A global analysis of following trajectories by redundant manipulators in the presence of obstacles

A global solution is given for the following problem: Given a continuous trajectory defined in a robotic site containing obstacles, and given a redundant manipulator in the site, is the manipulator´s end-effector able to track the trajectory? If yes, find all possible solutions (i.e., all feasible motions of the robot´s structure producing a continuous displacement of the end-effector along the trajectory) in order to further optimize the task. If not, find the feasible parts of the trajectory. Instead of considering the pre-image of the trajectory in the robot´s jointspace (which would involve handling n-dimensional spaces, where n is the number of degrees-of-freedom of the manipulator), the problem is stated in a (r+1)-dimensional space, where r is the degree of redundancy of the manipulator. Application examples are shown using 3-R planar manipulators