MEAM Ph.D. Thesis: “Geometric Methods for Efficient and Explainable Control of Underactuated Robotic Systems”

Robots are complex, high-dimensional systems, governed by nonlinear, underactuated dynamics and evolving on non-Euclidean manifolds, posing numerous challenges for control synthesis and analysis. While optimization-based methods of control can flexibly accommodate diverse dynamics, costs, and constraints, they often demand coarse approximations or powerful onboard processors (infeasible for many aerial and space systems) due to their relatively poor computational efficiency. Although learned controllers can generally cope with more moderate onboard resources, the computational burden of offline training is heavy, and both the training pipeline and the policy obtained are often brittle. Conversely, explicit control laws designed analytically often have miniscule computational overhead and perform robustly, but they are typically only applicable to individual systems or a narrow class, limiting their broader usefulness.

Nonetheless, robots are not black-box nonlinear control systems—rather, their dynamics enjoy powerful properties (e.g., symmetry and mechanical structure) that can be leveraged to gain traction on control design problems. In this thesis, we explore the role of geometric methods in mitigating many of the above drawbacks, across both analytical and data-driven methods. We study the role of symmetry in identifying effective abstractions for trajectory planning in underactuated mechanical systems (in particular, “flat outputs”) and explore applications to task space planning for aerial manipulation. We also develop methods for synthesizing tracking controllers for mechanical systems evolving on the general class of homogeneous Riemannian manifolds, and give certificates for the almost global asymptotic stability of cascades, which often appear in the closed-loop dynamics of hierarchical controllers for underactuated systems. Lastly, we leverage symmetry to accelerate training of tracking controllers via reinforcement learning (by constructing “continuous MDP homomorphisms”), also improving converged performance.

In all these methods, a geometric perspective enables us to explainably construct abstractions that reduce dimensionality, enforce structure, and capture essential properties, all the while representing the system or problem in a form more convenient for analysis or design. In contrast to ad hoc methods, such reduced representations typically improve computational efficiency, while also encouraging generality over a broader class of systems and affording insight into why prior handcrafted approaches were successful for particular cases. Sometimes, such realizations also guide mechanical design, closing the control-morphology feedback loop and leading to synergies between a robot’s embodiment and its controller. By combining explainable abstractions with scalable computation, such methods build towards a future in which robotic systems move through their surroundings as capably and dynamically as their counterparts in Nature.