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Autonomous underwater vehicle teams for adaptive ocean sampling: a data-driven approach

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Abstract

The current technological developments in autonomous underwater vehicles (AUVs) and underwater communication have nowadays allowed to push the original idea of autonomous ocean sampling network even further, with the possibility of using each agent of the network not only as an operative component driven by external commands (model-driven) but as a reactive element able to act in response to changing conditions as measured during the exploration (data-driven). With this paper, we propose a novel data-driven algorithm for AUVs team for adaptive sampling of oceanic regions, where each agent shares its knowledge of the environment with its teammates and autonomously takes decision in order to reconstruct the desired oceanic field. In particular, sampling point selection is made in order to minimize the uncertainty in the estimated field while keeping communication contact with the rest of the team and avoiding to repeatedly sampling sub-regions already explored. The proposed approach is based on the use of the emergent behaviour technique and on the use of artificial potential functions (interest functions) to achieve the desired goal at the end of the mission. In this way, there is no explicit minimization of a cost functional at each decision step. The oceanic field is reconstructed by the application of radial basis functions interpolation of irregularly spaced data. A simulative example for the estimation of a salinity field with sea data obtained using the Mediterranean Sea Forecasting System is shown in the paper, in order to investigate the effect of the different uncertainty sources, including sea currents, on the behaviour of the exploration team and ultimately on the reconstruction of the salinity field.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their constructive criticisms and suggestions. This work was supported in part by European Union, 7th Framework Programme, Project UAN—Underwater Acoustic Network under Grant no. 225669 and Project Co^3 AUV—Cognitive Cooperative Control for Autonomous Underwater Vehicles", Grant n. IST-231378.

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Correspondence to Andrea Munafò.

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This article is part of the Topical Collection on Maritime Rapid Environmental Assessment

Appendix A: Exploiting approximation properties of radial basis functions

Appendix A: Exploiting approximation properties of radial basis functions

The development of the approach described in the paper depends on the method employed to estimate the approximation error. In this section, we explain in more detail the approximation algorithms belonging to the class of Radial Basis Function (RBFs) and in particular we focus on how they can be used to derive an analytical formulation of the estimation error.

The reasons, in the present context, for choosing RBFs over other types of approximation methods are several: Radial basis functions, which have a long successful history of applications in the environmental field and in geostatistics (see the classic work of Hardy 1990), are ideally suited for interpolation and approximation of maps sampled on irregular grids (i.e., with samples not necessarily evenly spaced), as it is the case discussed in this paper. Moreover, the RBFs class is still fairly general, including multiquadric functions, thin-plate splines, B-splines, Gaussian functions, etc. The basic results on RBFs employed in the following of the section can be found in Schaback (1995) and (1997).

Let us select a family of RBFs \( \Phi :{\Re^d} \to \Re \), where d = 2 in our case; then the approximation algorithm S becomes:

$$ {S_{{{I^{{(j)}}}}}}\left( {\mathbf{x}} \right) = \sum\limits_{{h = 1}}^n {\sum\limits_{{i = 1}}^{{{k_h}}} {{\alpha_{{h,j}}}\Phi \left( {{\mathbf{x}} - {{\mathbf{x}}_{{h,j}}}} \right)} } $$
(10)

In Eq. 10, it is assumed that one basis function is centred at each sampled point: strictly speaking, this means that we are performing an interpolation of the measured data, and not an approximation. This assumption, which in some condition may lead to numerical difficulties, does not affect the generality of the discussion and it can be relaxed using approximation formulas (see Caiti et al. (2007), Iske (2003)).

Let \( \theta :{\Re^d} \to \Re \) be the true function approximated by S. It is assumed that \( \theta \left( {\mathbf{x}} \right) \) has Fourier transform \( \bar{\theta }\left( \omega \right) \), satisfying the following smoothness condition:

$$ \frac{{\overline \theta \left( \omega \right)}}{{\sqrt {{\overline \Phi \left( \omega \right)}} }} \in {L_2}\left( {{\Re^d}} \right) $$
(11)

where \( \bar{\Phi } \) is the generalized Fourier transform of the chosen RBF. Then θ belongs to a space H which has the structure of a Hilbert space with \( \Phi \left( {{\mathbf{x}},{\mathbf{y}}} \right) \) as reproducing kernel, and (semi-)norm:

$$ \left\| \theta \right\|_{\Phi }^2 = {\left( {2\pi } \right)^{{ - d}}}\int\limits_{{{\Re^d}}} {\frac{{{{\left| {\bar{\theta }\left( \omega \right)} \right|}^2}}}{{\bar{\Phi }\left( \omega \right)}}} d\omega $$
(12)

Note that the assumption of θ belonging to a specific reproducing kernel Hilbert space is an assumption on the regularity of the environmental map with respect to \( \left( {x,y} \right) \) coordinates. Note also that, depending on the specific choice amongst the RBF family, Φ can be positive definite, hence equation (12) is a norm, or conditionally positive definite, hence equation (12) is a semi-norm. If Φ is conditionally positive definite (of order m) the interpolation equation (10) must be complemented with a polynomial of degree m taking null values in the measured point and spanning the set of functions P m . The Hilbert space is then H\P m . In both cases the interpolation formulas reported in the following do not change, and the difference between the practical implications of the two cases is negligible.

