arXiv subjects

David W. Casbeer

Publications and source records attributed to David W. Casbeer.

At least 19 recordsLinked to original sources

arXiv2026

A Novel Model for 3D Motion Planning for a Generalized Dubins Vehicle with Pitch and Yaw Rate Constraints

In this paper, we propose a new modeling approach and a fast algorithm for 3D motion planning, applicable for fixed-wing unmanned aerial vehicles. The goal is to construct the shortest path connecting given initial and final configurations subject to motion constraints. Our work differs from existing literature in two ways. First, we consider full vehicle orientation using a body-attached frame, which includes roll, pitch, and yaw angles. However, existing work uses only pitch and/or heading angle, which is insufficient to uniquely determine orientation. Second, we use two control inputs to represent bounded pitch and yaw rates, reflecting control by two separate actuators. In contrast, most previous methods rely on a single input, such as path curvature, which is insufficient for accurately modeling the vehicle's kinematics in 3D. We use a rotation minimizing frame to describe the vehicle's configuration and its evolution, and construct paths by concatenating optimal Dubins paths on spherical, cylindrical, or planar surfaces. Numerical simulations show our approach generates feasible paths within 10 seconds on average and yields shorter paths than existing methods in most cases.

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arXiv2026

An Exact Algorithm for Load-Dependent Traveling Salesman Problem for Unmanned Aerial Vehicle Package Delivery

In this article, we present a novel formulation for the load-dependent traveling salesman problem (LD-TSP), in which travel cost (or energy expended) depends on the vehicle's current load. This problem is relevant for package delivery and urban air mobility, where vehicles must transport and drop cargo at specified locations. The challenge lies in modeling the cost, which varies with both route sequence and onboard load. Our key contributions are: (i) formulating an energy dissipation model and proving energy expenditure depends linearly on vehicle mass and distance; and (ii) formulating a mixed-integer nonlinear programming formulation and providing a novel relaxation to obtain a mixed-integer linear program. Extensive numerical results show that optimal solutions for most instances with up to 50 targets are obtained within one minute. For unsolved instances within a 10-minute limit, optimality gaps are under 13%, highlighting the formulation's tightness. We further benchmark our approach against three proposed baseline formulations and another algorithm from a related problem, and demonstrate that our formulation outperforms all baselines.

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arXiv2025

Mutual Support by Sensor-Attacker Team for a Passive Target

We introduce a pursuit game played between a team of a sensor and an attacker and a mobile target in the unbounded Euclidean plane. The target is faster than the sensor, but slower than the attacker. The sensor's objective is to keep the target within a sensing radius so that the attacker can capture the target, whereas the target seeks to escape by reaching beyond the sensing radius from the sensor without getting captured by the attacker. We assume that as long as the target is within the sensing radius from the sensor, the sensor-attacker team is able to measure the target's instantaneous position and velocity. We pose and solve this problem as a \emph{game of kind} in which the target uses an open-loop strategy (passive target). Aside from the novel formulation, our contributions are four-fold. First, we present optimal strategies for both the sensor and the attacker, according to their respective objectives. Specifically, we design a sensor strategy that maximizes the duration for which the target remains within its sensing range, while the attacker uses proportional navigation to capture the target. Second, we characterize the \emph{sensable region} -- the region in the plane in which the target remains within the sensing radius of the sensor during the game -- and show that capture is guaranteed {if and only if} the Apollonius circle between the attacker and the target is fully contained within this region. Third, we {derive a lower bound} on the target's speed below which capture is guaranteed, and an upper bound on the target speed above which there exists an escape strategy for the target, from an arbitrary initial orientation between the agents. Fourth, for a given initial orientation between the agents, we present a sharper upper bound on the target speed above which there exists an escape strategy for the target.

