Geometric properties of zero-torsion parallel kinematics machines

The advantages of tilt-and-torsion angles in analysis of zero-torsion parallel kinematics machines (PKM) have been reported by several literatures. However, geometric properties of tilt-and-torsion angles are not completely understood and fully utilized in synthesis of novel zero-torsion PKMs. In this paper, we study geometric properties of the so called zero-torsion motion types via differential geometry of Lie groups. We show that zero-torsion motion types admit simple representations under canonical coordinates of the first kind of the special Euclidean group SE(3). Using the proposed representation, we give a classification of zero-torsion PKMs. The synthesis condition for several well known zero-torsion PKMs are correctly identified. We will conduct type synthesis of zero-torsion PKMs in a separate paper.