RECURSIVE KINEMATICS AND DYNAMICS FOR PARALLEL STRUCTURED CLOSED-LOOP MULTIBODY SYSTEMS^*

Abstract

A kinematic formulation for the parallel structured closed-loop multibody mechanical systems, such as Stewart Platform and Hexapod machine tools, is presented in this paper, which is recursive in nature. This also leads to the minimum order representation of the dynamic equations of motion. The recursive algorithms are known for their efficiency when a system is large. They also provide many physical interpretations. On the other hand, the minimum 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 numerically stable integration algorithm that does not violate the kinematic constraints. Two recursive algorithms, one for the inverse and another for the forward dynamics, are proposed. The overall complexity of either problem is O(n) + n O(m), where n and m 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 such 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.

^*Communicated by J. McPhee.

Keywords:

• Parallel structure
• Closed loop
• Dynamics
• Recursive algorithm

ACKNOWLEDGMENT

We sincerely acknowledge the financial support of the Alexander von Humboldt Foundation of Germany that was granted to S. K. Saha, who also thanks the IIT Delhi for granting the necessary leave to allow him to carry out the work successfully at the University of Stuttgart, Germany. This research work was carried out during S. K. Saha's term as Humboldt Fellow at the Institut B für Mechanik, Universität Stuttgart, Germany.

Notes

^*Communicated by J. McPhee.

^*A simulation software in C at the Institute of B Mechanics, University of Stuttgart, Germany.