AN ELEMENTARY TREATISE ON THE DYNAMICS OF A SYSTEM OF RIGID BODIES
CONTENTS VECTORS AND THEIR MATRIX REPRESENTATIONS IN SELECTED REFERENCE FRAMES GENERAL FEATURES OF NOTATION VECTORS DE?NITION AND DESCRIPTION OF A VECTOR EQUALITY OF VECTORS OPPOSITE VECTORS VECTOR PRODUCTS DOT PRODUCT CROSS PRODUCT REFERENCE FRAMES REPRESENTATION OF A VECTOR IN A SELECTED REFERENCE FRAME MATRIX OPERATIONS CORRESPONDING TO VECTOR OPERATIONS CROSS PRODUCT AND SKEW SYMMETRIC CROSS PRODUCT MATRICES MATHEMATICAL PROPERTIES OF THE SKEW SYMMETRIC MATRICES EXAMPLES INVOLVING SKEW SYMMETRIC MATRICES ROTATION OF VECTORS AND ROTATION MATRICES VECTOR EQUATION OF ROTATION AND THE RODRIGUES FORMULA MATRIX EQUATION OF ROTATION AND THE ROTATION MATRIX EXPONENTIALLY EXPRESSED ROTATION MATRIX BASIC ROTATION MATRICES SUCCESSIVE ROTATIONS ORTHONORMALITY OF THE ROTATION MATRICES MATHEMATICAL PROPERTIES OF THE ROTATION MATRICES MATHEMATICAL PROPERTIES OF GENERAL ROTATION MATRICES MATHEMATICAL PROPERTIES OF THE BASIC ROTATION MATRICES EXAMPLES INVOLVING ROTATION MATRICES DETERMINATION OF THE ANGLE AND AXIS OF A SPECI?ED ROTATION MATRIX SCALAR EQUATIONS OF ROTATION DETERMINATION OF THE ANGLE OF ROTATION DETERMINATION OF THE AXIS OF ROTATION DISCUSSION ABOUT THE OPTIONAL SIGN VARIABLES DE?NITION AND PROPERTIES OF THE DOUBLE ARGUMENT ARCTANGENT FUNCTION MATRIX REPRESENTATIONS OF VECTORS IN DI?ERENT REFERENCE FRAMES AND THE COMPONENT TRANSFORMATION MATRICES MATRIX REPRESENTATIONS OF A VECTOR IN DI?ERENT REFERENCE FRAMES TRANSFORMATION MATRICES BETWEEN REFERENCE FRAMES DE?NITION AND USAGE OF A TRANSFORMATION MATRIX BASIC PROPERTIES OF A TRANSFORMATION MATRIX EXPRESSION OF A TRANSFORMATION MATRIX IN TERMS OF BASIS VECTORS COLUMN-BY-COLUMN EXPRESSION ROW-BY-ROW EXPRESSION EXPRESSION OF A TRANSFORMATION MATRIX AS A DIRECTION COSINE MATRIX DE?NITIONS OF DIRECTION ANGLES AND DIRECTION COSINES TRANSFORMATION MATRIX FORMED AS A DIRECTION COSINE MATRIX EXPRESSION OF A TRANSFORMATION MATRIX AS A ROTATION MATRIX CORRELATION BETWEEN THE ROTATION AND TRANSFORMATION MATRICES DISTINCTION BETWEEN THE ROTATION AND TRANSFORMATION MATRICES RELATIONSHIP BETWEEN THE MATRIX REPRESENTATIONS OF A ROTATION OPERATOR IN DI?ERENT REFERENCE FRAMES EXPRESSION OF A TRANSFORMATION MATRIX IN A CASE OF SEVERAL SUCCESSIVE ROTATIONS ROTATED FRAME BASED (RFB) FORMULATION INITIAL FRAME BASED (IFB) FORMULATION EXPRESSION OF A TRANSFORMATION MATRIX IN TERMS OF EULER ANGLES GENERAL DE?NITION OF EULER ANGLES IFB (INITIAL FRAME BASED) EULER ANGLE SEQUENCES RFB (ROTATED FRAME BASED) EULER ANGLE SEQUENCES REMARK 3 6: PREFERENCE BETWEEN IFB AND RFB SEQUENCES COMMONLY USED EULER ANGLE SEQUENCES EXTRACTION OF EULER ANGLES FROM A GIVEN