Approximation of Curvature-Constrained Shortest Paths through a Sequence of Points

Lee, Jaeha; Cheong, Otfried; Kwon, Woocheol; Chwa, Kyung-Yong

doi:10.1007/3-540-45253-2_29

Cited by 22 publications

(18 citation statements)

References 14 publications

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“…The idea has been used in the context of motion planning of robots by several authors including Jacobs and Canny (1992), Edison and Shima (2011), Cons et al (2014) and others. Alternative approaches for the DSPP are due to Goac et al (2013), Lee et al (2000), Rathinam and Khargonekar (2016) and Manyam et al (2017).…”

Section: Introductionmentioning

confidence: 99%

New heuristic algorithms for the Dubins traveling salesman problem

Babel

2020

J Heuristics

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The problem of finding a shortest curvature-constrained closed path through a set of targets in the plane is known as Dubins traveling salesman problem (DTSP). Applications of the DTSP include motion planning for kinematically constrained mobile robots and unmanned fixed-wing aerial vehicles. The difficulty of the DTSP is to simultaneously find an order of the targets and suitable headings (orientation angles) of the vehicle when passing the targets. Since the DTSP is known to be NP-hard there is a need for heuristic algorithms providing good quality solutions in reasonable time. Inspired by standard methods for the TSP we present a collection of such heuristics adapted to the DTSP. The algorithms are based on a technique that optimizes the headings of the targets of an open or closed subtour with given order. This is done by discretizing the headings, constructing an auxiliary network and computing a shortest path in the network. The first algorithm for the DTSP uses the order of the targets obtained from the solution of the Euclidean TSP. A second class of algorithms extends an open subtour by successively adding a new target and closes the tour if all targets have been added. A third class of algorithms starts with a closed subtour consisting of few targets and successively inserts a new target into the tour. The individual algorithms differ by the number of headings to be optimized in each iteration. Extensive simulation results show that the proposed methods are competitive with state-of-the-art methods for the DTSP concerning performance and superior concerning running time, which makes them applicable also to large-scale scenarios. Keywords Dubins traveling salesman problem • Curvature-constrained traveling salesman problem • Dubins vehicle • Heuristic methods B Luitpold Babel

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Section: Introductionmentioning

confidence: 99%

New heuristic algorithms for the Dubins traveling salesman problem

Babel

2020

J Heuristics

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“…The approximation algorithm was implemented on problem instances with 12,15,18,21,24,27 and 30 target points. For a given number of target points, we generated 100 instances.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Therefore, algorithms that can find feasible solutions with theoretical guarantees on the deviation of the solutions from the optimum are useful. The CSP was first considered by Lee et al in [21] where they provide a 5.03approximation algorithm. The approximation factor of this algorithm has been recently improved to (2 + 2 π + π 2 ≈ 4.21) in [22].…”

Section: Introductionmentioning

confidence: 99%

An Approximation Algorithm for a Shortest Dubins Path Problem

Rathinam

Khargonekar

2016

Volume 2: Mechatronics; Mechatronics and Controls in Advanced Manufacturing; Modeling and Control of Automotive Systems and Com

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The problem of finding the shortest path for a vehicle visiting a given sequence of target points subject to the motion constraints of the vehicle is an important problem that arises in several monitoring and surveillance applications involving unmanned aerial vehicles. There is currently no algorithm that can find an optimal solution to this problem. Therefore, heuristics that can find approximate solutions with guarantees on the quality of the solutions are useful. The best approximation algorithm currently available for the case when the distance between any two adjacent target points in the sequence is at least equal to twice the minimum radius of the vehicle has a guarantee of 3.04. This article provides a new approximation algorithm which improves this guarantee to 1 + π 3 ≈ 2.04. The developed algorithm is also implemented for hundreds of typical instances involving at most 30 points to corroborate the performance of the proposed approach. IntroductionAdvances in sensing, robotics and wireless networks have enabled the use of teams of small Unmanned Vehicles (UVs) for environmental sensing applications including crop monitoring [1, 2], ocean bathymetry [3], forest fire monitoring [4], ecosystem management [5,6], and civil security applications such as border surveillance [7,8] and disaster management [9]. These applications frequently require vehicles to collect data such as visible/infra-red/thermal images, videos of specified target sites, and environmental data such as temperature, moisture, humidity using onboard sensors, and deliver them to a base station. To accomplish this requirement, small unmanned vehicles with motion and fuel constraints commonly visit a set of target points.

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“…This path must intersect the line x = 0 in a point p and the line x = d in a point q. The distance |pq| is at least d. If the path from S to p intersects L S ∪ R S , then Ahn et al [18,Fact 1] showed that it has length at least π. Otherwise the path avoids L S ∪ R S and hence must have length at least π.…”

Section: Case Cmentioning

confidence: 99%

The cost of bounded curvature

Kim

Cheong

2013

Computational Geometry

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We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations σ, σ , let (σ, σ ) be the shortest bounded-curvature path from σ to σ . For d 0, let (d) be the supremum of (σ, σ ), over all pairs (σ, σ ) that are at Euclidean distance d. We study the function dub(d) = (d) − d, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that dub(d) decreases monotonically from dub(0) = 7π/3 to dub(d * ) = 2π, and is constant for d d * . Here d * ≈ 1.5874. We describe pairs of configurations that exhibit the worst-case of dub(d) for every distance d.this is a natural and fundamental question related to motion planning with bounded curvature, it is also a relevant question that has repeatedly appeared in the literature, with only partial answers so far.Dubins [11] showed that the shortest curvature-constrained path between two configurations consists of at most three segments, each of which is either a straight segment or a circular arc of radius one. Using ideas from control theory, Boissonnat et al. [7], in parallel with Sussmann and Tang [26], gave an alternative proof. Sussmann [25] extended the characterization to the 3-dimensional case. Bui et al.[10] discussed how the types of optimal paths partition the configuration space, and also proved that optimal paths for free final orientation have at most two segments [6]. Significant work has been done on the problems of deciding whether a bounded-curvature path exists between given configurations among different kinds of obstacles and finding the shortest such path [12,5,21,15,3,4,2,1,9,8].At least two interesting problems have been studied where not configurations but only locations for the robot are given. The first problem considers a sequence of points in the plane, and asks for the shortest curvature-constrained path that visits the points in this sequence. In the second problem, the Dubins traveling salesman problem, the input is a set of points in the plane, and asks to find a shortest curvature-constrained path visiting all points. Both problems have been studied by researchers in the robotics community, giving heuristics and experimental results [22,19,20]. From a theoretical perspective, Lee et al.[18] gave a linear-time, constant-factor approximation algorithm for the first problem. No general approximation algorithms are known for the Dubins traveling salesman problem (the approximation factor of the known algorithms depends on the smallest distance between points).All this work depends on some knowledge of the function dub. Lee et al. [18], for instance, prove that the approximation ratio of their algorithms is max(A, π/2 + B/π), where A = 1 + sup{dub(d)/d | d 2} and B = sup{dub(d) + d | d 2}. They claim without proof that dub(d) 2π for d 2 and derive from this that A = 1 + π and B 5π/2 + 3, leading to an approximation ratio of about 5.03. We ...

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Approximation of Curvature-Constrained Shortest Paths through a Sequence of Points

Cited by 22 publications

References 14 publications

New heuristic algorithms for the Dubins traveling salesman problem

New heuristic algorithms for the Dubins traveling salesman problem

An Approximation Algorithm for a Shortest Dubins Path Problem

The cost of bounded curvature

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