IMPROVING T H E DYNAMIC PERFORMANCE OF MULTIPLY-ARTICULATED VEHICLES by Michael R. Rempel B. A . Sc. (Mechanical Engineering), The University of British Columbia, 1999 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Applied Science in The Faculty of Graduate Studies Department of Mechanical Engineering We accept this mesj^^aj^onfomiing tojhe required standard: The University of British Columbia November 2001 © Michael R. RempeL 2001 In presenting this thesis in partial fulfillment of the requirements of an advanced degree at the University o f British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree the permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood the copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 2324 Main Mall Vancouver, B C Canada V 6 T 1Z9 Date: Abstract Current heavy vehicles having two or more trailers suffer from unique dynamic characteristics which limit both their stability and maneuverability at speed. The control o f these characteristics in A-train vehicles is the objective o f this work; specifically, the attenuation of rearward amplification and high-speed offtracking. Control is attained is through automatic steering of the A-dolly axle; the control system is localized entirely on the A-dolly, creating a modular and easily-implemented unit. The equations of motion were derived for a reference A-train vehicle, and the results simulated and compared to literature. A nonlinear two-dimensional yaw plane model with semi-static load transfer in the pitch and roll modes was found to adequately display the intended system characteristics. To apply control, a second linear state space model was created, based on the behavior of the A-dolly and the second semitrailer only. For high-speed, low amplitude maneuvers under feasible input frequencies, the results corresponded to the nonlinear simulation. Control was achieved using a state variable feedback controller to steer the dolly wheels; the gains were determined by moving the plant eigenvalues via Ackermann's method to the critically-damped locations of the "faster" mode. The controller was shown as to be robust to parameter estimation errors and to balance performance and required control inputs well. A n Extended Kalman Filter (EKF) was employed to estimate the unknown tire cornering properties and states not measurable direcdy from the A-dolly. Through simulation, it was found that the controller was effective in reducing both the observed rearward amplification and the dynamic offtracking, although the effectiveness decreased with increasing forward velocity. A t the nominal velocity for interstate highways in 11 the United States (24.6 m/s), the peak improvement in rearward amplification under control was reduced to a minimum of 25 percent of the uncontrolled value; the peak value of offtxacking was reduced up to 50 percent. Key Words: rearward amplification, ojftracking, A-train, nonlinear simulation, state space, feedback, Extended Kalman Filter Table of Contents Abstract ii Table of Contents iv List of Figures ix List of Tables xii Acknowledgements xiii Introduction and Objectives 1 1.1 Benefits of Twin Operations 2 1.2 Problems Associated With The Operation of Twins 3 1.2.1 Rearward Amplification 3 1.2.2 Rollover 4 1.2.3 Offtracking 5 1.3 Introduction to Equipment Used 7 1.3.1 The Fifth Wheel 7 1.3.2 The Pintle Hook 8 1.3.3 TheA-Dolly 9 1.4 Heavy Vehicle Configuration in the United States 10 1.5 Heavy Vehicle Operation in the United States 12 1.6 The Safety of Combination Vehicles 14 1.7 Conclusion - Subjects for Study 19 Review of Prior Research 22 2.1 Modeling Overview 22 2.2Reference Frames 23 2.3Tire Modeling 25 2.3.1.Slip Angle and Cornering Force 25 2.3.2.Linear Tire Modeling 29 iv 2.3.3 Tire Modeling Using A Look-Up Table 29 2.3.4 Tire Modeling Using Dedicated Functions 30 2.3.5 Combination of Braking and Tractive Force 32 2.4 Vehicle Modeling - Linear Two-Dimension 34 2.5 Vehicle Modeling - Non-Linear Two-Dimensional 34 2.6 Vehicle Modeling - Three Dimensional 36 2.7 Analysis Methods 38 2.7.1 Solution Using an Analog Computer 38 2.7.2 Solution Using Control Theory 39 2.7.3 Solution of Non-Linear Stability Using Lyapunov's Method 40 2.7.4 Simulation of a System of Non-Linear Equations of Motion 41 2.8 Results of Dynamic Modeling 41 2.9 Dedicated Offtracking Models 43 2.10 Methods of Reducing Rearward Amplification 44 2.10.1 C-Dollies and B-Trains 44 2.10.2 Shifted Instant Center Dollies 46 2.10.3 Forced-Steer and Skid-Steer Dollies 47 2.10.4 Liked Articulation Dollies 48 2.10.5 Roll Stiffened Pintle Hook Assembly 48 2.10.6 Locking A-Dolly 48 2.10.7 Steerable C-Dollies 50 2.10.8 Reduction of Rearward Amplification Through Active Yaw Control 50 2.10.9 Reduction of Rearward Amplification using Externally-Mounted Dampers 52 Modeling Multiply-Articulated Vehicles 53 3.1 Modeling Assumptions 53 3.2 Coordinate Systems 55 3.3 System of Equations 56 3.3.1 Yaw Equations of Motion 58 v 3.3.1.1 Tractor Yaw Equations 3.3.1.2 Lead Trailer Yaw Equations 3.3.1.3 Dolly Yaw Equations 3.3.1.4 Second Trailer Yaw Equations 3.3.1.5 Articulation Angles and Force Constraints 3.3.1.6 Velocity and Acceleration Constraints 3.3.1.7 Tire Forces and Aligning Moments 3.3.1.8 Manipulation into State Space Form 3.4 Load Transfer 58 59 59 60 61 63 64 65 66 3.4.1 Static Loading 66 3.4.2 Dynamic Loading Due to Roll 68 3.4.3 Dynamic Loading Due to Pitch 72 3.5 Tire Model Implementation 75 3.6 Driver Model 79 3.7 Simulation Protocols 81 3.8 Method Of Solution 84 3.9 Simulation Results 87 3.9.1 Verification of Results 87 3.9.2 Rearward Amplification Sensitivity Factors 91 Offtracking Results 95 - 3.9.3 3.10 Discussion 97 Control System Design 4.1 99 Linear Model Derivation 99 4.1.1 Preface to Modeling 100 4.1.2 Derivation of the Linear Equations of Motion.. 101 4.1.3 Constraint Equations 102 4.1.4 Slip Angles and Tire Modeling 104 4.2 4.2.1 State Space Formulation of the Equation of Motion Controllability and Observability 104 106 4.3 Comparison of Linear and Non-Linear Models 107 4.4 System Eigenvalues 110 vi 4.5 State Variable Feedback Control 112 4.5.1 State Feedback and Ackerman's Formula 113 4.5.2 Screening of Candidate Control Strategies 114 4.5.3 The Linear Quadratic Regulator 116 4.5.4 Prototype Control 117 4.5.5 Critical Damping Control of the A-Dolly 118 4.5.6 Critical Damping of Each Mode 120 4.6 Discussion 122 State And Parameter Estimation 5.1 124 State Augmentation and the E K F 124 5.2 Discretization of the Equations of Motion 126 5.3 Formulation of the Extended Kalman Filter 128 5.4 Estimating Performance of the E K F 130 5.4 Convergence of the E K F Algorithm 133 5.5 Evaluation of Miscellaneous Parameters 134 5.6 Discussion 135 Results 136 6.1 Tire Property Identification 136 6.2 Dynamic Performance Improvement 137 6.2.1 Testing Under S A E J2179 138 6.2.2 Testing Using the Frequency Response Method 144 6.3 Comparison of Proposed Controller With U M T R I Results 148 6.4 Steady-State Offtracking Performance 150 6.5 Parameter Sensitivity 151 6.6 A-Dolly Sensor and Actuator Requirements 152 6.7 Discussion 154 Conclusions and Recommendations 7.1 155 Contributions of the Present Work 156 vii 7.2 Recommendations for Future Work 157 References 160 Appendix A: Reference A-train Data 165 Appendix B: Tire Data 168 v viii List of Figures Figure 1-1 — Typical Twin Trailer Truck Figure 1-2 - Schematic of Rearward Amplification 2 4 Figure 1-3 — L o w Speed Offtracking [5] 6 Figure 1-4 - Fifth Wheel Connection Figure 1-5 - Pintle H o o k Connection (a) Pintle Hook, (b) Locking Eye 7 8 Figure 1-6 - Typical A-Dolly 9 Figure 1-7 — Representative Heavy Vehicle Types Figure 1-8 - National Average of Heavy Vehicle Usage, United States (1995) Figure 1-9 - Regions for Analysis Figure 1-10 — Combination Vehicle Usage By Region (Percentage o f Total Ton-Miles) Figure 1-11 — Fatal Crashes by Vehicle Class [9] Figure 1-12 — Fatal Crash Rates on Various Highway Classes [9] Figure 1-13 — Accident Locations for Singles and Doubles [10] Figure 1-14 - Accident Involvement for Singles Figure 1-15 - Accident Involvement for Doubles Figure 2-1 - ISO Standard Vehicle Reference Frame Figure 2-2 — Pneumatic Tire Under Lateral Loading [11] Figure 2-3 - Slip Angle Versus Lateral Force, Michelin 10.00x20 Figure 2-4 - Slip Angle Versus Self-Aligning Torque, Michelin 10.00x20 Figure 2-5 — Friction Ellipse Concept 11 12 13 14 15 16 17 18 18 24 26 27 28 32 Figure 2-6 — C-Dolly Configuration 45 Figure 2-7 - Trapezoidal Dolly Configuration 46 Figure 2-8 - Locking A - D o l l y (a) Locked, (b) Unlocked 49 Figure 3-1 - Inertia Reference Frames (black) and Tire Forces (red) and Hitch Forces (blue) for A-Train 55 Figure 3-2 Figure 3-3 Figure 3-4 Figure 3-5 - Tractor F B D Lead Semitrailer F B D A-Dolly F B D Second Semitrailer F B D 58 59 60 61 Figure 3-6 - Static Loading, Standard A-Train Figure 3-7 - Load Transfer Due to Roll for the A - D o l l y Second Semitrailer U n i t (a) A-Dolly, (b) Second Semitrailer 67 69 Figure 3-8 - Load Transfer Due to Roll for the Tractor-Lead Semitrailer Unit (a) Tractor, (b) Lead Semitrailer 71 — Load Transfer Due to Pitch (a) Second Semitrailer, (b) A-Dolly, (c) Lead Semitrailer, (d) Tractor 74 Figure 3-10 - Comparison of Tire Properties: (a) Lateral Force vs. Slip Angle, (b) Aligning Moment vs. Slip Angle 77 Figure 3-9 ix Figure 3-11 — Schematic o f the Look-Ahead Driver Model 80 Figure 3-12 - S A E J2179 Dynamic Stability Test Maneuver 82 Figure 3-13 — Solution Method Flow Chart (a) Overall, (b) State Equations 86 Figure 3-14 - Vehicle Lateral Accelerations for S A E J2179 88 Figure 3-15 - Comparison of Simulation and U M T R I Test Results 90 Figure 3-16 - Comparison of Reference and U M T R I Test A-Trains 91 Figure 3-17 - Effect of Velocity on Rearward Amplification 92 Figure 3-18 - Effect of Tire Type on Rearward Amplification 93 Figure 3-19 - Effect o f Frequency and Load Transfer on Rearward Amplification 94 Figure 3-20 - Offtracking During S A