General formulation for under-actuated manipulators

Under-actuated and macro-mini manipulators represent common characteristics in dynamics. This paper introduces a general formulation of kinematics and dynamics for manipulator systems which are composed of two types generalized coordinate, active and passive, or primary and secondary. The Generalized Jacobian Matrix and the Generalized Inertia Tensor, originally proposed for free-floating space manipulators, are shown to be redefined for general under-actuated manipulator systems. The Reaction Null-Space and reactionless motion which are recently studied focusing to flexible-base manipulators, the compensability which is originally discussed for flexible-arm manipulators, and the measure of dynamic coupling, are proposed as examples of common properties for general under-actuated systems.