# Deflection Reduction Shaping Commands with Asymmetric First-Order Actuators
## 2 Materials and Methods
### Analytical Deflection Reduction Shaping Command
The complete pure deflection resulting from a first-order actuator as shown in Figure 3a can be expressed with the superposition principle as follows:  
\[D(t)=\sum_{i=1}^{3}D_{i-i+1}(t)=\sum_{i=1}^{3}A_{i}\cdot D_{m}^{i}\Big{[}\sin \Big{\{}\omega_{n}t-\tan^{-1}(\omega_{n}\cdot\tau_{i})\Big{\}}\Big{]}\mathcal{ H}(t-t_{i}) \tag{6}\]  
where \(\mathcal{H}(t-t_{i})\) is a Heaviside step function with the impulse time location \(t_{i}\) of input shaper and the maximum deflection magnitude of each division \(D_{m}^{i}\) is defined as follows:  
\[D_{m}^{i}=\frac{V_{d}\sqrt{1+\omega_{n}^{2}\ \tau_{i}^{2}}}{L\ \omega_{n}\Big{(}1+\omega_{n}^{2}\ \tau_{i}^{2}\Big{)}} \tag{7}\]  
Figure 2: Pendulum system. and \(A_{i}=[1,\ -1,\ A_{3}]\) are defined and \(\tau_{i}=[\tau_{a},\ \tau_{d},\tau_{a}]\) are the time constants depending on first-order actuator dynamics as shown in Figure 3b. Equation (6) amounts to a piecewise continuous function consisting of 3 finite length segments with a limited range of applicability. Note that the magnitude of the deflection produced by a series of first-order response functions can exceed the maximum magnitude of \(D(t)\) if the deflection components from individual first-order response functions constructively interfere. From Equation (6), the final limited impulse magnitude \(A_{3}\) to maintain the transient deflection can be determined below a specific limit by satisfying the following constraint:  
\[\sum_{i=1}^{3}A_{i}D_{m}^{i}\Big{[}\sin\Big{(}\omega_{n}t-\tan^{-1}(\omega_{n }\tau_{i})\Big{)}\Big{]}\mathcal{H}(t-t_{i})\leq DEF_{film} \tag{8}\]  
where \(DEF_{film}\) is the first-order deflection limit. To meet the constraint Equation (8), the constraint can be adjusted by equating both sides and the final limited impulse magnitude \(A_{3}\) to control the transient deflection constraint can be solved as follows:  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]0  
where a non-dimensional ratio \(D_{film}=DEF_{film}/D_{m}^{1}\) is defined with respect to the largest magnitude \(D_{m}^{1}\) influencing the deflection response. In the case of time constants \(\tau_{a}=\tau_{d}=0\) for ideal actuators, Equation (9) is expressed as \(A_{3}=D_{film}\), implying a selection range \(0<D_{film}\leq 1\), and resulting in the relation \(D_{film}=2\cdot D_{lim}\) between the UMZV\({}_{\text{DBF}}\) and UMZV\({}_{\text{DBF}}\) shapers. Second, with a desired transient deflection limit Equation (9), the impulse times of the UMZV\({}_{\text{DBF}}\) shaper can be solved for zero residual oscillation by utilizing a phasor vector approach. With the modified input command instead of an on-off velocity command based on linear system theory, the command design procedure can be simplified with the transformation into an equivalent command process as shown in Figure 3a. The start portion of the velocity profile can be represented as a function of the step times \(t_{1}\), \(t_{2}\), and \(t_{3}\) as shown in Figure 1b and denoted as the following equation:  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]1  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]2 Command separation for start motion. (**a**) Equivalent command process and range division; (**b**) Segmented velocity profiles. where:  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]3  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]4  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]5  
Each term consists of a first-order response to a step command convolved with a time-delayed impulse.