Task space control for the whole arm

In this paper, task space control for the whole arm of a serial-chain manipulator is considered. The task space for the whole arm is interpreted in two ways, the space of rolled shapes or bunches of ordered n-frames. A rolled shape is a set of restricted points with rolling angles of links, and gives us intuitive viewpoint of the space. To represent the rolls of links, the concept of the canonical manipulator is introduced. A bunch of n-frames is a collection of frames each of which corresponds to the link orientation of the chain, and allows us to handle the space easily. A rolled spatial curve, which has a roll distribution over the curve, is used for specifying a desired rolled shape of the chain. A method generating the bunch of desired n-frames from a desired rolled curve is proposed. Based on the method, we derive a regulation law in the task space for the whole arm by Lyapunov design. The derived control law is very simple; recursive with respect to the link/joint number, and calculated by geometric operations in three-dimensional Euclidean space