Multiple Space Debris Collecting Mission - Debris selection and Trajectory optimization

A possible mean to stabilize the LEO debris population is to remove each year 5 heavy debris like spent satellites or launchers stages from that space region. This paper investigates the DeltaV requirement for such a Space Debris Collecting mission. The optimization problem is intrinsically hard since it mixes combinatorial optimization to select the debris among a list of candidates and functional optimization to define the orbital maneuvers. The solving methodology proceeds in two steps : firstly a generic transfer strategy with impulsive maneuvers is defined so that the problem becomes of finite dimension, secondly the problem is linearized around an initial reference solution. A Branch and Bound algorithm is then applied to optimize simultaneously the debris selection and the orbital maneuvers, yielding a new reference solution. The process is iterated until the solution stabilizes on the optimal path. The trajectory controls and dates are finally re-optimized in order to refine the solution. The method is applicable whatever the numbers of debris (candidate and to deorbit) and whatever the mission duration. It is exemplified on an application case consisting in selecting 5 SSO debris among a list of 11.

💡 Research Summary

The paper addresses the problem of planning a Space Debris Collecting (SDC) mission that must de‑orbit five heavy objects (e.g., spent satellites or upper stages) from low‑Earth orbit (LEO) each year. The authors formulate a combined optimisation problem that simultaneously selects which debris to service from a catalogue of candidates and determines the orbital transfers between successive targets. The transfer problem is a continuous optimal‑control problem: the spacecraft state includes mass, position and velocity (or orbital elements) and the control vector consists of thrust magnitude and direction. Fuel consumption is expressed as the integral of thrust over time, and each leg has a bounded duration. The path problem is a variant of the Travelling Salesman Problem (TSP). Unlike the classic TSP, only a subset (n) of the N candidate nodes must be visited, a global mission‑duration limit (Tmax) must be respected, and a minimum dwell time (Tdeorb) is required at each visited debris for capture, de‑orbiting and release. Binary variables s_ij indicate whether a leg i→j is selected; auxiliary binary variables x_k, y_k, z_k enforce at most one incoming and one outgoing edge per visited node and guarantee a single, loop‑free path with a unique start and end node.

The overall formulation is a mixed‑integer nonlinear program (MINLP) with infinite‑dimensional control functions. To make the problem tractable, the authors first prescribe a generic impulsive‑transfer strategy, reducing the control law to a finite set of ΔV vectors and associated departure/arrival epochs. An initial feasible solution is generated, and the cost and time functions are linearised around this reference. The resulting linearised model is a mixed‑integer linear program (MILP) that can be solved with a Branch‑and‑Bound (B&B) algorithm. At each B&B node the selected debris set is fixed, the optimal ΔV for each leg is recomputed, and lower/upper bounds are used to prune the search tree. Once the B&B process yields a candidate optimal path (debris selection and order), the authors return to the original continuous formulation and re‑optimise the thrust profiles and timing for each leg using direct or indirect optimal‑control methods, thereby refining the impulsive approximation. The whole procedure is iterated until the solution stabilises.

The methodology is demonstrated on a case study involving 11 Sun‑synchronous orbit (SSO) debris candidates. The authors compute pairwise transfer costs C_ij and durations T_ij using the impulsive model, then apply the B&B‑linearisation loop to obtain the optimal subset of five debris and the visiting sequence. The resulting total ΔV is significantly lower than that obtained by naïve heuristics (e.g., a simple nearest‑neighbour tour), and the total mission time respects the one‑year constraint.

Key contributions include: (1) a clear decomposition of a highly coupled combinatorial‑continuous problem into a finite‑dimensional surrogate that can be linearised; (2) the integration of a B&B scheme with iterative re‑optimisation of the underlying optimal‑control problem, enabling simultaneous handling of integer selection decisions and continuous trajectory design; (3) a demonstration that the approach scales to any number of candidates and any mission duration, provided a suitable impulsive transfer template is defined. Limitations are acknowledged: the impulsive assumption may be inaccurate for low‑thrust electric propulsion, and linearisation introduces approximation errors that must be mitigated by the final re‑optimisation step. Future work is suggested to incorporate realistic low‑thrust dynamics directly into the optimisation loop and to extend the framework to multi‑objective formulations (fuel, time, collision risk).