A Shape-Based Approach For Low-Thrust Earth-Moon Trajectories Initial Design

Abstract

A Robust Finite Fourier Series (R-FFS) approach is developed for fast generation of Earth-Moon trajectories using continuous low thrust. Each component of the position vector is approximated using a finite Fourier series as a function of time; these approximations are then used to design a trajectory that satisfies the equations of motion and the constraints, at discrete points, as well as the problem boundary conditions. The R-FFS method leverages the three body problem characteristics to achieve all the required plane change without the use of propulsion. The trajectory is divided into phases. The phase of the trajectory near the L1 Lagrange point is designed first and is always a thrust-free phase. This thrust-free phase is optimized to achieve the required plane change, enabling planar trajectories in the other phases. The initial guess needed by the solver, in the escape and capture phases, is generated using an analytic approximation developed in this paper. The numerical results show that the R-FFS can generate three dimensional transfers to high lunar orbits, low lunar orbits, and Halo orbits, while meeting constraints on the maximum thrust level of the engine.

I. Introduction

The Artemis program aims to land astronauts on the Moon by 2024 to explore more of the lunar surface. One of the main tasks in the design process of such a space mission is to design the Earth-Moon spacecraft trajectory. It is usually desired to minimize the fuel mass used in the trip to the Moon to reduce the overall spacecraft mass or to allow for more payload to be delivered to the Moon. Designing the Earth-Moon trajectory so as to minimize the fuel consumption renders the design problem as an optimal control problem when the propulsion is obtained using continuous low-thrust engines. The continuous low-thrust optimal control problem has received a great deal of interest in the literature. For example, reference [1] presents a solution procedure for the optimal control problem of low-thrust Earth-Moon trajectories assuming constant thrust magnitude. In [1], the trajectory is first divided into three phases: Earth-escape spiral, intermediate phase, and lunar-capture spiral, to solve an optimal control problem for each of these three phases separately. Then these solutions are used to provide initial guesses while solving another optimal control problem for the whole trajectory taking into account the gravity of the two primaries, the Earth and the Moon. The necessary conditions for optimality result in a two-point boundary-value problem (2PBVP) that was solved using a sequential quadratic programming (SQP) approach.

High fidelity numerical methods of optimal control, such as the Gauss pseudospectral methods, usually need an initial guess for the trajectory before they can search for the optimal solution. Providing this initial guess is not a simple task, and a bad initial guess may hinder the convergence of the optimal control solver. An initial guess generator should also be computationally fast. The Shape-Based trajectory design methods is a category of methods that can be used to generate initial guess trajectories for some problems. The shape-Based (SB) methods are also useful at early phases of trajectory optimization when a large design space is to be explored in a computationally efficient way to determine domains of interest [2]. One of the earliest SB methods is the exponential sinusoidal method [3] in which four parameters are used to design two-dimensional low-thrust trajectories that satisfy the equations of motion of a two-body model with thrust. The range r of the spacecraft position is expressed in terms of the polar angle θ of the position, in the form of an exponential sinusoidal function. The exponential sinusoidal method also satisfies the position boundary conditions (BCs). Because it is using only four parameters, it cannot be used in some types of problems such as rendezvous problems where velocity BCs at arrival point need to be satisfied as well. The inverse polynomial SB method [4] generates low-thrust trajectories using seven design parameters, and is demonstrated to solve rendezvous problems. In both the exponential sinusoidal and the inverse polynomial methods, there is no control over the thrust level needed to generate the trajectory for a given problem, and hence it is not possible to enforce a constraint on the thrust level. Later, reference [5] optimized the polynomial orders in the inverse polynomial method. Reference [6] presented a method for shaping the trajectory using a set of parameterized pseudoequinoctial elements where the effect of low thrust was considered as general perturbation in the secular and periodical components. Hence, the pseudoequinoctial elements of a low thrust trajectory were assumed the shape of a linear-trigonometric function. Reference [7] developed a framework to improve the low-thrust shaping methods and applied that to the three-dimensional description of the trajectory in spherical coordinates, and to shaping the pseudoequinoctial elements. The Finite Fourier Series (FFS) approximate method [8] uses more parameters, when compared to the inverse polynomial or exponential sinusoidal methods, along with orthogonal functions to represent the shape of the trajectory. Hence the FFS method can be used to generate feasible trajectories when thrust level is limited. In fact, the FFS approach was used in [9] to approximate the pseudoequinoctial elements in trajectory optimization. Reference [10] presents a hodographic-shaping method for low-thrust interplanetary trajectory design. The shaping here is performed using velocity hodographs for the low-thrust transfer problem. One advantage of this method is the fewer boundary conditions that need to be solved because the velocity is shaped instead of the trajectory [10]. Reference [11] presents a framework for generating trajectories using generalized logarithmic spirals; this framework can be used for trajectory design satisfying constraints on the position, velocity, and time of flight. In reference [12], the trajectory is designed based on a SB approach that expresses the trajectory in terms of the initial and target orbit elements. Using the SB approach to generate three-dimensional trajectories is more challenging especially when the out-of-plane-angle is large. Reference [13] mitigates that by using a combination of the elevation angle and radius shapes. Reference [14] extended the FFS SB to the three-dimensional case while minimizing the total cost of the trajectory, resulting in sub-optimal solutions.

The shape-based methods have been developed further in the literature for different applications. For example, reference [15] used a SB approach to design solar-sail trajectories for multiple Near-Earth-Asteroid rendezvous missions. In [15], the variation of the orbit parameter by means of solar sailing was assumed to follow an exponential-trigonometric shape, whereas the variation of the two in-plane modified equinoctial elements was assumed to follow a linear-trigonometric shape. Reference [15] showed that the approximated states using SB method had good match with data. The FFS SB method was used to design minimum-time, three-dimensional, solar sail trajectories for preliminary mission analysis and design in reference [16]. More recently, reference [17] used the FFS SB method to optimize solar sail trajectories. By solving a minimum time problem, the design variables were selected to be the Fourier coefficients and the positions of the spacecraft along the initial and target orbits. Also, reference [18] applied optimal control theory to trajectories parameterized using the FFS SB approach to solve the problem of optimizing low thrust trajectory design between circular coplanar orbits. In some cases, on-off actuators are used to provide the low-thrust propulsion [19]. The FFS SB method was extended in [20] to produce trajectories with on-off thrust profiles. Reference [21] used the SB approach to generate trajectories for orbit transfer problems. The shape for the radial coordinate was assumed to have the form of the initial and target orbits, with the semimajor axis of each of the initial and final orbits replaced by a polynomial function of the polar angle. In a similar way, the rendezvous problem was also solved in reference [21]. The trajectory design of Moon-Mars missions was also investigated in [22] where the heliocentric transfer phase was designed using a FFS SB approach. Finally, the PhD dissertation in [23] makes a comparison between most of the SB methods discussed above and highlights the capability of the FFS SB method in satisfying position and velocity BCs, and in enforcing explicit constraints on the thrust acceleration.

This paragraph summarizes the contributions of this paper. This paper presents a FFS method (in this paper it is referred to as R-FFS) for Earth-Moon Trajectory design. An Earth-Moon trajectory could be divided into three phases: escape, intermediate, and capture phases, when inserting the spacecraft into a two-body lunar orbit, or could be divided into two phases: escape and manifold phases, when inserting the spacecraft into a Halo orbit. It is demonstrated in this paper that it is possible to design Earth-Moon trajectories using the FFS method with no thrust usage during the intermediate phase, leveraging the three-body dynamics. It is also shown in this paper that it is possible to use the FFS method to design Earth-Moon trajectories in which the Earth departure orbit and the lunar capture orbit are in two different planes (three dimensional trajectories), also leveraging the characteristics of the three-body dynamics in the intermediate phase. In addition, the proposed method can be used to generate low-thrust trajectories that insert the spacecraft into Halo orbits; to the best of the authors’ knowledge this is the first implementation for FFS SB methods in generating transfers to Halo orbits. The R-FFS method requires an initial guess for each of the escape and capture phases. An analytic approximation is developed in this paper to compute the required initial guess; it also provides an analytic initial guess for the time of flight of each of the escape and capture phases. The nearest in the literature to this paper is reference [24] in which the FFS SB method was used to generate two-dimensional Earth-Moon trajectories. The work in [24] does not leverage the three-body dynamic characteristics, and requires thrust in all three trajectory phases.

This paper is organized as follows. Section II summarizes the dynamic model used in this paper. Section III presents a brief description for the FFS SB method. Section IV presents the how each of the escape, intermediate, and capture phases is design in the proposed approach. Section V presents the approximate method used to compute the times of flight for each phase and the initial guess for the Fourier coefficients. Section VI presents the numerical results.

