Using Lie Group Symmetries for Fast Corrective Motion Planning

For a mechanical system it often arises that its planned motion will need to be corrected either to refine an approximate plan or to deal with disturbances. This paper develops an algorithmic framework allowing for fast and elegant path correction for nonholonomic underactuated systems with Lie group symmetries, which operates without the explicit need for control strategies. These systems occur frequently in robotics, particularly in locomotion, be it ground, underwater, airborne, or surgical domains. Instead of reintegrating an entire trajectory, the method alters small segments of an initial trajectory in a consistent way so as to transform it via symmetry operations. This approach is demonstrated for the cases of a kinematic car and for flexible bevel tip needle steering, showing a prudent and simple, yet computationally tractable, trajectory correction.