# Deflection Reduction Shaping Commands with Asymmetric First-Order Actuators
## 2 Materials and Methods
### Optimized Deflection Reduction Shaping Commands  
The UMZV\({}_{\text{DRF}}\) shaper described in the previous section has the advantage of being derived in a closed-form solution by neglecting the exponential term. However, the approximated approach has an inevitable issue with respect to the control performance. For a problem outside of this approximation, a numerical optimization technique for deflection reduction shaping commands, the so-called UMZV\({}_{\text{DRF}}\), is developed with transient deflection reduction, zero residual oscillation, and time constant constraints. To constraint the maximum transient deflection, the local extrema of the deflection function should be located and limited below the desired deflection magnitude at these instances. For the location of the extrema point, the deflection response before the third impulse is compactly expressed by utilizing Equation (4) as follows:  
\[D_{o}(\mathfrak{t})=\sum_{i=1}^{2}\frac{A_{i}\,V_{d}}{L\,\omega_{n}(1+{\omega_ {n}}^{2}\tau_{i}^{2})}\bigg{[}\sqrt{1+{\omega_{n}}^{2}\tau_{i}^{2}}\sin\{ \omega_{n}\,t-\tan^{-1}(\omega_{n}\,\tau_{i})\}+\omega_{n}\,\tau_{i}\,e^{\frac {-1}{\tau_{i}}}\bigg{]}\mathcal{H}(\mathfrak{t}-t_{i}) \tag{22}\]  
Figure 4: Vector diagram of UMZV\({}_{\text{DRF}}\) shaper. where \(\mathcal{H}(\mathrm{t}-t_{i})\) is a Heaviside step function with the impulse time location \(t_{i}\) of input shaper. To determine the design parameters \(t_{2}\), \(t_{3}\), and \(A_{3}\) of the UMZVDRFO shaper, the maximum transient deflection of \(D_{o}(t)\) should be satisfied as the following constraint:  
\[\max[D_{o}(t)]\leq D_{max}\cdot D_{film} \tag{23}\]  
where \(D_{max}\) is computed by the UMZVDRF shaped velocity command with \(A_{3}=1\) and \(D_{film}=1\). Equation (24) is compactly expressed through the time derivative of Equation (22) and is used to determine the time location at the maximum deflection as follows:  
\[\sum_{i=1}^{2}\frac{A_{i}\,V_{d}}{L(1+\omega_{n}{}^{2}\tau_{i}^{2})}\left[ \sqrt{1+\omega_{n}{}^{2}\tau_{i}^{2}}\cos\left[\omega_{n}\ t\ -\tan^{-1}(\omega_{n}\,\tau_{i})\right]-e^{\frac{\pi i}{i}} \right]\mathcal{H}(\mathrm{t}-t_{i})=0 \tag{24}\]  
Then, \(\max[D_{o}(t)]\) can be determined using Equation (22). In the process of numerical optimization, the time is computed by the _fzero_ function in MATLAB(r). The constraints on zero residual oscillation are expressed with the x-y vector components of Equations (14)-(16) as follows:  
\[R_{x}=1-\frac{\sqrt{\left(\tau_{a}\omega_{n}\right)^{2}+1}}{\sqrt{\left(\tau_{ d}\omega_{n}\right)^{2}+1}}\cos\left(\omega_{n}t_{2}+tan^{-1}\left(\frac{1}{ \tau_{a}\omega_{n}}\right)-tan^{-1}\left(\frac{1}{\tau_{d}\omega_{n}}\right) \right)+A_{3}\cos(\omega_{n}t_{3})=0 \tag{25}\]  
\[R_{y}=-\frac{\sqrt{\left(\tau_{a}\omega_{n}\right)^{2}+1}}{\sqrt{\left(\tau_{ d}\omega_{n}\right)^{2}+1}}\sin\left(\omega_{n}t_{2}+tan^{-1}\left(\frac{1}{ \tau_{a}\omega_{n}}\right)-tan^{-1}\left(\frac{1}{\tau_{d}\omega_{n}}\right) \right)+A_{3}\sin(\omega_{n}t_{3})=0 \tag{26}\]  
Furthermore, an interval constraint between adjacent impulse times is included to maintain the acceptable control performance. Without collapsing the input commands, the following condition is enforced to ensure the rise time [29]:  
\[t_{i+1}-t_{i}>\frac{2.2}{\tau_{i}} \tag{27}\]  
where \(i=1\cdots 3\) for the index of the range division as shown in Figure 3. This condition is enforced to obtain acceptable control performance of the UMZVDRFO shaper without collapsing the command profile. Finally, the final impulse time \(t_{3}\) is minimized to ensure the shortest duration shaper as follows:  
\[J=min(t_{3}) \tag{28}\]  
where \(t_{1}=0\) is taken without losing the generality of the input shaper design. The computational procedure is summarized as follows:  
1. Find the maximum deflection \(D_{max}\) computed by Equation (22) at \(D_{film}=1\) and \(A_{3}=1\). 2.