E J2179 Test Protocol 95 Figure 3-21 - Effect o f Speed on Offtracking 96 Figure 3-22 - Steady-State Cornering Offtracking Performance 97 Figure 4-1 - F B D For Linear A-Dolly 101 Figure 4-2 - F B D For Linear Semitrailer 102 Figure 4-3 — Comparison o f Linear and Non-linear Results: Angular Velocity 108 Figure 4-4 — Comparison of Linear and Non-linear Results: Articulation Angles 109 Figure 4-5 — Sensitivity o f Eigenvalues to Velocity Ill Figure 4-6 - Uncontrolled Full-Trailer Behavior 115 Figure 4-7 — Effectiveness of Candidate Control Laws 123 Figure 5-1 - Comparison o f Estimated and Actual Parameters: Trailer Angular Velocity 130 Figure 5-2 — Comparison of Estimated and Actual Parameters: Trailer/A-dolly Articulation Angle 131 Figure 5-3 - Performance of the Tire Coefficient Estimator 132 Figure 6-1 — Required Steering Input 138 Figure 6-2 — Comparison Between Tractor and A-Dolly Steering Angles 139 Figure 6-3 — A-Train Lateral Acceleration Performance Under Control 140 Figure 6-4 - Comparison Between Controlled and Uncontrolled Second Trailer Lateral Acceleration 141 Figure 6-5 - A-train Displacement Performance Under Control 142 Figure 6-6 - Comparison Between Controlled and Uncontrolled Second Trailer Trajectory Following 143 Figure 6-7 - Controlled Versus Uncontrolled Response: V = 18.5m/s. 145 Figure 6-8 - Controlled Versus Uncontrolled Response: V - 24.6 m/s. 146 Figure 6-9 - Controlled Versus Uncontrolled Response: V - 30.8 m/s. 147 Figure 6-10 - Comparative Performance: Controlled and Uncontrolled A-train, Self-steer Cdolly, and Controlled Steer C-Dolly 149 Figure 6-11 - Steady-state Cornering Under Control 150 Figure 6-12 - Effect of Parameter Error in Rearward Amplification Attenuation 151 Figure 7-1 - Slow Subsequent Convergence of the Identified Tire Property Using the E K F 1 5 8 Figure B - l — Cornering Force Versus Slip Angle, Firestone 10.00x22F 168 Figure B-2 - Aligning Moment Versus Slip Angle, Firestone 10.00x22F 169 Figure B-3 - Cornering Force Versus Slip Angle, Firestone 10.00x22RIB 170 x3 x} x3 x Figure B-4 Figure B-5 Figure B-6 - Aligning Moment Versus Slip Angle, Firestone 10.00x22RIB Cornering Force Versus Slip Angle, Freuhauf 10.00x20 Aligning Moment Versus Slip Angle, Freuhauf 10.00x20 xi 171 172 173 List of Tables Table 3-1 - Force and Moment Naming Nomenclature Table 6-1 — Identified Tire Parameters Table 6-2 - Peak Reductions in Rearward Amplification Table 6-3 - Sensor and Actuator Requirements Table A-l - Tractor Data Table A-2 - Lead Semitrailer Data Table A-3 - A-dolly Data Table A-4 — Second Semitrailer Data Table A-5 - Miscellaneous Data xii 57 137 148 153 165 166 166 167 167 Acknowledgement I would like to express my gratitude towards my supervisor Dr. A . B. Dunwoody for his support, both financial and technical, throughout this project. His effort was invaluable towards the completion and success of this work. I would also like to acknowledge the assistance and encouragement of my friends and especially my family. Their support was much required and appreciated. T o them I dedicate this work. The research contained within this thesis was made possible through the generous support of Teleflex Canada, the Product Development Laboratory at U B C , and the National Science and Engineering Research Council ( N S E R Q under the Industrial Postgraduate Scholarship (IPS). xiii Chapter 1 Introduction and Objectives Following the passage of the Surface Transportation Assistance Act (SSAA) in 1982, the utilization of twin trailer trucks (a tractor pulling two trailing units) has increased gready in both the United States and Canada. The SSAA was important in that it provided provision for the operation of twin trailer trucks on all interstate highways as well as their connected arterials throughout the United States. Before the passage of the act, twin trailer trucks were illegal to operate in 14 eastern states [l] . Conversely, such vehicles were in common operation on the 1 West coast for several years prior to the act. The intent of the SSAA was to increase the efficiency o f freight transportation in general by allowing for an increase in volumetric cargo capacity, and greater flexibility in the handling and routing of cargo for trucking operators (since the width of heavy trucks is primarily limited with the width of lanes in the roadways in which they operate, the only option to allow for increased freight capacity is to increase the length of the trailer vans). Figure 1-1 depicts a typical twin trailer truck. Numbers refer to specific papers listed in the Resources Section 1 2 Chapter 1. Introduction and Objectives Figure 1-1 — Typical Twin Trailer Truck Unfortunately, the national use o f twin tractor trailers has been consistently tempered with concerns arising from their safety in everyday operation. Combination vehicles suffer from unique dynamic characteristics which limit both their maneuverability and their overall stability; such problems arise from a combination o f factors including: (a) Vehicle layout (b) Load type and distribution within the trailer(s) (c) Typical operating conditions (primarily speed) (d) Hitching method (e) Driver experience 1.1 Benefits of Twin Operation After the enactment of the SSAA, many freight carriers in the United States converted to twins. Between 1982 and early 1985, such carriers increased the total twin vehicle-miles in their fleets to nearly 48 percent; by 1990, their usage of twins jumped to 80 percent [1]. A t the forefront o f this conversion was less-than-truckload (LTL) carriers. A n L T L carrier is a general freight common carrier that specializes in hauling small shipments of such a size that Chapter 1. Introduction and Objectives 3 several shipments can make up a single load. Twins offer considerable operating efficiencies to L T L operators by reducing the number of times that freight is handled. Direct comparison of the productivity of 28-ft twin trailers to that o f standard 45-ft single trailers demonstrated an overall reduction in cost of approximately 11 percent [1]. It should be noted that such savings were realized only in companies who converted fully to twin tractor trailers. Companies with mixed fleets (that is, both single and twin trailer vehicles) experienced nominal changes in operating costs, due in large part to the logistics of operating two incompatible freight systems. 1.2 Problems Associated With The Operation of Twins Aside from logistics, the primary difficulty associated with the operation of twin are instabilities which may arise during maneuvering situations involved with avoidance (rearward amplification), and steady-state turning (offtracking, and rollover) [2]. 1.2.1 Rearward Amplification Rearward amplification refers to a situation in the operation of a multi-articulated vehicle in which the lateral acceleration of the towed unit(s) exceed those of the tractor. Rearward amplification is a dimensionless quotient of the lateral acceleration of the centers o f gravity of the last trailer and the tractor. Generally, such an event will occur as part as an avoidance maneuver or accident sequence. See Figure 1-2. 4 Chapter 1. Introduction and Objectives Figure 1-2 — Schematic of Rearward Amplification In Figure 1-2, the value of rearward amplification is defined as the quotient of B/A. Rearward amplification becomes dangerous as the lateral acceleration approaches the threshold values for rollover. In the limit, the last trailer may swing well off the path o f the tractor, with potentially serious consequences. In addition, i f the last trailer has a low rollover threshold (i.e. rollover can occur at relatively small values of vehicle lateral acceleration), it may roll over, even i f the leading trailer and tractor do not. 1.2.2 Rollover Rollover may occur for multi-articulated vehicles either as a primary event (as induced by a steady-state turn) or as a corollary effect to a severe transient maneuver (such as a rearward amplification event). Steady-state rollover generally occurs when the vehicle travels too quickly through a sweeping turn, such as those found on highway off ramps. The maneuver creates enough centrifugal force to exceed the vehicle's ability to counteract that force. This effect is particularly accentuated by the high center of gravity (CG) height typical of heavy trucks. Instead of a 5 Chapter 1. Introduction and Objectives skidding loss of control (as would be expected in a passenger vehicle), excessive lateral acceleration creates an overturning moment and the vehicle rolls over. In the case of steady-state cornering, roll stability in heavy vehicles is usually quantified using the static roll stability (SRS) [1]. SRS is defined as the maximum amount of lateral acceleration a given vehicle can withstand before rolling over. Higher values of SRS indicate superior dynamic stability. Depending on the vehicle and freight loading configurations, typical SRS values for twins range between 0.3 g and 0.45 g [3]. In comparison, a typical passenger car will have an SRS value between 0.8 g and 1.0 g. During transient maneuvers, a different rollover quantification metric is used. The measure used is the load transfer ratio (LTR) [4]. As the truck undergoes a dynamic maneuver, vertical loads are shifted between the tires at each axle. Under these conditions, the L T R represents the proportion of the loading which is carried by one side of the truck relative to the other. See Equation 1-1. Where: F is the vertical loading on the right side of the vehicle, and F is the vertical loading R L on the left side o f the vehicle. A vehicle which is perfecdy balanced will have a L T R value of 0.0. Values of L T R upwards o f 0.7 indicate impending rollover for most heavy truck configurations [1] and [4]; when the L T R value reached 1.0, the vehicle rolls. 