# Three-Dimensional Guidance Laws for Spacecraft Propelled by a SWIFT Propulsion System
## 2 Thrust Vector Model for Trajectory Design in a 3D Mission Scenario  
In this section, we discuss a simple thrust vector analytical model for the study and optimization of the SWIFT-based trajectories in a heliocentric 3D mission scenario. The proposed mathematical model will be employed in the next section to obtain the optimal guidance law during a typical interplanetary transfer between two assigned Keplerian orbits, without ephemeris constraints. In particular, the SWIFT-induced thrust vector illustrated in this section originates from the mathematical model, proposed by the authors in Ref. [9], for the more simple case of a 2D heliocentric orbit transfer. Note that the latter mathematical model (i.e., the model for 2D scenarios) has been briefly summarized in Appendix A for the sake of completeness and for convenience, because the approach proposed in this section begins from the expression of the propulsive acceleration vector \(a\) given by Equation (A1). In fact, bearing in mind the model illustrated in Appendix A and  
Figure 1: Artistic concept of a spacecraft equipped with a SWIFT. The axis of the conical structure coincides with the Sun–vehicle line during the flight. according to Equation (A1), at a given distance \(r\) from the Sun, the maximum value of the magnitude of \(\mathbf{a}\) is given by  
\[\max(\|\mathbf{a}\|)=a_{D_{\oplus}}\left(\frac{r_{\oplus}}{r}\right)^{2}(1+K) \tag{1}\]  
where the terms \(a_{D_{\oplus}}\), \(r_{\oplus}\), and \(K\) are defined in Appendix A. In particular, Equation (1) is obtained from Equation (A1) by enforcing the condition \(\mathbf{\hat{t}}=\mathbf{\hat{r}}\), that is, by assuming a configuration in which the direction of the steerable part of the SWIFT-induced propulsive acceleration (such that the direction is indicated by the unit vector \(\mathbf{\hat{t}}\)) coincides with the Sun-spacecraft direction. The latter is indicated by the radial unit vector \(\mathbf{\hat{r}}\), as discussed in Appendix A.  
Paralleling the terminology used in the literature concerning the trajectory design of spacecraft equipped with photonic solar sails [3; 4; 10; 11], such as the E-Sail [5; 6] or Zubrin's Magnetic Sail [12; 13; 14], for a SWIFT-propelled spacecraft, we introduce the concept of characteristic acceleration \(a_{c}\), that is, the maximum magnitude of \(\mathbf{a}\) when the distance of the spacecraft from the Sun is equal to 1 astronomical unit. Bearing in mind that, by definition, \(r_{\oplus}=1\) au, Equation (1) gives the expression of the characteristic acceleration for a SWIFT-propelled vehicle. In particular, \(a_{c}\) is a function of the two propulsion system design parameters \(\{a_{D_{\oplus}},K\}\), and it is written in a compact form as  
\[a_{c}=a_{D_{\oplus}}\left(1+K\right) \tag{2}\]  
Accordingly, using the previous equation to express \(a_{D_{\oplus}}\) as a function of \(a_{c}\) and \(K\), the vector \(\mathbf{a}\) given by Equation (A1) becomes  
\[\mathbf{a}=\frac{a_{c}}{1+K}\left(\frac{r_{\oplus}}{r}\right)^{2}\left(\mathbf{\hat{r }}+K\mathbf{\hat{t}}\right) \tag{3}\]  
in which, at a given solar distance \(r\), the propulsion system performance parameters are collected in the pair \(\{a_{c},\,K\}\), while the control term is given by the unit vector \(\mathbf{\hat{t}}\). Therefore, the thrust vector of the SWIFT-based spacecraft has two scalar control parameters, i.e., the two independent (scalar) terms that define the components of the unit vector \(\mathbf{\hat{t}}\). In this context, from the point of view of trajectory design and paralleling the classical solar sail-related literature [15; 16; 17], it is more convenient to introduce the two typical control angles \(\alpha\) and \(\delta\).