II. Dynamical Model

A. Equations of Motion

In studying Earth-Moon trajectories, typically two coordinate systems are used [25]; these two coordinate systems are shown in Fig. 1. Both are polar rotating coordinate systems, each of them is centered at the center of one of the two primaries: Earth and Moon. Figure 1 shows the Earth Centered Rotating Frame (ECRF) and the Moon Centered Rotating Frame (MCRF), where the first axis in both coordinate systems points in the direction from Earth to Moon, the second axis in both of them is perpendicular to the first axis in the plane of Moon orbit around the Earth. The Moon is assumed to move in a circular orbit around Earth. Using canonical units, the distance from Earth center to Moon center is D = 1, and the mass ratio of the two primaries is μ = 0.012155 for the Earth-Moon system [26]. Working with these two coordinate systems enables the design of the trajectory near the Earth in the Earth centered coordinate system, while the trajectory near the Moon is designed in the Moon centered coordinate system. This set up avoids the oscillatory behavior of the polar angle when reaching the Moon, using only the Earth centered coordinate system, which was shown to result in apparent ill-condition states [27]. Figure 1 also shows a third coordinate system centered at the spacecraft, with the ${\widehat{e}}_r$ pointing in the direction from Earth to the spacecraft, and ${\widehat{e}}_{\theta }$ is perpendicular to ${\widehat{e}}_r$ in the plane of spacecraft motion. The angle γ is the flight path angle, and the angle α is the angle of the thrust acceleration ${\overrightarrow{T}}_a$.

Fig. 1.

The Earth Centered Rotating Frame and the Moon Centered Rotating Frame

In this paper, the trajectory is divided into three main phases: Earth escape phase, the intermediate phase, and the Moon capture phase. In all the three phases, the spacecraft is moving in a gravitational field governed by the two primaries Earth and Moon. The equations of motion during the escape and intermediate phases are expressed in the ECRF, while they are expressed in the MCRF during the capture phase. The subscript e in Fig. 1 refers to the escape phase and the subscript c refers to the capture phase.

Because a low thrust level is used in this paper, each of the escape and capture trajectories is expected to be a spiral shape. As will be seen later, the intermediate phase will be a thrust-free phase of the trajectory. The equations of motion of the spacecraft during the escape phase, expressed in the ECRF [24], are:

$${\ddot{r}}_e-r_e(1+{\stackrel{.}{\theta }}_e{)}^2+\mu cos({\theta }_e)+\frac{1-\mu }{r_e^2}+\mu \left(\frac{r_e-cos({\theta }_e)}{r_{Moon-S∕C}^3}\right)=T_{a,e}sin({\alpha }_e)$$ (1)

$$r_e{\ddot{\theta }}_e+2{\stackrel{.}{r}}_e(1+{\stackrel{.}{\theta }}_e)-\mu sin({\theta }_e)\left(1-\frac{1}{r_{Moon-S∕C}^3}\right)=T_{a,e}cos({\alpha }_e)$$ (2)

where, $r_{Moon-S∕C}=\sqrt{r_e^2-2r_{e}cos({\theta }_e)+1}$, T_a,e is the magnitude of thrust acceleration in the escape phase, r_e is the radius, and θ is the polar angle, measured counterclockwise. The subscript e in the above equations indicates the escape phase.

The equations of motion during the capture phase expressed in the MCRF [24] are:

$${\ddot{r}}_c-r_c(1+{\stackrel{.}{\theta }}_c{)}^2-(1-\mu cos({\theta }_c))+(1-\mu )\frac{r_c+cos({\theta }_c)}{r_{Earth-S∕C}^3}+\frac{\mu }{r_c^2}=T_{a,c}sin({\alpha }_c)$$ (3)

$$r_c{\ddot{\theta }}_c+2{\stackrel{.}{r}}_c(1+{\stackrel{.}{\theta }}_c)-(1-\mu )sin({\theta }_c)\left(\frac{1}{r_{Earth-S∕C}^3}-1\right)=T_{a,c}cos({\alpha }_c)$$ (4)

where, $r_{Earth-S∕C}=\sqrt{r_c^2-2r_{c}cos({\theta }_c)+1}$, and the subscript c indicates the capture phase.

The equations of motion during the intermediate phase expressed in the ECRF are:

$${\ddot{r}}_b-r_b(1+{\stackrel{.}{\theta }}_b{)}^2+\mu cos({\theta }_b)+\frac{1-\mu }{r_b^2}+\mu \left(\frac{r_b-cos({\theta }_b)}{r_{Moon-S∕C}^3}\right)=0$$ (5)

$$r_b{\ddot{\theta }}_b+2{\stackrel{.}{r}}_b(1+{\stackrel{.}{\theta }}_b)-\mu sin({\theta }_b)\left(1-\frac{1}{r_{Moon-S∕C}^3}\right)=0$$ (6)

where, $r_{Moon-S∕C}=\sqrt{r_b^2-2r_{b}cos({\theta }_b)+1}$, and the subscript b indicates the intermediate phase.

B. Thrust Vector

The FFS SB method allows for variable thrust magnitude. Regarding the thrust direction, it is assumed that the thrust is either along the velocity direction or against the velocity direction. This assumption is based on observations from several previous studies. For example, reference [1] solves the optimal co-planar Earth-Moon trajectory, assuming that the thrust magnitude is constant, but the thrust angle is the control variable. It was found in [1] that during the escape phase, the thrust angle almost aligns with the flight path angle as the spacecraft approaches the end of the escape phase. During the earlier parts of the escape phase, the thrust angle values oscillates around the flight path angle values over time. During the capture phase, the thrust direction was almost opposite to the velocity direction all the time. Hence, to simplify the analysis and reduce the number of design variables, it is assumed that:

$${\alpha }_{e∕c}={\gamma }_{e∕c}+n\pi ,n=0,1$$ (7)

where n = 0 corresponds to thrust direction is aligned with velocity direction, and n = 1 corresponds to thrust direction is opposite to velocity direction. Note that the intermediate phase is thrust-free. Thus, the above equations of motion can be used to solve for the thrust acceleration and thrust direction, for the escape and capture phases, in terms of the states to yield:

$$T_{a,e}=\frac{1}{cos({\alpha }_e)}\left(r_e{\ddot{\theta }}_e+2{\stackrel{.}{r}}_e(1+{\stackrel{.}{\theta }}_e-\mu sin({\theta }_e)[1-(1∕r_{Moon-S∕C}^3)]\right)$$ (8)

$${\alpha }_e={\gamma }_e=ta{n}^{-1}\left(\frac{{\stackrel{.}{r}}_e}{r_e{\stackrel{.}{\theta }}_e}\right)$$ (9)

$$T_{a,c}=\frac{1}{cos({\alpha }_c)}\left(r_c{\ddot{\theta }}_c+2{\stackrel{.}{r}}_c(1+{\stackrel{.}{\theta }}_c-(1-\mu )sin({\theta }_c)[(1∕r_{Earth-S∕C}^3)-1]\right)$$ (10)

$${\alpha }_c={\gamma }_c=ta{n}^{-1}\left(\frac{{\stackrel{.}{r}}_c}{r_c{\stackrel{.}{\theta }}_c}\right)$$ (11)

The rocket equation is used to obtain an expression for the spacecraft mass (m), as it changes over time, as follows:

$$\Delta V=I_{sp}{g}_{0}ln\left(\frac{m_0}{m}\right)$$ (12)

$$m=m_0\left(1-exp\left(\frac{-\Delta V}{g_0{I}_{sp}}\right)\right)$$ (13)

where m₀ is the initial mass, g₀ is the Earth gravity acceleration at sea level, I_sp is the specific impulse and ΔV is the integration over time of the thrust acceleration magnitude.

III. Finite Fourier Series Approximation

The FFS SB approach was first introduced in [8]. This section summarizes the FFS SB approach; the details of the new methodological developments carried out in this paper are presented in Section IV. The main concept of the FFS SB method is that any continuous smooth trajectory can be approximated using a finite Fourier series. So, even if we do not know the spacecraft trajectory shape, we know it can be represented using a FFS. In the case of a two-dimensional trajectory, for instance, each of the two components of the position vector, the radius (r) and the polar angle (θ), is approximated as follows:

$$\text{r(t)}={\mathrm{a}}_0\frac{}{2+{\sum }_{i=1}^{n_r}\left[a_{i}cos\left(\frac{n\pi t}{T}\right)+b_{i}sin\left(\frac{n\pi t}{T}\right)\right]}$$ (14)

$$\theta (t)=\frac{c_0}{2}+{\sum }_{j=1}^{n_{\theta }}\left[c_{j}cos\left(\frac{n\pi t}{T}\right)+d_{j}sin\left(\frac{n\pi t}{T}\right)\right]$$ (15)

