Robots and Screw Theory

Davidson served as Associate Editor of the ASME Journal of Mechanisms, Transmissions, and Automation in Design from 1982-86. He also served one term as a member of the Executive Committee of the International Federation for the Theory of Machines and Mechanisms.Davidson is a Fellow of the American Society of Mechanical Engineers.Davidson is a member of the Robotics & Automation Society of the IEEE.Hunt served on the Honorary Editorial Advisory Board of the Pergamon journal Mechanism and Machine Theory from its inception in 1966 (as the Journal of Mechanisms) until his death in 2002. He was a Founding Executive Councillor of the International Federation for the Theory of Machines and Mechanisms, and he also was the Founding Chairman of the Australian National Committee for IFToMM.Hunt is a Fellow of the American Society of Mechanical EngineersHunt is a Fellow of the Institution of Mechanical EngineersHunt is a Fellow of the Institution of Engineers, AustraliaHunt is a Fellow of the Australian Academy of Technological Sciences

[Davidson and Hunt's] dedication to the analysis of the various cases considered is admirable and the degree of detail that they have expressed makes this work invaluable to those taking final mechanical degrees.

• 1.1 INTRODUCTION ; 1.2 FREEDOM OF THE END-EFFECTOR ; 1.3 THE INSTANTANEOUS CENTRES IN A PLANAR ROBOT-ARM 1.3.1 THE 'INVERSE VELOCITY-PROBLEM' SOLVED BY INSTANTANEOUS CENTRES ; 1.3.2 INSTANTANEOUS KINEMATICS AND STATIC EQUILIBRIUM ; 1.3.3 THE 'FORWARD VELOCITY-PROBLEM' SOLVED BY INSTANTANEOUS CENTRES ; EXERCISES 1A 7 1.4 VELOCITIES BY SUPERPOSITION ; 1.5 THE LINEAR SLIDING JOINT ; 1.6 TORQUES AT THE ACTUATED JOINTS ; 1.7 THE ASSEMBLY-CONFIGURATIONS OF A PLANAR ROBOT-ARM EXERCISES 1B ; 2. DESCRIBING THE SCREW ; 2.1 THE SCREW IN MECHANICS ; 2.1.1 THE SCREW IN STATICS ; 2.1.2 THE SCREW IN INSTANTANEOUS KINEMATICS ; 2.1.3 OTHER APPLICATIONS IN MECHANICS 2.2 THE FINITE TWIST 30 ; 2.3 FREEDOM AND CONSTRAINT OF A RIGID BODY ; 2.4 TWISTS, WRENCHES, AND SCREWS SUMMARIZED ; EXERCISES 2A ; 3.1 BACKGROUND ; 3.2 SCREW COORDINATES ; 3.2.1 THE COORDINATES ; 3.2.2 PHYSICAL INTERPRETATION OF THE COORDINATES ; 3.2.3 THE AXIS AND PITCH OF A SCREW; NORMALIZATION OF ITS COORDINATES ; 3.2.4 HOMOGENEITY OF SCREW COORDINATES ; 3.3 A LINE AS THE JOIN OF