MODERN ROBOTICS n
MECHANICS, PLANNING, AND CONTROL
n n n
KEVIN M. LYNCH
n n n AND n F RANK C. PARK
n n
Mayn 3,n 2017
Thisn documentn isn then preprintn versionn of
Modern Robotics
n
Mechanics,n Planning, nandnControl
⃝cn KevinnM.nLynchnandnFranknC.nPark
Thisnpreprintnisnbeingnmadenavailablenfornpersonalnusenonlynandnnotnfornfurther nd
istribution.n Then bookn willn ben publishedn byn Cambridgen Universityn Pressn innMa
yn2017,nISBNn9781107156302.n CitationsnofnthenbooknshouldncitenCambridgenUni
versitynPressnasnthenpublisher,nwithnanpublicationndatenofn2017.n Originalnfiguresnf
romnthisnbooknmaynbenreusednprovidednproper ncitationnisngiven.n Moreninformatio
nn onn then book,n includingn software,n videos,n andn an feedbackn formn cannben found
n atn http://modernrobotics.org.n n Commentsn are n welcome!
,
,Contents
Forewordn by nRoger nBrockett ix
Forewordn by n Matthew n Mason xi
Preface xiii
1 Preview 1
2 Configurationn Space 11
2.1 Degreesn ofn Freedomnofn an Rigidn Body.........................................................................12
2.2 DegreesnofnFreedomnofn anRobot .....................................................................................15
2.2.1 Robotn Joints...............................................................................................................16
2.2.2 Gr übler’s nFormula .................................................................................................17
2.3 Configurationn Space:n Topologyn andn Representation......................................23
2.3.1 Configurationn Spacen Topology......................................................................23
2.3.2 Configurationn Spacen Representation.........................................................25
2.4 Configurationn andn Velocityn Constraints...................................................................29
2.5 Taskn Spacen andn Workspace .............................................................................................32
2.6 Summary.......................................................................................................................................36
2.7 Notesn andn References...........................................................................................................38
2.8 Exercises........................................................................................................................................38
3 Rigid-Body n Motions 59
3.1 Rigid-Bodyn Motionsn inn then Plane ................................................................................62
3.2 Rotationsn andn Angularn Velocities ................................................................................68
3.2.1 Rotationn Matrices ..................................................................................................68
3.2.2 Angularn Velocities..................................................................................................76
3.2.3 Exponentialn Coordinaten Representationn ofn Rotation.....................79
3.3 Rigid-Bodyn Motionsn andn Twists...................................................................................89
i
, 3.3.1 Homogeneousn Transformationn Matrices............................................... 89
3.3.2 Twists............................................................................................................................. 97
3.3.3 Exponentialn Coordinaten Representationn ofn Rigid-Bodyn Mo-
ntions .............................................................................................................................104
3.4 Wrenches ...................................................................................................................................108
3.5 Summary.....................................................................................................................................111
3.6 Software ......................................................................................................................................113
3.7 Notesn andn References .......................................................................................................115
3.8 Exercises .....................................................................................................................................116
4 Forwardn Kinematics 137
4.1 Productn ofn Exponentialsn Formula ............................................................................140
4.1.1 Firstn Formulation:n Screwn Axesn inn then Basen Frame....................141
4.1.2 Examples....................................................................................................................143
×
4.1.3 Second n Formulation:n Screwn Axes n inn the n End-Effector n Frame148
4.2 Then Universaln Robotn Descriptionn Format ..........................................................152
4.3 Summary.....................................................................................................................................158
4.4 Software ......................................................................................................................................159
4.5 Notesn andn References .......................................................................................................160
4.6 Exercises .....................................................................................................................................160
5 Velocity n Kinematics n and n S tatics 171
5.1 Manipulator n Jacobian ........................................................................................................178
5.1.1 Spacen Jacobian .......................................................................................................178
5.1.2 Bodyn Jacobian ........................................................................................................183
5.1.3 Visualizingn then Spacen andn Bodyn Jacobian ..........................................185
5.1.4 Relationshipn betweenn then Spacen andn Bodyn Jacobian.................187
5.1.5 Alternativen Notionsn ofn then Jacobian .......................................................187
5.1.6 Lookingn Aheadn to n Inversen Velocityn Kinematics .............................189
5.2 Staticsn ofn Openn Chains .....................................................................................................190
5.3 Singularityn Analysis .............................................................................................................191
5.4 Manipulability ..........................................................................................................................196
5.5 Summary .....................................................................................................................................200
5.6 Software ......................................................................................................................................201
5.7 Notesn andn References .......................................................................................................201
5.8 Exercises .....................................................................................................................................202
6 InversenKinematics 219
6.1 Analyticn Inversen Kinematics ..........................................................................................221
6.1.1 6Rn PUMA-Type n Arm .........................................................................................221
6.1.2 Stanford-Typen Arms..........................................................................................225
6.2 NumericalnInversenKinematics .....................................................................................226
6.2.1 Newton–Raphsonn Method.............................................................................227
MECHANICS, PLANNING, AND CONTROL
n n n
KEVIN M. LYNCH
n n n AND n F RANK C. PARK
n n
Mayn 3,n 2017
Thisn documentn isn then preprintn versionn of
Modern Robotics
n
Mechanics,n Planning, nandnControl
⃝cn KevinnM.nLynchnandnFranknC.nPark
Thisnpreprintnisnbeingnmadenavailablenfornpersonalnusenonlynandnnotnfornfurther nd
istribution.n Then bookn willn ben publishedn byn Cambridgen Universityn Pressn innMa
yn2017,nISBNn9781107156302.n CitationsnofnthenbooknshouldncitenCambridgenUni
versitynPressnasnthenpublisher,nwithnanpublicationndatenofn2017.n Originalnfiguresnf
romnthisnbooknmaynbenreusednprovidednproper ncitationnisngiven.n Moreninformatio
nn onn then book,n includingn software,n videos,n andn an feedbackn formn cannben found
n atn http://modernrobotics.org.n n Commentsn are n welcome!
