1.5 LIST OF FIGURES
Number Figure Caption Page
2.1 The Problem Hierarchy of Robot Motion Planning .......................................... 5
2.2 An Illustration of a Path Plan vs. Trajectory Plan ............................................. 6
2.3 A Sample Robot Path ....................................................................................... 8
2.4 Point to Point and Continuous Paths ................................................................. 9
2.5 A Sample Trajectory Plan ............................................................................... 10
2.6 Examples of Geometric Path Planning Problems ........................................... 21
3.1 Basic Structure of an Artificial Neuron .......................................................... 28
3.2 Common Activation Functions ...................................................................... 30
3.3 Simple Feedforward Artificial Neural Networks ............................................ 31
3.4 Backpropagation of Errors.............................................................................. 32
4.1 A Single Input/Single Output Controller ........................................................ 38
4.2 A Self Tuning Controller ................................................................................ 39
4.3 A Model Referenced Adaptive Controller ...................................................... 41
4.4 A Feed Forward Controller ............................................................................ 41
4.5 An Example of a Neuromorphic Controller .................................................... 43
5.1 A Simple Planar Two Link Manipulator ......................................................... 52
5.2 Forward Kinematics ....................................................................................... 54
5.3 Inverse Kinematics (elbow up) ....................................................................... 55
5.4 Inverse Kinematics (elbow down) .................................................................. 55
5.5 Common Constants for Dynamics Equations (Algorithmic) .......................... 57
5.6 Equations for Inverse Dynamics (Algorithmic) ............................................. 57
5.7 Equations for Forward Dynamics (Algorithmic) ............................................ 58
5.8 Equations for Inverse Dynamics (Closed Form) ............................................ 59
5.9 Equations for Forward Dynamics (Closed Form Joint 2) ............................... 60
5.10 Equations for Forward Dynamics (Closed Form Joint 1) ............................... 61
5.11 A Sample Velocity Profile .............................................................................. 62
5.12 Block Diagram of Discrete Time Step Motion Planning/Control ................... 64
5.13 Robot Motion Over a Sample, Near-Optimal Torque Path ............................. 66
6.1 An example of a Velocity Profile ..................................................................... 72
6.2 Neural Network Motion Planner Architecture ................................................ 73
6.3 Basic Configuration for Neural Network Training .......................................... 74
6.4 Normalized Map of Test Path End Points ........................................................ 78
6.5 Flowchart for Illustration of Selection of Training/Testing Data ..................... 79
6.6 Network Training Convergence ...................................................................... 80
7.1 Maximum Velocity Profile .............................................................................. 84
7.2 A Maximum Velocity Path .............................................................................. 85
7.3 Maximum Velocity Profile with Oscillation .................................................... 86
7.4 Maximum Velocity Profile with Reduced Time Step ...................................... 87
7.5 Maximum Velocity Profile with Increased Time Step ..................................... 88
7.6 Case A: The last velocity step is scaled down to match the stop point ............. 89
7.7 Case B: Reduced Motion velocity is used. ...................................................... 90
7.8 Case C: The first and last time step velocities are reduced ............................... 91
7.9 Maximum Velocity Controller for Robot ........................................................ 91
7.10 Least Time Verses Unison Finish .................................................................... 92
7.11 Velocity Steps and Their Difference Equation ................................................ 93
7.12 Convergence of Neural Network for Maximum Velocity Motion ................... 94
7.13 Neural Network Convergence Statistics ......................................................... 95
7.14 Random Set of Path Endpoints For Testing Neural Network Control .............. 96
7.15 Algorithm and Neural Network Comparison for Test Path 1 ........................... 97
7.16 Path Times for Test Paths ................................................................................ 98
