Low Thrust, Minimum-Propellant Optimization for Multi-Orbit Rendezvous with Uncooperative LEO Space Debris using Indirect Methods

Since the beginning of space exploration, thousands of satellites have been launched into Earth orbit. Over the years, however, the increasing number of artificial satellites placed in orbit has been matched by an increase in the probability of collisions between the satellites themselves. Such events would produce orbiting fragments, each of which would exponentially increase the probability of further collisions occurring in the future. The criticality of the Kessler Syndrome, acknowledged by researchers since the 1980s, has led to the formation of a space debris belt around Earth, particularly in Low Earth Orbits (LEO) below 2000 km. This creates a severe threat to both the safety and functionality of space assets, as collisions with space debris can cause substantial damage or complete destruction of spacecraft; in recent decades, therefore, mission disposal operations have increasingly become relevant. Notably congested regions include the Sun-Synchronous Orbits (SSO) and Polar Orbits (PEO) at altitudes of 600-1200 km and inclinations of 80-105 degrees, particularly around 800-900 km. These orbits, critical for Earth observation, necessitate effective debris removal solutions for non-functional or undesirable objects. This paper proposes evaluating an optimal trajectory for a Space Debris Removal (SDR) mission utilizing electric propulsion (EP) to remove debris from such orbits. The indirect method is well suited for very high accuracy in solving the optimal EP trajectories, exploiting the Pontryagin’s minimum Principle and ensuring minimum-propellant expenditure through the extremization of the Hamiltonian and the application of bang-bang control. An essential aspect of SDR missions is the capability for controlled de-orbit and re-entry into the Earth’s atmosphere within 25 years, according to current space regulations. The mission is divided into two key phases: 1) a propulsive phase, where the spacecraft transitions from an initial parking orbit to the targeted debris using low-thrust EP arcs, and 2) a de-orbiting phase, which involves a controlled spiral descent and re-entry into Earth’s atmosphere at an altitude of 100 km, utilizing the remaining propellant with the throttle set at 10%. The dynamic model is based on the two-body problem, accommodating additional perturbations such as Earth’s non-sphericity and atmospheric drag during the de-orbiting phase.