A novel approach to Lie group structured configuration spaces of rigid bodies

Abstract

In rigid body mechanics, a body's configuration consisting of orientation and position have to be described, as well as angular velocity and velocity. There are several different ways to do this, leading to different configuration spaces. Popular choices include SO(3) × R 3 , SE(3) or unit dual quaternions. All these configuration spaces possess a Lie group structure. We present the novel approach S 3 R 3 , which is a configuration space similar to SE(3), but describes the orientation in terms of unit quaternions instead of rotation matrices. Furthermore we show its relation to and advantages over other configuration spaces.The configuration of a rigid body can be described in a configuration space G. It is well-known [6], that it forms a Lie group (G, •). The velocity of an object with configurationwhere v(t) ∈ R n describes the velocity of the object with configuration q(t) ∈ G, see [3].When large rotations are present, they are often described using SO(3) and translations usually by R 3 . The angular velocity is typically described with respect to the body-fixed frame, but the velocity can be described with respect to the inertial system (fixed frame of reference) or with respect to the body-fixed frame (moving frame of reference). Both are standard approaches [7]. The former leads to the direct product SO(3) × R 3 and the latter leads to the semidirect product SO(3) R 3 SE(3).It is well-known that unit quaternions S 3 are a singularity-free description of orientations, that can outperform SO(3) in numerical computations. When we want to describe the configuration of a rigid body using S 3 , we have again the choice to describe the velocities with respect to the inertial frame or with respect to the body-fixed frame. Again, the former leads to a direct product S 3 × R 3 . Our focus, however, is on the latter approach, which leads to the Lie-group structured semi-direct product configuration space S 3 R 3 . It is comparable to the approach of unit dual quaternions, as described below. We will then illustrate the benefits of the configuration space S 3 R 3 in Lie group time integration by numerical test results for a classical benchmark problem. Rotations with unit quaternionsIt is known [2] that rotations can be described by using the set of unit quaternions S 3 = {p ∈ R 4 : p 2 = 1} which is a Lie group together with the quaternion multiplication * : R 4 → R 4 . Let p = p 0 p , q = q 0 q ∈ S 3 , z = 0 z ∈ T e S 3 and w ∈ R 3 with scalar parts p 0 , q 0 , 0 ∈ R and vector partsp,q,ž ∈ R 3 . All of the following formulae for quaternions are well-known:

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

“…The performance of Lie group integrators may depend strongly on the specific Lie group formulation [5] with clear benefits for formulations that are based on the special Euclidean group [26,34]. For rotations being parametrized by unit quaternions, the corresponding Lie groups are multiples of the semi-direct product S 3 R 3 , see [11,19]. This semi-direct product (as well as the special Euclidean group) refers to rigid body motions in space and its application in beam theory helps to reduce the risk of locking phenomena [21,34], see also the last part of the present paper.…”

Staggered grid discretizations on Lie groups with applications in beam and shell theory

“…In order to describe the beam, we use the line of mass centroids, where at each point, we attach a frame that describes the orientation of the beam's cross section which is modeled to stay rigid. Thus, at each time instant t, the beam is represented by a framed curve q(•, t) : [0, L] → S 3 R 3 , see [2] for a description of S 3 R 3 .…”

“…This means we work with a micropolar model rather than with a director-based one. The velocities v(s (1) , s (2) , t) are defined in the same way as above. We now deal with two directional derivates that are represented by ∂ k q = dL q (e) w (k) for k = 1, 2.…”

“…We introduce a two-dimensional equidistant grid s (k) i = iL (k) /n (k) with step sizes h (k) for k = 1, 2. Again, the configurations are discretized by q i,j (t) ≈ q(s (1) i , s (2) j , t). The spatial derivatives w (k) are discretized inbetween the configurations in their respective direction.…”

“…Maximum of discrete L 2 errors in position, velocity and acceleration of the flying spaghetti benchmark s(2) …”