Mechanisms and Robots Analysis with MATLAB®

480 Pages·2009·2.62 MB·English

by  Dan B. Marghitu (auth.)

®
Mechanisms and Robots Analysis with MATLAB
Dan B. Marghitu
Mechanisms and Robots
®
Analysis with MATLAB

123
Dan B. Marghitu, Professor
Mechanical Engineering Department
Auburn University
270 Ross Hall
Auburn, AL 36849
USA

ISBN 978-1-84800-390-3 e-ISBN 978-1-84800-391-0
DOI 10.1007/978-1-84800-391-0
Springer Dordrecht Heidelberg London New York

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toStefania,toDaniela,
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Preface
Mechanismsandrobotshavebeenandcontinuetobeessentialcomponentsofme-
chanicalsystems.Mechanismsandrobotsareusedtotransmitforcesandmoments
and to manipulate objects. A knowledge of the kinematics and dynamics of these
kinematic chains is most important for their design and control. MATLAB(cid:2)R is a
modern tool that has transformed the mathematical calculations methods because
MATLAB not only provides numerical calculations but also facilitates analytical
calculations using the computer. The present textbook uses MATLAB as a tool to
solveproblemsfrommechanismsandrobots.Theintentistoshowtheconvenience
ofMATLABformechanismandrobotanalysis.UsingexampleproblemstheMAT-
LABsyntaxwillbedemonstrated.MATLABisveryusefulintheprocessofderiv-
ingsolutionsforanyprobleminmechanismsorrobots.Thebookincludesalarge
numberofproblemsthatarebeingsolvedusingMATLAB.Theprogramsareavail-
ableasappendicesattheendofthisbook.
Chapter 1 comments on the fundamentals properties of closed and open kine-
matic chains especially of problems of motion, degrees of freedom, joints, dyads,
and independent contours. Chapter 2 demonstrates the use of MATLAB in find-
ing the positions of planar mechanisms using the absolute Cartesian method. The
positions of the joints are calculated for an input driver angle and for a complete
rotation of the driver link. An external m-file function can be introduced to calcu-
late the positions. The trajectory of a point on a link with general plane motion is
plotted using MATLAB. In Chap. 3 the velocities and acceleration are examined.
MATLABisasuitabletooltodevelopanalyticalsolutionsandnumericalresultsfor
kinematicsusingtheclassicalmethod,thederivativemethod,andtheindependent
contourequations.InChap.4,thejointforcesarecalculatedusingthefree-bodydi-
agramofindividuallinks,thefree-bodydiagramofdyads,andthecontourmethod.
MATLABfunctionsareappliedtofindandsolvethealgebraicequationsofmotion.
Problems of dynamics using the Newton–Euler method are discussed in Chap. 5.
Theequationsofmotionareinferredwithsymbolicalcalculationandthesystemof
differentialequationsissolvedwithnumericaltechniques.Finally,thelastchapter
usescomputeralgebratofindLagrange’sequationsandKane’sdynamicalequations
forspatialrobots.
vii
Contents
1 Introduction................................................... 1
1.1 DegreesofFreedomandMotion.............................. 1
1.2 KinematicPairs ............................................ 3
1.3 Dyads .................................................... 8
1.4 IndependentContours....................................... 10
1.5 PlanarMechanismDecomposition ............................ 10
2 PositionAnalysis............................................... 15
2.1 AbsoluteCartesianMethod .................................. 15
2.2 Slider-Crank(R-RRT)Mechanism ............................ 16
2.3 Four-Bar(R-RRR)Mechanism ............................... 20
2.4 R-RTR-RTRMechanism .................................... 27
2.5 R-RTR-RTRMechanism:CompleteRotation ................... 31
2.5.1 MethodI:ConstraintConditions ....................... 31
2.5.2 MethodII:EuclidianDistanceFunction ................. 35
2.6 PathofaPointonaLinkwithGeneralPlaneMotion............. 37
2.7 CreatingaMovie........................................... 40
3 VelocityandAccelerationAnalysis ............................... 43
3.1 Introduction ............................................... 43
3.2 VelocityFieldforaRigidBody............................... 44
3.3 AccelerationFieldforaRigidBody ........................... 46
3.4 MotionofaPointthatMovesRelativetoaRigidBody ........... 50
3.5 Slider-Crank(R-RRT)Mechanism ............................ 53
3.6 Four-Bar(R-RRR)Mechanism ............................... 60
3.7 InvertedSlider-CrankMechanism............................. 65
3.8 R-RTR-RTRMechanism .................................... 71
3.9 DerivativeMethod.......................................... 79
3.10 IndependentContourEquations............................... 95
ix
x Contents
4 DynamicForceAnalysis ........................................109
4.1 EquationofMotionforGeneralPlanarMotion..................109
4.2 D’Alembert’sPrinciple......................................114
4.3 Free-BodyDiagrams........................................115
4.4 ForceAnalysisUsingDyads .................................116
4.4.1 RRRDyad..........................................116
4.4.2 RRTDyad ..........................................118
4.4.3 RTRDyad ..........................................119
4.5 ForceAnalysisUsingContourMethod.........................120
4.6 Slider-Crank(R-RRT)Mechanism ............................121
4.6.1 InertiaForcesandMoments ...........................124
4.6.2 JointForcesandDriveMoment ........................126
4.7 R-RTR-RTRMechanism ....................................147
4.7.1 InertiaForcesandMoments ...........................151
4.7.2 JointForcesandDriveMoment ........................154
