## 3 Minimum-Time Attitude Maneuver of Small Satellite Mounted with DRC Antenna and RW
### Converting the Optimization Problem into an NLP
\begin{array}{ll}\dot{\mathbf{q}}=\dfrac{1}{2}\left[ \begin{array}{cccc}0&-\omega_{s1}&-\omega_{s2}&-\omega_{s3}\\ \omega_{s1}&0&\omega_{s3}&-\omega_{s2}\\ \omega_{s2}&-\omega_{s3}&0&\omega_{s1}\\ \omega_{s3}&\omega_{s2}&-\omega_{s1}&0\end{array}\right]\mathbf{q}=\mathbf{\Omega}( \mathbf{\omega})\mathbf{q}=\dfrac{1}{2}\begin{bmatrix}-q_{1}&-q_{2}&-q_{3}\\ q_{0}&-q_{3}&q_{2}\\ q_{3}&q_{0}&-q_{1}\\ -q_{2}&q_{1}&q_{0}\end{bmatrix}\mathbf{\omega}=\mathbf{Q}(q)\mathbf{\omega}\\ \dot{\mathbf{\omega}}_{s}=-\mathbf{I}_{c}^{-1}\left(\begin{array}{cc}\dot{\mathbf{I}}_{ c}\mathbf{\omega}_{s}+\dot{\mathbf{I}}_{ac}\mathbf{e}_{\phi_{1}}\dot{\mathbf{\phi}}_{1}+\mathbf{I}_{ac} \mathbf{e}_{\phi_{1}}\frac{\tau_{\phi_{1}}}{P_{1}}+\dot{\mathbf{I}}_{ac}\mathbf{e}_{\phi_{ 2}}\dot{\mathbf{\phi}}_{2}+\mathbf{I}_{ac}\mathbf{e}_{\phi_{2}}\frac{\tau_{\phi_{2}}}{P_{2}} \\ +\mathbf{I}_{\text{RIV}}\mathbf{\Omega}+\mathbf{\omega}_{s}\times\left(\mathbf{I}_{ac}\mathbf{e}_{ \phi_{1}}\dot{\mathbf{\phi}}_{1}+\mathbf{I}_{ac}\mathbf{e}_{\phi_{2}}\dot{\mathbf{\phi}}_{2}+ \mathbf{I}_{c}\mathbf{\omega}_{s}+\mathbf{I}_{\text{RIV}}\mathbf{\Omega}\right)\\ \dot{\mathbf{\phi}}=\left[\tau_{\phi_{1}}/\left(\dot{a}_{2}\text{sin}^{2}\mathbf{\phi} _{2}+\dot{a}_{3}\text{cos}^{2}\phi_{2}+\mu\text{d}_{2}^{2}\text{cos}^{2}\phi_ {2}\right)\right]=\begin{bmatrix}\tau_{\phi_{1}}/P_{1}\\ \tau_{\phi_{2}}/P_{2}\end{bmatrix}\\ \mathbf{q}_{(t_{f})}=\mathbf{q}^{t},\ \mathbf{\omega}_{(t_{f})}=\mathbf{0},\ \mathbf{\phi}_{(t_{f})}=\mathbf{\phi}^{t},\ \dot{\mathbf{\phi}}_{(t_{f})}=\mathbf{0},\ \mathbf{u}_{(t_{f})}=\mathbf{0}\\ |\Omega|\leq\Omega^{\text{lim}}=2.85\,\text{rad}/\text{s}^{2},\ |\Omega|\leq\Omega^{\text{lim}}=2200\text{ rpm},\|\tau_{\phi}|\leq\tau_{\phi}^{ \text{lim}}=0.05\text{ Nm}\end{array}\right.\end{array}  \tag{21}\]  
where \(\mathbf{q}^{t}\) represents the target satellite quaternion, and \(\mathbf{\phi}^{t}\) represents the target antenna angle. We used MATLAB(r) SQP algorithm [21] to solve the NLP. The decision variable \(\mathbf{X}\) is repeatedly updated by iterative calculations until the end conditions are met. We realized the attitude maneuver and antenna motion, concurrently and rapidly, using the final \(\mathbf{X}\) as a reference for feedforward control input. Figure 6 shows the optimal control input that were calculated solving NLP. Figure 7 shows the time history of the satellite Euler angle and antenna angles with the application of optimization control input torques to the ideal model (Equation (12)) and real model (Equation (14)). In here, ideal model mean that we do not take modeling error into consideration and real model means that we take modeling error into consideration. We define the time history of the satellite Euler angle and antenna angles with the application of optimization control input torques to the ideal model as the reference trajectory. From Figure 7, we can confirm that the error of attitude exists in the case of the real model. Therefore, we should apply FB control to secure robustness in the model error. Figure 6: Reference control input calculated by SQP. (**a**) Reference control input of RW; (**b**) Reference control input of antenna motor.