Figure 22 - uploaded by Adam Wilmer
Content may be subject to copyright.
Source publication
![Figure 2: Models of Motion for the Restricted Four-Body Problem [1]](profile/Adam-Wilmer/publication/359909774/figure/fig1/AS:1144183582670848@1649806032157/Models-of-Motion-for-the-Restricted-Four-Body-Problem-1_Q320.jpg)
![Figure 4: Family of Lyapunov Orbits about the L 1 Lagrange Point [2]](profile/Adam-Wilmer/publication/359909774/figure/fig2/AS:1144183582670850@1649806032257/Family-of-Lyapunov-Orbits-about-the-L-1-Lagrange-Point-2_Q320.jpg)
The world is in the midst of the second Space Age. The continued development and support of NASA's Artemis program and similar international efforts has made frequent Earth-Moon travel more likely than ever in the coming decades. With this surge in traffic, it is paramount to have knowledge of the locations of all space objects in cislunar space to...
Citations
... (1) where the scalar distance of the spacecraft (sc) with respect to the Earth (e) and Moon (m) in the synodic reference frame is written as Eqs. (2) and (3) The variables for distance, time, and mass are non-dimensionalized according to the characteristic quantities given in Table 1. The quantity "DU" is equal to the distance between the Earth and Moon, "TU" is equal to the dimensional period of the system divided by 2π, and "MU" is equal to the sum of the Earth and Moon masses, or MU = m e + m m . ...
... Examples or these orbits are shown in Refs. [2,24]. ...
... Difficulties with performing the SSA mission, in particular the detection, orbit determination, tracking, and characterization functions, are accentuated when extended into the cislunar regime. Currently, ground-based radar and optical systems are the primary method for characterizing objects in space [2]. These sensors will likely be insufficient for detection and tracking in cislunar space due, not only to limitation with sensing abilities at large distances, but also issues related to the aforementioned solar, Earth, and lunar viewing exclusion angles. ...
... Therefore, a simplified version of the CR3BP may be used that excludes the z-terms in Eq. (5), formally known as the planar-CR3BP (PCR3BP). Using the grid search method, it is possible to have multiple periodic orbits lie in the same initial position, with varying velocities which is demonstrated in [41,42]. Examples of more robust grid search algorithms are showcased by [43] for the Earth-Moon three-body periodic orbits and [44] for periodic orbits involving triaxial rigid bodies. ...
Venus is Earth’s closest neighbor, hosting a similar size, density, and location in the Solar System. Due to these similarities, Venus has been suggested as a premier location for particular mission sets to include: near-Venus communications and positioning, navigation, and timing (PNT); planetary science; heliophysics; space weather monitoring; and planetary defense. This paper, for the first time in literature, presents 37 unique planar periodic orbits discovered in the Sun–Venus system that may be used for such missions. The Circular Restricted Three-Body Problem (CR3BP) is the primary dynamical model utilized to generate all periodic orbits in the Sun–Venus system. The discovered orbits are grouped into four categories based on their shape: Near-Venus periodic orbits, Sun–Venus touring periodic orbits, Sun–Venus touring periodic orbits featuring near-Sun flybys, and Sun–Venus touring periodic orbits featuring near-Mercury flybys. Each orbit is discussed in terms of its respective Jacobi constant and stability within the context of the CR3BP. The stability of the orbits provides a preliminary analysis of station-keeping costs related to propellant expenditure, thereby determining the feasibility of implementing these trajectories. Specifically, this research shows that most of the identified orbits possess stability indices below 2, suggesting that minimal propellant is necessary for station-keeping maneuvers to sustain the preferred trajectory. For all orbits, specific initial position and velocity states conditions are provided, accompanied by recommendations for their potential mission applications. This research aims to advance ongoing astrodynamics research by filling a catalog hole and providing a Sun–Venus CR3BP orbit baseline.
... Achieving superior inertial navigation performance is critical to ensure that cislunar surveillance activities such as orbit determination and sensor tasking can be accomplished effectively. Studies have shown that uncertainty in an observer's position can flow downstream into cislunar orbit determination solutions [18], that precise navigation estimates are required for planning stationkeeping maneuvers [19], and that certain equilibrium point orbits require more precise positioning and velocity knowledge in order to maintain periodicity of the orbit [20]. For these reasons, the performance of planned navigation algorithms is important to consider when optimizing a cislunar surveillance network. ...