TRANSFORMATION MATRIX POSITION OF A POINT EXPRESSED IN DI?ERENT REFERENCE FRAMES AND HOMOGENEOUS TRANSFORMATION MATRICES POSITION OF A POINT EXPRESSED IN DI?ERENT REFERENCE FRAMES HOMOGENEOUS, NONHOMOGENEOUS, LINEAR, NONLINEAR, AND A?NE RELATIONSHIPS A?NE COORDINATE TRANSFORMATION BETWEEN TWO REFERENCE FRAMES HOMOGENEOUS COORDINATE TRANSFORMATION BETWEEN TWO REFERENCE FRAMES MATHEMATICAL PROPERTIES OF THE HOMOGENEOUS TRANSFORMATION MATRICES VECTOR DI?ERENTIATION ACCOMPANIED BY VELOCITY AND ACCELERATION EXPRESSIONS DERIVATIVES OF A VECTOR WITH RESPECT TO DI?ERENT REFERENCE FRAMES DI?ERENTIATION AND RESOLUTION FRAMES COMPONENTS IN DI?ERENT DI?ERENTIATION AND RESOLUTION FRAMES EXAMPLE VECTOR DERIVATIVES WITH RESPECT TO DI?ERENT REFERENCE FRAMES AND THE CORIOLIS TRANSPORT THEOREM FIRST DERIVATIVES AND THE RELATIVE ANGULAR VELOCITY SECOND DERIVATIVES AND THE RELATIVE ANGULAR ACCELERATION COMBINATION OF RELATIVE ANGULAR VELOCITIES AND ACCELERATIONS COMBINATION OF RELATIVE ANGULAR VELOCITIES COMBINATION OF RELATIVE ANGULAR ACCELERATIONS ANGULAR VELOCITIES AND ACCELERATIONS ASSOCIATED WITH ROTATION SEQUENCES RELATIVE ANGULAR VELOCITIES AND ACCELERATIONS ABOUT RELATIVELY FIXED AXES ANGULAR VELOCITIES ASSOCIATED WITH THE EULER ANGLE SEQUENCES VELOCITY AND ACCELERATION OF A POINT WITH RESPECT TO DI?ERENT REFERENCE FRAMES VELOCITY OF A POINT WITH RESPECT TO DI?ERENT REFERENCE FRAMES ACCELERATION OF A POINT WITH RESPECT TO DI?ERENT REFERENCE FRAMES VELOCITY AND ACCELERATION EXPRESSIONS WITH SIMPLI?ED NOTATIONS KINEMATICS OF RIGID BODY SYSTEMS KINEMATIC DESCRIPTION OF A RIGID BODY SYSTEM BODY FRAMES AND JOINT FRAMES KINEMATIC CHAINS, KINEMATIC BRANCHES, AND KINEMATIC LOOPS JOINTS OR KINEMATIC PAIRS POSITION EQUATIONS FOR A KINEMATIC CHAIN OF RIGID BODIES RELATIVE ORIENTATION EQUATION BETWEEN SUCCESSIVE BODIES RELATIVE LOCATION EQUATION BETWEEN SUCCESSIVE BODIES ORIENTATION OF A BODY WITH RESPECT TO THE BASE OF THE KINEMATIC CHAIN LOCATION OF A BODY WITH RESPECT TO THE BASE OF THE KINEMATIC CHAIN LOOP CLOSURE EQUATIONS FOR A KINEMATIC LOOP VELOCITY EQUATIONS FOR A KINEMATIC CHAIN OF RIGID BODIES RELATIVE ANGULAR VELOCITY BETWEEN SUCCESSIVE BODIES RELATIVE TRANSLATIONAL VELOCITY BETWEEN SUCCESSIVE BODIES ANGULAR VELOCITY OF A BODY WITH RESPECT TO THE BASE TRANSLATIONAL VELOCITY OF A BODY WITH RESPECT TO THE BASE VELOCITY EQUATIONS FOR A KINEMATIC LOOP ACCELERATION EQUATIONS FOR A KINEMATIC CHAIN OF RIGID BODIES RELATIVE ANGULAR ACCELERATION BETWEEN SUCCESSIVE BODIES RELATIVE TRANSLATIONAL ACCELERATION BETWEEN SUCCESSIVE BODIES ANGULAR ACCELERATION OF A BODY WITH RESPECT TO THE BASE TRANSLATIONAL ACCELERATION OF A BODY WITH RESPECT TO THE BASE ACCELERATION EQUATIONS FOR