Control of a nonlinear underactuated system with adaptation, numerical stability verification, and the use of the LQR Trees algorithm

Abstract

Underactuated robotics, though surrounded by an established body of work, has certain limitations when nonlinear adaptive control principles are applied. This thesis applies a nonlinear adaptative controller that avoids many of these limitations using alterations inspired by the control of a similar underactuated system, the cart-pole. Due to the complexity of the system, a sums-of-squares MATLAB toolbox is used to generate a suitable Lyapunov Candidate used for proofs of stability, with claims of local stability made using Barbalat's Lemma. This provides us with a local domain of attraction for the altered classical nonlinear adaptive controller. In addition, the algorithm known as LQR Trees is applied to the system in order to create a controller with a larger region of attraction and lower torque requirements, though without an adaptive component. Both control systems are implemented in simulations using MATLAB.