# Robust Input Shapers for Acceleration-Limit Actuators
## 1 Introduction  
Input shaping technology is used to reduce residual oscillations in various flexible systems, including industrial cranes [1, 2] and flexible robot joint systems [3]. Input shapers enhance robustness with added impulses to ensure good control performance. However, conventional input shapers have achieved zero-residual-deflection control performance with accurate system modeling and ideal actuators, whereas acceleration-limit actuators are commonly employed owing to their cost-effective industrial applications. Therefore, new robust input shapers must be developed for practical industrial applications of input shaping techniques to flexible systems for maintaining control robustness under acceleration-limit actuators. Input shaping is implemented via the convolving of an input command with an impulse sequence comprising magnitudes and time locations determined by solving a set of constraint equations to generate a shaped input [4, 5, 6]. The constraint on the percentage residual oscillation amplitude (PRA) can be expressed as the ratio of the residual oscillation amplitudes with and without shaping. For an undamped model of a second-order linear system, \(\text{PRA}(\omega_{n})\) is expressed as:  
\[\text{PRA}(\omega_{n})=\sqrt{\left\{\sum_{i=1}^{n}A_{i}cos(\omega_{n}t_{i}) \right\}^{2}+\left\{\sum_{i=1}^{n}A_{i}sin(\omega_{n}t_{i})\right\}^{2}} \tag{1}\]  
where \(\omega_{n}\), \(A_{i}\), and \(t_{i}\) represent the natural frequency of the system, and the magnitude and time location of the \(i\)th impulse, respectively. By setting \(\text{PRA}(\omega_{n})=0\) regarding the actuator performance limit and the shortest duration, a zero vibration (ZV) shaper [7] was obtained with the least amount of robustness because the residual oscillation increases rapidly as the modeling parameter is deviated. To improve the robustness of input shapers against system modeling errors, a ZV and derivative (ZVD) shaper [8] was developed usingthe derivative with respect to the frequency of \(\text{PRA}(\omega_{n})=0\) as an additional constraint. To further improve the robustness by allowing a tolerable level of residual oscillation, an extra-insensitive (EI) shaper [9] was proposed. For the qualitative measurement of the robustness of input shapers, a sensitivity curve is displayed in Figure 1, indicating the change in residual oscillation as a function of the modeling error. Insensitivity is a measurement of the relative robustness of shapers and is defined as the width of the sensitivity curve at a tolerable percentage oscillation level \(V_{tol}\) with respect to the parameter of interest. Even if the EI and ZVD shapers produce longer insensitivity than the ZV shaper at \(V_{tol}=5\%\), these input shapers are developed under the assumption of linear system theory which means that their robustness performance is questionable for nonlinear dynamics. Considering the nonlinear dynamics of actuators, several input shapers have been proposed for ramp [10; 11], first-order [12; 13], and asymmetric 2 nd-order [14] actuators. However, nonlinear dynamics can be attributed to discontinuous nonlinearities within a system, such as backlash, saturation, rate-limiting, and dead-zone. When a hard nonlinearity distorts an input-shaped command, the oscillation reduction performance of the input shapers can be significantly degraded [15; 16]. The effects of backlash [17] and Coulomb friction [18] on an input-shaping control system were considered and techniques for the input shaper redesign to reduce the detrimental effects of nonlinearities were presented. However, all previous studies have focused on developing input shaping techniques against the nonlinearities of actuators rather than improving control robustness while addressing nonlinear dynamics. As an alternative approach to improve the control robustness of input shapers, nonlinear optimization is required with the typical constraints of transient oscillation [19], robustness [20], and time optimality [21; 22].