Fast Methods are very popular algorithms to compute time-of-arrival maps (distance maps measured in time units) solving the Eikonal equation. Since Fast Marching was proposed in 1995, it has been applied to many different applications such as robotics, medical computer vision, fluid simulation, etc. From then on, many alternatives to the original method have been proposed with two main objectives: reducing its computational time and improving its accuracy. In this paper, we collect the main single-threaded approaches, which improve the computational time of the standard Fast Marching Method and study them within a common mathematical framework. Then, they are evaluated using isotropic environments which are representative of their possible applications. The studied methods are: the Fast Marching Method with binary heap, Fast Marching Method with Fibonacci Heap, Simplified Fast Marching Method, Untidy Fast Marching Method, Fast Iterative Method, Group Marching Method, Fast Sweeping Method, Locking Sweeping Method and Double Dynamic Queue Method.

Path planning is general problem of mobile robots, which has special characteristics when applied to marine applications. In addition to avoid colliding with obstacles, in marine scenarios, environment conditions such as water currents or wind need to be taken into account in the path planning process. In this paper, several solutions based on the Fast Marching Method are proposed. The basic method focus on collision avoidance and optimal planning and, later on, using the same underlying method, the influence of marine currents in the optimal path planning is detailed. Finally, the application of these methods to consider marine robot formations is presented.

This research presents two novel approaches to nonholonomic motion planning. The methodologies presented are based on the standard fast marching square path planning method and its application to car-like robots. Under the first method, the environment is considered as a three-dimensional C-space, with the first two dimensions given by the position of the robot and the third dimension by its orientation. This means that we operate over the configuration space instead of the bi-dimensional environment map. Moreover, the trajectory is computed along the C-space taking into account the dimensions of the vehicle, and thus guaranteeing the absence of collisions. The second method uses the standard fast marching square, and takes advantage of the vector field of the velocities computed during the first step of the method in order to adapt the motion plan to the control inputs that a car-like robot is able to execute. Both methods ensure the smoothness and safety of the calculated paths in addition to providing the control actions to perform the trajectory.