THE EFFECTS OF SLAVE POSITION LOOP
BANDWIDTH, CONTROLLER SAMPLE RATE, AND
COMPUTATIONAL ARCHITECTURE UPON
EFFECTIVE ENVIRONMENT STIFFNESS IN A
SINGLE DEGREE-OF-FREEDOM TELEOPERATOR
SYSTEM
by
Peter Hall Argo
B.S., University of Colorado, 1981
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Electrical Engineering
1991
This thesis for the Master of Science
degree by
Peter Hall Argo
has been approved for the
Department of
Electrical Engineering
by
Jan T. Bialasiewicz
Edward T. Wall
Marvin F. Anderson
Date
Argo, Peter Hall (M.S., Electrical Engineering)
The Effects of Slave Position Loop Bandwidth, Controller Sample Rate, and
Computational Architecture Upon Effective Environment Stiffness in a
Single Degree-of-Freedom Teleoperator System
Thesis directed by Associate Professor Jan T. Bialasiewicz
Specifications required to assure the usability of force-reflecting teleoperated
systems are a point of contention in the robotics community. All agree that the
systems should provide 'good feel'; however, there is considerable debate
around what set of characteristics are required in order to provide the good feel.
This thesis proposes an analytical metric for good feel, which can also be
referred to as effective environment stiffness. Effective environment stiffness is
the stiffness reproduced by the teleoperation system at the user, or operator,
interface. This research analyzes the effects of the three characteristics slave
position loop bandwidth, controller sample rate, and computational architecture
- upon the analytical metric of effective environment stiffness.
This analysis also provides insight into some fundamental system
limitations, which are developed using simplified generic system models.
Although this research was performed with a set of simplified single degree-of-
ffeedom models, its results are of sufficient validity and generality to warrant
analytical consideration for the design of higher-order and multi-degree-of-
freedom systems. However, this topic is beyond the scope of this research.
This thesis is provided as a purely analytical discussion and does not
contain any hardware test data or results using physical systems.
The form and content of this abstract are approved. I recommend its
publication.
Jan T. Bialasiewicz
IV
This thesis is dedicated to my family Laureen, Jessica, Joshua, and
Benjamin who have accepted my dedication to furthering my education,
which, from time to time, has resulted in a minimal amount of family
participation.
This thesis is also dedicated to my parents Raymond and Patricia for
nurturing in me the curiosity and drive required to pursue graduate studies
CONTENTS
1.0 INTRODUCTION...............................................1
1.1 Teleoperation Characteristics..............................2
1.2 Teleoperation Performance..................................4
1.3 Thesis Subject Definition..................................7
2.0 PROBLEM DEFINITION.........................................8
2.1 Introduction...............................................8
2.1.1 Effective Environment Stiffness............................9
2.2 Specification of Thesis Parameters........................12
2.2.1 System Bandwidth..........................................13
2.2.2 Controller Sample Rates...................................13
2.2.3 Control Architectures.....................................13
2.2.3.1 Serial Computations Control Architecture..................13
2.2.3.2 Pipeline Control Architecture.............................14
2.2.4 Stability Requirements....................................16
2.3 Analytical Model Description..............................16
2.3.1 Slave Position Loop Bandwidth Model.......................16
2.3.1.1 Slave Position Loop Bandwidth Model Description...........16
2.3.1.2 Slave Environment Contact Model Description...............19
2.3.2 Operator Model............................................22
2.3.2.1 Operator Torque Command Model.............................27
2.3.2.2 Operator Torque Loop Model................................29
2.3.2.3 Operator Load Model.......................................30
2.3.2.4 Master Actuator Model................................... 30
2.3.2.5 Operator Perception Model.................................32
2.3.3 System Feedback Model.....................................33
2.3.4 System Command Model......................................34
2.4 System Analytical Model...................................35
3.0 RESULTS...................................................38
3.1 Continuous Analysis Results...............................38
3.1.1 Slave Position Loop Bandwidth Analysis Results............38
3.1.2 Slave Environment Contact Model Analysis Results..........43
3.1.3 Operator Model Analysis Results...........................45
3.1.4 Full-Up System Analysis Results...........................50
3.1.5 End-to-End Model Analysis Results...........................55
3.2 Hybrid Analysis Results.....................................57
3.2.1 Slave Position Loop Bandwidth Analysis Results..............57
3.2.2 Slave-Loop Analysis Results (Serial Control Architecture)...65
3.3 Pipelining Analysis Results.................................70
3.3.1 2-Stage Pipelining Analysis Results.........................75
3.3.2 4-Stage Pipelining Analysis Results.........................79
3.3.3 5-Stage Pipelining Analysis Results.........................81
4.0 CONCLUSIONS.................................................84
4.1 Continuous System Model Analysis............................84
4.2 Hybrid System Model Analysis................................86
4.2.1 Effects of Slave Position-Loop Bandwidth Upon Effective
Environment Stiffness..................................... 87
4.2.2 Effects of Controller Sample Rate Upon Effective Environment
Stiffness..................................................92
4.2.3 Effects of Pipeline Control Architecture Upon Effective
Environment Stiffness......................................97
4.3 Summary of Conclusions.....................................105
Appendix A TIME DELAY MODELING AND EFFECTS..............................107
A. 1 Time Delay Simulation Approach Taylor series.............107
A.2 Time Delay Simulation Evaluation Taylor series...........110
A.3 Time Delay Simulation Approach Pade'.....................115
A.4 Time Delay Simulation Evaluation Pade'...................117
A.5 Implications of Time Delay in Digital Systems..............124
A.6 Stability Boundaries Attributed to Pure Time Delay.........127
A. 7 Summary and Conclusions for Time Delay Modeling............127
Appendix B ADDITIONAL RESEARCH TOPICS...................................129
B. 1 Operator Position-Loop Modeling............................129
B. 2 High-Fidelity Actuator Modeling............................129
B. 3 Hardware Verification Testing..............................130
B.4 Operator Sensitivity to Reflected Stiffness................130
B. 5 Operator Sensitivity to Relative Reflected Stiffness.......131
Appendix C MODEL ANALYSIS...............................................132
C. l Continuous Domain Model Analysis...........................132
vii
C. 1.1 Slave Position Loop Bandwidth Model Analysis................132
C. 1.1.1 Slave Rate Loop Analysis....................................132
C. 1.1.2 Slave Position Loop Analysis................................136
C.1.2 Slave Environment Contact Model Analysis....................141
C.1.3 Operator Model Analysis.....................................156
C. 1.3.1 Operator Model Analysis Passive Load......................156
C. 1.3.2 Slave Loop Model Analysis Ideal Slave Actuator............161
C. 1.3.3 Slave Loop Model Analysis Active Operator Load............166
C. 1.3.4 Operator Model Analysis Active Slave Load.................172
C. 1.4 Full-Up System Analysis.....................................174
C. 1.4.1 Slave-Loop Model Analysis with Passive Operator Load........174
C. 1.4.2 Slave-Loop Model Analysis with Active Operator Load.........180
C.1.5 End-to-End Model Analysis...................................188
C. 1.5.1 End-to-End Model Analysis Free Space Motion...............188
C. 1.5.2 End-to-End Model Analysis Locked Slave Output Motion......191
C.2 Hybrid Model Analysis.......................................193
C.2.1 Slave Position Loop Bandwidth Model Analysis................193
C.2.1.1 Slave Rate Loop Analysis....................................196
C.2.1.2 Slave Position Loop Analysis................................199
C. 2.2 Full-Up System Analysis.....................................204
C.2.2.1 Slave Loop with Passive Operator Load.......................204
C.2.2.2 Slave Loop with Active Operator Load........................213
C. 3 Pipelining Architecture Analysis............................219
C. 3.1 2-Stage Pipeline Slave-Loop Analysis........................221
C.3.1.1 2-Stage Pipeline Slave-Loop Analysis Passive Operator Load... .221
C. 3.1.2 2-Stage Pipeline Slave-Loop Analysis Active Operator Load.226
C.3.2 4-Stage Pipeline Slave-Loop Analysis........................230
C.3.2.1 4-Stage Pipeline Slave-Loop Analysis Passive Operator Load... .232
C.3.2.2 4-Stage Pipeline Slave-Loop Analysis Active Operator Load.234
C.3.3 5-Stage Pipeline Slave-Loop Analysis........................236
C.3.3.1 5-Stage Pipeline Slave-Loop Analysis Passive Operator Load... .238
C.3.3.2 5-Stage Pipeline Slave-Loop Analysis Active Operator Load.242
REFERENCES.............................................................246
viii
FIGURES
Figure
1.2- 1 Pipelined Computer Architecture Example......................6
1.2- 2 Parallel Computer Architecture Example.......................6
2.1- 1 Simplified System Block Diagram..............................8
2.2.3- 1 Serial Computations Control Architecture Timing Diagram.....14
2.2.3- 2 Pipeline Control Architecture Timing Diagram................15
2.3.1- 1 Slave Position Loop Control Model...........................18
2.3.1 -2 Slave Position Loop Control Model with Environment Contact
Dynamics....................................................19
2.3.2- 1 Operator Model...............................................27
2.4- 1 End-to-End Analytical Model Part 1 of 2...................36
2.4- 1 End-to-End Analytical Model Part 2 of 2...................37
4.2.1 -1 Effect of Slave Bandwidth Upon Effective Environment
Stiffness Passive Operator Load...........................90
4.2.1 -2 Effect of Slave Bandwidth Upon Effective Environment
Stiffness Active Operator Load............................91
4.2.2- 1 Effect of Controller Sample Rate Upon Effective Environment
Stiffness Passive Operator Load...........................94
4.2.2- 2 Effect of Controller Sample Rate Upon Effective Environment
Stiffness Active Operator Load............................96
4.2.3- 1 Effect of Control Architecture Upon Effective Environment
Stiffness Passive Operator Load...........................103
4.2.3- 2 Effect of Control Architecture Upon Effective Environment
Stiffness Active Operator Load............................104
A.2-1 Polynomial Approximation Frequency Response 20 msec Time
Delay Gain Plot for Wide Frequency Band...................Ill
A.2-2 - Polynomial Approximation Frequency Response 20 msec Time
Delay Phase Plot for Wide Frequency Band..................112
A.2-3 Polynomial Approximation Frequency Response 20 msec Time
Delay Gain Plot for Narrow Frequency Band.................113
A. 2-4 Polynomial Approximation Frequency Response 20 msec Time
Delay Phase Plot for Narrow Frequency Band................114
A.4-1 Pade' Approximation Frequency Response 20 msec Time Delay
- Phase Plot for Wide Frequency Band........................118
Figure
A.4-2 Pade' Approximation Frequency Response 20 msec Time Delay
- Phase Plot for Narrow Frequency Band..................119
A.4-3 Pade' vs. Taylor Series Approximation Comparison Gain Plot
for Wide Frequency Band with 20 msec Time Delay.........121
A.4-4 Pade' vs. Taylor Series Approximation Comparison Phase Plot
for Wide Frequency Band with 20 msec Time Delay..........122
