Geometrically exact beam finite element formulated on the special Euclidean group

2014 ◽  
Vol 268 ◽  
pp. 451-474 ◽  
Author(s):  
V. Sonneville ◽  
A. Cardona ◽  
O. Brüls
Keyword(s):  
Finite Element ◽  
Euclidean Group ◽  
Geometrically Exact Beam ◽  
Special Euclidean Group

A Formulation on the Special Euclidean Group for Dynamic Analysis of Multibody Systems

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Valentin Sonneville ◽  
Olivier Brüls
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Rigid Bodies ◽  
Practical Case ◽  
Numerical Implementation ◽  
Euclidean Group ◽  
Finite Element Approach ◽  
Local Description ◽  
Element Approach ◽  
Special Euclidean Group

This paper presents a finite element approach of multibody systems using the special Euclidean group SE(3) framework. The development leads to a compact and unified mixed coordinate formulation of the rigid bodies and the kinematic joints. Flexibility in the kinematic joints is also easily introduced. The method relies on local description of motions, so that it provides a singularity-free formulation and exhibits important advantages regarding numerical implementation. A practical case is presented to illustrate the method.

scholarly journals Polynomial Invariants of the Euclidean Group Action on Multiple Screws

2021 ◽  
Author(s):  
◽  
Deborah Crook
Keyword(s):  
Group Action ◽  
Three Dimensions ◽  
Euclidean Group ◽  
Polynomial Invariants ◽  
Generating Sets ◽  
Special Euclidean Group ◽  
The One

<p>In this work, we examine the polynomial invariants of the special Euclidean group in three dimensions, SE(3), in its action on multiple screw systems. We look at the problem of finding generating sets for these invariant subalgebras, and also briefly describe the invariants for the standard actions on R^n of both SE(3) and SO(3). The problem of the screw system action is then approached using SAGBI basis techniques, which are used to find invariants for the translational subaction of SE(3), including a full basis in the one and two-screw cases. These are then compared to the known invariants of the rotational subaction. In the one and two-screw cases, we successfully derive a full basis for the SE(3) invariants, while in the three-screw case, we suggest some possible lines of approach.</p>

scholarly journals A Mobility Determination Method for Parallel Platforms Based on the Lie Algebra of SE(3) and Its Subspaces

2021 ◽  
pp. 1-12
Author(s):  
Alejandro de Jesús Sánchez-García ◽  
José M. Rico ◽  
J. Jesús Cervantes-Sánchez ◽  
Pablo Lopez-Custodio
Keyword(s):  
Lie Algebra ◽  
Screw Theory ◽  
Determination Method ◽  
Mobility Analysis ◽  
Euclidean Group ◽  
Second Stage ◽  
Direct Sum ◽  
Three Stages ◽  
Last Stage ◽  
Special Euclidean Group

Abstract This contribution presents a screw theory-based method for determining the mobility of fully parallel platforms. The method is based in the application of three stages. The first stage involves the application of the intersection of the subalgebras of Lie algebra, se(3), of the special Euclidean group, SE(3), associated with the legs of the platform. The second stage analyzes the possibility of the legs of the platform generating a sum or direct sum of two subalgebras of the Lie algebra, se(3). The last stage, if necessary, considers the possibility of the kinematic pairs of the legs satisfying certain velocity conditions; these conditions allow to reduce the platform's mobility analysis to one that can solved using one of the two previous stages.

scholarly journals Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

2021 ◽  
Author(s):  
Ivo Steinbrecher ◽  
Alexander Popp ◽  
Christoph Meier
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Lagrange Multiplier ◽  
Degrees Of Freedom ◽  
Element Discretization ◽  
Geometrically Exact Beam ◽  
Coupling Constraints ◽  
Rotational Degrees Of Freedom ◽  
3D Solid ◽  
Cosserat Continua

AbstractThe present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a $$L_2$$ L 2 -optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.

Discretely Actuated Manipulator Workspace Generation by Closed Form Convolution

1998 ◽  
Vol 120 (2) ◽  
pp. 245-251 ◽  
Author(s):  
Imme Ebert-Uphoff ◽  
Gregory S. Chirikjian
Keyword(s):  
Closed Form ◽  
Density Function ◽  
Unit Volume ◽  
Density Functions ◽  
Brute Force ◽  
Euclidean Group ◽  
Planar Case ◽  
Continuous Density ◽  
Group 3 ◽  
Special Euclidean Group

We determine workspaces of discretely actuated manipulators using convolution of real-valued functions on the Special Euclidean Group. Each workspace is described in terms of a density function that provides the number of reachable frames inside any unit volume of the workspace. A manipulator consisting of n discrete actuators each with K states can reach Kn frames in space. Given this exponential growth, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. However, if the manipulator is of macroscopically-serial architecture, the workspace can be generated by the following procedure: (1) partition the manipulator into B kinematically independent segments; (2) approximate the workspace of each segment as a continuous density function on a compact subset of the Special Euclidean Group; (3) approximate the whole workspace as a B-fold convolution of these densities. We represent density functions as finite Hermite-Fourier Series and show for the planar case how the B-fold convolution can be performed in closed form. If all segments are identical only O(log B) convolutions are necessary.

Discretely Actuated Manipulator Workspace Generation by Closed-Form Convolution

1996 ◽  
Author(s):  
Imme Ebert-Uphoff ◽  
Gregory S. Chirikjian
Keyword(s):  
Closed Form ◽  
Density Function ◽  
Computation Time ◽  
Density Functions ◽  
Brute Force ◽  
Euclidean Group ◽  
Planar Case ◽  
Group 3 ◽  
Special Euclidean Group

Abstract We discuss the determination of workspaces of discretely actuated manipulators using convolution of real-valued functions on the Special Euclidean Group. Each workspace is described in terms of a density function that provides for any unit taskspace volume of the workspace the number of reachable frames therein. A manipulator consisting of n discrete actuators each with K states can reach Kn frames in space. Given this exponential growth, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. However, if the manipulator is of macroscopically-serial architecture, the workspace can be generated by the following procedure: (1) partition the manipulator into segments; (2) approximate the workspace of each segment as a continuous density function on a compact subset of the Special Euclidean Group; (3) approximate the whole workspace as an n-fold convolution of these densities. We represent density functions as finite Hermite-Fourier Series and show for the planar case how the n-fold convolution can be performed in closed form requiring O(n) computation time. If all segments are identical the computation time reduces to O(logn).

scholarly journals SE-Sync: A certifiably correct algorithm for synchronization over the special Euclidean group

2018 ◽  
Vol 38 (2-3) ◽  
pp. 95-125 ◽  
Author(s):  
David M Rosen ◽  
Luca Carlone ◽  
Afonso S Bandeira ◽  
John J Leonard
Keyword(s):  
Computer Vision ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Likelihood Estimation ◽  
Semidefinite Relaxation ◽  
Optimization Approach ◽  
Optimal Solutions ◽  
Euclidean Group ◽  
Inference Problem ◽  
Special Euclidean Group

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown group elements [Formula: see text] given noisy measurements of a subset of their pairwise relative transforms [Formula: see text]. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a non-convex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation (MLE) whose minimizer provides an exact maximum-likelihood estimate so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so significantly faster than the Gauss–Newton-based approach that forms the basis of current state-of-the-art techniques.

Sign in / Sign up

Export Citation Format

Share Document