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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.

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... In the field of astrodynamics, Shang and Liu 25 assessed accessibility of Main-Belt asteroids by predicting the optimal bi-impulsive V costs to rendezvous with the asteroids via a GPR model trained on the family of such transfers to rendezvous states in a domain spanning the known main-belt asteroid region. Singh et al. 26 later demonstrated the effectiveness of a multi-input multioutput GPR in predicting the optimal family of Lambert transfer trajectories for cis-lunar transfers from Earth to a periodic L 1 Halo orbit; leveraging invariant manifolds. ...

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.

... In the field of astrodynamics, Shang and Liu [17] assessed accessibility of Main-Belt asteroids by predicting the optimal bi-impulsive ΔV costs to rendezvous with the asteroids via a GPR model trained on the family of such transfers to rendezvous states in a domain spanning the known main-belt asteroid region. Singh et al. [18] later demonstrated the effectiveness of a multi-input multi-output GPR in predicting the optimal family of Lambert transfer trajectories for cis-lunar transfers from Earth to a periodic 1 Halo orbit; leveraging invariant manifolds. ...

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.

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.

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 indirect formulation of optimal control theory. The end-to-end trajectories are achieved by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of employing such a hybrid optimization algorithm for solving end-to-end analogous fuel-optimal problems using indirect methods and leveraging a composite smooth control construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). A novel process is introduced to incorporate eclipse-induced coast arcs and their impact within optimization. The results quantify accurate Δ costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to find applications for analogous uncrewed missions.

In this work, end-to-end low-thrust transfers from a GTO orbit to a low-altitude lunar orbit by exploiting the manifolds of a chosen Earth-Moon L1 halo orbit was studied. The practicality of piece-wise, minimum-time transfers that exploit halo orbit manifolds is demonstrated, which offers more flexibility to meet mission objectives. It is known that the structure of the manifolds varies substantially due to the presence of the Sun and its contribution has to be considered to obtain more realistic trajectories. To incorporate Sun’s perturbation, we study (1) manifolds’ behavior within a Bi-Circular Problem (BCP) dynamics and (2) Sun’s impact on the previously converged trajectories obtained using the standard Circular Restricted Three-Body Problem (CR3BP). Comparisons of the resulting trajectories using the CR3BP and BCP are presented.

The stable distant retrograde orbits (DROs) around the Moon are considered as potential parking orbits for cis-lunar stations that are important facilities in cis-lunar space. Transfer orbits from DROs to lunar orbits will be fundamental and routine for operations of the cis-lunar stations. This paper studies transfer orbits from DROs to low lunar orbits with inclinations between 0 and 90 degrees. Ten DROs are selected for the construction of transfers. The planar transfer orbits from each DRO to the LLO with zero inclination are firstly obtained and compared in the planar circular restricted three-body problem (PCR3BP) to reveal basic characteristics of the transfer solutions. The planar transfers are classified into several types based on characteristics. Each type is discussed in details, especially their transfer cost and time. Based on the planar transfers, nonplanar transfer orbits are constructed in the circular restricted three-body problem (CR3BP). Some nonplanar transfers are selected and compared to show effects of the LLO inclination. Then, the planar transfer orbits are refined in the planar bicircular 2 restricted four-body problem (PBR4BP) with the gravity of the Sun. The comparison between results in PCR3BP and PBR4BP shows that the gravity of the Sun can increase transfer options and reduce the transfer cost. Further analysis is carried out based on the realistic results in PBR4BP, including the ballistic capture, departure and insertion locations, transfer cost and time, and etc. The results are useful for selecting parking DROs and designing transport systems to the Moon.

Near-Rectilinear Halo Orbits (NRHOs) are deemed to be favorable candidates for establishing a near-future crewed space station in the cis-lunar space. Although the 9:2 resonant southern $L_2$ NRHO has been earmarked as the working orbit for the Lunar Gateway Mission, a plethora of other neighboring resonant NRHOs are also viable options. The invariant manifolds of these periodic orbits provide natural pathways to a state in the vicinity of fixed points on the NRHOs. These manifolds can be leveraged while designing optimal low-thrust trajectories for both `NRHO-bound' and `Earth-bound' missions. In this work, the effects of the ephemeris model (JPLs DE436) on three NRHO manifolds derived based on the Circular Restricted Three Body (CR3BP) assumptions are characterized and presented. The three neighboring NRHOs are then investigated in the domain of the aforementioned mission categories for piece-wise minimum-time and minimum-fuel, low-thrust transfers facilitated by invariant manifolds of the NRHOs. The minimum-time and minimum-fuel trajectory optimization problems are formulated using the indirect formalism of optimal control and solved using a single-shooting solution scheme. The relative merits of the stable manifolds are studied with regard to minimizing either mission time of flight or minimization of fuel consumption, for a set of representative low-thrust family of transfers.

