force velocity control with neural network

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FORCE VELOCITY CONTROL WITH NEURAL NETWORK COMPENSATION FOR CONTOUR TRACKING WITH PNEUMATIC ACTUATION

BY

MOHAMMED ABU MALLOUH

A thesis submitted to the Department of Mechanical and Materials Engineering in conformity with the requirements for the degree of Doctor of Philosophy

Queen’s University Kingston, Ontario, Canada September, 2008

Copyright © Mohammed Abu Mallouh, 2008

ABSTRACT Mohammed Abu Mallouh: Force Velocity Control with Neural Network Compensation for Contour Tracking with Pneumatic Actuation. Ph.D. Thesis, Queen’s University, August, 2008.

Control of the contact force between a robot manipulator and a workpiece is critical for successful execution of tasks where the robot’s end effector must perform a contact operation along the contour of a workpiece. Representative tasks include polishing, grinding and deburring. Considerable research has been conducted on force control with electric robots. By contrast, little research has been conducted on force control with pneumatic robots. The later has the potential to be considerably cheaper. However, the compressible nature of air as the working fluid and relatively high friction means pneumatic robots are more difficult to control. The subject of this thesis is the design and testing of a controller that regulates the normal contact force and tangential velocity of the end effector of a pneumatic gantry robot while tracking the contour of a planar workpiece. Both experimental and simulation results are presented.

A PI Force Velocity (FV) controller for contour tracking was designed and tested experimentally. Three different workpiece edge geometries were studied: straight, inclined and curved. The tracking performance with the PI FV controller was comparable to the performance reported by other researchers with a similar controller implemented with an electric robot.

This result

confirms the potential of pneumatically actuated robots in force control applications.

A system model was developed and validated in order to investigate the parameters that affect performance. A good match between experiment and simulation was achieved when the friction of the z-axis cylinder was modeled with a Displacement Dependent Friction Model (DDFM) instead of a Velocity Dependent Friction Model (VDFM). Subsequently, a DDFM based friction compensator was designed and tested. However, it was found that performance could not be improved even with perfect friction compensation, due to the effects of system lag.

Two Neural Network (NN) compensators were designed to compensate for both the lag and friction in the system. Simulation results for straight and curved edges were used to examine the effectiveness of the NN compensators. The performance of the PI FV controller was found to improve significantly when a NN compensator was added. This result confirms the value of NN’s in control compensation for tracking applications with pneumatic actuation.

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ACKNOWLEDGEMENTS With a deep sense of gratitude, I wish to express my sincere thanks to my supervisor, Dr. Brian Surgenor, for his motivation and supervision through out the course of this research.

My sincere thanks are due to Dr. Ron Anderson, for the help extended to me when I approached him and the valuable discussion that I had with him during the course of my study at Queen’s University. I will neglect my duty if I do not mention my fellow research scholars, the laboratory staff and administrative staff of the Mechanical and Materials Engineering Department for their timely help, with special thanks to Mr. O. Oosten and Mr. A. McPhail for their quick and great help with my experimental setup.

The financial support provided by the Hashemite University and Queen’s University is greatly acknowledged.

I also want to thank my parents, who taught me the value of hard work by their own example. They rendered me enormous support during the whole tenure of my research. Also special thanks to my father and mother in law for their support.

Lastly, I am grateful to my wife Kholoud for the inspiration and moral support she provided throughout my research work. Without her loving support and understanding I would never have completed my present work. Particularly, I owe to my sons Karam and Yazan for their silent prayer for my work at the time when they needed my company most.

Finally, I would like to thank all whose direct and indirect support helped me complete my thesis.

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TABLE OF CONTENTS ABSTRACT.....................................................................................................................................ii ACKNOWLEDGEMENTS............................................................................................................iii TABLE OF CONTENTS................................................................................................................iv LIST OF FIGURES ......................................................................................................................viii LIST OF TABLES......................................................................................................................... xv LIST OF NOMENCLATURE.....................................................................................................xvii Chapter 1 INTRODUCTION........................................................................................................... 1 1.1 Background and Motivation .................................................................................................. 2 1.2 Objectives .............................................................................................................................. 4 1.3 Contributions ......................................................................................................................... 4 1.4 Thesis Outline ........................................................................................................................ 5 Chapter 2 LITERATURE REVIEW................................................................................................ 7 2.1 Force Control for Robots ....................................................................................................... 7 2.2 Force Control for Pneumatic Robots ................................................................................... 12 2.3 Contour Tracking ................................................................................................................. 20 2.4 Neural Networks .................................................................................................................. 21 2.4.1 Neural Networks in Control.......................................................................................... 23 2.5 Fuzzy Logic Systems ........................................................................................................... 25 2.5.1 Fuzzy Logic in Control ................................................................................................. 26 2.6 Neural Networks and Fuzzy Logic in Control ..................................................................... 28 2.6.1 Adaptive Neuro Fuzzy Inference System ..................................................................... 28 2.7 Friction and Friction Compensation .................................................................................... 31 2.7.1 Friction Models............................................................................................................. 31 2.7.2 Friction Compensation.................................................................................................. 32 2.8 Summary .............................................................................................................................. 36 Chapter 3 EXPERIMENTAL CONTROLLER DESIGN ............................................................. 38 3.1 Apparatus Description ......................................................................................................... 38 3.1.1 Pneumatic Cylinders ..................................................................................................... 40 3.1.2 Valves ........................................................................................................................... 41 3.1.3 Sensors .......................................................................................................................... 41 3.1.4 Data Acquisition and Control ....................................................................................... 42 3.2 Noise in Measured Signals................................................................................................... 44 3.3 Jatta and Benchmark FV Controller..................................................................................... 46

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3.4 Designing the PID FV Controller ........................................................................................ 51 3.4.1 Tuning Step 1................................................................................................................ 53 3.4.2 Tuning Step 2................................................................................................................ 54 3.4.3 Tuning Step 3................................................................................................................ 57 3.5 Effect of System Parameters ................................................................................................ 59 3.5.1 Effect of Surface Finish ................................................................................................ 59 3.5.2 Effect of Profile Shape and Angle ................................................................................ 62 3.5.3 Effect of Tracking Speed .............................................................................................. 64 3.6 Retuned PI FV Controller .................................................................................................... 68 3.6.1 Benchmark Results with Retuned Controller................................................................ 68 3.7 PI FV Controller with VDFM based Compensator.............................................................. 74 3.8 Summary .............................................................................................................................. 76 Chapter 4 MODELING AND SIMULATION RESULTS............................................................ 77 4.1 Dynamic Modeling .............................................................................................................. 77 4.1.1 General Dynamic Modeling.......................................................................................... 77 4.1.2 Contact Modeling ......................................................................................................... 79 4.1.3 System Lag Modeling ................................................................................................... 80 4.1.4 Signal Noise Modeling ................................................................................................. 84 4.1.5 Workpiece Surface Finish Modeling ............................................................................ 86 4.2 Model Validation ................................................................................................................. 87 4.2.1 Validation Stage 1......................................................................................................... 87 4.2.2 Displacement Dependent Friction Model ..................................................................... 90 4.2.3 Modeling and Validating the DDFM............................................................................ 94 4.2.4 ANFIS DDFM .............................................................................................................. 96 4.2.5 Validation Stage 2......................................................................................................... 98 4.3 Effect of System Parameters .............................................................................................. 107 4.3.1 Effect of Unidirectional PPC Valves .......................................................................... 107 4.3.2 Effect of Tracking Speed ............................................................................................ 109 4.3.3 Effect of Uncertainty in Workpiece Angle ................................................................. 110 4.3.4 Effect of Signal Measurement Noise .......................................................................... 111 4.3.5 Effect of System Lag .................................................................................................. 111 4.3.6 Effect of Workpiece Surface Finish............................................................................ 112 4.3.7 Effect of Coulomb Friction......................................................................................... 112 4.3.8 Effect of Workpiece Shape ......................................................................................... 114

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4.4 Workpiece Surface Finish Effect ....................................................................................... 114 4.4.1 Straight Edge Tracking ............................................................................................... 115 4.4.2 Inclined Edge Tracking............................................................................................... 117 4.4.3 Curved Edge Tracking ................................................................................................ 119 4.5 Summary ............................................................................................................................ 121 Chapter 5 NEURAL NETWORK BASED COMPENSATION ................................................. 123 5.1 PI FV Controller with Gain Scheduler............................................................................... 123 5.2 PI FV Controller with Friction Compensation................................................................... 126 5.3 System Lag Compensation ................................................................................................ 128 5.4 PI FV Controller with Neural Network Compensation...................................................... 132 5.4.1 EBPA Neural Network Compensator ......................................................................... 134 5.4.2 MBPM Neural Network Compensator ....................................................................... 136 5.4.3 Results with PI FV Controller and NN Compensation ............................................... 138 5.5 Summary ............................................................................................................................ 146 Chapter 6 CONCLUSIONS AND RECOMMENDATIONS...................................................... 148 6.1 Conclusions........................................................................................................................ 148 6.1.1 Summary Results ........................................................................................................ 149 6.1.2 Experimental Controller Design ................................................................................. 151 6.1.3 Modeling and Simulation Results............................................................................... 152 6.1.4 Neural Network Based Compensation........................................................................ 153 6.2 Recommendations.............................................................................................................. 155 References.................................................................................................................................... 157 Appendix A Technical Specifications.......................................................................................... 164 A.1 SMC VER PPC Valve....................................................................................................... 164 A.2 SMC MY1H Cylinder....................................................................................................... 165 A.3 Bimba Standard Cylinder.................................................................................................. 166 A.4 SMC VEA 250 Power Amplifier ...................................................................................... 167 A.5 FUTEK Bi-Axial Force Sensor......................................................................................... 168 A.6 FUTEK CSG (JM-2A) Amplifier ..................................................................................... 169 A.7 Celesco Position and Velocity Transducer ....................................................................... 170 A.8 Omega Differential Pressure Sensor ................................................................................. 171 A.9 DMD-465WB Signal Conditioner .................................................................................... 172 A.10 dSPACE DS1104 R&D Controller Board ...................................................................... 173 A.11 SKF Ball Bearing ............................................................................................................ 174

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Appendix B Sensor Calibration ................................................................................................... 175 B.1 Force Sensor Calibration ................................................................................................... 175 B.2 Position Sensor Calibration ............................................................................................... 177 B.3 Velocity Sensor Calibration .............................................................................................. 179 B.4 Differential Pressure Sensor Calibration........................................................................... 186 Appendix C System Parameter Identification.............................................................................. 188 C.1 Friction Parameter Identification....................................................................................... 188 C.1.1 Coulomb Friction Identification................................................................................. 188 C.1.2 Viscous Friction Coefficient Identification................................................................ 197 C.2 System Lag Identification ................................................................................................. 198 Appendix D Neural Network Program Details ............................................................................ 201 D.1 Neural Network Compensation with 14 Inputs................................................................. 201 D.2 Neural Network Compensation with 5 Inputs................................................................... 206 Appendix E Supplemental Tuning Results .................................................................................. 208 E.1 Tuning the x-axis Velocity Controller ............................................................................... 208 E.2 Tuning the z-axis Force Controller.................................................................................... 209 E.3 Tuning the x-axis Force Controller.................................................................................... 210 E.4 Tuning the z-axis Velocity Controller ............................................................................... 212

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LIST OF FIGURES Figure 1.1: Contour tracking of a planar glass workpiece during grinding process ........................ 1 Figure 1.2: Contour tracking of an engine part during polishing process........................................ 2 Figure 2.1: End effector design for passive stiffness control........................................................... 8 Figure 2.2: Impedance force control ................................................................................................ 9 Figure 2.3: Block diagram of FP controller ................................................................................... 10 Figure 2.4: Block diagram of parallel FP controller ...................................................................... 11 Figure 2.5: Tracking performance with short tubes (a) full order (b) reduced order ..................... 15 Figure 2.6: Tracking performance with long tubes (a) full order (b) reduced order...................... 15 Figure 2.7: Frequency response, full and reduced order, short tube .............................................. 16 Figure 2.8: Frequency response, full and reduced order, long tube............................................... 17 Figure 2.9: Force tracking with hybrid ANFMRC ........................................................................ 18 Figure 2.10: Labeled generic neuron ............................................................................................. 21 Figure 2.11: General mathematical neuron model......................................................................... 22 Figure 2.12: Some common choices for the activation function ................................................... 22 Figure 2.13: Multiple layers of neurons......................................................................................... 23 Figure 2.14: NN model reference control ...................................................................................... 24 Figure 2.15: Feedforward with NN inverse model control ............................................................ 24 Figure 2.16: NN internal model based control............................................................................... 25 Figure 2.17: NN generalized predictive control............................................................................. 25 Figure 2.18: Generic Fuzzy Logic system ..................................................................................... 26 Figure 2.19: Step response of a second order system with 3 different controllers ........................ 28 Figure 2.20: (a) Sugeno’s FL system and (b) equivalent ANFIS .................................................. 29 Figure 2.21: Friction models (a) Coulomb, (b) Coulomb plus viscous, (c) nonlinear, (d) exponential..................................................................................................................................... 32 Figure 2.22: Selection of friction compensation techniques.......................................................... 33 Figure 2.23: Neuro-fuzzy friction compensator architecture......................................................... 35 Figure 2.24: Experimental and NN estimated friction ................................................................... 35 Figure 3.1: Pneumatic gantry robot with main components labeled.............................................. 39 Figure 3.2: Close up of force sensor showing bearing in contact with edge of workpiece............ 40 Figure 3.3: Profile of the aluminum workpiece used in producing benchmark results ................. 40 Figure 3.4: A typical pneumatic system diagram with PPC valves ............................................... 41 Figure 3.5: Illustration of noise level in the measured signals....................................................... 45

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Figure 3.6: Block diagram of the Dual Force Velocity controller ................................................. 46 Figure 3.7: f n tracking response for iron rectangular edge workpiece.......................................... 47 Figure 3.8: f n tracking response for wooden curved edge workpiece.......................................... 47 Figure 3.9: f n tracking response for iron circular edge workpiece .............................................. 48 Figure 3.10: Adaptive neuro-fuzzy FV controller plus neuro-fuzzy friction compensator ........... 50 Figure 3.11: Block diagram PI FV controller with VDFM based compensator ............................ 50 Figure 3.12: Representation of the gantry robot operating on a planar curved workpiece. ........... 51 Figure 3.13: Block diagram of FV controller with four PID elements shown............................... 52 Figure 3.14: Results of the PID velocity control of x-axis when K iv x was changed (example of Step 1 tuning)................................................................................................................................. 54 Figure 3.15: Responses when tracking straight edge wood workpiece with z-axis force controller combined with the x-axis velocity controller (example of Step 2 tuning) ..................................... 56 Figure 3.16: Responses when tracking straight edge wood workpiece with z-axis velocity controller combined with the x-axis force controller (example of Step 2 tuning) ......................... 56 Figure 3.17: f n and vt responses when tracking wood workpiece with straight edge profile (example of Step 3 tuning)............................................................................................................. 59 Figure 3.18: Surface roughness and surface straightness .............................................................. 61 Figure 3.19: f n response when tracking wood workpiece with straight edge profile................... 61 Figure 3.20: f n response when tracking steel workpiece with straight edge profile .................... 61 Figure 3.21: f n response when tracking aluminum workpiece with straight edge profile ........... 62 Figure 3.22: f n response when tracking wood workpiece with θ = +3 deg ................................. 63 Figure 3.23: f n response when tracking wood workpiece with θ = -10 deg ................................ 63 Figure 3.24: f n response when tracking wood workpiece with θ = +92 deg ............................... 63 Figure 3.25: f n response when tracking steel workpiece with curved edge................................. 64 Figure 3.26: f n and vt responses when tracking aluminum curved edge, with medium vtd ........ 66 Figure 3.27: f n and vt responses when tracking aluminum curved edge, with low vtd ............... 66 Figure 3.28: f n and vt responses when tracking aluminum straight edge, with medium vtd ....... 67 Figure 3.29: f n and vt responses when tracking aluminum straight edge, with low vtd .............. 67 Figure 3.30: Responses when tracking aluminum straight edge ( θ = zero deg)........................... 70 Figure 3.31: Responses when tracking aluminum inclined edge ( θ = 7 deg)............................... 71 Figure 3.32: Responses when tracking aluminum inclined edge ( θ = 16 deg)............................. 72

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Figure 3.33: Responses when tracking aluminum curved profile edge ......................................... 73 Figure 3.34: Responses with PI only & PI with VDFM compensator, aluminum straight edge ... 75 Figure 4.1: Experimental result of m x identification test ............................................................. 79 Figure 4.2: Modeling the contact ................................................................................................... 80 Figure 4.3: Sample τ x identification test, when piston held still at x =1000 mm .......................... 83 Figure 4.4: Sample τ z identification test, when piston held still at z = 100 mm........................... 83 Figure 4.5: NDRNG noise added to θ .......................................................................................... 84 Figure 4.6: Signals with the noise model added ............................................................................ 85 Figure 4.7: θ with the NDRNG signal been added to model surface finish, for straight edge..... 86 Figure 4.8: x-axis simulation result for PI velocity tuning, responses when K PVx = 0.5 and K IVx = [8.0, 4.0, 2.0].................................................................................................................................. 88 Figure 4.9: x-axis experimental result for PI velocity tuning, responses when K PVx = 0.5 and K IVx = [8.0, 4.0, 2.0]............................................................................................................................... 88 Figure 4.10: z-axis simulation result for PI force tuning, responses when K PFz = 0.02 and K IFz = [0.32, 0.16, 0.08]............................................................................................................................ 89 Figure 4.11: z-axis experimental result for PI force tuning, responses when K PFz = 0.02 and K IFz = [0.32, 0.16, 0.08]......................................................................................................................... 89 Figure 4.12: DDFM (a) Dahl model, (b) Bowden and Tabor model ............................................. 91 Figure 4.13: Experimental open loop result for z axis DDFM identification test (with wire force) ....................................................................................................................................................... 92 Figure 4.14: Hysterisis loop of the z-axis DDFM (with wire force) .............................................. 93 Figure 4.15: Experimental open loop result for the z axis DDFM identification test (without wire force).............................................................................................................................................. 93 Figure 4.16: Experiment and simulation open loop results for z-axis DDFM identification......... 94 Figure 4.17: Experimental and simulation hysterisis loop of the z-axis DDFM............................ 95 Figure 4.18: Experimental and simulation result for z-axis PI stationary force control, responses when K PFz = 0.02 and K IFz = 0.08 ................................................................................................ 96 Figure 4.19: ANFIS DDFM for z-axis cylinder............................................................................. 97 Figure 4.20: Friction results of the experimental open loop test, the rule based DDFM and the ANFIS DDFM (off-line)................................................................................................................ 98 Figure 4.21: SIMULINK block diagram for PI FV controller, gantry robot and workpiece......... 99 Figure 4.22: Simulation responses when tracking straight edge ( θ = zero deg.)........................ 101

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Figure 4.23: Simulation responses when tracking inclined edge ( θ = 7 deg.)............................ 102 Figure 4.24: Simulation responses when tracking inclined edge ( θ = 16 deg.).......................... 103 Figure 4.25: Simulation responses when tracking curved edge................................................... 104 Figure 4.26: Experimental and simulation results for straight edge, with experimental θ ......... 105 Figure 4.27: Close up of the f n response for straight edge, with experimental θ ..................... 105 Figure 4.28: Experimental and simulation results for curved edge, with experimental θ .......... 106 Figure 4.29: Close up of the f n response for curved edge, with experimental θ ...................... 106 Figure 4.30: Responses of PI FV controller with bidirectional and unidirectional PPC valves .. 108 Figure 4.31: Responses when there is zero, 30 and 50 deg errors in θ ....................................... 110 Figure 4.32: Close up of the tracking period for straight edge workpiece, with circles showing significant f n errors .................................................................................................................... 116 Figure 4.33: Close up of the period from 23.6 to 24.0 sec for straight edge ............................... 117 Figure 4.34: Close up of the tracking period of inclined ( θ = 7 deg.) edge workpiece, with circles showing significant f n error ....................................................................................................... 118 Figure 4.35: Close up of the tracking period of inclined ( θ = 16 deg.) edge workpiece, with circles showing significant f n error............................................................................................ 119 Figure 4.36: Close up of the tracking period of curved edge workpiece, with circles showing significant f n errors .................................................................................................................... 120 Figure 5.1: Responses with PI FV controller and gain scheduler, curved edge........................... 124 Figure 5.2: Close up of the f n response in Figure 5.1 ................................................................ 125 Figure 5.3: Change in gains during contouring the curved edge ................................................. 125 Figure 5.4: Block diagram of the PI FV controller with friction compensation .......................... 127 Figure 5.5: f n response with PI FV and PI with friction compensators with/without lag in the system, showing effect of lag....................................................................................................... 127 Figure 5.6: Root locus of the z-axis force model with lag in the system ..................................... 129 Figure 5.7: Root locus of the z-axis force model without lag in the system ................................ 129 Figure 5.8: Smith predictor control scheme................................................................................. 130 Figure 5.9: NN Smith predictor and fuzzy-PID controller for pneumatic system ....................... 131 Figure 5.10: Block diagram of PD controller with NN compensation for pneumatic system ..... 132 Figure 5.11: Block diagram of PI FV controller with NN compensator...................................... 133 Figure 5.12: Structure of a three layered feedforward NN compensator with 14 inputs ............. 133

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Figure 5.13: Responses for straight edge tracking, with PI FV controller and EBPA NN compensator, zero initial weights and 14 inputs .......................................................................... 140 Figure 5.14: Close up of the f n response in Figure 5.13 ............................................................ 140 Figure 5.15: Responses for straight edge tracking, with PI FV controller and EBPA NN compensator, 0.1 initial weights and 14 inputs............................................................................ 141 Figure 5.16: Close up of the f n response in Figure 5.15 ............................................................ 141 Figure 5.17: Responses for straight edge tracking with PI FV and MBPM NN compensator, zero initial weights and 14 inputs ........................................................................................................ 142 Figure 5.18: Close up of the f n response in Figure 5.17 ............................................................ 142 Figure 5.19: Responses for curved edge tracking with PI FV controller and MBPM NN compensator, zero initial weights and 14 inputs .......................................................................... 143 Figure 5.20: Close up of the f n response in Figure 5.19 ............................................................ 143 Figure 5.21: General structure of a three layered feedforward NN compensator, with 5 inputs . 145 Figure A.1: Technical specification of VER PPC valve .............................................................. 164 Figure A.2: Technical specification of SMC MY1H cylinder..................................................... 165 Figure A.3: Technical specification of Bimba cylinder ............................................................... 166 Figure A.4: Technical specification of SMC VEA250 amplifier ................................................ 167 Figure A.5: Technical specification of FUTEK MBA400 force sensor ...................................... 168 Figure A.6: Technical specification of FUTEK CSG (JM-2A) amplifier.................................... 169 Figure A.7: Technical specification of Celesco cable extension transducer................................ 170 Figure A.8: Technical specification of Omega PX137 ................................................................ 171 Figure A.9: Technical specification of Omega DMD-460WB amplifier..................................... 172 Figure A.10: Technical specification of dSPACE DS1104 R&D controller board ..................... 157 Figure A.11: Technical specification of SKF ball bearing .......................................................... 174 Figure B.1: Force sensor calibration wooden jig ......................................................................... 175 Figure B.2: Curve fitting for the x-axis force sensor calibration data.......................................... 176 Figure B.3: Curve fitting for the z-axis force sensor calibration data.......................................... 176 Figure B.4: Curve fitting for the x-axis position sensor calibration data ..................................... 177 Figure B.5: Curve fitting for the z-axis position sensor calibration data ..................................... 178 Figure B.6: x-axis velocity sensor calibration, with 1.5 rad/sec sinusoidal setpoint. .................. 180 Figure B.7: x-axis velocity sensor calibration, with 2.5 rad/sec sinusoidal setpoint. .................. 181 Figure B.8: x-axis velocity sensor calibration, with 5.0 rad/sec sinusoidal setpoint. .................. 182 Figure B.9: z-axis velocity sensor calibration, with 1.5 rad/sec sinusoidal setpoint. .................. 183 xii

Figure B.10: z-axis velocity sensor calibration, with 2.5 rad/sec sinusoidal setpoint. ................ 184 Figure B.11: z-axis velocity sensor calibration, with 5.0 rad/sec sinusoidal setpoint. ................ 185 Figure B.12: Curve fitting for the x-axis differential pressure sensor calibration data................ 186 Figure B.13 Curve fitting for the z-axis differential pressure sensor calibration data ................. 187 Figure C.1: Block diagram of the program used for x-axis Coulomb friction identification....... 189 Figure C.2: Block diagram of the program used for z-axis Coulomb friction identification ....... 190 Figure C.3: Experimental results of Coulomb friction identification for x-axis, Case 1. ............ 191 Figure C.4: Experimental results of Coulomb friction identification for x-axis, Case 2. ............ 192 Figure C.5: Experimental results of Coulomb friction identification for x-axis, Case 3. ............ 193 Figure C.6: Experimental results of Coulomb friction identification for z-axis, Case 1.............. 194 Figure C.7: Experimental results of Coulomb friction identification for z-axis, Case 2.............. 195 Figure C.8: Experimental results of Coulomb friction identification for z-axis, Case 3.............. 196 Figure C.9: τ x identification test, when piston held still at x =1000 mm..................................... 198 Figure C.10: τ x identification test, when piston held still at x =500 mm..................................... 198 Figure C.11: τ x identification test, when piston held still at x =0 mm......................................... 199 Figure C.12: τ z identification test, when piston held still at z =100 mm ..................................... 199 Figure C.13: τ z identification test, when piston held still at z =0 mm ......................................... 200 Figure D.1: Over all model, NN compensator with 14 inputs added to main program ............... 201 Figure D.2: NN compensator model with 14 inputs .................................................................... 202 Figure D.3: MBPM NN compensator model ............................................................................... 203 Figure D.4: EBPA NN compensator model................................................................................. 204 Figure D.5: Screen captures of input parameters page for the EBPA NN, with 14 inputs .......... 205 Figure D.6: Screen captures of input parameters page for the MBPM NN, with 14 inputs ........ 205 Figure D.7: Over all model, NN compensator with 5 inputs added to main program ................. 206 Figure D.8: MBPM NN compensator model with 5 inputs ......................................................... 206 Figure D.9: Screen captures of input parameters page for the MBPM NN, with 5 inputs. ......... 207 Figure E.1: Block diagram of the PID v x controller ................................................................... 208 v

Figure E.2: Results of the PID velocity control of x-axis when K px was changed.................... 208 Figure E.3: Block diagram of PID f z controller......................................................................... 209 Figure E.4: Results of the PID force control of z-axis when K pf z was changed ........................ 209 Figure E.5: Results of the PID force control of z-axis when K i f z was changed ........................ 210

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Figure E.6: Block diagram of PID f x controller......................................................................... 210 Figure E.7: Results of the PID force control of x-axis when K pfx was changed .......................... 211 Figure E.8: Results of the PID force control of x-axis when K i fx was changed .......................... 211 Figure E.9: Results of the PID force control of x-axis when K dfx was changed .......................... 212 Figure E.10: Block diagram of vz controller............................................................................... 212 Figure E.11: Results of the PID velocity control of z-axis when K pvz was changed .................... 213 Figure E.12: Results of the PID velocity control of z-axis when K ivz was changed.................... 213 Figure E.13: Results of the PID velocity control of z-axis when K dvz was changed.................... 214

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LIST OF TABLES Table 2.1: Categorization of references by type of control and application .................................. 13 Table 3.1: Signal and channel assignments ................................................................................... 43 Table 3.2: Summary of main features of major components used in the apparatus ...................... 43 Table 3.3: Signal to noise ratio ...................................................................................................... 44 Table 3.4: Summary of Jatta et al’s results with DFV controller................................................... 48 Table 3.5: Gains after Step 1 tuning .............................................................................................. 54 Table 3.6: Gains after Step 2 tuning .............................................................................................. 55 Table 3.7: Gains after Step 3 tuning .............................................................................................. 59 Table 3.8: Summary of tracking performance for straight edge with different materials.............. 60 Table 3.9: Performance with curved aluminum workpiece and three different vtd values........... 65 Table 3.10: Performance with straight aluminum workpiece and three different vtd values........ 65 Table 3.11: Gains when straight aluminum workpiece ( θ = 30 deg) used in tuning. ................... 68 Table 3.12: Summary of the experimental tracking performance for different profiles ................ 69 Table 3.13: Identified friction model parameters........................................................................... 74 Table 3.14: Performance of the PI FV controller with/without VDFM based compensator ......... 75 Table 4.1: Summary of the τ x identification test, when x =1000 mm .......................................... 82 Table 4.2: Summary of the τ x identification test, when x = 500 mm ........................................... 82 Table 4.3: Summary of the τ x identification test, when x = zero mm .......................................... 82 Table 4.4: Summary of the τ z identification test, when z = 100 mm............................................ 82 Table 4.5: Summary of the τ z identification test, when z = zero mm ........................................... 82 Table 4.6: Gains of the PI FV controller used in simulation tests ................................................. 99 Table 4.7: Summary of simulation tracking performances, with DDFM and artificial θ .......... 100 Table 4.8: Summary of simulation tracking performances, with DDFM and experimental θ ... 100 Table 4.9: Simulation performance for curved edge, with three different v td values................. 109 Table 4.10: Simulation performance for straight edge, with three different v td values.............. 109 Table 4.11: Simulation tracking performances for straight edge................................................. 113 Table 4.12: Simulation tracking performances for inclined edge ( θ =7 deg.) ............................. 113 Table 4.13: Simulation tracking performances for inclined edge ( θ = 16 deg) .......................... 113 Table 4.14: Simulation tracking performances for curved edge .................................................. 113 xv

Table 4.15: f n and vt RMSE of the 5 cases for straight, inclined and curved edges ................. 114 Table 5.1: Gains lookup table ...................................................................................................... 124 Table 5.2: Performance of the PI FV controller with//without gain scheduler for curved edge .. 125 Table 5.3: Definition, initial value and final value of the EBPA NN design parameters ............ 139 Table 5.4: Definition, initial value and final value of the MBPM NN design parameters .......... 139 Table 5.5: Performance with different controllers and NN compensation .................................. 145 Table 6.1: Summary of experimental and simulation performances with different controllers .. 150 Table C.1: Experimental x-axis Coulomb friction identification tests summary......................... 197 Table C.2: Experimental z-axis Coulomb friction identification tests summary ......................... 197

