Low cost satellite constellations for nearly continuous global coverage

Satellite services are fundamental to the global economy, and their design reflects a tradeoff between coverage and cost. Here, we report the discovery of two alternative 4-satellite constellations with 24- and 48-hour periods, both of which attain nearly continuous global coverage. The 4-satellite constellations harness energy from nonlinear orbital perturbation forces (e.g., Earth’s geopotential, gravitational effects of the sun and moon, and solar radiation pressure) to reduce their propellant and maintenance costs. Our findings demonstrate that small sacrifices in global coverage at user-specified longitudes allow operationally viable constellations with significantly reduced mass-to-orbit costs and increased design life. The 24-hour period constellation reduces the overall required vehicle mass budget for propellant by approximately 60% compared to a geostationary Earth orbit constellation with similar coverage over typical satellite lifetimes. Mass savings of this magnitude permit the use of less expensive launch vehicles, installation of additional instruments, and substantially improved mission life.

sistent modeling assumptions and fidelity facilitated comparison with Chaos benchmarking and is shown in Section 2.2 of the Supplement to provide sufficient numerical accuracy to support early architecture design efforts. The work performed in pursuit of exploring these questions led to the constellation designs reported in this study, which represent an unreported region of the broader trade-space of constellation design alternatives for near global coverage.
Supplementary Tables 1 and 2 detail the orbital elements for the 48-and 24-hour period constellations reported in this study, respectively. The orbital elements reported here are Brouwer mean elements in a true-of-date frame. The epoch for the elements as presented is November 26, 1995 at midnight GMT in order to maintain consistency with prior published benchmark results 4 .
Although the reported designs represent only two of the many tradeoff alternatives identified in this study, they represent some of the most significant findings given their balance of nearly continuous global coverage and greatly reduced station-keeping propellant costs.
The constellation designs vary significantly from the classical benchmark Draim constellation 2 . One such significant difference is the eccentricity (e) of the reported orbits. Eccentricity describes the shape of the orbit. Satellites in orbit around a planet have eccentricity between 0 and 1; a value of 0 means the orbit is perfectly circular, and larger values mean the orbit is more elliptical in shape. The low eccentricity values of the orbits indicate that they are nearly circular for both the 24-and 48-hour period constellations, in contrast with the eccentricity of 0.263 for a Draim constellation.
As an orbit becomes more eccentric, the point of closest approach to the Earth (perigee) moves closer to the Earth, while the farthest point (apogee) moves farther away. In most cases, it is important to control the orientation of the perigee point. The argument of perigee (ω) describes the position of the perigee point relative to the point on the orbit which ascends across the equator, as illustrated in Figure 5. Orbits with low eccentricity require less propellant to maintain the argument of perigee than orbits with larger eccentricity. Because the constellations have small eccentricities, they essentially eliminate the need for argument of perigee maintenance through propulsive maneuvers.

Supplementary
The orientation of an orbit in space can be described by a series of Euler angle rotations; two of these angles are called the inclination (i) and the right ascension of the ascending node (RAAN).
The inclination is the angle at which the orbit is tilted out of the equatorial plane. Once the orbit is tilted out of the equatorial plane, it can be rotated around the Earth spin axis to define any one of a continuum of orbit planes. The RAAN is this angle of rotation around the Earth spin axis referenced by convention to the direction of the sun at the vernal equinox. The classical Draim constellation configuration places satellites in equally-spaced planes, such that the difference in the RAANs from one spacecraft to the next is always 90 • . However, the constellations proposed here do not share this uniform placement in RAAN.
In addition to the difference in RAAN spacing, the orbits exhibit much higher, nearly polar inclinations when compared with the Draim inclination of 31.3 • . To understand the advantage of these higher inclinations, it is necessary to understand a fundamental perturbation to orbital motion.
Anisotropies in the shape and mass distribution of the Earth create spatially-variant perturbations on the gravitational force experienced by a spacecraft. These perturbations cause the RAAN of orbits to change over time at a rate which is a function of the orbital parameters. The perturbations have a non-uniform effect on orbits with inclinations similar to the Draim inclination, causing the orbit planes to shift relative to one another over the course of the lifetime of the constellation.
However, at nearly polar inclinations the perturbations cause a smaller and more uniform motion of the orbit planes. This reduces the station-keeping maintenance required to maintain the shape of the constellation.
The constellation designs reported here represent just two of the most attractive designs identified by the MOEA during its exploration of the feasible design space. Supplementary Figure   1a illustrates the efficient frontier of constellation designs discovered throughout this work. The The presence of nearby options in the performance space highlights the diversity of constellation designs available to attain long-life constellations for nearly continuous global coverage.