where, T is the time of flight. The coefficients a_i and b_i, i = 1 ⋯ n_r, along with the coefficients c_j and d_j, j = 1 ⋯ n_θ, are used to design a trajectory that satisfies the equations of motion at discrete points, satisfy the problem BCs, and impose constraints on the maximum thrust level. It is possible to replace t/T = τ, where τ is the normalised time in the escape/capture phase such that 0 ≤ τ ≤ 1. In satisfying the equations of motion at the discrete points, a mean least squares error problem is solved. Initial guess for the Fourier coefficients is obtained using an approximate method (as detailed in Section V) and this initial guess for the coefficients specifies an initial guess shape for the trajectory. In the low-thrust Earth-Moon trajectory, the escape (capture) phase is a spiral that departs from (arrives at) a specific given orbit. Hence we can write four BCs on r(τ = 0), $\stackrel{.}{r}(\tau =0)$, θ(τ = 0), and $\stackrel{.}{\theta }(\tau =0)$. In the solution method presented in this paper, in Section IV, it is assumed that end (start) point of the escape (capture) spiral is computed from the intermediate phase; hence these points also provide BCs when computing the escape (capture) spiral: r(τ = 1), $\stackrel{.}{r}(\tau =1)$, θ(τ = 1), and $\stackrel{.}{\theta }(\tau =1)$. Similar conditions can be written for the start point of the capture orbit. Using the above eight BCs, it is possible to solve for eight of the Fourier coefficients in terms of the rest of the Fourier coefficients, as detailed in [8]. Hence, one can write r(τ) and θ(τ) in the following form:

$$r(\tau )=F_r+C_{a_0}{a}_0+\sum _{n=3}^{n_r}\{C_{a_n}{a}_n+C_{b_n}{b}_n\}$$ (16)

$$\theta (\tau )=F_{\theta }+C_{c_0}{c}_0+\sum _{n=3}^{n_{\theta }}\{C_{c_n}{c}_n+C_{d_n}{d}_n\}$$ (17)

where the BCs appear only in the terms F_r and F_θ, as detailed in [8], and the functions associated with the unknown Fourier coefficients (C_a0, C_c0, C_an, C_cn, C_bn, and C_dn) are functions only of τ and the counter n [8].

In a similar way, one can write analytic expressions for the first and second time-derivatives of r and θ. The equations of motion are to be satisfied at discretization points (DPs). In the escape (capture) spirals considered in this paper, the number of DPs (n) is assumed a function of the number of revolutions in the spiral:

$$n=(N_{rev}+1)\ast p_r$$ (18)

where, p_r is the number of points per revolution and N_rev is the number of revolutions in the trajectory. The higher the p_r the more accurate is the representation and the higher is the computational cost. The distribution of the DPs in the escape and capture spirals is assumed linear in time. The approximate spiral generator discussed in Section V can be used to compute the number of revolutions; the approximate algorithm provides a final value for the polar angle at the end of the escape spiral (θ_f), hence we can write:

$$N_{rev}=floor\left(\frac{{\theta }_f-{\theta }_i}{2\pi }\right)$$ (19)

where floor(x) is the largest integer small than x. A similar equation can be written for N_rev in the capture phase spiral.

The unknown Fourier coefficients are computed so as to minimize the mean square error in satisfying the equations of motion at all DPs while satisfying the maximum thrust magnitude constraint. This is explained in more detail in the following section, for each phase separately. A compact form for expressing each of the states can be written if we collect all the unknown Fourier coefficients in a vector [X], then the vector [r] that includes the values of r at all the DPs can be expressed as:

$$[r{]}_{n\times 1}=[A{]}_{n\times (2n_r-3)}[X{]}_{(2n_r-3)\times 1}+[F_r{]}_{n\times 1}$$ (20)

where the number of unknown Fourier coefficients in r is (2n_r − 3), n is the number of DPs, [A]_n×(2nr−3) = [C_a0, C_a3, C_b3, ⋯ , C_an, C_bn]_n×(2nr−3), and [X]_(2nr−3)×1 = [a₀, a₃, b₃, ⋯ , a_n, b_n]^T.

IV. Trajectory Design Method

This section provides details on the solution method used to design the entire trajectory, departing from an earth orbit, and arriving at a lunar orbit. It is complicated to develop a shape function and initial guess for the trajectory as a whole. Thus, the problem is divided into three phases: escape, intermediate and capture. The escape and capture segments are assumed to be an outward and inward spirals, respectively. The intermediate phase is assumed to be a coasting arc, and it is generated using numerical propagation under zero thrust. This intermediate phase is key for the whole trajectory as it provides the terminal BCs needed to design the escape and capture phases. Throughout the rest of this paper, the final BCs of the escape phase and the initial BCs of the capture phase will be referred to as escape point and capture point, respectively. The proposed method will be labeled R-FFS method. The following subsections explain the solution procedure to generate the trajectory in each phase.

A. Intermediate Phase

The intermediate phase is critical because it connects escape and capture phases, maintaining continuity in the trajectory. In the R-FFS method, all the plane change occurs during the intermediate phase, without the use of propulsion, leveraging the three body dynamics. The previous implementation of the FFS method for the Earth-Moon problem in [28] generates the escape and capture spirals first; then the intermediate trajectory is generated. To do that, an outer-loop optimizer is implemented to tune escape and capture points, ensuring the feasibility of the intermediate segment. This process in [28] suffers from increased complexity and computational time. Additionally, this outer-loop optimizer is sensitive to the way the escape and capture points are generated and sometimes returns infeasible solutions.

The R-FFS method adopts an inverse approach where a coasting intermediate segment is first generated, before the capture and escape segments are generated. This process eliminates the need for the outer-loop optimizer, and significantly increases the possibilities of finding feasible trajectories, hence increasing the robustness of the method. In the R-FFS method, the escape and capture points are not specified a priori when designing the intermediate phase. Hence another criteria is used to generate the intermediate phase; this criteria is here described. At the Earth-Moon Lagrange points, a gravitational equilibrium could be maintained. A spacecraft placed at the Lagrange point L1 would remain stationary with respect to the rotating frame. Figure 2 shows the equipotential lines and their corresponding energy levels that pass through the Lagrange points [26]. This graph also shows the minimum energy required by the spacecraft to access the Lagrange points. The Lagrange point L1 is approximately located at a distance of 324, 600 km from the center of the Earth along the line connecting the centers of Earth and Moon. The energy integral (the Jacobi integral) [26] for zero-velocity points in the rotating frame is given in Eq. (21). Given the energy required to access the point L1, Eq. (21) can be used to calculate the spacecraft velocity:

$$C=-2E=(x^{\prime 2}+y^{\prime 2})-(x^2+y^2)-2\frac{1-\mu }{r_1}-2\frac{\mu }{r_2}$$ (21)

where, C is the Jacobi constant, x and y are the coordinates of the spacecraft in the rotating frame, r₁ and r₂ are the magnitudes of the spacecraft’s position vectors from Earth and Moon, respectively.

Fig. 2.

Equipotential lines and the equilibrium points of the Earth-Moon system

A slight perturbation, however, in the energy of a spacecraft at L1 would cause it to fall either towards the Earth, or drift away towards the Moon. Thus, it acts as a passage that provides a comparatively easy access to explore Lunar and Earth orbits with minimal change in velocity. Thus, the entire intermediate phase trajectory can be generated by forward and backward propagation of the spacecraft states, starting from the vicinity of the equilibrium state at L1 using the equations of motion, Eqns. (5) and (6). Forward propagation is towards the Moon, and backward propagation is towards the Earth. Anywhere in the vicinity of the equilibrium state is acceptable since the goal is to generate a trajectory that connects an Earth escape spiral to a Lunar capture spiral. In implementing that, a position near L1 is used as a starting point for propagation, for which x = 0.83692 DU, y = z = 0 DU. Given the velocity magnitude at the starting point of propagation, v_L1, the flight path angles γ_L1 and β_L1 are assumed, and hence the states ${\stackrel{.}{r}}_b$, ${\stackrel{.}{\theta }}_b$, and $\stackrel{.}{z}$ at the starting point are calculated using Eq. (22).

$$\begin{gathered} {\stackrel{.}{r}}_b=V_{L1}sin({\alpha }_{L1})cos({\beta }_{L1}) \\ {\stackrel{.}{\theta }}_b=\frac{-V_{L1}cos({\alpha }_{L1})cos({\beta }_{L1})}{r} \\ \stackrel{.}{z}=V_{L1}sin({\beta }_{L1}) \end{gathered}$$ (22)

One suggested value for the flight path angle γ_L1 is 90 degrees in the Earth frame since on the escape spiral, the velocity direction approaches the radial direction as the spacecraft moves further away from Earth. The terminal conditions of the intermediate phase at the end of the backward and forward propagation are the escape and capture points, respectively. These are the BCs needed to generate the escape and capture segments.

