Underactuated Robotics

Algorithms for Walking, Running, Swimming, Flying, and Manipulation

Russ Tedrake

© Russ Tedrake, 2020
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Note: These are working notes used for a course being taught at MIT. They will be updated throughout the Spring 2020 semester. Lecture videos are available on YouTube.

Table of Contents


This book is about building robots that move with speed, efficiency, and grace. I believe that this can only be achieve through a tight coupling between mechanical design, passive dynamics, and nonlinear control synthesis. Therefore, these notes contain selected material from dynamical systems theory, as well as linear and nonlinear control.

These notes also reflect a deep belief in computational algorithms playing an essential role in finding and optimizing solutions to complex dynamics and control problems. Algorithms play an increasingly central role in modern control theory; these days even rigorous mathematicians consider finding convexity in a problem (therefore making it amenable to an efficient computational solution) almost tantamount to an analytical result. Therefore, the notes necessarily also cover selected material from optimization theory, motion planning, and machine learning.

Although the material in the book comes from many sources, the presentation is targeted very specifically at a handful of robotics problems. Concepts are introduced only when and if they can help progress the capabilities we are trying to develop. Many of the disciplines that I am drawing from are traditionally very rigorous, to the point where the basic ideas can be hard to penetrate for someone that is new to the field. I've made a conscious effort in these notes to keep a very informal, conversational tone even when introducing these rigorous topics, and to reference the most powerful theorems but only to prove them when that proof would add particular insights without distracting from the mainstream presentation. I hope that the result is a broad but reasonably self-contained and readable manuscript that will be of use to any enthusiastic roboticist.


The material in these notes is organized into a few main parts. "Model Systems" introduces a series of increasingly complex dynamical systems and overviews some of the relevant results from the literature for each system. "Nonlinear Planning and Control" introduces quite general computational algorithms for reasoning about those dynamical systems, with optimization theory playing a central role. Many of these algorithms treat the dynamical system as known and deterministic until the last chapters in this part which introduce stochasticity and robustness. "Estimation and Learning" follows this up with techniques from statistics and machine learning which capitalize on this viewpoint to introduce additional algorithms which can operate with less assumptions on knowing the model or having perfect sensors. The book closes with an "Appendix" that provides slightly more introduction (and references) for the main topics used in the course.

The order of the chapters was chosen to make the book valuable as a reference. When teaching the course, however, I take a spiral trajectory through the material, introducing robot dynamics and control problems one at a time, and introducing only the techniques that are required to solve that particular problem.

insert figure showing progression of problems here. pendulum -> cp/acro -> walking ... with chapter numbers associated.


All of the examples and algorithms in this book, plus many more, are now available as a part of our open-source software project: . is a C++ project, but in this text we will use Drake's Python bindings. I encourage super-users or readers who want to dig deeper to explore the C++ code as well (and to contribute back).

Please see the appendix for specific instructions for using along with these notes.

First chapter