There exists continued interest in building accurate models of wind turbine power curves for better understanding of performance or assessment of the condition of the turbine or both. Better predictions of the power curve allow increased insight into the operation of the turbine, aid operational decision making, and can be a key feature of online monitoring and fault detection strategies. This work proposes the use of a heteroscedastic Gaussian Process model for this task. The model has a number of attractive properties when modelling power curves. These include, removing the need to specify a parametric functional form for the power curve and automatic quantification of the variance in the prediction. The model exists within a Bayesian framework which exhibits built-in protection against over-fitting and robustness to noisy measurements. The model is shown to be effective on data collected from an operational wind turbine, returning accurate mean predictions ( normalised mean-squared error) and higher likelihoods than a corresponding homoscedastic model.

In this paper, neural networks are trained to learn the optimal time, the initial costates, and the optimal control law of time-optimal low-thrust interplanetary trajectories. The aim is to overcome the difficult selection of first guess costates in indirect optimization, which limits their implementation in global optimization and prevents on-board applications. After generating a dataset, three networks that predict the optimal time, the initial costate, and the optimal control law are trained. A performance assessment shows that neural networks are able to predict the optimal time and initial costate accurately, especially a 100% success rate is achieved when neural networks are used to initialize the shooting function of indirtect methods. Moreover, learning the state-control pairs shows that neural networks can be utilized in real-time, on-board optimal control.

In the near future, several space applications in the Earth-Moon system may require a spacecraft to hold a stable motion, but the transfer trajectory infrastructure to access such stable motions has not been fully investigated yet. The triangular libration points, L4 and L5, in the Earth-Moon system have long been thought of as potential locations for a communications satellite. Recently, Distant Retrograde Orbits (DROs) and Near-Rectilinear Halo Orbits (NRHOs) in the vicinity of the Moon have been identified as orbits of interest for manned and unmanned missions with a focus on operations in cislunar space. The triangular libration points, as well as lunar DROs and NRHOs describe special types of possible motion for a spacecraft/satellite that is influenced solely by the gravitational fields of the Earth and the Moon. What is common to the three types of solutions is that they are practically stable, that is, a spacecraft/satellite can naturally follow the solution for extended periods of time without requiring significant course adjustment maneuvers. This investigation proposes the lunar region as the central link to a transfer network that enables travel throughout the Earth-Moon system, connecting the lunar region to the vicinity of the Earth and the neighborhood of the triangular libration points. The work presented here also contributes to the infrastructure supporting such a network by expanding the transfer options available between these regions. Several new transfer options between regions of stability are presented and discussed, including transfer options between Low Earth Orbit (LEO) and lunar DRO, lunar DRO and periodic orbits near L4 and L5, as well as lunar DRO and L2 NRHOs. Underlying dynamical mechanisms enabling transfers between selected orbits are analyzed, and sample itineraries are provided.

Reverse-time migration (RTM) has shown its advantages over other conventional migration algorithms for ground-penetrating radar (GPR) imaging. RTM is preferred to be implemented in the computationally attractive 2-D domain, whereas a real measurement can only be conducted in a 3-D domain. Thus, we propose an asymptotic 3-D-to-2-D data conversion filter in the frequency domain for preprocessing of the recorded data for 2-D RTM. The accuracy of the data conversion filter is verified by two numerical tests on a homogeneous and a layered model. Then, we evaluate the effectiveness of the data conversion filter on the imaging result of 2-D RTM, which is applied to simulated multioffset GPR data from a buried pipe model. With the filter, subsurface image by the 2-D RTM matches better with the 3-D RTM result especially in the aspect of phase congruency. Therefore, we conclude that this data conversion filter is necessary for 2-D RTM. We also conducted a laboratory experiment on a volcanic ash pit using a multiinput–multioutput GPR system, which is adopted on the Chang-E 5 lunar exploration lander and works in a stationary mode. The 3-D-to-2-D data conversion filter is applied to the measured multioffset GPR data before the 2-D RTM. The imaging results demonstrate that three marble slabs buried at different depths up to 2 m are clearly imaged.

