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 extend TinyMPC, an open-source, high-speed solver targeting low-power embedded control applications, to provide support for second-order cones, as well as `C++` code generation from `Python`, `MATLAB`, and `Julia` for easy deployment. Microcontroller benchmarks show that our solver provides up to a two-order-of-magnitude speedup, ranging from 10.6x to 142.7x, over state-of-the-art embedded solvers on QP and SOCP problems, and enables us to fit order-of-magnitude larger problems in memory. We validate our solver's deployed performance through simulation and hardware experiments, including conically-constrained trajectory tracking on a 27g Crazyflie quadrotor. To get started with Conic-TinyMPC, visit our documentation, examples, and the open-source codebase at [tinympc.org](https://tinympc.org/).