Trajectory Optimization and Control Applied to Landing Maneuvers on Phobos from Mars-Phobos Distant Retrograde Orbits

Abstract

This paper presents research on the application of trajectory design, optimization, and control to an orbital transfer from Mars–Phobos Distant Retrograde Orbits to the surface of Phobos. Given a Distant Retrograde Orbit and a landing location on the surface of Phobos, landing trajectories for which total $\Delta v$ for a direct 2-burn maneuver is minimized are computed. This is accomplished through the use of Particle Swarm Optimization in which the required $\Delta v$ and time of flight are optimization parameters. The non-uniform gravitational environment of Phobos is considered in the computation. Results show how direct transfers can be achieved with $\Delta v$ on the order of ∼30 m/s.

1. Introduction

In order to facilitate the exploration of the Moon and Mars, intermediate staging locations have been proposed. The Lunar Gateway represents an intermediate orbital platform capable of linking, organizing, and supporting missions between the Earth and the Moon [1,2,3]. However, a lesser amount of literature regarding the use of similar platforms in the Martian environment exists. Such a Martian platform would be referred to as Mars Base Camp [4,5]. As well as the Lunar Gateway, Mars Base Camp is a proposed platform that would enable and aid the exploration of Mars and its surroundings, including its moons, Phobos and Deimos [6,7,8]. Although a final location for Mars Base Camp has not been chosen, it has been proposed that it could be located in the vicinity of Phobos or Deimos [9], such as a Mars–Phobos Distant Retrograde Orbit (MP DRO) [5]. This would avoid locating Mars Base Camp deep in Mars’ gravity well while keeping the Martian moons an accessible target. Some studies have shown the effectiveness of intermediate staging locations such as the aforementioned Mars Base Camp to facilitate missions in the vicinity of Mars and its moons [10,11]. Thus, the establishment of an infrastructure capable of performing In-Situ Resource Utilization (ISRU) of the material found in the regolith and under the surface of Phobos would prove to increase human and robotic exploration capabilities on Mars and its vicinity.

For the purpose of this research, we assumed the existence of a Phobos Base capable of resupplying Mars Base Camp located in a Distant Retrograde Orbit (DRO). We analyzed the necessary trajectories required to transport material from the surface of Phobos to Mars Base Camp. We applied trajectory design, optimization, and control to an orbital transfer in the Circular Restricted Three-Body Problem (CR3BP), specifically from an MP DRO to the surface of Phobos [12]. Given an MP DRO and a landing location on Phobos, the goal is to find the trajectory for which total $\Delta v$ for a direct two-burn maneuver is minimum [9,13]. Here, the total $\Delta v$ corresponds to the sum of $\Delta v$s required to initiate the transfer in the MP DRO and complete the transfer, i.e., land on the surface of Phobos. Considerations on Time-Of-Flight (TOF) were made in order to ensure that any necessary phasing and/or repositioning maneuvers are taken into account. A 2D analysis is sufficiently accurate since we considered the Mars–Phobos orbit plane as the reference frame, and motion will mostly happen in such a plane. However, further analysis would take into account a three-dimensional space problem in which the gravitational potential of Phobos is also considered [14]. Furthermore, the initial position among the departing DRO will be implemented as the third optimization parameter, given that the minimum $\Delta v$ does not necessarily occur at a fixed position along the orbit for a given landing location.

In this paper, a Particle Swarm Optimization (PSO) is used to determine the initial MP DRO given two parameters, $[A_x,V_y]$. Once the initial DRO is found, a trajectory optimization through a PSO is performed in order to determine the lowest $\Delta v$ trajectory to land a specific location on Phobos’ surface, given three parameters to optimize $[\alpha ,\varphi ,\Delta v]$. Here, we also discuss which PSO parameters are used for the analysis and which ones lead to the most efficient (i.e., the fastest) computations, that can be used for other optimization problems in the CR3BP. For further details on such parameters, refer to the nomenclature session at the beginning of the paper or to the detailed Section 2.3 and Section 3.

Phobos is the largest of two moons orbiting Mars, with an approximate size of $13.4$ km $\times 11.2$ km $\times 9.2$ km [15,16]. Phobos has an unusual shape (somewhat similar to a potato), which creates numerous difficulties for spacecraft to orbit and therefore land on the moon itself. In the literature, the gravitational potential and the density of Phobos have been of high interest, since it becomes essential for a precise and complete analysis to acknowledge such characteristics, especially for long-duration stability of orbits in the vicinity of the Moon [17,18,19].

