Quasi-static manipulation of a planar elastic rod using multiple robotic grippers

We consider the problem of manipulating a planar elastic rod using robotic grippers which grasp the rod at multiple points. Building upon previous work that considers a rod held only at its ends, we show that manipulating a rod held by n + 1 grippers is equivalent to planning a path on a smooth 3n-dimensional manifold. Using multiple grippers can be advantageous when manipulating around obstacles. We establish upper and lower bounds on the number of grippers needed for an equilibrium shape of the rod to pass between obstacles in a desired way. Finally, we consider manipulation planning when both ends of the rod are held fixed, and only the grippers located along the interior of the rod can move.