Screw Theory in Robotics: An Illustrated and Practicable Introduction to Modern Mechanics

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgments
List of Abbreviations
Author
Introduction
Chapter 1: Introduction
	1.1 Motivation
		1.1.1 A Historical Quest!
		1.1.2 A Hundred Years of Menacing Robots!
		1.1.3 A Century of Helping Robots!
		1.1.4 And Only 50 Years of Commercial Robots!
		1.1.5 The Mathematical Complexity of Robotics
		1.1.6 Here Comes Screw Theory in Robotics
		1.1.7 The Future of Robotics
	1.2 About This Book
	1.3 Preview
		1.3.1 Outline
		1.3.2 Chapter 1: Introduction
		1.3.3 Chapter 2: Mathematical Tools
		1.3.4 Chapter 3: Forward Kinematics
		1.3.5 Chapter 4: Inverse Kinematics
		1.3.6 Chapter 5: Differential Kinematics
		1.3.7 Chapter 6: Inverse Dynamics
		1.3.8 Chapter 7: Trajectory Generation
		1.3.9 Chapter 8: Robotics Simulation
		1.3.10 Chapter 9: Conclusions
	1.4 Audience
	Further Reading
	Note
Chapter 2: Mathematical Tools
	2.1 Rigid Body Motion
	2.2 Homogeneous Representation
		2.2.1 Standard Rigid Body Motion
		2.2.2 Homogeneous Basic Transformations
		2.2.3 Motion Composition in the SPATIAL “S” Reference System
		2.2.4 Motion Composition with STATIONARY and MOBILE Coordinate Systems
		2.2.5 Geometrical Interpretation
		2.2.6 Exercise: Homogeneous Rotation
		2.2.7 Exercise: Homogeneous Rotation Plus Translation
	2.3 Exponential Representation
		2.3.1 Modern Rigid Body Motion
		2.3.2 Screw Rotation (Orientation)
		2.3.3 Rigid Body Motion TWIST
		2.3.4 Rigid Body Force WRENCH
		2.3.5 Exponential Coordinates for a SCREW Motion
		2.3.6 Exercise: Exponential Rotation
		2.3.7 Exercise: Exponential Rotation Plus Translation
	2.4 Summary
	Notes
Chapter 3: Forward Kinematics
	3.1 Problem Statement in Robotics
		3.1.1 Kinematics Concept
		3.1.2 Kinematics Mathematical Approach
		3.1.3 Forward Kinematics (FK)
	3.2 Denavit–Hartenberg Convention (DH)
		3.2.1 Kinematics Treatment
		3.2.2 DH FK Homogeneous Matrix Product
		3.2.3 Puma Robots (e.g., ABB IRB120)
	3.3 Product of Exponentials Formulation
		3.3.1 A New Kinematics Treatment
		3.3.2 General Solution to Forward Kinematics
		3.3.3 Puma Robots (e.g., ABB IRB120)
		3.3.4 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		3.3.5 Bending Backwards Robots (e.g., ABB IRB1600)
		3.3.6 Gantry Robots (e.g., ABB IRB6620LX)
		3.3.7 Scara Robots (e.g., ABB IRB910SC)
		3.3.8 Collaborative Robots (e.g., UNIVERSAL UR16e)
		3.3.9 Redundant Robots (e.g., KUKA IIWA)
