This thesis studies structure-exploiting embedded optimization methods that extend cached Riccati-based MPC solvers for resource-constrained robotic systems. Building on first-order alternating direction method of multipliers (ADMM) solvers and offline caching, this thesis develops three complementary contributions. First, we develop conic extensions of embedded MPC solvers that support second-order cone constraints, allowing richer modeling of thrust, friction, and glideslope constraints while retaining real-time deployability on microcontrollers. Second, we present a semidefinite programming framework for embedded MPC that enables certifiable obstacle avoidance through lifted convex relaxations and an a posteriori rank-1 safety certificate. Third, we introduce First-Order Adaptive Caching, a sensitivity-based method for updating cached solver quantities online as the ADMM penalty parameter changes, improving convergence without sacrificing caching benefits. Together, these methods expand cached embedded MPC along three dimensions: expressiveness, safety, and robustness. Across simulation, microcontroller benchmarks, and Crazyflie hardware experiments, the resulting solvers demonstrate improvements in constraint handling, solve time, and safety-critical behavior under aggressive and dynamically changing conditions.
We introduce TinySDP, the first semidefinite programming solver designed for embedded systems, enabling real-time model-predictive control (MPC) with formal safety guarantees on microcontrollers for problems with nonconvex obstacle constraints. Our approach integrates positive-semidefinite cone projections into a cached-Riccati-based ADMM solver, leveraging computational structure for embedded tractability. We pair this solver with an a posteriori rank-1 certificate that converts relaxed solutions into explicit geometric guarantees at each timestep. On challenging benchmarks, e.g., cul-de-sac and dynamic obstacle avoidance scenarios that induce failures in local methods, TinySDP achieves collision-free navigation with up to 73% shorter paths than state-of-the-art baselines. We validate our approach on a Crazyflie quadrotor, demonstrating that certifiable semidefinite constraints can be enforced at real-time rates for agile embedded robotics.