1.2.3 Offtracking Depending on the vehicle velocity, trailers may either under- or overshoot the intended trajectory. Perhaps the most commonly-observed instance of this is low-speed offtracking (see Figure 1-3, below). Chapter 1. Introduction and Objectives 6 Figure 1-3 — Low Speed Offtracking [5] When a vehicle makes a low-speed turn, the trailer axles may track several meters to the inside of their intended trajectory. A high level of offtracking is undesirable as the vehicle may require more space for completing a given maneuver than is available. Decreasing trailer wheelbase and increasing the number o f articulations are effective in reducing the magnitude of low-speed offtracking. There exists a critical velocity at which the trailers will perfectiy follow the intended trajectory of the tractor. Above this velocity, a phenomenon known as high-speed offtracking occurs. During high-speed maneuvers, trailers track to the outside of the intended trajectory as defined by the tractor C G (they overshoot the intended path). This leads to encroachment of the trailers into the space reserved on the highway for the other lanes o f traffic, trailers running off the road, and collision with roadside objects, such as curbs and private property [6]. Such collisions may also serve to precipitate the subsequent rollover of the vehicle. Chapter 1. Introduction and Objectives 1.3 7 Introduction to Equipment Used Trucks having multiple trailers exist in a variety of configurations. These configurations consist of several basic components which are assembled to form the final vehicle. The primary units of any combination vehicles are the towing unit, or tractor, and the trailers. Trailers with axles at both ends of the vehicle are known as full trailers; conversely those with axles at one extremity are termed semitrailers. O f particular interest are the mechanisms utilized to combine the units together. 1.3.1 The Fifth Wheel The fifth wheel is the most common hitching method to combine semitrailers to a towing unit. See Figure 1-4 below. Figure 1-4 - Fifth Wheel Connection The fifth wheel consists of a clamping mechanism on the towing unit and a king pin on the trailing unit. The pinned connection allows the trailing unit to rotate relative to the towing unit 8 Chapter 1. Introduction and Objectives in the yaw and pitch. The kingpin allows the yaw degree o f freedom. A horizontal pivot pin under the landing plate provides the degree of freedom in pitch. However, motion in the roll direction is constrained by the landing plate. Such constraint is designed to take advantage of the fact that two adjacent units will roll in different directions during a dynamic maneuver. Since the coupling is rigid along the roll axis, a large counter-roll moment is produced and dynamic roll stability is improved. Note that roll is only purely constrained as the vehicle moves in a straight line; as the articulation angle between the two units increases, so also will the cross-coupling between the various rotational modes. 1.3.2 The Pintle Hook Like the fifth wheel, the pinde hook consists of two basic components: a hook on the leading unit and an eye and drawbar arrangement on the trailing unit Pintle hooks are used to connect full trailers to their leading units. Refer to Figure 1-5. (b) Figure 1-5 - Pintie Hook Connection (a) Pinde Hook, (b) Locking Eye Chapter 1. Introduction and Objectives The pintle hook differs from the fifth wheel in that in addition to allowing rotational freedom in yaw and pitch, motion is also permitted in roll. Thus, pinde hook connections do not have increased roll stability along the vehicle. 1.3.3 The A-Dolly T o create multi-unit tractor trailer vehicles, it is required that semitrailers be converted to the full trailer layout. This is achieved by a device known as a converter dolly. In the United States, the most common converter dolly is the A-dolly, given this name from the shape its frame members when seen from plan view. A typical A-dolly is depicted in Figure 1-6 below. ( Figure 1-6 — Typical A-dolly ^ 9 Chapter 1. Introduction and Objectives 10 The A-dolly has a fifth wheel hitch for interface with the semitrailer, and a drawbar and eye assembly for connection with the leading unit. Although the fifth wheel link allows for constrained roll motion between the trailing unit and the dolly itself, the pinde hook destroys the full trailer's yaw coupling to the remainder of the vehicle. The full trailer arrangement can also transmit significant portions of pitch motion due to braking, via vertical loading of the pinde hook hitch. Each time an A-dolly is introduced into a vehicle, an extra point of articulation is added. Although this extra degree of freedom is beneficial in reducing low-speed offtracking, it has a large influence on the high-speed dynamic stability of the resulting vehicle. 1.4 Heavy Vehicle Configuration in the United States The trucks considered in this thesis are limited to those used for carrying goods or special equipment on U.S. highways. Such vehicles are classified as class 7 or class 8 trucks [7j. They are defined as truck with as gross vehicle weights (GVW) of 12 000 to 15 000 kg for class 7 and over 15 000 kg for class 8. Class 8 trucks generally have a weight limit o f 36 000 kg, although Canada, Mexico, and various U.S. states allow for heavier vehicles on selected roads. There is great variability in the exact weights and configurations of heavy vehicles. A set of representative heavy vehicles is depicted in Figure 1-7, below. 11 Chapter 1. Introduction and Objectives CAR SMOLfrUNiriHUCK BOBTAIL TRACTOR-TRAILEH W E S T E R N D O U B L E (TWIN) ROCKY UOUNTAM I- 45'-48' 11 46-48' DOUBLE 1 TURNPIKE DOUBLE I—ZS-2B,'—I I—26-28'—I t-—28'-28'—I , TRIPLE Figure 1-7 - Representative Heavy Vehicle Types [8] Tractor trailers typically pull single trailers 12.2 to 16.2 m (40 — 53 ft) long. A tractor pulling two trailers, neither of which is longer than 8.7 m (28 ft), is termed a "Western Double" or twin configuration [8]. Additional types of multiple-trailer vehicles are referred to as longer combination vehicles (LCV). In general, L C V s have two or more trailers which have an overall length exceeding 17.4 m (57 ft). Both the tractor-semitrailer and the twin configurations were permitted nationwide by the STAA. The operation of the other L C V s is under debate, with their operation mandated in only 20 U.S. states [8]. In states which permit L C V s , their routes of operation are typically restricted. In 1991, the Surface Transport Efficiency Act prohibited the expansion of L C V s beyond the 20 states in which their operation is currently legal. This action was undertaken in part due to safety concerns arising from the crash experience of twins, since L C V s share certain handling characteristics with twins. 12 Chapter 1. Introduction and Objectives 1.5 Heavy Vehicle Operation in the United States The flexibility for motor carriers to choose both vehicles and payloads that meet their needs in an efficient manner and at minimum cost is of great importance. In an ideal case, the carrier loads each vehicle to its maximum payload. In the case o f high-density freight (such as sand, liquids and grains), the limiting factor is the difference between the maximum G V W o f the vehicle and its tare (empty) weight. For low-density cargo, the limiting factor is often the volumetric capacity of the trailers. In both cases, selection of the transport vehicle is made with the goal o f minimizing the unused freight capacity. Figure 1-8 depicts the distribution o f vehicle type used in 1995 within the United States [2]. The results are normalized by calculating the mass of freight moved per mile (ton-mile) for each configuration. 6+ Axle Figure 1-8 - National Average of Heavy Vehicle Usage, United States (1995) Chapter 1. Introduction and Objectives The majority of freight moved within the U.S. is via 5 axle, single trailer vehicles, with 79 percent of the cargo ton-miles. The usage of the twin configuration is second at 5.7 percent, and is the most common multiply-articulated vehicle by a large margin. Figure 1-8 represents a national average of heavy vehicle productivity. Further insight may be gained if the United States is subdivided into seven geographical/legislative regions. Figure 1-9 depicts the boundaries for analysis. combination vehicles by region. Figure 1-9 — Regions for Analysis Figure 1-10 shows the resulting distribution of 13 Chapter 1. Introduction and Objectives 14 10 Twins Others • Twins • Other Doubles! • Triples Twins 5? Twins B 4 Twins H Twins Twins Others 1 H Others Others Others Others ^^Ithers New England Middle Atlantic Tripled j South Atlantic Trip Triples Midwest South Central Western LCV Other Western Figure 1-10 — Combination Vehicle Usage By Region (Percentage of Total Ton-Miles) Combination vehicles, and in particular twins, are in the central and western United States at rates above the national average. Likewise, the usage in the New England region is minimal. In general, twins and other multiply-articulated vehicles are used for long, high-capacity transport between major cities, rather than delivery within them. 1.6 The Safety of Combination Vehicles Determining the relative safety of twin or L C V s against single tractor-trailers is difficult. A t first glance, a simple comparison o f fatal crash rates by vehicle class may seem to yield the answer. See Figure 1-11. Chapter 1. Introduction and Objectives 15 120 Personal Vehicle** Single Unit Single Trailer MultiTrailer Figure 1-11 - Fatal Crash Rates by Vehicle Class [9] Note that the comparative crash rates have been normalized relative to that o f the single trailer. Figure 1-11 seems to suggest that although single- and multi-unit heavy trucks have higher accident occurrences that personal vehicles, they have relatively the same rate o f fatal accidents. However, Section 1.5 implies that single- and multi-unit trucks are generally used in very different operating conditions. The results may be further skewed in that the drivers of twins are typically more experienced than those of singles, and typically work shorter shifts [1], Figure 1-12 shows the fatal crash rates of major heavy vehicle classes divided according to the type of road on which the accident occurred. Here again the resulting crash rates have been normalized relative to that of a single trailer combination unit operating on rural interstate. 