## 2 Trajectory Design and Optimization
### Heliocentric Dynamics of the Spacecraft's Center of Mass  
Based on the mathematical model of the propulsive acceleration vector presented in the previous subsection, and taking into account the results obtained in Ref. [1], the thrust vectoring of an SFGIS is obtained by rotating the sail nominal plane around the Sun-spacecraft line. This is equivalent to rotating the sail nominal plane \(\mathcal{P}\) around the direction of the unit vector \(\hat{n}\), that is, the direction of \(\hat{p}\). The problem of spacecraft thrust vectoring can be more easily studied by introducing an orbital reference frame. For this purpose, consider a classical Radial-Transverse-Normal (RTN) reference frame \(\mathcal{T}_{\text{RTN}}\), whose origin coincides with the spacecraft's center of mass \(C\), in which \(\hat{\mathbf{i}}_{\text{R}}\equiv\hat{\mathbf{r}}\) is the radial unit vector, \(\hat{\mathbf{i}}_{\text{T}}\) is the transverse unit vector, and \(\hat{\mathbf{i}}_{\text{N}}\) is the normal unit vector. In particular, the direction of \(\hat{\mathbf{i}}_{\text{N}}\) coincides with that of the spacecraft angular momentum vector, while \((\hat{\mathbf{i}}_{\text{R}},\hat{\mathbf{i}}_{\text{T}})\) coincides with the plane of the osculating orbit; see Figure 2.  
With reference to the sketch of Figure 2, we introduce the sail clock angle \(\delta\in[0,\,360]^{\circ}\) defined as the angle between the direction of the transverse unit vector \(\hat{\mathbf{i}}_{\text{T}}\) and the direction of the (body-fixed) unit vector \(\hat{\mathbf{m}}\). The clock angle is measured in the \(\mathcal{P}\)-plane counterclockwise from the direction of \(\hat{\mathbf{i}}_{\text{T}}\). According to Equation (1) and recalling the expressions of \(\{a_{p_{\text{R}}},\,a_{p_{\text{H}}}\}\) in Equations (4) and (5), the angle \(\delta\) allows the transverse (\(a_{p_{\text{T}}}\)) and normal (\(a_{p_{\text{N}}}\)) component of the propulsive acceleration vector to be written as  
\[a_{p_{\text{T}}}=\mathbf{a}_{p}\cdot\hat{\mathbf{i}}_{\text{T}}=a_{c}\left(\frac{r_{ \oplus}}{r}\right)^{2}\eta_{m}\,\cos\delta \tag{6}\]  
\[a_{p_{\text{N}}}=\mathbf{a}_{p}\cdot\hat{\mathbf{i}}_{\text{N}}=a_{c}\left(\frac{r_{ \oplus}}{r}\right)^{2}\eta_{m}\,\sin\delta \tag{7}\]  
with \(\sqrt{a_{p_{\text{N}}}^{2}+a_{p_{\text{T}}}^{2}}=a_{p_{\text{H}}}\), where \(a_{p_{\text{H}}}\) is given by Equation (5), so that  
\[[\mathbf{a}_{p}]_{\text{RTN}}=a_{c}\left(\frac{r_{\oplus}}{r}\right)^{2} \begin{bmatrix} \eta_{m}\,\cos\delta\\ \eta_{m}\,\sin\delta\end{bmatrix}  \tag{8}\]  
The orientation of the thrust vector is therefore defined by a single (scalar) control variable, namely, the sail clock angle \(\delta\), whose value indicates the direction of the horizontal component of the propulsive acceleration \(a_{p_{\text{H}}}\) with respect to the direction of \(\hat{\mathbf{i}}_{\text{T}}\). Note that, when \(\delta=\{0,\,180\}^{\circ}\) (or \(\delta=\{90,\,270\}^{\circ}\)) the thrust vector belongs to a plane coincident with (or perpendicular to) the plane of the osculating orbit. The ratio of the generic component \(\{a_{p_{\text{R}}},\,a_{p_{\text{T}}},\,a_{p_{\text{N}}}\}\) to the characteristic acceleration \(a_{c}\), when the solar distance is equal to \(r_{\oplus}\), is drawn in Figure 3 as a function of the clock angle. The expressions of the three components of \(\mathbf{a}_{p}\) in the RTN reference frame, see Equation (8), can be used to write the spacecraft equations of motion in a heliocentric scenario in which \(\mu_{\odot}\) is the Sun's gravitational parameter. To this end, the spacecraft state vector is described in terms of the six modified equinoctial orbital elements \(\{p,\ f,\ g,\ h,\ k,\ L\}\), the geometric interpretation of which is discussed in Refs. [8]. According to the mathematical model proposed by [PERSON] [19] and using Equations (9)-(12) of Ref.