机器人学几何基础与理论探索

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"《机器人几何基础》是一本深入探讨机器人学基本理论的著作，涵盖了机器人运动学、动力学、动量和能量等关键概念。该书由RAMIKHALIL撰写，作者提供了联系方式以便读者交流。此外，这本书还属于计算机科学系列的学术专著，旨在深入研究机器人学的几何原理及其在实际应用中的表现。" 本文将详细阐述《机器人几何基础》中涉及的核心知识点： 1. **机器人运动学**：这是研究机器人关节和连杆如何运动的学科。它包括笛卡尔坐标系和关节坐标系下的运动描述，以及正向运动学（如何从关节变量推导出末端执行器的位置和姿态）和逆向运动学（如何从期望的末端执行器位置反推出所需的关节角度）。 2. **动力学**：动力学是分析机器人受力和加速度之间关系的领域。它包括牛顿-欧拉方程，用于计算每个连杆的力和力矩，以及雅可比矩阵对力的线性近似。机器人动力学研究还包括刚体动力学和振动分析，这对于设计稳定和高效的控制系统至关重要。 3. **动量与能量**：在机器人系统中，动量和能量是衡量其动态行为的关键参数。动量涉及物体在运动中的惯性和相互作用，而能量则关注系统的总机械能（动能和势能）及其转换。了解这些概念有助于优化机器人的能效和动态性能。 4. **机器人几何**：这一部分可能涉及到机器人结构的几何特性，如连杆长度、关节角度以及它们如何影响机器人的工作空间和可达性。这包括对机器人臂的运动范围、奇异配置（导致无穷大或不可解情况的特定关节配置）的研究，以及通过几何方法求解运动学问题。 5. **计算机科学背景**：作为计算机科学系列的一部分，本书可能也探讨了与算法、数据结构和计算理论相关的方面，特别是在解决机器人学问题时的应用。例如，可能涉及到数值计算、优化算法和计算几何。 6. **软件配置管理与编程方法**：书中提及的其他书籍主题，如软件配置管理，可能与机器人控制系统的开发和维护有关。编程方法则可能涉及如何高效地编写和验证机器人控制软件。《机器人几何基础》不仅深入探讨了机器人学的基础理论，还与计算机科学的多个分支相结合，提供了全面理解机器人技术的视角。通过学习这些概念，读者可以掌握构建、控制和优化机器人系统所需的关键知识。

Contents xv

13.4 Lagrangian Dynamics of Robots 300

13.4.1 Euler Lagrange Equations 301

13.4.2 Derivatives of the Generalised Inertia Matrix 303

13.4.3 Small Oscillations 304

13.5 Hamiltonian Dynamics of Robots 306

13.6 Simplification of the Equations of Motion 309

13.6.1 Decoupling by Design 309

13.6.2 Ignorable Coordinates 312

13.6.3 Decoupling by Coordinate Transformation 316

14 Constrained Dynamics 321

14.1 Trees and Stars 321

14.1.1 Dynamics of Tree and Star Structures 323

14.1.2 Link Velocities and Accelerations 324

14.1.3 Recursive Dynamics for Trees and Stars 325

14.2 Serial Robots with End-Effector Constraints 327

14.2.1 Holonomic Constraints 327

14.2.2 Constrained Dynamics of a Rigid Body 330

14.2.3 Constrained Serial Robots 331

14.3 Constrained Trees and Stars 333

14.3.1 Systems of Freedom 333

14.3.2 Parallel Mechanisms 334

14.4 Dynamics of Planar 4-Bars 336

14.5 Biped Walking 340

14.6 The Stewart Platform 343

15 Differential Geometry 349

15.1 Metrics, Connections and Geodesies 349

15.2 Mobility of Overconstrained Mechanisms 355

15.3 Controlling Robots Along Helical Trajectories 360

15.4 Hybrid Control 363

15.4.1 What is Hybrid Control? 363

15.4.2 Constraints 364

15.4.3 Projection Operators 365

15.4.4 The Second Fundamental Form 369

References 373

Index 383

2 1. Introduction

carried by the robot's end-effector is more often than not a rigid body. Standard

analysis of the kinematics, dynamics and control of these robots all reflect this

rigid body approach.

In essence, this book is concerned with the geometry of the group of rigid

body motions SE{3), and its applications robotics. Before embarking on this

we look at a little history and background.

1.2 Robots and Mechanisms

Although modern industrial robots are only a few decades old, their antecedents

are mechanisms and linkages, which have centuries of history. Indeed, examples

like cranes and other lifting devices are among the oldest machines used by

humans.