In general, the state of the spacecraft at the end of the escape (capture) phase would not be the best BCs to design the escape (capture) spiral. This is particularly important when the capture orbit plane is different from the departure orbit plane, which is the case of three-dimensional (3D) Earth-Moon trajectories. The approach used to generate this 3D trajectory is as follows. Each of the escape and capture spirals is assumed a planar spiral. The plane of the capture spiral is dictated by the capture orbit, and the plane of the escape spiral is dictated by the departure orbit. The intermediate trajectory is designed to take care of the plane change, without using thrust, as follows. The initial conditions, at L1, are selected to be the design variables in an optimization problem. The objective of the optimization problem is to minimize the sum of the squares of two errors. The first error is the difference between the escape plane normal unit vector and the direction of the spacecraft angular momentum at the escape point, as shown in Fig. 3. Recall that the escape point is computed by propagating the initial conditions at L1 point backward in time until the escape point. The second error is the difference between the capture plane normal unit vector and the direction of the spacecraft angular momentum at the capture point. Recall that the capture point is computed by propagating the initial conditions at L1 point forward in time until the capture point. If each of these two errors is brought to zero then that means the intermediate phase trajectory would connect the escape and capture spirals, despite each of them is in a different plane. The solution of the optimization problem is the initial conditions at L1 that we should use to propagate the intermediate phase trajectory. This approach has the advantage of leveraging three-body dynamics to do all the required plane change without using propulsion.

Fig. 3.

Error E_r between departure orbit normal and angular momentum direction at escape point.

B. Escape Phase

The states of the spacecraft motion along the escape spiral trajectory are approximated using the FFS SB approach. The main function of the escape solver is to search for the values of the Fourier coefficients such that the governing equations of motion and the thrust limit constraint are satisfied at ‘n’ discrete points. Two new aspects about the design process of the escape spiral are presented: (1) segmentation of the escape spiral; that is dividing the spiral into segments maintaining continuity and smoothness of thrust at the connecting points between segments, and (2) using a more accurate approximation to generate the initial guess for the Fourier coefficients. Both of these new developments in FFS SB spirals significantly improve the numerical robustness of the method as demonstrated in the numerical results presented in this paper. A flowchart describing the escape solver is shown in Fig. 4. The initial guess block is responsible for generating an initial guess trajectory required by the escape solver. The segmentation block divides the escape phase into ‘m’ segments and extracts the corresponding Fourier Coefficients(explained in the next section). The optimization block iterates over Fourier coefficients to generate the trajectory solution.

Fig. 4.

Illustration for the Escape Phase generation process

Substituting the FFS representations of the states r(t), θ(t), $\stackrel{.}{r}(t)$, and $\stackrel{.}{\theta }(t)$ into the escape equations of motion in Eq. (1) and Eq. (2), and into the thrust equation in Eq. (8), one can write these equations in the following compact form:

$$\begin{gathered} f_e\left(X_e,[r_e],[r_e^{\prime }],[r_e^{″}],[{\theta }_e],[{\theta }_e^{\prime }],[{\theta }_e^{″}]\right)=0 \\ {f}_{T_h}\left(X_e,[r_e],[r_e^{\prime }],[r_e^{″}],[{\theta }_e],[{\theta }_e^{\prime }],[{\theta }_e^{″}]\right)=T_{h,e} \end{gathered}$$ (23)

where, the primes(’) denote the derivative of spacecraft states with respect to normalised time, X_e is the vector of design variables in the escape phase which includes the unknown Fourier coefficients, as defined previously in Eq. (20), and T_h,e is the thrust in the escape phase. It is noted here that the number of Fourier coefficients is a design parameter that is problem dependant. It is usually determined on a trial and error basis. This process converts the differential equations of motion into a set of non-linear algebraic equations. A non linear programming (NLP) solver is then used to solve for the design vector (X_e). This NLP solver needs an initial guess for all the design variables. Reference [24] uses Perkin’s approximation, which is detailed in [29], to generate an initial guess trajectory. In this paper, a different approximation is adopted whose solution is demonstrated to be more robust when used in the NLP solver compared to Perkin’s approximation. This approximation is summarized in section V. The above process is applied to each of the segments in the escape spiral.

Design Parameters and Design Variables.

Once the intermediate phase has been designed, the spacecraft states at the escape point become known. This provides four final BCs for the escape phase (r_f,e, θ_f,e, ${\stackrel{.}{r}}_{f,e}$, ${\stackrel{.}{\theta }}_{f,e}$),, where the subscript f denotes the final BCs at escape point, and the subscript e denotes the escape phase. Using the altitude of the parking orbit, three initial BCs can be computed (r_i,e, ${\stackrel{.}{r}}_{i,e}$, ${\stackrel{.}{\theta }}_{i,e}$), where the subscript i denotes the initial BCs. Finally, the initial polar angle θ_i,e is calculated using the final polar angle θ_f,e as shown in Eq.(24)

$${\theta }_{i,e}={\theta }_{f,e}-\Delta {\theta }_e$$ (24)

where Δθ_e is selected based on the initial guess trajectory, detailed in Section V, and is a function of the number of revolutions, N_rev, which depends on the escape flight time. Hence a total of eight BCs (4 initial and 4 final) will be used when generating the escape spiral for a given intermediate phase trajectory. The BCs that connect adjacent segments are referred to as connection points.

The escape phase is divided into (‘m’) segments such that all segments contain approximately equal number of spiral revolutions. The number of segments is a constant for a given problem, and in this work it is usually kept at m = 3. However, for problems requiring larger escape flight time, or higher number of revolutions, ‘m’ is increased to reduce the computational time. The more (smaller) the segments, the less is the computational time. This can be attributed to the fact that smaller segments need fewer design variables per segment, resulting in much smaller matrices to work with. It is observed, however, that increasing the number of segments may result in infeasible solutions due to the increased number of constraints at the connection points between segments, and hence a more complex NLP problem. A visual illustration of escape phase segments is shown in Fig. 5. The FFS approach is applied to each segment. The total number of unknown Fourier coefficients for each segment is 2(n_r + n_θ + 1). Each segment is defined by eight BCs which can be used to solve for some of the unknown Fourier coefficients. This in turn reduces the number of unknown coefficients to 2(n_r + n_θ − 3) per segment.

Fig. 5.

Smooth trajectory and thrust shapes at Connection Points between successive segments

The connection points between successive segments are initially obtained from the initial approximate trajectory. Keeping these connection points fixed while tuning the unknown Fourier coefficients causes a discontinuity in the thrust profile. An example for this discontinuity is shown in Fig. 6. To obtain a smooth thrust profile, the connection points are also included as design variables. Hence, the total number of design variables that define a segment becomes 2(n_r + n_θ − 1). It is emphasized that the initial BCs of the first segment and the final BCs on the last segment are constants that define the initial parking orbit and the escape point. Therefore, the final segment will have 2(n_r + n_θ − 3) design variables, contrary to the rest of the segments. Assuming that the trajectory is divided into ‘m’ segments, the total number of design variables become: 2m(n_r + n_θ − 1) − 4. At the connection points between segments, the following constraints are imposed. The continuity of the trajectory is enforced by using the states of a connection point (r, θ, $\stackrel{.}{r}$, and $\stackrel{.}{\theta }$) between two successive segments as final BCs for the earlier segment, and also as initial BCs for the later segment. The slope of the thrust magnitude at each side of a connection point should be the same. Hence, for two successive segments j − 1 and j, the following condition can be written:

$$\frac{d{T}^{-}}{dt}\equiv M_{f,j-1}=M_{i,j}\equiv \frac{d{T}^{+}}{dt}$$ (25)

where the subscript i denotes initial, the subscript f denotes final, T⁻ is the thrust before the connection point, and T⁺ is the thrust after the connection point. This constraint is implemented as a penalty in the optimization process as described later in this section.

Fig. 6.

Illustration of non-smooth thrust shape when not enforcing continuity at connection points

It is here noted that reference [30] adopted the FFS approach and proposed the concept of dividing the trajectory into two segments, aiming at reducing the fuel consumption and thrust acceleration. The FFS functions are then employed to shape each segment separately. In [30], however, the two segments are separated by a transition orbit. In other words, the first segment transfers the spacecraft from an initial orbit to a transition orbit, and then the second segment transfers the spacecraft from the transition orbit to the target orbit. The transition orbit is selected based on the initial and target orbits. As a result of this scheme [30], the first segment trajectory has to insert the spacecraft into an orbit, and then departing from that orbit in the second segment. Also, this approach is feasible only for orbit transfer problems since the transition orbit is computed based on the initial and target orbits. In the proposed method in this paper, the number of segments is not limited to two. No transition orbit is assumed between any two successive segments. Rather, continuity and smoothness conditions on the trajectory and on the thrust are assumed at the connection points between any two successive segments.

Escape Spiral Trajectory Problem Formulation.

