Robust Co-Design for Canonical Underactuated Systems

Optimal behaviours of a system to perform a specific task can be achieved by exploiting the coupling between trajectory optimization, stabilization and de- sign optimization. The main objective of this thesis work is to analyze a novel co-optimization approach, which aims to improve the optimization results ap- plicability to real world systems. This methodology has shown interesting advantages for underactuated systems, which are systems that have fewer ac- tuators than degrees of freedom and thus require to make use of the passive dynamics to compensate for their lack of control inputs. Two co-design algo- rithms, namely Robust Trajectory Control (RTC) and RTC with Design optimiza- tion (RTCD), have been concieved, implemented and evaluated. While the first method optimizes the trajectory behavior and the cost matrices of a stabilizing Time-varying Linear Quadratic Regulator (TVLQR) by fixing the model param- eters, the second algorithm adds a further optimization layer where a design optimization is performed. Both aim to maximize the system’s robustness, mea- sured by a time-varying Lyapunov-based Region of Attraction (ROA) analysis. This analysis provides an intuitive representation of the controller’s robustness to off-nominal states and can also result with a formal guarantee of stability for the entire stabilized trajectory. The proposed algorithms have been tested on two different underactuated sys- tems: the torque-limited simple pendulum and the cart-pole. The experiments demonstrate an increased volume of the stabilizable state-space region, indicat- ing improved robustness. Extensive simulations of off-nominal initial conditions have further validated the results, and real system experiments have shown an improved insensitivity to torque disturbances.