Optimization and deployment method of cislunar communication and navigation constellation based on libration point orbits

The invariant manifold structures of the libration points for restricted three-body problem will provide the framework for understanding complex dynamical phenomena. These stable and unstable invariant manifold tubes associated to libration point orbits can be used to construct new low-energy spacecraft trajectories. So, in this paper, we will introduce the libration point and its invariant manifolds structure. First, we introduce the restricted three-body problem, libration point and Halo orbit. Then, we discuss and provide the stable and unstable invariant manifold tubes associated to libration point orbits based on manifolds and dynamical system theory. Finally, we apply these dynamical properties to design the capture trajectory in planetary exploration.

Attitude stability analysis of a dual-spin spacecraft in halo orbits

Survey on advances in orbital dynamics and control for libration point orbits

<title>Abstract</title> The design of libration point orbits (LPO) in high-fidelity ephemeris model (HFEM) is essential for mission design in the vicinity of libration points. An important design challenge lies in ensuring the generated LPO in the HFEM inherits desirable geometric property exhibited by the counterpart LPO from simpler dynamic models such as the circular-restricted three-body problem. Another hurdle lies in generating fuel-optimal, long-duration LPOs, where small maneuvers are necessitated to maintain the orbital regime over extended periods of time. To address both the geometric preservation and fuel-optimality, we propose a nonlinear programming (NLP)-based shooting approach for generating LPO in HFEM. We adopt sequential convex programming to solve the NLP due to its theoretical guarantee of convergence to a local optimum. We demonstrate our approach by constructing 9:2 resonant L2 near rectilinear halo orbits, the proposed orbit for the Gateway, and 1:1 resonant L1 Lyapunov orbits, frequently proposed for space situational awareness missions equipped with optical sensors.

This paper is devoted to the study on applying numerical techniques to accurately compute and robustly extend the libration point orbits (LPOs). A new methodology is proposed exploiting the hyperbolic dynamics of the collinear libration points. Numerical tools are developed to facilitate the efficient computation process, which are applicable to realistic force models and inherently parallelizable. Extensive numerical explorations in the Earth–Moon system are carried out, revealing the delicate structures of nested island chains and bounded chaotic motions on the center manifold. Numerical results confirm that the proposed approach can handle the computations of various types of LPOs in a unified manner and is operational over a wide range of energy levels. LPOs obtained with this approach offer a broad range of future mission possibilities in an extended vicinity of the collinear libration points.

A note on libration point orbits, temporary capture and low-energy transfers

Techniques associated with stable manifold and lunar flyby have been applied to the construction of optimal transfers to Earth-Moon $L_{1} /L_{2}$ libration point orbits. Compared with traditional design methods and to reduce maneuver cost, the design process presents a detailed analysis on the effect of lunar proximity with multiple constraints. An accurate and fast design strategy for seeking an insertion point and modifying the stable manifold to satisfy these constraints is proposed. Combined this strategy with the differential correction algorithm, the optimal transfer trajectory can be determined from a low-Earth orbit to a halo orbit around the $L_{1} /L_{2}$ libration point within a little computational time. Different amplitudes and insertion points of halo orbit in conjunction with various constraint conditions about lunar flyby are considered to deeply examine the efficiency and reliability of the design algorithm. Preliminary results indicate that the required mission cost has a significant correlation with lunar proximity constraints, and demonstrate that the method of constructing impulsive lunar halo transfer trajectories with multiple constraints is feasible.

Preliminary designs of low-thrust transfer trajectories are developed in the Earth-moon three-body problem with variable specific impulse engines and fixed engine power. The solution for a complete time history of the thrust magnitude and direction is initially approached as a calculus of variations problem to locally maximize the final spacecraft mass. The problem is then solved directly by sequential quadratic programming, using either single or multiple shooting. The coasting phase along the transfer exploits invariant manifolds and, when possible, considers locations along the entire manifold surface for insertion. Such an approach allows for a nearly propellant-free final coasting phase along an arc selected from a family of known trajectories that contract to the periodic libration point orbit. This investigation includes transfer trajectories from an Earth parking orbit to some sample libration point trajectories, including L 1 halo orbits, L 1 and L 2 vertical orbits, and L 2 butterfly orbits. Given the availability of variable specific impulse engines in the future, this study indicates that fuel-efficient transfer trajectories could be used in future lunar missions, such as south pole communications satellite architectures.

A rapid and accurate method of calculating optimal impulsive transfers in the restricted problem of three bodies has been developed. The technique combines a multi-conic method of trajectory integration with primer vector theory and an accelerated gradient method of trajectory optimization. A unique feature is that the state transition matrix and the primer vector are found analytical without additional integrations or differentiations. The method has been applied to the determination of optimal two and three impulse transfers between the L2 libration point and circular orbits about both the earth and the moon.

