The cost of bounded curvature

Kim, H.S.,Cheong, O. Elsevier 2013 Computational geometry Vol.46 No.6

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<SUP>'</SUP>, let @?(σ,σ<SUP>'</SUP>) be the shortest bounded-curvature path from σ to σ<SUP>'</SUP>. For d≥0, let @?(d) be the supremum of @?(σ,σ<SUP>'</SUP>), over all pairs (σ,σ<SUP>'</SUP>) 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<SUP>@?</SUP>)=2π, and is constant for d≥d<SUP>@?</SUP>. Here d<SUP>@?</SUP>~1.5874. We describe pairs of configurations that exhibit the worst-case of dub(d) for every distance d.

Comparison theorems for the volumes of tubes about metric balls in CAT(κ)-spaces

이두한,김용일 충청수학회 2011 충청수학회지 Vol.24 No.3

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space R^n, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1<m<n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

Curvature estimates for a class of fully nonlinear elliptic equations with general right hand sides

Jundong Zhou 대한수학회 2024 대한수학회보 Vol.61 No.2

In this paper, we establish the curvature estimates for a class of curvature equations with general right hand sides depending on the gradient. We show an existence result by using the continuity method based on a priori estimates. We also derive interior curvature bounds for solutions of a class of curvature equations subject to affine Dirichlet data.

A Path-level Smooth Transition Method with Curvature Bound between Non-smoothly Connected Paths

최윤종(Yun Jong Choi),박부견(Poo Gyeon Park) 대한전자공학회 2008 電子工學會論文誌-SC (System and control) Vol.45 No.4

연속적인 경로 사이를 부드러운 곡선으로 잇기 위해서 기존의 로봇 제어기들은 일반적으로 연속적인 경로를 시간 축에서 합성하는 방법을 사용해 왔다. 하지만 이런 방법은 다음과 같은 두 가지 단점을 내재하고 있다. 천이 경로의 형태가 연접하게 생성될 수 없다는 점과 천이하는 동안 속력을 제어할 수 없다는 점이 그것이다. 이러한 문제점들을 극복하기 위해서 본 논문은 매끄럽지 않게 연결된 두 경로들을 부드럽게 잇기 위해 곡률이 제한된 새로운 천이 궤적 생성 방법을 제시하고자 한다. 실험 결과는 기존의 방법들보다 천이 궤적이 더 부드럽게 생성되는 것을 보여주며, 또한 보장된 곡률의 제한 수준은 0.02 ~ 1임을 보여준다. For a smooth transition between consecutive paths, conventional robot controllers usually generate a transition trajectory by blending consecutive paths in a time coordinate. However, this has two inherent drawbacks: the shape of a transition path cannot be designed coherently and the speed during transition is uncontrollable. To overcome these problems, this paper provides a path-level, rather than trajectory-level, smooth transition method with the curvature bound between non-smoothly connected paths. The experiment results show that the resultant transition trajectory is more smoothly connected than the conventional methods and the curvature is closely limited to the desired bound within the guaranteed level (0.02∼1).

원형튜브제품의 편심압출굽힘가공법에 관한 연구

김진훈,진인태 釜慶大學校 1998 釜慶大學校 論文集 Vol.3 No.2

The eccentric extrusion and bending process for the forming of the curved circular hollow tube is newly developed. Generally, the bending process of hollow tube is the secondary bending process followed by the extrusion process of the hollow tube from the round billet. But, many defects such as wrinkling and the difference of wall thichness can be occurred during the secondary bending process. On the basis of the facts about the curving phenomenon being able to occurred during the extrusion process, the new process for the bending is suggested by the control of the curvature of extruded parts during the extrusion process. From the results of the previous researches which the curving phenomenon during the extrusion process is caused by the eccentricity, the eccentric extrusion and bending process is applied to the U-bending of circular hollow tube. During being extruded to the circular hollow tube product from the circular hollow tube billet, the kinematically admissible velocity field between the dies surface and the internal plug boundary surface is developed. By the using this curving velocity field, the curvature of extruded products can be calculated with the parameters such as eccentricity, dies length, friction constant.

