The ability to avoid collisions with moving robots is critical in many applications. Moreover, if the robots have limited battery life, the goal is not only to avoid collisions but also to design efficient trajectories in terms of energy consumption and total mission time. This paper proposes a novel strategy for assigning turn angles for collision-free path planning in scenarios where a small team of robots cooperate in a certain mission. The algorithm allows each robot to reach a predetermined destination safely. It establishes consecutive, short time intervals, and at each interval, possible conflicts are solved centrally in an optimal manner. This is done by keeping constant speeds but generating a discrete set of possible directions for each robot, and solving efficiently the turn-angle allocation for a collision-free path that minimizes the path deviation from the shortest one. Due to the discretization, the final paths are not optimal, but the system can react to possible failures during execution, as conflicts are resolved at each time interval. Computational results and Software-In-The-Loop simulations are presented in order to evaluate the proposed algorithm. A comparison with a state-of-the-art approach shows that our algorithm is more energy-efficient and achieves lower mission completion time.

In the minimum-weight many-to-many point matching problem, we are given a set R of red points and a set B of blue points in the plane, of total size N , and we want to pair up each point in R to one or more points in B and vice versa so that the sum of distances between the paired points is minimized. This problem can be solved in O(N^3) time by using a reduction to the minimum-weight perfect matching problem, and thus, it is not fast enough to be used for on-line systems where a large number of tunes need to be compared. Motivated by similarity problems in music theory, in this paper we study several constrained minimum-weight many-to-many point matching problems in which the allowed pairings are given by geometric restrictions, i.e., a bichromatic pair can be matched if and only if the corresponding points satisfy a specific condition of closeness. We provide algorithms to solve these constrained versions in O ( N ) time when the sets R and B are given ordered by abscissa.

A group of cooperative aerial robots can be deployed to efficiently patrol a terrain, in which each robot flies around an assigned area and shares information with the neighbors periodically in order to protect or supervise it. To ensure robustness, previous works on these synchronized systems propose sending a robot to the neighboring area in case it detects a failure. In order to deal with unpredictability and to improve on the efficiency in the deterministic patrolling scheme, this paper proposes random strategies to cover the areas distributed among the agents. First, a theoretical study of the stochastic process is addressed in this paper for two metrics: the idle time, the expected time between two consecutive observations of any point of the terrain and the isolation time, the expected time that a robot is without communication with any other robot. After that, the random strategies are experimentally compared with the deterministic strategy adding another metric: the broadcast time, the expected time elapsed from the moment a robot emits a message until it is received by all the other robots of the team. The simulations show that theoretical results are in good agreement with the simulations and the random strategies outperform the behavior obtained with the deterministic protocol proposed in the literature.

Melodic similarity measurement is of key importance in music information retrieval. In this paper, we use geometric matching techniques to measure the similarity between two melodies. We represent music as sets of points or sets of horizontal line segments in the Euclidean plane and propose efficient algorithms for optimization problems inspired in two operations on melodies; linear scaling and audio compression. In the scaling problem, an incoming query melody is scaled forward until the similarity measure between the query and a reference melody is minimized. The compression problem asks for a subset of notes of a given melody such that the matching cost between the selected notes and the reference melody is minimized.

The objective of clustering is to discover natural groups in datasets and to identify geometrical structures which might reside there, without assuming any prior knowledge on the characteristics of the data. The problem can be seen as detecting the inherent separations between groups of a given point set in a metric space governed by a similarity function. The pairwise similarities between all data objects form a weighted graph adjacency matrix which contains all necessary information for the clustering process, which can consequently be formulated as a graph partitioning problem. In this context, we propose a new cluster quality measure which uses the maximum spanning tree and allows us to compute the optimal clustering under the min-max principle in polynomial time. Our algorithm can be applied when a load-balanced clustering is required.

Let {p1,…,pn} and {q1,…,qn} be two sets of n labeled points in general position in the plane. We say that these two point sets have the same order type if for every triple of indices (i,j,k), pk is above the directed line from pi to pj if and only if qk is above the directed line from qi to qj. In this paper we give the first non-trivial lower bounds on the number of different order types of n points that can be realized in integer grids of polynomial size.

This paper proposes a planning algorithm for autonomous media production with multiple Unmanned Aerial Vehicles (UAVs) in outdoor events.Given filming tasks specified by a media Director, we formulate an optimization problem to maximize the filming time considering battery constraints.As we conjecture that the problem is NP-hard, we consider a discretization version, and propose a graph-based algorithm that can find an optimal solution of the discrete problem for a single UAV in polynomial time.Then, a greedy strategy is applied to solve the problem sequentially for multiple UAVs.We demonstrate that our algorithm is efficient for small teams (3-5 UAVs) and that its performance is close to the optimum.We showcase our system in field experiments carrying out actual media production in an outdoor scenario with multiple UAVs.