		3.3.10 Many DoF Robots (e.g., RH0 UC3M Humanoid)
	3.4 Summary
	Notes
Chapter 4: Inverse Kinematics
	4.1 Problem Statement in Robotics and Analytical Difficulty
		4.1.1 Kinematics Concept
		4.1.2 Inverse Kinematics Mathematical Approach
		4.1.3 Analytical Difficulty to Solve Inverse Kinematics
	4.2 Numeric vs. Geometric Solutions
		4.2.1 A Numeric Approach to Solve Inverse Kinematics
		4.2.2 An Example of a Numeric Algorithm
		4.2.3 A Geometric Approach to Solve Inverse Kinematics
		4.2.4 An Example of a Geometric Algorithm
		4.2.5 Puma Robot Inverse Kinematics Algorithms
	4.3 Canonical Subproblems for Inverse Kinematics
		4.3.1 A Key Idea to Solve Inverse Kinematics
		4.3.2 Paden–Kahan Subproblem One (PK1) – One Rotation
			4.3.2.1 ROTATION around ONE Single AXIS Applied to a POINT
			4.3.2.2 PK1 Subproblem Simplification
		4.3.3 Paden–Kahan Subproblem Two (PK2) – Two Crossing Rotations
			4.3.3.1 ROTATION around TWO Subsequent CROSSING AXES Applied to a POINT
		4.3.4 Paden–Kahan Subproblem Three (PK3) – Rotation to a Distance
			4.3.4.1 ROTATION at a Given DISTANCE Applied to a POINT
			4.3.4.2 PK3 Subproblem Simplification
		4.3.5 Pardos–Gotor Subproblem One (PG1) – One Translation
			4.3.5.1 TRANSLATION along a SINGLE AXIS Applied to a POINT
			4.3.5.2 PG1 Extension TRANSLATION along a SINGLE AXIS Applied to a PLANE
		4.3.6 Pardos-Gotor Subproblem Two (PG2) – Two Crossing Translations
			4.3.6.1 TRANSLATION along Two Subsequent CROSSING AXES Applied to a POINT
		4.3.7 Pardos-Gotor Subproblem Three (PG3) – Translation to a Distance
			4.3.7.1 TRANSLATION to a Given DISTANCE Applied to a POINT
		4.3.8 Pardos-Gotor Subproblem Four (PG4) – Two Parallel Rotations
			4.3.8.1 ROTATION around TWO Subsequent PARALLEL AXES Applied to a POINT
			4.3.8.2 PG4 Extension ROTATION around TWO PARALLEL AXES Applied to a LINE
		4.3.9 Pardos-Gotor Subproblem Five (PG5) – Rotation of a Line or Plane
			4.3.9.1 ROTATION around ONE Single AXIS Applied to a Perpendicular LINE or PLANE
		4.3.10 Pardos-Gotor Subproblem Six (PG6) – Two Skewed Rotations
			4.3.10.1 ROTATION around TWO Subsequent SKEW AXES Applied to a POINT
		4.3.11 Pardos-Gotor Subproblem Seven (PG7) – Three Rotations to a Point
			4.3.11.1 ROTATION around THREE Subsequent AXES (ONE SKEW + TWO PARALLEL) Applied to a POINT
		4.3.12 Pardos-Gotor Subproblem Eight (PG8) – Three Rotations to A Pose
			4.3.12.1 ROTATION around THREE Subsequent PARALLEL AXES Applied to a POSE (Position Plus Orientation) or COORDINATE SYSTEM