A.4-5 Pade' vs. Taylor Series Approximation Comparison Phase Plot
for Narrow Frequency Band with 20 msec Time Delay........123
A.5-1 Phase Loss vs. Time Delay and System Bandwidth............125
A.5-2 Compensation Gain vs. Time Delay and System Bandwidth....126
A. 6-1 Time Delay Stability Limitations..........................128
C. 1.1-1 Rate Closed-Loop Frequency Response.......................135
C. 1.1-2 Position Closed-Loop Frequency Response...................139
C. 1.1 -3 Free-Space Position Step Response.........................140
C. 1.2-1 Environment Contact Feedback Loop Illustration............142
C. 1.2-2 Root Locus of Output Elements with Locked Output
Environment Contact......................................143
C. 1.2-3 Locked Output Position Closed-Loop Frequency Response -1.0
Hz. Model......................... .....................144
C. 1.2-4 Locked Output Position Closed-Loop Frequency Response -
10.0 Hz. Model...........................................145
C. 1.2-5 Continuous Domain Slave Position Loop with Environment
Contact..................................................147
C. 1.2-6 Locked Output Frequency Response Constant Stiffness.....153
C. 1.2-7 Locked Output Step Response Constant Stiffness..........155
C.l.3-1 Operator Stand-Alone Model Block Diagram..................156
C. 1.3 -2 Operator Model Closed-Loop Frequency Response.............159
C. 1.3-3 Operator Model Closed-Loop Step Response..................160
C. 1.3-4 Slave Loop Model Block Diagram............................161
C. 1.3-5 Slave Closed-Loop Step Response Ideal Actuator Model with
Passive Operator Load....................................165
C. 1.3-6 End-to-End System Block Diagram...........................167
Figure
C. 1.3 -7 Slave and Operator Closed-Loop Step Response Ideal
Actuator...................................................171
C.1.3-8 End-to-End System Step Response.............................175
C. 1.4-1 Closed Slave-Loop Passive Operator Load Time Response.....179
C. 1.4-2 Closed Slave-Loop Effective Environment Stiffness- Passive
Operator Load..............................................181
C. 1.4-3 Closed Slave-Loop Active Operator Load Time Response......185
C. 1.4-4 Closed Slave-Loop Effective Environment Stiffness- Active
Operator Load............................................ 187
C. 1.5-1 Free-Space End-to-End System Time Response..................190
C. 1.5-2 End-to-End Locked-Output Closed-Loop Time Response..........194
C.2.1-1 Slave Open Hybrid Rate-Loop Control Model...................196
C.2.1-2 Slave Open Hybrid Position-Loop Control Model...............201
C.2.2-1 Discrete Controller Block Diagram...........................205
C.2.2-2 Hybrid Slave Effective Stiffness Analysis Model Block
Diagram....................................................211
C. 3.1 -1 2-Stage Pipeline Discrete Controller Block Diagram..........222
C. 3.2-1 4-Stage Pipeline Discrete Controller Block Diagram..........231
C. 3.3 -1 5-Stage Pipeline Discrete Controller Block Diagram..........237
xi
TABLES
Table
2.3.2- 1 Operator Time Delay Parameters............................25
2.3.2- 2 Operator Impedance Model Parameters.......................26
2.3.2- 3 Operator Position Command Model Parameters.................29
2.3.2- 4 Operator Perception Model Parameters.......................33
3.1.1- 1 Implemented Slave Gain Parameters..........................39
3.1.1- 2 Theoretical Slave Gain Parameters..........................39
3.1.1 -3 Position Loop Bandwidth Results -1.0 Damping Ratio.........42
3.1.1- 4 Position Loop Bandwidth Results 0.707 Damping Ratio......43
3.1.2- 1 Slave Bandwidth Comparison Locked Output vs. Free-Space....44
3.1.3- 1 Operator Parameter Summaiy.................................47
3.1.4- 1 Locked-Output Slave-Loop Gain Values Passive Operator....51
3.1.4- 2 Effective Environment Stiffness Continuous Controller with
Passive Operator..........................................53
3.1.4- 3 Locked-Output Slave-Loop Gain Values Active Operator.....54
3.1.4- 4 Effective Environment Stiffness Continuous Controller with
Active Operator...........................................54
3.1.5- 1 End-to-End Free Space Bandwidth............................55
3.1.5- 2 Final Locked-Output System Parameters......................56
3.2.1 -1 a Predicted vs Simulation Rate Open-Loop Phase Margins for
Hybrid 1.0 Hz. Model......................................58
3.2.1- lb Predicted vs Simulation Rate Open-Loop Phase Margins for
Hybrid 2.0 Hz. Model......................................59
3.2.1 -lc Predicted vs Simulation Rate Open-Loop Phase Margins for
Hybrid 5.0 Hz. Model......................................59
3.2.1- ld Predicted vs Simulation Rate Open-Loop Phase Margins for
Hybrid 10.0 Hz. Model.....................................60
3.2.1- 2 Hybrid Rate Loop Bandwidth Results....................... 61
3.2.1- 3a Predicted vs Simulation Position Open-Loop Phase Margins for
Hybrid 1.0 Hz. Model......................................62
3.2.1 -3b Predicted vs Simulation Position Open-Loop Phase Margins for
Hybrid 2.0 Hz. Model......................................63
3.2.1 -3c Predicted vs Simulation Position Open-Loop Phase Margins for
Hybrid 5.0 Hz. Model......................................63
Table
3.2.1- 3d Predicted vs Simulation Position Open-Loop Phase Margins for
Hybrid 10.0 Hz. Model........................................63
3.2.1- 4 Hybrid Position Loop Bandwidth Results.......................64
3.2.1- 5 Simulation Control Rate Applicability........................65
3.2.2- 1 Predicted vs. Simulation Slave-Loop Feedback Gains for
Hybrid Controller Passive Operator Load...................68
3.2.2- 2 Effective Environment Stiffness Hybrid Controller with
Passive Operator.............................................71
3.2.2- 3 Predicted vs. Simulation Slave-Loop Feedback Gains for
Hybrid Controller Active Operator Load.....................72
3.2.2- 4 Effective Environment Stiffness Hybrid Controller with Active
Operator.....................................................73
3.3.1- 1 Predicted vs. Simulation Slave-Loop Feedback Gains for 2-
Stage Pipeline Passive Operator Load.........................77
3.3.1- 2 Effective Environment Stiffness 2-Stage Pipeline Controller
with Passive Operator........................................77
3.3.1- 3 Predicted vs. Simulation Slave-Loop Feedback Gains for 2-
Stage Pipeline Active Operator Load........................78
3.3.1- 4 Effective Environment Stiffness 2-Stage Pipeline Controller
with Active Operator.........................................78
3.3.2- 1 Predicted vs. Simulation Slave-Loop Feedback Gains for a 200
sps 4-Stage Pipeline Controller Passive Operator Load......79
3.3.2- 2 Effective Environment Stiffness 200 sps 4-Stage Pipeline
Controller with a Passive Operator...........................80
3.3.2- 3 Predicted vs. Simulation Slave-Loop Feedback Gains for a 200
sps 4-Stage Pipeline Controller Active Operator Load......80
3.3.2- 4 Effective Environment Stiffness 200 sps 4-Stage Pipeline
Controller with an Active Operator...........................80
3.3.3- 1 Predicted vs. Simulation Slave-Loop Feedback Gains for 5-
Stage Pipeline Controller Passive Operator Load...........82
3.3.3- 2 Effective Environment Stiffness 5-Stage Pipeline Controller
with a Passive Operator......................................82
xm
Table
3.3.3- 3 Predicted vs. Simulation Slave-Loop Feedback Gains for 5-
Stage Pipeline Controller Active Operator Load..........83
3.3.3- 4 Effective Environment Stiffness 5-Stage Pipeline Controller
with an Active Operator...................................83
4.1- 1 Effects Upon Slave Loop Feedback Gain Due to the Operator
Loop......................................................85
4.1 -2 End-to-End System Rise Time Data for Free-Space............86
4.2.1- 1 Effective Environment Stiffness Passive Operator Load....88
4.2.1- 2 Effective Environment Stiffness Passive Operator Load....89
4.2.2- 1 Effective Environment Stiffness Passive Operator Load....93
4.2.2- 2 Effective Environment Stiffness Active Operator Load.....95
4.2.3- 1 a Effective Environment Stiffness -1.0 Hz. Bandwidth Model
with Passive Operator Load and Pipelining.................99
4.2.3- lb Effective Environment Stiffness 2.0 Hz. Bandwidth Model
with Passive Operator Load and Pipelining.................100
4.2.3- lc Effective Environment Stiffness 5.0 Hz. Bandwidth Model
with Passive Operator Load and Pipelining.................100
4.2.3- Id Effective Environment Stiffness -10.0 Hz. Bandwidth Model
with Passive Operator Load and Pipelining.................101
4.2.3- 2a Effective Environment Stiffness -1.0 Hz. Bandwidth Model
with Active Operator Load and Pipelining..................101
4.2.3- 2b Effective Environment Stiffness 2.0 Hz. Bandwidth Model
with Active Operator Load and Pipelining..................102
4.2.3- 2c Effective Environment Stiffness 5.0 Hz. Bandwidth Model
with Active Operator Load and Pipelining..................102
4.2.3- 2d Effective Environment Stiffness -10.0 Hz. Bandwidth Model
with Active Operator Load and Pipelining..................103
C. 1.1-1 Slave Open Rate-Loop Parameters............................134
C.1.1-2 Slave Closed Rate-Loop Bandwidth...........................136
C. 1.1-3 Slave Position-Loop Position Error Gain Stability Limits...137
C. 1.1 -4 Slave Open Position-Loop Parameters........................137
C. 1.1 -5 Slave Closed Position-Loop Bandwidth.......................138
C. 1.1-6 Time Response Data for Free-Space Step Input.............138
xiv
Table
C. 1.2-1 Contact Resonance Summary....................................146
C. 1.2-2 Predicted Contact Resonance Results..........................150
C. 1.2-3 Contact Stability Stiffness Boundaries Absolute............152
C. 1.2-4 Stability Data for Constant Environment Stiffness............152
C. 1.2-5 Locked-Output Bandwidth Results..............................154
C. 1.2-6 Time Response Data for Constant Environment Stiffness........154
C. 1.4-1 Slave-Loop Absolute Stability Limits Passive Operator Load.176
C. 1.4-2 Slave-Loop Open-Loop Stability Data Passive Operator Load.... 177
C. 1.4-3 Closed Slave-Loop Characteristics Non-Ideal Slave Model
with Passive Operator Load..................................178
C. 1.4-4 Time Response Data for Closed Slave Loop with Passive
Operator Load...............................................178
C. 1.4-5 Closed Slave-Loop Effective Environment Stiffness Passive
Operator Load...............................................180
C. 1.4-6 Slave-Loop Absolute Stability Limits Non-Ideal Actuator
Model with Active Operator Load.............................183
C. 1.4-7 Slave-Loop Open-Loop Stability Data Active Operator Load...183
C. 1.4-8 Closed Slave-Loop Summary Data Active Operator Load........184
C. 1.4-9 Time Response Data for Closed Slave Loop with an Active
Operator Load...............................................186
C. 1.4-10 Closed Slave-Loop Effective Environment Stiffness Active
Operator Load...............................................186
C. 1.5-1 End-to-End Free Space Frequency Response Summary Data........189
C. 1.5-2 Time Response Data for Free-Space End-to-End System..........189
C. 1.5-3 Operator Absolute Stability Limits Closed Slave Loop.......191
C. 1.5-4 Operator Gain Values Closed Slave Loop.....................192
C. 1.5-5 End-to-End Closed Slave-Loop Characteristics.................192
C. 1.5-6 End-to-End Time Response Data for Locked-Output Slave........193
C.2.1 -1 Parameters for Hybrid Model Update...........................195
C.2.1-2a Hybrid Slave Open Rate-Loop Stability Margins -1.0 Hz. Slave
Bandwidth Design............................................197
C.2. l-2b Hybrid Slave Open Rate-Loop Stability Margins 2.0 Hz. Slave
Bandwidth Design............................................198
xv
Table
C.2.l-2c Hybrid Slave Open Rate-Loop Stability Margins 5.0 Hz. Slave
Bandwidth Design..............................................198
C.2. l-2d Hybrid Slave Open Rate-Loop Stability Margins -10.0 Hz.
