Simultaneous trajectory and shape optimization of flexible manipulators

In this paper, we develop a new method of simultaneous optimization of trajectory and shape of flexible manipulators. The method is based on trajectory planning of previous reports for rigid manipulators. We adopt approximate flexible manipulator model which consists of rigid beam elements and spring elements. B-spline function and mathematical programming method are utilized. We use B-spline function for approximation of trajectories of a joint angles. The joint angles are so approximated by 4th degree B-spline function that they are 2-times continuously differentiable with respect to time. There may exist too many trajectories which drive the end-effector of manipulator from the start point to the goal point. Therefore, it is necessary to choose the best one with reasonable performance index. In this method, to choose the best trajectory among all passible trajectories, we optimize the positions of control points which determine the trajectory. For shape optimization, we consider dimensions of each link as one of design variables. The performance index consists of 2 performance indices. The first is the driving energy, and the second is the deviation which is attributable to the flexibility of manipulator between end-effector's path of index trajectory and one of flexible manipulator's trajectory. By searching optimum positions of the control points and optimum dimensions of links which minimize the performance index, we design optimal trajectory and optimal shape of flexible manipulators. Through the simulations, we show the effectiveness of the developed method.