Equilibrium Configurations of a Kirchhoff Elastic Rod under Quasi-static Manipulation

Consider a thin, flexible wire of fixed length that is held at each end by a robotic gripper. The curve traced by this wire can be described as a local solution to a geometric optimal control problem, with boundary conditions that vary with the position and orientation of each gripper. The set of all local solutions to this problem is the configuration space of the wire under quasi-static manipulation. We will show that this configuration space is a smooth manifold of finite dimension that can be parameterized by a single chart. Working in this chart--rather than in the space of boundary conditions--makes the problem of manipulation planning very easy to solve. Examples in simulation illustrate our approach.