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LIST OF NOMENCLATURE ai , oi , ci

parameters of the activation function of i node at the first layer of ANFIS

ax , az

acceleration along x and z axis, respectively

bi2 , b13

bias of node i in the hidden and the output layers, respectively

C cn

contact damping coefficient along n direction

C fx , C fz

viscous friction coefficient a long x and z axis, respectively

D

robustifying control parameter in MBPM NN

e fn

difference between actual and desired normal contact force

e fx , e fz

difference between actual and desired contact force along x and z axis, respectively

evt

difference between actual and desired tangential velocity

ev x , ev z

difference between actual and desired velocity along x and z axis, respectively

f ax , f az

applied force by the x-axis and z-axis cylinders, respectively

f n , f nd

actual and desired normal contact force

ft

tangential friction force between bearing and workpiece

fx , fz

contact force along the x and z axes, respectively

fˆi

available estimation of friction force in joint i

Fdx , Fdz

dynamic Coulomb friction force a long x and z axis, respectively

Fx , Fz

Coulomb friction force along x and z axis, respectively

Fsx , Fsz

static Coulomb friction force a long x and z axis, respectively

Fwx , Fwz

position/velocity sensor wire force along the x and z axis, respectively

G

matrix of MBPM NN weights

G

maximum expected value of G

H1

learning rate for V

H2

learning rate for W

H3

learning rate for the node activation function parameters

k z , kv

gain terms in the MBPM NN

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K cn

contact stiffness along n direction

K p , K i , K d proportional, Integral and derivative gains

Kx , Kz

gains of the PPC valves

K df x , K df x

derivative gain of force controller for x and z axis, respectively

K dv x , K dv z

derivative gain of velocity controller for x and z axis, respectively

Kif x , K if z

integral gain of force controller for x and z axis, respectively

K iv x , K iv z

integral gain of velocity controller for x and z axis, respectively

K pf x , K pf z

proportional gain of force controller for x and z axis, respectively

K vpx , K vpz

proportional gain of velocity controller for x and z axis, respectively

L

layer number

mx , mz

effective mass along the x and z axis, respectively

nh

number of nodes in hidden layer

ni

number of inputs in input layer

no

number of nodes in output layer

net iL

sum of the inputs to node i in layer L

N

number of data samples

P

neural network input vector

p 1j

jth input of the input layer

pi , qi , ri

parameters of activation function of node i at the fourth layer of ANFIS

QiL

lower bound of the sigmoidal activation function of node i in layer L

R

coefficient of determination

Ra

average surface roughness

s iL

slope of the sigmoidal activation function of node i in layer L

S1

premise parameter set in ANFIS

S2

consequent parameter set in ANFIS

t

time

ux , uz

total controller signal to the x and z axes PPC valves, respectively

u xComp , u zComp compensator output signal for x and z axis, respectively

xviii

u xf x , u zf z

output signal of PI force controller for x and z axis, respectively

u xPI , u zPI

total output signal of PI force velocity controller for x and z axis, respectively

u xv x , u zv x

output signal of PI velocity controller for x and z axis, respectively

u zNN

output of the NN compensator

U iL

upper bound of the sigmoidal activation function of node i in layer L

vt , vtd

actual and desired tangential velocity, respectively

vx , vz

velocity of end effector along x and z axis, respectively

v xd , v zd

desired velocity along x and z axis, respectively

V

weights vector for the hidden layer

Vi ,2j

weight connecting node i in layer 2 and input p j

wi

output of node i at the second layer of ANFIS

wi

output of node i at the third layer of ANFIS

W

weights vector for the output layer

Wi ,3j

weight connecting node i in the output layer and node j in the hidden layer

x

position of end efector along x-axis

z

position of end effector along z-axis

zw

position of the workpiece along the z axis if there was no contact

xix

Greek letters

σ iL

activation function for node i in layer L

θ

angle between reference frame (x ,z) and task frame (t, n)

δn

spring deflection along n direction

λ

adaptation design parameter in MBPM NN

δ ,γ , β

tunable parameters of the DDFM

α

momentum term in the EBPA NN

Δf az

difference between current and previous value of f az

Δe f n

difference between current and previous value of e f n

ΔPx , ΔPz

differential pressure across the x-axis and z axis cylinders, respectively

ΔQiL

difference between current and previous values of QiL

ΔU iL

difference between current and previous values of U iL

ΔRMSE i

percentage of the RMSE change of case i

ΔsiL

difference between current and previous values of s iL

Δt

sampling time

ΔVi ,2j

difference between current and previous values of Vi ,2j

ΔWi ,3j

difference between current and previous values of Wi ,3j

Δz

difference between current and previous values of z

τ x ,τ z

time constants for time delay in x and z axes, respectively

Acronyms ANFIS

Adaptive Neuro-Fuzzy Inference System

ANFMRC

Adaptive Neuro-Fuzzy Model Reference Control

BPA

Back Propagation Algorithm

DDFM

Displacement Dependent Friction Model

DFV

Dual Force Velocity

DOF

Degrees Of Freedom

EBPA

Extended Back Propagation Algorithm

FC

Friction Compensator

xx

FL

Fuzzy Logic

FP

Force Position

FV

Force Velocity

GS

Gain Scheduling

ME

Maximum Error

MBPM

Modified Back Propagation Method

NA

Not Available

NDRNG

Normally Distributed Random Number Generator

NN

Neural Network

PI

Proportional Integral

PID

Proportional Integral Derivative

PPC

Proportional Pressure Control

RMSE

Root Mean Square Error

RMS

Root mean Square

SNR

Signal to Noise Ratio

SMC

Sliding Mode Controller

VDFM

Velocity Dependent Friction Model

ZN

Ziegler Nichols

xxi

Chapter 1 INTRODUCTION Control of the interaction between a robot manipulator and a workpiece is critical for successful execution of tasks where the robot’s end effector must perform a contact operation on the surface of a workpiece. Representative tasks include polishing, grinding and deburring. Figure 1.1 and Figure 1.2 are examples of contour tracking with an electrical robot during grinding and polishing processes, respectively. In these operations the robot’s end effector must track the contour of the workpiece while applying a specified force. An essential component for successful automation of these tasks is appropriate control of both the contact force and the motion of the end effector (tool). Even with the present state of technology in automation and manufacturing, most of these applications are still performed manually by (very expensive) human operators.

Annual

deburring costs alone are currently estimated at $3.9 billion in the United States and about 10 to 30 percent of total manufacturing operating costs (Pagilla and Yu, 2001).

Consequently,

automation of polishing, grinding and deburring operations is an active area of investigation.

Figure 1.1: Contour tracking of a planar glass workpiece during grinding process (http://www.pushcorp.com)

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Figure 1.2: Contour tracking of an engine part during polishing process (http://www.pushcorp.com)

1.1 Background and Motivation During contour tracking operations, the workpiece sets constraints on the geometric paths that can be followed by the end effector. This situation is generally referred as constrained motion. The use of a pure motion control strategy in such applications is usually unsuccessful, unless accurate models of both the robot manipulator (kinematics and dynamics) and the workpiece (geometry and mechanical features) are available. However, it is not easy to have an accurate model of a robot, especially for a pneumatically actuated robot. Furthermore, detailed knowledge about features of a workpiece is not always available, and if required, would limit robot applications to only those workpieces with known geometries. Thus, for contour tracking, there is a need for a force controller that does not require a formal model of the process or of the workpiece.

Considerable research has been conducted on force control with electric robots. By contrast, little research has been conducted on force control with pneumatic robots. This is understandable as pneumatic robots are much more difficult to control due to the various nonlinear effects. These

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nonlinearities include variable system parameters due to air compressibility, high Coulomb friction forces, lag in system response, variable air supply pressures and cylinder end effects. Over the years, better hardware (e.g., proportional valves and rodless cylinders) and control strategies (e.g., adaptive and sliding mode control) have led to a steady improvement in the performance of pneumatic servosystems. Relative to electrically actuated systems, pneumatically actuated systems are cheaper, cleaner, safer, easier to maintain and have a higher power to weight ratio (Bobrow and McDonell, 1998).

The main motivation for this research is to demonstrate that pneumatically actuated robots can be used in contact operations such as contour tracking. To accomplish this goal, a control system must be developed that can actively compensate for the nonlinearities inherent in a pneumatic gantry robot and simultaneously control both motion and contact force. Contour tracking with pneumatic actuators is a complex system that has unknown disturbances, hard to model dynamics and severe nonlinearities in terms of friction and lag. Usually advanced nonlinear controllers are used in such systems since classic linear controllers are normally unsuccessful in providing acceptable responses in terms of speed and accuracy. Most advanced nonlinear control strategies are based on a detailed mathematical modeling of the system to be controlled. If a simplified mathematical model is used, then performance is sensitive to uncertainties and parameter variations in the robot and the workpiece. This highlights the need for an adaptive controller that is not necessarily based on a mathematical model.

The so-called “intelligent control systems” are modeled after biological systems and human cognitive capabilities. They possess learning, adaptation, and classification capabilities that hold out the hope of improved control for today’s complex system. Well known types of intelligent controllers are Neural Network (NN) and Fuzzy Logic (FL) systems. NN captures the parallel processing and learning capability of biological nervous systems, and FL captures the decisionmaking capabilities of human linguistic and cognitive systems. It is expected that NN and FL are the most suitable systems to be used in compensating for the nonlinearities, disturbances and lag in the contour tracking problem and especially when implemented with a pneumatic robot.

FL controllers are rule based controllers that do not require a formal mathematical model of the system to be controlled. NN provides a means to generate an input/output model of system, again without requiring the user to develop a formal mathematical model. Thus, when used together, FL and NN are effective techniques for the control of complex systems. A number of different

3

neural-fuzzy systems have been developed. A novel example is the Adaptive Neuro-Fuzzy Inference System (ANFIS). ANFIS is a fuzzy inference system that is embedded in to the framework of a NN and trained by a hybrid-learning algorithm.

1.2 Objectives The thesis has three objectives. The first objective is to design and evaluate a Force Velocity (FV) controller for the pneumatic gantry robot as applied to contour tracking to demonstrate that pneumatically actuated robots can be used in such operations. The performance of the FV controller will be reported quantitatively and in sufficient detail that other researchers can make use of the performance results. The second objective is to examine the factors that influence performance in the context of this application. The third objective is to investigate different NN based algorithms as a means to improve performance. Specifically, two NN based approaches will be studied: 1) friction compensation and 2) system lag compensation.

1.3 Contributions This thesis will have the following main contributions: 1) Demonstration that pneumatically actuated robots can be used in challenging automation tasks such as contour tracking. This research will highlight the potential for automating operations such as polishing, deburring and grinding, which need relatively accurate contour tracking, with pneumatic robots. In the past, applications of pneumatic robots in either force or position control operations were limited due to their undesirable nonlinear characteristics. By contrast, there are numerous applications of electric robots in both force and motion control applications. But pneumatic robots are considerably cheaper. Thus, there is the potential to significantly reduce the cost of operations by using pneumatic robots instead of electric robots. One needs to clarify that the suggested applications are not highly precise operations, when compared to machining operations, for example. Thus, one is not suggesting that pneumatic robots could be used for machining. 2) Application and development of a FV controller for 2 Degrees Of Freedom (DOF) contour tracking. A review of the work done on contour tracking and force control with pneumatic actuators reveals that the work was not for contour tracking, where the point of contact is constantly changing. Instead, the work involved a stationary pneumatic

4

actuator applying a controlled force to a fixed point. Thus the problem of contour tracking with pneumatic actuators has not been investigated experimentally before, and this is the subject of this thesis. 3) Demonstration that NN is effective in dealing with system nonlinearity such as friction and lag, and as a result the tracking performance can be improved further. Proportional Integral Derivative (PID) controllers are widely used in industry and, when properly tuned, they can perform very well for linear and mildly nonlinear systems. However, when the system is highly nonlinear, the performance of a PID controller is significantly degraded. In this thesis, it will be shown that it is not necessary to discard a PID controller and replace it with advanced controller such as adaptive or sliding mode for challenging nonlinear applications. Instead, one can use a PID controller in combination with a NN compensator system. 4) This thesis will show the significant effect of friction and lag on force control in contour tracking application with pneumatic actuators.

1.4 Thesis Outline Chapter 2 will present a literature review on direct and indirect force control algorithms for robots, with examples of the most common control algorithms. Examples and performance of different force controllers implemented with pneumatic actuators in non contouring applications will be discussed in detail. Applications that require contour tracking will be discussed together with some examples of controllers that are designed for such applications. NN and FL systems are presented as adaptive non-model based controllers that compensate for nonlinearities and variation in system parameters. Backgrounds and applications of FL and NN in position and force control with electrical and pneumatic actuators will be discussed. Friction types, most common models, and compensation methods will be discussed with examples. The chapter ends with some conclusions and a review of the unique aspects of this research study. Elements of this chapter appear in Abu Mallouh and Surgenor (2006).

Chapter 3 will present the experimental apparatus and initial controller design. The general structure of the adopted PID based FV controller will be documented. The procedure for the design, tuning and testing of the controller will be described. The effect of surface finish, profile shape and tracking speed on contour tracking performance will be examined. Three different workpiece edge geometries will be studied: straight, inclined and curved. The performance of the

5

PI FV controller will be benchmarked against the performance of a similar FV controller but when implemented with an electrical robot. Finally, the effect on performance when a model based friction compensator is added to the PI FV controller will be studied. Elements of this chapter appear in Abu Mallouh and Surgenor (2007).

Chapter 4 will present the development and validation of a model of the system. The model will be used to better understand system behavior and to seek ways to improve system tracking performance. A PI FV controller similar to the one used in Chapter 3 will be adopted. The model will be validated against experimental results. The effects of eight parameters will be investigated: 1) unidirectional nature of the Proportional Pressure Control (PPC) valves, 2) tracking speed, 3) uncertainty in workpiece angle, 4) signal measurement noise, 5) system lag, 6) workpiece surface finish, 7) Coulomb friction and 8) workpiece shape. Elements of this chapter appear in Abu Mallouh and Surgenor (2007).

Chapter 5 will examine a number of techniques to improve tracking performance: 1) PI FV controller with a table look up based gain scheduler. 2) PI FV controller with friction compensation. 3) PI FV controller with Extended Back Propagation Algorithm (EBPA) Neural Network compensation. 4) PI FV controller with Modified Back Propagation Method (MBPM) Neural Network compensation.

Elements of this chapter can be found in Abu Mallouh and Surgenor (2008).

Finally, conclusions and recommendation for future work are given in Chapter 6.

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Chapter 2 LITERATURE REVIEW Control of the contact force between a robot manipulator and a workpiece is critical for successful execution of tasks where the robot’s end effector must perform a contact operation along the contour of a workpiece. Representative tasks include polishing, grinding and deburring. The control of a robotic manipulator can be conceptually divided into two approaches: position control and force control. Historically, position control was the main approach. Although force control has received a great deal of attention over the years, successful practical applications of force control are still relatively limited (Yoshikawa, 2000). In position control the goal is to control the end effector’s pose such that it follows a specified trajectory. By contrast, when the end effector is in contact with a workpiece, the workpiece sets constraints on the geometric paths that can be followed. This situation is referred to as constrained motion control and generally requires the implementation of a force controller. The use of a pure motion control strategy in such applications is usually unsuccessful, unless accurate models of both the robot manipulator (kinematics and dynamics) and the workpiece (geometry and mechanical features) are available.

In this chapter basic strategies of direct and indirect force control will be discussed, mainly in the context of pneumatically actuated systems. The contour tracking task and associated problems will be explained together with the most common controller approaches to such applications. An overview of FL and NN systems will be presented. The friction phenomena, the most common friction models and friction compensators will be discussed in some detail. Elements of this chapter can be found in Abu Mallouh and Surgenor (2006).

2.1 Force Control for Robots Robot force control is the term used to describe control schemes where measurements of the interaction forces are used to alter the trajectory commands given to the joint servos of a robot (Eppinger and Seering, 1992). Force control strategies can be divided into two types: those performing indirect force control and those performing direct force control (Siciliano and Villani, 1999). The main difference between the two types is that the indirect type achieves force control via motion control, without explicit closure of a force feedback loop. The direct type enables one to control the contact force to a desired value by means of explicit closure of the force feedback

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loop. To the indirect type belong stiffness (or compliance) and impedance controllers, where the position error is related to the contact force through an estimate of the stiffness or the impedance of the mechanical system.

Stiffness control can be passive or active. In passive stiffness control, the end effector of a robot is equipped with a mechanical device composed of springs (or springs and dampers). This is illustrated in Figure 2.1. Applications which require a robot to insert a peg in a hole or to write text can be achieved with a combination of passive stiffness and pure position control. This provides a simple, inexpensive solution for certain applications that otherwise could not be accomplished with position control alone. The disadvantage of passive stiffness devices is that they are limited in their range of applicability. To achieve a wider applicability, active control of end effector stiffness is necessary. Active stiffness control can be regarded as a programmable spring. This form of stiffness control is conceptually simple and easy to implement, but it can have stability and accuracy problems because it does not take into account the robot dynamics (Yoshikawa, 2000).

Figure 2.1: End effector design for passive stiffness control (Spong and Viyasagar, 1989) Impedance control is a generalization of stiffness control. Stiffness control is designed to achieve a desired static behavior of the interaction between the robot and the workpiece (Chiaverini et al, 1999). By contrast, impedance control is designed to achieve a desired dynamic behavior. The aim of impedance control is to control the position and force by adjusting the mechanical impedance of the end effector. The robot is described by an equivalent mass-spring-damper system with a contact force as an input. The impedance control block diagram is shown in Figure 2.2. With impedance control, it becomes possible to ensure limited values of contact force, given an estimate of workpiece mechanical properties, position and geometry.

8

However, some

applications require exact control of the contact force. This is possible, in theory, by tuning the impedance control parameters and ensuring accurate knowledge of the workpiece position and geometry. The selection of good impedance parameters that guarantee a satisfactory compliant behavior during the interaction may turnout to be inadequate to ensure accurate tracking of the desired position trajectory when the end effector moves in free space (Siciliano and Villani, 1999). To overcome these limitations, some researchers have used advanced techniques such as neural network control, predictive control and adaptive control to accommodate the lack of knowledge of the robot dynamics and the workpiece position and stiffness (Colbaugh et al, 1995 and Jung and Hsia, 1995).

Figure 2.2: Impedance force control (Siciliano and Villani, 1999) A different approach consists of designing a direct force controller that operates on the calculated error between the desired contact force and the measured contact force. Figure 2.3 illustrates a Force Position (FP) control as proposed by Raibert and Craig (1981). The aim of this controller is to control the position along the unconstrained task direction and the force along the constrained task direction. As shown in Figure 2.3, a control system based on this approach has two separate feedback loops, one for position and one for force. These loops are not implemented simultaneously. The matrix S in Figure 2.3 acts as a “constraint direction detector” and it switches between the force and the position controller. This switching action was shown to be a source of instability (Yoshikawa, 2000). Another disadvantage of this type of controller is that a detailed model of the workpiece is generally needed (Abu Mallouh and Surgenor, 2006).

9

Learning algorithms have been used to improve force control performance as in Jeon and Tomizuka (1993), Lange and Hirzinger (1992) and Lucibello (1993).

Figure 2.3: Block diagram of FP controller (Craig, 1986) Another approach is the parallel FP control strategy, which is considered more effective than the traditional FP control strategy since there is no hard switching between position and force control. In parallel FP control, an outer force control loop is closed around an inner motion or position control loop (Chiaverini et al, 1996). Such a controller is still considered to be a direct force control type as there is still explicit closure of a force feedback control loop. In order to enable motion control along the unconstrained task direction, the desired motion can be controlled by the inner loop of the parallel FP controller, with the force controlled by an outer loop. Figure 2.4 is an example of a design that provides parallel or simultaneous control of force and position. However, the algorithm is designed so as to ensure that force control dominates along the constrained task direction. The main advantages of parallel FP control are: 1) it does not need an accurate workpiece model and 2) there is no switching between the force and the position controller (which was identified as a source of potential instability).

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Figure 2.4: Block diagram of parallel FP controller (Siciliano and Villani, 1999) Yu and Kieffer (1999) proposed an adaptive Force Velocity (FV) controller for following unknown contours of granular materials. The controller’s concept is the same as for FP control but the position controller is replaced with a velocity controller. An adaptive algorithm is used to estimate the granular stiffness and the tangential kinetic friction coefficient. Experimental results show good performance even when there are errors in the initial estimation of the model’s parameters.

Most of the previously mentioned force control strategies include force sensors. A force control system that does not require a force sensor is advantageous as it avoids the expense and added hardware complexity of adding a force sensor to the apparatus. Minami et al, (1996) presented an approach that did not use a force sensor. Their approach estimated the contact force as a function of the state variables and desired contact force, by means of the dynamic equations of the robot and the constraint equation. The main disadvantage of this approach is that an accurate system model is required. They also neglected any friction forces.

Eom et al, (1998) designed an observer based force controller for an electric serial robot that also did not use a force sensor. One observer output is used to estimate internal forces on each robot joint when there are no external forces present. When external forces are present, the output of a second observer is used to estimate the total force, using the first observer as an input. The difference between the outputs of the two observers provides an estimate of the external force. Simulation and experimental results demonstrated that the observer based controller could effectively estimate and control the contact forces. But again, the disadvantage is that the design of the observers requires an accurate model of the system.

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Smith and Hashtrudi-Zaad (2005) took a neural network based approach to an inverse dynamics observer for force in order to avoid the need for complete determination of system dynamics. Experimental results using a twin-pantograph haptic device showed that the NN trained with the acceleration signal provided free motion torques with less than half the error of the model based approach.

2.2 Force Control for Pneumatic Robots Table 2.1 summarizes references cited in this thesis, as categorized by type of control and application. In examining this table, one notes that a lot of work has been done on force control with electrically actuated robots, but little work has been done with pneumatically actuated robots. Relative to electrically actuated systems, pneumatic servosystems are cheaper, cleaner, safer easier to maintain and have a higher power to weight ratio (Bobrow and McDonell, 1998). However, the compressible nature of air as the working fluid and relatively high cylinder friction means pneumatic robots are more difficult to control. The early motion controllers for pneumatic systems were based on linearized mathematical models and classical linear controllers (Situm et al, 2004). A modified pulse width modulation was used in Qi and Surgenor (2003) for servo control of a pneumatic cylinder. To achieve higher positioning accuracy, proportional flow control valves or Proportional Pressure Control (PPC) valves can be used. Subsequently, better mathematical models and nonlinear controllers (adaptive control, sliding mode control) were developed for pneumatic systems, and improved performance results were obtained as in Richer and Hurmuzlu (2000) and Surgenor and Vaughan (1997).

For pure servosystem applications, which do not require any form of force control, one can find multiple references on the subject of pneumatic position control as in Aziz and Bone (2000), Parnichkun and Ngaecharoenkul (2001), Schulte and Hahn (2004), Shibata et al (2002), Situm et al (2004), Ning and Bone (2002) and Qi and Surgenor (2003). On the other hand, references on pneumatic force control are limited as in Ben-Dov and Salcudean (1995), Bobrow and McDonell (1998), Kaitwanidvilai and Parnichkun (2005), Richer and Hurmuzlu (2000) and Ruihua et al (2004).

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Table 2.1: Categorization of references by type of control and application Type of application

Non-intelligent control system

Intelligent control system

Mathematical modeling of pneumatic systems

Modeling and compensation of friction

Pneumatic plus force control

[2], [10], [11], [67]

[3] ,[42], [68]

[10], [11], [42], [49], [50], [56], [66], [74], [82]

Non pneumatic plus force control

[1]*, [18], [19], [21], [22], [26], [28], [35], [36], [37], [38], [39], [45], [54]*, [59], [63], [86] [8], [56], [61], [61], [62], [64], [65], [81] [4], [24], [52]*, [78] There are a lot of papers about this subject [9]**, [22], [23]*,[27]*, [30]*, [51]**, [53]*, [72] [85]**, [79]

[12]*, [40]*, [41], [43]*, [44]*, [76]

[6]*, [7], [14]*, [15], [16]*, [24], [25], [29], [35], [38], [44]*, [49], [58], [78], [83]

Pneumatic plus motion control Non pneumatic plus motion control

Analysis and overview

[20], [49], [50], [60], [69], [70], [71], [74] [16]*, [29], [31]*, [32]*, [33]*, [46]*, [47], [48]*, [55], [61], [87]* [5]**, [13]*, [17]*, [31]*, [34]*, [47], [48]*,[57]*, [77]*, [84]

* Simulation results only presented. ** No simulation or experimental results presented, just theoretical

13

Richer and Hurmuzlu (2000) designed two different Sliding Mode Controllers (SMC’s) for a pneumatic force control application. One SMC used a detailed mathematical model of the system and several observers while the other SMC neglected the valve dynamic (spool position and valve delay) and the time delay due to connecting tubes. Unfortunately, the full order SMC required complex computations for implementation, which makes it unattractive for on-line applications. The authors investigated the effects of the connecting tube’s length as well as the degree of detail in the system model used by the controller. Figure 2.5 and Figure 2.6 show the force tracking performance for a sinusoidal setpoint of 75 N amplitude and 20 Hz frequency with short and long connecting tubes, respectively. In the short tubes case (Figure 2.5) both controllers performed very well especially in terms of tracking the force magnitude. The phase was slightly larger with the reduced order controller. In the long tubes case (Figure 2.6) the force tracking performances were almost unchanged for the full order SMC, but significantly worse for the reduced order controller, both in amplitude and phase. The frequency responses of the two controllers are shown in Figure 2.7 and Figure 2.8 for the short and long connecting tubes, respectively. The bandwidth for the full order SMC is 58 Hz and 30 Hz for short and long connecting tubes, respectively. While the bandwidth for the reduced order SMC is 53 Hz and 25 Hz for short and long tubes, respectively.

These results confirm that the length of the connecting tubes is

important, as well as the bandwidth of pneumatic systems. Bandwidth less than 100 Hz is considered low bandwidth (Thayer, 1984).

It is important to note that the experiments of Richer and Hurmuzlu were in fact for no motion cases, where the cylinder was fixed at the middle of its stroke. The air pressure was varied to give the variable force. A proportional flow valve and an ultra low friction cylinder were used in the experiments. The effect of cylinder friction on controller performance was not investigated. A true tracking application involves motion and subsequently cylinder friction cannot be neglected. It is not neglected in this thesis.

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Figure 2.5: Tracking performance with short tubes (a) full order (b) reduced order (Richer and Hurmuzlu, 2000)

Figure 2.6: Tracking performance with long tubes (a) full order (b) reduced order (Richer and Hurmuzlu, 2000)

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Figure 2.7: Frequency response, full and reduced order, short tube (Richer and Hurmuzlu, 2000)

16

Figure 2.8: Frequency response, full and reduced order, long tube (Richer and Hurmuzlu, 2000) In Ben-Dov and Salcudean (1995) a new pneumatic actuator for applications requiring moderate, but accurate, force control was presented. The actuator used voice coil flapper valves and low friction cylinders. Its force was controlled by manipulating differential pressure. A feedback controller was designed and implemented. Stationary force control tests were presented with good results for a sinusoidal setpoint of 2 N amplitude and 16 Hz frequency.

In Bobrow and McDonell (1998) a model based controller was designed to control the torque output of each of the three pneumatic actuators of a 3 DOF serial robot. A position controller was implemented in the horizontal plane and an open loop force controller in the vertical plane. There was no attempt to reduce the force error (because it was an open loop force controller). Good performance was achieved for a sinusoidal setpoint of 10 N.m magnitude at both 1 and 5 Hz frequencies. It is important to mention that the “measured” torque was in fact calculated from measuring the differential air pressure. Thus, the friction of the cylinder was ignored. A different approach for pneumatic force control is the non-model based force controller as in Kaitwanidvilai and Parnichkun (2005) and Ruihua et al (2004). Kaitwanidvilai and Parnichkun

17

proposed a hybrid Adaptive Neuro-Fuzzy Model Reference Control (ANFMRC) algorithm to control the contact force on a workpiece with frequently changing dynamics.

In hybrid

ANFMRC, multimode switching is used to switch between a ‘bang-bang’ controller (used when the controller error is large) and the ANFMRC controller (used when controller error is small or medium).

Simulation and experimental results showed that the hybrid ANFMRC could

effectively learn about the system dynamics and as a result improve the response.

Its

performance was better than the conventional (non-hybrid) ANFMRC. However, there was no comparison with other conventional (e.g., PID) or advanced controllers (adaptive or sliding mode). The force response of tracking a square wave setpoint with 0.1 Hz frequency and 5 N amplitude is shown in Figure 2.9. Although Figure 2.9 shows the learning aspect, it does not tell a lot about the performance in terms of the error and speed of the response because of the large time scale. It is important to mention that a proportional flow valve and a low friction cylinder were used in the experiments, which means the controller was not challenged by high cylinder friction effects.

Figure 2.9: Force tracking with hybrid ANFMRC (Kaitwanidvilai and Parnichkun, 2005) Ruihua et al (2004) develop a multi-region fuzzy tracking algorithm to control the squeezing force of a pneumatic servo system with unknown nonlinearity and frequently changing process dynamics. The proposed controller was verified experimentally in a textile sizing operation. The results showed that the controller could effectively deal with the unknown nonlinear dynamics of

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the process. However the performance of the controller was not tested extensively since only a low frequency square wave setpoint was considered. The range of the desired contact force was from 8 to 50 kN. The scale of the presented results was not sufficient to show the force error. Furthermore, the high forces involved also meant that cylinder friction effects were negligible. This work is also notable because PPC valves were used.

The papers in Table 2.1 that investigate force control experimentally with pneumatic actuators are Ben-Dov and Salcudean (1995), Bobrow and McDonell (1998), Kaitwanidvilai and Parnichkun (2005), Richer and Hurmuzlu (2000) and Ruihua et al (2004). All were used to control the contact force of a one DOF system, so it was not a contour tracking problem since the contact point was the same all the time. Thus, the problem of contour tracking with pneumatic actuation has not been reported experimentally.

The nonlinearity, air compressibility and significant cylinder friction in most pneumatic systems are expected to be the most dominant factors in the controller performance. Most of the papers that showed good force control performance (Ben-Dov and Salcudean, 1995 and Richer and Hurmuzlu, 2000) used ultra low friction cylinders, so the friction effects were reduced by hardware selection. Thus, the question of what level of tracking performance can be achieved with normal (high friction) actuators remains unanswered. It would be interesting to investigate the possibility of achieving good contour tracking performance with the presence of the regular friction available in most regular pneumatic cylinders.

The nonlinearity in force control problems with pneumatic actuators was considered in two ways: one is with a detailed model based controller as in Ben-Dov and Salcudean (1995), Bobrow and McDonell (1998) and Richer and Hurmuzlu (2000) and the other is a non-model based controller as in Kaitwanidvilai and Parnichkun (2005) and (Ruihua et al, 2004).

Even though the

experimental results presented for the model based controllers are better than the results for the non model based controllers, the latter has the advantage of adaptation and compensation of uncertainty in the system. It is expected that the non-model based controllers,such as FL and NN, are more suitable for a highly nonlinear systems such as pneumatic systems. So the problem of contour tracking of an unknown workpiece geometry with pneumatic actuators still needs to be investigated.

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2.3 Contour Tracking Contour tracking is an example of an application that encourages the development of force control. Contour tracking along planar paths while maintaining contact or holding a certain contact force is required in several industrial applications such as cleaning, grinding, polishing, deburring and shape recovery (Ahmad and Lee, 1990). In Raoufi (2003) a 3 DOF pneumatic gantry robot was used for grinding steel blanks with different shapes. It was shown that to improve the grinding quality the grinding operation should be conducted with a constant applied normal force independent of the workpiece geometry. Most of the force controllers reviewed in Section 2.1 and 2.2 require information about the workpiece geometry and properties in advance or at least an initial estimation if the adaptive approach was used.

