Impact of Pitch Angle Limitation on E-Sail Interplanetary Transfers

Abstract

The Electric Solar Wind Sail (E-sail) deflects charged particles from the solar wind through an artificial electric field to generate thrust in interplanetary space. The structure of a spacecraft equipped with a typical E-sail essentially consists in a number of long conducting tethers deployed from a main central body, which contains the classical spacecraft subsystems. During flight, the reference plane that formally contains the conducting tethers, i.e., the sail nominal plane, is inclined with respect to the direction of propagation of the solar wind (approximately coinciding with the Sun–spacecraft direction in a preliminary trajectory analysis) in such a way as to vary both the direction and the module of the thrust vector provided by the propellantless propulsion system. The generation of a sail pitch angle different from zero (i.e., a non-zero angle between the Sun–spacecraft line and the direction perpendicular to the sail nominal plane) allows a transverse component of the thrust vector to be obtained. From the perspective of attitude control system design, a small value of the sail pitch angle could improve the effectiveness of the E-sail attitude maneuver at the expense, however, of a worsening of the orbital transfer performance. The aim of this paper is to investigate the effects of a constraint on the maximum value of the sail pitch angle, on the performance of a spacecraft equipped with an E-sail propulsion system in a typical interplanetary mission scenario. During flight, the E-sail propulsion system is considered to be always on so that the entire transfer can be considered a single propelled arc. A heliocentric orbit-to-orbit transfer without ephemeris constraints is analyzed, while the performance analysis is conducted in a parametric form as a function of both the maximum admissible sail pitch angle and the propulsion system’s characteristic acceleration value.

1. Introduction

An interplanetary spacecraft equipped with an Electric Solar Wind Sail (E-sail) essentially consists of a large space structure made up of a series of long conducting tethers, which are capable of deflecting the charged particles of the solar wind through an artificial electric field that surrounds the vehicle [1]. The conducting tethers [2,3,4], which consist of a series of suitably interconnected metal wires [5,6], deploy from a central body [7,8,9,10] that contains the main subsystems of the spacecraft (such as the power generation system), including the payload whose mass affects the effective value of the available propulsive acceleration for an assigned total tether length [11]. Figure 1 shows the centrifugal deployment process [8] of an E-sail-propelled spacecraft, which is schematized by the main body, the conducting tethers, and the remote units at the very end of each cable. In particular, the deployment of the conducting tethers takes place substantially in a reference plane perpendicular to the direction of the spacecraft angular velocity vector (see the spin direction described by the curved orange arrows in the figure). That plane, which is indicated with a green shaded disk in Figure 1b, is the so-called “sail nominal plane”, which coincides with the plane that formally contains the conducting tethers during the interplanetary flight. Recall that the generic tether bends during flight [12,13], so the sail nominal plane must be considered a sort of reference plane useful for designing the guidance law required to complete the heliocentric transfer.

Over the years, the arrangement of the conducting tethers in a typical E-sail design has evolved from the grid configuration originally proposed by inventor Dr. Janhunen in his two seminal papers [14,15] that sparked the study of this fascinating propellantless propulsion system. In this regard, a review article by Bassetto et al. [16] describes the evolution of the E-sail concept (and the thrust vector mathematical model) up to the state of the art two years ago, while the more recent first edition of the special issue of the journal Aerospace entitled Advances in CubeSat Sails and Tethers [17] collects a series of interesting articles that provide a snapshot of the current state of scientific research regarding this advanced propulsion system, especially from the point of view of the (hopefully) upcoming flight tests [18,19,20]. In this context, Figure 2 shows an artistic impression of a single-tether E-sail that, after the first attempts unfortunately failed [21,22,23,24,25], could soon be used to achieve the first in situ evidence of the potentialities of this propulsion system concept through the use of a CubeSat [26,27,28] inserted in a high-elliptic lunar orbit, which allows the (single) conducting tether to interact with the solar wind flow [29,30]. Finally, from the point of view of the geometric arrangement of the E-sail conducting tethers, very recently, Bassetto et al. [31] analyzed the potentialities of a sort of “web-shaped” E-sail, in which a number of (closed) azimuthal conducting tethers connect the classical radial tethers deployed from the spacecraft central body. In particular, Ref. [31] analyzed the characteristics of the thrust vector of a web-shaped E-sail by using a semi-analytical approach, which allows the designer to evaluate the impact of the presence of the azimuthal tethers on the propulsive performance with a set of simple, closed-form equations.

