A micro Lie theory for state estimation in robotics

By Joan Solà, Jérémie Deray, Dinesh Atchuthan. arXiv.org, 2018

572 citations recorded by the source corpus.

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 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. We provide also a vast reference of formulas for the major groups used in robotics, including most jacobian matrices and the way to easily manipulate them.

Subfields: Aerospace Engineering, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/3a75252bab18250b8de8be28ec376db6cfc04084

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Aerospace Engineering

1. Yingdian Cao
2. Arnaud Doucet
3. Regina Barzilay
4. Youqi Pan
5. Wugen Zhou
6. Hongbin Zha
7. Simo Särkkä

In Computer Vision and Pattern Recognition

1. Jason Yim
2. Marco Tripodi
3. Yingdian Cao
4. Arnaud Doucet
5. Regina Barzilay
6. Ta-Chu Kao
7. Youqi Pan
8. Wugen Zhou

Citations in the corpus

• Manifold GPLVMs for discovering non-Euclidean latent structure in neural data (2020, background): “We can thus define a ‘capitalized’ exponential map Exp G : R n → G , which maps elements of R n to elements in G (Sola et al., 2018) (Appendix B).”
• Gaussian Splatting SLAM (2023, unknown): “Borrowing the notation from [30], let T ∈ SE (3) and τ ∈ se (3) .”
• Adaptive VIO: Deep Visual-Inertial Odometry with Online Continual Learning (2024, background): “The differentiation and backpropagation of Lie Group can refer to [29, 34, 39].”
• SE(3) diffusion model with application to protein backbone generation (2023, methodology): “We refer to Sola et al. (2018) for an introduction to Lie groups.”
• Broadly applicable and accurate protein design by integrating structure prediction networks and diffusion generative models (2022, unknown): “(6) ∇ r log IGSO3(r; r ^ , ε(2) ) = ∇ r ω(rr ^ ) · d dω log f(ω; ε (2))| ω=ω(rr ^ ) = 1 2 r log(rr ^ )/ω(rr ^ ) d dω log f(ω; ε (2))| ω=ω(rr ^ ) where is the logarithmic map from SO(3) to the Lie algebra of SO(3) [57], is the log(rr ^ ) ω(rr ^ )”