16 Chapter 1. Introduction and Objectives 600 Rural Interstate Other Rural Arterials Other Rural Roads Urban Interstates Other Urban Streets Figure 1-12 - Fatal Crash Rates on Various Highway Classes [9] There are several patterns of note. The first is that involvement rates on rural interstates are 300 to 400 percent lower than that of other rural roads for multi-trailer vehicles. In fact, the rural interstate is the safest of all road types for both single- and multi-unit vehicles. Second, accidents occur more frequently in the rural setting to multi-trailer vehicles than it does to their single counterparts. Third, the accident rates for multi-trailer trucks is much greater than both passenger vehicles, and single-unit trucks (SUT). Further, a new picture emerges in Figure 1-13 as the relative locations for all single and double accidents is shown. 17 Chapter 1. Introduction and Objectives Divided R o a d Undivided R o a d Unknown F i g u r e 1-13 — Accident Locations for Singles and Doubles [10] One observes that multi-trailer trucks have a much greater probability o f accident on divided (interstate-type) roads than do single tractor-trailers. However, it is known that interstates are highly-engineered and are among the safest of roads which heavy trucks traverse. Figure 1-13 thus implies that there exists a higher probability of high-speed accidents in multi-articulated vehicles over single tractor-trailer units. Next, the causes of said accidents will be investigated. Figures 1-14 and 1-15 depict the cause of accidents for singles and doubles as compiled by the Bureau of Motor Carrier Safety (BMCS)inl984[10L Chapter 1. Introduction and Objectives 18 R a n Off R o a d Collision 73.2% Figure 1-14 Accident Involvement for Singles — Jackknife 0.1% Figure 1-15 — Accident Involvement for Doubles Chapter 1. Introduction and Objectives 19 In the case of single tractor-trailer vehicles, the primary accident mode is collision at 76 percent. Secondary accident causes are running off the road, jacldcnife, and overturning, with magnitudes of 5,10, and 7 percent, respectively. Twin tractor-trailers are much different. Collision is again the most common accident cause, but its occurrence drops to 50 percent There is a marked increase in rollover accidents (29 percent), while the other causes remain approximately constant. Based on the data in Figures 1-14 and 1-15, it can be seen that the onset o f rollover-based accidents occurs much more frequentiy for twins than it does for single-trailer vehicles. Figure 1-12 shows that there is a strong probability that such an accident occurs in a high-speed setting. One may thus infer that there is a difference in the high-speed stability of multi-trailer vehicles compared to their single trailer counterpart. Winkler et al [10] have shown also that once an accident sequence had begun, doubles have a greater tendency to overturn than do singles. The same holds true for jackknife. The indications are clear that in certain areas of the performance envelope, conventional twintrailer vehicles do not perform as well as do singles. 1.7 Conclusion - Subjects for Study The preceding statistics show that economic and logistic forces are moving freight carriers in the United States to adopt twin tractor-trailer or L C V configurations. With the adaptation of such vehicles, there exist difficulties pertaining to both safety and to dynamic performance in the normal domain of operation. Current equipment has been shown relatively ineffective in combating these problems. The study herein, focused on improving the rearward amplification and offtracking ability of multiply-articulated vehicles (and in specific, the twin tractor-trailer configuration), investigated 20 Chapter 1. Introduction and Objectives the performance of a "smart" A-dolly using a variety of vehicle simulations. The following summarize the goals of such a system: (a) Reduce rearward amplification. (b) Improve the trajectory following behavior of trailers (offtracking). (c) Localize all related computing, sensing, and actuating components located on the "smart" A-dolly only. (d) The system must be compatible with existing vehicles. (e) The system should be as lightweight and simple as possible, implemented with a minimum of sensing and actuating devices. Further, the following was used to focus the study. 1. Literature Review Documentation pertaining to the modeling o f multi-trailer heavy vehicles is reviewed indepth in Chapter 2. The objective of this review was to obtain information regarding modeling techniques, tires, and previous and state-of-the-art methods for improving the dynamic performance of twins. Test data was also obtained from full-scale tests to verify the operation and validity of the models used in this thesis. 2. System Modeling A complex non-linear twin tractor-trailer vehicle was created and its dynamic performance noted. The model was run through a variety of lane change, steady-state cornering, and frequency response maneuvers to verify the validity of the output A "virtual driver" was instituted to more accurately control the trajectory of the vehicle through set maneuvers. Chapter 1. Introduction and Objectives 21 3. Control and Parameter Estimation To facilitate the implementation of various control strategies, a simplified linear model of the A-dolly and second semitrailer was created. This formulation was verified against the complex plant model to show reasonable fit in the desired operational domain. A control law was set with the intent of increasing the effective damping coefficients of the two primary modes. Due to environmental and system variations which routinely occur during the normal operation of twin tractor-trailer vehicles, a parameter estimation algorithm was created to account for changes in the engineering properties of the tires. Additionally, to decrease the required instrumentation, several states of the system were also estimated. The Extended Kalman Filter algorithm determines both of the former simultaneously and recursively. The effect of the controller on the full nonlinear plant was tested over a variety o f operational conditions and the improvement in dynamic performance noted. The background to the problem of dynamic performance in twin tractor-trailer and L C V s has been presented in this chapter. Objectives for meeting this deficiency have been shown; the following sections detail the procedures and specific results obtained to achieve these goals. Chapter 2 Review of Previous Research The aim o f this chapter is provide a comprehensive overview of developments pertaining to the improvement o f dynamic performance in multiply-articulated vehicles. main areas of information reviewed in this study. There are two The first concerns the modeling and mathematical representation of multiply-articulated vehicle systems. As the availability of computational power increased and became more accessible, so also did the accuracy and the allowable complexity o f the system representation. Subsequendy, the variety o f methods by which the desired information could be obtained from the equations of motion also increased. The second area of interest holds work completed in the area of improvement o f dynamic performance of heavy trucks. The solutions suggested in the literature have evolved from the simple mechanical systems o f the late 1960s to modem "intelligent" control strategies which constitute the current state-of-the-art. 2.1 Modeling Overview Creation of a system model is one of the central tasks in the study of dynamic systems. In the present case of multiply-articulated vehicles, this implies the distribution and discernment of vehicle physical characteristics and determining system behavior to both driver inputs (such as braking and steering) as well as external inputs (such as road conditions and other 22 Chapter 2. Review ofPrevious Research environmental factors). Decisions and assumptions pertinent to the modeling task may alternately enhance or constrain the results of successive mathematic analysis. The resulting model must be sufficiendy detailed to investigate the desired effect in the system (in addition to precluding conclusions based on insignificant behavior) while remaining at a minimum level of complexity. It is the task of the researcher to balance what assumptions can safely be made in the domain of operation of the model with the need for reasonable computational times and accuracy in the results. Due to the large variety of available heavy vehicle models and solution methods, the literature will be investigated systematically in terms of the major components in each model, the order, detail, and assumptions made in their derivations, and the methods by which the final analysis was completed. 2.2 Reference Frames The selection of reference frames is a critical step in the representation of any dynamic system in a system of equations. The reference frames must allow for full representation of the system kinematical (vehicle motion) and kinetic (tire forces) effects and still permit a final solution. In general, it is desired to utilize reference frames which facilitate the solution through elimination of excess variables. There are two general reference frame types used throughout the literature. The first involves inertial coordinates. These coordinates are fixed and the vehicle moves relative to their position. Inertial coordinates are particularly useful in describing the trajectory of the vehicle under study. However, a model created with respect to inertial coordinates suffers from increased complexity in the elimination of excess variables. Refer to subsequent sections for details. 