The complex gravitational potential and density distribution are the biggest challenges when it comes to orbiting and landing on and from the surface of Phobos. Various methods have been developed to calculate the gravitational potential around irregular-shaped asteroids, highlighting the efforts to overcome such challenges, such as the mascons method [20,21]. In such a method, the celestial body’s shape is approximated by a set of point masses. Other approaches evaluate the gravitational potential of a homogeneous polyhedron, employing a combination of planar triangles on its surface, to describe the gravitational potential of an unusual-shaped body [22,23,24]. This method is considered particularly effective in describing the gravitational field near or on the surface of a constant-density polyhedron [25,26]. As demonstrated in [27,28], the mascon gravity framework can be used with a polyhedral-shaped source. These studies have shown the effectiveness of modeling the gravitational field by calculating the mass of each tetrahedron in the polyhedral shape and assigning it to a point mass at the center of the tetrahedron. This method provides a more accurate estimation of the potential compared to harmonic coefficients and significantly reduces computational processing time [29]. Note that the polyhedral shape of Phobos is available in the Small Bodies Data Ferret [30]. A brief explanation of how the gravitational potential of Phobos is taken into account in this analysis can be found in Section 3.

In this paper, the following approximations are made. Phobos orbits Mars on an elliptical orbit with an eccentricity of $e=0.0151$. For this study, we modeled Mars and Phobos using the CR3BP, which assumes that the eccentricity is 0. The semi-major axis of this orbit (i.e., average distance between Mars and Phobos) is 9376 km. Finally, the orbit’s inclination with respect to Mars’ equatorial plane is around 1°. Since Phobos’ Sphere of Influence (SOI) is below its physical surface, it becomes impossible to orbit the moon in the classical Keplerian sense. In the CR3BP, considering Phobos and Mars as the two primary masses, some families of periodic orbits exist, such as Halo Orbits or Lyapunov Orbits, as shown in Figure 1. However, such periodic orbits are either highly unstable or come dangerously close to the surface [31]. That being considered, the family of DROs is of use in the following analysis. Such family of orbits proved to be “far” enough from Phobos’ surface and to be sufficiently stable as well, up to multiple hundreds of years, when considering a full-force model [31]. The word “far” is here used loosely, since these orbits are within tens of hundreds of km from Phobos. In fact, the orbit is “distant” in the sense that it passes above the Lagrange points, rather than being near the moon, and it is “retrograde” because the spacecraft travels around the moon in the opposite direction that the moon travels around the planet.

Several factors play an essential role while landing a celestial body, such as the Solar Radiation Pressure (SRP), the gravitational influence of the Sun, and the eventual interactions with asteroids approaching the Mars–Phobos system. However, additional information is required regarding the geometry and materials of the spacecraft to perform a complete analysis of how the SRP will affect landing trajectories. Without such information, SRP cannot be taken into account. Since this study only focuses on trajectory optimization and control rather than mission planning, this factor is not considered. Moreover, the effect of the Sun’s gravity becomes relevant in a time span of many months or years. The maneuvers here described have a time of flight of some hours, which is not enough to have the Sun’s gravity play a significant role in the analysis. Finally, it would be virtually impossible to predict asteroids’ flybys and approaches without enormously increasing the computational complexity of the analysis. In the event of an asteroid approaching, this would likely result in a quick passage in the vicinity of Mars and would hardly have any influence on both the DRO stability and the landing maneuvers.

Mission analysis for Earth to Mars–Phobos DROs is of high interest when associated with human and robotic exploration [4,10,11,16]. Phobos could prove extremely useful for future Mars missions as a popular location for future ISRU plants. It is widely recognized that having such in-orbit and in situ infrastructures would make space exploration much more affordable, sustainable, and effective [16,32,33]. Several advantages of such stations include the ability to not have to bring all the necessary material from Earth, including propellant, for deep space missions [34]. Such projects are already in development (such as NASA’s Artemis missions [35]) for what concerns Earth’s Moon: building a refueling in-orbit station along with several on-the-surface infrastructures can easily be imagined as an intermediate step to what would be a deep space exploration mission.

In the following analysis, optimization algorithms are developed in order to generate a specific DRO given a desired distance from Phobos and optimize a landing maneuver to a specific landing site on the surface of Phobos with respect to the $\Delta v$ needed, i.e., minimize the necessary impulsive maneuver applied for landing.

2. Background Theory

For the purpose of giving a better insight of the problem, a brief overview of the basic aspects regarding Distant Retrograde Orbits and the Particle Swarm Optimization method are given in the following sections.

