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Sums-of-squares (SOS) optimization is a promising tool to synthesizecertifiable controllers for nonlinear dynamical systems. Building upon priorworks, we demonstrate that SOS can synthesize dynamic controllers with boundedsuboptimal performance for various underactuated robotic systems by findinggood approximations of the value function. We summarize a unified SOS frameworkto synthesize both under- and over- approximations of the value function forcontinuous-time, control-affine systems, use these approximations to generateapproximate optimal controllers, and perform regional analysis on theclosed-loop system driven by these controllers. We then extend the formulationto handle hybrid systems with contacts. We demonstrate that our method cangenerate tight under- and over- approximations of the value function withlow-degree polynomials, which are used to provide stabilizing controllers forcontinuous-time systems including the inverted pendulum, the cart-pole, and thequadrotor as well as a hybrid system, the planar pusher. To the best of ourknowledge, this is the first time that a SOS-based time-invariant controllercan swing up and stabilize a cart-pole, and push the planar slider to thedesired pose.

Approximate Optimal Controller Synthesis for Cart-Poles and Quadrotors via Sums-of-Squares