# Input Shaping Control of a Flexible Structure for Rest-to-Rest and Non-Rest-to-Rest Maneuvers
## 3 Single Switch and ZVDD Input Shaping
### The Zero Vibration Derivative-Derivative (ZVDD) Input Shaper  
For a second-order system with natural frequency \(c_{3}\) and damping coefficient \(\zeta\), the impulse response equation for a series of \(N\) impulses can be written as [20]:  
\[y(T)=\sum_{i=1}^{N}\frac{A_{i}c_{3}}{\sqrt{1-\zeta^{2}}}e^{-\zeta c_{3}T_{i}} \sin{(c_{3}T_{i}\sqrt{1-\zeta^{2}})} \tag{19}\]  
where \(A_{i}\) is the amplitude of each impulse, and \(T_{i}\) indicates when it is applied. The equation for the residual vibration generated from a series of impulses can be stated as follows:  
\[V(c_{3},\zeta)=e^{-\zeta c_{3}T_{N}}\sqrt{C(c_{3},\zeta)^{2}+S(c_{3},\zeta)^{2}} \tag{20}\]  
where  
\[C(c_{3},\zeta) =\sum_{i=1}^{N}A_{i}e^{\zeta c_{3}T_{i}}\cos(c_{3}T_{i}\sqrt{1- \zeta^{2}}) \tag{21}\] \[S(c_{3},\zeta) =\sum_{i=1}^{N}A_{i}e^{\zeta c_{3}T_{i}}\sin(c_{3}T_{i}\sqrt{1- \zeta^{2}}) \tag{22}\]  
The system under consideration in this article is a \"spring-mass\" system, so the damping ratio \(\zeta=0\). The values for \(A_{i}\) and \(T_{i}\) can be obtained for various shapers [21; 22; 23]. Here, the ZVDD shaper is used because it offers more robustness than ZV and ZVD [10; 24], and its trajectory equations are less complicated than ZVDDD. However, depending on the requirements, the proposed shaper can be easily converted to ZV and ZVD by tuning one or more parameters. Similarly, by setting the impulse amplitude values and other parameters, the proposed input shaper can be used as an EI shaper (by specifying the vibration tolerance level), two-hump EI shaper (specifying the impulse amplitude values as mentioned in Ref. [25]), and Specified Insensitive (SI) shapers [24; 26; 27]. Thus, robustness  
Figure 3: Rest-to-rest case with single-switch, bang-bang torque. The plot units are degree (\(\theta_{1}(t)\) and \(\theta_{2}(t)\)), degree/second (\(\dot{\theta}_{1}(t)\) and \(\dot{\theta}_{2}(t)\)) and N.m (\(u(t)\)). can be controlled (improved) by choosing the appropriate shaper based on the complexity of the system. In the ZVDD shaper, the magnitudes of the impulses are constrained as defined in Equation (23):  
\[\sum_{i=1}^{N}A_{i}=1 \tag{23}\]  
The impulse time and amplitude matrix related to the ZVDD shaper can be written as  
\[ \begin{bmatrix}A_{i}\\ T_{i}\end{bmatrix} = \begin{bmatrix}\frac{1}{1+3+3K^{2}+k^{3}}&\frac{3K}{1+3+3K^{ 2}+k^{3}}&\frac{3K^{2}}{1+3K+3K^{2}+k^{3}}&\frac{K^{3}}{1+3K+3K^{2}+k^{3}}\\ 0&\frac{\pi}{\overline{w}_{d}}&\frac{2\pi}{\overline{w}_{d}}&\frac{3\pi}{ \overline{w}_{d}}\end{bmatrix} = \begin{bmatrix}\frac{1}{8}&\frac{3}{8}&\frac{3}{8 }&\frac{1}{8}\\ 0&\frac{\pi}{\overline{c}_{3}}&\frac{2\pi}{\overline{c}_{3}}&\frac{3\pi}{ \overline{c}_{3}}\end{bmatrix}  \tag{24}\]  
where \(K=\exp\left(\frac{-\pi c}{\sqrt{1-\xi^{2}}}\right)=1\) and \(w_{d}=c_{3}\sqrt{1-\xi^{2}}\) (\(w_{d}\) is the \"damped natural frequency\"). The values of \(A_{1}\), \(A_{2}\), \(A_{3}\) and \(A_{4}\) are \(1/8\), \(3/8\), \(3/8\), and \(1/8\) respectively, with \(\zeta=0\). For a unit torque, single-switch reference control, the convoluted torque after using the conventional ZVDD shaper will have the form shown in Figure 4.  
For any ZVDD shaper, the convoluted torque can be summarized as shown in Table 2.