Robotics for Beginners
一门面向大学生与深度爱好者的具身智能入门课。从线性代数、概率和李群讲起，一路走到规划、控制、机器人学习、VLA 与 Agentic Robotics。每一个概念都从直觉出发，每一页都可以动手拖一拖。 An introduction to robotics and embodied AI for university students and serious enthusiasts. It starts from linear algebra, probability and Lie groups and goes all the way to planning, control, robot learning, VLAs and agentic robotics. Every concept starts from intuition, and every page is something you can drag around.
为什么要做这门课Why this course exists
具身智能正在飞速发展：VLA 模型、世界模型、人形机器人、会使用工具的机器人智能体……但真正想入门的人很快会撞上两堵墙。一堵是数学：雅可比、协方差、SE(3)、李代数、KKT……教科书写得严谨却抽象，读完公式仍然"看不见"它在做什么。另一堵是知识的碎片化：经典机器人学、状态估计、优化、深度学习、强化学习各自有一套语言，散落在不同的课程和论文里。
这门课想做的，是一条连贯的路：先用可交互的图把核心数学变成直觉，再用同一套语言串起经典方法和学习方法，最后读懂并参与最前沿的研究。
Embodied AI is moving fast: VLA models, world models, humanoids, robot agents that use tools. Anyone who tries to get in soon hits two walls. The first is math: Jacobians, covariances, SE(3), Lie algebras, KKT conditions. Textbooks are rigorous but abstract, and you can follow every formula without ever seeing what it does. The second is fragmentation: classical robotics, state estimation, optimization, deep learning and RL each speak their own language, scattered across courses and papers.
This course aims to be one coherent path. First, interactive figures turn the core math into intuition. Then the same language ties classical and learning-based methods together. Finally, you will be able to read and take part in frontier research.
我们怎样讲How we teach
直觉先行，形式随后Intuition first, formalism second
先建立一幅图像：它在空间里做了什么？然后才写公式，并把公式里的每个符号指回图上。公式是对直觉的精确表达，而不是直觉的替代品。First build a picture: what does it do to space? Only then write the formula, and point every symbol back to the picture. A formula should make an intuition precise, not replace it.
可以拖动的概念Concepts you can touch
每一页都有多个交互图：拖动向量、调滑块、玩小游戏、完成"试一试"任务。核心概念会给最丰富的交互，次要知识点也尽量做到可交互。看懂一次，不如亲手把网格压扁一次。Every page has several interactive figures: drag vectors, move sliders, play mini-games, complete "Try this" tasks. Core ideas get the richest labs, and even minor points are made interactive where possible. Squashing a grid yourself beats reading about it ten times.
同一个概念，多种视角One idea, several views
矩阵既是"搬动点"也是"换尺子"；雅可比既是偏导表格，也是局部的线性变换；卡尔曼增益既是公式，也是"信任比例"。能在视角之间切换，才算真正理解。A matrix both "moves points" and "changes the ruler"; a Jacobian is both a table of partial derivatives and a local linear map; a Kalman gain is both a formula and a "trust ratio". Understanding means being able to switch views.
为机器人而选的数学Math chosen for robots
我们不追求数学的完备，而追求有用。每个知识点都从一个真实的机器人问题出发（"相机看到的杯子在世界哪里？"），也都回到一个机器人应用中去。We aim for usefulness, not mathematical completeness. Every topic starts from a real robot problem ("where in the world is the cup the camera sees?") and ends in a robot application.
螺旋上升A spiral, not a list
重要的概念会反复出现，每次多一层。比如雅可比会在下面这些地方一次次回来：Important ideas come back again and again, one layer deeper each time. The Jacobian, for example, returns in all of these places:
一套视觉语言One visual language
全课程统一配色和记号：x 轴永远是红色，y 是绿色，z 是蓝色，"我们关心的那个向量"是琥珀色。这样，后面章节里的图你一眼就能读懂。One color code and notation throughout: x is always red, y green, z blue, and "the vector we care about" amber. Figures in later chapters read at a glance.
从经典到前沿From classical to frontier
卡尔曼滤波和 VLA 并不对立。理解了经典方法在解决什么问题，才能看懂学习方法为什么这样设计、又在哪里会失效。课程从经典走向前沿，最后落在 Agentic Robotics。Kalman filters and VLAs are not rivals. Knowing which problem a classical method solves is what lets you see why a learning method is designed the way it is, and where it breaks. The course moves from classical to frontier and ends with agentic robotics.