2.1. Distant Retrograde Orbits

Distant Retrograde Orbits (DROs) are large orbits around the smaller primary body in the 3BP. Perfectly periodic DROs exist in the CR3BP and in-plane velocity perturbations create quasi-periodic orbits. Such orbits are favorable when looking for quarantine orbits since their stability demonstrates a noteworthy capability to resist perturbations thanks to their interactions with two Lagrange points (L${}_1$ and L${}_2$) of the planet–moon system [33]. Indeed, DROs can stay quasi-periodic when larger perturbations than other families of three-body orbits can withstand are applied [31]. On the other hand, both departing and entering a DRO require a significantly high $\Delta v$, since they are marginally stable. Additionally, a spacecraft in a DRO has a longer orbital period and a slower velocity compared to closer orbits. This makes docking and rendezvous challenging since a spacecraft missing its departure/arrival window would need to wait one full orbital period before another opportunity becomes available. Although DROs are highly stable, they are not entirely impervious to external influences. Significant perturbations or disturbances, such as variations in gravitational forces from other celestial bodies or non-uniform mass distributions, can impact the stability and longevity of DROs. However, their relative stability is demonstrated by Wallace et al. [31] using full-force ephemeris models. DROs have been recently proposed to be the ideal family of orbits when it comes to locating in-space infrastructures, and this is due to their outstanding stability and ease of access in terms of a gravity well [33]. DROs are usually characterized by the so-called x-amplitude ($A_x$), which represents the largest distance from $m_2$ in the CR3BP, i.e., Phobos, in the x-axis direction, given a $xyz$ rotating reference frame orbiting the primaries, as shown in Figure 2. In such analysis, it is typical to use a frame of coordinates with its center located at the barycenter of the primary masses [16]. According to this, several essential parameters can be determined in order to define a DRO. Such parameters are the radius at periapsis, and its corresponding velocity, required at the intersection of the $xz$-plane (i.e., where $y=0$) in order to create a periodic orbit. Since the radius has only one component along the x-axis, which is $A_x$-dependent, we can say that the essential parameters necessary to define a DRO are $[A_x,V_y]$, where $V_y$ indicates the y-component of the velocity vector at such location, which is, in fact, the only non-zero component; the x-component and z-component of the velocity vector at the considered starting location are zero. DROs have been identified and studied in the literature, for both Earth–Moon DROs and Mars–Phobos DROs [16,36], as shown in Figure 3. However, in this study, a PSO algorithm will be used to determine a DRO given its x-amplitude, which will be the starting point for the landing trajectory optimization.

2.2. Particle Swarm Optimization

The Particle Swarm Optimization method is used in this study as an optimization algorithm to generate periodic DROs and compute $\Delta v$-optimal landing trajectories. PSO is a heuristic method widely used in previous astrodynamics problems [37,38,39]. It is based on the unpredictable motion of bird flocks while searching for food, taking advantage of the mechanism of information sharing that affects the overall behavior of a swarm [40,41]. The initial population that composes a swarm is randomly generated at the first iteration of the algorithm. Each particle is associated with a position vector and a velocity vector. In the PSO terminology, the words “position” and “velocity” of a particle are not to be interpreted in the classical sense (i.e., position and velocity as intended in classical physics and mechanics). Instead, they indicate the position vector containing the unknown parameters to optimize and the velocity vector determining the position update. Each particle represents a possible solution to the problem. At the end of the iterations, the particle corresponding to the optimal solution is selected, i.e., the particle that minimizes the parameters of interest, such as $\Delta v$. At each iteration, the position vector’s elements move in the velocity vector’s corresponding direction, to a new position, which represents a new possible solution. The updating of position and velocity is based on the objective function evaluation at the end of every iteration. The objective function (henceforth referred to as “cost function”) is an expression that needs to be minimized (or maximized). Once this condition is satisfied or the maximum number of iterations set $N_{\max }$ is reached, the algorithm will have obtained an optimal solution. At every iteration, the best cost function is used to determine how to update both velocity and position. Indeed, with the approach formulated above, the algorithm will converge on the best solution in every iteration. For each particle, the formula for velocity update includes three components with stochastic weights. In the PSO terminology, these are known as the inertial ($c_I$), cognitive ($c_C$), and social ($c_S$) components. The inertial parameter is usually chosen randomly, but it has appeared to be proportional to each particle’s velocity in the previous iteration for some applications [39]; the cognitive parameter is based on the best position experienced by the particle; finally, the social component is direct toward the personal best position (i.e., the best location yet located by any particle in the swarm). The algorithm terminates when the maximum number of iterations (user decision) is reached. For this study, the following values are used in the algorithm. The function rand stands for a single uniformly distributed random number in the interval (0,1). Adopting the settings suggested in [42,43,44], the inertial, cognitive, and social (stochastic) weights have the following expressions:

$$\begin{array}{cc}\hfill c_I& =\frac{(1+rand)}{2}\hfill \\ \hfill c_C& =1.49445·rand\hfill \\ \hfill c_S& =1.49445·rand\hfill \end{array}$$

(1)

The constants in Equation (1) are derived from the literature and optimized for this problem in space trajectories [45,46]. The implementation of the PSO was run in MatLab where the function unifrnd was used to assign the initial values at each particle element for each particle. The function unifrnd creates a uniformly distributed array of random values included in a specific range as given by the boundary values in

Table 1. Boundary values for DRO optimization.

