Continuous Exploitative Measurement
       Trajectories Using Bayesian Optimisation

Rémi Delanghe1[0000−0002−6621−7291] , Tom Van Steenkiste2[0000−0002−3842−3151] ,
 Dirk Deschrijver3[0000−0001−6600−1792] , and Tom Dhaene4[0000−0003−2899−4636]

    Ghent University - imec, IDLab, Technologiepark-Zwijnaarde 126, Ghent 9052,
          Belgium Remi.Delanghe@UGent.be - www.ugent.be/ea/idlab/en



        Abstract. Line-based sampling strategies aim to capture as much in-
        formation as possible along a trajectory, whilst minimizing the trajec-
        tory’s length. The current state of the art primarily contains exploration
        techniques that focus on uniformly sampling the measurement space. In
        this work, Bayesian optimization is used to create a novel exploitative
        line-based sampling strategy, that is able to guide the sampling process
        towards interesting regions.

        Keywords: Bayesian Optimization · Line-Based Sampling · Design Of
        Experiments


1     Introduction
Automated data collecting, using robots and drones, is omnipresent in both sci-
entific studies and engineering applications. In such a context, the length of the
measurement trajectory is limited by battery restrictions or operating costs. As
such, a line-based sampling strategy is used, which aims at minimising the trajec-
tory length whilst maximising the information gained per step. The current state
of the art primarily contains explorative techniques that uniformly sample the
measurement space [1]. A model-driven approach allows the strategy to create a
representation of the measurement space which can guide the sampling process
towards interesting regions. In this work, Bayesian optimisation (BO) is used
to create a line-based sampling strategy for global optimization that naturally
deals with the exploration-exploitation trade off.


2     Line-based Bayesian Optimization
Classic BO aims to find the global optimum of the measurement space, using few
data-points. This is typically quantified by evaluating an Acquisition Function
(AF) over the entire space. Maximising the AF returns the sample location that
has most potential to improve on the current global optimum. The maximum
of the AF is iteratively acquired and recalculated in the search for the global
    Copyright 2019 for this paper by its authors. Use permitted under Creative Com-
    mons License Attribution 4.0 International (CC BY 4.0).
2         R. Delanghe et al.

optimum. The advantage of using an AF is that it naturally deals with the
exploration exploitation trade-off.
    In this work, BO is used to create a Line-based Bayesian optimisation (LBO)
strategy. The AF’s maximum is used as a waypoint for the measurement probe.
To reach this point, the probe moves in small steps, straight towards the way-
point and samples at every step until it is reached. At every step, it is allowed
to deviate from its trajectory by some angle if this results in the sampling of
a point with a higher acquisition score. The stepsize and angle of deviation are
parameters of the strategy. This LBO strategy can only be used in convex spaces
and is deterministic if the underlying BO is deterministic as well.

3      Experimentation, Results And Discussion




    (a) RMSE Between estimated space and   (b) Fraction of the maximum sampled
    the ground truth.                      value, compared to the global maximum
Fig. 1: Comparison of BO, LBO and Boustrophedon in terms of path length.
Mean and standard deviation are calculated using 8 runs with different starting
locations.

    Figure 1 shows the results of Comparing LBO, BO and the Boustrophedon
path [2] using the peaks function [3] as measurement space. Figure 1a compares
the rate at which the error of the estimated space decreases with respect to
the ground truth. Here the LBO outperforms the other strategies. In addition,
Figure 1b shows that the LBO is able to find the maximum of the measurement
space using a shorter path.
    In conclusion, introducing a Line-based element into BO, creates a sampling
strategy that is efficient and is able to explore and exploit the space using a
shorter path than the other common sampling strategies.

References
1. Van Steenkiste, T., J. van der Herten, D. Deschrijver, and T. Dhaene. 2019, Apr.
   ALBATROS: adaptive line-based sampling trajectories for sequential measurements.
   Engineering with Computers 35(2):537-550.
2. H. Choset and P. Pignon, Coverage path planning: The boustrophedon cellular
   decomposition, in Field and service robotics, pp. 203–209, Springer, 1998.
3. MathWorks, peaks. https://nl.mathworks.com/help/matlab/ref/peaks.html.