Motion Planning for Unlabeled Discs with Optimality Guarantees

We study the problem of path planning for unlabeled (indistinguishable) unit-disc robots in a planar environment cluttered with polygonal obstacles. We introduce an algorithm which minimizes the total path length, i.e., the sum of lengths of the individual paths. Our algorithm is guaranteed to find a solution if one exists, or report that none exists otherwise. It runs in time $\tilde{O}(m^4+m^2n^2)$, where $m$ is the number of robots and $n$ is the total complexity of the workspace. Moreover, the total length of the returned solution is at most $\text{OPT}+4m$, where OPT is the optimal solution cost. To the best of our knowledge this is the first algorithm for the problem that has such guarantees. The algorithm has been implemented in an exact manner and we present experimental results that attest to its efficiency

Optimal Routing and Control of Multiple Agents Moving in a Transportation Network and Subject to an Arrival Schedule and Separation Constraints

We address the problem of navigating a set of moving agents, e.g. automated guided vehicles, through a transportation network so as to bring each agent to its destination at a specified time. Each pair of agents is required to be separated by a minimal distance, generally agent-dependent, at all times. The speed range, initial position, required destination, and required time of arrival at destination for each agent are assumed provided. The movement of each agent is governed by a controlled differential equation (state equation). The problem consists in choosing for each agent a path and a control strategy so as to meet the constraints and reach the destination at the required time. This problem arises in various fields of transportation, including Air Traffic Management and train coordination, and in robotics. The main contribution of the paper is a model that allows to recast this problem as a decoupled collection of problems in classical optimal control and is easily generalized to the case when inertia cannot be neglected. Some qualitative insight into solution behavior is obtained using the Pontryagin Maximum Principle. Sample numerical solutions are computed using a numerical optimal control solver

Efficient Computation of Separation-Compliant Speed Advisories for Air Traffic Arriving in Terminal Airspace

A class of problems in air traffic management asks for a scheduling algorithm that supplies the air traffic services authority not only with a schedule of arrivals and departures, but also with speed advisories. Since advisories must be finite, a scheduling algorithm must ultimately produce a finite data set, hence must either start with a purely discrete model or involve a discretization of a continuous one. The former choice, often preferred for intuitive clarity, naturally leads to mixed-integer programs, hindering proofs of correctness and computational cost bounds (crucial for real-time operations). In this paper, a hybrid control system is used to model air traffic scheduling, capturing both the discrete and continuous aspects. This framework is applied to a class of problems, called the Fully Routed Nominal Problem. We prove a number of geometric results on feasible schedules and use these results to formulate an algorithm that attempts to compute a collective speed advisory, effectively finite, and has computational cost polynomial in the number of aircraft. This work is a first step toward optimization and models refined with more realistic detail

Multi-Robot Coordination with Environmental Disturbances

Multi-robot systems are increasingly deployed in environments where they interact with humans. From the perspective of a robot, such interaction could be considered a disturbance that causes a well-planned trajectory to fail. This dissertation addresses the problem of multi-robot coordination in scenarios where the robots may experience unexpected delays in their movements. Prior work by Čáp, Gregoire, and Frazzoli introduced a control law, called RMTRACK, which enables robots in such scenarios to execute pre-planned paths in spite of disturbances that affect the execution speed of each robot while guaranteeing that each robot can reach its goal without collisions and without deadlocks. We extend that approach to handle scenarios in which the disturbance probabilities are unknown when execution starts and are non-uniform across the environment. The key idea is to ‘repair’ a plan on-the-fly, by swapping the order in which a pair of robots passes through a mutual collision region (i.e. a coordination space obstacle), when making such a change is expected to improve the overall performance of the system. We introduce a technique based on Gaussian processes to estimate future disturbances, and propose two algorithms for testing, at appropriate times, whether a swap of a given obstacle would be beneficial. Tests in simulation demonstrate that our algorithms achieve significantly smaller average travel time than RMTRACK at only a modest computational expense. However, deadlock may arise when rearranging the order in which robots pass collision regions and other obstacles. We provide a precise definition of deadlock using a graphical representation and prove some of its important properties. We show how to exploit the representation to detect the possibility of deadlock and to characterize conditions under which deadlock may not occur. We provide experiments in simulated environments that illustrate the potential usefulness of our theory of deadlock

Multi-Agent Pathfinding in Mixed Discrete-Continuous Time and Space

In the multi-agent pathfinding (MAPF) problem, agents must move from their current locations to their individual destinations while avoiding collisions. Ideally, agents move to their destinations as quickly and efficiently as possible. MAPF has many real-world applications such as navigation, warehouse automation, package delivery and games. Coordination of agents is necessary in order to avoid conflicts, however, it can be very computationally expensive to find mutually conflict-free paths for multiple agents – especially as the number of agents is increased. Existing state-ofthe- art algorithms have been focused on simplified problems on grids where agents have no shape or volume, and each action executed by the agents have the same duration, resulting in simplified collision detection and synchronous, timed execution. In the real world agents have a shape, and usually execute actions with variable duration. This thesis re-formulates the MAPF problem definition for continuous actions, designates specific techniques for continuous-time collision detection, re-formulates two popular algorithms for continuous actions and formulates a new algorithm called Conflict-Based Increasing Cost Search (CBICS) for continuous actions