade information quality for faster execution.

  • Search-space representation: Branch-and-bound represents recursive-eSIP as a sum-max tree and uses bounds to reduce the search effort.Max nodes choose middle cells and budget allocations; sum nodes combine subpaths on either side of the selected cell.
  • Pruning and node ordering: Branches are pruned when their upper bound is below the current lower bound, and children are explored in decreasing upper-bound order to find strong solutions earlier.A child sum node is processed only when its upper bound exceeds the current best solution; higher bounds are treated as likely indicators of higher reward.
  • Bounding procedures: Upper bounds estimate the best reward attainable by relaxed subproblems, while lower bounds record known feasible solution rewards.Upper bounds are computed with GreedySubset after relaxing path constraints; lower bounds come from heuristic solutions and existing parent information.
  • Anytime search: Maintaining lower bounds makes recursive-eSIP anytime, so execution can stop early while returning the best path found so far.This is useful when path planning must finish within a hard upper bound on computation time.
  • Sub-approximation: Sub-approximation can explore only the top K sum nodes or scale lower bounds by α > 1, accelerating search at potential cost to collected information.Optimistic pruning may remove branches that should have been retained, although sufficiently small α values can substantially speed execution with little quality loss in practice.

8. Experimental Results

Experiments evaluated eMIP for aquatic monitoring and sensing datasets, showing accurate field predictions, faster computation, and improved multi-robot information collection under varied conditions.

  • In-field Experiments: 45.99 µS/cm RMS error against the pre-experiment raster scan was lower than the 57.55 µS/cm temporal variation between raster scans.The post-experiment comparison yielded 53.87 µS/cm RMS error.
  • In-field Experiments: 0.73 oC RMS error was obtained when eMIP predictions were compared with the post-experiment lake temperature raster scan.
  • Experiments on Sensing Datasets: eMIP improved running time by several orders of magnitude over recursive-greedy at higher budgets with almost the same collected reward.The comparison used 23 of 52 Berkeley temperature locations because recursive-greedy was very slow on the complete dataset.
  • Multi-robot Experiments: Increasing the number of robots increased collected reward and reduced RMS prediction error, while gains from one to two robots exceeded gains from two to three.The pattern was attributed to sequential allocation and the diminishing returns property of mutual information.
  • Multi-robot Experiments: Different starting locations produced a larger one-to-two-robot reward increase than using the same starting location for the second robot.The one-to-two-robot improvement still exceeded the two-to-three-robot improvement.

9. Related Work

Related work situates informative path planning within operations research and robotics, while distinguishing this paper’s submodular multi-robot setting and efficiency-oriented approach.

  • Operations Research: The paper studies submodular orienteering, extending path-planning problems beyond additive reward functions.
  • Known Approximations for the Orienteering Problem: Recursive-greedy provides approximation guarantees for submodular path planning but has quasi-polynomial running time, motivating efficient alternatives.
  • Known Approximations for the Orienteering Problem: The paper’s spatial decomposition approach limits eMIP to symmetric spaces, whereas directed orienteering concerns asymmetric spaces.
  • Sequential Allocation: Sequential allocation generalizes earlier multi-robot orienteering work from modular rewards to submodular reward functions.
  • Robotic Applications: Robotic exploration studies include information-gain and auction-based methods, but related multi-robot approaches may lack theoretical bounds or address narrower utility functions.

10. Conclusions and Future Work

The paper presents eSIP and sequential allocation as efficient, theoretically supported approaches for informative single- and multi-robot path planning, validated in environmental sensing applications.

  • Conclusions: eSIP near-optimally maximizes collected information under an upper bound on path cost while preserving recursive-greedy’s approximation guarantees.Spatial decomposition and branch-and-bound methods address recursive-greedy’s computational intractability.
  • Conclusions: Sequential allocation extends single-robot algorithms such as eSIP to multiple robots with a provably strong approximation guarantee.
  • Conclusions: In-field evaluations covered lake monitoring and river monitoring using the Networked Info Mechanical System.
  • Future Work: Future work includes applying the algorithm to SLAM and search and rescue, understanding static GP model limitations, and enabling online model adaptation.

APPENDIX

The appendix establishes approximation guarantees for sequential allocation and eSIP, while analyzing budget adjustments and recursive budget-splitting variants for informative path planning.

  • Sequential allocation: Sequential allocation achieves approximation guarantee (1 + η) for MIPP, improving to 1/(1 −exp (−1/η)) when robots share starting and finishing locations.The proof bounds the residual reward across stages using monotonicity and submodularity.
  • Proof structure: The sequential-allocation proof tracks cumulative selected nodes Ai and optimal nodes Oi, using telescoping sums, monotonicity, and submodularity to bound residual reward.These quantities support the stage-wise guarantee and its conversion into a total approximation factor.
  • Budget conversion: For SD-MIPP, the budget must account for sensing experiments near shared cell vertices and additional travel between cell corners.The construction adds 4L for four experiments near a common vertex and further travel cost when traversing cell edges.
  • Spatial decomposition: Greedy selection within each visited cell contributes the factor Fg = (1 −1/e) to the approximation analysis.The induction compares the selected sensing locations in each cell with the optimal SD-MIPP choices.
  • Budget splitting: Exponential experimental-budget splits require an increased budget, with the analysis showing a multiplicative growth bounded by (3/2)^log2 n relative to the linear allocation.The bound is established inductively over the height of the complete binary path-splitting tree.
Loading 1401.3462v1…