23 Chapter 2. Review ofPrevious Research 24 The second category of reference frames are termed bodyfixed frames. In this implementation, the coordinate frames are fixed relative to each vehicle in the system; the frames translate and rotate with the body to which they are fixed. T o ensure consistency throughout vehicle dynamics literature, an ISO standard has been created. The body-fixed reference frame is orientated such that the x-axis is positive forward in the vehicle's direction o f motion, the yaxis is positive to the driver's right, and the z-axis is positive downwards [11]; it yaws, pitches, rolls, and translates with the body. See Figure 2-1 below. PITCH VELOCITY Q. LATERAL VELOCITY V FORWARD VELOCITY V ^ x ROLL VELOCITY fi, VERTICAL VELOCITY V, YAW j VELOCITY Q t z Figure 2-1 - ISO Standard Vehicle Reference Frame Regardless of the reference frame chosen, it is common practice to express rotational variables in terms of Eulerian angles [12]. This facilitates conversion of information from one reference frame to another by simply utilizing a rotational transformation matrix (an idea borrowed from the robotics field). This convention also simplifies the mathematic connection of the various vehicle bodies in a multi-unit heavy truck. Translational behavior is usually modeled in terms of Cartesian coordinates. Cartesian coordinates offer a special convenience when Newtonian mechanics are used to derive the system equations of motion. In this instance, Cartesian coordinates allow methodical formulation of the problem in state variable form. certain solution strategies. This is a significant simplification for Translational movement may also be modeling in terms o f Chapter 2. Review ofPrevious Research 25 curvilinear coordinates in the instance where the system equations of motion are derived using techniques other than Newtonian mechanics. Some modeling strategies combine both types of reference frames. In these cases, the system equations of motion are derived with respect to as a body-fixed reference frame. The results are then converted into an inertial coordinate system for comparison and analysis. Additional details pertinent to such systems will be given in the following sections. 2.3 Tire Modeling The modeling of the mechanics of pneumatic tires is of critical importance in achieving an accurate model o f a multiply-articulated vehicle. Aside from aerodynamic and gravitational forces, nearly all other forces and moments affecting the motion of a ground vehicle are applied through the contact patch between the tire and the ground. In general, tires are expected to perform the following duties in heavy vehicles [11]: (a) Support the weight of the vehicle (b) Cushion the vehicle over surface irregularities (c) Provide sufficient traction from driving and braking (d) Provide adequate steering control and directional stability In our particular interest o f reducing rearward amplification and offtracking, the factors determining (c) and (d) will be investigated. The mechanical properties derived are valid for operation in on-road conditions only (operation on asphalt or concrete). 2.3.1 Slip Angle and Cornering Force In the absence of a cornering force, a pneumatic tire will roll in the direction parallel to a plane formed perpendicular to the wheel axle (the x-z plane in the ISO coordinates). This is referred to as the wheelplane. However, in the case of a side force being applied to the tire, the direction 26 Chapter 2. Review ofPrevious Research of motion will occur at an angle a to the wheel plane. Refer to Figure 2-2. The angle a formed between the plane through the wheel centerline and the direction of motion is referred to as the slip angle. The slip angle may be expressed as the angle formed between the forward velocity V and the lateral velocity V at the tire. x a = tan (2.1) 1 Chapter 2. Review ofPrevious Research 27 As can be seen in Figure 2-2, the force developed at the wheel center and the reaction force occurring at the contact patch are not collinear. The distance between the two forces t when p observed in plan view is termed the pneumatic trail, the product o f the pneumatic trail and the cornering creates a force couple known as the self-aligning torque. Vertical loading most strongly affects the cornering properties of tires. Other factors include tire construction, wear, inflation pressure, and environmental conditions. Figures 2-3 and 2-4 show representative slip angle versus lateral force and slip angle versus self-aligning torque for a typical heavy truck tire. Both of the former plots are based on data empirically measured at the University of Michigan Transport Research Institute (UMTRI) [10]. 0 Figure 0.02 0.04 0.06 0.08 Slip Angle (red) 2-3 - Slip Angle Versus Lateral Force, Michelin 10.00x20 0.1 0.12 Chapter 2. Review ofPrevious Research 0 0.02 0.04 28 0.06 0.08 Slip Angle (red) 0.1 0.12 0.14 Figure 2-4 - Slip Angle Versus Self-Aligning Torque, Michelin 10.00x20 It can be seen that cornering force generally increases in a nonlinear manner with increasing slip angle. The same effect occurs for the self-aligning torque. Both plots also show the nonlinear relationship between vertical loading and cornering force or self-aligning torque. The goal of tire modeling is to derive a function which represents the information contained in Figures 2-3 and 2-4 in the expected domain of operation for the model. The key methods of achieving this from the literature are described in the following sections. Chapter 2. Review ofPrevious Research 2.3.2 29 Linear Tire Modeling Linear tire modeling is the most widely-used and simplest technique of representing the relationship between slip angle and lateral tire force. This method is common to the approaches noted in references [13] — [20]. In the case of small slip angles (values less than approximately 0.02 rad) Figures 2-3 and 2-4 are relatively linear and independent of vertical tire loading. A parameter called the cornering coefficient C is introduced to represent the tire behavior. The cornering coefficient is defined as a the derivative o f the cornering force with respect to the slip angle a evaluated at zero slip angle: (2.2) da ia=0 A similar expression is created for the aligning moment coefficient Mj (2.3) da ! a=0 The value of both derivatives are assumed to be constant. The validity of Equations 2.2 and 2.3 decrease as the values of slip angle increase. Both linear tire coefficients perform acceptably in small slip-angle maneuvers with small amounts of load transfer; however, as the severity of the test and the load transfer increases, the linear approximation breaks down and quite drastically over-predicts the cornering and aligning moment coefficients. 2.3.3 Tire Modeling Using A Look-Up Table Tire models are usually generated from empirical data sets. As such, the lateral force function Fyxfa) and the aligning moment function My^a) are produced accurately at the measured points only. If such points have small enough spread between them, they can be connected using a line. The value of lateral force or aligning moment can then be uniquely defined using 30 Chapter 2. Review ofPrevious Research a two-dimensional interpolation scheme (the two dimensions being the slip angle and the vertical tire loading). The fit of the interpolated result is on the same order o f magnitude as curve- fitting results to the data [24]. Look-up tables are used extensively in models arising from U M T R I [1], [10], [21]-[24]. The look-up table is advantageous in the ease with which it can be applied to differing data sets; the interpolator can be designed modulady and test the effects of many different tire data sets, simply by reading different data tables. 2.3.4 Tire Modeling Using Dedicated Functions There are several schemes available to fit analytic functions for derived tire data. and M ja to empirically- For vehicle maneuvers in which the slip angle corresponding to the tire's maximum side force ct^ is not achieved, D'Souza and Eshleman [25] created a tire model relating the vertical tire loading and the tire slip angle a. (2.4) 9.0F 1 9.0 + 6800.0 57.27 Z F y 4 0.35F, a 1.011.0——10.0(sgn(«)) 6800.0 a + 3.0 a (2.5) Equations 2.4 and 2.5 are valid only for Goodyear 10.00-20 Super Hi-Miler truck tires; although the resulting function in this case has a very good fit to experimental data, it has limited applicability for conditions other than the original empirical tests, or for different tires. Bakkar et aL [26] expanded on the ideas set out by D'Souza and Echleman by creating a family of analytic functions to model the lateral force and aligning moment characteristics. Special effort was taken during the derivation to maximize the number o f physically meaningful parameters; that is, parameters which typify the behavior o f the tire. The general form of the derived tire functions is listed in Equation (2.6). Y(a) = Dsin[Carctan(5a-£(5a-arctan(5a)))] (2.6) Chapter 2. Review ofPrevious Research 31 The parameters B, C, D, and E are determined using experimental cornering and aligning moment data. parameters. Y(q) may represent either or M^, depending on the values given to the Once fully determined, Equation (2.6) represents an accurate fit to the given data [26]. A final approach was noted in Bernard et al [24] and Wong [11]. This method attempts to derive, from physical principles, the dynamic behavior of a given tire. There are two basic types of model. The first is based on the assumption that the tread of the tire can be represented by a stretched string restrained by lateral springs connected on one side to the stretched string, and on the other to the wheel rim. In the other model, the tread is considered equivalent to an elastic beam with continuous lateral elastic slope. Analysis of the two models reveals that the lateral force and aligning moment acting on the tire tread may be determined in terms of the tire lateral stiffness ky, the relaxation length //, and the contact length l Slip angle f may also be defined in terms of these variables. A n analytic relationship may be determined integrating the lateral force exerted on the tire over the contact length. Central to both theories is that the contact patch can be divided into regions of adhesion and sliding; shear forces developed in the adhesion region are dependant on the elastic properties of the tire while shear forces generated in the sliding region depend on the frictional properties at the tire/road interface [30]. While such an analytic relationship seems attractive, it is rarely used in simulation studies. The key difficulty lies in determining the required tire properties. Often, measurement of these properties is as complex as testing the overall tire behavior. Although initially used by Bernard et al [24] in his 1973 report for U M T R I , the approach was scrapped in favor of a look-up table for all subsequent U M T R I simulation studies. 