Control Design along Trajectories with Sums of Squares Programming
This provides formal guarantees for robot control tasks, addressing challenges like time-varying dynamics and input saturations, but is incremental as it builds on existing sums of squares methods.
The paper tackles the problem of ensuring stability and safety in robot control by designing controllers that maximize invariant funnels to a goal set, using sums of squares programming for certification, and demonstrates this on an underactuated double pendulum with simulation and hardware validation.
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.