Cartan’s Connection, Fiber Bundles and Quaternions in Kinematics and Dynamics Calculations

2015 ◽  
Author(s):  
Diego Colón
Keyword(s):  
Lie Group ◽  
Rotational Motion ◽  
Fiber Bundle ◽  
Reference Frames ◽  
Differentiable Manifold ◽  
Robotic Manipulators ◽  
Fiber Bundles ◽  
Kinematics And Dynamics ◽  
Future Directions ◽  
Principal Fiber Bundles

It is used the concept of Cartan’s connection and principal fiber bundles to obtain formulas for kinematics and dynamics calculations for robotic manipulators. A principal fiber bundle is a differentiable manifold formed by a base space B (in this case ℝ3)) plus all possible reference frames attached to a point p ∈ B (that is the fiber Sp). Cartan’s connections are the most general way to represent velocity of frames. In previous works, those ideas were applied to fiber bundles with fibers homomorphic to the Lie group SO(3) (or SE(3)). In this paper, it is applied to the case of fibers homomorphic either to the group SU(2) (for rotational motion) or to the group of unit dual quaternions (for translational plus rotational motion). It is also presented some results of calculations, and indicate future directions for research.

scholarly journals An Adaptive Controller for Robotic Manipulators with Unknown Kinematics and Dynamics

2020 ◽  
Vol 53 (2) ◽  
pp. 8796-8801
Author(s):  
Sihan Wang ◽  
Kaiqiang Zhang ◽  
Guido Herrmann
Keyword(s):  
Robotic Manipulators ◽  
Adaptive Controller ◽  
Kinematics And Dynamics

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

scholarly journals Invariant connections and invariant holomorphic bundles on homogeneous manifolds

2014 ◽  
Vol 12 (1) ◽  
pp. 1-13
Author(s):  
Indranil Biswas ◽  
Andrei Teleman
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Transitive Action ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Group A ◽  
Holomorphic Bundles ◽  
Invariant Gauge

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

scholarly journals Number of fiber bundles in the fetal anterior talofibular ligament

2021 ◽  
Author(s):  
Mutsuaki Edama ◽  
Tomoya Takabayashi ◽  
Hirotake Yokota ◽  
Ryo Hirabayashi ◽  
Chie Sekine ◽  
...  
Keyword(s):  
Fiber Bundle ◽  
Fiber Bundles ◽  
Anterior Talofibular Ligament ◽  
Type I ◽  
Type Ii ◽  
Crown Rump Length ◽  
Type Iii ◽  
The Difference ◽  
And Inversion

Abstract Background For the anterior talofibular ligament (ATFL), a three-fiber bundle has recently been suggested to be weaker than a single or double fiber bundle in terms of ankle plantarflexion and inversion braking function. However, the studies leading to those results all used elderly specimens. Whether the difference in fiber bundles is a congenital or an acquired morphology is important when considering methods to prevent ATFL damage. The purpose of this study was to classify the number of fiber bundles in the ATFL of fetuses. Methods This study was conducted using 30 legs from 15 Japanese fetuses (mean weight, 1764.6 ± 616.9 g; mean crown-rump length, 283.5 ± 38.7 mm; 8 males, 7 females). The ATFL was then classified by the number of fiber bundles: Type I, one fiber bundle; Type II, two fiber bundles; and Type III, three fiber bundles. Results Ligament type was Type I in 5 legs (16.7%), Type II in 21 legs (70%), and Type III in 4 legs (13.3%). Conclusions The present results suggest that the three fiber bundles of the structure of the ATFL may be an innate structure.

Multibody Dynamics on Differentiable Manifolds

2021 ◽  
Vol 16 (4) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differentiable Manifold ◽  
Kinematics And Dynamics ◽  
Variational Equations ◽  
Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

Actions That Fiber and Vector Semigroups

1972 ◽  
Vol 24 (1) ◽  
pp. 29-37 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Compact Space ◽  
Lie Group ◽  
Fiber Bundle ◽  
Closed Subset ◽  
Locally Compact ◽  
Locally Compact Space

From [2], we can derive a criterion for determining when an action of a Lie group on a locally compact space leads to a fiber bundle. Here, we present an equivalent criterion which can be stated purely in the language of actions of groups on spaces. This is Theorem I. Using this result, we are able to give a version of a result of Home [1] for dimensions greater than one. This is done in Theorem IV and Corollary IVA. In Theorem II, we show that if a vector semigroup acts on a space X, then whenever the map t ↦ tx is 1 — 1 from onto x, it is in fact a homeomorphism. Also, is a closed subset of X. This is also a version of a result in [1].

scholarly journals Dispersive semiflows on fiber bundles

2017 ◽  
Vol 37 (2) ◽  
pp. 85-99
Author(s):  
Josiney A. Souza ◽  
Hélio V. M. Tozatti
Keyword(s):  
Periodic Point ◽  
Fiber Bundle ◽  
Vector Bundles ◽  
Base Space ◽  
Total Space ◽  
Fiber Bundles ◽  
Almost Periodic ◽  
Special Result ◽  
Almost Periodic Point

This paper studies dispersiveness of semiflows on fiber bundles. The main result says that a right invariant semiflow on a fiber bundle is dispersive on the base space if and only if there is no almost periodic point and the semiflow is dispersive on the total space. A special result states that linear semiflows on vector bundles are not dispersive.

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