2 The relaxation length is defined as the lateral deflection required to reduce the lateralforceto l/e of its peak value [11]. Chapter 2. Review ofPrevious Research 2.3.5 Combination of Braking and Tractive Forces Eshleman et al [12] [29] and D'Souza [25] introduce the friction ellipse concept to tire modeling. The friction ellipse concept is based on the assumption that the tire may slide on the ground in any direction if the resultant of the longitudinal force (tractive or braking forces) and the lateral force (arising from cornering) reach a maximum value. See Figure 2-5 below. Figure 2-5 - Friction Ellipse Concept The maximum value of the resultant tire force is defined by the coefficient of road adhesion as well as the normal loading on the tire. However, as shown in Figure 2-5, the longimdinal and lateral forces must not exceed their respective maximum values. These maximum values (which constitute the major and minor axis of the ellipse, respectively) are determined using empirical tire data. 2.4 Vehicle Modeling - Linear Two-Dimensional The most common model for the study of multiply-articulated vehicles has traditionally been a two-dimensional linear model. Each unit is assumed to have a degree of freedom in the transverse direction ^-direction) and rotational freedom in yaw. To maintain linearity, all models in this category must stipulate the vehicle forward velocity to be a system parameter (i.e. constant) as the slip angle becomes non-linear (and by extension, thetirelateral forces) if the vehicle accelerates. Papers [13] — [20] involve linear vehicle models. 32 Chapter 2. Review ofPrevious Research 33 Two-dimensional linear models have in common the following assumptions in their derivation [20]: (a) The cornering forces and aligning moments are assumed to be linear functions o f the slip angle developed at the tires. (b) A l l rotational angles are assumed to be "small" (less than 15 degrees) such that small angle approximations hold for the trigonometric functions (sin(x) « x and cos(x) « /). (c) The motion of the vehicle is limited to the horizontal (yaw) plane. (d) There are no significant tire tractive or braking forces acting upon the tires. (e) Pitch and roll motions of the sprung masses are assumed small and not considered. (f) Load transfer and other effects of vehicle width are neglected. (g) Steering and suspension dynamics are neglected. Each unit in the vehicle train is assumed to be a rigid body; tire forces transfer directiy to the vehicle's sprung mass. (h) Gyroscopic forces due to rotating elements are neglected. The earliest work concerning the stability of multi-unit heavy trucks was completed in the late 1960s by Schmid [13] and Jindra [15]. In particular, Jindra pioneered the identification of the particular vehicle parameters leading to dynamic instability. Jindra's modeling was unique in that a single reference frame was attached to the tractor. Evoking the aforementioned small angle approximation, all system equations of motion were written relative to this single frame. The next step in model sophistication arose as authors attached local body-fixed reference frames to each unit in the vehicle train. T o maintain a linear model, forces and relative acceleration formulations were simplified and non-linear terms not considered. Fancher [17], El-Gindy [19], the U M T R I Y a w / R o l l Model [11], and Mallikarjunarao [20] all utilized such a model in similar capacities to facilitate their analyses. 34 Chapter 2. Review ofPrevious Research In his 1982 work, Fancher [16] investigated the motion o f the dolly-semitrailer (full trailer) only. H e proved that since the lateral force o f constraint was small at the pinde hook-eye connection compared to the lateral forces developed at the dolly tires, the trailer could be effectively " r e m o v e d " f r o m the vehicle train and its results studied separately. Because the full trailer was modeled in terms o f inertial coordinates, vehicle simulation along a desired trajectory was simplified and hence did not require a driver model. Comparison with experimental results shows reasonable agreement for small-magnitude maneuvers, such as sweeping cornering or lane changes under constant velocity, highway conditions (Ervin et al [28]). H o w e v e r , Mikulcik [27] determined that for movement along more complex or severe trajectories, rapid steering inputs, or other large amplitude maneuvers the linear model breaks d o w n ; the linear model may predict a stable m o t i o n w h i c h is far removed f r o m the actual behavior. In particular, it is impossible to model some types o f rollover, and the jackknife instability mode using a simple linear model. 2.5 Vehicle Modeling — Non-Linear Two-Dimensional In response to deficiencies in the simple linear model, several modifications can be made. B y removing the requirement o f vehicle linearity, accelerative and tractive considered in the direction o f motion o f the vehicle (along the x-axis). prerequisite o f constant velocity can also be lifted. B y augmenting the forces can be Additionally, the two-dimensional models with load transfer capabilities, results approaching that o f full three-dimensional analysis can be achieved with a significandy less modeling complexity and computational requirements. In the development o f the equations o f motion, the basic assumptions are similar to those o f the linear yaw plane model. T h e T B S model [11] and the A V D S I I / A V D S III models [12] [29] establish the following major improvements: 35 Chapter 2. Review ofPrevious Research (a) A n o n l i n e a r lateral f o r c e a n d a l i g n i n g m o m e n t tire m o d e l w a s i n t r o d u c e d . (b) F o r c e s p e r t i n e n t t o a c c e l e r a t i o n a n d b r a k i n g c a n b e d i s t r i b u t e d as d e s i r e d t h r o u g h o u t the vehicle system. (c) L o a d transfer (both longitudinal and lateral) are assigned through a quasi-static f u n c t i o n o f the lateral a n d f o r w a r d a c c e l e r a t i o n o f e a c h u n i t i n the v e h i c l e . B y a l l o w i n g dynamic loadings to be semi-static, the effects of the pitch and roll modes are d e c o u p l e d f r o m t h e y a w b e h a v i o r o f the v e h i c l e ; t h e i r effect is e n t e r e d t h r o u g h t h e n o n l i n e a r tire m o d e l (a) to d e t e r m i n e tire f o r c e s a n d a l i g n i n g m o m e n t s . A simplified s u s p e n s i o n m o d e l is g i v e n t o d e t e r m i n e t h e quasi-static l o a d transfer i n r o l l a n d p i t c h . (d) Steady a n d t r a n s i e n t a e r o d y n a m i c forces. (e) F r i c t i o n a n d d a m p i n g o c c u r r i n g w i t h i n the (f) fifth-wheel c o u p l i n g assembly. E f f e c t s o f g r a d e s , b a n k s , a n d o t h e r r o a d c h a n g e s are a c c o u n t e d f o r i n t h e i r c h a n g e o f l o a d transfer. T h e A V D S I I I m o d e l offers a n e x t e n s i o n f r o m d o u b l e t r a c t o r - s e m i t r a i l e r v e h i c l e s i n A V D S I I to t r i p l e trailer v e h i c l e s [29]. I n a d d i t i o n , t h e A V D S I I I m o d e l p e r m i t s tire w e a r , t a n d e m axles, a n d a n i m p r o v e d tire m o d e l t o b e i m p l e m e n t e d . A s i m i l a r m o d e l was created at U M T R I t o p r e d i c t t h e d i r e c t i o n a l a n d r o l l r e s p o n s e o f m u l t i p l y articulated v e h i c l e s n e a r i n g r o l l o v e r [11]. has u p to five degrees o f f r e e d o m F o r w a r d v e l o c i t y is h e l d c o n s t a n t . D e p e n d i n g o n h i t c h constraints, each s p r u n g mass (lateral a n d v e r t i c a l t r a n s l a t i o n , y a w , r o l l , a n d p i t c h ) . T h e s u s p e n s i o n is g i v e n f r e e d o m t o m o v e relative t o the s p r u n g m a s s , a n d is m o d e l e d u s i n g nonlineariti.es (such as s u s p e n s i o n lash). T i r e s are m o d e l e d u s i n g a l o o k - u p table, a n d are c o n s i d e r e d t o b e f u n c t i o n s o f s l i p angle a n d v e r t i c a l l o a d i n g o n l y . T h e t w o - d i m e n s i o n a l n o n - l i n e a r m o d e l s p e r f o r m m u c h better i n c o m p l e x m a n e u v e r s t h a n d o the linear models. E s h l e m a n et aL [12] d e m o n s t r a t e d that f o r a s t a n d a r d t w i n tractor-trailer v e h i c l e i n h i g h - s p e e d o p e r a t i o n , the o r d e r o f m a g n i t u d e f o r t h e y a w m o t i o n is a p p r o x i m a t e l y t w i c e that o f r o l l a n d t w e n t y t i m e s that o f p i t c h . w e l l a p p r o x i m a t e s a f u l l t h r e e - d i m e n s i o n a l analysis. I n t h i s case, quasi-static l o a d transfer v e r y Chapter 2. 