flexible manipulators, which comprise thin slender arms, can enable higher-speed operation and lower energy consumption because their weight is typically lower than that of a rigid manipulator.moreover,light-weight manipulators are beneficial for cutting down transports costs of industrial or space robots. Therefore, flexible manipulators are superior to rigid manipulators in the above respects.However,it is well known that flexible manipulators are easily deformed due to their low flexibility;therefore,unwanted vibrations,which have a harmful effect on working effectiveness,occur easily.Thus, to avoid the unwanted vibrations of flexible manipulators,many researchers have attacked the vibration problem and have presented various control schemes[1-3]. In particular,trajectory planning methods are one of the best ways to control the vibrations for point-to-point (PTP) motion tasks of flexible manipulators[4-19]. However, to the best of the author's knowledge, studies on reducing the operation energy required to run manipulators have been limited to rigid manipulations(e.g.,[20-22]). A trajectory planning method that simultaneously suppresses the residual vibration and driving energy of a flexible manipulator has not been presented. Energy savings for flexible manipulators are very important for space robots because there is a limited amount of energy available for tasks.
with this background in mind, a trajectory planning method was developed; it enables to simultaneously minimize residual vibrations and the driving energy for a single-link flexible manipulator[23] and a robotic arm mounted on a flexible link [24]. For this trajectory planning method, an artificial neural network (ANN) was employed to generate the joint angles