# Flattening the Curve of Flexible Space Robotics [PERSON]
## 2 Materials and Methods
### Classical Second-Order Structural Filtering  
The compensation of the system elaborated in Equation (32) is often augmented by structural filters whose transfer function is represented by Equation (33), where the frequencies of the zeros and poles (\(\omega_{z}\) and \(\omega_{p}\), respectively) and damping of the zeros and poles (\(\zeta_{z}\) and \(\zeta_{p}\) respectively) are designable, leading to the steady state gain, phase, and maximum gain, as revealed in Equations (34)-(36). Figure 11 illustrates how notch and bandpass filters, respectively, may be implemented by choice of the four designable parameters: \(\omega_{z}\), \(\omega_{p}\), \(\zeta_{z}\), and \(\zeta_{p}\), where the maximum phase lead and lag occur at respective frequencies found in Equations (37) and (38).  
\[\frac{Output(s)}{Input(s)}=\frac{\frac{s^{2}}{\omega_{z}^{2}}+\frac{2\zeta_{z} ^{2}}{\omega_{z}^{2}}s+1}{\frac{s^{2}}{\omega_{p}^{2}}+\frac{2\zeta_{p}}{ \omega_{p}^{2}}s+1} \tag{33}\]  
\[K_{\infty}=40 log_{10}\left(\omega_{p}/\omega_{z}\right) \tag{34}\]  
\[\phi_{max}=cos^{-1}\left[\frac{\left(2\zeta_{c}\sqrt{\omega_{p}/\omega_{z}} \right)^{2}-\left(\omega_{p}/\omega_{z}-1\right)^{2}}{\left(2\zeta_{c}\sqrt{ \omega_{p}/\omega_{z}}\right)^{2}+\left(\omega_{p}/\omega_{z}-1\right)^{2}}\right] \tag{35}\]  
\[K_{max}=20 log_{10}\left(\zeta_{z}/\zeta_{p}\right)dB \tag{36}\]  
\[\omega_{1}/\omega_{c}=\sqrt{2\zeta_{z}\zeta_{p}+1-\sqrt{\left(2\zeta_{z}\zeta _{p}+1\right)^{2}-1}} \tag{37}\]  
\[\omega_{2}/\omega_{c}=\sqrt{2\zeta_{z}\zeta_{p}+1+\sqrt{\left(2\zeta_{z}\zeta _{p}+1\right)^{2}-1}} \tag{38}\]  
Figure 11: Depictions of second-order structural filters. Graphics created by the author duplicating the depiction in [10]. **(a)** Notch filter named after the nature to “notch” out the response surrounding specific frequencies (e.g., \(\omega_{C}\)). **(b)** Bandpass filter named after the nature to only the response surrounding specific frequencies (e.g., \(\omega_{C}\) ). Filters are placed in series, as depicted in Figure 12.  
Incremental Application: Ensure Stability vs. Flatten the Curve  
Subsequent application of the filters expressed by Equation (33) is displayed in Figure 12's depiction of the simulation blocks used to produce the results in Section 3. A debatable design question is which filters are recommended for implementation. Utilization of both notch and bandpass filters at each identified node would seek to flatten the curve (the amplitude plot of the frequency response).  
Additional measures were investigated to control a presumably very highly flexible, lightweight space robot inspired by modern, optimal open-loop techniques, such as time-delay input shaping. A key strength is the avoidance of using step commands as the input trajectories, so several methods were compared, replacing the step input with the four alternatives depicted in Figure 13b: modern time-delayed input-shaping, rigid body optimal input trajectory, recently proposed whiplash compensation, and the presently proposed single sinusoidal input trajectory, where the single sine is chosen for both speed of performance and avoidance of flexible resonance and anti-resonance, as depicted in Figure 9b and the dashed line in Figure 13a. Each technique is described in the immediately following sections.