Motion Planning for Elastic, Non-Controlled, Hyper-Redundant Serial Mechanism

This paper introduces the kinematics and two motion planning schemes for a planar, hyper-elastic leaf-spring that is fully supported at the proximal end. The path of the leaf-spring's free-end depends on its zero-energy configuration and loaded initial configuration. These two configurations need to be determined such that the free-end's actual path will go through a desired path, which we define as a set of planar points. Our solutions minimize a weight function which compares the actual and desired paths. We show how our mathematical model fits the behavior of a shape memory alloy Nitinol wire under uniform heating and simulate our motion planning schemes. Finally, we calculate the shape of a real shape memory alloy wire so its free-end's path will follow a straight line.