Slave Bandwidth Design.........................................199
C.2.1 -3 Hybrid Slave Closed Rate-Loop Frequency Response
Characteristics................................................200
C.2.1-4 Hybrid Slave Open Position-Loop Stability Margins...................202
C.2.1 -5 Hybrid Slave Closed Position-Loop Frequency Response
Characteristics................................................203
C.2.2-1 a Hybrid Slave Locked-Output Feedback Gain Identification -1.0
Hz. Slave Bandwidth Design.....................................206
C.2.2- lb Hybrid Slave Locked-Output Feedback Gain Identification 2.0
Hz. Slave Bandwidth Design..........;..........................206
C.2.2-lc Hybrid Slave Locked-Output Feedback Gain Identification 5.0
Hz. Slave Bandwidth Design.....................................207
C.2.2- Id Hybrid Slave Locked-Output Feedback Gain Identification -
10.0 Hz. Slave Bandwidth Design...............................207
C.2.2-2 Hybrid Open Slave-Loop Stability Margins Passive Operator 208
C.2.2-3 Hybrid Closed Slave-Loop Step Response Characteristics -
Passive Operator Load..........................................210
C.2.2-4 Hybrid Slave Effective Environment Stiffness Characteristics -
Passive Operator...............................................212
C.2.2-5a Hybrid Slave Locked-Output Feedback Gain Identification -1.0
Hz. Slave Bandwidth Design with Active Operator Load...........214
C.2.2-5b Hybrid Slave Locked-Output Feedback Gain Identification 2.0
Hz. Slave Bandwidth Design with Active Operator Load...........214
C.2.2-5c Hybrid Slave Locked-Output Feedback Gain Identification 5.0
Hz. Slave Bandwidth Design with Active Operator Load...........215
C.2.2-5d Hybrid Slave Locked-Output Feedback Gain Identification -
10.0 Hz. Slave Bandwidth Design with Active Operator Load....215
C.2.2-6 Hybrid Open Slave-Loop Stability Margins Active Operator..........216
C.2.2-7 Hybrid Closed Slave-Loop Step Response Characteristics -
Active Operator Load...........................................217
xvi
Table
C.2.2-8 Hybrid Slave Effective Environment Stiffness Characteristics -
Active Operator................................................218
C.3.1-1 Slave-Loop Pipelining Time Delay......................................220
C. 3.1 -2a 2-Stage Open Slave-Loop Feedback Gain Identification -1.0
Hz. Bandwidth with Passive Operator Load.......................223
C.3. l-2b 2-Stage Open Slave-Loop Feedback Gain Identification 2.0
Hz. Bandwidth with Passive Operator Load.......................224
C.3.l-2c 2-Stage Open Slave-Loop Feedback Gain Identification 5.0
Hz. Bandwidth with Passive Operator Load.......................224
C.3.1-2d 2-Stage Open Slave-Loop Feedback Gain Identification -10.0
Hz. Bandwidth with Passive Operator Load.......................224
C.3.1-3 2-Stage Open Slave-Loop Stability Margins Passive Operator
Load.......................................................... 225
C.3.1-4 Hybrid Closed Slave-Loop Step Response Characteristics 2-
Stage Controller with Passive Operator Load....................225
C.3.1-5 Hybrid Slave Effective Environment Stiffness Characteristics -
2-Stage Pipeline Controller with Passive Operator..............226
C.3.l-6a 2-Stage Open Slave-Loop Feedback Gain Identification -1.0
Hz. Bandwidth with Active Operator Load........................227
C.3. l-6b 2-Stage Open Slave-Loop Feedback Gain Identification 2.0
Hz. Bandwidth with Active Operator Load........................228
C.3.l-6c 2-Stage Open Slave-Loop Feedback Gain Identification 5.0
Hz. Bandwidth with Active Operator Load.........................228
C.3. l-6d 2-Stage Open Slave-Loop Feedback Gain Identification -10.0
Hz. Bandwidth with Active Operator Load........................228
C.3.1-7 2-Stage Open Slave-Loop Stability Margins Active Operator
Load............................................................229
C.3.1-8 Hybrid Closed Slave-Loop Step Response Characteristics 2-
Stage Controller with Active Operator Load.....................229
C.3.1-9 Hybrid Slave Effective Environment Stiffness Characteristics -
2-Stage Pipeline Controller with Active Operator................230
C.3.2-1 4-Stage Open Slave-Loop Feedback Gain Identification -
Passive Operator Load with 200 sps Controller...................233
xvii
Table
C.3.2-2 4-Stage Open Slave-Loop Stability Margins Passive Operator
Load............................................................233
C.3.2-3 Hybrid Closed Slave-Loop Step Response Characteristics 4-
Stage 200 sps Controller with Passive Operator Load............233
C.3.2-4 Hybrid Slave Effective Environment Stiffness Characteristics -
4-Stage Pipeline Controller with Passive Operator..............234
C.3.2-5 4-Stage Open Slave-Loop Feedback Gain Identification Active
Operator Load with 200 sps Controller..........................235
C.3.2-6 4-Stage Open Slave-Loop Stability Margins Active Operator
Load........................................................... 235
C.3.2-7 Hybrid Closed Slave-Loop Step Response Characteristics 4-
Stage 200 sps Controller with Active Operator Load..............235
C.3.2-8 Hybrid Slave Effective Environment Stiffness Characteristics -
4- Stage Pipeline Controller with Active Operator..............236
C.3.3-la 5-Stage Open Slave-Loop Feedback Gain Identification -1.0
Hz. Slave Bandwidth Design with Passive Operator Load..........238
C. 3.3 -1 b 5-Stage Open Slave-Loop Feedback Gain Identification 2.0
Hz. Slave Bandwidth Design with Passive Operator Load..........239
C.3.3-lc 5-Stage Open Slave-Loop Feedback Gain Identification 5.0
Hz. Slave Bandwidth Design with Passive Operator Load..........239
C.3.3- Id 5-Stage Open Slave-Loop Feedback Gain Identification -10.0
Hz. Slave Bandwidth Design with Passive Operator Load..........239
C.3.3-2 5-Stage Open Slave-Loop Stability Margins Passive Operator
Load............................................................240
C.3.3-3 Hybrid Closed Slave-Loop Step Response Characteristics 5-
Stage Controller with Passive Operator Load....................241
C.3.3-4 Hybrid Slave Effective Environment Stiffness Characteristics -
5- Stage Pipeline Controller with Passive Operator.............241
C.3.3-5a 5-Stage Open Slave-Loop Feedback Gain Identification -1.0
Hz. Bandwidth with Active Operator Load.........................243
C.3.3-5b 5-Stage Open Slave-Loop Feedback Gain Identification 2.0
Hz. Bandwidth with Active Operator Load.........................243
xvm
Table
C.3.3-5c 5-Stage Open Slave-Loop Feedback Gain Identification 5.0
Hz. Bandwidth with Active Operator Load........................243
C.3.3-5d 5-Stage Open Slave-Loop Feedback Gain Identification -10.0
Hz. Bandwidth with Active Operator Load........................243
C.3.3-6 5-Stage Open Slave-Loop Stability Margins Active Operator
Load............................................................244
C.3.3-7 Hybrid Closed Slave-Loop Step Response Characteristics 5-
Stage Controller with Active Operator Load......................244
C.3.3-8 Hybrid Slave Effective Environment Stiffness Characteristics -
5-Stage Pipeline Controller with Active Operator................245
xix
ACKNOWLEDGEMENTS
The author wishes to make the following acknowledgements :
The Martin Marietta Corporation, without whose support this degree
would not have been possible,
My advisor, Dr. Jan T. Bialasiewicz, for guidance, counseling and
support throughout my master's program and thesis,
Thurston Brooks and Pete Spidaliere for providing the inspiration to
pursue this thesis topic, and
Lyle Cloud, Tom Depkovich, Jim Farler, and Dr. Bob Rice, whose
inspiration to pursue a master's degree and consultation on this
thesis are deeply appreciated.
1.0 INTRODUCTION
Robotics deals, in general, with the manipulation of objects; however, there are large
differences in the features of robotic systems. Some systems rely upon human
intelligence for task and path planning, while others strive for machine intelligence to
minimize human burden. Some systems are practical only in a localized area, e.g., 1 to
10 miles maximum, whereas other systems are designed to operate over very large
distances, e.g., Earth to Mars.
The specific area of robotics that this paper will deal with is referred to as tele-robotics,
or teleoperation. These robotics systems have the following characteristics :
1) The operator is an integral part of the system. In these systems, the operator
provides all the task level planning and trajectory generation. The operator
relies upon video and / or tactile feedback to perceive the activities at the task
site. The operator communicates commands to the system by holding onto a
device referred to as a master manipulator or a hand controller and producing
the desired motion at the task site, which the system then replicates. This
motion takes place real-time, as opposed to a teach and play-back approach. In
some systems the operator feels the forces created when an object at the task site
is touched, which is often referred to as force reflection, or force feedback.
2) The computer provides limited or no intelligence for operator support. In the
simplest of teleoperation systems, computers are used as the most useful way to
transmit information and / or the most convenient means to execute the required
control algorithms. In more complex systems, the computer is used to perform
algorithms designed to compensate for undesired forces created at the task site;
an example of this is impedance control.
3) Teleoperation is only practical for localized systems. Teleoperation relies upon
operator feedback, either visual or tactile, to allow perception of the activities at
the task site. As time delays grow with greater distance, this has been shown to
degrade the operator performance [1] [2] until teleoperation is no longer
practical. When teleoperation is no longer practical, other techniques utilizing
autonomous control must take over.
1.1 Teleoperation Characteristics
Even within the field of teleoperations, there is a wide range of system characteristics
available.
Basic teleoperation systems consist of a master manipulator, which the operator
positions, and a slave manipulator, which is located at the task. The master
manipulator is referred to as a hand controller in some systems. These systems
2
typically have a common kinematic configuration 1 for the master and slave
manipulators, referred to as replica master/slave systems. Control is accomplished by
sampling the position sensors in the master manipulator and commanding the slave
actuators to the same positions. Since the two manipulators have common kinematics,
no transformations are required. An additional capability sometimes added to these
systems is to command a torque to each master actuator based upon the position error at
each slave actuator. This is sometimes referred to as bilateral control, although this
term is often used in several different contexts.
More complex teleoperation systems involve non-replica master/slave systems, i.e. the
kinematics of the master and slave are not common. In these systems all operations
must be performed in Cartesian space, as opposed to actuator space for the replica
master/slave systems. Tactile feedback is provided in these systems typically by
sensing the contact forces between the slave and the task and reproducing the same
contact forces at the grip of the master manipulator.
In advanced teleoperation systems, various forms of shared control can be
implemented. Shared control covers a variety of control concepts; however, all are
intended to allow the operator to perform only a subset of the task. For example the
operator might desire to control the displacement degrees of freedom, while the
orientation degrees of freedom are controlled autonomously based upon a system
constraint such as tracking a target. Another form of shared control is autonomous
1 A common kinematic configuration requires that the two manipulators have the identical
rotational and/or longitudinal degrees of freedom applied in the identical order with
identical or proportional link displacements.
3
regulation of the forces applied to the task to a desired level, e.g., impedance control
techniques presented by Hogan [3,4, and 5].
1.2 Teleoperation Performance
The purpose of telerobotics is to allow human to remote task interaction as if the human
to task interaction was actually performed in a hands-on manner.
The ideal teleoperation system can be defined as one where there is no loss in
information due to the physical separation of operator and task, as discussed in Brooks
[6]. The ideal system can then be quantified as a system where the operator impedance
is applied at the task and the task impedance is applied at the operator through force
reflection. There are no teleoperation systems in the research community that are able
to completely achieve this; all are approximations to this approach.
One of the key characteristics of a teleoperated system is how effectively the operator
can perceive contact with the environment The operator's perception of environmental
contact is beyond the current analytical capability, due to insufficient operator
modelling. An analytical study can be made of the effective environment stiffness
presented at the user, or operator, interface. Succinctly stated, the greater the capability
of the system to reproduce the environment stiffness at the operator interface, the more
useable the system will prove to be.
4
However, several factors impact upon the effective environment stiffness provided by
the teleoperation system, including:
Controller Sample Rate The controller sample rate is the rate at which the control
data is acquired and the control laws are applied in the digital control system.
Control Law Properties The control law properties include the characteristics of
the control design and the constraints. This includes the system loop gain, the
various control loop bandwidths, phase and gain margin constraints, overshoot
specifications, etc.
System Data Latency The system data latency is the length of time required for a
system input to propagate through the complete digital control system, and can
also be referred to as the around the loop data latency
System Architecture The system architecture defines characteristics such as
pipelining, and parallel processing. Pipelining is a technique where a
processing task is segmented into multiple, nearly equal, dependant
processing elements, and then distributed to multiple processors (reference
Figure 1.2-1). The effect of pipelining, when implemented properly, is to
multiply the system data flow by the number of pipeline stages while producing
no impact upon the around the loop data latency. Parallel processing is a
technique where a processing task is segmented into multiple, nearly equal,
independent processing elements, and then distributed to multiple processors
(reference Figure 1.2-2). The effect of parallel processing, when implemented
properly, is to maintain the system data flow while reducing the around the loop
data latency by the number of parallel processes.
1 The term equal is intended to communicate that each process takes nearly the same amount of
time to execute, i.e. the computational load is distributed equally to each processor.
5
Process 1
Process 2
Process 3
Process 4
Time
tH
I 1 f
mmmm
t4H
T
Pipe
loop
DHL
WBBML
Output
Figure 1.2-1 Pipelined Computer Architecture Example
Input
Process 4 1
Time
Figure 1.2-2 Parallel Computer Architecture Example
The following notes apply to Figure 1.2-1:
1. The 'Critical Path' is illustrated in heavy lines.
2. Tpipe > max { Tj T2 ,..., Tn_j Tn }, where:
n is the number of pipeline stages,
Tj is the computational time required to complete the j* process, and
Tpipe is the pipeline clock period
3. Tjoop = (n-l)*Tpipe + Tn
The following notes apply to Figure 1.2-2:
1. The 'Critical Path' is illustrated in heavy lines.
2. TCyCie > max { Tj T2 ,...Tn-i Tn }, where:
n is the number of parallel processes,
Continued on Next Page
6
Mechanical Properties The system mechanical properties consist of master and
slave manipulator characteristics, operator characteristics, and task
characteristics. These characteristics include items like inertia,stiffness,
damping, force and/or torque authority, etc.
1.3 Thesis Subject Definition
This thesis will study the effects of several system characteristics upon the effective
environment stiffness of a telerobotic system. These characteristics will be dealt with
using a single degree-of-ffeedom control simulation. The following characteristics will
be studied:
1) Slave position-loop bandwidth
2) Controller sample rate
3) Computational architecture
Tj is the computational time required to complete the j* process, and
Tcyde is the process clock period
3. Ti00p = max { Tj T2 ,..., Tn_i Tn }
7
2.0 PROBLEM DEFINITION
2.1 Introduction
As stated in Section 1.2, the ideal teleoperation system can be defined as no loss
information due to the physical separation of the operator and the task.
A simple continuous system model can be analyzed, as shown in Figure 2.1-1.
Figure 2.1-1
Simplified System Block Diagram
Where the symbols illustrated in Figure 2.1-1 are defined as follows :
Hb(s) is the feedback path transfer function,
Hf(s) is the forward path transfer function,
Ke is defined as the environment stiffness,
K| is defined as the control loop gain,
K0 is defined as the operator stiffness,
0ref is the reference position signal, and
0Out is the output position signal
Let us assume, for the time being, that this system is stable. This condition will be a
system constraint applied later in the analysis.
2.1.1 Effective Environment Stiffness
If we break the system, reference Figure 2.1-1, at the operator interface, i.e. section A-
A, the resulting transfer function will define the effective stiffness of the environment
as seen by the operator.