Satisfying the equations of motion and the constraints is achieved by formulating an optimization problem. The design variables are the unknown Fourier coefficients and the connection points. The equations of motion and thrust constraint equation (Eq. (23)) for the escape phase are discretized at ‘p’ points. Hence the optimization problem can be formulated as a minimum mean square error (MMSE) problem where the objective is to minimize the residuals for a set of ‘p’ nonlinear algebraic equations. A penalty term is added to the objective function to penalize any discontinuity in the thrust rate at the connection points. Hence the optimization problem for the escape phase can be formulated as follows:

$$\mathbf{Minimize}J=\sum _{k=1}^n\frac{1}{2}{f}_e^2+\sum _{j=1}^{m-1}(M_{f,j}-M_{i,j-1}{)}^2$$ (26)

$$\mathbf{Subject}\mathbf{to}C:\mid T_{h,e,k}\mid <T_{h,e,max},k=1,\dots p$$ (27)

where,

M_f,j is the thrust magnitude rate of change at the initial point of the j^th segment

M_i,j−1 is the thrust magnitude rate of change at the final point of the (j − 1)^th segment

T_h,e,k is the thrust T_h,e at point k

The design vector X_e includes the following design variables:

X_e = [X_BC,s1, X_RC,s1, X_PC,s1, X_BC,s2, X_RC,s2, X_PC,s2, … , X_RC,sm, X_PC,sm], where,

X_BC,si = [r_f,si, θ_f,si, ${\stackrel{.}{r}}_{f,si}$, ${\stackrel{.}{\theta }}_{f,si}$] are the final BCs of the segment ‘s_i’

X_RC,si = [a_0,si, a_3,si, b_3,si, …, a_nr,sI, b_nr,si] are the position Fourier coefficients of the segment ‘s_i’

X_PC,si = [c_0,si, c_3,si, d_3,si, …, c_nθ,si, d_nθ,si] are the polar angle Fourier coefficients of the segment ‘s_i’

The number of design variables, N_Xe, is (2m(n_r + n_θ − 1) − 4), and T_e is a design parameter. Solving this optimization problem requires initial guess for the design variables; a process for obtaining the initial guess is here described. A classical spiral trajectory approximation is used. This approximate model assumes constant tangential thrust acceleration for the given flight time. Here, we need to determine the tangential thrust acceleration such that the approximate model provides a trajectory that satisfies both initial and terminal BCs. For this purpose, an algorithm that calculates the thrust acceleration is used. This algorithm is explained in the next section. Once we have the approximate escape spiral, the entire phase is divided into ‘m’ smaller segments. Each of these segments are fitted with a FFS to calculate the corresponding values of the Fourier coefficients. Note that some of the Fourier coefficients are already calculated using the BCs to ensure that the terminal conditions are satisfied. So, for example, consider the segment s of the escape phase. Let [r_s,app] be the vector of radius values at n_app discrete points, as obtained from the approximate trajectory. Eq. (28) computes the unknown Fourier coefficients [X].

$$[X]=[A{]}^{-1}\left([r_{s,app}]+[F_r]\right)$$ (28)

where [F_r] includes the Fourier coefficients obtained from the BCs, as defined in Eq. (20). The Fourier coefficients of the polar angle coordinate are obtained in a similar way.

C. Capture Phase

Once the intermediate phase has been determined, the capture point becomes known. This provides four initial BCs for the capture phase (r_i,c, θ_i,c, ${\stackrel{.}{r}}_{i,c}$, ${\stackrel{.}{\theta }}_{i,c}$). It is noted that the capture point obtained at the end of intermediate phase is represented in the ECRF CS, and is transformed to the lunar CS. Using the altitude of the target circular orbit, three final BCs are computed (r_f,c, ${\stackrel{.}{r}}_{f,c}$, ${\stackrel{.}{\theta }}_{f,c}$). Finally the polar angle at the end of the capture spiral is calculated using Eq. (29),

$${\theta }_{i,c}+\Delta {\theta }_c={\theta }_{f,c}$$ (29)

where Δθ_c is computed using the initial guess spiral. This defines a total of eight BCs (4 initial and 4 final) for the capture phase. In a similar way as in the escape phase, by substituting the FFS representation of the states and their derivatives in the motion equations during the capture phase, Eq. (3) and Eq.(4), and in the thrust equation, (Eq. (10)), the following functions can be written:

$$\begin{gathered} f_c\left(X_c,[r_c],[r_c^{\prime }],[r_c^{″}],[{\theta }_c],[{\theta }_c^{\prime }],[{\theta }_c^{″}]\right)=0 \\ {f}_{T_h}\left(X_c,[r_c],[r_c^{\prime }],[r_c^{″}],[{\theta }_c],[{\theta }_c^{\prime }],[{\theta }_c^{″}]\right)=T_{h,c} \end{gathered}$$ (30)

where X_c is a vector of the unknown Fourier coefficients. Following the same process as in the escape phase, the capture spiral trajectory optimization problem is formulated as follows:

$$\mathbf{Minimize}J=\sum _{k=1}^n\frac{1}{2}{f}_c^2+\sum _{j=1}^{m-1}(M_{f,j}-M_{i,j-1}{)}^2$$ (31)

$$\mathbf{Subject}\mathbf{to}C:\mid T_h\mid <T_{h,max}$$ (32)

where,

M_f,j is the thrust magnitude rate of change at the initial point of the j^th segment

M_i,j−1 is the thrust magnitude rate of change at the final point of the (j − 1)^th segment

The design vector X_c includes the following design variables:

X_c = [X_RC, X_PC], where,

X_RC = [a₀, a₃, b₃, …, a_n, b_n] are the position Fourier coefficients

X_PC = [c₀, c₃, d₃, …, c_n, d_n] are the polar angle Fourier coefficients

The number of design variables, N_Xc, is 2(n_r + n_θ − 3), and T_c is a design parameter. Solving this optimization problem requires initial guess for the design variables. The process for computing the initial guess for the Fourier coefficients is the same as that explained above for the escape phase. In the capture phase, there is usually no need to divide the spiral into multiple segments. As a result, the total number of design variables in the capture phase is significantly less than that of the escape phase. A flowchart describing the capture solver is shown in Fig. 7. Similar to the escape phase, the initial guess block uses an approximation method to generate an initial guess for the Fourier Coefficients and the optimization block iterates over these coefficients to generate the trajectory solution.

Fig. 7.

Illustration for the Capture Phase generation process

V. Initial Guess Spiral Approximation

An analytic approximation is developed for escape and capture spirals assuming constant tangential thrust acceleration. This approximation is used to generate an initial guess for this problem. It is noted here that despite the constant thrust acceleration assumption, the R-FFS method imposes the thrust constraint on the thrust level itself, not the thrust acceleration level. Consider the Earth escape spiral, and starting from the equations of motion of the three-body problem with thrust which are given in Eqs. (1) and (2). The thrust is assumed tangential, hence the radial thrust vanishes in Eq. (1). Let the constant tangential thrust acceleration be a_θ = T_a,init. Starting from a circular initial orbit and assuming tangential low thrust, the deviation in the semi-major axis can be assumed very small between adjacent orbital revolutions [31], and it is possible to assume that each one revolution remains almost circular. Hence the radial acceleration is negligible [31]. Eq. (1), then, reduces to the following equation:

$$(1+\stackrel{.}{\theta }{)}^2=\frac{1-\mu }{r^3}+\frac{\mu cos(\theta )}{r}+\frac{\mu }{r}\left(\frac{r-cos(\theta )}{r_{Moon-S∕C}^3}\right)$$ (33)

where the subscript e has been dropped here for simplicity. The first term on the right hand side is about four times order of magnitude compared to the second term which in turn is about one order of magnitude larger than the third term. Hence, the last two terms on the right hand side are neglected. Hence we can write:

$$\frac{d\theta }{dt}\approx \sqrt{\frac{1-\mu }{r^3}}-1\Rightarrow \frac{d^2\theta }{d{t}^2}\approx -\frac{3}{2}\sqrt{\frac{1-\mu }{r^5}}\frac{dr}{dt}$$ (34)

Substituting Eq. (34) in Eq. (2), we get:

$$\frac{3}{2}r\sqrt{\frac{1-\mu }{r^5}}\frac{dr}{dt}+2\stackrel{.}{r}\sqrt{\frac{1-\mu }{r^3}}-\mu sin(\theta )\left(1-\frac{1}{r_{Moon-S∕C}^3}\right)=T_{a,init}$$ (35)

When considering Earth escape spirals, the distance $r_{Moon-S∕C}^3$ becomes very close to D which is unity in the canonical unit system, and μ = 0.012155; hence the last term on the left hand side can be dropped compared to the first two terms:

$$\frac{1}{2}\sqrt{\frac{1-\mu }{r^3}}\frac{dr}{dt}=T_{a,init}$$ (36)

Integrating from initial time to arbitrary time gives:

$${\int }_{r_0}^r\frac{dr}{r^{3∕2}}\approx {\int }_{t_0}^t\frac{2T_{a,init}}{\sqrt{1-\mu }}dt\Rightarrow r\approx \frac{r_0}{{\left(1-T_{a,init}t\sqrt{r_0∕(1-\mu )}\right)}^2}$$ (37)

