Master’s Thesis Offer: Conservative Reachability Analysis for Uncertain Attitude Dynamics on SO(3)

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 reachableset 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 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.

References

[1] Pierre-Jean Meyer, Alex Devonport, and Murat Arcak. Interval reachability analysis: Bounding trajectories of uncertain systems with boxes for control and verification. Springer Nature, 2021.

[2] Matthew Abate and Samuel Coogan. “Robustly forward invariant sets for mixed-monotone systems”. In: IEEE Transactions on Automatic Control 67.9 (2022), pp. 4947–4954.

[3] Torsten Koller et al. “Learning-based model predictive control for safe exploration”. In: 2018 IEEE conference on decision and control (CDC). IEEE. 2018, pp. 6059–6066.

[4] Matthias Althoff, Olaf Stursberg, and Martin Buss. “Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization”. In: 2008 47th IEEE Conference on Decision and Control. IEEE. 2008, pp. 4042–4048.

[5] Xin Chen, Erika ´Abrah´am, and Sriram Sankaranarayanan. “Flow*: An analyzer for nonlinear hybrid systems”. In: International Conference on Computer Aided Verification. Springer. 2013, pp. 258–263.