  A Lie group is an old mathematical abstract object dating back to the XIX
century, when mathematician Sophus Lie laid the foundations of the theory of
continuous transformation groups. As it often happens, its usage has spread
over diverse areas of science and technology many years later. In robotics, we
are recently experiencing an important trend in its usage, at least in the
fields of estimation, and particularly in motion estimation for navigation. Yet
for a vast majority of roboticians, Lie groups are highly abstract
constructions and therefore difficult to understand and to use. This may be due
to the fact that most of the literature on Lie theory is written by and for
mathematicians and physicists, who might be more used than us to the deep
abstractions this theory deals with.
  In estimation for robotics it is often not necessary to exploit the full
capacity of the theory, and therefore an effort of selection of materials is
required. In this paper, we will walk through the most basic principles of the
Lie theory, with the aim of conveying clear and useful ideas, and leave a
significant corpus of the Lie theory behind. Even with this mutilation, the
material included here has proven to be extremely useful in modern estimation
algorithms for robotics, especially in the fields of SLAM, visual odometry, and
the like.
  Alongside this micro Lie theory, we provide a chapter with a few application
examples, and a vast reference of formulas for the major Lie groups used in
robotics, including most jacobian matrices and the way to easily manipulate
them. We also present a new C++ template-only library implementing all the
functionality described here.
