Recursive kinematics and dynamics for parallel structured closed-loop multibody systems

A kinematic formulation for the parallel structured closed-loop multibody m echanical systems, such as Stewart Platform and Hexapod machine tools, is p resented in this paper, which is recursive in nature. This also leads to th e minimum order representation of the dynamic equations of motion. The recu rsive algorithms are known for their efficiency when a system is large. The y also provide many physical interpretations. On the other hand, the minimu m order dynamic equations of motion are desired in control and simulation. For the latter a minimum set of dynamic equations of motion leads to a nume rically stable integration algorithm that does not violate the kinematic co nstraints. Two recursive algorithms, one for the inverse and another for th e forward dynamics, are proposed. The overall complexity of either problem is O(n) + nO(m), where n and in are the number of legs and the total number of rigid bodies in each leg, respectively. Hence the proposed formulation exploits the advantages of both minimum order representation and recursive algorithms, which earlier were available only for the open-loop systems suc h as serial manipulators. The method is illustrated with three examples: a one-degree-of-freedom (DOF) slider-crank mechanism, four-bar linkage, and a two-DOF five-bar planar parallel manipulator.