TWO FINITE POINTS ; EXERCISES 3A ; 3.4 HOMOGENEOUS COORDINATES OF A POINT ; 3.4.1 A POINT IN PROJECTIVE SPACE ; 3.4.2 A LINE AS THE JOIN OF TWO POINTS ; "FM" - 2004/1/22 - PAGE VIII - #8 ; VIII CONTENTS ; 3.5 HOMOGENEOUS COORDINATES OF A PLANE ; 3.5.1 A LINE AS THE MEET OF TWO PLANES ; 3.6 HOMOGENEITY, DIMENSIONS, AND UNITS ; 3.7 RAY- AND AXIS-COORDINATE ORDERS FOR SCREW COORDINATES ; 3.8 DUALITY AND LINES ; EXERCISES 3B ; A RIGID BODY ; 4.1 INTRODUCTION ; 4.1.1 COORDINATES ; 4.2 COORDINATE TRANSFORMATIONS FOR TWO DIMENSIONS ; 4.2.1 ROTATIONAL TRANSFORMATIONS WITH POINTS ; 4.2.2 GENERAL TRANSFORMATIONS WITH POINTS ON COPLANAR LAMINAE ; 4.2.3 DETERMINING FROM [AIJ ] THE AXIS AND ANGLE OF ROTATION ; 4.2.4 DETERMINING [AIJ ] FROM THE AXIS AND ANGLE OF ROTATION ; 4.2.5 TRANSFORMATIONS WITH FREE VECTORS AND PLANES ; 4.3 GENERAL ROTATIONAL TRANSFORMATIONS ; 4.3.1 SUCCESSIVE ROTATIONS ; 4.3.2 ROTATIONAL TRANSFORMATIONS WITH SCREWS, LINES, WRENCHES, AND TWISTS ; 4.4 INTERPRETATIONS OF A TRANSFORMATION ; 4.4.1 THE ACTIVE INTERPRETATION AND THE ACTIVE TRANSFORMATION ; EXERCISES 4A ; 4.5 COORDINATE TRANSFORMATIONS FOR THREE DIMENSIONS ; 4.5.1 THE GENERAL TRANSFORMATIONS WITH POINTS ; 4.5.2 TRANSFORMATIONS WITH VECTORS AND PLANES ; 4.5.3 GENERAL TRANSFORMATIONS WITH SCREWS, LINES, WRENCHES, AND TWISTS ; 4.6 THE FINITE TWIST ; 4.6.1 THE FINITE TWIST AND THE FINITE SCREW ; 4.6.2 THE PITCH H AND Q-PITCH Q OF A FINITE TWIST OR A FINITE SCREW ; 4.6.3 DETERMINING [AIJ ] FROM A FINITE TWIST $IJ (Q) ; 4.6.4 DETERMINING THE FINITE TWIST $IJ (Q) FROM [AIJ ] AND [$$IJ ] ; EXERCISES 4B ; LINEAR AND NON-LINEAR SCREW SYSTEMS ; 5.1 LINEAR DEPENDENCE OF POINTS AND PLANES ; 5.2 THE LINEAR TWO-SYSTEM OF SCREWS ; EXERCISES 5A ; 5.3 LINEAR SCREW SYSTEMS ; 5.3.1 THE ONE-SYSTEM ; 5.3.2 THE TWO-SYSTEM ; 5.3.3 THE THREE-SYSTEM ; 5.3.4 THE FOUR-SYSTEM ; "FM" - 2004/1/22 - PAGE IX - #9 ; CONTENTS IX ; 5.3.5 THE FIVE-SYSTEM ; 5.3.6 THE SIX-SYSTEM ; 5.3.7 SYSTEMS THAT ARE INVARIANT WITH FINITE JOINT-DISPLACEMENTS ; EXERCISES 5B ; 5.4 RECIPROCITY OF SCREWS ; 5.4.1 A ROTATING BODY ACTED ON BY A FORCE ; 5.4.2 A TWISTING BODY ACTED ON BY A WRENCH ; 5.5 RECIPROCITY AND LINEAR SCREW SYSTEMS ; EXERCISES 5C ; 5.6 LINEAR AND NON-LINEAR SCREW SYSTEMS ; 5.7 SOME FINITE DISPLACEMENTS AND THEIR SCREW SYSTEMS ; 5.7.1 THE SYSTEM OF FINITE