Efficient trajectory planning for two manipulators to deform flexible material

Coordinating two robot manipulators to handle flexible materials has a wide range of applications in the manufacturing industry. However, this problem has not been seriously addressed until recently. The two robot manipulators have to follow complicated trajectories to maintain a minimum interaction force with the flexible beam. These trajectories are very complicated and not suitable for real time systems. In this paper, three approximation methods of the optimal trajectories are introduced. The first one uses a piece-wise linear approximation of the optimal trajectories, while the second one uses adaptive piece-wise linear approximation. The third method utilizes the behavior of the optimal trajectories and applies a continuous approximation of the optimal trajectories using an ellipsoid. The interaction force under these approximations is investigated. The authors found that the first method produces a peak in the interaction force which can damage the object. The second method controls the peaks in the force to a prescribed limit. In the third method, the force is very close to the optimal force. Experimentation was used to verify the approximation methods