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arXiv2025

One-vs-one Threat-Aware Weaponeering with Basic Engagement Zones

In this paper we address the problem of 'weaponeering', i.e., placing the weapon engagement zone (WEZ) of a vehicle on a moving target, while simultaneously avoiding the target's WEZ. A WEZ describes the lethality region of a range-limited weapon considering both the range of the weapon along with the state of the target. The weapons are assumed to have simple motion, while the vehicles carrying the weapons are modeled with Dubins dynamics. Three scenarios are investigated and are differentiated in the assumptions that can be made about the target in the process of the vehicle control design: 1) no knowledge of target control, 2) avoid unsafe positions assuming the target's optimal control, 3) full knowledge of target's optimal control. The engagement is formulated as a stochastic optimal control problem with uncertainty in the target's control modeled using a noise parameter applied to the target's control input. After discretizing the Hamilton-Jacobi-Bellman equation, Value iteration is then used to obtain an approximate solution for the optimal vehicle control and time-to-go. Simulation results support usage of the first paradigm: assume no knowledge of the target's control.

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arXiv2024

Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation

In this paper, motion planning for a vehicle moving on a unit sphere with unit speed is considered, wherein the desired terminal location is fixed, but the terminal orientation is free. The motion of the vehicle is modeled to be constrained by a maximum geodesic curvature $U_{max},$ which controls the rate of change of heading of the vehicle such that the maximum heading change occurs when the vehicle travels on a tight circular arc of radius $r = \frac{1}{\sqrt{1 + U_{max}^2}}$. Using Pontryagin's Minimum Principle, the main result of this paper shows that for $r \leq \frac{1}{2}$, the optimal path connecting a given initial configuration and a final location on the sphere belongs to a set of at most seven paths. The candidate paths are of type $CG, CC,$ and degenerate paths of the same, where $C \in \{L, R\}$ denotes a tight left or right turn, respectively, and $G$ denotes a great circular arc.

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arXiv2024

Equivalence of Dubins Path on Sphere with Geographic Coordinates and Moving Frames

In this article, two methods of addressing path planning for a Dubins vehicle moving on a sphere are considered, wherein either spherical coordinates or a moving frame are considered to describe the vehicle's motion. The primary contribution of this article is to show the equivalence of these two approaches, which in turn shows that the results known for the moving frame-based description transfer to the model utilizing spherical coordinates.

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arXiv2024

Multi Agent Pathfinding for Noise Restricted Hybrid Fuel Unmanned Aerial Vehicles

Multi Agent Path Finding (MAPF) seeks the optimal set of paths for multiple agents from respective start to goal locations such that no paths conflict. We address the MAPF problem for a fleet of hybrid-fuel unmanned aerial vehicles which are subject to location-dependent noise restrictions. We solve this problem by searching a constraint tree for which the subproblem at each node is a set of shortest path problems subject to the noise and fuel constraints and conflict zone avoidance. A labeling algorithm is presented to solve this subproblem, including the conflict zones which are treated as dynamic obstacles. We present the experimental results of the algorithms for various graph sizes and number of agents.

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arXiv2024

Pursuit-Evasion on a Sphere and When It Can Be Considered Flat

In classical works on a planar differential pursuit-evasion game with a faster pursuer, the intercept point resulting from the equilibrium strategies lies on the Apollonius circle. This property was exploited for the construction of the equilibrium strategies for two faster pursuers against one evader. Extensions for planar multiple-pursuer single-evader scenarios have been considered. We study a pursuit-evasion game on a sphere and the relation of the equilibrium intercept point to the Apollonius domain on the sphere. The domain is a generalization of the planar Apollonius circle set. We find a condition resulting in the intercept point belonging to the Apollonius domain, which is the characteristic of the planar game solution. Finally, we use this characteristic to discuss pursuit and evasion strategies in the context of two pursuers and a single slower evader on the sphere and illustrate it using numerical simulations.

