# Damping of Oscillations of a Rotary Pendulum System
## 2 Input Shaping
### Shaper Constraints  
Due to the transcendental nature of the problem, there can be an infinite number of solutions. To choose a practical solution, it is necessary to define a set of different limiting equations (conditions), which are based on requirements, by which it is possible to determine the amplitudes and temporal positions of the pulses of the input shaper [28,29]. One of the fundamental limitations is the limitation of pulse amplitudes. If the pulse amplitudes are not limited, their values can reach positive or negative infinity due to time optimality [30]. The solution to this problem may be to limit the size of the pulses or require  
Figure 1: System response to a single and combined impulse. Figure 3: Block diagram of the system with input shaper. Figure 2: The internal structure of the (\(n\)th) order shaper. that all pulses have positive values. Limiting the pulse amplitudes is also necessary to ensure that the shaped input signal reaches the same final steady-state value as the original reference signal without shaping. This limitation is not explicitly stated in some filtered algorithm designs (especially in those where the filter has a non-linear character). It is often solved in an iterative way [31]. To prevent a change in the size of the control pulse, the sum of all amplitudes must equal one. The next limitation, perhaps the most important, is the condition for limiting residual vibrations (it was mentioned earlier as a vector condition). In many publications, this condition is associated with a dimensionless quantity of residual vibration amplitude, or the percentage ratio of vibrations (PRV) with input shaping to vibrations without input shaping. The mentioned ratio of amplitudes can be expressed as a function of the natural frequency, \(\omega\), and the damping coefficient, \(b\), of the system, which can be written in the form of an equation prescribing the zero value of the amplitude of the vector sum of all Dirac pulses forming the relevant type of shaper. The limitations mentioned so far are sufficient for the design of the basic (early) shaper (ZV), which has a very symmetrical structure (two appropriately shifted pulses in time), as demonstrated in Figure 1. The first impulse can be applied almost immediately (which is mainly desired). However, the second pulse is shifted in time by half of the oscillation period, which in most applications has proven to be a sufficient settling time but is directly related to the system frequency, which means that in the case of low-frequency dynamics, a high delay can occur. In reality, it is not possible to model this entirely accurately, or identify the parameters. Therefore, robustness to modeling errors is one of the other important factors. Suppose that small changes occur in the system, either due to inaccurate identification or in the case of a change in the length of the rope, the weight of the load, or any other property that can fundamentally affect the system parameters (mainly the frequency). In that case, the set parameters of this shaper will no longer be able to remove oscillations for the system, and with these changes, the residual oscillations will increase proportionally. Therefore, it is necessary to introduce another condition related to robustness. While the ZV shaper will theoretically provide zero vibration at the modeling frequency, it is sensitive to modeling errors [32]. This sensitivity makes the ZV shaper impossible to use practically on many systems. The robustness of input shapers is most commonly graphically represented by sensitivity curves. These curves show the percentage of residual vibration against the normalized vibration frequency. Insensitivity is one of the key measures of robustness derived from the sensitivity curve [6; 33]. Insensitivity is the width of the sensitivity curve at the tolerated percentage level (\(V_{TOL}\)) with respect to the monitored parameter. Generally, the more robust the shaper is, the more it can handle damping in the event of nonlinearities.