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Conservative Reachability Analysis for Uncertain Attitude Dynamics on SO(3)

24.03.2026, Abschlussarbeiten, Bachelor- und Masterarbeiten

This thesis investigates conservative reachability analysis for uncertain rigid-body attitude dynamics on SO(3), motivated by autonomous on-orbit servicing of tumbling satellites. It aims to develop mathematically rigorous methods for computing guaranteed over-approximations of future target spacecraft orientations under uncertainty. The work combines classical reachability theory with geometric modeling and validation on realistic orbital scenarios.

Master’s Thesis Offer – Reachability Analysis on SO(3)

At the Institute of Robotics and Mechatronics of the German Aerospace Center (DLR), we develop advanced autonomy methods for robotic proximity operations in orbit. The rapid growth of orbital debris and the increasing number of uncooperative satellites demand robust and reliable algorithms for safe on-orbit servicing missions. In such scenarios, servicer spacecraft must interact with tumbling targets whose orientation and inertia properties are uncertain.

To guarantee safety, it is essential to compute conservative over-approximations of the possible future orientations of a target. Classical reachability methods in Euclidean spaces construct convex over-approximations—such as hyperrectangles [1,2], ellipsoids [3], or zonotopes [4]—using conservative linearizations [4] or Taylor models [5]. Hyperrectangles have recently been extended to Lie groups [6]. However, extending these techniques to nonlinear attitude dynamics evolving on the Lie group SO(3) poses significant mathematical and computational challenges.

This thesis investigates conservative reachability methods that provably guarantee set containment of the true reachable states of a free-tumbling rigid body while maintaining tight bounds on the reachable set state approximation over long time horizons.

Your Contribution

We are seeking a highly motivated Master’s student to conduct research on conservative reachable-set estimation for uncertain rigid-body attitude dynamics in orbit.

You will:

  • Perform a comprehensive literature review of classical reachability methods (e.g., hyperrectangles, ellipsoids, zonotopes, Taylor models) and their extensions to nonlinear manifolds.
  • Study rigid-body attitude dynamics with parametric uncertainty (e.g., uncertain inertia matrices) evolving on SO(3).
  • Select and adapt one suitable conservative reachability method from the literature to the considered geometric setting.
  • Develop a mathematically rigorous formulation ensuring guaranteed set containment of the reachable orientations.
  • Evaluate the method on realistic tumbling satellite scenarios inspired by ESA's ENVISAT case studies.
  • Analyze tightness, computational efficiency, and long-horizon behavior of the resulting reachable-set over-approximations.

Project Scope

  • Literature review on conservative reachability analysis and set-propagation techniques.
  • Mathematical formulation of uncertain attitude dynamics on SO(3).
  • Adaptation of a classical conservative reachability framework to the SO(3) manifold.
  • Validation on simulated orbital tumbling debris scenarios.
  • Performance analysis with respect to tightness, scalability, and real-time applicability.

Your Qualifications

  • Enrolled in a Master’s program in Mathematics, Aerospace Engineering, Computer Science, Robotics, or a related field.
  • Strong interest in mathematics, especially geometry, Lie groups, and dynamical systems.
  • Solid background in control theory or nonlinear systems.
  • Programming experience in Python and/or MATLAB.
  • Strong analytical and problem-solving skills.
  • Ability to work independently and systematically.

We Offer

  • The opportunity to work on cutting-edge research in autonomous space robotics.
  • A mathematically rich topic at the intersection of geometry, control, and space applications.
  • Hands-on experience with high-performance numerical methods.
  • Collaboration within a leading research institute in space robotics.
  • The possibility to publish your results in a scientific journal or conference (subject to approval).

This thesis is ideal for students who enjoy rigorous mathematics, geometric reasoning, and translating theoretical guarantees into efficient computational tools for real-world space applications.

Supervisors: Alessandro Melone, Roberto Lampariello.

References

  1. Meyer, P.-J., Devonport, A., Arcak, M. Interval Reachability Analysis: Bounding Trajectories of Uncertain Systems with Boxes for Control and Verification. Springer, 2021.
  2. Abate, M., Coogan, S. “Robustly forward invariant sets for mixed-monotone systems.” IEEE Transactions on Automatic Control, 67(9), 4947–4954, 2022.
  3. Koller, T., Berkenkamp, F., Turchetta, M., Krause, A. “Learning-based model predictive control for safe exploration.” IEEE Conference on Decision and Control (CDC), 2018.
  4. Althoff, M., Stursberg, O., Buss, M. “Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization.” IEEE Conference on Decision and Control, 2008.
  5. Chen, X., Ábrahám, E., Sankaranarayanan, S. “Flow*: An analyzer for non-linear hybrid systems.” International Conference on Computer Aided Verification, 2013.
  6. Harapanahalli, A., Coogan, S. “Efficient reachable sets on Lie groups using Lie algebra monotonicity and tangent intervals.” IEEE Conference on Decision and Control (CDC), 2024.

Kontakt: alessandro.melone@dlr.de

Mehr Information

https://www.dlr.de/de/rm/jobs/studentische-arbeiten/masters-thesis-offer-conservative-reachability-analysis-for-uncertain-attitude-dynamics-on-so-3