This thesis investigates two fundamental challenges in robotics: ensuring reactive safety for agile systems and achieving scalable planning for contact-rich manipulation. Despite arising in distinct contexts, both problems hinge on reasoning about system dynamics and constraints. We argue that reachable sets—long studied in control theory—provide a unifying computational primitive for tackling both.

In the first part, we study reactive safety for uncertain and high-dimensional systems. We develop new methods for synthesizing control barrier functions (CBFs) using both formal optimization and learning-based techniques. Specifically, we introduce a robust-adaptive synthesis framework based on sum-of-squares programming, which guarantees safety under parametric uncertainty while reducing unnecessary conservatism. To address scalability, we propose an adversarial training framework that learns neural CBFs capable of handling higher-dimensional dynamics. Across cartpole, quadrotor, and other benchmarks, our controllers achieve 100% safety while interfering substantially less with nominal performance objectives than existing baselines.

In the second part, we extend reachable-set methods to long-horizon planning for contact-rich manipulation. We present a hierarchical planner for bimanual SE(2) reorientation tasks that uses mutual reachable sets as discrete motion primitives. By covering object space with a compact set of such primitives, we reduce the combinatorial complexity of hybrid contact planning to shortest-path search on a small graph. This yields efficient and expressive global plans that outperform sampling-based methods, with successful demonstrations on real robotic hardware.

Together, these contributions demonstrate how reachable sets can serve as a versatile foundation across the robotics stack: from low-level safety filters for agile systems to high-level planning abstractions for dexterous manipulation. This unifying perspective suggests broader opportunities for integrating reachability-based methods into robust and scalable robotic autonomy.