Study on the Existance Criteria and Synthesis of a Loop of Mechanisms.

Abstract

The algorithm for generating a loop of mechanisms utilizing the concept of the Lie algebra of the affine motion group has been developed. Firstly, the method to generate a loop from a mechanism with two loops are derived. The motion spaces of the loops are two kinds of Lie algebra. When one of connecting chains spans the intersection of the motion spaces of two kinds of Lie algebra, the single loop with the same degrees of freedom as a two loops mechanism can be obtained by removing this connecting chain. Another method utilizes a serial open loop kinematic chain composed of linear independent joint motors (screw coordinates). The serial chain is composed of the combination of Lie algebra of first several joint motors and another motion space spanned by remaining joint motors. When the latter motion space and one or more elements of the former Lie algebra span a second Lie algebra, by connecting the open loop by one or more joints in the motion space which is spanned by both the second Lie algebra and the sub-space of the former Lie algebra, a closed loop, the motion space of which has constant dimension while in motion, can be obtained. Some examples obtained by these methods are also shown in the paper.