# A micro Lie theory for state estimation in robotics

arXiv: https://arxiv.org/abs/1812.01537 · HTML: https://mlsanity.com/paper/1812.01537/

## Key facts

- Cited by 124 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2018.

## Built on by (most cited first, 30 of 124)

- [Gaussian Splatting SLAM](https://mlsanity.com/paper/2312.06741/) (2023)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [MASt3R-SLAM: Real-Time Dense SLAM with 3D Reconstruction Priors](https://mlsanity.com/paper/2412.12392/) (2024)
- [Motion Planning Diffusion: Learning and Planning of Robot Motions with Diffusion Models](https://mlsanity.com/paper/2308.01557/) (2023)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [Theseus: A Library for Differentiable Nonlinear Optimization](https://mlsanity.com/paper/2207.09442/) (2022)
- [Learning Robotic Manipulation Policies from Point Clouds with Conditional Flow Matching](https://mlsanity.com/paper/2409.07343/) (2024)
- [SE(3)-Equivariant Relational Rearrangement with Neural Descriptor Fields](https://mlsanity.com/paper/2211.09786/) (2022)
- [Tangent Space Backpropagation for 3D Transformation Groups](https://mlsanity.com/paper/2103.12032/) (2021)
- [Motion Planning Diffusion: Learning and Adapting Robot Motion Planning with Diffusion Models](https://mlsanity.com/paper/2412.19948/) (2024)
- [DSP-SLAM: Object Oriented SLAM with Deep Shape Priors](https://mlsanity.com/paper/2108.09481/) (2021)
- [Shelving, Stacking, Hanging: Relational Pose Diffusion for Multi-modal Rearrangement](https://mlsanity.com/paper/2307.04751/) (2023)
- [ActionFlow: Equivariant, Accurate, and Efficient Policies with Spatially Symmetric Flow Matching](https://mlsanity.com/paper/2409.04576/) (2024)
- [HARA: A Hierarchical Approach for Robust Rotation Averaging](https://mlsanity.com/paper/2111.08831/) (2021)
- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)
- [BARD-GS: Blur-Aware Reconstruction of Dynamic Scenes via Gaussian Splatting](https://mlsanity.com/paper/2503.15835/) (2025)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Confronting Ambiguity in 6D Object Pose Estimation via Score-Based Diffusion on SE(3)](https://mlsanity.com/paper/2305.15873/) (2023)
- [Composable Energy Policies for Reactive Motion Generation and Reinforcement Learning](https://mlsanity.com/paper/2105.04962/) (2021)
- [iFusion: Inverting Diffusion for Pose-Free Reconstruction from Sparse Views](https://mlsanity.com/paper/2312.17250/) (2023)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Morphological Symmetries in Robotics](https://mlsanity.com/paper/2402.15552/) (2024)
- [4DTAM: Non-Rigid Tracking and Mapping via Dynamic Surface Gaussians](https://mlsanity.com/paper/2505.22859/) (2025)
- [MimicFunc: Imitating Tool Manipulation from a Single Human Video via Functional Correspondence](https://mlsanity.com/paper/2508.13534/) (2025)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Orientation Probabilistic Movement Primitives on Riemannian Manifolds](https://mlsanity.com/paper/2110.15036/) (2021)
- [COMO: Compact Mapping and Odometry](https://mlsanity.com/paper/2404.03531/) (2024)
- [FUNCTO: Function-Centric One-Shot Imitation Learning for Tool Manipulation](https://mlsanity.com/paper/2502.11744/) (2025)
- [6D Object Pose Tracking in Internet Videos for Robotic Manipulation](https://mlsanity.com/paper/2503.10307/) (2025)
- [Homography Decomposition Networks for Planar Object Tracking](https://mlsanity.com/paper/2112.07909/) (2021)

## Abstract (arXiv)

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.