A catalog of inverse-kinematics planners for underactuated systems on matrix groups

This 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 $\text{SE}2$, $\text{SO}3$ and $\text{SE}2\times \mathrm{\Re }$ classified according to their controllability properties.