An axially symmetric E-sail is theoretically able to generate an outward radial thrust, that is, a thrust vector directed along the Sun–spacecraft line, when the sail nominal plane is substantially perpendicular to the direction of the solar wind flow [32]. This type of Sun-facing orientation, which can be passively maintained by the axially symmetric E-sail, is modified when the desired steering law requires the presence of a transverse (i.e., a non-radial) component of the propulsive acceleration [33]. In fact, the presence of a transverse thrust component is a necessary condition to achieve a transfer between two heliocentric Keplerian orbits that do not share the same value of the semilatus rectum. In this context, the recent work of the author [34] analyzes the optimal performance of an E-sail facing the Sun in a two-dimensional transfer between a generic Keplerian elliptic orbit and a circular one whose radius, in fact, has a value equal to the semilatus rectum of the initial (elliptic) parking orbit. Figure 3 shows the case of a zero pitch angle (i.e., the Sun-facing configuration), and the more generic situation in which the pitch angle is different from zero. Note in Figure 3b as the E-sail-induced thrust vector lies between the Sun–spacecraft line and the direction perpendicular to the sail nominal plane in the case of a non-zero value of the sail pitch angle.

The generation of a transverse thrust component is strictly connected to the possibility of tilting the sail nominal plane with respect to the Sun–spacecraft (radial) line, in order to obtain a value of the sail pitch angle different from zero [35,36]. This variation in the inclination of the sail nominal plane with respect to the (passively stable) Sun-facing configuration requires a specific attitude control law whose performance, in terms of system settling time, depends directly on the desired (target) value of the sail pitch angle [37]. The thrust vectoring of a spinning E-sail-propelled spacecraft is, in fact, a complex task, which requires a precise design of the attitude control system as thoroughly discussed in the two interesting papers by Toivanen and Janhunen [38,39] that analyzed this specific important problem by using an elegant analytical approach. In this regard, the E-sail-related scientific literature contains more recent and useful works [40,41] which investigate the connection between a generic E-sail attitude maneuver and the spacecraft response from the structural viewpoint, taking the intrinsic flexibility of such a large space structure into account [42,43,44].

From the perspective of attitude control system design, a constrained (and generally small) value of the target sail pitch angle could improve the effectiveness of the maneuver for the variation in the inclination of the sail nominal plane at the expense, however, of a worsening of the orbital transfer performance in terms of the required total flight time [45]. The aim of this paper is to investigate the effects of a constraint on the maximum value of the sail pitch angle, on the performance of a spacecraft equipped with an E-sail propulsion system in a typical interplanetary mission scenario [46]. In particular, the spacecraft mission performance is evaluated in terms of the minimum flight time for an assigned value of the maximum admissible sail pitch angle. In this context, the spacecraft guidance law is determined in a classical optimization framework by considering a mission scenario which is consistent with a three-dimensional, E-sail-based, interplanetary transfer from Earth to one of the other three terrestrial planets [47]. In particular, a heliocentric orbit-to-orbit transfer without ephemeris constraints is analyzed, while the performance analysis is conducted in a parametric form as a function of both the maximum admissible sail pitch angle and the propulsion system’s characteristic acceleration value. During flight, the E-sail propulsion system is considered to be always on so that the entire transfer can be considered a single propelled arc without any coasting phases. This specific situation allows the designer to further simplify the (optimal) guidance law because, during the whole transfer, the spacecraft control vector is formed only by the components of the propulsive acceleration unit vector. In other words, the guidance law for interplanetary transfer requires the evaluation, at each instant of time, of two independent scalar terms, one of which is precisely the constrained value of the pitch angle of the sail nominal plane.

In this context, the minimum (constrained) flight time is compared with the truly optimal (unconstrained) transfer time in order to evaluate the relative impact of the presence of a sail pitch angle limitation on the mission performance. In particular, the truly optimal transfer times, which are calculated to provide a benchmark, extend the E-sail literature because they are evaluated considering a single propulsion arc during the flight, i.e., they are the output of a simplified steering law which can be used to reduce the size of the E-sail control vector. Starting from the elegant thrust vector model proposed by Huo et al. [48], and taking the results of the optimization process described in a companion paper by the author [49], Section 2 discusses the E-sail control law in the presence of the constraint on the sail pitch angle. Section 3 illustrates the corresponding numerical simulations and a set of suitable graphs which allows the designer to quickly estimate the (percentage) performance degradation in the three interplanetary scenarios considered in this work. Finally, Section 4 concludes this work by summarizing the main results obtained.

