Singularity analysis of a four degree-of-freedom parallel manipulator based on an expanded 6 × 6 Jacobian matrix

This paper presents new constraint singularities of a four degree-of-freedom (DOF) parallel manipulator, H4, providing Schönflies motion (three translations and one rotation around a vertical axis). The H4 parallel manipulator equips two motors on the upper base while two other motors on the side base. Jacobian matrix, therefore, has a 4 × 4 dimension, which cannot provide all singular configurations. An expanded 6 × 6 Jacobian matrix that additionally includes constraints on the motion of the traveling plate is derived by the reciprocal screw theory. Three different constraint singularities can then be analyzed analytically from the last two rows in the expanded 6 × 6 Jacobian matrix. A discretization method is also used to find out all singular and near-singular configurations including three different constraint singularities in the whole workspace. In addition, singular configurations are compared for the 4 × 4 and 6 × 6 Jacobian matrices, which suggests that an expanded 6 × 6 Jacobian matrix should be used for the singularity analysis of limited-DOF parallel manipulators to find out all singular as well as near-singular configurations.