FM^{2} Path Planner for UAV Applications with Curvature Constraints: A Comparative Analysis with Other Planning Approaches
Abstract
:1. Introduction
2. Description of the Planning Approaches
2.1. $FMM$ and $F{M}^{2}$ Methods
Algorithm 1 Fast Marching Method 
Require: A grid map X of size $m\times n$, source point ${x}_{0}$. 
Ensure: The grid map X with the T value set for all cells. 
Initialization. 

 The first is the saturation parameter $\alpha $, which consists of setting a maximum value, between 0 and 1, from which all the values of the matrix W are equal to 1.In Figure 3 and Figure 4, for $\alpha $ = 0.8 and $\alpha $ = 0.6, respectively, it can be seen that the white area away from obstacles is more or less large. A saturation value closer to zero implies less curved trajectories, but with more angle at certain points and closer to obstacles. The trajectory of Figure 3c) also presents a curvature in some points bigger than that of Figure 4c).
 The second parameter, $\beta $, is an exponent between 0 and 1 to which each coefficient of matrix W of the first potential is raised. Figure 5 and Figure 6 show the behaviour of the algorithm for varying values of $\beta $. The closer the exponent to zero, the clearer the image and the less curvature of the trajectories. The closer the exponent to one, the more similar the curvature to the Medial Axis Transform (MAT), although the smoother the trajectories and the more separated from obstacles (safer paths).It is also possible to raise each coefficient of the matrix to numbers greater than one, but the results are not of interest because they are more and more similar to the trajectories obtained with a Voronoi diagram that have sharp edges.
2.2. Dubins, Euler–Mumford Elastica and Reeds–Shepp methods
3. Discussion of Simulation Results
3.1. Simulation Execution Conditions
 1.
 The map in Figure 7, corresponding to the city area of the Centre Pompidou, is used for the simulations.
 2.
 A total of 24 different trajectories are executed by each method, covering the main central area of the map.
 3.
 For the execution of the Dubins car, the Euler–Mumford Elastica and the Reeds–Shepp approaches, the codes proposed in [26,27] by Mirebeau et al. are used, which apply the Fast Marching with stencils method (see Section 2.2 and Equations (4) and (5) for a detailed description of the curvature cost functions). The initial and end points of the 24 trajectories are specified, and parameter $\xi $ is taken as 1.
 4.
 The $F{M}^{2}$ method is applied using the same initial and end points defined before, varing parameters $\alpha $ and $\beta $ as described in the following subsection.
 5.
 For the sake of comparison of the resulting paths and their curvatures using these four methods, two measures of similarity will be used: the Fréchet distance and the area between two curves, as defined before. The results are given and discussed in the following subsections.
3.2. Minimization of the Fréchet Distance
3.3. Minimization of the Area between Curves
3.4. Distance from Paths to Obstacles
3.5. Simulation Results Using a Quadcopter Model and a FixedWing UAV
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Dubins  Euler–Mumford Elastica  Reeds–Shepp  $F{M}^{2}$  

Computation time (secs)  18.936  19.199  19.577  0.087 
Dubins  Euler–Mumford Elastica  Reeds–Shepp  

$\alpha $  0.1  0.3  0.038 
$\beta $  0.98  0.96  0.92 
Fréchet cost function  8.1  11.1  9.2 
Dubins  Euler–Mumford Elastica  Reeds–Shepp  

$\alpha $  0.90  0.84  0.40 
$\beta $  0.18  0.20  0.50 
Area cost function  4.50  2.19  0.3352 
Dubins  Euler–Mumford Elastica  Reeds–Shepp  

$\alpha $  0.90  0.84  0.40 
$\beta $  0.18  0.20  0.50 
Original distance  0.0093  0.0169  0.0071 
Distance of the $F{M}^{2}$ approximation  0.0276  0.0191  0.0214 
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Garrido, S.; Muñoz, J.; López, B.; Quevedo, F.; Monje, C.A.; Moreno, L. FM^{2} Path Planner for UAV Applications with Curvature Constraints: A Comparative Analysis with Other Planning Approaches. Sensors 2022, 22, 3174. https://doi.org/10.3390/s22093174
Garrido S, Muñoz J, López B, Quevedo F, Monje CA, Moreno L. FM^{2} Path Planner for UAV Applications with Curvature Constraints: A Comparative Analysis with Other Planning Approaches. Sensors. 2022; 22(9):3174. https://doi.org/10.3390/s22093174
Chicago/Turabian StyleGarrido, Santiago, Javier Muñoz, Blanca López, Fernando Quevedo, Concepción A. Monje, and Luis Moreno. 2022. "FM^{2} Path Planner for UAV Applications with Curvature Constraints: A Comparative Analysis with Other Planning Approaches" Sensors 22, no. 9: 3174. https://doi.org/10.3390/s22093174