## 4 Spacecraft Attitude Maneuvers  
To address the problem of attitude maneuvers, it was assumed that the SFGIS was equipped with a pair of rectangular control vanes, mounted at the end of the sides of the SFGIS symmetrically to the diagonal of the sail; see Figure 15. These control vanes were reflective surfaces (essentially small solar sails) capable of generating a force orthogonal to the plane of their surface. They were also capable of generating a torque component along the axis normal to the sail nominal plane by rotating the control angle of the vane, i.e., \(\beta_{1}\) for surface 1 and \(\beta_{2}\) for surface 2, as shown in Figure 15. By construction, the rotations were positive (or negative) if the surfaces rotated toward the positive (or negative) semi-axis \(z\). Figure 14: Sun–spacecraft distance \(r\) as a function of time in a minimum-time Earth–Venus mission scenario when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\). The legend is reported in Figure 4. (**a**) Case of \(\mathcal{I}=\mathcal{I}_{\mathcal{D}}\); (**b**) Case of \(\mathcal{I}=\mathcal{I}_{\mathcal{D}}\). Using the reference system described in Figure 15 and with the sign convention adopted for the vane control angles, the vector arms of the forces due to the solar radiation pressure acting on the two vanes are  
\[\mathbf{p}_{1}=\frac{1}{2}\left[ \begin{array}{c}l+b\,\cos\beta_{1}\\ l\,(1-\gamma)\\ b\,\sin\beta_{1}\end{array} \right]\quad,\quad\mathbf{p}_{2}=-\frac{1}{2}\left[ \begin{array}{c}l+b\,\cos\beta_{2}\\ l\,(1-\gamma)\\ -b\,\sin\beta_{2}\end{array} \right] \tag{21}\]  
where \(l\) is the length of the SFGIS side, \(b\) is the width of the two control surfaces, and \(\gamma\,l\) (with \(\gamma\in(0,\,0.5]\)) represents their length. The two force vectors due to the solar radiation pressure acting on the control surfaces are given by  
\[F_{1}=2\,P\,A\,\cos^{2}\beta_{1}\left[ \begin{array}{c}\sin\beta_{1}\\ 0\\ -\cos\beta_{1}\end{array} \right]\quad,\quad\mathbf{F}_{2}=2\,P\,A\,\cos^{2}\beta_{ 2}\left[ \begin{array}{c}-\sin\beta_{2}\\ 0\\ -\cos\beta_{2}\end{array} \right] \tag{22}\]  
where  
\[P\triangleq P_{\oplus}\left(\frac{r_{\oplus}}{r}\right)^{2}\quad\text{with}\, \,P_{\oplus}=4.5391\times 10^{-6}\,\text{Pa} \tag{23}\]  
is the solar radiation pressure at a distance \(r\) from the Sun, \(A\triangleq\gamma\,l\,b\) is the area of each control surface, while the factor two in Equation (22) means that the control vanes are assumed perfectly reflective. Note that, in principle, \(\beta_{1}\) and \(\beta_{2}\) can take different values. The torque vectors are therefore  
Figure 15: Sketch of an SFGIS with two control vanes. The force due to the solar radiation pressure acting on the two moving surfaces is applied at the vane pressure center \(C_{p}\). The rotation of the surfaces, denoted by \(\beta\), is the vane control angle. \[\mathbf{M}_{1}=\mathbf{p}_{1}\times\mathbf{F}_{1}=P\,A\,\cos^{2}\beta_{1}\left[ \begin{array}{c} -\cos\beta_{1}\,l\,(1-\gamma)\\ b\,\sin^{2}\beta_{1}+\cos\beta_{1}\,(l+b\,\cos\beta_{1})\\ -\sin\beta_{1}\,l\,(1-\gamma)\end{array} \right] \tag{24}\]  
\[\mathbf{M}_{2}=\mathbf{p}_{2}\times\mathbf{F}_{2}=P\,A\,\cos^{2}\beta_{2}\left[ \begin{array} []{c}\cos\beta_{2}\,l\,(1-\gamma)\\ -b\,\sin^{2}\beta_{2}-\cos\beta_{2}\,(l+b\,\cos\beta_{2})\\ -\sin\beta_{2}\,l\,(1-\gamma)\end{array} \right] \tag{25}\]  
from which it is easily concluded that the first two components of \(\mathbf{M}\triangleq\mathbf{M}_{1}+\mathbf{M}_{2}\) are zero only if \(\beta_{1}=\beta_{2}=\beta\). Under this assumption, the total torque becomes  
\[\mathbf{M}=2\,P\,A\,\cos^{2}\beta\left[ \begin{array}{c}0\\ 0\\ -\sin\beta\,l\,(1-\gamma)\end{array}