arXiv subjects

David W. Casbeer

Publications and source records attributed to David W. Casbeer.

34 records · Page 2Linked to original sources

arXiv2022

Competitive Perimeter Defense of Conical Environments

We consider a perimeter defense problem in a planar conical environment in which a single vehicle, having a finite capture radius, aims to defend a concentric perimeter from mobile intruders. The intruders are arbitrarily released at the circumference of the environment and they move radially toward the perimeter with fixed speed. We present a competitive analysis approach to this problem by measuring the performance of multiple online algorithms for the vehicle against arbitrary inputs, relative to an optimal offline algorithm that has information about entire input sequence in advance. In particular, we establish two necessary conditions on the parameter space to guarantee (i) finite competitiveness of any algorithm and (ii) a competitive ratio of at least 2 for any algorithm. We then design and analyze three online algorithms and characterize parameter regimes in which they have finite competitive ratios. Specifically, our first two algorithms are provably 1, and 2-competitive, respectively, whereas our third algorithm exhibits different competitive ratios in different regimes of problem parameters. Finally, we provide a numerical plot in the parameter space to reveal additional insights into the relative performance of our algorithms.

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arXiv2022

Optimal Geodesic Curvature Constrained Dubins' Paths on a Sphere

In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius $r <1$, where $r$ depends on the bound, $U_{max}$. We show in this article that if $0 \frac{1}{2}$, while paths of the above type may cease to exist depending on the boundary conditions and the value of $r$, optimal paths may be concatenations of more than three circular arcs.

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arXiv2022

Optimal Strategies for the Game of Protecting a Plane in 3-D

A conflict between rational and autonomous agents is considered. The paper addresses a differential game of protecting a target in the 3-D space. This problem highlights the strong correlation between the highly dynamic scenario, the uncertainty on the behavior of the adversary, and the online and robust computation of state-feedback strategies which guarantee the required level of performance of each player. This work significantly expands previous results around this problem by providing the players' state-feedback saddle-point strategies. Additionally, the continuously differentiable Value function of the multi-agent differential game is obtained and it is shown to be the solution of the Hamilton-Jacobi-Isaacs equation. Finally, the Barrier surface is explicitly obtained and illustrative examples highlight the robustness properties and the guarantees provided by the saddle-point strategies obtained in this paper.

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arXiv2020

A Differential Game Approach for Beyond Visual Range Tactics

An operational relevant conflict between teams of autonomous vehicles in the Beyond Visual Range domain is addressed in this paper. Optimal strategies are designed in order for a team of air interceptors to protect a high value asset and block the attacking team at a safe distance from such asset. The attacking agents take specific roles of leader and wingman and also devise their own optimal strategies in order to launch an attack as close as possible from the asset. The problem is formulated as a zero-sum differential game between players with different speed over two stages: the attack and the retreat stages. For each stage the state-feedback optimal strategies of each player are derived in analytical form.

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arXiv2020

The Barrier Surface in the Cooperative Football Differential Game

This paper considers the blocking or football pursuit-evasion differential game. Two pursuers cooperate and try to capture the ball carrying evader as far as possible from the goal line. The evader wishes to be as close as possible to the goal line at the time of capture and, if possible, reach the line. In this paper the solution of the game of kind is provided: The Barrier surface that partitions the state space into two winning sets, one for the pursuer team and one for the evader, is constructed. Under optimal play, the winning team is determined by evaluating the associated Barrier function.

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arXiv2020

The Complete Differential Game of Active Target Defense

In the Target-Attacker-Defender (TAD) differential game, an Attacker missile strives to capture a Target aircraft. The Target tries to escape the Attacker and is aided by a Defender missile which aims at intercepting the Attacker before the latter manages to close in on the Target. The conflict between these intelligent adversaries has been suitably modeled as a zero-sum differential game. Optimal strategies have been synthesized covering the region of the state space where the Target/Defender team is able to win the game. However, the Game of Degree in the Attacker's region of win has not been fully addressed. Preliminary attempts at designing the players' strategies have not been proven to be optimal in the differential game sense. The main results of the paper present the optimal strategies of the Game of Degree in the Attacker's winning region of the state space. It is proven that the obtained strategies provide the saddle-point solution of the game; the Value function is obtained and it is shown to be continuous and continuously differentiable. It is also demonstrated that it is the solution of the Hamilton-Jacobi-Isaacs (HJI) equation. Finally, the obtained strategies are compared to recent results addressing the TAD differential game in [22]. It is shown by counterexample that the strategies proposed in [22] are not optimal. The unique regular solution of this differential game that actually provides a semipermeable Barrier surface is synthesized and verified in this paper.