Performing Space Domain Awareness (SDA) on the cislunar volume is infeasible with any single Electro-Optical or Infrared (EO/IR) sensor system. This difficulty results from factors such as the vast expanse of cislunar space, the chaotic dynamic conditions, and the time-varying illumination backgrounds across the volume. These complications necessitate the development of collaborative networks of distributed space-based sensors to close coverage gaps of existing Earth-based infrastructure and enable persistent cislunar SDA. We show progress on optimizing the distributed sensor optimization problem from the lens of not only the performance of technologies but also operational constraints that are unique to the Cislunar domain. Models representing both of these factors have been assembled into software packages to enable a Model-Based Systems Engineering (MBSE) analysis of the problem. In order to perform optimization studies across the library of potential models, we have further developed a rapidly configurable Multi-Disciplinary Analysis and Optimization (MDAO) modeling framework. The MDAO framework uses object-oriented programming techniques to standardize model interfaces and allow them to be integrated into a unified optimization environment that is extended from NASA's OpenMDAO package. At the optimization stage, this MDAO system leverages genetic algorithms to produce the optimal choice of technologies and designs with respect to desired operational performance metrics. The end result is a modular software package that can be used to execute optimizations across a range of present and future cislunar technologies and designs. We provide an overview of how the metrics for sensor detection performance and orbit maintenance costs can be integrated into a library of optimization metrics. These metrics were used to assess the performance of distributed sensor networks consisting of observers spanning the cislunar space. Sensor detection performance is quantified by the percentage of the coverage over time, whereas the maintenance costs are quantified by orbit instability and navigation uncertainty. These cost metrics are used to optimize the SDA architecture while considering the operational challenges of deploying and maintaining distributed sensors within the chaotic cislunar volume. We present our results in the form of a Pareto front of non-dominated solutions to each of the specified performance metrics and detail changes in how architecture optimization changes once our operational cost metrics are considered.
... [44]. Recent research by Wilmer et al. has also studied the use of periodic orbits for achieving surveillance of the Earth-Moon Lagrange points [45,46]. Their study showed that cislunar periodic orbits were highly effective at maintaining continuous surveillance coverage of halo and Lyapunov orbits about the Earth-Moon L 1 and L 2 Lagrange points. ...
Space is becoming increasingly congested every day and the task of accurately tracking satellites is paramount for the continued safe operation of both manned and unmanned space missions. In addition to new spacecraft launches, satellite break-up events and collisions generate large amounts of orbital debris dramatically increasing the number of orbiting objects with each such event. In order to prevent collisions and protect both life and property in orbit, accurate knowledge of the position of orbiting objects is necessary. Space Domain Awareness (SDA) used interchangeably with Space Situational Awareness (SSA), are the names given to the daunting task of tracking all orbiting objects. In addition to myriad objects in low-earth-orbit (LEO) up to Geostationary (GEO) orbit, there are a growing number of spacecraft in cislunar space expanding the task of cataloguing and tracking space objects to include the whole of the earth-moon system.
This research proposes a series of algorithms to be used in autonomous SSA for earth-orbiting and cislunar objects. The algorithms are autonomous in the sense that once a set of raw measurements (images in this case) are input to the algorithms, no human in the loop input is required to produce an orbit estimate. There are two main components to this research, an image processing and satellite detection component, and a dynamics modeling component for three-body relative motion.
For the image processing component, resident space objects, (commonly referred to as RSOs) which are satellites or orbiting debris are identified in optical images. Two methods of identifying RSOs in a set of images are presented. The first method autonomously builds a template image to match a constellation of satellites and proceeds to match RSOs across a set of images. The second method utilizes optical flow to use the image velocities of objects to differentiate between stars and RSOs. Once RSOs have been detected, measurements are generated from the detected RSO locations to estimate the orbit of the observed object. The orbit determination component includes multiple methods capable of handling both earth-orbiting and cislunar observations. The methods used include batch-least squares and unscented Kalman filtering for earth-orbiting objects. For cislunar objects, a novel application of a particle swarm optimizer (PSO) is used to estimate the observed satellite orbit. The PSO algorithm ingests a set of measurements and attempts to match a set of virtual particle measurements to the truth measurements. The PSO orbit determination method is tested using both MATLAB and Python implementations.
