# Three-Dimensional Guidance Laws for Spacecraft Propelled by a SWIFT Propulsion System
## 2 Thrust Vector Model for Trajectory Design in a 3D Mission Scenario
To this end, we introduce a classical, right-handed, Radial-Transverse-Normal (RTN) orbital reference frame [18], in which \(\mathbf{\hat{t}}_{\text{R}}\) is the radial unit vector, \(\mathbf{\hat{t}}_{\text{T}}\) is the transverse unit vector, and \(\mathbf{\hat{t}}_{\text{N}}\) is the normal unit vector; see the scheme of Figure 2, which has been adapted from Ref. [19]. Figure 2: Sketch of the RTN reference frame of unit vectors \(\mathbf{\hat{t}}_{\text{R}}\), \(\mathbf{\hat{t}}_{\text{T}}\), and \(\mathbf{\hat{t}}_{\text{N}}\). In particular, the unit vectors of the RTN reference frame are defined as  
\[\hat{\mathbf{t}}_{\rm R}\equiv\hat{\mathbf{r}}\quad,\quad\hat{\mathbf{t}}_{\rm N}=\frac{\mathbf{h }}{\|\mathbf{h}\|}\quad,\quad\hat{\mathbf{t}}_{\rm T}=\hat{\mathbf{t}}_{\rm N}\times\hat{ \mathbf{t}}_{\rm R} \tag{4}\]  
where \(\mathbf{h}\) is the spacecraft specific angular momentum vector. Note that the plane \((\hat{\mathbf{t}}_{\rm R},\hat{\mathbf{t}}_{\rm T})\) coincides with the plane of the spacecraft osculating orbit, see Figure 2, while the spacecraft inertial velocity \(\mathbf{v}\) has a positive component along the direction of the transverse unit vector, that is, \(\mathbf{v}\cdot\hat{\mathbf{t}}_{\rm T}>0\). Consider now the scheme of Figure 3 that shows the RTN reference frame and a generic configuration of the unit vector \(\hat{\mathbf{t}}\), which indicates the direction of the steerable part of the SWIFT-induced acceleration; see the discussion in Appendix A about the SWIFT thrust model. Figure 3 also shows the aperture angle \(\phi\in(0,180)\) deg of the conical mesh of (thin) conducting wires of the SWIFT propulsion system. The latter conical structure should be considered a sort of geometric constraint on the direction of \(\hat{\mathbf{t}}\), because the stream of charged particles accelerated out of the spacecraft cannot be directed inside the (conical) mesh of wires. From a mathematical viewpoint, that constraint can be expressed through the following inequality:  
Figure 3: Unit vector \(\hat{\mathbf{t}}\) in the RTN reference frame, definition of the SWIFT cone \(\alpha\), and clock \(\delta\) angle. The scheme also shows the aperture angle \(\phi\) of the conical mesh of conducting wires, which constitutes the large part of the propulsion system structure. \[\hat{\mathbf{t}}\cdot\hat{\mathbf{t}}_{\rm R}>-\cos(\phi/2) \tag{5}\]  
or  
\[\alpha\leq\alpha_{\rm max}\quad\text{with}\quad\alpha_{\rm max}<180\deg-\phi/2 \tag{6}\]  
where \(\phi\) has an assigned value. According to Figure 3, in the RTN reference frame, the unit vector \(\hat{\mathbf{t}}\) can be written as a function of the cone angle \(\alpha\in[0,\,a_{\rm max}]\) and the clock angle \(\delta\in[0,\,2\pi]\) rad as  
\[\hat{\mathbf{t}}=\cos\alpha\,\hat{\mathbf{t}}_{\rm R}+\sin\alpha\cos\delta\,\hat{\mathbf{ t}}_{\rm T}+\sin\alpha\sin\delta\,\hat{\mathbf{t}}_{\rm N} \tag{7}\]  
Therefore, bearing in mind that \(\hat{\mathbf{r}}\equiv\hat{\mathbf{t}}_{\rm R}\), the vector \(\mathbf{a}\) given by Equation (3) can be rewritten in a more convenient form as  
\[\max(\|\mathbf{a}\|)=a_{D_{\oplus}}\left(\frac{r_{\oplus}}{r}\right)^{2}(1+K) \tag{1}\]0  
Note that, in this case, the two angles \(\alpha\) and \(\delta\) are not defined by considering the direction of the propulsive acceleration (or the total thrust) vector. In fact, in a SWIFT-based scenario, the pair \(\{\alpha,\delta\}\) indicates the direction of \(\hat{\mathbf{t}}\), that is, the direction of only a part of the SWIFT-induced acceleration, as discussed in Appendix A. The expression of vector \(\mathbf{a}\), as a function of both the two control angles \(\{\alpha,\delta\}\) (or, equivalently, the unit vector \(\hat{\mathbf{t}}\)) and the SWIFT design characteristics \(\{a_{c},K\}\), will be employed in the next section to determine the 3D optimal guidance law.