This document provides a revision of the notation originally introduced in [20] for describing kinematics and dynamics quantities of mechanical systems composed by several rigid bodies. Relative to the first edition, this new version includes an expanded section on frame acceleration (Section 5.4), the correction of a few typos, and the change of the fonts used in the notation from single face to bold face.\u3cbr/\u3eThe notation detailed in this document is inspired by the well-known Featherstone notation introduced in [7], also used, with small adaptations, in the Handbook of Robotics [16]. Featherstone’s notation, while being extremely compact and pleasant for the eye, is not fully in accordance with Lie group formalism, with the potential of creating a misunderstanding between the robotics and geometric mechanics communities.\u3cbr/\u3eThe Lie group formalism is well established in the robotics literature [13, 14,\u3cbr/\u3e10]. However, it is less compact than Featherstone’s notation [7], leading to long expressions when several rigid bodies are present as in the case of a complete dynamic model of humanoid or quadruped robots. This report aims, therefore, at getting the best from these two worlds. The notation strives to be compact, precise, and in harmony with Lie Group formalism. The document furthermore introduces a flexible and unambiguous notation\u3cbr/\u3eto describe the Jacobians mapping generalized velocities of an arbitrary frame to Cartesian linear and angular velocities, expressed with respect to a reference frame of choice
