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# Robots and Screw Theory: Applications of kinematics and statics to robotics

## byJ. K. Davidson, K. H. Hunt

### Hardcover | April 7, 2004

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Robots and Screw Theory describes the mathematical foundations, especially geometric, underlying the motions and force-transfers in robots. The principles developed in the book are used in the control of robots and in the design of their major moving parts. The illustrative examples and theexercises in the book are taken principally from robotic machinery used for manufacturing and construction, but the principles apply equally well to miniature robotic devices and to those used in other industries. The comprehensive coverage of the screw and its geometry lead to reciprocal screwsystems for statics and instantaneous kinematics. These screw systems are brought together in a unique way to show many cross-relationships between the force-systems that support a body equivalently to a kinematic serial connection of joints and links. No prior knowledge of screw theory is assumed. The reader is introduced to the screw with a simple planar example yet most of the book applies to robots that move three-dimensionally. Consequently, the book is suitable both as a text at the graduate-course level and as a reference book for theprofessional. Worked examples on every major topic and over 300 exercises clarify and reinforce the principles covered in the text. A chapter-length list of references gives the reader source-material and opportunities to pursue more fully topics contained in the text.

### Details & Specs

Title:Robots and Screw Theory: Applications of kinematics and statics to roboticsFormat:HardcoverDimensions:476 pages, 9.45 × 6.61 × 1.19 inPublished:April 7, 2004Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198562454

