Table of Contents

1.1 Mathematical Modeling of Robots

1.1.1 Symbolic Representation of Robot Manipulators

1.1.2 The Configuration Space

1.1.3 The State Space

1.1.4 The Workspace

1.2 Robots as Mechanical Devices

1.2.1 Classification of Robotic Manipulators

1.2.2 Robotic Systems

1.2.3 Accuracy and Repeatability

1.2.4 Wrists and End Effectors

1.3 Common Kinematic Arrangements

1.3.1 Articulated Manipulator (RRR)

1.3.2 Spherical Manipulator (RRP)

1.3.3 SCARA Manipulator (RRP)

1.3.4 Cylindrical Manipulator (RPP)

1.3.5 Cartesian Manipulator (PPP)

1.3.6 Parallel Manipulator 18

1.4 Outline of the Text

1.4.1 Manipulator Arms

1.4.2 Underactuated and Mobile Robots

Problems

Notes and References

I The Geometry of Robots

2 Rigid Motions 

2.1 Representing Positions

2.2 Representing Rotations

2.2.1 Rotation in the Plane

2.2.2 Rotations in Three Dimensions

2.3 Rotational Transformations

2.4 Composition of Rotations

2.4.1 Rotation with Respect to the Current Frame

2.4.2 Rotation with Respect to the Fixed Frame

2.4.3 Rules for Composition of Rotations

2.5 Parameterizations of Rotations

2.5.1 Euler Angles

2.5.2 Roll, Pitch, Yaw Angles

2.5.3 Axis-Angle Representation

2.5.4 Exponential Coordinates

2.6 Rigid Motions

2.6.1 Homogeneous Transformations

2.6.2 Exponential Coordinates for General Rigid Motions

2.7 Chapter Summary

Problems

Notes and References

3.1 Kinematic Chains

3.2 The Denavit-Hartenberg Convention

3.2.1 Existence and Uniqueness

3.2.2 Assigning the Coordinate Frames

3.3 Examples

3.3.1 Planar Elbow Manipulator

3.3.2 Three-Link Cylindrical Robot

3.3.3 The Spherical Wrist

3.3.4 Cylindrical Manipulator with Spherical Wrist

3.3.5 Stanford Manipulator

3.3.6 SCARA Manipulator

3.4 Chapter Summary

Problems

Notes and References

4.1 Angular Velocity: The Fixed Axis Case

4.2 Skew-Symmetric Matrices

4.2.1 Properties of Skew-Symmetric Matrices

4.2.2 The Derivative of a Rotation Matrix

4.3 Angular Velocity: The General Case

4.4 Addition of Angular Velocities

4.5 Linear Velocity of a Point Attached to a Moving Frame

4.6 Derivation of the Jacobian

4.6.1 Angular Velocity

4.6.2 Linear Velocity

4.6.3 Combining the Linear and Angular Velocity Jacobians

4.7 The Tool Velocity

4.8 The Analytical Jacobian

4.9 Singularities

4.9.1 Decoupling of Singularities

4.9.2 Wrist Singularities

4.9.3 Arm Singularities

4.10 Static Force/Torque Relationships

4.11 Inverse Velocity and Acceleration

4.12 Manipulability

4.13 Chapter Summary

Problems

Notes and References

5.1 The General Inverse Kinematics Problem

5.2 Kinematic Decoupling

5.3 Inverse Position: A Geometric Approach

5.3.1 Spherical Configuration

5.3.2 Articulated Configuration

5.4 Inverse Orientation

5.5 Numerical Inverse Kinematics

5.6 Chapter Summary

Problems

Notes and References

II Dynamics and Motion Planning 

6.1 The Euler-Lagrange Equations

6.1.1 Motivation

6.1.2 Holonomic Constraints and Virtual Work

6.1.3 D’Alembert’s Principle

6.2 Kinetic and Potential Energy

6.2.1 The Inertia Tensor

6.2.2 Kinetic Energy for an n-Link Robot

6.2.3 Potential Energy for an n-Link Robot

6.3 Equations of Motion

6.4 Some Common Configurations

6.5 Properties of Robot Dynamic Equations

6.5.1 Skew Symmetry and Passivity

6.5.2 Bounds on the Inertia Matrix

6.5.3 Linearity in the Parameters

6.6 Newton-Euler Formulation

6.6.1 Planar Elbow Manipulator Revisited

6.7 Chapter Summary

Problems

Notes and References

7.1 The Configuration Space

7.1.1 Representing the Configuration Space

7.1.2 Configuration Space Obstacles

7.1.3 Paths in the Configuration Space

7.2 Path Planning for Q = ℝ2

7.2.1 The Visibility Graph

7.2.2 The Generalized Voronoi Diagram

7.2.3 Trapezoidal Decompositions

7.3 Artificial Potential Fields

7.3.1 Artificial Potential Fields for Q = ℝn 

7.3.2 Potential Fields for Q ≠ ℝn 

7.4 Sampling-Based Methods

7.4.1 Probabilistic Roadmaps (PRM)

7.4.2 Rapidly-Exploring Random Trees (RRTs)

7.5 Trajectory Planning

7.5.1 Trajectories for Point-to-Point Motion

7.5.2 Trajectories for Paths Specified by Via Points

7.6 Chapter Summary

Problems

Notes and References

8.1 Introduction

8.2 Actuator Dynamics

8.3 Load Dynamics

8.4 Independent Joint Model

8.5 PID Control

8.6 Feedforward Control

8.6.1 Trajectory Tracking

8.6.2 The Method of Computed Torque

8.7 Drive-Train Dynamics

8.8 State Space Design

8.8.1 State Feedback Control

8.8.2 Observers

8.9 Chapter Summary

Problems

Notes and References

9.1 Introduction

9.2 PD Control Revisited

9.3 Inverse Dynamics

9.3.1 Joint Space Inverse Dynamics

9.3.2 Task Space Inverse Dynamics