9
For example; 0ref and Gout are both expressed in units of radians, and Ke are in units
of ft-lbs per radian, K| is a unit-less gain, and Hf(s) and Hb(s) are unit-less transfer
functions, then the system can be described as follows:
Ke =
0A, where: (2.1.1)
/v
Ke is defined as the effective environmental stiffness,
0A is defined as the position input at the operator interface, and
is defined as the commanded torque output at the operator interface, and is
defined by the following equation :
ta = K|Hf(s)KeHb(s)0A (2.1.2)
thus, the effective environment stiffness can be described as :
Ke = K|Hf(s)KeHb(s) (2.1.3)
As stated previously, the ideal system has no information loss. In this case, no
information loss requires the effective and the actual environmental stiffness to be the
same, or:
K|Hf(s)Hb(s) = 1 (2.1.4)
Even in this simple model, this condition can not be met. The forward transfer
function, Hf(s)9 includes the actuator and associated control loops. The typical actuator
provides a torque output, which is integrated twice to define position output. The
10
additional compensation control laws required to assure actuator stability result in a
minimum of a third order system in the forward path. This results in, typically, a low-
pass filter that is unity for low frequencies and attenuated for high frequencies. The
time delay 1 will force a reduction in loop gain, in order to maintain a stability criteria.
The implications to the effective stiffness are that there will be some range of
frequencies over which the maximum loop gain is achieved.
It is important to note that the these limitations are a direct result of hardware selection.
The choice of hardware properties will not be examined by this thesis. Instead this
thesis will offer approximations to some critical hardware related parameters.
Thus we have concluded that the system will not be able to reconstruct the task
stiffness, Ke, at the operator interface to the system.
There is an important assumption made in this simple analysis; the environment can be
modelled as a pure stiffness. This assumption will need to be examined later. Even
this simple model provides useful insights into the features that have implications upon
the usefulness of teleoperation systems specifically, the loop transfer function proves
to be the limiting factor of the ability of the operator to perceive the environmental
stiffness. 1
1 Time delay modelling and effects are discussed in Appendix A.
11
2.2 Specification of Thesis Parameters
Recalling that the effects of the following characteristics, reference section 1.3, are to
be analyzed; (1) Slave Position-Loop Bandwidth, (2) Controller Sample Rate, and (3)
Computational Architecture, a more general system model is required. This model
must provide the following features:
1) Programmable slave position-loop bandwidth with a common damping ratio.
For this thesis, the following position bandwidths will be analyzed:
1.0,2.0,5.0 and 10.0 Hz.
2) Programmable controller sample rates and associated time delays. For this
thesis, the following position loop rates will be analyzed:
20 1,50,100,200,500 and 1000 samples per second (sps)
3) Multiple control stages, suitable for pipelining and/or parallelization. For this
thesis, the following architectures will be analyzed:
1) Serial Computations
2) Pipeline Computations
Not all control rates and position bandwidths are possible selections, as the lower control rates
will impose too great a phase lag on the higher bandwidth slave control loops. The
specific combinations available will be specified later in the analysis, Section 3.
12
2.2.1 System Bandwidth
The system bandwidth shall be defined as the -3 dB point on the free-space magnitude
frequency response plot. All other uses of bandwidth shall also reference a -3 dB
point, but may be taken from the steady-state value, if the steady state value is less than
zero dB.
2.2.2 Controller Sample Rates
The controller sample rate and the control period are direct inverses of each other. The
loop time and the control period are not typically one in the same. Only the serial
computational architecture provides a loop time that is less than the control period.
2.2.3 Control Architectures
2.2.3.1 Serial Computations Control Architecture
This control architecture is the most standard control architecture. In this architecture,
the loop time is less than the control period. This architecture is illustrated by the
timing diagram presented in Figure 2.2.3-1.
13
Figure 2.2.3-1 Serial Computations Control Architecture Timing
Diagram
The reader should be aware that the reference to 'forward computations' refers to the
process of converting the operator position command, 0opr, into the slave actuator
torque command, tCmd,s Likewise, the reference to 'feedback computations' refers to
the process of converting the sensed environment torque, Te > into the master actuator
torque command, Tcmd.m
2.2.3.2 Pipeline Control Architecture
The pipeline control architecture is best described as a multi-processor system, where
all processors share equally, or near-equally, in the computational load. All processors
are synchronized at the control period, also referred to as the pipe period.
14
The loop time for a symmetrical pipeline architecture is defined by the following
equation:
Tioop (n-1) Tpipe + Tn> where:
TBipe is the pipeline clock period,
n is the number of pipeline stages, and
Tn is the computational time required to complete the n* process
This architecture is illustrated by the timing diagram presented in Figure 2.23-2.
The reader should be aware that one complete cycle of the digital control system, i.e.
the loop time or the 'critical path', is illustrated in heavy lines in the preceding figure.
15
2.2.4 Stability Requirements
The stability requirements, for this research, are 30 degrees of phase margin and 6 dB
of gain margin. Since this research is concerned only with linear models, this is the
only specification for stability needed for this thesis.
2.3 Analytical Model Description
2.3.1 Slave Position Loop Bandwidth Model
2.3.1.1 Slave Position Loop Bandwidth Model Description
The slave hardware models, used in this thesis, will consist of the closed actuator
torque loop, torque to rate transfer function, rate to position transfer function, and anti-
aliasing filter transfer function.
16
In order to model a typical actuator, the simulation will utilize a torque loop model as
follows:
hTl(s)=---------------------ml-----------
tcmd,s(s) s2 + 2Â£tL(GTLS + COTL2 (2 3 T
where:
is the torque loop damping ratio, and is set for critical damping i.e. 1.0,
and
Â£&TL is the torque loop bandwidth, and is set to 314.16 rad/sec (50 Hz.)
based upon maintaining minimal separation, i.e. a factor of 5,
between the position and torque loops.
Modeling of the actuator load is accomplished by the output torque to output rate model
as follows:
Ht2cd(s) =
fl>out(s)
^OUt(S)
1
JLs + Bl
where:
Jl is the actuator load inertia, set to 1.0 ft-lb-sec2, and
Bl is the is the actuator load damping, set to 0.35 ft-lb-sec/rad
(2.3.2)
Finally, the output rate is converted to output position by adding a pure integrator, or:
Hco2e(s) =
Qout(s) _
put(s) s
(2.3.3)
17
Anti-aliasing filters are required in all physical implementations of digital control
systems, and will be provided by the following transfer function :
Halfas(s) =^m?7yr---------------------------------r
xoutt ) s2 + 2^a|jasC0a|iasS + alias (2.3.4)
where:
Calias is the anti-aliasing filter damping ratio, and is set at 0.707 to achieve a
slightly under-damped response, and
alias is the anti-aliasing filter bandwidth, and is set to one half of the
controller sample rate as converted to radians/sec.
The control law used in the simulation will be a simple Proportional-Derivitive (PD)
controller, using a position error gain of Kp and a feedback velocity gain of Kv.
Combining equations 2.3.1 through 2.3.4, the slave position control loop model will
be defined in Figure 2.3.1-1.
Figure 2.3.1-1 Slave Position Loop Control Model
18
2.3.1.2
Slave Environment Contact Model Description
In general, a structure is modelled as mass, stiffness, and damping, reference Hogan
[7] and Whitney [8]. The effects of environment modelling are seen as a force
disturbance into the system. In the simple model developed for this thesis this is
actually a disturbance torque input Although there are numerous contact types to
model, this research will consider only the locked-output type of environment contact.
Combining the environment model with the previous system model, reference Figure
2.3-1, defines the slave position model including environment contact dynamics as
shown in Figure 2.3.1-2.
Figure 2.3.1-2 Slave Position Loop Control Model with Environment
Contact Dynamics
19
In general, contact with a body is represented by the following transfer function :
He(s) = ^-=Mes2+Bes + Ke
9e(s) (2.3.5)
where:
Me is the environment mass,
Be is the environment damping,
Ke is the environment stiffness,
e(s) is the environment position error, which is defined by the type of
contact under consideration, and
'te(s) is the environment reaction torque
However, for the purpose of this thesis, the load is assumed to be locked to a physical
ground reference. This provides the following conditions :
1) There is no environment motion, i.e.:
4e.) 0
dt and
2) There is no environment acceleration, i.e.:
dt2
These two conditions are used to simplify the environment contact model to the
following:
He(s)
Te(s)
09(8)
= Ke
(2.3.6)
where:
Ke is the environment stiffness,
e(s) is the environment position error, which is defined by the type of
contact under consideration, and
Te(s) is the environment reaction torque
20
The locked output environment environment contact model, used in this research,
results in the following equation to define the environment position error:
9e(s) = 0output(s) (2.3.7)
where:
Qoutput(s) js output position of the actuator,
Combining this position error definition with the environment mode, He(s) from
equation 2.3.6, completely specifies the environment dynamics for this case.
21
2.3.2 Operator Model
A good historical summary of operator modeling activities is provided in section 3 of
Brooks [6], which concludes in the following operator model:
fTks+l)r
Tps+lj
C0n
where:
(2.3.8)
Kh is the operator steady-state gain,
Ki
ai
OTJ
is the non-linear operator response model,
e_sx is the operator time delay model,
(TjS+1) ^ moc^ t^ie operator adapatation to system dynamics,
gffl is a model for low-frequency neuromuscular loops, and
1
(Tns+1 (-M2-3s+l
Lv 1W C0h
is an analytical model for operator musclue impulse response
A brief summary of operator modelling activities, is available in Corliss [2] and Brooks
[6].
The operator models, provided in the reference material, are principally describing
function models. The describing function models are developed with the intention of
modeling aircraft pilots during interactions between the pilots and the airframe
instrumentation. Since the instrumentation is primarily visual, these models are
22
developed with the purpose of describing the force / torque reactions of pilots to a
visual representation of position error.
In this thesis, as in teleoperations, the operator model in concerned with the interaction
between tactile forces and the internal operator position control loop. Thus the operator
models provided in the reference material, are not completely applicable to teleoperation
analysis. There are significant results which are applicable to this thesis. These items
are included in the definition of the operator model.
One of the limitations of this analysis is the lack of adaptability of the operator
parameters. Experimental data has shown that operators adapt to stimuli after 1.2 to
1.5 seconds by changing their characteristics, reference Hogan [9]. This analysis will
assume the initial parameters.
Another limitation of operator modeling is the inability to examine the reference input.
Since the reference input is a 'signal' within the operator's neural system, the signal is
not available for examination. This prevents effective analysis of feedback and
command processing, which forces characterization of the operator from input to
output. Since the operator has numerous inputs, this is a very difficult task. These
issue will not be dealt with further in this document. I
I consider the primary elements to modeled as part of the operator to be as follows :
1) Time delay modeling,
2) Bandwidth modeling, and
3) Load dynamics modeling
23
In modeling the operator, the master actuator must also be considered. Although the
actuator is certainly not part of the operator, the actuator forms part of the operator load
and there it interacts between the actuator and operator torque commands.
From reviewing the available literature, the following operator characteristics are
available:
1) The operator command bandwidth varies widely between 1 and 5 Hz. This
bandwidth is interpreted as a position bandwidth. For this these, I desire to use
an operator position bandwidth of 3 Hz., reference [9]. The operator position
bandwidth will be obtained by defining, to the maximum extent possible, an
operator position error gain.
2) The operator sensory bandwidth has been a subject of considerable research. As
mentioned in Brooks [6], depending upon the type of receptor the sensory
bandwidth has been shown to be distributed from 10 Hz. to as high as 10 kHz.;
however, it is noted that signals can not be distinguished above 320 Hz. From
reviewing the literature, it seems that a reasonable value for the perception
bandwidth is 20 Hz. (125.66 rad/sec).
3) The operator muscular structure exhibits a torque (force) bandwidth of 10 Hz.,
reference [10].
4) The operator time delay has similarly been a subject of much research. Wargo,
in 1967, produced the minimum and maximum parameters shown in Table
24
2.3.2-1, referenced from reference [11]. Also shown in Table 2.3.2-1 are the
parameter values that I have chosen for this simulation. These values are
chosen primarily to be a median value, but are also chosen so that the total
operator time delay is close to the 250 millisecond value provided by Hogan
[9].
Table 2.3.2-1 Operator Time Delay Parameters
Item Minimum Maximum Model
Value
Receptor Delays 1 38 25
Afferent Transmission Delays 2- . 100 60
Central Process Delays 70 100 80
Efferent Transmission Delays 10 20 15
Muscle Latency and Activation Delays 30 70 50
Total Delay 113 318 230
Notes:
1. All times are specified in units of milliseconds
2. Afferent is derived from the Latin meaning 'carry to' (the brain)
3. Efferent is derived from the Latin meaning 'carry from' (the brain)
5) The operator impedance is approximated as a mass-spring-damper system:
M0pS1 2 3 + Bops + Kop (2.3.9)
where:
Mop is the effective operator mass,
Bop is the operator damping, and
Kop is the operator stiffness
25
Operator impedance has only recently been a subject of research. The only
literature, the author is aware of, in this subject matter has been conducted by
Hogan [9], which concludes the values shown in Table 2.3.2-2.
