arXiv subjects

Deepak Prakash Kumar

Publications and source records attributed to Deepak Prakash Kumar.

13 recordsLinked to original sources

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

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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arXiv2025

A Path Planning Algorithm for a Hybrid UAV Traveling in Noise Restricted Zones

This paper presents an integrated approach for efficient path planning and energy management in hybrid unmanned aerial vehicles (HUAVs) equipped with dual fuel-electric propulsion systems. These HUAVs operate in environments that include noise-restricted zones, referred to as quiet zones, where only electric mode is permitted. We address the problem by parameterizing the position of a point along the side of the quiet zone using its endpoints and a scalar parameter, transforming the problem into a variant of finding the shortest path over a graph of convex sets. We formulate this problem as a mixed-integer convex program (MICP), which can be efficiently solved using commercial solvers. Additionally, a tight lower bound can be obtained by relaxing the path-selection variable. Through extensive computations across 200 instances over four maps, we show a substantial improvement in computational efficiency over a state-of-the-art method, achieving up to a 100-fold and 10-fold decrease in computation time for calculating the lower bound and the exact solution, respectively. Moreover, the average gap between the exact cost and the lower bound was approximately 0.24%, and the exact cost was 1.05% lower than the feasible solution from the state-of-the-art approach on average, highlighting the effectiveness of our method. We also extend our approach to plan the HUAV route to visit a set of targets and return to its starting location in environments with quiet zones, yielding a Traveling Salesman Problem (TSP). We employ two methodologies to solve the TSP: one where the SOC at each target is discretized, and another where it is assumed to be the minimum allowable level upon departure. A comparative analysis reveals the second method achieves a cost within 1.02% of the first on average while requiring significantly less computational time.

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arXiv2025

Generation of Paths for Motion Planning for a Dubins Vehicle on Sphere

In this article, the candidate optimal paths for a Dubins vehicle on a sphere are analytically derived. In particular, the arc angles for segments in $CGC$, $CCC$, $CCCC$, and $CCCCC$ paths, which have previously been shown to be optimal depending on the turning radius $r$ of the vehicle by Kumar \textit{et al.}, are analytically derived. The derived expressions are used for the implementation provided in https://github.com/DeepakPrakashKumar/Motion-planning-on-sphere.

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arXiv2025

Time-optimal Convexified Reeds-Shepp Paths on a Sphere

This article studies the time-optimal path planning problem for a convexified Reeds-Shepp (CRS) vehicle on a unit sphere, capable of both forward and backward motion, with speed bounded in magnitude by 1 and turning rate bounded in magnitude by a given constant. For the case in which the turning-rate bound is at least 1, using Pontryagin's Maximum Principle and a phase-portrait analysis, we show that the optimal path connecting a given initial configuration to a desired terminal configuration consists of at most six segments drawn from three motion primitives: tight turns, great circular arcs, and turn-in-place motions. A complete classification yields a finite sufficient list of 23 optimal path types with closed-form segment angles derived. The complementary case in which the turning-rate bound is less than 1 is addressed via an equivalent reformulation. The proposed formulation is applicable to underactuated satellite attitude control, spherical rolling robots, and mobile robots operating on spherical or gently curved surfaces. The source code for solving the time-optimal path problem and visualization is publicly available at https://github.com/sixuli97/Optimal-Spherical-Convexified-Reeds-Shepp-Paths.

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arXiv2025

A New Approach to Motion Planning in 3D for a Dubins Vehicle: Special Case on a Sphere

In this article, a new model for 3D motion planning, applicable to aerial vehicles, is proposed to connect an initial and final configuration subject to pitch rate and yaw rate constraints. The motion planning problem for a curvature-constrained vehicle over the surface of a sphere is identified as an intermediary problem to be solved, and it is the focus of this paper. In this article, the optimal path candidates for a vehicle with a minimum turning radius $r$ moving over a unit sphere are derived using a phase portrait approach. We show that the optimal path is $CGC$ or concatenations of $C$ segments through simple proofs, where $C = L, R$ denotes a turn of radius $r$ and $G$ denotes a great circular arc. We generalize the previous result of optimal paths being $CGC$ and $CCC$ paths for $r \in \left(0, \frac{1}{2} \right]\bigcup\{\frac{1}{\sqrt{2}}\}$ to $r \leq \frac{\sqrt{3}}{2}$ to account for vehicles with a larger $r$. We show that the optimal path is $CGC, CCCC,$ for $r \leq \frac{1}{\sqrt{2}},$ and $CGC, CC_\pi C, CCCCC$ for $r \leq \frac{\sqrt{3}}{2}.$ Additionally, we analytically construct all candidate paths and provide the code in a publicly accessible repository.