Reachability by paths of bounded curvature in a convex polygon

Ahn, H.K.,Cheong, O.,Matousek, J.,Vigneron, A. Elsevier 2012 Computational Geometry Vol.45 No.1

Let B be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most 1, and let P be a convex polygon with n vertices. Given a starting configuration (a location and a direction of travel) for B inside P, we characterize the region of all points of P that can be reached by B, and show that it has complexity O(n). We give an O(n<SUP>2</SUP>) time algorithm to compute this region. We show that a point is reachable only if it can be reached by a path of type CCSCS, where C denotes a unit circle arc and S denotes a line segment.

Minimum-Time Trajectory of Three-Wheeled Omni-Directional Mobile Robots Following a Bounded Curvature Path without Heading Freedom

Kim Ki Bum,Kim Byung Kook 제어로봇시스템학회 2009 제어로봇시스템학회 국내학술대회 논문집 Vol.2009 No.9

We deal with minimum-time trajectory for three-wheeled omni-directional mobile robots (TOMRs) following walls with the bounded curvature path in this paper. We assume that TOMR motion is restricted to maintain the angle difference between the tangent of path and TOMR heading. Our studies are based on the dynamics of mobile robot actuated with battery voltage constraints. We suggest how to generate the bounded curvature path for TOMR motion with clothoid concept. We find that input voltage vector to three motors should have at least one extreme component. Also, the coupling terms of velocities in global coordinate frame vanish in the dynamics for the translational motion of TOMR, and the minimum time in our problem depends on an initial heading of TOMR.

단백질 접촉 영역의 기하학적 특성 가시화

김구진 한국정보처리학회 2019 정보처리학회논문지. 소프트웨어 및 데이터 공학 Vol.8 No.10

In this paper, we propose a method to visualize the geometric features of the contact region between proteins in a protein complex. When proteins or ligands are represented as curved surfaces with irregularities, the property that the two surfaces contact each other without intersections is called shape compatibility. Protein-Protein or Protein-Ligand docking researches have shown that shape complementarity, chemical properties, and entropy play an important role in finding contact regions. Usually, after finding a region with high shape complementarity, we can predict the contact region by using residual polarity and hydrophobicity of amino acids belonging to this region. In the research for predicting the contact region, it is necessary to investigate the geometrical features of the contact region in known protein complexes. For this purpose, it is essential to visualize the geometric features of the molecular surface. In this paper, we propose a method to find the contact region, and visualize the geometric features of it as normal vectors and mean curvatures of the protein complex. 본 논문에서는 단백질 복합체에서 단백질 사이의 접촉 영역이 갖는 기하학적 특징을 가시화하는 방법을 제안한다. 단백질 또는 리간드가 요철이 있는 곡면으로 표현될 때, 두 곡면이 서로 접하면서 교차하지 않는 성질을 형태 상보성이라 한다. 단백질-단백질 또는 단백질-리간드 도킹 연구에서 형태 상보성과 화학적인 성질, 엔트로피 등이 접촉 영역의 발견에 중요한 역할을 한다는 것을 볼 수 있다. 일반적으로 형태 상보성이 높은 영역을 발견한 뒤, 이 영역에 속한 아미노산들의 잔기 극성 및 소수성 등을 이용하여 접촉 영역을 예측한다. 접촉 영역을 예측하기 위한 연구에서는 기존에 알려진 복합체에서 접촉 영역이 갖는 기하학적인 특징을 조사하는 작업이 필요하며, 이를 위해 기하학적인 특징을 가시화하는 작업은 필수적이다. 본 논문에서는 단백질 복합체에서 접촉 영역을 발견하고, 두 개의 단백질 각각의 접촉 면에 속한 근거리의 정점들의 기하학적인 특징을 법선 벡터 및 평균 곡률로써 가시화하는 방법을 제안한다.