THEORY OF APPLIED ROBOTICS

1 Introduction 1
1.1 Historic al Development 1
1.2 Components and Mechanisms of a Robotic System 2
1.2.1 Link 3
1.2.2 Joint 3
1.2.3 Manipulator 5
1.2.4 Wrist  5
1.2.5 End-effector  6
1.2.6 Actuators 7
1.2.7 Sensors  7
1.2.8 Controller  7
1.3 Robot Classifications  7
1.3.1 Geometry  8
1.3.2 Worksp ace 11
1.3.3 Actuation 12
1.3.4 Control 13
1.3.5 Appli cation 13
1.4 Introduction to Robot 's Kinematics, Dynamics, and Control 14
1.4.1 Triad 15
1.4.2 Unit Vectors 16
1.4.3 Reference Frame and Coordinate System 16
1.4.4 Vector Function  19
1.5 Problems of Robot Dynamics  19
1.6 Preview of Covered Topics  21
1.7 Robots as Multi-disciplinary Machines 22
1.8 Summary 22

Exercises 25
I Kinematics
2 Rotation Kinematics
2.1 Rotation About Global Cartesian Axes .
2.2 Successive Rotation About Global Cartesian Axes
2.3 Global Roll-Pitch-Yaw Angles .
2.4 Rotation About Local Cartesian Axes .
2.5 Successive Rotation About Local Cartesian Axes
2.6 Euler Angles 48
2.7 Local Roll-Pitch-Yaw Angles 59
2.8 Local Axes Rotation Versus Global Axes Rotation 61
2.9 General Transformation . . . . . . 63
2.10 Act ive and Passive Transformation 71
2.11 Summary 73


3 Orientation Kinematics 81
3.1 Axis-angle Rotation 81
3.2 *Euler Parameters  88
3.3 *Determination of Euler Parameters 96
3.4 *Quaternions  99
3.5 *Spinors and Rot ators  102
3.6 * Problems in Representing Rotations  105
3.6.1 Rotation matrix 105
3.6.2 Angle-axis  106
3.6.3 Euler angles 107
3.6.4 Quat ernion  109
3.6.5 Euler parameters 111
3.7 *Composition and Decomposition of Rot ations 113
3.8 Summary 118

Exercises 119
4 Motion Kinematics 127
4.1 Rigid Body Motion  127
4.2 Homogeneous Transformation  131
4.3 Inverse Homogeneous Transformation 139
4.4 Compound Homogeneous Transformation 145
4.5 * Screw Coordinat es 154
4.6 * Inverse Screw  169
4.7 *Compound Screw Transformation 170
4.8 *The Plucker Line Coordinate  173
4.9 *The Geometry of Plane and Line 180
4.9.1 *Moment  180
4.9.2 *Angle and Distance  181
4.9.3 *Plane and Line  181
4.10 *Screw and Plucker Coordinate 186
4.11 Summary 188

Exercises 191
5 Forward Kinematics 199
5.1 Denavit-Hartenberg Notation  199
5.2 Transformation Between Two Adjacent Coordinate Frames 208
5.3 Forward Position Kinemat ics of Robots 226
5.4 *Coordinate Transformation Using Screws 242
5.5 *Sheth Method 247
5.6 Summary 253
Exercises 255
6 Inverse Kinematics 263
6.1 Decoupling Technique  263
6.2 Inverse Transformation Technique 270
6.3 It erat ive Technique 282
6.4 *Comparison of the Inverse Kinematics Techniques 287
6.4.1 *Existence and Uniqueness of Solution  287
6.4.2 * Inverse Kinemati cs Techniques 288
6.5 *Singular Configuration 289
6.6 Summary 291

Exercises 293
7 Angular Velocity 297
7.1 Angular Velocity Vector and Matrix 297
7.2 Time Derivative and Coordinat e Frames 310
7.3 Rigid Body Velocity  320
7.4 Velocity Transformation Matrix. 325
7.5 Derivative of a Homogeneous Transformation Matrix 330
7.6 Summary 336

Exercises 339
8 Velocity Kinematics 343
8.1 Rigid Link Velocity  343
8.2 Forward Velocity Kinematic s and the Jacobian Matrix 346
8.3 Jacobian Generating Vector s 351
8.4 Inverse Velocity Kinematics 363
8.5 Summary

Exercises
9 Numerical Methods in Kinematics
9.1 Linear Algebraic Equations
9.2 Matrix Inversion
9.3 Nonlinear Algebraic Equations
9.4 * Jacobian Matrix From Link Transformation Matrices
9.5 *Kinematics Recursive Equations
9.5.1 *Recursive Velocity in Base Frame
9.5.2 *Recursive Acceleration in Base Frame
9.6 Summary

Exercises
II Dynamics
10 Acceleration Kinematics
10.1 Angular Acceleration Vector and Matrix
10.2 Rigid Body Acceleration
10.3 Acceleration Transformation Matrix
10.4 Forward Accelerat ion Kinemat ics
10.5 * Inverse Acceleration Kinematics
10.6 Summary
Exercises
11 Motion Dynamics
11.1 Force and Moment
11.2 Rigid Body Translational Kinet ics
11.3 Rigid Body Rotational Kinetics
11.4 Mass Moment of Inertia Matrix
11.5 Lagrange's Form of Newton 's Equations of Motion
11.6 Lagrangian Mechanics
11.7 Summary

Exercises
12 Robot Dynamics
12.1 Rigid Link Recursive Acceleration
12.2 Rigid Link Newton-Euler Dynamics
12.3 Recursive Newton-Euler Dynamics
12.4 Robot Lagrange Dynamics
12.5 *Lagrange Equations and Link Transformation Matrices
12.6 Robot Statics
12.7 Summary
Exercises

III Control
13 Path Planning
13.1 Joint Cubic Path
13.2 Higher Polynomial Path
13.3 Non-Polynomial Path Planning
13.4 Manipulator Motion by Joint Path
13.5 Cartesian Pat h
13.6 *Rotat ional Path
13.7 Manipulator Motion by End-Effector Path
13.8 Summary

14 *Time Optimal Control
14.1 *Minimum Time and Bang-Bang Contro l
14.2 *Floating Time Method .
14.3 *Time-Opt imal Contro l for Robots
14.4 Summary

Exercises .
15 Control Techniques
15.1 Open and Closed-Loop Control
15.2 Computed Torque Contro l
15.3 Linear Control Technique
15.3.1 Proportional Contro l
15.3.2 Integral Control
15.3.3 Derivat ive Control
15.4 Sensing and Control
15.4.1 Position Sensors
15.4.2 Speed Sensors
15.4.3 Acceleration Sensors
15.5 Summary

Exercises .
References
A Global Frame Triple Rotation
B Local Frame Triple Rotation
C Principal Central Screws Triple Combination
D Trigonometric Formula

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