EA-Based ASV Trajectory Planner for Pollution Detection in Lentic Waters

This paper presents a new planner based on Evolutionary Algorithms (EAs) to optimize the trajectory of an Autonomous Surface Vehicle (ASV), equipped with a probe, that has to determine the location of a pollutant in lentic water bodies (e.g. reservoirs, dams). To achieve it, our planner 1) exploits the information provided by a simulator that determines the pollutant distribution based on the water currents and 2) is supported by an EA that optimizes the mission duration, the ASV trajectory length and the measurements taken by its probe in highly polluted areas. The current version of the planner also ensures that the trajectories are feasible from the ASV and water body perspective, and solves this constrained multi-objective problem as a mono-objective one that linearly combines the constraint and objective functions. The preliminary results over different scenarios show that the planner can already determine overall good solutions, but that needs to be modified (e.g. using a multi-objective intended EA) to improve them further.

1. 1.

   Each of them has advantages and disadvantages. For instance, the Finite Difference Method (FDM) is simple to understand and does not require to transform the model into a variational formulation, but its implementation in irregular meshes such as the one used in our case is exceedingly difficult. Alternatively, the Finite Elements Method (FEM) and Finite Volumes Method (FVM), which are often used in problems involving fluid mechanics, are harder to understand but can use irregular meshes easily. However, all of them often present convergence problems in the calculation process due to the non-linearity of the Navier-Stokes equation.

2. 2.

   As water is incompressible and the domain volume does not change, the outflow is forced to have the same magnitude as the inflow.

3. 3.

   The first of these accelerations is used in a Lagrangian frame of reference to obtain the velocity of a particle over time, while the second is calculated from the fluid flow in its Eulerian description at the position of the particle. Besides, Eq. (3) is also valid for a non-stationary flow, substituting \(\varvec{u}_{ss}(\varvec{w})\) by \(\varvec{u}(\varvec{w},t)\).

4. 4.

   Particles outside the water body are re-sampled to make them start at valid locations.

5. 5.

   As the only requirement for using this solver is to have a different sign inside and outside of the domain function, any mathematical function that meets that criterion (including those with non-continuous images such as boolean functions) is valid. In other words, our way of proceeding simplifies immensely the definition of domains with irregular geometry.

6. 6.

   We tested that it is quicker and more accurate to do it with Matlab than with our own trajectory discretization.

7. 7.

   In this case, we are forced to use the Matlab function, as \(\mathrm {OF}_2\) determines the value of L, which is required to perform the proportional trajectory discretization.

8. 8.

   In this case we do not need to slide the time vectors since the \(t_0\) of both parents always equals the initial mission time \(t_0^{mission}\).

9. 9.

   To obtain this average, for those runs that have finished earlier than in 500 iterations we extend the last obtained value to the remaining iterations up to 500.

10. 10.

    The center line of each shade represents the corresponding mean value while the shade width over/under the mean is the standard deviation.

11. 11.

    We avoid drawing the mean value over 25 runs of the EV of the best solution for each configuration and iteration, since the EV values of one configuration are not comparable with the values of others, due to the change of value in \(w_{obj,2}\).

12. 12.

    In fact, in the graphics in top row, the green lines of the particle trajectories appear after a blue section in the ASV horizontal trajectory.

13. 13.

    Particle vertical trajectories do not become green only for having the probe at its corresponding height, because it is also necessary to have the ASV located over them.

Authors and Affiliations

1. Universidad Complutense de Madrid, 28040, Madrid, Spain

   Gonzalo Carazo-Barbero, Eva Besada-Portas, José M. Girón-Sierra & José A. López-Orozco

Corresponding author

Correspondence to Eva Besada-Portas .

Editors and Affiliations

1. ETSIIT-CITIC, University of Granada, Granada, Spain

   Pedro A. Castillo

2. Université Le Havre Normandie, Le Havre, France

   Juan Luis Jiménez Laredo

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