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A novel methodology is proposed for designing low-thrust trajectories to quasi-periodic, near rectilinear Halo orbits that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs. The proposed methodology generates end-to-end, eclipse-conscious, fuel-optimal transfers in an ephemeris model using an...

## Citations

... The methodology used in the paper to generate D is based on the work of Izzo et al. 9 on "backward generation of optimal examples". A major stalling point in terms of database generation is that repeated solutions of hundreds or thousands of optimal control problems with underlying nonlinear dynamics is a tedious task even with the most state-of the art continuation and homotopy tools accessible to mission designers as shown in some previous investigations [32][33][34][35] . Spacecraft motion re-planning as well as autonomous guidance using learning methods are applications that require predictions with a high-level of accuracy. ...

A supervised stochastic learning method called the Gaussian Process Regression (GPR) is used to design an autonomous guidance law for low-thrust spacecraft. The problems considered are both of the time- and fuel-optimal regimes and a methodology based on “perturbed back-propagation” approach is presented to generate optimal control along neighboring optimal trajectories which form the extremal bundle constituting the training data-set. The use of this methodology coupled with a GPR approximation of the spacecraft control via prediction of the costate n-tuple or the primer vector respectively for time- and fuel-optimal trajectories at discrete time-steps is demonstrated to be
effective in designing an autonomous guidance law using the open-loop bundle of trajectories to-go. The methodology is applied to the Earth-3671 Dionysus time-optimal interplanetary transfer of a low-thrust spacecraft with of-nominal thruster performance and the resulting guidance law is evaluated under different design parameters using case-studies. The results highlight the utility and applicability of the proposed framework with scope for further improvements.

... As this is a geocentric case, the effect of J 2 and eclipses are also considered. Similarly to [38], the effect of eclipses is accounted for by switching the propulsion system off without deriving the midpoint boundary conditions. ...

This note develops easily applicable techniques that improve the convergence and reduce the computational time of indirect low thrust trajectory optimization when solving fuel- and time-optimal problems. For solving fuel optimal (FO) problems, a positive scaling factor -- $\Gamma_{TR}$ -- is introduced based on the energy optimal (EO) solution to establish a convenient profile for the switching function of the FO problem. This negates the need for random guesses to initialize the indirect optimization process. Similarly, another scaling factor-$\beta$-, is introduced when solving the time-optimal (TO) problem to connect the EO problem to the TO. The developed methodology for the TO problem was crucial for the GTOC11 competition. Case studies are conducted to validate the solution process in both TO and FO problems. For geocentric cases, the effect of eclipses and $J_2$ perturbations were also considered. The examples show that EO can provide a good guess for TO and FO problems and that introducing the constants can reduce the initial residuals and improve convergence. It is also shown that the equation for the Lagrangian multiplier of mass and the associated boundary condition can be ignored for both FO and TO cases without affecting optimality. This simplification reduces the problem dimensions and improves efficiency.

... The methodology used in the paper to generate D is based on the work of Izzo et al. [9] on "backward generation of optimal examples". A major stalling point in terms of database generation is that repeated solutions of hundreds or thousands of optimal control problems with underlying non-linear dynamics is a tedious task even with the most state-of the art continuation and homotopy tools accessible to mission designers [24][25][26]. Spacecraft motion re-planning as well as autonomous guidance using learning methods are applications that require predictions with a high-level of accuracy. Especially, for on-board re-planning of time-to go trajectories, the predictions provided for the unknown initial ad-joint n-tuple ( ( 0 )) should preferably converge in a low-single digit number of iterations. ...

A supervised stochastic learning method called the Gaussian Process Regression (GPR) is used to design an autonomous guidance law for low-thrust spacecraft. The problems considered are both of the time- and fuel-optimal regimes and a methodology based on ``perturbed back-propagation'' approach is presented to generate optimal control along neighboring optimal trajectories which form the extremal bundle constituting the training data-set. The use of this methodology coupled with a GPR approximation of the spacecraft control via prediction of the costate \textit{n}-tuple or the primer vector respectively for time- and fuel-optimal trajectories at discrete time-steps is demonstrated to be effective in designing an autonomous guidance law using the open-loop bundle of trajectories to-go. The methodology is applied to the Earth- 3671 Dionysus time-optimal interplanetary transfer of a low-thrust spacecraft with off-nominal thruster performance and the resulting guidance law is evaluated under different design parameters using case-studies. The results highlight the utility and applicability of the proposed framework with scope for further improvements.

