# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 1 Introduction  
A spacecraft propellantless propulsion system allows the design of space trajectories that are difficult to achieve using conventional thrusters, due to the well-known limitation related to the finite mass of propellant that can be carried on board the spacecraft. In the still very narrow field of propellantless propulsion systems for space vehicles, the electric solar wind sail (E-sail) [1, 2] and concepts derived from it, such as the interesting plasma brake [3, 4, 5], represent a fascinating and effective alternative to the more well-known and technologically mature photonic solar sails [6, 7, 8]. Unlike the latter, which essentially uses a large, lightweight mirror-like surface to reflect incoming photons, an E-sail uses a large rig of long, thin, conducting tethers to create an artificial electric field that deflects the chargedparticles carried by the solar wind, in order to provide a (propellantless) net propulsive acceleration to a spacecraft operating in a heliocentric mission scenario, i.e., outside the typical planetary magnetospheres [9; 10]. The radial rig of conducting tethers is deployed and stretched by rotating the vehicle about an axis perpendicular to the nominal plane of the sail [11; 12; 13] with a suitable angular velocity, which is consistent with the mechanical characteristics of the metallic wires that form the generic multiline tether [14; 15]. The conceptual scheme of a rotating E-sail is reported in Figure 1. The space vehicle's rotation imparts an inherent gyroscopic stiffness to the system, which complicates the thrust vectoring of the E-sail-based spacecraft, which is achieved by changing the orientation of the sail nominal plane with respect to an orbital reference frame [16; 17; 18]. For this reason, over the years, several control and guidance strategies have been proposed that are able to modify both the spin rate and the attitude of the E-sail-propelled spacecraft, in order to allow the achievement of a given heliocentric orbital transfer with acceptable performances in terms of total flight time [19]. In this regard, the interested reader is referred to the fundamental works of [PERSON] and [PERSON] [20; 21; 22; 23; 24], who, for the first time, addressed and discussed the considerable problem of the in-flight attitude variation of a large space structure as an E-sail of medium propulsive performances. In the context of a simplified E-sail guidance control during a typical interplanetary flight, some interesting steering techniques have recently been investigated which are based, for example, on maintaining the inertial direction of the spacecraft spin axis [25; 26] or on limiting the excursion of the so-called sail pitch angle [27]. The latter, borrowing the nomenclature introduced in the design of the photonic solar sail control law [28; 29], is defined as the angle formed by the unit vector perpendicular to the sail nominal plane (in direction opposite to the Sun) with the direction of propagation of the solar wind, which is usually assumed as coincident with the (radial) Sun-spacecraft line. Indeed, assuming a classical axially symmetric E-sail, the possibility of maintaining a constant value of the pitch angle of the vehicle during flight, as described in the interesting approach illustrated in ref. [30], allows the mission designer to theoretically conceive a simplified steering law in which the pitch angle can be freely selected in a finite set of possible values. The latter is, in fact, substantially the approach used by [PERSON] et al. [31; 32; 33] in their elegant works on the study of the heliocentric motion of an E-sail-propelled spacecraft with a fixed direction of the thrust vector (in the orbital reference frame) using an approximate semi-analytic approach, which is based on classical linear perturbation theory. In fact, according to the well-known analytical thrust model proposed by [PERSON] et al.