The tracking of unknown planar contours is a popular subject of research. An iterative (learning approach) was used in Jeon and Tomizuka (1993) and Lange and Hirzinger (1992).

In

Kaitwanidvilai and Parnichkun (2005) and Kiguchi and Fukuda (1995 and 1996) a neuro fuzzy approach was employed. A NN was used in Jung and Hsia (1995) and Jung et al (2001). In Mihaylova et al (2001) an interacting multiple model filter was implemented to estimate online the contour curvature and the orientation angle of the workpiece.

A novel approach to the problem of tracking of an unknown workpiece was proposed by Goddard et al (1992). In their Force Velocity (FV) controller, velocity control replaces position control in the classical FP controller. The FV control only needs local information, such as tangential velocity and normal contact force, in order to track an unknown profile, unlike FP control which needs a global model of the workpiece.

Jatta et al extensively investigated the properties of FV control experimentally. In Jatta et al (2002) and Jatta et al (2006) the experimental results show the importance of joint friction and how to compensate for it in order to achieve reasonable performance in terms of normal contact force and tangential velocity. The problem of the presence of elasticities in joint transmissions was addressed in Jatta et al (2003) and Jatta et al (2006). It was found that elasticity could cause large force oscillations depending on the robot configuration.

The FV control algorithm

presented in Jatta et al is considered the most suitable controller for planar tracking of an unknown workpiece geometry and consequently will be adopted in this thesis.

20

2.4 Neural Networks Neural Networks are closely modeled on biological processes for information processing, specifically the nervous system and its basic unit, the neuron. Figure 2.10 shows a generic neuron. The neuron receives multiple signals from other neurons through its dendrites, each signal multiplied by a weighting coefficient. These signals are added in the cell body. When the composite signal reaches a threshold value, a signal known as the action potential is sent through the axon to the synapses, where the signal is conveyed from one neuron to another. One description of a NN is a collection of mathematical formula with adjustable parameters, which are tuned or trained from a set of known inputs and outputs example (Norgaard et al, 2000). Figure 2.11 shows the equivalent mathematical model of Figure 2.10, where pi is the ith input,

V1,21 is the weight connecting p1 and node 1. σ 12 and b12 are the activation function and the bias of node 1, respectively, net12 is the sum of the inputs to node 1 and ni is the number of inputs. Figure 2.12 shows some other common choices for the activation function.

Figure 2.10: Labeled generic neuron (Anderson, 1995)

21

V1,21

net12 Vnh2 2 , ni

σ 12

b12

Figure 2.11: General mathematical neuron model (http://www.mathworks.com)

Figure 2.12: Some common choices for the activation function (Lewis et al, 2002) Figure 2.13 shows the structure of a multiple layer NN. Input and outputs nodes are connected by one or more hidden layers. When information goes from the input layer to the output layer it is called feedforward NN; there are no recurrent or backward connections.

The NN learns by modifying the weights through tuning. The NN weights may be tuned using several techniques; a common weight tuning algorithm is back propagation error.

22

Back

propagation is the most popular and widely used NN learning algorithm. It is similar; but more complex, to gradient descent method.

Because of the ability to model nonlinear functions and to learn from input-output data, NN’s have been used in many applications, such as function approximation, classification, data processing, systems identification and control. Layer 1

V1,21

Layer 3

Layer 2

net12

σ

2 1

W13,1

b12 net 22

2 netnh 2

Vnh2 2 , ni

net13 b13

σ

2 2

σ

2 nh2

bnh2 2

σ 13

W1,41

net14

σ 14

b14 net 23

3 net nh 3

Wnh3 3 , nh2

Layer 4

3 bnh 3

σ

σ

net24

3 2

4 netno

3 nh3

Wno4 , nh3

σ 24

σ no4

bno4

Figure 2.13: Multiple layers of neurons (http://www.mathworks.com)

2.4.1 Neural Networks in Control

Control of nonlinear systems is a major application area for NN’s. It is often seen that a NN is proposed as a tool for adaptive control of nonlinear systems with time varying dynamics. Implementation of a NN in control is usually done in one of two ways: 1) direct control and 2) indirect control. Direct NN control means that a NN is an active element of the controller; usually this type requires on-line training. Examples of direct NN’s are: NN model reference control, feedforward with NN inverse model control and NN internal model control as shown in Figure 2.14, Figure 2.15 and Figure 2.16, respectively.

The indirect methods represent a more conventional approach, where the design is based on NN model of the system to be controlled. In this case the controller is not itself a NN. The idea is to use a NN to model the system to be controlled, and then this model is employed in a more conventional controller design.

The NN is typically trained off-line. An example is NN

23

generalized predictive control, shown in Figure 2.17. Neural networks have been widely applied to system identification and indirect control of nonlinear systems, while less work had been presented about the use of NN in direct controllers (Huang and Lewis, 2003).

Figure 2.14: NN model reference control (Warwick et al, 1992)

Figure 2.15: Feedforward with NN inverse model control (Norgaard et al, 2000)

24

Figure 2.16: NN internal model based control (Hunt and Sbarbaro, 1991)

Figure 2.17: NN generalized predictive control (Soloway and Haley, 1996)

2.5 Fuzzy Logic Systems Fuzzy Logic (FL) control has found extensive applications in a wide variety of industrial systems (Parnichkun and Ngaecharoenkul, 2001). FL has two distinctive features: 1) human experience can easily be integrated, and 2) nonlinear relationships can be modeled without the need for a user to write a formal mathematical model. These features make FL attractive in control. FL controllers also have relatively low computational overhead (relative to model based controllers), which means they are attractive for on-line applications (Chiou and Huang, 2005).

As shown in Figure 2.18, a generic FL system is composed of five functional blocks: 1) A rule base containing a number of fuzzy if-then rules 2) A database, which defines the membership functions. 3) A decision-making unit, which performs the inference operation on the rules.

25

4) A fuzzification interface which transform the crisp input (numerical value) into degree of match (fuzzy value). 5) A defuzzification inference, which transform the fuzzy results into crisp output.

Furthermore, a generic FL system has three basic stages. 1) Compare the input variables with the membership functions on the premise part to obtain the membership values. This step is often called fuzzification. 2) Combine the membership values on the premise part to get firing strength of each rule. Next, generate the qualified consequence of each rule depending on the firing strength. These actions take place at the decision-making unit. 3) Aggregate the qualified consequence to produce a crisp output (numerical output). This third step is called defuzzification.

Figure 2.18: Generic Fuzzy Logic system (Jang, 1993)

2.5.1 Fuzzy Logic in Control

Motion and force control application examples with fuzzy controllers can be found in the literature. Bogdan and Kovacic (1993) used a FL rule based adaptive FP algorithm to control the contact force of a single DOF mechanism with electric actuation. The simulation results showed that the FL controller was more robust than a standard adaptive FP model reference-based controller in the presence of disturbances caused by the complex geometry of the workpiece. Ruihua et al (2004) developed a multi-region fuzzy tracking algorithm to control the squeezing force of a pneumatic servosystem with unknown linearities and frequently changing process dynamics. The proposed controller was verified experimentally in a textile sizing operation.

26

Parnichkun and Ngaecharoenkul (2001) proposed a hybrid FL PID controller for a pneumatic system. The control algorithm was separated into two parts: FL control and PID control. The FL control algorithm was used to control the cylinder when the cylinder was far away from the desired position. The PID controller was applied when the cylinder was closer to the desired position. This combination exploited the nonlinear characteristic of FL control for fast response and the precise nature of PID control for accurate response. The proposed hybrid FL PID controller showed better performance (faster settling time and lower steady state error) as compared to conventional P, PI, PD and PID controllers.

Zhao et al (1993) developed a FL PID Gain Scheduling (GS) scheme to replace the Ziegler Nichols (ZN) method for selecting the PID gains. Unlike ZN based PID gains which are fixed, the proposed controller would update the PID gains on-line based on the current position error and the difference between the current position error and the previously sampled position error. Figure 2.19 compares the step response of a second order system with the three different PID controller tuning methods: 1) the proposed fuzzy GS, 2) ZN method and 3) Kitamori method (which is based on partial model matching). The proposed scheme is seen to have less overshoot than the other two schemes, with roughly the same settling time.

Situm et al (2004) used a FL PID GS together with a friction compensator with a pneumatic servosystem to accommodate a variation in the air supply pressure. A tracking error of less than ± 1 mm was better than that obtained with a non-FL PID controller. Schulte and Hang (2004) used multi-local linear models of a pneumatic servosystem together with a fuzzy state feedback GS controller to accommodate variations in payload and cylinder position.

Based on the

cylinder’s position and payload size, the fuzzy GS selected the appropriate local linear model to identify the optimum PID gains.

The controller was tested on a single vertical pneumatic

cylinder. Experimental results demonstrated that the fuzzy controller had faster settling time, less oscillation and lower steady state error as compared with a fixed gain PI controller.

27

Figure 2.19: Step response of a second order system with 3 different controllers (Zhao et al, 1993)

2.6 Neural Networks and Fuzzy Logic in Control It was stated that one advantage of FL is that it provides a mechanism for the coding of human knowledge. However, it is difficult to design fuzzy controllers systematically (Hu et al, 2001). On the other hand, NN’s are able to learn from experience and adapt to uncertainties in a systematic highly structure fashion (Jeon and Tomizuka, 1993). The combination of FL and NN seems to be one of the best ways to solve the fuzzy logic controller design problem. Researchers have combined FL with NN in many different ways. Such systems are expected to be able to provide a mechanism for the coding of human knowledge, learn from experience and adapt to uncertainties in a systematic highly structured fashion (Jeon and Tomizuka, 1993).

2.6.1 Adaptive Neuro Fuzzy Inference System

Jang (1993) proposed a novel architecture, which can serve as a basis for the construction of a set of fuzzy rules, with appropriate membership functions to generate the needed input-output pairs. Jang called this algorithm an Adaptive Neuro Fuzzy Inference System (ANFIS). In ANFIS, a FL system is embedded in the framework of a NN and a hybrid algorithm is used for the learning stage.

Figure 2.20b shows the ANFIS equivalent to the FL system of Figure 2.20a. The ANFIS is seen to be a multilayered feedforward network in which each node performs a particular function on the inputs. The ANFIS network looks like any multilayer feedforward neural network, except that there are no weights associated with the links and not all the nodes are adaptive. The square

28

nodes in Figure 2.20b are adaptive, in that the node parameters can be updated. By contrast, the circle nodes are non adaptive. A hybrid learning algorithm that combines a gradient descent and least squares are used to identify the adaptive network’s parameters. A back propagation method is used in the backward pass and the least squares method is used in the forward pass.

Figure 2.20b shows that the ANFIS has five layers, which can be described as follows: Layer 1: Every node i in this layer is a square node (adaptive node) with output σ i1 ( x) . σ i1 ( x)

is the node i activation function. The node function σ i1 ( x) is usually a bell shape function with maximum equal to 1 and minimum equal to 0, such as:

σ i1 ( x) =

1 x − ci 2 oi 1 + [( ) ] ai

(2-1)

where {ai , oi , ci } is the parameter set S1 . Parameters in this layer are referred to as premise parameters. This layer corresponds to the fuzzification step of the FL system.

σ 11 σ 21 σ 31 σ 41 Figure 2.20: (a) Sugeno’s FL system and (b) equivalent ANFIS (Jang, 1993) Layer 2: Every node in this layer is a circle node (non adaptive node), which multiplies the

incoming signals and sends the product out as in:

29

wi = σ i1 ( x).σ i1 ( y )

(2-2)

This layer corresponds to the rules unit of the FL system. Each node output represents the firing strength of a rule.

Layer 3: Every node in this layer is a circle node (labeled N in Figure 2.20b). Output of this

layer is the normalized firing strengths. The ith node calculates the ratio of the ith rule’s firing strength with respect to the sum of the entire rule’s firing strength as in:

wi w1 + w2

wi =

(2-3)

Layer 4: Every node i in this layer is a square node with an activation function:

f i = ( pi x + qi y + ri )

(2-4)

where { pi , q i , ri } makeup the parameters set S 2 . Parameters in this layer are referred to as consequent parameters. Layer 5: The single node in this layer is a circle node labeled ∑

that calculates the overall

output as the summation of all incoming signals as:

∑w f i

i

∑w f = ∑w i

i

(2-5)

i

i

A hybrid learning method is proposed to identify the parameters sets S1 and S 2 . The hybrid learning algorithm combines the back propagation method and the least squares estimate method to identify the system parameters. The learning procedure is composed of a forward pass and a backward pass. In the forward pass, input data are supplied and the functional signal goes forward to calculate the output of each node. The parameters in S 2 are then identified by sequential least squares. After identifying the parameters in S 2 , the functional signal goes forward and the error measure (the difference between the actual output and the desired output) is calculated. In the backward pass, the error rates propagate from the output end toward the input end, and the parameters in S1 are identified by the Back Propagation Algorithm (BPA). This

30

forward to backward training continues until user specified criteria are satisfied and then the training is terminated. This hybrid learning method was found to be faster than the BPA alone.

Samhouri and Surgenor (2005) successfully used ANFIS for surface roughness prediction in a grinding application with a pneumatic robot.

Lee and Teng (2002) used ANFIS for PID

controller tuning.

2.7 Friction and Friction Compensation Friction is present in all machines incorporating parts with relative motion. Although friction may be a desirable property, as it is for brakes, it is generally an impediment for servo control. Friction can lead to tracking errors, limit cycles and undesired stick-slip motion (Armstronghelouvry et al, 1994). Friction is the tangential reaction force between two surfaces in contact. This force is the result of many different mechanisms, which depend on contact geometry and topology, properties of the bulk and surface materials of the bodies, displacement and relative velocity of the bodies and the presence of lubrication.

Control strategies that attempt to

compensate for the effects of friction, without resorting to high gain control loops, inherently require a suitable friction model to predict and to compensate for the friction (Canudas de wit et al, 1995).

2.7.1 Friction Models

Static friction opposes the movement of the body when the body is in a state of rest, and is usually a constant. Once the body is moving, dynamic friction opposes the motion and typically varies as a function of the velocity. Figure 2.21 shows four of the most common friction models. Figure 2.21a shows the Coulomb friction component with Fs = Fd , where Fs is the static friction force and Fd is the dynamic friction force. Figure 2.21b shows the Coulomb friction with viscous component. Figure 2.21c also has the Coulomb friction component, but with

Fs > Fd . Finally, Figure 2.21d adds exponential behavior (the Stribeck effect) for the transition from Fs to Fd . In Figure 2.21c, the sharp transition from Fs to Fd models the stick-slip behavior seen in high friction motion control applications, where one observes a “jerk” as the body to be moved breaks away from the Coulomb friction force.

31

Fs = Fd

Fs = Fd

Fs Fd

Fs

Figure 2.21: Friction models (a) Coulomb, (b) Coulomb plus viscous, (c) nonlinear, (d) exponential (Olsson et al, 1997)

2.7.2 Friction Compensation

Friction compensators address the need to correct for friction in motion control applications. There are many ways to compensate for friction. The effectiveness of a particular friction compensation technique depends strongly on the task and model used. Figure 2.22 connects task and compensation technique pairs based on more than one hundred papers in Armstrong-helouvry et al (1994).

Compensation techniques may be divided into three types: 1) friction problem avoidance, 2) non model based compensation and 3) model based compensation. The first type is not truly a friction compensator technique, but it is an attempt to replace the given system with one that is easier to control. The main difference between the second and third type is that the model based compensator type employs a friction model in a feedback or feedforward loop. The model parameters may be identified through a set of identification experiments. Otherwise, adaptive control techniques may be applied. Elhami and Brookfield (1996) present five friction parameter identification techniques and implement them on four different friction models. Examples of the

32

non model based compensation for friction are: 1) high gain feedback controller and 2) dither signal, that is a high frequency signal added to the control signal. The influence of the friction nonlinearities can to some extent be reduced by high gain linear feedback; however, this has some severe limitations. Because the friction nonlinearities will dominate any compensation for small errors, limit cycles may appears as a consequence of the dynamic interaction of the friction forces and the high gain controller (Canudas de wit et al, 1991) The effect of the dither signal is that it introduces extra forces that make the system move before the stiction level is reached.

Figure 2.22: Selection of friction compensation techniques (Armstrong-helouvry et al, 1994)

33

If a good friction model is available it is possible to use a model based friction compensator. The friction force is estimated using the model, and a signal that compensates for the estimated friction force is added to the control signal. In Canudas de wit et al (1991), an adaptive friction compensator was developed and the problems associated with inexact friction compensation were discussed.

Inexact friction compensation (undercompensation or overcompensation) usually

results in unwanted oscillations. However, the amplitude of these oscillations can be attenuated by the suitable choice of controller gains.

Southward et al (1991) designed a nonlinear friction compensator to supplement a PD controller applied to a one DOF mechanical system. The objective of the designed compensator was to achieve zero steady state position error without causing a limit cycle response as in PID controllers. It was shown that any system with Coulomb friction and a PID position controller will show a limit cycle in the neighborhood of the origin (desired set position). Southward et al’s simulation and experimental results show zero steady position error compared with a non-zero steady state error with PD. It is important to mention that the controller was implemented on a low friction system ( Fs = 2 N).

Cao (1993) designed a fuzzy compensator that is similar to the nonlinear compensator in Southward et al (1991) but with the advantages of the ability to use the SMC concept, higher robustness with respect to disturbances and ease of implementation.

In Gomes et al (2005) a neuro-fuzzy approach was used to design a friction compensator for FP control of a serial electric robot, as shown in Figure 2.23. Figure 2.24 shows the experimental friction torque and the friction torque estimated by the NN. The proposed compensator combined with PD controller shows zero steady state error compared with PD only. The disadvantages of the proposed compensator is that it dose not consider the presliding stage.

It is very important to consider friction effects when pneumatic actuators are used. In some applications, such as with rodless pneumatic cylinders, the static friction force in the cylinder can be up to 30% of the applied force. In motion control, anything more than 10% is considered a high friction application and acceptable performance is simply not achievable by conventional means (Qi, 2003).

34

Figure 2.23: Neuro-fuzzy friction compensator architecture (Gomes et al, 2005)

Figure 2.24: Experimental and NN estimated friction (Gomes et al, 2005)

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2.8 Summary A review of the literature on the subject of contour tracking and force control with pneumatic actuation has resulted in the following observations: •

Contour tracking is an example of applications that encourage the development of force control. Contour tracking along planar paths while applying constant contact force is required in several industrial applications such as cleaning, grinding, polishing, deburring and shape recovery.



A great deal of work has been done on force control with electrically actuated robots, but little work has been done with pneumatically actuated robots.



Pneumatic servosystems are cheaper, cleaner, safer easier to maintain and have a higher power to weight ratio. However, the compressible nature of air as the working fluid and relatively high cylinder friction means pneumatic robots are more difficult to control.



There is a fair amount of work reported on the use of individual pneumatic actuators for position and velocity control (i.e., servo pneumatics), but less so for force control. Force control with pneumatics is inherently more difficult as the required response times are much higher, and the compressibility of the air means system stiffness is low.



The work that is reported on pneumatic force control is not for contour tracking, where the point of contact is constantly changing. Instead, the work involved a stationary pneumatic actuator applying a controlled force to a fixed point.



Most studies on the subject of force control with pneumatic actuation have used low friction cylinders with proportional flow control valves.



The problem of force control with nonlinear pneumatic actuators was considered in one of two ways: the first is with a mathematical model based controller and the second is with a non-model based controller. Although both appear to be equivalent in terms of achievable performance, the non-model controllers appear to be more robust (e.g., ability to maintain performance in the face of changing operating conditions and system parameters).



Results of contour tracking are often reported without quantitative performance measures that can be used by other researchers for comparative purposes (e.g., Root Mean Square Error and Maximum Error).



NN and FL have found extensive applications in a wide variety of industrial systems, for both the control and the modeling of nonlinear systems.



The combination of FL and NN provides a mechanism for the coding of human

36

knowledge, learning from experiments and adapting to uncertainties in a systematic highly structured fashion. ANFIS is a novel NN and FL architecture. •

Friction is expected to have a significant effect on tracking performance especially in applications which use regular (high friction) pneumatic cylinders.

The lessons learned from previous work that will be incorporated into this research study are: •

The FV control approach is better than the FP control approach, when a priori knowledge of the workpiece geometry and material properties (i.e., stiffness) is not available.



There are two reasons that pneumatic systems are difficult to control: 1) compressibility of air and 2) relatively high Coulomb friction.



Common friction compensation techniques rely upon accurate friction models. It is not easy to model friction by conventional techniques due to its highly nonlinear and random nature. NN and neuro-fuzzy are recommended non-conventional techniques to model friction.

In summary, previous work in force control with pneumatic actuators used low friction cylinders with proportional flow control valves. Although the use of high friction cylinders with PPC valves represents a significant increase in the degree of difficulty of control, it is hypothesized that a well design FV PID controller in combination with a good friction compensator can achieve acceptable performance.

37

Chapter 3 EXPERIMENTAL CONTROLLER DESIGN Chapter 3 will present the experimental apparatus and initial controller design. The general structure of the adopted PID based FV controller will be documented. The procedure for the design, tuning and testing of the controller will be described. The effect of surface finish, profile shape and tracking speed on contour tracking performance will be examined. Three different workpiece edge geometries will be studied: straight, inclined and curved. The performance of a PI FV controller will be benchmarked against a similar FV controller proposed by Jatta et al (2006). Finally, the effect on performance when a Velocity Dependent Friction Model (VDFM) based compensator is added to the PI FV controller will be studied. Elements of this chapter can be found in Abu Mallouh and Surgenor (2007).

3.1 Apparatus Description The original mechanical design of the gantry robot was completed by Raoufi (2003). However, some changes were made for the purpose of conducting this research. The specific changes made were: z-axis cylinder replaced, manual flow control valves removed, surge tank, force sensor and differential pressure sensors were added. Figure 3.1 illustrates the gantry configuration of the pneumatic robot, with three DOF, labeled as the x-axis, y-axis and z-axis. The y-axis consists of two rodless pneumatic cylinders. The x-axis is a single rodless pneumatic cylinder, which also acts as the gantry bridge for the two y-axis cylinders. The mechanical coupling of the two y-axis cylinders with the x-axis cylinder was sufficient to avoid the synchronization problems that sometimes arise with this configuration. The z-axis is a rodded pneumatic cylinder with integral linear guides. As illustrated in Figure 3.2, the end effector of the z-axis is a 2-axis force sensor that ends with a bearing for contact with the workpiece. The z-axis is in the same direction as the y-axis, but it is labeled this way in order to highlight the availability of a third DOF.

PPC valves were used to manipulate the pressure of the air in the cylinders. The position and velocity of the cylinders were measured directly with wire linked potentiometers and tachometers, respectively. Differential pressure transducers measured the differential air pressure directly across each cylinder. Data acquisition and control was PC-based with dSPACE/DSP as

38

the data acquisition hardware/software and MATLAB/SIMULINK as the control software. Sampling time was 1 msec.

The workpiece was clamped in a horizontal orientation under the robot. Different workpiece shapes (straight, inclined and curved edges) and materials (wood, aluminum and steel) were used in the initial experiments. It was decided early on to benchmark results with the aluminum workpiece shown in Figure 3.3. The aluminum workpiece could be mounted in 5 configurations: straight edge with a zero, 7, 16 or 30 deg incline, or curved edge. The basic mode of operation was to use a FV controller to simultaneously control the tangential velocity and the normal contact force along the edge of the workpiece. The y-axis cylinders were not used in this set of experiments.

The full technical specifications of all the system components are given in

Appendix A.

PPC valves

force sensor

z-cylinder

x-cylinder workpiece

y-cylinder

Figure 3.1: Pneumatic gantry robot with main components labeled

39

force sensor

workpiece bearing

Figure 3.2: Close up of force sensor showing bearing in contact with edge of workpiece

Figure 3.3: Profile of the aluminum workpiece used in producing benchmark results

3.1.1 Pneumatic Cylinders

The x-axis cylinder is a SMC high precision guided mechanical joint rodless cylinder MY1H series. The bore diameter and stroke are 32 mm and 1000 mm, respectively. It is known that this type of cylinder has high Coulomb friction compared with rodded cylinders. The z-axis cylinder is a custom cylinder that was initially assembled by Raoufi (2003) by combining a SMC

40

MGCLB25-150 cylinder with a Parker ICN0.6 ND2 cylinder. The guide body of the SMC cylinder was used and the cylinder was replaced with the Parker cylinder. At the beginning of this research it was found that the Parker cylinder was worn out, so it was replaced with a Bimba nose mounted cylinder with a bore of 19 mm and stroke of 101 mm.

3.1.2 Valves

Two 5 port electro-pneumatic SMC VER2000 PPC valves were used to actuate the x and z cylinders. Figure 3.4 shows a typical pneumatic system with a PPC valve. The PPC valve can be used to switch and actuate a cylinder and to perform stepless pressure control of port A through electric signals. The amplifier (SMC VEA 250) converts the controller output voltage (0-5 volt) to a proportional current (0-1 ampere). In the initial apparatus designed by Raoufi (2003) there were manual flow valves, but in this research they were removed to enable the fastest possible response. The pressure supply was 551 kPa (80 psi) regulated by a manual pressure regulator (SMC NAL4000).

Figure 3.4: A typical pneumatic system diagram with PPC valves (SMC, 2000)

3.1.3 Sensors

Figure 3.2 shows the Bi-Axial FUTEK (model MBA400) force sensor which was used to measure the contact forces ( f x , f z ) between the end effector and the workpiece along the x and z axes. Each channel capacity is 222 N (50 lbs). The force sensor is based on bonded foil strain

gauge technology. When a force is applied, the sensor arm deflects. The voltage difference across the strain gauge bridge terminals is proportional to the applied force. This voltage is amplified by a universal strain gage signal conditioner (FUTEK CSG110) and fed to the

41

controller board. Two amplifiers were used, one for each channel. A bearing was attached to the end of the force sensor arm. The use of the bearing is intended to reduce the tangential friction between the end effector and the workpiece. A DV301 cable extension transducer measured the cylinder position ( x , z ) and velocity ( v x , v z ) along the x and z axes. The DV301 is a combination of position and velocity transducer. A precision plastic-hybrid potentiometer provides accurate position feedback while a self – generating DC tachometer provides a velocity signal that is proportional to the speed of the traveling stainless steel measuring cable. Each cylinder was connected to a DV301 with a range of 1270 mm for the x-axis and 127 mm for the z-axis. A differential pressure sensor was used to measure the differential pressure ( ΔPx , ΔPz ) between the two chambers of each cylinder along the x and z axes. Two PX137 silicon differential pressure sensors were used together with a DMD-465WB bridge sensor signal conditioner. Both have a differential pressure range of ± 689 kPa ( ± 100 psi). The signal conditioner was used to amplify the differential pressure sensor voltage output.

Multiplying ΔPx and ΔPz by each

cylinder’s cross sectional area gives the applied forces f ax and f az in Newton.

The calibration results for all sensors are presented in Appendix B.

3.1.4 Data Acquisition and Control

Data acquisition and control were PC-based with dSPACE/DSP as the data acquisition hardware/software and MATLAB/SIMULINK as the control software.

The DS1104 R&D

controller board is a real-time hardware based on PowerPC technology and its set of I/O interfaces makes the controller board an ideal solution for developing controllers in various fields. There are a total of 8 A/D channels (inputs): 4 are multiplexed with a resolution of 16 bits and conversion time of 2 msec and the other 4 are parallel channels with 12 bits resolution and 800 nsec conversion time. There are 8 channels of D/A (outputs) with 16 bits of resolution and a

settling time of 10 μsec. Table 3.1 shows the signal and channel assignments. Table 3.2 gives a summary of the main features of each of the major components used in the apparatus.

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Table 3.1: Signal and channel assignments Input Channel Ch1 (A/D) Ch2 (A/D) Ch3 (A/D)

Signal x-axis piston position ( x ) z- axis piston position ( z ) z- axis differential pressure ( f az )

Ch4 (A/D)

x-axis differential pressure ( f ax )

Ch5 (A/D)

x-axis piston velocity ( v x )

Ch6 (A/D)

z- axis piston velocity ( v z )

Ch7 (A/D)

x-axis contact force ( f x )

Ch8 (A/D) Output Channel Ch1 (D/A)

z- axis contact force( f z ) Signal z- axis control output ( u z )

Ch2 (D/A)

x-axis control output ( u x )

Table 3.2: Summary of main features of major components used in the apparatus No

Model

Axis

Description

Accuracy

1

DV301

x-axis

± 0.1% of full stroke = ±

2

DV301

z-axis

3

DV301

x-axis

4

DV301

z-axis

5

SMC VEA 250

6

VER2000

7

9

FUTEK MBA400 FUTEK CSG110 PX137

10

DMD-465WB

11

DS1104R&D

x&z axes x&z axes x&z axes x&z axes x&z axes x&z axes x&z axes

Cable extension for position Cable extension for position Cable extension for velocity Cable extension for velocity Valve’s amplifiers

8

Proportional pressure valve Force sensor Force sensor’s Amplifier Differential pressure sensor Differential pressure sensor’s amplifier Controller board & data acquisition

43

Response time (sec) NA

1.3 mm

± 0.1% of full stroke = ± 0.13 mm

± 3% at the rate 42 mm/sec = ± 1 mm/sec ± 3% at the rate 42 mm/sec = ± 1 mm/sec (Linearity) ± 1% of full scale= ± 0.01 A (Sensitivity) ± 0.5% of full scale = ± 5 kPa ± 0.1% of full capacity = ± 0.2 N ± 0.002% of rated output = 0.0002 V ± 0.1% of full scale = ± 0.7 kPa (Linearity) ± 0.05% of full scale = ± 0.005 V NA

NA NA NA 0.06 0.04 0.0006 0.001 NA NA NA

3.2 Noise in Measured Signals The performance of any control system depends on the level of electrical noise in the system. The overall noise level depends on the quality of the sensors, amplifiers and controller board. To obtain a measure of the noise level in the system, after all the sensors were calibrated, measurements were taken with no load or motion applied. Figure 3.5 plots the measured no-load values for the 8 measure variables: f x , f z , v x , v z , x , z , f ax and f az . Table 3.3 quantifies the noise levels seen in Figure 3.5 in terms of the maximum Signal to Noise Ratio (SNR) which was calculated as defined in Storey (1998):

⎛ maximum signal ⎞ maximum SNR = 20 log10 ⎜ ⎟ ⎝ RMS of noise ⎠

(3-1)

where RMS is the Root Mean Square. An acceptable maximum SNR is between 2 and 3 dB (Storey, 1998). As the maximum SNR levels were all above 3 dB the conclusion was that no active filtering was required.

