Introduction 引言

There has been a remarkable effort in the last years in the robotics community to formulate estimation problems properly.
在过去的几年里，机器人界做出了巨大的努力来正确地制定估算问题。

This is motivated by an increasing demand for precision, consistency and stability of the solutions.
这是由于对解决方案的精度、一致性和稳定性的不断追求。

Indeed, a proper modeling of the states and measurements, the functions relating them, and their uncertainties, is crucial to achieve these goals.
确实，状态与测量的合适建模，方程联系与不确定关系对于实现目标至关重要。

This has led to designs involving what has been known as ‘manifolds’, which in this context are no less
than the smooth topologic surfaces of the Lie groups where the state representations evolve.
这导致设计涉及到被称为“流形”的东西，在这种情况下，它不亚于状态表示演化的李群的光滑拓扑表面。

Relying on the Lie theory (LT) we are able to construct a rigorous calculus corpus to
handle uncertainties, derivatives and integrals with precisionand ease. Typically, these works have focused on the well known manifolds of rotation SO(3) and rigid motion SE(3).

依靠李理论（LT），我们能够构建一个严格的微积分语料库来精确容易地处理不确定性、导数和积分。通常，这些工作都集中在常见的旋转流形 SO(3) 和刚性运动 SE(3)。

When being introduced to Lie groups for the first time, it is important to try to regard them from different points of view.
当第一次接触Lie群时，它是从不同的角度看待他们是很重要的。

The topological viewpoint, see Fig. 1,
拓扑观点，见图1，

involves the shape of the manifold and conveys powerful intuitions of its relation to the tangent space and the exponential map.
涉及流形的形状，并传达其关系的强大直觉到切线空间和指数映射。

The algebraic viewpoint involves the group operations and their concrete
realization, allowing the exploitation of algebraic properties
to develop closed-form formulas or to simplify them.
代数观点涉及到群运算及其具体形式实现，允许利用代数性质发展或简化闭形式公式。

The geometrical viewpoint, particularly useful in robotics, associates group elements to the position, velocity, orientation and/or other modifications of bodies or reference frames.
几何观点在机器人学中特别有用，它将群元素与本体或参照标架的位置、速度、方向和/或其他变量相关联。

The origin frame may be identified with the group’s identity, and
any other point on the manifold represents a certain ‘local’ frame.
原始标架可以用群的标识来标识，并且流形上的任何其他点存在一个“局部”标架。

By resorting to these analogies, many mathematical abstractions of the LT can be brought closer to intuitive notions in vector spaces, geometry, kinematics and other more classical fields.
通过这些类比，许多Lie理论地数学抽象定义可以更接近直观的概念在向量空间中，几何学、运动学等更为经典领域。

Lie theory is by no means simple.
Lie理论绝不简单。

To grasp a minimum idea of what LT can be, we may consider the following
three references.
要了解什么是 LT，我们可以考虑以下三个参考资料。

First, Abbaspour’s “Basic Lie theory” [1] comprises more than 400 pages. With a similar title, Howe’s
“Very basic Lie theory” [2] comprises 24 (dense) pages, and is sometimes considered a must-read introduction. Finally, the more modern and often celebrated Stillwell’s “NaiveLie theory” [3] comprises more than 200 pages. With such precedents labeled as ‘basic’, ‘very basic’ and ‘naive’, the aim of this paper at merely 17 pages is to simplify Lie theory even more (thus our adjective ‘micro’ in the title).
首先，阿巴斯普尔的“基本谎言理论”[1] 超过 400 页。有一个类似的标题，豪的
“非常基本的李理论” [2] 包含 24（密集）页，有时被认为是必读的介绍。最后，更现代、更广为人知的 Stillwell 的“NaiveLie 理论”[3] 包含 200 多页。由于这些先例被标记为“基本”、“非常基本”和“天真”，（我们的）这篇仅 17 页的论文旨在进一步简化李理论（因此我们在标题中使用了形容词“微”）。

This we do in two ways. First, we select a small subset of material from the LT. This subset is so small that it merely explores the potential of LT.
我们通过两种方式做到这一点。首先，我们从 LT 中选择一小部分材料。这个子集非常小，它只是探索了 LT 的可能性。

However, it appears very useful for uncertainty
management in the kind of estimation problems we deal with in robotics (e.g. inertial pre-integration, odometry and SLAM,visual servoing, and the like), thus enabling elegant and rigorous designs of optimal optimizers.
但是，它在我们处理的机器人技术的不确定控制中的估计问题（例如惯性预积分、里程计和 SLAM，
视觉伺服等）非常有效，从而有助于实现优雅和精确的优化器的严格设计。

Second, we explain it in a didactical way, with plenty of redundancy so as to reduce the entry gap to LT even more, which we believe is still needed.
其次，我们解释它以一种教学的方式，具有大量的冗余，以进一步减少与 LT 的进入差距，我们认为这仍然是必要的。

That is, we insist on the efforts in this direction of, to name a paradigmatic title, Stillwell’s [3], and provide yet a more simplified version.
也就是说，我们坚持朝着这个方向努力为了命名一个范例性的标题，Stillwell的[3]，并提供但更简单的版本。

The main text body is generic,though we try to keep the abstraction level to a minimum.
尽管我们尽量将抽象级别保持在最低限度，但主文本体是通用的(generic)。

Inserted examples serve as grounding base for the general concepts when applied to known groups (rotation and motion matrices, quaternions, etc.).
当应用于已知群（旋转和运动矩阵、四元数等）时，插入的示例可作为一般概念的基础。

Also, plenty of figures with very verbose captions re-explain the same concepts once again.
此外，大量带有非常冗长标题的图再次重新解释了相同的概念。