Supplementary Methods
At the most general level, the performance of a constellation of satellites is assessed using metrics that quantify the specific performance characteristics that relate to the geometric access they provide for global observations. Geometric access represents time periods when a satellite in the constellation can perform its mission in sensing a specific location of interest. At its most general, access is calculated as time periods when any satellite within the constellation has an unobstructed line of sight to a target. When a mission is more defined, the locus of targets a satellite has access to at any instant may be constrained by a number of factors which inhibit the satellites ability to perform its mission. These constraints take the form of availability and visibility constraints.
Availability constraints limit the time during which a satellite is available to perform its mission, representing operational considerations such as duty cycle. Visibility constraints constrain the geometrical conditions between targets, satellites, and potentially other bodies relevant to performing a mission. Because of the general nature of this work, no availability or visibility constraints are levied in the calculation of access. However, the content in the Nearly Continuous Global Coverage section of Results in the main text studies the sensitivity of the reported constellation performance to the target elevation angle, which is the angle between the horizon and the satellite from the targets perspective. This is a commonly used visibility constraint to account for terrain masking on the ground when designing a constellation.
Constellations typically do not have continuous access or coverage to a target, so many access metrics measure the gaps in coverage. This work considers three gap-based access metrics: The lifetime simulations in this work employ the Aerospace Corporations SHARK propagator. SHARK is a general-purpose orbit propagator that can numerically simulate the motion of any Earth-orbiting spacecraft using models that have no known numerical singularities. Brouwer 6 and Chao 7 provide equations which approximate the secular motion that results from these pertur-bations. However, over longer time frames solutions derived from these approximations can drift potentially far away from the true solution. Although this drift is acceptable for some applications, it can lead to significant accuracy issues in the evaluation of constellation coverage performance due to the sensitivity of this performance to constellation phasing. Key technical features for the SHARK propagator are summarized below for (1) the integrator and (2) quantifying perturbations.
The default integrator is the eighth order DOPRI8 Runge-Kutta integrator 8 . An alternate option for conservative systems (no drag, no SRP) is the Runge-Kutta-Nystrom method, which is the primary method employed in this work 9 . Both of these algorithms are desirable for their accuracy and speed. While higher order integrators require more steps, they likewise allow larger steps to be taken. This directly translates to higher computational efficiency and speed, as noted by Montenbruck 10 . The RKN12(10)17M integrator described by Dormand 9 permits faster computational analysis than standard Runge-Kutta techniques for conservative systems, which is consistent with the perturbations modeled in this work. Both methods use an adaptive step size technique 11 to integrate in nonsingular cartesian coordinates (position and velocity) or in nonsingular equinoctial elements.
To quantify perturbations, a full geopotential model is available, supporting field sizes up to 80x80. The standard Legendre expansion can be used (which has singularities at the poles).
Alternatively, the user can select a non-singular technique described by Pines 12  The propagator provides the orbit states necessary to assess coverage performance. Evaluation of the measures discussed in Section 2.1 rely on computing the start and stop times of the visibility of a given satellite to a specified ground point. This process is complicated by a numerically generated orbit state history, which can be solved in a brute force manner via repeated root finding around times where satisfaction of the constraints associated with visibility change.
A more computationally efficient approach to accomplishing this is to store the spacecraft state history in a set of orbit tables. The process of creating and using these orbit tables is as follows. As the spacecraft state is propagated forwards in time, the integrated states are stored over a prescribed The station-keeping method used in the orbital simulations follows an anchor-chaser paradigm.
Although the anchor spacecraft performs maneuvers to maintain certain orbital parameters, it does not actively seek to maintain any specific in-track phasing with other members of the constellation. Similarly, the chaser spacecraft performs maneuvers to maintain specific orbital parameters, but it also is required to maintain an in-track position relative to the anchor spacecraft. The anchor spacecraft maintains its semi-major axis, perigee radius, and inclination. These orbital elements are constrained within pre-specified lower and upper limits. In the simulations, this is implemented in a simple manner that controls each orbital parameter separately. There is no effort to combine the maneuvers to reduce the required ∆V. The following algorithm describes the method employed to identify maneuver times and magnitudes: 1. Propagate the spacecraft until one of the specified elements (semi-major axis, perigee radius, inclination) crosses a lower/upper boundary.
2. Compute the maneuver that will return the offending element to its nominal value.
3. Execute the maneuver at the specified location.

