In the last three decades, there has been a phenomenal growth of activity in robotics,

both in terms of research and in capturing the general public’s illusion of endless

futuristic possibilities.

Growth in robotics coincides with an increasing number of courses at most major

research universities on various aspects of the subject. These courses are available in

many departments, such as Systems Engineering and Automation, Computer Science,

Mathematics, or Electrical and Mechanical Engineering. Several excellent textbooks

support this education, covering different topics in kinematics, dynamics, control,

sensing, vision, AI, planning, and navigation for robotic mechanisms. Given the state

of knowledge on the robot subject and the vast diversity of students interested in the

field, there is an inherent utility for a book that presents the more abstract mathematical

formulation given by the screw theory in robotics. This work attempts to provide

a visual approach of this formulation to an audience who perhaps did not have enough

examples of building up the necessary knowledge.

Although the advantages of screw theory are obvious, few teach it in engineering

courses, so there are still proportionally few postgraduates who know how to apply

it. Some numerous projects and publications demonstrate the crucial advances

achieved when applying the geometric formalisms of Lie’s algebra in various robotics

specialties. The screw theory has proven superior to other mathematical techniques

in robotics. Therefore, it is essential to communicate and disseminate these

methodologies among as many students as possible who might be working with

robots in the future.

This book will emphasize geometric techniques and provide a modern visual presentation,

so concepts are better understood. We best capture the key physical features

of a robot with a geometric description. However, these screw theory tools,

inaccessible to many students, require a new language (e.g., screws, twists, wrenches,

adjoint transformation, geometric Jacobian, spatial vector algebra). The rules for the

manipulation of this language can seem kind of obscure. In fact, at the heart of the

screw theory, there is a high-level but straightforward geometric interpretation of

mechanics. The alternative is the most standard algebraic alternatives, which unfortunately

make us often buried in the calculation details.

When Brockett (1983) showed how to use the Lie group structure to describe

mathematically the kinematic chains in rigid bodies’ motion, there was a breakthrough

in making classical screw theory accessible to modern robotics. In presenting

it, we must highlight the powerful Product of Exponentials (POE) as a fundamental

tool of the screw theory.

Overall, we can realize that a good theory is the fastest way to obtain a better

functional performance, and the only thing to do in exchange is to study this mathematics.

The selection of topics for this text targets a modern screw theory approach

for robot mechanics. The content is an excellent introduction to the subject (e.g.,

mathematical tools, kinematics, differential kinematics, dynamics, trajectory generation,

simulation) and the best way to appreciate this approach’s benefits. Other

xviii Preface

aspects such as control are briefly introduced and used in simulations but left out to

a thorough extent for later works.

This book covers the screw theory fundamentals with a very graphical approach.

Its contents are very focused, practical, and visual so that any student can quickly

grasp the advantages of the theory and the algorithmic definitions. In the future, we

will be able to extend those possibilities to other problems and applications. The

main goal of this text is to provide valuable tools for roboticists. We present all necessary

basic theoretical notions but with great emphasis on applications and exercises

with real industrial manipulators. This choice permits that once we get the concepts

for these robot mechanics, it will be possible to expand the approach to a great variety

of robotics architectures.

The examples and exercises in this work demonstrate that geometric solutions

based on the screw theory are more suitable for real-time robotics applications compared

to typical numerical iterative algorithms.

In short, this text aims to help students, engineers, scientists, researchers, and

practitioners of robotics who want to develop their advanced projects toward the

application of screw theory methodologies. We will facilitate the comprehension and

access to the topics addressed through convenient examples presented in a clear

graphical structure. This book’s design intends to build a functional bridge between

the mathematical fundamentals, which are extensive, and the practical technological

robotic applications.

Another component of this book is the additional material included to reinforce

the knowledge-making operative of the formulations. Here, we grant access to code,

examples, and simulations for real manipulators, which help many robotics structure

applications, even outside the field of industrial manipulators. The MATLAB® environment,

including Simulink® and Simscape™, is also a valuable supplement for

robotics simulation, permitting students to explore robots’ mechanics interactively.

This book presents a personal visual approach on how to explain the fundamental

concepts of screw theory. The text is more a companion to essential and excellent

textbooks that thoroughly cover the subject’s mathematical foundations. Particularly

influential are those written by Corke; Davidson & Hunt; Lynch & Park; Mason;

Murray, Li, and Sastry; Selig; Siciliano and Khatib. All these have contributed to

modern robotics in an inspirational way.

In reading this book, we hope that many will feel the enthusiasm about robotics’

technological and social prospects, with the elegance of the underlying screw theory

for developing effective and efficient robotics algorithms, solutions, and

applications.