2. Mathematical Preliminaries and Thrust Vector Analytical Model

This section discusses the E-sail thrust vector model in the presence of (1) a constraint on the maximum value of the sail pitch angle along the transfer orbit, and (2) a single propelled arc during the interplanetary flight between two assigned (Keplerian) heliocentric orbits. Furthermore, this section briefly indicates the mathematical approach used to obtain the optimal (constrained) transfer trajectory in the generic, orbit-to-orbit, heliocentric mission scenario. In the latter case, we avoid describing in detail the mathematical model used in the optimization process, to streamline the discussion of the proposed model. In any case, a detailed indication of the recent literature where the interested reader can find the related mathematical model is included in the text.

2.1. Analytical Model of the Propulsive Acceleration Vector

The expression of the propulsive acceleration vector ${\mathit{a}}_p$ used in this work is a simplified version of the one obtained by Huo et al. in 2018 [48] using a geometric (and analytical) approach. In particular, the formula obtained by Huo et al. is able to relate the propulsive acceleration vector generated by the E-sail both to the Sun–spacecraft unit vector $\widehat{\mathit{r}}$ and to the attitude of the sail nominal plane with respect to a classical orbital reference frame. This attitude is essentially described by the unit vector $\widehat{\mathit{n}}$ normal to the sail nominal plane in the direction opposite to the Sun.

In fact, bearing in mind that the E-sail propulsion system is always switched on during the flight, the elegant and compact Equation (17) by Ref. [48] simplifies as

$${\mathit{a}}_p={\displaystyle \frac{a_c{r}_{\oplus }}{2r}}\left[\widehat{\mathit{r}}+\left(\widehat{\mathit{r}}·\widehat{\mathit{n}}\right)\widehat{\mathit{n}}\right]$$

(1)

where r is the distance of the spacecraft from the Sun, $r_{\oplus }\triangleq 1\mathrm{AU}$ is a reference distance, and $a_c>0$ is the E-sail typical performance parameter, i.e., the sail characteristic acceleration [50]. The latter is defined, borrowing the nomenclature used in the field of solar sails [51,52,53], as the maximum value of the magnitude of the vector ${\mathit{a}}_p$ when the distance of the probe from the Sun is one astronomical unit, that is, when $r=r_{\oplus }$. Note that, at an assigned distance from the Sun, the maximum magnitude of the propulsive acceleration vector is obtained if $\widehat{\mathit{n}}\equiv \widehat{\mathit{r}}$, i.e., if the E-sail orientation is Sun-facing; see Figure 3a. To be precise, it should be noted that the expression of the propulsive acceleration vector given by Equation (1) is strictly valid if the distance of the vehicle from the Sun is around one astronomical unit since the mathematical model that underlies this expression of ${\mathit{a}}_p$ is consistent with the discussion reported in Refs. [1,38,39]. However, in the absence of a more suitable thrust model and taking into account the simplicity of the one proposed by Huo et al. [48], in the remainder of this work, the expression of ${\mathit{a}}_p$ given by the previous equation will be considered valid even when the distance of the spacecraft from the Sun is considerably different from $r_{\oplus }$.

As expected, Equation (1) indicates that $\parallel {\mathit{a}}_p\parallel$ cannot be zeroed during the flight (recall that $\widehat{\mathit{r}}·\widehat{\mathit{n}}\ge 0$ by assumption) except in the trivial case of $a_c=0$. The dependence of ${\mathit{a}}_p$ from the sail pitch angle can be highlighted by introducing a typical Radial-Transverse-Normal reference frame (RTN), like that sketched in Figure 4 in which ${\widehat{\mathit{i}}}_{\mathrm{R}}$ is the radial unit vector, ${\widehat{\mathit{i}}}_{\mathrm{T}}$ is the transverse unit vector, and ${\widehat{\mathit{i}}}_{\mathrm{N}}$ is the normal unit vector. This RTN reference system is quite conventional, but we point out that the origin of the frame coincides with the spacecraft’s center of mass ${\widehat{\mathit{i}}}_{\mathrm{R}}\equiv \widehat{\mathit{r}}$, while $({\widehat{\mathit{i}}}_{\mathrm{R}},{\widehat{\mathit{i}}}_{\mathrm{T}})$ coincides with the plane of the spacecraft osculating orbit during the interplanetary flight.