Table 3.3: Signal to noise ratio No. 1 2 3 4 5 6 7 8

Signal

fx fz vx vz

x

z f ax f az

Maximum signal 250 N

RMS of noise 0.05 N

Maximum SNR (dB) 74

250 N

0.05 N

74

1000 mm/s

1.0 mm/s

60

1000 mm/s

0.5 mm/s

66

1000 mm 101 mm 400 N

0.3 mm 0.1 mm 0.05 N

71 60 78

157 N

0.05 N

70

44

vx (mm/s)

4 2 0 -2 -4

vz (mm/s)

1 0 -1

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

fax (N)

0.2 0 -0.2

faz (N)

0.2 0

0.2 0 -0.2

0.2 0 -0.2

z (mm)

x (mm)

fz (N)

fx (N)

-0.2

1 0 -1

0.2 0 -0.2

Figure 3.5: Illustration of noise level in the measured signals

45

3.3 Jatta and Benchmark FV Controller It was concluded in the literature review that a suitable controller for the contour tracking application is the algorithm proposed by Jatta et al (2006). Figure 3.6 shows the block diagram of this controller, which consists of two PID controllers: one acts based on the tangential velocity error and the other acts based on the normal contact force error. Because it uses two PID controllers, it will be referred to as the Dual Force Velocity (DFV) controller. An estimation of the joint friction ( fˆ1 and fˆ2 ) is added to the PID controller signals. The objective of this

controller is to control the normal contact force ( f n ) and the tangential velocity ( vt ) when the end effector tracks the profile of a workpiece. This controller was successfully used by Jatta et al in a contour tracking application with an electrically actuated robot. Figure 3.7, Figure 3.8 and Figure 3.9 show tracking responses for rectangular, curved and circular edge workpieces, respectively.

Figure 3.6: Block diagram of the Dual Force Velocity controller (Jatta et al, 2006)

46

Figure 3.7: f n tracking response for iron rectangular edge workpiece (Jatta et al, 2006)

Figure 3.8: f n tracking response for wooden curved edge workpiece (Jatta et al, 2006)

47

Figure 3.9: f n tracking response for iron circular edge workpiece (Jatta et al, 2006) Table 3.4 shows the tracking performance achieved experimentally with the DFV controller in terms of Root Means Square Error (RMSE) and Maximum Error (ME), where RMSE is calculated as: N

RMSE =

∑ (error in f

n

or v t ) 2

(3-2)

1

N

where N is the number of the data samples.

Table 3.4: Summary of Jatta et al’s results with DFV controller (Jatta et al, 2003 and 2006) Case ME f n (N) ME vt (m/s) RMSE f n (N) RMSE vt (m/s) iron circular edge ( f nd = 20 N, vtd = 0.008 m/s)

14.17

NA

3.63

NA

wooden curved edge ( f nd = 20 N, vtd = 0.01 m/s) iron rectangle edge ( f nd = 20 N, vtd = 0.015 m/s) aluminum straight edge ( f nd = 30 N, vtd = 0.02 m/s)

16.56

NA

4.02

NA

14.00

NA

NA

NA

13.11

NA

2.95

NA

48

The work of Jatta et al provides two important findings: 1) a measure of achievable results (Table 3.4) and 2) a suggested control structure (Figure 3.6). For the purpose of this thesis, a more involved control structure is suggested as a benchmark controller, given the more demanding nature of the application. The benchmark controller, as shown in Figure 3.10, is an adaptive neuro fuzzy FV controller that is composed of three main parts: a PID FV controller, an ANFISbased PID gain scheduler and an ANFIS-based friction compensator. The benchmark controller is similar to the DFV controller, but there are three significant differences. First, the PID gains of the proposed FV controller would not be constant, as was the case with a DFV controller. Using ANFIS, the gains of the proposed PID controller would be updated on-line. The need for a variable gain structure is a reflection of the nonlinear nature of a pneumatically based system. Second, in the DFV an adaptive friction compensator, based upon recursively updating the friction compensator model parameters, was used.

However the proposed ANFIS friction

compensator for this project is a non model based adaptive friction compensator. It was shown in Raoufi and Surgenor (2003) that both the static and dynamic Coulomb friction change with the cylinder’s position and direction.

It was also shown in Qi and Surgenor (2003) that the

manipulated load would affect both the static and dynamic Coulomb friction. Thus, in order to achieve the highest possible performance, it may be necessary to assume variable Coulomb friction. It is expected that ANFIS will be able to identify the nonlinear relation between dynamic friction, static friction and other system parameters (cylinder position, load, direction of motion). Third, the DFV controller was implemented on a serial 2 DOF planar electric robot, while the proposed controller would be implemented on a gantry planar pneumatic robot.

49

Figure 3.10: Adaptive neuro-fuzzy FV controller plus neuro-fuzzy friction compensator It is hypothesized that the full complexity of by the control structure in Figure 3.10 will not be required to produce significant performance improvements. Thus, the first controller to be tested will represent only one component of that illustrated in Figure 3.10. Specifically, one will first set out to design a PID FV controller with a VDFM based compensator, as shown in Figure 3.11. This controller is similar to Jatta et al’s DFV controller. Tracking performance obtained with the FV controller was compared against the tracking performance obtained with DFV controller and reported in Table 3.4. Then the ANFIS controller shown in Figure 3.10 will be designed in order to further improve the tracking performance even more.

Figure 3.11: Block diagram PI FV controller with VDFM based compensator

50

3.4 Designing the PID FV Controller This section gives the steps for designing, tuning and testing the PID FV controller. A fixed reference frame (x, z) has the robot’s right side corner as its origin. All signals ( f x , f z , v x , v z , x , z , f ax and f az ) were measured in this frame. A moving task frame (t, n) has the point of contact between the force sensor and the workpiece as its origin, such that the n-axis is directed normal to the workpiece contour and the t-axis is along the tangential direction of motion as shown in Figure 3.12.

Figure 3.12: Representation of the gantry robot operating on a planar curved workpiece. Figure 3.13 provides more details of the PID controller’s components. Four PID controllers instead of two PID controllers as in the DFV controller were used. The four controllers are: 1) force controller to control the z-component of the normal contact force (or simply f z controller) 2) force controller to control the x-component of the normal contact force (or simply f x controller), 3) velocity controller to control the z-component of the tangential velocity (or simply v z controller), 4) velocity controller to control the x-component of the tangential velocity (or simply v x controller)

51

Figure 3.13: Block diagram of FV controller with four PID elements shown Each of the four PID controllers acts based on an error expressed in the joint domain (x and z) instead of the task domain (n and t) as in the DFV control algorithm. The reason behind using four controllers instead of two is to make the tuning safer and easier, where the x and z axes direction are fixed and known while the normal and tangential direction are not known easily and not fixed during the tracking especially for curved profiles. Because the PID tuning is basically a trial and error procedure, it is expected that using four controllers will be better in avoiding jerk motion when different values of PID gains are used. All tuning was done on a straight edge wooden workpiece.

Tuning of the FV controller; shown in Figure 3.13; was done in three steps:

Step 1: The four PID controllers were tuned independently. When one controller was being tuned, the other three controllers were turned off.

Step 2: The PID controllers were tuned in pairs. When one pair was being tuned, the other pair was turned off.

Step 3: The four PID controllers were tuned simultaneously. All four controllers were on and active during the tuning process.

52

3.4.1 Tuning Step 1

In this step the goal was not to tune the controllers to perform a contour tracking tasks, but rather to tune each controller individually to perform a velocity or stationary force control task along the (x, z) frame. Initially, all four controllers were full PID, with the following control equations: u xv x = K vpx evx + K iv x ∫ evx dt + K dv x (dev x dt )

(3-3)

u zv z = K vpz ev z + K iv z ∫ ev z dt + K dv z (dev z dt )

(3-4)

u xf x = K pf x e f x + K i f x ∫ e f x dt + K df x (de f x dt )

(3-5)

u zf z = K pf z e f z + K i f z ∫ e f z dt + K df z (de f z dt )

(3-6)

where u xvx , u zvz , u xf x and u zf z are the output of the v x , v z , f x and f z controllers, respectively. ev x and ev z are the difference between the desired and actual velocity of the end effector along the x and z axes, respectively. e f x and e f z are the difference between the desired and actual contact force along the x and z axes, respectively. The tuning procedures for all the controllers were the same; first the integral ( K i ) and differential ( K d ) gains were set to zero while the proportional gain ( K p ) was changed until good response was produced. Then K p was fixed and K d was set to zero while K i was changed until good response was produced. In the last step K p and K i were set to the values found previously and K d was changed until good response

was produced. Figure 3.14 gives the x-axis result for PI velocity tuning with a fixed K p and v

v

three values of K i . The response with K px and K i x equal to 0.2 and 4.0, respectively, was considered the tuned response. This response gives a good compromise between maximizing speed of response and minimizing the degree of oscillation. Similar tuning exercise was done for each of the four controllers. Additional tuning results are presented in Appendix E. Table 3.5 summaries the tuning results of Step 1.

53

Table 3.5: Gains after Step 1 tuning Kp Ki

Controller

Kd

vx

K vpx =0.2

K iv x = 4.0

K dv x =0.0

fz

K pf z =0.08

K i f z = 0.1

K df z =0.0

fx

K pf x =0.02

K i f x = 0.03

K df x = 0.0

vz

K vpz =0.25

K iv z = 6.0

K dv z =0.05

0.1

ux (volt)

vx (m/sec)

0 -0.1 Kp=0.2,Ki=4.0,Kd=0.0 Kp=0.2,Ki=8.0,Kd=0.0 Kp=0.2,Ki=2.0,Kd=0.0

-0.2

9

9.5

10

10.5 Time (sec)

11

11.5

12

9

9.5

10

10.5 Time (sec)

11

11.5

12

1

0.8

0.6

Figure 3.14: Results of the PID velocity control of x-axis when K iv x was changed (example of Step 1 tuning)

3.4.2 Tuning Step 2

At this step the controllers were tuned to perform contour tracking in the fixed reference frame (x, z). The PID controllers were tuned in pairs: the z-axis force controller was combined with the xaxis velocity controller. This pair controls the contact force along the z-axis and the velocity along the x-axis. The x-axis force controller was combined with the z-axis velocity controller.

54

This pair controls the contact force along the x-axis and the velocity along the z-axis. When one pair was being tuned, the other pair was turned off.

The gains shown in Table 3.5 were used as first guesses. The gains of controller pairs were tuned by trial and error until good response was achieved. Figure 3.15 and Figure 3.16 show the responses achieved with the first and second pairs, respectively. It is important to mention that because of the short stroke of the z-axis cylinder the first control pair gave better results than the second one. The tuned gains are listed in Table 3.6.

Figure 3.15 and Figure 3.16 show three stages: 1) approach, to establish a near contact condition between the end effector and the workpiece, 2) stationary contact, to control the contact force along one axis when there is no motion along the other axis and 3) contouring, both the contact force and the tracking velocity are controlled. One observes that the force controller works well in the stationary stages, but as the end effector starts moving the ability to regulate the contact force degrades. It is important to mention that the constrained motion phases (stationary contact and contouring) are the only phases that need to be dealt with by the controller. In other words, neither the unconstrained motion phase (no contact with workpiece), nor the unconstrained to constrained phase (approach) had to be considered. The approach stage was achieved by manual and open loop controllers. It is also important to mention that there was no major improvement in the responses when the K d component was added to the PI controller. Thus, it was decided to use the PI type.

Controller

Table 3.6: Gains after Step 2 tuning Kp Ki

Kd

vx

K vpx =0.05

K iv x = 1.0

K dv x =0.0

fz

K pf z =0.02

K i f z = 0.04

K df z =0.0

fx

K pf x =0.01

K i f x = 0.06

K df x = 0.0

vz

K vpz =0.5

K iv z = 3.0

K dv z =0.0

55

60

fz (N)

40

Approach in z both off

20

Stationary Contact, f z on, v x off

0 -20

Contouring f z on, v x on

0

5

10

15

20

25

30

35

0

5

10

15 20 time (sec)

25

30

35

0.1

vx (m/s)

0 -0.1 -0.2 -0.3

Figure 3.15: Responses when tracking straight edge wood workpiece with z-axis force controller combined with the x-axis velocity controller (example of Step 2 tuning) 0

fx (N)

-20 Approach in x

both off

-40 -60 -80

Stationary contact f x on, v z off

Contouring

f x on, v z on

0

5

10

15

20

25

30

35

40

0

5

10

15

20 time (sec)

25

30

35

40

0.2

vz (m/s)

0.15 0.1 0.05 0 -0.05

Figure 3.16: Responses when tracking straight edge wood workpiece with z-axis velocity controller combined with the x-axis force controller (example of Step 2 tuning)

56

3.4.3 Tuning Step 3

For this step the controllers were tuned to perform contour tracking in the task frame (n, t). The four PI controllers were tuned simultaneously. All four controllers were on and active during the tuning process. The force sensor measures the contact forces f x and f z along the x-axis and the z-axis, respectively. The velocity sensors measure the velocity of the end effector v x and v z along the x and z axes, respectively. The force and velocity signals were measured in the joint domain (x and z) and fed to a coordinate transformer. With the assumption that the tangential friction ( f t ) is negligible, the profile angle ( θ , angle between the task and reference frames) is calculated as:

tan θ =

fx ⇒ θ = a tan 2( f x , f z ) fz

(3-7)

Even thought the assumption of negligible tangential force is considered accurate and reasonable in the context of this particular application, it is expected to be a less than accurate assumption in surface finish operations. In Minami et al (1997) simulation results showed that the normal contact force and tangential velocity are the most important factors in surface finish operations. However, it has been shown also that tangential friction force has an effect on the normal contact force. Thus, when it comes to surface finish applications, it is believed that the tangential friction force should be considered.

f n and vt are calculated as:

f n = − f x sin θ + f z cosθ

(3-8)

vt = vx cosθ + vz sin θ

(3-9)

f n and vt are subtracted from the desired normal contact force ( f nd ) and desired tangential velocity ( vtd ), respectively, to produce errors as: e f n = f nd - f n

(3-10)

57

evt = vtd - vt

(3-11)

The errors are fed to another coordinate transformer which transforms the errors from the task domain to the joint domain using the following equations:

e f z = e f n cosθ

(3-12)

ev z = evt sin θ

(3-13)

e f x = −e f n sin θ

(3-14)

eVx = evt cosθ

(3-15)

Using the errors in the joint domain, the output signals of the PI controllers are calculated by Equations 3-3 to 3-6. The controller signals for each axis are then added together as:

u zPI = u zf z + u zvz

(3-16)

u xPI = u xf x + u xv x

(3-17)

where u xPI and u zPI are the total control signals for the PI FV controller along the x and z axes, respectively.

Based on this control algorithm, each of the actuators (along x and z axes)

contributes toward reaching the desired f n and vt at the same time.

The controllers were tuned again to give good response when tracking a wooden workpiece with a straight edge profile. The gains listed in Table 3.6 were used as first guesses. The gains listed in Table 3.7 are the final tuned gains. Figure 3.17 shows f n and vt responses when tracking a wooden workpiece with a straight edge profile using the gains listed in Table 3.7. The response in Figure 3.17 represents a compromise of minimum undershoot at the 15 sec mark and minimum oscillation between 15 and 25 sec during the contouring phase.

It is important to mention that in the three tuning steps there were multiple trials until the tuned gains were found. But there were no repeated trials under exactly the same conditions.

58

Table 3.7: Gains after Step 3 tuning Controller

Kp

Ki

vx

K vpx =1.0

K iv x =4.0

fz

K pf z =0.02

K i f z =0.04

fx

K pf x =0.01

K i f x =0.03

vz

K vpz =0.5

K iv z =5.0

40

fn (N)

30

Contouring f n on, vt on

20 10 0

0

5

10

15

20

10 time (sec)

15

20

0.1

vt (m/sec)

0 -0.1

f nd = 30 N

-0.2 -0.3

vtd = 0.06 m/sec 0

5

Figure 3.17: f n and vt responses when tracking wood workpiece with straight edge profile (example of Step 3 tuning)

3.5 Effect of System Parameters Before moving to the friction compensator design step, the effect of three system parameters was investigated: surface finish, profile shape and tracking speed. The gains listed in Table 3.7 were used for the results reported in this section.

3.5.1 Effect of Surface Finish

In this section the effect of the straightness of the workpiece profile on tracking response is reported. Three straight edge workpieces made of wood, aluminum and steel with different

59

surface finishes were tested. Figure 3.19, Figure 3.20 and Figure 3.21 show f n response when tracking wood, steel and aluminum workpieces, respectively. Table 3.8 summarizes the tracking performance in terms of RMSE and ME for f n . It is important to mention that the lengths of the workpieces were different.

A comparison of Figure 3.20 relative to Figure 3.19 and Figure 3.21 shows a significant degradation in performance for the steel workpiece. After investigating the profile of each of the workpieces it was noted that the profile of the steel workpiece was the roughest and the least straight. The profile of the wood workpiece was the smoothest and straightest of the three workpieces. The average surface roughness ( Ra ) of the profile of the three workpieces was measured with a HOMMEL T500 roughness tester. Ra was found to be 1.4, 0.7 and 4.0 μm for the wood, aluminum and steel workpieces, respectively.

It is important to distinguish between the surface roughness and the surface straightness. The surface roughness and straightness maybe be defined as the vertical deviations of a real surface from its ideal form. If these deviations have a high frequency and short wavelength, as shown in Figure 3.18c, then it is surface roughness and if the deviations have a low frequency and long wave length, as in Figure 3.18b, then this is surface straightness. Results in this section show that it is the surface straightness not the roughness which has a significant effect on the performance. The RMSE and ME of f n for the wooden edge were 76% and 42% less than for the steel edge, respectively. The RMSE and ME of f n for the aluminum edge were 50% and 33% less than for the steel edge, respectively. Because the difference in performance was more than 30%, one concludes that the surface finish has a significant effect on the tracking performance. The 30% threshold came from an industrial rule of thumb, whereby it’s not considered worth investing in a new control system, if the new system doesn’t improve the performance by 30%.

Table 3.8: Summary of tracking performance for straight edge with different materials Case ME f n (N) RMSE f n (N) wooden straight edge steel straight edge aluminum rectangle edge

9.0 15.6 10.4

60

1.0 4.2 2.0

a) surface profile

b) surface straightness (waviness) c) surface roughness

Figure 3.18: Surface roughness and surface straightness (http://www.hommel-etamic.de/) 40

fn (N)

30 20

f nd = 30 N

10 0

Contouring

vtd = 0.06 m/sec 0

5

10

15

20

Figure 3.19: f n response when tracking wood workpiece with straight edge profile 60

f nd = 30 N vtd = 0.06 m/sec

fn (N)

40

20

Contouring 0

0

5

10

15

20

25

time (sec)

Figure 3.20: f n response when tracking steel workpiece with straight edge profile

61

40

fn (N)

30 20

f nd = 30 N vtd = 0.06 m/sec

10 0

0

2

4

Contouring

6

8 time (sec)

10

12

14

16

Figure 3.21: f n response when tracking aluminum workpiece with straight edge profile

3.5.2 Effect of Profile Shape and Angle

In this section the effect of the profile shape of the workpiece on tracking response is reported. Tests were conducted on the straight wooden workpiece with four incline configurations and the steel workpiece with curved and straight edges. The steel curved edge profile is the same as the profile of the aluminum curved edge shown in Figure 3.3. Figure 3.19 (in previous section), Figure 3.22, Figure 3.23, and Figure 3.24 show f n responses when tracking a wood workpiece with θ = zero, +3, -10 and +92 deg., respectively. One observes that the performance degrades as θ increases.

This was expected since the FV

controller was tuned using a straight ( θ = zero deg.) wooden workpiece. Figure 3.20 (in previous section) and Figure 3.25 show f n responses when tracking straight and curved steel workpieces, respectively. Surprisingly the f n response when tracking the curved profile was better than the response when tracking the straight profile. It was expected that the straight edge tracking would not be as challenging as the curved edge. It was also expected that the controller would perform the best when θ = zero deg. From these results one concludes that both the shape and the angle θ of the workpiece has an effect on tracking performance. But when a comparison is made with the results in the previous section, the effect is seen to be less important than the effect of the surface finish. When tracking inclined edges with negative angles it was noted that the performance degraded as the angle increased in the negative direction. The PPC valves used in the experiments were unidirectional action valves, which means they control the pressure of the air flow through only one of the

62

valve’s output ports (port A as seen in Figure 3.4). As a result, the valve can only control the pressure on one side of the cylinder. This will limit the range of profile shapes that can be tracked to the one with positive angles only. The effect of the unidirectional valves on tracking performance will be studied in Chapter 4. 40

fn (N)

30 20

Contouring

f nd = 30 N

10 0

vtd = 0.06 m/sec 0

5

10

15 time (sec)

20

25

Figure 3.22: f n response when tracking wood workpiece with θ = +3 deg 40

fn (N)

30

Contouring

20

f nd = 30 N

10 0

vtd = 0.06 m/sec 0

5

10 time (sec)

15

20

Figure 3.23: f n response when tracking wood workpiece with θ = -10 deg

fn (N)

60

f nd = 30 N

Contouring

vtd = 0.06 m/sec

40 20 0 0

2

4

6

8 time (sec)

10

12

14

Figure 3.24: f n response when tracking wood workpiece with θ = +92 deg

63

16

40

fn (N)

30 20

f nd = 30 N

10 0

Contouring

vtd = 0.06 m/sec 0

2

4

6

8

10 12 time (sec)

14

16

18

20

22

Figure 3.25: f n response when tracking steel workpiece with curved edge

3.5.3 Effect of Tracking Speed

In general it is expected that the slower the tracking speed, the better the performance. In this section the effect of the tangential tracking speed on f n response is investigated. Figure 3.26 and Figure 3.27 show the tracking response for the aluminum curved edge when vtd was set to 0.06 m/s and 0.03 m/s, respectively. Table 3.9 lists the tracking performance in terms of RMSE with three vtd values. The RMSE of f n decreased by 25%, and the RMSE of v t decreased by 13% and 15% when v td was decreased from 0.06 m/s to 0.03 m/s and from 0.09 m/s to 0.06 m/s, respectively. Because the improvements when slowing down were less than 30% it was not considered significant. Also with the low vtd the end effector went through stall stages as shown in Figure 3.27. It is expected that this happened because of the Coulomb friction.

Figure 3.28 and Figure 3.29 show the responses when tracking the straight aluminum edge with vtd set to 0.06 m/s and 0.03 m/s, respectively. Table 3.10 lists the tracking performance in terms of RMSE with three vtd values. The RMSE of f n decreased by 23% and 6% and the RMSE of

v t decreased by 20% and by 6% when v td was decreased from 0.06 m/s to 0.03 m/s and from 0.09 m/s to 0.06 m/s, respectively. Again the improvement when slowing down was less than 30% and was not considered significant. From all these results one concludes that there was no significant effect of the vtd on the tracking performance. It was also found that a good value for

vtd was 0.06 m/s since it gave acceptable performance and prevented piston stalling.

64

Table 3.9: Performance with curved aluminum workpiece and three different vtd values.

vtd (m/s) RMSE RMSE

0.03

0.06

0.09

f n (N)

2.4

3.0

3.6

vt (m/s)

0.053

0.060

0.069

Table 3.10: Performance with straight aluminum workpiece and three different vtd values.

vtd (m/s) RMSE RMSE

0.03

0.06

0.09

f n (N)

2.6

3.2

3.4

vt (m/s)

0.015

0.018

0.019

65

40

fn (N)

30 20

Contouring 10 0

0

2

4

6

8 10 time (sec)

12

14

16

18

6

8 10 time (sec)

12

14

16

18

0.1

vt (m/s)

0 -0.1

f nd = 30 N

-0.2

vtd = 0.06 m/sec 0

2

4

Figure 3.26: f n and vt responses when tracking aluminum curved edge, with medium vtd 40

fn (N)

30 20 10 0

Contouring 0

5

10

15

20

25

20

25

time (sec) 0.1

vt (m/s)

0 -0.1

f nd = 30 N

vtd = 0.03 m/sec

-0.2

0

5

stall 10

15 time (sec)

Figure 3.27: f n and vt responses when tracking aluminum curved edge, with low vtd

66

40

fn (N)

30 20

Contouring

10 0

0

2

4

6

8 10 time (sec)

12

14

16

6

8 10 time (sec)

12

14

16

0.1

vt (m/s)

0 -0.1

f nd = 30 N

-0.2 -0.3

vtd = 0.06 m/sec 0

2

4

Figure 3.28: f n and vt responses when tracking aluminum straight edge, with medium vtd 40

fn (N)

30 20 10 0

Contouring 0

5

10

15

20

25

15

20

25

time (sec) 0.1

vt (m/s)

0 -0.1

f nd = 30 N

-0.2

vtd = 0.03 m/sec 0

5

10 time (sec)

Figure 3.29: f n and vt responses when tracking aluminum straight edge, with low vtd

67

3.6 Retuned PI FV Controller The results in the previous sections led to the conclusion that the PI FV controller needed to be retuned. The gains given in Table 3.7 were obtained with the straight wood workpiece, which has a very smooth profile. Better overall performance would be achieved if the gains were obtained with the more challenging case of a straight aluminum workpiece mounted on a 30 deg incline and with medium surface finish.

Table 3.11 shows the tuned gains when the aluminum

workpiece ( θ = 30 deg) was used.

Table 3.11: Gains when straight aluminum workpiece ( θ = 30 deg) used in tuning. Controller

Kp

Ki

vx

K vpx

fz

K pf z = 0.02

K i f z = 0.08

fx

K pf x = 0.01

K i f x = 0.03

vz

K vpz = 0.5

K iv z = 2.0

= 1.0

K iv x

= 4.0

3.6.1 Benchmark Results with Retuned Controller

In this section benchmark results are presented for four cases: straight, inclined with 7 deg, inclined with 16 deg, and curved edge as shown in Figure 3.30, Figure 3.31, Figure 3.32 and Figure 3.33, respectively. All cases are for the aluminum workpiece. Table 3.12 summarizes the experimental tracking performance in terms of RMSE and ME of f n and vt during the tracking stage.

Comparing Table 3.12 and Table 3.4 one notes that for the straight edge and curved edge tracking cases, the f n RMSE and f n ME values are very close. This highlights the potential of using a pneumatic actuated robot in contour tracking applications, since the results presented in Table 3.4 were obtained with an electrically actuated robot with a similar controller, and the performance results in Table 3.12 are seen to be comparable. One notes that the f n tracking response of the straight edge has the worst performance in terms of the RMSE. This observation is similar to the observation made in Section 3.5.2 for the steel workpiece case. Investigating the aluminum workpiece profile showed that the surface finish was

68

not as straight and not as smooth as the wood workpiece. In the next chapter, a model of the system will be developed in order to answer the question as to why this is the case.

Results presented in this section indicate the performance that one can achieve with a fixed gain PI FV controller.

The tracking performance results given in Table 3.12 will be used to

benchmark any further improvements in performance, for example through the addition of a neuro-fuzzy compensator.

Table 3.12: Summary of the experimental tracking performance for different profiles Case ME f n (N) ME vt (m/s) RMSE f n (N) RMSE vt (m/s) aluminum straight ( f nd = 30 N, vtd = -0.06 m/s) aluminum inclined (7 deg) ( f nd = 30 N, vtd = -0.06 m/s) aluminum inclined (16 deg) ( f nd = 30 N, vtd = -0.06 m/s) aluminum curved ( f nd = 30 N, vtd = -0.06 m/s)

12.4

0.118

3.0

0.039

9.6

0.135

1.2

0.041

5.6

0.192

1.0

0.047

16.8

0.170

2.6

0.047

69

50

contact

40 fn (N)

30 20

retreat

approach

10

contouring

0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

10

15

20 25 Time (sec)

30

35

vt (m/s)

0

-0.1

-0.2

θ (deg)

2

0

-2 0

5

40

45

z (mm)

100

50

0 300

400

500

600

700

800

900

1000

x (mm)

Figure 3.30: Responses when tracking aluminum straight edge ( θ = zero deg)

70

50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

10

15

20 25 Time (sec)

30

35

vt (m/s)

0

-0.1

-0.2

θ (deg)

10

5

0

0

5

40

45

z (mm)

100

50

0 300

400

500

600

700

800

900

1000

x (mm)

Figure 3.31: Responses when tracking aluminum inclined edge ( θ = 7 deg)

71

50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

10

15

20 25 Time (sec)

30

35

vt (m/s)

0

-0.1

-0.2

θ (deg)

20

15

10

0

5

40

45

z (mm)

100

50

0 300

400

500

600

700

800

900

1000

x (mm)

Figure 3.32: Responses when tracking aluminum inclined edge ( θ = 16 deg)

72

50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1

-0.2

60

θ (deg)

40 20 0 -20 -40

z (mm)

100

50

0 300

400

500

600 700 x (mm)

800

900

Figure 3.33: Responses when tracking aluminum curved profile edge

73

1000

3.7 PI FV Controller with VDFM based Compensator In this section the design of the VDFM based compensator is presented.

The following

parameters need to be identified by experiments:



static Coulomb friction force of x-axis cylinder ( Fsx )



dynamic Coulomb friction force of x-axis cylinder ( Fdx )



viscous friction coefficient of x-axis cylinder ( C fx )



static Coulomb friction force of z-axis cylinder ( Fsz )



dynamic Coulomb friction force of z-axis cylinder ( Fdz )



viscous friction coefficient of z-axis cylinder ( C fz )

Various experimental tests were conducted in order to identify the cylinder’s friction model parameters. Table 3.13 shows the average of the identified friction model parameters. The static friction forces were found to be 83% and 67% of the desired normal contact force for the x and z cylinders, respectively. Such levels of friction forces are considered high (Qi, 2003). Details about the identification of the friction model parameters are given in Appendix C.

Fdx ( N) 11

Table 3.13: Identified friction model parameters C fx (N / (m/s)) C fz (N / (m/s)) Fsx ( N) Fdz ( N) Fsz ( N) 25

10

20

74

40

The Coulomb friction was calculated for x and z axes using:

⎧− sgn(v x ) Fdx Fx = ⎨ ⎩ − sgn( f ax ) max( f ax , Fsx )

v x > 0.01⎫ ⎬ v x ≤ 0.01⎭

(3-18)

⎧− sgn(v z ) Fdz Fz = ⎨ ⎩ − sgn( f az ) max( f az , Fsz )

v z > 0.01⎫ ⎬ v z ≤ 0.01⎭

(3-19)

The viscous friction was added to the coulomb friction and the total was divided by a gain to convert from Newton to volt. The compensator outputs ( u xComp and u zComp ) were calculated by:

u xComp =

[ Fx + v x C fx ]

(3-20)

Kx

74

u zComp =

[ Fz + v z C fz ]

(3-21)

Kz

where K x and K z are the gains of the PPC valves. The gains take into account the bore of the cylinder to provide the appropriate conversion from air pressure to applied force. u xComp and u zComp were added to the outputs of the PI FV controller ( u xPI , u zPI ) to produce the total controller signals ( u x , u z ) as shown in Figure 3.11.

