JACKKNIFE STABILITY OF ARTICULATED TRACTOR SEMITRAILER VEHICLES WITH HIGH-OUTPUT BRAKES AND JACKKNIFE DETECTION ON LOW COEFFICIENT SURFACES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Ashley Liston Dunn, M.S.M.E.
* * * * *
The Ohio State University
2003
Ph.D. Examination Committee: Approved by Dr. Dennis Guenther , Adviser Dr. Giorgio Rizzoni, Adviser AdviserAdviser Dr. Gary Heydinger Co-Adviser Department of Mechanical Engineering
Copyright 2003
by
Ashley Liston Dunn, M.S.M.E.
ABSTRACT
This dissertation includes a detailed study of the effects on jackknife stability when using brakes that have higher than standard torque output on the prime mover. This research was initiated by the National Highway Traffic Safety Administration (NHTSA) to investigate the long-term effects of reducing the government mandated stopping distances for heavy commercial vehicles (i.e., FMVSS 121 standards). A significant reduction in stopping distance requirements would result in the need for higher torque brakes on the prime mover (tractor) steer and drive axles. The brake type might change from drum to disc, changing not only the torque output in magnitude but the dynamic transfer function characters as well.
To investigate this phenomenon, a sophisticated nonlinear model for a complete pneumatic braking system, typically found on Class 8 heavy trucks in North America, was developed. Modern brake torque measurements were used to develop brake torque output relationships. The modeled pneumatic and anti-lock braking (ABS) systems were verified over a wide variety of load/surface µ conditions using experimental data taken at
NHTSA’s Vehicle Research and Test Center. The nonlinear model was then run in
ii parallel with a proven commercial vehicle simulation package, as a hybrid computer
model.
The rigorous investigations showed no detrimental effects on jackknife stability,
with the prime mover ABS “on” or “off,” due to higher-torque brakes that were
electronically controlled (deleting the time lag of the traditional pneumatic control
system).
To signal a hypothetical vehicle stability system of impending jackknife
instability, a 15-state model was rigorously developed using Lagrange’s method, then
integrated into an Extended Kalman Filter (EKF). The assumed measured states of longitudinal slip ratios, prime mover lateral acceleration and yaw rate are used, in
conjunction with the EKF, to estimate the states of trailer hitch angle and its rate of
change. The hitch angle parameters were used to construct state phase plots, whose
output was shown to agree with sophisticated vehicle modeling programs over a range of
low-load and low-µ conditions.
iii
DEDICATION
Dedicated to my wife, Melissa, and to my family, including my mother, Blanche, for her tireless love and dedication to my success.
Also, to Michael and Malika Drexel for their supporting love and wonderful friendship.
iv
ACKNOWLEDGMENTS
I wish to express deep appreciation to Dr. Don Houser, Dr. Dennis Guenther, Dr.
Giorgio Rizzoni, and Dr. Gary Heydinger for their infinite support and patience in this
team effort. Dr. Houser has helped me since the beginning of my graduate career, in
September 1996. I am quite fortunate for the honor of being a doctoral candidate in the
Department of Mechanical Engineering at The Ohio State University.
The National Highway Traffic Safety Administration (of the United States
Department of Transportation) has funded this research in full. I am eternally grateful for
the support from the knowledgeable management, engineers, and staff at NHTSA’s
Vehicle Research and Test Center, in East Liberty, Ohio. The VRTC has been for me one of the most pleasant work environments one could ask for. Their technical support has greatly complemented the quality of this research.
Thanks also to Dr. Kamel Salaani for his efforts in keeping me focused on the goals of import, meticulous editing of my technical documents, and most importantly, stimulation of technical thought regarding this research. I am deeply grateful for the boundless and enduring support of my wonderful wife, Melissa Wolfe, who herself is a
Ph.D. candidate.
v Finally, many thanks are necessary to the very generous engineering staff at
DANA Corporation for their technical assistance in understanding the mechanisms behind torque generation for heavy commercial vehicle brakes.
vi
VITA
Ashley Liston Dunn, M.S.M.E
March 3, 1961 ...... Born – Morristown, Tennessee
June, 1986 ...... Bachelor of Science in Mechanical Engineering, North Carolina State University, Raleigh, North Carolina
June, 1999 ...... Master of Science in Mechanical Engineering, The Ohio State University, Columbus, Ohio
publications
Research Publications 1. A.L. Dunn, D.R. Houser, and T.C. Lim, “A New Metric for Rating In-Vehicle Gear Whine Levels,” Noise-Con 98, 1998.
2. A.L. Dunn, D.R. Houser, and T.C. Lim, “Methods for Researching Gear Whine in Automotive Transaxles,” SAE 99NV-167, 1999.
3. A.L. Dunn, “The Effects of Cornering Force Variation on Articulated Vehicle Predictions,” NHTSA / VRTC internal report, September 2000.
vii
4. A.L. Dunn, “Brake-In-Turn Study Comparing Disc/Drum to Drum/Drum Truck Brake Combinations,” NHTSA / VRTC internal report, 19 September 2000.
5. A.L. Dunn, “The Effects of Tire Free Rolling Cornering Properties on Analytical Predictions for Heavy Truck Roll and Yaw Predictions”, NHTSA / VRTC internal report, 24 September 2001.
6. A.L. Dunn, G.J. Heydinger, G. Rizzoni, and D.A. Guenther, “New Model for Simulating the Dynamics of Pneumatic Heavy Truck Brakes with Integrated Anti- Lock Control,” SAE 2003-01-1322, 2003.
7. A.L. Dunn, G.J. Heydinger, G. Rizzoni, and D.A. Guenther, “Methods for Modeling Brake Torque from Experimental Data,” SAE 2003-01-1325, 2003.
8. A.L. Dunn, “Heavy Truck Tire Load Transfer Sensitivity,” NHTSA / VRTC internal report, 28 March 2003.
FIELDS OF STUDY
Major Field: Mechanical Engineering
viii
TABLE OF CONTENTS
ABSTRACT...... ii DEDICATION...... iv ACKNOWLEDGMENTS ...... v VITA...... vii TABLE OF CONTENTS...... ix LIST OF FIGURES ...... xv LIST OF TABLES...... xxiv CHAPTER 1 INTRODUCTION TO THE PROBLEM OF MODELING HEAVY TRUCK BRAKING SYSTEMS AND PREDICTING JACKKNIFE STABILITY...... 1 1.1 Abstract and Motivation...... 1
1.2 Problem Background ...... 2
1.2.1 The Operation of Pneumatically Controlled Brakes on Tractor- Semitrailer Vehicles...... 2 1.2.2 Electronic Controlled Braking Systems (ECBS) ...... 3 1.2.3 Compatibility of Vehicles Equipped with ECBS and Those Equipped with Pneumatically Controlled Brakes...... 4 1.2.4 Pneumatic Disc Brakes on Class 8 Trucks in the United States...5 1.3 Intent and Scope of This Research ...... 8
1.3.1 Jackknife Instability During Brake-in-Turn (B.I.T.) Maneuvers..8 1.4 Detection of the Jackknife Event While It Occurs ...... 12
1.5 Chapter 1 References...... 14
ix CHAPTER 2 EMPIRICAL MODELS FOR COMMERCIAL VEHICLE BRAKE TORQUE FROM EXPERIMENTAL DATA...... 16 2.1 Abstract...... 16
2.2 Motivation...... 16
2.3 Model Development in General...... 17
2.3.1 Model Limits and Assumptions...... 17 2.3.2 Linear Brake Torque Model...... 19 2.4 Nonlinear Response of Some Brake Torques...... 22
2.5 Overall View of Brake Torque Test Data...... 30
2.6 Brief Discussion on Dynamic Axle Weights and Their Impact on These Models ...... 30
2.7 Addressing Brake Fade During a Stop ...... 36
2.8 Brake Torque Model in Operation...... 40
2.9 Conclusions...... 45
2.10 Chapter 2 References...... 46
CHAPTER 3 DEVELOPMENT OF AN ANALYTICAL MODEL FOR SIMULATING THE DYNAMICS OF PNEUMATIC HEAVY TRUCK BRAKES WITH INTEGRATED ANTI-LOCK CONTROL ...... 47 3.1 Abstract...... 47
3.2 Motivation...... 48
3.3 Background...... 49
3.4 Model Overview in General ...... 50
3.5 Dynamics of the Pneumatic Brake System...... 51
3.5.1 Modeling the Dynamics of the Pneumatic Brake Chambers...... 51 3.5.2 Torque Properties of the Drum and Disc Brakes...... 53 3.6 Brake System Parameters ...... 58
x 3.6.1 Brake Simulation Physical Parameters ...... 58 3.7 The Hysteresis Element of the Brake Model...... 60
3.7.1 The Influence of Hysteresis on Simulated ABS-Assisted Stopping Distance...... 63 3.8 The Integrated 4s/4m ABS Controller...... 64
3.8.1 Brief Discussion of Tire Traction Theory and Priorities of an ABS System...... 64 3.8.2 The Simulated 4s/4m ABS Controller ...... 65 3.9 The Simulation of ECBS ...... 71
3.10 Comparisons of Simulation to Experimental Data ...... 73
3.11 The Build-Hold-Build Algorithm ...... 83
3.11.1 Additional Data Results...... 87 3.12 Conclusions...... 97
3.13 Chapter 3 References...... 98
CHAPTER 4 THE EFFECTS OF USING ECBS-DISC BRAKE SYSTEMS ON THE JACKKNIFE STABILITY OF TRACTOR-SEMITRAILER VEHICLES IN BRAKE-IN-TURN MANEUVERS...... 99 4.1 Abstract...... 99
4.2 Intent and Scope of This Research ...... 100
4.2.1 Previous Study Results, Summarized ...... 101 4.3 Simulation Conditions for This Study ...... 103
4.3.1 Simulated Variable µ Levels...... 105 4.3.2 Brake Simulation Physical Parameters ...... 106 4.3.3 Simulated Brake Antilock Control...... 106 4.3.4 TruckSim™ v. 5.0 Run Screens and Files...... 108 4.3.5 Pass-Fail Criteria...... 108 4.4 Simulation Findings...... 109
4.5 State Plots and Observed Behaviors ...... 114
4.6 Conclusions...... 117
xi 4.7 Chapter 4 References...... 119
CHAPTER 5 DERIVATION AND VALIDATION OF 3-AXLE AND 5-AXLE PLANAR ARTICULATED VEHICLE MODELS...... 120 5.1 Abstract...... 120
5.2 3-Axle Planar Articulated Vehicle Model Derivation ...... 121
5.3 Kinematic Equations and Constraints...... 122
5.4 Derivation of Forcing Functions via Virtual Work Expressions ...... 132
5.4.1 Alternate Methods for Determining the Virtual Work Expressions ...... 138 5.5 The Equations of Motion for the 3-Axle Planar Model...... 141
5.6 Linearized Tire Forces...... 142
5.7 State-Space Representation of the 3-Axle Planar Cornering Model ....148
5.8 Verification of the 3-Axle Model ...... 149
5.8.1 Initial Verification...... 149 5.8.2 Verification by Comparison of the Planar Model Response with that from TruckSim™ Vehicle Dynamics Software...... 149 5.8.3 Verification of the 3-Axle Planar Model by TruckSim™ Simulation Output via Light Handling Maneuvers...... 152 5.8.4 Discussion of 3-Axle Simulation Outputs ...... 153 5.9 Adapting the 3-Axle Model to the 5-Axle Configuration...... 159
5.10 Validation of 5-Axle Model and Comparisons to the 3-Axle Model ...162
5.11 The Effects of Tire Dynamic Lag on Model Response ...... 171
5.11.1 Linear Model Response to a Jackknife-Producing Input...... 171 5.12 Development and Validation for the 5-Axle Nonlinear Model ...... 178
5.13 Performance of the 5-Axle Planar Nonlinear Model for Simulating the Jackknifing Event ...... 185
5.13.1 Successful Simulation of the Jackknife Event...... 192 5.14 Conclusions and Recommendations ...... 194
xii 5.15 Chapter 5 References...... 195
CHAPTER 6 IMPLEMENTATION OF THE EXTENDED KALMAN FILTER TO AN ARTICULATED VEHICLE MODEL FOR THE PURPOSE OF JACKKNIFE PREDICTION ...... 196 6.1 Abstract...... 196
6.2 Linear Observer Theory and How It Applies to Kalman Filters ...... 196
6.2.1 Linear State Observers...... 197 6.3 The Linear Observer as the Kalman Filter...... 201
6.4 Background...... 202
6.5 The Kalman Filter Equations and Operation Philosophy ...... 202
6.6 Extrapolation of the Kalman Filter into the Extended Kalman Filter for Nonlinear Systems ...... 207
6.7 Application of the Linear and Extended Kalman Filter to the Planar Articulated Vehicle Model...... 211
6.7.1 Stated Goals...... 211 6.7.2 Outline of the Extended Kalman Filter Algorithm ...... 211 6.8 Tuning the Extended Kalman Filter to Optimize Its Performance ...... 214
6.8.1 Comparisons of Three Estimates for Error Covariance...... 215 6.9 Various Runs – Discussion...... 228
6.9.1 Medium µ (µ=0.55), ½ GVW Load ABS ON, Resulting in a Stable Stop...... 228 6.9.2 Medium µ (µ=0.55), ½ Load ABS OFF, Resulting in a Jackknife ...... 235 6.9.3 Low µ (µ=0.30), ½ GVW Load ABS ON, Resulting in Controlled Stop...... 239 6.9.4 Low µ (µ=0.30), ½ GVW Load ABS OFF, Resulting in a Jackknife ...... 243 6.9.5 Medium µ (µ=0.55), 0 Payload ABS ON, Resulting in a Stable Stop ...... 247 6.9.6 Sensitivity of the EKF to Improper Load Estimates...... 251 6.9.7 Medium µ (µ=0.55), 0 Load ABS OFF, Resulting in a Jackknife ...... 256
xiii 6.9.8 Low-µ (µ = 0.30), 0-Load, ABS ON, Resulting in a Stable Stop ...... 260 6.9.9 EKF Sensitivity to Improper Estimation of Surface µ...... 265 6.9.10 Low-µ (µ = 0.30), 0-Payload, ABS OFF, Resulting in a Jackknife ...... 273 6.9.11 Operation of the EKF in a Double Lane Change While in a 152.4 m (500-ft) Diameter Turn, No Braking, µ = 0.55 ...... 278 6.10 Jackknife Detection Warning Lead Time Estimates...... 283
6.11 Conclusions...... 286
6.12 Chapter 6 References...... 289
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS...... 290 7.1 Summary of Fundamental Contributions to the Engineering Community ...... 290
7.2 Summary of Dissertation Topics ...... 291
7.3 Recommended Actions...... 294
7.4 Chapter 7 References...... 296
APPENDIX A TruckSim™ v.5.0 Parsfile Numbers and import / expport variables used in Chaper 5 ...... 297 APPENDIX B State-space coefficients for 3-axle planar model...... 303 APPENDIX C State-space coefficients for 5-axle planar model...... 307 APPENDIX D Runge-kutta integration routine in matlab® ...... 313 BIBLIOGRAPHY...... 315
xiv
LIST OF FIGURES
Figure 1.1 Schematic of the actuation mechanism of a pneumatically operated s-cam drum brake ...... 3 Figure 1.2 Modeled and experimental data for drive axle drum brakes. Broken lines (with icons) indicate experimental data at a simulated axle load of 21,000 lb...... 7 Figure 1.3 Modeled and experimental data for drive axle disc brakes at a simulated GAWR of 23,000 lb. Color and type schemes are identical to those of Figure 1.2...... 7 Figure 2.1 Drum brake torque measurements at 20, 50 (2 reps), and 60 mph, for a wide variety of s-cam brakes used for drive axle applications...... 18 Figure 2.2 Brake torque model output shown as a function of chamber pressure for various hub speeds. Solid lines (without symbols) indicate linear brake model output for 20, 25, 30, 40, 50, and 60 mph. Dashed lines (with symbols) indicate experimental data at an equivalent axle load of 13,200 lb for three test speeds (20, 50, 60 mph)...... 20 Figure 2.3 Comparisons of trailer axle brake torque models to experimental data. Output from the linear model is shown (with experimental data) in the top panel; output from the quadratic model (with the same experimental data) is shown in the bottom panel. The low-pressure linear fit is included in the quadratic model, but not the linear model. Dashed lines with symbols indicate the experimental data...... 24 Figure 2.4 Trailer axle drum brake model error, in percent of test output, for linear (solid lines) and quadratic (broken lines) models at 20, 50 and 60 mph. . 25 Figure 2.5 Steer axle drum brake model error, in percent of test output, for linear (indicated by solid lines) and quadratic (dashed lines) models at 20, 50, and 60 mph...... 25 Figure 2.6 Quadratic model output for drum brakes. Steer axle brake model is in the top panel; drive axle brake model is in the bottom panel. Dashed lines with symbols indicate the experimental data...... 27 Figure 2.7 Quadratic model output for disc brakes. Steer axle brake model (based on experimental data at 12,000-lb GAWR, actuated by a 20-in2 brake chamber) is in the top panel; drive axle brake model (based on
xv experimental results at 23,000-lb GAWR, actuated by a 30-in2 brake chamber) is in the bottom panel. Dashed lines with symbols indicate the experimental data...... 29 Figure 2.8 Steer axle drum brake torque measurements at 20 (top panel), 50 mph (middle panel), and 60 mph (bottom panel)...... 32 Figure 2.9 Drive axle drum brake experimental measurements 20 (top panel), 50 mph (middle panel), and 60 mph (bottom panel)...... 33 Figure 2.10 Air disc brake torque measurements at 20 (top panel), 50 mph (middle panel), and 60 mph (bottom panel)...... 35 Figure 2.11: Brake dynamometer experimental output showing step input response to the brake pressure command for a steer axle s-cam drum brake and a drive axle s-cam drum brake. Traces are brake chamber pressure (right axis) and measured torque (left axis)...... 40 Figure 2.12 Brake dynamometer simulation and experimental data: response to a step input in command pressure. Top panel: pressures, simulated command (treadle), simulated chamber, experimental chamber, with simulated speed output. Bottom panel: simulated torque output, experimental dynamometer torque output...... 42 Figure 2.13 Brake dynamometer simulation. Top panel: pressures, simulated command (treadle), simulated chamber, simulated dynamometer speed (ft/s). Bottom panel: simulated brake torque output...... 43 Figure 3.1 Schematic illustration of s-cam drum brake assembly showing crucial working parts [12]...... 53 Figure 3.2 Brake Torque model output shown as a function of chamber pressure for various application speeds. Solid lines (without symbols) indicate brake model output for 20, 25, 30, 40, 50, and 60 mph. Dashed lines (with symbols) indicate experimental data at an equivalent axle load of 20,000 lbs for three test speeds (20, 50, and 60 mph). The thicker gray line models the low-pressure torque response for all wheel speeds (Pc < 20 psi)...... 55 Figure 3.3 Time response of the pressure chamber model [top panel] and brake torque model [bottom panel] to a theoretical step input in P(treadle) at t=0 seconds to 80 psi, then back to 0 psi at t=2 seconds. Experimental dynamometer torque data are included in the bottom panel for comparison...... 57 Figure 3.4 Brake Dynamometer Simulation. Top panel: Simulated command (treadle) and chamber pressures (psi), and simulated dynamometer speed (ft/s). Bottom panel: Simulated brake torque output (lb-ft)...... 58 Figure 3.5 Dynamometer output and estimated hysteresis loop resulting from application and release of an S-cam brake...... 62 Figure 3.6 Simulation output showing the effect of hysteresis level on the response to a theoretical step input in treadle (reference) pressure...... 62 Figure 3.7 Increase in simulated stopping distance on a wet surface (broken line) and dry pavement (solid line) (µ=0.40 and µ=0.75, respectively), showing a
xvi significant effect on stopping distance resulting from adjusting the level of hysteresis in the drum brake models...... 63 Figure 3.8 Normalized lateral and longitudinal traction coefficients versus longitudinal slip ratio magnitude for a steer axle truck tire at rated load. Experimental measurements were taken at 0°, 2°, and 4° lateral slip angles...... 65 Figure 3.9 ABS control logic demonstration, showing (from top) wheel slip, “slip optimum” trigger, “slip high” trigger, wheel tangential acceleration, “accel. high” trigger, ABS logic output, and chamber pressure (Pc)...... 70 Figure 3.10 Experimental brake pressures (top panel) and longitudinal slip levels (bottom panel) for one wheel position from the steer, drive, and trailer axle. Note the successive time delays for the brake signal to reach each axle. All data have been low-pass filtered at 2 Hz...... 72 Figure 3.11 Simulation output showing treadle pressure with wheel chamber pressure (top panel), wheel slip (middle), and ABS control outputs (bottom) for steer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, loaded to GVW...... 74 Figure 3.12 Experimental data showing treadle pressure with wheel chamber pressure (top) and wheel slip (bottom) for steer axle (L). Maneuver is a straight- ahead stop from 30 mph on a wet Jennite surface, loaded to GVW...... 75 Figure 3.13 Simulation output showing treadle pressure with wheel chamber pressure (top panel), wheel slip (middle), and ABS control outputs (bottom) for the leading drive axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load...... 76 Figure 3.14 Experimental data showing treadle pressure with wheel chamber pressure (top) and wheel slip (bottom) for the leading drive axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load. 77 Figure 3.15 Showing recent testing with an unloaded tractor with 48-ft. van trailer on a wet Jennite surface, stopping from 30 mph. This experimental data shows the different control characteristics for the tractor drive axle ABS controllers (Meritor-Wabco D-Type 4s/4m)...... 78 Figure 3.16 Simulation output showing treadle pressure with wheel chamber pressure (top panel), wheel slip (middle), and ABS control outputs (bottom) for the leading trailer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load...... 80 Figure 3.17 Experimental data showing treadle pressure with wheel chamber pressure (top) and wheel slip (bottom) for the leading trailer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load...... 81 Figure 3.18 Showing recent testing with an unloaded tractor with 48-ft. van trailer on a wet Jennite surface, stopping from 30 mph. The experimental data shows the different control characteristics for the tractor and trailer ABS controllers (here, Meritor-Wabco D-Type 4s/4m, Eaton 2000 4s/2m,
xvii respectively). Note also the longer release time for the trailer brake, due to lag and higher system restriction...... 82 Figure 3.19 Simulation (left) versus experimental (right) output illustrating the build- then-hold algorithm during a stop from 30 mph (48.3 kph) on wet pavement (µ=0.375) at GVW load. Brake pressure, wheel slip, and tangential velocity are shown for the steer axle...... 84 Figure 3.20 Simulation (left) versus experimental (right) output illustrating the build- then-hold algorithm during a stop from 30 mph (48.3 kph) on wet pavement (µ=0.375) at GVW load. Brake pressure, wheel slip, and tangential velocity are shown for the drive axle...... 85 Figure 3.21 Simulation (left) versus experimental (right) output illustrating the build- then-hold algorithm during a stop from 30 mph (48.3 kph) on wet pavement (µ=0.375) at GVW load. Brake pressure, wheel slip, and tangential velocity are shown for the semitrailer axle...... 86 Figure 3.22 Simulation (left panel) and experimental data (right panel) for drive axle (L), 30 mph (48.3 kph), on wet Jennite, loaded to GVW. ABS control is via a Meritor Version C 4s/4m controller...... 88 Figure 3.23 Repeat of simulated drive axle (L), 30 mph (48.3 kph), on wet Jennite, loaded to GVW (left panel) compared to experimental data (right panel) for leading drive axle (L & R sides) at 30 mph (48.3 kph), on wet Jennite, loaded to GVW. Tractor ABS control is via a Wabco-Meritor Version D 4s/4m controller. Note that , with Wabco-Meritor controller, Pc behavior is closer to the simulation...... 89 Figure 3.24 Simulated steer axle (L), 30 mph (48.3 kph), on wet Jennite, 0 payload (left panel), compared to experimental data (right panel) for steer axle (L), 30 mph (48.3 kph), on wet Jennite, 0 payload...... 90 Figure 3.25 Simulated (left panel) lead drive axle (L), 50 mph (80.5 kph), on dry concrete, loaded to GVW compared to experimental data (right panel) for lead drive axle (L), 50 mph (80.5 kph), on dry concrete, loaded to GVW...... 91 Figure 3.26 Simulated leading semitrailer axle (left panel), 30 mph (48.3 kph), on wet Jennite, loaded to GVW compared to experimental data (right panel) for leading semitrailer axle (L), 30 mph (48.3 kph), on wet Jennite, loaded to GVW. ABS control is via a Midland-Grau (currently Haldex) 2s/1m controller...... 92 Figure 3.27 Simulated leading drive axle (left panel), 50 mph (80.5 kph), on dry concrete, loaded to GVW compared to experimental data (right panel) for leading drive axle (L), 50 mph (80.5 kph), on dry concrete, loaded to GVW...... 93 Figure 3.28 Simulink® model for the enabled subsystem representing each of the modulator controlled brake chambers. Treadle (reference) pressure and previous chamber pressure are imported. Chamber pressure is exported to the model. This module cannot be multiplexed...... 94
xviii Figure 3.29 Simulink® model for brake torque generation. Multiplexed chamber pressures are imported. Multiplexed brake torques are exported. Coefficients for brake torque calculation are adjusted for brake application speed...... 95 Figure 3.30 Simulink® model for the 2s/2m ABS controller used on the trailer tandem for the simulations. Multiplexed wheel velocities and acceleration rates, with vehicle CG longitudinal velocity and acceleration, are imported. ABS logic commands of –1 (dump pressure), 0 (hold pressure constant), or +1 (allow pressure to build following Pt) are exported to the model. .. 96 Figure 4.1 Top panel: µ-level output for the variable coefficient simulator. Bottom panel: vehicle speed in kph during the same maneuver...... 107 Figure 4.2 Close-up of variable µ computer from the simulation...... 108 Figure 4.3 Excerpt from TruckSim™ animator showing a typical jackknife event with the ABS off for the tractor and ABS on for the semitrailer. The trailer appears to remain on the tangent for the curve. Simulation: file 776, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS OFF.”...... 112 Figure 4.4 Excerpt from TruckSim™ animator showing a typical non-jackknife event with the ABS off for both the tractor and semitrailer. The entire vehicle appears to remain on the tangent for the curve. Simulation: file 775, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS OFF.”...... 112 Figure 4.5 Excerpt from TruckSim™ animator showing a typical non-jackknife event with the ABS on for both the tractor and semitrailer. The entire vehicle remained within the prescribed lane during the stop. Simulation: file 774, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS ON.” ...... 113 Figure 4.6 Another excerpt from same TruckSim™ animator file shown above, which shows a typical non-jackknife event with the ABS on for both the tractor and semitrailer. This view better shows the maintenance of lane position by the entire vehicle. Simulation: file 774, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS ON.”...... 113 Figure 4.7 Phase plane diagram showing relationships between model state variables during a maneuver in which the hitch articulation angle remained under control, i.e., no jackknife occurred. The conditions for this simulation were: full treadle brake application, no ABS (either vehicle), medium traction (µ=0.55), and air-drum configuration. The boxes on the state plots are arbitrary operation limits on the respective states...... 115 Figure 4.8 Phase plane diagram showing relationships between model state variables during a maneuver in which the hitch a jackknife occurred. The conditions for this simulation were: half treadle brake application, no ABS (either vehicle), medium traction (µ=0.55), and air-drum configuration. All state traces exceed the same limits applied to the non- jackknife example in Figure 4.6...... 116 Figure 5.1 Schematic of the 3-axle planar model as derived ...... 122 Figure 5.2 Virtual displacement of the model in the direction δψ...... 138 Figure 5.3 Virtual displacement of the model in the direction δγ ...... 139
xix Figure 5.4 Measured normalized cornering forces for a drive axle truck tire in free rolling (i.e., no braking slip) ...... 150 Figure 5.5 3-axle planar model response to sinusoidal steer input...... 155 Figure 5.6 3-axle model response to 1 degree step input...... 156 Figure 5.7 3-axle planar model response to a 2-degree step input...... 157 Figure 5.8 3-axle planar model response to a slow ramp input...... 158 Figure 5.9 Schematic of the 5-axle planar truck model...... 160 Figure 5.10 Comparison of 3- and 5-axle linear model responses to the reference for sinusoidal steer input...... 165 Figure 5.11 Comparison of 3- and 5-axle linear model responses to the reference for a 1-degree step input...... 166 Figure 5.12 Comparison of 3- and 5-axle linear model responses to the reference for a 2-degree step steer input...... 167 Figure 5.13 Comparison of 3- and 5-axle linear model responses to the reference for a 3-degree step steer input...... 168 Figure 5.14 Comparison of 3- and 5-axle linear model responses to the reference for a 4-degree step steer input...... 169 Figure 5.15 Comparison of the 3- and 5-axle linear model responses to a ramp steer input...... 170 Figure 5.16 Linear model response compared to the reference (for 2 values of tire response lag) for a ± 2-degree amplitude sinusoidal steer input...... 172 Figure 5.17 Linear model response compared to the reference (for 2 values of tire response lag) for a 2-degree step input to steer angle...... 173 Figure 5.18 Linear model response compared to the reference (for 2 values of tire response lag) for a 4-degree step input to steer angle...... 174 Figure 5.19 Linear model response compared to the reference (for 2 values of tire response lag) for a 4.5-degree step input to steer angle...... 175 Figure 5.20 State-model response to the same path-following steering input showing inaccuracies at the point of jackknife (occurs at t ≈ 6.5 seconds). The “input” and TruckSim™ comparison data are from Run #836...... 176 Figure 5.21 Comparison of linear gain (left) and linear gain with saturation (right). 177 Figure 5.22 Experimentally measured and modeled truck tire free rolling cornering force for all five axle applications at a single load. The model is labeled “combinator” in the legend...... 184 Figure 5.23 Tire lateral force at various lateral slip angles, modeled as a function of longitudinal wheel slip...... 185 Figure 5.24 Nonlinear model response to sinusoidal steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison...... 187 Figure 5.25 Nonlinear model response to 1-degree step steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison...... 188 Figure 5.26 Nonlinear model response to 2-degree step steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison...... 189 Figure 5.27 Nonlinear model response to 4-degree step steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison...... 190 xx Figure 5.28 Nonlinear model response to 4.4-degree ramp steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison...... 191 Figure 5.29 5-axle nonlinear model response to steering input that resulted in a jackknife...... 193 Figure 6.1 Schematic of a linear state observer...... 199 Figure 6.2 Flow diagram illustrating the implementation of the Kalman Filter...... 206 Figure 6.3 Schematic for the Extended Kalman Filter operation...... 210 Figure 6.4 Optimal R: initial 2.75 seconds shown for each state...... 217 Figure 6.5 Optimal R: initial 2.75 seconds shown for each state, along with model parameters and values for R...... 218 Figure 6.6 Optimal R: tire lateral slip angles forces shown for the entire run...... 219 Figure 6.7 Phase plot for case of optimal R...... 220 Figure 6.8 Low R – initial 2.75 seconds shown for each state...... 221 Figure 6.9 Low R – initial 2.75 seconds shown for each state, along with model parameters and values for R...... 222 Figure 6.10 Low R – tire lateral slip angles forces shown for the entire run...... 223 Figure 6.11 Phase plot for condition of very low R...... 224 Figure 6.12 High R...... 225 Figure 6.13 High R...... 226 Figure 6.14 High R. tire lateral slip angles forces shown for the entire run...... 227 Figure 6.15 State phase plot for very high values of R...... 228 Figure 6.16 Output showing EKF states and steering input signals...... 231 Figure 6.17 Showing model states and configuration parameters...... 232 Figure 6.18 Showing EKF states of lateral slip and tire forces. Note that the bold lines are for the EKF, and the lighter lines are of the TruckSim™ simulation...... 233 Figure 6.19 Vehicle forward speed and hitch angle phase plot, showing very good agreement in the state phase plane plot for hitch articulation angle, γ. .. 234 Figure 6.20 Steer angle input along with EKF states, showing very good agreement with the comparison TruckSim™ model...... 235 Figure 6.21 Steer angle input, model states of lateral velocity and yaw rate, along with model parameter settings...... 236 Figure 6.22 Note the good agreement for model tire lateral slip angles, αi, and tire lateral forces, Fyi...... 237 Figure 6.23 Vehicle forward speed and hitch angle phase plot. Note the good agreement in state phase plots for hitch articulation angle, γ...... 238 Figure 6.24 Steer input and model states. Note that the EKF had some difficulty tracking on this low-µ surface with the high-frequency forces of the ABS system...... 239 Figure 6.25 Same run, note change in load scale factors and slight increase in µ level used by EKF to get optimal agreement...... 240 Figure 6.26 Tire lateral slip and lateral force output from EKF and TruckSim™. Note the unusually high lateral slip angles that the EKF modeled for the trailer, in spite of the lateral forces for the trailer being modeled quite accurately.
xxi This behavior indicates that saturation in Fy has been exceeded by the semitrailer axles, i.e., they are sliding...... 241 Figure 6.27 Vehicle forward speed and state phase plot for hitch articulation angle, γ...... 242 Figure 6.28 Steer angle input and model states shown below steering input...... 243 Figure 6.29 Model states and run conditions...... 244 Figure 6.30 Lateral slip angles and tire lateral forces for both EKF and TruckSim™ models; bold lines indicate EKF output...... 245 Figure 6.31 Vehicle longitudinal speed and state phase plot for hitch angle, γ...... 246 Figure 6.32 Steer angle input and EKF model states...... 247 Figure 6.33 Model states and EKF running parameters...... 248 Figure 6.34 Tire lateral slip angles and tire lateral forces...... 249 Figure 6.35 Simulation vehicle longitudinal speed and state phase plot for hitch angle (γ)...... 250 Figure 6.36 Same run as previous, but EKF run @ ½ load instead of 0 load...... 252 Figure 6.37 Same run, but EKF run @ ½ GVW load, not 0-payload...... 253 Figure 6.38 Same run, but EKF run @ ½ GVW load, instead of 0-payload...... 254 Figure 6.39 Same run, but EKF run @ ½ -GVW load instead of proper 0-payload.. 255 Figure 6.40 Steer angle input and EKF states...... 256 Figure 6.41 Steer angle input, model states, and EKF parameters...... 257 Figure 6.42 Tire lateral slip angles and lateral forces...... 258 Figure 6.43 Vehicle forward speed and hitch angle phase plot. Note the good agreement in the state phase plot for γ...... 259 Figure 6.44 Steer angle input and EKF states...... 261 Figure 6.45 Showing steer input, model states, and EKF parameters...... 262 Figure 6.46 Tire lateral slip angles and forces...... 263 Figure 6.47 Vehicle forward speed and state phase plane plot for hitch articulation angle, γ, showing good agreement between EKF and TruckSim™...... 264 Figure 6.48 Same run, load and inertias scaled to 1.0...... 266 Figure 6.49 Same run, load and inertia scaled to 1.0...... 267 Figure 6.50 Same run, using load and inertia scales at 1.0...... 268 Figure 6.51 Similar conditions as previous, lower speed entry speed...... 270 Figure 6.52 Similar conditions, but slightly lower speed. Note the very high dynamic loading (high frequency dynamics) in the tractor tire lateral forces, as simulated by TruckSim™...... 271 Figure 6.53 Similar conditions, slightly lower speed...... 272 Figure 6.54 Steer angle input and EKF states...... 274 Figure 6.55 Model states and EKF parameter settings...... 275 Figure 6.56 Showing the EKF outputs (BOLD lines) and TruckSim™ (narrow lines) for tire lateral slip angles and forces...... 276 Figure 6.57 Vehicle forward speed and phase plane plot for hitch articulation angle, γ...... 277 Figure 6.58 Steer angle input and EKF states during a DLC while in a curve. The vertical line @ t=6 seconds indicates the onset of system instability..... 279 xxii Figure 6.59 Model states and EKF parameters for DLC in a curve...... 280 Figure 6.60 Tire lateral slip angles and forces during DLC in a curve...... 281 Figure 6.61 State phase plot and model speed...... 282 Figure 6.62 Time traces and state phase plot for hitch angle (γ) and its rate of change, showing the points of EKF detection of instability (“A”) and loss of control (“B”). The data were presented in Section 6.10...... 284 Figure 6.63 Time traces and state phase plot for hitch angle (γ) and its rate of change, showing the points of EKF detection of instability (“A”) and loss of control (“B”). The data were presented in Section 6.12...... 285
xxiii
LIST OF TABLES
Table 1.1 Initial Brake-in-Turn Simulation Study Results ...... 10 Table 2.1 Dynamic Steer Axle Loads from Best-Effort Braking Simulations ...... 34 Table 2.2 Total Work and Average Power Consumed per Brake During a Stop ..... 38 Table 2.3 Experimental Brake Fade, in Terms of Torque Decrease, for Various Brake Types ...... 41 Table 2.4 Model Fit Coefficients for the Steer Axle Brake Torque Model...... 44 Table 2.5 Model Fit Coefficients for the Drive Axle Brake Torque Model ...... 44 Table 2.6 Model Fit Coefficients for the Trailer Axle Brake Torque Model ...... 44 Table 2.7 Model Fit Coefficients for the ADB Brake Torque Model...... 44 Table 3.1 Hysteresis and Delay Time Parameters for Brake Models, ECBS-Disc and Air-Drum Configurations...... 60 Table 3.2 Brake Chamber Time Constant Values for Brake Model, ECBS-Disc and Air-Drum Configurations...... 60 Table 3.3 ABS Logic Truth Table ...... 67 Table 4.1 Initial Brake-In-Turn Simulation Study Results...... 102 Table 4.2 Simulation Entry Speeds Determined from Maximum Drive–Through Speeds ...... 104 Table 4.3 Jackknife Stability Simulation Result Matrix – Phase I: Full ABS and No ABS, Either Vehicle...... 110 Table 4.4 Jackknife Stability Simulation Result Matrix – Phase II: No ABS on Tractor, ABS Fully Functional on Semitrailer...... 110 Table 5.1 Physical Parameters Used for the 3-Axle Articulated Vehicle Model at the ½ GVW Load Condition...... 152 Table 5.2 Physical Parameters Used for the 5-Axle Articulated Vehicle Model at the ½ GVW Load Condition...... 164
xxiv CHAPTER 1
INTRODUCTION TO THE PROBLEM OF MODELING HEAVY TRUCK BRAKING SYSTEMS AND PREDICTING JACKKNIFE STABILITY
1.1 Abstract and Motivation
The motivation for this dissertation comes primarily from questions regarding heavy truck stability, rising from interest in government and private industry in reducing the stopping distances for heavily loaded commercial Class 8 trucks.
Reducing the stopping distance of a given mass at a given speed involves simply increasing the stopping forces – and therefore power dissipated during the stop. This study sought to answer two questions:
1. would significant increases in braking forces result in increased jackknife propensity for articulated tractor-semitrailer rigs, and
2. would such an event be detectible using currently available on-board vehicle dynamics measurement instrumentation, without the need to monitor articulation angle explicitly?
This extensive study shows that the simulated presence of Electronically Controlled Braking System with pneumatically actuated disc brakes (ECBS-disc) on the tractor results in no significant degradation of the performance of the rig in terms of jackknife stability while braking in a turn. Furthermore, elaborate simulations of vehicles
1 equipped with disc brakes and electronically controlled brakes systems (ECBS) show significant reduction in the tractor maximum yaw rate and hitch articulation angle.
These studies were conducted by simulating a brake-in-turn maneuver for a tractor-semitrailer at the speed corresponding to 90% of the theoretical maximum lateral acceleration in the constant radius curve that would allow a drive-through without the vehicle leaving the 3.66 m (12-ft) lane. All simulations were conducted on a 152.4 m (500 ft) radius corner with the mean friction coefficient levels set at either µ=0.30 or µ=0.55.
1.2 Problem Background
1.2.1 The Operation of Pneumatically Controlled Brakes on Tractor-Semitrailer Vehicles
The vast majority of heavy trucks on the road today have simple but proven pneumatic drum brakes. Most – or all – use a pneumatic low-flow-volume control line that delivers the command signal (from the “treadle valve” at the brake pedal) to activate the brakes. This command signal is usually terminated at each axle or tandem into a modulator valve, which allows flow of the actuating air to the brake assembly. The brake is then actuated via pneumatics from a local reservoir. One problem that exists with pneumatic control is that for today’s longer trailers, up to 16.15 m (53 ft.) in length, changes in the pneumatic brake control signal could take 400 ms or longer, even for a properly maintained system, to reach the semitrailer tandem. That delay alone translates to an additional 27 m (89 ft) covered at 96.6 kph (60 mph) before the trailer brakes – approximately 40% of the braking power on a five-axle rig – actuate. Figure 1.1 is a schematic of the most prevalent heavy truck brake used in North America today [1].
2
Figure 1.1 Schematic of the actuation mechanism of a pneumatically operated s- cam drum brake
1.2.2 Electronic Controlled Braking Systems (ECBS)
One solution to the problems presented by pneumatic control is the advent of Electronic Controlled Braking Systems (ECBS), which use electrical signals to “fire” solenoids that are located near the brake chamber, thus minimizing delays associated with the pneumatic control signal. The utility and flexibility of using electronic control of pneumatic braking in heavy trucks (ECBS) is well documented [2, 3] and has been proven by the general acceptance, from industry as well as regulatory entities, of these systems in Europe [4]. Market penetration in North America, however, remains sparse.
Although the implementation of ECBS in some tractor/trailer combinations has enjoyed good press, its full contribution to the dynamic stability of heavy trucks is yet to be realized. Benefits currently appearing in some prototypes and a few production vehicles in North America are:
1. the ability to control each brake separately for the purposes of traction control and vehicle stability control,
3 2. the ability for dynamic roll stability control, especially for double and triple semitrailer combinations [5], and
3. the ability of brake-by-wire systems to provide the vehicle operator with realistic, repeatable force feedback through the brake pedal, similar to that of a properly designed hydraulic braking system.
The research in this dissertation addresses a primary concern in the area of vehicle dynamics that accompanies the implementation of ECBS on new tractors. One question that this research was intended to answer is how the elimination of the “treadle signal” delay to the tractor, while substantial delay remains on the traditionally controlled semitrailer braking system, will affect jackknife stability for brake-in-turn maneuvers.
1.2.3 Compatibility of Vehicles Equipped with ECBS and Those Equipped with Pneumatically Controlled Brakes
Fleet operated tractors have a typical useful life of about three years, versus ten or more years for trailers. Fleet owners do not purchase or operate these rigs as single vehicles, but “mix and match” tractors and trailers of infinite variety. Therefore, fleet consumers of heavy trucks need interchangeability between “modern” ECBS-controlled tractors or semitrailers with “traditional” tractors or semitrailers, equipped with pneumatically controlled braking systems, and vice-versa.
The treadle signal, which traditionally controls the modulator valves on tractor- semitrailer rigs, is a pneumatic signal, having a pressure range of 0 to 7.58 bar (110 psi ). For an EBS system, the pneumatic signal is replaced by an electronic control signal, which varies line voltage, current, or transmission frequency to signal the brake modulator valve to control the amount of braking effort demanded by the driver and/or brake controller.
ABS control for tractor-semitrailer vehicles is currently required by law in the U.S. (the roll-in began for prime movers in March 1997) [6]. In order to be flexible for tractor-semitrailer interchangeability, ABS controls for current production tractors and semitrailers are not electronically coupled. Hence, the ABS unit for the tractor is
4 separate, and it operates separately from that for the semitrailer. The same is true for semitrailer ABS systems, in that they measure wheel speeds, estimate semitrailer velocities, and operate the brakes independently of the ABS unit on the tractor. The only ABS interconnection between the tractor and semitrailer is the ABS power circuit, which provides electrical power only to the semitrailer ABS controller. Therefore, both ABS controllers not only work independently, but can fail independently as well.