The propulsive acceleration vector can be written as a function of the sail pitch angle ${\alpha }_n$ by using Equation (1) and the scheme of Figure 5, which shows the generic orientation of the normal unit vector $\widehat{\mathit{n}}$ in the RTN reference frame. In particular, Figure 5 indicates two important angles that define the E-sail control vector, namely, the (already defined) sail pitch angle ${\alpha }_n$ and the sail clock angle $\delta \in [0,360]\mathrm{deg}$, which gives the orientation of the projection of $\widehat{\mathit{n}}$ in the plane $({\widehat{\mathit{i}}}_{\mathrm{T}},{\widehat{\mathit{i}}}_{\mathrm{N}})$.

Accordingly, using Equation (1) to express vector ${\mathit{a}}_p$ and bearing in mind the scheme of Figure 5, one obtains the radial ($a_{p_{\mathrm{R}}}\triangleq {\mathit{a}}_p·{\widehat{\mathit{i}}}_{\mathrm{R}}$), transverse ($a_{p_{\mathrm{T}}}\triangleq {\mathit{a}}_p·{\widehat{\mathit{i}}}_{\mathrm{T}}$), and normal ($a_{p_{\mathrm{N}}}\triangleq {\mathit{a}}_p·{\widehat{\mathit{i}}}_{\mathrm{N}}$) components of the propulsive acceleration vector as

$$\begin{array}{cc}\hfill a_{p_{\mathrm{R}}}& ={\displaystyle \frac{a_c{r}_{\oplus }}{2r}}\left(1+{cos}^2{\alpha }_n\right)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill a_{p_{\mathrm{T}}}& ={\displaystyle \frac{a_c{r}_{\oplus }}{2r}}sin{\alpha }_{n}cos{\alpha }_{n}cos\delta \hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill a_{p_{\mathrm{N}}}& ={\displaystyle \frac{a_c{r}_{\oplus }}{2r}}sin{\alpha }_{n}cos{\alpha }_{n}sin\delta \hfill \end{array}$$

(4)

which can be more conveniently rewritten in a dimensionless form (the tilde superscript will be used in the rest of the paper to indicate the dimensionless version of a given term) as

$$\begin{array}{cc}\hfill {\tilde{a}}_{p_{\mathrm{R}}}& ={\displaystyle \frac{r_{\oplus }}{2r}}\left(1+{cos}^2{\alpha }_n\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\tilde{a}}_{p_{\mathrm{T}}}& ={\displaystyle \frac{r_{\oplus }}{2r}}sin{\alpha }_{n}cos{\alpha }_{n}cos\delta \hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\tilde{a}}_{p_{\mathrm{N}}}& ={\displaystyle \frac{r_{\oplus }}{2r}}sin{\alpha }_{n}cos{\alpha }_{n}sin\delta \hfill \end{array}$$

(7)

In a typical unconstrained case, the sail pitch angle ranges from zero to $90\mathrm{deg}$, i.e., ${\alpha }_n\in [0,90]\mathrm{deg}$, so that the variation in the three dimensionless components $\{{\tilde{a}}_{p_{\mathrm{R}}},{\tilde{a}}_{p_{\mathrm{T}}},{\tilde{a}}_{p_{\mathrm{N}}}\}$ of the propulsive acceleration vector gives the so-called “force bubble” [54] shown in Figure 6, that is, the bubble-shaped surface (in this specific case, it is a sphere) on which the tip of the dimensionless vector ${\tilde{\mathit{a}}}_p$ is constrained to lie. In particular, the force bubble drawn in Figure 6 refers to the special case where $r=r_{\oplus }$, while the presence of a solar distance different from $1\mathrm{AU}$ does not modify the shape of the bubble. In Figure 6, recall that the direction of propagation of the solar wind coincides with the direction of the z-axis, that is, the Sun belongs to the vertical axis and is located in the negative half-space.

In a constrained case in which the sail pitch angle ranges in the interval ${\alpha }_n\in [0,{\alpha }_{n_{max}}]$, where ${\alpha }_{n_{max}}<90\mathrm{deg}$ is the assigned maximum value of the pitch angle, the shape of the force bubble changes with respect to the scheme of Figure 6. In particular, the reduction in the value of ${\alpha }_{n_{max}}$ results in a reduction in the size of the force bubble as highlighted in Figure 7, which shows, for illustrative purposes, the four cases of ${\alpha }_{n_{max}}\in \{30,45,60,75\}\mathrm{deg}$. Note from Figure 7 that in the (rather limiting) case of ${\alpha }_{n_{max}}=30\mathrm{deg}$, the force bubble becomes a small spherical cap, which indicates that, as expected, for a small value of ${\alpha }_{n_{max}}$, the propulsive acceleration vector ${\mathit{a}}_p$ is essentially radial with a very small transverse component. This last aspect, as discussed above, is a limiting factor in the heliocentric transfer between two orbits with different values of the semilatus rectum.