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arXiv2019

Multiple Pursuer Multiple Evader Differential Games

In this paper an N-pursuer vs. M-evader team conflict is studied. The differential game of border defense is addressed and we focus on the game of degree in the region of the state space where the pursuers are able to win. This work extends classical differential game theory to simultaneously address weapon assignments and multi-player pursuit-evasion scenarios. Saddle-point strategies that provide guaranteed performance for each team regardless of the actual strategies implemented by the opponent are devised. The players' optimal strategies require the co-design of cooperative optimal assignments and optimal guidance laws. A representative measure of performance is proposed and the Value function of the game is obtained. It is shown that the Value function is continuous, continuously differentiable, and that it satisfies the Hamilton-Jacobi-Isaacs equation - the curse of dimensionality is overcome and the optimal strategies are obtained. The cases of N=M and N>M are considered. In the latter case, cooperative guidance strategies are also developed in order for the pursuers to exploit their numerical advantage. This work provides a foundation to formally analyze complex and high-dimensional conflicts between teams of N pursuers and M evaders by means of differential game theory.

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arXiv2019

Cooperative Routing for an Air-Ground Vehicle Team -- Exact Algorithm, Transformation Method, and Heuristics

This article considers a cooperative vehicle routing problem for an intelligence, surveillance, and reconnaissance mission in the presence of communication constraints between the vehicles. The proposed framework uses a ground vehicle and an Unmanned Aerial Vehicle (UAV) that travel cooperatively and visit a set of targets while satisfying the communication constraints. The problem is formulated as a mixed-integer linear program, and a branch-and-cut algorithm is developed to solve the problem to optimality. Furthermore, a transformation method and a heuristic are also developed for the problem. The effectiveness of all the algorithms is corroborated through extensive computational experiments on several randomly generated instances.

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arXiv2017

Multi-UAV Routing for Persistent Intelligence Surveillance & Reconnaissance Missions

We consider a Persistent Intelligence, Surveillance and Reconnaissance (PISR) routing problem, which includes collecting data from a set of specified task locations and delivering that data to a control station. Each task is assigned a refresh rate based on its priority, where higher priority tasks require higher refresh rates. The UAV team's objective is to minimize the maximum of the delivery times of all the tasks' data to the control station, while simultaneously, satisfying each task's revisit period constraint. The centralized path planning problem for this PISR routing problem is formulated using mixed integer linear programming and solved using a branch-and-cut algorithm. Heuristics are presented to find sub-optimal feasible solutions that require much less computation time. The algorithms are tested on several instances and their performance is compared with respect to the optimal cost and computation time.

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arXiv2016

Pursuit on a Graph under Partial Information from Sensors

We consider a class of pursuit-evasion problems where an evader enters a directed acyclic graph and attempts to reach one of the terminal nodes. A pursuer enters the graph at a later time and attempts to capture the evader before it reaches a terminal node. The pursuer can only obtain information about the evader's path via sensors located at each node in the graph; the sensor measurements are either green or red (indicating whether or not the evader has passed through that node). We first show that it is NP-hard to determine whether the pursuer can enter with some nonzero delay and still be guaranteed to capture the evader, even for the simplest case when the underlying graph is a tree. This also implies that it is NP-hard to determine the largest delay at which the pursuer can enter and still have a guaranteed capture policy. We further show that it is NP-hard to approximate (within any constant factor) the largest delay at which the pursuer can enter. Finally, we provide an algorithm to compute the maximum pursuer delay for a class of node-sweeping policies on tree networks and show that this algorithm runs in linear-time for bounded-degree trees.

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arXiv2015

Decentralized Event-Triggered Consensus over Unreliable Communication Networks

This article studies distributed event-triggered consensus over unreliable communication channels. Communication is unreliable in the sense that the broadcast channel from one agent to its neighbors can drop the event-triggered packets of information, where the transmitting agent is unaware that the packet was not received and the receiving agents have no knowledge of the transmitted packet. Additionally, packets that successfully arrive at their destination may suffer from time-varying communication delays. In this paper, we consider directed graphs, and we also relax the consistency on the packet dropouts and the delays. Relaxing consistency means that the delays and dropouts for a packet broadcast by one agent can be different for each receiving node. We show that even under this challenging scenario, agents can reach consensus asymptotically while reducing transmissions of measurements based on the proposed event-triggered consensus protocol. In addition, positive inter-event times are obtained which guarantee that Zeno behavior does not occur.