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arXiv2024

Heterogeneous Min-Max Multi-Vehicle Multi-Depot Traveling Salesman Problem: Heuristics and Computational Results

In this paper, a heuristic for a heterogeneous min-max multi-vehicle multi-depot traveling salesman problem is proposed, wherein heterogeneous vehicles start from given depot locations and need to cover a given set of targets. In the considered problem, vehicles can be structurally heterogeneous due to different vehicle speeds and/or functionally heterogeneous due to different vehicle-target assignments originating from different sensing capabilities of vehicles. The proposed heuristic for the considered problem has three stages: an initialization stage to generate an initial feasible solution, a local search stage to improve the incumbent solution by searching through different neighborhoods, and a perturbation/shaking stage, wherein the incumbent solution is perturbed to break from a local minimum. In this study, three types of neighborhood searches are employed. Furthermore, two different methods for constructing the initial feasible solution are considered, and multiple variations in the neighborhoods considered are explored in this study. The considered variations and construction methods are evaluated on a total of 128 instances generated with varying vehicle-to-target ratios, distribution for generating the targets, and vehicle-target assignment and are benchmarked against the best-known heuristic for this problem. Two heuristics were finally proposed based on the importance provided to objective value or computation time through extensive computational studies.

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arXiv2024

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

In this paper, motion planning for a Dubins vehicle on a unit sphere to attain a desired final location is considered. The radius of the Dubins path on the sphere is lower bounded by $r$. In a previous study, this problem was addressed, wherein it was shown that the optimal path is of type $CG, CC,$ or a degenerate path of the same for $r \leq \frac{1}{2}.$ Here, $C = L, R$ denotes an arc of a tight left or right turn of minimum turning radius $r,$ and $G$ denotes an arc of a great circle. In this study, the candidate paths for the same problem are generalized to model vehicles with a larger turning radius. In particular, it is shown that the candidate optimal paths are of type $CG, CC,$ or a degenerate path of the same for $r \leq \frac{\sqrt{3}}{2}.$ Noting that at most two $LG$ paths and two $RG$ paths can exist for a given final location, this article further reduces the candidate optimal paths by showing that only one $LG$ and one $RG$ path can be optimal, yielding a total of seven candidate paths for $r \leq \frac{\sqrt{3}}{2}.$ Additional conditions for the optimality of $CC$ paths are also derived in this study.

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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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arXiv2023

Heuristic for Min-Max Heterogeneous Multi-Vehicle Multi-Depot Traveling Salesman Problem

In this article, a heuristic is proposed for a min-max heterogeneous multi-vehicle multi-depot traveling salesman problem (TSP), wherein heterogeneous vehicles start from given depot positions and need to cover a given set of targets. The vehicles should cover given targets such that the maximum tour time is minimized. In the considered problem, vehicles considered can be functionally heterogeneous, wherein specific targets must be covered by a particular vehicle, structurally heterogeneous, wherein the vehicles can travel at different speeds, or both. The proposed heuristic generalizes the MD heuristic for the min-max homogeneous multi-vehicle multi-depot TSP and has three stages: an initialization stage to generate a feasible solution, a local search stage in which the vehicle with the maximum tour time is improved, and a perturbation stage to break from a local minimum. The proposed heuristic is benchmarked with the optimal solution obtained by solving a mixed integer linear program using branch and cut for instances considering three vehicles covering thirty targets. Variations in the percentage of vehicle-target assignments and the number of vehicles starting at the same depot are studied to show the heuristic's effectiveness in producing high-quality solutions. It was observed that the heuristic generated feasible solutions within 4% of the optimum on average for the considered instances.

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arXiv2023

UAV Routing for Enhancing the Performance of a Classifier-in-the-loop

Some human-machine systems are designed so that machines (robots) gather and deliver data to remotely located operators (humans) through an interface in order to aid them in classification. The performance of a human as a (binary) classifier-in-the-loop is characterized by probabilities of correctly classifying objects of type $T$ and $F$. These two probabilities depend on the dwell time, $d$, spent collecting information at a point of interest (POI or interchangeably, target). The information gain associated with collecting information at a target is then a function of dwell time $d$ and discounted by the revisit time, $R$, i.e., the duration between consecutive revisits to the same target. The objective of the problem of routing for classification is to optimally route the vehicles and determine the optimal dwell time at each target so as to maximize the total discounted information gain while visiting every target at least once. In this paper, we make a simplifying assumption that the information gain is discounted exponentially by the revisit time; this assumption enables one to decouple the problem of routing with the problem of determining optimal dwell time at each target for a single vehicle problem. For the multi-vehicle problem, we provide a fast heuristic to obtain the allocation of targets to each vehicle and the corresponding dwell time.

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arXiv2022

The Weighted Markov-Dubins Problem

In this article, a variation of the classical Markov-Dubins problem is considered, which deals with curvature-constrained least-cost paths in a plane with prescribed initial and final configurations, different bounds for the sinistral and dextral curvatures, and penalties $μ_L$ and $μ_R$ for the sinistral and dextral turns, respectively. The addressed problem generalizes the classical Markov-Dubins problem and the asymmetric sinistral/dextral Markov-Dubins problem. The proposed formulation can be used to model an Unmanned Aerial Vehicle (UAV) with a penalty associated with a turn due to a loss in altitude while turning or a UAV with different costs for the sinistral and dextral turns due to hardware failures or environmental conditions. Using optimal control theory, the main result of this paper shows that the optimal path belongs to a set of at most $21$ candidate paths, each comprising of at most five segments. Unlike in the classical Markov-Dubins problem, the $CCC$ path, which is a candidate path for the classical Markov-Dubins problem, is not optimal for the weighted Markov-Dubins problem. Moreover, the obtained list of candidate paths for the weighted Markov-Dubins problem reduces to the standard $CSC$ and $CCC$ paths and the corresponding degenerate paths when $μ_L$ and $μ_R$ approach zero.

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