We put special attention to the computation of Jacobians (a topic that is not treated in [3]), which are essential for most optimal optimizers and the source of much trouble when designing new algorithms.
我们特别关注雅可比矩阵的计算（[3] 中未涉及的主题），这对于大多数最优优化器来说是必不可少的，也是设计新算法时许多麻烦的根源。

We provide a chapter with some applicative examples for robot localization and mapping, implementing EKF and nonlinear optimization algorithms based on LT.
我们提供了一个章节，其中包含一些机器人定位和映射的应用示例，并在LT的基础上实现了EKF和非线性优化算法。

And finally, several appendices contain ample reference for the most relevant details of the most commonly used groups in robotics:
unit complex numbers, quaternions, 2D and 3D rotation matrices, 2D and 3D rigid motion matrices, and the
trivial translation groups.
最后，几个附录包含对机器人技术中最常用组的最相关细节的充分参考：
单位复数、四元数、2D 和 3D 旋转矩阵、2D 和 3D 刚性运动矩阵以及平凡的平移组。

到现在为止还是介绍的部分、

用g代替G

Yet our most important simplification to Lie theory is in terms of scope. The following passage from Howe [2] may
serve us to illustrate what we leave behind:
然而，我们对李理论最重要的简化是在范围方面。Howe [2] 的以下段落可能为我们展示我们落下的东西：

“The essential phenomenon of Lie theory is that one may associate in a natural way to a Lie group G its Lie algebra g. The Lie algebra g is first of all a vector space and secondly is endowed with a bilinear nonassociative product called the Lie bracket […].
Amazingly, the group G is almost completely determined by g and its Lie bracket.
Thus for many purposes one can replace G with g.
Since G is a complicated nonlinear object and g is just a vector space, it is usually vastly simpler to work with g.
[…] This is one source of the power of Lie theory.”

“李理论的基本现象是，人们可以以一种自然的方式将李群 G 与李代数 g 联系起来。李代数 g 首先是一个向量空间，其次被赋予一种称为李括号的双线性非关联乘积 […]
令人惊讶的是，群G 几乎完全由 g 及其李括号决定。
因此，出于许多目的，可以用g代替G。
由于 G 是一个复杂的非线性对象，而 g 只是一个向量空间，因此使用 g 通常要简单得多。
[…] 这是李理论力量的来源之一。”

In [3], Stillwell even speaks of “the miracle of Lie theory”.
在[3]中，斯蒂尔韦尔甚至谈到了“李理论的奇迹”。

In this work we will effectively relegate the Lie algebra to a second plane in favor of its equivalent vector space Rn,and will not introduce the Lie bracket at all.
在这项工作中，我们将有效地将李代数降级到第二个平面，以支持其等效向量空间 Rn，并且根本不会引入李括号。

Therefore, the connection between the Lie group and its Lie algebra will not be made here as profound as it should.
因此，在这李群与其李代数之间的联系不会像它应该的那样深刻。

Our position is that,given the target application areas that we foresee, this material is often not necessary.
我们的立场是，鉴于我们预见的目标应用领域，这种材料通常不是必需的。

Moreover, if included, then we would fail in the objective of being clear and useful, because the reader would have to go into mathematical concepts that, by their abstraction or subtleness, are unnecessarily complicated.
此外，如果包括在内，那么我们将无法实现清晰和有用的目标，因为读者将不得不进入数学概念，这些概念由于其抽象或微妙而不必要地复杂。

Our effort is in line with other recent works on the subject [4], [5], [6], which have also identified this need of bringing the LT closer to the robotician.
我们的努力与最近关于该主题的其他工作一致 [4]、[5]、[6]，这些工作也确定了使 LT 更接近机器人技术的必要性。

Our approach aims at appearing familiar to the target audience of this paper: an audience that is skilled in state estimation (Kalman filtering, graph-based optimization, and the like), but not yet familiar with the theoretical corpus of the Lie theory.
我们的方法旨在让本文的目标受众看起来很熟悉：精通状态估计（卡尔曼滤波、基于图的优化等）但还不熟悉李理论的理论语料库的受众。

We have for this taken some initiatives with regard to notation, especially in the definition of the derivative, bringing it close to the vectorial counterparts, thus making the chain rule clearly visible.
为此，我们在符号方面采取了一些举措，特别是在导数的定义中，使其接近向量对应物，从而使链式法则清晰可见。

As said, we opted to practically avoid the material proper to the Lie algebra,
and prefer instead to work on its isomorphic tangent vector space Rn, which is where we ultimately represent uncertainty or (small) state increments.
如上所述，我们实际上选择避免使用李代数固有的材料，

而是更喜欢在其同构切向量空间 Rn 上工作，这是我们最终表示不确定性或（小）状态增量的地方。

All these steps are undertaken with absolutely no loss in precision or exactness,and we believe they make the understanding of the LT and the manipulation of its tools easier.
所有这些步骤都是在绝对不损失精度或准确度的情况下进行的，我们相信它们使 LT 的理解及其工具的操作更容易。

This paper is accompanied by a new open-source C++ header-only library, called manif [7], which can be found
at https://github.com/artivis/manif. manif implements the widely used groups SO(2), SO(3), SE(2) and SE(3), with
support for the creation of analytic Jacobians. The library is designed for ease of use, flexibility and performance.

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A micro Lie theory for state estimation in robotics002

Introduction 引言

到现在为止还是介绍的部分、

用g代替G

而是更喜欢在其同构切向量空间 Rn 上工作，这是我们最终表示不确定性或（小）状态增量的地方。

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