[Paper Review] A micro Lie theory for state estimation in robotics

Joan Solà, Jérémie Deray+1 more|arXiv (Cornell University)|Dec 4, 2018

Robotics and Sensor-Based Localization11 references158 citations

TL;DR

This paper presents a minimal, practical micro Lie theory tailored for robotics state estimation, focusing on intuitive concepts, useful Jacobians, and a C++ template library for common groups used in SLAM and visual odometry.

ABSTRACT

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.

Motivation & Objective

• Motivate the use of Lie group concepts to improve estimation precision, consistency, and stability in robotics.
• Offer a simplified, didactic subset of Lie theory that is accessible to roboticists without deep algebraic prerequisites.
• Provide actionable tools and formulas (Jacobian computation, exponential/log maps) for common robotics estimation problems.
• Demonstrate applicability to EKF, nonlinear optimization, and SLAM tasks.
• Deliver an open-source implementation framework to facilitate adoption in practice.

Proposed method

• Introduce the basic notions of Lie groups and Lie algebras in a robotics context without full algebraic depth.
• Explain the Jacobian-centric calculus on manifolds and the role of exponential and logarithmic maps for state increments.
• Present concrete examples on SO(2), SO(3), SE(2), and SE(3) with their tangent spaces and adjoint mappings.
• Define convenient hat/vee and Exp/Log operators to move between Lie algebra vectors and group elements.
• Offer guidelines for deriving analytic Jacobians essential for optimal estimators.
• Provide a C++ header-only library (manif) implementing the key groups and Jacobian support.

Experimental results

Research questions

• RQ1 How can a minimal Lie theory be formulated to aid uncertainty management in robotics estimation tasks?
• RQ2 What practical tools (Jacobian derivations, maps, and templates) enable stable and efficient estimation on common motion groups?
• RQ3 How can one apply micro Lie theory to EKF, optimization, and SLAM pipelines in robotics?
• RQ4 What are the concrete representations and transformations between Lie groups, their algebras, and Cartesian spaces that are most useful to practitioners?

Key findings

• A compact introduction to Lie theory tailored for robotics aids estimation with rigorous yet approachable tools.
• The paper emphasizes Jacobian computation and provides practical mappings (Exp/Log, hat/vee) to move between representations.
• It covers typical robotics groups (SO(2), SO(3), SE(2), SE(3)) with their tangents, adjoints, and group actions.
• The approach promotes representing uncertainty in the tangent space using a real vector space for ease of linear algebra.
• An open-source C++ header-only library, manif, is released to implement the discussed groups and analytic Jacobians.
• The material supports applying LT concepts to EKF, visual odometry, and SLAM within a structured, didactic framework.