This paper presents a numerical technique for path planning inside the workspace of parallel manipulators avoiding singularity. A generic numerical algorithm for generating the reachable workspace of parallel manipulators is described. The singularity points are determined inside the workspace. These points are grouped into several clusters and modeled as obstacles. Subsequently, a path planning algorithm is used to find an optimal path avoiding these obstacle. If any singularity point lies on or very close to the path, the path is restructured to avoid the singularity point by a local routing method based on Grassmann's line geometry. The path planning algorithm is uniformly applicable to parallel manipulators with any combinations of revolute and prismatic joints. An example is demonstrated for the effectiveness of the algorithm.

This paper presents a numerical algorithm for the analysis of reachable workspace and singularity representation of three-legged parallel manipulators. After finding out the approximate maximum workspace, a radial and equal area discretization is done. Then, a multitasking search is performed to determine the exact workspace boundary. The volume of the workspace is determined easily by a numerical integration method. Any void inside the workspace is found out. information about its position and stretch is determined and singularity curves inside the workspace is also drawn. A three-legged modular parallel manipulator is considered to demonstrate the effectiveness of the algorithm and found to work quite satisfactorily. The model is based on Product-of-Exponential scheme of formulation and, hence, the algorithm is uniformly applicable to any combinations of revolute and prismatic joints in the configuration of the leg.

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.