Path planning trajectories in fluid environments

This paper deals with the problem of finding a mission path that minimizes acceleration and drag while a vehicle moves from an initial position to a final target in fluid environments. A variational problem will be formulated in the general context of manifolds, where the energy functional depends on acceleration and drag forces. The corresponding Euler-Lagrange equations will be derived. Questions regarding the integrability of the Euler-Lagrange equations are the main challenge of this problem even when the geometry of the configuration space is not taken into consideration. This is mainly due to the fact that the power needed to overcome the drag forces is proportional to the cube of the speed. A numerical optimization approach will be presented in order to obtain approximate solutions for the problem in some particular configuration spaces.