1.2.4 Pneumatic Disc Brakes on Class 8 Trucks in the United States
Although offered by some heavy truck brake manufacturers, pneumatically actuated disc brakes (ADB) have enjoyed a very slow infiltration into the North American market, now accounting for only a few percent of the brakes supplied with new prime movers. However, disc brakes are standard equipment on all Class 8 trucks currently sold in Europe [7]. Although passenger cars are federally required to come to a stop in 65.8 m (216 ft) from 96.6 kph (60 mph) [8], most can accomplish 45.7 m (150 ft) stops on dry pavement. The current requirement for heavy trucks, loaded to tractor GVW, is 108.2 m (355 ft) [9]. However, most modern rigs can stop from 96.6 kph (60 mph) within 91.4 m (300 ft). The long-term goals for NHTSA include revising the FMVSS 121 requirements such that heavy trucks will be required to stop from (96.6 kph) 60 mph within 76.2 m (250 ft), optimally as low as 61.0 m (200 ft) [10].
A significant contributor to these improvements in stopping distance compatibility between loaded tractor-semitrailers and light vehicles is the implementation of modern air disc brakes (ADB). One physical advantage of disc brakes is that the inevitable expansion resulting from thermal stress causes the ferrous brake surface to expand; this is disadvantageous for drums, taking the braking surface away from the pad, but better for discs, forcing the metal disc into the pad. Another performance improvement may result from the significant reduction in operating hysteresis seen with the less flexible air disc brake design. Recent studies have shown that for a given level of ABS-controlled braking power, a system hysteresis level of 10% of full scale capability can result in a 20% increase in wet stopping distances for a loaded truck versus a system
5 with negligible hysteresis [11]. This dissertation presents a theory suggesting that reduction of this phenomenon is an important contributor to improved ABS-controlled stopping distances.
Manufacturers of commercially available disc brakes tout a potential increase in effective braking torque of 20% over existing drum brakes. Other manufacturer’s claims include:
1. Meritor claims 16% shorter stopping distance (40 ft. off of a 250-ft. stop) with their “disc Plus” disc brake system,
2. The Mercedes-Benz Actros claims a 30% improvement in stopping performance due to the use of a prototype ECBS/disc system at all brake positions, and
3. Bendix claims a 10% weight savings and 30% stopping distance improvement using their disc brakes, as well as 1,000,000-mile disc life and 500,000-mile pad life for highway use [12].
My study at NHTSA’s Vehicle Research and Test Center (VRTC) of contemporary heavy truck brakes revealed that the performance gains from disc brakes are not necessarily from the availability of significantly higher ultimate torque, but instead from the consistency of their brake torque generation with respect to application speed. Figures 1.2 and 1.3 illustrate the differences in brake torque generation for pneumatic drums and ADB, respectively. The drive axle drum brakes shown in Figure 1.2 show much more sensitivity to application speed than do the drive axle disc brakes shown in Figure 1.3. For the drum brake, the slope of the relationship of brake torque
(Tb) with respect to chamber pressure (Pc) decreases significantly, as does the maximum torque, with increasing application speed. The same is not true for the ADB results shown in Figure 1.3. Further information on these evaluations and the models resulting from them can be found in [13]. Other concerns regarding disc brake systems include initial and maintenance costs, which will eventually improve due to economies of scale.
The analytical models used in this study have been in use at NHTSA since 2000, when Ed Milich reported on the correlation between elaborate rigid-body simulation program output and experimental vehicle data for a broad series of maneuvers [14]. 6
Figure 1.2 Modeled and experimental data for drive axle drum brakes. Broken lines (with icons) indicate experimental data at a simulated axle load of 21,000 lb.
Figure 1.3 Modeled and experimental data for drive axle disc brakes at a simulated GAWR of 23,000 lb. Color and type schemes are identical to those of Figure 1.2.
7 1.3 Intent and Scope of This Research
1.3.1 Jackknife Instability During Brake-in-Turn (B.I.T.) Maneuvers
The purpose of the B.I.T. study was two-fold. The first goal was to develop advanced analytical models for electro-pneumatic disc braking systems (“ECBS-disc”) and traditional pneumatically controlled drum braking systems (“air-drum”) that surpass current public domain models for accuracy in simulating dynamic brake behaviors. The second goal was to use these in-house developed brake system models, in parallel with a sophisticated vehicle simulation package, to analyze the potential impact of advanced high-torque ECBS-disc equipped tractors on the jackknife stability of tractor-trailer rigs.
The vehicle simulation package used for this research was TruckSim™ version 5.0, by Mechanical Simulation Corporation of Ann Arbor [15]. The TruckSim™ software is a nonlinear solver that treats the vehicle chassis, suspension, and drivetrain masses as rigid bodies. This software package uses linear and nonlinear force and moment relationships to simulate the applied forces to the vehicle and internal forces between the vehicle components. The TruckSim™ software simulates the dynamics of the vehicle, including highly nonlinear aspects such as the tire force models, suspension deflection models, leaf spring models, and the hitch model. The brake system dynamics, brake torque outputs, and brake hysteresis were modeled with a nonlinear Simulink® model developed for use within NHTSA’s Vehicle Research and Test Center (VRTC) [16]. The brake simulation software received the brake system command from the vehicle simulation, then calculated brake chamber pressures and torques accordingly. The brake simulation software also simulated a 4s/4m ABS system for the tractor, and a separate 2s/2m system for the semitrailer.
The term “4s/4m” refers to the number of sensors (4) and modulator valves (also 4 for this system) that the ABS system uses to sense wheel speed, and then control brake pressure. A typical “4s/4m” system senses the front wheel speeds (2) and the leading drive axle hub speeds (2), for a total of 4 sensed wheel speeds. The designation
8 following the slash refers to the four modulators, which in this configuration control the steer axle brake chambers independently (requiring two modulators), then control each side of the drive tandem independently (requiring the other two modulators). Thereby both the leading and trailing axle brake position on each side of the drive tandem is controlled via one modulator valve.
This study, therefore, compares the ECBS-disc equipped tractor to an otherwise identical air-drum tractor, each coupled to the same simulated air-drum equipped semitrailer. The rigs are evaluated for jackknife stability under brake-in-turn maneuvers. Brake actuation is simulated to occur at either the onset of negotiating a curve, or after the vehicle is fully established in the curve, depending on simulation setup. The test parameters were derived by analyzing a prior study that used similar vehicle comparisons on simulated wet and dry surfaces at load conditions of GVW and no payloads [17]. The previous study concluded that:
1. disc brake equipped tractors should not exacerbate jackknife problems for tractor-semitrailer rigs in brake-in-turn maneuvers, regardless of ABS functionality, and
2. for disc or drum equipped tractors, the combination of vehicle configuration / simulation condition that proved most troublesome for jackknife stability was that of vehicles with low trailer loads maneuvering on low coefficient (µ) pavement surfaces.
Table 1.1 shows the results of the initial study.
9 No Load GVW Load Hi µ Low µ Hi µ Low µ Drum Disc / Drum Disc / Drum Disc / Drum Disc / / drum drum / drum drum / drum drum / drum drum Full ABS Slow ABS No
Half Treadle Half Treadle ABS Full ABS Slow ABS No
Full Treadle ABS
Indicates near jackknife (high hitch articulation angle), and/or high hitch forces Indicates jackknife
Table 1.1 Initial Brake-in-Turn Simulation Study Results
10 The following are explanations of the terms used in Table 1.1.
“Full ABS” indicates functioning ABS for both tractor and semitrailer, as per the vehicle simulation software, TruckSim™ version 4.6.
“Slow ABS” indicates simulated partial failure of the ABS system, on both the tractor and semitrailer, by increasing the slip threshold at which the ABS signals the brake chamber to dump pressure, and increasing the slip threshold at which the ABS signals the brake chamber to allow pressure to increase again.
“No ABS” indicates no simulated ABS operation on either the tractor (prime mover) or the semitrailer.
“Full Treadle” and “Half Treadle” correspond to the magnitude of control line pressure demanded by the driver after the vehicle was fully established in the 152.4 m (500 ft) radius corner. It is the experience and observation of this researcher that not all drivers apply full braking power after realizing that they may be going too fast for the traction level in a corner. Slow, cautious brake application is arguably the best tactic under some emergency situations.
“Drum/drum” refers to the traditional configuration of air-drum brakes on the tractor, and air-drum brakes on the trailer. “Disc/drum” refers to ECBS-disc brakes on the tractor, and air-drum brakes on the trailer.
Although experimental studies with disc brake equipped tractors and semitrailers are ongoing at NHTSA, the simulations conducted as part of this dissertation were intended to expand upon the limited number of test conditions available to test engineers. Not only is this dissertation interested in the effects that significantly increased braking torque would have on tractor-trailer stability (i.e., Will the increase in braking force on the tractor make it more susceptible to jackknifing due to higher kingpin force magnitudes?), but how will controllability be influenced by the response time improvements promised with ECBS and disc brakes?
11 1.4 Detection of the Jackknife Event While It Occurs
Detection – or even prediction – of the jackknife as it occurs would be quite useful for an on-board vehicle stability system, which could use the incoming information about an impending jackknife to initiate execution of a procedure to correct for the instability. Such on-board vehicle stability systems are becoming quite common in many passenger vehicles at the time of this dissertation. Vehicle stability systems currently available on production passenger vehicles mitigate vehicle yaw instability by manipulation of the vehicle’s brakes (independently) and/or throttle. A fully operational system for heavy Class 8 trucks is not only inevitable, but certain to save fleet owners millions of dollars in repair, medical, and legal costs each year.
Detection of the jackknife event would be quite easy if hitch articulation angle could be directly measured. However, such measurements become difficult when employing expensive and delicate angular measurement systems. More difficulty is encountered when the engineer tries to find a way to have the device fit and work on many different tractor-semitrailer rigs, without causing great inconvenience to the driver or maintenance crews.
A more elegant solution is to estimate hitch angle and rate, using other state variables that we can assume are measured by a vehicle stability / ABS system. In Chapter 5, a complete planar model for an articulated vehicle is developed “from the ground up,” then linearized to produce a four-state constant-speed model, having the states of:
v = lateral velocity
r = yaw rate
q = hitch articulation angle
γ = hitch articulation rate
12 The four-state model serves as the basis for a far more elaborate fifteen-state model that is necessary to accurately model the vehicle dynamics during the onset of instability.
The Kalman Filter [18] has proven to be a popular and very flexible state observer since its introduction by Kalman in 1961 [19]. Further utility with highly nonlinear systems exists in the Extended Kalman Filter, which expands the linear Kalman Filter equations into differential equations that are integrated with each time step. Inspiration for this approach, as well as a thorough discussion on the history and implementation of the Kalman and Extended Kalman Filters are covered by Chrstos in [20].
Chapter 6 of this dissertation discusses the development of an Extended Kalman Filter for the purpose of estimating hitch articulation angle and rate, based on state measurements that are assumed to be available in any vehicle stability system – steer angle (the input), vehicle lateral acceleration, and vehicle yaw rate.
13 1.5 Chapter 1 References
1. Figure from D. Yanakiev, J. Eyre, I. Kannellakopoulos, “Longitudinal Control of Heavy Duty Vehicles: Experimental Evaluation,” California PATH research report, UCLA Electrical Engineering, 1998. 2. Scania Trucks press release, (www.scanio.com/ms/events/press/wwwtxt/n97111en.htm). 3. Heavy Truck Magazine, Roemer Insurance, Inc., (http://www.roemer- insurance.com/rr22000.html). 4. “Technology Review for Electronically Controlled Braking Systems,” NHTSA, 1998. 5. “Mechanics of Heavy Truck Systems,” short course material, University of Michigan Transportation Research Institute (UMTRI), July 2000. 6. Q&A: ANTILOCK BRAKES: CARS, TRUCKS, MOTORCYCLES, Insurance Institute for Highway Safety, (http://www.hwysafety.org/safety_facts/qanda/antilock.htm). 7. Land Line Magazine (April 11, 2000). 8. FMVSS 135 braking standards, Oct. 2001. 9. FMVSS 121 braking standards, Oct. 2001. 10. NHTSA Vehicle Safety Rulemaking priorities 2002-2005, (http://www.nhtsa.dot.gov/cars/rules/rulings/PriorityPlan/Index.html#secIV). 11. A.L. Dunn, G. J. Heydinger, G. Rizzoni, and D. Guenther, “New Model for Simulating the Dynamics of Pneumatic Heavy Truck Brakes with Integrated Anti- Lock Control,” SAE 2003-01-1323. 12. Bendix Truck Brake System Advertisement, (http://www.bendix.com/products/SellSheet?p=ADB225). 13. A.L. Dunn, G. J. Heydinger, G. Rizzoni, and D. Guenther, “Empirical Models for Commercial Vehicle Brake Torque from Experimental Data,” SAE 2003-01-1325. 14. E. Milich, “An Evaluation of VDM-Road and VDANL Vehicle Dynamics Software for Modeling Tractor-Trailer Dynamics” (master’s thesis, The Ohio State University, 1999). 15. TruckSim version 5.0 User Manual (Ann Arbor, MI: Mechanical Simulation Corporation, 2003); and (http://www.trucksim.com). 16. A.L. Dunn, “Simulink Heavy Truck Brake Model Simulation Manual,” NHTSA / VRTC internal report, 2003.
14 17. A.L. Dunn, “Brake-In-Turn Study Comparing Disc/Drum to Drum/Drum Truck Brake Combinations,” NHTSA / VRTC internal report, 19 September 2000. 18. G. Welch and G. Bishop, “An Introduction to the Kalman Filter,” TR95-041, Department of Computer Science, University of North Carolina at Chapel Hill. 19. R.E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” ASME Journal of Basic Engineering, 1960. 20. Chrstos, Jeffrey P., “Use of Vehicle Dynamics Modeling to Quantify Race Car Handling Behavior” (Ph.D. Dissertation, The Ohio State University, 2000).
15 CHAPTER 2
EMPIRICAL MODELS FOR COMMERCIAL VEHICLE BRAKE TORQUE FROM EXPERIMENTAL DATA
2.1 Abstract
This chapter introduces a new series of empirical mathematical models developed to characterize brake torque generation of pneumatically actuated Class-8 vehicle brakes. The brake torque models, presented as functions of brake chamber pressure and application speed, accurately simulate steer axle, drive axle, and trailer tandem brakes, as well as air disc brakes (ADB). The contemporary data that support this research were collected using an industry standard inertial brake dynamometer, which is routinely used for verification of FMVSS 121 commercial vehicle brake standards.
2.2 Motivation
The goal for these analyses and model development was to accurately characterize brake torque versus chamber pressure and application speed at the hub, for use with multi-body vehicle dynamics simulations. Information on the topic of pneumatic brake torque output has not been published in this form since 1975 [1]. Since then, new developments, such as improved lining materials and air disc brakes (ADB), along with the availability of numerous advanced PC-based vehicle dynamics modeling 16 programs, have resulted in a renewed demand for accurate, publicly available information regarding brake torque output of pneumatically operated commercial vehicle brakes.
The data discussed herein are used to develop empirical mathematical models, which are implemented in the form of multiple quadratic relationships that relate torque output to a real-time dynamic brake chamber pressure model and tire tangential speed. These torque outputs are part of newly developed nonlinear mathematical models that use a systems modeling approach to simulate a pneumatic commercial heavy vehicle braking system (discussed in Chapter 3 and [2]). The newly developed brake system models covered in Chapter 3 and [2] use the torque relations discussed here, as well as additional relevant source information regarding hysteresis and response lag times inherent in the actuation of pneumatic-over-mechanical Class-8 vehicle brakes [1, 3-5].
2.3 Model Development in General
2.3.1 Model Limits and Assumptions
These models were intended to very accurately describe the dynamic behavior of commercial braking systems during stops when the temperature limits dictated in the FMVSS 121 test procedures were not exceeded. Also, these models were developed from experimental brake dynamometer data at application speeds from 20 to 60 mph. Hence, valid simulation results cannot be guaranteed when tire tangential speeds significantly exceed this speed range.
As seen in Figure 2.1, different linings or test conditions produce vastly different brake torque (Tb) versus chamber pressure (Pc) relationships. For each condition of speed
and test load (inertia), the curves representing Tb versus Pc can be estimated using a number of methods, such as a linear, quadratic, third order, or an exponential approximation.
17 Drive Axle Air Drum Brake Torques by Speed for Linings 'A,B,C,D,E' 25000
20000
15000
10000 Torque (ft-lbs)
5000
0 020406080100120 chamber pressure (psi) 20 mph 50 mph 60 mph Lining E 26k GAWR drive-1 20 mph 50 mph 60 mph Lining D 29k GAWR drive-2 20 mph 50 mph 60 mph Lining C 22k GAWR drive-3 20 mph 50 mph 60 mph Lining B 21k GAWR drive-3 20 mph 50 mph 60 mph Lining B 20k GAWR drive-4 20 mph 50 mph 60 mph Lining B 20k GAWR drive-4 20 mph 50 mph 60 mph Lining A 20k GAWR drive-3
Figure 2.1 Drum brake torque measurements at 20, 50 (2 reps), and 60 mph, for a wide variety of s-cam brakes used for drive axle applications.
For the purpose of modeling brake torque versus chamber pressure and speed, experiments conforming to the “Brake Retardation Force test” (NHTSA §5.4.1.1, FMVSS 121 standards) were conducted from initial speeds of 20, 50, and 60 mph (32.2, 80.5, and 96.6 kph). Brake chamber pressure was applied at a constant value of 20 psi (1.38 bar), and brake torque output measured, and averaged, during the stop. The stops were repeated as the constant chamber pressure was increased in 10-psi increments, up to 100 psi. Also, data from a modified version of the FMVSS “Brake Retardation Force test,” having an initial brake chamber pressure of 7 psi (0.48 bar), then 10 psi (0.69 bar), followed by 10-psi increments to 100 psi (6.89 bar), were used to verify the data at 50 mph (80.5 kph). The nature by which the supporting data were gathered should make it clear that these models are not intended for use in simulating the brake fade phenomenon
18 that occurs during long descents or repeated hard stops that would cause brake temperatures to exceed those mandated in FMVSS 121.
2.3.2 Linear Brake Torque Model
In previous formulations of the quasi-static relationship of torque to brake chamber pressure and speed, the models were postulated from experimental data to have
piecewise linear relationships of brake torque (Tb) versus chamber pressure (Pc) [1, 6]. The nominal slope of these relationships differed for each brake application speed, and generally decreased with increasing speed.
Beginning with steer axle brake torques, various mathematical models were applied, then compared, to determine which model most efficiently provided an acceptable estimate of the available experimental data. For the steer axle brakes, the relationship between Tb and Pc was relatively linear in nature, and a series of linear
equations were derived using linear regression of Tb with respect to Pc. at each application speed. With the assumption that the linear coefficients for each line fitting a
Tb versus Pc relationship at a single application speed (i.e., tire-wheel tangential speed at which the brake was applied) may have a quantifiable relationship to brake application speed, an empirical mathematical model was developed using linear regression to
describe the interaction of the coefficients of each Tb versus Pc linear relationship, as
functions of speed. For the steer axle brakes, the linear model of Tb versus Pc, whose coefficients vary linearly with respect to application speed, provided good results, with model error being within 6% over the entire range of application pressures. This linear model response is co-plotted in Figure 2.2 with actual brake torque data.
19 Steer Axle Air Drum Linear / Linear Brake Torque vs. Pressure for Various Speeds
9000
Vhub = 20 mph Vhub = 25 mph 8000
Vhub = 30 mph
7000 Vhub = 40 mph
Vhub = 50 mph 6000
Vhub = 60 mph 5000
4000
Brake Torque (ft-lb) Torque Brake 3000
2000
1000
0 0 20406080100 Pc = chamber pressure (psi)
20 25 30 40 50 60 20 mph test 50 mph test 60 mph test
Figure 2.2 Brake torque model output shown as a function of chamber pressure for various hub speeds. Solid lines (without symbols) indicate linear brake model output for 20, 25, 30, 40, 50, and 60 mph. Dashed lines (with symbols) indicate experimental data at an equivalent axle load of 13,200 lb for three test speeds (20, 50, 60 mph).
The linear torque versus pressure relationships for the entire pressure operating range are expressed in equation (2.1). For the pressure range of [7 < Pc ≤ 20 psi], torque output for all application speeds has been modeled using a single linear model between 0 psi and the convergent speed-averaged torques at 20 psi (P0). Brake torque is usually negligible below the “push-out pressure” (Ppo, usually around 7 psi), where brake chamber motion is prevented by stiction.
20 ≤≤ ⎧⎫0 0 PPcpo ⎪⎪ ⎪⎪TbP =<≤0 TPbc⎨⎬ PPP poc0 (2.1) ⎪⎪P0 ⎪⎪+<≤ ⎩⎭CCP01ccres PPP 0
TCP==+C TPP ( ) (2.2) bP0 010 b c 0
=+ ≤ Ccii01 cV ihub PP c po ≤≤ (2.3) 20Vmphhub 60
For equations (1-3):
Tb = torque available at the brake
C0, C1 = coefficients for linear torque relationship
Pc = brake chamber pressure (psi)
Pres = brake pneumatic pressure reservoir pressure, taken as constant, usually under 110 psi.
P0 = pressure at which the torque relationships converge, with respect to speed, here P0 = 20 psi.
TbPo = speed averaged torque at P0
Ppo = pop-out pressure, here Ppo = 7 psi. cij = speed regressed coefficients, i.e., Ci = f[cij, Vhub]
Vhub = tangential wheel speed (mi/hr)
21 2.4 Nonlinear Response of Some Brake Torques
After application of the aforementioned linear model to the trailer brake torque outputs, the agreement was less than satisfactory. In fact, for the trailer brakes for which information was available, the Tb versus Pc relationship was far less linear, and actually more quadratic in character.
Applying the procedure described above to a higher degree of equation, the Tb versus Pc relationship was first modeled at each test speed (20, 50, and 60 mph) as a
quadratic function having a non-zero offset at Pc = 0. The three coefficients for each
“constant-application-speed” Tb versus Pc relationship model were then trended with respect to speed, resulting in what is called here a quadratic model. The trailer brake models, showing both the linear and the quadratic relationships for comparison, are presented in Figure 2.3.
The quadratic model is described in equations (2.4) through (2.6), and is similar to equations (2.1) through (2.3) except for the additional higher degree term. Comparison of the two panels in Figure 2.3 shows the improvement in the model accuracy (at the speeds for which data exist) when the quadratic model is employed, as opposed to a linear model or combinations of “linear /quadratic” models (not shown). Note that the phrase “quadratic / quadratic” refers to both sets of the empirical model coefficients being modeled as quadratic responses (of pressure, then speed) instead of linear. The description “quadratic/quadratic” is used here interchangeably with “quadratic” for simplicity. The same usage applies to “linear/linear” models.
2 =≤j adjust only if : CcVi∑ ij hub PP c po (2.4) j=0
22 ⎧⎫ ⎪⎪0 0 ≤≤PP ⎪⎪cpo ⎪⎪T =<≤bP0 TPbc⎨⎬ PPP poc0 (2.5) ⎪⎪P0 ⎪⎪2 i <≤ ⎪⎪∑CPic( ) P0 P c P res ⎩⎭i=0
2 TCP==( )i TPP ( ) (2.6) bP0 ∑ i00 b c i=0
Consistent with figures comparing modeled versus experimental brake torque in this report, the dashed lines in Figure 2.3 represent brake dynamometer output (for the specific brake and lining type modeled) at 20, 50, and 60 mph. The solid lines represent the model output at 20, 25, 30, 40, 50, and 60 mph. On the bottom panel, the quadratic nature of the Tb versus Pc relationship for the trailer brakes is clearly shown.
Figure 2.4 contains the error curves (model error versus Pc) for both the linear and quadratic models for the trailer brakes. Error for the best fit linear model (shown using solid lines) ranges from –15% at low pressure to +7% at medium chamber pressure. The error for the quadratic model (shown using dashed lines) is much friendlier, at ±2.5%.
A quadratic/linear model (where the coefficients of the quadratic relationship between Tb and Pc are modeled linearly with speed) was tried, but the model fit did not improve satisfactorily (error was generally 5%, up to 15% for some speed-chamber pressure combinations).
23 Lin e ar M odels for T raile r Axle Air D rum B rake T orque v s. Pre ssure for Various S pe ed s
14000
12000
10000
8000
6000 Brake Torque (ft-lbi) 4000
2000
0 020406080100 Pc = chamber pre ssure (psi)
20 25 30 40 50 60 20 m ph test 50 mph test 60 m ph tes t
Quad ratic /Qua dratic M od els fo r T ra iler Axle Air D rum B rake To rq ue v s. P res sure for V ario us Sp eeds
14000
12000
10000
8000
6000 Brake Torque (ft-lbi) 4000
2000
0 020406080100 Pc = chamber pressure (psi)
20 25 30 40 50 60 20 m ph tes t 50 mph test 60 mph test 0-20 psi, all speeds
Figure 2.3 Comparisons of trailer axle brake torque models to experimental data. Output from the linear model is shown (with experimental data) in the top panel; output from the quadratic model (with the same experimental data) is shown in the bottom panel. The low-pressure linear fit is included in the quadratic model, but not the linear model. Dashed lines with symbols indicate the experimental data.
24 Quadratic and Linear Model Error Plot for Trailer Axle Brake (sheet: trailer_6)
10.0%
5.0%
0.0%
-5.0% Brake Torque ERROR (%) Brake Torque ERROR -10.0%
-15.0% 0 20406080100 Pc = chamber pressure (psi)
quad-20 quad-50 quad-60 linear-20 linear-50 linear-60
Figure 2.4 Trailer axle drum brake model error, in percent of test output, for linear (solid lines) and quadratic (broken lines) models at 20, 50 and 60 mph.
Quadratic and Linear Model Error Plot for Steer Axle Drum Brake for Steer Axle @ 13.2 k# GAWR
6.0%
4.0%
2.0%
0.0% 0 20 40 60 80 100 Brake Torque (%)) ERROR -2.0%
-4.0%
-6.0% Pc = chamber pressure (psi)
quad-20 quad-50 quad-60 linear-20 linear-50 linear-60
Figure 2.5 Steer axle drum brake model error, in percent of test output, for linear (indicated by solid lines) and quadratic (dashed lines) models at 20, 50, and 60 mph.
25 Figure 2.5 shows the improvement in model error when the steer axle model was converted to a quadratic model. The error magnitude for the steer axle drum brakes dropped from a high of over 6% for the “linear/linear” model to 2.5% for the “quadratic/quadratic” model. The improvement is significant despite the much more linear character of the steer axle brakes than that of the trailer brakes.
The results displayed in Figures 2.3, 2.4, and 2.5 show that the “quadratic/quadratic” model provides superior fit, as compared to the “linear/linear” model. The “quadratic/quadratic” model was therefore applied to each of the brake types analyzed, including drum brakes for steer (15” diameter by 4” wide drum – 381 x 102 mm), drive, and trailer axles (both drums at 16.5” x 7” – 419 x 178 mm), and 430 mm (16.9 inch) diameter disc brakes for steer and drive axles. Note that air disc brake models were regressed from data taken under two vastly different applications. To simulate use on the steer axle, the brake was tested at an equivalent gross axle weight rating (GAWR) of 12,000 lb (5,443 kg) and applied using a 20-in2 (129.0-cm2) brake chamber; for drive axle applications, the brake was tested at 23,000 lb (10,433 kg) GAWR and applied using a 30-in2 (193.5-cm2) brake chamber. An additional benefit of using one model for all brake types was convenience of implementation. Hence, the standard model equations can be applied to any brake type for a tractor-semitrailer vehicle, by varying only the nine coefficients for different brake types.
The two panels in Figure 2.6 show the quadratic model output for drum brakes on the steer and drive axles. The quadratic steer axle brake model is shown in the top panel; the quadratic drive axle brake model is shown in the bottom panel.
26 Quadratic /Quadratic Models for Steer Axle Air Drum Brake Torque vs. Pressure for Various Speeds (13.2 k# GAWR)
14000
12000
10000
8000
6000 Brake Torque (ft-lbi)
4000
2000
0 0 20406080100 Pc = chamber pressure (psi)
20 25 30 40 50 60 20 mph test 50 mph test 60 mph test 0-20 psi, all speeds
Quadratic /Quadratic Models for Drive Axle Air Drum Brake Torque vs. Pressure Model for Various Speeds
14000
12000
10000
8000
6000 Brake Torque (ft-lbi)
4000
2000
0 020406080100 Pc = chamber pressure (psi) 20 25 30 40 50 60 20 mph test 50 mph tes t 60 mph test 0-20 psi, all speeds Figure 2.6 Quadratic model output for drum brakes. Steer axle brake model is in the top panel; drive axle brake model is in the bottom panel. Dashed lines with symbols indicate the experimental data.
27 The two panels in Figure 2.7 show the output from the quadratic model for disc brakes. A disc brake model for the steer axle (based on experimental data taken at 12,000 lb GAWR, using a 20-in2 chamber) is shown in the top panel; a disc brake model for the drive axle (based on experimental data taken at 23,000 lb GAWR, using a 30-in2 chamber) is shown in the bottom panel. Note in Figure 2.7, that although the ultimate torque output (at Pc = 100 psi) for the disc brake (drive axle application) is no higher than that of the drive axle drum brakes of equivalent power capacity (both are near 14,000 lb- ft – 18.98 kN), sensitivity of the torque-pressure slope to hub speed is vastly reduced with the disc brake. Furthermore, it might be appropriate to apply a single second-order model for all speeds of the 23,000-lb GAWR disc brake application.
28 Quadratic /Quadratic Models for Air DISC Brake Torque vs. Pressure for Various Speeds on the Steer Axle (12 k# GAWR)
14000
12000
10000
8000
6000 Brake Torque (ft-lbi)
4000
2000
0 020406080100 Pc = chamber pressure (psi)
20 25 30 40 50 60 20 mph test 50 mph test 60 mph test 0-20 psi, all speeds
Quadratic /Quadratic Models for Air DISC Brake Torque vs. Pressure for Various Speeds on the Drive Axle (23 k# GAWR) 14000
12000
10000
8000
6000 Brake Torque (ft-lbi) 4000
2000
0 020406080100 Pc = chamber pressure (psi)
20 25 30 40 50 60 20 mph test 50 mph test 60 mph test 0-20 psi, all speeds
Figure 2.7 Quadratic model output for disc brakes. Steer axle brake model (based on experimental data at 12,000-lb GAWR, actuated by a 20-in2 brake chamber) is in the top panel; drive axle brake model (based on experimental results at 23,000-lb GAWR, actuated by a 30-in2 brake chamber) is in the bottom panel. Dashed lines with symbols indicate the experimental data.
29 2.5 Overall View of Brake Torque Test Data
Figures 2.8, 2.9, and 2.10 show torque outputs for ADB, steer axle drum, and drive axle drum brakes, respectively. Although the amount of information can be overwhelming, the figures are presented to illustrate the vast possibility of torque outputs from a properly functioning brake assembly. It can be concluded that the brake size, air chamber (actuator) size, and pad lining material have the most significant effects (at a given speed) on torque output, whereas the assembly type produces secondary effects on torque output for a new, properly operating brake assembly. Figure 2.1 is a compilation of the drive axle drum brake data, and again is presented only to illustrate the spectrum of response character and magnitude available for the same size and type of brake. In the interest of brevity, the trailer brake output is not presented in this form. Note that Figures 2.1, 2.8, 2.9, and 2.10 show two individual tests for the same conditions at 50 mph (80.46 kph).
2.6 Brief Discussion on Dynamic Axle Weights and Their Impact on These Models
Dynamic axle loads can vary significantly around the static load during braking and cornering maneuvers. This natural response of the vehicle dynamics might suggest that the brake torque model output be adjusted to the changes in load that the axle experiences as a result of dynamic forces on the entire vehicle.
Thus, vehicle dynamics simulation outputs were reviewed to summarize the range of vertical loads that could be experienced on an axle during straight-ahead braking, and high-speed cornering under braking. The following results were extracted from simulations on dry asphalt (µ ≈ 0.75). The simulations have been vigorously verified using experimental vehicle data [7, 8], and were run using the TruckSim™ version 4.6 vehicle dynamics analysis package. Verification was against experimental vehicle data
30 taken at NHTSA’s Vehicle Research and Test Center (VRTC) in support of vehicle models developed for the National Advanced Driving Simulator [9].
The information in Table 2.1 gives some interesting insight into the effects of braking dynamics on vertical tire load for heavy trucks. The first two rows in Table 2.1 show an increase of 24% to 33% in vertical tire load under straight-ahead braking, regardless of whether the tractor/trailer is unloaded or loaded to GVW.
The situation becomes more complex as three-dimensional vehicle dynamics are considered. The bottom two rows of Table 2.1 contain results from a simulated best- effort stop while in a 500-ft radius corner and loaded to GVW. For the inside tire there is roughly a 3,600-lb (76%) maximum increase in vertical load while braking. For the outside tire, which experiences a higher dynamic load before braking, there is an increase of roughly 1,700 lbs (or 23%) during braking. As expected, the vertical loads on the inside and outside steer axle tires converge near the end of the stop, as lateral acceleration approaches zero.
From this summary, it is clear that the loads on the axle change dramatically during braking. This fact raises the consideration of how the model should treat the load effect revealed by the simulation, based on an assumption that the amount of load being arrested affects torque generation during one stop. The subject of brake fade is addressed in the following section.
31 Steer Axle Air Drum Brake Torques at 20 mph for Lining 'A,B,C, D'
9000
8000
7000
6000
5000
4000
Torque (ft-lbs) Torque 3000
2000
1000
0 0 20406080100 chamber pressure (psi)
Lining A 12k GAW R Lining B 13.2k GAW R Lining C 13.2 k GAW R Lining D 12k GAW R
S te er A xle Air D ru m B rake To rq ues at 50 m p h fo r L inin g 'A ,B ,C , D '
9000
8000
7000
6000
5000
4000
Torque (ft-lbs) 3000
2000
1000
0 020406080100120 chamber pressure (psi)
50 mph Lining A 12k GAW R 50 mph Lining B 13.2k GAW R 50 mph Lining C 13.2 k GAW R 50 mph Lining D 12k GAW R
S te er A xle Air D ru m B rake T o rq u es at 6 0 m p h fo r L in in g 'A ,B ,C , D ' 9000
8000
7000
6000
5000
4000
Torque (ft-lbs) 3000
2000
1000
0 0 20406080100120 chamber pressure (psi)
Lining A 12k GA W R Lining B 13.2k GAW R Lining C 13.2 k GAW R Lining D 12k GAW R
Figure 2.8 Steer axle drum brake torque measurements at 20 (top panel), 50 mph (middle panel), and 60 mph (bottom panel).
32 Drive Axle Air Drum Brake Torques at 20 m ph for Linings 'A,B,C,D,E' 18000
16000
14000
12000
10000
8000
Torque (ft-lbs) 6000
4000
2000
0 0 20406080100120 chamber pressure (psi)
Lining E 26k GAWR drive-1 Lining D 29k GA W R drive-2 Lining C 22k GA WR drive-3 Lining B 21k GAWR drive-3 Lining B 20k GAWR drive-4 Lining B 20k GA W R drive-4 Lining A 20k GA W R drive-3
Drive Axle Air Drum Brake Torques at 50 mph for Linings 'A,B,C,D,E'
18000
16000
14000
12000
10000
8000
Torque (ft-lbs) 6000
4000
2000
0 0 20406080100120 chamber pressure (psi)
Lining E 26k GAWR drive-1 Lining D 29k GA W R drive-2 Lining C 22k GA WR drive-3 Lining B 21k GAWR drive-3 Lining B 20k GAWR drive-4 Lining B 20k GA W R drive-4 Lining A 20k GA W R drive-3
Drive Axle Air Drum Brake Torques at 60 m ph for Linings 'A,B,C,D,E' 18000
16000
14000
12000
10000
8000
Torque (ft-lbs) 6000
4000
2000
0 0 20406080100120 cham ber pressure (psi)
Lining E 26k GAWR drive-1 Lining D 29k GA W R drive-2 Lining C 22k GA WR drive-3 Lining B 21k GAWR drive-3 Lining B 20k GAWR drive-4 Lining B 20k GA W R drive-4 Lining A 20k GA W R drive-3 Figure 2.9 Drive axle drum brake experimental measurements 20 (top panel), 50 mph (middle panel), and 60 mph (bottom panel).
33 Maximum Tire Vertical dynamic tire Change in Percent Vehicle load before load under load change in Load Road braking (lbs) braking (lbs) (lbs) load 0 load Straight 5,665 7,025 1,360 + 24 % GVW “ 6,182 8.206 2,024 + 33 %
GVW 500-ft 4,721 8,318 3,597 + 76% (inside tire) radius GVW (outside “ 7,194 8,880 1,686 + 23 % tire)
Table 2.1 Dynamic Steer Axle Loads from Best-Effort Braking Simulations
Although the results presented in Table 2.1 show that the load effect may be quite significant, it is unclear that adding more complexity to the brake model would contribute significantly to its accuracy. In straight-ahead braking, axle vertical load changes about 30%. Even during a braking-in-corner maneuver, where the wheel load can vary as much as 76%, this difference is the maximum difference during the stop, and it does not occur over a wide range of speeds. Note also, referring to Figure 2.10, that for a 15% change in test load for the ADB, the effect of load change on brake torque production appears minimal.
34 Air Disc Drum Brake Torques @ 20 mph for Lining 'A' 23,000 GAW R 14000 30 in2 air chamber
20,000 GAW R 12000 30 in2 air chamber
10000 12,000 GAW R 2 8000 20 in air chamber
6000 Torque (ft-lbs) Torque 4000
2000
0 0 20 40 60 80 100 120 chamber pressure (psi)
Lining A 23k GAW R Lining A 12k GAW R Lining A 20 k GAW R
Air Disc Brake Torques @ 50 mph for Lining 'A' 23,000 GAW R (3 tests) 14000 30 in2 air chamber
12000 20,000 GAW R (2 tests) 30 in2 air chamber 10000
8000 12,000 GAW R (2 tests) 20 in2 air chamber 6000 Torque (ft-lbs) 4000
2000
0 0 20 40 60 80 100 120 chamber pressure (psi)
Lining A 23k GAW R 50 mph Lining A 23k GAW R 50 mph Lining A 12k GAW R 50 mph Lining A 20 k GAW R
Air Disc Brake Torques by Speed for Lining 'A' 14000 23,000 GAW R 30 in2 air chamber 12000 20,000 GAW R 30 in2 air chamber 10000
8000 12,000 GAW R 20 in2 air chamber 6000 Torque (ft-lbs) Torque
4000
2000
0 0 20 40 60 80 100 120 chamber pressure (psi)
Lining A 23k GAW R Lining A 12k GAW R Lining A 20 k GAW R
Figure 2.10 Air disc brake torque measurements at 20 (top panel), 50 mph (middle panel), and 60 mph (bottom panel).
35 Since in this study there were no tests of drum brakes designed to show the effects of significant changes in dynamometer load (inertia) on Tb, we cannot determine precisely the effect of load on Tb. However, this topic should be explored in future research.
2.7 Addressing Brake Fade During a Stop
Brake fade that occurs during a single hard stop is also of concern to this study. In general, brake fade results from many phenomena including thermal expansion of the brake drum (away from the pad / mechanism), thermal or mechanical distortion of the mounting brackets and actuation mechanisms, and, probably most important, heat-related degradation of the friction between the brake pad (or shoe) and the (usually) ferrous surface that the brake pad or shoe works against [10].
For the purposes of analyzing vehicle stability during a stop, it was initially assumed that modeling the amount of brake fade (torque degradation) for both disc and drum brakes during a single stop would be of significant importance. To that end, real- time torque data during a stop would be most useful. In the absence of real-time data, the use of the FMVSS 121 standard “Brake Power” Test data was explored to quantify brake fade. As stated in FMVSS § 5.4.2, the test is performed by executing ten snubs that decelerate the inertial load from 50 to 15 mph (80.5 to 24.1 kph), at an average tangential (tire-wheel) rate of 9 ft/s2 (2.75 m/s2) on a 72-second duty cycle. Brake torque degradation (or fade) can be determined by the change (usually an increase) in brake chamber pressure required to maintain the prescribed 9-ft/s2 average deceleration rate for each of the ten consecutive snubs.
A second test, the “Brake Recovery” Test, utilizes twenty stops that decelerate the test inertia from 30 to 0 mph, at an average rate of 12 ft/s2 (3.66 m/s2) repeating on a 60- second duty cycle (FMVSS §5.4.3). A factor in deciding to use Brake Power Test, as opposed to the Brake Recovery Test to quantify brake fade was the amount of energy
36 consumed by the brake assembly during either test, compared to the total work done by an axle, during a typical hard stop executed by an actual vehicle.
The first two rows in Table 2.2 apply to simulated work and average horsepower per brake during a 0.37-G (12 ft/s2) stop from 60 mph, and a “snub” from 60 to 30 mph. The following row applies to a 0.37-G stop from 30 to 0 mph. The next two rows give the same data for one brake during the Brake Power and Brake Recovery Tests conducted at the same GAWR. The final three rows repeat the comparison at a simulated 23,000-lb GAWR. Here, we set our goal of using experimental data wherein the energy dissipated by the brakes, as computed in equation (2.7), is as close as possible to that dissipated by an actual vehicle brake at similar GAWR. Given the stated goal, it would appear that the work and average power for the Brake Power Test (0.28-G snub from 50 to 15 mph) are somewhat more applicable than those from the Brake Recovery Test for estimating the fade for a heavy truck brake during an actual vehicle stop.
1 ∆=WmVV()22 − (2.7) 2 f i
37
Work GAWR Speed during stop Average (lbs) Decel level range (k-lbs-ft) horsepower Comment 12 ft/s2 Typical vehicle 12,000 60-0 mph 721 179 (0.37 G) stop “ “ 60-30 mph 541 268 “ Dyno Recovery “ “ 30-0 mph 180 89 Test 9 ft/s2 Dyno Power “ 50-15 mph 455 145 (0.28 G) Test 12 ft/s2 Typical vehicle 23,000 60-0 mph 1263 313 (0.37 G) stop “ “ 60-30 mph 947 470 “ 9 ft/s2 Dyno Power “ 50-15 mph 798 254 (0.28 G) Test
Table 2.2 Total Work and Average Power Consumed per Brake During a Stop
Furthermore, two snubs from 50 to 15 mph (from the Brake Power Test conducted at 12,000-lb GAWR) result in a total work performed of 910 klb-ft (= 2 x 455) (1.234 kJ), which corresponds fairly closely to the 721 klb-ft (0.977 kJ) of work done by a typical steer axle brake during a 0.37-G vehicle stop (see the first row). Therefore, for this exercise, the first several repetitions of all Brake Power Test data available for each brake type were used to roughly estimate the amount of degradation of brake torque that one brake might experience during one stop from 50 mph to rest.
The method for determining the torque change (usually degradation for a given pressure) during a stop was straightforward. For the first three applications (snubs) of the Brake Power Test, the required increase in pressure (if any) was converted to a decrease in torque at the vicinity of chamber pressure necessary to perform the test. In other
38 words, a linear interpolation was used to estimate the amount of torque change near the operating point experienced by the brake during those first applications of the Brake Power Test. From the estimated torque loss during one Brake Power Test snub, a linear brake degradation relationship was formulated in the terms given in equation (2.8). The estimates of brake fade expected during a hard stop are listed in Table 2.3.
∆T R = b (2.8) fade ∆W
where
Rfade = brake fade ratio (% / 100 ft-lb)
The results from Table 2.3 reveal that steer axle drum brakes experience the most fade during a stop, the drive and trailer axle drum brakes (of similar design and identical dimension) show similar rates of fade, and the disc brakes (on either axle) show negligible fade. However, no brake demonstrated enough torque fade during a series of FMVSS 121 compliant stops or snubs to justify additional model complication to account for fade during one or two stops from highway speed.
