The Design of a Lunar Global Navigation Satellite System Tekilanand Persaud University of Florida, Gainesville, Florida, 32612
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The Design of a Lunar Global Navigation Satellite System Tekilanand Persaud University of Florida, Gainesville, Florida, 32612 An optimal Lunar navigation system was designed by finding the minimum total Delta-V required to launch and maintain a system over its entire lifetime for a design space of constellations. Considerations were made for the two largest orbital perturbations, those from third gravitational bodies, and from anisotropic Lunar gravity. The results show that the first two optimal constellations from the set of those tested are two Walker-Delta configurations with a semimajor axis of 5700 km and five orbital planes and a semimajor axis of 6700 km with three planes, both with a 75° inclination. The third was a Walker-Star configuration with a semimajor axis of 9200 km and three planes. All optimal configurations had 15 satellites. Key words – satellite constellation design, GNSS mean anomaly in their orbit. Additional configurations were I. Introduction investigated using star and delta configurations supplemented S space exploration continues into the mid-21st century, with satellites in halo orbits around the Earth-Moon Lagrange Athe need for navigation systems in the vicinity of other points. Examples of star and delta configurations are shown in celestial bodies arises from motivations such as servicing the appendix. human operations and precisely locating unmanned systems for When choosing between star (polar) and delta (inclined) economic and scientific purposes. The peculiarities of the configurations, it is important to consider the primary Earth-Moon system are both detrimental and advantageous to drawbacks between each: star patterns provide strong polar the design of a Lunar Global Navigation Satellite System coverage due to the intersection of all of the orbits above the (GNSS); the low mass of the Moon means that satellites in its poles. However, this makes coverage at the equator orbit are heavily perturbed by third bodies, and mass comparatively sparse. Both delta and star configurations were concentrations (mascons) caused by magma upwellings in the tested. The presence of water ice at the Moon’s poles makes Moon create a highly anisotropic gravity field. However, the them attractive sites for exploration and future Lunar bases; star presence of the Earth may be exploited to place spacecraft at configurations with redundant polar coverage would thus be the nearby Earth-Moon Lagrangian points to reduce the total more desirable [4]. number of spacecraft needed for a GNSS constellation. The spacecraft bus was allowed to vary from masses Previous work has determined the minimum number of typically seen in modern navigation systems [5] to spacecraft as satellites required for total GNSS coverage of an arbitrary small as CubeSats. The cost of the spacecraft itself and its spherical body [1]. Some literature even addresses the particular corresponding launch cost to obtain the mission orbit are non- case of the minimum satellites required for coverage of the negligible. Since recent advances in microelectronics have Moon using CubeSats [2]. Still, more general studies [3] have allowed for the use of CubeSats for navigation [2], both determined the optimal constellations for a spherical body using CubeSats in the < 6U range and traditional spacecraft bus sizes algorithms such as symmetric/inclined-plane orbits and streets were considered. of coverage, including the time required to find the optimal When designing orbits around the Moon, two major solution for total coverage for each algorithm. However, these considerations must be made for the semimajor axis of the orbit: approaches are implicitly predicated on the satellite count itself perturbations from Lunar mascons and gravitational contributing to the majority of the cost; over the lifetime of the perturbations from third bodies. The former’s strength relative constellation, other factors, such as launch Delta-V (ΔV) costs, to the first-order term of the Moon’s gravity increases with stationkeeping, and disposal, may contribute costs high enough decreasing semimajor axis [6] and the latter’s relative strength to warrant their exploration and minimization, especially for to the Moon’s gravity increases with increasing semimajor axis, constellations on other celestial bodies. This paper constitutes a making the two conflicting parameters to be optimized. Forces minimization of the first two of these costs for traditional from third bodies such as the Sun and Earth are included as symmetric, inclined orbits. perturbations due to the orbits being within the sphere of influence of the Moon [7]. Lunar gravity models have been developed by multiple