  The motions of mechanisms can be described in terms of screw coordinates by
means of an exponential mapping. The product of exponentials (POE) describes
the configuration of a chain of bodies connected by lower pair joints. The
kinematics is thus given in terms of joint screws. The POE serves to express
loop constraints for mechanisms as well as the forward kinematics of serial
manipulators. Besides the compact formulations, the POE gives rise to purely
algebraic relations for derivatives wrt. joint variables. It is known that the
partial derivatives of the instantaneous joint screws (columns of the geometric
Jacobian) are determined by Lie brackets the joint screws. Lesser-known is that
derivative of arbitrary order can be compactly expressed by Lie brackets. This
has significance for higher-order forward/inverse kinematics and dynamics of
robots and multibody systems. Various relations were reported but are scattered
in the literature and insufficiently recognized. This paper aims to provide a
comprehensive overview of the relevant relations. Its original contributions
are closed form and recursive relations for higher-order derivatives and Taylor
expansions of various kinematic relations. Their application to kinematic
control and dynamics of robotic manipulators and multibody systems is
discussed.
