An instantaneous kinematics and singularity analysis of a class of three-legged, 6-DOF parallel manipulators are addressed. The kinematic relation between the actuator joint rates and end-effector velocity is established using twist annihilators. Then the singularity analysis is performed using the concepts of reciprocal screw and Grassmann line geometry. The instantaneous kinematics model derived using the product-of-exponential formulation, is uniform to any combination of revolute and prismatic joints in the leg. It is shown that reciprocal screws can be easily constructed by using twist annihilators. Based on the line geometry, the geometric conditions are proposed to identify each of the case of singularity configurations for the considered class of parallel manipulators. These geometric conditions are simple and thus, the singularity configurations are readily conceived.
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