Convexifying Monotone Polygons

Biedl, Thérèse; Demaine, Erik D.; Lazard, Sylvain; Robbins, Steven; Soss, Michael

doi:10.1007/3-540-46632-0_42

Cited by 9 publications

(3 citation statements)

References 2 publications

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“…In order to plan a set of constraints circumscribing a desired homotopy, metrics describing homotopy goodness must be defined and ascribed to individual triangles and 1 Methods for convexifying non-convex polygons described in [21], [22]. …”

Section: B Homotopy Evaluationmentioning

confidence: 99%

The intelligent copilot: A constraint-based approach to shared-adaptive control of ground vehicles

Anderson

Karumanchi

Iagnemma

et al. 2013

IEEE Intell. Transport. Syst. Mag.

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This paper presents a new approach to semi-autonomous vehicle hazard avoidance and stability control, based on the design and selective enforcement of constraints. This differs from traditional approaches that rely on the planning and tracking of paths and facilitates "minimally-invasive" control for human-machine systems. Instead of forcing a human operator to follow an automation-determined path, the constraint-based approach identifies safe homotopies, and allows the operator to navigate freely within them, introducing control action only as necessary to ensure that the vehicle does not violate safety constraints. This method evaluates candidate homotopies based on "restrictiveness," rather than traditional measures of path goodness, and designs and enforces requisite constraints on the human's control commands to ensure that the vehicle never leaves the controllable subset of a desired homotopy. This paper demonstrates the approach in simulation and characterizes its effect on human teleoperation of unmanned ground vehicles via a 20-user, 600-trial study on an outdoor obstacle course. Aggregated across all drivers and experiments, the constraintbased control system required an average of 43% of the available control authority to reduce collision frequency by 78% relative to traditional teleoperation, increase average speed by 26%, and moderate operator steering commands by 34%.

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Section: B Homotopy Evaluationmentioning

confidence: 99%

The intelligent copilot: A constraint-based approach to shared-adaptive control of ground vehicles

Anderson

Karumanchi

Iagnemma

et al. 2013

IEEE Intell. Transport. Syst. Mag.

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“…A classical problem in computational geometry is that of convexifying simple polygons; that is, using a given fixed set of transformations that can be applied to the vertices and edges of P , try to transform P into a convex polygon in such a way that some properties of P are preserved. The first formulation of a problem of this kind was proposed by Erdős [4], who proposed a strategy to convexify a non-convex polygon by using flips; see also [1,2,3,5,9,10].…”

Section: Introductionmentioning

confidence: 99%

Convexifying Monotone Polygons while Maintaining Internal Visibility

Aichholzer

Cetina

Fabila-Monroy

et al. 2012

Lecture Notes in Computer Science

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Abstract. Let P be a simple polygon on the plane. Two vertices of P are visible if the open line segment joining them is contained in the interior of P . In this paper we study the following questions posed in [7,8]: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously moved such that during the process no vertex-vertex visibility is lost and some vertex-vertex visibility is gained? (2) Can every simple polygon be convexified by continuously moving only one vertex at a time without losing any internal vertex-vertex visibility during the process? We provide a counterexample to (1). We note that our counterexample uses a monotone polygon. We also show that question (2) has a positive answer for monotone polygons.

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“…These questions have been in the math community since the 1970's [22] and in the computational geometry community since 1991 [18,19], but first appeared in print in 1993 and 1995: [20] and [16, p. 270]. Initial computational geometry results focused on certain classes of configurations such as "visible" chains [4], star-shaped polygons [9] and monotone polygons [3]. Connelly, Demaine, and Rote have recently proved that in the plane, no chain or polygon is locked [7]; Streinu [28] provides an alternative proof.…”

Section: Introductionmentioning

confidence: 99%

A note on reconfiguring tree linkages: trees can lock

Biedl

Demaine

et al. 2002

Discrete Applied Mathematics

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It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an N -link tree can have 2 Ω(N) equivalence classes of configurations.

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Convexifying Monotone Polygons

Cited by 9 publications

References 2 publications

The intelligent copilot: A constraint-based approach to shared-adaptive control of ground vehicles

The intelligent copilot: A constraint-based approach to shared-adaptive control of ground vehicles

Convexifying Monotone Polygons while Maintaining Internal Visibility

A note on reconfiguring tree linkages: trees can lock

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