Robots and screw theory : applications of kinematics and statics to robotics

TLDR: In this paper, the authors describe a number of different types of transformations in a planar robotic arm, such as the following: 1.1.1 The SCREW in MECHANICS 2.2 The FINITE TWIST 30 2.3 The AXIS and PITCH OF A RIGID BODY 2.4 The AXis and Pitch OF ROTATION 4.5.

Abstract: 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