Baker–Campbell–Hausdorff–Dynkin Formula for the Lie Algebra of Rigid Body Displacements

Condurache, Daniel; Ciureanu, Ioan-Adrian

doi:10.3390/math8071185

“…( 22), the symbol ⋄ denotes the composition operation for rotation parameters, whose explicit form depends on the respective rotation parameters. For LGIM, the composition operation for Euler angles is given in [11], for the rotation vector in, e.g., [11,34], and for Euler parameters in, e.g., [3,12].…”

Section: Rigid Body Kinematics On R 3 × So(3)mentioning

confidence: 99%

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

Holzinger

¹

,

Arnold

²

,

Gerstmayr

³

2023

Preprint

0

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As commonly known, standard time integration of the kinematic equations of rigid bodies modelled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using several typical rigid multibody systems. It is shown, that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turned out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the average number of Newton iterations required per time step is most often smaller for the Lie group integration method. It also turns out, that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly.

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“…(e) The Jacobian J An important quantity in rotational geometry and kinematics is the (left) Jacobian J of SO(3), which relates angular velocity to the Euler axis-angle variables: ω = J φ. The Jacobian is furthermore related to the rotation matrix by C = 1 + φ × J = 1 + Jφ × and can be given explicitly as [17,33,34] royalsocietypublishing.org/journal/rspa Proc. R. Soc.…”

Section: (D) the Inverse Of Cmentioning

confidence: 99%

“…An important quantity in rotational geometry and kinematics is the (left) Jacobian J of SOfalse(3false), which relates angular velocity to the Euler axis-angle variables: bold-italicω=Jbold-italicϕ˙. The Jacobian is furthermore related to the rotation matrix by C=1+ϕ×J=1+Jϕ× and can be given explicitly as [17,33,34] J=∑k=0∞bold-italicϕ×kfalse(k+1false)!=1+1−cos⁡ϕϕbolda×+ϕ−sin⁡ϕϕa×bolda×.But it can also be determined with the help of the foregoing eigendecomposition. Note that the Jacobian is equally obtained by [17,35] J=∫01boldCα dα.Using complex decomposition, …”

Section: Rotations—sofalse(3false)mentioning

confidence: 99%

On the eigenstructure of rotations and poses: commonalities and peculiarities

D’Eleuterio

¹

,

Barfoot

²

2022

Proc. R. Soc. A.

1

0

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Locating vehicles, targets and objects in three-dimensional space is key to many fields of science and engineering such as robotics, aerospace, computer vision and graphics. Rotations and poses (position plus orientation) of bodies can be expressed in a variety of ways. Rotation matrices constitute one of the classic matrix Lie groups, the special orthogonal group— SO ( 3 ) . Poses can likewise be represented by matrices. One such representation is embodied in a 4 × 4 matrix establishing another famous matrix Lie group, the special Euclidean group— SE ( 3 ) . An alternative representation of pose uses 6 × 6 matrices and is referred to as the group of pose adjoints— Ad ( SE ( 3 ) ) . The eigenstructures of these representations reveal much about them from Euler’s theorem for rotations to the Mozzi–Chasles theorem for the general displacement of a rigid body. While the eigenstructure of SO ( 3 ) has been extensively studied, those of SE ( 3 ) and Ad ( SE ( 3 ) ) have hardly received the same scrutiny yet their structure is much richer. Motivated by their importance in kinematics and dynamics, we provide here a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations. An eigendecomposition of pose matrices reveals that they can be cast into a form similar to that of rotations although the structure of the former can vary depending on the nature of the pose involved. In particular, the pose matrices of SE ( 3 ) and Ad ( SE ( 3 ) ) cannot generally be diagonalized as can rotation matrices but they of course do yield to a Jordan normal form, from which we can identify a principal-axis pose in much the same manner that we can a principal-axis rotation. We also address the minimal polynomials for poses and derive a novel expression for the Jacobian in Ad ( SE ( 3 ) ) . Finally, we argue that the true counterpart to SO ( 3 ) for poses is not SE ( 3 ) but Ad ( SE ( 3 ) ) .

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“…for some a, b, and c such that the result is an element of SE(3). We know that higher-order powers of ξ ∧ are unnecessary owing to (13), which comes from the Cayley-Hamilton theorem.…”

Section: B Vector Mappings For Posesmentioning

confidence: 99%

“…For SE(3), Condurache and Ciureanu [18] recently discussed a closed-form solution for compounding two pose vectors in the case that φ = ϕa. Bauchau and Choi (2003) showed that we can easily generalize this result to any of our vectorial pose parameterizations.…”

Section: Compounding Posesmentioning

confidence: 99%

Vectorial parameterizations of pose

Barfoot

¹

,

Forbes

²

,

D’Eleuterio

³

2021

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Robotics and computer vision problems commonly require handling rigid-body motions comprising translation and rotation – together referred to as pose. In some situations, a vectorial parameterization of pose can be useful, where elements of a vector space are surjectively mapped to a matrix Lie group. For example, these vectorial representations can be employed for optimization as well as uncertainty representation on groups. The most common mapping is the matrix exponential, which maps elements of a Lie algebra onto the associated Lie group. However, this choice is not unique. It has been previously shown how to characterize all such vectorial parameterizations for SO(3), the group of rotations. Some results are also known for the group of poses, where it is possible to build a family of vectorial mappings that includes the matrix exponential as well as the Cayley transformation. We extend what is known for these pose mappings to the $4 \times 4$ representation common in robotics and also demonstrate three different examples of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem. In the pointcloud alignment problem, our results lead to a new algorithm based on the Cayley transformation, which we call CayPer.

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Baker–Campbell–Hausdorff–Dynkin Formula for the Lie Algebra of Rigid Body Displacements

Cited by 8 publications

References 48 publications

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

On the eigenstructure of rotations and poses: commonalities and peculiarities

Vectorial parameterizations of pose

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