Path Trajectory Verification for Robot Manipulators in a Manufacturing Environment

In a manufacturing environment where a robotic arm is programmed to follow a specified trajectory such as in welding, painting, and soldering, it is often the case that the arm reaches a singular configuration, where the program is stopped, the arm is switched to a new configuration, and the motion is continued. This difficulty has been a long standing problem. This paper presents a mathematical formulation for verifying whether a trajectory can be completed, uninterrupted, avoiding halting of a planned path. In some cases, this formulation also allows the selection of an initial configuration to ensure a smooth path trajectory. The paper presents an analytical formulation for determining barriers to motion inside the workspace of manipulator arms. Crossability analysis of the end-effector on a barrier are addressed. A criterion for selecting an initial configuration that would result in an un-interrupted motion is introduced. The mathematical theory is validated through numerical examples of planar and spatial manipulator arms.