Using the Matrix Method to Compute the Degrees of Freedom of Mechanisms

Abstract

In this paper, some definitions and traditional formulas for calculating the mobility of mechanisms are represented, e.g. Grubler formula, Somov - Malyshev formula, and Buchsbaum - Freudenstei. It is discussed that there are certain cases in which they are too ambiguous and incorrect to use. However, a matrix method is suggested based on the rank of the Jacobian of the mechanism and its application is investigated. It is shown that the matrix method will definitely lead to a correct answer; however, it is lengthy and consumes more computational effort. It is shown that in the cases the traditional formulas give a wrong answer and the matrix method gives the correct mobility. To compare the methods, several examples are given including the four bar planar linkage, the augmented four bar linkage, University of Maryland manipulator, Cartesian parallel manipulator (CPM), delta robot, orthoglide robot, and H4 parallel robot.