Figure 2.11 shows time-based torque data from a dynamometer test, resulting from an application at constant brake chamber pressure of 80 psi (5.515 bar). The ordinate of Figure 2.11 gives the torque output in terms of the peak measured for that brake application. The data shown in Figure 2.11 confirm the results from the previous, more involved method, in which it is shown that is that there is little appreciable brake fade during a single stop. The 10% deviation in measured brake torque levels (after peak) in Figure 2.11 compare favorably to the predicted fade of around 6% in Table 2.3.
39 50 MPH Brake Power Test at 80 psi 16.5x7 (drive axle S-cam drum brake) 15x4 (steer axle S-cam drum brake)
100% 100
90% 95 80% 90 70% brake torques 85 60%
50% 80
40% chamber pressures 75 30%
16.5x7 Torque 70 Chamber Pressure [psi]
% of Maximum Brake Torque 20% 15x4 Torque 16.5x7 Chamber Pressure 65 10% 15x4 Chamber Pressure 0% 60 0.02 0.42 0.82 1.22 1.62 2.02 2.42 2.82 3.22 3.62 4.02 4.42 Time [Sec]
Figure 2.11: Brake dynamometer experimental output showing step input response to the brake pressure command for a steer axle s-cam drum brake and a drive axle s-cam drum brake. Traces are brake chamber pressure (right axis) and measured torque (left axis).
2.8 Brake Torque Model in Operation
Figures 2-12 and 2-13 show dynamic output of the brake torque model, as coupled to the pneumatic model discussed at length in reference [2].
Figure 2.12 shows the response of a dynamic model of the brake dynamometer to a step input in the command signal at t=0 seconds. The top panel shows both simulated and actual dynamometer brake chamber time responses, along with the theoretical step command. The bottom panel shows torque output from the simulation containing the models developed herein, along with experimental dynamometer torque output. Note
40 that the deviations in torque after stabilization of chamber pressure (and torque) were not modeled in the simulation.
Rfade Percentage Work done Brake type Decrease in Fade Ratio Comments during snub Tb (% per 100 ft- lbs of work) Steer axle 6.2 % 455 k-ft-lbs 1.363 drum Drive axle 6.5 % 797.9 k-ft-lbs 0.814 drum Trailer Axle 6.0 % “ 0.752 Drum * = negligible Steer axle disc 0 % * 455 k-ft-lbs 0 increase in Pc Drive axle * = negligible 0 % * 797.9 k-ft-lbs 0 disc increase in Pc
Table 2.3 Experimental Brake Fade, in Terms of Torque Decrease, for Various Brake Types
41 DRIVE Axle Brake Simulation: Treadle & Chamber Pressure
100
step command 80
simulated dyno speed (ft/s) 60
40 simulated treadle (or command) simulated brake chamber experimental DYNO brake chamber dyno speed (ft/s) pressure (psi) and speed (ft/s) 20
0 -1 0 1 2 3 4 5
DRIVE Axle Brake Simulation with Experimental Dynamometer Torque Output 12000
10000
8000
brake system model response to step up and step down 6000 experimental DYNO Torque Response to step input
4000
Brake Torque Magnitude (ft-lb) Magnitude Torque Brake 2000
0 -1 0 1 2 3 4 5 time (s)
Figure 2.12 Brake dynamometer simulation and experimental data: response to a step input in command pressure. Top panel: pressures, simulated command (treadle), simulated chamber, experimental chamber, with simulated speed output. Bottom panel: simulated torque output, experimental dynamometer torque output.
42 Figure 2.13 shows the response to several cycles of a simulated step-up-then- release application using the same dynamometer model. As the brake pressure is momentarily released and falls below Ppo, the torque versus speed relationship is re- applied as per equation (2.6). As the hub speed decreases, the increasing torque magnitude can be seen rising in the bottom panel of Figure 2.13, as a consequence of the change in Tb versus Pc coefficients with respect to application speed.
DRIVE Axle Brake Simulation: Treadle & Chamber Pressure
step up/down command simulated treadle (or command) 100 simulated brake chamber dyno speed (ft/s)
80
60 simulated dyno speed (ft/s)
40
pressure (psi) and speed (ft/s) 20
0 -1 0 1 2 3 4 5
DRIVE Axle Brake Simulation with Experimental Dynamometer Torque Output 12000
10000
8000
6000
4000
Brake Torque Magnitude (ft-lb) 2000
0 -1 0 1 2 3 4 5 tim e (s) Figure 2.13 Brake dynamometer simulation. Top panel: pressures, simulated command (treadle), simulated chamber, simulated dynamometer speed (ft/s). Bottom panel: simulated brake torque output.
43 Tables 2.4 through 2.7 contain the model coefficients described in equations (2.4) through (2.6).
Cij i = 0 i = 1 i = 2 j = 1 0.000331 -0.0259 0.4262 j = 2 -0.0433 2.4617 53.66 j = 3 0.6582 -31.738 -201.3
Table 2.4 Model Fit Coefficients for the Steer Axle Brake Torque Model
Cij i = 0 i = 1 i = 2 j = 1 -0.000620 0.0455 -0.7757 j = 2 0.0311 -2.9823 207.33 j = 3 -0.3425 38.555 -1286.1
Table 2.5 Model Fit Coefficients for the Drive Axle Brake Torque Model
Cij i = 0 i = 1 i = 2 j = 1 -0.00023 0.0191 -0.7024 j = 2 0.0186 -2.431 204.02 j = 3 0.1065 7.205 -1087.7
Table 2.6 Model Fit Coefficients for the Trailer Axle Brake Torque Model
Cij i = 0 i = 1 i = 2 j = 1 -0.00099 0.0814 -1.7314 j = 2 0.0986 -8.4106 324.4 j = 3 -0.6748 69.392 -2533.7
Table 2.7 Model Fit Coefficients for the ADB Brake Torque Model
44 2.9 Conclusions
In this chapter, the development of a new empirical brake torque model was presented. The model is intended for simulating heavy truck brake torque as a function of brake chamber pressure and application (hub) speed, during a single hard stop. The model can easily be applied to brake simulations or integrated with heavy vehicle dynamic simulations. Having characteristics of models developed for similar purposes in the past, this model uses empirical equations that describe the torque versus pressure relationship; the coefficients of those equations adjust with respect to application speed. For many modeling applications, the empirical formulas may be easier to apply than other methods, such as lookup tables.
The empirical brake torque model developed was derived using contemporary dynamometer data from representative examples of s-cam drum brakes currently available for steer, drive, and trailer applications. Modern production air disc brakes are also characterized and show robust behavior with respect to speed, versus drum brakes of similar capacity.
Also, this chapter addressed the option of correcting the brake torque output as a function of the vertical load (or work history) of the brake during a single stop, concluding that further investigation was needed.
Finally, it was shown that models such as this, that are designed and developed to model heavy vehicle brakes during single stop simulations (as opposed to long braking- on-hill simulations), do not need to be adjusted for brake fade.
45 2.10 Chapter 2 References
1. T.M. Post, P.S. Fancher, and J.E. Bernard, “Torque Characteristics of Commercial Vehicle Brakes,” SAE 750120. 2. A.L. Dunn, G.J. Heydinger, G. Rizzoni, and D.A. Guenther, “New Model for Simulating the Dynamics of Pneumatic Heavy Truck Brakes with Integrated Anti- Lock Control,” SAE 2003-01-1325. 3. D. Yanakiev, J. Eyre, and I. Kanellakopoulos, “Longitudinal Control of Heavy Duty Vehicles: Experiment Evaluation,” California PATH Research Report, UCB-ITS- PRR-98-15. 4. M.A. Flick, “An Overview of Heavy Vehicle Brake System Test Methods,” SAE 96225. 5. C. Hatipoğlu, T. Acarman, and Ü. Özgüner, “Pneumatic Pressure Control: Blending Simulations to Implementation,” Proceedings of ESDA 2002 Conference, 2002. 6. TruckSim version 5.0 User Manual (Ann Arbor, MI: Mechanical Simulation Corporation, 2003). 7. E. Milich, “An Evaluation of VDM Road and VDANL Vehicle Dynamics Software for Modeling Tractor-Trailer Dynamics” (master’s thesis, The Ohio State University, 1999). 8. A.L. Dunn, “The Effects of Cornering Force Variation on Articulated Vehicle Predictions,” NHTSA / VRTC internal report, September 2000. 9. W.R. Garrott, P.A. Grygier, J.P. Chrstos, G.J. Heydinger, M.K. Salaani, J.G. Howe, and D.A. Guenther, “Methodology for Validating the National Advanced Driving Simulator's Vehicle Dynamics (NADSdyna),” NHTSA report, 2001. 10. “Mechanics of Heavy Truck Systems,” short course material, University of Michigan Transportation Research Institute (UMTRI), July 2000.
46 CHAPTER 3
DEVELOPMENT OF AN ANALYTICAL MODEL FOR SIMULATING THE DYNAMICS OF PNEUMATIC HEAVY TRUCK BRAKES WITH INTEGRATED ANTI-LOCK CONTROL
3.1 Abstract
This chapter introduces a new nonlinear model for simulating the dynamics of pneumatic-over-mechanical commercial vehicle braking systems. The model employs an effective systems approach to accurately reproduce forcing functions experienced at the hubs of heavy commercial vehicles under braking. The model, which includes an on-off type ABS controller, was developed to accurately simulate the steer, drive, and trailer axle drum (or disc) brakes on modern heavy commercial vehicles. This model includes parameters for the pneumatic brake control and operating systems, a 4s/4m (four sensor, four modulator) ABS controller for the tractor, and a 2s/2m ABS controller for the trailer. The dynamics of the pneumatic control (treadle system) are also modeled. Finally, simulation results are compared to experimental data for a variety of conditions.
47 3.2 Motivation
The goal for this research and development was to create a model via a systems approach that can accurately simulate the brake system dynamics during ABS-assisted braking. The component level model allows detailed study of the influence from individual component parameters on system performance.
Although vehicle modeling experts have developed algorithms intended to simulate braking systems (with varying degrees of detail), many of the published models lack certain elements, such as brake torque hysteresis, which have a significant influence on the character and magnitude of forcing functions experienced by the vehicle chassis due to braking forces. And, although previous models have provided various depths of complexity, the description of their components in technical forums is often vague due to the necessary protection of a manufacturer’s competitive edge. In addition, perennially increasing computer power allows researchers the ability to add more sophistication to new dynamic models used to simulate the forcing functions at vehicle hubs (which ultimately control the vehicle dynamics). Therefore, this “new generation” brake model contributes significantly to the levels of detail and accuracy available in pneumatically actuated commercial vehicle brake models accessible in the public domain.
The brake models discussed herein were developed to be used as part of larger nonlinear multi-body vehicle dynamics simulations. Specifically, the brake system models are used as system components; these models run in the Simulink® environment, in parallel with TruckSim™ version 5.0 heavy truck dynamics simulation software. When the models are running in parallel, the TruckSim™ software effectively models the dynamic behavior of non-brake related hardware, such as suspension deflection, aerodynamics, hitch deflection parameters, and the very complex force and moment dynamics from the tires.
These complex vehicle dynamics and brake system simulations are combined to study the effects on stability of using ECBS (electronically controlled braking system) actuated, high-torque brakes, such as air disc brakes (ADB), on the prime mover (tractor) 48 while keeping traditional pneumatically controlled drum brake technology on the trailer. The results of the analytical studies on braking stability are discussed in detail in Chapter 4.
3.3 Background
A history of various published brake models for commercial vehicles is as follows. • During the 1970’s, Fancher, Post, et. al., from the University of Michigan’s UMTRI, published several technical papers that reported and discussed brake torque measurements [1] and brake models [2].
• During the 1980’s, the National Highway Traffic Safety Administration’s Vehicle Research and Test Center, led by Richard Radlinski, published many papers discussing measured braking performance [3] and brake system compatibility [4] for heavy trucks.
• A significant portion of the current literature on heavy vehicle brakes is summarized in the University of Michigan “Mechanics of Heavy Duty Truck Systems” short course [5].
• A model similar to the one presented herein was published in 1998, by Yanakiev et al., at UCLA [6].
• ABS models exist, but many are understandably simple or vague in the interest of protecting complex proprietary control algorithms.
• The default ABS model in TruckSim™ v. 5.0 models the ABS system by setting “on” and “off” thresholds for the vehicle brakes [7].
• Recent contributions have been made by Hatipoğlu, et. al., with regard to modeling pneumatic brake systems and ABS systems [8, 9, 10].
49 3.4 Model Overview in General
This new generation model includes the following significant features to simulate pneumatic brake system dynamics.
• First-order differential equations model the pneumatic system dynamics for the control (treadle) circuit and main brake actuation circuits
• Time delays for control (treadle) signals based on physical location of the associated modulator valve
• 4s/4m (four sensor, four modulator) integrated ABS control system for the tractor
• 2s/2m integrated ABS control for the semitrailer
• Simulated ABS controller calculation lag
• ABS control strategy based on longitudinal wheel slip level and tangential acceleration, tuned to match actual vehicle performance on wet and dry surfaces
• ABS system integration and control that commands each of six modulator valves in the 10-brake-position system to build (apply), hold, or dump (release) brake pressure
• Quadratic model of brake torque output as a function of application speed and chamber pressure (also discussed in [11]).
• Brake system hysteresis, as seen in the modern s-cam drum type brakes, common on Class-8 heavy commercial vehicles.
• The option of simulated ECBS (electronically controlled braking system) control
• The option of simulated torque output for disc brakes with various sizes of pneumatic brake chambers based on experimental brake dynamometer results
50
3.5 Dynamics of the Pneumatic Brake System
The forces and moments transmitted by an actual brake system, whether pneumatic or hydraulic, are capable of relatively high-frequency load inputs. The frequency and magnitude of these dynamic braking forces become all the more important when abrupt brake application and release, inherent with ABS control, is applied to the system.
3.5.1 Modeling the Dynamics of the Pneumatic Brake Chambers
Some inspiration for the mechanical systems models contained herein came from a paper by the UCLA School of Engineering, which addressed the use of tractor semitrailer brakes to assist speed and following distance control of trucks traveling in convoy [6]. In their work, Yanakiev et. al., modeled the first-order system response of the pneumatic pressure, converting said pressure into force along the chamber pushrod, then coupling that force with a linear transfer function which considered slack adjuster length, S-cam effective radius, brake drum diameter, and brake lining material coefficient of friction to produce a brake torque.
It is obvious from their and other vehicle system models, including TruckSim™ version 5.0, that the dynamic effects due to the pneumatics are highly important in accurately modeling heavy truck brake behaviors.
This model therefore uses the traditional first-order ordinary differential equation to model the dynamics of the pneumatic pressure within each brake pressure chamber, which in turn drives the brake torque output for each brake. As seen in equation (3.1), the rate of change of the chamber pressure is directly proportional to the difference between that chamber pressure and the command (treadle) pressure.
51 1 PPP =−() ctcτ (3.1)
and
ττ=≤> filling when PP c po and P c 0 ττ=>> rising when PP c po and P c 0 ττ=< falling when P c 0
where
Pc = brake chamber pressure
τ = time constant (brake chamber)
τt = time constant (treadle system)
Pt = treadle (command) system pressure
Pref = command reference from driver input
PPo = “push-out pressure”, usually 7 psi
The treadle pressure is itself subject to first-order system behavior, which is similarly modeled in equation (3.2). Note in equation (3.2) that the driver command
(from the parent vehicle model) is the intended treadle pressure, Pref. The time constants for the brake chambers (τ) and treadle system (τt) are determined from experimental data. Finally, the time constants for each brake chamber type are state-derivative dependent, i.e. they are differ, based on whether the pressure is rising or falling.
1 PPP =−() treftτ (3.2) t
The mechanical system model is discussed in two parts, addressing the chamber pressure (Pc) to brake torque (Tb) relationship then system hysteresis. 52 3.5.2 Torque Properties of the Drum and Disc Brakes
Referring to Figure 3.1, to generate brake torque, a non-ECBS pneumatic drum brake system uses a pneumatic command (or treadle) signal, which controls the modulator valve (not shown), which in turn allows air to flow to the brake (pressure) chamber at each brake. As the brake chamber fills to the pressure referenced to the modulator valve by the treadle (command), the pressure is converted into a force, pushing on one end of the slack adjuster via a push rod. The moment created by the force on the end of the slack adjuster rotates an S-cam. The S-cam in turn pushes one end of both leading and trailing brake shoes outward as they contact the inside surface of the drum.
Figure 3.1 Schematic illustration of s-cam drum brake assembly showing crucial working parts [12].
53 Detailed modeling of this numerically complex system is possible by formulating kinematic equations of motion for all the mechanical parts of the typical braking system. Mathematical relationships would be necessary to describe, among other phenomenon, free play in the mechanical linkages, torsional stiffness in the slack adjuster and torsion arm to the S-cam, deflection of the S-cam assembly, compressibility of the brake lining material, and – most difficult – the relationships between brake shoe displacement and actual pad force (or lining pressure) onto the drum, as well as the actual coefficient of lining-drum friction with respect to the pressure, wheel speed, and duration of application.
However, for this research, such detailed modeling of every mechanical aspect between chamber pressure and brake torque would not offer a substantially improved model in terms of accuracy and convenience. Instead, it was decided to solve the problem from a systems engineering approach, thus deriving the mathematical relationship between chamber pressure and torque output from many combinations of pneumatic drum brakes, including steer axle, drive axle, and semitrailer axle. Also studied were air disc brakes, set up for use on both the steer axle (using 20-in2 pressure chambers) and the drive axle (using 30-in2 pressure chambers). From those extensive studies, quadratic relationships of brake torque output (Tb) versus chamber pressure (Pc) were formulated. The coefficients of those relationships were also modeled using quadratic relationships to hub speed at time of brake application. Detailed description of the model is included in Chapter 2 and [11]. Figure 3.2 shows a typical drive axle drum brake model from that research, co plotted with brake dynamometer data.
The benefits of using algebraic relationships relating brake torque output to chamber pressure and hub speed include computational efficiency and model simplicity, as well as ease of simulating brake torque output of different brakes and linings by simply adjusting the nine coefficients that are used to describe the quadratic relationships.
54 Quadratic Models for Trailer Axle Air Drum Brake Torque for Various Speeds
Vhub = 20 mph 10,000 Vhub = 40 mph
Vhub = 50 mph 8,000 Vhub = 60 mph (ft-lb) 6,000
4,000 Brake Torque Torque Brake
2,000
0 020406080100 Pc = chamber pressure (psi)
Figure 3.2 Brake Torque model output shown as a function of chamber pressure for various application speeds. Solid lines (without symbols) indicate brake model output for 20, 25, 30, 40, 50, and 60 mph. Dashed lines (with symbols) indicate experimental data at an equivalent axle load of 20,000 lbs for three test speeds (20, 50, and 60 mph). The thicker gray line models the low-pressure torque response for all wheel speeds (Pc < 20 psi).
The equations that govern the torque response depicted graphically in Figure 3.2 are given in equations (3.3) through (3.5).
The quadratic coefficients calculated in equation (3.5) are allowed to readjust only when the pressure is below the “push-out” pressure (Ppo), effectively “clamping” the Tb versus Pc relationship at the initial application speed. This feature has been used in previous models, since brake dynamometer data show little change in available brake torque as speed decreases during a single brake application, as seen in the bottom panel of Figure 3.3 and referenced in [1, 2, 5, 7, and 11]. The term “push-out pressure” refers to a very low chamber pressure (usually around 7 psi – 0.483 bar) below which stiction causes the system to remain stationary, resulting in a constant volume in the pressure
55 chamber (and no torque output from a properly adjusted brake) until that stiction is overcome.
⎧⎫ ⎪⎪0 0 ≤≤PP ⎪⎪cpo ⎪⎪T =<≤bP0 TPbc⎨⎬ PPP poc0 (3.3) ⎪⎪P0 ⎪⎪2 i <≤ ⎪⎪∑CPic( ) P0 P c P res ⎩⎭i=0
2 i TCP==( ) TPP ( ) (3.4) bP0 ∑ i00 b c i=0
2 =≤j adjust only if : CcVi∑ ij hub PP c po (3.5) j=0
Figure 3.3 shows simulation output in response to a theoretical step input in Pt. The command steps up to 80 psi (5.52 bar) at t=0, then back to 0 psi at t=2 seconds. Experimental brake dynamometer torque data in response to a step input of 80 psi is shown in the bottom panel for comparison.
The effects of readjusting the coefficients to a new application speed can be seen in Figure 3.4, showing a simulated brake dynamometer with a brake chamber command signal stepping to 80 psi, then back 0 psi, for four cycles. Note the increase in maximum available brake torque for each cycle (bottom panel) that results from a decrease in dynamometer speed (top panel).
56 DRIVE Axle Brake Simulation: Treadle & Chamber Pressure
brake chamber 100 treadle (or command)
80
60 Theoretical Step Input to Command (or treadle) Signal. Response of actual treadle circuit pressure (psi) 40 is actually a 1st order response
20
0 -1 0 1 2 3 4 5
DRIVE Axle Brake Simulation with Experimental Dynamometer Torque Output 12000
10000
8000
6000
4000
Brake Torque Magnitude (ft-lb) 2000 brake system model response to step up and step down experimental DYNO Torque Response to step input
0 -1 0 1 2 3 4 5
Figure 3.3 Time response of the pressure chamber model [top panel] and brake torque model [bottom panel] to a theoretical step input in P(treadle) at t=0 seconds to 80 psi, then back to 0 psi at t=2 seconds. Experimental dynamometer torque data are included in the bottom panel for comparison.
57 DRIVE Axle Brake Simulation: Treadle & Chamber Pressure
step up/down command simulated treadle (or comm and) 100 simulated brake cham ber dyno speed (ft/s)
80
60 sim u la te d d yno sp e e d (ft/s)
40
pressure (psi) and speed (ft/s) speed and (psi) pressure 20
0 -1 0 1 2 3 4 5
D RIVE Axle B rake Simulation with Experimental D ynamomete r T orque O utput 12000
10000
8000
6000
4000
Brake Torque Magnitude (ft-lb) Magnitude Torque Brake 2000
0 -1 0 1 2 3 4 5 tim e (s)
Figure 3.4 Brake Dynamometer Simulation. Top panel: Simulated command (treadle) and chamber pressures (psi), and simulated dynamometer speed (ft/s). Bottom panel: Simulated brake torque output (lb-ft).
3.6 Brake System Parameters
3.6.1 Brake Simulation Physical Parameters
The pneumatic brake simulation hysteresis levels and treadle delay times are listed in Table 3.1. Parameters for the brake chamber time constants (τ) are given in Table 3.2. The delay times and time constants were calculated using experimental data from VRTC’s VR-5 tractor with various test trailers. VR5 is a 1995 Volvo 6x4 tractor
58 used as a primary test vehicle at TRC. Although many brake ABS controllers have been on VR-5 for experimental testing, the configuration for this testing was standard s-cam drums and Wabco ABS control..
The brake chamber dynamic model was implemented such that the first-order system time constants were varied depending on the actual state of the chamber. If the pressure was below the push out pressure (usually taken as 48.2 kPa or 7 psi), the dynamic model used τ = τfilling. As the chamber was filling (i.e., the derivative of chamber pressure was positive), the model used τ = τrising. When the brake chamber pressure was decreasing (i.e., chamber pressure was negative), the model used τ = τfalling. The time constants were derived using the linear system theory estimate given in equation (3.6). The time constant values for the semitrailer brake chambers were revised downward slightly to yield more accurate dynamic response under ABS braking conditions. This revision is justified since the path of air release through the ABS release valve is usually far less restrictive than release through the modulation. The time constant for the condition of falling brake chamber pressure (τfalling) was originally estimated to be around 380 ms from experimental data, but was revised to 200 ms while tuning the ABS controller to obtain better agreement with experimental stopping data.
t τ ≈ (3.6) 2.2
where
∆t = time corresponding to the chamber pressure reaching 90% of full scale
59
Simulation Setting ECBS-disc air-drum Steer axle brake hysteresis 113 N-m (83.3 lb-ft) 565 N-m (416.7 lb-ft) drive brake hysteresis 113 N-m (83.3 lb-ft) 565 N-m (416.7 lb-ft) trailer brake hysteresis 565 N-m (416.7 lb-ft) 565 N-m (416.7 lb-ft) treadle delay at steer axle 0 20 ms treadle delay at drive tandem 0 60 ms treadle delay at trailer 300 ms 300 ms tandem
Table 3.1 Hysteresis and Delay Time Parameters for Brake Models, ECBS-Disc and Air-Drum Configurations
Simulation Setting τfilling τrising τfalling treadle n.a. 110 ms 60 ms steer axle (20 in2 chamber) 30 ms 214 ms 146 ms drive axle (30 in2 chamber) 53 ms 264 ms 155 ms trailer axle (30 in2 chamber) 53 ms 212 ms 200 ms
Table 3.2 Brake Chamber Time Constant Values for Brake Model, ECBS-Disc and Air-Drum Configurations
3.7 The Hysteresis Element of the Brake Model
Figure 3.5 uses experimental brake dynamometer data for the brake application phase and simulated data for the brake release phase to demonstrate brake system hysteresis. We know that considerable hysteresis exists as a result of compliance of various components in the chain that converts brake chamber pressure into torque, such as flexibility of the mounting and actuation hardware and lining material, and self- actuation of the leading brake shoe. Modeling these cumulative effects is essential in faithfully simulating the forcing functions that the vehicle experiences from the brakes during ABS modulated stops. 60 Visual comparison of an air drum brake with an air-disc brake (ADB) reveals fewer potential compliances between the brake chamber and friction interface. Less compliance leads to the deduction that the ADB would demonstrate less hysteresis in operation than S-cam drum brakes. At the time of this writing, studies that will quantify the brake hysteresis are ongoing..
Extrapolation from the application loop in the experimental brake dynamometer data in Figure 3.5 (left) leads to the deduction that hysteresis for S-cam type drum brake is significant. Hysteresis levels of at least 100-200 lb-ft (135.6 – 271.2 N-m) can be roughly estimated for the steer axle drum brake shown. Furthermore, since we estimate that air disc brakes would exhibit less hysteresis between the pressure chamber and the lining-metal interface, we can also infer that the ADB configuration may indeed lend itself to more efficient cycling operation and controllability under ABS control. Also in Figure 3.5 is the simulated hysteresis loop, using the Simulink® hysteresis block [12].
Figure 3.6 shows the effects of two levels of system hysteresis on the simulated step response of the Tb versus Pc transfer function. Although the effects of hysteresis on a single brake application may appear insignificant, the hysteresis of repetitive application and release cycles can combine to produce a significant influence on performance of the overall brake system.
61 5 7000 x 10 Brake Torque vs. Chamber Pressure for Step Input 2.5
6000 Hysteresis “Return Loop” from decreasing 2 5000 pressure data. ft] - 4000 1.5
3000
Brake Torque [lb 1
2000 brake torque [in-lbs]
0.5 1000
0 0102030405060708090 0 0 10 20 30 40 50 60 70 80 90 100 Pressure [psi] chamber pneumatic pressure [lbs / in2]
Figure 3.5 Dynamometer output and estimated hysteresis loop resulting from application and release of an S-cam brake.
DRIVE Axle Brake DYNAMOMETER Simulation: Effect of Hysteresis on Torque Output 11000
10000 200 ft-lb hysteresis 9000 1000 ft-lb hysteresis
8000
7000
6000
5000
4000
3000 brake torque magnitude (lb-ft)
2000
1000
0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 time (s)
Figure 3.6 Simulation output showing the effect of hysteresis level on the response to a theoretical step input in treadle (reference) pressure.
62 3.7.1 The Influence of Hysteresis on Simulated ABS-Assisted Stopping Distance
Figure 3.7 shows the effect of brake system hysteresis on simulated stopping distances of a 4x2 straight truck loaded with 12,235 kg (26,917 lb). The simulation used the models discussed herein for drum brakes on the steer and drive axles, with the 4s/4m ABS controller discussed below. Note that the effect of having 1,500 ft-lb (2034 N-m) of hysteresis in the system (roughly 10% of system full scale capability) can increase wet stopping distances by almost 20%.
Percent Change in Stopping Distance vs. Drum Brake Hysteresis Level
25%
20%
15%
10% % Change in Stopping Distance 5%
0% 0 500 1000 1500 2000 2500 3000 3500 Hysteresis level [lb-ft]
Dry % increase Wet % increase
Figure 3.7 Increase in simulated stopping distance on a wet surface (broken line) and dry pavement (solid line) (µ=0.40 and µ=0.75, respectively), showing a significant effect on stopping distance resulting from adjusting the level of hysteresis in the drum brake models.
63 3.8 The Integrated 4s/4m ABS Controller
3.8.1 Brief Discussion of Tire Traction Theory and Priorities of an ABS System
The current popular opinion is that ABS control was originally developed to improve stopping distances. To the contrary, ABS controls were originally developed to ensure directional control and stability of the vehicle under braking, by avoiding full tire lockup (i.e., –100% longitudinal slip). With modern controllers, however, developers have been able to achieve both high levels of directional stability and stopping distances that are significantly shorter than those achievable with manual brake modulation.
Figure 3.8 shows experimental traction measurements for a steer axle truck tire at rated vertical load (at 125-psi inflation pressure, or 8.62 bar), for longitudinal slip magnitudes from 0 to 75%. Normalized longitudinal traction is expressed for lateral slip angles of 0°, 2°, and 4°, and normalized lateral traction is shown for slip angle magnitudes of 2° and 4°. Figure 3.8 shows that beyond 20% slip magnitude, the magnitude of tire lateral traction that was available at free rolling has been approximately halved. Therefore, for operating slip magnitudes over 20% (generally speaking, for wet and dry paved surfaces) the tire may very well cease to provide the adequate lateral force required to maintain directional control (for the steered tires) or stability (for tires located behind the vehicle CG). Additionally, as longitudinal braking slip increases towards the level corresponding to peak longitudinal traction (usually about 15-20% magnitude for moderate-to-high µ surfaces), traction quickly decreases. The decreasing traction, coupled with constant or increasing available brake torque, results in a “positive feedback” situation wherein control of longitudinal slip becomes more difficult. Hence, the second priority for the controller is to compel the tire-wheel assembly to remain at longitudinal slip magnitudes below that where peak longitudinal traction occurs, thus facilitating easier system control. During the 1990’s, passenger car ABS development priority was shifted to maximize the longitudinal traction efficiency, while maintaining lateral efficiency, as a result of consumer and media demands for shorter ABS-assisted stopping distances and advances in ABS control software and hardware. 64 Longitudinal and Lateral Traction vs. Slip Ratio at Rated Load for Steer Axle tire, 45 mph, dry surface
1.0
0 deg SA longitudinal 0.9
0.8
0.7
0.6
0.5 4 deg SA longitudinal
0.4
0.3 4 deg SA lateral
0.2
Normalized Traction Coefficients (Fx / Fz AND Fz) Fz Fy / / (Fx Coefficients Traction Normalized 0.1 2 deg SA lateral
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
longitudinal slip ratio
0 deg SA MU 2 deg SA MU 2 deg SA TAU 4 deg SA MU 4 deg SA TAU
Figure 3.8 Normalized lateral and longitudinal traction coefficients versus longitudinal slip ratio magnitude for a steer axle truck tire at rated load. Experimental measurements were taken at 0°, 2°, and 4° lateral slip angles.
3.8.2 The Simulated 4s/4m ABS Controller
Previous on-vehicle testing in support of the National Advanced Driving Simulator [13] utilized a conventional tractor with a 4s/4m antilock braking system controller. The designation “4s/4m” refers to a system that uses four wheel speed sensors and four modulator valves that control the brake chamber pressures on the vehicle, as commanded by the ABS controller. For the steer axle, each front hub speed is sensed, and separate modulators control pressure for each of the two steer axle brake chambers. For the drive tandem, only the leading axle hub speeds are sensed, and each modulator valve controls both the leading and trailing brake chambers on its respective side of the vehicle. This configuration is often referred to as “side-to-side” control.
65 The simulated controller uses wheel speed and vehicle CG longitudinal speed signals, available from the parent model, to calculate wheel slip for each wheel position, as expressed in equation (3.7):
rVω − κ = ii ref op (3.7) V ref
The CG longitudinal acceleration from the parent model is also monitored. Hence, the ABS controller operates on the assumption that accurate, low-noise signals for
Vref and aCG (estimated CG velocity and acceleration rates) already exist. Brake chamber pressure control experiences three modes – build (following Pt), hold, or dump – based on the regimes of wheel slip and comparison of vehicle CG with the tangential acceleration of each sensed wheel position.
Steer and drive axle slip limits are set separately, allowing for the commonly accepted control strategy of having more aggressive control on the front axle to maximize available braking potential while keeping a more conservative strategy on the rear axle(s) to ensure yaw stability during brake-in-turn situations. The controller operating parameters were chosen based on exhaustive comparative study using data from the National Advanced Driving Simulator testing, performed in 1996; the resulting hold and dump thresholds vary with surface coefficient and vehicle load, but are typically between –5% and –10% for the steer, drive, or trailer axle brakes. Slip thresholds for the drive axle are typically the most conservative at –4% (hold) and –6.5% (dump). Similar to production ABS systems, a truth table is used to facilitate command decisions for each brake modulator valve (see Table 3.3).
66 Inputs 1-3 slip slip alpha Row high optimum high output remarks 1 0 0 0 1 increase pressure 2 0 0 1 -1 dump pressure 3 0 1 0 0 hold pressure 4 0 1 1 -1 dump pressure 5 1 0 0 -1 dump pressure 6 1 0 1 -1 dump pressure 7 1 1 0 -2 fault, but dump anyway 8 1 1 1 -2 fault, but dump anyway
Note: "FAULT" means that mutually exclusive conditions are reported.
Table 3.3 ABS Logic Truth Table
Using the hypothetical situation where operating slip (κop) lies between the design limits of
[κ1 κ2] = [-5% -10%],
and assuming that wheel braking slip magnitude for a braked hub remains below 5% (refer to equation (3.7)), then brake pressure follows the treadle circuit command signal, as expressed in equation (3.8). If wheel slip magnitude for the same hub enters the region between 5% and 10%, the chamber pressure is held constant (assuming Ptr ≥
Pc) as expressed in equation (3.9). The operating slip range of [-5% -10%] is considered a desirable longitudinal slip range for optimum straight-ahead braking while maintaining adequate lateral force capability. Finally, if wheel slip magnitude for the same hub exceeds 10%, then the modulator receives the command to dump all pressure in that wheel chamber (see equation (3.10)).
67 == κκ ≤≤ κ PPtc(,i.e. P c 0) when 21 op (3.8)
==ττ Pt 0 and falling κκ< (3.9) when op 2
where
κop = model operating longitudinal slip ratio for the sensed wheel position
κ1 = model lower set point operating longitudinal slip ratio design limit, below which system pressure is allowed to build
κ2 = model upper set point operating longitudinal slip ratio design limit, above which system pressure is dumped
1 PPP =−() and ττ = ctcτ rising (3.10) κκ> when op 1
To give the ABS some derivative control, the controller also signals the pressure chamber to dump pressure if the tire-wheel tangential acceleration for that wheel position (as defined by the estimated tire rolling radius and equation (3.11)) exceeds that of the vehicle C.G. by a significant amount. For the acceleration threshold, choosing a proper control algorithm proved problematic due to divide-by-zero (singularity) errors at low speeds, and signal noise at all speeds. These problems would only get worse in real systems where signal noise, time lag and signal distortion resulting from filtering, and computation time only exacerbate the problems of noise in this signal. To avoid false triggers, a robust algorithm was chosen such that any modulator valve was commanded to
68 dump pressure whenever the inequality in equation (3.11) was satisfied for a wheel position under its control.
Hence, the control for the same hypothetical braked hub, based additionally on relative tangential tire deceleration exceeding vehicle CG deceleration by a significant amount (indicating tire lockup is imminent), signals the modulator valve to dump pressure regardless of wheel slip estimates discussed above. To restate, this segment of the control algorithm proved quite sensitive to noise in the vehicle acceleration signal, which resulted in a conservative control strategy (i.e., numerically high values for K1). The tunable constants in this model are the same for both the steer and drive axles but may vary slightly depending on conditions.
The separate semitrailer ABS system consisted of a 2s/2m controller, with side-to- side control for the trailer tandem, similar to the drive axles. ABS control parameters for the semitrailer are set, and operate, independent of those of the tractor.
Finally, the ABS algorithm output is shown (and the corresponding pressure chamber state) in response to arbitrary inputs for longitudinal wheel slip and tangential acceleration signals in Figure 3.9.
a α ≥+cg i1KK 2 (3.11) ri
where
αi = wheel rotation speed in rad/s
2 aCG = vehicle CG acceleration in ft/s
ri = estimated tire/wheel rolling radius in ft
K1, K2 = tunable constants
69
Figure 3.9 ABS control logic demonstration, showing (from top) wheel slip, “slip optimum” trigger, “slip high” trigger, wheel tangential acceleration, “accel. high” trigger, ABS logic output, and chamber pressure (Pc).
70 3.9 The Simulation of ECBS
By design, this model is easily tunable to simulate electronic braking systems (ECBS), where driver control to the brake chamber pressure modulator occurs via an electronic signal (having relatively insignificant time lag) in place of a pneumatic signal (having significant time lag). ECBS dovetails conveniently with ABS and stability control [9].
Experimental data in Figure 3.10 show that a standard pneumatically controlled system experiences first-order system behavior, even in the control system (treadle circuit). This model uses an integrator to simulate the first-order behavior of pressure building in the treadle circuit. In addition, a pure time lag simulates the finite time that elapses between actuation of the treadle valve (at the brake pedal) and arrival of the initial pressure front at the steer, drive, and semitrailer modulator valves. Experimental measurements show considerable time lag for drive and semitrailer circuits, with about 50-100 ms for the drive tandem, and 200 ms for the semitrailer. Test vehicles have shown delay times as high as 500 ms for the treadle valve signal to reach the semitrailer tandem modulator valve. The time lags are clear in the experimental brake chamber pressure data shown in the top panel of Figure 3.10. In this model, an ECBS system can be easily simulated by removing the time delays and the first-order integrator for the treadle circuit, then using the “reference” signal from the parent simulation. That “reference” or “treadle” signal, from the parent simulation, directly becomes the reference level for the modulator, which in turn controls the first-order integrators that still simulate the pressure chamber dynamics at each brake position.
71 Treadle & Brake Chamber Pressures During a Best-Effort Stop on Dry Pavement
100 control line steer axle 80 ]
2 drive axle semi-trailer axle 60
40 pressure [lbs/in
20
0 1.4 1.6 1.8 2 2.2 2.4
Longitudinal Slip Rates for Each Axle During Best-Effort Braking on Dry Pavement 2
0
-2
-4
-6 steer axle - left -8 leading drive axle - left leading semi-trailer axle - left slip ratio [%] ratio slip -10
-12
-14 1.4 1.6 1.8 2 2.2 2.4 time [s]
Figure 3.10 Experimental brake pressures (top panel) and longitudinal slip levels (bottom panel) for one wheel position from the steer, drive, and trailer axle. Note the successive time delays for the brake signal to reach each axle. All data have been low-pass filtered at 2 Hz.
.
72 3.10 Comparisons of Simulation to Experimental Data
As stated before, there exist ample experimental vehicle data taken on the Transportation Research Center’s skid pad and wet Jennite test surfaces. The skid pad surface is a dry concrete, having an ASTM coefficient of µ ≈ 0.90; the wet Jennite is a wetted epoxy-type surface, resulting in an ASTM coefficient of µ ≈ 0.40. For the development of these ABS controlled brake models, straight-ahead braking was examined on both surfaces at unloaded and GVW load conditions.
Figures 3.11 and 3.12 compare the simulation to the experimental data for the steer axle (vehicle loaded to GVW) during a stop from 30 mph (48.3 kph) on the wet Jennite surface. The simulated stop is shown in Figure 3.11 and the experimental data is shown in Figure 3.12. The operating pressure and slip behaviors have been accurately captured with the simulation. Note that ongoing efforts are currently being invested to simulate the gradual “build-hold-build” cycle experienced under most conditions with the Meritor Version C 4s/4m controller on the tractor.
The bottom panel of Figure 3.11 shows the simulated 4s/4m ABS controller output, which in turn signaled the modulator valve to either build (1), hold (0), or dump (- 1) brake pressure in the controlled wheel chambers. Note that “build” should be interpreted literally as “build following Pt.”
73 Steer Axle: Reference, Treadle & L1 Wheel Chamber Pressures ---- Simulation Cond: wet surface, loaded
100 treadle REF treadle (actual) 80 L1 brake chamber
60
40 Pressure (psi) Pressure 20
0 0 1 2 3 4 5 6 7 8 9
L1 Slip Ratio 0
-0.05
-0.1
Long. slipratio -0.15
-0.2 0 1 2 3 4 5 6 7 8 9
Steer Axle L1: ABS Logic Output 1.5
1
0.5
0
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5
-1
-1.5
ABS Command 0 1 2 3 4 5 6 7 8 9 time (s)
Figure 3.11 Simulation output showing treadle pressure with wheel chamber pressure (top panel), wheel slip (middle), and ABS control outputs (bottom) for steer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, loaded to GVW.
74 Tractor Steer Axle Brake Line Pressures : (data file:v3rsg04) 100 control line steer axle 80
60
40 line pressure [psi] 20
0 0 1 2 3 4 5 6 7 8
Tractor Steer Axle Longitudinal Wheel Slip (data file:v3rsg04) 0
-0.05
-0.1
slip ratio slip steer axle (L) steer axle (R) -0.15
-0.2 0 1 2 3 4 5 6 7 8 time [s]
Figure 3.12 Experimental data showing treadle pressure with wheel chamber pressure (top) and wheel slip (bottom) for steer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, loaded to GVW.
Figures 3.13 and 3.14 compare simulated and experimental results for the leading drive axle (with no vehicle payload) during a stop from 30 mph on the wet Jennite surface. Again, the operating chamber pressure and slip ranges are quite realistic for the simulated data.
Referring to Figure 3.13, the ABS hold regime occurs in the simulation for time between 1.5 and 2.5 seconds. The operating slip range for the simulation is approximately –2.5% to –9.0%, with a mean around –6%. Operating slip for the actual vehicle was a slightly more conservative 0 to –6.5%, with a mean operating slip around –2.5%. Again, the 60-ms build-hold-build cycle can be seen for the ABS controller used
75 for experimental tests. In contrast, however, note the behavior of a more modern controller (Meritor-Wabco D-type) under similar conditions, shown in Figure 3.15. Chamber pressures from a stop using the D-type Meritor-Wabco controller (Figure 3.15) demonstrate behavior much more similar to the simulation than the earlier C-type controller.
Drive Axle: Reference, Treadle & L2 Wheel Chamber Pressures ---- Simulation Cond: wet surface, NO load
100 treadle REF treadle (actual) 80 L2 brake chamber
60
40 Pressure (psi) 20
0 0 1 2 3 4 5 6 7 8 9
Drive Axle: L2 Slip Ratio 0
-0.05
-0.1
Long. slip ratio -0.15
-0.2 0 1 2 3 4 5 6 7 8 9
Drive Axle L2: ABS Logic Output 1.5
1
0.5
0
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5
-1
-1.5
ABS Command 0 1 2 3 4 5 6 7 8 9 time (s)
Figure 3.13 Simulation output showing treadle pressure with wheel chamber pressure (top panel), wheel slip (middle), and ABS control outputs (bottom) for the leading drive axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load.