Continue with the propagation until the next violation.
The allowable ranges of the orbital elements of the lifetime simulation are shown in Supplementary Table 3. As an example, with a nominal initial inclination of 86.49 • , the allowable range would be 83.99 • to 88.99 • . When the inclination exceeds either of these bounds, the model executes a maneuver to return the inclination to its nominal value of 86.49 • .
There are three primary goals of maintaining these specific parameters (semi-major axis, perigee radius, and inclination): (1) By controlling the semi-major axis, the orbital period is maintained, (2) By controlling the perigee radius, the eccentricity is maintained, and (3) By maintaining all three of these parameters, a secondary effect is to approximately control the secular rate of the RAAN. Supplementary Equation 1 describes this rate as a function of the three controlled parameters, as well as the Earth's J 2 term, and the planet's radius, R e . Controlling the three orbital parameters in this equation thus implicitly leads to control of the secular RAAN rate as well. For the anchor spacecraft, there is no explicit effort to directly maintain the RAAN, the argument of perigee, or the longitude. It is possible that more sophisticated approaches to coordinated stationkeeping of the constellation could lead to designs which further improve on the results presented Regarding the three chaser spacecraft, they are required to maintain the perigee radius and inclination, as well as the in-track phase (position relative to the anchor). For the former two elements, the techniques used are the same as for the anchor. For in-track phasing, each chaser maintains a station-keeping cycle of 14 days. At the beginning of each cycle, the spacecraft performs a maneuver that targets a particular value of semi-major axis. This value is chosen such that after 14 days, the spacecraft will arrive at the nominal phase offset. During the 14-day cycle, the phase value drifts freely, but at the end of the cycle, the phase returns to the nominal value. The maximum phase excursions during the station-keeping cycle can be controlled by increasing or decreasing the station-keeping cycle duration, which will increase or decrease the excursions, respectively. For these constellations, the 14-day station-keeping cycle provides phase control within approximately 0.5 • .
Controlling the semi-major axis indirectly maintains the phase of the chasers. Thus, the chasers also maintain control of the three key parameters: semi-major axis, perigee radius (eccen-tricity), and inclination. This provides control of the secular RAAN rate for the chasers. As with the anchor, there is no direct control of the RAAN, the argument of perigee, or the longitude.
The Borg algorithm is used to find the Pareto-approximate (or nondominated) set of solutions for each problem formulation. These solutions are not inferior in any objective with regard to all other feasible solutions 22 . The term Pareto-approximate reflects the fact that the optimal set of solutions remains unknown, but we aim to provide as close an approximation as possible. The results reported here were obtained from search evaluations of over 5 million simulated orbital designs. High-fidelity orbit propagation and careful simulation of station-keeping significantly increased the computational demands of simulating each candidate design. More details on the simulation-optimization framework can be found in 15 .
Supplementary Table 4     The 24-hour period case is summarized in Supplementary Table 5 and incorporates lessons learned from trends identified during the 48-hour search. The semi-major axis is given more freedom to vary from the ideal 24-hour value to better quantify the desirability of groundtrack repeatability. As with the 48-hour case, the MOEA converged to the ideal 24-hour value. The allowable initial eccentricity for the anchor satellite ranges from 0.02 to 0.7628, and the eccentricities of the remaining satellites were allowed to vary 0.01 from the anchor value. As with the 48-hour case, these ranges preclude the eccentricities from entering restricted ranges. The 48-hour case did not Dashed lines indicate orbit radii at significant points. In addition to a broad range of eccentricities, the MOEA could explore a wide range of inclinations, RAANs, arguments of perigee, and true anomalies at epoch. anomaly variables all vary over the full available range. For both 24-and 48-hour cases, the field of view of each spacecraft is the area under the spacecraft with elevation angle greater than 0 • ; this carries with it an implicit assumption neglecting any terrain blockage.