Variable	Value	Description
$B_{\mathrm{lo}}$($V_y$)	$\frac{3}{2}\frac{t^{*}}{l^{*}}{V}_y$	Lower limit for $V_y$ (km/s)
$B_{\mathrm{up}}$($V_y$)	$\frac{t^{*}}{l^{*}}{V}_y$	Upper limit for $V_y$ (km/s)
$B_{\mathrm{lo}}$(T)	T/$t^{*}$	Lower limit for T (s)
$B_{\mathrm{up}}$(T)	$2\pi$	Upper limit for T (s)
$N_{\max }$	80	Maximum number of iterations
$N_{\mathrm{particles}}$	40	Number of particles

. Once that is implemented, the initial position is assigned

$${\overrightarrow{r}}_0=[A_x+(1-\lambda )]\widehat{i}+0\widehat{j}$$

(2)

where

$$\lambda =\frac{m_{\mathrm{Phobos}}}{(m_{\mathrm{Mars}}+m_{\mathrm{Phobos}})}$$

(3)

represents the mass ratio. Note that, as already mentioned, the following analysis will be conducted assuming 2D motion. Therefore, the $i\mathrm{th}$ particle (i.e., the particle corresponding to the $j\mathrm{th}$ iteration) is

$$P_i=[V_y\left(i\right);T\left(i\right)]$$

(4)

so that

$${\overrightarrow{v}}_0=0\widehat{i}+P_i\left(1\right)\widehat{j}$$

(5)

The PSO algorithm here used is summarized in the flowchart in Figure 4. The coordinate system used is represented in Figure 5. For a matter of simplicity, in Figure 5, $m_1=m_{\mathrm{Mars}}$ and $m_2=m_{\mathrm{Phobos}}$ as they are indicated in Equation (3). Note that these two parameters are picked for the MP DRO analysis described in the Mars–Phobos DRO section below and are here used to better describe the general problem and give the reader a full comprehension of the general approach. Once the Initial Conditions (ICs) are determined (${\overrightarrow{r}}_0$ and ${\overrightarrow{v}}_0$), numerical integration is performed in order to compute the resulting trajectory using the CR3BP equations, as derived in [47]:

$$\begin{array}{cc}\hfill \ddot{x}-2\dot{y}-x& =-\frac{(1-\lambda )(x+\lambda )}{r_1^3}-\frac{\lambda (x-1+\lambda )}{r_2^3}\hfill \\ \hfill \ddot{y}+2\dot{x}-y& =-\frac{(1-\lambda )y}{r_1^3}-\frac{\lambda y}{r_2^3}\hfill \\ \hfill \ddot{z}& =-\frac{(1-\lambda )z}{r_1^3}-\frac{\lambda z}{r_2^3}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill r_1& =\sqrt{{(x+\lambda )}^2+y^2+z^2}\hfill \\ \hfill r_2& =\sqrt{{(x+\lambda -1)}^2+y^2+z^2}\hfill \end{array}$$

(7)

Once the integration is finished, if the final position and velocity of the resulting trajectory are within a small tolerance (here ${10}^{-10}$) to the initial conditions ${\overrightarrow{r}}_0$ and ${\overrightarrow{v}}_0$, then a periodic orbit is achieved and the optimization algorithm has reached a final solution.

Therefore, the cost function, i.e., function to minimize, is

$$\begin{array}{cc}\hfill J& =\mid {\overrightarrow{r}}_{\mathrm{final}}-{\overrightarrow{r}}_0\mid +\mid {\overrightarrow{v}}_{\mathrm{final}}-{\overrightarrow{v}}_0\mid \hfill \\ \hfill & =\mid r_{\mathrm{final}}\widehat{i}-[A_x+(1-\lambda )]\mid +\mid v_{\mathrm{final}}\widehat{j}-V_y\left(i\right)\mid \hfill \end{array}$$

(8)

Satisfying the condition in Equation (8) results in having a periodic orbit, i.e., $\overrightarrow{r}\left(0\right)=\overrightarrow{r}\left(t_f\right)$ and $\overrightarrow{v}\left(0\right)=\overrightarrow{v}\left(t_f\right)$. Once the cost function is evaluated, each particle velocity (intended in the PSO terminology) is updated as follows:

$$\begin{array}{c}\hfill V_i=c_I{V}_i+c_C(P_{\mathrm{best}}-P_i)+c_S(G_{\mathrm{best}}-P_i)\end{array}$$

(9)

where $P_{\mathrm{best}}$ is the personal best of each particle, and $G_{\mathrm{best}}$ is the overall best position between all particles and all iterations as of the $i\mathrm{th}$ iteration. The coefficients $c_S,c_C,c_I$ are defined in Equation (1). Note that the subscript i in the Equations (4), (8) and (9) indicates the $i\mathrm{th}$ element at the $j\mathrm{th}$ iteration of the algorithm. Naturally, the larger the number of particles initialized, the more probability the algorithm has to achieve an optimal result. Similarly, a larger number of maximum iterations has more probability to converge to a solution. Nevertheless, a compromise between computational complexity and the number of iterations and particles needs to be made. In fact, increasing either the number of particles or the maximum iterations allowed can make the computational time increase significantly. However, due to the random nature of the algorithm, one cannot guarantee that an absolute optimal solution can be reached. On the other hand, a ‘good’ initial guess is not necessary in order to initiate the algorithm, unlike gradient-based methods. Furthermore, heuristic optimization methods are capable of finding absolute maxima/minima regardless of ICs [45]. On the other hand, the heuristic method of a PSO does not guarantee that a final solution will be found since finding absolute maxima/minima is not guaranteed. Hence, the final solution is sub-optimal.

