# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 1 Introduction
in 2018 [34], the direction of the thrust vector induced by the E-sail is strictly connected to the direction of the unit vector normal to the sail nominal plane. In particular, a constant value of the pitch angle  
Figure 1: Diagram of a spinning E-sail with the nominal plane highlighted in yellow. corresponds to a constant (and usually very small) value of the sail cone angle, defined as the angle formed by the thrust vector with the propagation direction of the solar wind. Note that the term pitch angle does not coincide with the one usually used to indicate one of the three rotations between the body axes and the orbital frame's axes, i.e., one of the three Euler angles. In this paper, a different control strategy is examined based on keeping the sail pitch angle value constant during the whole interplanetary flight, and using an optimization approach to derive the actual, optimal form of the spacecraft transfer trajectory. In this context, the nonlinear equations of motion of the vehicle in the heliocentric frame are considered and numerically integrated, through an automated routine based on a classical predictor-corrector method [35; 36]. In this specific, pitch angle-constrained, E-sail configuration, the spin axis of the axially symmetric spacecraft can rotate freely around the direction of solar wind propagation, performing a sort of conical motion, allowing the sail nominal plane (and then the propulsive acceleration vector) to vary its azimuthal orientation in an orbital reference frame in order to obtain a desired thrust vectoring. In fact, in an orbital reference frame, the spacecraft spin axis, which is perpendicular to the nominal plane of the sail, describes a circular cone having an aperture equal to twice the (fixed) sail pitch angle, and whose axis coincides with the Sun-spacecraft radial line. This situation is schematically illustrated in the diagram shown in Figure 2, where the circular cone on whose external surface the unit vector normal to the sail nominal plane is constrained to lie contains, within itself, the thrust vector of the E-sail. This interesting guidance law reduces to one the number of control angles that can be freely selected during interplanetary flight, and therefore significantly simplifies, at least from a conceptual point of view, the problem of the sail attitude variation during a heliocentric orbital transfer. The mathematical description of the constrained thrust model of a spinning E-sail is illustrated in Section 2. The fixed sail pitch angle-based control strategy, which has been shown in the past to be really effective in a photonic solar sail-propelled heliocentric mission scenario [37], is analyzed here by considering a classical  
Figure 2: Conceptual sketch of an E-sail-propelled spacecraft, in a heliocentric scenario, with a constant pitch angle. The unit vector normal to the sail nominal plane (in the direction opposite to the Sun) is constrained to lie on the surface of a circular cone whose aperture is equal to twice the fixed sail pitch angle. three-dimensional interplanetary orbit-to-orbit transfer between the Earth and two of the three remaining terrestrial planets, namely Venus and Mars. The two interplanetary mission scenarios, together with some notes on the design of the optimal transfer trajectory, allowing the potential reader to refer to recent literature, are illustrated in Section 3. Accordingly, the work extends to an E-sail-based mission scenario the literature results regarding this potential simplified guidance law. The numerical results, in terms of the minimum flight time as a function of the propulsive (reference) characteristics of the E-sail, are then compared with a typical unconstrained optimal transfer, with the aim of quantifying the performance degradation due to the assumed constraint of the pitch angle value during the whole transfer.