Robust Path Planning and Control For Polygonal Environments via Linear Programming Mahroo Bahreinian 1 , Erfan Aasi 2 and Roberto Tron 3 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 Func- tion (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. I. INTRODUCTION Path planning is a major research area in the context of mobile robots, as it deals with the problem of finding a path from an initial state toward a goal state while considering collision avoidance. Traditional path planning methods focus on finding single nominal paths in a given known map, and the majority of them makes the implicit assumption that the agent possesses a lower-level state feedback controller for following such nominal path in the face of external disturbances and imperfect models. Biological system do not rely on the same restrictive assumptions; for instance, consider a person navigating in an unfamiliar room: despite the fact that the person does not have a precise blueprint of the floor, and does not know its precise location, they can reliably and robustly navigate toward a desired door, from any location in the room. While this ability in biological systems is the result of complex and not fully understood mechanisms, in this paper we aim to narrow the gap between planning algorithms and biological systems by synthesizing output-feedback controllers that This work was supported by ONR MURI N00014-19-1-2571 “Neuro- Autonomy: Neuroscience-Inspired Perception, Navigation, and Spatial Awareness” 1 Mahroo Bahreinian is with Division of Systems Engineering at Boston University, Boston, MA, 02215 USA. Email: mahroobh@bu.edu 2 Erfan Aasi is with Department of Mechanical Engineering at Boston University, Boston, MA, 02215 USA. Email: eaasi@bu.edu 3 Roberto Tron is with Faculty of Department of Mechanical Engineering at Boston University, Boston, MA, 02215 USA. Email: tron@bu.edu are robust to imprecise map knowledge. By synthesizing controllers instead of specific paths, we more tightly integrate the high-level path planning and the low-level regulation tasks, allowing the agent to cope with disturbances without replanning; moreover, the focus on the controllers allows us to plan by directly using measurements (outputs) available to the agent, instead of assuming full state knowledge; finally, since the controllers depend on the environment indirectly (through measurements that are taken online), we empirically show that such controllers are robust to (often very significant) changes in the map. In order to pursue strong theoretical guarantees, in this paper we assume agents with controllable linear dynamics, and environments that admit a polygonal convex cell decomposition (e.g., via Delaunay triangulations [9] or trapezoidal decompositions [22]). Methods to address these limitations are planned as part of our future work (see also the Conclusions section). Previous work. Existing works on path planning can be roughly classified into two categories: combinatorial path planning methods, and sample-based path planning methods [11]. Some of the path planning methods consider a continuous model for the environment and therefore provide a continuous path, such as potential fields [18], [20] and navigation functions [27], while the other group solves the planning problem by abstracting the environment to a finite representation and find a discrete path, such as probabilistic roadmaps [16] and cell decomposition methods [25]. One of the well known combinatorial path planning algo- rithms is cell decomposition where a complex environment is decomposed into a set of cells, avoiding obstacles by planning straight paths in individual cells; for each individual step, traditional methods use midpoints [7], [23], [28], while more recent solutions aim to optimize path length [19]. Our work can be seen as a descendant of previous work that handles the cell decomposition vis- ´ a-vis the continuous dynamic through a hybrid system perspective by synthesizing a state-feedback controller for each cell. Initial work proposed potential-based controllers [8], while others characterize the theoretical conditions [13] and closed-form solutions [3] for linear affine controllers. Although the latter approaches were extended to nonlinear systems in [10] and to uncertain maps [32] (using intelligent re-planning), they all assume that each cell in the decomposition is a simplex (a polytope in R d with d +1 vertices, e.g., a 2-D triangle). In contrast, our method can handle arbitrary convex polytopes, and design output-feedback controllers (instead of state-feedback). Sampling-based planning algorithms, such as rapidly ex- ploring random tree (RRT), have become popular in last arXiv:1910.07976v2 [eess.SY] 12 Oct 2020