Polygonal Environment Research Articles

Concealing Robots in Environments: Enhancing Navigation and Privacy through Stealth Integration

With the continuous advancement of robotics technology, the integration of robots into diverse human environments has become increasingly prevalent. However, the presence of robots in public spaces can often elicit discomfort or unease among individuals. To address this concern, the concept of concealing robots in various settings has emerged as an innovative approach to improve robot navigation and interaction while minimizing intrusion on human privacy. This paper explores the motivations, challenges, and potential benefits of hiding robots in different environments, particularly within the context of swarm robotics where multiple interconnected robots form a cohesive swarm. Equipped with onboard processing, communication, and sensing capabilities, these robots can autonomously interact with each other and adapt to the environment.The paper investigates the problem of maximizing the number of hidden orthogonal swarm robots, considering scenarios in which robots need to navigate and operate within polygonal environments. Specifically, it presents a 4-approximation algorithm for computing a maximum hidden robot set in such environments. The algorithm offers a practical solution for determining an efficient arrangement of robots that minimizes their visibility while ensuring effective swarm operation.By concealing robots in diverse environments, several benefits can be achieved. First, it helps to alleviate discomfort or unease among individuals, allowing for smoother integration of robots into public spaces. Additionally, concealing robots enhances their navigation capabilities by leveraging stealth techniques, allowing them to move seamlessly and unobtrusively within the environment. This approach also promotes improved human-robot interaction, as the reduced visibility of the robots can alleviate concerns and foster a more natural and comfortable interaction between humans and robots. The paper sheds light on the current state of the field, discussing the motivations behind concealing robots in different settings and highlighting the challenges that need to be addressed. Furthermore, it presents insights into future directions, including the development of more advanced stealth technologies, ethical frameworks for integrating hidden robots, and the potential impact on urban planning and infrastructure. In conclusion, hiding robots in diverse environments offers a promising approach to enhancing robot navigation, interaction, and privacy. The research presented in this paper contributes to the understanding of this emerging field and provides a foundation for further exploration and development of hiding strategies for swarm robotics in various settings.

Multi-robot motion planning for unit discs with revolving areas

We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R′ that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm.On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(log⁡nlog⁡log⁡n)-approximation algorithm for this problem.

A Graph-Based Approach to Generate Energy-Optimal Robot Trajectories in Polygonal Environments

As robotic systems continue to address emerging issues in areas such as logistics, mobility, manufacturing, and disaster response, it is increasingly important to rapidly generate safe and energy-efficient trajectories. In this article, we present a new approach to plan energy-optimal trajectories through cluttered environments containing polygonal obstacles. In particular, we develop a method to quickly generate optimal trajectories for a double-integrator system, and we show that optimal path planning reduces to an integer program. To find an efficient solution, we present a distance-informed prefix search to efficiently generate optimal trajectories for a large class of environments. We demonstrate that our approach, while matching the performance of RRT* and Probabilistic Road Maps in terms of path length, outperforms both in terms of energy cost and computational time by up to an order of magnitude.

Моделирование поведения мобильных роботов с использованием генетических алгоритмов

The article is devoted to the analysis of the behavior of a mobile robot using finite state machine algorithms in order to find a way to the goal and avoid obstacles. After justifying the use of such methods, the analysis of a standard deterministic finite automaton is done. Further, the theory of Markov processes is applied to this algorithm, as a result of which the state machine becomes part of the hidden Markov model. This allows you to apply probabilistic methods to modeling the behavior of the robot. This probabilistic behavior is most promising in complex environments with unpredictable obstacle configurations. To compare the efficiency of deterministic and probabilistic finite state machine, we applied a genetic algorithm. In the numerical experiment that we conducted in the Scilab software, we considered two main types of environments in which a mobile robot can move - an office-type environment and a polygonal-type environment. For each type of environment, we alternately applied each of the indicated behavior algorithms. For the genetic algorithm, we used one hundred individuals who were trained over 1000 generations to find the most optimal path in the specified environments. As a result, it was found that the deterministic finite state machine algorithm is the most promising for movement in an office-type environment, and the probabilistic finite state machine algorithm gives the best result in a complex polygonal environment.

