Analytical synthesis of least curvature 2D paths for underwater applications

Calculating smooth tra jectories of bounded curvature has received a very wide attention in the robotic literature of the last 40 years. Most results focus on the study of wheeled non-holonomic vehicles and are concerned with 2D path generation algorithms. The problem of smooth and least curvature 3D path planning is addressed with a variational approach and the general 3D Euler-Poisson equation is derived. The 2D solution is calculated as the plane projection of the general 3D solution and some special 2D cases are analyzed. It is shown that if some special conditions are satisfied along the 2D path the optimal solution is approximated by the well known Cornu spiral; moreover in these same conditions the optimal solution (and the Cornu spiral) are approximated by the more familiar cubic polynomial. Applications to rigid body underwater dynamics are discussed.