# Three-Dimensional Guidance Laws for Spacecraft Propelled by a SWIFT Propulsion System
## 4 Potential Transfer Trajectories and Case Study  
The optimal control law described in the previous section is now employed to analyze the minimum time transfer of a SWIFT-propelled spacecraft in two typical, heliocentric, 3D mission scenarios. In this regard, and with the aim also to validate the mathematical model discussed in this paper, we consider the propulsion system characteristics and the two interplanetary mission scenarios analyzed in Ref. [9]. More precisely, Section 4 of Ref. [9] reports the results of a 2D optimization problem in which a spacecraft equipped with a SWIFT propulsion system of performance parameters \(a_{\text{D}_{\oplus}}=0.035\,\text{mm}/\text{s}^{2}\) and \(K=1\) [so that the characteristic acceleration is \(a_{c}=0.07\,\text{mm}/\text{s}^{2}\); see Equation (2)], moves from a circular (ecliptic) orbit of radius \(r=1\,\text{au}\) to a target, coplanar, circular orbit of radius \(r\in\{0.723,\,1.524\}\,\text{au}\). In particular, the scenario in which the radius of the target circular orbit is \(0.723\,\text{au}\) (or \(1.524\,\text{au}\)) is consistent with a simplified, 2D Earth-Venus (or Earth-Mars) interplanetary transfer in which both the eccentricity and the inclination of the planetary orbits are neglected. In those mission scenarios, the 2D analysis discussed in Ref. [9] gives a minimum value of the (circle-to-circle) flight time of roughly \(3.62\,\text{years}\) for the Earth-Venus mission and \(8.09\,\text{years}\) for the Earth-Mars mission. In this section, instead, we consider the actual 3D shape of the heliocentric orbits of the three planets involved in the transfer by using the ephemerides data obtained on July 1 st 2024 obtained by the well known JPL Horizon system. In particular, the characteristics of the planet orbits used in the trajectory optimization process are summarized in terms of classical orbital elements \(\{a,e,i,\Omega,\omega\}\) in Table 1. Recall that the angular position of the spacecraft along the two Keplerian orbits, both at the beginning and the end of the transfer, is left free. The boundary value problem connected to the optimization process is solved using a numerical method based on a classical shooting procedure [42] with a tolerance of \(10^{-8}\). The initial guess of the boundary value problem solution is obtained by adapting the semianalytical procedure described in Refs. [43,44]. Finally, the differential equations, that is, the EoMs and the Euler-Lagrange equations, are integrated using the Adams-Bashforth methods [45] with a relative tolerance of \(10^{-10}\). In the case of an Earth-Venus orbit-to-orbit 3D transfer, the solution of the optimization problem gives a minimum flight time of about 3.93 years (+8.5% with respect to the value obtained in a rapid 2D circle-to-circle transfer [9]). The time variation of the components of the control unit vector \(\mathbf{\hat{t}}\) in the RTN reference frame and the optimal transfer trajectory in the Cartesian heliocentric ecliptic reference frame are reported in Figure 8. In particular, the scale of the \(z\) axis in Figure 8b is different from that of the other two axes to highlight the three-dimensionality of the transfer. Finally, in the Earth-Mars case, the flight time is 8.76 years (+8.3% with respect to the 2D case [9]), while the optimal guidance law and the transfer orbit are shown in Figure 9. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Orbital Parameter** & **Earth** & **Venus** & **Mars** \\ \hline semimajor axis \(a\) [au] & \(1.0004\) & \(0.7233\) & \(1.5236\) \\ eccentricity \(e\) & \(1.6335\times 10^{-2}\) & \(6.73308\times 10^{-3}\) & \(9.3289\times 10^{-2}\) \\ inclination \(i\) [deg] & \(4.1705\times 10^{-3}\) & \(3.3944\) & \(1.8478\) \\ long. of the asc. node \(\Omega\) [deg] & 146.5136 & 76.6118 & \(49.4893\) \\ arg. of perihelion \(\omega\) [deg] & 317.9011 & 55.1947 & \(286.6906\) \\ \hline \hline \end{tabular}
\end{table}