Session B6: GNSS Applications in Space

Methods for Accurate Ephemeris Modeling of the Lunar Near Rectilinear Halo Orbit
Garvin Saner and Kirsten Strandjord, University of Minnesota
Alternate Number 2

I. BACKGROUND
As public agencies, governments, and private companies continue to invest time and money into cislunar projects, the need to develop secure, lunar-focused navigation services has become apparent. The Deep Space Network (DSN) is currently the primary source for navigation and communication for spacecraft in that region. However, the DSN is overburdened, and may not be able to handle the amount of cislunar and lunar surface assets that have been proposed [NASA, 2023]. To remedy this, both NASA and the ESA are working on LunaNet, a framework for building a network for both navigation and communication services [NASA, 2021]. In conjunction with the use of GNSS beyond the altitude of the constellation [Strandjord and Cornish, 2023], the navigation subsystem of LunaNet is proposed to operate similarly to how GNSS operates around Earth, with some key differences. The objective of this research is to investigate the challenges and the set of possible solutions related to the difficult task of representing the broadcast ephemeris of the Near-Rectilinear Halo Orbit (NRHO). Not only will this be important for the operations related to cislunar exploration, but also for many position, navigation, and timing (PNT) applications. The knowledge gained from previous investigations of terrestrial navigation orbits inform the way in which the positions along the NRHO can be compactly described by a set of methodically chosen parameters. And though the NRHO poses many challenges in comparison to many of the more stable orbits (e.g. LLO, ELFO, etc.), the benefits of constructing an accurate and precise ephemeris to describe this orbit is all the more valuable and will serve many navigation and space situational purposes.
A conventional GNSS user needs at least four pseudorange measurements and a model of the motion of the corresponding GNSS space vehicles in order to instantaneously estimate their position. Most GNSS systems in use today broadcast ephemeris based on the Keplerian orbital elements, also known as classical orbital elements (COEs). Throughout an ideal, two-body orbit, all but one of these parameters will remain constant. To account for perturbations caused by lunisolar accelerations, solar radiation pressure, and other disturbances, linear and harmonic corrective terms are also calculated and included in the ephemeris [Vallado, 2001]. While this parameterization works well for the current GNSS systems, efforts to modify the structure of the broadcast ephemeris have been ongoing. Recently, Dobbin and Axelrad have proposed using B-splines as a framework for platform-agnostic transmission of ephemeris [Dobbin and Axelrad, 2023]. But the topic of creating alternative broadcast ephemerides for MEO constellations as well as other orbital regimes has been studied for several decades [Montenbruck et al., 2002]. While one may be tempted to implement a similar broadcast ephemeris structure for a lunar system, the highly chaotic nature of cislunar orbits will make this approach more difficult.
For many orbits currently in use, including the orbits of GNSS satellites, a two-body approximation is usually appropriate. When looking at cislunar orbits, however, the third-body effects from the Earth are too large to ignore. This is especially true of the NRHO, which is of particular interest due to its ability to stay uneclipsed by the Moon, and therefore provide a near-constant line of sight to Earth [Davis et al., 2017]. The orbit is able to do this by taking advantage of the prominent third-body forces of the Earth. The downside to this, however, is that the dynamics of the orbit become highly nonlinear, which makes it difficult to describe the orbit in a way that an LNSS satellite could convey to its users.
To account for this, Cortinovis et al. have examined the use of both standard and Chebyshev polynomials to approximate the inertial coordinates of propagated state values in Low Lunar and Elliptical Lunar Frozen Orbits [Cortinovis et al., 2023]. While their method is able to accurately approximate the vehicle’s state for most of the orbit, it struggles with the dynamically challenging region around perilune. Additionally, the cartesian coordinates used are unintuitive compared to the Keplerian orbital elements. While GNSS uses a mix of these mostly-constant elements and additional corrective terms, they vary too greatly in an NRHO for a GNSS-like parameterization to be applied. These elements, however, have been shown to be periodic in nature, much like the NRHO itself [Machuca et al., 2022]. If modeled appropriately, this periodicity may provide the framework for constructing accurate and concise broadcast ephemeris for the NRHO.
II. METHODOLOGY
The purpose of the proposed work is to investigate methods for efficiently packaging ephemeris information for an LNSS satellite in an NRHO. This includes the examination of several different types of curve fits with the classical orbital elements, a modified set of orbital elements, as well as the cartesian states, in several different frames. The accuracy of these methods is assessed throughout the course of the orbit, as well as varying update periods. The true trajectory of the LNSS spacecraft is generated using NASA’s General Mission Analysis Tool, an open source, high-fidelity program used for modeling orbits. It is this trajectory that needs to be efficiently packaged and sent to a user.
