# Energy-Saving Breakthrough in the Point-to-Point Control of a Flexible Manipulator
## 1 Introduction  
Robotic manipulators, designed for repetitive pick-and-place tasks, must be lightweight to minimize energy costs. However, their reduced weight often leads to increased vibrations due to decreased rigidity. Addressing this issue, various approaches have been developed to mitigate vibrations in flexible manipulators [1; 2; 3; 4; 5; 6]. These approaches fall into two primary categories: feedback and feedforward control schemes. Feedback control is effective in managing disturbances and variable parameters in the objects being controlled. Feedforward control, which does not require sensors to measure vibrations (as demonstrated in [7; 8]), is cost-effective and has received significant attention from researchers aiming to refine vibration control in flexible manipulators [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. Recent studies on feedforward vibration control include the application of the finite element method to investigate vibration control in single-link flexible planar and curved manipulators, particularly focusing on triangular and trapezoidal velocity motion commands [23]. [PERSON] et al. [23] reported that the residual vibrations of flexible manipulators were suppressed by selecting appropriate deceleration times in the trapezoidal velocity profile. [PERSON] et al. [24] introduced a method for nonlinear dynamic modeling and trajectory planning in a flexure-based macro-micro manipulator, significantly reducing residual microscopic-level vibrations. [PERSON] et al. [25] presented an optimization strategy for minimizing residual vibrations in space manipulator systems, utilizing an absolute coordinate model. Enhancements in existing input shaping techniques were reported by [PERSON] and [PERSON] [26], enabling the determination of multiple vibration modes in industrial robots with joint elasticity through the optimal S-curve trajectory, robust zero-vibration shaper, and dynamic zero-vibration shaper. [PERSON] et al. [27] examined the timing of jerks in trapezoidal motion profiles to minimize vibrations in a six-degrees-of-freedom (DOF) commercial articulated robot with a cantilevered beam. Further advancements included a trajectoryplanning method based on quintic polynomials for suppressing vibrations in spatial flexible manipulators with curved links [28], where a dynamic model was established as a differential algebraic equation using the absolute nodal coordinate formulation and Lagrange equation. [PERSON] et al. [29] studied vibration suppression in a planar multi-DOF serial manipulator with flexible links, proposing an online vibration-suppression path-planning method using backpropagation neural networks. A zero residual vibration position control strategy combining motion planning and optimization algorithms for two-DOF flexible systems was introduced by [PERSON] et al. [30]. This method was applied to various systems, including a car-spring-car system, a planar two-dimensional overhead crane, and a planar single-link flexible manipulator, and its effectiveness was demonstrated through numerical simulations. Lastly, [PERSON] et al. [31] presented an improved method for suppressing transient and residual vibrations in flexible industrial robots by adjusting acceleration/deceleration times in trapezoidal velocity profiles and incorporating a decision tree classification algorithm (C4.5) to refine input pre-shaping. However, the aforementioned studies focused only on vibration control, and methods targeting energy conservation were limited to manipulator systems with rigid links. [PERSON] et al. [32] conducted a comprehensive review of the literature on the optimization of energy consumption in industrial robots, in which they reviewed 136 papers published between the years 2004 and 2023. Focusing on industrial robots, autonomous vehicles, and embedded systems, [PERSON] et al. [33] provided a systematic review of the classification and analysis of various methodologies and solutions developed to improve the energy performance of robotic systems.