ISBN - 13:9780198562450

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### Extra Content

Table of Contents

1. THE PLANAR SERIAL ROBOT-ARM1.1 Introduction1.2 Freedom of the End-effector1.3 The Instantaneous Centres in a Planar Robot-arm 1.3.1 The 'Inverse Velocity-problem' Solved by Instantaneous Centres1.3.2 Instantaneous Kinematics and Static Equilibrium1.3.3 The 'Forward Velocity-problem' Solved by Instantaneous CentresExercises 1A 7 1.4 Velocities by Superposition1.5 The Linear Sliding Joint1.6 Torques at the Actuated Joints1.7 The Assembly-configurations of a Planar Robot-arm Exercises 1B1.8 Foreshadowing the Spatial Serial Robot-arm 212. DESCRIBING THE SCREW2.1 The Screw in Mechanics2.1.1 The Screw in Statics2.1.2 The Screw in Instantaneous Kinematics2.1.3 Other Applications in Mechanics 2.2 The Finite Twist 302.3 Freedom and Constraint of a Rigid Body2.4 Twists, Wrenches, and Screws SummarizedExercises 2A3. ANALYSING THE SCREW3.1 Background3.2 Screw Coordinates3.2.1 The Coordinates3.2.2 Physical Interpretation of the Coordinates3.2.3 The Axis and Pitch of a Screw; Normalization of its Coordinates3.2.4 Homogeneity of Screw Coordinates3.3 A Line as the Join of Two Finite PointsExercises 3A3.4 Homogeneous Coordinates of a Point3.4.1 A Point in Projective Space3.4.2 A Line as the Join of Two Points"fm" - 2004/1/22 - page viii - #8viii Contents3.5 Homogeneous Coordinates of a Plane3.5.1 A Line as the Meet of Two Planes3.6 Homogeneity, Dimensions, and Units3.7 Ray- and Axis-coordinate Orders for Screw Coordinates3.8 Duality and LinesExercises 3B4. TRANSFORMATIONS FOR COORDINATES THAT LOCATEA RIGID BODY4.1 Introduction4.1.1 Coordinates4.2 Coordinate Transformations for Two Dimensions4.2.1 Rotational Transformations with Points4.2.2 General Transformations with Points on Coplanar Laminae4.2.3 Determining from [Aij ] the Axis and Angle of Rotation4.2.4 Determining [Aij ] from the Axis and Angle of Rotation4.2.5 Transformations with Free Vectors and Planes4.3 General Rotational Transformations4.3.1 Successive Rotations4.3.2 Rotational Transformations with Screws, Lines, Wrenches, and Twists4.4 Interpretations of a Transformation4.4.1 The Active Interpretation and the Active TransformationExercises 4A4.5 Coordinate Transformations for Three Dimensions4.5.1 The General Transformations with Points4.5.2 Transformations with Vectors and Planes4.5.3 General Transformations with Screws, Lines, Wrenches, and Twists4.6 The Finite Twist4.6.1 The Finite Twist and the Finite Screw4.6.2 The Pitch h and q-Pitch q of a Finite Twist or a Finite Screw4.6.3 Determining [Aij ] from a Finite Twist $ij (q)4.6.4 Determining the Finite Twist $ij (q) from [Aij ] and [$$ij ]Exercises 4B5. LINEAR DEPENDENCE, RECIPROCITY OF SCREWS:LINEAR AND NON-LINEAR SCREW SYSTEMS5.1 Linear Dependence of Points and Planes5.2 The Linear Two-System of ScrewsExercises 5A5.3 Linear Screw Systems5.3.1 The One-system5.3.2 The Two-system5.3.3 The Three-system5.3.4 The Four-system"fm" - 2004/1/22 - page ix - #9Contents ix5.3.5 The Five-system5.3.6 The Six-system5.3.7 Systems that are Invariant with Finite Joint-displacementsExercises 5B5.4 Reciprocity of Screws5.4.1 A Rotating Body Acted on by a Force5.4.2 A Twisting Body Acted on by a Wrench5.5 Reciprocity and Linear Screw SystemsExercises 5C5.6 Linear and Non-linear Screw Systems5.7 Some Finite Displacements and Their Screw Systems5.7.1 The System of Finite Screws for the Twists that Displace a Point5.7.2 The System of Finite Screws for the Twists that Displace a DirectedLine a5.7.3 The System of Finite Screws for the Twists that Displace a Point ona Directed Line5.7.4 Commutativity and Sequential Finite TwistsExercises 5D6. SPATIAL SERIAL ROBOT-ARMS6.1 Introduction6.2 Some Typical Six-actuator Arms6.3 A Gantry Arm6.3.1 Axes of the Actuated Joints and the Jacobian6.3.2 Det [J] and Special Configurations6.3.3 The Reciprocal Screw at a Special Configuration6.3.4 The Ubiquity of Special Configurations6.3.5 The Inverse of the Jacobian6.3.6 [J]-1 and Special Configurations6.3.7 The Gantry Arm with an 'Offset Roll-pitch-roll' Wrist6.3.8 The 'Pitch-yaw-roll' Wrist6.3.9 The Spherical '3-Roll Wrist'6.3.10 Other Wrist DesignsExercises 6A6.4 The CM T3-566 Arm (Elbow Manipulator)6.4.1 The Forward and Inverse Rate-problems6.4.2 Special Configurations: Individual Conditions6.4.3 Transversals and Reciprocal Screws6.4.4 Special Configurations: Combinations of Conditions6.5 A Unimate PUMA Arm6.6 A Manipulator with Rotary Joints in Just Three Directions6.7 General Features of Special Configurations6.8 Workspace6.8.1 Geometrical Constructions6.8.2 Configurations of a Robot-arm when B is at the Boundary"fm" - 2004/1/22 - page x - #10x Contents6.8.3 Transversals and Reciprocal Screws inWorkspace Identification6.8.4 Influence of Excursion-limits at the Joints6.8.5 Subspaces within the Reachable Point-workspace6.8.6 Workspaces of Reference Planes and Lines on the End-effector6.9 Five-actuator ArmsExercises 6B6.10 Control6.10.1 Joint Control and Cartesian Control6.10.2 Closing the Feedback Loop on the Task6.10.3 Wrench Control and Hybrid Control6.11 Torques (Forces) at the Joints of a Six-actuator ArmExercises 6C7. THE ASSEMBLY-CONFIGURATIONS OF SERIALROBOT-ARMS7.1 Introduction7.1.1 