Once generated, the trajectory is divided into prediction windows of a set length. Over this window, a least-squares method will be applied to a desired parameterization, fit to a desired function, in a desired frame. For example, the Cartesian coordinates in the Moon-centered inertial frame could be fit to a fifth order polynomial over a window of two hours. The details of each of these variables is described below.
Window
This is the length (in time) of the prediction window. This will dictate how much of the trajectory will be fit to, and therefore the length of time the broadcast ephemeris is expected to be valid for. It is expected that the shorter this window is, the more accurate the broadcast ephemeris will become. Window lengths of 15 minutes to 4 hours will be tested.
Parameters
Parameters are the coordinates or elements that are being fit to. Since the NRHO is in a third-body dominated orbit, regular Keplerian orbital elements can not be used. Instead, quasi-Keplerian orbital elements will be calculated using the positions and velocities of each point on the trajectory. These, in addition to Cartesian coordinates, will be tested. Recent work by Peterson & Scheeres have produced local orbital elements for the circular restricted three-body problem (CR3BP) [Peterson and Scheeres, 2023]. These will also be investigated.
Frames
The parameters used can be resolved in multiple different frames. Obviously the Cartesian coordinates vary from frame to frame, but also the Keplerian orbital elements will also be different when calculated in different frames. The Moon-centered inertial (MCI) and synodic frames will be tested. The radial, in-track, cross-track (RIC) frame, relative to a reference CR3BP orbits will also be investigated.
Fits
The largest dimension of the design space belongs to the functions used for the least-squares fit. Options for these functions that will be tested include polynomial, Chebyshev, and Fourier series. These functions will also be tested at various orders.
III. PRELIMINARY RESULTS AND ANALYSIS
As a first pass, the process outlined in the previous section was performed using a trajectory known to be stable [Williams et al., 2017]. A two hour window and third order polynomial fit was used for Keplerian elements calculated from the synodic frame. As expected, both the position and velocity errors are the highest for the few hours around perilune, peaking at 10 km and 10 m/s. However, for most of the orbit the errors stay below 10 m and 0.1 mm/s.
This work will not only serve to show innovative ways in which the NRHO can be accurately described by a set of compact terms, but will also demonstrate the complexity of this orbit in comparison to those that are more readily described in terms of perturbations away from a relative-two body orbit. In this way, the research presented here will show the accuracy and precision of many of the current methods, but also eliminate methods that do not accurately represent the physics governing this orbit.
REFERENCES
[Cortinovis et al., 2023] Cortinovis, M., Iiyama, K., and Gao, G. (2023). Satellite ephemeris approximation methods to support lunar positioning, navigation, and timing services. In Proceedings of the 36th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2023), pages 3647–3663.
[Davis et al., 2017] Davis, D., Bhatt, S., Howell, K., Jang, J.-W., Whitley, R., Clark, F., Guzzetti, D., Zimovan, E., and Barton, G. (2017). Orbit maintenance and navigation of human spacecraft at cislunar near rectilinear halo orbits. In AAS/AIAA Space Flight Mechanics Meeting, number JSC-CN-38626.
[Dobbin and Axelrad, 2023] Dobbin, M. and Axelrad, P. (2023). A flexible ephemeris representation for gnss and alternative pnt signal sources using b-splines. NAVIGATION: Journal of the Institute of Navigation, 70(4).
[Machuca et al., 2022] Machuca, P., Rosengren, A. J., and Ross, S. D. (2022). xgeo space domain awareness: Parametrization and characterization of cislunar space.
[Montenbruck et al., 2002] Montenbruck, O., Gill, E., and Lutze, F. (2002). Satellite orbits: models, methods, and applications. Appl. Mech. Rev., 55(2):B27–B28.
[NASA, 2021] NASA (2021). Empowering artemis with communications and navigation interoperability. NASA.
[NASA, 2023] NASA (2023). Audit of NASA’s Deep Space Network.
[Peterson and Scheeres, 2023] Peterson, L. T. and Scheeres, D. J. (2023). Local orbital elements for the circular restricted three-body problem. Journal of Guidance, Control, and Dynamics, 46(12):2275–2289.
[Strandjord and Cornish, 2023] Strandjord, K. and Cornish, F. (2023). Direct positioning estimation beyond the constellation using falcon gold data collected on highly elliptical orbit. In Proceedings of the 2023 International Technical Meeting of The Institute of Navigation, pages 679–691.
[Vallado, 2001] Vallado, D. (2001). Fundamentals of astrodynamics and applications. Space Technology Library, pages 303–323.
[Williams et al., 2017] Williams, J., Lee, D. E., Whitley, R. J., Bokelmann, K. A., Davis, D. C., and Berry, C. F. (2017). Targeting cislunar near rectilinear halo orbits for human space exploration, 27th aas. In AIAA Space Flight Mechanics Meeting, pages 1–19