Catch Me If You Can: Pursuit and Capture in Polygonal Environments with Obstacles

We resolve a several-years old open question in visibility-based pursuit evasion: how many pursuers are needed to capture an evader in an arbitrary polygonal environment with obstacles? The evader is assumed to be adversarial, moves with the same maximum speed as pursuers, and is "sensed'' by a pursuer only when it lies inline-of-sight of that pursuer. The players move in discrete time steps, and the capture occurs when a pursuer reaches the position of the evader on its move. Our main result is that O(√h + log n) pursuers can always win the game with a deterministic search strategy in any polygon with n vertices and h obstacles (holes). In order to achieve this bound, however, we argue that the environment must satisfy a minimum feature size property, which essentially requires the minimum distance between any two vertices to be of the same order as the speed of the players. Without the minimum feature size assumption, we show that Ω < ( √(n/log n)) pursuers are needed in the worst-case even for simply-connected (hole-free) polygons of n vertices! This reveals an unexpected subtlety that seems to have been overlookedin previous work claiming that O(log n) pursuers can always win insimply-connected n-gons. Our lower bound also shows that capturing an evader is inherently more difficult than just "seeing" it because O(log n) pursuers are provably sufficient for line-of-sight detection even against an arbitrarily fast evaderin simple n-gons.

Symmetric Rendezvous in Planar Environments With and Without Obstacles

We study the symmetric rendezvous search problem in which two robots that are unaware of each other’s locations try to meet as quickly as possible. In the symmetric version of this problem, the robots are required to execute the same strategy. First, we present a symmetric rendezvous strategy for the robots that are initially placed on the open plane and analyze its competitive performance. We show that the competitive complexity of our strategy is O(d/R) where d is the initial distance between the robots and R is the communication radius. Second, we extend the symmetric rendezvous strategy for the open plane to unknown environments with polygonal obstacles. The extended strategy guarantees a complete coverage of the environment. We analyze the strategy for square, translating robots and show that the competitive ratio of the extended strategy is O(d/D) where D is the length of the sides of the robots. In obtaining this result, we also obtain an upper bound on covering arbitrary polygonal environments which may be of independent interest.

A visibility-based approach to computing non-deterministic bouncing strategies

Inspired by motion patterns of some commercially available mobile robots, we investigate the power of robots that move forward in straight lines until colliding with an environment boundary, at which point they can rotate in place and move forward again; we visualize this as the robot “bouncing” off boundaries. We define bounce rules governing how the robot should reorient after reaching a boundary, such as reorienting relative to its heading prior to collision, or relative to the normal of the boundary. We then generate plans as sequences of rules, using the bounce visibility graph generated from a polygonal environment definition, while assuming we have unavoidable non-determinism in our actuation. Our planner can be queried to determine the feasibility of tasks such as reaching goal sets and patrolling (repeatedly visiting a sequence of goals). If the task is found feasible, the planner provides a sequence of non-deterministic interaction rules, which also provide information on how precisely the robot must execute the plan to succeed. We also show how to compute stable cyclic trajectories and use these to limit uncertainty in the robot’s position.

Blocking an Evader in a Polygon by a 2-Modem Searcher

In this paper, we introduce a version of Pursuit-Evasion problem in which the pursuer is a 2-modem which pursues an unpredictable evader in a polygonal environment. A 2-modem searcher is a wireless device whose radio signal can penetrate two walls. Furthermore, a query path inside the polygon is given so that the 2-modem moves on it. We want to find out that when the 2- modem moves on this path if evaders can move from an invisible region to another one without detecting by the pursuer or not. In the other word, we want to find out if the invisible regions merge together or not.

