Approximate Optimal Controller Synthesis for Cart-Poles and Quadrotors Via Sums-Of-Squares

Sums-of-squares (SOS) optimization is a promis- ing tool to synthesize certifiable controllers for nonlinear dynamical systems. Building upon prior works [1], [2], we demonstrate that SOS can synthesize dynamic controllers with bounded suboptimal performance for various underactuated robotic systems by finding good approximations of the value function. We summarize a unified SOS framework to syn- thesize both under- and over- approximations of the value function for continuous-time, control-affine systems, use these approximations to generate approximate optimal controllers, and perform regional analysis on the closed-loop system driven by these controllers. We then extend the formulation to han- dle hybrid systems with contacts. We demonstrate that our method can generate tight under- and over- approximations of the value function with low-degree polynomials, which are used to provide stabilizing controllers for continuous- time systems including the inverted pendulum, the cart-pole, and the quadrotor as well as a hybrid system, the planar pusher. To the best of our knowledge, this is the first time that a SOS-based time-invariant controller can swing up and stabilize a cart-pole, and push the planar slider to the desired pose. Videos at https://youtu.be/QQR pPNPeyg; demo code at https://deepnote.com/workspace/lujieyang/project/hjb-sos.