Geometry

Millman, Richard S.; Parker, George D.

doi:10.1007/978-1-4684-0130-1

Cited by 14 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Near γ 0 (·), a family of curves γ λ (·), called the Bertrand family c.f. [25], can be constructed as γ λ (s) = γ 0 (s) + λ y 0 (s) where λ is a real number. The tan-…”

Section: A3) the Function Z Is Smooth On The Open Setmentioning

confidence: 99%

Coordinated patterns of unit speed particles on a closed curve

Zhang

Leonard

2007

Systems & Control Letters

123

View full text Add to dashboard Cite

“…Near γ 0 (·), a family of curves γ λ (·), called the Bertrand family c.f. [25], can be constructed as γ λ (s) = γ 0 (s) + λ y 0 (s) where λ is a real number. The tan-…”

Section: A3) the Function Z Is Smooth On The Open Setmentioning

confidence: 99%

Coordinated patterns of unit speed particles on a closed curve

Zhang

Leonard

2007

Systems & Control Letters

123

View full text Add to dashboard Cite

“…, and at any given point, the unit tangent vector to the curve, denoted by x 1 (s), satisfies x 1 (s) · y 1 (s) = 0. Then we have the following Frenet-Serret equations [49]:…”

Section: B Cooperative Estimation Of Hessianmentioning

confidence: 99%

Cooperative Filters and Control for Cooperative Exploration

Zhang

Leonard

2010

IEEE Trans. Automat. Contr.

248

149

View full text Add to dashboard Cite

Autonomous mobile sensor networks are employed to measure large-scale environmental fields.Yet an optimal strategy for mission design addressing both the cooperative motion control and the cooperative sensing is still an open problem. We develop strategies for multiple sensor platforms to explore a noisy scalar field in the plane. Our method consists of three parts. First, we design provably convergent cooperative Kalman filters that apply to general cooperative exploration missions. Second, a novel method is established to determine the shape of the platform formation to minimize error in the estimates and a cooperative formation control law is designed to asymptotically achieve the optimal formation shape. Third, we use the cooperative filter estimates in a provably convergent motion control law that drives the center of the platform formation to move along level curves of the field. This control law can be replaced by control laws enabling other cooperative exploration motion, such as gradient climbing, without changing the cooperative filters and the cooperative formation control laws.Performance is demonstrated on simulated underwater platforms in simulated ocean fields.

show abstract

“…If two vectors, defined at some point, undergo parallel transport along the same curve, the angle between two vectors is preserved: the angle before the parallel transport is equal to the angle after the parallel transport [15]. Therefore, the cross equivalence class structure is preserved, and the parallel transport of crosses is well-defined.…”

Section: Cross-field and φ-Manifold Definitionsmentioning

confidence: 99%

“…The line traced by the curve at a given θ is known as a meridian. The circles at given (r, z) values are known as circles of revolution [15]. The principle axis directions at a point are the directions of the circle of revolution and meridian passing through that point.…”

Section: A Curved Surface Examplementioning

confidence: 99%

A continuum theory for unstructured mesh generation in two dimensions

Bunin

2008

Computer Aided Geometric Design

View full text Add to dashboard Cite

A continuum description of unstructured meshes in two dimensions, both for planar and curved surface domains, is proposed. The meshes described are those which, in the limit of an increasingly finer mesh (smaller cells), and away from irregular vertices, have ideally-shaped cells (squares or equilateral triangles), and can therefore be completely described by two local properties: local cell size and local edge directions. The connection between the two properties is derived by defining a Riemannian manifold whose geodesics trace the edges of the mesh. A function φ, proportional to the logarithm of the cell size, is shown to obey the Poisson equation, with localized charges corresponding to irregular vertices. The problem of finding a suitable manifold for a given domain is thus shown to exactly reduce to an Inverse Poisson problem on φ, of finding a distribution of localized charges adhering to the conditions derived for boundary alignment. Possible applications to mesh generation are discussed.Key words: Unstructured mesh generation, differential geometry. PACS:1 Overview A mesh is a partition of a domain into smaller parts, typically with simpler geometry, called cells. In two dimensions, both on the plane and on curved surfaces, cells are usually triangles or quadrilaterals. The shapes of the cells may be important; for many applications, cells with shapes similar to an equilateral triangle or a square are preferred. The problem of mesh generation can then be seen as an optimization problem: to find a partition of a domain into well-shaped cells, possibly under additional demands, such as cell size requirements.

show abstract

Geometry

Cited by 14 publications

References 0 publications

Coordinated patterns of unit speed particles on a closed curve

Coordinated patterns of unit speed particles on a closed curve

Cooperative Filters and Control for Cooperative Exploration

A continuum theory for unstructured mesh generation in two dimensions

Contact Info

Product

Resources

About