We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified Chebyshev-Picard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions, however we show that using modified Chebyshev-Picard iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert’s problem using the method of particular solutions with modified Chebyshev-Picard iteration is about an order of magnitude faster compared with the classical shooting method and a tenth-twelfth order Runge-Kutta integrator. It is well known that the solution to Lambert’s problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.

The low-thrust propulsion will be one of the most important propulsion in the future due to its large specific impulse. Different from traditional low-thrust trajectories (LTTs) yielded by some optimization algorithms, the gradient-based design methodology is investigated for LTTs in this paper with the help of invariant manifolds of \(\mathit{LL}_{1}\) point and Halo orbit near the \(\mathit{LL}_{1}\) point. Their deformations under solar gravitational perturbation are also presented to design LTTs in the restricted four-body model. The perturbed manifolds of \(\mathit{LL}_{1}\) point and its Halo orbit serve as the free-flight phase to reduce the fuel consumptions as much as possible. An open-loop control law is proposed, which is used to guide the spacecraft escaping from Earth or captured by Moon. By using a two-dimensional search strategy, the ON/OFF time of the low-thrust engine in the Earth-escaping and Moon-captured phases can be obtained. The numerical implementations show that the LTTs achieved in this paper are consistent with the one adopted by the SMART-1 mission.

Recent research on deep learning, a set of machine learning techniques able to learn deep architectures, has shown how robotic perception and action greatly benefits from these techniques. In terms of spacecraft navigation and control system, this suggests that deep architectures may be considered now to drive all or part of the on-board decision making system. In this paper this claim is investigated in more detail training deep artificial neural networks to represent the optimal control action during a pinpoint landing, assuming perfect state information. It is found to be possible to train deep networks for this purpose and that the resulting landings, driven by the trained networks, are close to simulated optimal ones. These results allow for the design of an on-board real time optimal control system able to cope with large sets of possible initial states while still producing an optimal response.

A lunar standstill, also named a lunistice in resonance of the solar ‘solstice’ (the Sun standing still), is the moment of the lunar month when the Moon is seen farthest north or south with respect to other positions of that particular swinging motion from a given position on Earth.

In this paper, a technique for the analysis and the design of low-energy interplanetary transfers, exploiting the invariant manifolds of the restricted three-body problem, is presented. This approach decomposes the full four-body problem describing the dynamics of an interplanetary transfer between two planets, in two three-body problems each one having the Sun and one of the planets as primaries; then the transit orbits associated to the invariant manifolds of the Lyapunov orbits are generated for each Sun-planet system and linked by means of a Lambert's arc defined in an intermediate heliocentric two-body system. The search for optimal transit orbits is performed by means of a dynamical Poincaré section of the manifolds. A merit function, defined on the Poincaré section, is used to optimally generate a transfer trajectory given the two sections of the manifolds. Due to the high multimodality of the resulting optimization problem, an evolutionary algorithm is used to find a first guess solution which is then refined, in a further step, using a gradient method. In this way all the parameters influencing the transfer are optimized by blending together dynamical system theory and optimization techniques. The proposed patched conic-manifold method exploits the gravitational attractions of the two planets in order to change the two-body energy level of the spacecraft and to perform a ballistic capture and a ballistic repulsion. The effectiveness of this approach is demonstrated by a set of solutions found for transfers from Earth to Venus and to Mars.

This paper presents the multidisciplinary optimization of an aircraft carried sub-orbital spaceplane. The optimization process focused on three disciplines: the aerodynamics, the structure and the trajectory. The optimization of the spaceplane geometry was coupled with the optimization of its trajectory. The structural weight was estimated using empirical formulas. The trajectory was optimized using a pseudo-spectral approach with an automated mesh refinement that allowed for increasing the sparsity of the Jacobian of the constraints. The aerodynamics of the spaceplane was computed using an Euler code and the results were used to create a surrogate model based on a nonstationary Gaussian process procedure that was specially developed for this study.