76
Tractor Drive Axle Brake Line Pressures : (data file:v2rsf09) 50 control line drive axle 40
30
20 line pressure[psi] 10
0 0 1 2 3 4 5 6 7 8
Tractor Drive Axle Longitudinal Wheel Slip (data file:v2rsf09)
0
-0.05
-0.1 slip ratio slip lead drive axle (L) lead drive axle (R) -0.15
-0.2 0 1 2 3 4 5 6 7 8 time [s]
Figure 3.14 Experimental data showing treadle pressure with wheel chamber pressure (top) and wheel slip (bottom) for the leading drive axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load.
77 Wet Jennite Straight-Ahead Braking from 30 mph, 0-Load TBIC file: N:\1TBIC\VR5\VR5V\VR5VMT\CMT387.mat
treadle leading drive axle (L) leading drive axle (R) 100
80
60
Brake PressuresBrake (psi) 40
20
0
0 1 2 3 4 5 6
time (s)
Figure 3.15 Showing recent testing with an unloaded tractor with 48-ft. van trailer on a wet Jennite surface, stopping from 30 mph. This experimental data shows the different control characteristics for the tractor drive axle ABS controllers (Meritor-Wabco D-Type 4s/4m).
Comparisons of the operating slip ranges from Figures 3.12 and 3.13 to experimental tire traction data in Figure 3.8 show that although the optimum simulation settings resulted in a slightly higher operating slip than seen experimentally, both operating slip regimes are well below 15% slip, where lateral traction becomes significantly inferior to that near free rolling (0% longitudinal slip). Longitudinal traction also remains close to its peak, after which control of longitudinal slip by any means becomes far more difficult.
Finally, Figures 3.16 and 3.17 compare simulated and experimental results for the leading trailer axle (with no vehicle payload) during a stop from 30 mph on the wet Jennite surface. The higher longitudinal slip magnitudes seen by the trailer axle tires, versus the steer and drive axle tires, are apparent. The higher operating slip magnitudes
78 result from three differences in the trailer braking system versus the tractor braking system. The first reason is the aggressiveness of the trailer ABS control algorithm. Although it is still essential to maintain lateral traction for the trailer tires to prevent swing-out, the trailer tandem is responsible neither for steering the vehicle, as with the steer axle, nor for maintaining yaw stability for the tractor, as is the drive tandem. Therefore, my research concluded that emphasis for the trailer ABS is shifted to maximize longitudinal stopping efficiency, at the obvious sacrifice of some lateral tire force potential and controllability for the ABS controller. Hence the trailer tires experience relatively aggressive longitudinal slip ratios (–10%, up to –20%) for wet stopping. Referring again to Figure 3.8, it can be seen that – generally speaking – the trailer tires are therefore closer to the peak longitudinal traction at the expense of some lateral traction, which is still sufficient to prevent swing-out.
79 Trailer Axle: Reference, Treadle & L4 Wheel Chamber Pressures ---- Simulation Cond: wet surface, NO load
100 treadle REF treadle (actual) 80 L4 brake chamber
60
40 Pressure (psi) 20
0 0 1 2 3 4 5 6 7 8 9
Trailer Axle: L4 Slip Ratio 0
-0.05
-0.1
Long. slip ratio -0.15
-0.2 0 1 2 3 4 5 6 7 8 9
Trailer Axle L4: ABS Logic Output 1.5
1
0.5
0
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5
-1
-1.5
ABS Command 0 1 2 3 4 5 6 7 8 9 time (s)
Figure 3.16 Simulation output showing treadle pressure with wheel chamber pressure (top panel), wheel slip (middle), and ABS control outputs (bottom) for the leading trailer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load.
80 Trailer Brake Line Pressures : (data file:v2rsf05) 50 control line trailer axle 40
30
20 line pressure [psi] 10
0 0 1 2 3 4 5 6 7 8
Trailer Longitudinal Wheel Slip (data file:v2rsf05)
0
-0.05
-0.1 slip ratio slip lead trailer axle (L) -0.15 lead trailer (R)
-0.2 0 1 2 3 4 5 6 7 8 time [s]
Figure 3.17 Experimental data showing treadle pressure with wheel chamber pressure (top) and wheel slip (bottom) for the leading trailer axle (L). Maneuver is a straight-ahead stop from 30 mph on a wet Jennite surface, with no load.
Another significant reason for the deeper cycles seen by the operating slip of the trailer tires during an ABS stop is the significantly larger time constants (τ) that appear in the model for the more pneumatically restrictive trailer system than exist for the tractor. It can be seen in Figure 3.18 that for both brake application and release, the trailer pneumatic system was significantly slower to respond versus the drive axle brakes. Hence, in the simulation, as with the actual vehicle, the trailer brakes, having a higher
81 time constant (τ), were inherently more difficult to control than either the steer or drive axle brakes.
Finally, and perhaps the most significant factor affecting trailer slip magnitudes, is the vertical tire load. The simulated vertical tire load for the leading trailer axle (shown in this report) was about 50% of that seen by the drive tandem tires – before braking. The substantially lower load results in a much lower magnitude forcing function available to accelerate the similar tire-wheel inertias up to speed.
Section 3.12.1 offers additional comparisons of simulated results to experimental results for other combinations of axle, load, speed, and surface µ for straight-ahead best- effort stopping; in the same sections are Simulink® schematics of key modules in the pneumatic braking simulation.
Wet Jennite Straight-Ahead Braking from 30 mph, 0-Load TBIC file: N:\1TBIC\VR5\VR5V\VR5VMT\CMT387.mat
treadle leading drive axle (L) leading trailer axle (L) 100
80
60
Brake PressuresBrake (psi) 40
20
0 0 1 2 3 4 5 6 time (s) Figure 3.18 Showing recent testing with an unloaded tractor with 48-ft. van trailer on a wet Jennite surface, stopping from 30 mph. The experimental data shows the different control characteristics for the tractor and trailer ABS controllers (here, Meritor-Wabco D-Type 4s/4m, Eaton 2000 4s/2m, respectively). Note also the longer release time for the trailer brake, due to lag and higher system restriction.
82 3.11 The Build-Hold-Build Algorithm
Inspection of the chamber pressure traces in the experimental data presented in the previous section reveals a more sophisticated algorithm for the reapplication of brake pressure during an ABS-assisted stop. Note in the experimental data presented in Figure 3.19 that after the initial cycle of the ABS, the on-vehicle ABS computer allows the chamber pressure (Pc) to rebuild via a succession of “build-then-hold” cycles. The cycle periods are typically about 60 ms. The purpose of this part of the ABS algorithm is most likely to allow stability for the physical and/or measurement systems as brake torque is being applied. Note also that the second reapplication (from t ≈ 3.55s to t ≈ 4.25s) occurs at a different overall rate than the first one.
An approximation of the “build-then-hold” cycle was constructed in the simulated ABS algorithms for both the tractor and semitrailer. The output of the simulation algorithm is compared to corresponding experimental data for the steer, drive, and semitrailer axles during a stop from 30 mph (48.3 kph), at a GVW load, on a wet Jennite surface (µ ≈ 0.375) in Figures 3.19 through 3.21.
The simple addition of a “build-then-hold” sequence, having a 60-ms period, does not accomplish duplication of the reapplication sequence in the Meritor-Wabco type C controller that was on the experimental vehicle during acquisition of the data presented in Figures 3.19, 3.20, and 3.21. Note that the experimental reapplication sequence has a macro-linear appearance, while the simulated system reapplication has significant concavity. Adding a simple “build-then-hold” sequence, with no further modifications, appears to have a similar macro effect as increasing the time constant during the pressure rebuilding phase (τrising). Comparison of the simulation to the experimental chamber pressures in Figures 3.19 through 3.21 reinforces the need for further work in this area.
At the time of this writing, ongoing experiment research at NHTSA is focused on better understanding of current production ABS algorithms. As understanding grows, the advanced concepts will be added to these simulated ABS systems.
83 84
Figure 3.19 Simulation (left) versus experimental (right) output illustrating the build-then-hold algorithm during a stop from 30 mph (48.3 kph) on wet pavement (µ=0.375) at GVW load. Brake pressure, wheel slip, and tangential velocity are shown for the steer axle.
85
Figure 3.20 Simulation (left) versus experimental (right) output illustrating the build-then-hold algorithm during a stop from 30 mph (48.3 kph) on wet pavement (µ=0.375) at GVW load. Brake pressure, wheel slip, and tangential velocity are shown for the drive axle.
86
Figure 3.21 Simulation (left) versus experimental (right) output illustrating the build-then-hold algorithm during a stop from 30 mph (48.3 kph) on wet pavement (µ=0.375) at GVW load. Brake pressure, wheel slip, and tangential velocity are shown for the semitrailer axle.
3.11.1 Additional Data Results
Figures 3.22 through 3.27 in this section contain additional data comparing simulated brake system pressure, wheel slip, and ABS logic output to experimental brake chamber pressures and slip ratios. The data plots are placed side-by-side, facilitating easier comparison of the simulation to experiment, at the same conditions. Figures 3.28 through 3.30 show some pertinent Simulink® schematics of the brake model.
87 Drive Axle: Reference, Treadle & L2 Wheel Chamber Pressures ---- Simulation Cond: wet surface, loaded Tractor Drive Axle Brake Line Pressures : (data file:v3rsg04) 100 100 treadle REF control line treadle (actual) drive axle 80 L2 brake chamber 80 60
40 60 Pressure (psi) 20 40 0 0 1 2 3 4 5 6 7 8 9 line pressure [psi] Drive Axle: L2 Slip Ratio 20 0
0 -0.05 0 1 2 3 4 5 6 7 8
-0.1
Tractor Drive Axle Longitudinal Wheel Slip (data file:v3rsg04)
Long. slip ratio Long. -0.15 0 -0.2 0 1 2 3 4 5 6 7 8 9
Drive Axle L2: ABS Logic Output -0.05 1.5 88 1 -0.1 0.5 slip ratio slip lead drive axle (L) 0 lead drive axle (R) -0.15 : 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5
-1
-1.5 -0.2
ABS Command 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 time (s) time [s]
Figure 3.22 Simulation (left panel) and experimental data (right panel) for drive axle (L), 30 mph (48.3 kph), on wet Jennite, loaded to GVW. ABS control is via a Meritor Version C 4s/4m controller.
TBIC Test file: N:\1TBIC\VR5\VR5F\VR5FGV\CGV015.mat Drive Axle: Reference, Treadle & L2 Wheel Chamber Pressures ---- Simulation Cond: wet surface, loaded Wet Jennite Straight Ahead Braking from 30 mph
100 treadle REF treadle (actual) 100 80 L2 brake chamber drive 3 drive 4 60 90 40 Pressure (psi) Pressure 20 80 0 0 1 2 3 4 5 6 7 8 9
Drive Axle: L2 Slip Ratio 70 0
-0.05 60
-0.1 50
Long. slipratio -0.15 40 -0.2 0 1 2 3 4 5 6 7 8 9 unfiltered Chamber Pressures (psi) Drive Axle L2: ABS Logic Output 30 1.5
89 1 20 0.5
0 10
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5
-1 0 -1.5
ABS Command 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 time (s) time (s)
Figure 3.23 Repeat of simulated drive axle (L), 30 mph (48.3 kph), on wet Jennite, loaded to GVW (left panel) compared to experimental data (right panel) for leading drive axle (L & R sides) at 30 mph (48.3 kph), on wet Jennite, loaded to GVW. Tractor ABS control is via a Wabco-Meritor Version D 4s/4m controller. Note that , with Wabco- Meritor controller, Pc behavior is closer to the simulation.
Steer Axle: Reference, Treadle & L1 Wheel Chamber Pressures ---- Simulation Cond: wet surface, NO load Tractor Steer Axle Brake Line Pressures : (data file:v2rsf05) 50 100 treadle REF control line treadle (actual) steer axle 80 L1 brake chamber 40 60
40 30 Pressure (psi) Pressure 20 20 0 0 1 2 3 4 5 6 7 8 9 line pressure [psi] L1 Slip Ratio 10 0
0 -0.05 0 1 2 3 4 5 6 7 8
-0.1
Tractor Steer Axle Longitudinal Wheel Slip (data file:v2rsf05) Long. slipratio -0.15 0 -0.2 0 1 2 3 4 5 6 7 8 9
Steer Axle L1: ABS Logic Output -0.05 1.5 steer axle (L) 90 steer axle (R) 1 -0.1 0.5 slip ratio slip 0
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5 -0.15
-1
-1.5 -0.2
ABS Command 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 time (s) time [s]
Figure 3.24 Simulated steer axle (L), 30 mph (48.3 kph), on wet Jennite, 0 payload (left panel), compared to experimental data (right panel) for steer axle (L), 30 mph (48.3 kph), on wet Jennite, 0 payload.
Tractor Drive Axle Brake Line Pressures : (data file:v3rsd00) Drive Axle: Reference, Treadle & L2 Wheel Chamber Pressures ---- Simulation Cond: dry surface, loaded
100 100
80 80 treadle REF control line 60 treadle (actual) L2 brake chamber drive axle 40 60 Pressure (psi) 20 40 0
0 1 2 3 4 5 6 7 8 9 [psi] pressure line
Drive Axle: L2 Slip Ratio 20 0 0 -0.05 0 1 2 3 4 5 6 7 8
-0.1 Tractor Drive Axle Longitudinal Wheel Slip (data file:v3rsd00)
Long. slip ratio -0.15 0 lead drive axle (L) -0.2 0 1 2 3 4 5 6 7 8 9 lead drive axle (R) -0.05 Drive Axle L2: ABS Logic Output 91 1.5
1 -0.1
0.5 slip ratio
0 -0.15
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5
-1 -0.2 -1.5 0 1 2 3 4 5 6 7 8
ABS Command 0 1 2 3 4 5 6 7 8 9 time (s) time [s]
Figure 3.25 Simulated (left panel) lead drive axle (L), 50 mph (80.5 kph), on dry concrete, loaded to GVW compared to experimental data (right panel) for lead drive axle (L), 50 mph (80.5 kph), on dry concrete, loaded to GVW.
Trailer Axle: Reference, Treadle & L4 Wheel Chamber Pressures ---- Simulation Cond: wet surface, loaded Trailer Brake Line Pressures : (data file:v3rsg04) 100 100 treadle REF control line treadle (actual) trailer axle 80 L4 brake chamber 80 60
40 60 Pressure (psi) 20 40 0 0 1 2 3 4 5 6 7 8 9 line pressure [psi]
Trailer Axle: L4 Slip Ratio 20 0
0 -0.05 0 1 2 3 4 5 6 7 8
-0.1
Trailer Longitudinal Wheel Slip (data file:v3rsg04)
Long. slip ratio -0.15 0 -0.2 0 1 2 3 4 5 6 7 8 9
Trailer Axle L4: ABS Logic Output -0.05 1.5 92 1 -0.1 0.5 slip ratio slip 0 lead trailer axle (L) lead trailer (R)
: 1=build; 0=hold; -1=dump 0=hold; 1=build; : -0.5 -0.15
-1
-1.5 -0.2
ABS Command 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 time (s) time [s]
Figure 3.26 Simulated leading semitrailer axle (left panel), 30 mph (48.3 kph), on wet Jennite, loaded to GVW compared to experimental data (right panel) for leading semitrailer axle (L), 30 mph (48.3 kph), on wet Jennite, loaded to GVW. ABS control is via a Midland-Grau (currently Haldex) 2s/1m controller.
Trailer Axle: Reference, Treadle & L4 Wheel Chamber Pressures ---- Simulation Cond: dry surface, loaded Trailer Brake Line Pressures : (data file:v3rsd00)
100 100 80 80 60 treadle REF treadle (actual) 40 L4 brake chamber 60 Pressure (psi) Pressure 20
0 40 control line 0 1 2 3 4 5 6 7 8 9 trailer axle line pressure [psi] Trailer Axle: L4 Slip Ratio 20 0
-0.05 0 0 1 2 3 4 5 6 7 8
-0.1
Trailer Longitudinal Wheel Slip (data file:v3rsd00) Long. slip ratio -0.15 0 lead trailer axle (L) -0.2 0 1 2 3 4 5 6 7 8 9 lead trailer (R)
Trailer Axle L4: ABS Logic Output -0.05 1.5 93 1
0.5 -0.1 slip ratio slip 0
: 1=build; 0=hold; -1=dump 0=hold; : 1=build; -0.5 -0.15
-1
-1.5 -0.2 ABS Command 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 time (s) time [s]
Figure 3.27 Simulated leading drive axle (left panel), 50 mph (80.5 kph), on dry concrete, loaded to GVW compared to experimental data (right panel) for leading drive axle (L), 50 mph (80.5 kph), on dry concrete, loaded to GVW.
Pneumatic Drum Brake Pressure Chamber Inluding Brake Chamber Dynamics ONLY Inputs: treadle ("REFERENCE") pressure (0-120 psi) equivalent wheel speed (mi / hr) previous value (used for initial condition) Output: chamber pressure (psi)
TAU - rising Pc
Tau_drive_rising
TAU - filling Tau 1 (under 7 psi u Enable popout pressure) Comparator Inverse of TAU1 1 / Tau if P > Ppo Comparator Tau_drive_filling (pop-out pressure), if P_dot > 0, then use Tau-filling then use Tau-filling TAU - falling Pc1 Velocity2 Tau_drive_falling Memory Treadle (Ref erence) to EACH wheel's control v alv e 1 P_chamber_dot P_chamber [psi] P_chamber [psi]
94 1 REFERENCE 1 Pc Ptreadle or 0, s Product1 Ooutput1 depending 2 Sum2 xo on Control1 Previous Value From Output MEMORY1
Figure 3.28 Simulink® model for the enabled subsystem representing each of the modulator controlled brake chambers. Treadle (reference) pressure and previous chamber pressure are imported. Chamber pressure is exported to the model. This module cannot be multiplexed.
1
4 Constant4 4 1 4 Chamber Pressure IN 0
Compare Chamber Pressure Output to Ppo (popout pressure for chamber stiction), threshold for minimum brake Torque Output. Effectively keeps 7 <= Pc <= P_reservoi r psi. Thi s is done si nce the Interpol ati on for Low Pressure i s Onl y Valid for 7 psi <= Pc <= 20 psi
2 4 u 4 Speed Relationship Section to Chamber Pressure Squared 4 Compute Brake Torque vs. Chamber Pressure, 4 4 4 with coefficients being fcns. of speed Ppo 4 4 4 Wheel Speed I N Torque Coefficient OUT Quadratic Tb vs. Pc Coeff. Pc - Pswitch 4 Quadrati c Term on Tb vs. Pc2 4 QUADRATIC Term Cal cul ator 4 4 4 4 4 4 4 4 Wheel Speed IN Torque Coefficient OUT Li near Tb vs. Pc Coeff. Tb vs. Pc1 4 Li near Term on Tb vs. Pc1 Li near Term Cal cul ator 4 4 4 2 95 4 Input (Equi vl ant Wheel Speed) [mph] Wheel Speed I N Torque Coefficient OUT OFFSET Tb vs. Pc Coeff. Saturati on 4 Li mit Brake Torque 4 4 Tb_L1 Control Speed Feedback Offsect Cal c. 4 4 4 4 to Limits from whi ch 4 4 1 model was deri ved Tb Saturati on S-cam Hystereis swi thces Tb on/off, Brake Torque Ouput 4 KeepsT_brake >= 0 model ed vi a a Backlash bl ock based on Pc & Ppo control Tb 4 Tb_dri ve_dr_20 / 20 Compare Pc to Pswi tch to deci de i f eventual l y shoul d be a fcn (veh. speed)2 (pop-out pressure) for 4 program shoul d use speed rel s. Brake Torque (here = 20 psi) Sign Gai n1 = If Pc > Pswi tch, then use speed rel ati onshi ps. sl ope of Tb vs. Pc rel ati onshi p If Pc < Pswi tch, then use l i near approxi mati on -1 at Pressures between Ppo & 20 psi for Tb vs. Pc. Invert Si gnal Sign
Figure 3.29 Simulink® model for brake torque generation. Multiplexed chamber pressures are imported. Multiplexed brake torques are exported. Coefficients for brake torque calculation are adjusted for brake application speed.
Ashley Dunn [NHTSA] 2-Channel ABS Computer
controller type: 2s / 2m
Thi s model is the property of NHTSA and the Ohi o State University. DOCUMENTATION In1 Use ot this model outside of those institutions is prohibited 2 L1 ABS Logi c Output unless explicit permission is granted. In2Out 1 2 In3 [1x2] L1 ABS Logi c Output L1 Logi c Tabl e 2 In1 2 2 R1 ABS Logi c Out put 2 In2Out 1 R1 ABS Logi c Output Trai l er ABS Lead Trailer Axle 2 Wheel Speeds (mph) SLIP Outputs In3 2 Wheel Speeds (mph) [1x2] 4 (4 wheel pos) trailer_sl i p_hol d 2 =1 i f Sl i p rat i o > min level R1 Logi c Tabl e Mux 2 Demux 2 >= AND 2 ABS Logi c Signals Vehi cl e 2 Wheel Speeds (mph) 2 2 Trailer ABS Cont rol l er SLI P Out put Opti mum Slip =1 i f Sl i p rat i o opt i mum Termi nator Rel ati onal 2 lower threshol d Logi cal for trailing Operator2 Operator 2 True Logi c Out put 2 Trailer ABS LOGIC OUTPUTS Trailer Axl e 2 1 Trailer ABS LOGIC OUTPUTS Wheel Speeds Slip Rat i os Wheel pos 7 & 8 ABS Wheel Speed [ kph] Saturati on 2 2 2 Control l er Slip Rat i o Speed Li mit Force Control Output 2 OUTPUTS Vehi cl e Speed (kph) (in mph) to l i e between -1 & 1, Ti me Del ay for trailer_sl i p_dump 2 2 <= NOT i gnori ng faul t codes1 Ti me Del ay for Slip Cal cul ati on 2 =1 i f Sl i p rat i o < max l evel 2 =1 i f Slip > max l ev el Slip Rati o Computer Hi gh Slip 2 Vehicle 2 Speed Cal cul ati on Rel ati onal threshol d Logi cal Operator1 Operator1 1
Forced BUILD Command if Speed i s very l ow
Vehi cl e Speed [ mph]
1 [1x2] [0 0] Vehicle 2 Speed (mph) [1x2] Keep Flag OFF Out put = 1 when omega_dot MAG > (2 * omega_dot _cri t i cal ) [1x2] Vehi cl e Decel Rat e (m/s^2) 3 Manual Switch Vehi cl e 2 Decel Rate (m/s^2) Omega_dot _cri t i cal cal cul at ed (rad/ s^2) Ti me Del ay for Vehi cl e 2 2 * Omega_dot _cri t i cal (rad / s^2) trailer_ABS_gai n >= [1x2] Decel Rate Cal cul ati on [1x2] Calculate omega_dot_cri ti cal Gai n2 Rel ati onal for tire-wheel assembl y (rad/s^2) Operator [1x4] 510 1/1000 [1x4] [1x4] 96 Trailer Wheel Decel Rat es (real or si mul at ed) APPROXIMATE Gai n FORCE al pha TOO HIGH 4 Ti re Rol l i ng Radi us (mm) Manual Switch1 2 2 3 OFFSET Trai l er al pha - al pha_cri t for vehi cl e Decel Rate Li mit (rad / s^2) [1x2] Omega_dot_cri t - Omega_dot
Wheel Decel erat i on Rat es f or Wheel pos 7 & 8 (rad/ s^2) Lead Trailer Axle Wheel Trailer Wheel Decel Rat es [1x2] Decel Rat es (rad/ s^2) [1x2] (rad/sec^4 2) f or 4 wheel posi t i ons 4 Demux [1x4] [1x2] Ti re-Wheel Assembl y Ti me Del ay for Wheel Omega_dot Decel Rate Cal cul ati on 2 (rad/s^2) [1x2] Termi nator for trailing Trai l er Axl e Wheel Speeds1
Figure 3.30 Simulink® model for the 2s/2m ABS controller used on the trailer tandem for the simulations. Multiplexed wheel velocities and acceleration rates, with vehicle CG longitudinal velocity and acceleration, are imported. ABS logic commands of –1 (dump pressure), 0 (hold pressure constant), or +1 (allow pressure to build following Pt) are exported to the model.
3.12 Conclusions
In this chapter, a fully developed system model that simulates the dynamics of a pneumatic brake system was presented. The model was shown to accurately simulate the dynamics of a pneumatic heavy truck braking system, as well as the actual brake torque magnitudes, under many diverse operating conditions and configurations.
This brake system model can be tuned at multiple levels to simulate a broad range of functioning heavy vehicle pneumatic brake systems. The resulting simulation is one that can be tailored to accurately reflect brake torque output for a wide spectrum of operating loads, speeds, and surface coefficients.
An ABS system was developed, and was shown to be capable of simulating the control behavior seen by an experimental vehicle’s pneumatic braking system under a wide variety of conditions. The ABS system is tunable to allow simulation of varied responses of the steer, drive, and trailer brakes to a wide spectrum of simulated operating conditions.
Although some development was shown on an improved simulated ABS controller that can implement the “gradual build” or “step-up-and-hold,” further studies into the operation of these systems is essential to faithfully reproduce their behavior during simulation.
Finally continued positive inter-reliant relationships with industry technical representatives could help researchers understand certain control strategies. Engineers from DANA corporation contributed generously to my understanding of dynamometer measured brake torque for these studies.
97 3.13 Chapter 3 References
1. T.M. Post, P.S. Fancher, and J.E. Bernard, “Torque Characteristics of Commercial Vehicle Brakes,” SAE 750120. 2. L.K. Johnson, et. al., “An Empirical Model for the Prediction of the Torque Output of Commercial Vehicle Air Brakes,” Final Technical Report, MVMA Project #1.26, Highway Safety Research Institute (HSRI), University of Michigan, December 1978. 3. R.W. Radlinski, “Braking Performance of Heavy U.S. Vehicles,” SAE 870492. 4. R.W. Radlinski, and M.A. Flick, “Tractor and Trailer Brake System Compatibility,” SAE 861942. 5. “Mechanics of Heavy Truck Systems,” short course material, University of Michigan Transportation Research Institute (UMTRI), July 2000. 6. D. Yanakiev, J. Eyre, and I. Kanellakopoulos, “Longitudinal Control of Heavy Duty Vehicles: Experiment Evaluation,” California PATH Research Report, UCB-ITS- PRR-98-15. 7. TruckSim version 5.0 User Manual (Ann Arbor, MI: Mechanical Simulation Corporation, 2003. 8. C. Hatipoğlu, and T. Acarman, “Pneumatic Brake System Modeling for Systems Analysis,” SAE 2000-01-3414. 9. C. Hatipoğlu, and A. Malik, “Simulation Based ABS Algorithm Development,” SAE 1999-01-3714. 10. C. Hatipoğlu, T. Acarman, Ü. Özgüner, “Pneumatic Pressure Control: Blending Simulations to Implementation,” Proceedings of ESDA 2002 Conference, 2002. 11. A.L. Dunn, G.J. Heydinger, G. Rizzoni, and D.A. Guenther, “Methods for Modeling Brake Torque from Experimental Data,” SAE 2003-01-1322. 12. Using Simulink®, version 5 (Natick, MA: The Mathworks, Inc., 2002). 13. W.R. Garrott, P.A. Grygier, J.P. Chrstos, G.J. Heydinger, M.K. Salaani, J.G. Howe, and D.A. Guenther, “Methodology for Validating the National Advanced Driving Simulator's Vehicle Dynamics,” NHTSA report, 2001. .
98
CHAPTER 4
THE EFFECTS OF USING ECBS-DISC BRAKE SYSTEMS ON THE JACKKNIFE STABILITY OF TRACTOR-SEMITRAILER VEHICLES IN BRAKE-IN-TURN MANEUVERS
4.1 Abstract
The research presented in this chapter shows that the simulated presence of ECBS-disc brakes on the tractor results in no significant degradation of the performance of the rig, in terms of jackknife stability, while braking in a turn. Furthermore, the three- dimensional vehicle simulations showed significant reduction in the tractor maximum yaw rate and hitch articulation angle, for the simulated vehicles equipped with disc brakes and electronically controlled brakes systems (ECBS).
These studies were conducted by simulating a brake-in-turn maneuver using the newly developed brake models covered in Chapter 2 and 3 of this dissertation. The simulation conditions for the tractor-semitrailer were conducted on a 152.4 m (500 ft) radius corner with the mean friction coefficient levels set at either µ=0.30 or µ=0.55. Vehicle speeds corresponded to 90% of the theoretical maximum lateral acceleration in
99 the constant radius curve that would allow a drive-through without the vehicle leaving the 3.66 m (12-ft) lane.
4.2 Intent and Scope of This Research
The goal of this study was to use these in-house developed brake system models, in parallel with a sophisticated vehicle simulation package, to analyze the potential impact of advanced higher-torque ECBS-disc equipped tractors on the jackknife stability of tractor-trailer rigs.
The vehicle simulation package used for this research was TruckSim™ version 5.0, by Mechanical Simulation Corporation of Ann Arbor [1]. The TruckSim™ software utilizes a nonlinear solver that treats the vehicle chassis, suspension, and drivetrain masses as rigid bodies. This software package uses linear and nonlinear force and moment relationships to simulate the applied forces to the vehicle and internal forces between the vehicle components. The TruckSim™ software simulates the dynamics of the vehicle, including highly nonlinear aspects such as the tire force models, suspension deflection models, and the hitch deflection model. The pneumatic brake system dynamics, brake torque outputs, and brake hysteresis were modeled with a nonlinear Simulink® model developed for use within NHTSA’s Vehicle Research and Test Center (VRTC), in East Liberty Ohio. The brake simulation software received the brake system command from the vehicle simulation, then calculated brake chamber pressures and subsequent torques accordingly. The brake simulation model also simulated a 4s/4m ABS system for the tractor, and a separate 2s/2m system for the semitrailer.
The term “4s/4m” refers to the number of sensors (4) and modulator valves (also 4 for this system) that the ABS system uses to sense wheel speed, and then control brake pressure. A typical “4s/4m” system senses the front wheel speeds (2) and the leading drive axle hub speeds (2), for a total of 4 sensed wheel speeds. The designation following the slash refers to the four modulators, which in this configuration control the
100 steer axle brake chambers independently (requiring two modulators), then control each side of the drive tandem independently (requiring the other two modulators). Thereby both the leading and trailing axle brake position on each side of the drive tandem is controlled via one modulator valve.
This study compared the ECBS-disc equipped tractor to an otherwise identical air-drum tractor, each coupled to the same simulated air-drum equipped semitrailer. The rigs were evaluated for jackknife stability under brake-in-turn maneuvers. Brake actuation was simulated to occur at either the onset of negotiating a curve, or after the vehicle is fully established in the curve, depending on simulation setup. The test parameters were derived by analyzing a prior study that used similar vehicle comparisons on simulated wet and dry surfaces at load conditions of GVW and no payloads [2].
4.2.1 Previous Study Results, Summarized
This previous study concluded that:
1. disc brake equipped tractors did not appear to exacerbate jackknife problems for tractor-semitrailer rigs in brake-in-turn maneuvers, regardless of ABS functionality, and
2. for disc or drum equipped tractors, the combination of vehicle configuration / simulation condition that proved most troublesome for jackknife stability was that of vehicles having low trailer loads that are maneuvering on low coefficient (µ) pavement surfaces.
Table 4.1 shows the results of the initial study.
101 No Load GVW Load Hi µ Low µ Hi µ Low µ Drum / Disc / Drum / Disc / Drum / Disc / Drum / Disc / drum drum drum drum drum drum drum drum Full ABS Slow ABS No Half Treadle ABS Full ABS Slow ABS No Full Treadle ABS
Indicates near jackknife (high hitch articulation angle), and/or high hitch
forces Indicates jackknife
Table 4.1 Initial Brake-In-Turn Simulation Study Results
The following are explanations of the terms used in Table 4.1.
1. “Full ABS” indicates functioning ABS for both tractor and semitrailer, as per the vehicle simulation software, TruckSim™ version 4.6.
2. “Slow ABS” indicates simulated partial failure of the ABS system, on both the tractor and semitrailer, by increasing the slip threshold at which the ABS signals the brake chamber to dump pressure, and increasing the slip threshold at which the ABS signals the brake chamber to allow pressure to increase again.
3. “No ABS” indicates no simulated ABS operation on either the tractor (prime mover) or the semitrailer.
4. “Full Treadle” and “Half Treadle” correspond to the magnitude of control line pressure demanded by the driver after the vehicle was fully established in the 152.4 m (500 ft) radius corner. It is the experience and observation of this researcher that not all drivers apply full braking power after realizing that they may be going too fast for the traction level in a corner. Slow, cautious brake
102 application, arguably the best tactic under some emergency situations, can result in a different stability control problem versus full-treadle braking.
5. “Drum/drum” refers to the traditional configuration of air-drum brakes on the tractor, and air-drum brakes on the trailer. “Disc/drum” refers to ECBS-disc brakes on the tractor, and air-drum brakes on the trailer.
Although experimental studies with disc brake equipped tractors and semitrailers are ongoing at VRTC, these simulations were intended to expand upon the limited number of test conditions available to test engineers. Not only are we interested in the effects that significantly increased braking torque would have on tractor-trailer stability (i.e., Will the increase in braking force on the tractor make it more susceptible to jackknifing due to higher kingpin force magnitudes?), but how will controllability be influenced by the response time improvements promised with ECBS and disc brakes?
4.3 Simulation Conditions for This Study
The following conditions were simulated for this study.
1. Load conditions (2):
a. no payload and
b. ½ of GVW [3]
2. Surfaces (2):
a. µ = 0.55 (“wet”) and
b. µ=0.30 (“snow-ice”)
c. Realistic traction surfaces were simulated by having the levels of adhesion vary slightly around the means listed above. Variance of the surface coefficient about that mean was deemed necessary to simulate “real- world” surface coefficients, which are not constant.
3. Speeds (4, dependent on load and surface): determined to attain 90% of the lateral acceleration level seen (theoretically) for a successful drive-through of a 3.66 m (12 ft) lane at each load-µ condition, on a 152.4 m (500 ft) radius
103 curve, having no crown or super elevation (refer to Table 4.2 for simulation entry speeds). Equations (4.1) and (4.2) describe the procedure for determining the simulation initial speed:
0.90V 2 a = crit (4.1) entry R
= 2 VVentry0.90 crit (4.2)
where:
2 aentry = lateral acceleration desired at maneuver entry (m/s )
Vcrit = maximum drive-through speed through corner, determined by simulation (m/s)
Ventry = corner entry speed for brake-in-turn simulation (m/s)
R = corner radius in (m)
theoretical successful G at speed (kph) drive- successful 90% of corresponding Surface through drive- drive- to 90%-G µ load speed (kph) through through G drive-thru 0.30 0 68 0.239 0.215 64.5 0.55 0 82 0.347 0.312 77.8 0.30 ½ GVW 66 0.225 0.202 62.6 0.55 ½ GVW 77 0.306 0.275 73.0
Table 4.2 Simulation Entry Speeds Determined from Maximum Drive–Through Speeds
104 4. ABS (3 conditions) for either brake configuration (air-drum or ECBS-disc) on tractor:
a. Fully operational for tractor and semitrailer (“Phase I”)
b. Fully non-operational for tractor and semitrailer (also “Phase I”)
c. Non-functioning (tractor) with fully functional (semitrailer) (“Phase II”)
5. Brake configurations (2), for the tractor / semitrailer:
a. Pneumatically controlled drum / Pneumatically controlled drum
b. ECBS-controlled disc / Pneumatically controlled drum
6. Brake Application for Phase I: brakes were applied at the time corresponding to 1.0 second beyond that when the initial steering input occurs
a. Full Treadle: ramp from 0 to 0.6895 MPa (100 psi) in 0.3 seconds
b. Half treadle: ramp from 0 to 0.345 MPa (50 psi) in 0.5 seconds
7. Brake Application for Phase II: brakes were applied at time corresponding to the entire tractor-semitrailer reaching the maximum lateral acceleration for that entry speed that would be attained without braking, having the same brake ramp characteristics as in Phase I:
a. Full Treadle: ramp from 0 to 0.6895 MPa (100 psi) in 0.3 seconds
b. Half treadle: ramp from 0 to 0.345 MPa (50 psi) in 0.5 seconds
4.3.1 Simulated Variable µ Levels
The Simulink® simulation for the vehicle brake and ABS systems simulated a variable µ level. The effect was accomplished via a Band Limited White Noise Block, whose output was passed through a second order transfer function (filter), followed by a gain. The phasing of the µ-level between steer axle, drive axles, and semitrailer axles was critical for accurate effects. The phasing was dependent upon vehicle speed and distance between axles. The effect was accomplished via a variable transport delay,
105 using the ratio of li / Vref, where li is the distance between the steer axle and each successive axle, and Vref is the vehicle speed, in compatible units. The output for the µ- level generator, operating about a mean of µ=0.375, is seen in the upper panel of Figure 4.1. The lower panel of Figure 4 shows the tractor CG speed during the simulation. The coefficient levels have similar shapes, but variable phasing, resulting from the varying speed during the stop. Also note the instability in the calculation as speed approaches 0. Figure 4.2 shows part of the schematic for the surface µ generator block from the simulation.
4.3.2 Brake Simulation Physical Parameters
Brake physical parameters used, such as hysteresis levels and treadle delay times are discussed in Chapter 3, covering development of the brake system simulation in detail.
4.3.3 Simulated Brake Antilock Control
A brake antilock system was also simulated, and is discussed in Chapter 3, covering development of the brake system simulation in detail.
106 Variable Traction Coefficient Model 0.5 steer axle mu lead drive axle mu 0.45 lead trailer axle mu
0.4
0.35
0.3 traction coefficient traction
0.25
0.2 -2 0 2 4 6 8 10 12
CG speed vs. time 90
80
70
60
50
40 speed (kph) 30
20
10
0 -2 0 2 4 6 8 10 12 time (s)
Figure 4.1 Top panel: µ-level output for the variable coefficient simulator. Bottom panel: vehicle speed in kph during the same maneuver.
107 1 0.25 band limited white noise 0.5s 2 +1.0s+5 bl white noise thru f ilter Steer: Const mu modif ied w/ bl white noise Band-Limited Transfer Fcn White Noise Gain
mu_nom Variable TRAILER Transport Delay Constant
tr DRIVE: Const m
Variable Transport Delay1 1 Vx 1 (kph)
ld TRAILER: Con Variable Transport Delay2 1000/3600
kph to m/s 4.126
Constant1 Product trl TRAILER: Con Variable Transport Delay3
5.452 speed (kph) Constant2 Product1
Figure 4.2 Close-up of variable µ computer from the simulation.
4.3.4 TruckSim™ v. 5.0 Run Screens and Files
Each condition simulated is associated with its own TruckSim™ Control Screen and Parsfile to retain the ability to easily retrieve the information and results for post- processing. The TruckSim™ Parsfile numbers are listed in Appendix A, having similar format to the results tables. Matlab routines exist that allow the conversion of all of the TruckSim™ binary data files into Matlab’s binary format, as well as some routines for plotting their information. Appendix A also contains printouts of the export/import setup files (from the TruckSim™ v.5.0 environment), explicitly listing the variables imported from Simulink® to TruckSim™ and vice versa.
4.3.5 Pass-Fail Criteria
Large articulation angle was the criteria for “failure.” Some caveats apply, as in the case of locked-wheel behavior. With all wheels locked on the tractor, the only non- inertial forces on the tractor are those from sliding friction at the contact patches – only
108 opposite their direction vector – and the hitch forces. For this reason, the net moment around the tractor CG, in the yaw plane, is small enough that the yaw rate was somewhat low. However, if allowed to run until a full jackknife occurred, the tractor would have collided with the trailer. For those simulations where tractor yaw rotation was deliberate and substantial enough to create loss of control, the vehicle was deemed to “jackknife.”
The simulations were begun with the software controlling the tractor steering to maintain its lateral position in the middle of a 3.66 m (12 ft) lane. The turn was conducted on a simulated roadway that was not superelevated or crowned. If any tire from the vehicle exited the lane, as per the TruckSim™ animator, the event was deemed a “lane excursion.” Although lane excursions were documented, they did not appear to reveal much about jackknife stability, but were recorded nonetheless. Following a lockup at high speed (for all conditions with no ABS on the tractor) the rig experienced a lane excursion. For all simulations where the ABS was functional, the rig stayed within the proper lane.
4.4 Simulation Findings
The results are summarized in Tables 4.3 and 4.4. Table 4.3 shows Phase I simulation results for ABS “on” and ABS “off” for the entire tractor-semitrailer vehicle. Table 4.4 shows the Phase II simulation results, having the ABS “off” for the tractor only. Maximum tractor yaw rates, in degrees/second, are listed numerically in the respective cell. Cell background color indicates jackknife stability for that condition set.
109 No Load ½ GVW Load Medium µ Low µ Medium µ Low µ (µ~0.55) (µ~0.30) (µ~0.55) (µ~0.30) Drum / Disc / Drum / Disc / Drum / Disc / Drum / Disc / drum drum drum drum drum drum drum drum Full 6.9 6.2 5.8 5.6 7.2 6.7 7.5 6.7 ABS
Half No
Treadle 49.7 19.3 4.7 2.2 50.9 34.9 7.8 2.6 ABS Full 6.4 6.6 6.0 6.2 6.9 6.4 7.5 7.6 ABS
Full No
Treadle 8.8 3.5 2.0 1.2 12.5 6.0 2.3 1.3 ABS
Table 4.3 Jackknife Stability Simulation Result Matrix – Phase I: Full ABS and No ABS, Either Vehicle
No Load ½ GVW Load Medium µ Low µ Medium µ Low µ (µ~0.55) (µ~0.30) (µ~0.55) (µ~0.30) Drum / Disc / Drum Disc / Drum Disc / Drum Disc / drum drum / drum drum / drum drum / drum drum No ABS (tractor 60.0 35.8 18.0 8.0 63.5 42.8 25.0 24.5 Half
Treadle only)
No ABS (tractor 30.4 18.9 9.0 7.9 39.5 34.1 25.5 25.6 Full
Treadle only)
Indicates near-jackknife (high hitch articulation angle), and/or high hitch forces Indicates jackknife
Table 4.4 Jackknife Stability Simulation Result Matrix – Phase II: No ABS on Tractor, ABS Fully Functional on Semitrailer
110 1. There was no clear effect on the outcome of a brake-in-turn event that can be correlated with the presence of ECBS-disc or air-drum brakes on the tractor.
a. For most of the conditions, the presence of ECBS-disc brakes on the tractor significantly reduced the tractor maximum yaw rate and hitch articulation angle seen during the simulation.
2. Having the tractor ABS “off,” while leaving the trailer ABS “on” (the most likely scenario for ABS failure for the prime mover for contemporary rigs), resulted in much further road departures after tractor lockup during the simulation. For this condition, the trailer largely remained “locked” on a heading more-or-less consistent with that at tractor brake lockup due to its long wheelbase, high yaw moment of inertia, and low magnitude external forces versus the available tire cornering force. Figure 4.3 shows the simulation animator output for the condition stated. Figure 4.4 shows output for the condition ABS “off” for both vehicles.
3. Also true for the condition of tractor ABS “off”-trailer ABS “on,” jackknifing (or very high articulation angles, over 60 degrees) was ubiquitous over all simulation conditions.
4. For the condition of ABS “off” (both tractor and semi-trailer), jackknifing was rare, occurring in 3 out of 16 simulated brake-in-turn maneuvers. This result was a consequence of brake lockup on both tractor and semi-trailer being less than 0.5 second apart, therefore not allowing the very high magnitude dynamic hitch forces that often lead to jackknifing. This condition is illustrated in the animator output shown in Figure 4.4.
5. For both ABS-failure modes of simulation, vehicle lane excursion was quite complete, being worse for the case of ABS “off” (tractor) - ABS “on” (semi- trailer).
6. For the cases where the ABS was functional for both the tractor and the semitrailer, neither lane excursion nor jackknife occurred during the brake-in- turn maneuver. Figures 4.5 and 4.6 illustrate a typical animator output, and show the vehicle remaining within the 3.66 m (12 ft) lane.