2.3. Mars–Phobos DRO

Several DROs have been defined in the literature [12,16] for given $A_x$ values. Here, a PSO algorithm has been implemented to generate DROs. The dimensionality of the trade space needs to be minimized to optimize the computational effort of the analysis. Since a 2D analysis is performed, considering the state vector $[x,y,z,\dot{x},\dot{y},\dot{z}]$, all the z-components are zero, along with the y-component and the $\dot{x}$-component. Hence, two components are left to be found, knowing the initial value for one of those ($\dot{y}$). This reduces the minimum parameters necessary to optimize and characterize a DRO to 2. Hence, the optimization was implemented starting from the two parameters [T,$V_y$], where $V_y$ was defined in Section 2 and T represents the orbital period of the DRO. The initial data are summarized in

Table 2. Orbital parameters for Mars–Phobos and initial values for DRO optimization.

Variable	Value	Description
${\mu }_{\mathrm{Phobos}}$ (km${}^3$s${}^{-2}$)	$7.11358812096305\times {10}^{-4}$	Gravitational parameter for Phobos
${\mu }_{\mathrm{Mars}}$ (km${}^3$s${}^{-2}$)	$\mathrm{42,828.375214}$	Gravitational parameter for Mars
$\lambda$	$1.660952106463386\times {10}^{-8}$	Mass ratio
R (km)	9376	Average distance Mars-Phobos
$A_x$ (km)	125	DRO Amplitude

.

Starting from data taken from [16], the value of $V_y$ for $A_x=100$ km is known to be $V_y=-0.045620256764708$ km/s, which identifies a DRO with an orbital period $T=2.731044880670166\times {10}^4$ s [48]. Knowing such parameters helps the implementation of boundary values for the two parameters used in this analysis (i.e., the maximum and minimum values that the parameters can assume in the particles). In particular, the initial conditions for the optimization are listed in Table 1. Here, non-dimensionalization parameters were initialized to implement boundary values for the parameters used. The characteristic length $l^{*}$ and the characteristic time $t^{*}$ were implemented as

$$\begin{array}{c}\hfill l^{*}=R=9376 \mathrm{km}\end{array}$$

(10)

$$\begin{array}{c}\hfill t^{*}=\sqrt{\frac{R^3}{({\mu }_{\mathrm{Phobos}}+{\mu }_{\mathrm{Mars}})}}\end{array}$$

(11)

Hence, the entire analysis was conducted using non-dimensional values. Similarly, any other DRO of interest can be derived, although only one is here chosen for the results shown.

The parameters to optimize are the initial velocity $v_0$ (Equation (5)) at position $r_0$ (Equation (2)) and the orbital period T (Equation (4)). In the literature, different DROs have been identified for $A_x\in [15,300]$ km [16]. DROs with $A_x$ values below 15 km result in a certain impact with Phobos’ surface and the ones with higher $A_x$ values than 300 km are considered too far away to maintain the characteristics in which we are interested (vicinity to Phobos and low-$\Delta v$ for landing). Given the initial orbital parameters for Mars–Phobos and the initial values for the DRO, specified in Table 2, the resulting DRO is shown in Figure 6, and its reference values for position and velocity are summarized in Equation (12).

$$\begin{array}{cc}\hfill \overrightarrow{r}& =[9500.920971466154,-0.006046323363597]\mathrm{km}\hfill \\ \hfill \overrightarrow{v}& =[0.0007100917133235669,-0.056800560873732]\mathrm{km}/\mathrm{s}\hfill \\ \hfill T& =\mathrm{27,310.98049729574}\mathrm{s}\hfill \end{array}$$

(12)

where T represents the total orbital period, and v is the velocity vector at position r. Given the $N_{\max }$ and $N_{\mathrm{particles}}$ specified in Table 1, the result was achieved with a minimum cost function J in the order of ${10}^{-4}$ approximately, as shown in Figure 7, so the final values are such that $\mid \overrightarrow{r_f}-\overrightarrow{r_0}\mid \approx$ 0.1 mm and $\mid \overrightarrow{v_f}-\overrightarrow{v_0}\mid \approx$ 0.1 mm/s. The total average computation time was ∼20 s. Several other approaches were attempted, in terms of the number of particles and the number of maximum iterations, but these parameters proved to give a sufficiently accurate result in a reasonable amount of time. If, for instance, we doubled the $N_{\max }$ up to 160, the computation time increases up to ∼40 s, generating a minimum cost function J in the order of ${10}^{-5}$. So doubling the computational time will generally give a result that is just one order of magnitude more accurate than the initial one. As already outlined, one has to compromise between computational efficiency, computational speed, and the results’ accuracy.

