Article (Scientific journals)
Highly accurate differentiation of the exponential map and its tangent operator
Todesco, Juliano; Bruls, Olivier
2023In Mechanism and Machine Theory, 190, p. 105451
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Todesco J and Bruls 2O202 Highly accurate differentiation of the exponential map and its tangent operator (Author postprint).pdf
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The .rar file provides functions essential for computing the gradients and derivatives of the tangent operator of the SO(3) and SE(3) groups, as detailed in the paper. Notably, the archive encompasses a MATLAB code for an algorithm that determines the series expansion up to an order of N for both the gradient and derivative of the tangent operator. Additionally, a test run is incorporated to reproduce the graphs showcased in the paper. Copyright (C) 2011-2023 University of Liege Author: Olivier Bruls (ULiege, Multibody & Mechatronic Systems Lab) Contact: Author: Valentin Sonneville (ULiege, Multibody & Mechatronic Systems Lab) Author: Juliano Todesco (ULiege, Multibody & Mechatronic Systems Lab) Bibliographic reference: J. Todesco and O. Bruls. Highly accurate differentiation of the exponential map and its tangent operator. Mechanism and Machine Theory, 190, p.105451, 2023. This code is related with the open source code GECOS (GEometric toolbox for COnstrained mechanical Systems) available on This code is licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. Cite: Olivier Brüls and Valentin Sonneville. GECOS. GitLab repository., 2019.

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Keywords :
Derivatives; Lie group; Numerical methods; Special Euclidean group; Special orthogonal group; Exponential map
Abstract :
Exponential coordinates are widely used in simulation codes for flexible multibody systems based on a Lie group approach. The accurate and efficient evaluation of the exponential map, the tangent operator and its higher order derivatives is crucial. This paper presents a systematic derivation process based on the matrix series expansion of the exponential map and of the tangent operator. This approach is general as it can be applied to any matrix Lie group. For the Lie groups SO(3) and SE(3), the closed form expression of these operators can also be established and is summarized in the paper. It is shown that the closed form of the operators is affected by round-off errors for small rotation amplitudes, whereas the series form is affected by truncation errors at high rotation amplitudes. The computational efficiency of the two approaches is also discussed. A switching strategy between the closed form and the series form is then proposed to obtain an adjustable compromise between accuracy and computational cost.
Disciplines :
Mechanical engineering
Aerospace & aeronautics engineering
Author, co-author :
Todesco, Juliano  ;  Université de Liège - ULiège > Aérospatiale et Mécanique (A&M)
Bruls, Olivier  ;  Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Language :
Title :
Highly accurate differentiation of the exponential map and its tangent operator
Publication date :
31 July 2023
Journal title :
Mechanism and Machine Theory
Publisher :
Elsevier BV
Volume :
Pages :
Peer reviewed :
Peer Reviewed verified by ORBi
Funders :
EU - European Union [BE]
Funding text :
This work has received financial support from the EI-OPT and ORFI projects funded by the Walloon Region (Pôle Skywin) and from the Robotix Academy project funded by the Interreg Greater Region program, which are gratefully acknowledged. It has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 860124. The present paper only reflects the author's view. The European Commission and its Research Executive Agency (REA) are not responsible for any use that may be made of the information it contains.
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