Parallel unit tangent vector computation for homotopy curve tracking on a hypercube

1990 ◽  
Author(s):  
A. Chakraborty ◽  
D. C. S. Allison ◽  
C. J. Ribbens ◽  
L. T. Watson
Keyword(s):  
Tangent Vector ◽  
Unit Tangent Vector ◽  
Curve Tracking

Note on unit tangent vector computation for homotopy curve tracking on a hypercube

1991 ◽  
Vol 17 (12) ◽  
pp. 1385-1395 ◽  
Author(s):  
A. Chakraborty ◽  
D.C.S. Allison ◽  
C.J. Ribbens ◽  
L.T. Watson
Keyword(s):  
Tangent Vector ◽  
Unit Tangent Vector ◽  
Curve Tracking

On a Generalization of the First Curvature of a Curve in a Hypersurface of a Riemannian Space

1954 ◽  
Vol 6 ◽  
pp. 210-216 ◽  
Author(s):  
T. K. Pan
Keyword(s):  
Riemannian Space ◽  
Tangent Vector ◽  
Curvature Vector ◽  
Tangential Component ◽  
Unit Tangent Vector

The unit tangent vector at a point of a curve in a hypersurface of a Riemannian space has two derived vectors along the curve, one with respect to the Riemannian space in which the hypersurface is imbedded and one with respect to the hypersurface itself. When the former vector is decomposed along the directions normal and tangent to the hypersurface, its tangential component, which is called the first curvature vector of the curve at the point in the hypersurface, is exactly the latter vector.

Bounded geodesics and Hausdorff dimension

1994 ◽  
Vol 116 (3) ◽  
pp. 505-511 ◽  
Author(s):  
C. S. Aravinda
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Bundle ◽  
Negative Curvature ◽  
Tangent Vector ◽  
Unit Tangent Vector ◽  
Constant Negative Curvature ◽  
Compact Closure ◽  
Riemannian Volume ◽  
Unit Tangent Bundle ◽  
Almost All

Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all (p, v)∈ SM, where p ∈M and v is a unit tangent vector at p, the geodesic through p in the direction of v is dense in M. A theorem of Dani [Dl] says that the set of all (p, v)∈SM for which the corresponding geodesic is bounded (namely those with compact closure in M) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

scholarly journals On the concept of gravitational force and Gauss's theorem in general relativity

1937 ◽  
Vol 5 (2) ◽  
pp. 93-102 ◽  
Author(s):  
J. L. Synge ◽  
H. S. Ruse
Keyword(s):  
General Relativity ◽  
Physical Meaning ◽  
Gravitational Force ◽  
Normal Force ◽  
Free Particle ◽  
Tangent Vector ◽  
Curvature Vector ◽  
Unit Tangent Vector ◽  
Normal Congruence ◽  
Definition Of

Whittaker and Ruse have developed forms of Gauss's theorem in general relativity, their theorems connecting integrals of normal force taken over a closed 2-space V2 with integrals involving the distribution of matter taken over an open 3-space bounded by V2. The definition of force employed by them involves the introduction of a normal congruence (with unit tangent vector λi), the “force” relative to the congruence being the negative of the first curvature vector of the congruence (– δλi/δs). This appears at first sight a natural enough definition, because – δλi/δs at an event P represents the acceleration relative to the congruence of a free particle travelling along a geodesic tangent to the congruence at P. In order to give physical meaning to this definition of force it is necessary to specify the congruence λi physically, and it would seem most natural to choose the congruence of world-lines of flow of the medium. Supposing certain conditions satisfied by this congruence (cf. Ruse, loc. cit.), the theory of Ruse is applicable, and from this follows a form of Gauss's theorem.

Three-Dimensional Curve Tracking for Particles Using Gyroscopic Control

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Chuanfeng Wang
Keyword(s):  
Tracking Control ◽  
Three Dimensional ◽  
Smooth Curve ◽  
Autonomous Agent ◽  
Control Law ◽  
Steering Control ◽  
3D Space ◽  
Moving Direction ◽  
Tangent Direction ◽  
Curve Tracking

Curve-tracking control is challenging and fundamental in many robotic applications for an autonomous agent to follow a desired path. In this paper, we consider a particle, representing a fully actuated autonomous robot, moving at unit speed under steering control in the three-dimensional (3D) space. We develop a feedback control law that enables the particle to track any smooth curve in the 3D space. Representing the 3D curve in the natural Frenet frame, we construct the control law under which the moving direction of the particle will be aligned with the tangent direction of the desired curve and the distance between the particle and the desired curve will converge to zero. We demonstrate the effectiveness of the proposed 3D curve-tracking control law in simulations.

Adaptive control for planar curve tracking under controller uncertainty

Automatica ◽  
2013 ◽  
Vol 49 (5) ◽  
pp. 1411-1418 ◽  
Author(s):  
Michael Malisoff ◽  
Fumin Zhang
Keyword(s):  
Adaptive Control ◽  
Planar Curve ◽  
Curve Tracking
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