... The HTS is also a key component of a novel framework -Composite Smooth Control (CSC) -proposed for solving optimal control problems with discrete and multiple modes of operations [9], [10]. The CSC framework is used for co-optimization of spacecraft propulsion system and its trajectory [11] and for eclipse-conscious cislunar lowthrust trajectory design [12]. The HTS is also used for fueloptimal asteroid landing problems [13]. ...

Indirect formalism of optimal control theory is used to generate minimum-time and minimum-fuel trajectories for formation of two spacecraft (deputies) relative to a chief satellite. For minimum-fuel problems, a hyperbolic tangent smoothing method is used to facilitate numerical solution of the resulting boundary-value problems by constructing a one-parameter family of smooth control profiles that asymptotically approach the theoretically optimal, but non-smooth bang-bang thrust profile. Impact of the continuation parameter on the solution of minimum-fuel trajectories is analyzed. The fidelity of the dynamical model is improved beyond the two-body dynamics by including the perturbation due to the Earth’s second zonal harmonic, J 2 . In addition, a particular formation is investigated, where the deputies are constrained to lie diametrically opposite on a three-dimensional sphere centered at the chief.

... When planning a manifold-aided mission to the Moon, the most important step is to pick target patch points for insertion, which facilitates a ballistic capture transfer to the target through invariant space curves. A number of heuristic methods [6,7,14] for a priori evaluations of patch points have been studied previously. These approaches primarily use osculating element-space phase portraits of the entire set of piercing points, which are defined as the points at which the periodic-orbit manifolds intersect the Earth's plane in the Earth-Moon synodic frame. ...

... Therefore, in this study, the MAE was coupled with the relative error percentage (RE%) as a secondary metric, which quantifies the number of testing points within acceptable bounds around the truth. RE% is defined as (14) and captures the local prediction behavior of discrete testing/prediction points, rather than quantifying a mean measure. A GPR model was trained using uniformly distributed points from the entire domain, as described in Table 3. ...

In this study, a supervised machine learning approach called Gaussian process regression (GPR) was applied to approximate optimal bi-impulse rendezvous maneuvers in the cis-lunar space. We demonstrate the use of the GPR approximation of the optimal bi-impulse transfer to patch points associated with various invariant manifolds in the cis-lunar space. The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solutions of the optimal bi-impulsive Lambert transfer because the learned map is computationally efficient. This approach promises to be useful for aiding in preliminary mission design. The use of invariant manifolds as part of the transfer trajectory design offers unique features for reducing propellant consumption while facilitating the solution of trajectory optimization problems. Long ballistic capture coasts are also very attractive for mission guidance, navigation, and control robustness. A multi-input single-output GPR model is presented to represent the fuel costs (in terms of the ΔV magnitude) associated with the class of orbital transfers of interest efficiently. The developed model is also proven to provide efficient approximations. The multi-resolution use of local GPRs over smaller sub-domains and their use for constructing a global GPR model are also demonstrated. One of the unique features of GPRs is that they provide an estimate of the quality of approximations in the form of covariance, which is proven to provide statistical consistency with the optimal trajectories generated through the approximation process. The numerical results demonstrate our basis for optimism for the utility of the proposed method.