Collaborative Pursuit-Evasion Strategy of UAV/UGV Heterogeneous System in Complex Three-Dimensional Polygonal Environment

The UAV/UGV heterogeneous system combines the air superiority of UAV (unmanned aerial vehicle) and the ground superiority of UGV (unmanned ground vehicle). The system can complete a series of complex tasks and one of them is pursuit-evasion decision, so a collaborative strategy of UAV/UGV heterogeneous system is proposed to derive a pursuit-evasion game in complex three-dimensional (3D) polygonal environment, which is large enough but with boundary. Firstly, the system and task hypothesis are introduced. Then, an improved boundary value problem (BVP) is used to unify the terrain data of decision and path planning. Under the condition that the evader knows the position of collaborative pursuers at any time but pursuers just have a line-of-sight view, a worst case is analyzed and the strategy between the evader and pursuers is studied. According to the state of evader, the strategy of collaborative pursuers is discussed in three situations: evader is in the visual field of pursuers, evader just disappears from the visual field of pursuers, and the position of evader is completely unknown to pursuers. The simulation results show that the strategy does not guarantee that the pursuers will win the game in complex 3D polygonal environment, but it is optimal in the worst case.

Planar max flow maps and determination of lanes with clearance

One main challenge in multi-agent navigation is to generate trajectories minimizing bottlenecks in environments cluttered with obstacles. In this paper we approach this problem globally by taking into account the maximum flow capacity of a given polygonal environment. Given the difficulty in solving the continuous maximum flow of a planar environment, we present in this paper a GPU-based methodology which leads to practical methods for computing maximum flow maps in arbitrary two-dimensional polygonal domains. Once a flow map representation is obtained, lanes can be extracted and optimized in length while keeping constant the flow capacity achieved by the system of trajectories. This work extends our previous work on max flow maps by presenting a clearance-based flow generation method which takes into account the size of the agents at the flow generation phase. In this way we ensure that the maximum possible number of lanes with the needed clearance is always obtained, a property that was found to not be always obtained with our previous method. As a result we are able to generate trajectories of maximum flow from source to sink edges across a generic set of polygonal obstacles, enabling the deployment of large numbers of agents utilizing the maximum flow capacity of a continuous description of the environment and eliminating bottlenecks.

Line-of-Sight Pursuit in Monotone and Scallop Polygons

We study a turn-based game in a simply connected polygonal environment [Formula: see text] between a pursuer [Formula: see text] and an adversarial evader [Formula: see text]. Both players can move in a straight line to any point within unit distance during their turn. The pursuer [Formula: see text] wins by capturing the evader, meaning that their distance satisfies [Formula: see text], while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader [Formula: see text] always knows the location of [Formula: see text]. Meanwhile, [Formula: see text] only has line-of-sight visibility: [Formula: see text] observes the evader’s position only when the line segment connecting them lies entirely within the polygon. Therefore [Formula: see text] must search for [Formula: see text] when the evader is hidden from view. We provide a winning strategy for [Formula: see text] in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line [Formula: see text] can be moved continuously over [Formula: see text] so that (1) [Formula: see text] is a line segment and (2) every point on the boundary [Formula: see text] is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook’s strategy during its pursuit phase, rather than the well-known lion’s strategy. The rook’s strategy is crucial for obtaining a capture time that is linear in the area of [Formula: see text]. For both monotone and scallop polygons, our algorithm has a capture time of [Formula: see text], where [Formula: see text] is the number of polygon vertices.

Stochastic Multi-Robot Patrolling with Limited Visibility

Patrolling an environment with multiple robots is a problem with applicability to both military activities and other areas requiring security. In an adversarial environment, wireless communication between the robots may be jammed, and their sensor ranges may be limited to visibility. This increases the difficulty of the problem, but solutions will be widely applicable regardless of the environment. Robot paths that are deterministic can be observed and predicted by an adversary, permitting exploitation of known gaps in coverage. We also wish to avoid requirements for synchronization or a particular form of the environment. We, therefore, propose a method of finding patrolling policies for multiple robots that monitor any polygonal environment using limited visibility regions and non-deterministic patrolling paths. First, visibility regions are calculated for a subset of locations that cover the whole environment or a part of the environment. Then, we find distributed patrolling policies in the form of Markov chains, using convex optimization to minimize the average expected commute time for the subset of the locations permitting each robot to cover the whole environment independently. We also find centralized and Markov chain based patrolling policies that minimize the average expected commute time for the subset of locations permitting each robot to cover a part of the environment while communicating with a central base station. Finally, we evaluate the vulnerability of our patrolling policies by finding the probability of capturing an adversary and the maximum unguarded time for a location in our proposed patrolling scenarios. We present multiple simulation results and a physical implementation to show the effectiveness of our visibility-based non-deterministic patrolling method.