111
Figure 4.3 Excerpt from TruckSim™ animator showing a typical jackknife event with the ABS off for the tractor and ABS on for the semitrailer. The trailer appears to remain on the tangent for the curve. Simulation: file 776, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS OFF.”
Figure 4.4 Excerpt from TruckSim™ animator showing a typical non-jackknife event with the ABS off for both the tractor and semitrailer. The entire vehicle appears to remain on the tangent for the curve. Simulation: file 775, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS OFF.”
112
Figure 4.5 Excerpt from TruckSim™ animator showing a typical non-jackknife event with the ABS on for both the tractor and semitrailer. The entire vehicle remained within the prescribed lane during the stop. Simulation: file 774, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS ON.”
Figure 4.6 Another excerpt from same TruckSim™ animator file shown above, which shows a typical non-jackknife event with the ABS on for both the tractor and semitrailer. This view better shows the maintenance of lane position by the entire vehicle. Simulation: file 774, “Disc/Drum, ½ treadle, ½ GVW load, low µ, ABS ON.” 113 4.5 State Plots and Observed Behaviors
To better understand the relationships between important state variables during maneuvers where a jackknife did or did not occur, phase plane diagrams were examined for most of the simulation conditions. Figures 4.7 and 4.8 show typical phase plane diagrams for a non-jackknife and jackknife maneuver, respectively.
In Figure 4.7, the upper left hand pane shows the hitch articulation angle exceeding 15° during the brake-in-turn maneuver, but not growing to magnitudes that would indicate a jackknife. This maximum articulation angle is a significantly larger than that expected during maneuvers where control would have been optimum. For those maneuvers where the ABS was functional for both the tractor and semitrailer, the hitch articulation angle did not exceed 5°. The rectangular windows drawn around the origin of the phase plane diagrams were sized based on the stable behavior seen in this example. The two top rows of Figures 4.7 and 4.8 show hitch articulation angle (top left) and vehicle side slip angles (top right). Panes in rows two through four of Figures 4.7 and 4.8 show the following phase plane diagrams.
1. Hitch articulation rate versus. angle
2. Side Slip rate versus angle (tractor and semitrailer)
3. Difference in Side slip rates versus difference in side slip angles (tractor - semitrailer)
4. Yaw rate versus yaw angle (tractor and semitrailer)
5. Difference in yaw rates versus difference in yaw angles (tractor - semitrailer)
6. Difference in yaw rates versus tractor, then semitrailer yaw angles
114
Figure 4.7 Phase plane diagram showing relationships between model state variables during a maneuver in which the hitch articulation angle remained under control, i.e., no jackknife occurred. The conditions for this simulation were: full treadle brake application, no ABS (either vehicle), medium traction (µ=0.55), and air-drum configuration. The boxes on the state plots are arbitrary operation limits on the respective states.
115
Figure 4.8 Phase plane diagram showing relationships between model state variables during a maneuver in which the hitch a jackknife occurred. The conditions for this simulation were: half treadle brake application, no ABS (either vehicle), medium traction (µ=0.55), and air-drum configuration. All state traces exceed the same limits applied to the non-jackknife example in Figure 4.6.
116 Note that the phase plane diagrams in Figure 4.8 all display bounded behaviors. The phase plane diagrams in Figure 4.7 and Figure 4.8 are presented on the same scale for comparison. Figure 4.8 shows the same time and phase plane diagrams, but for a maneuver that clearly resulted in a jackknife, as indicated by the hitch articulation angle exceeding 60°. All of the phase plane traces exceed the outer limits set using the data in Table 4.4, with the expected exception of the trailer side slip.
Comparing the number of evenly spaced observations (temporally) required for the states to exceed the boundaries arbitrarily set for this exercise could help determine which states more quickly alert an “observer” to impending hitch articulation instability. Comparison with very low speed, high articulation maneuvers, such as those seen in a parking lot maneuver, would be necessary to filter out “false alarms.”
The interested observer may notice that the apparently less abrupt “half-treadle” brake application (in Figure 4.8) resulted in a jackknife whereas the more abrupt “full treadle” brake application (in Figure 4.7) did not. Based on the author’s personal experiences with testing motor vehicles at high speeds and under severe maneuvers, this phenomenon can be expected. The slower, and lower amplitude, brake application for the “half treadle” braking allows more time for dynamic phenomena to take place before all the tires are locked, resulting in a system in pure sliding, and having far lower variable tire lateral forces applied. Such phenomena may include dynamic weight transfer, suspension deflection steering, kingpin forces, and higher time lag between brake torque being realized at the front and rear axles.
4.6 Conclusions
This study indicates that the simulated presence of ECBS-disc brakes on the tractor, while leaving pneumatically operating s-cam drum brakes on the semitrailer, will not lead to a significant degradation of jackknife stability while braking in a turn. The vehicle simulations showed significant reduction in the tractor maximum yaw rate and
117 hitch articulation angle seen during the simulation, for those simulated vehicles equipped with air disc brakes and ECBS.
Although all of the simulated configurations wherein the tractor ABS was disabled while keeping the semitrailer ABS operable resulted in a jackknife, the ECBS- disc equipped tractor produced significantly lower yaw rates versus the tractor equipped with air-drum brakes. This result was due to the ECBS-disc braked vehicles reaching lockup sooner, thus resulting in a lower magnitude yaw rate being established by the time that lockup had occurred.
The simulations were run at simulated light loads (0-payload and ½ GVW) and low surface coefficients (µ=0.3 and µ=0.55) that have in previous simulation-based studies been found to be most troublesome for jackknife stability. Although a large spectrum of braking situations exist under which a jackknife sequence can be triggered, the author currently believes that the net result of the initial conditions and boundary conditions will bring the system near the same state, in terms of inertial, internal, and external forces that existed at the onset of the jackknife incidents seen in this and previous analytical studies. Therefore, it is currently felt that these finding are – in that respect – comprehensive.
118 4.7 Chapter 4 References
1. TruckSim version 5.0 User Manual (Ann Arbor, MI: Mechanical Simulation Corporation, 2003); and (http://www.trucksim.com). 2. A.L. Dunn, “Brake-In-Turn Study Comparing Disc/Drum to Drum/Drum Truck Brake Combinations,” NHTSA / VRTC internal report, 19 September 2000. 3. The simulated trailer was the VRTC 53-foot box trailer with the tandem in the forward most position. For the “1/2 GVW” load condition, two simulated concrete blocks were at the front of the trailer, three blocks were at the rear, each block having a mass of 4250 lbs (1928 kgs). 4. A.L. Dunn, G.J. Heydinger, G. Rizzoni, and D.A. Guenther, “New Model for Simulating the Dynamics of Pneumatic Heavy Truck Brakes with Integrated Anti- Lock Control,” SAE 2003-01-1325.
119
CHAPTER 5
DERIVATION AND VALIDATION OF 3-AXLE AND 5-AXLE PLANAR ARTICULATED VEHICLE MODELS
Abstract
This chapter discusses the derivation and validation of planar models of articulated vehicles for analyses of jackknifing. The equations of motion are rigorously derived using Lagrange’s method, then linearized for use in state-space models. The models are verified using a commercially popular nonlinear solid body vehicle dynamics modeling package, TruckSim™, the models therein being previously verified using extensive on-vehicle experimental data [1, 2]. A 3-axle articulated model is expanded to contain five axles to avoid lumping the parameters for the drive and semitrailer tandems necessary in 3-axle models. Compromises inherent in using the linearized models are discussed and evaluated.
Finally, a nonlinear tire cornering force model is coupled with the 5-axle model, and its ability to simulate the jackknife event is demonstrated.
120 5.2 3-Axle Planar Articulated Vehicle Model Derivation
The system to be represented is depicted in Figure 5.1. There are four inertial bases as described below:
{E} = standard fixed inertial basis i {}′ = θ eEEii rotate standard basis { } by angle about 3 ′′ eei remains oriented such that 21 is always pointed at CG {}= ′′ψ eeii rotate basis { } by angle about e3 , fixed to vehicle 1 at CG1
eei remains oriented such that 1 is always pointeed in the direction of travel {}eeˆ = rotate basis { } by angle γ about e , fixed to vehicle 2 at CG ii 32 eeˆˆi remains oriented such that 1 is always parallel to semitrailer long. axis
also note :
===′ Eeee3333ˆ
The articulated vehicle has a forward velocity, denoted by ue , and lateral 1 velocity denoted by ve . Note that, from this point forward, the under-tilde ( e ), used 2 i to designate vectors, will be dropped for convenience. Also note that as shown in Figure 5.1, positive forward velocity for CG1 results in a negative rotational velocity, θ . The angle ψ denotes the vehicle yaw angle, and the angle γ denotes the hitch articulation angle. For this derivation only the front axle is allowed to steer, and as per usual convention, the steer angle, relative to the body centerline, is denoted by δ.
121
Figure 5.1 Schematic of the 3-axle planar model as derived
5.3 Kinematic Equations and Constraints
The only constraint seen by the system is the hitch constraint requiring the position, velocity, and acceleration of both vehicles to be identical at that point. Since I am not interested in dealing with the hitch forces at this time, using Lagrange’s equations is an attractive choice, since they provide the unique ability to ignore internal constraint forces when deriving the equations of motion for multi-bodied systems.
In the case where constraint forces are of interest, the constraint force information could be revealed by using Lagrange multipliers [3]. This study is, however, not
122 interested in analyzing hitch forces at this time, but is interested in the states of hitch angular rate and hitch angle, to be used in phase plots to investigate jackknife prediction.
Equation (5.1) is the generally accepted expression for Lagrange’s equations. To use Lagrange’s method, I will derive expressions for the location of the both CG’s (that of the tractor and that of the semitrailer), then differentiate the location vector to derive their velocities. From the velocities, I will derive expressions for kinetic energy, T, in terms of the generalized coordinates. Potential energy expressions are zero (0) for all conditions.
∂∂∂ =−+=dT⎛⎞ T V Lqq⎜⎟ Q (5.1) dt⎝⎠∂∂∂ q q q
where:
V = system Kinetic Energy term
T = system Potential Energy term
Qq = potential work expression for generalized variable, q
Q = generalized displacement variable
The generalized coordinates will be:
y = centripetal location of CG1 with respect to the center of the turn
θ = rotational location of CG1 with respect to the fixed basis, Ei
ψ = rotational orientation of the tractor (CG1) (yaw), and
γ = rotation orientation of semitrailer (CG2) (hitch articulation), with respect to the tractor.
123 As a cross-check, I will verify that the generalized variables above are sufficient to describe the system of interest, i.e., can the system represented in Figure 5.1 be fully described (with respect to the inertial reference frame) by the generalized coordinates listed above? The answer is, of course, “yes.”
Equations (5.2) and (5.3) are the position vectors for the respective CG’s (ρ1 and
ρ2), expressed in the most convenient bases.
ρ =+'' 1120eye (5.2)
ργγ=− + − 212(cos)sincd e d e (5.3)
where:
ρ1 = position vector of CG1
ρ2 = position vector of CG2
The respective velocity expressions can be derived by either using the relative velocity in a rotating reference frame expression given in equation (5.4), or by using the change of bases in equations (5.5) through (5.7) to transfer the velocity vectors to the standard (fixed) reference basis {Ei}, then simply differentiating the position equations ′ with respect to time. Equation (5.5) is the change-of-basis from the { ei } basis to the standard basis {Ei}. Equation (5.6) is the change-of-basis from the { ei } basis to the ′ rotating basis{ ei }. Combining equations (5.5) and (5.6) yield equation (5.7), which is used to transfer vectors from the { ei } basis (of CG1) to the standard fixed reference,
{Ei}.
124 ρρ=+×ω ρ iiii (5.4) where
ρ th i = velocity vector for the i center of mass,
ρ th i = position vector for the i center of mass,
ρ th i = co-rotational velocity vector for the i center of mass, and
ω th i = total rotational velocity vector for the i center of mass, with respect to fixed inertial data base (Ei)
θθ ′ ⎡⎤⎡Ee11cos sin 0 ⎤⎡⎤ ⎢⎥⎢Ee=−sinθθ cos 0 ⎥⎢⎥′ (5.5) ⎢⎥⎢22 ⎥⎢⎥ ′ ⎣⎦⎣⎢⎥⎢Ee33001 ⎦⎣⎦⎥⎢⎥
′ ψψ ⎡⎤ee11 ⎡cos sin 0 ⎤⎡⎤ ⎢⎥ee′ =− ⎢sinψψ cos 0 ⎥⎢⎥ (5.6) ⎢⎥22 ⎢ ⎥⎢⎥ ′ ⎣⎦⎢⎥ee33 ⎣⎢001 ⎦⎣⎦⎥⎢⎥
ψ θ−+ ψθ ψθ ψ θ ⎡⎤⎡Ee1 cos cos sin sin cos sin sin cos 0 ⎤⎡⎤1 ⎢⎥⎢Ee=−(cosψθψθ sin sin cos ) cos ψθ cos − sin ψθ sin 0 ⎥⎢⎥ (5.7) ⎢⎥⎢2 ⎥⎢⎥2 ⎣⎦⎣⎢⎥⎢Ee3 001 ⎦⎣⎦⎥⎢⎥3
125 Translating the position vectors to the standard basis is not only cumbersome, but yields equations that prove unwieldy to differentiate with respect to time unless a software package like Maple™ [4] is used. Although the code structure in Maple™ is often felt to be obtuse, the symbolic mathematics software can prove useful in reducing algebraic errors that frequently appear when deriving equations of motion for multi-body dynamical systems.
Both methods for differentiating the CG coordinate vectors were used as a mode of verification. This approach is strongly recommended.
Equations (5.8) and (5.9) represent the velocity of CG1 in the {ei} basis and the
{Ei} basis, respectively. Equations (5.10) and (5.11) represent the velocity of CG2 in the
{ei} basis and the {Ei} basis, respectively. Note the added complication of the hitch articulation angle γ, especially to the velocity of CG2 in the fixed inertial basis, equation (5.11).
ρψθψψθψ =−⎡⎤⎡⎤ ++ 11⎣⎦⎣⎦yysin cos eyy cos sin e 2 (5.8)
ρ =−⎡⎤⎡⎤ θθ − θ + θθθ − 112⎣⎦⎣⎦yysin cos Eyy cos sin E (5.9)
where:
ρ 1 [eq. (5.8)] = velocity of CG1 expressed in the body-centered { ei }basis
ρ 1 [eq. (5.9)] = velocity of CG1 expressed in the standard {Ei} basis
126 ρψγωθψ =−⎡ − ⎤ 221⎣ ydsin ( sin ) y cos )⎦ e (5.10) +−−+⎡⎤ ψω γωθψ ⎣⎦ycdycos12 cos sin ) e 2
ωψθψθ+ ⎡⎤c 1(cos sin sin cos ) ⎢⎥ ρωθψγψγθψγψγ =+ − + + 22⎢⎥dE(sin (cos cos sin sin ) cos (sin cos cos sin ) 1 ⎢⎥ ⎣⎦−−yy sinθθ cos θ ωψθ− ψθ ⎡c 1(sin sin cos cos ) ⎤ ++⎢ ωθψγ + ψγ + θψγψγ − ⎥ ⎢ d 2 (sin (sin cos cos sin ) cos (sin sin cos cos )⎥ E2 ⎢ ⎥ ⎣+−yy cosθθθ sin ⎦
(5.11)
where:
ρ 2 [eq. (5.10)] = velocity of CG2 expressed in the standard {Ei} basis
ρ { } 2 [eq. (5.11)] = velocity of CG2 expressed in the body-centered eˆi basis, and
ωθ=+ ψ 1 (5.12) ωθ=++ ψγ 2
Terms for the system kinetic energy are expressed in equations (5.13) through (5.15), with equation (5.13) representing the translational kinetic energy for each ith mass, 127 equation (5.14) representing the rotational kinetic energy for each ith mass, and equation (5.15) representing the total system kinetic energy at any time.
1 Tm=•()ρρ (5.13) iiiitrans 2
1 TJ=•()ωω (5.14) iiiirot 2
TT=+ T (5.15) ∑ iitrans∑ rot
Combining equations (5.8) through (5.15) lead to the expression for system kinetic energy, given in equation (5.16). Note that, conveniently, the total system kinetic energy expression should be the same regardless of the basis that the individual velocities were derived in.
11 TJmcJmd=+()()22ωω ++ 22 2212 1 22 2 +−+ωω γ ψ θ ψ ω mcd212cos mc 2 ( y cos y sin ) 1 +−−+ψγ ψ γ θ ψγ ψγω md2 [ y (sin sin cos cos ) y (sin cos cos sin )] 2 1 ++()[()]mmy 22 + yθ 2 12
(5.16)
where the system rotation velocity scalars, ωi, are expressed in equation (5.12).
128 I next applied Lagrange’s equations (equation (5.1)) to the kinetic energy expression in equation (5.16), for each generalized coordinate, q, of {y, θ, ψ, γ}. The system potential energy, V, is zero for all time. The generalized work expressions, Qi, will be derived below, where two methods are compared.
Applying equation (5.1) to equation (5.16) for the first generalized coordinate of
{q1 = y} yields equation (5.17).
=+ −θωψωψ 22 + − Lmmyymcy ()[](sincos)12 21 1 (5.17) +++−ω 2 ψγ ψγω ψγ ψ γ md22[ (sin cos cos sin ) 2 (sin sin cos cos )]
Assuming constant velocity (resulting in equation (5.18)), and applying small- angle relationships in equation (5.19) yields a simpler, yet still nonlinear version of equation (5.17), given in equation (5.20).
θ ≈ 0 (5.18)
⎧sinθθ≈ ⎫ ⎪ θ ≈ ⎪ ⎪cos 1 ⎪ ⎪sinψψ≈ ⎪ ⎨ ψ ≈ ⎬ (5.19) ⎪cos 1 ⎪ ⎪sinγγ≈ ⎪ ⎪ ⎪ ⎩⎭cosγ ≈ 1
Lmmyymc=+()[]( −θω 22 +ψ −ω ) ysmall angle 12 21 1 (5.20) +++−ωψ2 γ ωψγ md22[( ) 2 ( 1)]
129
Equation (5.20) can be further simplified by elimination of higher order terms (H.O.T.), resulting in equation (5.21). Furthermore, the method listed above for the first generalized coordinate (y) is applied for the remaining three generalized coordinates to yield equations (5.22) through (5.24), for the linearized system dynamic forces.
Lmmyymcd=+()[]() −θωω 2 − + (5.21) ysmall angle 12 21 2
LJdmcdyyθ =+[( +−−γψγ ] small angle 22 +++ +22 + + −ψψγ + + θ [()()2()]JJ12 mmymcd 1 2 2 ycd d +++ +2 −ψψγψ + + [()()]JJmcd12 2 ycd d −+−ψγ [(mc2 d d )] y ++ −ψψγ ++ θ (5.22) [(mmymcd12 ) 2 ( d )]2 y +−−ψγ γ ψγ [(md2 y y c )]2 −+−ψγ ψ 2 [(myc2 d d )] − +−γψγγ2 [md2 ()]yc y
2 Lmcdyyψ =−()[] + − θ small angle 2 (5.23) +++ +2 ψγ + + + [()][()]JJmcd12 2 Jmdcd 2 2
22 LmdyyJmdcdJmdγ =−[][()][] −θ + + +ψγ + + (5.24) small angle 22222
130 Note, finally, that since I intend for this model to be pseudo-steady-speed, the dynamics equation for θ (equation (5.22)) can be neglected from this point forward. In order to drop θ from the formulation, another meaningful variable needs to be substituted. In this problem, the forward velocity of the lead vehicle, as shown in equations (5.25) through (5.28), will be substituted.
−=−yyθθθ 2 () (5.25)
−=()yuθ (5.26)
θψ ≈= r (5.27)
−≈yurθ 2 (5.28)
where
u = forward velocity, and
r = ψ = vehicle (1) yaw rate.
Equations (5.29) and (5.30) exist to convert the system into a first order state- space system. Substituting equations (5.25) through (5.30) into equations (5.21), (5.23), and (5.24) yield the Lagrangian expressions in equations (5.31) through (5.33).
⎧ψ = r⎫ ⎨ ⎬ (5.29) ⎩⎭ψ = r
⎧γ = q⎫ ⎨ ⎬ (5.30) ⎩⎭γ = q
131 =+ +− +− Ly ()[]()mmvurmcdrmdq12 2 2 (5.31)
=− + + Lmcdvurψ 2 ()[] (5.32) +++ +2 ++ + [(JJmcdrJmdcdq12 2 ( ) ] [ 2 2 ( )]
=− + + + + + + 2 Lγ mdvurJmdcdrJmdq22222[][()][ ] (5.33)
5.4 Derivation of Forcing Functions via Virtual Work Expressions
The system forcing functions are first derived using the virtual displacement method, as described by Török [3]. Virtual displacements for each generalized coordinate qi, of {y, θ, ψ, γ} are shown below. The equation for virtual displacements is given for the general case in equation (5.34), where the virtual displacement is expressed th as δri for the i generalized coordinate. The equation for the virtual displacement of the
“steer axle,” or the point on the system where the first lateral force, F1, is applied, is shown in equation (5.35).
n ⎛⎞∂r δδrq= i ij∑⎜⎟∂ (5.34) j=1 ⎝⎠q j
where:
δ th ri = the virtual displacement vector at the i applied force,
132 ∂r i = th th ∂ the partial derivative of the i applied force vector w.r.t. each j q j generalized coordinate, and
∂ th q j = the virtual displacement icon for the j generalized coordinate.
Addressing the tire lateral force at the steer axle, and denoting the position vector of force F1 as r1, the first generalized displacement, δr1, is derived as expressed in equation (5.35).
⎛⎞∂∂∂⎛⎞ ⎛⎞ ∂ δδδθδ=++rrr111⎛⎞ ψ + r 1δγ ry1 ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ (5.35) ⎝⎠∂∂∂y ⎝⎠θψ⎝⎠ ⎝⎠ ∂ γ
The virtual work done by the forces at the ith point of application is simply the dot product of the forces at the ith point of force application and the infinitesimally small virtual displacement, expressed in equation (5.36).
δδ=•i WFrii∑ (5.36) i
After calculating the virtual work for each (ith) point of force application, they are combined into one expression for the total virtual work done by the forces acting on the system, and can be grouped as shown in equation (5.37).
133 δδ= WWTi∑ i =+++[]δδ [] [] δ qq12 qn (5.37) =+[]δδ [] ++ [] δ Qq11 Qq 2 2 Qqnn
Finally, the individual virtual force terms become the forcing functions, or the
“right-hand-side,” for each Lagrangian, Li, derived above, are extracted as the terms “Qi” of equation (5.37). The resultant grouped system of equations make up the differential equations of motion for the system being analyzed.
LQ= qqii = (5.38) f ()qgFij ( )
Next, considering the lateral force of the steer axle tire in Figure 5.1, denoted as
F1, the equations for the virtual forces on the system are derived.
First, the position vector for the lateral force at the steer axle, F1 in the model, is expressed in the { ei } basis for computation convenience:
=+′ ryeae121 =++−ψψ (5.39) [sinyaeye ]12 [ cos]
134 Equation (5.35) is then applied to the position vector expressed in equation (5.39).
Then, as expressed in general form in equation (5.34), the virtual displacement for F1 is shown in equation (5.40). The right hand side of equation (5.40) is grouped by virtual displacement (δqi) before applying equations (5.36) and (5.37) above.
δ =+[ ψψ]δ +−+++ψψδθ δψ +δγ reeyyeyaeae112sin cos [ cos 1 ( sin ) 22 ] [ ] [0]
(5.40)
δδ=•i WFr111∑ i δδ=•++−++++[]()()()()ψψδ ψψδθ δψ δγ WF11cos e 2⎣⎦⎡⎤ sin e 1 cos ey 2 y cos ey 1 ( sin ae ) 2 ae 2 0 =•−++++[][δψδψδθψδψδθδψ ] Fe12cos (sin yye cos ) 1 (cos yyaae ( sin ) ) 2 =+++δ ψδ ψ δθ δψ Fyyaa1 cos (cos ( sin ) )
(5.41)
In a similar fashion, the virtual work expressions for the other forcing functions (i.e., lumped tire cornering forces at each axle) are given below.
=−′ ryebe221 =−+−ψψ (5.42) [sinybeye ]12 [ cos]
δ =+[ ψψ]δ +−+−+−+[ ψψ]δθ[ ] δψ [ ]δγ reeyyeybebe212sin cos cos 1 ( sin ) 2 2 0 =−[][]ψδ ψδψ ++−− ψδ ψ δθ δψ sinyy cos e12 cos y ( y sin b ) b e
135 (5.43)
δδ=•i WFr222∑ i =•[] ()()ψδ − ψδψ + ψδ + ψ −− δθ δψ F22eyyeyybbe⎣⎦⎡⎤sin cos 1 cos ( sin ) 2 =+−−()ψδ ψ δθ δψ Fyybb2 cos ( sin )
(5.44)
=′ − −+γγ −+ ryecede321()cos()sin ede 1 e 2 =ψγψγ −−+ +− −+ (5.45) [sin()cos][cos()sin]y c de e12 y de e
δ =+[ ψψ]δ ++−[ γψ +−−+ ψ γ]δθ r312sin e cos e y (( de )sin y cos ) e 1 ( y sin c ( de )cos ++[]γγ − − δψ ()(sincos)de e122 e ce ++[]γγδγ − ()(sincos)de e12 e
=−[]ψδ ψδθ ++++ γ δθ δψ δγ sinyy cos ( de )sin ( ) e1 ++[]ψδ ψδθ −+++−+ γ δθ δψ δγ δθ δψ cosyy sin ( de )cos ( ) c ( ) e2
(5.46)
136 δδ=•i WFr333∑ i ⎡⎤⎡⎤sinψδyy− cos ψδθ e ⎢⎥⎢⎥++γδθδψδγ + + 1 ⎢⎥⎣⎦(de )sin ( ) =−[]FeFesinγγ + cos •⎢⎥⎡⎤cosψδyy+ sin ψδθ 3132⎢⎥ +−⎢⎥()cos(de +γδθδψδγ + + ) e ⎢⎥⎢⎥2 ⎢⎥−+δθ δψ ⎣⎦⎣⎦⎢⎥c()
=−γψδψδθ[] − + + γδθδψδγ + + Fyyde3 sin sin cos ( )sin ( ) + γψ[]δ +−+++−+ψδθ γ δθ δψ δγ δθ δψ F3 cos cos yysin ( de )cos ( ) c ( )
(5.47)
Combining equations (5.41), (5.44), and (5.47) results in the total virtual work
expression in equation (5.48), sorted by generalized displacement, δqi. The expressions within each bracket are therefore the virtual force expressions for each Lagrangian equation, Lqi, expressed in equations 6-20 through 6-23 above. I have not invested time into the “δθ” expression, since nearly constant vehicle speed is assumed, and can forgo using θ as a generalized coordinate in the final problem expression.
δδ= WWTi∑ i =+−+[]δ ψψψγψγδ F1233cos cosFF cos sin sin F cos cos y +−−++[]δγδψ aF123cos bF F ( c cos d e ) (5.48) +−[] + δγ Fd3 () e +[] δθ
137 =++−δ ψψ ψγψγ QFy 123cos cos F cos F (cos cos sin sin ) =−−++δγ QaFψ 123cos bFFc ( cos de ) (5.49) =− + QFdeγ 3 ()
5.4.1 Alternate Methods for Determining the Virtual Work Expressions
First, it can be shown that determining the virtual work can be performed more simply by using a simple graphic and literally displacing the system, by each generalized coordinate individually, and deriving algebraic expressions for the work done as the result of the “infinitesimally small” displacement of each generalized coordinate, qi.
The sketch in Figure 5.2 shows the individual generalized coordinate of ψ as slightly displaced. Figure 5.3 illustrates the “infinitesimal” displacement of the generalized coordinate for hitch articulation angle γ. By applying equation (5.36) to each individual point of applied force, one will arrive at the same group of equations for the generalized forces as shown in equation (5.49).
Figure 5.2 Virtual displacement of the model in the direction δψ
138 Finally, for systems that do not readily allow themselves to be graphically analyzed in the method described above, there exists a mathematic approach that is far simpler than the one using virtual displacement derivations discussed above. As outlined in full detail in Ginsberg [5], the scientist can use virtual velocities at each point of force application for an algebraically neater and simpler way to derive the virtual work expressions. The method is outlined below.
Again, beginning with the steer axle “hub,” or the point where the tire cornering force, F1, is applied (r1, as expressed in equation (5.39)), the velocity at that point is expressed below in equation (5.50).
Figure 5.3 Virtual displacement of the model in the direction δγ
=−ψθψ ++++ ψ ψ θψ ry 11[ sin y cos ] ey [ cos ( y sin a ) ae ] 2 (5.50)
Manipulating equation (5.50) to allow the velocities, in terms of the generalized coordinates (qi), to become virtual displacements, as outlined in equation (5.51), results
139 in equation (5.52), which compares quite exactly with equation (5.40), but through less extensive means.
yy ⇒ δ θδθ ⇒ (5.51) ψ ⇒ δψ γ ⇒ δγ
δδ=−ψ δθ ψ ++++δ ψψδθ δψ ry11[ sin y cos ] ey [ cos ( y sin a ) ae ] 2 =+[][][]ψψδ +−+++ψψδθ δψ (sin )eeyyeyaeae12 (cos ) ( cos ) 1 ( sin ) 22
(5.52)
Likewise, the other force location velocities expressed in equations (5.53) and (5.54) can be manipulated (using equation (5.51), above) to yield equations (5.55) and (5.56), as well as the virtual displacement expressions associated with the other forcing functions, noted as F2 and F3.
=−ψθψ ++−− ψ ψ θψ ry 21[sin y cos] ey [cos (sin y bbe ) ] 2 (5.53)
=−ψθψ ++++ γθψγ ry 3 [ sin y cos ( de )sin ( )] e1 (5.54) ++ψθψ −+−+++ θψ γθψγ [cosyy sin c ( ) ( de )cos( )] e2
140
δ =+[][][][]ψψδ +−+−+−+ψψδθ δψ δγ reeyyeybebe212sin cos cos 1 ( sin ) 2 2 0 =−[][]ψδ ψδψ ++−− ψδ ψ δθ δψ sinyy cos e12 cos y ( y sin b ) b e
(5.55)
δψδψδθγδθδψδγ=−[] ++++ ryyde3 sin cos ( )sin ( ) e1 ++[]ψδ ψδθ −+++−+ γ δθ δψ δγ δθ δψ cosyy sin ( de )cos ( ) c ( ) e2
(5.56)
5.5 The Equations of Motion for the 3-Axle Planar Model
Equating the linearized Lagrangian expressions in equations (5.31) through (5.33) with the virtual work expressions grouped in equation (5.49), and small-angle relationships in equations (5.19), yields the system equations of motion, given in equations (5.57) through (5.59). The system expressed in equations (5.57) through (5.59) agrees with Pacejka’s equations in [6].
++−+−=++ψγ ()[]()mmvurmcd12 2 mdFFF 2 123 (5.57)
−++++++2 mc2122()[][( dv ur J J mc ()] d r (5.58) ++2 = − − ++ []J22 m d q aF 1 bF 23 F () c d e
−++++++=−+2 mdv222223[][()][ ur J md c d r J md ] q F () d e (5.59) 141
5.6 Linearized Tire Forces
The forces, Fi, in equations (5.57) through (5.59) are the tire cornering forces, which can be roughly estimated as linear functions of the tire slip angle, as discussed in great detail by Gillespie[7], Milliken [8], and Pacejka[9].
Estimating the tire cornering forces simply as linear functions of slip angle, we can relate the forces using equation (5.60).
=− α FCiiiα (5.60)
This leaves the task of finding a hopefully linear expression for the slip angle for the “tire” (or in this case, the axle) in terms of the state variables in equations (5.57) through (5.59), above.
The slip angle is defined as the angle between the heading of the tire centerline and its vector. The respective heading and velocity can be taken with respect to the standard (fixed) reference basis {Ei}. In nonlinear form, the slip angles can be described by equations (5.61) through (5.63). Linearizing the relationships (using small angle relationships, eliminating higher order terms (H.O.T.), assuming θ <<ψ , and using equation (5.26)) yields the simpler expressions in equations (5.64) through (5.66). Note, in equation (5.66), how the semitrailer hitch articulation angle acts as a steer angle (similar to the actual steer angle, δ equation (5.64)).
142 − ⎡⎤varqycosψθψ+++ ( ) sin αδ=−tan 1 1 ⎢⎥ (5.61) ⎣⎦vysinψθψ− cos
− ⎡vbcosψψ−++ ( θθ ) y sinψ ⎤ α = tan 1 3 ⎢ ⎥ (5.62) ⎣ vysinψθψ− cos ⎦
− ⎡⎤vdecosψγθψγθψθψ−+ ( )cos ( ++− ) c ( ++ ) y sin α =−tan 1 γ 3 ⎢⎥ ⎣⎦vdesinψγθψγθψ++ ( )sin ( ++− ) y cos
(5.63)
+ αδ=−−1 ⎡⎤var 1 tan ⎢⎥ (5.64) ⎣⎦u
− α = −1 ⎡vbr⎤ 2 tan ⎢ ⎥ (5.65) ⎣ u ⎦
−++ − + α =−−1 ⎡vcderdeq()()⎤ γ 3 tan ⎢ ⎥ (5.66) ⎣ u ⎦
143 Substitution of equations (5.64) through (5.66) into equations (5.57) through (5.59), combined with the small angle relationship in equation (5.67), yields the equations of motion in equations (5.68) through (5.70).
− tan1 (λλ ) ≈ (5.67)
++−+−= ()[]()mmvurmcdrmdq12 2 2 +−−++−+ −−−−⎡⎤⎡⎤⎡varδ vbr v()() cderdeq −γ ⎤ CCC123 ⎣⎦⎣⎦⎣⎢⎥⎢⎥⎢uu u ⎦⎥
(5.68)
−++++++2 ++ + mc212222()[][( dv ur J J mc ()][ d r J mdcdq ()] +− −++−+ =−⎡⎤⎡⎤var −δ − vbr − + + ⎡ v()() cderdeq −γ ⎤ aC12⎢⎥⎢⎥ bC() c d e C 3 ⎢ ⎥ ⎣⎦⎣⎦uu ⎣ u ⎦
(5.69)
−++ − + −++++++=−2 ⎡⎤vcderdeq()() −+γ mdv222223[][()][ ur J md c d r J md ] q C () d e ⎣⎦⎢⎥u
(5.70)
144 From this point, manipulation of the system equations of motion is simply a matter of algebra and simplification to suit the user’s needs. At first, the equations of motion can be manipulated to represent a first-order system in matrix form, as depicted in equation (5.71).
=+ MxKxBu 1 (5.71)
For the representation in equation (5.71) above, the state vector is given in equation (5.72). The system matrices are then represented in equations (5.74) through (5.76). The input is simply front axle steer angle, denoted as δ.
⎡⎤yv ⎡⎤ ⎡vehicle lateral velocity ⎤ ⎢⎥ψ ⎢⎥r ⎢vehicle yaw rate ⎥ x ===⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥γ ⎢⎥q ⎢hitch articulation rate ⎥ (5.72) ⎢⎥ ⎢⎥ ⎢ ⎥ ⎣⎦γγ ⎣⎦ ⎣hitch articulation angle ⎦
⎡⎤ yv ⎡⎤ ⎡vehicle lateral acceleration ⎤ ⎢⎥ψ ⎢⎥r ⎢vehicle yaw acceleration ⎥ x ===⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥γ ⎢⎥q ⎢hitch articulation acceleration ⎥ (5.73) ⎢⎥ ⎢⎥ ⎢ ⎥ ⎣⎦γγ ⎣⎦ ⎣hitch articulation rate ⎦
145 +−+− ⎡⎤mm12() cdm 2 dm 2 0 ⎢⎥−+()cdmJ + J + () cdm +2 J + dcdm ()0 + M = ⎢⎥212 22 2 −+++2 (5.74) ⎢⎥dm22 J d() c d m 222 J d m 0 ⎢⎥ ⎣⎦00 01
146
−++ − + +++ − +2 + ⎡ ()CCC123 aCbCcdeCmmu 1 2 ()()() 3 12 deC 3 uC 3 ⎤ ⎢− + +++ −22 − −++ 2 ++ 2 −+ ++ −++ ⎥ 1 ⎢ aCbCcdeC12() 3 aCbC 1 2 ()()()()() cdeC 3 cdum 2 decdeC 3 cdeuC 3⎥ K = u ⎢ ()deC+ −+ ()() decdeCdmu ++ +22 −+ () deC −+ () deuC⎥ ⎢ 33233⎥ ⎣⎢ 0010⎦⎥
(5.75)
147
⎡ C1 ⎤ ⎢aC ⎥ B = ⎢ 1 ⎥ (5.76) 1 ⎢ 0 ⎥ ⎢ ⎥ ⎣ 0 ⎦
5.7 State-Space Representation of the 3-Axle Planar Cornering Model
In order to observe the dynamics of the model, tune the model response, and implement model controllers and/or observers, the model is manipulated to be represented in state-space format, shown in general form in equation (5.77). The formulation in equation (5.71) is converted into state-space format using the transformation in equations (5.78) and (5.79).
x =+Ax Bu (5.77)
where:
− A = MK1 (5.78)
= −1 B MB1 (5.79)
Performing the manipulation on the system to convert the system into a state- space format yields the system matrices A and B, whose coefficients are listed in Appendix B.
.
148 5.8 Verification of the 3-Axle Model
5.8.1 Initial Verification
Initially, the model coefficients were verified, after transcribing from the symbolic (Maple™) model into the numeric program (Matlab®), by comparing coefficients of the system dynamics matrix, A, with those resulting from substitution of the model constants into the system equations in Maple™. By doing so, one can readily compare model coefficients, the determinant of the initial “mass” matrix, M, as well as eigenvalues between programs, in this case Maple™ and Matlab®.
5.8.2 Verification by Comparison of the Planar Model Response with that from TruckSim™ Vehicle Dynamics Software
After verification that the coefficient expressions had been transcribed as needed, the model was coded in state-space form and verified to follow a prescribed set of steering inputs that were moderate and intended to keep the model operation within the tire’s “linear” range, and where load transfer on an actual vehicle (or three-dimensional model, for that matter) would not result in tire vertical forces reaching zero (0), which would indicate axle and wheel lift-off.
The maneuvers were studied to understand the limitations of the highly simplified linear model, as well as to verify its behavior, vis-à-vis a far more sophisticated, and proven, vehicle dynamics model.
Truck physical parameters, such as masses, inertias, and physical dimensions were taken directly from the TruckSim™ simulation input files. No physical dimensions or properties were adjusted to improve model compliance with the TruckSim™ reference.
Tire cornering coefficients, Cαi, however presented a different case. Tire lateral force development properties are complex. In addition to the tire’s force output varying significantly with surface type, a tire’s force output is a highly nonlinear function of 149 vertical load on the tire and slip angle that the tire is experiencing at the time. Additionally, all these tire force determinants are coupled. Given such factors, reducing a tire’s cornering force properties to a linear constant, devoid of coupling to vertical load, has obvious limitations.
Figure 5.4 shows the tire force vs. slip angle data for the tires used in the TruckSim™ simulation. The tire force and moment data were measured from a moving test laboratory at 72.4 kph (45 mph) over actual dry test pavement. The source of the information is well chronicled in [10]. Tire lateral force data presented in Figure 5.4 have been averaged over positive and negative slip angles, and normalized by load for clearer representation of the tire’s vertical load effect on its cornering efficiency.
Figure 5.4 Measured normalized cornering forces for a drive axle truck tire in free rolling (i.e., no braking slip)
150 Note in Figure 5.4 that any estimate of a linear cornering coefficient must depend on where it is presumed that vehicle “corner” will be operating, in terms of vertical load and slip angle. Linearizations about that point prove quite sufficient for many applications. However, many simulations use a sophisticated algorithm such as Pacejka’s “magic tire formula” [11] or Pottinger’s nonlinear regression [12]. Both models consist of a complex fit of experimental data to an empirical formula derived from sines and cosines to reproduce tire behaviors in a continuous algebraic relationship. Simulations that do not use continuous formulas such as Pacejka’s or Pottinger’s use lookup tables – which often prove quite efficient, in terms of software performance – to reproduce the tires’ output forces as functions of slip angle (α) and load (Fz).
Initially, the linear representation of an articulated heavy vehicle used approximations of the cornering coefficient taken from regimes on the tire cornering force diagrams (such as the one in Figure 5.4) that were deemed most likely for the truck axles to be operating. Tire cornering coefficients were derived for steer, drive, and semitrailer axle locations based on slip angle and vertical force outputs from the larger TruckSim™ simulations. It was realized that the cornering coefficients may need to be scaled, either to tune the model up front, or to update the model computer program periodically, depending on other model states at that time. The lumped cornering coefficients are listed, along with the other important vehicle parameters, in Table 5.1. Vehicle mass and inertia parameters listed in Table 5.1 originated from static vehicle parameters as output by the TruckSim™ simulation, and obviously include the lumped masses and inertias of vehicle payload, as well as the drive and semitrailer tandem axles. The reader is reminded that the TruckSim™ vehicle parameters originate from exhaustive measurements of an actual vehicle, experimental measurements from which were used to validate the TruckSim™ models.
151
Parameter Value Comments m1 8812 Mass – tractor, kg m2 16484 Mass – semitrailer, kg J1 46100 Yaw inertia – tractor, kg-m2 J2 4.5201e+005 Yaw inertia – semitrailer, kg-m2 a 2.062 Steer axle to tractor CG, m b 2.723 CG to drive axle, m c 2.539 Steer axle to kingpin, m d 7.483 Kingpin to trailer CG, m e 3.760 Kingpin to trailer axle, m Lumped cornering stiffness – steer C 3.8193e+005 1 axle, N/rad Lumped cornering stiffness – drive C 7.3339e+005 2 axle, N/rad Lumped cornering stiffness – C 8.8144e+005 3 semitrailer axle, N/rad
Table 5.1 Physical Parameters Used for the 3-Axle Articulated Vehicle Model at the ½ GVW Load Condition
5.8.3 Verification of the 3-Axle Planar Model by TruckSim™ Simulation Output via Light Handling Maneuvers
Verification of the planar model was performed by putting light handling maneuver inputs into the model steering, then comparing the response with the response of a TruckSim™/Simulink® hybrid simulation to the identical inputs. The inputs used were:
1. 0.5 Hz sinusoid at ± 2 degrees of amplitude. Note that the amplitude was slowly ramped up (at a rate of 0.75 steering degrees / second) to avoid abrupt dynamics.
2. 1 degree step input
3. 2 degree step input
4. 4 degree step input 152 5. slow ramp input to 2.11 degrees at a rate of 0.75 degrees / second
Note that the larger steering inputs of 4 degrees for the step input and 4.5 degrees for the ramp input were chosen because they were indeed the upper limits of steer angle input that the more sophisticated TruckSim™/Simulink® simulated vehicle could tolerate without significant nonlinear responses, such as wheel lift-off for the semitrailer. The TruckSim™/Simulink® simulation was run on a medium-µ surface having a simulated coefficient of µ=0.55. Loads and dimensions were as listed in Table 5.1. Also note that under this linear configuration, surface coefficient has little effect on the 4-state model; having constant cornering coefficients, there is mathematically no limit to the force producing capability of the “tires” in this simple model. The amplitude of 2.11 degrees for the ramp input was chosen due to the result that the TruckSim™/Simulink® simulation reached saturation – with respect to front axle cornering forces – on the medium-µ surface with steering inputs of that amplitude.