Table 2.3.2-2 Operator Impedance Model Parameters
Item Symbol Model Value Units
Operator Mass M0p 0.8 Kg
Operator Damping B0p 5.5 N-sec/m
Operator Stiffness K0p 550 N/m
Notes:
1. The parameters specified in this table will be converted to the proper
system of units when performing the analysis.
Considering these factors, the operator model shown in Figure 2.3.2-1 will be used in
this thesis. The following s-domain transfer function are illustrated in Figure 2.3.2-1,
and are defined in subsequent sections:
The operator torque command model, Hopc(s) section 2.3.2.1,
The operator torque loop model, HTL,opr(s) section 2.3.2.2,
The operator load model, Hioad,opr(s) section 2.3.2.3,
The actuator load model, Hioad,act(s) section 2.3.2.4,
The integrator model, H^eCs) section 2.3.2.4, and
The operator perception model, Hopp(s) section 2.3.2.5,
26
Ht.
TL,m
/Tout, mi!
' S % s \%v NSW A
, ' i/
Master ^
- Actuator
H, w >(s)
load, act
err
0
H (3 H_
opc TL.opr
opr
ref
0
sense H (s) opp
2.3.2-1 Operator Model
H, h (s)
load, opr
opr
HWS)
2.3.2.1 Operator Torque Command Model
The principal effects to be considered in this model are the operator command time
delay, and the operator command bandwidth. As a result, the operator position
command model used will be as follows:
Li fc,s 'tcmd.opr(s)
nopcW
6err(s)
= KH
Â£-STc
where:
Kh is the operator steady-state gain,
(ii
\TnS+1 / is the operator command time delay model, and
e'SXc is the neuro-muscular lag model
(2.3.10)
27
The operator command bandwidth has been, and is still, a subject of considerable
research. Since this parameter is primarily responsible for defining the operator
bandwidth, this is one of the most critical and contested parameters in the system.
Operator bandwidth parameters have been identified over a range from 1 Hz. to 10 Hz.
Most recently, Hogan [9] has identified an operator bandwidth at 3 Hz. Hogan's work
is applied to the operator arm, as opposed to an operator joint, and appears to be
reasonable, given the research base. As a result, it is desired to achieve the 3 Hz.
(18.85 rad/sec) bandwidth number by specifying the operator steady-state gain, Kh, as
part of the analysis in Appendix C.
As noted in Section 2.3.2.2, there does not appear to be any significant data available
on operator damping ratios. As a result, I will assume a critically damped operator
model, i.e. Â£ = 1.0.
The last three delays from Table 2.3.2-1, i.e. central process, efferent transmission,
and muscle latency and activation delays, will all be lumped to form the operator
command time delay, tc .
The parameters values for the operator command model are specified in Table 2.3.2-2.
28
Table 23.2-3 Operator Position Command Model Parameters
Item Symbol Model Value Units
Steady-State Gain Kh (Specify from Analysis) N/A
Command Time Delay 145 msec
Neuro-Muscular Lag Constant Tn 150 msec
2.3.2.2 Operator Torque Loop Model
The principal effects to be considered in this model are the operator torque command
bandwidth, which was referred to earlier as the operator force bandwidth. This item
defines the bandwidth of the muscular system. As a result, the operator torque loop
model used will be as follows :
HTL,opr(s)
'Eopr(s)
'Ecmd,opr(s)
\TL,oprJ
JL
!2+2CtL,
opr
\\TL,opr/ TL,opr
5+1
(2.3.11)
where:
h>TL,opr is the operator torque bandwidth, and is set to 62.832
rad/sec, i.e. 10 Hz.
Cu.opr is the operator damping ratio, and is set for critical damping
i.e. 1.0
29
2.3.2.3 Operator Load Model
The principal effects to be considered in this model are the effective operator inertia and
damping. These items were identified previously in section 2.3.2, as part of the
operator impedance model. The operator load model used will be as follows :
H,oad,opr(s) = t4t= T-------
^netW Joprs + **opr (2.3.12)
where:
JPr is the operator load inertia, and
Pr is the is the operator load damping
Note: Both of these values will be based upon the operator dynamics data
provided by Hogan, reference [7],
2.3.2.4 Master Actuator Model
The master actuator model is comprised of a torque loop model and a load dynamics
model.
For simplicity purposes, I will assume a common configuration between the master and
slave actuator torque loop models. While this is not practical for multiple degree-of-
ffeedom teleoperation systems, this configuration is practical for a laboratory test
system.
30
This results in the following transfer function models :
Hu,m(s) ----------------------
T'errtSJ s2 + 2Â£n_CQTL_S + COjL2 (2 3 13'
where:
^TL is the torque loop damping ratio, and is set for critical damping i.e. 1.0,
and
On. is the torque loop bandwidth, and is set to 314.16 rad/sec (50 Hz.).
The actuator load dynamics will be represented by the load damping and inertia as
follows:
^net(s) Jacts + Bact (2.3.14)
where:
Jact is the actuator load inertia, set to 0.025 ft-lb-sec2, and
Bact is the is the actuator load damping, set to 0.35 ft-lb-sec/rad
The output rate is converted to output position by adding a pure integrator, or:
Hto2e(s) =
Qopr(s) i
C0out(s) s
(2.3.15)
31
2.3.2.5
Operator Perception Model
The principal effects to be considered in this model are the operator perception time
delay, and the operator perception bandwidth. The operator perception model used will
be as follows:
Qsense(S)
^out(s)
Hopp(s) = i^sensem e-sxp
S \2,2Cp
-h-^-S+1
\\COp / COp
(2.3.16)
where:
e'^p is the operator perception time delay model, and
1
[_s_r_3p.s+l
P
is the operator perception bandwidth model,
There are three parameters required to complete the operator perception model; operator
sensory bandwidth, operator sensory damping, and operator sensory time delay.
The first two delays from Table 2.3.2-1, i.e. receptor and afferent transmission delays,
will all be lumped to form the operator perception time delay, xp .
The parameter values for the operator perception model are specified in Table 2.3.2-4.
32
Table 2.3.2-4 Operator Perception Model Parameters
Item Symbol Model Value Units
Perception Time Delay Tp 85 msec
Operator Perception Bandwidth COp 125.66 rad/sec
Operator Perception Damping Ratio cp 1.0 N/A
2.3.3 System Feedback Model
In more complex teleoperation systems, this element performs sampled data filtering,
sensor decoupling, coordinate frame conversions, etc. In addition this element
provides the force reflection scaling and, as a side effect, a time delay.
In the single degree-of-freedom teleoperation system, there is limited need for these
computations; however, it is desirable to model the important effects of a more complex
systems in order to accurately reflect system performance capabilities. The last two
features of the complex systems, i.e. time delay and force reflection scaling, are the
features that are applicable to this system and will be retained for this model.
33
The system feedback model provides a programmable time delay model, and is defined
as follows:
Hfb(s) = Gbe-sTfb (2.3.17)
where:
Gb is defined as the backward (or feedback) path gain, and
Tfb is defined as the time delay in the feedback path of the system model
2.3.4 System Command Model
In more complex teleoperation systems, this element performs forward and inverse
kinematic computations, transformation scaling, coordinate frame conversions, etc. In
addition this element provides the position command scaling and, as a side effect, a
time delay.
In the single degree-of-ffeedom teleoperation system, there is no need for complex
kinematic equations; however, it is desirable to model the important effects of a more
complex systems in order to accurately reflect system performance capabilities. The
last two features of the complex systems are the features that are applicable to this
system and will be retained for this model.
34
The system command model provides the loop gain and a programmable time delay
model, and is defined as follows :
HC(S) = G|e-sTc (2.3.18)
where:
Gf is defined as the forward path gain, and
Tc is defined as the time delay in the command path of the system model
2.4 System Analytical Model
By assembling the individual models described in Section 2.3, the end-to-end analytical
model can be defined, as illustrated in Figure 2.4-1.
35
9\
Figure 2.4-1 End-to-End Analytical Model Part 1 of 2
u>
Figure 2.4-1 End-to-End Analytical Model Part 2 of 2
3.0 RESULTS
3.1 Continuous Analysis Results
3.1.1 Slave Position Loop Bandwidth Analysis Results
The purpose of section C. 1.1 is to define a set of position error and velocity feedback
gains that would achieve the four required slave position loop bandwidths (i.e. 1,2, 5,
and 10 Hz.) while satisfying the stability margin requirements of 30 degrees phase and
6 dB gain.
The slave position bandwidth is programmable, by specifying the position error and
rate feedback gains. This is not a surprising result. The gains specified in Section
C.1.1 are summarized in Table 3.1.1-1.
Table 3.1.1-1 Implemented Slave Gain Parameters
Bandwidth (Hz.) Velocity Feedback Gain (Kv) Position Error Gain (Kn)
1.0 15 71
2.0 28 252.5
5.0 62.832 1285
10.0 85 2775
The one issue that results from the analysis, in Appendix C, is these gains did not
match the theoretical gain values. The theoretical gain values are provided in Table
3.1.1-2, as computed from the following equations :
KP = (cObw f (3.1.1)
and
Kv = 2*^*C0bw (3 i 2)
where:
Wbw is the desired position bandwidth in radians per second, and
C is the desired damping ratio
Table 3.1.1-2 Theoretical Slave Gain Parameters
Bandwidth (Hz.) Velocity Feedback Gain (Kv) Position Error Gain (Kn)
1.0 12.566 39.478
2.0 25.133 157.914
5.0 62.832 986.96
10.0 125.66 3947.842
39
This section will attempt to explain the obvious discrepancies, by deriving and
analyzing the defining equations.
The closed rate-loop transfer function is defined as follows :
, HTl(s) HT2o(s)
1 +Kv*Ht(s)*hWs) (3.1
where:
Htl(s) is the torque loop transfer function (defined by equation 2.3.1),
Ht2
2.3.2), and
Kv is the velocity feedback gain
The closed position-loop transfer function is defined as follows:
Hgc(s)
Kp Htl(s) HT2to(s) H(q29(s)
1 +KV*
Htl(s) Ht2w(s) + Kp Hjl(s) H^s) H0320(5) (3.1.4)
where:
Htl(s)
- is the torque loop transfer function (defined by equation 2.3.1),
Ho)2e(s) is the rate to position integrator, and
Kp- is the position error gain
40
Substituting the following relationships :
Htl(s) =
cotl
s2 + 2Â£ti_g>tls + tl reference equation 2.3.1
Hx2(o(s)
Jls + B|_ reference equation 2.3.2
Hco2e(s) s ^ reference equation 2.3.3
into equation 3.1.4 produces the following equation :
Hec(s) =
kP4l
_____________________________________Jl_______________________________________
ZA |2CtlO>TlJl + BL|, t + 2CtlCQTlBl| |KvC0tL + BlCOtl^ | KpCO^
(3.1.5)
The following parameters are defined:
Torque loop damping ratio :
Torque loop bandwidth:
Actuator load inertia:
Actuator load damping:
Ctl=1,
cotl = 314.16,
Jl = 1, and
BL = 0.35,
41
The actual position closed-loop transfer function is then simplified to:
Hec(s) =
_________________________98696.04 Kp__________________________
s4 + 314.51 s3 + 98915.95s2 + 98696.04Kv + 0.35)s + 98696.04 Kp (3A 6)
Evaluating the position closed-loop transfer function model, defined in equation 3.1.6,
in Matlab provides the data summarized in Table 3.1.1-3.
Table 3.1.1-3 Position Loop Bandwidth Results 1.0 Damping
Ratio
Bandwidth (Hz.) Damping Ratio Kv Kp -3 dB Bandwidth (Hz.) -6 dB Bandwidth (Hz.)
1.0 1.0 12.566 39.478 0.621 0.979
2.0 1.0 25.133 157.914 1.274 2.008
5.0 1.0 62.832 986.96 3.284 5.242
10.0 1.0 125.66 3947.842 6.819 11.343
Repeating the analysis with a damping ratio of 0.707 produces the data shown in Table
3.1.1-4.
42
Table 3.1.1-4 Position Loop Bandwidth Results 0.707 Damping
Ratio
Bandwidth (Hz.) Damping Ratio Kv Kp -3 dB Bandwidth (Hz.) -6 dB Bandwidth (Hz.)
1.0 0.707 8.886 39.478 0.973 1.302
2.0 0.707 17.771 157.914 2.016 2.675
5.0 0.707 44.429 986.96 5.364 7.101
10.0 0.707 88.858 3947.842 9.948 13.109
The conclusions reached by analyzing these sets of data are as follows :
1. The rules, equations 3.1.1 and 3.1.2, used to define the position error gain,
Kp, and the velocity feedback gain, Kv, support a -6 dB gain criteria for
defining bandwidth with a damping ratio of 1.0, i.e. critical damping.
2. The rules, equations 3.1.1 and 3.1.2, used to define the position error gain,
Kp, and the velocity feedback gain, Kv, support a -3 dB gain criteria for
defining bandwidth with a damping ratio of 0.707.
