Higher derivatives of the kinematic mapping and some applications

A thorough analysis of mechanisms requires higher derivatives of the kinematic relations between its members. Such a relation is the (forward) kinematic mapping of a kinematic chain that relates the joint motions to the motion of the links. A first-order motion approximation is provided by the instantaneous joint screws. Higher-order approximations thus require higher derivatives of this screw system. Since the representation of screw entities is not unique a particular representation gives rise to a particular explicit form of the derivatives. Two commonly used variants are the spatial and body-fixed representation. Here a closed form expression for the v-th partial derivatives of the joint screws within a kinematic chain w.r.t., the joint variables is presented for the spatial and body-fixed representation. The form of the final expressions makes them ideal for computer implementations. The expressions are given explicitly for derivatives of up to 4th order. The paper concludes with a brief discussion of applications where higher derivatives are relevant. These are the kinematic analysis and determination of motion spaces of serial mechanisms, the higher-order mobility analysis, and the algebraic formulation of motion equations.