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 transport 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 manipulators
(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.