The PI FV controller with the VDFM based

compensator was tested on the straight edge aluminum workpiece. Experimental results with PI FV controller only and with PI FV and VDFM based compensator are shown in Figure 3.34. Tracking performance is reported in Table 3.14 in terms of RMSE and ME. According to these results, addition of the friction compensator resulted in no significant improvement. This was not expected and warrants further investigation. 50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1 PI only PI + VDFM Comp

-0.2

desired 0

5

10

15

20 25 Time (sec)

30

35

40

45

Figure 3.34: Responses with PI only & PI with VDFM compensator, aluminum straight edge

Table 3.14: Performance of the PI FV controller with/without VDFM based compensator Case

ME f n

PI FV controller with VDFM compensator

(N) 17.4

PI FV controller

12.4

75

RMSE f n

RMSE vt

0.160

(N) 3.2

(m/s) 0.028

0.118

3.0

0.039

ME vt (m/s)

3.8 Summary This chapter set out to present the experimental apparatus and initial controller design. The general structure of the adopted PID based FV controller was documented. The procedure for the design, tuning and testing of the controller was described. Two PI controllers were assigned to each axis: one controller was used to control the contact force and the other to control the velocity. The effect of surface finish, profile shape and tracking speed on contour tracking performance was examined. Three different workpiece edge geometries were studied: straight, inclined and curved. The performance of a PI FV controller was benchmarked against the DFV controller proposed by Jatta et al (2006). Finally, the effect on performance when a VDFM based compensator was added to the PI FV controller was studied.

The main observations from this chapter are:



It was assumed that the performance when tracking the straight edge profile would be better than the performance when tracking the curved profile. However, the reverse was found to be true. Further work is required to explain this effect.



Both the profile shape and the surface finish had a significant effect on the tracking performance. Further work is required to understand their relative effects.



The tracking speed did not have a significant effect on the tracking performance.



The addition of the VDFM based friction compensator did not improve performance.

For the PI FV controller, the tracking performance with the straight and curved edge workpieces was found to be comparable to the performance achieved with a similar PI FV controller on an electrically actuated robot. This result confirms the potential of pneumatically actuated robots in force control applications. It also highlights the ability of a traditional fixed gain PI controller to match the performance of a more sophisticated PI controller (Jatta et al), when properly tuned.

Based upon the experimental results given in this chapter, it is concluded that a model of the system needs to be developed in order to answer the following questions:



Why was performance for the curved profile better than that for the straight edge profile?



What are the relative effects of the profile shape and the surface finish?



Why was there no improvement when the friction compensator was added?



What can be done to improve the performance?

The development and validation of a system model is the subject of the next chapter.

76

Chapter 4 MODELING AND SIMULATION RESULTS Chapter 4 will present the development and validation of a model of the system. The model will be used to better understand system behavior and to seek ways to improve system tracking performance. A PI FV controller similar to the one used in Chapter 3 will be adopted. The model will be validated against experimental results. The effects of eight parameters will be investigated: 1) unidirectional nature of the PPC valves, 2) tracking speed, 3) uncertainty in workpiece angle, 4) signal measurement noise, 5) system lag, 6) workpiece surface finish, 7) Coulomb friction and 8) workpiece shape. Elements of this chapter can be found in Abu Mallouh and Surgenor (2007).

4.1 Dynamic Modeling The gantry robot is tasked to track the profile of a workpiece while applying a constant normal force and moving with a constant tangential velocity. In the following sections a dynamic model of the robot and its contact with the workpiece is derived. System parameters are estimated using different experimental tests.

4.1.1 General Dynamic Modeling

As was illustrated in Figure 3.12, the gantry robot can be represented as a mechanism with two prismatic joints. The fixed frame (x, z) has the right side corner of the robot as its origin. The moving task frame (t, n) has the point of contact between the force sensor and the workpiece as its origin.

Given a force balance where the forces applied are resisted by the contact forces, the wire forces, viscous friction forces and the Coulomb friction forces, then the general dynamic equations for the system can be given as: f ax − f x − v x C fx − Fx − Fwx = m x a x f az − f z − v z C fz − Fz − Fwz = m z a z

77

(4-1)

where m x and m z are the effective masses of the cylinders assemblies. a x and a z are the end effector accelerations along the x and z axis, respectively. The same friction model used in designing the VDFM in Chapter 3 was used as the friction model in this chapter.

m z was measured with a digital scale and found to be 2.25 kg. It was hard to disassemble the moving parts of the x-axis cylinder in order to measure m x . As a result it was decided to estimate m x experimentally. To do so, a velocity PID controller was designed to track a linear velocity setpoint. While the piston was moving with linear velocity the acceleration was constant and m x was calculated from:

mx =

f ax − Fdx − v x C fx − Fwx ax

(4-2)

The velocity was measured directly and the acceleration was calculated by differentiating the velocity signal. A second order low pass filter was designed to filter the acceleration signal. Figure 4.1 shows the velocity and the acceleration responses. The average value of f ax and a x during the time interval from 11.0 to 12.5 sec were calculated. These values were used to calculate m x from Equation 4-2, m x was found to be 7.3 kg.

78

vx

vx (m/s)

0

vxd -0.2

-0.4 8

8.5

9

9.5

10

10.5 11 time (sec)

11.5

12

12.5

13

1 actual ax

ax (m/s2)

0.5

filtered ax

0 -0.5 -1

8

8.5

9

9.5

10

10.5 11 time (sec)

11.5

12

12.5

13

Figure 4.1: Experimental result of m x identification test

4.1.2 Contact Modeling

The workpiece is considered rigid, but the force sensor is a semi-rigid cantilever as was seen in Figure 3.2. The contact force between the workpiece and the end effector is modeled as a springdamper system as:

f n = K cnδ n + C cnδ&n

(4-3)

where K cn and Ccn are the contact stiffness and damping coefficients along the n direction, respectively. δ n is the spring deflection along the normal direction as shown in Figure 4.2 and calculated from:

δ n = ( z − z w ) cos θ

(4-4)

where z is the position of the end effector and z w is the position of the workpiece as if there is no contact with the end effector along the z-axis.

79

Figure 4.2: Modeling the contact

4.1.3 System Lag Modeling

The compressible nature of air in a pneumatic system causes a delay (lag) in response. The valves also take time to respond to their command signals. Consequently, the air pressure takes time to build up in the cylinder’s chamber. The forces applied to the cylinder due to the action of the PPC valve are modeled as a first order lag, with u x and u z as the control signals:

τ x f&ax + f ax = K x u x

(4-5)

τ z f&az + f az = K z u z

where τ x and τ z are the time constants that model the effect of air compressibility and the overall lag in the applied forces along the x and z axes, respectively.

τ x and τ z were identified experimentally by stepping up the input signals to the PPC valves while the pistons were held still. The input signal was stepped up with a rate of 0.25 volt every 6 seconds, and then the input signal was stepped down with the same rate. The time the applied force took to reach 65 % of its steady state value at each voltage step was calculated to give τ x and τ z . It is important to mention that the identified lag values represent the overall lag due to the time delay in each of the system components. Thus, the lag in the pneumatic gantry robot system includes: the time the valve takes to response to the command signal, the time the air takes to flow through the connecting tubes, the time the pressure takes to build up in the cylinder chamber and the time it takes for the cylinder to move in response to the increase in pressure.

80

The possibility that the time constants were not constant was investigated. The piston position and the condition of the cylinder as being pressurized or depressurized were assumed to be the only two variables that might cause a variable lag. Three tests were conducted for the x-axis cylinder by holding the piston still at three different positions. Because of the short stroke for the z-axis cylinder only two tests were done. In the first part of the test the input signal was stepped up (pressurizing the cylinder) and during the second part it was stepped down (depressurizing the cylinder). Figure 4.3 is an example of the test results when x-axis piston was held still at x = 1000 mm. Table 4.1, Table 4.2 and Table 4.3 summarize the τ x identification test results when the x-axis piston was held still at x = 1000, 500 and zero mm, respectively. Investigating the tables one notes that the effect of the piston position on τ x is significant and the effect of the condition of the cylinder as being pressurized or depressurized is not. A similar test was done for the z-axis. Figure 4.4 is an example of the test results when the z-axis piston was held still at z = 100 mm. Table 4.4 and Table 4.5 summarize the τ z identification test results when the z-axis piston was held still at z = 100 and zero mm, respectively. Investigating the tables one notes that the effect of the piston position and the condition of the cylinder as being pressurized or depressurized on τ z are not significant. The average value of τ x and τ z were found to be 0.3 and 0.15 sec, respectively. It was decided to use the average value of τ x and τ z in the initial contour tracking tests. Additional τ x and τ z identification results are presented in Appendix C.

81

Table 4.1: Summary of the τ x identification test, when x =1000 mm volt stepped up

pressurizing (N)

0.65-0.9 0.9-1.15 1.15-1.35 τ x average

-4 to -46 -46 to - 98 -98 to -132

τx

(65%) sec 0.15 0.12 0.12 0.13

volt stepped down

depressurizing (N)

1.3-1.15 1.15-0.9 0.9-0.65 τ x average

-132 to -105 -105 to -54 -54 to -6

τx

(65%) sec 0.20 0.12 0.15 0.16

Table 4.2: Summary of the τ x identification test, when x = 500 mm volt stepped up

pressurizing (N)

0.65-0.9 0.9-1.15 1.15-1.35 τ x average

-6 to -41 -41 to –96 -96 to –130.5

τx

(65%) sec 0.25 0.25 0.25 0.25

volt stepped down

depressurizing (N)

1.3-1.15 1.15-0.9 0.9-0.65 τ x average

-130.5 to -108 -108 to –55 -55 to -6

τx

(65%) sec 0.25 0.26 0.25 0.25

Table 4.3: Summary of the τ x identification test, when x = zero mm volt stepped up

pressurizing (N)

0.65-0.9 0.9-1.15 1.15-1.35 τ x average

-6 to -40 -40 to -97 -97 to –131

τx

(65%) sec 0.45 0.60 0.45 0.50

volt stepped down

depressurizing (N)

1.3-1.15 1.15-0.9 0.9-0.65 τ x average

-131 to –109 -109 to -55 -55 to -7

τx

(65%) sec 0.45 0.55 0.45 0.48

Table 4.4: Summary of the τ z identification test, when z = 100 mm volt stepped up

pressurizing (N)

0.85-1.1 1.1-1.35 1.35-1.6 τ z average

12 to 36 36 to 61 61 to 87

τz

(65%) sec 0.20 0.15 0.12 0.16

volt stepped down

depressurizing (N)

1.6-1.35 1.35-1.1 1.1-0.85 τ z average

87 to 62 62 to 37 37 to 12

Table 4.5: Summary of the τ z identification test, when z = zero mm volt stepped up

pressurizing (N)

0.85-1.1 1.1-1.35 1.35-1.6 τ z average

11 to 35.5 35.5 to 61 61 to 87

τz

(65%) sec 0.16 0.13 0.12 0.14

82

volt stepped down

depressurizing (N)

1.6-1.35 1.35-1.1 1.1-0.85 τ z average

87 to 62 62 to 37 37 to 12

τz

τz

(65%) sec 0.12 0.14 0.21 0.16

(65%) sec 0.11 0.15 0.16 0.14

0

fax (N)

-50 -100 -150 0

5

10

15

20 Time (sec)

25

30

35

40

0

5

10

15

20 Time (sec)

25

30

35

40

ux (volt)

1.2 1 0.8 0.6

Figure 4.3: Sample τ x identification test, when piston held still at x =1000 mm

faz (N)

100

50

0

0

10

20

30

40 Time (sec)

50

60

70

80

0

10

20

30

40 Time (sec)

50

60

70

80

uz (volt)

1.6 1.4 1.2 1 0.8

Figure 4.4: Sample τ z identification test, when piston held still at z = 100 mm

83

4.1.4 Signal Noise Modeling

In the FV control algorithm the measured f x and f z signals are used to calculate θ and f n . Thus, the noise in the f x and f z signals is transferred to θ and f n , and as a result might affect the tracking performance. So it was decided to model the signal noise and investigate its effect on the performance.

To model the signal noise a Normally Distributed Random Number Generator (NDRNG) available in the SIMULINK toolbox was used. The mean and variance of the NDRNG was set such that the generated noise was close to the actual noise in the measured signals that was identified in Chapter 3 and shown in Figure 3.5. Figure 4.5 shows an example of how the NDRNG was added to the θ signal, similarly a NDRNG was added to the f n , v x and v z signals. Figure 4.6 shows the f n , θ , v x and v z signals with the added noise.

It is essential to highlight an important difference between the simulation model and the actual system. In the control algorithm that was implemented experimentally in Chapter 3, the f x and f z signals were measured on-line and used to calculate θ and f n . However, in the simulation θ is set by the user and not calculated on-line. Setting θ by the user was the easiest way to model workpieces with different profile shapes. For example, if θ = zero deg then this is a straight edge configuration and if θ = 7 deg then this is an inclined edge configuration. The curved edge shape was modeled with a θ as a function of x (variable θ ).

Figure 4.5: NDRNG noise added to θ

84

vx (mm/sec)

4 2 0 -2 -4

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

vz (mm/sec)

2 1 0 -1 -2

θ (deg)

0.5

0

-0.5

0.2

fn (N)

0.1 0 -0.1 -0.2

Figure 4.6: Signals with the noise model added

85

4.1.5 Workpiece Surface Finish Modeling

The effect of the workpiece surface finish (straightness) was studied experimentally in Chapter 3. It was found that the surface of the workpieces used in the experiments was not perfectly smooth and straight. For example, the aluminum straight edge used in the experimental tests was not perfectly straight, but there were small bumps (waves) along the length of the edge. It was shown in Chapter 3 that the surface finish of the tracked edges has a significant effect on performance, especially when θ = zero deg (straight edge case).

Based on this observation it is considered important to model the workpiece surface finish. The experimental benchmark results with the aluminum straight edge presented in Chapter 3 showed that θ fluctuates between ± 3 deg. Based on this fluctuating, a NDRNG signal bounded by ± 3 deg was added to θ as a model of the edge surface finish. Figure 4.7 shows the θ response when the noise and surface finish models were added, where θ started to fluctuate between ± 3 deg when the end effector started contouring the workpiece profile. 3

2

contouring

θ (deg)

1

0

-1

-2

-3

0

5

10

15

20 25 Time (sec)

30

35

40

45

Figure 4.7: θ with the NDRNG signal been added to model surface finish, for straight edge

86

4.2 Model Validation The objective of this section is to tune the parameters of the controller, gantry robot and workpiece models until a good match between the experimental and simulation results is achieved. The model of the gantry robot and the workpiece were developed in the previous section. Similar to the controller presented in Chapter 3, the PI FV controller used in the simulation consists of the same four controllers. A series of tuning experiments were conducted to validate the system model. The model validation consisted of two stages:

Validation Stage 1: The four PID controllers with the system model are tuned and validated independently. When one controller is turned on, the other three are turned off. Validation Stage 2:

The four PID controllers with the system model are tuned

simultaneously. All four controllers are on during the tuning and validation process.

4.2.1 Validation Stage 1

In this stage the goal is not to tune and validate the system in contour tracking tasks, but rather to tune and validate the model when only one of the four controllers is turned on to perform a velocity or stationary force control tasks along the (x, z) frame. Initially the velocity tuning tests were conducted on the x-axis with the z-axis stationary and no contact with the workpiece. The force tuning tests were conducted with the x-axis cylinder stationary and the z-axis cylinder pressing against the workpiece. Figure 4.8 gives the x-axis simulation result for PI velocity tuning with a fixed K vpx and three values of K iv x . Figure 4.9 gives the equivalent experimental result. One notes a good match between Figure 4.8 and Figure 4.9 (with the same gains). Figure 4.10 gives the z-axis simulation result for PI force tuning with a fixed K pf z and three values of K i f z . Figure 4.11 gives the equivalent experimental result. One notes a poor match between Figure 4.10 and Figure 4.11 (with the same gains). It is concluded that there is something wrong with the system model. This will be investigated in the next section. Tuning exercises conducted for the z-axis velocity controller and the x-axis force controller did not alter this conclusion.

87

0.2 Ki=8.0 Ki=4.0 Ki=2.0 vxd

vx (m/s)

0.15 0.1 0.05 0

0

2

4

6 Time (sec)

8

10

12

0

2

4

6 Time (sec)

8

10

12

ux (v)

0.9 0.8 0.7 0.6 0.5

Figure 4.8: x-axis simulation result for PI velocity tuning, responses when K PVx = 0.5 and K IVx = [8.0, 4.0, 2.0] 0.2

Ki=8.0 Ki=4.0 Ki=2.0 vxd

vx (m/s)

0.15 0.1 0.05 0

0

2

4

6 Time (sec)

8

10

12

0

2

4

6 Time (sec)

8

10

12

ux(v)

0.9 0.8 0.7 0.6 0.5

Figure 4.9: x-axis experimental result for PI velocity tuning, responses when K PVx = 0.5 and K IVx = [8.0, 4.0, 2.0]

88

40 Ki=0.32 Ki=0.16 Ki=0.08 fzd

fz (N)

30 20 10 0

2

4

6

8 Time (sec)

10

12

14

2

4

6

8 Time (sec)

10

12

14

2

uz (v)

1.5

1

0.5

Figure 4.10: z-axis simulation result for PI force tuning, responses when K PFz = 0.02 and K IFz = [0.32, 0.16, 0.08] 40 Ki=0.32 Ki=0.16 Ki=0.08 fzd

fz (N)

30 20 10 0

2

4

6

8 Time (sec)

10

12

14

2

4

6

8 Time (sec)

10

12

14

2

uz (v)

1.5

1

0.5

Figure 4.11: z-axis experimental result for PI force tuning, responses when K PFz = 0.02 and K IFz = [0.32, 0.16, 0.08]

89

4.2.2 Displacement Dependent Friction Model

In the previous section the match between the simulation and experimental results of the z-axis force control was not as good as the match between the simulation and experimental results of the x-axis velocity control. After a series of tests, it was found that the model for friction is responsible for this mismatch. The simulation results presented in Figure 4.8 and Figure 4.10 were obtained using the VDFM. The problems with the VDFM are the difficulties in detecting a zero velocity and the switching between different state equations for sticking and sliding. All the friction models discussed in Chapter 2 were velocity dependent type. So there is a need to investigate other types of friction models, especially the non VDFM type.

To overcome the problems associated with the VDFM, Karnopp developed a model that defines a zero velocity interval (Olsson et al, 1997). Within this interval the velocity is considered to be zero and friction is force dependent, and outside the interval it is velocity dependent. With Kanopp’s approach the complexity of the friction model increases as the complexity of the dynamic systems increases (D.A Haessig and Friedland, 1991).

Another approach to friction modeling and based on experimental observation of presliding displacement is the Dahl Displacement Dependent Friction Model (DDFM). As shown in Figure 4.12 the model shows a hysterisis behavior of the friction when the velocity is close to zero. Bowden and Tabor modified the Dahl model to show that the first force/displacement curve is not the same as subsequence ones (Threlfall, 1977). It was shown that the DDFM is more accurate than the VDFM for very small displacements (Armstrong-helouvry et al, 1994).

It is expected that the DDFM will be more suitable than the VDFM, especially in applications that include velocities close to zero or very small displacements such as the stationary force control test presented in Stage 1. It was decided to initially design a DDFM for the z-axis only and to test it on the stationary z-axis force control, since in such a test the displacement of the zaxis cylinder is very small.

The VDFM parameters were identified in Chapter 3 using

unconstrained motion tests, where the end effector was not in contact with the workpiece. To investigate the potential of using the DDFM in the system model, an open loop test was conducted along the z-axis with the end effector in contact with the workpiece (constrained motion test).

90

b

a

Figure 4.12: DDFM (a) Dahl model, (b) Bowden and Tabor model (Threlfall, 1977) In the open loop test, u z was stepped up at a rate of 0.1 volt/6 sec. 6 sec was enough time to enable the system to reach steady state. When u z reached 1.6 volt the controller signal was stepped down at the same rate.

f z and f az were measured and used to calculate the Coulomb

friction with:

Fz = −( f az − f z − Fwz )

(4-6)

Figure 4.13 shows u z , f z and f az responses of the experimental constrained open loop test used in z-axis DDFM identification. From Figure 4.13 one notices that there is no contact force until the applied force reaches a value high enough to overcome the wire force and the static friction. When z is plotted against Fz and f az (as in Figure 4.14) one sees a hysterisis behavior similar to the one described in the Dahl model and shown in Figure 4.12. Figure 4.14 shows that the friction did not reach a constant value once the applied force was enough to overcome the resisting forces (static Coulomb friction and the wire forces), but it did change as the applied force and displacement changed. This of friction behavior can be predicted with the DDFM not the VDFM. These observations show that the friction in the z-axis cylinder can be modeled better with the DDFM than the VDFM, particularly for the stationary force control cases. Based on Equation 4-6 there is a doubt that Fz might be constant and Fwz might be variable. To investigate this possibility, a similar open loop test was repeated without connecting the position/velocity sensor’s wire. This makes Fwz equal to zero and Fz is calculated by:

91

Fz = −( f az − f z )

(4-7)

Figure 4.15 shows how the friction is not actually constant but varies as f az changes, and this confirms the validity of the DDFM. In the next section the DDFM is constructed and tested in a stationary force control condition along the z-axis.

uz(v)

1.5

1

0.5

0

20

40

60

80

100 120 Time (sec)

140

160

180

200

60

force, (N)

faz(N) 40

fz(N)

20 0 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

Figure 4.13: Experimental open loop result for z axis DDFM identification test (with wire force)

92

10

FZ (N)

5 0 -5 -10 -15

0

5

10

15

20

25 30 faz (N)

35

40

45

50

55

41.8

41.9

42

10

Fz (N)

5 0 -5 -10 -15 41

41.1

41.2

41.3

41.4

41.5 41.6 z (mm)

41.7

Figure 4.14: Hysterisis loop of the z-axis DDFM (with wire force)

uz(v)

1.5

1

0.5

0

20

force, (N)

60

40

60

80

100 120 Time (sec)

140

160

180

200

faz(N) fz(N)

40 20 0 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

Figure 4.15: Experimental open loop result for the z axis DDFM identification test (without wire force)

93

4.2.3 Modeling and Validating the DDFM

In this section a DDFM for the z-axis is constructed, validated and tested in a stationary force control condition along the z-axis. The DDFM proposed by Dahl is adopted. Based on this model the friction is calculated by:

Fz (t + Δt ) = Fz (t ) +

dFz (t + Δt ) .Δt dt

(4-8)

dFz (t + Δt ) ⎛ dz (t + Δt ) ⎞ 1 = ⎜ γ ( Fz (t ) − β sign (v z (t + Δt ))) 2 . ⎟. dt dt ⎝ ⎠δ

where δ , β and γ are tunable parameters, t is the current time and Δt is the sampling time. An open loop test that is similar to the one used in the previous section was tested in simulation. The DDFM tunable parameters ( δ , β and γ ) were tuned until a good match between the simulation and experimental results was obtained as shown in Figure 4.16 and Figure 4.17.

uz(v)

1.5

1 Experiment Simulation 0.5

0

20

40

0

20

40

60

80

100 120 Time (sec)

140

160

180

200

faz (N)

40

20

0

60

80

100 120 Time (sec)

140

160

180

200

Figure 4.16: Experiment and simulation open loop results for z-axis DDFM identification

94

10 Experiment Simulation

FZ (N)

5 0 -5 -10 -15

0

5

10

15

20

25 30 faz (N)

35

40

45

50

55

41.8

41.9

42

10

Fz (N)

5 0 -5 -10 -15 41

41.1

41.2

41.3

41.4

41.5 41.6 z (mm)

41.7

Figure 4.17: Experimental and simulation hysterisis loop of the z-axis DDFM Figure 4.18 shows the experimental and simulation results of the z-axis stationary force control test when K PFz = 0.02 and K IFz = 0.08. One notes that the simulation results with the DDFM are much closer to the experimental results than the results obtained with the VDFM as was shown in Figure 4.10.

95

40 Experiment Simulation

fz (N)

30 20 10 0

0

2

4

6 8 Time (sec)

10

12

14

0

2

4

6 8 Time (sec)

10

12

14

2

uz (v)

1.5

1

0.5

Figure 4.18: Experimental and simulation result for z-axis PI stationary force control, responses when K PFz = 0.02 and K IFz = 0.08

4.2.4 ANFIS DDFM

The designing and tuning of the rule based DDFM was not an easy process because of the highly nonlinear nature of the DDFM. It is expected that ANFIS can be used to model the friction and it will be easier to tune and design. In this section ANFIS is used to model the friction in the z-axis cylinder.

Based on the governing equation and the observed behavior of the rule based DDFM, an ANFIS DDFM was designed as shown in Figure 4.19. The 5 inputs to the ANFIS DDFM are f az , Δf az ,

f n , z and Δz and the only output is Fz . The ANFIS library function available in the SIMULINK fuzzy logic toolbox was used to build and train the ANFIS DDFM.

The

experimental data presented in Section 4.2.2 was used in the ANFIS DDFM training. The training was implemented off-line. Three Gaussian membership functions were assigned to each of the 5 inputs. A grid partition method was used to automatically build the structure of the ANFIS. A hybrid learning method was adopted with 10 epochs and zero tolerance. After

96

training, the ANFIS DDFM was tested and it showed a better match with the experimental results than the rule based DDFM as shown in Figure 4.20.

Despite these good off-line results with the ANFIS DDFM, the ANFIS DDFM did not behave as well as the rule based DDFM when implemented in on-line tracking tests.

Thus further

investigation is required in order to improve the ANFIS DDFM performance and confirm its potential in friction modeling for on-line applications. Since the rule based DDFM was well tuned and ready to use, it was decided not to invest more time in the ANFIS DDFM approach.

In the next section, the system model (including the rule based DDFM) is validated when tracking workpieces with different profile shapes.

Δf az (t )

Δz (t )

Figure 4.19: ANFIS DDFM for z-axis cylinder

97

10 Experiment rule based DDFM ANFIS DDFM

FZ (N)

5 0 -5 -10 -15

0

5

10

15

20

25 30 faz (N)

35

40

45

50

55

41.8

41.9

42

10

FZ (N)

5 0 -5 -10 -15 41

41.1

41.2

41.3

41.4

41.5 41.6 z (mm)

41.7

Figure 4.20: Friction results of the experimental open loop test, the rule based DDFM and the ANFIS DDFM (off-line)

4.2.5 Validation Stage 2

In Stage 1 each of the four controllers with the system model was tuned and validated individually to perform a velocity or stationary force control task along the (x, z) frame. In this stage the four controllers with the system model (including the rule based DDFM) were tuned and validated simultaneously to perform contour tracking tasks along the (n, t) frame

Figure 4.21 illustrates the SIMULINK block diagram for the PI FV controller, gantry robot and workpiece. Similar to the PI FV controller used in Chapter 3, the PI FV controller used in the simulation is made up of four controllers and two coordinate transformers. In this section the task is to track workpieces with almost the same profile, sequence and desired setpoints as in the experimental benchmark results presented in Chapter 3. Initially the workpiece profile was defined with an artificial θ (modeled, not experimental).

Table 4.6 lists the PI FV controller gains, which are the same as the gains used in the experimental tests in Chapter 3.

98

Figure 4.21: SIMULINK block diagram for PI FV controller, gantry robot and workpiece

Table 4.6: Gains of the PI FV controller used in simulation tests Controller

Kp

Ki

vx

K vpx =1.0

K iv x =4.0

fz

K pf z =0.02

K i f z =0.08

fx

K pf x =0.01

K i f x =0.03

vz

K vpz =0.5

K iv z =2.0

Figure 4.22, Figure 4.23, Figure 4.24 and Figure 4.25 show the tracking responses for straight ( θ = zero deg), inclined ( θ = 7 and 16 deg) and curved edges with DDFM and artificial θ , respectively. Table 4.7 summarizes the simulation tracking performance in terms of RMSE and ME of f n and vt during the tracking stage.

Even though these results were not close

quantitatively to the experimental results but they were very close qualitatively. The simulation results agree with the experimental results in the unexpected degradation in tracking performance when tracking the straight edge workpiece. Figure 4.26 and Figure 4.27 shows the experimental and simulation f n and v t responses when the actual θ (obtained experimentally) for the straight edge was used. Figure 4.26 illustrate a very good match between experiment and simulation. Figure 4.27 gives a close-up of the constant velocity stage as the important stage for the operation. Figure 4.28 and Figure 4.29

99

illustrate a very good match between the experiment and simulation, when the actual experimental θ of the curved edge was used. Figure 4.29 gives a close-up of the constant velocity. Table 4.8 summarizes the simulation tracking performance in terms of RMSE and ME of f n and vt during the tracking stage. Based on the results obtained with the actual angle of the tracked profile, one is more confident about the validity of the system model. The next step is to investigate the effect of system parameters.

Table 4.7: Summary of simulation tracking performances, with DDFM and artificial θ Case Max f n (N) Max vt (m/s) RMSE f n (N) RMSE vt (m/s) aluminum straight ( f nd =30 N, vtd =-0.06 m/s) aluminum inclined (7 deg) ( f nd =30 N, vtd =-0.06 m/s) aluminum inclined (16 deg) ( f nd =30 N, vtd =-0.06 m/s) aluminum curved ( f nd =30 N, vtd =-0.06 m/s)

17.8

0.119

4.4

0.042

10.4

0.113

1.4

0.042

10.2

0.118

1.6

0.045

16.4

0.154

3.2

0.053

Table 4.8: Summary of simulation tracking performances, with DDFM and experimental θ Case Max f n (N) Max vt (m/s) RMSE f n (N) RMSE vt (m/s) aluminum straight ( f nd =30 N, vtd =-0.06 m/s) aluminum inclined (7 deg) ( f nd =30 N, vtd =-0.06 m/s) aluminum inclined (16 deg) ( f nd =30 N, vtd =-0.06 m/s) aluminum curved ( f nd =30 N, vtd =-0.06 m/s)

11.8

0.121

3.0

0.042

9.4

0.116

1.2

0.043

7.0

0.118

1.2

0.045

14.8

0.282

2.6

0.062

100

50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

5

10

15

20 25 Time (sec)

30

35

40

vt (m/s)

0

-0.1

-0.2

θ (deg)

2

0

-2 0

45

z (mm)

100

50

0 300

400

500

600 700 x (mm)

800

900

1000

Figure 4.22: Simulation responses when tracking straight edge (θ = zero deg.)

101

50 40 fn (N)

30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1

-0.2

θ (deg)

10

5

0

0

z (mm)

100

50

0 300

400

500

600 700 x (mm)

800

900

1000

Figure 4.23: Simulation responses when tracking inclined edge (θ = 7 deg.)

102

50 40 fn (N)

30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1

-0.2

θ (deg)

20

15

10

0

z (mm)

100

50

0 300

400

500

600 700 x (mm)

800

900

1000

Figure 4.24: Simulation responses when tracking inclined edge (θ = 16 deg.)