5.8.4 Discussion of 3-Axle Simulation Outputs
Simulation output is shown in Figure 5.5 through Figure 5.8 for the inputs outlined in the previous section. Co-plotted are the TruckSim™/Simulink® simulation (as the reference), and 3-axle model output with the cornering coefficients scaled by 1:1 and 0.75:1. Below are some discussion points about the output.
Agreement with the sinusoidal input, with the more sophisticated TruckSim™/Simulink® simulation is very good for all model states. Note the increase in lateral velocity response magnitude to a 25% decrease in cornering stiffness.
The model’s sensitivity to (and limitations resulting from) a constant linear tire cornering stiffness can be seen when comparing the 1-degree and 2-degree step inputs. Note in Figure 5.6 (response to 1-degree step input) that all model states, including that of lateral velocity, agree well with the reference for Cαi = 1. However, when the input amplitude doubles (see Figure 5.7), the lateral velocity output prefers the 75% scaled cornering coefficients instead of full-scale. Further understanding of the data in Figure
153 5.4 explains the disparity; for actual pneumatic tires, the cornering stiffness “rolls off” at higher steer amplitudes, reducing the “slope,” for the tire’s response.
Response to the slow ramp input only verifies the results seen above, i.e., the lateral velocity output is quite sensitive to tire cornering coefficient magnitude.
154
Figure 5.5 3-axle planar model response to sinusoidal steer input.
155
Figure 5.6 3-axle model response to 1 degree step input.
156
Figure 5.7 3-axle planar model response to a 2-degree step input.
157
Figure 5.8 3-axle planar model response to a slow ramp input.
158
5.9 Adapting the 3-Axle Model to the 5-Axle Configuration
The schematic sketch for the 5-axle planar model is shown in Figure 5.9; and the system matrices for the 5-axle system are represented below in equations (5.80) and (5.81).
These equations show that expanding the 3-axle planar model to a more representative 5-axle model does not significantly affect the model complexity. In theory, adapting the model to a more representative (of the vehicle simulated in TruckSim™) configuration of 5 axles reduces the model’s sensitivity to the tire cornering coefficients. The more accurate positioning along the vehicle centerline for the lateral forces should result in a wider range of operating regimes within which the model could “operate” with reasonable accuracy.
Inspection of the 3-axle system equations of motion (equations (5.68) through (5.70)) reveals that a simple transformation to the newer coordinate system (necessary for describing the 5-axle system) is all that is needed on the left-hand side of the equations. Although straightforward, slightly more work is needed on the right-hand side, since additional cornering force relationships must be added to the model.
The state-space 5-axle planar system coefficients are in Appendix C.
159
Figure 5.9 Schematic of the 5-axle planar truck model.
+−−− ⎡⎤mm12() acdm 2 dm 2 0 ⎢⎥()acdmJ−− + J + () acdm −−2 J − dacdm ()0 −− M = ⎢⎥212 22 2 5 −−−−+2 ⎢⎥dm22 J d() a c d m 2 J 22 d m 0 ⎢⎥ ⎣⎦00 01
(5.80)
160
−++++ −++++−+2 + + ⎡ ()CCCCC12345 aClClClClCmmu 12233445512 ()() fCfC 1425 uCC 45⎤ ⎢ ⎥ 1 −+aC lC + lC + lC + lC − aC22222 − l C − lC − lC − lC + lum 2 −()() flC + flC − ulC + lC K = ⎢ 122334455 1223344556 2 144255 4455⎥ 5 u ⎢ fC+−++− fC() flC flC dmu22 ( f C +−+fC2 )( ufC fC )⎥ ⎢ 14 25 144 255 2 1 4 25 14 25⎥ ⎣⎢ 0010⎦⎥
where :
=− lba21 =− lba32 =− lfa41 =− 161 lfa52 =+− lcda6
and new variables of: = b1 distance, steer axle to leading drive axle, = b2 distance, steer axle to trailing drive axle, = f1 distance, kingpin to leading semitrailer axle, = f2 distance, kingpin to trailing semitrailer axle.
(5.81)
and finally,
⎡ C1 ⎤ ⎢aC ⎥ CC==⎢ 1 ⎥ (5.82) 53⎢ 0 ⎥ ⎢ ⎥ ⎣ 0 ⎦
5.10 Validation of 5-Axle Model and Comparisons to the 3-Axle Model
The 5-axle model was validated in a method similar to that used to validate the 3- axle model, with several exceptions. First, a more aggressive ramp input to 4.5 degrees was added to the list of inputs. Also, tractor lateral acceleration was added to the model outputs, for comparison to the larger TruckSim™/Simulink® model. Lateral acceleration was not a state variable, so it was computed as shown in equation (5.83). This will prove useful later since lateral acceleration is more easily measured and more universally referenced then lateral velocity.
d xˆˆ=+()xur () (5.83) dt
where
xˆ = total vehicle lateral acceleration,
xˆ = model estimated vehicle lateral velocity, and
ur = estimated vehicle lateral centripetal acceleration only (product of forward velocity and yaw rate
162 Similar to the 3-axle model, verification of the 5-axle planar model was accomplished by comparing the model’s response to the response of a TruckSim™/Simulink® hybrid simulation with identical inputs. For all simulations, forward speed was held as constant at 60 kph (37.3 mph). The inputs used were:
1. 0.5 Hz sinusoid at ± 2 degrees of amplitude. Note that the amplitude was slowly ramped up (at a rate of 0.75 steering degrees / second) to avoid abrupt dynamics.
2. 1 degree step input
3. 2 degree step input
4. 3 degree step input
5. 4 degree step input
6. ramp input to 4.5 degrees at a rate of 6 degrees / second.
Note that the larger steering inputs of 4 degrees for the step and 4.5 degrees for the ramp input were chosen because they were indeed the upper limits of steer angle input that the more sophisticated TruckSim™/Simulink® simulated vehicle could tolerate without significant nonlinear responses, including wheel lift-off for the semitrailer. The TruckSim™/Simulink® simulation was run on a medium-µ surface having a simulated coefficient of µ=0.55. Lumped vehicle mass and inertia values, as well as dimensions for the simulated 5-axle tractor-semitrailer are as listed in Table 5.2. Also note that under the current linear configuration that surface coefficient has little effect on the 4-state model; having constant cornering coefficients, there is mathematically no limit to the force producing capability of the “tires” in this simple model.
The validation simulations shown in Figures 5.10 through 5.15 reveal that the model states agree very well with the TruckSim™ reference. The states of interest for estimation in Chapter 6, hitch articulation angle (γ ) and rate (γ ), show extremely good agreement with the reference and insensitivity to adjustments in the cornering coefficient scalars (Cα ). Also, the use of the added nonlinear state of total lateral acceleration, i
163 calculated in equation (5.83), improves model insensitivity to cornering stiffness adjustments over the linear state of lateral velocity
Parameter Value Comments m1 8812 Mass – tractor, kg m2 16484 Mass – semitrailer, kg J1 46100 Yaw inertia – tractor, kg-m2 J2 4.5201e+005 Yaw inertia – semitrailer, kg-m2 a 2.062 Steer axle to tractor CG, m b1 4.126 Steer axle to lead drive axle, m b2 4.452 Steer axle to trailing drive axle, m c 4.601 Steer axle to kingpin, m d 7.483 Kingpin to trailer CG, m f1 10.599 Kingpin to lead trailer axle, m f2 11.887 Kingpin to trailing trailer axle, m Lumped cornering stiffness – steer C 3.8193e+005 1 axle, N/rad Lumped cornering stiffness – drive C , C 3.6669e+005 2 3 axle, N/rad Lumped cornering stiffness – C , C 4.4072e+005 4 5 semitrailer axle, N/rad
Table 5.2 Physical Parameters Used for the 5-Axle Articulated Vehicle Model at the ½ GVW Load Condition
164
Figure 5.10 Comparison of 3- and 5-axle linear model responses to the reference for sinusoidal steer input.
165
Figure 5.11 Comparison of 3- and 5-axle linear model responses to the reference for a 1-degree step input.
166
Figure 5.12 Comparison of 3- and 5-axle linear model responses to the reference for a 2-degree step steer input.
167
Figure 5.13 Comparison of 3- and 5-axle linear model responses to the reference for a 3-degree step steer input.
168
Figure 5.14 Comparison of 3- and 5-axle linear model responses to the reference for a 4-degree step steer input.
169
Figure 5.15 Comparison of the 3- and 5-axle linear model responses to a ramp steer input.
170 5.11 The Effects of Tire Dynamic Lag on Model Response
Some discrepancies between the dynamic portions of the planar model and the TruckSim™/Simulink® simulations were thought to be caused by the tire dynamic lag [12, 13], engineered into the more sophisticated TruckSim™/Simulink®. To test this theory, the tire dynamic lag was reduced from 600 mm to 60 mm in the TruckSim™/Simulink® model and the simulation was run and compared to the planar model simulations. Figures 5.16 through 5.19 show that the tire dynamic lag has only a minor effect on the TruckSim™/Simulink® simulation, and thus the addition of tire dynamic lag to the simpler planar model would not appreciably close the gap between the two models.
Note the excellent agreement between the linear 5-axle model and the elaborate TruckSim™ nonlinear simulation for identical inputs. Note also that although agreement improves with the 5-axle model over the 3-axle model, the magnitude of improvement is very slight.
5.11.1 Linear Model Response to a Jackknife-Producing Input
Figure 5.20 shows the 5-axle linear model response to brake-in-turn maneuver on a 152.4m (500-ft) radius curve, resulting in a jackknife. The run was also simulated with the TruckSim™ / Simulink® hybrid simulation, for reference, at conditions of 0 payload, µ=0.30, and the tractor ABS system disabled. The 5-axle linear model is constantly integrated at each time step using a Runge-Kutta integrator (the code for which appears in Appendix D).
It can be seen in Figure 5.20 that the linear model closely follows the more elaborate hybrid nonlinear model until the onset of instability, where the two diverge at t≈6.5 seconds.
171 SINE Steer Response for model states -- Input frequency = 0.5 Hz Amplitude ramps to 2 deg @ 0.75 deg/s -- Forward speed (u) = 60 2 [deg] δ
0 TSim ref, original dynamic tire lag (L = 600 mm) INPUT -2 TSim ref, LOW dynamic tire lag (by factor of 10) 3-AXLE bicycle model best case - unscaled 0 1 2 3 4 5 6 5-axle model scales: C1: 1 C2: 1 C3: 1 C4: 1 C5: 1 0.2
0 lat velocity lat [m/s] 1
CG -0.2 0 1 2 3 4 5 6 ] 2 2
0 Lat. Accel. Lat. [m/s 1 -2 VEH 0 1 2 3 4 5 6
5
0 yaw rate [deg/s] 1 -5
VEH 0 1 2 3 4 5 6
5
0
-5
hitch rate angle [deg/s] 0 1 2 3 4 5 6
2
0
-2 hitch [deg] angle 0 1 2 3 4 5 6 time [sec]
3 and 5-AXLE BICYCLE MODELS source file: truck obser v8e.m prrint time: 15:19 date: 7 / 10
Figure 5.16 Linear model response compared to the reference (for 2 values of tire response lag) for a ± 2-degree amplitude sinusoidal steer input.
172 STEP Steer Response for model states -- step size = 2 deg Forward speed (u) = 60 3
2 [deg] δ
1 TSim ref, original dynamic tire lag (L = 600 mm)
INPUT TSim ref, LOW dynamic tire lag (by factor of 10) 0 3-AXLE bicycle model best case - unscaled 0 1 2 3 5-axle4 model scales: C1: 15 C2: 1 C3: 1 C4: 1 C5:6 1
0.1
0 lat velocity lat [m/s]
1 -0.1 CG 0 1 2 3 4 5 6 ] 2 2
1 Lat. Accel. Lat. [m/s 1 0 VEH 0 1 2 3 4 5 6
6 4 2 yaw rate [deg/s] 1 0
VEH 0 1 2 3 4 5 6
0
-2
-4
hitch rate angle [deg/s] 0 1 2 3 4 5 6
0
-2
-4 hitch [deg] angle 0 1 2 3 4 5 6 time [sec]
3 and 5-AXLE BICYCLE MODELS source file: truck obser v8e.m prrint time: 15:19 date: 7 / 10
Figure 5.17 Linear model response compared to the reference (for 2 values of tire response lag) for a 2-degree step input to steer angle.
173 STEP Steer Response for model states -- step size = 4 deg Forward speed (u) = 60
4 TSim ref, original dynamic tire lag (L = 600 mm) [deg] δ 2 TSim ref, LOW dynamic tire lag (by factor of 10) 3-AXLE bicycle model best case - unscaled
INPUT 5-axle model scales: C1: 1 C2: 1 C3: 1 C4: 1 C5: 1 0 0 1 2 3 4 5 6
0.2 0 -0.2 lat velocity lat [m/s] 1 -0.4 CG 0 1 2 3 4 5 6 ] 2 4
2 Lat. Accel. [m/s Lat. 1 0 VEH 0 1 2 3 4 5 6 15
10
5 yaw rate[deg/s] 1 0 VEH 0 1 2 3 4 5 6
0
-5
-10 hitch rate angle [deg/s] 0 1 2 3 4 5 6
0
-5
hitch[deg] angle -10 0 1 2 3 4 5 6 time [sec]
3 and 5-AXLE BICYCLE MODELS source file: truck obser v8e.m prrint time: 15:19 date: 7 / 10
Figure 5.18 Linear model response compared to the reference (for 2 values of tire response lag) for a 4-degree step input to steer angle.
174 FAST RAMP Steer Response for model states Amplitude ramps to 4.5 deg @ 6 deg/s -- Forward speed (u) = 60 kph
6
[deg] 4 δ
2 TSim ref, original dynamic tire lag (L = 600 mm)
INPUT TSim ref, LOW dynamic tire lag (by factor of 10) 0 3-AXLE bicycle model best case - unscaled 0 1 2 3 45- ax le model s c ales5 : C1: 1 C2: 1 C3: 1 6C4: 1 C5: 1 0.2 0 -0.2 -0.4 lat velocity lat [m/s] 1 CG 0 1 2 3 4 5 6 ] 2 4
2 Lat. Accel. Lat. [m/s 1 0 VEH 0 1 2 3 4 5 6
15
10
5 yaw rate [deg/s] 1 0 VEH 0 1 2 3 4 5 6
0
-5
-10 hitch rate angle [deg/s] 0 1 2 3 4 5 6
0
-5
hitch [deg] angle -10 0 1 2 3 4 5 6 time [sec]
3 and 5-AXLE BICYCLE MODELS source file: truck obser v8e.m prrint time: 15:19 date: 7 / 10
Figure 5.19 Linear model response compared to the reference (for 2 values of tire response lag) for a 4.5-degree step input to steer angle.
175
Figure 5.20 State-model response to the same path-following steering input showing inaccuracies at the point of jackknife (occurs at t ≈ 6.5 seconds). The “input” and TruckSim™ comparison data are from Run #836.
176 The response seen in Figure 5.20 leads to the conclusion (quite obvious to the seasoned dynamicist) that the linear cornering coefficients used in this model (Figure 5.21, left panel) are insufficient in detail to reproduce the desired jackknife phenomenon. Thus, using a tire model that includes the natural lateral force saturation is necessary (re, Figure 5.22, right panel). The appropriate tire models are developed, explained, and validated in the following section.
Figure 5.21 Comparison of linear gain (left) and linear gain with saturation (right).
177 5.12 Development and Validation for the 5-Axle Nonlinear Model
The results in Figure 5.20 show that although the purely linear model is sufficient for the spectrum of maneuvers up to those that approach the tires’ limit of lateral adhesion, the model clearly falls short of accurately following the more sophisticated TruckSim™ simulation. Hence it is clear that the model needs to accurately reflect the highly nonlinear large slip angle response of the pneumatic tire in order to accurately track that aspect of vehicle operation.
The use of a nonlinear component model leaves the modeler with two options, either the use of lookup tables and their associated algorithms or the use of a continuous albeit nonlinear mathematical expression for the nonlinear components. For this application, the continuous mathematical model approach was chosen due to its ability to be quickly tuned and the promise of stability during high frequency, high amplitude model response.
The model used to calculate the tire cornering force, as a function of vertical load and lateral slip, comes from Pottinger [14] and is shown below for the radial steer and drive axle radial truck tires modeled in his paper.
=+ +23 + D ccFcFcF01 11zz 21 31 z =+ +23 + Sctzzz04 cFcFcF 14 24 34 =+ +23 + B ccFcFcF05 15zz 25 35 z (5.84) =+ +23 + Ec06 cFcFcF 16zz 26 36 z S C = t BD
178 In modifying Pottinger’s original model, the coefficients for residual lateral and longitudinal forces (at δ = 0) are set to 0 for model stability at zero input:
== SSxy0 (5.85)
Finally, equations (5.84) and (5.85) are combined for the actual tire lateral force output:
FE=−tan−1 [ B (α S )] yxa F =−−+CBSEFtan−1 [ (α )(1 ) ] yxyba (5.86) FSD=+sin[ F ] yy yb
In equations (5.84) through (5.86), the tire vertical load, Fz, and lateral force output, Fy, are in lb-f, and the tire lateral slip angle, α, is in degrees. The corresponding coefficients, cij, for the steer axle model are listed in equation (5.87). Likewise, the coefficients for the drive axle model are given in equation (5.88) for the drive axle tire model.
D Sx Sy St B E c0j =[ -1.0337e+02 -1.1188 -8.7463e+01 3.8691e+01 5.5313e-02 8.8461e-01] c=[1j -1.1302 -5.0940e-04 -4.0421e-02 1.7837e-01 -1.2251e-05 2.7795e-04] c2j =[-6.7475e-05 -6.3361e-08 -2.3053e-06 8.8465e-06 -1.7103e-09 6.4848e-08] c3j =[-2.6407e-09 -2.4844e-12 -1.3419e-10 1.3960e-10 -7.2049e-14 4.2541e-12]
(5.87) 179
D Sx Sy St B E c0j =[ 9.6131e+01 1.8946e-01 1.3765e+01 2.7054e+01 6.4179e-02 4.9194e-01] c1j =[-8.9545e-01 1.0704e-04 -3.1592e-02 1.6133e-01 -9.5343e-06 -7.9762e-05] c2j =[-3.7244e-05 1.3082e-08 -9.7336e-07 7.4003e-06 -8.7104e-10 -2.0456e-08] c3j =[-1.0775e-09 4.0276e-13 1.3974e-11 8.6558e-11 -2.0294e-14 -1.1298e-12]
(5.88)
The model expressed in equations (5.84) through (5.86) does not correct for the loss of lateral tire force as the tire longitudinal slip increases from the free-rolling value of zero (0). For that application, the following exponential mathematic model was found to be quite accurate and worked well in simulating the tires’ loss of lateral tractive force at the high levels of longitudinal slip that occur during braking.
FFccecorr =+()cSR3 (5.89) yyii12
where
F corr = lateral tire force, corrected for longitudinal slip corrected for the ith axle, yi
F = free rolling lateral force calculated for the free-rolling tire, yi
ci = empirically derived constants, and
SR = longitudinal slip ratio.
180 Finally, the tire lateral force was corrected for the appropriate surface coefficient using simple linear scaling:
⎛⎞µ FF= corr ⎜⎟act (5.90) yi yi ⎜⎟µ ⎝⎠ref
where µ act = actual surface coefficient desired, and
µ act = the ASTM measured coefficient on a dry surface.
µ = ref 0.95 for the all simulations discussed.
The results from implementing the combination of tire lateral traction models expressed in equations (5.84) through (5.89) are presented below. Figure 5.22 shows the tire model output under free-rolling conditions, at vertical loads for each of the five model axles corresponding to the “1/2 GVW” total vehicle weight. Note how the lateral traction offset, at steer angle δ = 0, is deliberately set to zero to ensure that the planar model has the intended steady-state response at 0 input. Including “ply steer” effects, producing essentially an offset in lateral force at δ = 0, is more accurate than the experimental data, but will result in a non-zero response with a zero input for the planar model. Accuracy of the free-rolling tire force models are quite acceptable.
Figure 5.23 shows the output of the compensated lateral force at three lateral slip angles (α = -2 and -4) after being compensated for slip ratio, as expressed in equation (5.89). The constants used for this demonstration were
= c1 0.05 = c2 0.95 (5.91) = c3 3.471512
181
The empirically derived coefficients used in running most of the simulations shown here and in Chapter 7 were slightly different:
= c1 0.15 = c2 0.85 (5.92) = c3 3.471512
The coefficients shown in equation (5.92) were developed more thoroughly, using normalized tire lateral force and several vertical loads, for both steer axle and drive axle truck tire lateral force. Finally, note that the sum of c1 and c2 should be unity, corresponding to “full” lateral force at 0 longitudinal slip (κ = 0).
Implementing the complete tire model derived above also results in a far simpler set of state-space equations, as listed in equation (5.93) through (5.95). Note that the “mass” matrix remains unchanged from the 5-axle model using a linear tire model, and that the input matrix is simply the identity matrix.
182
+−−− ⎡⎤mm12() acdm 2 dm 2 0 ⎢⎥()acdmJ−− + J + () acdm −−2 J − dacdm ()0 −− M = ⎢⎥212 22 2 (5.93) 5−non −−−−+2 linear ⎢⎥dm22 J d() a c d m 2 J 22 d m 0 ⎢⎥ ⎣⎦00 01
−+ ⎡0(mmu12 )00⎤ ⎢0(−−−ma c du )00⎥ = ⎢ 2 ⎥ K5−nonlinear (5.94) ⎢000mdu2 ⎥ ⎢ ⎥ ⎣0010⎦
⎡1000⎤ ⎢0100⎥ C − = ⎢ ⎥ (5.95) 5 nonlinear ⎢0010⎥ ⎢ ⎥ ⎣0001⎦
183
Figure 5.22 Experimentally measured and modeled truck tire free rolling cornering force for all five axle applications at a single load. The model is labeled “combinator” in the legend.
184 Modeled Drive Axle Tire Cornering Force vs. Longitudinal Slip Tire Load = 41204 N (9263 lb ) f 18000 experimental @ α =-2 deg model @ α =-2 deg 16000 experimental @ α =-4 deg model @ α =-4 deg -4 degrees lateral slip 14000
12000
10000
8000 cornering force (N)
6000
4000 -2 degrees lateral slip
2000
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 longitudinal braking slip, κ
Figure 5.23 Tire lateral force at various lateral slip angles, modeled as a function of longitudinal wheel slip.
5.13 Performance of the 5-Axle Planar Nonlinear Model for Simulating the Jackknifing Event
The modeling techniques discussed in the previous sections were combined, resulting in a robust nonlinear model that is well suited for simulating articulated vehicle behavior, even for highly nonlinear operating regimes.
Lateral load transfer was taken into account by implementing a simple scalar onto the static tire vertical forces. The tire vertical load scalars were empirically adjusted to maximize model performance over the spectrum of steer angle inputs and lateral acceleration regimes for a given vehicle mass / surface coefficient operating regime. In operation, these values could be scaled in real time as function of measured (and observed) lateral acceleration. The empirically derived vertical load corrections are left for future research, and would take on the appearance of
185 CfuA= ( , ,...) (5.96) Fyzi
where
u = forward vehicle speed, and
Ay = vehicle lateral acceleration.
Following are several model validation run outputs wherein the 5-axle model was compared to the far more complex TruckSim™ / Simulink® hybrid model output. Note that vehicle lateral velocity is the model state most sensitive to vertical load, F . Also zi note that the model remains a compromise in that lateral velocity is better predicted with the vertical load scales C = 1.10 for the 1-degree step input (Figure 5.25), but is better Fzi predicted with C = 0.90 for the 4-degree step input (Figure 5.26) as well as the 4.4- Fzi degree ramp input (Figure 5.27). Finally, note that the auxiliary state of lateral acceleration, expressed in equation (5.97) below, is much less sensitive to tire vertical load adjustments than is lateral velocity, and is a much more standard and reachable measurement for actual vehicles.
d AVur=+() (5.97) yydt
where
Error! Objects cannot be created from editing field codes. = model lateral velocity,
u = model forward velocity, and
r = model yaw rate.
186 SINE Steer Response for NON-Linear model states -- Input frequency = 0.5 Hz Amplitude ramps to 2 deg @ 0.75 deg/s -- Forward speed (u) = 60 2 [deg] δ
0
INPUT -2 steer input - both models 0 1 2 3 4 5 6 0.2
0
Truc kSim/Simulink Model lat velocity lat [m/s] 1 90% Fz-drive CG -0.2 100% Fz-drive 0 1 2 3 4 5 6 110% Fz-drive ] 2 2
0 Lat. Accel. Lat. [m/s 1 -2 VEH 0 1 2 3 4 5 6
5
0 yaw rate [deg/s] 1 -5
VEH 0 1 2 3 4 5 6
5
0
-5
hitch angle rate [deg/s]hitch angle 0 1 2 3 4 5 6
2
0
-2 hitch[deg] angle 0 1 2 3 4 5 6 time [sec]
5-AXLE NON-Linear BICYCLE MODEL source file: truck obser v10.m prrint time: 11:56 date: 7 / 11 Figure 5.24 Nonlinear model response to sinusoidal steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison.
187 Step Steer Response for NON-Linear model states -- Input Magnitude = 1 deg Forward speed (u) = 60 2 [deg]
δ 1
steer input - both models INPUT 0 0 1 2 3 4 5 6 TruckSim/Simulink Model 0.05 90% Fz-drive 100% Fz-drive 0 110% Fz-drive
-0.05 lat velocity lat [m/s] 1
CG -0.1 0 1 2 3 4 5 6 ] 2 1
0.5 Lat. Accel. [m/s 1 0 VEH 0 1 2 3 4 5 6
4
2 yaw rate [deg/s] 1 0 VEH 0 1 2 3 4 5 6
0
-1
-2
-3 hitch rate angle [deg/s] 0 1 2 3 4 5 6
0
-1
-2 hitch [deg] angle -3 0 1 2 3 4 5 6 time [sec]
5-AXLE NON-Linear BICYCLE MODEL source file: truck obser v10.m prrint time: 11:56 date: 7 / 11
Figure 5.25 Nonlinear model response to 1-degree step steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison.
188 Step Steer Response for NON-Linear model states -- Input Magnitude = 2 deg Forward speed (u) = 60 3
2 [deg] δ steer input - both models 1 INPUT 0 0 1 2 3 4 5 6 TruckSim/Simulink Model 0.1 90% Fz-drive 100% Fz-drive 0 110% Fz-drive
-0.1 lat velocity lat [m/s] 1
CG -0.2 0 1 2 3 4 5 6 ] 2 2
1 Lat. Accel. Lat. [m/s 1 0 VEH 0 1 2 3 4 5 6 8 6 4 2 yaw rate [deg/s] 1 0 VEH 0 1 2 3 4 5 6
0
-2
-4
-6 hitch rate angle [deg/s] 0 1 2 3 4 5 6
0
-2
-4 hitch [deg] angle -6 0 1 2 3 4 5 6 time [sec] 5-AXLE NON-Linear BICYCLE MODEL source file: truck obser v10.m prrint time: 11:56 date: 7 / 11
Figure 5.26 Nonlinear model response to 2-degree step steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison.
189 Step Steer Response for NON-Linear model states -- Input Magnitude = 4 deg Forward speed (u) = 60
4 [deg]
δ steer input - both models 2 INPUT 0 0 1 2 3 4 5 6 TruckSim/Simulink Model 0.2 90% Fz-drive 100% Fz-drive 0 110% Fz-drive -0.2 lat velocity lat [m/s] 1 -0.4 CG 0 1 2 3 4 5 6 ] 2 4
2 Lat. Accel. Lat. [m/s 1 0 VEH 0 1 2 3 4 5 6 15
10
5 yaw rate [deg/s] 1 0 VEH 0 1 2 3 4 5 6
0
-5
-10 hitch angle rate [deg/s] angle hitch 0 1 2 3 4 5 6
0
-5 hitch angle [deg] hitch angle -10 0 1 2 3 4 5 6 time [sec]
5-AXLE NON-Linear BICYCLE MODEL source file: truck obser v10.m prrint time: 11:56 date: 7 / 11 Figure 5.27 Nonlinear model response to 4-degree step steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison.
190 FAST RAMP Steer Response for model states Amplitude ramps to 4.42 deg @ 6 deg/s -- Forward speed (u) = 60 6 steer input - both models 4 [deg] δ
2 INPUT 0 0 1 2 3 4 5 6 0.2 TruckSim/Simulink Model 0 90% Fz-drive 100% Fz-drive -0.2 110% Fz-drive -0.4 lat velocity lat [m/s] 1 CG 0 1 2 3 4 5 6 ] 2 4
2 Lat. Accel. [m/s 1 0 VEH 0 1 2 3 4 5 6
15 10 5 yaw rate[deg/s] 1 0 VEH 0 1 2 3 4 5 6
0
-5
-10 hitch angle ratehitch [deg/s] angle 0 1 2 3 4 5 6
0
-5
-10 hitch angle [deg] hitch angle 0 1 2 3 4 5 6 time [sec]
5-AXLE NON-Linear BICYCLE MODEL source file: truck obser v10.m prrint time: 11:56 date: 7 / 11 Figure 5.28 Nonlinear model response to 4.4-degree ramp steer input. The tire vertical load scale factors have been adjusted ± 10% for comparison.
191 5.13.1 Successful Simulation of the Jackknife Event
Finally, in Figure 5.29, the nonlinear model is subjected to identical parameters and inputs used for the linear model (in Figure 5.20). The revised nonlinear model responds far more accurately to the maneuver than does the linear model, and the states we wish to track, hitch angle and hitch angular rate, are tracked accurately with the nonlinear model.
192 NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 836 3
2 Steer Axle Scale: 1.00 Steer Axle Load: -49951 N [deg] δ
Drive Axle Scale: 1.10 Drive Axle Load: -31426 N 1 Trailer Axle Scale: 1.00 Trailer Axle Load: -26085 N INPUT
0 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.2 0 -0.2 simulation µ level: 0.30 TruckSim signal -0.4 reference µ level: 0.95 model estimate lat velocity lat [m/s]
1 -0.6
CG -0.8 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 10
5 Yaw Rate [deg/s] Rate Yaw 1 0 VEH 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
0
-2
-4 articulation rate [deg/s] articulation 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
0
-2
-4 point of divergence for linear model articulation angle [deg] angle articulation 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v12.m prrint time: 12:35 date: 7 / 11
Figure 5.29 5-axle nonlinear model response to steering input that resulted in a jackknife.
193 5.14 Conclusions and Recommendations
A 3-axle planar dynamic model was thoroughly derived using Lagrangian techniques, then linearized for ease of application. The 3-axle model was shown to agree satisfactorily with a more sophisticated 3-dimensional model for the same boundary conditions, initial conditions, and system inputs, for mild handling maneuvers.
The 3-axle planar model was then evolved into a linear 5-axle planar model to improve model integrity. The 5-axle model was found to be quite accurate up to, but not including, inputs and initial conditions that resulted in a jackknife instability. Since jackknife instability is the response we wish most to detect, further development was deemed necessary.
A nonlinear tire model was used in conjunction with the 5-axle planar model to enhance the model performance in operating regimes where tire lateral force is near the point of saturation. Further, the vehicle model operating regimes necessitated the addition of an exponential field to simulate loss of tire lateral traction at high levels of longitudinal slip
The nonlinear model worked very well to extend the ranges of the original linear model into the nonlinear regimes of operation.
194 5.15 Chapter 5 References
1. E. Milich, “An Evaluation of VDM Road and VDANL Vehicle Dynamics Software for Modeling Tractor-Trailer Dynamics” (master’s thesis, The Ohio State University, 1999). 2. A.L. Dunn, “The Effects of Tire Free Rolling Properties on Analytical Predictions for Heavy Truck Roll and Yaw,” NHTSA internal report, 24 September 2001. 3. J.S. Török, Analytical Mechanics with an Introduction to Dynamical Systems (New York: John Wiley and Sons, 2000), 102-107. 4. Maple version 8 was used for this work, (http://www.mapleapps.com/maplelinks/sh_resources.shtml). 5. J.S. Ginsberg, Advanced Engineering Dynamics (Cambridge, Eng: Cambridge University Press, 1995), chap. 6. 6. H.B. Pacejka, Tire and Vehicle Dynamics (Warrendale, PA: Society of Automotive Engineers, 2002), 55-59. 7. T. Gillespie, Fundamentals of Vehicle Dynamics (Warrendale, PA: Society of Automotive Engineers, Inc., 1992), 199, 348-353, 360-361. 8. W.F. Milliken and D.L. Milliken, Race Car Vehicle Dynamics (Warrendale, PA: Society of Automotive Engineers, 1995), 25. 9. H.B. Pacejka, Tire and Vehicle Dynamics (Warrendale, PA: Society of Automotive Engineers, 2002). 10. M.G. Pottinger and W. Pelz, “Free-Rolling Cornering, Straight-Line Braking, and Combined Cornering and Braking for a 295/75 | 275/80R22.5 Drive, Axle Tire,” Smithers Scientific Services Inc., Ravenna, Ohio. (Prepared for SAE Cooperative Research under NHTSA contract number DTNH22-92-C-17189). 11. H.B. Pacejka, Tire and Vehicle Dynamics (Warrendale, PA: Society of Automotive Engineers, 2002), 181-182, 210. 12. TruckSim™ Version 5.0 User Manual (Ann Arbor, MI: Mechanical Simulation Corporation, 2003). 13. G.J. Heydinger, W. R. Garrott, and J. P. Chrstos, “The Importance of Tire Lag on Simulated Transient Vehicle Response,” SAE SP-861. 14. D.J. Schuring, W. Pelz, M.G. Pottinger, “A Model for Combined Tire Cornering and Braking Forces,” SAE 960180.
195 CHAPTER 6
IMPLEMENTATION OF THE EXTENDED KALMAN FILTER TO AN ARTICULATED VEHICLE MODEL FOR THE PURPOSE OF JACKKNIFE PREDICTION
6.1 Abstract
In this chapter, a basic overview of the theories and principles behind the Kalman Filter (KF) – also referred to as an “optimal estimator” in various texts – and the Extended Kalman Filter (EKF) are discussed and presented. The EKF is applied to the articulated vehicle model developed in Chapter 6 to predict jackknifing. Tuning, performance, and compromises for the EKF are discussed.
6.2 Linear Observer Theory and How It Applies to Kalman Filters
An overview of linear observer theory is helpful to understand the theory and utility of Kalman Filters. Simply stated, linear observers perform the task of estimating system state behaviors – which cannot be directly measured – by employing the combination of a linear system model with information from model states that are
196 available for direct measurement. Basic linear observer theory states that, if the system is observable [1, 2], then the unmeasured states can be expressed as a linear combination of the measured and model-estimated states. The purpose of this section is not to thoroughly discuss the theories and mathematics behind linear observers, but rather review their principles.
6.2.1 Linear State Observers
A linear observer estimates states that cannot be directly measured, by assuming that the unmeasurable states can be expressed as a linear combination of the measured states (Cy), system inputs (u), and model estimated states ( xˆ ). For a state-space represented system shown in equation (6.1) wherein the state derivatives can be expressed as linear combinations of the previous states (x), system inputs (u), and external excitations (eTd), the state outputs (y) can also be expressed as a linear combination of the previous system states.
=++ x Ax Bu eTd (6.1) yCx=
For the system illustrated in Figure 6.1, it can be shown that estimator output is a linear combination of the previous state estimate, and the error between state estimates and measured states, system input, and modeled external disturbance as expressed in equation (6.2). Equation (6.3) shows the state estimate error as the difference between the actual and estimated states. Rearranging equation (6.2) leads to a modified state- space representation of the linear estimator, which will have its own unique dynamics.
Combination of equations (6.1), (6.2), and (6.3) results in equation (6.4), which simplifies to equation (6.5), yielding a simple dynamic system for the state measurement
197 error, the dynamics of which rely on the system matrix (A), the measurement matrix (C, which is degenerate), and the state estimate error or observer gains, (L).
=+ −++ xˆˆAx L() yy ˆ eTd Bu =+ − + + AxLCxCxeTBuˆˆ[]d (6.2) =− + + + xˆˆ[]ALCxLCxeTBud
x =−xxˆ (6.3) x =−xx ˆ
xx −= []() ALCxx − − + LCxeTBu + + d (6.4) −= / − − − + + + − + + x [][]ALCxALCxLCxeTBuAxBueTdd ( )
x =−[]ALCx (6.5)
198
199
Figure 6.1 Schematic of a linear state observer.
Finding the determinant of the error system matrix in equation (6.5) and setting it to 0 leads to system characteristic equations (via equations (6.6), (6.7), and (6.8), shown here for a two-state system). The designer proceeds by choosing system eigenvalues (λ*) to give the desired observer dynamics (rise time, settling time, etc.) and equating the coefficients of the characteristic equation in equation (6.9) to the polynomial for the λ desired eigenvalues ( *), solving for the observer gains, lij .
detλIALC−− [ ] =/ 0 (6.6)
⎡⎤λ 0/ ⎛⎞⎡⎤⎡⎤aa l det−−11 12 1 []cc =/ 0 (6.7) ⎢⎥/ λ ⎜⎟⎢⎥⎢⎥12 ⎣⎦0 ⎝⎠⎣⎦⎣⎦aa21 22 l 2
⎡⎤⎡⎤λ −−aa lclc det11 12−=/ 1 1 1 2 0 ⎢⎥⎢⎥−−λ (6.8) ⎣⎦⎣⎦aalclc21 22 2 1 2 2
ρλ* ≡= ρλ ()iiijii (,,,)0alc (6.9)
200 6.3 The Linear Observer as the Kalman Filter
The Kalman Filter (KF) is an extension of linear observer theory with the addition of updating the linear filter gains, for every observation. The Kalman Filter is recursive,1 unbiased,2 and minimum variance.3 The optimal gain updates are a function of the system dynamics and the known, or estimated, measurement system and dynamic system noise covariances. Hence, the user is assumed to have some prior knowledge (or estimate) of the process noise (or variation) and the measurement noise for the measured states. The observer output state vector adjusts for each iteration (time step) by comparing the model estimated states with the observer estimated states vis-à-vis the estimated and actual process noise.
A good summary of optimal estimation, which is what Kalman Filters do, can be found in Gelb [3]: “An optimal estimator is a computational algorithm that processes measurements to deduce a minimum error estimate of the state of a system by unitizing: knowledge of system and measurement dynamics, assumed statistics of system noises and measurement errors, and initial condition information.”
The Kalman Filter works well in practice as well as being theoretically attractive, due to the fact that it can be shown that the Kalman Filter has the unique property of minimizing the variance of the estimation error [4].
1 does not require previous data to be kept in storage; i.e., the system is reprocessed every time a new measure state vector becomes available.
2 Having a mean error of 0
3 variance of the estimate can be the minimum achievable
201 6.4 Background
Recent literature on applications of the Kalman Filter to estimate processes is varied and plentiful. The “smart observer” is being used in every engineering field, from monitoring chemical reactions in plant environments [4, 5] to estimating states in an operating gas turbine engine [6]. An extremely thorough and well referenced discussion of the development and early application of the Kalman and Extended Kalman Filters are in Chrstos’ Ph.D. dissertation [7]. Academic and industry resources are plentiful in their discussion of the topic, many of which are well documented in [7].
The first published discussion of the Kalman estimator (actually a predictor) occurred in [8] where R. E. Kalman published his mathematical modification to the Bucy filter in 1961.
Beneficial instructional resources are the published documents by Welch and Bishop [9, 10], as well as textbooks by Sorenson [11] and Brookner [12]. A very informative basic discussion on the topic is covered by Professor Dan Simon of the Department of Electrical Engineering at Cleveland State University [13, 14].
6.5 The Kalman Filter Equations and Operation Philosophy
The Kalman (or Kalman-Bucy) filter is the most common type of optimal filter being employed today. One advantage to its real-world application is its recursive abilities, i.e., previous measurements need not be stored to compute the current estimate. The basic constructs behind the Kalman Filter are presented here. The discussion will begin with a dynamic system in state-space form, as shown in equation (6.10).
202 x =++Ax Bu Gv (6.10) zCxw=+
where:
x = the model state vector,
u = known vector of inputs into the modeled system,
z = available (and incomplete) state measurement, and
w, v = “white” noise having known spectral densities.
Therefore, for a linear system Kalman observer, the observed states can be expressed as in equation (6.11).
xˆˆ =++Ax Bu K[] z − Cx ˆ (6.11)
where:
xˆ = observed state vector,
u = known vector of inputs,
z = available (and incomplete) state measurement
C = state measurement matrix (not full rank), and
K = the optimal gain matrix.
Given that the error can be defined as in equation (6.12), equations (6.10) through (6.12) can be combined to derive a differential equation for the error, given in equation (6.13).
203 exx=−ˆ (6.12)
e =−[] AKCeFvKw +− (6.13)
Allowing P(t) (the expected value of the error, e) to be the estimate error covariance as described in equation (6.14), P(t) can be differentiated (from Gelb [3]) as follows in equation (6.15).
PEee= []T (6.14)
− Pˆˆˆˆ=+AP PATT − PC W1 CP ˆ + GVG T (6.15)
where:
P = estimate error covariance matrix,
A = state-space system matrix,
G = system noise covariance estimate matrix, and
V = spectral density of the process noise, v
From equation (6.15), P(t) can be minimized, as expressed in equation (6.16).
PAPPAGVG =+TT + (6.16)
where the innovation or residue (W) is expanded in equation (6.17).
204 WCPCR=+T (6.17)
And finally, the optimal gain is expressed in equation (6.18).
− Kˆˆ= PCT W 1 (6.18)
The process operation is illustrated Figure 6.2 in discrete loop form in which the state estimate variables are denoted by the following:
− xˆk = observer predicted states, integrated from previous observer estimate and inputs,
xˆk −1 = the previous observer state estimate, and
xˆk = the updated filter estimate.
205 0) Initialize state vector and error covariance matrix
xPˆ00//
Time Update 5) Update error covariance using “Prediction” Phase Kalman gains and previous error covariance
1) Project the state ahead by =− − numerical integration of the linear Pkkk[]IKCP model − =+ xˆˆkkAx−1 Bu k
4) Update state estimates using measured states zk and Kalman gains 2) Project the error covariance ahead ˆˆ=+−−⎡ − ˆ⎤ using the previous covariance and xkkxKzCx kkk⎣ ⎦ estimated process noise covariance − =+T PkkAP−1 A Q
Measurement Update “Correction” Phase
3) Compute the Kalman gain using projected error covariance and measurement noise covariance
=+−−TT −1 KPCCPCRkk[] k
Figure 6.2 Flow diagram illustrating the implementation of the Kalman Filter.
206 6.6 Extrapolation of the Kalman Filter into the Extended Kalman Filter for Nonlinear Systems
A natural extension of the linear filter, the Extended Kalman Filter (EKF) has been applied to many nonlinear system models since its introduction. One impressive (and inspiring) example is Chrstos’ application of the EKF to a three-dimensional model of a ground-effects, open-suspension race car to estimate the states of tire lateral slip and body side-slip. The differences between the linear KF and nonlinear EKF lie in the fact that in the EKF the error covariance matrix, P, which is updated for each integration, is actually integrated with respect to time. The system and measurement matrices (“F” and “H,” respectively, presumed nonlinear) are also linearized about their operating point for each iteration. For this exercise, the linearization was accomplished via a routine that numerically computes the Jacobian of both matrices at each iteration. It is apparent that algebraically applying the Jacobians (in closed form) for the system and measurement matrices would indeed save computation time during EKF execution.
