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6.1. Recapitulative The underwater world is a very demanding environment for trajectory planning algorithms. Great efforts are currently being made to develop autonomous systems as underwater technology becomes more mature. Several key issues for the three dimensional underwater trajectory planning problem have been addressed in this chapter. Reliability of trajectory planners has been improved by introducing the Fast Marching algorithm as a new basis for sampling based trajectory planning methods in the continuous domain. First, we have introduced the trajectory planning framework and the basic concepts shared by all the deterministic sampling based planning algorithms. The Fast Marching method, as one of these trajectory planning technique is similar in spirit to classical grid-search algorithms such as the A* algorithm. This led us to develop a new algorithm, called FM*, that combines the exploration efficiency of the A* algorithm with the accuracy of the Fast Marching method. For these reasons, the FM* algorithm opens new possibilities for planning trajectories in wide and continuous underwater environments. Second, even if they are implemented on a discretized perception of the world, Fast Marching based planning methods have the property to extract derivable trajectories. By applying mathematical tools from differential geometry, it has been proved that smoothing input data results in smoother trajectories. A technique has been proposed that insures the feasibility of a trajectory for a mobile robot with a given turning radius. This technique iteratively smoothes input data until a formal criterion is satisfied. The method is efficient because the Fast Marching algorithm is eventually launched only when input data are compliant with the curvature constraints of the vehicle. Third, another approach has been developed to speed up the exploration process in the case of partially-known or dynamic environments. A dynamic version of the Fast Marching |