Two methods for deriving first-order partial derivatives of the outputs with respect to the inputs of the Lambert boundary value problem are presented. The first method assumes the Lambert problem is solved via the universal vercosine formulation. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed in a computationally efficient form. The typical added cost of computing these partials is found to be ∼15 to 35% of the Lambert computed cost. A second set of the same partial derivatives is derived from the fundamental perturbation matrix, also known as the state transition matrix of the Keplerian initial value problem. The equations are formulated in terms of Battin's partitions of the state transition matrix and its adjoint. This alternative approach works with any Lambert formulation, including one that solves a perturbed Lambert problem, subject to the availability of the associated state transition matrix. The analytic partial derivatives enable fast trajectory optimization formulations that implicitly enforce continuity constraints via embedded Lambert problems.

The Battin, Gooding, and Sun algorithms are able to converge for nearly 100% of all orbit combinations. Both the Gooding and Sun algorithms have approximately the same performance, whereas the Battin algorithm is consistently slower. The performance for each Lambert algorithm, when run on the graphics processing unit (GPU), has been compared. Each Lambert solution algorithm is able to compute tens of millions of solutions per second, with Sun's method having the best performance at nearly 32 million solutions per second. This represents an increase in performance of two orders of magnitude when using GPUs over standard CPU algorithms. Although even the grid search CPU run times are not high when compared with common high-performance computing algorithms, mission optimization algorithms often require run times ranging from hours to multiple days. By offloading computations for solutions to Lambert's problem to GPU(s) and sufficiently parallelizing optimization algorithms, performance increases of up to two orders of magnitude can be realized when compared with standard CPU algorithms. This will allow mission designers to quickly compute complex trajectories when evaluating potential mission architectures.

Resonant orbits have been widely employed in mission design for planetary flyby trajectories (Jupiter Europa Orbiter) and, more recently, as a source of long-term stability (Interstellar Boundary Explorer). Yet, resonant orbits have not been explored extensively as transfer mechanisms between nonresonant orbits in multibody systems. To highlight the benefit of employing resonant orbits for transfers and given the increased interest in employing libration-point orbits for a variety of purposes, planar and three-dimensional transfers from a low Earth orbit to the vicinity of the Earth-moon libration points via resonant arcs are constructed. Solutions are efficiently generated in the circular restricted three-body model, and transitioned to a higher-fidelity model for validation. Direct optimization techniques are applied to further reduce the propellant requirements, and a system translation process is defined to allow for the quick translation of Earth-moon transfers to other three-body systems.

Response surfaces are frequently used in airfoil design due to less resource requirement compared with direct slow simulations. In many cases, multiple responses need to be modeled to achieve multiple objectives. Considering the correlations between multiple responses in modeling the nonlinear relationship between airfoil shapes and aerodynamic performance, the authors construct multiresponse surfaces for airfoil design with multiple-output-Gaussian-process-regression model. The authors simulate computational data to evaluate the prediction accuracy and stability of the multiple-output Gaussian process in airfoil design, compared with other popular alternative approaches, kriging, and backpropagation and radial-basis-function neural networks. In the experiments, response surfaces from the airfoil shapes, parameterized by the class/shape-function-transformation method, to lift, drag, and pitching-moment coefficients are constructed. The results indicate that the multiple-output Gaussian process receives higher prediction accuracy and stability in modeling multiresponse surfaces than other popular methods when there are significant correlations between responses.

In this paper, we combine the multiple-input–multiple-output (MIMO) array antenna technology with a multipolarization component in a ground penetrating radar (GPR) system to improve target detection accuracy. The MIMO technology introduced in previous literature is widely applied in radar and other wireless communication fields. Here, we apply the MIMO technology with a “plane-wave like” (PWL) source that uses array antennas with small spacing to emit a pulse source at the same time in GPR detection. First, we analyze the physical mechanism of the MIMO GPR system with a “PWL” source to improve the target detection resolution. Then, we carry out a numerical simulation with a finite-difference time-domain method in 1-D and 2-D array antennas to compare the imaging results of the MIMO and traditional GPR systems. Finally, the synthetic data MIMO GPR experiment with a step-frequency GPR system is implemented. Compared with the traditional GPR system, our results demonstrate that the MIMO GPR system with a multipolarization detection mode can overcome the influence of target radar cross sections and antenna radiation directions, and improve target detection accuracy effectively. Meanwhile, the synthetic MIMO GPR system also provides a good idea to improve the system performance and reduce system design requirements and the manufacture cost.