Many computational issues may occur while implementing the algorithm. However, parallel programming and computing is a suitable solution to speed up the computation time. In fact, every possible trajectory computed at each iteration is independent with respect to all the other trajectories, until every particle is updated. Note that for this study a 2016 MacBook Pro is used. Hardware and Software details are summarized in

Table 3. Computer system characteristics.

Item	Description
Operating System	MacOS Monterey Version 12.6.7
Processor	3.3 GHz Dual-Core Intel Core i7
RAM	16 GB 2133 MHz
MatLab version	R2020b Update 3
System architecture	64-bit operating system, x-64-based processor

.

3. Landing Trajectory Optimization

In this section, the trajectory optimization for multiple landing maneuvers is analyzed through the use of PSO. Given an initial stable parking DRO around Phobos like the example computed in Section 2.3, and a desired landing location, the analysis aims to find the minimum $\Delta v$ necessary to encounter that specific location. Here, the gravitational parameters $J_2$ and $J_3$ of Phobos are taken into consideration during the CR3BP equations of motion integration. Although a simplified geometry of Phobos is considered, taking the gravitational distribution into account gives more accurate results to the presented analysis. The problem is computationally more complex than the one described in the MP DRO Section and it presents several more difficulties in the formulation. First of all, the choice of the parameters to optimize is non-trivial and there are different ways to approach the optimization. Here, the following parameters were chosen, minimizing the dimensionality of the trade space to three, as explained in Section 2.3:

$$P_i=[\alpha \left(i\right);\Delta v\left(i\right);\varphi \left(i\right)]$$

(13)

where the element $\Delta v\left(i\right)$ represents the $\Delta v$ to apply, and $\alpha \left(i\right)$ indicates the angle with respect to the x-axis at which the $\Delta v$ is applied. The parameter $\varphi \left(i\right)$ represents the angle with respect to the x-axis, which consequently identifies a location on the DRO $r\left(\varphi \right)$, as shown in Figure 8. In this figure, a graphical representation of the optimization parameters is shown. As represented, $\varphi$ indicates the position of the spacecraft along the initial DRO (we can compare $\varphi$ to a sort of true anomaly). Once the location of the spacecraft is known, the entire space vector of the spacecraft at such location is also known. Here, a $\Delta v$ needs to be applied to depart the DRO and initiate the landing maneuver. Hence, the other parameters to optimize are the $\Delta v$ magnitude and the angle $\alpha$ at which the impulsive maneuver is applied (no second out-of-plane angle is necessary since this is purely a 2D problem). This $\alpha$ angle is essential to calculate the resulting trajectory and perform a vector analysis. Note that in this analysis, the index i represents the $i\mathrm{th}$ particle at the $j\mathrm{th}$ iteration.

This analysis requires more computational time than the DRO optimization because there are three parameters to optimize instead of two, and we do not have specific analytic expressions to relate them to each other. Hence, a larger number of iterations was performed and a larger amount of particles was initialized, increasing the computational time and the overall complexity of the analysis. Computational times for the landing trajectory optimization are reported in

Table 4. Computation times for landing optimization algorithm.

Landing Location Sector	Computation Time
1	2 min, 56 s
2	2 min, 21 s
3	3 min, 20 s
4	2 min, 17 s
5	2 min, 42 s
6	2 min, 15 s
7	2 min, 56 s
8	3 min, 17 s
9	2 min, 21 s
10	2 min, 19 s
11	2 min, 53 s
12	4 min, 8 s
Average	2 min, 54 s

. The starting DRO is equal to the one defined in the optimization described in Section 2.3. Therefore, as obtained in Equation (12), the initial values and the particle boundary values are summarized in

Table 5. Initial values for landing trajectory optimization.

Variable	Value	Description
$\overrightarrow{r_0}$ (km)	$[9500.920971466154,-0.006046323363597]$	Initial position of parking DRO
$\overrightarrow{v_0}$ (km/s)	$[0.0007100917133235669,-0.056800560873732]$	Orbit velocity at position $r_0$
$T_{\mathrm{orbit}}$ (s)	27,310.98049729574	Orbital period

and

Table 6. Boundary values for landing optimization using PSO.