103

50 40 fn (N)

30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1

-0.2

60

θ (deg)

40

20

0 20

25

30

35

Time (sec)

z(mm)

100

50

0 300

400

500

600 700 x (mm)

800

900

Figure 4.25: Simulation responses when tracking curved edge

104

1000

50 Experiment Simulation Desired

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1

-0.2

Figure 4.26: Experimental and simulation results for straight edge, with experimental θ 50 Experiment Simulation Desired

45 40

fn (N)

35 30 25 20 15 10 5 22

23

24

25

26

27 28 Time (sec)

29

30

31

32

Figure 4.27: Close up of the f n response for straight edge, with experimental θ

105

50 Experiment Simulation Desired

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

vt (m/s)

0

-0.1

-0.2

Figure 4.28: Experimental and simulation results for curved edge, with experimental θ 50 Experiment Simulation Desired

45 40 35

fn (N)

30 25 20 15 10 5 0 22

23

24

25

26

27 28 Time (sec)

29

30

31

32

33

Figure 4.29: Close up of the f n response for curved edge, with experimental θ

106

4.3 Effect of System Parameters In the previous sections a model of the system was constructed, validated and tested on straight, inclined and curved workpieces. experimental results.

Simulation results were very close to the equivalent

In this section the effect of eight parameters is investigated: 1)

unidirectional nature of the PPC valves, 2) tracking speed, 3) uncertainty in workpiece angle, 4) signal noise, 5) system lag, 6) workpiece surface finish, 7) Coulomb friction and 8) workpiece shape. It is expected that these eight parameters have the most significant effect on tracking performance. In order to improve the performance one needs to understand how each of these parameters is affecting the f n and vt responses.

4.3.1 Effect of Unidirectional PPC Valves

The PPC valves used in the experiments were unidirectional action valves, which means they control the pressure of the air flow through only one of the valve’s output ports (port A as seen in Figure 3.4). As a result, the valve can only control the pressure of one side of the cylinder. This section illustrates the effect of using unidirectional action valves on the performance.

Figure 4.30 shows the tracking responses with unidirectional and bidirectional action valves. From Figure 4.30 one notes that with the unidirectional action valves the performance degrades as θ becomes negative, but with bidirectional action valves the performance significantly improves. This is expected because with the unidirectional action valve there is active control of the cylinder pressure along the negative x-axis and positive z-axis directions only. Controlling the pressure along these two directions gives full control on f n and vt as long as θ is positive, but when θ is negative there is no active control on f n and vt .

Based on these results there were two options: 1)

To use another two PPC valves and connect them on the non pressure controlled axis directions (positive x-axis and negative z-axis), so at the end there are a total of four PPC valves.

2) To use the current two PPC valves but limit the contour tracking on the part of the profile where θ is positive. This means that f n and vt are controlled only while tracking the profile with positive θ .

107

It was decided to adopt the second option, due to limitations in time and resources. Furthermore, if performance can be improved with the existing PPC valve setup, then it follows that performance can be improved even further if two additional PPC valves are purchased and installed.

fn (N)

0

bidirectional valve unidirectional valve desired

-20

-40

0

5

10

15

10

15

10

15

Time (sec)

vt (m/s)

0

-0.1

-0.2

0

5 Time (sec)

θ (deg)

20

0

-20

0

5 Time (sec)

Figure 4.30: Responses of PI FV controller with bidirectional and unidirectional PPC valves

108

4.3.2 Effect of Tracking Speed

In this section the effect of the tracking speed ( v td ) on performance when tracking straight and curved edge workpieces is studied in simulation. Three values for v td were tested: 0.03, 0.06 and 0.09 m/s. The RMSE was used to measure the tracking performances. Table 4.9 lists the RMSE of f n and v t for the curved edge tracking case. The results show that the RMSE of f n decreased by 3% and by 16%, and the RMSE of v t decreased by 15% and 25% when v td was decreased from 0.06 m/s to 0.03 m/s and from 0.09 m/s to 0.06 m/s, respectively.

The

improvement when slowing down is not considered significant since the improvement is less than 30%. Table 4.10 lists the RMSE of f n and v t for the straight edge tracking case. The results show that the RMSE of f n decreased by 16% and by 9%, and the RMSE of v t decreased by 5% and 26% when v td was decreased from 0.06 m/s to 0.03 m/s and from 0.09 m/s to 0.06 m/s, respectively. Again the improvement when slowing down is not considered significant.

From these results one concludes that there is no significant effect to using a low tracking speed, matching the conclusion made in Chapter 3 based on experimental results.

Table 4.9: Simulation performance for curved edge, with three different v td values

v td (m/s)

0.03

0.06

0.09

RMSE f n

3.0

3.2

3.8

RMSE v t

0.046

0.053

0.066

Table 4.10: Simulation performance for straight edge, with three different v td values

v td (m/s)

0.03

0.06

0.09

RMSE f n

3.8

4.4

4.8

RMSE v t

0.040

0.042

0.051

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4.3.3 Effect of Uncertainty in Workpiece Angle

In experimental tests θ was calculated in real time based on the force sensor readings and the assumption that the tangential friction between the bearing and the workpiece was negligible ( f t =0).

Because there is a certain amount of error in the force sensor readings and the

assumption might not be totally true, there is some degree of uncertainty in the calculated θ . This section investigates the effect of the error in the calculated θ on the f n and v t responses.

Figure 4.31 shows the responses with the PI FV controller when tracking the straight edge workpiece with an angle error of zero, 30 and 50 deg. One notes how the performance degrades in the presence of error in θ , especially with the error =50 deg. Further tests showed that if the error in θ is more than 10 deg, then the f n response starts to degrade significantly. Hence, to ensure that uncertainty in the calculated θ has no major effect on the responses, the error in the calculated θ should be less than 10 deg. 10

θ error=0 deg θ error=30 deg θ error=50 deg

0

fn (N)

-10 -20

desired

-30 -40 -50 -60

0

0.2

0.4

0.6

0.8

1 Time (sec)

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 Time (sec)

1.2

1.4

1.6

1.8

2

vt (m/sec)

0.2

0

-0.2

-0.4

Figure 4.31: Responses when there is zero, 30 and 50 deg errors in θ

110

4.3.4 Effect of Signal Measurement Noise

In this section the effect of the signal measurement noise on the tracking performance is investigated. It is expected that the noise will not have a major effect on performance since the noise levels found in Chapter 3 were low.

Table 4.11, Table 4.12, Table 4.13 and Table 4.14 show the effect of four parameters; noise in measured signals, workpiece surface finish, system lag and Coulomb friction; on the tracking performance for straight, inclined with θ = 7 deg, inclined with θ = 16 deg and curved edges, respectively. There are a total of 5 cases in each of the tables: Case 1 is when all of the four parameters models are included in the system model. The remaining four cases are for when the models for each of the four parameters are removed one at a time. Specifically, Cases 2, 3, 4 and 5 correspond to the removal of noise from the measured signals, elimination of system lag, removing the workpiece surface finish model (presented in Figure 4.7) and elimination of Coulomb friction, respectively. Taking out the parameter models one at a time helps to identify the parameter that has the most significant effect on performance. The RMSE and ME are used for performance measurement and the values reported for Case 1 are considered as the reference values to which the other cases are compared. The improvement in tracking performance for case i is measured using the percentage of the RMSE change ( ΔRMSE i ) of f n and v t , as calculated from:

ΔRMSE i ( f n or vt ) =

RMSE 1 ( f n or vt ) - RMSE i ( f n or vt ) .100% RMSE 1 ( f n or vt )

(4-9)

Investigating Case 2 in Table 4.11 to Table 4.14 shows that the effect of the noise on performance is not significant at all and the noise has the least effect among the four parameters. Hence, there is no need for filtering the measured signals, which agrees with the conclusion made in Chapter 3. Also one can safely drop the noise model from the system.

4.3.5 Effect of System Lag

Investigating Case 3 in Table 4.11 to Table 4.14 shows that the effect of lag on tracking performance is not significant. The RMSE of f n was improved by 23%, 14%, 13% and 19 % when the lag model was taken out of the system model for straight, 7 deg incline, 16 deg incline

111

and curved edge tracking cases, respectively. The RMSE of v t was improved by 12%, 4%, 4% and 11 % when the lag model was taken out of the system model for straight, 7 deg incline, 16 deg incline and curved edge tracking cases, respectively. Because the improvement in all cases was less than 30%, the effect of lag on performance is not considered significant.

4.3.6 Effect of Workpiece Surface Finish

Investigating Case 4 in Table 4.11 to Table 4.14 shows that the effect of the surface finish on tracking performance is the most significant when tracking the straight edge only. The RMSE of

f n was improved by 90%, 0%, 0% and 43 % when the surface finish model was taken out of the system model for the straight, 7 deg incline, 16 deg incline and curved edge tracking cases, respectively. The RMSE of v t was improved by 12%, 4%, 4% and 11 % when the surface finish model was taken out of the system model for the straight, 7 deg incline, 16 deg incline and curved edge tracking cases, respectively. It is important to notice how the effect of the surface finish is reduced significantly when tracking non straight edges.

4.3.7 Effect of Coulomb Friction

Investigating Case 5 in Table 4.11 to Table 4.14 shows that the effect of the friction on tracking performance is really significant when tracking all profile shapes.

The RMSE of f n was

improved by 86%, 57%, 50% and 69 % when the friction model was taken out of the system model for the straight, 7 deg incline, 16 deg incline and curved edge tracking cases, respectively. The RMSE of v t was improved by 26%, 79%, 78% and 36 % when the friction model was taken out of the system model for the straight, 7 deg incline, 16 deg incline and curved edge tracking cases, respectively.

Based upon these results one concludes that the friction has the most

significant effect on tracking performance for all profile shapes and it is strongly recommended to compensate for it.

112

Table 4.11: Simulation tracking performances for straight edge Case

1 2 3 4 5

noise in signals Yes No Yes Yes Yes

surface finish

lag

Yes Yes Yes No Yes

Yes Yes No Yes Yes

friction

Yes Yes Yes Yes No

RMSE

RMSE

ME

ME

fn

vt

fn

vt

4.4 4.2 3.4 0.4 0.6

0.042 0.042 0.037 0.042 0.031

17.8 17.0 10.4 1.0 5.6

0.119 0.119 0.061 0.121 0.061

ΔRMSE i

ΔRMSE i

fn

vt Reference

5 23 90 86

0 12 0 26

Table 4.12: Simulation tracking performances for inclined edge ( θ =7 deg.) Case

1 2 3 4 5

noise in signals Yes No Yes Yes Yes

surface finish

lag

Yes Yes Yes No Yes

Yes Yes No Yes Yes

friction

Yes Yes Yes Yes No

RMSE

RMSE

ME

ME

fn

vt

fn

vt

1.4 1.4 1.2 1.4 0.6

0.042 0.042 0.038 0.042 0.009

10.4 10.4 6.4 10.4 5.2

0.113 0.105 0.062 0.113 0.061

ΔRMSE i

ΔRMSE i

fn

vt Reference

0 14 0 57

0 4 0 79

Table 4.13: Simulation tracking performances for inclined edge ( θ = 16 deg) Case

1 2 3 4 5

noise in signals Yes No Yes Yes Yes

surface finish

lag

Yes Yes Yes No Yes

Yes Yes No Yes Yes

friction

Yes Yes Yes Yes No

RMSE

RMSE

ME

ME

fn

vt

fn

vt

1.6 1.6 1.4 1.6 0.8

0.045 0.045 0.040 0.045 0.010

10.2 10.2 6.4 10.2 8.8

0.118 0.117 0.062 0.118 0.061

ΔRMSE i

ΔRMSE i

fn

vt Reference

0 13 0 50

0 11 0 78

ΔRMSE i

ΔRMSE i

fn

vt

Table 4.14: Simulation tracking performances for curved edge Case

1 2 3 4 5

noise in signals Yes No Yes Yes Yes

surface finish

lag

Yes Yes Yes No Yes

Yes Yes No Yes Yes

friction

Yes Yes Yes Yes No

RMSE

RMSE

ME

ME

fn

vt

fn

vt

3.2 3.0 2.6 1.8 1.0

0.053 0.053 0.038 0.053 0.034

16.4 21.4 19.6 21.6 20.4

0.154 0.160 0.061 0.161 0.074

113

Reference 6 19 43 69

0 28 0 36

4.3.8 Effect of Workpiece Shape

Table 4.15 reports the RMSE of f n and vt of all the 5 cases when tracking straight, inclined and curved edges. It is expected that the curved edge tracking cases will give the worst tracking performance. However a careful look at Table 4.15 shows that this is true only when the model of the surface finish or the Coulomb friction was not included in the system model (Cases 4 and 5). For the straight edge case, the f n response was the worst only when the models of both the surfaces finish and friction were included in the system model (Cases 1, 2 and 3). When the surface finish or the Coulomb friction models were not included (Cases 4 and 5) the performance significantly improved. This means that the surface finish has a significant effect only for the straight edge tracking case and somehow the surface finish and Coulomb friction are linked together. In the next section the effect of surface finish and its link with friction are investigated in more detail.

Table 4.15: f n and vt RMSE of the 5 cases for straight, inclined and curved edges Case

straight RMSE RMSE

+7 degrees RMSE RMSE

+16 degrees RMSE RMSE

curve RMSE RMSE

fn

vt

fn

vt

fn

vt

fn

vt

1

4.4

0.042

1.4

0.042

1.6

0.045

3.2

0.053

2 3 4 5

4.2 3.4 0.4 0.6

0.042 0.037 0.042 0.031

1.4 1.2 1.4 0.6

0.042 0.038 0.042 0.009

1.6 1.4 1.6 0.8

0.045 0.040 0.045 0.010

3.0 2.6 1.8 1.0

0.053 0.038 0.053 0.034

4.4 Workpiece Surface Finish Effect Based on the observations from the previous section, there are two questions that need to be answered: 1) What is the link between the surface finish and the friction when tracking a straight edge? 2) Why does the surface finish have a significant effect on performance only when tracking a straight edge?

In this section the surface finish is investigated in more detail using the experimental benchmark results from Chapter 3 in order to answer these two questions.

114

4.4.1 Straight Edge Tracking

Figure 4.32 shows a close up of the experimental tracking responses produced in Chapter 3 for the straight edge case, with the significant errors in the

f n response circled. A careful look at

the z and f n responses during the tracking period shows that the circled f n errors occurred whenever the end effector changed its direction of motion along the z-axis (see the arrows in Figure 4.32). Based on these observations, the explanation for the f n response shown in Figure 4.32 is as follows: the straight edge is not perfectly straight. Therefore, θ does not equal zero all the time, but it oscillates about zero by ± 2 deg (as shown in Figure 4.32). Whenever the value of θ changes in a way that makes the end effector change its direction of motion along the z-axis, the friction in the z-axis cylinder switches its direction too and as a result increases or decreases f az . This instantaneous increase/decrease in f az cannot be compensated with the controller action because of the lag in the system. Figure 4.33 is a close up of the tracking period from 23.6 to 24.0 sec; it shows that the controller responds almost immediately to the f n error but the actual pressure build up was delayed by about 0.15 sec because of the lag in the system. τ z was identified experimentally in Chapter 3 and the average value was found to be 0.15 sec.

Based on this explanation one understands the link between the friction and the surface finish and how they combine to cause such an unexpected f n response when tracking the straight edge.

115

z (mm)

41 40.5 40 23

24

25

26 27 Time (sec)

28

29

30

24

25

26 27 Time (sec)

28

29

30

24

25

26 27 Time (sec)

28

29

30

fn (N)

40 30 20

θ (deg)

23

2 0 -2 23

Figure 4.32: Close up of the tracking period for straight edge workpiece, with circles showing significant f n errors

116

fn (N)

40 30 23.6

23.65

23.725

23.87 Time (sec)

uz (volt)

1.4 1.2 1 23.6

23.65

23.725

23.87 Time (sec)

faz (N)

40 35 30 25 23.6

0.15 sec

23.65

23.725

23.87 Time (sec)

Figure 4.33: Close up of the period from 23.6 to 24.0 sec for straight edge

4.4.2 Inclined Edge Tracking

Figure 4.34 and Figure 4.35 are close ups of the experimental tracking responses produced in Chapter 3 for the inclined edge with θ = 7 deg and 16 deg, respectively, with the only significant error in the f n response circled. The f n response is much better than the one for the straight edge tracking. There was no significant error in f n along the tracking period, other than the error at the beginning of tracking (circled in Figure 4.34 and Figure 4.35). This is expected since the end effector did not switch its direction of motion along the z-axis and as a result the friction did not switch its direction. It is important to mention that the workpiece used in the inclined edge tracking tests was the same workpiece used in the straight edge tracking test, but the workpiece was mounted in a configuration that made θ equal to 7 or 16 deg. It was found that the value of

θ determines how much the effect of the surface finish has on the tracking performance. The f n error in Figure 4.35 was less than the error in Figure 4.34. Thus, it was concluded that the effect of surface finish was reduced as θ was increased.

117

z (mm)

40

20

0 23

24

25

26

27

28

29

30

27

28

29

30

27

28

29

30

Time (sec)

fn (N)

-20 -30 -40 23

24

25

26 Time (sec)

θ (deg)

10 8 6 4 23

24

25

26 Time (sec)

Figure 4.34: Close up of the tracking period of inclined ( θ = 7 deg.) edge workpiece, with circles showing significant f n error

118

z (mm)

100

50

0 23

24

25

26 27 Time (sec)

28

29

30

24

25

26 27 Time (sec)

28

29

30

24

25

26 27 Time (sec)

28

29

30

fn (N)

-20 -30 -40 23

θ (deg)

20

15

10 23

Figure 4.35: Close up of the tracking period of inclined ( θ = 16 deg.) edge workpiece, with circles showing significant f n error

4.4.3 Curved Edge Tracking

Figure 4.36 is a close up of the experimental responses when tracking the curved edge. The significant errors in f n are circled in Figure 4.36. At the beginning of the tracking the edge is straight, so the first circled f n error is due to the surface finish effect. The second and third circled f n errors are due to the workpiece profile shape, where the end effector changed its direction along the z-axis based on the profile shape of the workpiece. When the end effector changed its direction of motion (along the z-axis) the friction changed its direction correspondingly, and this with the presence of the lag cause the observed errors in f n . From Figure 4.36 one notes that the curved edge profile did not cause frequent changes in the end effector direction and as a result the errors in f n due to the workpiece shape are not as significant as the errors in f n due to the surface finish as in Figure 4.32. This is why the tracking performance for the straight edge is worse than for the curved edge. It is important to mention 119

that the surface finish of the straight and curved edges are similar. It was found that when θ is close to zero the surface finish effect is dominant, but when θ is larger than 5 deg the workpiece shape effect is dominant.

80 z (mm)

60 40 20 0 23

24

25

26

27 28 Time (sec)

29

30

31

32

24

25

26

27 28 Time (sec)

29

30

31

32

24

25

26

27 28 Time (sec)

29

30

31

32

fn (N)

-20 -30 -40 23 50

θ (deg)

25 0 -25 -50 23

Figure 4.36: Close up of the tracking period of curved edge workpiece, with circles showing significant f n errors

120

4.5 Summary This chapter set out to develop and validate a model of the system, in order to better understand system behavior and to seek ways to improve system tracking performance. A PI FV controller similar to the one used in Chapter 3 was adopted. The model was validated against experimental results. The effects of eight parameters were investigated: 1) unidirectional nature of the PPC valves, 2) tracking speed, 3) uncertainty in workpiece angle, 4) signal measurement noise, 5) system lag, 6) workpiece surface finish, 7) Coulomb friction and 8) workpiece shape.

The observations from this chapter are:



Switching from a unidirectional to bidirectional valve will significantly improve system tracking performance and will widen the range of profile shapes that can be tracked.



The potential of ANFIS in friction modeling requires more investigation. The ANFIS design tool available in MATLAB enables the user to implement only off-line training, it is expected that this will be the main limitation when using the MATLAB ANFIS designing tool in friction modeling.



There was no improvement in the tracking performance when v td was reduced.



If the estimated angle of the workpiece is within ± 10 deg, performance will not be adversely affected.



There was no significant effect due to the noise in the measured signals on the tracking performance.



The system lag prevents the controller from acting quickly and consequently limits the ability of the system to properly regulate the contact force.



Surface finish has a significant effect on performance when tracking a straight edge, because it was observed to induce frequent changes in the end effector and friction directions along the z-axis. This also explains the unexpected f n response when tracking a straight edge.



Coulomb friction has the most significant effect when tracking the three different profiles, and consequently is the main limitation to better force control.



Both the experimental and simulation results confirm that tracking performance with the straight edge is worse than with the inclined and curved edge workpieces.

121

Another key observation from this chapter is that the match between experimental and simulation results with the DDFM is better than the match with the VDFM. In fact, simulation results with the DDFM match the experimental results to a very good degree. Thus, a DDFM compensator should be adopted to improve overall tracking performance. It is concluded that a DDFM compensator is more suitable than a VDFM compensator for tracking applications. This is especially true when the displacement is small as in the case of straight edge tracking or when significant system lag is present.

In the next chapter a number of techniques will be investigated that will help deal with system nonlinearities such as lag and friction, and consequently further improve contour tracking performance.

122

Chapter 5 NEURAL NETWORK BASED COMPENSATION Chapter 5 will examine a number of techniques to improve tracking performance: 1) PI FV controller with a table look up based gain scheduler. 2) PI FV controller with friction compensation. 3) PI FV controller with Extended Back Propagation Algorithm (EBPA) Neural Network compensation. 4) PI FV controller with Modified Back Propagation Method (MBPM) Neural Network compensation. It was found in Chapter 4 that system nonlinearities such as the system lag and Coulomb friction had significant negative effects on tracking performance. To compensate for these nonlinearities, a gain scheduler approach will first be examined because of its relative simplicity.

Next,

assuming that friction is the dominant factor, a DDFM based compensator for the z-axis and VDFM based compensator for the x-axis will be studied. The rule based DDFM developed in Chapter 4 will be used instead of the ANFIS based DDFM because the former is easier to implement on-line. Finally, two different NN compensators will be designed to compensate for both the system lag and friction in the system. Simulation results for straight and curved edges

will be presented to demonstrate the effectiveness of the techniques. Elements of this chapter can be found in Abu Mallouh and Surgenor (2008).

5.1 PI FV Controller with Gain Scheduler The need for a variable gain structure is a reflection of the nonlinear nature of lag and friction in the pneumatically based system. With the gain scheduling approach the PI FV controller gains are not constant, but they are updated on-line. It is expected that for each value of θ there is a set of gains that gives the best tracking performance. The gain scheduling approach based on θ is investigated in this section. The PI FV controller has a total of 8 gains. Each of the 8 gains was retuned to give the best performance when tracking edges with θ = 0, 15, 30 and 45 deg, retuned gains are presented in Table 5.1. Table 5.1 shows 4 sets of gains categorized by θ value. The gain scheduling approach was tested on a curved edge with θ ranging between 0 to 50 deg. While tracking the curved edge the controller selects the gain setting from Table 5.1 that is

123

associated with the current value of θ . If the gain scheduling approach is effective it is expected to have a better performance with the gain scheduler than with the fixed gains. Figure 5.1 and Figure 5.2 show the tracking response for curved edge workpiece with and without the gain scheduler. Figure 3.33 shows the profile of the curved edge. Figure 5.3 shows how four of the eight gains change with time during the contouring stage. Tracking performance is reported in Table 5.2 in terms of RMSE and ME. According to these results, the gain scheduling approach resulted in no improvement, most likely because of lag in the system. Feedback control in the presence of lag leads to particular difficulties since the lag places a limit on the achievable bandwidth and allowed maximum gain (Huang and Lewis, 2003)

θ

gains set No.

(deg.)

0 ≤ θ < 15 15 ≤ θ < 30 30 ≤ θ < 45 45 ≤ θ

1 2 3 4

K vpx

Table 5.1: Gains lookup table K iv x K pf z K if z

0.5 0.5 1 2

2 2 4 16

0.02 0.01 0.02 0.04

0.08 0.08 0.08 0.32

K pf x

K if x

K vpz

K iv z

0.025 0.05 0.01 0.025

0.12 0.06 0.03 0.01

0.1 0.4 0.8 1.0

2.0 2.0 2.0 0.5

50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

30

35

40

45

0

vt (m/s)

-0.1 -0.2

PI FV PI FV + gain scheduler Desired

-0.3 0

5

10

15

20 25 Time (sec)

Figure 5.1: Responses with PI FV controller and gain scheduler, curved edge

124

50 45 40 35

fn (N)

30 25 20 15 10

PI FV PI FV + gain scheduler Desired

5 0 22

23

24

25

26

27 28 Time (sec)

29

30

31

32

33

Figure 5.2: Close up of the f n response in Figure 5.1

Kifx Kpfx

0.3

Kifz Kpfz

gain value

0.25

0.2

0.15

0.1

0.05

0 22

23

24

25

26

27 28 Time (sec)

29

30

31

32

33

Figure 5.3: Change in gains during contouring the curved edge

Table 5.2: Performance of the PI FV controller with//without gain scheduler for curved edge Case

ME f n (N)

ME vt (m/s)

RMSE f n (N)

RMSE vt (m/s)

PI FV controller

14.8

0.282

2.6

0.062

PI FV controller with gain scheduler

28.0

0.282

3.4

0.063

125

5.2 PI FV Controller with Friction Compensation In Chapter 4 the rule based DDFM for the z-axis and VDFM for the x-axis cylinders had been designed and validated.

In this section the PI FV controller with a friction compensation

approach is tested on tracking a straight edge workpiece first. If the controller shows good performance then it will be implemented on workpieces with other profile shapes.

It was

concluded in Chapter 4 that the DDFM is more accurate than the VDFM when the displacement is small as in the case of the end effector displacement along the z-axis when tracking a straight edge. Thus, the DDFM was adopted for the z-axis and the VDFM for the x-axis. If the PI FV controller with friction compensation approach is to be implemented on tracking workpieces with any profile shape, it is recommended to have a DDFM and VDFM for each axes (x and z axes). The DDFM will be active when the displacement is very small and the VDFM will be active when there is a significant displacement. The outputs of the x and z axes compensators are the inverse of the VDFM and DDFM, respectively. The outputs of the friction compensators are added to the outputs of the PI FV controller as shown in Figure 5.4. When the PI FV controller with friction compensation was implemented on a straight edge tracking test, surprisingly the tracking performance did not improve. This was investigated intensively and it was found that the lag in the system was responsible for this result. Figure 5.5 shows the f n simulation response with a: 1) PI FV controller alone and with lag in the system, 2) PI FV controller with friction compensation and with lag in the system and 3) PI FV controller with friction compensation and without lag in the system. From Figure 5.5 one notes that the performance of the PI FV controller with friction compensation and with the presence of lag in the system did not improve at all, but even got worse. To confirm the effect of the lag and the functioning of the friction compensator, Figure 5.5 shows that the friction compensator was able to effectively negate the friction effect when there is no lag in the system. Thus, it is concluded that it is not possible to improve the f n response, even with perfect friction compensation, without dealing with the lag in the system. The next section will be on the subject of system lag compensation.

126

Figure 5.4: Block diagram of the PI FV controller with friction compensation 50 PI only, with Lag PI+Comp, with Lag PI+Comp, without Lag Desired

45 40

fn (N)

35 30 25 20 15 10 5 22

23

24

25

26

27 28 Time (sec)

29

30

31

32

Figure 5.5: f n response with PI FV and PI with friction compensators with/without lag in the system, showing effect of lag

127

5.3 System Lag Compensation It was shown in the previous section that the tracking performance did not improve, even with perfect friction compensation, because of the lag in the system.

In order to improve the

performance, the lag must be dealt with. In this section a distinction is made between the terms time delay and lag. The effects of lag on system stability and the main approaches to control systems with lag are reviewed and presented.

Time delay is frequently encountered in various dynamic systems, such as pneumatics, telerobotics, or long distance satellite transmissions (Huang and Lewis, 2003). Time delay systems are also known as systems with time lag (Gu and Niculescu, 2003). It was found in the literature that the term time delay is more common used to describe the delay in transmission of information between different parts of the system. It is important to distinguish between the use of the time delay and lag terms in this thesis. From the literature it was concluded that the time delay term describes a delay in a singular component of the system such as a delay in solenoid valves, sensors, and slow acoustic connections. In the context of this thesis, lag could be described as “apparent time delay”, following the practice in chemical process control (Singhal and Salsbury, 2005). Thus, the lag in the pneumatic gantry robot system includes the time the valve takes to response to the command signal, the time the air takes to flow through the connecting tubes, the time the pressure takes to build up in the cylinder chamber and the time it takes for the cylinder to move in response to the increase in pressure. Figure 5.6 and Figure 5.7 show the root locus of the z-axis force model with and without lag, respectively. As shown in Figure 5.7, the open loop transfer function of the z-axis force model has two poles at -15 ± 65j and one zero at -333. The poles of the P-only closed loop transfer function are always in the stable side of the S-plane. But when the lag model is included in the system, a third pole at -3.33 is added and the poles move to the unstable side as the gain increases. As shown in Figure 5.6 most of the poles are located on the unstable side, which means the gains that can be used without driving the system to the unstable region are limited.

128

2500

0.115

0.085

0.056

0.036

0.016 2e+003

2000 0.17

1.5e+003

1500

G(s) =

0.26 1e+003

1000 Imaginary Axis

500

0.5

500

0.5

500

0 -500 -1000

1e+003 0.26

-1500

1.5e+003

-2000 0.17 0.115 -2500 -350

-300

-250

0.085 -200

0.056 -150

2e+003 0.016

0.036 -100

-50

0

50

100

150

Real Axis

Figure 5.6: Root locus of the z-axis force model with lag in the system 400 0.95

0.9

0.82

0.7

0.52

0.3

300 0.978 200 0.994 Imaginary Axis

100

0

1e+003

800

600

400

200

-100 0.994 -200 0.978 -300 0.95 -400 -1200

0.9 -1000

-800

0.82 -600

0.7 -400

0.52 -200

0.3 0

200

Real Axis

Figure 5.7: Root locus of the z-axis force model without lag in the system

129

Many kinds of lag compensation methods have been presented for linear systems (Huang and Lewis, 2003). One of the earliest controllers for systems with lag and still relevant today is the Smith predictor. In Smith’s scheme, Figure 5.8, a predictor is equivalent to the lag free part of the plant. Control is applied based on feedback information from the predictor. However, the Smith predictor controller requires a good model of the plant. If there is a mismatch between the model and the actual plant, the closed loop performance may be very poor. Moreover if the plant is nonlinear or there is a disturbance during the control process, it is difficult to use a linear model based strategy to obtain satisfactory control performance. Because of these limitations there have been many modifications to the conventional Smith predictor. Matausek and Micic (1996) modified the Smith predictor to improve the disturbance rejection and to reduce the number of tuning parameters. In Bahill, (1983) an adaptive control loop was added to the Smith predictor in order to adjust the model parameters on-line.

Figure 5.8: Smith predictor control scheme (Bahill, 1983) More recently NN’s have been applied to compensate for the lag effects. As shown in Figure 5.9, a NN has been implemented in the Smith predictor in order to compensate for the lag and the mismatch between the model and actual pneumatic system as in Lin et al (2005). A similar approach was implemented by Lewis and Huang (2003) in telerobot control with lag caused by communication channel transmission delays.