Life prediction of bearing is the urgent demand in engineering practice, and the effective bearing degradation assessment technique is beneficial to predictive maintenance. This paper presents an application of an important Bayesian machine learning method named Gaussian Process Regression (GPR) for bearing degradation assessment. The Gaussian Process (GP) model holds many advantages such as easy coding, prediction with probability interpretation and self-adaptive acquisition of hyper-parameters. In this study, the GPR model with different kinds of covariance functions is applied for assessment of bearing state of health (SOH). Two common covariance functions and a composite covariance function of GPR which is obtained by additive single standard covariance functions are discussed. The dynamic model is introduced to realize a better assessment by analyzing some important features. From the experimental results, it can be concluded that using GPR model for prognosis can achieve a high performance, and the composite covariance function can improve the prediction precision. In addition, compared with wavelet neural network (WNN), GPR model shows more excellent features. So the purposed model can be utilized in bearing degradation analysis, and meanwhile can serve as a reference for similar data-mining projects.

In minimum-fuel impulsive spacecraft trajectories, long-duration coast
arcs between thrust impulses can occur. If the coast time is long enough
to allow more than one complete revolution of the central body, the
solution becomes complicated. Lambert's Problem, which is the
determination of the orbit, given the terminal radius vectors and the
transfer time, has a multiplicity of solutions. For a transfer time long
enough to allow N revolutions of the central body there exist 2N + 1
trajectories which satisfy the boundary value problem. An algorithm
based on the classical Lagrange formulation for an elliptic orbit is
developed and demonstrated which determines all the trajectories.

Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincar and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the Sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the Earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32 379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a halo orbit of the Moon.

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium pointL
1 of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total v required, the figures obtained are similar to the ones given by the standard procedures of optimization.

The combination of low-thrust propulsion and gravity assists to enhance deep space missions has proven to be a formidable task. While trajectories generated by methods based on optimal control theory are typically close to the required initial guess, recently investigated global evolutionary programming techniques often necessitate the successive use of different methods. In this paper, we present a new method that is based on evolutionary neurocontrollers. The advantage lies in its ability to explore the solution space autonomously
to find optimal trajectories, without requiring an initial guess and the permanent attendance of an expert. A steepest ascent algorithm is introduced that acts as
a navigator during the planetary encounter, providing the neurocontroller with the optimal insertion parameters. Results are presented for a Mercury rendezvous with a Venus gravity assist and for a Pluto flyby with a Jupiter gravity assist. They show very good agreement with the reference trajectories, in particular virtually no further refinement of the solution is required.

The design and optimization of interplanetary trans- fer trajectories is one of the most important tasks during the analysis and design of a deep space mis- sion. Due to their larger V -capability, low-thrust propulsions systems can significantly enhance or even enable those missions. Searching low-thrust trajec- tories that are optimal with respect to transfer time or propellant consumption is usually a dicult and time-consuming task that involves much experience and expert knowledge, because the convergence be- havior of traditional optimizers that are based on nu- merical optimal control methods depends strongly on an adequate initial guess, which is often hard to find. Even if the optimizer finally converges to an "opti- mal" trajectory, this trajectory is typically close to the initial guess that is rarely close to the (unknown) global optimum. Within this paper, trajectory op- timization is attacked from the perspective of arti- ficial intelligence and machine learning, which is a perspective quite dierent from that of optimal con- trol theory. Inspired by natural archetypes, a novel smart method for global low-thrust trajectory opti- mization is presented that fuses artificial neural net- works and evolutionary algorithms to so-called evo- lutionary neurocontrollers. This paper outlines how evolutionary neurocontrol works and how it could be implemented. Using evolutionary neurocontrol, low-thrust trajectories are optimized without an ini- tial guess and without the attendance of an expert in astrodynamics and optimal control theory. For an exemplary mission to a near-Earth asteroid, its performance for low-thrust trajectory optimization and interplanetary mission analysis is assessed. It is demonstrated that evolutionary neurocontrollers are able to find spacecraft steering strategies that gener- ate better trajectories - closer to the global optimum - because they explore the search space more ex- haustively than a human expert can do by using tra- ditional optimal control methods. Finally, the use of evolutionary neurocontrol for the analysis of a piloted Mars mission using a spacecraft with a nuclear elec- tric propulsion system is demonstrated within this paper. 1 LOW-THRUST TRAJECTORY OPTIMIZATION