As per the usual definition, if the nonlinear system equations can be expressed as
x = Fxut(,,) (6.19)
for the system dynamics matrix, and for the measurement system matrix:
hHxut= (,,) (6.20)
then the Jacobian for each can be expressed as
∂f F = i ij ∂ (6.21) x j
and likewise
207
∂h H = i ij ∂ (6.22) x j
where from equations (6.19) and (6.20) the nonlinear functions can be expressed as
f11(,,)xutj 1 f (,,)xut Fxut(,,)= 22j 2 (6.23)
fnnjn(,,)xut
and
hx11(,,)j ut 1 hx(,,) ut Hxut(,,)= 22j 2 (6.24)
hxnnjn(,,) ut
and the differential equation for the now time-dependant error covariance matrix becomes
− PFPPFQ =+T + (6.25)
or, for discrete systems
− =+T + PFPPFQkkkkk−−11 (6.26)
208 where P− indicates the a priori, or system predicted value for the error covariance matrix. The Kalman gain matrix computation also changes somewhat from equations (6.17) and (6.18), which evolves to
=+−−TT −1 KPHHPHRkk() k (1.27)
and, finally, the observed state vector update equation becomes
=+− − xˆˆkkxKzz kkk()ˆ (6.28)
where
zk = vector of measured states, and
zˆk = vector of the system estimated states.
The EKF system flow diagram evolves to that displayed in Figure 6.3.
209 0) Initialize state vector and error covariance matrix
xPˆ00//
Time Update 7) Update error covariance using “Prediction” Phase Kalman gains and previous error covariance 1) Project the state ahead using the =− − Pkkk[]IKHP linear system model and inputs
− =+ xˆˆkkAx−1 Bu k 6) Update state estimates using measured states zk and Kalman gains ˆˆ=+− ⎡⎤ −ˆ 2) Calculate the estimated xkkxKzz kkk⎣⎦ measurement vector = zhxutˆkk(,,)ˆ 5) Compute the Kalman gain using projected error covariance and 3) Retrieve numerical Jacobian for measurement noise covariance the system and measurement system =+−−TT −1 matrices KPHHPHRkk[] k ∂∂f h F ==ii, and H ij∂∂ ij x jjx Measurement Update “Correction” Phase
4) Project the error covariance ahead using the previous error covariance and estimated process noise covariance − =+T + PkkkFP−−11 P F Q
Figure 6.3 Schematic for the Extended Kalman Filter operation.
210 6.7 Application of the Linear and Extended Kalman Filter to the Planar Articulated Vehicle Model
6.7.1 Stated Goals
Real-time on-board jackknife prediction is the critical first step in triggering a vehicle stability system, which in turn manipulates the vehicle brakes and/or throttle to assist the driver in stabilizing the vehicle.
As shown in Chapter 5, the jackknife event can clearly be observed using state phase plots of the hitch articulation angle, wherein both states of hitch angle and rate grow unbounded. The 5-state model discussed in Chapter 5 has the capability to model the states of vehicle articulation angle and its rate of change.
The goal of this chapter’s research is to apply the Extended Kalman Filter algorithm, discussed above, to the 16-state nonlinear planar model developed and verified in Chapter 5, to predict a jackknife event of a “heavy” articulated vehicle on a low-µ surface.
6.7.2 Outline of the Extended Kalman Filter Algorithm
The Extended Kalman Filter Algorithm illustrated in
Figure 6.3 was employed using a Matlab routine to estimate the state variables of hitch angle (γ) and hitch angle rate (γ ).
The filter algorithm uses a Runge-Kutta matrix integration routine, along with an Adams-Bashforth predictor-corrector, which works quite efficiently, in terms of routine run time. A copy of the integration routine is located in Appendix D.
The Matlab EKF routine process proceeds as follows.
1. Estimate model parameters to match those of the actual “vehicle”
a. mass and inertia for tractor and semitrailer
211 b. longitudinal C.G. placement for tractor and semitrailer
c. estimate tire normal loads
2. Load in data file containing TruckSim™ / Simulink® simulation output, which in this experiment substitutes for the “vehicle”
a. apply white noise using Matlab “randn” function to the “measured” states of vehicle 1 (tractor) forward speed, vehicle 1 forward velocity, and vehicle 1 yaw rate
b. Low-pass filter white noise at 10 Hz
3. Estimate measurement error covariance matrix (R) and process noise covariance matrix (Q)
4. Initialize state estimate ( xˆ ), and state error covariance matrix (P)
⎡⎤0 ⎢⎥0 a. xˆ = ⎢⎥, and 0 ⎢⎥0 ⎢⎥ ⎣⎦0
b. P0 = R
5. Begin loop (for k = 2 : end of data file) –
6. Calculate lateral wheel slip values based on previous model states, xˆk −1
7. Use select-low strategy to determine longitudinal wheel slip for each axle position
8. Call subroutine to calculate tire lateral forces based on estimated vertical load and lateral slip
9. Scale tire lateral force based on minimum level (maximum magnitude) longitudinal slip at each axle position
10. Scale tire lateral force to surface µ estimate
11. Calculate the state-space input function, u = f(Fyi)
12. Retrieve state matrices (A, B, C, D) from the 5-axle planar model (re: Chapter 5) with the updated vehicle speed, u
212 13. Call integrator to integrate the state model and update the predicted (or a − priori) state estimate, xˆk , to the next time step (from k-1 to k)
− =+ a. xˆˆkkAx−1 Bu
14. Differentiate previous predicted lateral velocity states, with respect to time
15. Calculate full lateral acceleration
a. AVu=+ ψ yykkkk
16. Update estimated measurement vector, zˆk
⎡⎤Ay ⎢⎥k ψ a. zˆ = ⎢⎥k k ⎢⎥0 ⎢⎥ ⎣⎦⎢⎥0
17. Retrieve numeric Jacobian for the system and measurement matrices (Linearizing system about current operating point)
∂f a. F = i ij ∂ x j
∂h b. H = i ij ∂ x j
18. Update the predicted (or a priori) measurement covariance matrix (P-) via Adams-Bashforth numeric integrator
− =+T + a. PFPPFQkkk−−11
19. Compute the Kalman gains for the current time step
=+−−TT −1 a. KPHHPHRkk() k
20. Update the state estimate vector using a linear combination of the predicted (or a priori) state estimate and current measured states, along with the current Kalman gain
=+− − a. xˆˆkkxKzz kkk()ˆ 213 21. Finally, update the state covariance matrix (to the a posteriori state) using the current Kalman gains and the predicted (or a priori) state covariance matrix
=− − a. PIKHPkkk( )
22. End of Loop ( k = k + 1), return
6.8 Tuning the Extended Kalman Filter to Optimize Its Performance
As stated by Welch and Bishop [9], tuning the EKF is a compromise between speed and accuracy of tracking the states of interest. In actual operation, there are two system noise magnitudes that must be known or estimated to an acceptable level of accuracy.
The first noise matrix is the measurement error covariance matrix, R, which should accurately reflect the spectral content (in terms of covariance about the true measurement value) of the measurement system error for each channel. Most references deal with determining only the diagonal of the matrix R, which are admittedly the variance of each measurement signal (i.e., the square of the standard deviation, χ). However, there are some references that discuss calculating the off-diagonal terms as functions of the states which are co-related by that matrix entry [12, 15]. In general terms, the equation for R would look like
χ 2 ⎡⎤(xˆˆˆˆˆˆˆ1121314 ) cov(xx , ) cov( xx , ) cov( xx , ) ⎢⎥ cov(xˆˆ ,x )χ 2 ( x ˆ ) cov( xx ˆˆ , ) cov( xx ˆˆ , ) R = ⎢⎥12 2 23 24 (6.29) ⎢⎥cov(xˆˆ ,xxxx ) cov( ˆˆ , )χ 2 ( ˆ ) cov( xx ˆˆ , ) ⎢⎥13 23 3 34 χ 2 ⎣⎦⎢⎥cov(,)cov(,)cov(,)xxˆˆ14 xx ˆˆ 24 xx ˆˆ 34 () x ˆ 4
For this application, the entries for the matrix R were estimated as follows:
214 χχχψ222 ⎡⎤()AAyy ()() 0 0 ⎢⎥ χχψχψχγχψ22(Ay ) () 2 () 22 () () 0 R = ⎢⎥ (6.30) ⎢⎥0()()()0χγχψ22 χγ 2 ⎢⎥2 ⎣⎦⎢⎥000()χγ
The off-diagonal terms were justified by understanding their relationship in the mathematical model. For those state relationships not explicitly expressed in the mathematical model, the related off-diagonal term for R remained “0”.
For this application the system deviation (or noise) was not well understood, so the system noise covariance matrix Q, was simply estimated along the diagonal as a fraction of the state measurement variances:
χ 2 ⎡ ()xˆ1 0 0 0 ⎤ ⎢ ⎥ 0()00χ 2 xˆ Qe=1 −5 ⎢ 2 ⎥ (6.31) ⎢ 00()0χ 2 xˆ ⎥ ⎢ 3 ⎥ χ 2 ⎣⎢ 000()xˆ4 ⎦⎥
Below are several simulations using widely varied magnitudes for the measurement error covariance matrix to illustrate the effects that the matrix can have on performance of the EKF.
6.8.1 Comparisons of Three Estimates for Error Covariance
The EKF algorithm was further checked for integrity by varying the measurement error covariance matrix, R, to verify that the state estimates corresponded logically, as discussed by Welch and Bishop [9]. The entire matrix was scaled by factors of 106 and the simulation/EKF rerun to gage the effects.
215 The maneuver discussed in this section is a double lane change in a corner, conducted on a µ = 0.55 surface, with an entry speed of 73.0 kph (45.4 mph). No speed control or brakes were applied, allowing the simulation speed to drop slightly from 73 to about 67 kph.
The measurement error covariance matrix magnitude was adjusted to reflect more or less faith in the system measurements. A higher value in the error covariance matrix would essentially tell the EKF that there was less faith in the measurements – a lower value would indicate stronger faith in the measurements [9].
The optimal and high (by a factor of 106) settings for R were quite similar in the way the EKF tracked the actual simulation, initially and longer term. However, the EKF run which used a low estimate for R (also by a factor of 106) showed higher initial error plus higher dispersion from the original simulation. The noise in the EKF force predictions was also significantly higher in comparison to the other two choices for R.
6.8.1.1 The Case of Optimal Settings for R
Initially, the EKF performance is shown for what should be the optimal setting for R (i.e., variances that were superimposed upon the TruckSim™ outputs to simulate system noise). In each case, only close-ups of the estimated states for the initial few seconds of each simulation are presented to show the effects of the error covariance magnitudes on speed of response and tracking accuracy for the EKF.
216
Figure 6.4 Optimal R: initial 2.75 seconds shown for each state.
217
Figure 6.5 Optimal R: initial 2.75 seconds shown for each state, along with model parameters and values for R.
218
219
Figure 6.6 Optimal R: tire lateral slip angles forces shown for the entire run.
Figure 6.7 Phase plot for case of optimal R.
220
6.8.1.2 Low Error Covariance Values in R-Matrix by Factor of 106
Figure 6.8 Low R – initial 2.75 seconds shown for each state.
221
Figure 6.9 Low R – initial 2.75 seconds shown for each state, along with model parameters and values for R.
222
223
Figure 6.10 Low R – tire lateral slip angles forces shown for the entire run.
Figure 6.11 Phase plot for condition of very low R.
224
6.8.1.3 High Values for R-Matrix by Factor of 106
Figure 6.12 High R.
225
Figure 6.13 High R.
226
227
Figure 6.14 High R. tire lateral slip angles forces shown for the entire run.
Figure 6.15 State phase plot for very high values of R.
6.9 Various Runs – Discussion
The following sections illustrate and explain the behavior of the optimally tuned EKF under many varied simulation conditions. Note that the label “ABS ON” means that the tractor and semitrailer ABS systems are fully operational. Otherwise, the label “ABS OFF” indicates that the tractor ABS is not functional, while the semitrailer ABS remains fully functional, as discussed in Chapter 4.
6.9.1 Medium µ (µ=0.55), ½ GVW Load ABS ON, Resulting in a Stable Stop
This series is a good example of how well the filter can work over a large speed range. Except for the highly dynamic parts of the signal where longitudinal acceleration is not constant (i.e., longitudinal motion is exacerbated by brake actuation) and/or lateral
228
load transfer is extreme, the EKF tracks very well from the entry speed of 73.0 kph (45.4 mph, for these conditions) down to 5 kph (3.1 mph). This accuracy is one considerable advantage of using a nonlinear system model that allows speed to update for each observation. These results also suggest that additional degrees of freedom are not necessary to capture the longitudinal dynamic loads on the hitch, which result from longitudinal tire forces. The addition of longitudinal (tire braking) forces could increase the model accuracy, and their estimation is readily available since the ABS computer is assumed to be providing tire-wheel longitudinal slip information. Furthermore, the only significant model disagreement (versus the original TruckSim™ simulation) occurs when d jerk is not zero, i.e. ()0A ≠ and/or lateral load transfer is quite significant. However, dt x I argue that the added cost and complexity of having a model with five more degrees of freedom may be avoided with further development of the existing model’s robustness. For instance, correction for the extreme load transfer is discussed in the Conclusion section of this chapter.
During this run, some separation in the data (EKF vs. TruckSim™) occurs at t ≈ 5.5 seconds, but the EKF recovers by t ≈ 8.0 seconds. Note also that the estimated channels of γ and γ appear to actually have a time lag induced during this period, which disappears by t ≈ 8.0 seconds (refer to Figure 6.16 and Figure 6.17 Note also that the maximum discrepancy in α and Fy also occurs at t ≈ 6-8 seconds, when α and Fy are at their maximum for that run. Additional work with the filter and model parameters could indeed improve these dispersions. The contribution of the dynamics of the EKF model itself, along with the longitudinal degrees of freedom missing, are also factors.
In these and subsequent plots the model input is steer angle for the front axle, in degrees, with additional noise superimposed onto the signal. Steer angle is presented as the input, as opposed to hand wheel angle, since the steer angle (δ) is the actual model input. However, it should be clear that for actual applications, the measured on-vehicle input parameter would indeed be hand wheel angle, which is much more easily measured in production vehicles. Therefore, in practice, the hand wheel angle measurement would 229
have to be further conditioned to faithfully estimate the actual steer angle. Faithful estimation of the steer input (δ) would result from the addition of dynamic factors to account for hysteresis and time lag, as well as the application of a gain value, at or near the steering gear ratio (which in this case is 25:1). Note that for all of the plots comparing model estimates for tire lateral slip angle and force, the bold lines represent EKF output, and the thinner lines indicate the TruckSim™ comparison. Limits for γ and γ are set by comparing phase plots of successful (no jackknife) and unsuccessful (jackknife) negotiations of the brake-in-turn maneuver.
230
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 850 5 4 3 [deg] δ 2 1 INPUT 0 0 2 4 6 8 10 12 14 4 ] 2 3 TruckSim "signal" EKF estimated 2 1 lat accel lat [m/s 1 0 CG -1 0 2 4 6 8 10 12 14
10 TruckSim actual signal EKF estimate
5 Yaw Rate [deg/s]Yaw 1 0 VEH 0 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 0 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT
articulation angle [deg] angle articulation -10 0 2 4 6 8 10 12 14 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:29 date: 6 / 26
Figure 6.16 Output showing EKF states and steering input signals.
231
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 850 5
4
3 [deg] δ 2
INPUT 1
0 0 2 4 6 8 10 12 14
0
-0.5 TruckSim "signal" EKF estimate lat velocity lat [m/s]
1 -1 CG
-1.5 0 2 4 6 8 10 12 14
12 10 TruckSim "signal" EKF estimate 8 6 4 Yaw Rate Yaw [deg/s] 1 2
VEH 0 0 2 4 6 8 10 12 14 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.30 Steer Axle Load: -66655 N simulation µ level: 0.55 Drive Axle Scale: 1.30 Drive Axle Load: -58172 N reference µ level: 0.95 Trailer Axle Scale: 1.30 Trailer Axle Load: -69943 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:29 date: 6 / 26
Figure 6.17 Showing model states and configuration parameters.
232
Tru c k S im /S im u lin k B .I.T. R u n = 8 5 0 Tra c to r S lip Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i α α -2 -2 α , model 4 α , model 5 α , TS im -3 -3 4 α , model α , TS im 1 5 α , model 2 α , model -4 3 -4 tire slipangles, α , TS im tire slipangles, 1 α , TS im 2 -5 α , TS im -5 3
-6 -6 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Trac tor Tire L ate ra l F o rc e S e m itra iler Tire L ate ral F o rc e 10000 5000 model axle 1 m o d e l a x le 4 model axle 2 m o d e l a x le 5 model axle 3 TSIM axle 4 avg 8000 TSIM axle 1 avg 4000 TSIM axle 5 avg TSIM axle 2 avg
233 TSIM axle 3 avg 6000 3000
4000 2000 tire lat force(N) tire lat force (N)
2000 1000
0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
NON-Linear 5-A XLE BICY CLE MODEL source file: kalman-truck-obser-v14.m prrint time: 17:29 date: 6 / 2 6
Figure 6.18 Showing EKF states of lateral slip and tire forces. Note that the bold lines are for the EKF, and the lighter lines are of the TruckSim™ simulation.
TruckSim/Simulink B.I.T. Run = 850 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph] Speed
20
0 0 2 4 6 8 10 12 14
TruckSim/Simulink B.I.T. Run = 850 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10 TruckSim 8 EKF estimate JACKKNIFE LIMIT 6
4 dot γ 2
0
-2 hitch ratee, angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 17:29 date: 6 / 26
Figure 6.19 Vehicle forward speed and hitch angle phase plot, showing very good agreement in the state phase plane plot for hitch articulation angle, γ.
234
6.9.2 Medium µ (µ=0.55), ½ Load ABS OFF, Resulting in a Jackknife
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 843 5 4 3 [deg] δ 2 1 INPUT 0 2 3 4 5 6 7 8 4 ] 2 3 2 1 lat accel lat [m/s
1 TruckSim "signal" 0
CG EKF estimated -1 2 3 4 5 6 7 8
10
5 TruckSim actual signal EKF estimate Yaw Rate Yaw [deg/s] 1 0 VEH 2 3 4 5 6 7 8
0
-5 γ jackknife LIMIT dot
articulation rate [deg/s]articulation -10 2 3 4 5 6 7 8
0
-5 γ jackknife LIMIT
articulation [deg] angle -10 2 3 4 5 6 7 8 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:36 date: 6 / 26
Figure 6.20 Steer angle input along with EKF states, showing very good agreement with the comparison TruckSim™ model.
235
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 843 5 4 3 [deg] δ 2
INPUT 1 0 2 3 4 5 6 7 8
0 TruckSim "signal" EKF estimate
-0.5 lat velocity lat [m/s]
1 -1 CG
-1.5 2 3 4 5 6 7 8
12 10 8 6 4 TruckSim "signal" Yaw Rate Yaw [deg/s] 1 2 EKF estimate
VEH 0 2 3 4 5 6 7 8 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.30 Steer Axle Load: -66655 N simulation µ level: 0.55 Drive Axle Scale: 1.30 Drive Axle Load: -58172 N reference µ level: 0.95 Trailer A xle Scale: 1.30 Trailer Axle Load: -69943 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:36 date: 6 / 26
Figure 6.21 Steer angle input, model states of lateral velocity and yaw rate, along with model parameter settings.
236
Truc k S im /S im ulink B .I.T. Run = 843 Tractor S lip Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i
α -2 α -2
-3 -3 α , model α , model 1 4 α , model α , model -4 2 -4 5 tire slip angles, tire slip angles, α , model α , TSim 3 4 α , TSim α , TSim 1 5 α , TSim -5 2 -5 α , TSim 3 -6 -6 2 3 4 5 6 7 8 2 3 4 5 6 7 8
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 model axle 4 model axle 2 model axle 5 8000 model axle 3 4000 TSIM axle 4 avg 237 TSIM axle 1 avg TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 6000 3000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 3 4 5 6 7 8 2 3 4 5 6 7 8
NON-Linear 5-A XLE BICY CLE MODEL s ourc e file: kalman-truc k-obs er-v 14.m prrint time: 17:36 date: 6 / 26
Figure 6.22 Note the good agreement for model tire lateral slip angles, αi, and tire lateral forces, Fyi.
TruckSim/Simulink B.I.T. Run = 843 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph] Speed
20
0 2 3 4 5 6 7 8
TruckSim/Simulink B.I.T. Run = 843 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 TruckSim JACKKNIFE LIMIT 6 EKF estimate
4 dot γ 2
0
-2 hitch ratee, angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 17:36 date: 6 / 26
Figure 6.23 Vehicle forward speed and hitch angle phase plot. Note the good agreement in state phase plots for hitch articulation angle, γ.
238
6.9.3 Low µ (µ=0.30), ½ GVW Load ABS ON, Resulting in Controlled Stop
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 834 5 4 3 [deg] δ 2 1 INPUT 0 3 4 5 6 7 8 9 10 4 ] 2 3 2 1
lat accel lat [m/s TruckSim "signal" 1 0 EKF estimated CG -1 3 4 5 6 7 8 9 10
10
5
Yaw Rate [deg/s] Rate Yaw TruckSim actual signal 1 0 EKF estimate VEH 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT
articulation angle [deg] angle articulation -10 3 4 5 6 7 8 9 10 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:42 date: 6 / 26
Figure 6.24 Steer input and model states. Note that the EKF had some difficulty tracking on this low-µ surface with the high-frequency forces of the ABS system.
239
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 834 5
4 3 [deg] δ 2
INPUT 1
0 3 4 5 6 7 8 9 10
0
-0.5 lat velocity lat [m/s] 1 -1 TruckSim "signal"
CG EKF estimate -1.5 3 4 5 6 7 8 9 10
12 10 8 6 4 Yaw Rate [deg/s] Yaw
1 TruckSim "signal" 2 EKF estimate VEH 0 3 4 5 6 7 8 9 10 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.30 Steer Axle Load: -66655 N simulation µ level: 0.35 Drive Axle Scale: 1.30 Drive Axle Load: -58172 N reference µ level: 0.95 Trailer Axle Scale: 1.00 Trailer Axle Load: -53802 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:42 date: 6 / 26
Figure 6.25 Same run, note change in load scale factors and slight increase in µ level used by EKF to get optimal agreement.
240
Tru c k S im /S im u lin k B .I.T. R u n = 8 3 4 Tractor S lip Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i
α -2 α -2
-3 -3 α , m o d e l 1 α , m o d e l 2 -4 α , m o d e l -4 α , model
tire slip angles, 3 tire slip angles, 4 α , TS im α , model 1 5 α , TS im α , TS im -5 2 -5 4 α , TS im α , TS im 3 5
-6 -6 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10
Trac tor Tire L a te ral F orc e S em itrailer Tire L ate ral F orc e 10000 5000 m o d e l a x le 1 model axle 4 m o d e l a x le 2 model axle 5 m o d e l a x le 3 8000 4000 TSIM axle 4 avg TSIM axle 1 avg TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 6000 3000 241
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10
NO N-L ine a r 5 -A X L E B ICY CL E M O DEL s o u rc e f ile: ka lma n -tru c k-o b s e r-v 1 4 .m p rrint tim e : 1 7 :4 2 d a te : 6 / 2 6
Figure 6.26 Tire lateral slip and lateral force output from EKF and TruckSim™. Note the unusually high lateral slip angles that the EKF modeled for the trailer, in spite of the lateral forces for the trailer being modeled quite accurately. This behavior indicates that saturation in Fy has been exceeded by the semitrailer axles, i.e., they are sliding.
TruckSim/Simulink B.I.T. Run = 834 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph]
20
0 3 4 5 6 7 8 9 10
TruckSim/Simulink B.I.T. Run = 834 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10 TruckSim 8 EKF estimate JACKKNIFE LIMIT 6
4 dot γ 2
0
-2 hitch angle ratee, hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 17:42 date: 6 / 26
Figure 6.27 Vehicle forward speed and state phase plot for hitch articulation angle, γ.
242
6.9.4 Low µ (µ=0.30), ½ GVW Load ABS OFF, Resulting in a Jackknife
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 844 5 4 3 [deg] δ 2 1 INPUT 0 2 3 4 5 6 7 8 9 10 4 ] 2 3 TruckSim "signal" EKF estimated 2 1 lat accel lat [m/s 1 0 CG -1 2 3 4 5 6 7 8 9 10
10 TruckSim actual signal EKF estimate
5 Yaw Rate [deg/s] Rate Yaw 1 0 VEH 2 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 2 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT
articulation angle [deg] angle articulation -10 2 3 4 5 6 7 8 9 10 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:47 date: 6 / 26
Figure 6.28 Steer angle input and model states shown below steering input.
243
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 844 5
4 3 [deg] δ 2
INPUT 1
0 2 3 4 5 6 7 8 9 10
0 TruckSim "signal" EKF estimate
-0.5 lat velocity lat [m/s]
1 -1 CG
-1.5 2 3 4 5 6 7 8 9 10
12 10 TruckSim "signal" EKF estimate 8 6 4 Yaw Rate Rate [deg/s] Yaw 1 2
VEH 0 2 3 4 5 6 7 8 9 10 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.55 Steer Axle Load: -79473 N simulation µ level: 0.30 Drive Axle Scale: 1.10 Drive Axle Load: -49223 N reference µ level: 0.95 Trailer Axle Scale: 1.80 Trailer Axle Load: -96844 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:47 date: 6 / 26
Figure 6.29 Model states and run conditions.
244
Truc k S im /S im ulink B .I.T. R un = 844 Tractor S lip Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i α α -2 -2
-3 -3 α , model 4 α , model α , model 5 1 α , TS im α , model 4 -4 2 -4 α tire slip angles, tire slip angles, , TS im α , model 5 3 α , TSim 1 -5 α , TSim -5 2 α , TSim 3 -6 -6 2 4 6 8 10 2 4 6 8 10
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 model axle 4 model axle 2 model axle 5 model axle 3 TSIM axle 4 avg 8000 TSIM axle 1 avg 4000 TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 245 6000 3000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 4 6 8 10 2 4 6 8 10
NON-Linear 5-A XLE BICY CLE MODEL source file: kalman-truck-obser-v14.m prrint time: 17:47 date: 6 / 26
Figure 6.30 Lateral slip angles and tire lateral forces for both EKF and TruckSim™ models; bold lines indicate EKF output.
TruckSim/Simulink B.I.T. Run = 844 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph]
20
0 2 3 4 5 6 7 8 9 10
TruckSim/Simulink B.I.T. Run = 844 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 JACKKNIFE LIMIT 6
4 TruckSim dot
γ EKF estimate 2
0
-2 hitch angle ratee, hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 17:47 date: 6 / 26
Figure 6.31 Vehicle longitudinal speed and state phase plot for hitch angle, γ.
246
6.9.5 Medium µ (µ=0.55), 0 Payload ABS ON, Resulting in a Stable Stop
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 851 5 4 3 [deg] δ 2 1 INPUT 0 1 2 3 4 5 6 7 8 9 10 4 ] 2 3 TruckSim "signal" EKF estimated 2 1 lat accel lat [m/s 1 0 CG -1 1 2 3 4 5 6 7 8 9 10
10 TruckSim actual signal EKF estimate
5 Yaw Rate [deg/s] Rate Yaw 1 0 VEH 1 2 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 1 2 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT
articulation angle [deg] angle articulation -10 1 2 3 4 5 6 7 8 9 10 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:50 date: 6 / 26
Figure 6.32 Steer angle input and EKF model states.
247
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 851 5
4 3 [deg] δ 2
INPUT 1
0 1 2 3 4 5 6 7 8 9 10
0
-0.5 TruckSim "signal" EKF estimate lat velocity lat [m/s]
1 -1 CG
-1.5 1 2 3 4 5 6 7 8 9 10
12 10 TruckSim "signal" EKF estimate 8 6 4 Yaw Rate Rate [deg/s] Yaw 1 2
VEH 0 1 2 3 4 5 6 7 8 9 10 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.30 Steer Axle Load: -66655 N simulation µ level: 0.55 Drive Axle Scale: 1.30 Drive Axle Load: -58172 N reference µ level: 0.95 Trailer Axle Scale: 1.30 Trailer Axle Load: -69943 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:50 date: 6 / 26
Figure 6.33 Model states and EKF running parameters.
248
Truc k S im /S im ulink B .I.T. R un = 851 Tractor S lip Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i α α -2 -2
α , model -3 1 -3 α , model α , model 2 4 α , model α , model 3 5 -4 α , TSim -4 α , TS im tire slip angles, tire slip angles, 1 4 α , TSim α , TS im 2 5 α , TSim -5 3 -5
-6 -6 2 4 6 8 10 2 4 6 8 10
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 m o d e l a x le 4 model axle 2 m o d e l a x le 5 model axle 3 TSIM axle 4 avg 8000 TSIM axle 1 avg 4000 TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 6000 3000 249
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 4 6 8 10 2 4 6 8 10
NON-Linear 5-A XLE BICY CLE MODEL source file: kalman-truck-obser-v14.m prrint time: 17:50 date: 6 / 26
Figure 6.34 Tire lateral slip angles and tire lateral forces.
TruckSim/Simulink B.I.T. Run = 851 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph] Speed
20
0 1 2 3 4 5 6 7 8 9 10
TruckSim/Simulink B.I.T. Run = 851 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 TruckSim JACKKNIFE LIMIT EKF estimate 6
4 dot γ 2
0
-2 hitch ratee, angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 17:50 date: 6 / 26
Figure 6.35 Simulation vehicle longitudinal speed and state phase plot for hitch angle (γ).
250
6.9.6 Sensitivity of the EKF to Improper Load Estimates
Following are images from the SAME run, but with the EKF running with model loads and inertias assumed from the ½ load condition, instead of the proper 0-load condition. Surprisingly, the EKF accuracy is not harmed as badly as predicted with such a large change in parameters.
251
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim /S im ulink B .I.T. Run = 851 5 4 3 [deg] δ 2
INPUT 1 0 1 2 3 4 5 6 7 8 9 10
3 TruckSim "signal" ] 2 EKF estimate 2 post calculated
1 lat accel lat [m/s 1 0 CG -1 1 2 3 4 5 6 7 8 9 10
15 TruckSim actual signal EKF estimate 10
5 Yaw Rate [deg/s] 1
VEH 0 1 2 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT dot articulation rate[deg/s] articulation -10 1 2 3 4 5 6 7 8 9 10
0
-5 γ jackknife LIMIT articulation [deg] angle -10 1 2 3 4 5 6 7 8 9 10 time [sec]
NON-Linear 5-A XLE BICY CLE MODEL source file: truck-truck-obser-v14.m prrint time: 14:19 date: 6 / 19
Figure 6.36 Same run as previous, but EKF run @ ½ load instead of 0 load.
252
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 851 5
4
3 [deg] δ 2
INPUT 1
0 1 2 3 4 5 6 7 8 9 10
0
TruckSim "signal" EKF estimate -0.5 lat velocity lat [m/s] 1 CG
-1 1 2 3 4 5 6 7 8 9 10
15 TruckSim "signal" EKF estimate 10
5 Yaw Rate [deg/s] Yaw 1 VEH 0 1 2 3 4 5 6 7 8 9 10 time [s]
Simulation and EKF Parameters for This Run
Steer Axle Scale: 1.30 Steer Axle Load: -66655 N simulation µ level: 0.55 Drive Axle Scale: 1.30 Drive Axle Load: -58172 N reference µ level: 0.95 Trailer Axle Scale: 1.30 Trailer Axle Load: -69943 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 14:19 date: 6 / 19
Figure 6.37 Same run, but EKF run @ ½ GVW load, not 0-payload.
253
TruckSim/Simulink B.I.T. Run = 851 Tractor S lip Semitrailer Slip Angles 1 1
0 0 -1 [deg] [deg] i i α α -1 -2 α , model 1 α , model 2 -3 α , model 3 -2 α , TSim 1 α , TSim -4 2 α , model α 4 tire slip angles, tire slip angles, , TSim 3 α , model 5 -3 α , TSim 4 -5 α , TSim 5
-6 -4 2 4 6 8 10 2 4 6 8 10
Tractor Tire Lateral Force Semitrailer Tire Lateral Force 10000 5000 model axle 1 model axle 4 model axle 2 model axle 5 model axle 3 TSIM axle 4 avg 8000 4000 TSIM axle 1 avg TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 254 6000 3000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 4 6 8 10 2 4 6 8 10
NON-Linear 5-AXLE BICYCLE MODEL source file: kalman-truck-obser-v14.m prrint time: 14:19 date: 6 / 19
Figure 6.38 Same run, but EKF run @ ½ GVW load, instead of 0-payload.
TruckSim/Simulink B.I.T. Run = 851 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 80 TruckSim vehicle speed
60
40 Speed [kph] Speed
20
0 1 2 3 4 5 6 7 8 9 10
TruckSim/Simulink B.I.T. Run = 851 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10 TruckSim 8 non-Linear EKF JACKKNIFE LIMIT 6
4 dot γ 2
0
-2 hitch angle ratee,hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v12.m prrint time: 14:19 date: 6 / 19
Figure 6.39 Same run, but EKF run @ ½ -GVW load instead of proper 0-payload.
255
6.9.7 Medium µ (µ=0.55), 0 Load ABS OFF, Resulting in a Jackknife
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 852 5 4 3 [deg] δ 2 1 INPUT 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 4 ] 2 3 2
1 TruckSim "signal" lat accel lat [m/s 1 0 EKF estimated CG -1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
10
5 TruckSim actual signal Yaw Rate [deg/s] Rate Yaw
1 EKF estimate 0 VEH 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
-5
γ jackknife LIMIT
articulation angle [deg] angle articulation -10 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:56 date: 6 / 26
Figure 6.40 Steer angle input and EKF states.
256
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 852 5
4 3 [deg] δ 2
INPUT 1
0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0 TruckSim "signal" EKF estimate
-0.5 lat velocity lat [m/s]
1 -1 CG
-1.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
12 10 8 6 4 TruckSim "signal" Yaw Rate Rate [deg/s] Yaw 1 2 EKF estimate
VEH 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.30 Steer Axle Load: -64935 N simulation µ level: 0.55 Drive Axle Scale: 1.30 Drive Axle Load: -36920 N reference µ level: 0.95 Trailer Axle Scale: 1.30 Trailer Axle Load: -33911 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:56 date: 6 / 26
Figure 6.41 Steer angle input, model states, and EKF parameters.
257
Truc k S im /S im ulink B .I.T. R un = 852 Tractor S lip Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i α α -2 -2
-3 -3 α , model 1 α , model -4 2 -4
tire slip angles, slip angles, tire α , model slip angles, tire 3 α , model α , TS im 4 1 α , model α , TS im 5 -5 2 -5 α , TS im α , TS im 4 3 α , TS im 5 -6 -6 1 2 3 4 5 6 1 2 3 4 5 6
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 model axle 4 model axle 2 model axle 5 TSIM axle 4 avg 8000 model axle 3 4000 TSIM axle 1 avg TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 258 6000 3000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 1 2 3 4 5 6 1 2 3 4 5 6
NO N-Linear 5-A X LE B ICY CLE MO DEL s ourc e f ile: kalman-truc k-obs er-v 14.m prrint time: 17:56 date: 6 / 26
Figure 6.42 Tire lateral slip angles and lateral forces.
TruckSim/Simulink B.I.T. Run = 852 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph]
20
0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
TruckSim/Simulink B.I.T. Run = 852 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 JACKKNIFE LIMIT TruckSim 6 EKF estimate
4 dot γ 2
0
-2 hitch angle ratee, hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 17:56 date: 6 / 26
Figure 6.43 Vehicle forward speed and hitch angle phase plot. Note the good agreement in the state phase plot for γ.
259
6.9.8 Low-µ (µ = 0.30), 0-Load, ABS ON, Resulting in a Stable Stop
At first, this configuration proved most difficult to arrive at a good agreement between TruckSim™ and the EKF. It is therefore apparent that low-µ, low-load conditions are quite challenging to duplicate and would probably consume considerable effort to tune for robustness.
Many varying EKF configurations (in terms of adjusting µ level and Fz-load scale factors) were run. The best agreement for the EKF settled on an EKF µ level of 0.35 (overstating µ). Tire vertical force scale factors of 1.40 / 1.65 / 1.65 were used for the steer, drive, and semitrailer axles, respectively. The mass and yaw inertia for the semitrailer were also adjusted 60 and 75% of their actual values, respectively. I am not comfortable with these adjustments, as they suggest some problems with the model which are yet to be understood.
Fairly good agreement continues throughout the simulation comparison, however the agreement fidelity suffers at the onset of braking where jerk (time rate of change of acceleration) is non-zero, at t ≈ 6.4-6.5 seconds. This phenomenon is consistent across simulation conditions.
Note in the tire lateral force plots that the stick-slip phenomenon is quite significant in the TruckSim™ simulation.
For all of these runs at low-µ / low-load, the simulation takes considerably more time to run.
260
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 853 5 4 3 [deg] δ 2 1 INPUT 0 2 4 6 8 10 12 14 4 ] 2 3 TruckSim "signal" EKF estimated 2 1 lat accel lat [m/s 1 0 CG -1 2 4 6 8 10 12 14
10 TruckSim actual signal EKF estimate
5 Yaw Rate [deg/s] Rate Yaw 1 0 VEH 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT
articulation angle [deg] angle articulation -10 2 4 6 8 10 12 14 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:30 date: 6 / 26
Figure 6.44 Steer angle input and EKF states.
261
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 853 5
4 3 [deg] δ 2
INPUT 1
0 2 4 6 8 10 12 14
0
-0.5 TruckSim "signal" EKF estimate lat velocity lat [m/s]
1 -1 CG
-1.5 2 4 6 8 10 12 14
12 10 TruckSim "signal" EKF estimate 8 6 4 Yaw Rate Rate [deg/s] Yaw 1 2
VEH 0 2 4 6 8 10 12 14 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.40 Steer Axle Load: -69930 N simulation µ level: 0.35 Drive Axle Scale: 1.65 Drive Axle Load: -46860 N reference µ level: 0.95 Trailer Axle Scale: 1.65 Trailer Axle Load: -43040 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-008 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-007 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-007 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-007
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:30 date: 6 / 26
Figure 6.45 Showing steer input, model states, and EKF parameters.
262
Truc k S im /S im ulink B .I.T. R un = 853 Tractor Slip Angles Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] α i i , model 4 α α -2 α , model -2 α , model 1 5 α , model α , TSim 2 4 α , model α , TSim -3 3 -3 5 α , TSim 1 α , TSim 2 α , TSim -4 3 -4 tire slip angles, tire slip angles,
-5 -5
-6 -6 2 4 6 8 10 12 14 2 4 6 8 10 12 14
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 m o d e l a x le 4 model axle 2 m o d e l a x le 5 model axle 3 TSIM axle 4 avg 8000 TSIM axle 1 avg 4000 TSIM axle 5 avg TSIM axle 2 avg
263 TSIM axle 3 avg 6000 3000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14
NO N-Linear 5-A XLE BICY CLE MO DEL s ourc e file: kalman-truc k-obs er-v 14.m prrint time: 18:30 date: 6 / 26 .
Figure 6.46 Tire lateral slip angles and forces.
TruckSim/Simulink B.I.T. Run = 853 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph]
20
0 2 4 6 8 10 12 14
TruckSim/Simulink B.I.T. Run = 853 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 JACKKNIFE LIMIT 6 TruckSim EKF estimate 4 dot γ 2
0
-2 hitch angle ratee, hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 18:30 date: 6 / 26
Figure 6.47 Vehicle forward speed and state phase plane plot for hitch articulation angle, γ, showing good agreement between EKF and TruckSim™.
264
6.9.9 EKF Sensitivity to Improper Estimation of Surface µ
Figures 6.48 through 6.50 show the SAME maneuver, run with the unit 2 mass and inertias scaled as 1.0 (but µ still modeled at 0.35). The EKF states remain in acceptable agreement with TruckSim™. In Figure 6.50, however tire lateral slip agreement becomes less than acceptable.
265
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 853 5 4 3 [deg] δ 2 1 INPUT 0 2 4 6 8 10 12 14 4 ] 2 3 TruckSim "signal" EKF estimated 2 1 lat accel lat [m/s 1 0 CG -1 2 4 6 8 10 12 14
10 TruckSim actual signal EKF estimate
5 Yaw Rate [deg/s] Rate Yaw 1 0 VEH 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT dot
articulation ratearticulation [deg/s] -10 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT
articulation angle [deg] angle articulation -10 2 4 6 8 10 12 14 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:37 date: 6 / 26
Figure 6.48 Same run, load and inertias scaled to 1.0.
266
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 853 5
4 3 [deg] δ 2
INPUT 1
0 2 4 6 8 10 12 14
0
-0.5 TruckSim "signal" EKF estimate lat velocity lat [m/s]
1 -1 CG
-1.5 2 4 6 8 10 12 14
12 10 TruckSim "signal" EKF estimate 8 6 4 Yaw Rate Rate [deg/s] Yaw 1 2
VEH 0 2 4 6 8 10 12 14 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.40 Steer Axle Load: -69930 N simulation µ level: 0.35 Drive Axle Scale: 1.65 Drive Axle Load: -46860 N reference µ level: 0.95 Trailer Axle Scale: 1.65 Trailer Axle Load: -43040 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-008 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-007 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-007 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-007
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:37 date: 6 / 26
Figure 6.49 Same run, load and inertia scaled to 1.0.
267
Truc k S im /S im ulink B .I.T. R un = 853 Tractor Slip Angles Semitrailer Slip Angles 1 1
0 0
-1 -1 [deg] [deg] i i
α -2 α -2
α , model 1 -3 α , model -3 2 α , model α , model 3 4 α , TSim α , model -4 1 -4 5 tire slip angles, tire slip angles, α , TSim α , TSim 2 4 α , TSim α , TSim 3 5 -5 -5
-6 -6 2 4 6 8 10 12 14 2 4 6 8 10 12 14
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 m o d e l a x le 4 model axle 2 m o d e l a x le 5 model axle 3 TSIM axle 4 avg 8000 TSIM axle 1 avg 4000 TSIM axle 5 avg
268 TSIM axle 2 avg TSIM axle 3 avg 6000 3000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14
NON-Linear 5-A XLE BICY CLE MODEL source file: kalman-truck-obser-v14.m prrint time: 18:37 date: 6 / 26
Figure 6.50 Same run, using load and inertia scales at 1.0.
Following are plots from a similar maneuver was run at a lower speed of 60 kph, in place of the 64.5 kph of the previous run (conditions and parameters otherwise identical), lowering the simulation speed from 64.5 kph to 60 kph to reduce dynamic loading (the premise being that the trailer tires may be experiencing lift-off in the full TruckSim™ simulation. This simulation did not work any better than the previous attempt. The operating µ level had to be set at µ = 0.40 (instead of the varying µ = 0.30± used by the TruckSim™ simulation), and the tire vertical loading scales were still high at 1.40 / 1.65 / 1.65. However, the state phase plot for γ did show slightly better agreement than seen in the higher speed simulations.
269
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 854 5 4 3 [deg] δ 2
INPUT 1 0 2 4 6 8 10 12 14
3
] TruckSim "signal" 2 EKF estimate 2 post calculated 1 lat accel [m/s 1 0 CG -1 2 4 6 8 10 12 14
15 TruckSim actual signal EKF estimate 10
5 Yaw Rate [deg/s] Rate Yaw 1
VEH 0 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT dot articulation rate [deg/s] -10 2 4 6 8 10 12 14
0
-5 γ jackknife LIMIT articulation [deg] angle -10 2 4 6 8 10 12 14 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 17:40 date: 6 / 19
Figure 6.51 Similar conditions as previous, lower speed entry speed.