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arXiv2015

Escape Regions of the Active Target Defense Differential Game

The active target defense differential game is addressed in this paper. In this differential game an Attacker missile pursues a Target aircraft. The aircraft is however aided by a Defender missile launched by, say, the wingman, to intercept the Attacker before it reaches the Target aircraft. Thus, a team is formed by the Target and the Defender which cooperate to maximize the separation between the Target aircraft and the point where the Attacker missile is intercepted by the Defender missile, while the Attacker simultaneously tries to minimize said distance. This paper focuses on characterizing the set of coordinates such that if the Target's initial position belong to this set then its survival is guaranteed if both the Target and the Defender follow their optimal strategies. Such optimal strategies are presented in this paper as well.

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arXiv2015

Periodic Event-Triggered Synchronization of Linear Multi-agent Systems with Communication Delays

Multi-agent systems cooperation to achieve global goals is usually limited by sensing, actuation, and communication issues. At the local level, continuous measurement and actuation is only approximated by the use of digital mechanisms that measure and process information in order to compute and update new control input values at discrete time instants. Interaction with other agents or subsystems takes place, in general, through a digital communication channel with limited bandwidth where transmission of continuous-time signals is not possible. Additionally, communication channels may be subject to other imperfections such as time-varying delays. This paper considers the problem of consensus (or synchronization of state trajectories) of multi-agent systems that are described by general linear dynamics and are connected using undirected graphs. An event-triggered consensus protocol is proposed, where each agent implements discretized and decoupled models of the states of its neighbors. This approach not only avoids the need for continuous communication between agents but also provides a decentralized method for transmission of information in the presence of time-varying communication delays where each agent decides its own broadcasting time instants based only on local information. This method gives more flexibility for scheduling information broadcasting compared to periodic and sampled-data implementations. The use of discretized models by each agent allows for a periodic event-triggered strategy where continuous actuation and continuous measurement of the states are not necessary.

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arXiv2015

Decentralized Event-Triggered Consensus of Linear Multi-agent Systems under Directed Graphs

An event-triggered control technique for consensus of multi-agent systems with general linear dynamics is presented. This paper extends previous work to consider agents that are connected using directed graphs. Additionally, the approach shown here provides asymptotic consensus with guaranteed positive inter-event time intervals. This event-triggered control method is also used in the case where communication delays are present. For the communication delay case we also show that the agents achieve consensus asymptotically and that, for every agent, the time intervals between consecutive transmissions is lower-bounded by a positive constant.

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arXiv2015

Active Target Defense Differential Game with a Fast Defender

This paper addresses the active target defense differential game where an Attacker missile pursues a Target aircraft. A Defender missile is fired by the Target's wingman in order to intercept the Attacker before it reaches the aircraft. Thus, a team is formed by the Target and the Defender which cooperate to maximize the distance between the Target aircraft and the point where the Attacker missile is intercepted by the Defender missile, while the Attacker tries to minimize said distance. The results shown here extend previous work. We consider here the case where the Defender is faster than the Attacker. The solution to this differential game provides optimal heading angles for the Target and the Defender team to maximize the terminal separation between Target and Attacker and it also provides the optimal heading angle for the Attacker to minimize the said distance.

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arXiv2015

Bridge Consensus: Ignoring Initial Inessentials

In this paper, the problem of bridge consensus is presented and solved. Bridge consensus consists of a network of nodes, some of whom are participating and others are non-participating. The objective is for all the agents to reach average consensus of the participating nodes initial values in a distributed and scalable manner. To do this, the nodes must use the network connections of the non-participating nodes, which act as bridges for information and ignore the initial values of the non-participating nodes. The solution to this problem is made by merging the ideas from estimation theory and consensus theory. By considering the participating nodes has having equal information and the non-participating nodes as having no information, the nodes initial values are transformed into information space. Two consensus filters are run in parallel on the information state and information matrix. Conditions ensuring that the product of the inverse information matrix and the information state of each agent reaches average consensus of the participating agents' initial values is given.

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