The invariant manifold structures of the collinear libration points for the
spatial restricted three-body problem provide the framework for understanding
complex dynamical phenomena from a geometric point of view.
In particular, the stable and unstable invariant manifold \tubes" associated
to libration point orbits are the phase space structures that provide a
conduit for orbits between primary bodies for separate three-body systems.
These invariant manifold tubes can be used to construct new spacecraft
trajectories, such as a \Petit Grand Tour" of the moons of Jupiter. Previous
work focused on the planar circular restricted three-body problem.
The current work extends the results to the spatial case.

An initial study of techniques to be used in understanding how invariant manifolds are involved in low thrust trajectory design for the Jovian moon missions was performed using a baseline trajectory from the Europa Orbiter (EO) studies. Poincare sections were used in order to search for unstable resonant orbits. The unstable manifolds of these resonant orbits were computed, and they were found to provide an indication of how the EO trajectory was able to transition between resonances. A comparison with the stable and unstable manifolds of Lissajous orbits around the Jupiter-Europa L(sub 2) Lagrange point provided evidence that the Europa capture utilizes invariant manifolds of quasi-periodic orbits.

A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously specify the spacecraft position and velocity vectors, as well as the thrust history. To mitigate this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP) are investigated as candidate solutions to seed initial low-thrust trajectory designs. The addition of a low-thrust force to the CR3BP modifies the locations and stability of the equilibria, offering novel geometries for mission applications. Transfers between these novel equilibria are constructed by leveraging the associated stable and unstable manifolds and insights from the low-thrust CR3BP.

The rapid developments of artificial intelligence in the last decade are influencing aerospace engineering to a great extent and research in this context is proliferating. We share our observations on the recent developments in the area of spacecraft guidance dynamics and control, giving selected examples on success stories that have been motivated by mission designs. Our focus is on evolutionary optimisation, tree searches and machine learning, including deep learning and reinforcement learning as the key technologies and drivers for current and future research in the field. From a high-level perspective, we survey various scenarios for which these approaches have been successfully applied or are under strong scientific investigation. Whenever possible, we highlight the relations and synergies that can be obtained by combining different techniques and projects towards future domains for which newly emerging artificial intelligence techniques are expected to become game changers.

Nickel-based superalloys, used extensively in advanced gas turbine engines, exhibit complex microstructures that evolve during exposure to high temperatures (i.e., aging treatments). In this work, we examine critically if the principal component (PC) representation of rotationally invariant 2-point spatial correlations can adequately capture the salient features of the microstructure evolution in the thermal aging of the superalloys. For this purpose, an experimental study involving microstructure characterization of 27 differently aged (i.e., different combinations of temperature and time of exposure) samples was designed and conducted. Of these, 23 samples were employed for training a Gaussian Process Regression (GPR) model that took the aging temperature and the aging time as inputs, and predicted the microstructure statistics as output. The viability of the approach described above was evaluated critically by comparing the predictions for the four samples that were not used in the training of the GPR model. Furthermore, a new strategy was developed and implemented to generate digital microstructures corresponding to the predicted microstructure statistics. The predicted microstructures were found to be in good agreement with the experimentally measured one, validating the novel framework presented in this work.

This paper highlights natural transport pathways between in-plane and out-of-plane states associated with the vertical instability of planar Lyapunov orbits around the Lagrange points L1 and L2 in the Earth-Moon circular restricted three-body problem. Computations of invariant manifolds associated with the vertical instability of planar periodic orbits, ``vertically" stable and unstable manifolds, enable quantitative analyses of inclination changes. This study finds that multiple lunar flybys gradually change the orbital elements of vertically stable and unstable manifolds, and that the distributions of the affected orbital elements depend on the Jacobi constant and on the associated Lagrange point L1 or L2 of the planar Lyapunov orbits. As an application, this study uses the vertically stable manifolds of the planar Lyapunov orbits as initial guesses for optimizing transfers from near rectilinear halo orbits to planar distant retrograde orbits. Significant delta-v savings as compared with the known solutions demonstrate the usefulness of the vertical instability in spacecraft trajectory designs.

