  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.