One of the most interesting and challenging aspects of formation guidance algorithm design is the coupling of the orbit design and the science return. The effectiveness of the formation as a science instrument is intimately coupled with the relative geometry and evolution of the collection of spacecraft. Therefore, the science return can be maximized by optimizing the orbit design according to a performance metric relevant to the science mission goals. In this work, we present a general method for optimal formation guidance that is applicable to missions whose performance metric, requirements, and constraints can be cast as functions that are explicitly dependent upon the orbit states and spacecraft relative positions and velocities. The approach is fully nonlinear, accommodates orbital perturbations, and is applicable to multiple flight regimes including highly eccentric, hyperbolic, interplanetary, or libration point orbits. Furthermore, the method is applicable to small formations, as well as large formations and constellations. We present a general form for the cost and constraint functions, and derive their semi-analytic gradients with respect to the formation initial conditions. The gradients are broken down into two types. The first type are gradients of the mission-specific performance metric with respect to formation geometry. The second type are derivatives of the formation geometry with respect to the orbit initial conditions. The fact that these two types of derivatives appear separately allows us to derive and implement a general framework that requires minimal modification to be applied to different missions or mission phases. To illustrate the applicability of the approach, we conclude with applications to two missions: the Magnetospheric Multiscale Mission, and the Laser Interferometer Space Antenna.

The application of dynamical systems techniques to trajectory design has demonstrated that employing invariant manifolds and resonant flybys expands the trajectory options and potentially reduces the requirements. Low-cost transfer trajectories from two- and three-dimensional resonant orbits are explored in the Saturn–Titan system via Poincaré mapping techniques and the application of dynamical systems theory. Natural maneuver-free transfers between resonant orbits and libration point orbits are constructed with the aid of a trajectory design tool that blends manifold arcs associated with unstable resonant orbits. Homoclinic-type resonant connections exist in two and three dimensions and are computed employing the transfer design tool. As an application of resonant transitions to preliminary mission design, the accessibility of Hyperion from orbits that are resonant with Titan is examined, and two transfer trajectories to the orbit of Hyperion are constructed by exploiting the invariant manifolds associated with unstable resonant orbits. A local optimization algorithm is employed to further decrease the cost of insertion and to illustrate the relationship between the local minimum and the invariant manifolds.

Libration point orbits may be ideal locations for satellite imaging formations. Therefore, control of these arrays in multi-body regimes is critical. A continuous feedback control algorithm is developed that maintains a formation of satellites in motion that is bounded relative to a halo orbit. This algorithm is derived based on the dynamic characteristics of the phase space near periodic orbits in the circular restricted three-body problem (CR3BP). By adjusting parameters of the control algorithm appropriately, satellites in the formation follow trajectories that are particularly advantageous to imaging arrays. Image reconstruction and coverage of the ( u , v ) plane are simulated for interferometric satellite configurations, demonstrating potential applications of the algorithm and the resulting motion.

Driven by the surmounting interest for dedicated infrastructure in cislunar space, this work considers the satellite constellation design for cislunar space situational awareness (CSSA). We propose a mixed-integer linear programming (MILP)-based formulation that simultaneously tackles the constellation design and sensor-tasking subproblems surrounding CSSA. Our approach generates constellation designs that provide coverage with considerations for the field of view of observers. We propose a time-expanded p-median problem (TE-p-MP) that considers the optimal placement of p space-based observers into discretized locations based on orbital slots along libration point orbits, simultaneously with observer pointing directions across discretized time. We further develop a Lagrangian method for the TE-p-MP, where a relaxed problem with an analytical solution is derived, and customized heuristics leveraging the orbital structure of candidate observer locations are devised. The performance of the proposed formulation is demonstrated with several case studies for CSSA constellations monitoring the cislunar Cone of Shame and a periodic time-varying transit window for low-energy transfers located in the Earth–moon L2 neck region. The proposed problem formulation, along with the Lagrangian method, is demonstrated to enable a fast assessment of near-optimal CSSA constellations, equipping decision-makers with a critical technique for exploring the design trade space.

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.

This paper proposes a semi-analytical method combining Poincaré Contraction Mapping (PCM) with Conditional Variational Autoencoder (CVAE) for efficient screening of bounded relative orbit across perturbed environments. The PCM projects high-dimensional system parameters to a two-dimensional (2D) feature parameter pair of crossing period and separation angle, which is surjective but non-injective. To address the one-to-many inverse mapping challenge, a CVAE-based deep learning model is developed to raise the dimensionality from the 2D feature parameters back to the state space, enabling rapid and diverse generation of long-duration relative orbits with bounded amplitude. And the PCM-CVAE method is validated in both Earth-centered displaced orbits (DO) and Earth-Moon libration point orbits (LPO), demonstrating consistent generality. Moreover, the accuracy and dispersion of CVAE method are evaluated and compared with ANN and KNN-GMM algorithms. Results of the scenario of DO demonstrate that the time-angle accuracy of PCM-CVAE method realizes an average value (0.53%, 2.48%) and an optimal value (1.92×10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup>, 5.94×<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup>). Meanwhile, compared to the previous traversal search method by finding intersection of contour maps, PCM-CVAE reduces the search time from 6.44 hours to 7.83 minutes, achieving 98% reduction in computational cost with high accuracy.