The constrained thrust vector model given by Equations (2)–(4) with ${\alpha }_n\in [0,{\alpha }_{n_{max}}]$ and $\delta \in [0,360]\mathrm{deg}$ can be used to describe the influence of the E-sail-induced acceleration in an optimized (from the flight time point of view) interplanetary orbit transfer. In this regard, the recent literature already contains the complete (unconstrained) mathematical model for trajectory optimization in the heliocentric scenario. This latter model can be easily adapted to this constrained case as briefly discussed in the next section.

2.2. Brief Notes on Trajectory Optimization in Presence of Pitch Angle Constraint

The optimization of the three-dimensional transfer trajectory in the presence of a geometric constraint on the maximum value of the sail pitch angle can be studied by easily adapting the mathematical model illustrated by the author in an open-access companion paper [49] appearing in this same Special Issue.

More precisely, the very recent Ref. [49] analyzes the three-dimensional, heliocentric, rapid orbit cranking maneuver of an E-sail propelled spacecraft by assuming a circular parking and target orbit. In particular, the mathematical model proposed in [49] uses the dimensionless modified equinoctial orbital elements $\{\tilde{p},f,g,h,k,L\}$ [55] to describe the spacecraft dynamics, a classical indirect approach [56,57] to obtain the rapid transfer trajectory, and the Pontryagin’s maximum principle [58] to derive the optimal guidance law during the interplanetary flight.

As regards the mathematical description of the E-sail-based spacecraft heliocentric dynamics, the model discussed in [49] can be applied directly with the only difference concerning the initial and final conditions, which, in the case treated in this paper, must be consistent with the specific interplanetary mission scenario analyzed. As discussed in the next section, the constrained thrust model is used here to analyze the E-sail performance in a set of interplanetary orbit-to-orbit transfers, in which the spacecraft parking (or final) orbit coincides with the heliocentric trajectory of the Earth (or the target planet). Accordingly, when the initial value of the time is set to zero, Equation (21) of Ref. [49] is substituted by

$$\tilde{p}\left(0\right)={\tilde{p}}_{\oplus },f\left(0\right)=f_{\oplus },g\left(0\right)=g_{\oplus },h\left(0\right)=h_{\oplus },k\left(0\right)=k_{\oplus },{\lambda }_L\left(0\right)=0$$

(8)

where $\{{\tilde{p}}_{\oplus },f_{\oplus },g_{\oplus },h_{\oplus },k_{\oplus }\}$ are the five dimensionless modified equinoctial elements that define the shape and orientation of the Earth’s heliocentric orbit, while ${\lambda }_L$ is the variable adjoint to the state L. In particular, the value of the set $\{{\tilde{p}}_{\oplus },f_{\oplus },g_{\oplus },h_{\oplus },k_{\oplus }\}$ was obtained using the data retrieved from the well-known JPL Horizon system on 1 August 2024. Note that in Equation (8), the condition ${\lambda }_L\left(0\right)=0$ indicates that the starting point on the parking orbit is not constrained, and is an output of the optimization process. The spacecraft final conditions, that is, the conditions to be reached at the final time $t_f$, which in Ref. [49] are given by Equations (33) and (34), are now expressed as

$$\tilde{p}\left(t_f\right)={\tilde{p}}_t,f\left(t_f\right)=f_t,g\left(t_f\right)=g_t,h\left(t_f\right)=h_t,k\left(t_f\right)=k_t,{\lambda }_L\left(t_f\right)=0,\mathcal{H}\left(t_f\right)=1$$

(9)

where $\{{\tilde{p}}_t,f_t,g_t,h_t,k_t\}$ depend on the orbital characteristics of the target planet (the values have been retrieved again from the JPL Horizon system on 1 August 2024), while $\mathcal{H}\left({\tilde{t}}_f\right)$ is the final value of the Hamiltonian function. Even at the final time, the spacecraft angular position along the target orbit is left free [indeed, one has ${\lambda }_L\left({\tilde{t}}_f\right)=0$ in Equation (9)] so that the interplanetary transfer is considered ephemeris-free.

