## Modeling and Control of Manipulators

2.3.6. Properties of homogeneous transformation matrices . ...... 46 Modeling, identification and control of robots. U5 = 5T6. 4. 4. 4. 6. 5. 5. C5C6. C5S6. S5 0. S6. C6. 0. 0. = ...... [Raibert 77] Raibert M.H., "Analytic equations vs. table look-up for ...
Modeling and Control of Manipulators Part I: Geometric and Kinematic Models

Wisama KHALIL Ecole Centrale de Nantes

Master I EMARO European Master on Advanced Robotics

Master I ASP Master Automatique, et Systèmes de production Master in Control Engineering and Production systems 2010-2011

Table of Contents Chapter 1: Terminology and general definitions....................................................1 1.1. Introduction ............................. .................... .................... .................... ............. 1 1.2. Mechanical components of a robot..................................... ................................ 2 1.3. Definitions ................................................................................ .......................... 4 1.3.1. Joints .......................................................................................... ..................... 4 1.3.2 Degrees of Mobility or Number of degrees of freedom of a body ...... ............ 5 1.3.3. Joint space........................................................................................ ................ 5 1.3.4. Task space.......................................................................... .............................. 5 1.3.5. Redundancy............................................................................ ...........;............. 6 1.3.6. Singular configurations .............................................................. .................... 6 1.4. Choosing the number of degrees of freedom of a robot ................. ................... 7 1.5. Architectures of robot manipulators ................................................... ............... 7 1.6. Characteristics of a robot ........................................................................ ......... 12 1.7. Conclusion........................................................................................... ............. 13 Chapter 2: Transformation matrix between vectors, frames and screws........................................................................................................................ 2.1. Introduction ........................................................................................ .............. 2.2. Homogeneous coordinates................................................................... ............. 2.2.1. Representation of a point ............................................................... ............... 2.2.2. Representation of a direction.............................................. ........................... 2.2.3. Representation of a plane ................................................................. ............. 2.3. Homogeneous transformations [Paul 81]...................................... .................... 2.3.1. Transformation of frames................................................................ .............. 2.3.2. Transformation of vectors ................................................................ ............. 2.3.3. Transformation of planes ................................................................ ............. 2.3.4. Transformation matrix of a pure translation...................................... ........... 2.3.5. Transformation matrices of a rotation about the principle axes ........ ........... 2.3.6. Properties of homogeneous transformation matrices ..................... .............. 2.3.7. Transformation matrix of a rotation about a general vector located at the origin ...................................................................................... ........... 2.3.8. Equivalent angle and axis of a general rotation................................. ............. 2.4. Kinematic screw....................................................................................... ......... 2.4.1. Definition of a screw....................................................................... ............... 2.4.2. Representation of velocity (kinematic screw) ......................................... ...... 2.4.3. Tra