I read about the exponential map of SE(3), in which the left Jacobian of SO(3) is used. I found two different definitions.
Using the a common notation (rotation axis: $\boldsymbol{\omega}$, rotation angle: $\theta$), the first definition is
$$\boldsymbol{J}_A(\theta) = \frac{\sin{\theta}}{\theta} \boldsymbol{I} + \left( 1 - \frac{\sin{\theta}}{\theta} \right) \boldsymbol{\omega}\boldsymbol{\omega}^T + \frac{1 - \cos{\theta}}{\theta} [\boldsymbol{\omega}]$$
and it is used by
- T. D. Barfoot and P. T. Furgale, "Associating Uncertainty With Three-Dimensional Poses for Use in Estimation Problems," in IEEE Transactions on Robotics, vol. 30, no. 3, pp. 679-693, June 2014, doi: 10.1109/TRO.2014.2298059. URL: http://ncfrn.mcgill.ca/members/pubs/barfoot_tro14.pdf
- Exponential and logarithmic map of SE(3) are defined in Appendix B.
- Jacobian in Equation 98.
- J. Solà, J. Deray, and D. Atchuthan, "A micro Lie theory for state estimation in robotics". Technical report. URL: https://arxiv.org/abs/1812.01537
- Exponential and logarithmic of SE(3) map are defined in Appendix D.
- Jacobian in Equation 145.
A second definition is
$$\boldsymbol{J}_B(\theta) = \theta \boldsymbol{I} + \left( \theta - \sin{\theta} \right) [\boldsymbol{\omega}]^2 + \left( 1 - \cos{\theta} \right) [\boldsymbol{\omega}]$$
from
K. M. Lynch and F. C. Park, "Modern Robotics: Mechanics, Planning, and Control". URL: http://hades.mech.northwestern.edu/images/2/25/MR-v2.pdf
- Exponential map of SE(3) is defined in Section 3.3.3.1
- Jacobian in Equation 3.87.
Both representations probably have their justification. Where are they used? What are the differences?
edit: It seems that $J_A(\theta) = \frac{J_B(\theta)}{\theta}$
edit2: In fact, the Jacobians are exactly the same, but the definition of the exponential map of SE(3) is different. In the first definition, the translation components are not multiplied by the angle. In the second definition they are. The second definition originates from screw theory.