The second main component of this research develops a novel linear dynamics model of relative motion for satellites in cislunar space. A set of novel linear relative equations of motion are developed with a semi-analytical matrix exponential method. The motion models are tested on various cislunar orbit geometries for both the elliptical restricted three-body problem (ER3BP) and the circular restricted three-body problem (CR3BP) through MATLAB simulations. The linear solution method's accuracy is compared to the non-linear equations of relative motion and are seen to hold to meter level accuracy for deputy position for a variety of orbits and time-spans.
Two applications of the linearized motion models are then developed. The first application defines a differential corrector to compute closed relative motion trajectories in a relative three-body frame. The second application uses the exponential matrix solution for the linearized equations of relative motion to develop a method of initial relative orbit determination (IROD) for the CR3BP.
... Nevertheless, the complex interactions of multi-body dynamics result in the formation of unique and, depending on the initial conditions, arguably aesthetic trajectories that could "fill a gallery with postmodern art" as described by Cixin [21]. Table 3. Cislunar Periodic Orbit Initial Conditions in the CR3BP (µ = 0.012150584673414) [42] ...
The Earth–Moon gravitational system is naturally chaotic, and mathematical models such as the Circular Restricted Three-Body Problem (CR3BP) employ layers of ordered assumptions in order to enable both trajectory generation and analysis. However, even within this simplified mathematical construct, the underlying chaos of multi-body gravitational systems introduces a pseudo-instability that yield a dynamic and sometimes drastic evolution of trajectory geometry despite seemingly insignificant changes in the initial position and/or velocity of an object. Three-body trajectories feature a natural artistry reminiscent of artistic styles and spiritual motifs originating from diverse time periods and cultures, ranging from sacred geometry and mandalic-like designs indicative of Buddhist and Hindu symbolism, to Spirograph art, and modernist and post-modernist minimalism. This work will analyze the artistic qualities of two classes of three-body trajectories within the Earth–Moon system: (1) cislunar periodic and near-periodic orbits, and (2) the mixture of quasi-periodic and chaotic trajectories of debris particles following a catastrophic breakup event. Overall, this work sets to prove that even in a chaotic multi-body environment, beauty and order emerges through a fine balance of initial conditions.
... where the scalar distance of the spacecraft (sc) with respect to the Earth and Moon in the synodic reference frame is written as Eqs. (2) and (3), respectively: ...
Cislunar space is a region of growing interest with nations investing resources to cultivate long presence habitations on the lunar surface. With this increased attention, cislunar orbital pathways must be established and example rendezvous and proximity (RPO) operations must be simulated and analyzed. There is limited analysis on the effectiveness of spacecraft in the cislunar domain to perform, not only search and rescue missions, but RPO missions as a whole. The present study adds to the limited work in the cislunar RPO field to investigate and analyze a search and rescue mission comprising a spacecraft in a touring cislunar periodic orbit (TCPO) that will rendezvous with an impaired spacecraft in a near-rectilinear halo orbit (NRHO) about the $L_2$ Lagrange point. This work will simulate a variety of rendezvous' in which the impaired spacecraft's location within the NRHO and the rescuer spacecraft in the TCPO are varied. Psuedospectral methods within the dynamics of the circular restricted three-body problem (CR3BP) are utilized to find optimized least time solutions given a max $\Delta V$. The results of this work show the TCPO used in this work to be advantageous in the timely rendezvous with an impaired spacecraft in a NRHO, hosting response times as fast as a few hours. However, to ensure the rendezvous is timely, priority should be placed on creating a constellation large enough to ensure that at least one rescuer spacecraft is near the area of interest (i.e. where the impaired spacecraft is located)
... (1) was presented in Refs. [15,16]. The variables for distance, time, and mass were nondimensionalized according to the characteristic quantities given in Table 1. ...
... Orbits 1-9 were generated or "discovered" using methods presented by Wilmer et al. [15,20]. The initial conditions and approximate periods for orbits 1-9 are presented in Table 5. ...