2.6 36 Review of Previous Research Vehicle Modeling - Three Dimensional There exist operational conditions in which a non-linear two-dimensional model will not predict accurately the motion of a multiply-articulated vehicle. In addition, although the general output may follow the trend in the experimental data, it may be desired to more precisely match the model to the real vehicle. For these reasons, full three-dimensional models have been created. Pioneering work on three-dimensional models was done by Mikulcik [27] in 1971. This work formed the basis for specification of coordinate frames in three-dimensional models. Each unit in the vehicle has both a body-fixed and an inertial reference ISO frame located through the center of gravity which are coincident at the outset of the simulation. Once started, the body-fixed reference frame translates and rotates with its body. In addition to the frame at the center of gravity, each "sub-component" (suspension point, fifth wheel, kingpin, pinde hook, tires, etc.) have their own ISO reference frame orientated at known locations from the center of gravity. Forces are defined relative to their local reference frames, and the results transformed into the "main" reference frame at the center of gravity of each unit The vehicle is assembled by eliminating the forces of constraint at hitching locations. The majority of three-dimensional modeling has been undertaken as part of an initiative at UMTRI, starting in 1971 [24]. The goal of the "Phase" series of models was to create a comprehensive computer model for simulating the braking and steering response of heavy trucks. Motions of the vehicles are represented by differential equations derived from Newtonian mechanics. The resulting mathematical model is very complex, having up to 71 degrees of freedom (the number of degrees of freedom is dependant on vehicle configuration). Each sprung mass in the train was given a full six degrees of freedom. Degrees of freedom were removed from trailers according to the hitching method (a fifth wheel removes 3 translational degrees of Chapter 2. Review ofPrevious Research • freedom, a pinde hook-eye assembly removes a single translational degree of freedom). Each axle was allowed motion in bounce and roll while thetiresare given rotational freedom to turn about its axle only. The Phase models [24] [11] [22] incorporated the following features: (a) A semi-empirical non-lineartiremodel was included in early versions (Phase II). All later versions incorporated a look-up table method of modeling tires. (b) Modeling of deflection and compliance occurring in the steering mechanism. (c) Inclusion of Ackerman steering geometry as well as steering effects arising from roll, pitch, and bounce in the suspension. (d) Vehicle frame compliance in torsion. (e) Wheel directional dynamics. (f) Accurate modeling of main suspension types (single-axle, 4-spring, walking beam). The Phase IV model [22] included load-leveling action in tandem suspensions. All suspension motions were modeled using a small-angle approximation. (g) Brake torque and temperature (relevant to brake fade); also modeling of the brake hysteresis phenomenon. (h) Aerodynamic loadings. (I) Roadway effects. Full-scale experimentation reveals the accuracy of the Phase models in predicting vehicle motion [24]. The Phase IV model was used extensively in UMTRI's investigation of rearward amplification in 1986 [10]. More recently, the mathematics of the Phase TV model have been included in a Microsoft Windows application known as ArcSIM [56]. ArcSIM retains the capabilities of the Phase TV model, but includes a graphical interface, graphing tools, and a simplified interface for input and modification of vehicle data. Similady, Day [30] created a similar graphical simulation 37 Chapter 2. Review of Previous Research package E D V D S . 38 E D V D S is based primarily on the Phase I V model, but the interface has been ported into the H V E simulation environment. Differences between the E D V D S and Phase IV models consist of the removal of all small angle approximations, inclusion of a drive train model, and the addition o f a facility to simulate, vehicular motion on a three-dimensional surface of arbitrary complexity. Unfortunately, mree-dimensional models are extremely complex and require a large amount of physical data about the vehicle under review. The full Phase TV model incorporates up to 2300 lines of input data [11], while a linear two-dimensional analysis of the same vehicle would require only 35. In addition, time required to run a given simulation is over five times greater than what is required in a non-linear two-dimensional model [11]. 2.7 Analysis Methods The complexity of the mathematic models arising from the former modeling techniques has given rise to a variety of solution methods. Although solutions pertaining to the relative stability of the vehicle are possible for the simple linear two-dimensional models, higher order models require in many cases the "simulation" of the vehicle equations o f motion to establish stability rather than to "solve" for the system stability in the classical sense. 2.7.1 Solution Using an Analog Computer Eady simulations using the system equation of motion were achieved through the construction of an analog computer [15]. This strategy stipulates writing the linear equations of motion in state form: x = Ax + Bu (2-7) Where: x is the state vector, x is the derivative of the state vector, u is the plant input (steering angle), and matrices A and J3 define the system. Chapter 2. Review ofPrevious Research Using the known system parameters, the coefficients of the matrices A and B are calculated. The elements o f the state vector and plant input are "programmed" through applying a voltage through a potentiometer (the potentiometers "multiply" each element of the state vector or input by its corresponding element in the matrix). The voltages representing the state and the input are connected to a summing amplifier, and the result passed through an integrating amplifier. The ensuing voltage on the integrating amplifier output is representative o f the derivative o f a state variable. Note that the amplitudes of the state variables and their derivatives must be scaled so the voltages are of the same order of magnitude; time scaling must also be performed to control the speed of solution. Solution using an analog computer is both difficult and complex. The resulting hard-wired computer is limited to step steering inputs [15]. 2.7.2 Solution Using Control Theory If the mathematical model o f the system is linear, concepts from control engineering may be used to solve for stability. This approach is interesting in that the structure undedying the system is investigated, instead of the specific output of the system in response to a given input. Fancher [16] Laplace transformed the linear equations of motion and determined a transfer function relationship for a full trailer. Using this transfer function, the maximum gain for a given input was determined. The same procedure was completed for the lead tractor- semitrailer unit. The product of the maximum gain of the full trailer and the maximum gain of the tractor-semitrailer was demonstrated to give a good indication of the overall amplification in lateral acceleration along a double tractor-trailer. Investigation of the characteristic equation of the full trailer also demonstrated an equivalent damping ratio for the unit. 39 Chapter 2. Review ofPrevious Research 40 Similarly, the system equations o f motion can be written in state space form and the system eigenvalues found (which are themselves simply the roots of the system characteristic equation). In the nomenclature of Equation (2.7): \U-A\ = 0 (-) 2 8 Both Mallikarjunarao [20] and Fancher [17] utilized this technique to plot the variation in the system stability with both forward velocity and vehicle parameters. Both studies demonstrated movement of the eigenvalues to less negative real parts with smaller effective damping coefficients. See Section 2.8 for a complete discussion of the results. 2.7.3 Solution of Non-Linear Stability Using Lyapunov's Method Eshleman et al [29] utilized Lyapunov's direct (second) method to quantify the stability of their nonlinear multi-articulated vehicle model. Stability was investigated for two types of nominal motions: (a) Stationary maneuver — The forward velocity , state variables, and system inputs are held constant (example: circular trajectory) (b) Non-stationary maneuver — The system inputs and state variables are considered to vary with time (example: lane-change trajectory). Eshleman et al determined that although Lyapunov's direct method yielded acceptable results in the stationary maneuver case, the method was too restrictive for time varying responses. This conclusion was attributed to the fact that Lyapunov implies stability over an infinite time frame, while an articulated vehicle can be instantaneously unstable in the sense of the mathematics and yet recover at a new nominal motion. In response to this, Eshleman and his researchers attempted to define a "finite time" stability criterion, but concluded that "acceptable results were beyond the state-of-the-art (for 1973)" [29]. The techniques used were successful in defining stability limits for contrived non-linear examples, but had difficulty Chapter 2. Review ofPrevious Research 41 with more practical problems. Physical testing and non-linear simulations o f the equations of motion were found to give more accurate results in the latter situations. 