## 2 Mathematical Preliminaries
### Thrust Vector Mathematical Model  
Consider a spacecraft propelled by a diffractive solar sail without EOCPs. Assuming a Sun-facing configuration and using the model discussed in Refs. [53; 54], the spacecraft propulsive acceleration vector \(\mathbf{a}\) may be written as  
\[\mathbf{a}=\frac{a_{c}}{\sqrt{2}}\left(\frac{r_{\oplus}}{r}\right)^{2}\left(\mathbf{ \hat{n}}-\mathbf{\hat{K}}\right) \tag{1}\]  
where \(r\) is the Sun-spacecraft distance, \(r_{\oplus}=1\) au is a reference distance, \(a_{c}\) is the characteristic acceleration defined as the magnitude of \(\mathbf{a}\) when \(r=r_{\oplus}\), \(\mathbf{\hat{n}}\) is the unit vector normal to the sail nominal plane in the direction opposite to the Sun, and \(\mathbf{\hat{K}}\) is the grating momentum unit vector [46]; see Figure 5. The characteristic acceleration, which is the usual performance parameter used in solar sail design, depends on the sail characteristics and the vehicle's total mass. The direction of \(\mathbf{\hat{K}}\) is considered to be fixed relative to a body reference frame when the diffractive sail is designed without EOCPs. Finally, Equation (1) states that the angle between the thrust vector direction and the radial line is \(45^{\circ}\), as illustrated in Figure 4.  
According to [PERSON] [46], the possible presence of a set of EOCPs allows the in-plane component of the thrust vector to be flipped (with respect to the Sun-spacecraft line) as described schematically in Figure 4. In this case, that is, in the presence of a set of EOCPs that simultaneously change their state, the diffractive sail propulsive acceleration can be simply obtained from Equation (1) by introducing a dimensionless switching parameter \(\tau\in\{-1,\,1\}\), which varies the direction of the in-plane components of \(\mathbf{a}\). Accordingly, when a set of EOCPs is considered in the diffractive sail design, the expression of \(\mathbf{a}\) becomes  
\[\mathbf{a}=\frac{a_{c}}{\sqrt{2}}\left(\frac{r_{\oplus}}{r}\right)^{2}\left(\mathbf{ \hat{n}}-\tau\mathbf{\hat{p}}\right) \tag{2}\]where  
\[\tau=\begin{cases}1&\text{for state \textcircled{1}}\\ -1&\text{for state \textcircled{2}}\end{cases} \tag{3}\]  
and \(\hat{p}\) is an auxiliary unit vector, belonging to the sail nominal plane, whose direction is fixed in a spacecraft body reference frame. In particular, the auxiliary unit vector \(\hat{p}\) coincides with the grating momentum unit vector \(\hat{K}\) when the EOCPs are on state \(\textcircled{1}\); see the scheme of Figure 6. According to Equation (2), the dimensionless term \(\tau\) can be thought of as a sort of (binary) control parameter whose value can be changed during the flight in order to perform an assigned orbit transfer, as described later on in this section. Note that the assignment of \(\tau=1\) in correspondence of state \(\textcircled{1}\) (and, consequently, \(\tau=-1\) when the EOCPs are on state \(\textcircled{2}\)) is arbitrary. However, such a specific assumption does not affect the transfer performance in terms of total flight time, although it influences the shape of the optimal control law.  
The propulsive acceleration expression given by Equation (2) can be used in a general three-dimensional mission, because the orientation of the auxiliary unit vector \(\hat{p}\) relative to an orbital reference frame can be varied by rotating the sail nominal plane around the Sun-spacecraft line. Note also that Equation (1) can be recovered from Equation (2) by selecting \(\tau=1\) along the whole flight; compare Figures 5 and 6a. In this sense, the thrust  
Figure 5: Propulsive acceleration vector (and its components in a body reference frame) for an ideal diffractive sail, without EOCPs, in a Sun-facing configuration.  
Figure 6: Propulsive acceleration components as a function of \(\tau\) and the state of EOCPs; see also Equations (2) and (3).  
mathematical model based on Equations (2) and (3) extends and completes the model proposed in the recent literature [53].