Control Design along Trajectories with Sums of Squares Programming

Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant “funnel” that leads to a predefined goal set. Our certificates of invariance are given in terms of sums of squares proofs of a set of appropriately defined Lyapunov inequalities. These certificates, together with our proposed polynomial controllers, can be efficiently obtained via semidefinite optimization. Our approach can handle time-varying dynamics resulting from tracking a given trajectory, input saturations (e.g. torque limits), and can be extended to deal with uncertainty in the dynamics and state. The resulting controllers can be used by space-filling feedback motion planning algorithms to fill up the space with significantly fewer trajectories. We demonstrate our approach on a severely torque limited underactuated double pendulum (Acrobot) and provide extensive simulation and hardware validation.

By: Anirudha Majumdar, Amir Ali Ahmadi, Russ Tedrake

Published in: Proceedings of 2013 IEEE International Conference on Robotics and Automation (ICRA)Piscataway, NJ,IEEE,, p.4054-61 in 2013


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