DIRTREL: Robust Nonlinear Direct Transcription with Ellipsoidal Disturbances and LQR Feedback

Many critical robotics applications require robustness to disturbances arising from unplanned forces, state uncertainty, and model errors. Motion planning algorithms that explicitly reason about robustness require a coupling of trajectory optimization and feedback design, where the system’s closedloop response to bounded disturbances is optimized. Due to the often-heavy computational demands of solving such problems, the practical application of robust trajectory optimization in robotics has so far been limited. We derive a tractable robust optimization algorithm that combines direct transcription with linear-quadratic feedback to efficiently reason about closed-loop responses to disturbances. In the case of ellipsoidal disturbance sets, the state and input deviations along a nominal trajectory can be computed locally in closed form, thus allowing for fast evaluations of robust cost and constraint functions. The resulting algorithm, called DIRTREL, is an extension of classical direct transcription that demonstrably improves tracking performance over non-robust formulations while incurring only a modest increase in computational cost. We evaluate the algorithm in several simulated robot planning and control tasks.