A novel formalisation of the Markov-Dubins problem

The Markov-Dubins problem requires to find the shortest path that connects an initial point and angle to a final point and angle with bounded turning radius. Formally, this is equivalent to solve an interpolation problem with continuity up to the first derivative and with bounded curvature. We propose a mathematical framework that models with a single equation the different cases that arise, i.e., we can represent with the same function an arc of circle or a line segment by smoothly blending from one to the other. This allows us to restate the problem as a standard Mixed Integer Nonlinear Programming (MINLP), which can be relaxed into a standard Nonlinear Programming (NLP) and therefore opens the way to solve it using off-the-shelf solvers. Moreover, our formalism captures the symmetries of the problem in a more intuitive way with respect to previous works, thanks to the considered conformal bipolar transform. This approach is suitable for an effective solution of the extended problem of connecting multiple points, that will be addressed in future research.