3.1.2 Slave Environment Contact Model Analysis Results
Section C.1.2 presents the linear analysis associated with the locked output
environment model, reference Section 2.3.I.2. The intent of Section C.1.2 is to
provide a bound on environment stiffness for stable system response. A stiffness value
of 100 foot-pounds per radian provides stable results for all four slave bandwidths.
43
In addition, a simplified position loop model is used to generate predictions on steady-
state gain and the resonant frequency. The results of these equations, equations C.1.20
and C.1.22, are presented in Table C. 1.2-2.
One characteristic to examine is the relationship between free-space and locked-output
bandwidths. The locked-output condition creates a steady-state attenuation in the
position output. The free-space bandwidth to the attenuated locked-output response
level is summarized in Table 3.1.2-1.
Table 3.1.2-1 Slave Bandwidth Comparison Locked Output vs.
Free-Space
Bandwidth (Hz.) Locked-Output Steady-State Value (dB) GCF Free Space (Hz.) GCF Locked Output (dB)
1.0 -7.635 2.2781 2.52
2.0 -2.898 2.9759 2.96
5.0 -0.651 5.6415 5.52
10.0 -0.307 10.54 10.51
By inspection of the data in Table 3.1.2-1, the following conclusion can be reached:
The locked-output position bandwidth can be approximated by examining the
bandwidth of the free-space system to the steady-state value of the locked-
output system.
44
3.1.3 Operator Model Analysis Results
The purpose of section C.1.3 is to define operator position error gains that would
satisfying the stability margin requirements of 30 degrees phase and 6 dB gain for both
the free-space and locked-output slave conditions. The resulting operator bandwidth
was desired to be 3 Hz., but is not a constraint upon the gain.
This section initiated analysis of the full-up system, which is illustrated by Figure
C. 1.3-6 as follows :
The complete system is formed by two loops; Li is the slave loop, and L2 is the
operator loop. Opening the slave loop at the command input to the slave position
45
closed-loop model and examining the relationship between the reference and operator
output positions allows characterization of the stand-alone operator characteristics.
The operator model results were somewhat disappointing, but then there is a vast lack
of data defining the details of the operator; the operator control loops can not be
properly analyzed
Given the limitations, the results presented in this thesis are acceptable. In fact, as
shown later the limitations were not significant.
The disappointment results from the 0.609 Hz. bandwidth that resulted from the
operator model chosen for this thesis. The relevant operator parameters are as shown
in Table 3.1.3-1.
The operator gain is defined as 0.4594, based upon a 30 degree phase margin and 6 dB
gain margin stability requirement imposed on the free-space open position-loop
frequency response. This results in an operator position bandwidth of 0.609 Hz.
The slave loop can be examined by opening up the operator torque command output
and examining the relationship between the operator torque output and the slave
position output.
46
Table 3.1.3-1 Operator Parameter Summary
Item Symbol Model Value Units
Operator Mass M0p 0.8 Kg
Operator Damping Bop 5.5 N-sec/m
Command Time Delay *c 145 msec
Neuro-Muscular Lag Constant Tn 150 msec
Operator Torque Loop Damping Ratio Ctl 1.0 N/A
Operator Torque Loop Bandwidth con. 314.16 rad/sec
Perception Time Delay Tp 85 msec
Operator Perception Bandwidth COp 125.66 rad/sec
Operator Perception Damping Ratio Cp 1.0 N/A
In Section C.1.3, the slave loop analysis is performed utilizing an ideal slave actuator
model. I have defined the ideal slave as infinite bandwidth at unity gain, or:
He(s)=l
Under this condition, the root locus plot of the open-loop slave transfer function
provided a loop gain stability limit of 0.5992. This gain is formed by the product of the
position command gain and the torque feedback gain, and defines a zero margin gain
limit. I
I proceeded to define the slave loop gain that satisfies the 30 degree phase and 6 dB
gain margins. This gain limit is 0.149.
47
The effective environment stiffness can be predicted by examining the steady-state gain
around the slave-loop, or:
& = = G, Gb He(s) He(s) Hn,(!)
0opr(s) (3.1.7)
where:
Ke is the effective environment stiffness
Gf is the forward, i.e. command, path gain,
Gb is the backward, i.e. feedback, path gain,
H0(s)
is the locked output slave transfer function,
He(s)
is the environment transfer function, and
HTL,m(s) is the master torque-loop transfer function
All analysis has been performed with the forward path gain set to unity, or:
Gf = 1
For this section of analysis, the slave position-loop transfer function is modelled as:
He(s) = 1
The locked output environment model reduces to a simple equation, reference equations
2.3.6 and 2.3.7, or :
He(0) = Ke
48
The master torque-loop model was defined, reference equation 2.3.13, as a second
order transfer function:
________(OIL2______
S2 + 2Â£nO>TLS + COJL2
which reduces to unity for the steady-state condition.
Thus the steady-state effective environment stiffness is defined as follows :
Ke=Gb*Ke (3.1.8)
where:
Ke is the effective environment stiffness
Gb is the backward, i.e. feedback, path gain, and
Ke is the environment stiffness
The predicted stiffness for the ideal actuator model is the product of the environment
stiffness and the feedback path gain, or 14.9 foot-pounds / radian. The simulation
result for effective environment stiffness is 14.903.
The next set of analysis is performed to examine the coupling effects between the slave
loop and the operator loop. This is performed by opening the slave loop and closing
the operator position loop. This is found to have slight impact on the stability of the
slave loop, and causes a small change in the feedback gain. This results in an effective
environment stiffness of 14.762, with a feedback gain of 0.148.
HTL,m(s) =
(
49
When the operator is analyzed with the slave loop closed, the operator gain is found to
be modifiable to a higher value. This results from the effective load caused by the
locked-output slave actuator. The increased gain is limited so as to maintain the 30
degree phase and 6 dB gain margins on the open position-loop operator transfer
function, which results in an operator gain of 11.246.
3.1.4 Full-Up System Analysis Results
Section C.1.4 repeats most of the work in the previous section, except that the locked
output slave position transfer functions, derived in Section C.1.2, are used in place of
the ideal actuator transfer function.
The first step was to examine the loop stability of the open slave loop with the operator
loop open. By examining the open-loop frequency response, the feedback gain is
defined to satisfy the required 30 degree phase and 6 dB gain margins. This results in
the gains defined in Table 3.1.4-1
The effective environment stiffness can be predicted by examining the steady-state gain
around the slave-loop, which is presented in equation C.1.9.
Ka = Tl-m~~
Oopr(s)~~
50
Table 3.1.4-1
Locked-Output Slave-Loop Gain Values Passive
Operator
Pos BW (Hz.) Feedback Path Gain (N/A)
1.0 3.1551x10-02
2.0 2.6593x10-02
5.0 3.4651x10-02
10.0 4.3558x10-02
The steady-state locked-output slave transfer function is derived from the locked output
slave position transfer function, defined in Section C.1.2 equation C.1.16 :
i_l ________________KpH-riH.2 a)Hg)29__________
9 1 + H^tofKpHTLH^e + KvHtl + HeH^e}
where the following transfer functions are defined:
htl(s) = -ieaS-------------------------
tcind.s(s) s2 + 2CTLm.s + TL2 from equation 2.3.1,
Xout(s) Jls + Bl from equation 2.3.2, and
u 0out(s) 1
out(s) s ffom equation 2.3.3, and
He = Ke from equations 2.3.6 and 2.3.7
51
Combining like terms produces the following transfer function defining the frequency
response of the locked-output slave model:
He(s) =
_________________________KpCOjL2________________________
JlS4 + (B|_ + 2^tLTlJl) S3 + (cOtA + 2CtlCTlBl + Ke)s2 +
(ci>TL2Bl + KvC0tl2 + 2^TLTLKe)s + (KpC0n_2 + KeC0TL2)
Evaluating the transfer function at s = 0, i.e. the steady-state value, produces the
steady-state slave transfer function:
He(0) =
KP
(Kp + Ke)
where:
is the slave position error gain, and,
Ke is the environment stiffness
(3.1.9)
The locked output environment model, He, reduces to the environment stiffness, Ke,
and the master torque-loop model reduces to unity for the steady-state condition, as
shown previously.
52
Thus the steady-state effective environment stiffness is defined as follows :
g Gb Ke Kp
(Kp + Ke) (3.1.10)
where:
Ke is the effective environment stiffness
Gb is the backward, i.e. feedback, path gain,
Kp is the slave position error gain, and,
Ke is the environment stiffness
The predicted and simulation effective environment stiffness results are presented in
Table 3.1.4-2.
Table 3.1.4-2 Effective Environment Stiffness Continuous
Controller with Passive Operator
Pos BW (Hz.) Predicted Stiffness (foot-pounds / radian) Simulation Stiffness (foot-pounds / radian) Stiffness Error (percent)
1.0 1.3099 1.2914 1.41
2.0 1.9048 1.9050 -0.01
5.0 3.2149 3.2149 0.0
10.0 4.2042 4.2044 0.0 (note 1)
Notes:
1. Error percentage was less than lxlO'2.
53
The analysis is repeated with the operator position loop closed. This results in the
gains defined in Table 3.1.4-3, and the predicted and simulation effective environment
stiffness results presented in Table 3.1.4-4.
Table 3.1.4-3 Locked-Output Slave-Loop Gain Values Active
Operator
Pos BW (Hz.) Feedback Path Gain (N/A)
1.0 2.4641x10-02
2.0 1.9549x10-02
5.0 3.5886x10-02
10.0 4.7211x10-02
Table 3.1.4-4 Effective Environment Stiffness Continuous
Controller with Active Operator
Pos BW (Hz.) Predicted Stiffness (foot-pounds / radian) Simulation Stiffness (foot-pounds / radian) Stiffness Error (percent)
1.0 1.0231 1.0234 - 0.03
2.0 1.4003 1.4003 0.0
5.0 3.3295 3.3295 0.0
10.0 4.5568 4.5569 0.0 (note 1)
Notes:
1. Error percentage was less than lxlO-2.
54
3.1.5 End-to-End Model Analysis Results
The purpose of this section is to examine the end-to-end system model responses for
both the free-space and locked output slave conditions. For the ffee-space condition,
the operator gain has been defined, and the remaining work was to provide the end-to-
end step response. For the locked-output condition, modified operator position error
gains are developed. The results are provided as frequency response plots and a step
(time) response plot.
The free-space model produced the system bandwidths identified in Table 3.1.5-1
Table 3.1.5-1 End-to-End Free Space Bandwidth
Slave BW (Hz.) End-to-End System Bandwidth (Hz.)
1.0 0.5812
2.0 0.6013
5.0 0.6075
10.0 0.6086
55
The time domain characteristics are provided in Table C. 1.5-2, and are also included
here:
Time Response Data for Free-Space End-to-End System
Pos BW (Hz.) Rise Time (sec) Percent Overshoot Peak Response Settling Time Steady-State Value
1 1.036 33.99 1.3286 3.2432 0.9915
2 0.9209 36.31 1.3464 4.1241 0.9877
5 0.8759 36.78 1.3497 4.1091 0.9868
10 0.8859 36.82 1.3503 4.1191 0.9869
The locked-output analysis requires definition of the operator position error gain under
the locked-output condition. The zero margin operator gains are defined, via the root
locus technique. The gain required to meet the stability criteria for the four conditions
is defined by examining the frequency response of the open operator position loop via
the bode technique. This results in the parameters provided in Table 3.1.5-2
Table 3.1.5-2 Final Locked-Output System Parameters
Slave BW (Hz.) Operator Position Error Gain (N/A) End-to-End System Bandwidth (Hz.) Steady-State Gain (dB)
1.0 0.6098 1.5054 -17.516
2.0 0.815 1.723 -13.375
5.0 1.7627 2.408 -9.667
10.0 2.6008 3.109 -8.662
56
3.2 Hybrid Analysis Results
3.2.1 Slave Position Loop Bandwidth Analysis Results
The purpose of section C.2.1 is to define the applicability of a set of predefined
position error and velocity feedback gains, from Section C.1.1, required to achieve the
four slave position loop bandwidths (i.e. 1,2, 5, and 10 Hz.) to the available six
discrete controller sample rates (i.e. 20, 50,100, 200, 500, and 1000 sps) while
satisfying the stability margin requirements of 30 degrees phase and 6 dB gain.
If we consider a phase loss equation :
^lag = -^{Halias(bw)}, (3.2.1)
where:
^la9 is defined as the additional phase lag (in degrees),
bw is defined as the system bandwidth (in radians / second), and
^{Haiias(t0bw)} is phase lag due to the anti-aliasing filter evaluated at the
system bandwidth
then we can estimate the loss in phase margin due to the anti-aliasing filter, combined
with the knowledge of the continuous domain open-loop phase margins to define the
estimated hybrid controller open-loop phase margins. This estimate does not include
any phase loss due to transport delay, reference Appendix A. The transport delay to be
observed inside the rate loop would be principally due to three effects rate input
57
processing, torque command computation and torque command output processing in a
hardware implementation. This time is assumed insignificant for this analysis, but
would need to be analyzed and bounded for a hardware implementation.
