A micro Lie theory for state estimation in robotics (1812.01537v9)

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.

• The paper presents a micro Lie theory that distills complex Lie group concepts to practical state estimation for robotics.
• It employs selective theory and clear didactics, providing a C++ library (manif) for applications in SLAM, visual odometry, and EKF.
• The approach enhances robotic navigation by reducing mathematical complexity while ensuring robust and efficient performance.

Micro Lie Theory for State Estimation in Robotics

The paper "A micro Lie theory for state estimation in robotics" by Joan Solà, Jeremie Deray, and Dinesh Atchuthan outlines an approach to simplify and apply Lie group theory to state estimation in robotics. Lie groups, fundamental since their mathematical introduction in the 19th century, are crucial in fields like robotics for their ability to describe and manipulate continuous transformation groups such as rotations and motions. Despite their importance, the inherent complexity of Lie theory makes it daunting for practitioners outside of pure mathematics. This paper aims to bridge that gap by proposing a streamlined and educationally accessible version of Lie theory tailored to roboticists engaged in state estimation.

Key Contributions and Approach

The paper offers a "micro" version of Lie theory, focusing on the elements directly applicable to robotic navigation and estimation challenges. This involves truncating the more complex aspects of Lie theory to render it more comprehensible and directly useful for robotics applications, particularly in Simultaneous Localization and Mapping (SLAM), visual odometry, and Extended Kalman Filters (EKF). The authors achieve this simplification through a series of educational tools and conventions:

• Selective Theory: The authors concentrate on selecting the most pertinent segments of Lie theory, which include the exponential map for transferring elements from Lie algebras to Lie groups and the adjoint matrix for managing transformations within tangent spaces.
• Clear Didactics: The theoretical exposition is accompanied by didactic examples and graphical illustrations intended to facilitate understanding through visualization. This includes extensive visual content aimed at demystifying abstract concepts.
• Practical Tools: The paper introduces a suite of practical tools, including a new C++ library called manif, which implements the theory for convenient deployment in robotic simulations and real-world applications. This library emphasizes usability, performance, and includes groups like $\SO(2)$, $\SO(3)$, $\SE(2)$, and $\SE(3)$, which are prevalent in robotics.

Practical Implications

The implications of this work are extensive for practical robotics, particularly in areas requiring precise estimation and navigation in real-time systems. The simplification of Lie theory aids roboticists in modeling and solving complex navigational problems with confidence, utilizing mathematically sound frameworks without deep diving into the complexity traditionally required. This could lead to more robust SLAM solutions, better odometry measurements, and ultimately, more reliable autonomous robotic operations.

Theoretical Insights and Challenges

While the paper focuses on practical application, it also implicitly asks whether fundamental elements of Lie theory can be translated into simplified computational models without significant loss of generality. This challenges the community to rethink approaches to complex mathematical frameworks, promoting accessibility without sacrificing precision or correctness.

Future Developments

Future work building on this paper may involve expanding the capabilities of the manif library, encompassing additional Lie groups or further optimizations for computational efficiency. Another avenue could be the extension and verification of this "micro" Lie theory's applicability to more diverse fields within robotics, such as humanoid robotics or multi-agent systems. Moreover, bridging further mathematical fields into such practical domains may lead to similarly accessible frameworks, significantly reducing the barriers to effective applied research.

In summary, this paper represents a significant contribution to robotics by re-evaluating and distilling complex mathematical theory into tools and methodologies more easily digestible and applicable by the robotics community, facilitating advancements in state estimation methods.