International attention is being increasingly placed on the cislunar region of space, which offers to serve as the new high ground for land- and space-based operations. With an anticipated surge in cislunar traffic, it is paramount to be able to detect and track all objects in cislunar space to optimize mission readiness and prevent catastrophic collisions. Cislunar periodic orbits provide an elegant means to fill the observational capability gaps that are present in ground-based and/or near-Earth space-based sensors. In this research, cislunar periodic orbits are identified that provide frequent-to-consistent coverage of the L1 and L2 regions. Specifically, select orbits are analyzed, both individually and within a multiorbit sensor architecture, for their effectiveness (in terms of percent of visibility time) at monitoring target satellites in Lyapunov and halo orbits about the Earth–moon L1 and L2 Lagrange points. The results of this research show cislunar periodic orbits to be highly effective in monitoring Lyapunov and halo orbits about the Earth–moon L1 and L2 Lagrange points. Particular figures and the overall methodology are derived from a paper presented at the 2021 ASCEND Conference (Wilmer, A. P., Bettinger, R. A., and Little, B. D., “Cislunar Periodic Orbit Constellation Assessment for Space Domain Awareness of L1 and L2 Halo Orbits,” 2021 ASCEND Conference, 2021.).
... The periodic orbits used for this study come from previous research conducted into their effectiveness in conducting an SDA mission [11,12]. Previous work focused on visibility analysis of target spacecraft from the observer constellation, while this work focuses on orbit prediction from two of the periodic orbits analyzed in [12]. ...
... Previous work focused on visibility analysis of target spacecraft from the observer constellation, while this work focuses on orbit prediction from two of the periodic orbits analyzed in [12]. The target spacecraft orbit used in this work, also from [11], is a L1 Halo orbit. These orbits were chosen based on the results of [11] detailing the high observability rates of the target from the observer constellation. ...
... The target spacecraft orbit used in this work, also from [11], is a L1 Halo orbit. These orbits were chosen based on the results of [11] detailing the high observability rates of the target from the observer constellation. While periodic orbits were picked for the observer constellation in this study, their are a multitude of different orbits that could and should be investigated. ...
With an increased interest in the cislunar domain for both civilian and military applications comes a need for Space Domain Awareness (SDA) of objects in cislunar space. Space-based SDA in cislunar space is challenging in part due to difficulties associated with accurately estimating the position of the observer satellite, which is a requirement for effectively performing the SDA mission. Using multiple observation satellites with lower-fidelity equipment helps alleviate these concerns by aggregating together multiple data sets with higher variance to achieve the same level or better accuracy as compared to fewer higher-quality measurement systems. Earth-Moon periodic orbits are used for the observer constellation with a target spacecraft in a L1 Halo orbit. All orbits are modeled using the circular restricted three body problem (CR3BP). Systems Tool Kit (STK) is used to calculate orbit geometries and angles-only measurements are extracted to simulate observing spacecraft with optical sensors. The measurement data is then processed utilizing an Extended Kalman Filter to estimate the position of the target spacecraft. The analysis focuses on comparing the effectiveness of different numbers of observer spacecraft. The results of this simulation found that the use of a low-fidelity constellation can match the performance achieved by a constellation with higher-fidelity systems.
... Recently, Vendl [10] has studied periodic orbits for their ability to detect and track objects within cislunar space. Touring Cislunar periodic orbits (CPOs) have been extensively studied by Wilmer [12,13,14] for their effectiveness in surveillance type missions commonly referred to as space domain awareness (SDA) mission sets. The allure of touring CPOs is that they provide mission-related benefits in terms of their ability to traverse wide expanses of cislunar space capturing multiple different viewing angles of targets. ...
... SDA effectiveness is determined by the percentage of time with which a sensor bearing satellite is able to have a visual site of the Target. The initial conditions of the touring CPOs used in this work were obtained through the works of Wilmer [12,13] who used a µ value of 0.012150584673414. The initial conditions of these orbits may be seen in Table 5: The initial conditions of the halo orbit was also obtained through the works of Wilmer [12,13] who followed a method laid out by Grebow [5] in which the bifurcation with the planar Lyapunov orbits was used as the starting point for forming halo orbits. ...
... The initial conditions of the touring CPOs used in this work were obtained through the works of Wilmer [12,13] who used a µ value of 0.012150584673414. The initial conditions of these orbits may be seen in Table 5: The initial conditions of the halo orbit was also obtained through the works of Wilmer [12,13] who followed a method laid out by Grebow [5] in which the bifurcation with the planar Lyapunov orbits was used as the starting point for forming halo orbits. The initial conditions of the halo orbits used in this analysis is shown in Table 6: The initial conditions for the NRHO are obtained through the work of Bucchioni and Innocenti [3] and adjusted to fit a standardized non-dimensional mass parameter. ...