Tables 3.2.1-la, lb, lc, and Id provide a comparison of the predicted versus the
simulation phase margins for the four slave bandwidth models of 1,2,5, and 10 Hz.
respectively.
As stated previously, this analysis utilized the position error and the velocity feedback
gains that resulted from the continuous domain analysis. A side affect from this is
variations between the continuous domain and hybrid system closed-loop bandwidths.
The variation in the closed rate-loop bandwidths are presented in Table 3.2.1-2.
Table 3.2.1-la Predicted vs Simulation Rate Open-Loop Phase
____________Margins for Hybrid 1.0 Hz. Model_________________
Sample Rate (sps) Predicted Phase Margin (degrees) Simulation Phase Margin (degrees) Phase Error
20 66.245 48.151 18.094
50 78.148 69.545 8.603
100 82.026 77.724 4.302
200 83.958 81.718 2.240
500 85.115 84.121 0.994
1000 85.501 84.951 0.550
58
Table 3.2.1-lb Predicted vs Simulation Rate Open-Loop Phase
____________Margins for Hybrid 2.0 Hz. Model ____
Sample Rate (sps) Predicted Phase Margin (degrees) Simulation Phase Margin (degrees) Phase Error
20 42.768 11.073 31.695
50 66.142 49.961 16.181
100 73.433 65.344 8.089
200 77.031 72.962 4.069
500 79.182 77.412 1.770
1000 79.899 78.945 0.954
Table 3.2.1-lc Predicted vs Simulation Rate Open-Loop Phase
Margins for Hybrid 5.0 Hz. Model_______________
Sample Rate (sps) Predicted Phase Margin (degrees) Simulation Phase Margin (degrees) Phase Error
20 -18.546 -48.843 30.297
50 35.856 5.686 30.170
100 52.689 35.098 17.591
200 60.666 51.846 8.820
500 65.376 61.338 4.038
1000 66.939 64.706 2.233
59
Table 3.2.1-Id Predicted vs Simulation Rate Open-Loop Phase
__________Margins for Hybrid 10.0 Hz. Model
Sample Rate (sps) Predicted Phase Margin (degrees) Simulation Phase Margin (degrees) Phase Error
20 -47.161 -73.766 26.605
50 17.627 -16.092 33.719
100 40.714 18.211 22.503
200 51.381 39.227 12.154
500 57.609 52.514 5.095
1000 59.671 56.929 2.742
60
Table 3.2.1-2____Hybrid Rate Loop Bandwidth Results
Model Bandwidth (Hz.) Continuous Bandwidth (Hz.) Sample Rate (sps) Hybrid Bandwidth (Hz.) Bandwidth Change (Hz.) Percentage Change
1.0 2.70 20.0 5.192 4.492 166.37
1.0 2.70 50.0 4.353 1.653 61.22
1.0 2.70 100.0 3.363 0.663 24.55
1.0 2.70 200.0 3.01 0.31 11.48
1.0 2.70 500.0 2.831 0.131 4.85
1.0 2.70 1000.0 2.785 0.085 3.15
2.0 5.509 50.0 9.677 4.168 75.65
2.0 5.509 100.0 8.664 3.155 57.27
2.0 5.509 200.0 6.944 1.435 26.05
2.0 5.509 500.0 6.116 0.607 11.02
2.0 5.509 1000.0 5.823 0.314 5.69
5.0 15.723 200.0 20.067 4.344 27.63
5.0 15.723 500.0 18.413 2.69 17.11
5.0 15.723 1000.0 17.104 1.381 8.78
10.0 22.493 500.0 25.037 2.544 11.31
10.0 22.493 1000.0 23.835 1.342 5.97
Similar to estimating the phase margin for the rate open-loop frequency response of the
hybrid model is estimating the phase margin for the position open-loop frequency
response. This estimate does not include any phase loss due to transport delay,
61
reference Appendix A. The transport delay to be observed inside the position loop
would be due to position input processing, rate command computation and the rate loop
processing, in a hardware implementation. This time is assumed insignificant for this
analysis, but would need to be analyzed and bounded for a hardware implementation.
Tables 3.2.1-3a, 3b, 3c, and 3d provide a comparison of the predicted versus the
simulation phase margins for the four slave bandwidth models of 1,2,5, and 10 Hz.
respectively. The variation in the closed position-loop bandwidths are presented in
Table 3.2.1-4.
Table 3.2.1-3a Predicted vs Simulation Position Open-Loop Phase
___________Margins for Hybrid 1.0 Hz. Model______________
Sample Rate (sps) Predicted Phase Margin (degrees) Simulation Phase Margin (degrees) Phase Error
20 67.811 71.423 - 3.612
50 71.281 72.697 - 1.416
100 72.435 73.113 - 0.678
200 73.012 72.827 0.185
500 73.358 73.488 - 0.130
1000 73.474 73.511 - 0.037
62
Table 3.2.1-3b Predicted vs Simulation Position Open-Loop Phase
____________Margins for Hybrid 2.0 Hz. Model
Sample Rate (sps) Predicted Phase Margin (degrees) Simulation Phase Margin (degrees) Phase Error
50 68.266 71.029 - 2.763
100 70.501 71.794 - 1.293
200 71.617 72.193 - 0.576
500 72.287 72.382 - 0.095
1000 72.510 72.423 0.087
Table 3.2.1-3c Predicted vs Simulation Position Open-Loop Phase
Margins for Hybrid 5.0 Hz. Model
Sample Predicted Phase Simulation Phase Phase
Rate (sps) Margin (degrees) Margin (degrees) Error
200 69.064 70.733 - 1.669
500 70.622 70.986 -0.364
1000 71.141 71.292 -0.151
Table 3.2.1-3d Predicted vs Simulation Position Open-Loop Phase
___________Margins for Hybrid 10.0 Hz. Model_____________
Sample Predicted Phase Simulation Phase Phase
Rate (sps) Margin (degrees) Margin (degrees) Error
500 65.914 66.171 - 0.257
1000 66.757 66.909 - 0.152
63
Table 3.2.1-4 Hybrid Position Loop Bandwidth Results
Model Bandwidth (Hz.) Continuous Bandwidth (Hz.) Sample Rate (sps) Hybrid Bandwidth (Hz.) Bandwidth Change (Hz.) Percentage Change
1.0 1.006 20.0 1.494 0.488 48.51
1.0 1.006 50.0 1.098 0.092 9.14
1.0 1.006 100.0 1.046 0.04 3.98
1.0 1.006 200.0 1.033 0.027 2.68
1.0 1.006 500.0 1.025 0.019 1.89
1.0 1.006 1000.0 1.008 0.002 0.19
2.0 2.003 50.0 2.662 0.659 32.90
2.0 2.003 100.0 2.214 0.211 10.53
2.0 2.003 200.0 2.108 0.105 5.24
2.0 2.003 500.0 2.039 0.036 1.79
2.0 2.003 1000.0 2.016 0.013 0.65
5.0 5.008 200.0 6.217 1.209 24.14
5.0 5.008 500.0 5.365 0.357 7.13
5.0 5.008 1000.0 5.157 0.149 2.97
10.0 10.003 500.0 13.34 3.337 33.36
10.0 10.003 1000.0 11.248 1.245 12.44
64
In summary, Table 3.2.1-5 provides the applicability of control rates to the simulation
position loop bandwidths.
Table 3.2.1-5 Simulation Control Rate Applicability
Sample Control 1.0 2.0 5.0 10.0
Rate (sps) Loop Hz. Hz. Hz. Hz.
20 rate Pass Fail Fail Fail
20 position Pass N/A N/A N/A
50 rate Pass Pass Fail Fail
50 position Pass Pass N/A N/A
100 rate Pass Pass Fail Fail
100 position Pass Pass N/A N/A
200 rate Pass Pass Pass Pass
200 position Pass Pass Pass Fail
500 rate Pass Pass Pass Pass
500 position Pass Pass Pass Pass
1000 rate Pass Pass Pass Pass
1000 position Pass Pass Pass Pass
3.2.2 Slave-Loop Analysis Results (Serial Control Architecture)
The purpose of section C.2.3 is to define the reduction in feedback gain required to
stabilize the applicable combinations of the four slave position loop bandwidths (i.e.
1,2,5, and 10 Hz.) and the available six discrete controller sample rates (i.e. 20,50,
65
100,200,500, and 1000 sps) to assure the stability margin requirements of 30 degrees
phase and 6 dB gain.
If we consider a phase loss equation:
<|>|ag 57.3*Tsamp|e C0gCf + ^{Ha|jas(t0gCf)| ^ (3.2.2)
where:
is defined as the additional phase lag (in degrees),
Tsample is the controller sample period (in seconds),
gcf is defined as the gain crossover frequency (in radians / second), and
^(Haiias(Wgcf)} js phase lag due to the anti-aliasing filter evaluated at the
system gain crossover frequency
then we can estimate the loss in phase margin due to the anti-aliasing filter and the
transport delay of the slave-loop, reference Appendix A. The effect of this phase loss
in the previous section was to eliminate the controller slave bandwidth combination.
In this section, the effect of the phase loss is cause a gain reduction such that the slave-
loop is stable.
The gain reduction can be defined by computing the difference between the estimated
phase margin and the required phase margin. The resulting error can be used to
estimate the required gain crossover frequency and, intern, the required gain reduction.
This estimate is based upon the phase and gain slopes at both the gain crossover
frequency and the reduced gain crossover frequency. The accuracy of the estimates is a
result of the accuracy of the slopes and the linearity of the slopes.
66
There is an equation related to equation 3.2.2, but defined at the phase crossover
frequency, as follows:
^lag = 57.3 Tsample*COpcf + ^(Ha|ias(pcf)) j (3.2.3)
where:
is defined as the phase crossover frequency (in radians / second), and
then we can estimate the loss in phase at the phase crossover frequency due to the anti-
aliasing filter and the transport delay of the slave-loop, reference Appendix A. The
effect of this phase loss is a reduction in the phase crossover frequency. Reducing the
phase crossover frequency causes the gain margin to decrease.
The gain reduction can be defined by computing the estimated phase crossover
frequency. The estimated gain margin is then defined as the gain margin at the
estimated gain crossover frequency. The difference between the estimated gain margin
and the required gain margin is then used to define the gain reduction. This estimate is
based upon the phase and gain slopes at both the phase crossover frequency and the
estimated phase crossover frequency. The accuracy of the estimates is a result of the
accuracy of the slopes and the linearity of the slopes.
Equations 3.2.2 and 3.2.3 were used in defining predicted slave-loop feedback gains,
based upon the discrete controller frequency and continuous domain stability
parameters for the four slave bandwidth models of 1,2, 5, and 10 Hz., respectively,
with a passive operator load. Table 3.2.2-1 provides a comparison of the predicted and
simulation results.
67
Table 3.2.2-1 Predicted vs. Simulation Slave-Loop Feedback Gains
for Hybrid Controller Passive Operator Load
Bandwidth Sample Predicted Simulation Gain
Model (Hz.) Rate (sps) Gain Gain Error
1.0 20 1.7822x10-02 1.7918x10-02 - 0.54 %
1.0 50 2.5118x10-02 2.4157x10-02 3.83 %
1.0 100 2.8152x10-02 2.7214x10-02 3.33 %
1.0 i 200 2.9803x10-02 2.8983x10-02 2.75 %
1.0 500 3.0840x10-02 3.0219x10-02 2.01 %
1.0 1000 3.1193x10-02 3.0668x10-02 1.68 %
2.0 50 1.8299x10-02 1.7913x10-02 2.11 %
2.0 100 2.2061x10-02 2.1388x10-02 3.05 %
2.0 200 2.4222x10-02 2.3655x10-02 2.34 %
2.0 500 2.5618x10-02 2.5335x10-02 1.10 %
2.0 1000 2.6101x10-02 2.5961x10-02 0.54 %
5.0 200 3.0752x10-02 2.9889x10-02 2.81 %
5.0 500 3.3035x10-02 3.2559x10-02 1.44 %
5.0 1000 3.3834x10-02 3.3502x10-02 0.98 %
10.0 500 4.0801x10-02 3.9982x10-02 2.01 %
10.0 1000 4.2157x10-02 4.1674x10-02 1.15 %
68
The effective environment stiffness can be predicted by examining the steady-state gain
around the slave-loop, including the anti-aliasing filter transfer functions, or
Â£ = = Gf Gb He(s) H(s) HT1..rn(s) Haiias(s) fWs)
0opr(s) (3.2.4)
where:
Ke is the effective environment stiffness
Gf is the forward, i.e. command, path gain,
Gb is the backward, i.e. feedback, path gain,
He(s)
is the locked output slave transfer function,
He(s) is the environment transfer function,
HTL,m(s) is the master torque-loop transfer function, and
Halias(s) is the anti-aliasing filter transfer function (2 are shown in the
equation since two filters are present in the system, reference Figure
2.4-1)
The steady-state locked-output slave transfer function was derived previously,
reference equation 3.1.9 as :
He(0) =
Kp
(Kp + K0)
The locked output environment model reduces to the environment stiffness, Ke, and the
master torque-loop model reduces to unity for the steady-state condition, as shown
previously.