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arXiv2024

Shortest Trajectory of a Dubins Vehicle with a Controllable Laser

We formulate a novel planar motion planning problem for a Dubins-Laser system that consists of a Dubins vehicle with an attached controllable laser. The vehicle moves with unit speed and the laser, having a finite range, can rotate in a clockwise or anti-clockwise direction with a bounded angular rate. From an arbitrary initial position and orientation, the objective is to steer the system so that a given static target is within the range of the laser and the laser is oriented at it in minimum time. We characterize multiple properties of the optimal trajectory and establish that the optimal trajectory for the Dubins-laser system is one out of a total of 16 candidates. Finally, we provide numerical insights that illustrate the properties characterized in this work.

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arXiv2024

Noise Aware Path Planning and Power Management of Hybrid Fuel UAVs

Hybrid fuel Unmanned Aerial Vehicles (UAV), through their combination of multiple energy sources, offer several advantages over the standard single fuel source configuration, the primary one being increased range and efficiency. Multiple power or fuel sources also allow the distinct pitfalls of each source to be mitigated while exploiting the advantages within the mission or path planning. We consider here a UAV equipped with a combustion engine-generator and battery pack as energy sources. We consider the path planning and power-management of this platform in a noise-aware manner. To solve the path planning problem, we first present the Mixed Integer Linear Program (MILP) formulation of the problem. We then present and analyze a label-correcting algorithm, for which a pseudo-polynomial running time is proven. Results of extensive numerical testing are presented which analyze the performance and scalability of the labeling algorithm for various graph structures, problem parameters, and search heuristics. It is shown that the algorithm can solve instances on graphs as large as twenty thousand nodes in only a few seconds.

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arXiv2023

Assisted Path Planning for a UGV-UAV Team Through a Stochastic Network

In this article, we consider a multi-agent path planning problem in a stochastic environment. The environment, which can be an urban road network, is represented by a graph where the travel time for selected road segments (impeded edges) is a random variable because of traffic congestion. An unmanned ground vehicle (UGV) wishes to travel from a starting location to a destination while minimizing the arrival time at the destination. UGV can traverse through an impeded edge but the true travel time is only realized at the end of that edge. This implies that the UGV can potentially get stuck in an impeded edge with high travel time. A support vehicle, such as an unmanned aerial vehicle (UAV) is simultaneously deployed from its starting position to assist the UGV by inspecting and realizing the true cost of impeded edges. With the updated information from UAV, UGV can efficiently reroute its path to the destination. The UGV does not wait at any time until it reaches the destination. The UAV is permitted to terminate its path at any vertex. The goal is then to develop an online algorithm to determine efficient paths for the UGV and the UAV based on the current information so that the UGV reaches the destination in minimum time. We refer to this problem as Stochastic Assisted Path Planning (SAPP). We present Dynamic $k$-Shortest Path Planning (D*KSPP) algorithm for the UGV planning and Rural Postman Problem (RPP) formulation for the UAV planning. Due to the scalability challenges of RPP, we also present a heuristic based Priority Assignment Algorithm (PAA) for the UAV planning. Computational results are presented to corroborate the effectiveness of the proposed algorithm to solve SAPP.

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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

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

Development of Linear Battery Model for Path Planning with Mixed Integer Linear Programming: Simulated and Experimental Validation

Mixed Integer Linear Programs (MILPs) are often used in the path planning of both ground and aerial vehicles. Such a formulation of the path planning problem requires a linear objective function and constraints, limiting the fidelity of the the tracking of vehicle states. One such parameter is the state of charge of the battery used to power the vehicle. Accurate battery state estimation requires nonlinear differential equations to be solved. This state estimation is important in path planning to ensure flyable paths, however when using MILPs to formulate the path planning problem these nonlinear equations cannot be implemented. Poor accuracy in battery estimation during the path planning runs the risk of the planned path being feasible by the estimation model but in reality will deplete the battery to a critical level. To the end of higher accuracy battery estimation within a MILP, we present here a simple linear battery model which predicts the change in state-of-charge (SOC) of a battery given a power draw and duration. This model accounts for changes in battery voltage due to applied electrical load and changes in battery SOC. The battery model is presented and then tested against alternate battery models in numerical and in experimental tests. Further, the effect the proposed linear model has over a simpler SOC estimation on the time-to-solve a resource constrained shortest path problem is also evaluated, implemented in two different algorithms. It is seen that the linear model performs well in battery state estimation while remaining implementable in a Linear Program or MILP, with little affect on the time-to-solve.