诚实的难度Honest about difficulty
直觉不是"糊弄"。每一页都给出精确的数学和常见误区，告诉你直觉在哪里会失灵，并附上进一步阅读的经典教材与论文。Intuition is not hand-waving. Every page gives the precise math and the common pitfalls, says where the intuition breaks down, and points to the classic textbooks and papers for further reading.
谁适合读、怎么读Who it is for, and how to use it
适合For
- 想进入机器人 / 具身智能方向的本科生、研究生；Undergraduate and graduate students heading into robotics or embodied AI;
- 有编程基础、想系统补齐数学与机器人基础的工程师和爱好者；Engineers and enthusiasts who can program and want a systematic foundation in the math and the robotics;
- 做学习方法、却想真正理解经典机器人学的研究者。Learning-focused researchers who want to really understand classical robotics.
前置要求：高中数学，加一点编程经验。其余的我们一起补。Prerequisites: high-school math and a little programming. We build the rest together.
怎么读How to use it
- 按顺序读：Part I 是后面所有部分的共同语言；In order: Part I is the shared language for everything after it;
- 按需跳读：每页都标注了前置知识，可以从感兴趣的问题倒着补；On demand: every page lists its prerequisites, so you can start from a question you care about and backfill;
- 一节约 15–25 分钟。不要只看，一定要动手完成"试一试"任务和小练习。Each lesson takes 15–25 minutes. Don't just read: do the "Try this" tasks and the quiz.
右上角可以随时切换中文 / English 和明暗主题。Switch 中文 / English and light / dark at the top right at any time.
每一节课的结构Anatomy of a lesson
- 为什么机器人需要它Why robots need it — 一个具体的机器人问题作为钩子a concrete robot problem as the hook
- 直觉Intuition — 先讲图像和类比，暂不出现公式pictures and analogies, no formulas yet
- 动手玩Play — 核心交互实验室，配有会自动打勾的"试一试"任务the core interactive lab, with "Try this" tasks that tick themselves
- 再看数学The math — 公式与图联动：悬停在符号上，图中对应元素高亮formulas linked to figures: hover a symbol and its counterpart lights up
- 机器人中的它In robotics — 一个真实应用的小演示a small demo of a real application
- 常见误区Pitfalls — 直觉在哪里会失灵where intuition breaks down
- 小练习Check yourself — 即时判分并附讲解instant feedback with explanations
- 延伸阅读Further reading — 经典教材与论文classic textbooks and papers
视觉与记号约定Visual and notation conventions
| 记号Notation | 含义Meaning |
|---|---|
| \(\mathbf v\), \(\mathbf v^\top\) | 向量默认是列向量；\(\top\) 表示转置Vectors are column vectors by default; \(\top\) is the transpose |
| 坐标系Frames | 右手系；三维中 z 轴朝上（与 ROS 一致）Right-handed; z points up in 3D (as in ROS) |
| \({}^{A}\mathbf p\) | 点 p 在坐标系 A 中的坐标Coordinates of point p in frame A |
| \({}^{A}T_{B}\), \({}^{A}R_{B}\) | 坐标系 B 相对于 A 的位姿 / 姿态，满足 \({}^{A}\mathbf p = {}^{A}T_{B}\,{}^{B}\mathbf p\)Pose / orientation of frame B relative to A, so that \({}^{A}\mathbf p = {}^{A}T_{B}\,{}^{B}\mathbf p\) |
| \(\skew{\boldsymbol\omega}\), \(\Exp\), \(\Log\) | 叉积矩阵（hat）；向量与李群之间的指数 / 对数映射Cross-product (hat) matrix; exponential / logarithm maps between vectors and the Lie group |
| \(\boldsymbol\xi = (\boldsymbol\omega, \mathbf v)\) | twist：角速度在前、线速度在后（注意有些教材顺序相反）Twist: angular part first, linear part second (some texts use the opposite order) |
完整大纲The full outline
Part I（58 节）、Part II（14 节）与 Part III（15 节）已全部上线，★ 标记的是核心节；其余部分会陆续发布。All 58 lessons of Part I, 14 of Part II and 15 of Part III are live; ★ marks core lessons. The other parts will follow.