2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67327This paper presents motion planning algorithms for underactuated systems evolving on rigid rotation and displacement groups. Motion planning is transcribed into (low-dimensional) combinatorial selection and inverse-kinematics problems. We present a catalog of solutions for all underactuated systems on $\SE{2}$, $\SO{3}$ and $\SE{2}\times\real$ classified according to their controllability properties.8 pages, double column, 3 figuresOptimization and Control70Q05; 70B15; 70G65text