arXiv subjects

Alexander Von Moll

Publications and source records attributed to Alexander Von Moll.

30 records · Page 2Linked to original sources

arXiv2023

Virtual Target Selection for a Multiple-Pursuer Multiple-Evader Scenario

This paper considers an M-pursuer N-evader scenario involving virtual targets. The virtual targets serve as an intermediary target for the pursuers, allowing the pursuers to delay their final assignment to the evaders. However, upon reaching the virtual target, the pursuers must decide which evader to capture. It is assumed that there are more pursuers than evaders and that the pursuers are faster than the evaders. The objective is two-part: first, assign each pursuer to a virtual target and evader such that the pursuer team's energy is minimized, and second, choose the virtual targets' locations for this minimization problem. The approach taken is to consider the Apollonius geometry between each pursuer's virtual target location and each evader. Using the constructed Apollonius circles, the pursuer's travel distance and maneuver at a virtual target are obtained. These metrics serve as a gauge for the total energy required to capture a particular evader and are used to solve the joint virtual target selection and pursuer-evader assignment problem. This paper provides a mathematical definition of this problem, the solution approach taken, and an example.

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arXiv2023

Basic Engagement Zones

This paper establishes a more formal definition for an Engagement Zone (EZ) and derives some basic EZs associated with fundamental engagement models associated with pursuit-evasion and turret-evasion. The basic EZs presented in this paper capture the most salient aspects of the Pursuer-Agent and Turret-Agent engagements: namely the geometry of the aspect angle and the relative differences in capability (i.e., maximum speeds, range, etc.). One of the main advantages of utilizing EZs for path planning is that they encode an overall desire for Agent to go somewhere without requiring an aggressive maneuver or active evasion should the Pursuer or Turret begin its pursuit. It is shown that there is some advantage, in terms of time savings, in EZ-based navigation around a single range-limited Pursuer as compared with circumnavigating the capturability region.

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arXiv2023

Escape from an Orbiting Pursuer with a Nonzero Capture Radius

This paper explores a multi-agent containment problem, where a fast evader, modeled having constant speed and using constant heading, attempts to escape a circular containment region that is orbited by a slower pursuer with a nonzero capture radius. The pursuer is constrained to move along the edge of the containment region and seeks to capture the evader. This paper presents an in-depth analysis of this pursuer-evader containment scenario. First, multiple types of capture conditions for a single-pursuer case are analyzed defining the worst-case initial position for the pursuer. Second, a parametric study is performed to demonstrate the effects of speed ratio, capture radius, and initial location of the evader. Finally, a reachability analysis is performed to investigate the viable escape headings and reachable regions by the evader. This work provides a foundation for the analysis of escape under more general evader inputs as well as a multiple-pursuer version of the scenario.

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arXiv2023

Perimeter Defense using a Turret with Finite Range and Service Times

We consider a perimeter defense problem in a planar conical environment comprising a single turret that has a finite range and non-zero service time. The turret seeks to defend a concentric perimeter against $N\geq 2$ intruders. Upon release, each intruder moves radially towards the perimeter with a fixed speed. To capture an intruder, the turret's angle must be aligned with that of the intruder's angle and must spend a specified service time at that orientation. We address offline and online versions of this optimization problem. Specifically, in the offline version, we establish that in general parameter regimes, this problem is equivalent to solving a Travelling Repairperson Problem with Time Windows (TRP-TW). We then identify specific parameter regimes in which there is a polynomial time algorithm that maximizes the number of intruders captured. In the online version, we present a competitive analysis technique in which we establish a fundamental guarantee on the existence of at best $(N-1)$-competitive algorithms. We also design two online algorithms that are provably $1$ and $2$-competitive in specific parameter regimes.

