Parametrised collision-free optimal motion planning algorithms in Euclidean spaces

We present optimal parametrised motion planning algorithms which can be used in designing practical systems controlling objects that move in Euclidean $d$-space, with $d\geq 2$ even, without collisions and in the presence of two obstacles with unknown a priori positions. Our algorithms are optimal in a very concrete sense, namely, they have the minimal possible number of local planners.