Area coverage and communication are fundamental concerns in networks of cooperating robots. The goal is to address the issue of how well a group of collaborating robots having a limited communication range is able to monitor a given geographical space. Typically, an area of interest is partitioned into smaller subareas, with each robot in charge of a given subarea. This gives rise to a communication network that allows robots to exchange information when they are sufficiently close to each other. To be effective, the system must be resilient, i.e., be able to recover from robot failures. In a recent paper Bereg et al. (J Comb Optim 36(2):365–391, 2018), the concept of k-resilience of a synchronized system was introduced as the cardinality of a smallest set of robots whose failure suffices to cause that at least k surviving robots operate without communication, thus entering a state of starvation. It was proven that the problem of computing the k-resilience is NP-hard in general. In this paper, we study several problems related to the resilience of a synchronized system with respect to coverage and communication on realistic topologies including grid and cycle configurations. The broadcasting resilience is the minimum number of robots whose removal may disconnect the network. The coverage resilience is the minimum number of robots whose removal may result in a non-covered subarea. We prove that the three resilience measures can be efficiently computed for these configurations.

This paper addresses a synchronization problem that arises when a team of aerial robots (ARs) need to communicate while performing assigned tasks in a cooperative scenario. Each robot has a limited communication range and flies within a previously assigned closed trajectory. When two robots are close enough, a communication link may be established, allowing the robots to exchange information. The goal is to schedule the flights such that the entire system can be synchronized for maximum information exchange, that is, every pair of neighbors always visit the feasible communication link at the same time. We propose an algorithm for scheduling a team of robots in this scenario and propose a robust framework in which the synchronization of a large team of robots is assured. The approach allows us to design a fault-tolerant system that can be used for multiple tasks such as surveillance, area exploration, searching for targets in a hazardous environment, and assembly and structure construction, to name a few.

A group of UAVs can be used to efficiently patrol a terrain, in which each robot flies around an assigned area and shares information with the neighbors periodically in order to protect or supervise it. To ensure robustness, previous works propose sending a robot to the neighboring area in case it detects a failure. In order to add unpredictability and to improve on the efficiency in the deterministic patrolling scheme, this paper presents random strategies to cover the areas distributed among the agents. We evaluate these strategies using three metrics: the idle-time, the isolation-time and the broadcast-time. The idle-time is the expected time between two consecutive observations of any point of the terrain. The isolation-time is the expected time that a robot is isolated (that is, without communication with any other robot). The broadcast-time is the expected time elapsed from the moment a robot emits a message until it is received by all the other robots of the team. Simulations show that the random strategies outperform the results obtained with the deterministic protocol.

Given a finite set of weighted points in ℝ^d (where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in d=1 these computations are #P-hard, and give pseudo polynomial-time algorithms in the case where the weights are integers in a bounded interval. For d=2 , we consider that each point is colored red or blue, where red points have weight +1 and blue points weight -∞ . The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane.

We study an optimization problem that arises in the design of covering strategies for multi-robot systems. Consider a team of n cooperating robots traveling along predetermined closed and disjoint trajectories. Each robot needs to periodically communicate information to nearby robots. At places where two trajectories are within range of each other, a communication link is established, allowing two robots to exchange information, provided they are "synchronized", i.e., they visit the link at the same time. In this setting a communication graph is defined and a system of robots is called synchronized if every pair of neighbors is synchronized. If one or more robots leave the system, then some trajectories are left unattended. To handle such cases in a synchronized system, when a live robot arrives to a communication link and detects the absence of the neighbor, it shifts to the neighboring trajectory to assume the unattended task. If enough robots leave, it may occur that a live robot enters a state of starvation, failing to permanently meet other robots during flight. To measure the tolerance of the system under this phenomenon we define the k-resilience as the minimum number of robots whose removal may cause k surviving robots to enter a state of starvation. We show that the problem of computing the k-resilience is NP-hard if k is part of the input, even if the communication graph is a tree. We propose algorithms to compute the k-resilience for constant values of k in general communication graphs and show more efficient algorithms for systems whose communication graph is a tree.