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TruckS im /Sim ulink B.I.T. Run = 854 Tractor S lip Semitrailer Slip Angles 1 1
0 0 -1 [deg] [deg] i i
α α , model α -1 -2 1 α , model α , model 2 4 α , model α , model 3 5 -3 α , TSim α , TSim 1 -2 4 α , TSim α , TSim 2 5 α , TSim -4 3 tire slipangles, tire slipangles, -3 -5
-6 -4 2 4 6 8 10 12 14 2 4 6 8 10 12 14
Tractor Tire Lateral Force S em itrailer Tire Lateral Force 10000 5000 model ax le 1 model axle 4 model ax le 2 model axle 5 model ax le 3 TSIM axle 4 avg 8000 4000 TSIM axle 1 avg TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg 3000
271 6000
4000 2000 tire lat force (N) force lat tire (N) force lat tire
2000 1000
0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14
NON-Linear 5-A XLE BICY CLE MODEL source file: kalman-truck-obser-v14.m prrint time: 17:40 date: 6 / 19
Figure 6.52 Similar conditions, but slightly lower speed. Note the very high dynamic loading (high frequency dynamics) in the tractor tire lateral forces, as simulated by TruckSim™.
TruckSim/Simulink B.I.T. Run = 854 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 80 TruckSim vehicle speed
60
40 Speed [kph] Speed
20
0 2 4 6 8 10 12 14
TruckSim/Simulink B.I.T. Run = 854 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 TruckSim JACKKNIFE LIMIT non-Linear EKF 6
4 dot γ 2
0
-2 hitch angle ratee,hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v12.m prrint time: 17:40 date: 6 / 19
Figure 6.53 Similar conditions, slightly lower speed.
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6.9.10 Low-µ (µ = 0.30), 0-Payload, ABS OFF, Resulting in a Jackknife
Here, once again, the very low coefficient (µ) made the model very difficult to tune, in terms of parameters. The final set of parameters used in this maneuver was µ = 0.30, and load scales = 1.00 / 1.00 / 1.00.
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NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 855 5 4 3 [deg] δ 2 1 INPUT 0 3 4 5 6 7 8 9 4 ] 2 3 TruckSim "signal" 2 EKF estimated 1 lat accel [m/s lat 1 0 CG -1 3 4 5 6 7 8 9
10 TruckSim actual signal EKF estimate 5 Yaw Rate Rate [deg/s] Yaw 1 0 VEH 3 4 5 6 7 8 9
0
-5 γ jackknife LIMIT dot articulation rate [deg/s] -10 3 4 5 6 7 8 9
0
-5 γ jackknife LIMIT
articulation [deg] angle -10 3 4 5 6 7 8 9 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:40 date: 6 / 26
Figure 6.54 Steer angle input and EKF states.
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NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 855 5 4 3 [deg] δ 2
INPUT 1 0 3 4 5 6 7 8 9
0
-0.5 TruckSim "signal"
lat velocity [m/s] lat EKF estimate
1 -1 CG
-1.5 3 4 5 6 7 8 9
12 10 TruckSim "signal" EKF estimate 8 6 4 Yaw Rate [deg/s] Rate Yaw 1 2
VEH 0 3 4 5 6 7 8 9 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.00 Steer Axle Load: -49950 N simulation µ level: 0.30 Drive Axle Scale: 1.00 Drive Axle Load: -28400 N reference µ level: 0.95 Trailer Axle Scale: 1.00 Trailer Axle Load: -26085 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:40 date: 6 / 26
Figure 6.55 Model states and EKF parameter settings.
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Truc k S im /S im ulink B .I.T. R un = 855 Tractor Slip Angles Semitrailer Slip Angles 1 1 α , model 1 α , model 0 2 0 α , model 3 α , TS im 1 α , TS im -1 2 -1 α , TS im 3 [deg] [deg] i i α α -2 -2
-3 -3
α , model 4 α , model -4 -4 5 tire slip angles, angles, slip tire tire slip angles, angles, slip tire α , TS im 4 α , TS im -5 -5 5
-6 -6 3 4 5 6 7 8 9 3 4 5 6 7 8 9
Trac tor Tire Lateral F orc e S em itrailer Tire Lateral F orc e 10000 5000 model axle 1 m o d e l a x le 4 model axle 2 m o d e l a x le 5 model axle 3 TSIM axle 4 avg 8000 TSIM axle 1 avg 4000 TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg
276 6000 3000
4000 2000 tirelat force (N) tirelat force (N)
2000 1000
0 0 3 4 5 6 7 8 9 3 4 5 6 7 8 9
NO N-Linear 5-A X LE B ICY CLE MO DEL s ourc e f ile: kalman-truc k-obs er-v 14.m prrint time: 18:40 date: 6 / 26
Figure 6.56 Showing the EKF outputs (BOLD lines) and TruckSim™ (narrow lines) for tire lateral slip angles and forces.
TruckSim/Simulink B.I.T. Run = 855 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph]
20
0 3 4 5 6 7 8 9
TruckSim/Simulink B.I.T. Run = 855 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 10
8 TruckSim JACKKNIFE LIMIT 6 EKF estimate
4 dot γ 2
0
-2 hitch angle ratee, hitch angle -4
-6
-8
-10 -10 -5 0 5 10 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 18:40 date: 6 / 26
Figure 6.57 Vehicle forward speed and phase plane plot for hitch articulation angle, γ.
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6.9.11 Operation of the EKF in a Double Lane Change While in a 152.4 m (500-ft) Diameter Turn, No Braking, µ = 0.55
Entry speed for this maneuver was 68.0 kph (42.25 mph). Braking was not applied. Though no jackknife occurred, high levels of hitch articulation angle and rate occurred during the maneuver, which eventually stabilized. The TruckSim™ steering wheel angle was also limited to ± 240 degrees to simulate a more realistic driver’s response limit during an emergency maneuver. Path deviation was designed to be a 1.4- m sinusoidal half-wave deviation from the 152.4-m constant-radius path of curvature. The first-order “driver” controller in TruckSim™ was left to attempt to follow this path.
Although vertical tire loads as simulated in the three-dimensional TruckSim™ model varied significantly for this relatively abrupt maneuver, the TruckSim™ simulated semitrailer tires did not experience lift-off (of the ground). Hence Fz scales could be kept at a conservative 1.30 / 1.30 / 1.30.
Although the state phase plot (Figure 6.61) shows significant motion about the origin, the rig did not jackknife and the EKF tracked the phase plot quite accurately. The phase plot significantly exceeds the arbitrary limits of ± 8 degrees set previously. This is not at all a condemnation of the ± 8 degree limit, since it may be assumed that any vehicle stabilization system triggering off of the EKF output would have been activated during this abrupt maneuver. Real-world applications of this tool could indeed use different limits for both γ and γ . The limits on the hitch angle and rate could also be dynamically adjusted in the controller domain, based on ABS system estimated µ levels, forward speed, hand wheel angle or rate, or many other parameters. Theoretically, the magnitudes seen in the phase plot (Figure 6.61) could have been controlled to far smaller magnitudes by a vehicle stability system.
Before the onset of instability (at t ≈ 6 seconds), the EKF tracks quite well during this abrupt maneuver. The model disagreement is significant only when the tire and slip angle estimates are compromised by the lack of compensation during the significant weight transfer. 278
NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 847 10
5 [deg] δ
0
INPUT -5 Onset of Instability -10 1 2 3 4 5 6 7 8 9 10
5 ] 2
0
lat accel [m/s lat TruckSim "signal" 1
CG EKF estimated -5 1 2 3 4 5 6 7 8 9 10
20 10 0 -10
Yaw Rate Yaw [deg/s] TruckSim actual signal 1 -20 EKF estimate VEH 1 2 3 4 5 6 7 8 9 10
20 γ jackknife LIMIT 10 dot 0 -10 -20 articulation rate [deg/s] 1 2 3 4 5 6 7 8 9 10
20
10 γ jackknife LIMIT 0
-10
articulation angle [deg]articulation angle -20 1 2 3 4 5 6 7 8 9 10 time [sec]
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 13:32 date: 7 / 6
Figure 6.58 Steer angle input and EKF states during a DLC while in a curve. The vertical line @ t=6 seconds indicates the onset of system instability.
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NON-Linear Model Predictions During Brake-in-Turn Maneuver TruckSim/Simulink B.I.T. Run = 847 10
5 [deg] δ
0
INPUT -5
-10 1 2 3 4 5 6 7 8 9 10
2
1
0 lat velocity lat [m/s] 1 -1 TruckSim "signal" CG EKF estimate -2 1 2 3 4 5 6 7 8 9 10
20
10
0
TruckSim "signal" Yaw Rate Rate [deg/s] Yaw -10 1 EKF estimate
VEH -20
1 2 3 4 5 6 7 8 9 10 time [s]
Simulation and EKF Parameters for This Run Steer Axle Scale: 1.30 Steer Axle Load: -66655 N simulation µ level: 0.55 Drive Axle Scale: 1.30 Drive Axle Load: -58172 N reference µ level: 0.95 Trailer Axle Scale: 1.30 Trailer Axle Load: -69943 N
Measurement Error Covariance Matrix [R] System Noise Covariance Matrix [Q]
2.5000e-003 1.2656e-004 0.0000e+000 0.0000e+000 2.5000e-009 0.0000e+000 0.0000e+000 0.0000e+000 1.2656e-004 5.0625e-002 2.5629e-002 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 2.5629e-002 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-001 0.0000e+000 0.0000e+000 0.0000e+000 5.0625e-008
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-truck-obser-v14.m prrint time: 18:44 date: 6 / 26
Figure 6.59 Model states and EKF parameters for DLC in a curve.
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Truc k S im /S im ulink B .I.T. R un = 847 Tractor Slip Angles Semitrailer Slip Angles 10 10 α , model α , model 1 4 α , model α , model 2 5 α , model α , TS im 3 4 α , TS im α , TS im 5 1 5 5 α , TS im 2 α , TS im 3 [deg] [deg] i i α α
0 0
tire slip angles, angles, slip tire -5 angles, slip tire -5
-10 -10 2 4 6 8 10 2 4 6 8 10
4 4 x 10 Trac tor Tire Lateral F orc e x 10 S em itrailer Tire Lateral F orc e 1.5 1
1 0.5 0.5 281
0 0
tire lat force (N) force lat tire -0 .5 (N) force lat tire model axle 1 model axle 4 model axle 2 -0.5 model axle 5 model axle 3 -1 TSIM axle 4 avg TSIM axle 1 avg TSIM axle 5 avg TSIM axle 2 avg TSIM axle 3 avg -1 .5 -1 2 4 6 8 10 2 4 6 8 10 NO N-Linear 5-A X LE B ICY CLE MO DEL s ourc e f ile: kalman-truc k-obs er-v 14.m prrint time: 18:44 date: 6 / 26
Figure 6.60 Tire lateral slip angles and forces during DLC in a curve.
TruckSim/Simulink B.I.T. Run = 847 Calculated Semitrailer Yaw Rate - Non-linear Extended Kalman Filter vs. TruckSim 100 TruckSim vehicle speed 80
60
40 Speed [kph]
20
0 1 2 3 4 5 6 7 8 9 10
TruckSim/Simulink B.I.T. Run = 847 Hitch Angle Phase Plot - Non-linear Extended Kalman Filter vs. TruckSim 30
TruckSim 20 EKF estimate
10 dot γ
0
hitch angle ratee, hitch angle -10 JACKKNIFE LIMIT
-20
-30 -30 -20 -10 0 10 20 30 hitch angle, γ
NON-Linear 5-AXLE BICYCLE MODEL source file: truck-obser-v14.m prrint time: 18:44 date: 6 / 26
Figure 6.61 State phase plot and model speed.
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6.10 Jackknife Detection Warning Lead Time Estimates
A remaining question is whether or not the successful detection of instability seen in the preceding sections could occur within a time frame necessary for a modern vehicle stability system to effectively react. Inspection of the phase plane plots for hitch articulation angle in the preceding sections led to the conclusion that detection of a change of inflection for the state trajectories might indeed be sufficient to mark the beginning of the jackknife event. For the phase plane plot for the double lane change maneuver discussed in Section 6.15.2, however, a method might rely on proximity of the state trajectory to preset limits on γ and γ , although a change of inflection does occur early in the maneuver.
Given either method, the elapsed time from the point where EKF detected the change of inflection in the state trajectory and the virtual vehicle was declared “out of control” by preset limits on the phase plane plot usually exceeded 500 ms.
The method for estimating the elapsed time between EKF detection and a proclaimed “loss of control” is demonstrated in detail in Figure 6.62. The point of EKF detection, labeled “A,” is indicated on the phase plane and time plots. The point corresponding to “loss of control” is denoted as point “B.” The procedure is illustrated again in Figure 6.63.
Comparison of Figure 6.58 to Figure 6.61 (for the double lane change maneuver) show at least a 1-second elapsed time between the point of inflection in the phase plane and initial crossing of the “loss of control” boundary.
Note that the “loss of control” limits were set comparing maneuvers under a fairly narrow set of similar operating conditions. There would, in practice, be many such limit sets predetermined during the “robustness tuning” phase of such a device. Determination of whether 500 ms is sufficient to permit effective intervention by a stability control system is left for future research.
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Figure 6.62 Time traces and state phase plot for hitch angle (γ) and its rate of change, showing the points of EKF detection of instability (“A”) and loss of control (“B”). The data were presented in Section 6.10.
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Figure 6.63 Time traces and state phase plot for hitch angle (γ) and its rate of change, showing the points of EKF detection of instability (“A”) and loss of control (“B”). The data were presented in Section 6.12.
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6.11 Conclusions
In this chapter an Extended Kalman Filter (EKF) algorithm was devised and adapted to the planar vehicle dynamics model developed and discussed in Chapter 5. Application of the EKF to the articulated vehicle planar model proved effective in predicting the onset of instability for the articulated vehicle under low-µ, low-load conditions. It can readily be argued that low-µ, low-load conditions are not only those which render the heavy articulated vehicle the most vulnerable to jackknife instability (refer to Chapter 4), but they are also the conditions that prove to be the most challenging for the EKF, due to the rates at which the states evolve.
The EKF was shown to predict reliably the onset of jackknife instability, and it was also shown to be robust in predicting when the system hitch angle and rate would remain bounded (i.e., stable).
As predicted, the EKF-model system was most vulnerable to errors and most difficult to tune for the conditions of very low surface coefficient (µ=0.30) and very low load (i.e., 0 payload). The causes for these difficulties were many, but included the following:
1. more significant tire model nonlinearity at very low loads,
2. more abrupt vehicle response when traction limits were exceeded (µpeak vs. µslide for varying µ),
3. trailer wheel-tire lift-off during abrupt maneuvers, and
4. significant load transfer from one side to another in the actual vehicle, which is difficult to capture in a planar model.
The planer model can still benefit from further refinement that will bring its response even closer to the “actual vehicle.” On initial inspection, one might conclude that adding the longitudinal degrees of freedom (i.e., tire longitudinal forces and vehicle longitudinal dynamics) would eliminate the discrepancies between the EKF and
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TruckSim™ simulation during the transition from constant speed to constant deceleration under braking. Inspection of any run presented earlier in this chapter could lead to the conclusion that the addition of longitudinal forces to the model would indeed eliminate (for instance) the discrepancy between EKF and TruckSim™ models as shown in the EKF states in Figure 6.32 through Figure 6.35. However, closer inspection of longitudinal speed in Figure 6.35 reveals that the duration of non-zero jerk (or rate of change of longitudinal acceleration) lasts between t=4.5 and t=5.5 seconds; and, inspection of the model states in Figure 6.32 reveals that the model discrepancies persist from roughly t=4.5 to t=7.0 seconds. The time of occurrence for the discrepancies in states are dependant upon the individual states. Lateral acceleration and yaw rate agree closely up to t=5 seconds, whereas the estimated states of hitch angular rate and hitch angle begin to disagree at t=4.5 seconds. In the following Figure 6.33, lateral velocity also shows significant disagreement from the onset of braking, at t = 4.5 seconds.
Although one might correctly conclude that the absence of longitudinal degrees of freedom contribute to model discrepancies, these discrepancies are results of other root causes. One potential source is the presence of significant load transfer at all vehicle axles during these maneuvers. Although the three-dimensional TruckSim™ program can readily predict such load transfer, the planar model can not inherently account for load transfer. A reasonable workaround was shown by the method of assigning maximum load transfer predictions to the EKF prior to running, then tuning these inputs off line. However, a more elegant solution exists that would allow the model to faithfully simulate the roll degrees of freedom, with little complexity to the model or algorithm.
The model accuracy during highly dynamic situations may be improved by allowing the tire vertical force scale factors to be adjusted as a function of lateral acceleration during filter operation. Due to the relatively high roll stiffness for the tractor-semitrailer, lateral load transfer correction may be quite straightforward. Scale factors would then become load transfer factors, scaling the overall effectiveness of the tires’ lateral force producing ability as a function of lateral acceleration, as proposed in equation (6.32)
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CfA= () (6.32) Fyz
or more specifically,
n CCAy= i (6.33) Fiz ∑ i=1
where the coefficient can be expressed as an nth-order polynomial based on the already available vehicle lateral acceleration.
Also this chapter discussed the performance of the EKF during the very high magnitude inputs into the system seen during a double lane change avoidance maneuver while negotiating a 152.4 m (500 ft). radius corner. The EFK performed very well in duplicating the magnitude and direction of the hitch angle phase plot during this maneuver, up to and exceeding the onset of instability (at t ≈ 6 seconds).
Finally, elapsed time between an EKF-detected impending jackknife and actual “loss of control,” as defined by previously set limits on hitch articulation angle and rate, was determined to exceed 500 ms for the simulations studied.
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6.12 Chapter 6 References
1. B. Friedland, Control System Design, an Introduction to State-Space Methods (New York: McGraw-Hill, Inc., 1986). 2. J.S. Bay, Fundamentals of Linear State-Space Systems (New York: McGraw-Hill, Inc, 1999). 3. A. Gelb, Applied Optimal Estimation (Cambridge, MA: The M.I.T. Press, 1974). 4. S. Papastratos, C. Hart, K. Speaks, P. Hayot, “State Estimation for On-Line First Principle Models,” (http://www.aspentech.com/publication_files/TA25.pdf). 5. “Chemical Process Optimization,” course number ENCH 737, University of Maryland Graduate School Catalog, Fall 2002. 6. G. Alag and G.B. Gilyard, “A Proposed Kalman Filter Algorithm for Estimation of Unmeasured Output Variables for an F100 Turbofan Engine,” NASA Technical Memorandum, October 1990. 7. J.P. Chrstos, “Use of Vehicle Dynamics Modeling to Quantify Race Car Handling Behavior” (Ph.D. Dissertation, The Ohio State University, 2000). 8. R.E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” ASME Journal of Basic Engineering, 1960. 9. G. Welch and G. Bishop, “An Introduction to the Kalman Filter,” TR95-041, Department of Computer Science, University of North Carolina at Chapel Hill. 10. G. Welch website, (http://www.cs.unc.edu/~welch, http://www.cs.unc.edu/~gb). 11. H.W. Sorenson, Kalman Filtering, Theory and Application (New York: I.E.E.E. Press, 1985). 12. E. Brookner, Tracking and Kalman Filtering Made Easy (New York: John Wiley & Sons, 1998). 13. D. Simon, “Kalman Filtering,” Embedded Systems Programming, June 2001. 14. D. Simon website, (http://academic.csuohio.edu/simond/). 15. F. Bouttier and F. Coutier, “Data Assimilation Concepts and Methods,” March 1999, (http://www.ecmwf.int/newsevents/training/rcourse_notes/DATA_ASSIMILATION/ ASSIM_CONCEPTS/Assim_concepts.html).
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CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1 Summary of Fundamental Contributions to the Engineering Community
1. A family of nonlinear brake torque versus pressure versus speed models has developed using contemporary brake dynamometer data on pneumatically- operated s-cam drum and disc brakes for heavy commercial vehicles.
2. A Simulink® model of a 10-position pneumatically actuated brake system, with anti-lock control – commonly found on heavy commercial vehicles in North America was developed from the ground up. The model response can be changed via constants in an “init” program, to simulate pneumatic or electronically controlled, as well as s-cam drum or air-disc brake systems. The anti-lock control (“ABS”) is tunable, and separate autonomous “systems” exist for the tractor and semitrailer, as is common industry practice today.
3. Linear equations of motion for a planar model of a 5-axle articulated vehicle were thoroughly developed, beginning with fundamental derivation of the equations of motion using Lagrange’s equations. The linearized mathematical model was then expanded to accommodate a nonlinear tire model, which is
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necessary to simulate vehicle behavior at and beyond the tires’ limits of lateral adhesion.
4. An Extended Kalman Filter was applied in conjunction with the nonlinear planar model (discussed above) to predict jackknifing of articulated commercial vehicles by estimating the vehicle articulation angle and its rate of change.
7.2 Summary of Dissertation Topics
Chapter 1 of this dissertation discusses the “initial conditions and boundary conditions,” that led to the need for this research into the behavior and consequences of using modern pneumatically operated disc brakes on Class 8 articulated road vehicles. Engineering professionals at the National Highway Traffic and Safety Administration (NHTSA) have proven that revised stopping distance regulations are possible for Class 8 trucks. Regulations reducing heavy truck stopping distances by 20-30% would close the gap in braking potential between heavy trucks and passenger cars in the U.S.
The remainder of this dissertation covers the development of modern brake torque models, demonstration of how significant increases in braking torque can be expected to affect jackknife stability, and finally dynamic models of the heavy truck system employed in a nonlinear observer to warn a hypothetical stability control system of impending jackknife as it occurs.
In Chapter 2, brake torque models were developed to simulate brake torque vs. air chamber pressure vs. application speed. These models were developed as quadratic relationships from contemporary brake dynamometer data for pneumatically operated s- cam drum brakes as well as modern “air disc brakes.” Brake models were developed that very accurately simulate steer axle, drive axle, and semitrailer axle brake applications. The torque vs. pressure relationship was shown to consistently be a function of brake application speed, not brake-to-pad relative speed.
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Chapter 3 presents the development, tuning, and application of a dynamic model of a modern Class 8 heavy truck braking system, built around the mathematical relationships discussed in Chapter 2. A single Simulink® system model was tuned to simulate disc or drum, pneumatic or electronically controlled, 20-in2 or 30-in2 brake chambers by simple editing of the initialization program. The autonomous ABS systems were tuned to simulate real vehicle brake behavior over a wide range of load and surface conditions, using existing on-vehicle experimental data. The need for a more robust ABS algorithm was indicated by a significant portion of on-vehicle experimental brake pressure chamber data showing an incremental step-up-and-hold algorithm which forces the brake pressure to slowly build upon and ABS controlled brake application. The step- up-and-hold algorithms were presumably developed to allow the physical and measurement systems to stabilize after each incremental increase in brake pressure as it is reapplied by the ABS system after a release cycle.
The models described in Chapters 2 and 3 were then run in parallel with a commercial rigid-body nonlinear vehicle dynamics simulation package developed specifically to simulate heavy commercial vehicle dynamics (TruckSim™). The hybrid vehicle simulation was used to analyze the impact of employing electronically actuated high torque brakes, on the prime mover only, to jackknife stability. The study, discussed in Chapter 4, revealed that whether ABS was functional on the prime mover (tractor) or not, the higher torque brakes displayed no negative effects on jackknife stability for the brake-in-turn maneuvers simulated. The simulations were conducted over low- coefficient (i.e., “low µ”) surfaces and for lightly loaded commercial vehicles because the low-load, low-µ condition was demonstrated in previous studies to make the articulated vehicles most vulnerable for jackknifing.
Chapter 5 describes the development of a planar model of an articulated vehicle sufficient to drive an observer that, using vehicle steer angle, yaw rate, and lateral acceleration measurements available, will reveal an impending jackknife situation. The planar model was developed from the ground up using Lagrange’s equations, to avoid having the internal hitch forces in the equations of motion. The equations of motion were
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linearized using the usual assumption of small angles and pseudo-constant vehicle forward velocity. The fully linear planar model, along with linear tire models, was shown to very accurately reproduce vehicle state responses to a range of sinusoidal, step, and ramp inputs when compared to a sophisticated three-dimensional rigid body nonlinear vehicle simulation program. However, the linear model was unable to simulate a jackknife event, due to the highly nonlinear saturation properties of pneumatic tires not being captured by the linear simplification. The linear vehicle model was then improved by adding a highly nonlinear tire model that reproduces tire lateral force as functions of vertical load and lateral slip angle using trigonometric relationships developed by Pacejka in [1] and presented by Pottinger, et. al. in [2]. The tire model was adapted to simulate loss of lateral traction at high levels of longitudinal slip by using an exponential relationship of lateral traction as a function of longitudinal slip ratios. The three relationship coefficients were derived and verified using experimentally measured tire forces under combined steering-braking conditions. Slip ratio measurements can be assumed present due to the mandatory existence of ABS systems on all heavy trucks since 1996 [3-6]. The nonlinear vehicle-tire model was then demonstrated to show the ability to accurately simulate the vehicle states of lateral acceleration, yaw rate, hitch articulation rate, and hitch articulation angle during a jackknife event.
Chapter 6 covers the implementation of the Extended Kalman Filter (EKF) to the articulated vehicle model developed in Chapter 5 to observe the otherwise difficult to measure states of hitch articulation rate and angle. The effects of various states of filter tuning (vis-à-vis the model error covariance matrix magnitude) were demonstrated. The EKF was further demonstrated to accurately track the unmeasured states of hitch angle and rate in low and high severity maneuvers. The resulting phase plots faithfully reproduce the extensive TruckSim™ model outputs for many low-coefficient, low-load conditions. The EKF was shown to accurately predict jackknife stability/instability during brake-in-turn maneuvers through the application of traditional nonlinear system theory techniques of detecting unbounded state response. Further model capability was shown by the states’ operating regime on the phase diagram for a double lane change
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maneuver conducted while in a 154.2 m (500 ft) curve. The EKF demonstrated some lag when attempting to track the states of articulation angle and rate for conditions wherein the tire vertical loads are abruptly and significantly changing. Although the filter tracking shortfall may be partially credited to rapid changes in vehicle longitudinal acceleration, it was shown, by varying tire vertical load scale factors, that adding a roll degree of freedom might indeed improve the EKF tracking ability during stages of rapid transition.
7.3 Recommended Actions
The author recommends that further brake dynamometer testing be conducted to evaluate the amount of brake system hysteresis that occurs during an application-release cycle. The existence of antilock braking makes mechanical efficiency of the brake assembly under cyclic operation an imperative. In Chapter 3, very high levels of brake hysteresis are shown to significantly degrade stopping distances due to reduced system efficiency. “Brake system hysteresis” can result from mechanical lash, as well as nonlinear compliances of brake system components, such as the brake linings.
The author theorizes that air disc brakes will provide significant improvements in brake system hysteresis due to the reduction of highly compliant parts between the high pressure actuating air in the brake chamber and the disc-pad interface. Brake dynamometer testing, currently being conducted by NHTSA, can demonstrate the differences in hysteresis between s-cam drum and air disc brake assemblies under cyclic operation.
It is strongly recommended that future on-vehicle braking experiments include measurement of all the state inputs, and control outputs, of the ABS controllers. This information is necessary to continue development of the ABS system algorithms discussed in Chapter 3.
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Finally, the author proposes a simple, possibly linear, relationship that might be used to adjust tire vertical load scaling to account for vehicle weight transfer in a corner. Making assumptions for the model, such as minimal roll damping and very high roll stiffness (resulting in moderate to low sprung mass roll angles), a linear, quadratic, or exponential function of vehicle lateral acceleration should be explored to improve EKF integrity during very severe maneuvers and on high-coefficient surfaces.
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7.4 Chapter 7 References
1. H.B. Pacejka, Tire and Vehicle Dynamics (Warrendale, PA: Society of Automotive Engineers, 2002). 2. D.J. Schuring, W. Pelz, M.G. Pottinger, “A Model for Combined Tire Cornering and Braking Forces,” SAE 960180. 3. Q&A: ANTILOCK BRAKES: CARS, TRUCKS, MOTORCYCLES, Insurance Institute for Highway Safety, (http://www.hwysafety.org/safety_facts/qanda/antilock.htm). 4. Land Line Magazine (April 11, 2000). 5. Federal Motor Vehicle Safety Standard (FMVSS) Section 135 braking standards, Oct. 2001. 6. Federal Motor Vehicle Safety Standard (FMVSS) Section 121 braking standards, Oct. 2001.
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APPENDIX A
TRUCKSIM™ V.5.0 PARSFILE NUMBERS AND IMPORT / EXPPORT VARIABLES USED IN CHAPER 5
No Load ½ GVW Load 297 Medium µ Low µ Medium µ Low µ (~0.55) (~0.30) (~0.55) (~0.30) Drum / Disc / Drum Disc / Drum Disc / Drum Disc / drum drum / drum drum / drum drum / drum drum Full 750 766 756 761 747 769 742 774 ABS
Half No
Treadle 749 767 757 760 748 768 743 775 ABS Full 752 765 755 762 746 770 740 773 ABS
Full No
Treadle 753 764 754 763 745 771 744 772 ABS
Table A1 Jackknife Stability Simulation Parsfile Numbers – Phase I: Full ABS and No ABS, either vehicle
No Load ½ GVW Load Medium µ Low µ Medium µ Low µ (~0.55) (~0.30) (~0.55) (~0.30) Drum / Disc / Drum Disc / Drum Disc / Drum Disc / drum drum / drum / drum / drum drum drum drum No ABS 786 780 785 782 789 779 790 776
Half (tractor Treadle only)
No ABS 787 781 784 783 788 778 791 777 (tractor only) Full Treadle Notes: numbers in each matrix entry correspond to the TruckSim™ v. 5.0 parsfile number corresponding to each simulation “run.” Refer to the TruckSim™ v. 5.0 manual for a complete description of parsfiles.
Table A2 Jackknife Stability Simulation Parsfile Numbers – Phase II: No ABS on tractor, ABS fully functional on semitrailer
Import/Export Variable Lists
Following is the list of variables imported into TruckSim™ version 5.0 from Simulink®, then the list of variables exported from TruckSim™ to Simulink®, during each simulation. Bold type indicates information from the TruckSim™ work screen. Variables are identified following each variable name, some are in the form of footnotes due to their length.
Thirty variables were exported from the Simulink® environment to the TruckSim™ environment. Sixty-two variables were imported into the Simulink®
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environment from TruckSim™. The import/export operation occurred in “real time,” as the simulation ran.
The variable type format is as follows (refer to TruckSim™ manual for further information):
IMP_MYBK_L1: “IMP” only signals TruckSim™ that this variable is to be imported. “MYBK_L1” is the actual TruckSim™ v.5.0 variable name for, braking moment, left side, 1st axle (i.e., left front).
Or
IMP_MUY_R5o: import surface coefficient (µ level) (“MU”) corresponding to the body y-direction (“Y), for the right side, axle 5 (“R5”), outer wheel position (“o”).
PARSFILE
#FullDataName Generic Data Lists`Brake Torque Import/Export v.7 (var MU)`Simulink®
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Import / Export Variable Set: IMP_MUX_R2i IMP_MUX_R2o
IMP_MUY_L2i IMP_MUY_L2o IMP_MUY_R2i OPT_ALL_IMPORT 0 IMP_MUY_R2o OPT_ECHO_IMPORT 1 IMP_MUX_L3i
IMP_MUX_L3o begin_set_import – follows is the list of variables imported into TruckSim™ IMP_MUX_R3i IMP_MUX_R3o IMP_MYBK_L1…1 IMP_MUY_L3i IMP_MYBK_R1 IMP_MUY_L3o IMP_MYBK_L2 IMP_MUY_R3i IMP_MYBK_R2 IMP_MUY_R3o IMP_MYBK_L3 IMP_MUX_L4i IMP_MYBK_R3 IMP_MUX_L4o IMP_MYBK_L4 IMP_MUX_R4i IMP_MYBK_R4 IMP_MUX_R4o IMP_MYBK_L5 IMP_MUY_L4i 300 IMP_MYBK_R5 IMP_MUY_L4o IMP_MUY_R4i IMP_MUX_L1i…2 IMP_MUY_R4o IMP_MUX_L1o IMP_MUX_L5i IMP_MUX_R1i IMP_MUX_L5o IMP_MUX_R1o IMP_MUX_R5i IMP_MUY_L1i IMP_MUX_R5o IMP_MUY_L1o IMP_MUY_L5i IMP_MUY_R1i IMP_MUY_L5o IMP_MUY_R1o IMP_MUY_R5i IMP_MUX_L2i IMP_MUY_R5o IMP_MUX_L2o end_set_import
1 begin_set_export– the list of variables exported brake moment_brake position from TruckSim™ to Simulink® 2 surface µ_wheel position
EXP_Vx_L1…3 EXP_VxCenL5o EXP_Vx_R1 EXP_VxCenR5o EXP_Vx_L2 EXP_Vx_R2 EXP_Vx … tractor long. speed @ C.G. EXP_Vx_L3 EXP_Ax… tractor long. accel. @ C.G. EXP_Vx_R3 EXP_Vx_2… trailer long. speed, C.G. EXP_Vx_L4 EXP_Ax_2… trailer long. accel, C.G. EXP_Vx_R4 EXP_Pbk_Con… treadle pressure EXP_Vx_L5 EXP_AVz… yaw acceleration, tractor EXP_Vx_R5 EXP_AVz_2… yaw acceleration, trailer EXP_Yaw … yaw rate, tractor EXP_AAy_L1…4 EXP_Yaw_2… yaw rate, trailer EXP_AAy_R1 EXP_X_Target…6 EXP_AAy_L2 EXP_Y_Target… 7 EXP_AAy_R2 EXP_XCG_TM…. CG coords, various EXP_AAy_L3 EXP_YCG_TM EXP_AAy_R3 EXP_XCG_TM2 EXP_AAy_L4 EXP_YCG_TM2 EXP_AAy_R4 EXP_Station…vehicle distance traveled EXP_AAy_L5 EXP_Steer_L1… steer angle, left front EXP_AAy_R5 EXP_Steer_L2… “, left lead drive EXP_ArtR_H… hitch articulation rate EXP_VxCenL1i…5 EXP_Art_H….hitch articulation angle EXP_VxCenR1i EXP_Beta…. side slip, tractor EXP_VxCenL2o EXP_Beta_2…side slip, trailer EXP_VxCenR2o EXP_AlphaL1i…8 EXP_VxCenL3o EXP_AlphaR1i EXP_VxCenR3o EXP_AlphaL2o EXP_VxCenL4o EXP_AlphaR2o EXP_VxCenR4o EXP_AlphaL3o
3 tangential wheel speed_hub position 6 driver’s target path, x-coord. 4 tangential wheel acceleration_hub position 7 driver’s target path, x-coord.
5 global hub x-velocity_hub position 8 tire-wheel lateral slip angle_wheel position
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EXP_AlphaR3o EXP_AlphaR5o EXP_AlphaL4o end_set_export EXP_AlphaR4o EXP_AlphaL5o
Title Brake Torque Import/Export v.7 (var MU)
#DataSet : Brake Torque Import/Export v.7 (var MU)
#Category: Simulink
#FileID : GList156
#Product : TruckSim™ Version 5 (August 2002)
#Last update: 12-09-2002 14:53:29
#LinkCat Generic Lists
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APPENDIX B
STATE-SPACE COEFFICIENTS FOR 3-AXLE PLANAR MODEL
The state-space system matrix is in the form:
⎡aaaa11 12 13 14 ⎤ 303 ⎢aa ii⎥ ⎢ 21 22 ⎥ A = ⎢aa31ii 34 ⎥ ⎢ ⎥ ⎣ 0010⎦
The determinant of the non-state-space system “mass” matrix, K, is:
=+ +22 + det(KmmJJmmdJcJ ) (1212121 ) ( 2 )
Row 1: − 1 2 aCCCJJdCCdeCmJacCbcCcmJ=++++−+++−+⎡⎤()(())(()()) 11uKdet( ) ⎣⎦ 1 2 3 1 2 1 2 3 2 1 1 2 2 2
−−++ +22 − + ++ −1 ⎡⎤(aC12 bC ( c d e )) C 3121 J J ( ad C bd C 2 de ( c d e )) C 321 m J a = ⎢⎥ 12 uKdet( ) ++−−+ ++2222 + + ⎣⎦⎢⎥((acacC )122212121212 cbbcCmJ ( )) ( m mJJu ) mmuJd ( Jc )
304
Jd()+ e aJdemC=−1 [] 13uKdet( ) 2 2 3
J aJdemC=−1 [] 14det(K ) 2 2 3
Row 2: − 1 22 a=−−+−++++−+⎡()()(()()) aC bC cC m J ad C bd C cdeC m m a c C b c C m J ⎤ 21uKdet( ) ⎣ 1 2 3 1 2 1 2 3 1 2 1 2 2 2 ⎦
−1 ⎡(())(()aC22++++ bC cc d eC mJ + d 222 aC + bC ⎤ a = 12 31212 22 ⎢−++ +++−+ ⎥ uKdet( ) ⎣ cde())(()()) c d e C312 m m a a c C 1 b b c C 2 m 22 J ⎦
305
cd()+ em ademJC=−1 [] 23uKdet( ) 2 2 3
cm ademJC=−1 [] 24det(K ) 2 2 3
Row 3:
1 ⎡()()()(()())aC−− bC cC m J ++ c d adC − bdC + ceC m m +++−+ a c C b c C m J ⎤ a = 12312 1 2 312 1 222 31 ⎢+− − + + + ⎥ uKdet( ) ⎣ ()()dC12321 dC eC m J d e m 113 J C ⎦
1 ⎡(aCbCccdeCmJcdadCbdCcecdeCmm22+ + ( ++ )) + ( + )( 2 + 2 − ( ++ )) ⎤ a = 12 312 1 2 312 32 ⎢++−−+ +−+ −++ ⎥ uKdet( ) ⎣ ((a a c ) C122212321 b ( b c ) C ) m J ( adC bdC e ( c d e ) C ) m J ⎦ 306
−+()de ademJemJcmJcecdmmC=++−++[]() () 33uKdet( ) 1 1 2 1 1 2 1 2 3
1 a=−+++−[] cm J() d e m J ce () c d m m em J C 34det(K ) 1 2 1 1 1 2 2 1 3
APPENDIX C
STATE-SPACE COEFFICIENTS FOR 5-AXLE PLANAR MODEL
The state-space system matrix is in the form:
⎡aaaa11 12 13 14 ⎤ ⎢aa ii⎥ = ⎢ 21 22 ⎥ A5 ⎢aa31ii 34 ⎥ ⎢ ⎥ ⎣ 0010⎦
The determinant of the non-state-space system “mass” matrix, K5, is:
=+ +22 +− det(KmmJJmmdJcaJ51212121 ) ( ) ( ( ) 2 )
307
Row 1:
−++[ −−] −1 ⎡()caGmJ22 ( fCfCmdJcaJ 14 25 ) 2 1 () 2 ⎤ a = ⎢ ⎥ 11 det(K ) −++++ +22 +− 5 ⎣⎢ ()(())C123451221 C C C C JJ dmJ c a mJ 22⎦⎥
222−++− −1 ⎡udet( K5122122 ) GJJ ( dmJ ( c a ) mJ )⎤ a = ⎢ ⎥ 12 uKdet( ) +−[] − −− 5 ⎣⎢ H() c amJ22 dmJ 21 Nc () amJ 22 ⎦⎥
308
=− +− + +− + + − + + − + G( aCabC112231425 ( )( abCcafC )( )( cafC ))
=−22 − − − − 2 − − + 2 − − + 2 N (()()()())Ca112231425 abC ab C ca f C ca f C
=− − + − − + H ((fc114225 a fC ) fc ( a fC ))
+ +22 +− +− 1 ⎡()[(()]()f14C fC 25 JJ 12 md 2 J 1 c a mJ 22 c amJH 22 ⎤ a = ⎢ ⎥ 13 uKdet( −+22 −− 5) ⎣⎢ ()(())fC14 fC 25 dmJ 21 c amJ 22 ⎦⎥
+++−−+22 −− 1 ⎡⎤()(CCJJdmJcamJfCfCdmJcamJ4 5 12 21 ())()(()) 22 1 4 2 5 21 22 a = ⎢⎥ 14 det(K ) +−[] −− + −− 5 ⎣⎦⎢⎥()(ca ac fC14 )( ac fCmJ 25 ) 22
Row 2: 309
+ +2 +− ++++ 1 ⎡GmmJdmmcaCCCCCmJ((122 ) 12 ) ( )( 1234522 ) ⎤ a = ⎢ ⎥ 21 uKdet( ) −+[] + +−++ 5 ⎣⎢ ()()()fC14 fC 25 m 1 m 2 J 2 d a c d mm 12 ⎦⎥
1 ⎡()()()N −++−HJ m m c aGmJ⎤ a = 21 2 22 22 ⎢+−−++ ⎥ uKdet(5 ) ⎣ dm12 m(( Nd a c d )) H ⎦
++222 + +[ ++−++ ] 1 ⎡⎤HJmm((21 2 ) dmmfCfCJmm 12 )( 14 25 ) 21 ( 2 ) dacdmm ( ) 12 a = ⎢⎥ 23 uKdet( ) ++ − 5 ⎣⎦⎢⎥()()fC14 fC 25 c a mJ 22
310
⎡⎤()()(CCcamJfCfCJmm+− ++ )()[ ++−++ dmmacd ( )] = 1 45 221425212 12 []a24 ⎢⎥ det(K ) +−− +−− + +2 5 ⎣⎦⎢⎥()()(())ac fC14 ac fC 25 Jm 21 m 2 dmm 12
Row 3:
− ⎡ []+ + −++ − +⎡⎤ + + +−++ 2 ⎤ 1 GJmm21()() 2 dacdmmfCfCJ 12 ( 1425 )()()()⎣⎦ 1 Jmm 2 1 2 acdmm 12 a = ⎢ ⎥ 31 uKdet( ) −++++ −− 5 ⎢⎣ ()(())C12345 C C C C dmJ 21 c amJ 22 ⎦⎥
⎡HmmJ⎡⎤()()()+++−+++ J acdmmGdmJcamJ2 [ −− ()]⎤ = 1 ⎣⎦1212 12 21 22 []a32 ⎢ ⎥ uKdet( ) ⎢−++−++ ⎥ 5 ⎣ NJ21()() m m 2 d a c dmm 12 ⎦ 311
⎡−++−++−+[]22⎡⎤ + ++−++ 2⎤ 1 Hm()(122 mJ dacdmm ) 1214251212 ( fC fC )()()()⎣⎦ m m J J acdmm 12 a = ⎢ ⎥ 33 uKdet( ) ++ −− 5 ⎢⎣ ()(())fC14 fC 25 dmJ 21 c a mJ 22 ⎦⎥
− ⎡()()()()fC++++−++ fC⎡⎤ m m J J a c d2 mm ⎤ = 1 14 25⎣⎦ 1 2 1 2 12 a34 ⎢ ⎥ det(K ) ⎢−+[][ −− +−− +−− + +−++ ][⎥ ] 5 ⎣ ()CCmdJcaJ4521 ()( 2 acfCacfCmmJdacdmm 14 )()()( 25122 ) 12⎦
312
APPENDIX D
RUNGE-KUTTA INTEGRATION ROUTINE IN MATLAB®
function Xout = runge_kutta_v2(dynamic_function, X, h)
% dynamic_function
% X
% h
%
% Runge-Kutte Matrix Integration Routine
%
% Code modified from Laurie Ray
%
%
% Euler predictor-one-third step
%
funct_1 = feval(dynamic_function, X);
% "feval(...) evaluates the state-space system for one iteration at the
% operating point "X"
313
X1 = X + h .* funct_1 ./ 2;
%
% Trapezoid corrector-one-third step % funct_2 = feval(dynamic_function, X1);
X2 = X + h .* funct_2 ./ 2; % % Adams-Bashforth predictor-half step % funct_3 = feval(dynamic_function, X2);
X3 = X + h .* funct_3; % % Adams-Bashforth predictor-full step % funct_4 = feval(dynamic_function, X3);
Xout = X + h .* (funct_1 + 2 .* (funct_2 + funct_3) + funct_4) ./ 6;
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