A KINEMATIC LOOP EXAMPLE 5 1 : A SERIAL MANIPULATOR WITH AN RRP ARM KINEMATIC DESCRIPTION OF THE SYSTEM POSITION ANALYSIS VELOCITY ANALYSIS ACCELERATION ANALYSIS EXAMPLE 5 2 : A SPATIAL SLIDER-CRANK (RSSP) MECHANISM KINEMATIC DESCRIPTION OF THE MECHANISM LOOP CLOSURE EQUATIONS DEGREE OF FREEDOM OR MOBILITY JOINTS AND THEIR KINEMATIC CHARACTERISTICS KINEMATIC DETAILS OF THE JOINTS DESCRIPTION OF A JOINT AS A KINEMATIC PAIR DEGREE OF FREEDOM OR MOBILITY OF A JOINT NUMBER OF DISTINCT JOINTS BETWEEN TWO RIGID BODIES CLASSI?CATION OF THE JOINTS TYPICAL LOWER ORDER JOINTS SINGLE-AXIS JOINTS UNIVERSAL JOINT SPHERICAL JOINT PLANE-ON-PLANE JOINT HIGHER ORDER JOINTS WITH SIMPLE CONTACTS LINE-ON-PLANE JOINT POINT-ON-PLANE JOINT POINT-ON-SURFACE JOINT TYPICAL MULTI-JOINT CONNECTIONS FORK-ON-SURFACE JOINT TRIANGLE-ON-SURFACE JOINT ROLLING CONTACT JOINTS WITH POINT CONTACTS SURFACE-ON-SURFACE JOINT CURVE-ON-SURFACE JOINT CURVE-ON-CURVE JOINT ROLLING CONTACT JOINTS WITH LINE CONTACTS CONE-ON-CONE JOINT CONE-ON-CYLINDER JOINT CONE-ON-PLANE JOINT CYLINDER-ON-CYLINDER JOINT CYLINDER-ON-PLANE JOINT EXAMPLES EXAMPLE 6 1: AN RRRSP MECHANISM EXAMPLE 6 2: A TWO-LINK MECHANISM WITH THREE POINT-ON-PLANE JOINTS EXAMPLE 6 3: A SPATIAL CAM MECHANISM EXAMPLE 6 4: A SPATIAL CAM MECHANISM THAT ALLOWS ROLLING WITHOUT SLIPPING KINEMATIC FEATURES OF SERIAL MANIPULATORS KINEMATIC DESCRIPTION OF A GENERAL SERIAL MANIPULATOR DENAVIT–HARTENBERG CONVENTION D–H CONVENTION FOR SUCCESSIVE INTERMEDIATE LINKS AND JOINTS ASSIGNMENT AND DESCRIPTION OF THE LINK FRAMES D–H PARAMETERS RELATIVE POSITION FORMULAS BETWEEN SUCCESSIVE LINKS ALTERNATIVE MULTI-INDEX NOTATION FOR THE D–H CONVENTION D–H CONVENTION FOR THE FIRST JOINT D–H CONVENTION FOR THE LAST JOINT D–H CONVENTION FOR SUCCESSIVE JOINTS WITH PERPENDICULARLY INTERSECTING AXES D–H CONVENTION FOR SUCCESSIVE JOINTS WITH PARALLEL AXES D–H CONVENTION FOR SUCCESSIVE JOINTS WITH COINCIDENT AXES POSITION AND MOTION ANALYSES OF GENERIC SERIAL MANIPULATORS FORWARD KINEMATICS COMPACT FORMULATION OF FORWARD KINEMATICS DETAILED FORMULATION OF FORWARD KINEMATICS MANIPULATORS WITH OR WITHOUT SPHERICAL WRISTS INVERSE KINEMATICS INVERSE KINEMATIC SOLUTION FOR A REGULAR MANIPULATOR REGULAR MANIPULATOR WITH A SPHERICAL WRIST REGULAR MANIPULATOR WITH A NONSPHERICAL WRIST INVERSE KINEMATIC SOLUTION FOR A REDUNDANT MANIPULATOR SOLUTION BY SPECIFYING THE VARIABLES OF CERTAIN JOINTS SOLUTION BY OPTIMIZATION INVERSE KINEMATIC SOLUTION FOR A DE?CIENT MANIPULATOR COMPROMISE IN ORIENTATION IN FAVOR OF A COMPLETELY SPECI?ED LOCATION COMPROMISE IN LOCATION IN FAVOR OF A COMPLETELY SPECI?ED ORIENTATION FORWARD KINEMATICS OF MOTION FORWARD KINEMATICS OF VELOCITY RELATIONSHIPS