Solution Manual
For
Robot Dynamics and Control
Second Edition
Mark W. Spong, Seth Hutchinson, and M. Vidyasagar
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and Writing Skills: Busin
,Contents
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and Writing Skills: Busin
1 INTRODUCTION 5
1.1 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 History of Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Components and Structure of Robots . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Symbolic Representation of Robots . . . . . . . . . . . . . . . . . . . 8
1.3.2 Degrees of Freedom and Workspace . . . . . . . . . . . . . . . . . . 9
1.3.3 Classification of Robots ................................................................................... 10
1.3.4 Common Kinematic Arrangements................................................................ 11
1.3.5 Robotic Systems ............................................................................................... 15
1.3.6 Accuracy and Repeatability ............................................................................ 16
1.3.7 Wrists and End-Effectors ................................................................................ 18
1.4 Outline of the Text ....................................................................................................... 20
2 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS 29
2.1 Representing Positions ................................................................................................. 29
2.2 Representing Rotations ................................................................................................ 31
2.2.1 Rotation in the plane....................................................................................... 32
2.2.2 Rotations in three dimensions ....................................................................... 34
2.3 Rotational Transformations ........................................................................................ 36
2.3.1 Summary ........................................................................................................... 40
2.4 Composition of Rotations ............................................................................................ 40
2.4.1 Rotation with respect to the current coordinate frame .............................. 40
2.4.2 Rotation with respect to a fixed frame ......................................................... 42
2.4.3 Summary ........................................................................................................... 44
2.5 Parameterizations of Rotations ................................................................................... 45
2.5.1 Euler Angles...................................................................................................... 45
2.5.2 Roll, Pitch, Yaw Angles................................................................................... 47
2.5.3 Axis/Angle Representation ............................................................................. 48
2.6 Homogeneous Transformations ................................................................................... 51
3
3 FORWARD KINEMATICS: THE DENAVIT-HARTENBERG CONVEN-
TION 57
3.1 Kinematic Chains ........................................................................................................... 57
3.2 Denavit Hartenberg Representation ............................................................................ 60
, 3.2.1 Existence and uniqueness issues ..................................................................... 61
3.2.2 Assigning the coordinate frames ..................................................................... 63
3.2.3 Summary ............................................................................................................ 66
3.3 Examples ......................................................................................................................... 67
4 INVERSE KINEMATICS 79
4.1 The General Inverse Kinematics Problem .................................................................. 79
4.2 Kinematic Decoupling ................................................................................................... 81
4.3 Inverse Position: A Geometric Approach ................................................................... 83
4.4 Inverse Orientation ........................................................................................................ 89
5 VELOCITY KINEMATICS – THE MANIPULATOR JACOBIAN 95
5.1 Angular Velocity: The Fixed Axis Case...................................................................... 96
5.2 Skew Symmetric Matrices ............................................................................................. 97
5.3 Angular Velocity: The General Case ......................................................................... 100
5.4 Addition of Angular Velocities................................................................................... 101
5.5 Linear Velocity of a Point Attached to a Moving Frame ....................................... 102
5.6 Derivation of the Jacobian .......................................................................................... 103
5.6.1 Angular Velocity.............................................................................................. 104
5.6.2 Linear Velocity................................................................................................. 104
5.7 Examples ....................................................................................................................... 109
5.8 The Analytical Jacobian ............................................................................................. 111
5.9 Singularities .................................................................................................................. 113
5.9.1 Decoupling of Singularities ............................................................................ 114
5.9.2 Wrist Singularities ........................................................................................... 115
5.9.3 Arm Singularities ............................................................................................ 115
5.10 Inverse Velocity and Acceleration .............................................................................. 119
5.11 Redundant Robots and Manipulability ..................................................................... 120
5.11.1 Redundant Manipulators ................................................................................ 120
5.11.2 The Inverse Velocity Problem for Redundant Manipulators ..................... 121
5.11.3 Singular Value Decomposition (SVD) ........................................................... 122
5.11.4 Manipulability .................................................................................................. 124
6 COMPUTER VISION 127
6.1 The Geometry of Image Formation ........................................................................... 127
6.1.1 The Camera Coordinate Frame .................................................................... 128
6.1.2 Perspective Projection .................................................................................... 128
6.1.3 The Image Plane and the Sensor Array ....................................................... 129
6.2 Camera Calibration ...................................................................................................... 130
6.2.1 Extrinsic Camera Parameters....................................................................... 130
6.2.2 Intrinsic Camera Parameters ........................................................................ 131
6.2.3 Determining the Camera Parameters .......................................................... 131
6.3 Segmentation by Thresholding.................................................................................. 134
6.3.1 A Brief Statistics Review .............................................................................. 134
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and
Writing Skills: Busin
6.3.2 Automatic Threshold Selection .................................................................... 136
, 6.4 Connected Components ............................................................................................. 140
6.5 Position and Orientation ............................................................................................ 143