Cislunar space is anticipated to become increasingly congested in the coming decades with both nations and private companies building up infrastructure to support missions to the Moon and Mars. With this anticipated increase in space traffic, it is paramount to conduct space surveillance, commonly referred to as Space Situational/Domain Awareness (SSA/SDA), missions in such a way as to encapsulate the entirety of the Earth-Moon system. Performing SSA/SDA utilizing classical ground-and/or near Earth space-based sensors becomes increasingly challenging when applied to the cislunar orbit regime. Therefore, orbits which reside in cislunar space such as cislunar periodic orbits (CPOs) provide an elegant means to fill the observational capability gaps which exist in current near-Earth sensors. This work seeks to compare the effectiveness between a touring class of CPOs, herein referred to as "touring" CPOs and L1/L2 halo orbits in a sample surveillance mission. Specifically, these orbits will be evaluated on their ability to detect and track satellites in a Near Rectilinear Halo Orbit (NRHO) trajectory, such as one intended for NASA's Lunar Gateway that aims to support colonization efforts on the lunar surface for future ventures to Mars. Visual magnitude is used in determining if a target satellite is visible. Notional space-to-space sensors will be used to determine limitations of orbit geometry for the SDA mission as a function of sensor range, capability, and Sun/Earth/Moon exclusion angles. Simulations will consist of 12 sensor satellites in either a touring CPO or an L1/L2 halo orbit, depending on the scenario. These sensor satellites will be monitoring two target satellites in the NRHO. Results show the halo orbits to be more effective than the touring CPOs at NRHO surveillance, with the L1 halo orbit hosting a visual of the Target for an average of 99.28% of the 30 day simulation. This is a result of the conclusion that closeness to the target of interest is one of the most influential factors for successful space surveillance missions.
... The Circular Restricted Three-Body Problem (CR3BP), is a trajectory model which aims at predicting the motion of a spacecraft under the gravitational force of the Earth and Moon with no other perturbation considered. The main assumptions to this model are that the Moon is in a circular orbit about the Earth and that the spacecraft's mass is small compared to that of the Earth and Moon [11]. Through the gravitational interactions between the Earth and Moon, points of stability emerge referred to as Lagrange points. ...
... Using CR3BP dynamics, example CPO were generated that correspond to the missions discussed in the preceding sub-sections. Figure 2 illustrates two proposed orbits for each mission type, with the orbits depicted with respect to the Earth-Moon synodic reference frame, an orthogonal system that rotates about the Earth-Moon barycenter [11,12]. ...
... In addition, the ability to search, detect, track, and characterize resident space objects is pivotal for identifying new objects to catalogue, minimize cross-tagging of new objects with those currently catalogued, and monitoring if spacecraft are adhering to posted flight plans for transiting potentially congested regions of cislunar space. The touring CPOs proposed for this mission set have been previously studied [11,14,15], and preliminary analysis reveals that these orbits perform well at monitoring objects near the L1 and L2 points, two regions interest due to their proximity to the Moon. Accordingly, it is reasonable to assume that touring CPOs will also perform well in a similar nearlunar surveillance mission. ...
The world is arguably in the beginning of the second Space Age. The continued development and support of NASA's Artemis program and similar international efforts has made frequent Earth-Moon travel more likely than ever in the coming decades. With this surge in traffic, it is paramount to establish common cislunar routes and celestial highways for managing the flow of space travel and logistical operations between the Earth and Moon, a highly chaotic system with complex dynamics. One class of orbits, identified herein as cislunar periodic orbits (CPOs), may act as an efficient means of establishing these common routes whilst utilizing natural perturbations to minimize propellant consumption. Two types of CPOs are discussed in the work: "touring" CPOs and halo orbits. Touring CPOs are defined as orbits which periodically return to their initial conditions while they traverse a wide expanse of cislunar space-the spherical volume of space extending from geostationary orbit to and including the Earth-Moon Lagrange points. Such orbits could provide support for a wide swath of logistical missions to include re-supply, personnel transport, tourism, and space-based infrastructure development. In particular, touring CPOs which feature close proximity passes of both the Earth and Moon, identified herein as cycler orbits, may be extremely useful for such mission sets. As with any new system or policy, legislation for common regulations will be forthcoming as implementation comes to fruition and plans are set for establishment. With the anticipation for the need of common celestial highways, space-faring nations have a vested interest in research focused on implementation and law surrounding these orbits. In this work, orbit feasibility, logistical challenges, patentability and rights of orbits, and international space law and policy will be explored to investigate the practicability of establishing these international routes using various forms of shared CPOs.
