[Paper Review] A Time-Dependent TSP Formulation for the Design of an Active Debris Removal Mission using Simulated Annealing

Lorenzo Federici, Alessandro Zavoli+1 more|arXiv (Cornell University)|Sep 23, 2019

Optimization and Search Problems29 references9 citations

TL;DR

This paper formulates Active Debris Removal (ADR) mission design as a Time-Dependent Traveling Salesman Problem (TDTSP) with time-dependent costs based on departure time and transfer duration, solving it via Simulated Annealing to jointly optimize target sequence and rendezvous epochs. The method achieves near-optimal ∆V performance with significantly reduced computational cost, demonstrating solutions for up to 20 debris targets with total ∆V as low as 0.74495 km/s.

ABSTRACT

This paper proposes a formulation of the Active Debris Removal (ADR) Mission Design problem as a modified Time-Dependent Traveling Salesman Problem (TDTSP). The TDTSP is a well-known combinatorial optimization problem, whose solution is the cheapest mono-cyclic tour connecting a number of non-stationary cities in a map. The problem is tackled with an optimization procedure based on Simulated Annealing, that efficiently exploits a natural encoding and a careful choice of mutation operators. The developed algorithm is used to simultaneously optimize the targets sequence and the rendezvous epochs of an impulsive ADR mission. Numerical results are presented for sets comprising up to 20 targets.

Motivation & Objective

• To address the growing urgency of space debris mitigation by optimizing multi-target ADR mission trajectories.
• To model the ADR problem as a Time-Dependent TSP (TDTSP) where transfer costs depend on both departure time and transfer duration.
• To jointly optimize the sequence of debris targets and their rendezvous epochs to minimize total propellant consumption (∆V).
• To develop a computationally efficient meta-heuristic solution using Simulated Annealing with a tailored mutation strategy and ∆V heuristic.
• To validate the approach on sets of up to 20 debris objects, demonstrating near-optimal performance with minimal computational overhead.

Proposed method

• Formulates the ADR mission as a TDTSP variant where cost depends on both departure time and transfer duration (approach B), capturing time-dependent orbital dynamics.
• Employs Simulated Annealing with a natural permutation encoding to explore the solution space of target sequences and encounter times.
• Uses a fast, analytic ∆V heuristic for single-impulse rendezvous to estimate transfer costs without solving the full trajectory problem.
• Applies a two-level optimization: first, optimize sequence and epochs via SA; second, refine each leg using a parameterized multi-impulse transfer model to minimize individual ∆V.
• Implements time-discretization with a factor of 3 to balance solution accuracy and computational efficiency, enabling tractable search over time windows.
• Validates the approach by comparing results against full mixed-integer nonlinear programming (MINLP) solutions, showing close agreement with far less computation.

Experimental results

Research questions

• RQ1 Can a TDTSP formulation effectively model the time-dependent nature of multi-rendezvous ADR missions with variable transfer durations and epoch-dependent costs?
• RQ2 How well does Simulated Annealing with a custom mutation operator perform in jointly optimizing target sequence and rendezvous timing for ADR missions?
• RQ3 To what extent can a fast ∆V heuristic approximate the true optimal transfer cost while reducing computational load in ADR trajectory design?
• RQ4 How does time-discretization affect the quality and computational cost of the solution in a large-scale ADR mission with up to 20 targets?
• RQ5 Can the proposed two-level optimization strategy—sequence/epoch optimization followed by individual leg refinement—achieve near-optimal ∆V with significantly reduced computational effort compared to full MINLP?

Key findings

• For a 10-target mission, the best solution achieved a total ∆V of 0.44876 km/s, with a sequence and epoch schedule that minimized propellant use.
• For a 20-target mission, the best solution achieved a total ∆V of 0.74495 km/s, demonstrating scalability of the method to complex ADR scenarios.
• The method achieved solutions within close proximity of the full MINLP solution, with a 90% reduction in computational cost due to time-discretization and heuristic evaluation.
• The use of a time-discretization factor of 3 enabled efficient search while maintaining high solution quality, as confirmed by comparison with higher-resolution benchmarks.
• The Simulated Annealing approach with tailored mutation operators consistently outperformed simpler meta-heuristics in convergence speed and solution quality for the TDTSP formulation.
• Refining individual legs using impulse parameterization after sequence optimization led to a 5–10% improvement in ∆V efficiency compared to fixed-epoch solutions.