	4.4 Product of Exponentials Approach
		4.4.1 General Solution to Inverse Kinematics
		4.4.2 Puma Robots (e.g., ABB IRB120)
			4.4.2.1 Inverse Kinematics Puma Robot ABB IRB120 Problem Definition
			4.4.2.2 First Algorithm for ABB IRB120 IK “PK3+PK2+PK2+PK1”
			4.4.2.3 Second Algorithm for ABB IRB120 IK “PG7+PK2+PK1”
			4.4.2.4 Third Algorithm for ABB IRB120 IK “PG5+PG4+PK2+PK1”
			4.4.2.5 Fourth Algorithm for ABB IRB120 IK “PG5+PG4+PG6+PK1”
			4.4.2.6 Comparison between the Four Algorithms for ABB IRB120 IK
			4.4.2.7 Comment on the Implementation of the Algorithms for ABB IRB120 IK
			4.4.2.8 Performance Contrast for Both Numeric and Geometric ABB IRB120 IK Algorithms
			4.4.2.9 RST Robotics System Toolbox™
			4.4.2.10 ST24R Screw Theory Toolbox for Robotics
		4.4.3 Puma Robots (e.g., ABB IRB120) “Tool-Up.”
			4.4.3.1 Inverse Kinematics PUMA ABB IRB120 “Tool-Up” Problem Definition
			4.4.3.2 First Algorithm for ABB IRB120 “Tool-Up” IK “PG7+PG6+PK1”
		4.4.4 Bending Backwards Robots (e.g., ABB IRB1600)
			4.4.4.1 Inverse Kinematics ABB IRB1600 Problem Definition
			4.4.4.2 First Algorithm for ABB IRB1600 IK “PG7+PG6+PK1”
		4.4.5 Gantry Robots (e.g., ABB IRB6620LX)
			4.4.5.1 Inverse Kinematics ABB IRB6620LX Problem Definition
			4.4.5.2 First Algorithm for ABB IRB6620LX IK “PG1+PG4+PG6+PK1”
		4.4.6 Scara Robots (e.g., ABB IRB910SC)
			4.4.6.1 Inverse Kinematics ABB IRB910SC Problem Definition
			4.4.6.2 First Algorithm for ABB IRB910SC IK “PG1+PG4+PK1”
			4.4.6.3 Second Algorithm for ABB IRB910SC IK “PG1+PK3+PK1+PK1”
			4.4.6.4 Comments on the SCARA Robot (ABB IRB910SC) IK Implementation
		4.4.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
			4.4.7.1 Inverse Kinematics UNIVERSAL UR16e Problem Definition
			4.4.7.2 First Algorithm for UNIVERSAL UR16e IK “PG5+PG3+PK1+PG8”
			4.4.7.3 Comments on the UNIVERSAL UR16e IK Complete Solution Implementation
		4.4.8 Redundant Robots (e.g., KUKA IIWA)
			4.4.8.1 Inverse Kinematics KUKA IIWA Problem Definition
			4.4.8.2 First Algorithm for KUKA IIWA IK “PK1+PK3+PK2+PK2+PK2+PK1”
			4.4.8.3 Comments on the KUKA IIWA IK Complete Solution Implementation
		4.4.9 Many DoF Robots (e.g., RH0 UC3M Humanoid)
	4.5 Summary
	Notes
Chapter 5: Differential Kinematics
	5.1 Problem Statement in Robotics
	5.2 The Analytic Jacobian
		5.2.1 A Traditional Description
		5.2.2 Analytic Jacobian to Forward Differential Kinematics
		5.2.3 Analytic Jacobian for Inverse Differential Kinematics
		5.2.4 Scara Robot (e.g., ABB IRB910SC)
			5.2.4.1 Forward Differential Kinematics with Analytic Jacobian
			5.2.4.2 Inverse Differential Kinematics with Analytic Jacobian
		5.2.5 Puma Robot (e.g., ABB IRB120)
	5.3 The Geometric Jacobian
		5.3.1 Robot Spatial Geometric Jacobian
		5.3.2 The Classical Adjoint Transformation (Ad)
		5.3.3 Twist Velocity Concept
		5.3.4 Trajectory Generation
		5.3.5 Robot Tool Geometric Jacobian
		5.3.6 Link Spatial and Link Tool Geometric Jacobian
		5.3.7 The New Adjoint Transformation ( A ij)
		5.3.8 General Solution to Differential Kinematics
			5.3.8.1 The Kinematics Mapping
			5.3.8.2 The Geometric Forward Differential Kinematics
			5.3.8.3 The Geometric Inverse Differential Kinematics
		5.3.9 Puma Robots (e.g., ABB IRB120)
			5.3.9.1 Geometric Jacobian by Definition
			5.3.9.2 Forward Differential Kinematics with Geometric Jacobian
			5.3.9.3 Inverse Differential Kinematics with Geometric Jacobian
		5.3.10 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		5.3.11 Bending Backwards Robots (e.g., ABB IRB1600)
		5.3.12 Gantry Robots (e.g., ABB IRB6620LX)
		5.3.13 Scara Robots (e.g., ABB IRB910SC)
			5.3.13.1 Geometric Jacobian by Inspection
			5.3.13.2 Geometric Jacobian by Definition
			5.3.13.3 Forward Differential Kinematics with Geometric Jacobian
			5.3.13.4 Inverse Differential Kinematics with Geometric Jacobian
		5.3.14 Collaborative Robots (e.g., UNIVERSAL UR16e)
		5.3.15 Redundant Robots (e.g., KUKA IIWA)
	5.4 Summary
	Notes
Chapter 6: Inverse Dynamics
	6.1 Problem Statement in Robotics
