Canonical Decomposition of the Instantaneous Kinematics of Darboux Frame

This paper gives a screw theoretic insight into the instantaneous kinematics of the Darboux frame (D-frame) moving on a smooth surface. It is established that the velocity state of D-frame is described by a three screw system. The principal screws of this system depend only upon the local shape of the surface. The twist of the D-frame is decomposed into twisting about surface normal line and a screw which is a ruling of a cylindroid. The properties of this cylindroid are studied. It is also shown that the twist of the D-frame can be obtained by a linear combination of pure rotations about three axes. This result is used to arrive at an equivalent 3R (revolute) serial chain whose end-effector is the D-frame. An illustrative numerical example is presented.