On Approach Based on Lie Groups and Algebras to the Structural Synthesis of Parallel Robots

Abstract

This paper proposes an approach to the structural synthesis of parallel robots based on the Lie algebras of the displacement groups. This approach allows the automation of the synthesis and eliminates the disadvantages of both methods based on the using of displacement subgroups, and methods based on the screw theory. Parallel robot is presented consisting of the moving platform and the stationary base connected via several serial kinematic chains. It is shown that movement of kinematic chain correspond to Lie subgroups of displacement, called displacement subgroups generators. These generators play an important role in determining the possible architectures of parallel mechanisms. Structural synthesis was performed for the case in which the mobile platform speed relative to the fixed basement forms a subalgebra of the Lie algebra of the displacement group of Euclidean space. In this case, we obtain three types of parallel robots. We also consider the case when the speed of mobile platform relative to the fixed basement forms a subspace, which is not a subalgebra of the displacement group of Euclidean space. It is divided into two types of robots. Is discussed case of parallel robot with serial chains generates Lie subalgebras in displacement space synthesis.