5 DirectDynamics:Newton–EulerEquationsofMotion..............183
5.1 CompoundPendulum .......................................183
5.2 DoublePendulum ..........................................192
5.3 One-LinkPlanarRobotArm .................................201
5.4 Two-LinkPlanarRobotArm .................................204
6 AnalyticalDynamicsofOpenKinematicChains ...................209
6.1 GeneralizedCoordinatesandConstraints.......................209
6.2 LawsofMotion............................................211
6.3 Lagrange’sEquationsforTwo-LinkRobotArm .................213
6.4 RotationTransformation.....................................225
6.5 RRTRobotArm ...........................................228
6.5.1 DirectDynamics.....................................228
6.5.2 InverseDynamics....................................246
6.5.3 Kane’sDynamicalEquations ..........................250
6.6 RRTRRobotArm ..........................................257
7 Problems......................................................275
7.1 ProblemSet:Mechanisms ...................................275
7.2 ProblemSet:Robots ........................................291
A ProgramsofChapter2:PositionAnalysis.........................301
A.1 Slider-Crank(R-RRT)Mechanism ............................301
A.2 Four-Bar(R-RRR)Mechanism ...............................303
A.3 R-RTR-RTRMechanism ....................................306
A.4 R-RTR-RTRMechanism:CompleteRotation ...................309
A.5 R-RTR-RTRMechanism:CompleteRotationUsingEuclidian
DistanceFunction ..........................................312
A.6 PathofaPointonaLinkwithGeneralPlaneMotion:R-RRT
Mechanism................................................314
Contents xi
A.7 PathofaPointonaLinkwithGeneralPlaneMotion:R-RRR
Mechanism................................................315
B ProgramsofChapter3:VelocityandAccelerationAnalysis .........317
B.1 Slider-Crank(R-RRT)Mechanism ............................317
B.2 Four-Bar(R-RRR)Mechanism ...............................322
B.3 InvertedSlider-CrankMechanism.............................326
B.4 R-RTR-RTRMechanism ....................................331
B.5 R-RTR-RTRMechanism:DerivativeMethod ...................339
B.6 InvertedSlider-CrankMechanism:DerivativeMethod............344
B.7 R-RTRMechanism:DerivativeMethod ........................347
B.8 R-RRRMechanism:DerivativeMethod........................349
B.9 R-RTR-RTRMechanism:ContourMethod .....................354
C ProgramsofChapter4:DynamicForceAnalysis ..................363
C.1 Slider-Crank(R-RRT)Mechanism:Newton–EulerMethod........363
C.2 Slider-Crank(R-RRT)Mechanism:D’Alembert’sPrinciple .......368
C.3 Slider-Crank(R-RRT)Mechanism:DyadMethod ...............372
C.4 Slider-Crank(R-RRT)Mechanism:ContourMethod .............378
C.5 R-RTR-RTRMechanism:Newton–EulerMethod................382
C.6 R-RTR-RTRMechanism:DyadMethod .......................396
C.7 R-RTR-RTRMechanism:ContourMethod .....................408
D ProgramsofChapter5:DirectDynamics .........................423
D.1 CompoundPendulum .......................................423
D.2 CompoundPendulumUsingtheFunctionR(t,x) ..............425
D.3 DoublePendulum ..........................................426
D.4 DoublePendulumUsingtheFileRR.m ........................428
D.5 One-LinkPlanarRobotArm .................................430
D.6 One-LinkPlanarRobotArmUsingthem-FileFunction
Rrobot.m ...............................................432
D.7 Two-Link Planar Robot Arm Using the m-File Function
RRrobot.m ..............................................433
E ProgramsofChapter6:AnalyticalDynamics .....................437
E.1 Lagrange’sEquationsforTwo-LinkRobotArm .................437
E.2 Two-LinkRobotArm:InverseDynamics.......................442
E.3 RRTRobotArm ...........................................444
E.4 RRTRobotArm:InverseDynamics ...........................453
E.5 RRTRobotArm:Kane’sDynamicalEquations..................457
E.6 RRTRRobotArm ..........................................462
References.........................................................475
Index .............................................................477
Chapter 1
Introduction
1.1 DegreesofFreedomandMotion
The number of degrees of freedom (DOF) of a mechanical system is equal to the
numberofindependentparameters(measurements)thatareneededtouniquelyde-
fineitspositioninspaceatanyinstantoftime.ThenumberofDOFisdefinedwith
respecttoareferenceframe.
Figure1.1showsarigidbody(RB)lyinginaplane.Thedistancebetweentwo
particlesontherigidbodyisconstantatanytime.Ifthisrigidbodyalwaysremains
in the plane, three parameters (three DOF) are required to completely define its
position:twolinearcoordinates(x,y)todefinethepositionofanyonepointonthe
rigidbody,andoneangularcoordinateθ todefinetheangleofthebodywithrespect
totheaxes.Theminimumnumberofmeasurementsneededtodefineitspositionare
showninthefigureasx,y,andθ.Arigidbodyinaplanethenhasthreedegreesof
freedom. The particular parameters chosen to define its position are not unique.
Any alternative set of three parameters could be used. There is an infinity of sets
of parameters possible, but in this case there must always be three parameters per
set,suchastwolengthsandanangle,todefinethepositionbecausearigidbodyin
planemotionhasthreeDOF.
Sixparametersareneededtodefinethepositionofafreerigidbodyinathree-
dimensional (3-D) space. One possible set of parameters that could be used are
z
θ
RB
Fig.1.1 Rigidbodyinplanar
y
motion with three DOF:
translationalongthex-axis,
translationalongthey-axis,
and rotation, θ, about the x
z-axis
1