6.5.1 Moments.......................................................................................................... 143
6.5.2 The Centroid of an Object ............................................................................ 144
6.5.3 The Orientation of an Object........................................................................ 144
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and
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7 PATH PLANNING AND COLLISION AVOIDANCE 147
7.1 The Configuration Space ........................................................................................... 148
7.2 Path Planning Using Configuration Space Potential Fields .................................. 151
7.2.1 The Attractive Field ...................................................................................... 152
7.2.2 The Repulsive field ........................................................................................ 153
7.2.3 Gradient Descent Planning ........................................................................... 154
7.3 Planning Using Workspace Potential Fields ............................................................ 155
7.3.1 Defining Workspace Potential Fields ........................................................... 156
7.3.2 Mapping workspace forces to joint forces and torques.............................. 158
7.3.3 Motion Planning Algorithm .......................................................................... 162
7.4 Using Random Motions to Escape Local Minima................................................... 163
7.5 Probabilistic Roadmap Methods............................................................................... 164
7.5.1 Sampling the configuration space ................................................................ 165
7.5.2 Connecting Pairs of Configurations.............................................................. 165
7.5.3 Enhancement .................................................................................................. 167
7.5.4 Path Smoothing .............................................................................................. 167
7.6 Historical Perspective................................................................................................. 168
8 TRAJECTORY PLANNING 169
8.1 The Trajectory Planning Problem............................................................................ 169
8.2 Trajectories for Point to Point Motion..................................................................... 170
8.2.1 Cubic Polynomial Trajectories ..................................................................... 172
8.2.2 Multiple Cubics............................................................................................... 175
8.2.3 Quintic Polynomial Trajectories................................................................... 175
8.2.4 Linear Segments with Parabolic Blends (LSPB) ......................................... 180
8.2.5 Minimum Time Trajectories ......................................................................... 183
8.3 Trajectories for Paths Specified by Via Points ....................................................... 185
8.3.1 4-3-4 trajectories ............................................................................................. 186
9 DYNAMICS 187
9.1 The Euler-Lagrange Equations.................................................................................. 187
9.1.1 One Dimensional System............................................................................... 188
9.1.2 The General Case ........................................................................................... 190
9.2 General Expressions for Kinetic and Potential Energy .......................................... 196
9.2.1 The Inertia Tensor......................................................................................... 197
9.2.2 Kinetic Energy for an n-Link Robot ............................................................ 198
9.2.3 Potential Energy for an n-Link Robot ......................................................... 199
9.3 Equations of Motion ................................................................................................... 199
For
Robot Dynamics and Control
Second Edition
Mark W. Spong, Seth Hutchinson, and M. Vidyasagar
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and Writing Skills: Busin
,Contents
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and Writing Skills: Busin
1 INTRODUCTION 5
1.1 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 History of Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Components and Structure of Robots . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Symbolic Representation of Robots . . . . . . . . . . . . . . . . . . . 8
1.3.2 Degrees of Freedom and Workspace . . . . . . . . . . . . . . . . . . 9
1.3.3 Classification of Robots ................................................................................... 10
1.3.4 Common Kinematic Arrangements................................................................ 11
1.3.5 Robotic Systems ............................................................................................... 15
1.3.6 Accuracy and Repeatability ............................................................................ 16
1.3.7 Wrists and End-Effectors ................................................................................ 18
1.4 Outline of the Text ....................................................................................................... 20
2 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS 29
2.1 Representing Positions ................................................................................................. 29
2.2 Representing Rotations ................................................................................................ 31
2.2.1 Rotation in the plane....................................................................................... 32
2.2.2 Rotations in three dimensions ....................................................................... 34
2.3 Rotational Transformations ........................................................................................ 36
2.3.1 Summary ........................................................................................................... 40
2.4 Composition of Rotations ............................................................................................ 40
2.4.1 Rotation with respect to the current coordinate frame .............................. 40
2.4.2 Rotation with respect to a fixed frame ......................................................... 42
2.4.3 Summary ........................................................................................................... 44
2.5 Parameterizations of Rotations ................................................................................... 45
2.5.1 Euler Angles...................................................................................................... 45
2.5.2 Roll, Pitch, Yaw Angles................................................................................... 47
2.5.3 Axis/Angle Representation ............................................................................. 48
2.6 Homogeneous Transformations ................................................................................... 51
3
3 FORWARD KINEMATICS: THE DENAVIT-HARTENBERG CONVEN-
TION 57
3.1 Kinematic Chains ........................................................................................................... 57
3.2 Denavit Hartenberg Representation ............................................................................ 60
, 3.2.1 Existence and uniqueness issues ..................................................................... 61
3.2.2 Assigning the coordinate frames ..................................................................... 63
3.2.3 Summary ............................................................................................................ 66
3.3 Examples ......................................................................................................................... 67
4 INVERSE KINEMATICS 79
4.1 The General Inverse Kinematics Problem .................................................................. 79
4.2 Kinematic Decoupling ................................................................................................... 81
