Orbital Facility Location Problem for Satellite Constellation Servicing Depots

TL;DR

The paper tackles the problem of optimally locating on-orbit servicing depots for high-altitude satellite constellations by formulating the Orbital Facility Location Problem (OFLP), a discrete facility-location problem that uses Equivalent Mass to Low Earth Orbit (EMLEO) as a unified cost metric. It deploys Q-Law-based low-thrust transfers to compute the propulsion costs between depots and clients, discretizes the candidate depot slots in orbital element space, and solves the OFLP to determine both the number and placement of depots as well as allocation of clients. A continuous-space refinement follows the discrete OFLP solution, implemented as a nonlinear program per depot and solved with gradient-free optimization to further reduce EMLEO. An a posteriori analysis of multiclient servicing trips is also conducted to assess potential cost savings from bundling visits. The methodology is demonstrated on GPS and Galileo constellations, revealing notable EMLEO reductions when serving both constellations together and highlighting design insights such as clustering depots by orbital planes and the benefits of relaxing the depot locations through refinement.

Abstract

This work proposes an adaptation of the Facility Location Problem for the optimal placement of on-orbit servicing depots for satellite constellations in high-altitude orbit. The high-altitude regime, such as Medium Earth Orbit (MEO), is a unique dynamical environment where existing low-thrust propulsion systems can provide the necessary thrust to conduct plane-change maneuvers between the various orbital planes of the constellation. As such, on-orbit servicing architectures involving servicer spacecraft that conduct round-trips between servicing depots and the client satellites of the constellation may be conceived. To this end, orbital facility location problem is a binary linear program, where the costs of operating and allocating the facility(ies) to satellites are considered in terms of the sum of Equivalent Mass to Low Earth Orbit (EMLEO), is proposed. The low-thrust transfers between the facilities and the clients are computed using a parallel implementation of a Lyapunov feedback controller. The total launch cost of the depot along with its servicers, propellant, and payload are taken into account as the cost to establish a given depot. The proposed approach is applied to designing on-orbit servicing depots for the Galileo and the GPS constellations.

Figures (13)

• Figure 1: Proposed procedure involving preparing the costs (turquoise), solving the OFLP (navy), and refining in continuous space (red)
• Figure 2: Approximate lifetime of currently active GPS Block IIF (blue/red lines, circle markers) and III (green lines, square markers), assuming a lifetime of 15 years from launch, except for Block IIF-2 (red line), which experienced a clock anomaly on 10 July 2023. RAAN corresponds to TLE values queried in December, 2022.
• Figure 3: Return trip conducted by servicer between facility $j$ and client $i$. Once the servicer arrives to the client $i$, the payload $m_{s,L}$ is deposited.
• Figure 4: Distribution of GPS and Galileo satellites in inclination and RAAN
• Figure 5: Contour of mass ratio $\phi$ for candidate orbital slots (left) and of transfer propellant mass $\tilde{c}_{ij}$ corresponding to round trip between candidate orbital slots and hypothetical client satellite on circular orbit with $a = 1.0 \, \mathrm{DU}$ and identical orbital plane ($i$ and $\Omega$) as the depot (right)