# Damping of Oscillations of a Rotary Pendulum System
## 1 Introduction  
The issue of controlling flexible structures is an extensive area of research, as in many applications, unwanted oscillations are generated due to various elasticities, especially in the case where nowadays there is a demand for the implementation of increasingly faster and more aggressive movements of mechanical systems with flexible elements. Input shaping is a feedforward filtering method that modifies the frequency characteristics of the input signal, thereby reducing the dynamic-motion-induced vibrational response of the system (extracting the input energy around the system's natural frequency). Input shaping schemes first appeared in the 1950s, when early implementations were challenging without digital computers. This method began to be widely applied in the 1980s and 1990s when controlling the movements of various types of cranes by the operator [1; 2; 3], which created the need to generate a trajectory in real time, which again limited the number of available algorithms. These methods were often designed for linear movements, and even whenthe load moved in different axes (especially in the case of gantry cranes), each could be controlled separately due to multiple actuators.  
Sensitivity to modeling errors caused significant residual vibrations [4]. Input shaping became helpful in many natural systems when a robust method was developed [5]. Later, improved methods were introduced to increase robustness further [6]. Due to its robustness, input shaping has been implemented on various systems where precise positioning is required, including robotic manipulators [7], elevators [8], conveyors, throttle control [9], MEMS [10], liquid filling machines [11], etc. While the basic shaping methods assume specific, unchanging characteristics of the system, in the case of various dynamic changes in the system, it is possible to use an extension with adaptive methods, which can also help eliminate nonlinearities [12].  
Since the basic principle of shaping techniques is to delay the input signal, this technique can be combined with feedforward control actions, which are crucial to guaranteeing the delay-independent stability of time-delayed systems. Such an integrated approach not only tries to suppress oscillations but also ensures robust stability that takes into account intentional delays caused by shaping techniques and inherent delays of the system, which may also have multiple inputs and outputs [13]. Another possible extension (very popular today), which we also owe to the rapid development of computing technology, is the extension of heuristic methods, which are often based on experience, intuition, or experimental observation and are used to solve problems where a direct and exact calculation can be too complex, difficult, or even impossible. They include various techniques including genetic algorithms [14], neural networks [15; 16; 17; 18; 19], PSO algorithms [20; 21], and fuzzy [22; 23; 24].  
The remainder of this paper is structured as follows. Section 2 discusses the time and frequency properties of basic shapers. Section 3 demonstrates the description and simulation of a dynamical system. Section 4 presents a shaper design. The simulation and experimental results, which were performed to verify the proposed design method, are presented in Section 5. The article then ends with pertinent conclusions.