Position Analysis of the Rhombic Assur Group

Abstract

The paper presents a position analysis of a special higher Assur group, the homogeneous rhombic Assur group of class 4 and order 4. For an Assur group, i.e., a kinematic chain with a certain number of inputs (the class of the group) which becomes rigid if the input parameters are held constant, the position analysis means the determination of all possible positions of this chain which correspond to a given set of input parameters. By solving an algebraic equation of order 20 and two equations of order 18 it is found that the rhombic Assur group 4,4 can theoretically occupy 56 different (real or complex) positions. To know these positions becomes vital if an Assur group for exemple serves as basic mechanism for a redundant planar manipulator whose singular positions must be carefully avoided. Though we were not able to establish an equation which would determine the positions of the general Assur group 4,4, (it would be an algebraic equation of order 56), we nevertheless can conclude that the number of its possible positions must also be 56.