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arXiv2022

Target Defense against Sequentially Arriving Intruders

We consider a variant of the target defense problem where a single defender is tasked to capture a sequence of incoming intruders. The intruders' objective is to breach the target boundary without being captured by the defender. As soon as the current intruder breaches the target or gets captured by the defender, the next intruder appears at a random location on a fixed circle surrounding the target. Therefore, the defender's final location at the end of the current game becomes its initial location for the next game. Thus, the players pick strategies that are advantageous for the current as well as for the future games. Depending on the information available to the players, each game is divided into two phases: partial information and full information phase. Under some assumptions on the sensing and speed capabilities, we analyze the agents' strategies in both phases. We derive equilibrium strategies for both the players to optimize the capture percentage using the notions of engagement surface and capture circle. We quantify the percentage of capture for both finite and infinite sequences of incoming intruders.

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arXiv2022

Surveillance of a Faster Fixed-Course Target

The maximum surveillance of a target which is holding course is considered, wherein an observer vehicle aims to maximize the time that a faster target remains within a fixed-range of the observer. This entails two coupled phases: an approach phase and observation phase. In the approach phase, the observer strives to make contact with the faster target, such that in the observation phase, the observer is able to maximize the time where the target remains within range. Using Pontryagin's Minimum Principle, the optimal control laws for the observer are found in closed-form. Example scenarios highlight various aspects of the engagement.

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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

One Apollonius Circle is Enough for Many Pursuit-Evasion Games

This paper investigates obstacle-free simple motion pursuit-evasion problems where the pursuer is faster and game termination is point capture. It is well known that the interior of the Apollonius Circle (AC) is the evader's dominance region, however, it was unclear whether the evader could reach outside the initial AC without being captured. We construct a pursuit strategy that guarantees the capture of an evader within an arbitrarily close neighborhood of the initial AC. The pursuer strategy is derived by reformulating the game into a nonlinear control problem, and the guarantee holds against any admissible evader strategy. Our result implies that the evader can freely select the capture location, but only inside the initial AC. Therefore, a class of problems, including those where the payoff is determined solely based on the location of capture, are now trivial.

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arXiv2021

An Optimal Engagement Zone Avoidance Scenario in 2-D

In this paper, an optimal control problem is considered where a target vehicle aims to reach a desired location in minimum time while avoiding a dynamic engagement zone. Using simple motion, four potential approaches are considered. First, the min-time strategy which ignores the engagement zone is posed and solved. Second, the min-time strategy which avoids the engagement zone entirely is considered. Third, the min-time strategy which allows for some time in the engagement zone; but, still strives to stay away from the center of the engagement zone is posed. Lastly, a fixed final-time strategy is considered, wherein the target tries to avoid the engagement zone; but, is required to arrive at the desired location at a specific time. Using a nonlinear program solver, the optimal strategies are numerically solved. From the results of the numeric solutions, the optimal strategies are discussed and comparisons are drawn.

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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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arXiv2018

Optimal Target Intercept Paths for Vehicles with Turn Radius Constraints

We present a path planning problem for a pursuer to intercept a target traveling on a circle. The pursuer considered here has limited yaw rate, and therefore its path should satisfy the kinematic constraints. We assume that the distance between initial position of the pursuer and any point on the target circle is greater than four times the minimum turn radius of the pursuer. We prove the continuity of the Dubins paths of type Circle-Straight line-Circle with respect to the position on the target circle. This is used to prove that the optimal interception path is a Dubins path, and we present an iterative algorithm to find the optimal interception point on the target circle.

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arXiv2018

Shortest Dubins Path to a Circle

The Dubins path problem had enormous applications in path planning for autonomous vehicles. In this paper, we consider a generalization of the Dubins path planning problem, which is to find a shortest Dubins path that starts from a given initial position and heading, and ends on a given target circle with the heading in the tangential direction. This problem has direct applications in Dubins neighborhood traveling salesman problem, obstacle avoidance Dubins path planning problem etc. We characterize the length of the four CSC paths as a function of angular position on the target circle, and derive the conditions which to find the shortest Dubins path to the target circle.

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