FORWARD KINEMATICS OF ACCELERATION RELATIONSHIPS JACOBIAN MATRICES ASSOCIATED WITH THE WRIST AND TIP POINTS RECURSIVE POSITION, VELOCITY, AND ACCELERATION FORMULATIONS ORIENTATIONS OF THE LINKS LOCATIONS OF THE LINK FRAME ORIGINS LOCATIONS OF THE MASS CENTERS OF THE LINKS ANGULAR VELOCITIES OF THE LINKS VELOCITIES OF THE LINK FRAME ORIGINS VELOCITIES OF THE MASS CENTERS OF THE LINKS ANGULAR ACCELERATIONS OF THE LINKS ACCELERATIONS OF THE LINK FRAME ORIGINS ACCELERATIONS OF THE MASS CENTERS OF THE LINKS INVERSE MOTION ANALYSIS OF A MANIPULATOR BASED ON THE JACOBIAN MATRIX INVERSE VELOCITY ANALYSIS OF A REGULAR MANIPULATOR INVERSE ACCELERATION ANALYSIS OF A REGULAR MANIPULATOR INVERSE MOTION ANALYSIS OF A REDUNDANT MANIPULATOR INVERSE VELOCITY ANALYSIS INVERSE ACCELERATION ANALYSIS INVERSE MOTION ANALYSIS OF A DE?CIENT MANIPULATOR INVERSE MOTION ANALYSIS OF A REGULAR MANIPULATOR USING THE DETAILED FORMULATION INVERSE VELOCITY SOLUTION INVERSE ACCELERATION SOLUTION KINEMATIC ANALYSES OF TYPICAL SERIAL MANIPULATORS PUMA MANIPULATOR KINEMATIC DESCRIPTION ACCORDING TO THE D–H CONVENTION FORWARD KINEMATICS IN THE POSITION DOMAIN INVERSE KINEMATICS IN THE POSITION DOMAIN MULTIPLICITY ANALYSIS SINGULARITY ANALYSIS IN THE POSITION DOMAIN FORWARD KINEMATICS IN THE VELOCITY DOMAIN INVERSE KINEMATICS IN THE VELOCITY DOMAIN SINGULARITY ANALYSIS IN THE VELOCITY DOMAIN STANFORD MANIPULATOR ELBOW MANIPULATOR SCARA MANIPULATOR AN RP2R3 MANIPULATOR WITHOUT AN ANALYTICAL SOLUTION AN RPRPR2 MANIPULATOR WITH AN UNCUSTOMARY ANALYTICAL SOLUTION A DE?CIENT PUMA MANIPULATOR WITH FIVE ACTIVE JOINTS SOLUTION IN THE CASE OF FULLY SPECI?ED TIP POINT LOCATION SOLUTION IN THE CASE OF FULLY SPECI?ED END-E?ECTOR ORIENTATION MULTIPLICITY ANALYSIS IN THE POSITION DOMAIN ANALYSIS IN THE CASE OF FULLY SPECI?ED TIP POINT LOCATION ANALYSIS IN THE CASE OF FULLY SPECI?ED END-E?ECTOR ORIENTATION SOLUTION IN THE CASE OF FULLY SPECI?ED TIP POINT VELOCITY SOLUTION IN THE CASE OF FULLY SPECI?ED END-E?ECTOR ANGULAR VELOCITY ANALYSIS IN THE CASE OF FULLY SPECI?ED TIP POINT VELOCITY ANALYSIS IN THE CASE OF FULLY SPECI?ED END-E?ECTOR ANGULAR VELOCITY A REDUNDANT HUMANOID MANIPULATOR WITH EIGHT JOINTS CONSISTENCY OF THE INVERSE KINEMATICS IN THE POSITION AND VELOCITY DOMAINS POSITION AND VELOCITY ANALYSES OF PARALLEL MANIPULATORS GENERAL KINEMATIC FEATURES OF PARALLEL MANIPULATORS POSITION EQUATIONS OF A PARALLEL MANIPULATOR VELOCITY EQUATIONS OF A PARALLEL MANIPULATOR STEWART–GOUGH PLATFORM AS A 6UPS SPATIAL PARALLEL MANIPULATOR KINEMATIC DESCRIPTION POSITION EQUATIONS VELOCITY EQUATIONS DELTA ROBOT: A 3RS2S2 SPATIAL PARALLEL MANIPULATOR INDEPENDENT KINEMATIC LOOPS AND THE ASSOCIATED EQUATIONS