Within this setting, the approximation error in a ball of radius ρ centred in a point x is given by:

$$ \varepsilon \left( {\mathbf{x}} \right) = \left| {\theta \left( {\mathbf{x}} \right) - S\left( {\mathbf{x}} \right)} \right| \leqslant {\left\| \theta \right\|_{\Phi }}{F_{\Phi }}\left( {{h_{\rho }}} \right) $$
(13)

where the explicit dependence of ε and S from the information set I has been omitted for the sake of simplicity. The quantity h p is the so-called local fill distance, in the RBFs jargon, and it depends on the density of the sampling points:

$$ {h_{\rho }}(y) = \mathop{{\sup }}\limits_{{w \in B\left( {y,\rho } \right)}} \mathop{{\min }}\limits_{{{\mathbf{x}} \in {M^{{(j)}}}}} {\left\| {w - {\mathbf{x}}} \right\|_2} $$
(14)

while F Ф() (the power function) is a known function that depends exclusively on the specific RBF choice (gaussian, multiquadric, etc.); some typical forms, given by Schaback (1995), are reported in Table 1. Under some additional technical assumptions - decay to zero of the RBF Fourier transform, and uniform interior cone condition holding on the domain of interest A (Iske (2003))—Eq. (13) can be extended to the whole domain A by replacing the local fill distance with the global fill distance:

$$ {h_{{A,{M^{{(j)}}}}}}(y) = \mathop{{\sup }}\limits_{{w \in A}} \mathop{{\min }}\limits_{{{\mathbf{x}} \in {M^{{(j)}}}}} {\left\| {w - {\mathbf{x}}} \right\|_2} $$
(15)
Table 1 Expression of the bounds on the power function (Eq. 13) as a function of the RBF family chosen (from Schaback 1995)

Note that the technical assumptions for the existence of a global fill distance are respected by the RBFs of Table 1, and that a compact, convex domain A is sufficient to guarantee the interior cone condition.

As evident, the approximation error in Eq. 13 depends on the unknown norm of the true \( {\left\| \theta \right\|_{\Phi }} \) and cannot be evaluated from the data; however, by assuming that the following condition holds:

$$ \frac{{\overline \theta \left( \omega \right)}}{{\sqrt {{\overline \Phi \left( \omega \right)}} }} \leqslant \frac{{\overline S \left( \omega \right)}}{{\sqrt {{\overline \Phi \left( \omega \right)}} }} $$
(16)

the following bound holds true (see Schaback (1995) for a proof):

$$ \varepsilon \left( {\mathbf{x}} \right) = \left| {\theta \left( {\mathbf{x}} \right) - S\left( {\mathbf{x}} \right)} \right| \leqslant {\left\| {S\left( {\mathbf{x}} \right)} \right\|_{\Phi }}{F_{\Phi }}\left( {{h_{\rho }}\left( {\mathbf{x}} \right)} \right) $$
(17)

The error bound can now be incrementally computed with the available data using the current approximation of the environmental map at the place of the map itself.

It is worth noticing that, crucial to this development, is the assumption of Eq. 16, which, in practical terms, implies that the environmental map θ is smoother than its approximation S. This regularity condition is indeed much stronger than the assumption of Eq. 12, and it may be more difficult to guarantee a priori. Nevertheless, Eq. 17 can always be used as an approximation of Eq. 13, since as the number of sampling points increase the two expression will eventually converge; however, the bound on the approximation error is not strictly guaranteed anymore at each new sampling stage of the algorithm, causing possibly repeated explorations of the same sub-areas.

Assuming Eq. 16) to hold, the jth vehicle can incrementally determine the radius \( p_{{k + 1}}^{{(j)}} \) at each new step in the planning as the local fill distance to be inserted in Eq. 17 to satisfy the error requirements of the mission.

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Munafò, A., Simetti, E., Turetta, A. et al. Autonomous underwater vehicle teams for adaptive ocean sampling: a data-driven approach. Ocean Dynamics 61, 1981–1994 (2011). https://doi.org/10.1007/s10236-011-0464-x

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