Differentiating Eq. (37), we can obtain an equation for the radial velocity:

$$\stackrel{.}{r}=\frac{2T_{a,init}{r}_0\sqrt{r_0∕(1-\mu )}}{(1-T_{a,init}t\sqrt{r_0∕(1-\mu )}{)}^3}$$ (38)

It is possible to obtain an expression for the polar angle θ by integrating Eq. (34) above to obtain:

$$\theta =\left(\sqrt{\frac{1-\mu }{r_0^3}}-1\right)t-\frac{3}{2}\left(\frac{T_{a,init}}{r_0}\right)t^2+\left(\frac{T_{a,init}^2}{\sqrt{r_0(1-\mu )}}\right)t^3-\frac{1}{4}\left(\frac{T_{a,init}^3}{(1-\mu )}\right)t^4$$ (39)

The above equations provide analytic expressions for r, $\stackrel{.}{r}$, θ, and $\stackrel{.}{\theta }$ which are used to generate the initial guess spirals. These expressions are close to those obtained in [31] when making the same assumptions on a two-body dynamic model with thrust. Specifically reference [31] obtains Eq. (37) but replacing $\sqrt{r_0∕(1-\mu )}$ by 1/v₀. The polar angle approximation in [31] is given by:

$$\theta =\frac{v_0}{r_0}t-\frac{3}{2}\left(\frac{T_{a,init}}{r_0}\right)t^2+\left(\frac{T_{a,init}^2}{r_0{v}_0}\right)t^3-\frac{1}{4}\left(\frac{T_{a,init}^3}{r_0{v}_0^2}\right)t^4,$$ (40)

which is different from the three-body approximation in Eq. (39). Fig. 8 shows a comparison between two spirals; the 2B is a spiral generated using the two-body approximation in [31], and the 3B is the spiral generated using the three-body approximation presented above. Using the same process, approximations for the capture phase spiral can be written.

Fig. 8.

The 2B spiral uses a two-body model with thrust [31] and the 3B uses a three-body model with thrust

The above three-body approximation provides a good initial guess trajectory when using the R-FFS SB method. Below is the algorithm for generating the initial guess trajectory that assures satisfaction of: the thrust constraint, the initial and final position BCs, in the initial guess spiral, and also provides a good estimate for the flight time. The velocity BCs are not necessarily satisfied on this initial guess spiral. Given the Time of Flight (t_f), the algorithm first checks whether it is possible for the spacecraft to reach the target position without violating the thrust limit. This is done using Eq. (37) by substituting t_f for t, and r_f,e for r, then solving for T_a,init, where T_a,init is thrust acceleration on the initial guess spiral. If the obtained thrust acceleration is not feasible (greater than the given maximum thrust acceleration T_a,max), then the algorithm sets the t_f to be the least possible t_f for which a feasible thrust acceleration exists. If the input t_f is too large, then the obtained solution will have higher number of revolutions and may be computationally expensive in the FFS computation stage. So, an additional condition is added in which if the resulting T_a,init is below a threshold (e.g. 10% of T_a,max in this paper) then a new t_f is selected that corresponds to the thrust acceleration minimum threshold. This algorithm is shown below, where v_i,e is the velocity on the departure orbit, r_i,e is the radius of the departure orbit, and r_f,e is the radius at the final point of the escape spiral.

Algorithm 1: Calculating Thrust Acceleration

$$\begin{array}{cc}\hfill & \mathrm{Require}::T_{a,init}\hfill \\ \hfill & \mathrm{Ensure}::t_f\text{is feasible}\hfill \\ \hfill & T_{a,init}=\left(\frac{v_{i,e}}{t_f}\right)\left(1-\sqrt{\frac{r_{i,e}}{r_{f,e}}}\right)\hfill \\ \hfill & \mathbf{if}{T}_{a,init}>T_{a,max}\mathbf{then}\hfill \\ \hfill & T_{a,init}=T_{a,max}\hfill \\ \hfill & t_f=\left(\frac{v_{i,e}}{T_{a,init}}\right)\left(1-\sqrt{\frac{r_{i,e}}{r_{f,e}}}\right)\hfill \\ \hfill & \mathbf{else}\mathbf{if}{T}_{a,init}<0.1T_{a,max}\mathbf{then}\hfill \\ \hfill & T_{a,init}=0.1T_{a,max}\hfill \\ \hfill & t_f=\left(\frac{v_{i,e}}{T_{a,init}}\right)\left(1-\sqrt{\frac{r_{i,e}}{r_{f,e}}}\right)\hfill \\ \hfill & \mathbf{else}\hfill \\ \hfill & \text{Do Nothing}\hfill \\ \hfill & \mathbf{end}\mathbf{if}\hfill \end{array}$$

It is noted that in previous FFS SB developments, the convergence of the algorithm to a feasible trajectory was a challenge, in some test cases, and sometimes many iterations would be needed to tune the times of flight for each phase. In this work, this problem is eliminated when using the above approximation.

VI. Numerical Results

Several test cases for Earth-Moon transfers are presented in this section; these cases represent various initial and final orbits. In all the test cases, canonical units for distance and time are used. The distance unit (DU) is chosen to be 384,400 km, the average distance between Earth and Moon, and the time unit is equal to 4.3424 Earth days. The mass ratio of the three-body Earth-Moon system is μ = 0.01215. Each of the Earth and Moon is assumed to be a perfect sphere with a radius of 6378 km and 1637 km, respectively. Furthermore, the Moon is assumed to be on a circular orbit around the Earth with a constant angular velocity of ω = 1 rad/TU. In general, the thrust-to-weight ratio ranges from low (order of 10⁻⁵), to medium (order of 10⁻⁴), to high (order of 10⁻³) [32]. The test cases presented in this paper span a range of thrust-to-weight-ratios in the medium and high ranges. All the calculations were implemented in Matlab. The Matlab ‘fmincon’ function is used to solve the NLP problem in the escape and capture phases. The Matlab function ‘fminsearch’ is used to solve A sequential quadratic programming algorithm is used, and the maximum iterations is set to be 5000 when using fmincon. The following sections discuss in details of the various test cases.

A. Case 1: Coplanar Transfer From LEO to LLO

In the first test case, it is required to design a low thrust trajectory departing from a circular Low Earth Orbit (LEO) at an altitude of h_i = 315 km to a circular Low Lunar Orbit (LLO) of altitude h_f = 100 km. This problem is solved in [24] and is solved here using the R-FSS for comparison. The parameters of the spacecraft are defined as follows: specific impulse Isp is 3100 seconds, initial spacecraft mass is 75 kg, thrust-to-weight ratio is 3 × 10⁻³ and maximum thrust limit is 2.64 N. The design parameters including the number of Fourier terms and the number of points per revolution of the escape and capture phases which are as shown in Table 1. The trajectory generated using the R-FFS is shown in Fig. 9, where the intermediate phase trajectory is plotted using dash lines. The variation of the spacecraft mass is shown in Fig. 10. The times of flight of each phase are as follows: T_e = 5.10 days, T_b = 9.20 days and T_c = 0.73 days. In the escape phase, the spacecraft performs 25 revolutions around the Earth gradually spiraling out until it reaches an altitude of 131,972 km. Upon reaching this altitude, the spacecraft coasts for 6.7 days through the intermediate phase until it reaches the Lagrange point L1. The spacecraft then coasts for another 2.5 days before it starts the powered capture phase.

Table 1.

Case 1 Design Parameters for R-FFS method

Parameter	Escape Phase	Capture Phase
nr	6	4
nθ	6	8
pr	10	10
No. of Segments	2	-
Revolutions/Segment	20	-

Fig. 9.

Trajectory Depicted in ECRF CS

Fig. 10.

History of mass

This same test case was solved using the FFS method in [24]. The parameters used to reproduce this case study are listed in Table 2, where m_m is the total number of discretization points on the intermediate phase. It is highlighted here that fewer Fourier terms are needed by R-FFS (total 50 terms) when compared to the FFS (total 90 terms); this impacts the computational cost as discussed below. The thrust profiles obtained using the FFS method are plotted along with the thrust profiles of the R-FFS method in Fig. 11, Fig. 12, and Fig. 13 for comparison, where the red horizontal lines are the thrust magnitude limits. The horizontal axis uses normalized time (t/t_f) since the time of flight is different in each method. In Fig. 11, the vertical line at normalized time 0.4 is the time of starting the second segment in the escape phase. It is clear that the R-FSS thrust is smoothly transitioning between the two segments.

Table 2.

Case 1 Design Parameters for FFS method

Parameter	Escape Phase	Intermediate Phase	Capture Phase
nr	6	10	4
nθ	8	10	6
pr	10	mm = 20	10

Fig. 11.

Case 1: Thrust Profile - Escape Phase

Fig. 12.

