Galerkin Lie-group variational integrators based on unit quaternion interpolation

2018 ◽  
Vol 338 ◽  
pp. 333-361 ◽  
Author(s):  
Thomas Leitz ◽  
Sigrid Leyendecker
Keyword(s):  
Lie Group ◽  
Variational Integrators ◽  
Unit Quaternion ◽  
Quaternion Interpolation

C2-continuous orientation trajectory planning for robot based on spline quaternion curve

2018 ◽  
Vol 38 (3) ◽  
pp. 282-290 ◽  
Author(s):  
Xuejuan Niu ◽  
Tian Wang
Keyword(s):  
Spherical Surface ◽  
Industrial Applications ◽  
Practical Case ◽  
Unit Quaternion ◽  
Content Type ◽  
Mapping Algorithm ◽  
C2 Continuity ◽  
Quaternion Interpolation ◽  
Teaching Orientations ◽  
Continuous Orientation

Purpose To realize the smooth interpolation of orientation on robot end-effector, this paper aims to propose a novel algorithm based on the unit quaternion spline curve. Design/methodology/approach This algorithm combines the spherical linear quaternion interpolation and the cubic B-spline quaternion curve. With this method, a C2-continuous smooth trajectory of multiple teaching orientations is obtained. To achieve the visualization of quaternion curves on a unit sphere, a mapping algorithm between a unit quaternion and a point on the spherical surface is given based on the physical meaning of the unit quaternion. Findings Finally, the curvature analysis of a practical case shows that the orientation trajectory (OT) constructed by this algorithm satisfied the C2-continuity. Originality/value This OT satisfies the requirement of smooth interpolation among multiple orientations on robots in industrial applications.

scholarly journals Lie Group Spectral Variational Integrators

2015 ◽  
Vol 17 (1) ◽  
pp. 199-257 ◽  
Author(s):  
James Hall ◽  
Melvin Leok
Keyword(s):  
Lie Group ◽  
Variational Integrators

scholarly journals Lie group variational integrators for the full body problem

2007 ◽  
Vol 196 (29-30) ◽  
pp. 2907-2924 ◽  
Author(s):  
Taeyoung Lee ◽  
Melvin Leok ◽  
N. Harris McClamroch
Keyword(s):  
Lie Group ◽  
Body Problem ◽  
Variational Integrators ◽  
Full Body

scholarly journals Multisymplectic variational integrators and space/time symplecticity

2016 ◽  
Vol 14 (03) ◽  
pp. 341-391 ◽  
Author(s):  
François Demoures ◽  
François Gay-Balmaz ◽  
Tudor S. Ratiu
Keyword(s):  
Lie Group ◽  
Numerical Scheme ◽  
Variational Integrators ◽  
Beam Model ◽  
Structure Preserving ◽  
Dimensional Spacetime ◽  
Geometrically Exact Beam ◽  
Temporal And Spatial ◽  
Group Symmetries

Multisymplectic variational integrators are structure-preserving numerical schemes especially designed for PDEs derived from covariant spacetime Hamilton principles. The goal of this paper is to study the properties of the temporal and spatial discrete evolution maps obtained from a multisymplectic numerical scheme. Our study focuses on a (1+1)-dimensional spacetime discretized by triangles, but our approach carries over naturally to more general cases. In the case of Lie group symmetries, we explore the links between the discrete Noether theorems associated to the multisymplectic spacetime discretization and to the temporal and spatial discrete evolution maps, and emphasize the role of boundary conditions. We also consider in detail the case of multisymplectic integrators on Lie groups. Our results are illustrated with the numerical example of a geometrically exact beam model.

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