The optimal control law is a simplified version of the one described in Ref. [49] because the E-sail propulsion system is always switched on during the transfer. In this case, in fact, we must determine the (optimal) time variation in the two control angles ${\alpha }_n$ and $\delta$, taking into account the constraint ${\alpha }_n\le {\alpha }_{n_{max}}$. As for the optimal value of the clock angle, the results of the unconstrained case are still valid, and one can directly take the value given by Equations (27) and (28) of Ref. [49]. On the other hand, the expression of the optimal pitch angle ${\alpha }^{★}$ given by Equation (30) of Ref. [49] cannot be used in this context because it does not take into account the presence of the constraint on its maximum value. However, Equation (30) of Ref. [49] is still a good starting point to obtain the optimal sail pitch angle. In fact, the presence of the constraint ${\alpha }_n\le {\alpha }_{n_{max}}$ gives the following expression of the optimal value of ${\alpha }_n$:

$${\alpha }_n=\left\{\begin{array}{cc}{\alpha }^{★},\hfill & \mathrm{if}{\alpha }^{★}\le {\alpha }_{n_{max}}\hfill \\ {\alpha }_{n_{max}},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(10)

where ${\alpha }^{★}$ is given by Equation (30) of Ref. [49]. The boundary value problem associated to the optimization procedure has been solved using a shooting method, in which the initial value of the unknown costates have been estimated by adapting the recent procedure proposed by the author in Ref. [59]. The numerical results, for the three mission scenario selected, are illustrated in the next section.

3. Orbital Simulations and Numerical Results

The impact of the constraint on the maximum value of the sail pitch angle is evaluated considering three interplanetary scenarios, namely, three-dimensional Earth–Mars, Earth–Venus, and Earth–Mercury transfers. The orbital elements of the four celestial bodies involved in the numerical analysis are summarized in

Table 1. Classical orbital elements of celestial bodies involved in interplanetary transfers analyzed in this section. The astronomical data were retrieved from the JPL Horizon system on 1 August 2024.

Planet	Semi-Major Axis	Eccentricity	Inclination	Argument of Perihelion	RAAN
Earth	$0.999\mathrm{AU}$	$1.670\times {10}^{-2}$	$3.235\times {10}^{-3}\mathrm{deg}$	$288.784\mathrm{deg}$	$174.301\mathrm{deg}$
Mars	$1.523\mathrm{AU}$	$9.331\times {10}^{-2}$	$1.848\mathrm{deg}$	$286.699\mathrm{deg}$	$49.489\mathrm{deg}$
Venus	$0.7233\mathrm{AU}$	$6.729\times {10}^{-3}$	$3.394\mathrm{deg}$	$55.206\mathrm{deg}$	$76.611\mathrm{deg}$
Mercury	$0.3871\mathrm{AU}$	$0.2056$	$7.003\mathrm{deg}$	$29.195\mathrm{deg}$	$48.300\mathrm{deg}$

. In all selected mission scenarios, the spacecraft has a canonical value of the characteristic acceleration, namely, $a_c$ = 1 mm/s², which corresponds to approximately $17\%$ of the gravitational acceleration of the Sun at one astronomical unit distance from the star. The procedure, of course, can be applied for a generic value of $a_c$ as will be discussed in the last part of this section.

The first step of this analysis is to determine the optimal transfer performance in the unconstrained case, which essentially corresponds to selecting ${\alpha }_{n_{max}}=90\mathrm{deg}$ in the automatic routines that optimize the constrained version of the interplanetary transfer. In this important case, the numerical results in terms of the minimum (unconstrained) flight time $t_f$ and the maximum (or minimum) value of the sail pitch angle reached during the orbit-to-orbit interplanetary transfer $max({\alpha }_n)$ [or $min({\alpha }_n)$] are summarized in

Table 2. Simulation results for an unconstrained, three-dimensional, optimal orbit-to-orbit transfer when $a_c$ = 1 mm/s². The term $t_f$ is the minimum flight time, while $max({\alpha }_n)$ [or $min({\alpha }_n)$] is the maximum (or minimum) value of the sail pitch angle reached during the transfer. Recall that, by assumption, the propulsion system is always switched on during the flight, and the sail pitch angle is unconstrained.

Scenario	${\mathit{t}}_{\mathit{f}}\left[\mathbf{days}\right]$	$min({\mathit{\alpha }}_{\mathit{n}})\left[\mathbf{deg}\right]$	$max({\mathit{\alpha }}_{\mathit{n}})\left[\mathbf{deg}\right]$
Earth–Mars	$495.66$	$12.90$	$87.87$
Earth–Venus	$303.62$	$12.24$	$64.86$
Earth–Mercury	$577.05$	$39.17$	$51.05$

. Recall that the propulsion system is always switched on during the flight.