7.17 Position Graphs for Algorithm and Neural Network (Case 2) ........... 99
7.18 Position Graphs for Algorithm and Neural Network (Case 3) ........... 99
7.19 Position Graphs for Algorithm and Neural Network (Case 4) ......... 100
7.20 Position Graphs for Algorithm and Neural Network (Case 5) ......... 100
7.21 Position Graphs for Algorithm and Neural Network (Case 6) ......... 101
7.22 Position Graphs for Algorithm and Neural Network (Case 7) ......... 101
7.23 Position Graphs for Algorithm and Neural Network (Case 8) ......... 102
7.24 Position Graphs for Algorithm and Neural Network (Case 9) ......... 102
7.25 Position Graphs for Algorithm and Neural Network (Case 10) ....... 103
8.1 Velocity and Acceleration for Maximum Acceleration Motion .................... 105
8.2 Position Diagram for Maximum Acceleration .............................................. 105
8.3 Case A: The first and last time steps are scaled to match the stop point ......... 107
8.4 Case B: Smooth Midpoint Inflection ............................................................. 108
8.5 Case C: Reduced Velocity ............................................................................. 108
8.6 Case D: Acceleration Hybrid Case ................................................................ 109
8.7 Case E: Acceleration Hybrid Case ................................................................ 109
8.8 Case F: A Reduced Inflection with an odd number of steps ........................... 110
8.9 Case G: The Inverse of Case E ...................................................................... 110
8.10 Case H: Smoothed start and inflection points (even numbers of steps) .......... 111
8.11 Case I: Smoothed start and inflection (odd numbers of steps) ........................ 111
8.12 Case 1: Optimal Path (even numbers of steps) ............................................... 112
8.13 Case 2: Optimal Path (odd numbers of steps) ................................................ 114
8.14 Maximum Acceleration Controller for Robot ............................................... 115
8.15 Training Convergence for Maximum Acceleration Network ........................ 116
8.16 Neural Network Convergence Statistics ........................................................ 117
8.17 Comparison of Algorithm and Neural Network for Test Path 1 ..................... 117
8.18 Path Times for Test Paths .............................................................................. 118
8.19 Comparison of Algorithm and Neural Network for Test Path 2 ..................... 119
8.20 Comparison of Algorithm and Neural Network for Test Path 3 ..................... 120
8.21 Comparison of Algorithm and Neural Network for Test Path 4 ..................... 120
8.22 Comparison of Algorithm and Neural Network for Test Path 5 ..................... 121
8.23 Comparison of Algorithm and Neural Network for Test Path 6 ..................... 121
8.24 Comparison of Algorithm and Neural Network for Test Path 7 ..................... 122
8.25 Comparison of Algorithm and Neural Network for Test Path 8 ..................... 122
8.26 Comparison of Algorithm and Neural Network for Test Path 9 ..................... 123
8.27 Comparison of Algorithm and Neural Network for Test Path 10 ................... 123
9.1 Spline Continuity .......................................................................................... 125
9.2 Some First order B-Splines are shown for Constructing An Example Spline. 126
9.3 Bezier Spline Approximation of a Curve ...................................................... 127
9.4 A Hermite Spline Segment ............................................................................ 129
9.5 A Bezier Spline Segment .............................................................................. 131
9.6 A Bezier Spline Segment for Motion Modelling ........................................... 135
9.7 A Flowchart for Bezier Spline Modelling of a Motion .................................. 137
10.1 A Motion Plan With A Bezier Spline ............................................................ 139
10.2 The Use of Cosine Functions for Path Planning ............................................ 140
10.3 Robot Motions in Joint Space ........................................................................ 142
10.4 Joint Space Effect of Coupling Factors ......................................................... 145
10.5 Sample Paths for Varied Coupling Values ..................................................... 147
10.6 Approximate Graph of Effect of Coupling Factor on Motion Time ............... 149
10.7 Final Path Produced by Coupling, and Iterative Reduction ........................... 150
11.1 Hookes and Jeeves Search Method ............................................................... 159
11.2 An Example of a Hookes and Jeeves Search Deadlock ................................. 161
11.3 Example of a Random Walk Search .............................................................. 163
11.4 Near-Optimal Path Plan for Previous Test Case ........................................... 165