... The 9:2 resonant Near Rectilinear Halo Orbits are extensively analyzed by Zimovan et al. [8], both as possible candidates for the hosting the Gateway and in terms of the transfer possibilities toward other cislunar orbits by McGuire et al. with and without the aid of low-thrust propulsion [9]. Singh et al. [10] investigate eclipse-aware low-thrust transfer strategies to such orbits, proposing a method whose concept resembles the one of this work, leveraging the perturbation effects through the use of high-fidelity analogues of the invariant manifolds of the Circular Restricted Three Body Problem. Other applications also regard pure interplanetary orbits, for instance the ESA/NASA mission A. Masat and C. Colombo On the other side, the Picard-Chebyshev method is a semi-analytical technique to globally integrate the evolution of a generic dynamical system accounting for a generic perturbation source. ...

... Nonetheless, the modified Picard-Chebyshev method has been continuously developed in the past few years, both in its formulation and implementation side and outlining possible applications for Earth orbits where it contributed to increase the efficiency of the numerical analyses. Junkins et al. [21] analyzed the performances of the method comparing the efficiency against the Runge-Kutta-Nystrom 12 (10) integrator, proposing also a second order version. Later, Koblick and Shankar [22] extended the analysis to the propagation of accurate orbits testing difference force models with NASA's Java Astrodynamics toolkit. ...

... Because of the definition of Chebyshev nodes in Eqs.(10) and(11). ...

Orbital resonances have been exploited in different contexts, with the latest interplanetary application being the ESA/NASA mission Solar Orbiter, which uses repeated flybys of Venus to change the ecliptic inclination with low fuel consumption. The b-plane formalism is a useful framework to represent close approaches at the boundaries of the sphere of influence of the flyby planet. In the presented work, this representation is exploited to prune the design of perturbed resonant interplanetary trajectories in a reverse cascade, replacing the patched conics approximation with a continuity link between flybys and interplanetary legs. The design strategy splits the flyby time and state variables in a two-layer optimization problem. Its core numerically integrates the perturbed orbital motion with the Picard-Chebyshev integration method. The analytical pruning provided by the b-plane formalism is also used as starting guess to ensure the fast convergence of both the numerical integration and the trajectory design algorithm. The proposed semi-analytical strategy allows to take advantage of complex gravitational perturbing effects optimizing artificial maneuvers in a computationally efficient way. The method is applied to the design of a Solar Orbiter-like quasi-ballistic first resonant phase with Venus.

... The 9:2 resonant Near Rectilinear Halo Orbits are extensively analyzed by Zimovan et al. [8], both as possible candidates for the hosting the Gateway and in terms of the transfer possibilities toward other cis-lunar orbits by McGuire et al., with and without the aid of low-thrust propulsion [9]. Singh et al. [10] investigate eclipse-aware low-thrust transfer strategies to such orbits, proposing a method whose concept resembles the one of this work, leveraging the perturbation effects through the use of high-fidelity analogues of the invariant manifolds of the Circular Restricted Three Body Problem. Other applications also regard pure interplanetary orbits, for instance the ESA/NASA mission Solar Orbiter [11] as the latest example: resonant trajectories with Venus are exploited to raise the orbital inclination up to almost 30 degrees [12] over the ecliptic, to better observe the near-polar regions of the Sun. ...

... Nonetheless, the modified Picard-Chebyshev method has been continuously developed in the past few years, both in its formulation and implementation side and outlining possible applications for Earth orbits where it contributed to increase the efficiency of the numerical analyses. Junkins et al [21] analyzed the performances of the method comparing the efficiency against the Runge-Kutta-Nystrom 12 (10) integrator, proposing also a second order version. Later, Koblick and Shankar [22] extended the analysis to the propagation of accurate orbits testing difference force models with NASA's Java Astrodynamics toolkit. ...

Orbital resonances have been exploited in different contexts, with the latest interplanetary application being the ESA/NASA mission Solar Orbiter, which uses repeated flybys of Venus to change the ecliptic inclination with low fuel consumption. The b-plane formalism is a useful framework to represent close approaches at the boundaries of the sphere of influence of the flyby planet. In the presented work, this representation is exploited to prune the design of perturbed resonant interplanetary trajectories in a reverse cascade, replacing the patched conics approximation with a continuity link between flybys and interplanetary legs. The design strategy splits the flyby time and state variables in a two-layer optimization problem. Its core numerically integrates the perturbed orbital motion with the Picard-Chebyshev integration method. The analytical pruning provided by the b-plane formalism is also used as starting guess to ensure the fast convergence of both the numerical integration and the trajectory design algorithm. The proposed semi-analytical strategy allows to take advantage of complex gravitational perturbing effects optimizing artificial maneuvers in a computationally efficient way. The method is applied to the design of a Solar Orbiter-like quasi-ballistic first resonant phase with Venus.

