AN INTRODUCTION TO ROBOTICS: MECHANICAL ASPECTS Pierre DUYSINX and Michel GERADIN University of Liege` Novembre 2004 2 Contents 1INTRODUCTION 1 1.1Originofthenamerobot............................... 2 1.2 Mechanical role of a robot manipulator . 2 1.3 Generalstructureofarobotmanipulator...................... 3 1.4Structureofthecontrolunit.............................. 4 1.5 Industrial robots at the present day . 4 1.6 The mechanical aspects of robotics . 5 1.7 Multidisciplinaryaspectsofrobotics......................... 6 2 BASICS OF ROBOTICS TECHNOLOGY 1 2.1TheMechanicalStructureofaRobot......................... 2 2.1.1 Degreesoffreedomofarigidbody...................... 2 2.1.2 Jointsandkinematicconstraints....................... 2 2.1.3 Generalizedcoordinates............................ 3 2.2KinematicPairs..................................... 3 2.2.1 Number of degrees of freedom of the joint . 3 2.2.2 Classificationofjoints............................. 3 2.2.3 Higherandlowerpairs............................. 5 2.2.4 Graphicrepresentationofjoints........................ 5 2.2.5 Joints used in robots . 5 2.3TopologyofKinematicChains............................ 8 2.3.1 Classification of robot topologies . 8 2.3.2 Description of simple open-tree structures . 11 2.4 Mobility Index and Number of dof for a Simple Open-Tree Manipulator . 11 2.4.1 Mobility index and GRUBLERformula...................¨ 11 2.4.2 Numberofdofofasimpleopen-treestructure................ 13 2.4.3 Number of DOF of a manipulator . 13 2.4.4 Jointspace................................... 13 2.4.5 Taskspace................................... 13 2.4.6 Redundancy . 13 2.4.7 Singularity................................... 14 2.4.8 Examples.................................... 16 2.4.9 Exercices.................................... 18 2.5Numberofdofofthetask............................... 20 2.6Robotmorphology................................... 22 2.6.1 Numberofpossiblemorphologies....................... 22 2.6.2 General structure of a manipulator . 22 2.6.3 Possiblearmarchitectures........................... 23 i ii CONTENTS 2.6.4 Kinematic decoupling between effector orientation and position . 23 2.7 Workspace of a robot manipulator . 25 2.7.1 Definition.................................... 25 2.7.2 Comparison of the workspaces with different arm configurations . 25 2.7.3 Workspaceoptimisation............................ 26 2.8 Accuracy, repeatability and resolution . 32 2.8.1 Staticaccuracy................................. 32 2.8.2 Repeatability . 32 2.8.3 Resolution.................................... 32 2.8.4 Normalisation.................................. 33 2.9Robotactuators.................................... 33 2.9.1 Distributedmotorization............................ 33 2.9.2 Centralizedmotorization............................ 34 2.9.3 Mixedmotorization.............................. 35 2.10 Mechanical characteristics of actuators . 35 2.10.1Powertomassratio.............................. 35 2.10.2 Maximum acceleration and mechanical impedance adaptation . 36 2.11Differenttypesofactuators.............................. 41 2.11.1Stepmotors................................... 41 2.11.2Directcurrentmotors............................. 42 2.11.3Hydraulicactuators.............................. 42 2.12Thesensors....................................... 43 2.13Integrationofsensorsinthemechanicalstructure.................. 43 2.13.1Positionsensors................................. 44 2.13.2Velocitysensors................................. 47 2.14TherobotmanipulatorASEA-IRb-6......................... 51 2.15 Technical sheets of some industrial robot manipulators . 57 2.15.1 Industrial robot ASEA IRb-6/2 . 58 2.15.2 Industrial robot SCEMI 6P-01 . 61 2.15.3 Industrial robot PUMA 560 . 63 2.15.4 Industrial robot ASEA IRB1400 . 65 3 BASIC PRINCIPLES OF ROBOT MOTION CONTROL 1 3.1Objectivesofrobotcontrol.............................. 2 3.1.1 Variables under control . 2 3.1.2 Robotmotioncontrol............................. 3 3.1.3 Level 1 of control: artificial intelligence level . 5 3.1.4 Level 2 of control or the control mode level . 5 3.1.5 Level 3 of control or servo-system level . 6 3.2Kinematicmodelofarobotmanipulator....................... 7 3.3Trajectoryplanning.................................. 11 3.3.1 Jointspacedescription............................. 12 3.3.2 Cartesianspacedescription.......................... 12 3.3.3 Descriptionofoperationalmotion....................... 13 3.4Dynamicmodelofarobotmanipulator....................... 14 3.4.1 The concept of dynamic model and its role . 14 3.4.2 Dynamicmodelofatwo-linkmanipulator.................. 15 3.5Dynamicmodelinthegeneralcase.......................... 18 3.6Dynamiccontrolaccordingtolinearcontroltheory................. 