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, perhaps by their academic formation, to the deep abstractions this theory
deals with.
In estimation for robotics, it is not necessary to exploit the full capacity of the theory,
and therefore an effort of selection of materials is required. In this twosessions course,
I will walk you through the basic principles of the Lie theory, with the aim of conveying
clear and useful ideas. I have taken a particular approach which I consider didactical,
based on some principles:
The fist principle is pertinence: we will present the key aspects of the theory that are
pertinent for the needs of estimation in robotics. We will therefore discard a significant
part of the theoretical corpus of the Lie theory.
The second principle is concretion. We will ground every abstract concept with examples
from our common background. We will illustrate the Lie theory through explanations on the
groups of rotation matrices, rigid motion matrices,
quaternions, and complex numbers.
The third principle is connectedness. We will present the key elements of the Lie theory
in a way that connects with easier concepts in vector spaces and linear algebra. This will
render the ideas more intuitive and easier to grasp.
The fourth principle is usefulness. We will present the concepts in a way that allow us
to build tools that are useful for our tasks. In particular, we will define operators,
derivatives, Jacobian matrices, perturbations, covariances matrices, and timeintegrals.
Through connectedness, they will resemble what we know from linear algebra in vector spaces.
Also, we will be able to manipulate them in very familiar ways.
The course is organized in two sessions of two hours each. As the name tells, these sessions
are designed as courses, not seminars. I therefore encourage assistants to take an active role,
possibly taking notes and above all making questions and debate. It is through some effort that
nontrivial concepts are acquired.

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