11.5 Near-Optimal Path Plan for Case 1 ............................................................... 165
11.6 Near-Optimal Path Plan for Case 2 ............................................................... 167
11.7 Near-Optimal Path Plan for Case 3 ............................................................... 167
11.8 Near-Optimal Path Plan for Case 4 ................................................................ 168
11.9 Near-Optimal Path Plan for Case 5 ................................................................ 168
11.10 Near-Optimal Path Plan for Case 6 ................................................................ 169
11.11 Near-Optimal Path Plan for Case 7 ................................................................ 169
11.12 Near-Optimal Path Plan for Case 8 ................................................................ 170
11.13 Near-Optimal Path Plan for Case 9 ................................................................ 170
11.14 Near-Optimal Path Plan for Case 10 .............................................................. 171
12.1 Maximum Torque Controller for Robot ........................................................ 174
12.2 Neural Network Convergence for Maximum Torque Planner ....................... 177
12.3 Neural Network Convergence Statistics ....................................................... 178
12.4 Path Times for Test Paths .............................................................................. 179
12.5 Maximum Torque Control With a Neural Network, Case 1 ........................... 181
12.6 Maximum Torque Control With a Neural Network, Case 2 ........................... 181
12.7 Maximum Torque Control With a Neural Network, Case 3 ........................... 182
12.8 Maximum Torque Control With a Neural Network, Case 4 ........................... 182
12.9 Maximum Torque Control With a Neural Network, Case 5 ........................... 183
12.10 Maximum Torque Control With a Neural Network, Case 6 ........................... 183
12.11 Maximum Torque Control With a Neural Network, Case 7 ........................... 184
12.12 Maximum Torque Control With a Neural Network, Case 8 ........................... 184
12.13 Maximum Torque Control With a Neural Network, Case 9 ........................... 185
12.14 Maximum Torque Control With a Neural Network, Case 10 ......................... 185
12.15 Comparison of Neural Network to Ideal Solution ......................................... 186
12.16 Test Case With Time Step of 0.05 seconds .................................................... 189
12.17 Test Case With Time Step of 0.10 seconds .................................................... 190
12.18 Test Case With Time Step of 0.20 seconds .................................................... 190
13.1 Output Scaling for Maximum Torque Control .............................................. 192
13.2 Path Times for Test Paths, Including Scaled Neural Network Outputs .......... 193
13.3 Maximum Torque Control With a Scaled Neural Network, Case 1 ............... 193
13.4 Maximum Torque Control With a Scaled Neural Network, Case 2 ............... 194
13.5 Maximum Torque Control With a Scaled Neural Network, Case 3 ............... 194
13.6 Maximum Torque Control With a Scaled Neural Network, Case 4 ............... 195
13.7 Maximum Torque Control With a Scaled Neural Network, Case 5 ............... 195
13.8 Maximum Torque Control With a Scaled Neural Network, Case 6 ............... 196
13.9 Maximum Torque Control With a Scaled Neural Network, Case 7 ............... 196
13.10 Maximum Torque Control With a Scaled Neural Network, Case 8 ............... 197
13.11 Maximum Torque Control With a Scaled Neural Network, Case 9 ............... 197
13.12 Maximum Torque Control With a Scaled Neural Network, Case 10 ............. 198
14.1 Use of Maximum Torque Controller for Path Tracking ................................. 203
14.2 Direct Application of Maximum Torque Controller ...................................... 204
14.3 Application of Maximum Torque Controller to Acceleration Control .......... 205
14.4 Direct Application of Maximum Acceleration Controller ............................. 205
14.5 Maximum Acceleration Controller Applied to Torque Controllers ............... 205
14.6 Maximum Acceleration Controller Applied to Velocity Control .................. 206
14.7 Direct Application of Velocity Controller ..................................................... 206
A.1 A Typical Two Link Manipulator ................................................................. 210
A.2 Inverse Kinematics for a Two Link Manipulator .......................................... 211
A.3 Derivation of Inverse Kinematics for a Two Link Manipulator .................... 212
B.1 Schematic Diagram of Two Link Manipulator .............................................. 214
L.1 Main Window for User Interface ................................................................... 362
L.2 Auxiliary Window for Graphing Function .................................................... 367