Robust Planning and Control For Polygonal Environments via Linear Programming (1910.07976v2)

Abstract: We propose a novel approach for navigating in polygonal environments by synthesizing controllers that take as input relative displacement measurements with respect to a set of landmarks. Our algorithm is based on solving a sequence of robust min-max Linear Programming problems on the elements of a cell decomposition of the environment. The optimization problems are formulated using linear Control Lyapunov Function (CLF) and Control Barrier Function (CBF) constraints, to provide stability and safety guarantees, respectively. The inner maximization problem ensures that these constraints are met by all the points in each cell, while the outer minimization problem balances the different constraints in a robust way. We show that the min-max optimization problems can be solved efficiently by transforming it into regular linear programming via the dualization of the inner maximization problem. We test our algorithm to agents with first and second-order integrator dynamics, although our approach is in principle applicable to any system with piecewise linear dynamics. Through our theoretical results and simulations, we show that the resulting controllers: are optimal (with respect to the criterion used in the formulation), are applicable to linear systems of any order, are robust to changes to the start location (since they do not rely on a single nominal path), and to significant deformations of the environment.

• The paper introduces a novel controller synthesis method using linear programming and dualization techniques to achieve robust path planning and control.
• It employs Control Lyapunov and Barrier Functions with relative position measurements to ensure stability and safety in complex polygonal environments.
• Simulation results validate the framework’s efficiency and robustness under severe environmental deformations across various dynamic systems.

Robust Path Planning and Control For Polygonal Environments via Linear Programming

The paper presented addresses the development of a robust path planning and control framework for navigating polygonal environments, utilizing Linear Programming (LP). The focus is on synthesizing controllers through relative displacement measurements with respect to landmarks, a marked departure from traditional methods that assume complete map and path information.

Core Contributions

1. Controller Synthesis via Linear Programming: The authors propose solving robust min-max optimization problems, converting these into linear programs via dualization techniques. This approach involves using Control Lyapunov Functions (CLFs) for stability assurance and Control Barrier Functions (CBFs) for safety guarantees. The method is flexible enough to be applied to systems with any piecewise linear dynamics.
2. Environment and System Model: The system models comprise of agents with linear dynamics navigating environments that can be decomposed into convex polygonal cells using techniques such as Delaunay triangulation. The controllers are based on relative position measurements, providing robustness against variations in initial conditions and environmental changes without the need for a nominal trajectory.
3. Path Planning and Implementation: The framework accommodates different planning objectives such as point stabilization and patrolling, which are demonstrated through simulations. The authors showcase the robustness of their method by testing it on environments with significant deformations.
4. Handling Constraints and Optimization: By transforming the min-max problem into a min-min formulation using dual forms, the authors bypass the complexities involved in bi-level optimization, rendering the framework computationally efficient for solving robust control problems.

Numerical Results and Implications

In the empirical evaluation, the authors demonstrate that their controllers achieve the desired objectives effectively, even under adverse conditions like significant environment alterations. The tests conducted for both first-order and second-order integrator dynamics emphasize the robustness attribute, thus suggesting practical viability for real-world applications.

The theoretical insight provided by the stability and safety conditions (via CLF and CBF) reconciles the requirement for robust path planning in dynamically varying environments, sidestepping typical reliance on precise models or exhaustive replanning phases. This indication is pivotal for implementing navigation solutions in autonomous robotic systems, where adaptability and robustness are imperative.

Future Prospects and Implications

The paper posits future directions such as integrating smoothness constraints across cell boundaries and coupling with sampling-based methods to overcome discrete convex cell assumption limitations. Moreover, leveraging this approach for nonlinear systems represents a promising extension, potentially increasing the applicability of the framework to a broader class of dynamic and complex environments.

This research thus gives serious momentum to the development of robust autonomous navigation frameworks. By proving that LP formulations can be both efficient and robust without full state feedback, the authors pave the way for practical, scalable implementations in robotics and autonomous systems research arenas.