On the Application of Lagrange's Method to the Description of Dynamic Systems

Abstract
Open-loop kinematic chains are used to describe industrial manipulators which represent a particular class of interconnected jointed mechanisms. Before constructing such mechanisms it is necessary to carry out analytical and synthesis studies on their mathematical models. Although Lagrange's method has certain advantages over other model formulation techniques, a straightforward application of this method to obtain the dynamic equations of interconnected mechanisms and open-loop kinematic chains is not entirely free of drawbacks. The procedure has a large computational redundancy resulting in lengthy and time-consuming derivations with a large probability of error. These factors also make the procedure unsuitable for implementation on digital computers when computing time is critical. It is shown that Lagrangian formulation of open-loop kinematic chains can be considerably simplified by perfornting differentiation prior to performing scalar product of vectors. Computational redundancy is eliminated and the procedure can also be adapted as a more efficient automated method than others, especially when the computing time is critical. The power of the method is demonstrated by using it to obtain the dynamic equations of an anthropomorphic industrial manipulator.