69
The anti-aliasing filter transfer function was define in Section 2.3.1.1, reference
equation 2.3.4:
Halias(s) = -----------------------------
xoutisf Â§2 + 2^aliasC0aliasS + 0)a|ias
The steady-state value of this transfer function, Haiias(0), is unity. Thus, the steady-
state effective environment stiffness is as defined by equation 3.1.10:
^ Gb Ke Kp
6= (Kp + Ke)
The predicted and simulation effective environment stiffness results are presented in
Table 3.2.2-2. Tables 3.2.2-3 and 3.2.2-4 repeat the previous process for the active
operator load, i.e. the closed-loop operator model
3.3 Pipelining Analysis Results
The purpose of section C.3 is to define the reduction in feedback gain required to
stabilize the applicable combinations of the four slave position loop bandwidths (i.e.
1,2, 5, and 10 Hz.), the available six discrete controller sample rates (i.e. 20, 50,
100,200, 500, and 1000 sps), and 2, 3,4, or 5 stage pipeline controller to assure the
stability margin requirements of 30 degrees phase and 6 dB gain.
70
Table 3.2.2-2 Effective Environment Stiffness Hybrid Controller
with Passive Operator
Bandwidth Model (Hz.) Sample Rate (sps) Predicted Stiffness Simulation Stiffness Stiffness Error (percent)
1.0 20.0 0.7439 0.7447 -0.11
1.0 50.0 1.0029 1.0030 -0.01
1.0 100.0 1.1299 1.1303 -0.03
1.0 200.0 1.2034 1.2048 -0.12
1.0 500.0 1.2547 1.2553 - 0.05
1.0 1000.0 1.2733 1.2739 -0.05
2.0 50.0 1.2831 1.2896 -0.51
2.0 100.0 1.5320 1.5351 -0.20
2.0 200.0 1.6944 1.6947 -0.02
2.0 500.0 1.8147 1.8156 -0.05
2.0 1000.0 1.8596 1.8617 -0.11
5.0 200.0 2.7731 2.7766 -0.13
5.0 500.0 3.0208 3.0284 -0.25
5.0 1000.0 3.1083 3.1135 -0.17
10.0 500.0 3.8591 3.8652 -0.16
10.0 1000.0 4.0224 4.0248 -0.06
71
Table 3.2.2-3 Predicted vs. Simulation Slave-Loop Feedback Gains
for Hybrid Controller Active Operator Load
Bandwidth Sample Predicted Simulation Gain
Model (Hz.) Rate (sps) Gain Gain Error
1.0 20 1.0088x10-02 1.1246x10-02 - 11.48 %
1.0 50 1.7242x10-02 1.6177x10-02 6.18 %
1.0 100 2.0613x10-02 1.9251x10-02 6.61 %
1.0 200 2.2537x10-02 2.1825x10-02 3.16 %
1.0 500 2.3777x10-02 2.2696x10-02 4.55 %
1.0 1000 2.4205x10-02 2.4641x10-02 - 1.80 %
2.0 50 1.1364x10-02 1.1134x10-02 2.02 %
2.0 100 1.4905x10-02 1.4133x10-02 5.18 %
2.0 200 1.7070x10-02 1.6033x10-02 6.08 %
2.0 500 1.8517x10-02 1.7443x10-02 5.80%
2.0 1000 1.9026x10-02 1.8920x10-02 0.56 %
5.0 200 3.0192x10-02 2.6337x10-02 12.77 %
5.0 500 3.3490x10-02 3.0132x10-02 10.03 %
5.0 1000 3.4667x10-02 3.1869x10-02 8.07 %
10.0 500 4.4059x10-02 4.1285x10-02 6.30 %
10.0 1000 4.5733x10-02 4.3440x10-02 5.01 %
72
Table 3.2.2-4 Effective Environment Stiffness Hybrid Controller
with Active Operator
Bandwidth Model (Hz.) Sample Rate (sps) Predicted Stiffness Simulation Stiffness Stiffness Error (percent)
1.0 20.0 0.4669 0.4681 - 0.26
1.0 50.0 0.6717 0.6719 -0.03
1.0 100.0 0.7993 0.7993 0.0
1.0 200.0 0.9062 0.9078 - 0.18
1.0 500.0 0.9423 0.9445 - 0.23
1.0 1000.0 1.0231 1.0255 -0.23
2.0 50.0 0.7975 0.7976 -0.01
2.0 100.0 1.0123 1.0128 -0.05
2.0 200.0 1.1484 1.1489 -0.04
2.0 500.0 1.2494 1.2496 - 0.02
2.0 1000.0 1.3552 1.3564 - 0.09
5.0 200.0 2.4435 2.4468 - 0.13
5.0 500.0 2.7956 2.7979 - 0.08
5.0 1000.0 2.9568 2.9628 -0.20
10.0 500.0 3.9848 3.9849 0.0 (note 1)
10.0 1000.0 4.1928 4.2464 - 1.28
Notes:
1. Error percentage was less than lxlO'2.
73
The controller sample periods were chosen from the available control rates for the
applicable bandwidth model, reference Table 3.2.1-5. The number of pipeline stages is
chosen such that the slave-loop data latency was common to the sample period for a
given serial architecture system. This is done to provide an analytical basis for gain
comparisons.
If we consider a phase loss equation:
^lag 57.3 N Tsamp|e (OgCf ^ (3.3.1)
where:
is defined as the additional phase lag (in degrees),
N is the number of pipeline stages,
Tsample is the controller sample period (in seconds), and
get is defined as the gain crossover frequency (in radians / second)
then we can estimate the loss in phase margin due to the additional transport delay of
the slave-loop, reference Appendix A, caused by adding the pipeline stage(s). The
effect of this phase loss is to cause a gain reduction such that the slave-loop is stable.
The gain reduction can be defined as shown, previously, in Section 3.2.2 Again, the
accuracy of the estimates is a result of the accuracy of the slopes and the linearity of the
slopes.
74
There is an equation related to equation 3.3.1, but defined at the phase crossover
frequency, as follows:
^lag 57.3 N Tsamp|e COpcf ^ (3.3.2)
where:
pcf is defined as the phase crossover frequency (in radians / second)
then we can estimate the loss in phase at the phase crossover frequency due to the
additional transport delay of the slave-loop, reference Appendix A. The effect of this
phase loss is a reduction in the phase crossover frequency. Reducing the phase
crossover frequency causes the gain margin to decrease.
The gain reduction can be defined as shown, previously, in Section 3.2.2 Again, the
accuracy of the estimates is a result of the accuracy of the slopes and the linearity of the
slopes.
3.3.1 2-Stage Pipelining Analysis Results
The controller sample rate / bandwidth model combinations analyzed for the two stage
pipeline in Section C.3.1 are as follows:
1.0 Hz. Model: 100,200, and 1000 samples per second
2.0 Hz. Model: 100,200, and 1000 samples per second
5.0 Hz. Model: 1000 samples per second
10.0 Hz. Model: 1000 samples per second
75
Equations 3.3.1 and 3.3.2 were used in defining predicted slave-loop feedback gains,
based upon the discrete controller frequency and continuous domain stability
parameters for the four slave bandwidth models of 1, 2,5, and 10 Hz., respectively,
with a passive operator load. Table 3.3.1-1 provide a comparison of the predicted and
simulation results.
The predicted effective environment stiffness is again defined by applying equation
3.1.10 :
p Gb Ke Kp
8 (Kp + Ke)
The predicted and simulation effective environment stiffness values are presented in
Table
3.3.1- 2. Repeating the previous analysis for the active operator load condition, i.e.
operator position loop closed, provides the data presented in Tables 3.3.1-3 and Table
3.3.1- 4.
76
Table 3.3.1-1 Predicted vs. Simulation Slave-Loop Feedback Gains
____________for 2-Stage Pipeline Passive Operator Load
Bandwidth Sample Predicted Simulation Gain
Model (Hz.) Rate (sps) Gain Gain Error
1.0 100 2.5291x10-02 2.5487x10-02 - 0.77 %
1.0 200 2.7889xlO-02 2.7993x10-02 - 0.37 %
1.0 1000 3.0399x10-02 3.0416x10-02 - 0.06 %
2.0 100 1.9094x10-02 1.9481x10-02 - 2.03 %
2.0 200 2.2291x10-02 2.2424x10-02 - 0.60 %
2.0 1000 2.5692x10-02 2.5631x10-02 0.24 %
5.0 1000 3.4085x10-02 3.2963x10-02 3.29 %
10.0 1000 4.0754x10-02 4.0784x10-02 - 0.10 %
Table 3.3.1-2 Effective Environment Stiffness 2-Stage Pipeline
Controller with Passive Operator______________________________
Bandwidth Model (Hz.) Sample Rate (sps) Predicted Stiffness Simulation Stiffness Stiffness Error (percent)
1.0 100.0 1.0582 1.0596 -0.13
1.0 200.0 1.1623 1.1649 -0.22
1.0 1000.0 1.2629 1.2633 - 0.03
2.0 100.0 1.3954 1.3963 -0.06
2.0 200.0 1.6062 1.6085 - 0.14
2.0 1000.0 1.8359 1.8369 -0.05
5.0 1000.0 3.0583 3.0638 -0.18
10.0 1000.0 3.9365 3.9433 - 0.17
77
Table 3.3.1-3 Predicted vs. Simulation Slave-Loop Feedback Gains
for 2-Stage Pipeline Active Operator Load_____________________
Bandwidth Sample Predicted Simulation Gain
Model (Hz.) Rate (sps) Gain Gain Error
1.0 100 1.8381x10-02 1.7915x10-02 2.54 %
1.0 200 2.1118xlO-02 2.0757x10-02 1.71 %
1.0 1000 2.4852x10-02 2.3848x10-02 4.04 %
2.0 100 1.2702x10-02 1.2689x10-02 0.10 %
2.0 200 1.6400x10-02 1.5294x10-02 1.95 %
2.0 1000 1.9256x10-02 1.8476x10-02 4.05 %
5.0 1000 3.2429x10-02 3.1913x10-02 1.73 %
10.0 1000 4.3055x10-02 4.2964x10-02 - 0.89 %
Table 3.3.1-4 Effective Environment Stiffness 2-Stage Pipeline
Controller with Active Operator_______________________________
Bandwidth Model (Hz.) Sample Rate (sps) Predicted Stiffness Simulation Stiffness Stiffness Error (percent)
1.0 100.0 0.7438 0.7447 -0.12
1.0 200.0 0.8618 0.8635 -0.19
1.0 1000.0 0.9902 0.9915 - 0.13
2.0 100.0 0.9089 0.9096 -0.08
2.0 200.0 1.0955 1.0957 - 0.02
2.0 1000.0 1.3234 1.3245 - 0.08
5.0 1000.0 2.9609 2.9628 -0.06
10.0 1000.0 4.1469 4.1489 - 0.05
78
3.3.2 4-Stage Pipelining Analysis Results
The controller sample rate / bandwidth model combinations analyzed for the two stage
pipeline in Section 3.3.1 are as follows :
1.0 Hz. Model: 200 samples per second
2.0 Hz. Model: 200 samples per second
Table 3.3.2-1 provides a comparison of the predicted and simulation gain results. The
effective environment stiffness can again be predicted by applying equation 3.1.10.
The results of predicted and simulation effective stiffness values are presented in Table
3.3.2- 2. Repeating the previous analysis for the active operator load condition, i.e.
operator position loop closed, provides the data presented in Table 3.3.2-3, and Table
3.3.2- 4.
Table 3.3.2-1 Predicted vs. Simulation Slave-Loop Feedback Gains
for a 200 sps 4-Stage Pipeline Controller Passive
____________Operator Load__________________________
Bandwidth Predicted Simulation Gain
Model (Hz.) Gain Gain Error
1.0 2.5888xl0-2 2.6227xl0-2 - 1.31 %
2.0 1.9780x10-02 2.0333x10-02 - 2.79 %
79
Table 3.3.2-2 Effective Environment Stiffness 200 sps 4-Stage
_________Pipeline Controller with a Passive Operator
Bandwidth Model (Hz.) Predicted Stiffness (foot-pounds / radian) Simulation Stiffness (foot-pounds / radian) Stiffness Error (percent)
1.0 1.0889 1.0894 -0.05
2.0 1.4564 1.4574 - 0.07
Table 3.3.2-3 Predicted vs. Simulation Slave-Loop Feedback Gains
for a 200 sps 4-Stage Pipeline Controller Active Operator
Load
Bandwidth Predicted Simulation Gain
Model (Hz.) Gain Gain Error
1.0 1.8834x10-02 1.8842x10-02 - 0.04 %
2.0 1.3255x10-02 1.3544x10-02 - 2.18 %
Table 3.3.2-4 Effective Environment Stiffness 200 sps 4-Stage
Pipeline Controller with an Active Operator
Bandwidth Model (Hz.) Predicted Stiffness (foot-pounds / radian) Simulation Stiffness (foot-pounds / radian) Stiffness Error (percent)
1.0 0.7823 0.7837 -0.18
2.0 0.9702 0.9716 -0.14
80
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