Variable	Value	Description
$B_{\mathrm{lo}}$($V_y$)	0	Lower limit for $\Delta v$ (km/s)
$B_{\mathrm{up}}$($V_y$)	$-V_y\frac{t^{*}}{l^{*}}$	Upper limit for $\Delta v$ (km/s)
$B_{\mathrm{lo}}$($\alpha$)	0	Lower limit for $\alpha$ (rad)
$B_{\mathrm{up}}$($\alpha$)	$2\pi$	Upper limit for $\alpha$ (rad)
$B_{\mathrm{lo}}$($\varphi$)	0	Lower limit for $\varphi$ (rad)
$B_{\mathrm{up}}$($\varphi$)	$2\pi$	Upper limit for $\varphi$ (rad)
$N_{\max }$	200	Maximum number of iterations
$N_{\mathrm{particles}}$	100	Number of particles

, respectively. As explained in Section 2.3, the dimensionality of the trade space is here minimized, and only three parameters are necessary to define a landing trajectory since most of the components of the state vector are either known or null.

In this particular optimization, a penalty scaling factor $\gamma$ was introduced. The addition of such a factor is necessary in order to prioritize the terms of the cost function [49]. In fact, the essential condition for a valid outcome of the optimization is to land at the desired location on Phobos. Indeed, a trajectory is considered valid only if it intersects the surface of Phobos at the desired coordinates. Secondarily, given that a valid trajectory to accomplish such a condition has been found, the total $\Delta v$ needs to be minimized. Hence, a penalty scaling factor plays an essential role in prioritizing the parameters and improving the algorithm’s reliability to obtain a close-to-optimal solution. The expression for $\gamma$ was derived by Conte et al. [49] and it is designed to be an adaptive non-linearly decreasing function of the iterations number i [50]. The following proved to be an effective value for this factor:

$$\gamma \left(i\right)=\beta \left[{\gamma }_{\mathrm{initial}}-\frac{{\gamma }_{\mathrm{initial}}-{\gamma }_{\mathrm{final}}}{N_{\max }}\sqrt{(2N_{\max }-i)i}\right]$$

(14)

where i is the current iteration number, $N_{\max }$ is the maximum number of iterations and ${\gamma }_{\mathrm{initial}}$ and ${\gamma }_{\mathrm{final}}$ represent the initial and the final weighting values. In this analysis, they are equal to 1 and 0.1, respectively. The factor $\beta$ is an order scaling factor, set equal to 100 for this optimization, which is used to make sure that the location checking term has a higher order than the $\Delta v$ term. Therefore, the cost function becomes

$$J=\left\{\begin{array}{cc}\gamma \mid {\overrightarrow{r}}_{\mathrm{final}}-{\overrightarrow{r}}_{\mathrm{landing}}\mid +\Delta v_{\mathrm{total}},\hfill & \mathrm{if}\mid {\overrightarrow{r}}_{\mathrm{final}}-{\overrightarrow{r}}_{\mathrm{landing}}\mid >\mathrm{tol}\hfill \\ \Delta v_{\mathrm{total}},\hfill & \mathrm{if}\mid {\overrightarrow{r}}_{\mathrm{final}}-{\overrightarrow{r}}_{\mathrm{landing}}\mid <\mathrm{tol}\hfill \end{array}\right.$$

(15)

For this optimization, the tolerance (tol) is set to 0.1 km (100 m). It is clear how the scaling factor prioritizes a specific term in Equation (15), given that the $\Delta v$ term is a relatively small value.

For a matter of simplicity, Phobos was ideally divided into 12 sectors, and a landing location within each sector was picked as a reference location for such an area. This means that the $\Delta v$ required to achieve landing in such a sector can be considered quantitatively similar to the reference location’s $\Delta v$. Results for the landing locations in each sector are summarized in

Table 7. Final results for landing optimization at one landing location for each sector.

Landing Loc. Sector	Time of Flight	Total $\mathbf{\Delta }\mathit{v}$ (m/s)	$\mathit{\alpha }$ (°)	$\mathit{\varphi }$ (°)
1	4 h, 41 min	27.35	246.10	147.14
2	3 h, 51 min	29.24	205.53	108.16
3	5 h, 10 min	31.02	347.27	256.35
4	7 h, 33 min	32.42	101.32	89.22
5	4 h, 54 min	27.27	88.41	337.07
6	3 h, 29 min	32	0	270.49
7	5 h, 13 min	27.41	100.38	0
8	6 h, 6 min	28.72	154.32	62.75
9	3 h, 41 min	31.93	0	270.84
10	4 h, 24 min	31.34	168.82	73.85
11	5 h, 36 min	27.66	260.27	204.36
12	5 h, 13 min	30.25	209.12	161.04
Average	4 h, 59 min	29.72 m/s

. Computation times for each example in Table 7 are indicated in Table 4. As expected, the average computation time is slightly higher than the one for the DRO optimization described in the MP DRO analysis (which is in a range between ∼30 s and ∼1 min).

