Shortest paths of bounded curvature in the plane

Boissonnat, Jean‐Daniel; Cérézo, André; Leblond, Juliette

doi:10.1007/3-540-57132-9_1

Cited by 109 publications

(130 citation statements)

References 5 publications

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“…Theorem 1 was later proved again by Boissonnat et al [1] by formulating it as a control problem and applying Pontryagin's Minimum Principle. A more detailed version of [1] can be found in [13].…”

Section: Curvature-constrained Pathsmentioning

confidence: 91%

“…The major tools employed in tackling this problem are adapted from Dubins [5] and Boissonnat [1]. By transforming the minimum weighted length problem into a minimum time problem, it becomes clear that the reciprocal of the directional-cost function becomes the object of interest in characterising the forms of optimal paths that have to be considered.…”

Section: Introductionmentioning

confidence: 99%

“…The first step is to apply Pontryagin's Minimum Principle as in [1] to conclude that any optimal path must only consist of maximum curvature arcs and straight segments, given any strict velocity function. We then prove some basic properties of the cost of paths subjected to SPC velocity functions.…”

Section: Introductionmentioning

confidence: 99%

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Curvature-constrained directional-cost paths in the plane

et al. 2011

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This paper looks at the problem of finding the minimum cost curvatureconstrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We conclude that if the reciprocal of the directional-cost function is strictly polarly convex, then the forms of the optimal paths are of the same forms as Dubins paths. If we relax the strict polar convexity to weak polar convexity, then we show that there exists a Dubins path which is optimal. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint and the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground.

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Section: Curvature-constrained Pathsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Curvature-constrained directional-cost paths in the plane

et al. 2011

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“…We interpret γ : I → R 3 as the deformed configuration of some material curve in the body, the so-called base curve, with length parameter s. d 1 A deformed configuration of the rod is thus determined by mappings γ : I → R 3 and D : I → SO (3). It is reasonable to assume that γ ∈ W 1,q (I, R 3 ) and D ∈ W 1, p (I, R 3×3 ) for p, q ≥ 1.…”

Section: Constrained Elastic Rodsmentioning

confidence: 99%

Locking constraints for elastic rods and a curvature bound for spatial curves

Schuricht

2005

Calc. Var.

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“…Recently, the shortest-path problem of [13] and [14] has been revisited subject to the optimal-control framework. As a result, Sussmann and Tang and Boissonnat et al independently provided a new proof and solution based on the minimum principle of Pontryagin [18,19]. For Dubins' car, an optimal path consists of at most three segments including arcs of a circle with minimal radius and straight line segments (or degenerate forms).…”

mentioning

confidence: 99%

Optimal-control approach to trajectory planning for a class of mobile robotic manipulations

Payandeh

2009

J Eng Math

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Optimal-control theory is used in the manipulation planning for an ability-limited robot (i.e., a mobile robot with limited maneuverability). The goal of manipulation is to push an object from a given initial configuration to a goal configuration-a common task could be found in the application of mobile robots to manufacturing, in adaptive fixturing for robotic assembly and in robotic gaming. The ability-limited robot is restricted here to perform only two types of motion in a plane. One is pushing the object forward with a constant and fixed velocity, the other one is rotating the object counter-clockwise along a fixed-radius circle with a constant angular velocity. First, the controllability of the proposed manipulation system is proved. Then, the optimal-planning problem is studied within the framework of a switched system. By the aid of optimal-control theory, it is proved that an optimal trajectory involves at most two switchings between these two simple actions. After evaluating the existence and structure of the candidate optimal trajectories, an optimal planner is implemented to find the optimal trajectory. Finally, manipulation examples are provided to demonstrate the proposed manipulation method.

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Shortest paths of bounded curvature in the plane

Cited by 109 publications

References 5 publications

Curvature-constrained directional-cost paths in the plane

Curvature-constrained directional-cost paths in the plane

Locking constraints for elastic rods and a curvature bound for spatial curves

Optimal-control approach to trajectory planning for a class of mobile robotic manipulations

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