II. Configurations and Considerations gravity mapping missions. In particular, the Gravity Recovery Regular, periodic configurations were selected using the and Interior Laboratory (GRAIL) mission provides the most Walker-Delta and Walker-Star configurations commonly used recent and comprehensive Lunar gravity model. GRAIL has for contemporary navigation systems. For delta configurations, determined that the Lunar surface gravity field varies by as orbit planes with fixed-inclinations have their ascending nodes much as 1.913 Gals (1.913 × 10−3 g) [6]. This corresponds to equally spaced about a central body. For star configurations, a value within 2% of the perturbation force strength from Earth 90° inclination orbits are equally-spaced by their ascending node [1]. Within each plane, all satellites are equally spaced by 1 University of Florida Table 1 - Parameters for Constellations Variable Value 15 : 15 : 90 푖 (inclination, degrees) 90 3500 : 100 : 10000 푎 (semimajor axis, km) (푅푀표표푛 + 200) : 50 : (푅푀표표푛 + 2000) 2 : 1 : 10 푝 (orbit planes) 2 : 1 : 10 10 : 1 : 50 푛 (satellite count) 10 : 1 : 1000 0 : 4 : 4 퐿 (number of libration point satellites) 0 : 4 : 4 The format of the parameter values is given in (initial value):(step):(final value). Permutation sets of these five variables were taken using the respective line from each variable (i.e. permutation set one consists of the first line of 푎, 푖, 푝, 푛, and 퐿). Inclinations only 90 degrees correspond to star configurations. A set of low- altitude star configurations with many satellites was tested. 푅푀표표푛 = 1738 푘푚 is the equatorial radius of the Moon. These extremely low altitude orbits were tested at an inclination of 90° because of previous literature which found stable orbits within 2° of this inclination [22]. Three additional constellations corresponding to the Moon-radius scaled versions of GPS, GLONASS, and Galileo were also tested. at a Lunar altitude of 200 km, and decays to less than 1.0% the Earth vector will rotate at the same rate as this rotation. perturbative strength of Earth’s gravity at a semimajor axis of Over a ten-year propagation, the Sun vector will 5500 km. similarly rotate around the selenocentric frame ten Perturbations from non-gravitational sources such as solar times. radiation pressure (SRP), infrared emission, and Lunar albedo 3. Non-gravitational perturbation forces are negligible at radiation pressure are finite but small enough to ignore, using all altitudes. These include SRP, albedo pressure, data from GRAIL as a reference. SRP was the largest non- infrared emission, and the Lunar atmosphere. The gravitational perturbation for this mission and introduced an perturbative strength of harmonic terms is reduced error on the order of 1% of the range of gravitational forces on below 1.0% of Earth’s perturbative strength at the Lunar surface [6]. For CubeSats, this error is expected to be semimajor axes greater than 5500 km and may be proportionally smaller based on the spacecraft’s surface area. neglected beyond this point to improve computation Although further from the Lunar surface, these forces become times. more significant due to the decreased strength of variations felt 4. Effects such as apsidal precession, evection, libration, due to anisotropies in Lunar gravity, they remain the same precession of the line of nodes, and other perturbations relative to Sun and Earth gravitational perturbations, which are to the Moon’s orbit are negligible due to their small on the order of Lunar gravity field variations at the Lunar magnitude relative to their mean value [8]. surface. Thus, these non-gravitational perturbations may be 5. Line of sight is enough to provide coverage to a point. neglected without significant discrepancies between the real No considerations were made towards limitations on and computed total optimal ΔV for stationkeeping. the cone of contact for instrumentation and antennae. Ground segment optimization and placement was not Furthermore, points of contact on the Lunar surface considered in computation due to the current uncertainty in the were assumed to have no elevation mask limitations due practicalities and details of such an establishment. For the same to irregular terrain. reasons, considerations for the end-of-life disposal of all 6. Halo orbits around Earth-Moon libration points will spacecraft were withheld. have small enough deviations from the libration point and have periods short enough that satellites at these III. Computation points may be modelled as being at the libration point itself. This removes the need to solve for the halo orbits A. Approximations in the circular restricted three-body problem. Simplifying assumptions were made to expedite the 7. The differences in ΔV needed to cause commercial computation process: rocket booster stages carrying the satellites to reach the 1. The