Construction of Generalized Coordinates�� Basis Functions in Lagrangian Dynamics of Flat Manipulators

Abstract

Second order Lagrange equations are used for describing dynamics of planar mechanism with rotation joints. For calculating kinetic energy of the links local coordinates of velocity vectors are used as well as recursive matrix transformations. Kinetic energy quadratic form coefficients are represented by linear combinations of seven independent trigonometric functions of generalized coordinates, i.e. basis functions. A number of these functions are connected to number of links by quadratic dependence. Constant coefficients in expansions in basic functions are determined from linear equation systems, representing kinetic energy of the mechanism in its several nonrecurring configurations with non-zero values for one or two generalized velocities. The resulting system of dynamics differential equations is integrated numerically with Runge-Kutta method in software environment Mathcad. Efficiency of the proposed method of creating and solving dynamic equations is demonstrated by example of numerical solution the direct dynamic problem of three-link mechanism.