Let S be a point set in the plane such that each of its elements is colored either red or blue. A matching of S with rectangles is any set of pairwise-disjoint axis-aligned closed rectangles such that each rectangle contains exactly two points of S. Such a matching is monochromatic if every rectangle contains points of the same color, and is bichromatic if every rectangle contains points of different colors. We study the following two problems: (1) Find a maximum monochromatic matching of S with rectangles. (2) Find a maximum bichromatic matching of S with rectangles. For each problem we provide a polynomial-time approximation algorithm that constructs a matching with at least 1 / 4 of the number of rectangles of an optimal matching. We show that the first problem is \(\mathsf {NP}\)-hard even if either the matching rectangles are restricted to axis-aligned segments or S is in general position, that is, no two points of S share the same x or y coordinate. We further show that the second problem is also \(\mathsf {NP}\)-hard, even if S is in general position. These \(\mathsf {NP}\)-hardness results follow by showing that deciding the existence of a matching that covers all points is \(\mathsf {NP}\)-complete in each case. Additionally, we prove that it is \(\mathsf {NP}\)-complete to decide the existence of a matching with rectangles that cover all points in the case where all the points have the same color, solving an open problem of Bereg et al. (Comput Geom 42(2):93–108, 2009).

We study the problem of optimally inspecting an underground (underwater) gallery with k agents. We consider a gallery with a single opening and with a tree topology rooted at the opening. Due to the small diameter of the pipes (caves), the agents are small robots with limited autonomy and there is a supply station at the gallery’s opening. Therefore, they are initially placed at the root and periodically need to return to the supply station. Our goal is to design off-line strategies to efficiently cover the tree with k small robots. We consider two objective functions: the covering time (maximum collective time) and the covering distance (total traveled distance). The maximum collective time is the maximum time spent by a robot needs to finish its assigned task (assuming that all the robots start at the same time); the total traveled distance is the sum of the lengths of all the covering walks. Since the problems are intractable for big trees, we propose approximation algorithms. Both efficiency and accuracy of the suboptimal solutions are empirically showed for random trees through intensive numerical experiments.

This paper addresses a synchronization problem that arises when a team of robots needs to communicate while repeatedly performing assigned tasks in a cooperative scenario. Each robot has a limited communication range and moves along a previously defined closed trajectory. When two robots are close enough, a communication link may be established, allowing the robots to exchange information. The goal is to schedule the motions such that the entire system can be synchronized for maximum information exchange; that is, every pair of neighbors always visit the feasible communication link at the same time. An algorithm for scheduling the team of robots in this scenario is proposed and a robust framework that assures the synchronization of a large team of robots is presented. Simulations, experiments, and computational results demonstrate the applicability of the algorithm. The approach allows the design of fault-tolerant systems that can be used for multiple tasks, such as surveillance, area exploration, and searching for targets in hazardous environments, among others.

A new paradigm for task allocation in cooperative multi-robot systems is proposed in this paper. The block-information-sharing (BIS) strategy is a fully distributed approach, where robots dynamically allocate their tasks following the principle of share & divide to maintain an optimal allocation according to their capabilities. Prior studies on multi-robot information sharing strategies do not formally address the proof of convergence to the optimal allocation, nor its robustness to dynamic changes in the execution of the global task. The BIS strategy is introduced in a general framework and the convergence to the optimal allocation is theoretically proved. As an illustration of the approach, the strategy is applied to the automatic construction of truss structures with aerial robots. In order to demonstrate the benefits of the strategy, algorithms and simulations are presented for a team of heterogeneous robots that can dynamically reallocate tasks during the execution of a mission. (C) 2017 Elsevier B.V. All rights reserved.

We say that a polygon inscribed in the circle is asymmetric if it contains no two antipodal points being the endpoints of a diameter. Given n diameters of a circle and a positive integer k < n, this paper addresses the problem of computing a maximum area asymmetric k-gon having as vertices k < n endpoints of the given diameters. The study of this type of polygons is motivated by ethnomusiciological applications.

Fault tolerance is increasingly important for unmanned autonomous vehicles. For example, in a multi robot system the agents need the ability to effectively detect and tolerate internal failures in order to continue performing their tasks without the need for immediate human intervention. The system must react to unplanned events in order to optimize the task allocation between the robots. In a broad sense, the resilience of a system can be defined as the ability to maintain or recover a stable state when subject to disturbance and it is related to the concept of robustness in industrial systems. In this paper, we study the resilience in a synchronized multi-robot system stated as follows:Consider a team of $n$ (ground or aerial) robots each moving along predetermined periodic closed trajectories. Each of the agents needs to communicate informationabout its operation to other agents, but the communication links have a limited range. Hence, when two agents are within communication range, a communication link is established, and information is exchanged. Thus, two neighbors are synchronized if they visit the communication link at the same time and a multi-robot system is called synchronized if each pair of neighbors is synchronized. If a set of robots left the system, then some trajectories has no robots. In these cases, when an alive robot detects no neighboring robot then it pass to this neighboring trajectory to assume the unattended task. In this framework, a fault-tolerance measure is introduced: the resilience of the system is the largest number of robots that can fail while executing the global task. Interesting combinatorial properties of the resilience are showed that allow to know its value for some usual scenarios.