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arXiv2022

Cooperative Coverage with a Leader and a Wingmate in Communication-Constrained Environments

We consider a mission framework in which two unmanned vehicles (UVs), a leader and a wingmate, are required to provide cooperative coverage of an environment while being within a short communication range. This framework finds applications in underwater and/or military domains, where certain constraints are imposed on communication by either the application or the environment. An important objective of missions within this framework is to minimize the total travel and communication costs of the leader-wingmate duo. In this paper, we propose and formulate the problem of finding routes for the UVs that minimize the sum of their travel and communication costs as a network optimization problem of the form of a binary program (BP). The BP is computationally expensive, with the time required to compute optimal solutions increasing rapidly with the problem size. To address this challenge, here, we propose two algorithms, an approximation algorithm and a heuristic algorithm, to solve large-scale instances of the problem swiftly. We demonstrate the effectiveness and the scalability of these algorithms through an analysis of extensive numerical simulations performed over 500 instances, with the number of targets in the instances ranging from 6 to 100.

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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

Path Planning and Energy Management of Hybrid Air Vehicles for Urban Air Mobility

A novel coupled path planning and energy management problem for a hybrid unmanned air vehicle is considered, where the hybrid vehicle is powered by a dual gas/electric system. Such an aerial robot is envisioned for use in an urban setting where noise restrictions are in place in certain zones necessitating battery only operation. We consider the discrete version of this problem, where a graph is constructed by sampling the boundaries of the restricted zones, and develop a path planning algorithm. The planner simultaneously solves the path planing along with the energy mode switching control, under battery constraints and noise restrictions. This is a coupled problem involving discrete decision making to find the path to travel, and determining the state of charge of the battery along the path, which is a continuous variable. A sampling based algorithm to find near optimal solution to this problem is presented. To quantify the efficacy of the solution, an algorithm that computes tight lower bounds is also presented. The algorithms presented are verified using numerical simulations, and the average gap between the feasible solutions (upper bounds) and the lower bounds are, empirically, shown to be within 15% of each other.

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arXiv2022

Assisted Shortest Path Planning for a Convoy through a Repairable Network

In this article, we consider a multi-agent path planning problem in a partially impeded environment. The impeded environment is represented by a graph with select road segments (edges) in disrepair impeding vehicular movement in the road network. A convoy wishes to travel from a starting location to a destination while minimizing some accumulated cost. The convoy may traverse an impeded edge for an additional cost (associated with repairing the edge) than if it were unimpeded. A second vehicle, referred to as a service vehicle, is simultaneously deployed with the convoy. The service vehicle assists the convoy by repairing an edge, reducing the cost for the convoy to traverse that edge. The convoy is permitted to wait at any vertex to allow the service vehicle to complete repairing an edge. The service vehicle is permitted to terminate its path at any vertex. The goal is then to find a pair of paths so the convoy reaches its destination while minimizing the total time (cost) the two vehicles are active, including any time the convoy waits. We refer to this problem as the Assisted Shortest Path Problem (ASPP). We present a generalized permanent labeling algorithm to find an optimal solution for the ASPP. We also introduce additional modifications to the labeling algorithm to significantly improve the computation time and refer to the modified labeling algorithm as $GPLA^*$. Computational results are presented to illustrate the effectiveness of $GPLA^*$ in solving the ASPP. We then give concluding remarks and briefly discuss potential variants of the ASPP for future work.

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