# Damping of Oscillations of a Rotary Pendulum System
## 2 Input Shaping
### Shaper Constraints
[PERSON] and [PERSON] were the first to develop an input shaping technique with an extended region around the system's natural frequency, which is robust enough to be used in most cases. To reduce the sensitivity of the input shaper to natural frequency errors, the shaper must meet another constraint, which consists of setting the derivative of the vibration equation with respect to the natural frequency to zero at the modeling frequency (ZVD). Limiting the zero derivative places additional zeros on the flexible poles, flattening the sensitivity curve at the modeled frequency and increasing insensitivity. This shaper has one pulse more, and the time lead equals one period of the vibration frequency. This shaper is based on ZV, and we paid for the increase in robustness by increasing the transition time of the system, which means that input shaping has a fundamental compromise between robustness and the duration of the shaper. An input shaper with even greater insensitivity than the ZVD shaper can be obtained by setting the second derivative to zero. This shaper is called the ZVDD shaper [34]. We could thus modify the algorithm by using an indefinitely repeated derivation of the vibration percentage equation (ZVDDDM...). For each derivative, an additional pulse is added to the shaper. Thus, the shaper is extended by half the frequency period. It is generally proven that in real systems, a particular modeling error will occur, so it is not advisable to try to get the vibrations around the natural frequency to zero, but rather to limit the vibrations to a certain low but at the same time acceptable level [6]. The extra-insensitive shaper extends the robustness of the ZVD shaper. When modeling this type of shaper, the constant \(V_{TOL}\) is given, which is generally equal to the upper limit of the acceptable residual oscillations, and thus, with an increase in the allowed size of oscillations, the insensitivity to modeling errors increases. The El shaper has the exact same pulse times as the ZVD shaper but with different amplitude values, resulting in greater robustness. Like zero-derivative methods, this idea can be extended from a basic shaper with a single hump in the sensitivity curve to a higher level of robustness by adding constraints that create multiple humps in the curve, called Multi-Hump El Shapers. Given the wide variety of systems that can benefit from input shaping, it seems desirable to develop a shaping method that specifies insensitivity and tailors the shaper's robustness to the specific system for which it may be designed. There are several ways to design this type of shaper, for example, using an approximation method where the vibrations are limited below a certain tolerable level at several points in the range of the required parameters, but this method provides approximate solutions. The amplitudes and times of the shaping pulses are then generated using optimization procedures. In practice, it is possible to use a small number of points to suppress vibrations in a wide range of parameters. The second procedure is more challenging to implement but brings accurate solutions. The advantage is that they can be designed to have non-symmetrical sensitivity curves so that the shaper is more robust and resistant to frequency increases than frequency decreases or vice versa. One disadvantage of the SI shaper is that optimization is required [35; 36]. An alternative limitation for constraints on time optimality allows the pulses to acquire negative but finite values, which causes a reduction in the time of individual pulses and thereby speeds up the entire process of ramping up to the desired value of the control pulse. With this type of shaper (UMZV), one can expect favorable properties concerning the overall delay, but at the expense of less robustness and possible high-mode excitation. Significant distinctions can be made between input-shaped commands and smooth commands based on their shape, design approach, and system response.