The landing trajectories for each sector are represented in Figure 9 and a heat map of all the landing locations is shown in Figure 10. To have a better insight into the results, a fully loaded SpaceX Starship is considered. SpaceX’s Starship is expected to have an empty mass of ∼120 t (260,000 lb or 120,000 kg) and a gross mass of ∼1320 t (2,910,000 lb or 1,320,000 kg) with a propellant capacity of ∼1200 t (2,600,000 lb or 1,200,000 kg) [51]. Recall Tsiolkovsky’s rocket equation:

$$\Delta v=v_{e}ln\left(\frac{m_0}{m_f}\right)=g_0{I}_{sp}ln\left(\frac{m_0}{m_f}\right)$$

(16)

where $m_0$ is the total mass of the spacecraft, including the propellant, $m_f$ is the final total mass assuming all propellant to be consumed (also known as dry mass), $I_{sp}$ is the spacecraft’s specific impulse and $g_0$ is the standard gravity at Earth’s sea level. Note that the SI-metric ton (t) is used, where 1 t = 1000 kg ≃ 2205 lb.

We can now assume to land on Phobos’ surface at sector 7, the latitude of which corresponds to the Stickney Crater on the surface of Phobos and which corresponds to a total $\Delta v$ of 27.41 m/s (see Table 7). From Equation (16), we can estimate the necessary propellant mass to land the full-load Starship on Phobos.

We consider the spacecraft to empty its tanks at landing. Given a specific impulse of 380 s in a vacuum and a payload of 100 t (100,000 kg), Starship will achieve landing with a total propellant mass equal to

$$\begin{array}{cc}\hfill m_p& =m_f(e^{\frac{\Delta v}{g_0{I}_{sp}}}-1)\hfill \\ \hfill & =(120+100)·(e^{\frac{27.41}{9.81·380}}-1)\hfill \\ \hfill & \approx 1.62\mathrm{t}\equiv 1620\mathrm{kg}\hfill \end{array}$$

(17)

$$\frac{m_p}{m_f}=0.7\%$$

(18)

where $m_p$ is the total propellant mass used to achieve the indicated $\Delta v$. Since the landing $\Delta v$ is quantitatively equal to the $\Delta v$ required to go from the surface of Phobos to the initial DRO (the trajectory will just look different from a qualitative point of view), and we assumed Phobos to have a re-fueling station on its surface, the total propellant mass necessary to leave and arrive at the DRO is simply two times the mass indicated in Equation (17). Therefore, assuming one travel per terrestrial day for a year (365 days) back and forth the Stickney Crater, the total propellant mass per year necessary to achieve such transfers is ∼1185.22 t of propellant, which is equal to ∼1,185,220 kg.

If we consider a 1200 t propellant capacity, Starship can achieve a back-and-forth transfer ∼369 times before refueling. Although it is challenging to precisely estimate the propellant price per ton, SpaceX claims that refueling Starship would cost approximately USD 900,000. This means that a complete transfer would cost around USD 2435, transporting a 100 t payload.

4. Conclusions

This study led to valuable results for both DRO implementations and landing trajectory optimization, proving the feasibility of landing trajectories to the surface of Phobos. Several DROs have been defined in the literature for given $A_x$ values, but not much exists about landing trajectories from DROs to the surface of Phobos. Regarding the landing trajectory optimization, which is the main purpose of this study, it is clear from Table 7 how the $\Delta v$ is on average equal to ∼30 m/s. This value is reasonably low and makes this orbit feasible for an in-orbit refueling station or in general for a stationary orbit around Phobos, which one can easily reach and leave. Several landing locations were reached, and the impulsive $\Delta v$ was always found to be in approximately the same range. Therefore, nearly every point on Phobos’ surface can be reached with a $\Delta v$ of approximately ∼30 m/s. Furthermore, as summarized in Table 4, the average computation time for such a PSO algorithm is reasonable and makes the optimization feasible for future mission designs.

Mars’ moons exploration has been of high interest within the scientific community in the past years. The existence of periodic orbits around Phobos, such as DROs, makes the future design of in-orbit refueling and supporting stations more than feasible. Moreover, the ease of departing and arriving in such orbits with low $\Delta v$s gives the possibility of exploring Phobos and establishing a Mars Base Camp, in order to explore Mars and its surroundings with great versatility and with lower flight times between transfers from and to Mars itself. Having stations or spacecrafts orbiting around Mars (in Low-Mars Orbits for example) would require continuous trajectory adjustments because of Mars’ gravity potential acting on the spacecraft/station, and will then result in significant $\Delta v$ used over the course of a long-term mission. Besides, given the distance between Earth and Mars, it would not be convenient to perform frequent trajectory adjustments or remote control operations, given an average transmission delay of ∼20 min between Mars and Earth. Hence, having a long-term stability solution is crucial, and DROs appear to be feasible for such an objective. The method here presented can be used for multiple Mars–Phobos DROs optimizations at different amplitudes. Plus, a Mars–Deimos DRO can utilize this very same approach.