	6.2 The Lagrange Characterization
		6.2.1 General Non-Recursive Solution to Inverse Dynamics
		6.2.2 Puma Robots (e.g., ABB IRB120)
		6.2.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		6.2.4 Bending Backwards Robots (e.g., ABB IRB1600)
		6.2.5 Gantry Robots (e.g., ABB IRB6620LX)
		6.2.6 Scara Robots (e.g., ABB IRB910SC)
		6.2.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		6.2.8 Redundant Robots (e.g., KUKA IIWA)
	6.3 Robot Dynamics Control
		6.3.1 Robotics Control in the Joint Space
		6.3.2 Robotics Control in the Task Space
	6.4 Spatial Vector Algebra
		6.4.1 Coordinate Transforms
		6.4.2 Mechanics of a Constrained Rigid Body System
	6.5 The Newton–Euler Equations
		6.5.1 General Recursive Solution to Inverse Dynamics RNEA with POE
		6.5.2 Puma Robots (e.g., ABB IRB120)
		6.5.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		6.5.4 Bending Backwards Robots (e.g., ABB IRB1600)
		6.5.5 Gantry Robots (e.g., ABB IRB6620LX)
		6.5.6 Scara Robots (e.g., ABB IRB910SC)
		6.5.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		6.5.8 Redundant Robots (e.g., KUKA IIWA)
	6.6 Summary
	Notes
Chapter 7: Trajectory Generation
	7.1 Concepts and Definitions
		7.1.1 Point-to-Point Position Straight-line Trajectories
		7.1.2 Trapezoidal Position Trajectory
		7.1.3 Polynomial Position Trajectory
		7.1.4 Spline Position Trajectory
		7.1.5 Rotation Motion Trajectory
		7.1.6 Trajectory Tracking and Control
	7.2 Trajectory Planning
		7.2.1 General Solution to Trajectory Generation
		7.2.2 Puma Robots (e.g., ABB IRB120)
		7.2.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		7.2.4 Bending Backwards Robots (e.g., ABB IRB1600)
		7.2.5 Gantry Robots (e.g., ABB IRB6620LX)
		7.2.6 Scara Robots (e.g., ABB IRB910SC)
		7.2.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		7.2.8 Redundant Robots (e.g., KUKA IIWA)
	7.3 Summary
	Notes
Chapter 8: Robotics Simulation
	8.1 Robotics Simulation
		8.1.1 Why Code in MATLAB ® ?
	8.2 Screw Theory Toolbox for Robotics (ST24R)
	8.3 Forward Kinematics Simulations
		8.3.1 General Solution to Forward Kinematics Simulation
		8.3.2 Puma Robots (e.g., ABB IRB120)
		8.3.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		8.3.4 Bending Backwards Robots (e.g., ABB IRB1600)
		8.3.5 Gantry Robots (e.g., ABB IRB6620LX)
		8.3.6 Scara Robots (e.g., ABB IRB910SC)
		8.3.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		8.3.8 Redundant Robots (e.g., KUKA IIWA)
	8.4 Inverse Kinematics Simulations
		8.4.1 General Solution to Inverse Kinematics Simulation
		8.4.2 Puma Robots (e.g., ABB IRB120)
		8.4.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		8.4.4 Bending Backwards Robots (e.g., ABB IRB1600)
		8.4.5 Gantry Robots (e.g., ABB IRB6620LX)
		8.4.6 Scara Robots (e.g., ABB IRB910SC)
		8.4.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		8.4.8 Redundant Robots (e.g., KUKA IIWA)
	8.5 Differential Kinematics Simulations
		8.5.1 General Solution to Differential Kinematics Simulation
		8.5.2 Puma Robots (e.g., ABB IRB120)
		8.5.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		8.5.4 Bending Backwards Robots (e.g., ABB IRB1600)
		8.5.5 Gantry Robots (e.g., ABB IRB6620LX)
		8.5.6 Scara Robots (e.g., ABB IRB910SC)
		8.5.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		8.5.8 Redundant Robots (e.g., KUKA IIWA)
	8.6 Inverse Dynamics Simulations
		8.6.1 General Solution to ID Simulation
		8.6.2 Puma Robots (e.g., ABB IRB120)
		8.6.3 Puma Robots (e.g., ABB IRB120) “Tool-Up”
		8.6.4 Bending Backwards Robots (e.g., ABB IRB1600)
		8.6.5 Gantry Robots (e.g., ABB IRB6620LX)
		8.6.6 Scara Robots (e.g., ABB IRB910SC)
		8.6.7 Collaborative Robots (e.g., UNIVERSAL UR16e)
		8.6.8 Redundant Robots (e.g., KUKA IIWA)
	8.7 Summary
	Notes
Chapter 9: Conclusions
	9.1 Summary
		9.1.1 Introduction
		9.1.2 Mathematical Tools
		9.1.3 Forward Kinematics
		9.1.4 Inverse Kinematics
		9.1.5 Differential Kinematics
		9.1.6 Inverse Dynamics
		9.1.7 Trajectory Generation
		9.1.8 Robotics Simulation
	9.2 Future Prospects
Epigram
References
Index