SCREWS FOR THE TWISTS THAT DISPLACE A POINT ; 5.7.2 THE SYSTEM OF FINITE SCREWS FOR THE TWISTS THAT DISPLACE A DIRECTED ; LINE A ; 5.7.3 THE SYSTEM OF FINITE SCREWS FOR THE TWISTS THAT DISPLACE A POINT ON ; A DIRECTED LINE ; 5.7.4 COMMUTATIVITY AND SEQUENTIAL FINITE TWISTS ; EXERCISES 5D ; 6.1 INTRODUCTION ; 6.2 SOME TYPICAL SIX-ACTUATOR ARMS ; 6.3 A GANTRY ARM ; 6.3.1 AXES OF THE ACTUATED JOINTS AND THE JACOBIAN ; 6.3.2 DET [J] AND SPECIAL CONFIGURATIONS ; 6.3.3 THE RECIPROCAL SCREW AT A SPECIAL CONFIGURATION ; 6.3.4 THE UBIQUITY OF SPECIAL CONFIGURATIONS ; 6.3.5 THE INVERSE OF THE JACOBIAN ; 6.3.6 [J]-1 AND SPECIAL CONFIGURATIONS ; 6.3.7 THE GANTRY ARM WITH AN 'OFFSET ROLL-PITCH-ROLL' WRIST ; 6.3.8 THE 'PITCH-YAW-ROLL' WRIST ; 6.3.9 THE SPHERICAL '3-ROLL WRIST' ; 6.3.10 OTHER WRIST DESIGNS ; EXERCISES 6A ; 6.4 THE CM T3-566 ARM (ELBOW MANIPULATOR) ; 6.4.1 THE FORWARD AND INVERSE RATE-PROBLEMS ; 6.4.2 SPECIAL CONFIGURATIONS: INDIVIDUAL CONDITIONS ; 6.4.3 TRANSVERSALS AND RECIPROCAL SCREWS ; 6.4.4 SPECIAL CONFIGURATIONS: COMBINATIONS OF CONDITIONS ; 6.5 A UNIMATE PUMA ARM ; 6.6 A MANIPULATOR WITH ROTARY JOINTS IN JUST THREE DIRECTIONS ; 6.7 GENERAL FEATURES OF SPECIAL CONFIGURATIONS ; 6.8 WORKSPACE ; 6.8.1 GEOMETRICAL CONSTRUCTIONS ; 6.8.2 CONFIGURATIONS OF A ROBOT-ARM WHEN B IS AT THE BOUNDARY ; "FM" - 2004/1/22 - PAGE X - #10 ; X CONTENTS ; 6.8.3 TRANSVERSALS AND RECIPROCAL SCREWS INWORKSPACE IDENTIFICATION ; 6.8.4 INFLUENCE OF EXCURSION-LIMITS AT THE JOINTS ; 6.8.5 SUBSPACES WITHIN THE REACHABLE POINT-WORKSPACE ; 6.8.6 WORKSPACES OF REFERENCE PLANES AND LINES ON THE END-EFFECTOR ; 6.9 FIVE-ACTUATOR ARMS ; EXERCISES 6B ; 6.10 CONTROL ; 6.10.1 JOINT CONTROL AND CARTESIAN CONTROL ; 6.10.2 CLOSING THE FEEDBACK LOOP ON THE TASK ; 6.10.3 WRENCH CONTROL AND HYBRID CONTROL ; 6.11 TORQUES (FORCES) AT THE JOINTS OF A SIX-ACTUATOR ARM ; EXERCISES 6C ; ROBOT-ARMS ; 7.1 INTRODUCTION ; 7.1.1 PLACEMENT OF CARTESIAN COORDINATE FRAMES ON LINKS ; 7.1.2 FORWARD AND INVERSE KINEMATICS FOR POSITION ; 7.1.3 THE SCALAR EQUATION A COS F + B SIN F = C ; 7.2 THE ASSEMBLY-CONFIGURATIONS OF SIX-ACTUATOR ROBOT-ARMS ; 7.2.1 A GANTRY ARM ; 7.2.2 THE CM T3-566 ARM (ELBOW MANIPULATOR) ; 7.2.3 A UNIMATE PUMA ARM ; 7.2.4 THE INVERTED CM T3-566 ARM WITH AN EQUIVALENT SPHERICAL JOINT ; 7.3 A FIVE-ACTUATOR ARM ; EXERCISES 7A ; 7.4 SIX-ACTUATOR ROBOT-ARMS WITH GENERALLY PLACED AXES ; 7.4.1 A STANDARD PLACEMENT OF CARTESIAN COORDINATE FRAMES ON LINKS ; 7.4.2 THE FUNDAMENTAL EQUATIONS ; 7.4.3 TWO ALTERNATIVE METHODS ; 7.4.4 THE MOTOMAN-V6 ROBOT-ARM ; 7.4.5 CONTINUATION METHODS ; 7.5 ROBOT-ARMS WITH CLOSED-FORM SOLUTIONS ; EXERCISES 7B ; 8.1 INTRODUCTION ; 8.2 THE 6-6 FULLY IN-PRALLEL MANIPULATOR ; 8.2.1 THE BRICARD-BOREL PHENOMENA ; 8.2.2 ASSEMBLY CONFIGURATIONS ; 8.2.3 SPECIAL CONFIGURATIONS AND OTHER LIMITATIONS: GENERALITIES ; 8.3 THE OCTAHEDRAL MANIPULATOR: GEOMETRY ; 8.3.1 POLYHEDRA AND CAUCHY'S THEOREM ; 8.3.2 ASSEMBLY-CONFIGURATIONS AND CONCAVITY ; EXERCISES 8A ; "FM" - 2004/1/22 - PAGE XI - #11 ; CONTENTS XI ; 8.4 TRANSITORY KINEMATIC EQUIVALENCE: SERIAL VERSUS IN-PARALLEL ; 8.4.1 THE GENERAL 'CANONICAL' WRENCH-APPLICATOR AND THE UNACTUATED ; SCREW-SUPPORT ; 8.4.2 SERIES-PARALLEL COMPARISONS ; 8.4.3 THE WRENCH-APPLICATOR FOR A PURE COUPLE ; 8.4.4 THE WRENCH-APPLICATOR FOR A PURE FORCE ; 8.4.5 SOME VARIANTS OF WRENCH-APPLICATORS ; EXERCISES 8B ; 8.5 STATICS AND KINEMATICS OF FULLY IN-PARALLEL ROBOTS ; 8.5.1 CHARTS OF ANALOGUES ; 8.6 THE OCTAHEDRAL MANIPULATOR: PROPORTIONS AND CONFIGURATIONS ; 8.6.1 THE DATUM CONFIGURATION ; 8.6.2 DEPARTURES FROM THE DATUM CONFIGURATION ; 8.6.3 A SUBSTITUTION FOR THE DOUBLE-SPHERICAL JOINTS ; 8.6.4 SEPARATION OF THE DOUBLE-SPHERICAL JOINTS ; 8.6.5 ACTUATION OF FORCE-APPLICATORS ; 8.6.6 OTHER POSSIBLE SEPARATION ARRANGEMENTS FOR DOUBLE-SPHERICAL JOINTS ; 8.6.7 AN ACTUATED RECIPROCAL CONNECTION ; 8.6.8 COGNATE OCTAHEDRAL MANIPULATORS ; EXERCISES 8C ; 8.7 SPECIAL CONFIGURATIONS: FURTHER OBSERVATIONS ; 8.7.1 A CASE STUDY ; 8.7.2 SERIES-PARALLEL COMPARISONS ; EXERCISES 8D ; SERIAL DEVICES ; 9.1 INTRODUCTION ; 9.2 TWO COMPOSITE ROBOTS ; 9.3 THE FORCE-APPLICATOR: SOME VARIANTS IN SIX-ACTUATOR ROBOTS ; 9.4 MOBILITY, CONNECTIVITY, AND OVER-CONSTRAINT ; 9.4.1 THE GENERAL MOBILITY CRITERION ; 9.4.2 CONNECTIVITY CIJ ; 9.4.3 ONE CLASS OF OVER-CONSTRAINED DEVICES ; EXERCISES 9A ; 9.5 THE ADJUSTABLE TRIPOD AS A MANIPULATOR ; 9.5.1 STRUCTURE, MOBILITY, AND KINEMATIC SUBSTITUTIONS ; 9.5.2 PERFORMANCE AND PROPORTIONS OF THE TRIPOD ; EXERCISES 9B ; 9.6 GENERALIZED RECIPROCAL CONNECTIONS: SOME DERIVED ROBOTS ; 9.6.1 THREE-FREEDOM PLANAR-MOTION ROBOTS ; 9.6.2 HOMOKINETIC SHAFT COUPLINGS FOR