# Intersatellite and Downlink Agile Attitude Maneuvers
## 4 Actuation
### Actuator Dynamics  
Let \(\underline{W_{0}}^{(j)}\) denote the vectors associated with the nominal orientation of the \(j\)-th wheel, which is the configuration for which \(\theta_{1}^{(j)}=0\),  
\[\underline{\underline{W_{0}}^{(j)}}=\left[\hat{w}_{1}^{(j)}\quad\hat{w}_{2}^{( j)}\quad\hat{w}_{3}^{(j)}\right] \tag{18}\]  
Vectrix \(\underline{\underline{W_{1}}^{(j)}}\) denotes the vectors associated with the instantaneous orientation of the \(j\)-th wheel,  
\[\underline{\underline{W_{1}}^{(j)}}=\left[\hat{g}_{j}\quad\hat{f}_{j}\quad \hat{h}_{j}\right] \tag{19}\]  
Let \(R_{A}^{(j)}\) represent the constant rotation matrix that describes the orientation of \(\underline{W_{0}}^{(j)}\) with respect to \(\underline{\underline{B}}\) (i.e., the geometry of the mounting of the \(j\)-th gyroscope inside the spacecraft),  
\[{R_{A}^{(j)}}^{T}=\left[\underline{L}_{A1}^{(j)}:\ L_{A2}^{(j)}:\ L_{A3}^{(j) }\right] \tag{20}\]  
\[\underline{\underline{W_{0}}^{(j)}}^{T}=R_{A}^{(j)}\underline{\underline{B}}^ {T} \tag{21}\]  
The space vehicle is equipped with an array of four SGCMGs, which are mounted in a pyramidal arrangement (cf. Figure 2), where the angle \(\beta\) between the faces and the base is assumed to equal 54.74\({}^{\circ}\). This configuration is widely used in space missions thanks to the fact that it is associated with a nearly spherical momentum envelope [29]. The standard methodology of Eulerian dynamics [24; 30] leads to describing the rotational dynamics of a system composed of a spacecraft with 4 SGCMGs, all characterized by constant rotor speed, denoted with \(\omega_{R}^{(j)}\).  
The dynamics equation is  
\[\begin{split}&\int_{C}^{(\text{\tiny E})}\dot{\underline{\omega}} =\underline{M}_{C}-\tilde{\omega}\,J_{C}^{(\text{\tiny E})}\underline{\omega} -\int_{C}^{(\text{\tiny E})}\underline{\omega}+\tilde{\omega}\sum_{j=1}^{4}I _{S}^{(j)}\,\omega_{R}^{(j)}\big{[}\sin\left(\theta_{1}^{(j)}\right)\underline {L}_{A2}^{(j)}-\cos\left(\theta_{1}^{(j)}\right)\underline{L}_{A3}^{(j)} \big{]}+\\ &+\sum_{j=1}^{4}\{I_{S}^{(j)}\,\omega_{R}^{(j)}\big{[}\cos\left( \theta_{1}^{(j)}\right)\underline{L}_{A2}^{(j)}+\sin\left(\theta_{1}^{(j)} \right)\underline{L}_{A3}^{(j)}\big{]}-\tilde{\omega}I_{T}^{(j)}\,\underline{ L}_{A1}^{(j)}\}\dot{\theta}_{1}^{(j)}-\sum_{j=1}^{4}I_{T}^{(j)}\,\underline{L}_{A1}^{(j)} \,\bar{\theta}_{1}^{(j)}\end{split} \tag{22}\]  
Figure 2: Pyramid array of SGCMGs used in the simulations.  
In Equation (22), \(I_{T}^{(j)}\) and \(I_{S}^{(j)}\) are the transverse and axial momenta of inertia of wheel \(j\). The three terms with summation and the term related to the time derivative of the inertia matrix represent the actual control torque \(\underline{T}_{a}\) applied to the system, due to the presence of the array of SGCMGs. In fact, from Equation (22), it is straightforward to recognize that if these terms vanish, the preceding equation reduces to the [PERSON]'s equation of a single rigid body. The following quantities are introduced:  
\[\dot{\underline{\theta}} = [\dot{\theta}_{1}^{(1)}\cdot\ldots\ldots\dot{\theta}_{1}^{(4)}]^{T} \tag{23}\] \[\ddot{\underline{\theta}} = [\dot{\theta}_{1}^{(1)}\cdot\ldots\dot{\theta}_{1}^{(4)}]^{T}\] (24) \[A = [I_{S}^{(1)}\omega_{R}^{(1)}\underline{f}_{1}\cdot\ldots\ldots I_ {S}^{(4)}\omega_{R}^{(4)}\underline{f}_{4}]\] (25) \[B = [I_{T}^{(1)}\underline{I}_{A1}^{(1)}\cdot\ldots\ldots I_{T}^{(4) }\underline{I}_{A1}^{(4)}] \tag{26}\]  
where \(\underline{f}_{j}\) is a (\(3\times 1\))-vector that includes the components of \(\hat{f}_{j}\) in \(\underline{B}\). The actual torque is written in a compact way as  
\[\underline{T}_{a}=A\dot{\underline{\theta}}-B\dot{\underline{\theta}}-I_{C}^{ (\underline{\theta})}\underline{\omega}-\tilde{\omega}B\dot{\underline{ \theta}}+\tilde{\omega}\sum_{j=1}^{4}I_{S}^{(j)}\omega_{R}^{(j)}[\sin{(\theta _{1}^{(j)})}\underline{r}_{A2}^{(j)}-\cos{(\theta_{1}^{(j)})}\underline{r}_{ A3}^{(j)}] \tag{27}\]