# Novel Pointing and Stabilizing Manipulator for Optical Space Payloads
## 4 Simulation Validation
### Isolation Algorithm Verification  
The proposed manipulator capability to stabilize the LOS is tested next. The floor disturbance transmissibility is first compared in the frequency domain. Since the feedforward controller (55) is associated with the manipulator orientation, four pointing directions were considered, whose RPY angles are, respectively, (0\({}^{\circ}\), 0\({}^{\circ}\), 0\({}^{\circ}\)), (8.5\({}^{\circ}\), 0\({}^{\circ}\), 0\({}^{\circ}\)), (0\({}^{\circ}\), \(-\)7\({}^{\circ}\), 0\({}^{\circ}\)), and (0\({}^{\circ}\), 0\({}^{\circ}\), 14\({}^{\circ}\)). The low-pass filters are both 9-order Chebyshev type II filters whose stop frequencies are 1000 Hz. The high-pass filters are both 2-order filters whose corner frequencies are 10\({}^{-4}\) Hz. As shown in Figure 8, when the feedforward controllers are acted, the maximum attenuation exceeds 50 dB in all four cases. Since the phase and gain mismatch, the attenuation effect would degenerate in the transition frequency band of high-pass and low-pass filter, and floor disturbance in the band of 100-700 Hz would be amplified by an active feedforward system, but the amplifications do not exceed 9 dB. The theoretical results derived from Formula (38) are also shown in the figures. It is seen that the theoretical results are well consistent with the passive transmissibility of the virtual system in the low-frequency band. Thus the validation of Formula (38) could be accepted. Furthermore, the vibration isolation effect is compared in the time domain. Disturbance force generated by 0.1-1000 Hz white noise is acted on the support module. The platform orientation jitter is shown in Table 2. It is seen that the jitter levels of all the channels in all four cases sharply go down, the magnitude of platform angular displacement falls to 10\({}^{-7}\) rad from 10\({}^{-5}\) rad, and the root-mean-square (RMS) level of platform angular displacement reduces by over 95%. The jitter trajectories of the image point and their convex hull are shown in Figure 9. It is seen that the scattered areas in all four cases sharply go down, too. Additionally, the scattered radius standard deviation falls to 10\({}^{-7}\) meters from 10\({}^{-5}\) meters, which could meet the pointing stability requirements of common optical payloads. Combining the frequency-domain and time-domain results, it could be accepted that the feedforward controller (55) could effectively improve the pointing stability. Figure 7: The simulation trajectories and the corresponding algorithm error on each plane. (**a**). The quadrilateral trajectory comparison and the algorithm error on the roll-pitch plane, (**b**). those on the roll-pitch plane, (**c**). those on the roll-pitch plane, (**d**). the random trajectory comparison and the algorithm error on the roll-pitch plane. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Orientation** & \multicolumn{2}{c}{**Passive Case (Rad)**} & **Active Case (Rad)** & **Rejection Ratio** \\ \hline \multirow{3}{*}{\((0^{\circ},0^{\circ},0^{\circ})\)} & \(\theta_{p1}\) & \(1.65\times 10^{-5}\) & \(1.14\times 10^{-7}\) & 99.3\% \\  & \(\theta_  
Figure 9: _Cont_. The stability of IPFL consisting of a feedforward controller (55) and the virtual prototype is tested last. The stability is shown by the bode diagrams of the open-loop transfer function matrices (OLTFM). For convenience, only the elements in the first row and first column of OLTFM and the case of (\(0^{\circ}\), \(0^{\circ}\), \(0^{\circ}\)) are shown. Besides the Original Support Module (OSM), the manipulator is also installed on a Lighter Support Module (LSM) weighted 9.33 \(\times\) 10\({}^{2}\) kg to simulate the case of a larger ratio of the payload mass to the support module mass. A delay of 0.02 s is introduced into all the IPFLs to increase uncertainty. As shown in Figure 10, the amplitude-frequency curve (AFC) corresponding to OSM is totally below the 0 dB line, i.e., the clockwise encirclement number of the Nyquist plot about the \(-1\) point is zero.