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.
In this work, we introduce First-Order Adaptive Caching, which precomputes not only select matrix operations but also their sensitivities to hyperparameter variations, enabling online hyperparameter updates without full recomputation of the cache. We demonstrate the effectiveness of our approach on a number of dynamic quadrotor tasks, achieving up to a 63.4% reduction in ADMM iterations over the use of optimized fixed hyperparameters and approaching 70% of the performance of a full cache recomputation, while reducing the computational cost from O(n^3) to O(n^2) complexity. This performance enables us to perform figure-eight trajectories on a 27g tiny quadrotor under wind disturbances.