# Robust Input Shapers for Acceleration-Limit Actuators
## 2 Robust Input Shapers for Acceleration-Limit Actuators  
In this section, two types of robust input shapers were analytically developed with only three impulse sequences for the oscillation reduction of flexible system operations with acceleration-limit actuators during point-to-point maneuvers. In the derivation process, the impulse magnitudes of the linear robust input shapers that satisfied the actuator performance limit were utilized to determine the impulse time locations. Two types of robust input shapers are presented besides a phasor vector representation of a ramp function to approximate the distorted input command profile, as described below. In the case of ideal actuators with infinite acceleration and braking performances, ideal-shaped input commands were generated by convolving the pulse input and three impulse sequences, where \(A_{i}\) and \(t_{i}\) are the amplitude and time location of the impulse, respectively, as shown in Figure 2a. An acceleration-limit actuator that generates ramp-shaped input commands changes its own input commands, as shown in Figure 2b. These distorted input commands degrade the control performance of flexible systems and produce large residual oscillations. In Figure 2, \(V_{d}\) represents the desired velocity of the actuator and \(v_{s}\) represents the velocity shaped by the input shaper. To develop an input shaper for such a nonlinear input command profile, the input command is transformed and simplified into a ramp shape, as depicted in Figure 3. This approach, with an equivalent constraint, provides a solution process that can be reformulated to develop an input shaper for the distorted command [11]. To present the two types of robust input shapers using the modified process described above, a single-pendulum system was utilized with ramp-type input commands, as shown in Figure 4. The equation of motion is given as:  
\[L\ddot{\theta}(t)+\mathrm{g}\ sin\theta(t)=u(t)cos\theta(t) \tag{2}\]  
Figure 2: Effects of ideal and acceleration-limit actuators. (**a**) Ideal actuator effect. (**b**) Acceleration-limit actuator effects. where \(u(t)=\dot{v_{a}}(t)\) represents the velocity input command, \(L\) represents the system string length from the trolley to the payload, and \(g\) represents the gravitational acceleration. Assuming that \(\theta(t)\) is sufficiently small, Equation (2) is Laplace transformed as:  
\[\theta(s)=G_{p}(s)V_{a}(s)=-\frac{sV_{a}(s)}{L}\frac{1}{s^{2}+{\omega_{n}}^{2}} \tag{3}\]  
where \(V_{a}(s)\) represents the Laplace transform of \(v_{a}(t)\). The output \(\theta(s)\) of the system is reformulated as the system input \(H(s)\) and the sine input to be expressed in phasor form as:  
\[\theta(s)=\underbrace{-\frac{s.V_{a}(s)}{L\omega_{n}}}_{H(s)}\underbrace{ \frac{\omega_{n}}{s^{2}+{\omega_{n}}^{2}}}_{\text{sine input}} \tag{4}\]  
As a key point in the input shaping technique, the steady-state equation can be expressed according to Equation (4) as:  
\[\theta_{\text{ss}}(t)=\frac{|V_{a}(j\omega_{n})|}{L}sin(\omega_{n}t-\frac{\pi} {2}+\angle V_{a}(j\omega_{n})) \tag{5}\]  
For no residual oscillation, \(\theta_{\text{ss}}(t)\) in Equation (5) must be zero. Therefore, \(|V_{a}(j\omega_{n})|\) can be determined by dividing a command by the impulse time of the input shaper, as presented in Figure 5b, in an acceleration-limit input command in rising mode, as shown in Figure 5a. Figure 4: Pendulum system. Figure 3: Equivalent transformation. (**a**) Original command process. (**b**) Equivalent command process. As shown in Figure 5, the acceleration limit of the actuator must be considered. The equivalent shaper formulation under the acceleration rate limit can be used with the following restrictions:  
\[t_{i+1}-t_{i}\geq\frac{A_{i}V_{d}}{\kappa_{i}}\quad(\kappa_{a}\text{ if }A_{i}\text{ is positive, }\kappa_{d}\text{ if }A_{i}\text{ is negative}) \tag{6}\]  
Here, \(\kappa_{a}\) and \(\kappa_{d}\) represent the acceleration and deceleration constants, respectively.