130

du/dt Actual output Desired setpoint Fuzzy Logic Controller

k

1 s

Plant

Actual output -1 Z -2 Z Estimated output Time delay

NN Smith predictor NN compensator

Figure 5.9: NN Smith predictor and fuzzy-PID controller for pneumatic system (adapted from Lin et al, 2005) In order to get good performance with a NN Smith predictor, it must be trained to model the system off-line. In tracking applications modeling the dynamics of the overall system depends not only on the actuation system, but also on the workpiece, which is usually not fixed. This means if the NN Smith predictor is to be used, one needs to train the NN (identify the overall system model) off-line every time the workpiece is changed.

As a conclusion, NN Smith

predictor is not considered a good candidate for tracking applications.

It is expected that the direct NN non model based controller approach (presented in Chapter 2) is the best candidate for contour tracking applications. Unlike the Smith predictor and the indirect NN model based controller, the direct NN controller implements on-line training based on the actual error in the controlled variables ( f n and vt ) and does not require a model of the system. An example of this type is presented in Li and Asakura, (2003), where a NN was applied to compensate for the lag and nonlinear friction effects in position control of a 2-link pneumatic manipulator. Figure 5.10 shows the block diagram of the system and the PD controller with the NN compensator. Simulation and experimental results show good performance for position control with a lag range of 0.012 to 0.12 sec. As this is the level of delay experienced in the gantry robot application, this NN approach was adopted. This approach will be used in the next section to develop a NN compensator.

131

Short tube Desired joint angles

Joint angles PD controller

Robot Long tube NN compensator du/dt Joint angles

Figure 5.10: Block diagram of PD controller with NN compensation for pneumatic system (adapted from Li and Asakura, 2003)

5.4 PI FV Controller with Neural Network Compensation In this section a NN compensator is developed to correct for the system lag and friction. Figure 5.11 shows the block diagram of the PI FV controller with the NN compensator. The NN compensator output is added as an offset to the z-axis PI controller output. As shown in Figure 5.12 the NN compensator is a feedforward three layered NN with 14 inputs and one output. The NN compensator acts as a dynamic multiple input single output model of the system. Inputs were selected to provide information about the system lag and e f n . On-line training is adopted. Thus, neuron weights are constantly updated to minimize the e f n .

Two types of NN’s were studied. The first is the Extended Back Propagation Algorithm (EBPA) NN where the NN is a feedforward NN with a modified sigmoidal activation function. The second is a feedforward three layered NN with a standard sigmoidal activation function and a Modified Back Propagation Method (MBPM) that is designed expressly for on-line training and controlling. Details about both NN’s are presented in the next sections.

132

u xPI

u zPI

u zNN

Figure 5.11: Block diagram of PI FV controller with NN compensator

input layer

output layer

hidden layer

hidden layer weight vector V

e f n (t)

V1,21

output layer weight vector W

σ

e f n (t-1) e f n (t-2)

σ

Δe f n (t) input vector P

W13,1

f nd (t) f nd (t-1) f nd (t-2) f nd (t-3) f nd (t-4) f n (t) f n (t-1) f n (t-2) f n (t-3) f n (t-4)

u zNN

σ

σ

W13,nh

σ Vni2,nh

Figure 5.12: Structure of a three layered feedforward NN compensator with 14 inputs

133

5.4.1 EBPA Neural Network Compensator

Chen and Nutter (1992) proposed that each neuron in the hidden and output layers of a multilayer feedforward NN has the capability of updating the activation function parameters as well as the connection weights. Their relative learning algorithm was named the EBPA. In a separate study, an EBPA NN demonstrated substantial performance improvements with respect to the standard Back Propagation Algorithm (BPA) in terms of accuracy and learning speed, when applied to a fault tolerant flight control system (Napolitano et al, 2000). For a three layered sigmoidal NN, the activation function σ iL for node i in layer L is commonly given as:

σ iL (net iL ) =

1

(5-1)

L

1 + e − neti

where the function net iL is the sum of the inputs to node i in layer L and is defined for nodes in the hidden layer and the output layer, respectively, as:

⎞ ⎛ ni net i2 = ⎜⎜ ∑ Vi ,2j p 1j ⎟⎟ + bi2 , i = 1, 2, ..., nh ⎠ ⎝ j =1

(5-2)

⎛ nh ⎞ net i3 = ⎜⎜ ∑ Wi ,3j σ 2j ⎟⎟ + bi3 , i = 1, 2, ..., no ⎝ j =1 ⎠

(5-3)

where ni is number of inputs, nh and no are number of nodes in the hidden and output layers, respectively, p 1j is jth input in the input layer, Vi ,2j is the weight connecting node i in the hidden layer and the input p 1j , Wi ,3j is the weight connecting node i in the output layer and the output of node j in the hidden layer, bi2 and bi3 are the bias of node i in the hidden and output layers, respectively.

In EBPA, Equation 5-1 is replaced by a more flexible sigmoidal activation function:

σ (net ) = L i

L i

U iL − QiL 1+ e

− ( netiL / siL )

+ QiL

134

(5-4)

where U iL and QiL are the upper and lower bounds of the sigmoidal activation function for node i in layer L , respectively, and s iL is the slope of the activation function. The output of the only node in the output layer of the EBPA NN u zNN is given as:

⎛ nh ⎞ u zNN = σ 13 ⎜⎜ ∑ W13,j σ 2j + b13 ⎟⎟ ⎝ j =1 ⎠

(

)

(5-5)

For back propagation based NN’s, the updates to the tuning weights and the sigmoidal activation function parameters in the hidden layer are given as:

∂σ i2 ∂σ 13 ΔV (t + Δt ) = H 1 p W e fn + ΔVi ,2j (t )α 3 2 ∂net1 ∂net i 2 i, j

1 j

3 1,i

(5-6)

ΔU i2 (t + Δt ) = H 3W13,i

∂σ i2 ∂σ 13 e + ΔU i2 (t )α f ∂U 13 n ∂U i2

(5-7)

ΔQi2 (t + Δt ) = H 3W13,i

∂σ i2 ∂σ 13 e + ΔQi2 (t )α fn 3 2 ∂Q1 ∂Qi

(5-8)

Δsi2 (t + Δt ) = H 3W13,i

∂σ i2 ∂σ 13 e + Δs i2 (t )α fn 3 2 ∂s1 ∂s i

(5-9)

where H1 is the learning rate for Vi ,2j and H 3 is the learning rate for U i2 , Qi2 and s i2 . α is a momentum term, which determines the effect of past weight changes on the current direction of movement in weight space, to increase the learning speed without leading to oscillation.

The updates to tuning weights and sigmoidal activation function parameters of the only node in the output layer are given as:

ΔW13,j (t + Δt ) = H 2σ 2j

ΔU13 (t + Δt ) = H 3

∂σ 13 e + ΔW13,j (t )α 3 fn ∂net1

∂σ 13 e + ΔU13 (t )α 3 fn ∂U1

135

(5-10)

(5-11)

ΔQ13 (t + Δt ) = H 3

Δs13 (t + Δt ) = H 3

∂σ 13 e + ΔQ13 (t )α 3 fn ∂Q1

∂σ 13 e f + Δs13 (t )α ∂s13 n

(5-12)

(5-13)

The derivative terms in Equations 5-6 to 5-13 are given as:

∂σ iL − (σ iL − U iL )(σ iL − QiL ) = ∂net iL siL (U iL − QiL )

(5-14)

∂σ iL 1 = L − net iL ∂U i 1+ e

(5-15)

siL

∂σ iL ∂σ iL 1 = − ∂QiL ∂U iL

(5-16)

∂σ iL − netiL ∂σ iL = ∂siL siL ∂netiL

(5-17)

The advantage of EBPA relative to standard BPA is that it is faster and the potential for misleading local minima of the error function is minimized. EBPA NN program details are available in Appendix D.

5.4.2 MBPM Neural Network Compensator

A NN used in a dynamic control loop poses problems not present when it is used for traditional classification or system identification. Persistent problems include the inability to guarantee satisfactory performance of the systems in terms of small tracking errors and bounded NN weights (Lewis et al, 1996).

Lewis et al (1996) developed a multilayer nonlinear NN based controller for a serial link robot. Their algorithm is referred to as MBPM. In MBPM, an improved weight tuning method is used to correct for deficiencies that result when a NN with standard BPA is used in control

136

applications in the presence of unmodeled disturbances. The advantage of an MBPM NN is that it lends itself to on-line training. The weights can be easily initialized and tuned on-line. No offline training is required.

The standard sigmoid activation function given in Equation 5-1 is used as the basis for MBPM. In MBPM, Equations 5-6 and 5-10 become:

ΔVi ,2j (t + Δt ) = H 1 p 1jW13,i

∂σ i2 ∂σ 13 e − λH 1Vi ,2j (t ) e f n fn 3 2 ∂net1 ∂net i

ni ∂σ 2j ∂σ 13 ∂σ 13 1 2 ΔW (t + Δt ) = H 2σ e − H ( p V ) e − λH 2W13,j (t ) e f n 2∑ f i j ,i fn 2 3 ∂net13 n ∂ net ∂ net i =1 1 j

3 1, j

2 j

(5-18)

(5-19)

where λ is an adaptation design parameter. Examination of Equations 5-18 and 5-19 reveal that they consist of a standard back propagation term (first term in the equations), plus the error modification term taken from adaptive control (last term in the equations), plus a novel second order forward propagation term taken from the back propagation network (second term in Equation 5-19). In MBPM the NN output u zNN is given as:

⎛ nh ⎞ u zNN = σ 13 ⎜⎜ ∑ W13,j σ 2j + b13 ⎟⎟ + D ⎝ j =1 ⎠

(

)

(5-20)

where D is a robustifying control parameter added to the standard NN output to overcome higher order modeling errors. The parameter D is given by:

D = − k z ( G + G )e f n − k v e f n

(5-21)

where G is the maximum expected value of G, k z and k v are gain terms and the matrix G is given by:

⎡W 0 ⎤ G=⎢ ⎥ ⎣0 V⎦

(5-22)

The advantage of an MBPM relative to EBPA is that it is designed expressly for on-line training. Thus, the weights can be easily initialized and tuned on-line. No off-line training is required. MBPM NN program details are available in Appendix D.

137

5.4.3 Results with PI FV Controller and NN Compensation

Both the EBPA and the MBPM NN’s were implemented with MATLAB’s SIMULINK and are based upon a library of Adaptive Neural Networks originally developed by Campa et al (2002). This approach was considered easier to implement on-line than working with MATLAB’s traditional Neural Network Toolbox. The EBPA and MBPM NN’s were designed as a three layer feedforward NN with 14 inputs and one output. Tuning of both NN’s was done by trial and error and trying different combination of inputs and tuning parameters values. Initially 30 inputs were used but later this number was dropped to 14 as shown in Figure 5.12. It was found that one of the problems with NN was the lack of a systematic method for tuning. The NN output was added as an offset to the control signal of the z-axis as shown in Figure 5.11.

Figure 5.13 and Figure 5.14 illustrate the tracking responses for a straight edge with PI FV controller and EBPA NN compensator with zero initial value for the weights. There was no improvement in the tracking performance with the EBPA NN compensator. It was found that the EBPA NN is sensitive to the initial value of the node weights. After a series of trials it was found that the response was improved significantly when weights were initialized at 0.1 as shown in Figure 5.15 and Figure 5.16. Table 5.3 shows the definition, initial value and final value of the design parameters.

Figure 5.17 and Figure 5.18 illustrate the tracking responses for a straight edge with PI FV controller and MBPM NN compensator with zero initial value for the weights. The tuning of the MBPM NN was conducted by trial and error. Table 5.4 shows the definition, initial value and final value of the design parameters.

One notes a significant improvement in tracking

performance with the MBPM NN compensator. As expected the MBPM NN was not sensitive to initial weight values. The MBPM NN required fewer nodes in the hidden layer and in general it was easier to tune than the EBPA NN. Thus the MBPM NN is considered better and was adopted for the next test. Figure 5.19 and Figure 5.20 illustrate the tracking responses for a curved edge with PI FV controller and MBPM NN compensator with zero initial value for the weights. Once again one notes a significant improvement in performance.

138

Table 5.3: Definition, initial value and final value of the EBPA NN design parameters No.

Parameter

Definition

Initial value

Final value

1

ni

30

14

2

nh

1

50

3

no

number of inputs number of nodes in hidden layer number of outputs

1

1

4 5

H1 H2

1 1

0.0006 0.0006

6

H3

learning rate of V learning rate of W learning rate of activation function parameters

1

0.0006

7

LimV

limit of V

8

LimW

1× 10 37 1× 10 37

9

LimP

limit of W limit of activation function parameters Momentum Initial node weight

1× 10 37 1× 10 37

1× 10 37

1× 10 37

0 0

0.1 0.1

10 11

α NA

Table 5.4: Definition, initial value and final value of the MBPM NN design parameters No.

Parameter

Definition

Initial value

Final value

1

ni

30

14

2

nh

1

20

3 4 5 6

no H1 H2

1 1 1 1

1 0.4 0.4 0.1

7

s

1

1

8

bias

number of inputs number of nodes in hidden layer number of outputs learning rate of V learning rate of W adaptation parameter slop of sigmoid activation function activation function bias

9

LimV

limit of V

10

LimW

11

λ

0

1 37

10

limit of W

1× 10 1× 10 37

kz

Robustifying gain term

0

0.01

12

kv

Robustifying gain term

0

0.1

13

NA

Initial node weight

0

0

139

10

50

fn (N)

40 30 20 10 0

0

5

10

15

20 25 time (sec)

30

35

40

45

30

35

40

45

vt (m/s)

0

-0.1 PI FV controller PI FV + EBPA NN comp. Desired

-0.2 0

5

10

15

20 25 time (sec)

Figure 5.13: Responses for straight edge tracking, with PI FV controller and EBPA NN compensator, zero initial weights and 14 inputs 50 PI FV controller PI FV + EBPA NN comp. Desired

45 40

fn (N)

35 30 25 20 15 10 5 22

23

24

25

26

27 28 time (sec)

29

30

31

Figure 5.14: Close up of the f n response in Figure 5.13

140

32

50

fn (N)

40 30 20 10 0

0

5

10

15

20 25 time (sec)

30

35

40

45

30

35

40

45

vt (m/s)

0

-0.1 PI FV controller PI FV + EBPA NN comp. Desired

-0.2 0

5

10

15

20 25 time (sec)

Figure 5.15: Responses for straight edge tracking, with PI FV controller and EBPA NN compensator, 0.1 initial weights and 14 inputs 50 PI FV controller PI FV + EBPA NN comp. Desired

45 40

fn (N)

35 30 25 20 15 10 5 22

23

24

25

26

27 28 time (sec)

29

30

31

Figure 5.16: Close up of the f n response in Figure 5.15

141

32

50

fn (N)

40 30 20 10 0

0

5

10

15

20 25 time (sec)

30

35

40

45

30

35

40

45

vt (m/s)

0

-0.1 PI FV controller PI FV + MBPM NN comp. Desired

-0.2 0

5

10

15

20 25 time (sec)

Figure 5.17: Responses for straight edge tracking with PI FV and MBPM NN compensator, zero initial weights and 14 inputs 50 PI FV controller PI FV + MBPM NN comp. Desired

45 40

fn (N)

35 30 25 20 15 10 5 22

23

24

25

26

27 28 time (sec)

29

30

31

Figure 5.18: Close up of the f n response in Figure 5.17

142

32

50

fn (N)

40 30 20 10 0 0

5

10

15

20 25 Time (sec)

30

35

40

45

30

35

40

45

vt (m/s)

0

-0.1 PI FV controller PI FV + MBPM NN comp. Desired

-0.2 0

5

10

15

20 25 Time (sec)

Figure 5.19: Responses for curved edge tracking with PI FV controller and MBPM NN compensator, zero initial weights and 14 inputs 50 PI FV controller PI FV + MBPM NN comp. Desired

45 40

fn (N)

35 30 25 20 15 10 5 22

23

24

25

26

27 28 Time (sec)

29

30

31

Figure 5.20: Close up of the f n response in Figure 5.19

143

32

Table 5.5 provides quantitative measures of the relative performance of the controllers, for both straight and curved edge workpieces. Tracking performance is given by the RMSE and ME of

f n and vt . The superiority of the PI FV controller with MBPM NN compensator, relative to the PI FV controller with EBPA NN compensator and PI FV controller, is confirmed by the results given as Cases 1, 2, 3 and 4 for the straight edge, and Cases 6 and 7 for the curved edge workpieces. For the straight edge tracking case the ME and RMSE of f n were improved with the PI FV controller and MBPM NN compensator by 36% and 60%, respectively, when compared to PI only results. For the curved edge tracking case the ME and RMSE of f n were improved by 55% and 69%, respectively, when compared to PI only results.

It is expected that the MBPM NN compensator with the 14 inputs will impose significant computational load when implemented experimentally. Also it was found that some of the 14 inputs are redundant. Thus, further testing was conducted in order to reduce the number of inputs without significantly reducing the performance efficiency. Initial investigation showed that a good performance could be achieved with only 5 inputs. Figure 5.21 shows the MBPM NN compensator with the 5 inputs. For the straight edge tracking case the ME and RMSE of f n were improved with the PI FV controller and MBPM NN compensator by 41% and 73%, respectively when compared to PI only results. For the curved edge tracking case the ME and RMSE of f n were improved by 50% and 62%, respectively when compared to PI only results. Even though there was no significant change in tracking performance when the MBPM NN with 5 inputs was used, reducing the number of inputs from 14 to 5 is considered important as it reduces the computational load when implemented experimentally.

144

input layer

hidden layer

output layer

σ e f n (t)

σ σ

e f n (t-1) input vector P

σ

e f n (t-2) uz

u zNN

σ

σ

(t-1)

σ uz

(t-2)

Figure 5.21: General structure of a three layered feedforward NN compensator, with 5 inputs

Table 5.5: Performance with different controllers and NN compensation Case 1

Straight edge

2 3 4

5

Curved edge

6

7

8

PI FV controller PI FV + EBPA NN compensator, (with zero initial weights and 14 inputs) PI FV + EBPA NN compensator, (with 0.1 initial weights and 14 inputs) PI FV + MBPM NN compensator, (with zero initial weights and 14 inputs) PI FV + MBPM NN compensator, (with zero initial weights and 5 inputs) PI FV controller

ME f n (N)

ME vt (m/s)

RMSE f n (N)

RMSE vt (m/s)

11.8

0.121

3.0

0.042

13.6

0.122

4.0

0.025

9.6

0.122

1.4

0.025

7.6

0.123

1.2

0.025

7.0

0.122

0.8

0.025

14.8

0.282

2.6

0.062

6.6

0.159

0.8

0.048

7.4

0.159

1.0

0.048

PI FV + MBPM NN compensator, (with zero initial weights and 14 inputs) PI FV+ MBPM NN compensator, (with zero initial weights and 5 inputs)

145

5.5 Summary The objective of this chapter was to examine ways to improve the tracking performance. Initially a PI FV controller with a table look up based gain scheduler was investigated. Next, a PI FV controller with DDFM based compensator was tested.

Finally, two Neural Network based

compensators were designed for the PI FV controller. Simulation tracking results for straight and curved edges were presented to demonstrate the effectiveness of the different techniques.

The observations from this chapter are: •

The table look up gain scheduling approach based on workpiece angle did not improve the performance, most likely because of the lag in the system. Feedback control in the presence of lag leads to particular difficulties since a delay places a limit on the achievable bandwidth and allowable maximum gain.



Because of the lag in the system, the force regulation response could not be improved even with perfect DDFM based friction compensation.



Smith predictor and indirect NN model based controllers are not considered good candidates for contour tracking applications, since they require off-line modeling for both the pneumatic apparatus and the workpiece. Direct NN model based controllers are recommended instead.



The PI FV controller tracking performance with the MBPM NN compensator was better than with the EBPA NN compensator. MBPM NN was also easier to tune than EBPA NN. However, both NN based PI FV controllers, when trained properly to learn the system dynamics, showed significant improvement over the PI only FV controller.



For the straight edge tracking case the ME and RMSE for force regulation were improved with the PI FV controller and MBPM NN compensator by 36% and 60%, respectively, when compared to PI only FV controller results. For the curved edge tracking case, the ME and RMSE for force regulation were improved by 55% and 69%, respectively.



The number of inputs to the MBPM NN was reduced from 14 to 5 without significant changes on the performance. Reducing the number of the inputs is important when it comes to experimental implementation of the controller, since this is expected to reduce the computational load and thus enable easier real time training.

The key observation from this chapter is that the lag in the system must be accounted for in order for friction compensation to be effective. The NN compensation approach was the most effective

146

in improving tracking performance. The MBPM NN compensator is better than the EBPA NN because: 1) it has fewer nodes; 2) it is less sensitive to the initial value of the weights and 3) it is easier to tune. However, further work is required to provide more formal guidance as to the optimum number of inputs for a MBPM NN.

147

Chapter 6 CONCLUSIONS AND RECOMMENDATIONS Control of the contact force between a robot manipulator and a workpiece is critical for successful execution of tasks where the robot’s end effector must perform a contact operation along the contour of a workpiece. Representative tasks include polishing, grinding and deburring. Considerable research has been conducted on force control with electric robots. By contrast, little research has been conducted on force control with pneumatic robots. The latter has the potential to be considerably cheaper. However, the compressible nature of air as the working fluid and the relatively high cylinder friction means that pneumatic robots are more difficult to control.

The subject of this thesis was the design and testing of a controller that regulates the normal contact force and tangential velocity of the end effector of a pneumatic gantry robot while tracking the contour of a planar workpiece. A PI FV controller for contour tracking was designed and tested experimentally. The performance of the PI FV controller was benchmarked against a similar PID FV implemented with an electrically actuated robot. The effect on performance when a Velocity Dependent Friction Model (VDFM) based compensator was added to the PI FV controller was studied.

A system model was developed and validated in order to investigate the parameters that affected performance. Two Neural Network (NN) compensators were designed to compensate for both the lag and friction in the system. Simulation results for straight and curved edges were used to examine the effectiveness of the NN compensators.

6.1 Conclusions In Chapter 2, the contour tracking task and associated problems were reviewed together with common force control approaches for such applications. Basic strategies of direct and indirect force control were discussed, mainly in the context of pneumatically actuated systems. An overview of FL and NN systems was presented. The friction phenomena, the most common friction models and friction compensators were discussed in some detail.

148

From the literature review it was concluded that this research may be considered as a first step towards the application of pneumatic actuated robots in contour tracking with the uniqueness of being the first reported study of the application of pneumatic actuators to contour tracking with the use of proportional pressure valves and high friction cylinders.

6.1.1 Summary Results

In this thesis both experimental and simulation tests were conducted with the pneumatic gantry robot and different type of controllers.

Table 6.1 summarizes the quantitative tracking

performances, as taken from Table 3.4, Table 3.12, Table 4.8 and Table 5.5, for the main experimental and simulation tracking tests conducted on straight, inclined and curved edges. The tracking performances were reported in terms of ME and RMSE, the RMSE of the tracking responses was calculated based on a similar but not exactly the same time horizon. Cases 1 and 8 in Table 6.1 represent the experimental tracking performance of the PID DFV controller, as reported in (Jatta et al, 2003), when conducted with an electrically actuated robot on straight and curved edge workpieces. Cases 2, 9, 13 and 15 represent the experimental tracking performance with the PI FV controller for straight, 7 deg inclined, 16 deg and curved edge workpieces, respectively. Cases 3, 10, 14 and 16 represent the simulation tracking performance with the PI FV controller for straight, 7 deg inclined, 16 deg and curved edge workpieces, respectively. Cases 4, 5, 6 and 7 represent the simulation tracking performance with the PI FV controller and the NN compensation for straight edge workpiece. Cases 11 and 12 represent the simulation tracking performance with the PI FV controller and the NN compensation for curved edge workpiece.

149

Table 6.1: Summary of experimental and simulation performances with different controllers

Straight edge

Case

(m/s)

ME vt (m/s)

RMSE f n (N)

RMSE vt (m/s)

30

0.02

13.11

NA

2.95

NA

2

Experimental PI FV controller

30

0.06

12.4

0.118

3.0

0.039

3

Simulation PI FV controller

30

0.06

11.8

0.121

3.0

0.042

30

0.06

13.6

0.122

4.0

0.025

30

0.06

9.6

0.122

1.4

0.025

30

0.06

7.6

0.122

1.2

0.025

30

0.06

7.0

0.122

0.8

0.025

20

0.01

16.56

NA

4.02

NA

30

0.06

16.8

0.170

2.6

0.047

30

0.06

14.8

0.282

2.6

0.062

30

0.06

6.6

0.159

0.8

0.048

30

0.06

7.4

0.159

1.0

0.048

4

7

8

9 Curve edge

(N)

ME f n (N)

Experimental Jatta et al [37]

6

10

11

12 Inclined edge (7 deg.)

vtd

1

5

Inclined edge (16 deg.)

f nd

Simulation PI FV controller with EBPA NN compensator 14 inputs and zero initial weights Simulation PI FV controller with EBPA NN compensator 14 inputs and 0.1 initial weights Simulation PI FV controller with MBPM NN compensator 14 inputs and zero initial weights Simulation PI FV controller with MBPM NN compensator 5 inputs and zero initial weights Experimental Jatta et al [37]

Experimental PI FV controller

Simulation PI FV controller Simulation PI FV controller with MBPM NN compensator 14 inputs and zero initial weights Simulation PI FV controller with MBPM NN compensator 5 inputs and zero initial weights

13

Experimental PI FV controller

30

0.06

9.6

0.135

1.2

0.041

14

Simulation PI FV controller

30

0.06

9.4

0.116

1.2

0.043

15

Experimental PI FV controller

30

0.06

5.6

0.192

1.0

0.047

16

Simulation PI FV controller

30

0.06

7.0

0.118

1.2

0.045

150

6.1.2 Experimental Controller Design

Chapter 3 set out to present the experimental apparatus and initial controller design.

The

procedure for the design, tuning and testing of the controller was described. The effect of surface finish, profile shape and tracking speed on contour tracking performance was examined. Three different workpiece edge geometries were studied: straight, inclined and curved.

The

performance of a PI FV controller was benchmarked against the PID DFV controller with FC proposed by Jatta et al (2006).

Finally, the effect on performance when a VDFM based

compensator was added to the PI FV controller was studied.

The main conclusions from this chapter were: •

As shown in Table 6.1, it was concluded that the PI FV controller tracking performance with the straight and curved edge workpieces (Cases 2 and 9, respectively) was comparable to the performance achieved with a similar FV controller on an electrically actuated robot (Cases 1 and 8, respectively).

This result confirms the potential of

pneumatically actuated robots in force control applications. •

It was assumed that the performance when tracking the straight edge workpiece (Case 2) would be better than the performance when tracking the curved workpiece (Case 9). However, the reverse was found to be true.



Both the profile shape and the surface finish had a significant effect on the tracking performance.



The tracking speed did not have a significant effect on the tracking performance.



The addition of the friction compensator did not improve performance.

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6.1.3 Modeling and Simulation Results

Chapter 4 set out to develop and validate a model of the system, in order to better understand system behavior and to seek ways to improve system tracking performance. A PI FV controller similar to the one used in Chapter 3 was adopted. The model was validated against experimental results. The effects of eight parameters were investigated: 1) unidirectional nature of the PPC valves, 2) tracking speed, 3) uncertainty in workpiece angle, 4) signal measurement noise, 5) system lag, 6) workpiece surface finish, 7) Coulomb friction and 8) workpiece shape.

The conclusions from this chapter were: •

The validity of the system model was confirmed by the good match between the experimental and simulation results for the straight edge workpiece (Case 2 versus Case 3, respectively), the curved edge workpiece (Case 9 versus Case 10, respectively), the 7 deg inclined edge workpiece (Case 13 versus Case 14, respectively) and the 16 deg inclined edge workpiece (Case 15 versus Case 16, respectively).



Switching from a unidirectional to bidirectional valve will significantly improve system tracking performance and will widen the range of profile shapes that can be tracked.



The DDFM is more suitable than a VDFM in tracking applications. This is especially true when the displacement is small as in the case of straight edge tracking.



The potential of ANFIS in friction modeling requires more investigation. The ANFIS design tool available in MATLAB software enables the user to implement only off-line training. It is expected that this will be the main limitation when using the MATLAB ANFIS toolbox in friction modeling.



There was no improvement in the tracking performance when v td was reduced.



If the estimated angle of the workpiece is within ± 10 deg, performance will not be adversely affected.



There was no significant effect due to the noise in the measured signals on the tracking performance.



The system lag prevents the controller from acting quickly and consequently limits the ability of the system to properly regulate the contact force.



Surface finish has a significant effect on performance when tracking a straight edge, because it was observed to induce frequent changes in the end effector and friction directions along the z-axis. This also explains the unexpected f n response when tracking

152

a straight edge. •

Coulomb friction has the most significant effect when tracking the three different profiles, and consequently is the main limitation to better force control.

6.1.4 Neural Network Based Compensation

The objective of Chapter 5 was to examine ways to improve the tracking performance. Initially a PI FV controller with a table look up based gain scheduler was investigated. Next, a PI FV controller with DDFM based compensator was tested. Finally, two NN based compensators were designed for the PI FV controller. Simulation tracking results for straight and curved edges were presented to demonstrate the effectiveness of the different techniques.

The conclusions from this chapter were: •

The table look up gain scheduling approach based on workpiece angle did not improve the performance, most likely because of the lag in the system. Feedback control in the presence of lag leads to particular difficulties since a delay places a limit on the achievable bandwidth and allowable maximum gain.



Because of the lag in the system, the force regulation response could not be improved even with perfect DDFM based friction compensation.



Smith predictor and indirect NN model based controllers are not considered good candidates for contour tracking applications, since they require offline modeling for both the pneumatic apparatus and the workpiece. Direct NN model based controllers are recommended instead.



Simulation results showed that the NN compensation approach was the most effective in improving tracking performance.



The PI FV controller tracking performance with the MBPM NN compensator (Case 6) was better than with the EBPA NN compensator (Cases 4 and 5) when tracking the straight edge workpiece. However, both NN based PI FV controllers when trained properly to learn the system dynamics, showed significant improvement over the PI only FV controller (Cases 3 and 10 for straight and curved edges, respectively).



The MBPM NN compensator is better than the EBPA NN because: 1) it has fewer nodes; 2) it is less sensitive to the initial value of the weights and 3) it is easier to tune.



For the straight edge tracking, the ME and RMSE for force regulation were improved with the PI FV controller and MBPM NN compensator (Case 6) by 36% and 60%,

153

respectively, when compared to PI only FV controller results (Case 3). For the curved edge tracking, Case 11, the ME and RMSE for force regulation were improved by 55% and 69%, respectively, when compared to PI only FV controller results (Case 10). •

The number of inputs to the MBPM NN was reduced from 14 to 5 without significant changes in the performance, as shown when comparing the performance of Cases 6 and 7 for the straight edge and Cases 11 and 12 for the curved edge. Reducing the number of the inputs is important when it comes to experimental implementation of the controller, since this expected to reduce the computational load and thus enable easier real time training.

The experimental implementation of the PI FV controller with MBPM NN compensation is expected to be challenging. Anticipated problems are: 1) Computational load due to the NN implementation. Even though a great effort was made to select the best inputs and the right NN parameters, it is believed that there is more to do. The number of inputs to the MBPM NN was reduced from 14 to 5 without significant changes on the performance. Reducing the number of the inputs is important when it comes to experimental implementation of the controller. However, it is not clear what the optimum number of inputs is in terms of balancing the computational load with system performance. 2) Compatibility of the Adaptive Neural Networks algorithm.

