Control Design along Trajectories with Sums of Squares Programming

TL;DR

This paper introduces a formal control design method using sums of squares programming to ensure stability and safety for complex robot tasks, capable of handling uncertainties, input limits, and time-varying dynamics.

Contribution

It presents a novel polynomial control synthesis approach that maximizes invariant funnels for trajectory tracking, with efficient semidefinite optimization and practical validation on a torque-limited underactuated robot.

Findings

Successfully designed controllers for a torque-limited Acrobot

Achieved larger invariant funnels compared to previous methods

Validated effectiveness through extensive simulations and hardware tests

Abstract

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.