On the Distance between a Point and a Clothoid Curve

We consider the problem of computing the mini-mum distance between a given point and an assigned clothoid curve. This models, for instance, the tracking error of a robotic vehicle such as a car-like one, from a prescribed trajectory at a certain instant. The proposed algorithm finds this minimum distance (not unique, in general) up to a specified tolerance with a few iterations of a robust hybrid Dekker/Brent numerical scheme with cubic order of convergence. We guarantee the convergence to the global optimum proving a theorem that gives the intervals where there are all the local solutions so that it is easy to select the global one. We show the performance of the method with a test case and compare the results with state of the art solutions.