9.3.3 Robust Inverse Dynamics

9.3.4 Adaptive Inverse Dynamics

9.4 Passivity-Based Control

9.4.1 Passivity-Based Robust Control

9.4.2 Passivity-Based Adaptive Control

9.5 Torque Optimization

9.6 Chapter Summary

Problems

Notes and References

10.1 Coordinate Frames and Constraints

10.1.1 Reciprocal Bases

10.1.2 Natural and Artificial Constraints

10.2 Network Models and Impedance

10.2.1 Impedance Operators

10.2.2 Classification of Impedance Operators

10.2.3 Thévenin and Norton Equivalents

10.3 Task Space Dynamics and Control

10.3.1 Impedance Control

10.3.2 Hybrid Impedance Control

10.4 Chapter Summary

Problems

Notes and References

11.1 Design Considerations

11.1.1 Camera Configuration

11.1.2 Image-Based vs. Position-Based Approaches

11.2 Computer Vision for Vision-Based Control

11.2.1 The Geometry of Image Formation

11.2.2 Image Features

11.3 Camera Motion and the Interaction Matrix

11.4 The Interaction Matrix for Point Features

11.4.1 Velocity Relative to a Moving Frame

11.4.2 Constructing the Interaction Matrix

11.4.3 Properties of the Interaction Matrix for Points

11.4.4 The Interaction Matrix for Multiple Points

11.5 Image-Based Control Laws

11.5.1 Computing Camera Motion

11.5.2 Proportional Control Schemes

11.5.3 Performance of Image-Based Control Systems

11.6 End Effector and Camera Motions

11.7 Partitioned Approaches

11.8 Motion Perceptibility

11.9 Summary

Problems

Notes and References

12.1 Background

12.1.1 Manifolds, Vector Fields, and Distributions

12.1.2 The Frobenius Theorem

12.2 Feedback Linearization

12.3 Single-Input Systems

12.4 Multi-Input Systems

12.5 Chapter Summary

Problems

Notes and References

13.1 Introduction

13.2 Modeling

13.3 Examples of Underactuated Robots

13.3.1 The Cart-Pole System

13.3.2 The Acrobot

13.3.3 The Pendubot

13.3.4 The Reaction-Wheel Pendulum

13.4 Equilibria and Linear Controllability

13.4.1 Linear Controllability

13.5 Partial Feedback Linearization

13.5.1 Collocated Partial Feedback Linearization

13.5.2 Noncollocated Partial Feedback Linearization

13.6 Output Feedback Linearization

13.6.1 Computation of the Zero Dynamics

13.6.2 Virtual Holonomic Constraints

13.7 Passivity-Based Control

13.7.1 The Simple Pendulum

13.7.2 The Reaction-Wheel Pendulum

13.7.3 Swingup and Balance of The Acrobot

13.8 Chapter Summary

Problems

Notes and References

14.1 Nonholonomic Constraints

14.2 Involutivity and Holonomy

14.3 Examples of Nonholonomic Systems

14.4 Dynamic Extension

14.5 Controllability of Driftless Systems

14.6 Motion Planning

14.6.1 Conversion to Chained Forms

14.6.2 Differential Flatness

14.7 Feedback Control of Driftless Systems

14.7.1 Stabilizability

14.7.2 Nonsmooth Control

14.7.3 Trajectory Tracking

14.7.4 Feedback Linearization

14.8 Chapter Summary

Problems

Notes and References

A Trigonometry 

A.1 The Two-Argument Arctangent Function

A.2 Useful Trigonometric Formulas

B Linear Algebra 

B.1 Vectors

B.2 Inner Product Spaces

B.3 Matrices

B.4 Eigenvalues and Eigenvectors

B.5 Differentiation of Vectors

B.6 The Matrix Exponential

B.7 Lie Groups and Lie Algebras

B.8 Matrix Pseudoinverse

B.9 Schur Complement

B.10 Singular Value Decomposition (SVD)

C Lyapunov Stability 

C.1 Continuity and Differentiability

C.2 Vector Fields and Equilibria

C.3 Lyapunov Functions

C.4 Stability Criteria

C.5 Global and Exponential Stability

C.6 Stability of Linear Systems

C.7 LaSalle’s Theorem

C.8 Barbalat’s Lemma

D Optimization 

D.1 Unconstrained Optimization

D.2 Constrained Optimization

E Camera Calibration 

E.1 The Image Plane and the Sensor Array

E.2 Extrinsic Camera Parameters

E.3 Intrinsic Camera Parameters

E.4 Determining the Camera Parameters

Bibliography

Index

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Covering the theoretical fundamentals and latest technological advances in robot kinematics. With so much advancement in technology, from robotics to motion planning, society can implement more powerful and dynamic algorithms than ever before. This in-depth guide educates readers in the fundamentals of robotic and motion control with in-depth control theory and nonlinear system analysis.

New case studies covering topics such as: 

  • Motion-planning, collision avoidance, trajectory optimization, and control of robots
  • Popular topics within the robotics industry and how they apply to various technologies
  • An expanded set of examples, simulations, problems, and case studies
  • Open-ended suggestions for students to apply the knowledge to real-life situations

Authors

Mark W. Spong,

Seth Hutchinson

M. Vidyasagar