Rendezvous in planar environments with obstacles and unknown initial distance

In the rendezvous search problem, two or more robots at unknown locations should meet somewhere in the environment as quickly as possible. We study the symmetric rendezvous search problem in unknown planar environments with polygonal obstacles. In the symmetric version of the problem, the robots must execute the same rendezvous strategy. We consider the case where the initial distance between the robots is unknown, and the robot is unaware of its and the other robots' locations in the environment. We first design a symmetric rendezvous strategy for two robots and perform its theoretical analysis. We prove that the competitive ratio of our strategy is O(d/D). Here, d is the initial distance between the robots and D is the length of the sides of the square robots. In unknown polygonal environments, robots should explore the environment to achieve rendezvous. Therefore, we propose a coverage algorithm that guarantees the complete coverage of the environment. Next, we extend our symmetric rendezvous strategy to n robots and prove that its competitive ratio is O(d/(nD)). Here, d is the maximal pairwise distance between the robots. Finally, we validate our algorithms in simulations.

A discrete-time pursuit–evasion game in convex polygonal environments

This paper studies a discrete-time pursuit–evasiongame within a convex polygonal environment. Building on solutions of the classic lion-and-man problem, two strategies are proposed for the pursuer, which guarantee exact capture in finite time and provide upper bounds on the time-to-capture at each move of the game. A numerical procedure for updating the so-called center of the game, which is instrumental for computing the lion’s move, is devised. Numerical simulations show that optimizing the center position, with respect to a suitable cost function taking into account the structure of the environment, allows one to remarkably reduce the number of moves required to capture the evader.

Adaptive target tracking with a mixed team of static and mobile guards: deployment and activation strategies

This work explores a variation of the art gallery problem in which a team of static and mobile guards track a mobile intruder with unknown maximum speed. We consider the special case when the mobile guards are restricted to move along the diagonals of a polygonal environment. First, we present an algorithm to identify candidate vertices in a polygon at which either static guards can be placed or they can serve as an endpoint of the segment on which mobile guards move. Next, we present a technique to partition the environment based on the triangulation of the environment, and allocate guards to each partition to track the intruder. The allocation strategy leads to a classification of the mobile guards based on their task and coordination requirements. Finally, we present a strategy to activate/deactivate static guards based on the speed of the intruder. Simulation results are presented to validate the efficacy of the proposed techniques.

3-D Decentralized Prioritized Motion Planning and Coordination for High-Density Operations of Micro Aerial Vehicles

This paper presents a decentralized motion planning method for multiple aerial vehicles moving among 3-D polygonal obstacles resembling an urbanlike environment. The algorithm combines a prioritized $A^\star$ algorithm for high-level planning, along with a coordination method based on barrier functions for low-level trajectory generation and vehicle control. To this end, we extend the barrier functions method developed in our earlier work so that it treats 2-D and 3-D polygonal obstacles, and generates collision-free trajectories for the multiagent system. We furthermore augment the low-level trajectory generation and control with a prioritized $A^\star$ path planning algorithm, in order to compute waypoints and paths that force the agents of lower priority to avoid the paths of the agents of higher priority, reducing thus congestion. This feature enhances further the performance of the barrier-based coordination, and results in shorter paths and time to the goal destinations. We finally extend the proposed control design to the agents of constrained double-integrator dynamics, compared with the single-integrator case in our earlier work. We assume that the obstacles are known to the agents, and that each agent knows the state of other agents lying in its sensing area. Simulation results in 2-D and 3-D polygonal environments, as well as experimental results with micro aerial vehicles (quadrotors) in an indoor lab environment demonstrate the efficacy of the proposed approach.