Such machines consist of rigid links connected by joints. The inspiration for

these devices may well have come from animal skeletons. The slow but steady

development and innovation of such mechanisms has continued throughout his-

tory and in all parts of the world. However, the industrial revolution in Europe

in the late eighteenth century created a huge demand for these devices, a de-

mand that was met in excess.

One of the most pressing problems was how to turn the reciprocating motion

generated by piston engines into rotary motion. Many people, including Watt,

proposed approximate straight-line linkages. However, it was not until 1864

that a French naval officer named Peaucellier invented an eight-bar mechanism

that does the job exactly. By that time, machining methods had become more

accurate and lubrication technology had improved, so the practical importance

of the discovery had disappeared. However, mathematicians were intrigued by

the machine since the design was easily adapted to mechanically invert curves.

Hence, it could be used to study 'inversion geometry'. It also begged the ques-

tion. Just which curves can be traced by linkages?

In 1876, Kempe, the London solicitor and amateur mathematician, proved

that all algebraic curves can be traced by mechanisms. Later, Koenigs proved a

similar theorem for curves in space. At around this time, many mathematicians

were interested in the theory of mechanisms—Chebychev, Schonflies and Dar-

boux, to name but a few. By far the most intensively studied machine was and

probably remains the 4-bar mechanism. The device is ubiquitous in mechanical

engineering, as it is an extraordinary design element. It has been used for ev-

erything from door hinges to the tilting mechanism for high speed trains. Even

so,

there still remain some unanswered questions concerning the geometrical

capabilities of this mechanism.

Mathematicians were also interested in more general problems. Clifford de-

veloped geometric algebras, modelled on Hamilton's quaternions.

Around the turn of the twentieth century, Ball developed screw theory, which

dealt with infinitesimal rigid body motions and was mainly used to look at

problems concerning statics and dynamics.

1.2 Robots and Mechanisms 3

A little later, Study looked at the geometry of the set of all finite rigid body

motions. For this work he invented dual quaternions, which could represent

translations as well as rotations.

After the first world war, mathematicians seem to have turned away from

the study of mechanisms. There have been notable exceptions; see, for example,

Thurston and Weeks

[120].

However, it would be true to say that the subject

remains a backwater, away from the main development of modern mathematics.

Mechanical engineers also neglected mechanisms during this period. Although

the machines remained of great practical importance, academic engineers were

more interested in other areas. The notable exception here was the Moscow

school, led by Artobolevskii. In particular, Dimentberg was using screw theory

in 1948 to analyse closed kinematic chains.

In the 1950s, Freudenstein, at Columbia University, began a revival of the

subject in the west. More or less simultaneously Hunt and Phillips, at Monash

and Sydney respectively, championed Ball's screw theory for analysing spatial

mechanisms.

Perhaps the main difference between robots and mechanisms is that mecha-

nisms are usually designed for particular functions and hence usually have only

a few, normally just one, degree of freedom. Robots are supposed to be general

purpose machines with, consequently, many degrees of freedom. The geometri-

cal analyses of mechanisms and robots are to all intents and purposes the same.

Geometry here refers to both kinematics and dynamics. Kinematics studies the

possible movements the machine can make, irrespective of any forces or inertias,

while dynamics looks at how the machine will move, taking forces and inertias

into account.

In 1954, Devol patented what he called a "programmed articulated trans-

fer device". This machine derived from telechirs used in the nuclear industry

for the remote handling of radioactive materials and computer (numerically)

controlled machine tools. Engleberger, a Columbia University student, realised

that Devol's machine was essentially a robot. In 1956, he bought the patent and

set up Unimation. The first industrial robot was installed by General Motors

in 1961. In 1968, Kawasaki bought a licence from Unimation to manufacture

robots in Japan. In 1978, Unimation introduced the PUMA, an acronym for

'programmable universal machine for assembly'. This robot was the result of

a study for General Motors on automating assembly tasks. In 1979, SCARA

(selective compliance assembly robot arm) was introduced in Japan for assem-

bly of printed circuit boards. The mechanisms community was well placed to

contribute to the study and design of the new machines.

In the 1980s, the subject of robotics attracted a lot of funding and hence

the attention of other researchers. In particular, electrical engineers became

interested in the problem of controlling robots. Computer scientists saw robots

as vehicles for testing their ideas about artificial intelligence. The recession

of the late 1980s meant a decline in the funding for robotics. However, there

remained a substantial interest in the subject in many disciplines.

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