... However, direct methods can lead to large-scale NLP problems and still require "good" initial guesses for their performance to be improved. Indirect optimization methods lead to high-resolution solutions (i.e., states and control inputs), which is of importance for certain aerospace problems, e.g., low-thrust trajectory design [14][15][16][17][18][19]. ...

This paper presents an efficient indirect optimization method to solve time-and fuel-optimal asteroid landing trajectory design problems. The gravitational field of the target asteroid is approximated with two methods: 1) a simple two-body (point-mass) model, and 2) a high-fidelity polyhedral model. A homotopy approach, at the level of the gravity-model, is considered to solve the resulting boundary-value problems. Moreover, in fuel-optimal trajectories, the difficulties associated with control discontinuities (i.e., the so-called bang-off-bang profile of thrust) are overcome by employing a recently introduced hyperbolic tangent smoothing method. We generate time-and fuel-optimal trajectories for spacecraft landing on asteroid (101955) Bennu with low-thrust propulsion. The trajectories are validated through comparison with the trajectories obtained by other methods. The results indicate that the two-body gravity model can be used for generating high-quality solutions to "warm-start" numerical methods when a high-fidelity gravity model is used.

... Transfers in the lunar domain, especially involving periodic orbits like NRHOs and DROs among others, have also been studied extensively by various researchers [4,21,43]. In a recent paper, a novel methodology to study eclipse conscious transfers to an 'ephemeris-corrected' 9:2 Southern L 2 NRHO in the Earth-Moon system has been discussed [32]. Recently, polyhedral interpolative techniques were used by Pontani et al. [26] to study and classify orbits in the neighborhood of interior collinear libration points and time-optimal low thrust transfers were solved between a low Earth orbit and a Lyapunov orbit using stable manifolds. ...

In this paper, we investigate the manifolds of three Near-Rectilinear Halo Orbits (NRHOs) and optimal low-thrust transfer trajectories using a high-fidelity dynamical model. Time- and fuel-optimal low-thrust transfers to (and from) these NRHOs are generated leveraging their ‘invariant’ manifolds, which serve as long terminal coast arcs. Analyses are performed to identify suitable manifold entry/exit conditions based on inclination and minimum distance from the Earth. The relative merits of the stable/unstable manifolds are studied with regard to time- and fuel-optimality criteria, for a set of representative low-thrust family of transfers.

... In addition to the aforementioned solution artifices, one can take advantage of the most natural set of coordinates and elements to represent spacecraft motion. The regularity of coordinates and elements chosen to describe the motion of spacecraft has been shown to have a significant impact on the convergence performance of optimization algorithms [33,36,37] and remains an active area of research [38,39]. For more complex low-thrust trajectory design tasks (e.g., transfer from the Earth to the Moon), a common practice is to use dual (or multiple) coordinate frames to achieve enhanced convergence properties [40,41]. ...

Numerical solutions of optimal control problems are influenced by the appropriate choice of coordinates. The proposed method based on the variational approach to map costates between sets of coordinates and/or elements is suitable for solving optimal control problems using the indirect formalism of optimal control theory. The Jacobian of the nonlinear map between any two sets of coordinates and elements is a key component of costate vector mapping theory. A new solution for the class of planar, free-terminal-time, minimum-time, orbit rendezvous maneuvers is also presented. The accuracy of the costate mapping is verified, and its utility is demonstrated by solving minimum-time and minimum-fuel spacecraft trajectory optimization problems. KEYWORDS indirect optimization spacecraft trajectory optimization optimal control theory minimum time minimum fuel interplanetary Earth-centered low thrust Research Article