A Linear Objective Function-Based Heuristic for Robotic Exploration of Unknown Polygonal Environments

This work presents a heuristic for describing the next best view location for an autonomous agent exploring an unknown environment. The approach considers each robot as a point mass with omnidirectional and unrestricted vision of the environment and line-of-sight communication operating in a polygonal environment which may contain holes. The number of robots in the team is always sufficient for full visual coverage of the space. The technique employed falls in the category of distributed visibility-based deployment algorithms which seek to segment the space based on each agent’s field of view with the goal of deploying each agent into the environment to create a visually connected series of agents which fully observe the previously unknown region. The contributions made to this field are a technique for utilizing linear programming methods to determine the solution to the next best observation (NBO) problem as well as a method for calculating multiple NBO points simultaneously. Both contributions are incorporated into an algorithm and deployed in a simulated environment built with MATLAB for testing. The algorithm successfully deployed agents into polygons which may contain holes. The efficiency of the deployment method was compared with random deployment methods to establish a performance metric for the proposed tactic. It was shown that the heuristic presented in this work performs better the other tested strategies.

Visibility-Based Target-Tracking Game: Bounds and Tracking Strategies

The art gallery problem or museum problem is a well-known visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. This letter deals with a variant of the art gallery problem in which guards are deployed to track a mobile intruder. We consider a team of free guards equipped with omnidirectional cameras deployed to track a bounded speed intruder inside a simply connected polygonal environment. We present a partitioning technique for any simply connected polygon using its diagonals such that each partition is at most a nonagon. Next, we propose an event-triggered strategy for a single guard to track an intruder around a reflex vertex. Based on the proposed strategy, we formulate the problem of finding the minimum speed of the guard and its corresponding trajectory inside any partition as a convex-optimization problem. The solution to the convex-optimization problem provides an upper bound on the speed of the mobile guard required for persistent tracking of the mobile intruder. Furthermore, we show that the number of guards deployed by the proposed technique belongs to $\mathcal {O}(\frac{2n}{7})$ and $\Omega (\frac{n}{6})$ , which is strictly less than $\lfloor {\frac{n}{3}} \rfloor$ (sufficient and sometimes necessary number of guards required for coverage). Finally, we extend the analysis to orthogonal polygons, and show that the upper bound on the number of guards deployed for tracking is strictly less than $\lfloor {\frac{n}{4}} \rfloor$ which is sufficient and sometimes necessary for the coverage problem.

Decentralized Hybrid Control for Multi-Agent Motion Planning and Coordination in Polygonal Environments

This paper presents a decentralized hybrid control scheme for the motion planning and coordination of teams of mobile agents in known obstacle environments with both convex and non-convex obstacles. A mathematical analysis using tools from switched systems theory is carried out to establish the convergence of the system trajectories under certain modeling assumptions on the surrounding environment. The design resolves a class of deadlock situations arising in earlier work, and allows for a wider class of obstacles (both convex and non-convex) to be considered in the environment. Simulation results demonstrate the efficacy of the algorithm.

Complete and optimal visibility-based pursuit-evasion

This paper computes a minimum-length pursuer trajectory that solves a visibility-based pursuit-evasion problem in which a single pursuer moving through a simply-connected polygonal environment seeks to locate an evader which may move arbitrarily fast, using an omni-directional field-of-view that extends to the environment boundary. We present a complete algorithm that computes a minimum-cost pursuer trajectory that ensures that the evader is captured, or reports in finite time that no such trajectory exists. This result improves upon the known algorithm of Guibas, Latombe, LaValle, Lin, and Motwani, which is complete but makes no guarantees about the quality of the solution. Our algorithm employs a branch-and-bound forward search that considers pursuer trajectories that could potentially lead to an optimal pursuer strategy. The search is performed on an exponential graph that can generate an infinite number of unique pursuer trajectories, so we must conduct meticulous pruning during the search to quickly discard pursuer trajectories that are demonstrably suboptimal. We describe an implementation of the algorithm, along with experiments that measure its performance in several environments with a variety of pruning operations.