The data reported in Table 2 are consistent with the graphs of Figure 8, Figure 9 and Figure 10, which show the time variation in the two control angles $\{{\alpha }_n,\delta \}$ in the three selected interplanetary mission scenarios. The corresponding optimal transfer trajectory is reported (black line) in Figure 11, Figure 12 and Figure 13 by employing a typical heliocentric–ecliptic reference frame $\mathcal{T}(O;x,y,z)$ with the origin O in the center-of-mass of the Sun [60] and the distances of the axes expressed in astronomical units. Note that the z-axis in Figure 11, Figure 12 and Figure 13 is intentionally exaggerated to better highlight the three-dimensionality of the interplanetary transfer analyzed by the optimization routines. According to Figure 13, it is noted that, even with a medium-high value of the characteristic acceleration such as the one used in the simulations, the optimal transfer without constraints towards the orbit of Mercury is rather complex and requires more than two complete revolutions around the Sun before being completed. On the other hand, the corresponding optimal guidance law is rather simple, and during the flight, the (optimal) value of the sail pitch angle “oscillates” around the value ${\alpha }_n=45\mathrm{deg}$ that maximizes the non-radial component of the propulsive acceleration vector; see Equations (6) and (7).

The impact of a constraint on the maximum value of the sail pitch angle is analyzed by calculating the optimal (constrained) interplanetary transfers when the pitch angle ranges in the interval ${\alpha }_{n_{max}}\in [30,90]\mathrm{deg}$. Indeed, the limiting case of ${\alpha }_{n_{max}}=90\mathrm{deg}$ gives the same results as the unconstrained scenario, while a value of ${\alpha }_{n_{max}}<30\mathrm{deg}$ gives an extremely high flight time when compared to the one obtained in the unconstrained case. Note that the other limiting case of ${\alpha }_{n_{max}}=0\mathrm{deg}$ refers to a Sun-facing flight, where the propulsive acceleration vector is purely radial at every instant of time. In that case, interplanetary transfer is physically impossible because the Sun-facing flight preserves the value of the semilatus rectum, whose value coincides with that of the heliocentric parking orbit (i.e., of the Earth). In any case, an analytical study of the heliocentric motion of a Sun-facing E-sail is presented in Ref. [61].

The numerical results of the constrained optimization process are summarized in Figure 14 for the three interplanetary mission scenarios selected in this work.

In particular, the relative variation in the flight time with respect to the unconstrained case is evaluated by introducing a new dimensionless parameter D defined as

$$D\left({\alpha }_{n_{max}}\right)\triangleq 100\times {\displaystyle \frac{{\left.t_f\left({\alpha }_{n_{max}}\right)\right|}_{\mathrm{con}}-{\left.t_f\right|}_{\mathrm{unc}}}{{\left.t_f\right|}_{\mathrm{unc}}}}$$

(11)

where ${\left.t_f\right|}_{\mathrm{unc}}$ is the minimum flight time obtained in the unconstrained case (the value of such time is summarized in Table 2), while ${\left.t_f\left({\alpha }_{n_{max}}\right)\right|}_{\mathrm{con}}$ is the value of the optimal flight time obtained in the constrained case as a function of ${\alpha }_{n_{max}}$. For example, a value of $D\left({\alpha }_{n_{max}}\right)=30$ indicates that, for the selected value of ${\alpha }_{n_{max}}$, the constrained flight time is $30\%$ greater than the value obtained when the sail pitch angle is free to change during the transfer.

According to Figure 14, we note that the shape of the function $D=D\left({\alpha }_{n_{max}}\right)$ is similar for the three orbit-to-orbit transfers, while the actual value of D for a given maximum sail pitch angle is strictly dependent on the specific scenario. More precisely, the value of D remains equal to zero until the value of ${\alpha }_{n_{max}}$ becomes smaller than the value of $max\left({\alpha }_n\right)$ given in Table 2. Indeed, from the numerical results viewpoint, a case in which ${\alpha }_{n_{max}}>max\left({\alpha }_n\right)$ is substantially unconstrained because the constraint is not active. On the other hand, we observe that when ${\alpha }_{n_{max}}$ approaches the value of $30\mathrm{deg}$, the degradation of the transfer performance becomes very evident and the flight time increases by more than thirty percent in the Earth–Mars scenario, while D settles at about sixteen percent in the Earth–Mercury scenario.

