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Near-optimal probabilistic search using spatial Fourier sparse set

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Abstract

Autonomous search is an essential research topic for rescue and other robotic applications. However, searching for targets efficiently is still an unsolved problem. To achieve this objective, a robot needs to simultaneously maximize environmental coverage, maximize probability of detection (PD) and minimize motion cost. The problems associated with these objectives are NP-hard. This research reformulates the three objective functions as a maximum cumulative PD problem with motion cost. Since the PD function depends on the environment, the robot needs to both learn the PD function and the cost-to-go (CTG) function. This research proposes a reinforcement learning algorithm to learn the PD and CTG functions simultaneously. Since the PD function is sparse in the Fourier domain under certain subgoal patterns, spatial Fourier sparse set is proposed to learn PD functions based on the compressed sensing technique. The learned PD and CTG functions can then be used to generate subgoals that achieve \((1-1/e)\) of the optimum due to the submodularity. Experiments conducted with this algorithm demonstrate that the robot can search for the target faster than prior learning approaches (e.g., PMAC and FSS) and the benchmark model (e.g., PD).

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Notes

  1. The conditional probability table is computed based on the detection outcome. There are four cases: true positive, false negative, false positive, and true negative. For example, if the target is there and the sensor cannot detect, this is called false negative.

  2. The H-area is defined as the \(20 \times 20\) cm\(^2\) area around the highest probability cell. If the target is in the H-area and \(P(A)>90\%\), it is a positive decision.

  3. The value of glimpse function can be decided from the conditional probability table of the given detector. \(g=\frac{q-pq}{1-pq}\), where p denotes the probability of detected area. q denotes the probability of that the robot detects the target if the target is at the detected area. The derivation of glimpse function can be found in the appendix of Tseng and Mettler (2015).

  4. The number of episodes means the number of trials in reinforcement learning.

  5. In EX1, 12 subgoals can cover the environments over \(80\%\). In EX2, since the motion cost is considered, the robot needs more subgoals to cover over \(80\%\).

  6. \(3\%\) is a noisy measurement since the maximal coverage of one subgoal is less than \(15\%\).

  7. SFSS can converge within 2000–3000 samples but it needs to explore the environment. Hence, it takes 10+ episodes.

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Acknowledgements

This research was completed thanks to the financial support from ONR Grant 1361538 and NSF CAREER CMMI 1254906. Kuo-Shih would like to thank his daughter, Chin-Chun Tseng. The hide-and-seek game they played inspires the learning concept for search problems.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, College of Science and Engineering, University of Minnesota, Minneapolis, MN, USA

    Kuo-Shih Tseng

  2. Department of Aerospace Engineering and Mechanics, College of Science and Engineering, University of Minnesota, Minneapolis, MN, USA

    Bérénice Mettler

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  1. Kuo-Shih Tseng
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  2. Bérénice Mettler
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Correspondence to Bérénice Mettler.

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This is one of several papers published in Autonomous Robots comprising the Special Issue on Active Perception.

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Tseng, KS., Mettler, B. Near-optimal probabilistic search using spatial Fourier sparse set. Auton Robot 42, 329–351 (2018). https://doi.org/10.1007/s10514-017-9616-2

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