Placement of Cartesian Coordinate Frames on Links7.1.2 Forward and Inverse Kinematics for Position7.1.3 The Scalar Equation a cos f + b sin f = c7.2 The Assembly-configurations of Six-actuator Robot-arms7.2.1 A Gantry Arm7.2.2 The CM T3-566 Arm (Elbow Manipulator)7.2.3 A Unimate PUMA Arm7.2.4 The Inverted CM T3-566 Arm with an Equivalent Spherical Joint7.3 A Five-actuator ArmExercises 7A7.4 Six-actuator Robot-arms with Generally Placed Axes7.4.1 A Standard Placement of Cartesian Coordinate Frames on Links7.4.2 The Fundamental Equations7.4.3 Two Alternative Methods7.4.4 The Motoman-V6 Robot-arm7.4.5 Continuation Methods7.5 Robot-arms with Closed-form SolutionsExercises 7B8. IN-PARALLEL ACTUATION I : SIMPLE AND DIRECT8.1 Introduction8.2 The 6-6 Fully In-prallel Manipulator8.2.1 The Bricard-Borel Phenomena8.2.2 Assembly Configurations8.2.3 Special Configurations and Other Limitations: Generalities8.3 The Octahedral Manipulator: Geometry8.3.1 Polyhedra and Cauchy's Theorem8.3.2 Assembly-configurations and ConcavityExercises 8A"fm" - 2004/1/22 - page xi - #11Contents xi8.4 Transitory Kinematic Equivalence: Serial versus In-parallel8.4.1 The General 'Canonical' Wrench-applicator and the UnactuatedScrew-support8.4.2 Series-parallel Comparisons8.4.3 The Wrench-applicator for a Pure Couple8.4.4 The Wrench-applicator for a Pure Force8.4.5 Some Variants of Wrench-applicatorsExercises 8B8.5 Statics and Kinematics of Fully In-parallel Robots8.5.1 Charts of Analogues8.6 The Octahedral Manipulator: Proportions and Configurations8.6.1 The Datum Configuration8.6.2 Departures From the Datum Configuration8.6.3 A Substitution for the Double-spherical Joints8.6.4 Separation of the Double-spherical Joints8.6.5 Actuation of Force-applicators8.6.6 Other Possible Separation Arrangements for Double-spherical Joints8.6.7 An Actuated Reciprocal Connection8.6.8 Cognate Octahedral ManipulatorsExercises 8C8.7 Special Configurations: Further Observations8.7.1 A Case Study8.7.2 Series-parallel ComparisonsExercises 8D9. IN-PARALLEL ACTUATION I I : COMBINATIONS WITHSERIAL DEVICES9.1 Introduction9.2 Two Composite Robots9.3 The Force-applicator: Some Variants in Six-actuator Robots9.4 Mobility, Connectivity, and Over-constraint9.4.1 The General Mobility Criterion9.4.2 Connectivity Cij9.4.3 One Class of Over-constrained DevicesExercises 9A9.5 The Adjustable Tripod as a Manipulator9.5.1 Structure, Mobility, and Kinematic Substitutions9.5.2 Performance and Proportions of the TripodExercises 9B9.6 Generalized Reciprocal Connections: Some Derived Robots9.6.1 Three-freedom Planar-motion Robots9.6.2 Homokinetic Shaft Couplings for Parallel Shafts9.7 Two Planar In-parallel Robots9.7.1 The Planar In-parallel Robot with Three Linear Actuators9.7.2 A Planar In-parallel Robot with Three Rotary Actuators"fm" - 2004/1/22 - page xii - #12xii ContentsExercises 9C9.8 Homokinetic Coupling Robots and Derivative9.8.1 A Translatory Robot Based on a Homokinetic Coupling9.8.2 The Three Translatory Freedoms of the DELTA Robot9.9 The Inverse Kinematics for Position of Composite and Planar In-parallelRobots9.9.1 The Planar In-parallel Robot with Three Linear Actuators9.9.2 A Planar In-parallel Robot with Three Rotary Actuators9.9.3 A Coupling Robot and the Translatory Freedoms of the DELTA Robot9.10 Two Over-constrained Translatory ManipulatorsExercises 9D10. REDUNDANT ROBOTIC SYSTEMS10.1 Introduction10.1.1 Kinematic Redundancy10.2 Pseudoinverse Control10.2.1 The Coordinates of a Screw and the Jacobian [J]10.2.2 The Pseudoinverse of [J] and Other Solutions to eqns (10.3)10.2.3 Solutions to eqns (10.3) by Augmenting [J]10.2.4 Comparison of [J]# to [J]-110.3 The Control of a Four-axis Spherical Wrist10.3.1 Overspeeding in the Three-axis Orthogonal Spherical Wrist10.3.2 Pseudoinverse Control of the Four-axis Orthogonal Spherical Wrist10.3.3 Redundant Serial Arms with Rotary Joints in Just Four Directions10.4 Actuator-torques (Forces) at the Joints of Redundant Serial ArmsExercises 10A10.5 Statically Redundant Robots and Manipulators10.5.1 Screw Systems at Localized Contacts10.5.2 The Equilibrating and Interacting Force Fields10.5.3 Frictional Contacts10.5.4 The Jacobian of Force-components for Frictional Contacts10.5.5 The Pseudoinverse Solution and the Equilibrating System10.5.6 The Frictional Grasp of a Disc10.5.7 Optimization of a Grasp Using Interacting Systems of ForcesExercises 10B11. STATIC STABILITY IN LEGGED VEHICLES11.1 Introduction11.2 Wheeled and Legged Vehicles11.3 Margin of Static Stability11.3.1 The Principle of Normalized Virtual Power11.3.2 Other Measures for Margin of Stability11.4 Application to General Locations of the Contacts11.4.1 Four Contacts with the Ground11.4.2 Three Contacts with the Ground11.4.3 Comparison with a Horizontal ProjectionContents xiii11.5 Virtual Power Used in Control11.6 A Display for Margin of Static Stability11.6.1 The Rectangular Display11.6.2 Three Contacts with the Ground11.7 ConclusionExercises 11AA. APPENDIX A SOME USEFUL EXPRESSIONS FOR LINESB. APPENDIX B THE SCREW AS A POINT IN PROJECTIVE FIVE-SPACEC. APPENDIX C THE FINITE TWIST AND EDUARD STUDY'S COORDINATESD. APPENDIX D COMPUTER FILE FOR CHAPTER 10ANSWERS TO EXERCISESREFERENCESINDEXD. The Screw Axis for a Finite Displacement