PARALLEL SHAFTS ; 9.7 TWO PLANAR IN-PARALLEL ROBOTS ; 9.7.1 THE PLANAR IN-PARALLEL ROBOT WITH THREE LINEAR ACTUATORS ; 9.7.2 A PLANAR IN-PARALLEL ROBOT WITH THREE ROTARY ACTUATORS ; "FM" - 2004/1/22 - PAGE XII - #12 ; XII CONTENTS ; EXERCISES 9C ; 9.8 HOMOKINETIC COUPLING ROBOTS AND DERIVATIVE ; 9.8.1 A TRANSLATORY ROBOT BASED ON A HOMOKINETIC COUPLING ; 9.8.2 THE THREE TRANSLATORY FREEDOMS OF THE DELTA ROBOT ; 9.9 THE INVERSE KINEMATICS FOR POSITION OF COMPOSITE AND PLANAR IN-PARALLEL ; ROBOTS ; 9.9.1 THE PLANAR IN-PARALLEL ROBOT WITH THREE LINEAR ACTUATORS ; 9.9.2 A PLANAR IN-PARALLEL ROBOT WITH THREE ROTARY ACTUATORS ; 9.9.3 A COUPLING ROBOT AND THE TRANSLATORY FREEDOMS OF THE DELTA ROBOT ; 9.10 TWO OVER-CONSTRAINED TRANSLATORY MANIPULATORS ; EXERCISES 9D ; 10.1 INTRODUCTION ; 10.1.1 KINEMATIC REDUNDANCY ; 10.2 PSEUDOINVERSE CONTROL ; 10.2.1 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]-1 ; 10.3 THE CONTROL OF A FOUR-AXIS SPHERICAL WRIST ; 10.3.1 OVERSPEEDING IN THE THREE-AXIS ORTHOGONAL SPHERICAL WRIST ; 10.3.2 PSEUDOINVERSE CONTROL OF THE FOUR-AXIS ORTHOGONAL SPHERICAL WRIST ; 10.3.3 REDUNDANT SERIAL ARMS WITH ROTARY JOINTS IN JUST FOUR DIRECTIONS ; 10.4 ACTUATOR-TORQUES (FORCES) AT THE JOINTS OF REDUNDANT SERIAL ARMS ; EXERCISES 10A ; 10.5 STATICALLY REDUNDANT ROBOTS AND MANIPULATORS ; 10.5.1 SCREW SYSTEMS AT LOCALIZED CONTACTS ; 10.5.2 THE EQUILIBRATING AND INTERACTING FORCE FIELDS ; 10.5.3 FRICTIONAL CONTACTS ; 10.5.4 THE JACOBIAN OF FORCE-COMPONENTS FOR FRICTIONAL CONTACTS ; 10.5.5 THE PSEUDOINVERSE SOLUTION AND THE EQUILIBRATING SYSTEM ; 10.5.6 THE FRICTIONAL GRASP OF A DISC ; 10.5.7 OPTIMIZATION OF A GRASP USING INTERACTING SYSTEMS OF FORCES ; EXERCISES 10B ; 11.1 INTRODUCTION ; 11.2 WHEELED AND LEGGED VEHICLES ; 11.3 MARGIN OF STATIC STABILITY ; 11.3.1 THE PRINCIPLE OF NORMALIZED VIRTUAL POWER ; 11.3.2 OTHER MEASURES FOR MARGIN OF STABILITY ; 11.4 APPLICATION TO GENERAL LOCATIONS OF THE CONTACTS ; 11.4.1 FOUR CONTACTS WITH THE GROUND ; 11.4.2 THREE CONTACTS WITH THE GROUND ; 11.4.3 COMPARISON WITH A HORIZONTAL PROJECTION ; CONTENTS XIII ; 11.5 VIRTUAL POWER USED IN CONTROL ; 11.6 A DISPLAY FOR MARGIN OF STATIC STABILITY ; 11.6.1 THE RECTANGULAR DISPLAY ; 11.6.2 THREE CONTACTS WITH THE GROUND ; 11.7 CONCLUSION ; EXERCISES 11A ; ANSWERS TO EXERCISES ; REFERENCES ; INDEX