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.
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.