The main-belt asteroids are of great scientific interest and have become one of the primary targets of planetary exploration. In this paper, the accessibility of more than 600,000 main-belt asteroids is investigated. A computationally efficient approach based on Gaussian process regression is proposed to assess the accessibility. Two transfer models consisting of globally optimal two-impulse and Mars gravity-assist transfers are established, which would serve as a source of training samples for Gaussian process regression. The multistart and deflection technologies are incorporated into the numerical optimization solver to avoid local minima, thereby guaranteeing the quality of the training samples. The covariance function, as well as hyperparameters, which dominate the regression process, are chosen elaborately in terms of the correlation between samples. Numerical simulations demonstrate that the proposed method can achieve the accessibility assessment within tens of seconds, and the average relative error is only 1.33%. Mars gravity assist exhibits significant advantage in the accessibility of main-belt asteroids because it reduces the total velocity increment by an average of 1.23 km/s compared with the two-impulse transfer. Furthermore, it is observed that 3976 candidate targets have potential mission opportunities with a total velocity increment of less than 6 km/s. © 2016 by the American Institute of Aeronautics and Astronautics, Inc.

Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based
on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure
and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy
efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on
the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design
can be suitably augmented and compute approximations to trajectories for a mission to Venus.

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L3-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L1-family and L2-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.

A procedure is described that provides a universal solution for Lambert's problem. Based on the approach of Lancaster and his colleagues, the procedure uses Halley's cubic iteration process to evaluate the unknown parameter, x, at the heart of the approach, initial estimates for x being selected so that three iterations of the process always suffice to yield an accurate value. The overall procedure has been implemented via three Fortran-77 subroutines, listings of which are appended to the paper, and the way in which the subroutines have been tested is outlined.

Transmission spectroscopy, which consists of measuring the
wavelength-dependent absorption of starlight by a planet's atmosphere during a
transit, is a powerful probe of atmospheric composition. However, the expected
signal is typically orders of magnitude smaller than instrumental systematics,
and the results are crucially dependent on the treatment of the latter. In this
paper, we propose a new method to infer transit parameters in the presence of
systematic noise using Gaussian processes, a technique widely used in the
machine learning community for Bayesian regression and classification problems.
Our method makes use of auxiliary information about the state of the
instrument, but does so in a non-parametric manner, without imposing a specific
dependence of the systematics on the instrumental parameters, and naturally
allows for the correlated nature of the noise. We give an example application
of the method to archival NICMOS transmission spectroscopy of the hot Jupiter
HD 189733, which goes some way towards reconciling the controversy surrounding
this dataset in the literature. Finally, we provide an appendix giving a
general introduction to Gaussian processes for regression, in order to
encourage their application to a wider range of problems.

The International Sun-Earth Explorer (ISEE) scientific satellite to be stationed in 1978 in the vicinity of the sun-earth interior libration point to continuously monitor the space between the sun and the earth, including the distant geomagnetic tail is described. Orbit selection considerations for the ISEE-C are discussed along with stationkeeping requirements and fuel-optimal trajectories. Due to the alignment of the interior libration point with the sun as viewed from the earth, it will be necessary to place the satellite into a 'halo orbit' around the libration point, in order to eliminate solar interference with down-link telemetry. Parametric data for transfer trajectories between an earth parking orbit (altitude about 185 km) and a libration-point orbit are presented. It is shown that the insertion magnitude required for placing a satellite into an acceptable halo orbit is rather modest.

The role of cis-lunar space in future global space exploration

- M Bobskill
- M Lupisella

Bobskill, M., Lupisella, M. The role of cis-lunar space in
future global space exploration. In: Proceedings of the
Global Space Exploration Conference, Washington, DC,
USA, 2012: GLEX-2012.05.5.4x12270.

Autonomous time-optimal many-revolution orbit raising for electric propulsion GEO satellites via neural networks

- H Y Li
- F Topputo
- H X Baoyin

Li, H. Y., Topputo, F., Baoyin, H. X. Autonomous
time-optimal many-revolution orbit raising for electric
propulsion GEO satellites via neural networks. arXiv
preprint, 2019, https://doi.org/10.48550/arXiv.1909.
08768.

USA. He recently received the highest honor in his field, the AIAA Robert H

and Astronautics (AIAA), USA. He recently received the
highest honor in his field, the AIAA Robert H. Goddard
Astronautics Award (2019). E-mail: junkins@tamu.edu.