# Intersatellite and Downlink Agile Attitude Maneuvers
## 6 Attitude Maneuvering for Intersatellite Connection
### Numerical Simulations in Nominal Conditions  
The two spacecraft are assumed to be placed in two orbits with identical semimajor axes, eccentricities, and inclinations, set to the values of Section 5.2. Instead, the two orbits have different values of RAAN,  
\[\Omega_{S}=0^{\circ} \tag{75}\] \[\Omega_{T}=90^{\circ} \tag{76}\]  
At the initial time, both satellites are at the ascending node.  
Figure 18: Geometry of the commanded attitude (intersat link).  
Figure 19 depicts \(\theta\) and \((\theta_{S}+\theta_{T})\) during a time interval of 10,000 s, in which the two spacecraft are mutually visible three times. The first and third visibility arc correspond to transits in the North latitudinal regions, the second (shorter) visibility arc is associated with a transit in the South emisphere. The initial attitude conditions are  
\[\underline{\omega}(t_{0})=\begin{bmatrix}0&0&0\end{bmatrix}^{T} \tag{77}\] \[\underline{\theta}(t_{0})=\begin{bmatrix}0&\frac{\pi}{4}&\frac{ \pi}{2}&\frac{3\pi}{4}\end{bmatrix}^{T}\] (78) \[\underline{\dot{\theta}}(t_{0})=\begin{bmatrix}0&0&0&0\end{bmatrix}^ {T}\] (79) \[R(t_{0})=\begin{bmatrix}1&0&0\\ 0&-0.1736&-0.9848\\ 0&0.9848&-0.1736\end{bmatrix} \tag{80}\]  
For simulating these attitude maneuvers, \(Q\) is set to  
\[Q=10^{-7}I_{6\times 6} \tag{81}\]  
The gains used in the numerical simulations are reported in Appendix A.  
Figure 20 illustrates the eigenangle time history, which provides a clear indication of misalignment. This time history exhibits some spikes immediately after the end of tracking intervals, when S must reorient to point toward the direction where T will appear in the subsequent visibility arc. Instead, during visibility arcs, the eigenangle reaches modest values, e.g., below 0.1\({}^{\circ}\), as shown in Figure 21, referred to the 2 nd tracking interval. Figure 22 portrays the time histories of the components of the error angular velocity, which reach again their maximum values at the beginning of reorientation arcs. In contrast, in visibility arcs these components quickly drop to zero, as illustrated in Figure 23. Precision in pointing is testified by the two mean values \(\bar{\Phi}_{e_{i}}\) and \(\bar{\omega}_{e_{i}}\), defined in Equations (5) and (6),  
\[\bar{\Phi}_{e_{i}} =4.90\cdot 10^{-2\circ} \tag{82}\] \[\bar{\omega}_{e_{i}} =2.60\cdot 10^{-4\circ}/\text{s} \tag{83}\]  
The actual torque components, depicted in Figures 24 and 25, reach values of order of tens of Nm and component \(T_{a3}\) is larger than the other ones. The angular rates of the gimbals are portrayed in Figures 26 and 27, occasionally exceed 20 \({}^{\circ}\)/s, and never exceed 35 \({}^{\circ}\)/s. The time histories of net motor torques, shown in Figure 28, turn out to be of order of 1 Nm, never exceeding 2 Nm. For the sake of completeness, Figure 29 portrays the time histories of \(\bar{J}_{11}\), \(\bar{J}_{22}\), \(\bar{J}_{33}\), \(\bar{J}_{12}\), \(\bar{J}_{13}\), and \(\bar{J}_{23}\), which represent the displacement of the inertia momenta and products from the respective actual values.  
Figure 21: Pointing error \(\Phi_{e}(\mathfrak{h})[^{\circ}]\) during the second tracking interval.  
Figure 22: Components of the error angular velocity \(\omega_{e1}(\mathfrak{h})\), \(\omega_{e2}(\mathfrak{h})\), \(\omega_{e3}(\mathfrak{h})[^{\circ}/\mathsf{s}]\) during the entire time interval.  
Figure 23: Components of the error angular velocity \(\omega_{c1}(t)\), \(\omega_{c2}(t)\), \(\omega_{c3}(t)[^{\circ}/s]\) during the second tracking interval.  
Figure 24: Actual torque components \(T_{a1}(t)\), \(T_{a2}(t)\), \(T_{a3}(t)[\text{Nm}]\) during the entire time interval.  
Figure 25: Actual torque components \(T_{a_{1}}(t)\),\(T_{a_{2}}(t)\),\(T_{a_{3}}(t)[\text{Nm}]\) during the second tracking interval.