Another expected

challenge when implementating the controller experimentally is the compatibility of the Adaptive Neural Networks originally developed by Campa et al (2002) with the dSPACE controller board and software. Specifically, it is not clear that the software algorithm used in the SIMULINK implementation of the MBPM will be compatible with the dSPACE software interface. It is possible that the MBPM will have to be rewritten in straight MATLAB code.

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6.2 Recommendations In this research there were two main objectives: 1) To confirm the potential of pneumatically actuated robots in force control and contour tracking applications. This was proved experimentally with the PI FV controller. 2) To improve the tracking performance with intelligent algorithms such as NN and FL. Simulation results show that this can be achieved with a NN compensator.

Even though both objectives were achieved, it is believed that the subject still needs more investigation. The following are recommendations for further work in this area: •

Switch from a unidirectional to bidirectional proportional pressure valve.

This is

expected to significantly improve system tracking performance and widen the range of profile shapes that can be tracked with the gantry robot. •

Investigate further the potential of ANFIS in friction modeling. In Chapter 4 it was shown that ANFIS had good potential in friction modeling.

The match between

experimental and simulation results with the ANFIS was better than the match with the rule based DDFM when tested off-line. The rule based DDFM was used in the thesis instead of the ANFIS because it was easier to implement on-line. •

In order to accurately model the tracking of workpieces with any profile shape, it is recommended to have a DDFM and VDFM for each axes (x and z axes). The DDFM will be active when the displacement is very small and the VDFM will be active when there is a significant displacement. It was concluded in Chapter 4 that the DDFM is more accurate than the VDFM when the displacement is small as in the case of the end effector displacement along the z-axis when tracking straight edge. Thus, the DDFM was adopted for the z-axis and the VDFM for the x axis.



One notes that the output of the MBPM NN was added to the z-axis control signal only. It was done this way, since the MBPM NN was designed explicitly to improve the tracking performance of the straight edge. Thus, the friction and lag in the z-axis were considered as the source of the problem not the x-axis friction and lag. It is believed that this consideration is true for straight edge tracking only. In the straight edge tracking case surface finish has a significant effect on performance, because it was observed to induce frequent changes in the end effector and friction directions along the z-axis. In this research the approach was to improve the straight edge tracking performance first,

155

since it was the worst. In the curved edge tracking case, it is the shape of the profile, not the surface finish, which induces frequent changes in the end effector and friction directions. Such effects could happen along both axes depending on the profile angle. Even though the performance with the current PI FV controller and the MBPM NN for the curved edge was improved significantly, it is expected it can be improved even more if the controller is designed such that the MBPM NN output contributes to control in both axes. •

It was shown that the type of NN used as a compensator has important effects on the tracking performance. Tracking performance with the MBPM NN was better than with the EBPA NN. It is recommended that other types of NN should be investigated in order to further improve the performance and reduce the tuning effort.



Even though improvements in performance with NN compensation in simulation were significant, it is believed that equally significant improvements could be achieved by switching from pressure control valves to flow control valves with investigating air pressure as a feedback signal in the control. With a view to industrial implementation, it is recommended that further work to improve tracking performance experimentally be directed at the hardware aspects of the apparatus.

156

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2

Abu Mallouh, M. and Surgenor, B. (2007) “Hybrid Force/Velocity Control of a Pneumatic Gantry Robot for Contour Tracking: Tuning and Model Validation”, Proceedings of ASME IDETC’07, Las Vegas, Nevada, September 4-7.

3

Abu Mallouh, M. and Surgenor, B. (2008) “Force/Velocity Control of a Pneumatic Gantry Robot for Contour Tracking with Neural Network Compensation”, Proceedings of ASME MSEC2008, Evanston, Illinois, USA, October 7-10.

4

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Appendix A Technical Specifications This appendix gives the technical specifications of the components of the pneumatic gantry robot.

A.1 SMC VER PPC Valve

Figure A.1: Technical specification of VER PPC valve (SMC, 2000)

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A.2 SMC MY1H Cylinder

Figure A.2: Technical specification of SMC MY1H cylinder (SMC, 2000)

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A.3 Bimba Standard Cylinder

Figure A.3: Technical specification of Bimba cylinder (www.bimba.com)

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A.4 SMC VEA 250 Power Amplifier

Figure A.4: Technical specification of SMC VEA250 amplifier (SMC, 2000)

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A.5 FUTEK Bi-Axial Force Sensor

Figure A.5: Technical specification of FUTEK MBA400 force sensor (www.futek.com/)

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A.6 FUTEK CSG (JM-2A) Amplifier

Figure A.6: Technical specification of FUTEK CSG (JM-2A) amplifier (www.futek.com/)

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A.7 Celesco Position and Velocity Transducer

Figure A.7: Technical specification of Celesco cable extension transducer (www.celesco.com)

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A.8 Omega Differential Pressure Sensor

Figure A.8: Technical specification of Omega PX137 (www.omega.com)

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A.9 DMD-465WB Signal Conditioner

Figure A.9: Technical specification of Omega DMD-460WB amplifier (www.omega.com)

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A.10 dSPACE DS1104 R&D Controller Board

Figure A.10: Technical specification of dSPACE DS1104 R&D controller board (www.dspace.de/ww/en/pub/start.cfm)

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A.11 SKF Ball Bearing

Figure A.11: Technical specification of SKF ball bearing (www.skf.com)

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Appendix B Sensor Calibration This appendix gives the calibration results for the force, position, velocity and differential pressure sensors that were used in the pneumatic gantry robot.

B.1 Force Sensor Calibration The x and z axes channels of the force sensor were calibrated. The force sensor was attached to a wooden jig as shown in Figure B.1. Dead weights were applied gradually at the end point of the end effector. The load was increased from zero up to 175 N. The sensor output voltage was recorded for each step increment in the applied load. Figure B.2 and Figure B.3 show the curve fitting for the x and z axes results, respectively. It was found that the contact forces can be calculated with:

f x = −17 f xVolt

(B-1)

f z = 17 f zVolt

where f xVolt and f zVolt are the output voltage of the x and z axes force channels, respectively. The coefficient of determination (R) equals 0.99 for both figures.

Figure B.1: Force sensor calibration wooden jig

175

200 150 100

load (N)

50 0 -50 -100 -150 -200 -10

-8

-6

-4

-2 0 2 output voltage (volt)

4

6

8

10

Figure B.2: Curve fitting for the x-axis force sensor calibration data 200 150 100

load (N)

50 0 -50 -100 -150 -200 -10

-8

-6

-4

-2 0 2 output voltage (volt)

4

6

8

Figure B.3: Curve fitting for the z-axis force sensor calibration data

176

10

B.2 Position Sensor Calibration The calibration of the position sensors was done by moving the piston of the x and z axes cylinders to known positions and then recording the output voltage. Figure B.4 and Figure B.5 show the curve fitting results for the x and z axes, respectively. It was found that the piston position can be calculated with:

x = 180 xvolt − 44

(B-2)

z = 56 z volt − 24

where xvolt and zvolt are the output voltage of the x and z axes position sensors, respectively, with R equals 1.0 for both figures. 1000 900 800

Position (mm)

700 600 500 400 300 200 100 0

0

1

2

3 output voltage (volt)

4

5

Figure B.4: Curve fitting for the x-axis position sensor calibration data

177

6

90 80 70

Position (mm)

60 50 40 30 20 10 0 0.4

0.6

0.8

1

1.2 1.4 1.6 output voltage (volt)

1.8

2

2.2

Figure B.5: Curve fitting for the z-axis position sensor calibration data

178

2.4

B.3 Velocity Sensor Calibration The calibration of the x and z axes velocity sensors was done by designing a P-only position controller with a sinusoidal position setpoint.

When the cylinder piston was tracking the

sinusoidal position setpoint, the velocity response was very close to a square wave. The velocity signal was obtained by differentiating the position signal. The velocity sensor output voltage was recorded during the test. The velocity signal was filtered with a Butterworth second order filter. Figure B.6 shows the position, velocity sensor output voltage and the corresponding piston velocity responses for the x-axis with a 1.5 rad/sec sinusoidal setpoint frequency. For each interval where the velocity was constant, the velocity was divided by the recorded output voltage, and as a result a constant that relates the output voltage to the actual velocity was found. This was done for all intervals where the velocity was constant, and at the end the overall average was found.

The test was repeated with three different sinusoidal setpoint frequencies. Figure B.6 to Figure B.8 show the calibration results of the x-axis velocity sensor with 1.5, 2.5 and 5 rad/sec sinusoidal setpoint frequencies, respectively. The overall average was found to be 333.3 (

mm / sec ). volt

Similar tests were conducted to calibrate the velocity sensor of the z-axis. Figure B.9 to Figure B.11 show the calibration results of the x-axis velocity sensor with 1.5, 2.5 and 5 rad/sec sinusoidal setpoint frequencies, respectively.

(

The overall average was found to be 132.3

mm / sec ). volt

179

800

position setpoint actual position

x (mm)

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time (sec) 3 ux (volt)

2 1 0 0

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1

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-0.5 -1

0

5

10 time (sec)

vx (mm/sec)

500 filtered signal actual signal 0

-500

0

5

10

15

20

25

time (sec)

Figure B.6: x-axis velocity sensor calibration, with 1.5 rad/sec sinusoidal setpoint.

180

1000 x (mm)

800 600 position setpoint1 actual position

400 200

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8 10 time (sec)

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1 0

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12

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vx (mm/sec)

1000 500 0 filtered signal actual signal

-500

-1000

0

2

4

6

8 10 time (sec)

12

14

Figure B.7: x-axis velocity sensor calibration, with 2.5 rad/sec sinusoidal setpoint.

181

16

18

800

x (mm)

600 400 position setpoint actual position

200 0

0

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2 1 0

vx (volt)

0.5 filtered signal actual signal 0

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0

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40 time (sec)

50

60

70

80

0

10

20

30

40 time (sec)

50

60

70

80

vx (mm/sec)

400 200 0

-200 -400

Figure B.8: x-axis velocity sensor calibration, with 5.0 rad/sec sinusoidal setpoint.

182

100 position setpoint actual signal

z (mm)

80 60 40 20

0

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10

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25

15

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time (sec) 3 uz (volt)

2 1 0 0

5

10 time (sec)

vz (volt)

0.5

0 filtered signal actual signal

-0.5

0

5

10

15

20

25

15

20

25

time (sec)

vz (mm/sec)

100 50 0 -50

-100

0

5

10 time (sec)

Figure B.9: z-axis velocity sensor calibration, with 1.5 rad/sec sinusoidal setpoint.

183

100

position setpoint actual position

z (mm)

80 60 40 20

0

2

4

6

8

10 12 time (sec)

14

16

18

20

0

2

4

6

8

10 12 time (sec)

14

16

18

20

3 uz (volt)

2 1 0

vz (volt)

1 filtered signal actual signal

0.5 0

-0.5

0

2

4

6

8

10 12 time (sec)

14

16

18

20

0

2

4

6

8

10 12 time (sec)

14

16

18

20

vz (mm/sec)

200 100 0

-100

Figure B.10: z-axis velocity sensor calibration, with 2.5 rad/sec sinusoidal setpoint.

184

100 actual position position setpoint

z (mm)

80 60 40 20

0

1

2

3

4

5 time (sec)

6

7

8

9

10

0

1

2

3

4

5 time (sec)

6

7

8

9

10

3 uz (volt)

2 1 0

vz (volt)

2 filtered signal acual signal

1 0

-1

0

1

2

3

4

5 time (sec)

6

7

8

9

10

0

1

2

3

4

5 time (sec)

6

7

8

9

10

vz (mm/sec)

200 100 0

-100 -200

Figure B.11: z-axis velocity sensor calibration, with 5.0 rad/sec sinusoidal setpoint.

185

B.4 Differential Pressure Sensor Calibration To calibrate the differential pressure sensors one end of the sensor was left open to atmospheric pressure while the other end was connected to a measured pressure air line. The pressure was stepped up gradually and the sensor output voltage was recorded at each pressure step. The same sequence was repeated but this time by connecting the measured pressure supply line to the sensor port that was initially left open to atmospheric pressure. Figure B.12 and Figure B.13 show the curve fitting results for the x and z axes, respectively. It was found that the differential pressure can be calculated with:

ΔPx = 48ΔPxVolt − 3.9

(B-3)

ΔPz = 49ΔPzVolt − 3.9

where ΔPxVolt and ΔPzVolt are the output voltage of the x and z axes differential pressure sensors, respectively, with R equals 0.99 for both figures. 500 400

differential pressure (kpa)

300 200 100 0 -100 -200 -300 -400 -500 -10

-8

-6

-4

-2 0 2 output voltage (volt)

4

6

8

10

Figure B.12: Curve fitting for the x-axis differential pressure sensor calibration data

186

500 400

differential pressure (kpa)

300 200 100 0 -100 -200 -300 -400 -500 -10

-8

-6

-4

-2 0 2 output voltage (volt)

4

6

8

10

Figure B.13 Curve fitting for the z-axis differential pressure sensor calibration data

187

Appendix C System Parameter Identification This appendix presents the experimental friction parameter and system lag identification tests for the x and z axes cylinders. The two components of the Coulomb friction; the static Coulomb friction ( Fsx , Fsz ) and the dynamic Coulomb friction ( Fdx , Fdz ) for both axes cylinders are identified using open loop tests. The coefficients of the viscous friction ( C fx , C fz ) for both axes are identified using closed loop tests. System lag ( τ x ,τ z ) are identified using open loop tests.

C.1 Friction Parameter Identification C.1.1 Coulomb Friction Identification

For the Coulomb friction identification a saw teeth voltage signal was used as an input to the PPC valves in an open loop test. Initially the piston was at rest. Then the voltage was ramped up and the applied force increased until the piston started to move. The force at which the cylinder piston started to move was calculated as the static Coulomb friction using:

Fsx = f ax − Fwx

(C-1)

Fsz = f az − Fwz

Next, the voltage signal was ramped down and the applied force dropped until the piston stopped. The force at which the piston stopped was calculated as the dynamic Coulomb friction using:

Fdx = f ax − Fwx

(C-2)

Fdz = f az − Fwz

Figure C.1 and Figure C.2 show the block diagrams of the open loop programs used for x and z axes Coulomb friction identification tests, respectively. The test was repeated three times using three different ramp input voltage. The over all average static and dynamic Coulomb friction forces were calculated. 0.01 m/sec was considered as a threshold value that indicates motion beginning. Figures C.3 to Figure C.5 shows the results of the tests for the x-axis using three different ramp input voltage and Table C.1 summaries the results. The average Fsx and Fdx were found to be 25 N and 11 N, respectively. 188

Similar tests were conducted on the z-axis cylinder.

Figures C.6 to Figure C.8 show the

experimental results for the 3 cases (3 different ramp input voltage signals).

Table C.2

summarizes the results. The average Fsz and Fdz were found to be 20 N and 15 N, respectively.

Figure C.1: Block diagram of the program used for x-axis Coulomb friction identification

189

Figure C.2: Block diagram of the program used for z-axis Coulomb friction identification

190

0.1

vx (m/sec)

0 -0.1 -0.2 -0.3

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

300

fax (N)

200 100 0 -100

x (mm)

1000

500

0

-500

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

1

ux (volt)

0.9 0.8 0.7

Figure C.3: Experimental results of Coulomb friction identification for x-axis, Case 1.

191

0.1

vx (m/sec)

0 -0.1 -0.2 -0.3

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

300

fax (N)

200 100 0 -100

x (mm)

1000

500

0

-500

1

ux (volt)

0.9 0.8 0.7

Figure C.4: Experimental results of Coulomb friction identification for x-axis, Case 2.

192

0.1

vx (m/sec)

0 -0.1 -0.2 -0.3

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

300

fax (N)

200 100 0 -100

x (mm)

1000

500

0

1

ux (volt)

0.9 0.8 0.7

Figure C.5: Experimental results of Coulomb friction identification for x-axis, Case 3.

193

0.6

vz (m/sec)

0.4 0.2 0 -0.2

0

0.5

1

1.5 2 time (sec)

2.5

3

3.5

0

0.5

1

1.5 2 time (sec)

2.5

3

3.5

0

0.5

1

1.5 2 time (sec)

2.5

3

3.5

0

0.5

1

1.5 2 time (sec)

2.5

3

3.5

faz (N)

50

0

-50

-100

z (mm)

100

50

0

1.4

uz (volt)

1.2 1 0.8

Figure C.6: Experimental results of Coulomb friction identification for z-axis, Case 1.

194

0.6

vz (m/sec)

0.4 0.2 0 -0.2

0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

4.5

5

50

faz (N)

0

-50

-100

z (mm)

100

50

0

1.4

uz (volt)

1.2 1 0.8

Figure C.7: Experimental results of Coulomb friction identification for z-axis, Case 2

195

0.3

vz (m/sec)

0.2 0.1 0 -0.1

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

50

faz (N)

0

-50

-100

z (mm)

100

50

0

0

5

10

15 time (sec)

20

25

30

0

5

10

15 time (sec)

20

25

30

1.4

uz (volt)

1.2 1 0.8

Figure C.8: Experimental results of Coulomb friction identification for z-axis, Case 3.

196

Table C.1: Experimental x-axis Coulomb friction identification tests summary Case

Fdx (N)

Fsx (N)

Average Fdx (N)

Average Fsx (N)

1 2 3

12 11 11

30 25 21

11

25

Table C.2: Experimental z-axis Coulomb friction identification tests summary Case

Fdz (N)

Fsz (N)

Average Fdz (N)

Average Fsz (N)

1 2 3

14 14 16

17 20 23

15

20

C.1.2 Viscous Friction Coefficient Identification

The viscous friction coefficient for the x and z axes cylinders were identified by tracking a position sinusoidal setpoint using a P-only position controller. The data collected from section B.3 was used in this section. While the pistons were moving with constant velocities the viscous friction coefficients were calculated with:

C fx =

f ax − Fdx − Fwx vx

(C-3)

f − Fdz − Fwz C fz = az vz

It was found that the overall average for C fx and C fz were 74 N/ (m/s) and 40 N/ (m/s), respectively.

197

C.2 System Lag Identification

faX (N)

0 -50 -100 -150 0

5

10

15

20 Time (sec)

25

30

35

40

0

5

10

15

20 Time (sec)

25

30

35

40

ux (volt)

1.2 1 0.8 0.6

Figure C.9: τ x identification test, when piston held still at x =1000 mm

faX (N)

0 -50 -100 -150 0

5

10

15

20 25 Time (sec)

30

35

40

45

0

5

10

15

20 25 Time (sec)

30

35

40

45

ux (volt)

1.2 1 0.8 0.6

Figure C.10: τ x identification test, when piston held still at x =500 mm

198

0

faX (N)

-50 -100 -150 0

10

20

30 Time (sec)

40

50

60

0

10

20

30 Time (sec)

40

50

60

ux (volt)

1.2 1 0.8 0.6

Figure C.11: τ x identification test, when piston held still at x =0 mm

faz (N)

100

50

0

0

10

20

30

40 Time (sec)

50

60

70

80

0

10

20

30

40 Time (sec)

50

60

70

80

uz (volt)

1.6 1.4 1.2 1 0.8

Figure C.12: τ z identification test, when piston held still at z =100 mm

199

faz (N)

100

50

0

0

5

10

15

20

25 30 Time (sec)

35

40

45

50

0

5

10

15

20

25 30 Time (sec)

35

40

45

50

uz (volt)

1.6 1.4 1.2 1 0.8

Figure C.13: τ z identification test, when piston held still at z =0 mm

200

Appendix D Neural Network Program Details This appendix presents details of the simulation program and the screen captures of input parameters pages for the two Neural Networks used in Chapter 5. Details will be given for the EBPA and MBPM Neural Networks with 14 and 5 inputs.

D.1 Neural Network Compensation with 14 Inputs Below are the details of the EBPA and MBPM Neural Network, with 14 inputs, when added to the PI FV controller.

Vt(m/sec )

Vx Error (m/sec )

Vx Error (m/s)

UVx(volt ) 0:0

Ux(volt ) Fx Error (N)

Fx Error (N)

UFx(volt )

Fz Error (N)

Fz Error (N)

Theta(rad)

Vt(m/s)

Saturation

Fn(N)

X(m)

theta(rad)

Theta(rad)

Vz Error (m/sec )

Vz Error (m/s)

theta(rad)

Fn(N)

UFz(volt ) 0:0

Uz(volt )

Workpiece angle (theta) Theta(rad)

Theta (rad)

UVz(volt )

Vtd(m/sec )

PI Controller X(mm)

Vtd

x(m)

Saturation Gantry Robot, Workpiece & fram conversion

Fn Error (N) Fn error

0 Clock 1

clock 1

Fnd

desired force & velocity

Fnd(N) Fn(N)

fram conversion 1

Fn

UzNN

Fnd

NN compensator

Figure D.1: Over all model, NN compensator with 14 inputs added to main program

201

-1 Z Integer Delay

3 Fnd

EBPA & MBPM NN

-2 Z Integer Delay -3 Z Integer Delay -4 Z Integer Delay

x ys e LE

2 Fn

X

EBPA NN tuned

-2 Z Integer Delay

-1 Z Integer Delay x

0:0

ys e

-3 Z Integer Delay7

Saturation2

LE X

-4 Z Integer Delay20

NN on/of

MBPM NN tuned

1 Fn error -1 Z Integer Delay

-2 Z Integer Delay

Figure D.2: NN compensator model with 14 inputs

202

1 UzNN

V' [V]

[1 0 0:0 ]'

Y=X' Z=X*Y

[Vtx]

bias0:0

[x]

mu

V'x a [Vtx]

Switch

mu

a

[sig]

a mu 1 0:0

[sigp]

asig(1-sig) -1

2 e

[sigp ]

[z] 3

[Vtx] sigpVtx

LE

Z=X*Y

[sig]

[z]

(sig-sigp )z

error in

-F

[Wdt]

[W] 1 x

[x]

[W] lam n (z) W Dot Prod

2 X

[V]

[z]

sqrt

lam

[V] lam n (z) V [x]

-G

[Vdt]

Z=X*Y

Z=X*Y xzW'sigp zW' [W]

[sigp ]

zM'sigp

Y=X' W' K Ts

[sig]

W Limiter 0:0

W'sig

[W]

[Wdt]

[z]

z-1

K Ts z-1

0:0

[V]

vad

1/z 1 ys

Unit Delay

V Limiter [Vdt ]

Z=X*Y

sqrt -K-

0:0 Z_bar

vbar

Product

Kz0:0

Figure D.3: MBPM NN compensator model

203

[Lh ] [Uh]

[uhlh ] [dNETh ] [ha ]

[hn ] -1

[Th ] 1

e

1

u

1

0:0

-1

[dUh ]

u

0:0

-1 [uhlh ]

[dTh ]

[Uh]

[hn]

[dLh ]

[Th ]

[Lo ] [Uo]

[uolo ] oa

[on ] -1

[To ] 1

e

1

u

u

[dNETo ] 1 ys

-1 -1

[dUo ]

[uolo ] [on]

0:0

1

0:0

[dTo ]

[Uo] [To ] [dLo ]

[dloa ]

Z=X*Y

[W] [V]

[e]

Z=X*Y

1 x

[hn ]

dNETo ]

[x]

[e]

[dloa ]

[dlUo ]

[Te ]

Z=X*Y

[ha ]

[dhu ]

[dUh ]

[on ]

[dlLo ]

[e] [Gm ]

Z=X*Y

[W] [dlUo ]

[dUo ] [W]

[dha ]

dNETh ]

[dlLo ]

[dLo ]

Z=X*Y

[W]

[dhl ]

[dLh ] [e]

e 2

[dlTo ]

[dTo ]

[e]

[dlTo ]

Z=X*Y

[W]

[dht ]

[dTh ]

0:0

[dha ] [x]

etaV

[DV]

Z=X*Y 0:0

[dloa ] [ha ]

etaW Z=X*Y

[DV]

DV

V(k)

[DW]

DW

W(k) Gm(k)

[Gm]

DUo

Uo(k)

[Uo ]

DLo

Lo(k)

[Lo ]

DTo

To(k)

DTe

Te(k)

[Te ]

[dha ]

DUh

Uh(k)

[Uh ]

[dhu ]

DLh

Lh(k)

[Lh ]

[dhl ]

DTh

Th(k)

[Th ]

[dloa ] [DW]

[V] [W]

DGm

[dlUo ] [dlLo ]

0:0

etaP

[dlTo ]

[dht ]

mu

3 LE

Figure D.4: EBPA NN compensator model

204

LE

X

[To ]

2 X

Figure D.5: Screen captures of input parameters page for the EBPA NN, with 14 inputs

Figure D.6: Screen captures of input parameters page for the MBPM NN, with 14 inputs

205

D.2 Neural Network Compensation with 5 Inputs Below are the details of the MBPM Neural Network, with 5 inputs, when added to the PI FV controller.

Vt(m/sec )

Vx Error (m/sec )

Vx Error (m/s)

UVx(volt ) 0:0

Ux(volt ) Fx Error (N)

Fx Error (N)

UFx(volt )

Fz Error (N)

Fz Error (N)

Theta(rad)

Vt(m/s)

Saturation

Fn(N)

X(m)

Theta(rad)

theta(rad)

Vz Error (m/sec )

Vz Error (m/s)

theta(rad)

Fn(N)

UFz(volt ) 0:0

Uz(volt )

Workpiece angle (theta) Theta(rad)

Theta (rad)

UVz(volt )

Vtd(m/sec )

Gantry Robot, Workpiece & fram conversion

PI Controller X(mm)

0

clock 1

Vtd

x(m)

Saturation

Fn Error (N)

Fn error

Fnd(N)

Fnd

Fn(N)

Clock 1

UzNN

desired force & velocity

Uz

fram conversion 1

NN compensator

Figure D.7: Over all model, NN compensator with 5 inputs added to main program

2 Uz

-1 Z Integer Delay7

-2 Z Integer Delay20

x

1 Fn error

0:0

ys

-2 Z Integer Delay

e

-1 Z Integer Delay

LE X

NN on/of

MBPM NN tuned

Figure D.8: MBPM NN compensator model with 5 inputs

206

Saturation2

1 UzNN

Figure D.9: Screen captures of input parameters page for the MBPM NN, with 5 inputs.

207

Appendix E Supplemental Tuning Results This appendix presents supplemental tuning results for Step 1 tuning in Chapter 3.

E.1 Tuning the x-axis Velocity Controller

Figure E.1: Block diagram of the PID v x controller 0.1

vx (m/sec)

0 -0.1 -0.2

9.5

10

10.5

11

11.5

12

ux (volt)

time (sec)

1 Kp=0.4,Ki=4.0,Kd=0.0 Kp=0.1,Ki=4.0,Kd=0.0 Kp=0.2,Ki=4.0,Kd=0.0

0.8

0.6 9.5

10

10.5

11

11.5

12

time (sec) v

Figure E.2: Results of the PID velocity control of x-axis when K px was changed

208

E.2 Tuning the z-axis Force Controller

Figure E.3: Block diagram of PID f z controller 15

fz (N)

10 5 0 -5

18

20

22

24 time (sec)

26

28

30

uz (volt)

1.2 1 Kp=0.08,Ki=0.1,Kd=0.0 Kp=0.16,Ki=0.1,Kd=0.0 Kp=0.04,Ki=0.1,Kd=0.0

0.8 0.6

18

20

22

24 time (sec)

26

28

30

Figure E.4: Results of the PID force control of z-axis when K pf z was changed

209

15

fz (N)

10 5 0 -5

18

20

22

24 time (sec)

26

28

30

uz (volt)

1.2 1 Kp=0.08,Ki=0.1,Kd=0.0 Kp=0.08,Ki=0.2,Kd=0.0 Kp=0.08,Ki=0.05,Kd=0.0

0.8 0.6

18

20

22

24 time (sec)

26

28

f

30

Figure E.5: Results of the PID force control of z-axis when K i z was changed

E.3 Tuning the x-axis Force Controller

Figure E.6: Block diagram of PID f x controller.

210

fx (N)

0 -10 -20

-30 30

31

32

33

34

35 time (sec)

36

37

38

39

40

uz (volt)

1.5

1

Kp=0.02,Ki=0.03,Kd=0.0 Kp=0.04,Ki=0.03,Kd=0.0 Kp=0.01,Ki=0.03,Kd=0.0

0.5

0 30

31

32

33

34

35 time (sec)

36

37

38

39

40

Figure E.7: Results of the PID force control of x-axis when K pfx was changed

fx (N)

0 -10 -20

-30 30

32

34

36

38

40 time (sec)

42

44

46

48

50

uz (volt)

1.5

1 Kp=0.02,Ki=0.03,Kd=0.0 Kp=0.02,Ki=0.06,Kd=0.0 Kp=0.02,Ki=0.01,Kd=0.0

0.5

0 30

32

34

36

38

40 time (sec)

42

44

46

48

Figure E.8: Results of the PID force control of x-axis when K i fx was changed

211

50

fx (N)

0 -10 -20

-30 30

32

34

36

38

40 time (sec)

42

44

46

48

50

uz (volt)

1.5

1 Kp=0.02,Ki=0.03,Kd=0.0 Kp=0.02,Ki=0.03,Kd=0.01 Kp=0.02,Ki=0.03,Kd=0.005

0.5

0 30

32

34

36

38

40 time (sec)

42

44

46

48

Figure E.9: Results of the PID force control of x-axis when K dfx was changed

E.4 Tuning the z-axis Velocity Controller

Figure E.10: Block diagram of vz controller

212

50

0.1

vz (m/sec)

0.05 0 -0.05 -0.1

4

4.5

5

5.5

6

6.5

time (sec)

uz (volt)

1

0.8

0.6

Kp=0.25,Ki=6.0,Kd=0.0 Kp=0.50,Ki=6.0,Kd=0.0 Kp=0.125,Ki=6.0,Kd=0.0 4

4.5

5

5.5

6

6.5

time (sec)

Figure E.11: Results of the PID velocity control of z-axis when K pvz was changed

vz (m/sec)

0.2 0.1 0 -0.1

4

4.5

5

5.5

6

6.5

uz (volt)

time (sec)

1 Kp=0.25,Ki=6.0,Kd=0.0 Kp=0.25,Ki=12.0,Kd=0.0 Kp=0.25,Ki=3.0,Kd=0.0

0.8 0.6

4

4.5

5

5.5

6

6.5

time (sec)

Figure E.12: Results of the PID velocity control of z-axis when K ivz was changed

213

vz (m/sec)

0.1

0.05

0 4

4.5

5

5.5

6

6.5

uz (volt)

time (sec)

1 Kp=0.25,Ki=6.0,Kd=0.0 Kp=0.25,Ki=6.0,Kd=0.1 Kp=0.25,Ki=6.0,Kd=0.05

0.8 4

4.5

5

5.5

6

6.5

time (sec)

Figure E.13: Results of the PID velocity control of z-axis when K dvz was changed

214