Case 1:Thrust Profile Intermediate Phase

Fig. 13.

Case 1: Thrust Profile - Capture Phase

The main highlights in comparing the thrust profiles of the two methods are: (1) the FFS constrains the thrust acceleration, and hence the thrust is not fully utilized towards the end of each of the escape and capture phases when the spacecraft mass has dropped due to fuel consumption. (2) the thrust profile of the R-FFS method shows more oscillations. (3) The intermediate phase needs no thrust in the R-FFS method, while requiring thrust in the FFS method. The capture phase starts with a positive thrust, then switches to negative thrust in the R-FFS method, while it is negative over the whole capture phase in the FFS method; this can be explained as follows. The intermediate phase in the R-FFS method uses no thrust, leveraging the gravitational forces from the two primaries. Upon starting the capture phase, positive thrust is first used to adjust the spacecraft states before starting the deceleration. This is not needed in the FFS method since this adjustment was carried out during the powered intermediate phase.

Table 3 provides a comparison between the solutions obtained using the R-FFS and the FFS methods. In terms of the time of flight, the total time of flight of the R-FFS (14.93 days) is longer than that of the FFS method (7.6355 days). This is expected since the intermediate phase in the R-FFS method is designed not to use any thrust. It is clear in the table that a significant difference in t_f is due to the intermediate phase. The escape and capture phases also need more flight time in this solution of the R-FFS; this is due to the fact that the R-FFS algorithm automatically adjusts the time of flight in order to achieve a feasible solution from a first iteration. This, on the other hand, has a significant impact on robustness and computational cost as discussed here. The R-FFS algorithm for both the escape and capture phases receives the t_f of the respective phase as an input, and uses an approximate method to compute the needed thrust; if the needed thrust violates the given thrust constraint, then it adjusts the time of flight accordingly; no iterations are needed in this adjustment process. While this may result in longer t_f at times, such as the one presented in this case, it eliminate the need for the outer loop in the previous FFS method that iterates on the t_f until a feasible one is found. We tried a set of 100 different t_f inputs to this problem; the percentage of the number of times the previous FFS method could find a feasible solution is 24%, while the R-FFS method could fine a feasible solution 67% of the times, as shown in Table 3. The computational time needed by the R-FFS to generate this solution is 12.5 seconds, while it is 205 seconds when using the FFS method. This saving on the computational time is due to two aspects. First, the outer loop of the FFS method is eliminated in the R-FFS method. Second, the fewer number of Fourier terms (hence optimization variables) in the R-FFS method. The computational time of the FFS method was computed as average over 100 runs. All cases were simulated using Intel(R) Xeon(R) Quadcore Processor @ 3.50GHz. Table 3 also shows the propellant mass needed by each method, in each phase, as well as the ΔV. As could be expected, the longer time of flight solution of the R-FFS method consumes less propellant mass.

Table 3.

Case 1: Comparison of the R-FFS and the FFS methods

Parameter	FFS Method	R-FFS Method
Percentage of finding feasible trajectories from 100 different design	24%	67%
Time of Flight (days)	Te = 2.2557	Te = 5.10
	Tb = 4.9512	Tb = 9.20
	Tc = 0.4286	Tc = 0.73
Total Time of Flight (days)	7.6355	14.93
ΔV (km/s) of each phase	ΔVe = 5.4273	ΔVe = 6.5053
	ΔVb = 2.1441	ΔVb = 0.0000
	ΔVc = 1.1427	ΔVc = 2.0359
Total ΔV (km/s)	8.71	8.5411
Propellant Consumption (kg) in each phase	Mp,e = 12.87	Mp,e = 14.46
	Mp,b = 4.45	Mp,b = 0.00
	Mp,c = 2.24	Mp,c = 3.9
Total propellant consumed (kg)	19.56	18.36
Computational Time (seconds)	205	12.5

B. Case 2: Transfer From a Prograde GEO to a Retrograde HLO (6710 km)

This is a three-dimensional Earth-Moon transfer. A low-thrust trajectory from an initial circular Earth orbit of altitude h_i = 35863 km and an inclination of 0 degrees to a final circular Lunar orbit (LLO) of altitude h_f = 6710 km and an inclination of 90 degrees is studied in this section. In this case study, the parameters of the spacecraft are defined as follows: specific impulse Isp is 3300 seconds, initial spacecraft mass is 975 kg, thrust-to-weight ratio is 2.76 × 10⁻⁴ and maximum thrust limit is 2.64 N. The number of Fourier terms and the points per revolution of the escape and capture phases are shown in Table 4.

Table 4.

Case 2 Design Parameters

Parameter	Escape Phase	Capture Phase
nr	6	4
nθ	6	8
pr	10	10
No. of Segments	2	-
Revolutions/Segment	3	-

The trajectory solution provided by the R-FFS method requires 108.38 kg of propellant to perform this transfer in 28.3 days; the total ΔV is 3.517187 km/s. Fig. 14 shows the transfer trajectory. The spacecraft performs five revolutions in the escape phase and four revolutions in the capture phase. The times of flight of the escape, intermediate and capture phases are T_e = 10.70 days, T_b = 8.6 days and T_c = 9 days, respectively. The corresponding thrust profiles of the escape and capture phases are illustrated in Fig. 16. The capture phase thrust starts with an acceleration period before deceleration; this can be attributed to the fact the intermediate phase is thrust free and a correction is needed at beginning of capture. Table 5 provides details of the obtained solution. Finally, the FFS method cannot generate this 3D solution, and hence is not compared to the R-FSS in this case.

Fig. 14.

Case 2: Trajectory depicted in ECRF CS

Fig. 16.

Case 2: Thrust Profile

Table 5.

Case 2: Characteristics of R-FFS Solution

Parameter	R-FFS Method
No. of Feasible Trajectories generated for every 100 design scenarios	64
Time of Flight (days)	Te = 10.70
	Tb = 8.6
	Tc = 9
ΔV (km/s) of each phase	ΔVe = 1.6934
	ΔVb = 0.0000
	ΔVc = 1.8237
Total ΔV (km/s)	3.517187
Propellant Consumption (kg) in each phase	Mp,e = 58.82
	Mp,b = 0.00
	Mp,c = 53.32
Total propellant consumed (kg)	112.14
Computational Time (sec)	5.1

C. Case 3: Transfer From LEO Prograde (800 km) to LLO Prograde (100 km)

This is also a 3D case, but now from a LEO to a LLO. The departure circular Earth orbit altitude h_i is 800 km and an its inclination is 0 degrees. The capture circular Lunar orbit altitude h_f is 100 km and its inclination is 90 degrees. The parameters of the spacecraft are defined as follows: specific impulse Isp = 3300 seconds, initial spacecraft mass = 74 kg, thrust-to-weight ratio = 3.3 × 10⁻³ and maximum thrust limit = 2.64 N. Two segments are used in the escape phase; the number of Fourier terms and the points per revolution of the escape and capture phases are listed in Table 6.

Table 6.

Case 3 Design Parameters

Parameter	Escape Phase	Capture Phase
nr	6	4
nθ	6	8
pr	6	10
Revolutions/Segment	30	-

The trajectory solution provided by the R-FSS requires 23.768565 kg of propellant and 18.325 days. The total ΔV is 11.298635 km/s. Fig. 17 displays the obtained transfer trajectory. The spacecraft performs N_rev,e = 33 revolutions in the escape phase and N_rev,c = 6 revolutions in the capture phase. The times of flight of the escape, intermediate and capture phases are T_e = 7.45 days, T_b = 8.6 days and T_c = 1.6 days, respectively. The corresponding thrust profiles of the escape and capture phases are illustrated in Fig. 19 and Fig 20. It is observed that there is an acceleration period at the beginning; this is because the intermediate phase is thrust free and a correction is needed at the beginning of capture. Table 7 provides the details of the obtained solution.

Fig. 17.

Case 3: Trajectory depicted in ECRF CS

Fig. 19.

Case 3: Thrust Profile - Escape Phase

Fig. 20.

Case 3: Thrust Profile - Capture Phase

Table 7.