,
,Contents
Forewordn by nRoger nBrockett ix
Forewordn by n Matthew n Mason xi
Preface xiii
1 Preview 1
2 Configurationn Space 11
2.1 Degreesn ofn Freedomnofn an Rigidn Body.........................................................................12
2.2 DegreesnofnFreedomnofn anRobot .....................................................................................15
2.2.1 Robotn Joints...............................................................................................................16
2.2.2 Gr übler’s nFormula .................................................................................................17
2.3 Configurationn Space:n Topologyn andn Representation......................................23
2.3.1 Configurationn Spacen Topology......................................................................23
2.3.2 Configurationn Spacen Representation.........................................................25
2.4 Configurationn andn Velocityn Constraints...................................................................29
2.5 Taskn Spacen andn Workspace .............................................................................................32
2.6 Summary.......................................................................................................................................36
2.7 Notesn andn References...........................................................................................................38
2.8 Exercises........................................................................................................................................38
3 Rigid-Body n Motions 59
3.1 Rigid-Bodyn Motionsn inn then Plane ................................................................................62
3.2 Rotationsn andn Angularn Velocities ................................................................................68
3.2.1 Rotationn Matrices ..................................................................................................68
3.2.2 Angularn Velocities..................................................................................................76
3.2.3 Exponentialn Coordinaten Representationn ofn Rotation.....................79
3.3 Rigid-Bodyn Motionsn andn Twists...................................................................................89
i
, 3.3.1 Homogeneousn Transformationn Matrices............................................... 89
3.3.2 Twists............................................................................................................................. 97
3.3.3 Exponentialn Coordinaten Representationn ofn Rigid-Bodyn Mo-
ntions .............................................................................................................................104
3.4 Wrenches ...................................................................................................................................108
3.5 Summary.....................................................................................................................................111
3.6 Software ......................................................................................................................................113
3.7 Notesn andn References .......................................................................................................115
3.8 Exercises .....................................................................................................................................116
4 Forwardn Kinematics 137
4.1 Productn ofn Exponentialsn Formula ............................................................................140
4.1.1 Firstn Formulation:n Screwn Axesn inn then Basen Frame....................141
4.1.2 Examples....................................................................................................................143
×
4.1.3 Second n Formulation:n Screwn Axes n inn the n End-Effector n Frame148
4.2 Then Universaln Robotn Descriptionn Format ..........................................................152
4.3 Summary.....................................................................................................................................158
4.4 Software ......................................................................................................................................159
4.5 Notesn andn References .......................................................................................................160
4.6 Exercises .....................................................................................................................................160
5 Velocity n Kinematics n and n S tatics 171
5.1 Manipulator n Jacobian ........................................................................................................178
5.1.1 Spacen Jacobian .......................................................................................................178
5.1.2 Bodyn Jacobian ........................................................................................................183
5.1.3 Visualizingn then Spacen andn Bodyn Jacobian ..........................................185
5.1.4 Relationshipn betweenn then Spacen andn Bodyn Jacobian.................187
5.1.5 Alternativen Notionsn ofn then Jacobian .......................................................187
5.1.6 Lookingn Aheadn to n Inversen Velocityn Kinematics .............................189
5.2 Staticsn ofn Openn Chains .....................................................................................................190
5.3 Singularityn Analysis .............................................................................................................191
5.4 Manipulability ..........................................................................................................................196
5.5 Summary .....................................................................................................................................200
5.6 Software ......................................................................................................................................201
5.7 Notesn andn References .......................................................................................................201
5.8 Exercises .....................................................................................................................................202
6 InversenKinematics 219
6.1 Analyticn Inversen Kinematics ..........................................................................................221
6.1.1 6Rn PUMA-Type n Arm .........................................................................................221
6.1.2 Stanford-Typen Arms..........................................................................................225
6.2 NumericalnInversenKinematics .....................................................................................226
6.2.1 Newton–Raphsonn Method.............................................................................227
