Generalized Jacobian analysis of lower mobility manipulators

Exploring screw theory through the formalities of linear algebra, this paper presents a general approach for Jacobian analysis of lower mobility manipulators. Given the definitions of twist/wrench spaces and their subspaces of the end-effector, the underlying relationships amongst these subspaces are identified using the virtual work principle. Using the orthogonal and dual properties of these subspaces and variational representations to account for the permitted and restricted instantaneous motions of the end-effector, a rigorous general and systematic procedure for the formulation of a generalized Jacobian is proposed. The merit of the generalized Jacobian is that it allows the first order kinematic and static modeling (velocity, accuracy, force and stiffness) to be integrated into a unified mathematical framework, so standardizing the modeling procedure and improving the efficiency of design and analysis. The generalized Jacobians for the three well-known parallel manipulators are derived as examples to illustrate the generality and effectiveness of this approach.