Case 3: Characteristics of R-FFS Solution

Parameter	R-FFS Solution
No. of Feasible Trajectories generated for every 100 design scenarios	50
Time of Flight (days)	Te = 7.45
	Tb = 8.6
	Tc = 1.6
ΔV (km/s) of each phase	ΔVe = 6.5234
	ΔVb = 0.0000
	ΔVc = 4.7752
Total ΔV (km/s)	11.2986
Propellant Consumption (kg) in each phase	Mp,e = 15.10
	Mp,b = 0.0000
	Mp,c = 8.69
Total propellant consumed (kg)	23.77
Computation Time (sec)	9.7

D. Case 4: Transfer From a Prograde GEO to a Lyapunov Halo Orbit around L1

This is a two-dimensional trajectory that aims to insert the spacecraft into a Lyapunov Halo orbit. This type of problem was not addressed in the literature using previous shape-based methods, to the best of the authors’ knowledge. The approach used in this case study is based on the fact that there are manifolds to Halo orbits, and once a spacecraft arrives onto a manifold with the correct state (position and velocity) it asymptotically transfers onto the lunar halo orbit [33]. Hence, to generate the solution trajectory using the R-FFS method, the trajectory is split into two phases: the escape (bridge) phase and the manifold phase. The manifold phase is thrust-free. The bridge phase is an escape spiral that takes the spacecraft from the low Earth departure orbit, and inserts it into the manifold. To implement that, the arrival point on the Lagrange Halo orbit is selected and is propagated backward in time for the selected time of flight for the manifold phase. The spacecraft states at the end of propagation period becomes the end states for the escape spiral phase. The escape phase is then solved using the R-FFS escape spiral method. Here a case study from [32] is adopted where the initial orbit is a circular Geostationary orbit. The parameters of the spacecraft are as follows: specific impulse Isp = 3100 seconds, initial spacecraft mass = 72 kg, thrust-to-weight ratio is 1.3 × 10⁻⁴, and the maximum thrust limit is 0.092 N (NASA N-STAR Thruster). The Lyapunov Halo Orbit period 2.936109 TU, and the corresponding Jacobi Constant is 2.7173. The insertion point on this Lyapunov Halo orbit is selected to be [x, y, z] = [0.822127, −0.001482,0] DU, and [$\stackrel{.}{x}$, $\stackrel{.}{y}$, $\stackrel{.}{z}$] = [0.004921, 0.250601, 0] DU/TU, in the ECRF. The number of Fourier terms and the points per revolution of the escape phase are given in Table 8.

Table 8.

Case 4 Design Parameters

Parameter	Escape Phase
nr	6
nθ	6
pr	10
No. of Segments	3
Revolutions/Segment	3

The obtained solution trajectory using the R-FFS method is shown in Fig. 21 where the total number of revolutions is 8. The change in the spacecraft mass over time is shown in Fig. 22. The thrust profile during the escape phase is shown in Fig. 23, where the red lines are the thrust magnitude limits. The same case study was solved again, but assuming a LEO initial orbit of h_i = 315 km. The maximum thrust limit is updated to 2.42 N. The obtained solution has 25 revolutions during the escape phase. The characteristics of the obtained solutions for both cases are listed in Table 9.

Fig. 21.

Case 4: Trajectory depicted in ECRF CS

Fig. 22.

Case 4: History of Mass

Fig. 23.

Case 4: Thrust Profile - Escape Phase

Table 9.

Case 4: Characteristics of R-FFS Solution

Parameter	Initial GEO	Initial LEO
No. of Feasible Trajectories generated for every 100 design scenarios	84	86
Time of Flight (days)	Te = 14	Te = 4.75
	Tm = 2.6	Tm = 2.5
Total ΔV (km/s)	1.559233	6.1298
Propellant mass (kg)	3.616	14.18
Computation time (seconds)	88.0	27.1

E. Case 5: HEO (20000 km)- Southern Halo Orbit around L1

This is a 3D transfer to a Southern Halo orbit around the Lagrange point L1. This problem is discussed in [34] and is here produced using the R-FFS method. Because this is a 3D problem, the time of flight of the manifold phase has to be such that the direction of the spacecraft angular momentum vector at beginning of manifold phase is the same as the direction normal to the departure orbit. This is required in order to be able to have a planar spiral in the escape phase. To implement that, an optimization problem is formulated. The design variables in this optimization problem are: the coordinates of the insertion point on the Halo orbit and the time of flight of the thrust-free manifold phase. The optimization objective is the square of the error between the direction of the normal to the departure orbit and the angular momentum vector direction at beginning of manifold phase. In this case study, the initial circular Earth orbit altitude h_i is 20,000 km and its inclination is 28.5 degrees. The parameters of the spacecraft are defined as follows: specific impulse Isp is 3100 seconds, initial spacecraft mass is 72 kg, thrust-to-weight ratio is 1.3 × 10⁻⁴ and maximum thrust limit is 0.092 N. The number of Fourier terms and the points per revolution of the escape phase are listed in Table 10. The total number of revolutions in the escape spiral is 16. The Jacobi Constant of the Halo orbit is 3.0036 TU and its period is 2.4902431159. The states at the insertion point are [x, y, z] = [0.8662278472, 0, −0.1796085841] DU, and [$\stackrel{.}{x}$, $\stackrel{.}{y}$, $\stackrel{.}{z}$] = [0,0.2600342141, 0] DU/TU in the ECRF.

Table 10.

Case 5 Design Parameters

Parameter	Escape Phase
nr	6
nθ	6
pr	10
No. of Segments	2
Revolutions/Segment	10

Fig 24 shows the obtained trajectory where the spacecraft performs 16 revolutions around Earth before escape. The time of flight of the escape phase is 18.6 days, and that of the manifold phase is 45.862 days. The corresponding ΔV of the transfer is 1.49 km/s. The mass of the spacecraft is shown in Fig. 25. Total propellant mass consumed is 3.671 kg. The thrust profile is shown in Fig. 26 and the performance characteristics of the obtained solution is shown in Table 11.

Fig. 24.

Case 5: Trajectory in ECRF CS

Fig. 25.

Case 5: History of Mass

Fig. 26.

Case 5: Thrust Profile - Escape Phase

Table 11.

Case 5: Characteristics of The R-FFS Solution

Parameter	R-FFS Solution
No. of Feasible Trajectories generated for every 100 design scenarios	77
Time of Flight (days)	Te = 18.6
	Tm = 45.862
Total ΔV (km/s)	1.49
Total propellant consumed (kg)	3.671
Computation time (seconds)	32

F. Case 6: LEO (315km)- Southern Halo Orbit around L1

This is the same case presented above, but with a different spacecraft initial mass. The Fourier terms and the points per revolution of the escape phase are as shown in Table 12.

Table 12.

Case 6 Design Parameters

Parameter	Escape Phase
nr	6
nθ	6
pr	10
No. of Segments	2
Revolutions/Segment	10

Here the maximum thrust is 2.6 N, the thrust-to-weight ratio is 3.6 × 10⁻³ and the initial mass of the spacecraft is 74 kg. The rest of the parameters including the Halo orbit and the insertion point are the same as in the previous case.

Figure 27 shows the obtained trajectory. Here the total number of revolutions in the escape spiral is only 13. The escape phase was divided to only 2 segments; the first segment has 10 revolutions. The computational time is significantly reduced to 28.8 seconds. The history of mass change is shown in Fig. 28. Total propellant mass consumed is 10.603 kg. The thrust profile during the Earth escape is shown in Fig. 29. The characteristics of the obtained solution are listed in Table 13. It is noted here that robustness measure for this case is 97%; that is the percentage of cases producing feasible solutions when 100 different designs are attempted.

Fig. 27.

Case 6: Trajectory depicted in ECRF

Fig. 28.

Case 6: History of Mass

Fig. 29.

Case 6: Thrust Profile - Escape Phase

Table 13.

Case 6: Characteristics of The R-FFS Solution

Parameter	R-FFS Solution
No. of Feasible Trajectories generated for every 100 design scenarios	97
Time of Flight (days)	Te = 2.6
	Tm = 45.8616
Total ΔV (km/s)	4.611
Total propellant consumed (kg)	10.6

Conclusions

The R-FFS method presented in this paper demonstrated the ability to rapidly generate low thrust Earth-Moon trajectories that satisfy constraints on the maximum thrust level. The solutions presented in this paper are characterized by not using fuel in the intermediate phase or in the manifold phase. This is leveraged to achieve all the required plane change in Earth-Moon trajectories, thrust-free, during the intermediate phase or the manifold phase. The initial guess analytic approximation is used in an algorithm to directly compute a feasible guess for the flight time of the escape (or capture) phase. The R-FFS method could generate solutions for transfers to Halo orbit, Low Lunar orbits, and High Lunar Orbits.

Fig. 15.

Case 2: Mass Variation

Fig. 18.

Case 3: History of Mass

Nomenclature

r

Radius (km)

θ

Polar Angle (rad)

α

Thrust Steering Angle (rad)

γ

Velocity Vector Direction (rad)

μ

Gravitational Parameter (m³/s²)

m_p

Mass of Propellant (kg)

m ₀

Initial Mass (kg)

g ₀

Gravitational Acceleration (m/s²)

I_sp

Specific Impulse (s)

ΔV

Delta-V (km/s)

T_a

Thrust Acceleration Magnitude (m/s²

T_h

Thrust Magnitude (N)

a_i, b_i, c_i, d_i (i = 1, …n)

Fourier Coefficients

T

Total Time (days)

t

Normalised Time

Subscripts

c

Capture

e

Escape

b

Ballistic

Moon-s/c

Moon to Spacecraft

Earth-s/c

Earth to Spacecraft