A geometric approach for modelling of unfixed-base manipulators

In this article, we establish a coordinate-free description of the kinematic and dynamic models of unfixed-base manipulator by using standard ideas from Lie group and Lie algebra. We use directly the exponential product formula to formulate the kinematic equation of an unfixed-base manipulator. In order to derive the dynamic equation of an unfixed-base manipulator, geometric variations, Riemannian metric, and Christoffel symbol are introduced. And, a concise, closed-form and coordinate-free dynamic equation is given according to the Hamilton´s variational principle. This derived model has two characteristics. On one hand, the dynamic model of the fixed-base manipulator is obtained naturally from this dynamic model of the unfixed-base manipulator. On the other hand, it is a control-oriented model and has some important structural properties which may be used to construct the tracking control law.