Nonlinear Robust Optimization for Planning and Control

TL;DR

This work addresses robust trajectory optimization for constrained nonlinear discrete-time systems subject to unknown bounded disturbances, seeking policies that remain feasible for all realizations. It introduces a bi-level framework with an outer trust-region successive convexification to linearize dynamics and robust constraints and an inner convex robust optimization solved via ADMM/augmented Lagrangian methods, including a novel treatment of linearization errors as additional uncertainty. A NRTO-LE variant extends the framework by explicitly modeling linearization errors within ellipsoidal sets, yielding guaranteed constraint satisfaction in simulations. The approach demonstrates improved robustness and scalability on nonlinear dynamical systems, with potential applications to autonomous vehicles, UAVs, and multi-agent planning under uncertainty.

Abstract

This paper presents a novel robust trajectory optimization method for constrained nonlinear dynamical systems subject to unknown bounded disturbances. In particular, we seek optimal control policies that remain robustly feasible with respect to all possible realizations of the disturbances within prescribed uncertainty sets. To address this problem, we introduce a bi-level optimization algorithm. The outer level employs a trust-region successive convexification approach which relies on linearizing the nonlinear dynamics and robust constraints. The inner level involves solving the resulting linearized robust optimization problems, for which we derive tractable convex reformulations and present an Augmented Lagrangian method for efficiently solving them. To further enhance the robustness of our methodology on nonlinear systems, we also illustrate that potential linearization errors can be effectively modeled as unknown disturbances as well. Simulation results verify the applicability of our approach in controlling nonlinear systems in a robust manner under unknown disturbances. The promise of effectively handling approximation errors in such successive linearization schemes from a robust optimization perspective is also highlighted.

Figures (5)

• Figure 1: NTO vs. NRTO: Only 2 out of 1000 trajectory realizations obtained using NTO satisfy all the constraints. While only 3 out of 1000 trajectory realizations obtained using NRTO violate the constraints.
• Figure 2: Performance Comparison between NRTO and NRTO-LE: Uncertainty level is set to $\tau = 0.1$. (a) and (b) correspond to the NRTO, with $92.1 \%$ constraint satisfaction. (c) and (d) correspond to the NRTO-LE, with $100 \%$ constraint satisfaction.
• Figure 3: Uncertainty vs. Constraint Satisfaction: Comparison of percentage of constraint satisfaction over 1500 realizations using NRTO and NRTO-LE. The constraint satisfaction decreases with increase in uncertainty using the NRTO, while NRTO-LE provides $100 \%$ constraint satisfaction in all the cases.
• Figure 4: Performance Comparison between NRTO and NRTO-LE for car model: Uncertainty level is set to $\tau = 0.01$. (a) and (b) correspond to the NRTO, with $84.2 \%$ constraint satisfaction. (c) and (d) correspond to the NRTO-LE, with $100 \%$ constraint satisfaction.
• Figure 5: Complex scenario for unicycle with ten obstacles using NRTO-LE: Uncertainty level is set to $\tau = 0.05$. $100 \%$ constraint satisfaction considering $1500$ trajectory realizations.