4.3 Inverse Position: A Geometric Approach ................................................................... 83
4.4 Inverse Orientation ........................................................................................................ 89
5 VELOCITY KINEMATICS – THE MANIPULATOR JACOBIAN 95
5.1 Angular Velocity: The Fixed Axis Case...................................................................... 96
5.2 Skew Symmetric Matrices ............................................................................................. 97
5.3 Angular Velocity: The General Case ......................................................................... 100
5.4 Addition of Angular Velocities................................................................................... 101
5.5 Linear Velocity of a Point Attached to a Moving Frame ....................................... 102
5.6 Derivation of the Jacobian .......................................................................................... 103
5.6.1 Angular Velocity.............................................................................................. 104
5.6.2 Linear Velocity................................................................................................. 104
5.7 Examples ....................................................................................................................... 109
5.8 The Analytical Jacobian ............................................................................................. 111
5.9 Singularities .................................................................................................................. 113
5.9.1 Decoupling of Singularities ............................................................................ 114
5.9.2 Wrist Singularities ........................................................................................... 115
5.9.3 Arm Singularities ............................................................................................ 115
5.10 Inverse Velocity and Acceleration .............................................................................. 119
5.11 Redundant Robots and Manipulability ..................................................................... 120
5.11.1 Redundant Manipulators ................................................................................ 120
5.11.2 The Inverse Velocity Problem for Redundant Manipulators ..................... 121
5.11.3 Singular Value Decomposition (SVD) ........................................................... 122
5.11.4 Manipulability .................................................................................................. 124
6 COMPUTER VISION 127
6.1 The Geometry of Image Formation ........................................................................... 127
6.1.1 The Camera Coordinate Frame .................................................................... 128
6.1.2 Perspective Projection .................................................................................... 128
6.1.3 The Image Plane and the Sensor Array ....................................................... 129
6.2 Camera Calibration ...................................................................................................... 130
6.2.1 Extrinsic Camera Parameters....................................................................... 130
6.2.2 Intrinsic Camera Parameters ........................................................................ 131
6.2.3 Determining the Camera Parameters .......................................................... 131
6.3 Segmentation by Thresholding.................................................................................. 134
6.3.1 A Brief Statistics Review .............................................................................. 134
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and
Writing Skills: Busin
6.3.2 Automatic Threshold Selection .................................................................... 136
, 6.4 Connected Components ............................................................................................. 140
6.5 Position and Orientation ............................................................................................ 143
6.5.1 Moments.......................................................................................................... 143
6.5.2 The Centroid of an Object ............................................................................ 144
6.5.3 The Orientation of an Object........................................................................ 144
knowledge and practical application. Depending on the subject area, these exams can vary significantly in format. Common types of business exams and
Writing Skills: Busin
7 PATH PLANNING AND COLLISION AVOIDANCE 147
7.1 The Configuration Space ........................................................................................... 148
7.2 Path Planning Using Configuration Space Potential Fields .................................. 151
7.2.1 The Attractive Field ...................................................................................... 152
7.2.2 The Repulsive field ........................................................................................ 153
7.2.3 Gradient Descent Planning ........................................................................... 154
7.3 Planning Using Workspace Potential Fields ............................................................ 155
7.3.1 Defining Workspace Potential Fields ........................................................... 156
7.3.2 Mapping workspace forces to joint forces and torques.............................. 158
7.3.3 Motion Planning Algorithm .......................................................................... 162
7.4 Using Random Motions to Escape Local Minima................................................... 163
7.5 Probabilistic Roadmap Methods............................................................................... 164
7.5.1 Sampling the configuration space ................................................................ 165
7.5.2 Connecting Pairs of Configurations.............................................................. 165
7.5.3 Enhancement .................................................................................................. 167
7.5.4 Path Smoothing .............................................................................................. 167
7.6 Historical Perspective................................................................................................. 168
8 TRAJECTORY PLANNING 169
8.1 The Trajectory Planning Problem............................................................................ 169
8.2 Trajectories for Point to Point Motion..................................................................... 170
8.2.1 Cubic Polynomial Trajectories ..................................................................... 172
8.2.2 Multiple Cubics............................................................................................... 175
8.2.3 Quintic Polynomial Trajectories................................................................... 175
8.2.4 Linear Segments with Parabolic Blends (LSPB) ......................................... 180
8.2.5 Minimum Time Trajectories ......................................................................... 183
8.3 Trajectories for Paths Specified by Via Points ....................................................... 185
8.3.1 4-3-4 trajectories ............................................................................................. 186
9 DYNAMICS 187
9.1 The Euler-Lagrange Equations.................................................................................. 187
9.1.1 One Dimensional System............................................................................... 188
9.1.2 The General Case ........................................................................................... 190
9.2 General Expressions for Kinetic and Potential Energy .......................................... 196
9.2.1 The Inertia Tensor......................................................................................... 197
9.2.2 Kinetic Energy for an n-Link Robot ............................................................ 198
9.2.3 Potential Energy for an n-Link Robot ......................................................... 199
9.3 Equations of Motion ................................................................................................... 199