18 3.6.1 Theobjectiveofcontroltheory........................ 19 CONTENTS iii 3.6.2 Open-loop equations for motion of a physical system . 19 3.6.3 Closed-loopequationofmotion........................ 19 3.6.4 Stability, damping and natural frequency of the closed-loop system . 21 3.6.5 Positioncontrol................................. 23 3.6.6 Integralcorrection............................... 23 3.6.7 Trajectoryfollowing.............................. 24 3.6.8 Controllawpartitioning............................ 24 3.7 Motion Control of Non-Linear and Time-Varying Systems . 26 3.7.1 Designofnonlinearcontrollaws....................... 26 3.8Multi-VariableControlSystems............................ 27 3.9Multi-variableProblemManipulators......................... 28 3.10PracticalConsiderations................................ 30 3.10.1 Lack of knowledge of parameters . 30 3.11TimeEffectsinComputingtheModel........................ 30 3.12 Present Industrial Robot Control Systems . 31 3.12.1IndividualjointPIDcontrol.......................... 31 3.12.2 Individual joint PID control with effective joint inertia . 32 3.12.3Inertialdecoupling............................... 32 3.13CartesianBasedControlSystems........................... 32 3.13.1Comparisonwithjointbasedschemes.................... 33 4 KINEMATICS OF THE RIGID BODY 1 4.1Introduction....................................... 2 4.2Therotationoperator................................. 2 4.2.1 Propertiesofrotation............................. 3 4.2.2 Remark..................................... 3 4.3Positionandorientationofarigidbody....................... 3 4.4 Algebraic expression of the rotation operator . 4 4.5Theplanerotationoperator.............................. 5 4.6 Finite rotation in terms of direction cosines . 6 4.7Finiterotationintermsofdyadicproducts..................... 7 4.8CompositionofFiniteRotations........................... 8 4.8.1 Composition rule of rotations . 8 4.8.2 Non commutative character of finite rotations . 8 4.9Eulerangles....................................... 9 4.9.1 Singularvalues................................. 10 4.9.2 Inversion.................................... 10 4.10FiniterotationsintermsofBryantangles...................... 11 4.10.1Singularities................................... 12 4.10.2Inversion.................................... 12 4.11 Unique rotation about an arbitrary axis . 12 4.11.1inversion..................................... 15 4.12FiniterotationsintermsofEulerparameters.................... 16 4.13Rodrigues’parameters................................. 17 4.14Translationandangularvelocities........................... 18 4.15Explicitexpressionforangularvelocities....................... 19 4.15.1IntermsofEulerAngles............................ 20 4.15.2IntermsofBryantangles........................... 20 4.15.3 In terms of Euler parameters . 21 4.16 Infinitesimal displacement . 21 iv CONTENTS 4.17 Accelerations . 21 4.18Screworhelicoidalmotion............................... 22 4.19 Homogeneous representation of vectors . 24 4.20 Homogeneous representation of frame transformations . 25 4.21Successivehomogeneoustransformations....................... 27 4.22Objectmanipulationinspace............................. 28 4.23Inversionofhomogeneoustransformation...................... 30 4.24 Closed loop of homogeneous transformation . 30 4.25Homogeneousrepresentationofvelocities....................... 32 4.26 Homogeneous representation of accelerations . 33 5 KINEMATICS OF SIMPLY CONNECTED OPEN-TREE STRUCTURES 1 5.1Introduction....................................... 2 5.2LinkdescriptionbyHartenberg-Denavitmethod.................. 2 5.3 Kinematic description of an open-tree simply connected structure . 5 5.3.1 Linkcoordinatesystemassignment...................... 6 5.4GeometricmodelofthePUMA560.......................... 7 5.5LinkdescriptionusingtheShethmethod....................... 11 5.6Sheth’sgeometrictransformation........................... 14 5.7Relativemotiontransformations........................... 15 5.7.1 Therevolutepair ............................... 15 5.7.2 Theprismaticpair............................... 15 5.7.3 Thecylindricalpair.............................. 17 5.7.4 Thescrewjoint ................................ 17 5.7.5 Thegearpair ................................. 18 5.8Sheth’sdescriptionofthePUMA560......................... 18 5.9Inversionofgeometrickinematicmodel....................... 20 5.10 Closed form inversion of PUMA 560 robot . 22 5.10.1Decouplingbetweenpositionandorientation................ 22 5.10.2Pieper’stechnique............................... 22 5.10.3 General procedure to determine jointangles................. 23 5.10.4 Calculation of θ1 ................................ 24 5.10.5 Calculation of θ2 and