For example, in the Earth–Mars scenario where the constrained minimum flight time is roughly $652\mathrm{days}$ when ${\alpha }_{n_{max}}=30\mathrm{deg}$, the time variation in the two control angles is shown in Figure 15. Note that the sail pitch angle is equal to the maximum allowable value of $30\mathrm{deg}$ throughout almost the entire interplanetary transfer (the $83\%$ to be right), while the clock angle varies over the entire allowable range during the flight. Similar considerations can be made for the other two mission scenarios considered in this work.

Numerical simulations also highlight that the worsening of the performance obtained in the passage from the unconstrained to the constrained model also depends on the value of the characteristic acceleration considered. This aspect is briefly illustrated in the next subsection.

Effect of the Value of the Characteristic Acceleration

The value of the spacecraft characteristic acceleration $a_c$ influences the performance degradation (in terms of flight time) due to the presence of a constraint on the maximum sail pitch angle. To investigate this aspect, we use a parametric approach to study the Earth–Mars mission scenario, where both the maximum value of the sail pitch angle and the characteristic acceleration vary in the range ${\alpha }_{n_{max}}\in [30,90]\mathrm{deg}$ and $a_c\in [0.6,1]\mathrm{mm}/{\mathrm{s}}^2$, respectively. Note that the case of $a_c=0.6\mathrm{mm}/{\mathrm{s}}^2$ corresponds to a decrease of $40\%$ from the value used in the simulations discussed in the first part of this section, i.e., $a_c$ = 1 mm/s².

The optimal flight time in the unconstrained case is a function of $a_c$ only, while $t_f$ in the constrained case is a function of both ${\alpha }_{n_{max}}$ and $a_c$. In other terms, the dimensionless performance parameter D defined by Equation (11) is now a function of the two terms $\{{\alpha }_{n_{max}},a_c\}$, that is, one has $D=D({\alpha }_{n_{max}},a_c)$. Accordingly, the function $D=D({\alpha }_{n_{max}},a_c)$ is a surface whose shape can be numerically obtained by solving a suitable number of optimization problems. The results of the numerical simulations of the optimal transfer trajectories are summarized in Figure 16 and Figure 17, where the dependence of the degradation parameter D on the value of $a_c$ (for an assigned ${\alpha }_{n_{max}}$) can be observed.

4. Conclusions

The thrust vectoring of an interplanetary spacecraft whose primary propulsion system is an Electric Solar Wind Sail (E-sail) is obtained by changing the orientation of the sail nominal plane (i.e., the reference plane that ideally contains the conducting tethers) with respect to a classical orbital reference frame. In particular, a transverse thrust, that is, a component of the E-sail-induced thrust vector perpendicular to the radial direction, can be introduced in the spacecraft dynamics by tilting the sail nominal plane with respect to the Sun–vehicle line in order to reach a value of the sail pitch angle different from zero. This paper has discussed the impact of an active constraint on the maximum value of the sail pitch angle on the performance of an E-sail-propelled spacecraft in a set of typical heliocentric mission scenarios. In particular, using a classical indirect approach and taking into account the results of the recent literature on E-sail trajectory optimization, the minimum time transfers between Earth and one of the remaining terrestrial planets of the Solar System have been simulated by considering a typical orbit-to-orbit three-dimensional transfer without an ephemeris constraint. In this context, the E-sail propulsion system has been considered always switched on during the transfer that, in its turn, can be considered a single propelled arc without any coasting phases.

Numerical results obtained from orbital simulations show that the impact of a constraint on the maximum value of the sail pitch angle is not negligible, especially when the orbital transfer requires a considerable variation in the magnitude of the (specific) angular momentum vector. This is the case, for example, of an Earth–Mars or an Earth–Mercury interplanetary transfer in which the E-sail-powered spacecraft must significantly vary both the eccentricity and the semi-major axis of the osculating orbit during the heliocentric flight. Furthermore, in a given mission scenario, the actual impact of the sail pitch angle constraint on the transfer performance depends, in a rather complex way, on the selected value of the characteristic acceleration that defines the propulsive capabilities of the propellantless propulsion system installed on board. In this respect, the (constrained) optimal performance given by the orbital simulations gives a snapshot of the dependence of the minimum flight time on both the control law constraint and the propulsion system typical characteristics.

The obtained value of the minimum flight time is influenced by the hypothesis of the presence of a single propelled arc. In fact, if on one hand the absence of a switch function simplifies the optimal guidance law, then on the other hand, the absence of any coasting arcs (in general) worsens the transfer performance. In this sense, therefore, the evaluation of the minimum (constrained) flight time in the presence of a time-varying switching function, which allows the thrust vector to be set to zero during the transfer, appears to be the natural extension of this work.