# Three-Dimensional Guidance Laws for Spacecraft Propelled by a SWIFT Propulsion System
## 1 Introduction  
This paper analyzes the optimal control law of a spacecraft equipped with a Solar Wind Ion Focusing Thruster (SWIFT) [1, 2] in a heliocentric, three-dimensional (3D) orbit transfer. The SWIFT is an interesting propellantless propulsion system concept, which has been proposed as a possible alternative in a typical interplanetary orbit transfer to more conventional propellantless thrusters such as, for example, classical solar salts [3, 4] or the more recent E-sail [5, 6].  
From a conceptual viewpoint, a SWIFT-based spacecraft uses the solar wind stream to produce a net thrust in the interplanetary space, as in the case of the E-sail propulsion system proposed by Dr. [PERSON] two decades ago [7, 8]. However, different from the more studied E-sail, a SWIFT propulsion system captures the solar wind charged particles through a large conical space structure formed by a mesh of conducting (thin) cables, which are maintained at a high positive electric potential, as sketched in Figure 1. The ion stream is then collected by a suitable spacecraft subsystem and accelerated out of the spacecraft in a manner that resembles the acceleration process of a classical ion electric thruster. Accordingly, a part of the SWIFT-induced thrust vector can be steered within a prescribed range without varying the axis direction of the mesh-formed conical structure. The latter, in fact, nominally remains aligned with the Sun-spacecraft line during the interplanetary flight, as indicated in Figure 1.  
This paper considers a 3D heliocentric mission scenario, wherein the mathematical results discussed in this work extend the two-dimensional (2D) model recently proposed by the authors in Ref. [9]. In particular, considering a SWIFT-propelled spacecraft in the context of a 3D heliocentric orbit transfer, the optimal control law is obtained in analytical form by solving an optimization problem in which the performance index is the total flight time. To this end, the problem is analyzed by using the calculus of variations, which allows for obtaining the (optimal) direction of the SWIFT-induced thrust vector as a function of both the propulsion system design characteristics and the components of the costate vector.  
The obtained optimal guidance laws are used to simulate the SWIFT-propelled spacecraft's optimal performance in some typical 3D orbit-to-orbit transfers within a heliocentric framework. In this regard, a classic Earth-Venus and Earth-Mars interplanetary transfer is analyzed in detail, with the aim of comparing the results of the literature with those obtained using the mathematical model proposed in this work. Starting from the thrust model used to study a 2D scenario, which is briefly described in Appendix A, the next section discusses a simplified thrust model for the analysis of a general 3D heliocentric transfer. The latter is used in Section 3 to obtain the optimal guidance law with the aid of [PERSON]'s maximum principle. In particular, a number of different forms of the spacecraft state vector are considered in the trajectory optimization process, and the validity of the obtained optimal control law is discussed in Section 4 by comparing the outputs of the (3D) numerical simulations with the results of the recent literature [9]. More precisely, Section 4 compares the Earth-Venus and Earth-Mars minimum transfer times obtained in a 3D mission scenario with the results obtained with the 2D simplified approach proposed in Ref. [9].