2.7.4 Simulation of a System of Non-Linear Equations of Motion The majority of studies whose models are nonlinear have chosen to simulate, rather than solve, their system equations of motion. The A V D S II [12] [25] and A V D S III [29] models were compiled in BASIC and utilized a unique computational strategy. The solution procedure is as follows: The value of all the variables is assumed known at time /. For time t+At, the steering angle was replaced by S+AS and the total tractive/braking force by D+AD. The equations of motion are then expanded in terms of AS and AD, and terms higher than second order dropped. Subroutines are called to calculate properties such as semi-static load transfer and tire lateral forces. Since the translational velocities and accelerations are known, the procedure yields equations in the unknown angular accelerations o f each unit as well as AS and AD. This set of linear equations is solved using Gaussian elimination, and the- resulting angular accelerations were integrated using a Taylor scheme with no intermediate computation and fourth-order error to determine the angular velocities and position. The results are then backsubstituted to improve the value of the other variables for the next iteration. In contrast, the U M T R I models [11] [22] [24] were solved in a much more conventional manner. The equations in these cases were programmed in either F O R T R A N or C, and employed a fourth-order Runge-Kutta technique to solve the equations of motion. 2.8 Results of Dynamic Modeling Wong [11] compiled a comparison o f the behavior o f both linear and nonlinear twodimensional (TBS) models as well as full three-dimensional simulations (UMTRI Phase IV). The results were also compared to test data for a standard five-axle semitrailer undergoing a standard lane-change maneuver. Chapter 2. Review ofPrevious Research It was found that all of the tested simulation models had varying degrees of discrepancy with the experimental data, and that there are no significant differences in the steady-state steering responses for lateral accelerations below approximately 0.25g. However, the TBS and U M T R I Phase I V models in particular had peak acceleration values closest to that of the real vehicle. Because the Phase I V and T B S models take into account the effects o f load transfer and nonlinearities in tire behavior, such models can predict changes in handling due to lateral acceleration; because of assumptions made in its formulation, the linear two-dimensional model can not. Load transfer effects were demonstrated to become increasingly important as the seventy of the test increased. Irrespective of the model used, the literature reveals several key contributing factors to rearward amplification in double or triple articulated vehicles. Jindra [15] and Fancher [16] noted the following trends in their analysis: (a) Rearward amplification increases with increasing vehicle velocity. Fancher [17] showed that the effective damping ratio o f the last trailer in a twin tractor-trailer decreased by more than 50 percent as the velocity was increased from 22.3 m/s to 31.3 m/s. (b) Rearward amplification decreases as the trailer wheelbase is increased. This in one reason why rearward amplification is predominandy a problem in double tractor-trailer vehicles (which use shorter units) as opposed to single-trailer trucks, which tend to use longer vans. (c) Rearward amplification increases as the tire properties (cornering stiffness and aligning moment) become more compliant (d) Rearward amplification increases as the number of articulations in a vehicle increases; it also increases as the total number of units in the combination vehicle becomes larger (Ervin [28]). (e) As the dolly drawbar length is shortened, the observed rearward amplification increases. 42 Chapter 2. Review ofPrevious Research 43 (f) In vehicles with loads having a high center of gravity, increased weight transfer during a given maneuver can increase rearward amplification. (g) Rearward amplification is sensitive to the frequency o f the steering input U p to the ergonomic constraint on human input frequency (3 tp 3.25 rad/s (Ervin [28])), more rapid steering causes the rearward amplification to rise. Beyond this threshold, rearward amplification decreases as the movements become too rapid for the vehicle to fully respond. A smaller effect is noted due to the magnitude o f the steering input (arising from increased load transfer). Mallikarjunarao [20] studied the effects of load distribution on rearward amplification. He concluded that the best dynamic performance occurred when both trailers in a twin configuration were full and uniformly loaded. Surprisingly, the condition in which both trailers were empty did not yield the poorest behavior. Instead, the condition in which the lead trailer was empty and the last trailer loaded produced the worst results. In addition, localizing the load over the rear axle of the trailer aggravated further the rearward amplification condition. 2.9 Dedicated Offtracking Models In most simulations, it is possible to determine the tracking behavior simply through inspection of the trajectory of the vehicle in the yaw plane. However, for more in-depth analysis of offtracking behavior, two dedicated models have been proposed. Jindra [31] proposed a simplified mathematical formulation involving a semitrailer and dolly combination. He described the motion of the trailer's rear axle as following a "general txactrix" with respect to the original steering curve front axle (located on the dolly). The difference between the trajectory of the front axle and the general tractrix defines the offtracking performance of the full trailer. In order to find the offtracking of a vehicle train, Jindra proposed reducing the problem to finding a series o f consecutive tractrix curves. The tractrix was defined mathematically for straight-line tracking and a simple 90 degree comer and the results compared to experimental data derived from scale model testing. Experimental Chapter 2. Review ofPrevious Research 44 results matched well to the model, but results were limited by assumptions made in his formulation to low-speed, simplified maneuvers. Bernard [18] also studied the tracking of full trailers. In his. 1980 analysis, a semitrailer was modeled assuming linear conditions and constant forward velocity. Input to the trailer kingpin was taken to be the angular velocity of the leading fifth wheel, expressed in the semitrailer frame of reference. In this way, arbitrary maneuvers could be tested by applying the appropriate angular velocity input By comparing the position of the last axle of the trailer to that of the kingpin, the offtracking behavior was quantified. Both studies arrived at essentially the same conclusions. The first conclusion is that all trailers have a critical speed at which no offtracking will occur. For typical trailers in use in the United States and Canada, this speed ranges between 9 m/s and 21 m/s (32.4 k m / h to 75.6 km/h), depending on the cornering properties of the trailer tires and the wheelbase o f the trailer [18]. Second is that tandem axles serve to aggravate low speed offtracking. Last, trailers will undershoot (track to the inside) of the intended curvature of the maneuver at speeds below the critical speed. Likewise, the same units overshoot (track to the outside) intended curvature of the maneuver at speeds above critical. 2.10 Methods of Reducing Rearward Amplification There are several approaches laid out in the literature to suppress dynamic instability in heavy trucks. Due to the cost and size of the required sensors and actuators, early solutions involved purely mechanical means. Conversely, more modem approaches involve modem control theory and are more "intelligent"-type systems. 2.10.1 C-Dollies and B-Trains C-Dollies represent the most common solution to the rearward amplification problem. Their use in Canada is further propagated by the fact that companies employing C-dollies are Chapter 2. Review ofPrevious Research permitted to carry higher axle loads than conventional A-dollies [21] [7]. Like A-dollies, Cdollies are given their name from the shape of the frame members when seen from plan view. See Figure 2-6. ( — : — > j Figure 2-6 — C-Dolly Configuration Each drawbar of the C-dolly is connected via pinde hook to the leading unit. In this way, the dolly is rigidly attached to the frame of the leading unit while still affording a degree of relative freedom in pitch [21]. A common modification of this concept is to incorporate the fifth-wheel assembly direcdy into the frame of the leading unit. The resulting vehicle is then created of only the tractor and semitrailers and is called a B-train. 45 Chapter 2. Review ofPrevious Research Regardless of the implementation, C-Dollies 46 and B-trains share the same essential characteristics. The reduction of one articulation joint in doubles (and two in triples) along with increased roll coupling reduces the observed rearward amplification. Unfortunately, the design also features increased frame stresses (although this may be negated somewhat by utilizing a more compliant "compensating" fifth wheel assembly), tire scuffing, and low-speed offtracking. B-trains are additionally limited by increased weight, cost, and compatibility issues. 2.10.2 Shifted Instant Center Dollies In Section 2.8 it was stated that increasing the length of an A-dolly's drawbar can reduce rearward amplification. Since generally it is not desirable to increase the overall length of the vehicle, the effective length of the drawbar may be increased by shifting the dolly's instant center of rotation (1Q to a point in front of the pinde hook. A n implementation of this concept is the trapezoidal dolly [21]. Figure 2-7 — Trapezoidal Dolly Configuration In the trapezoidal dolly, the two drawbars and the frames of the dolly and lead trailer form a four-bar linkage; the drawbars are connected to the lead trailer and dolly using pin joints. The 47 Chapter 2. Review ofPrevious Research IC is located using classical linkage theory at the point where lines connecting the pins on each drawbar intersect The resulting device maintains a relatively constant IC location during use in its intended highway operation (implying low motion amplitude of the vehicle). There are several variations on this basic design. A n asymmetric design was proposed to simplify the hitching procedure and reduce weight A European solution moved the instant center behind the pinde hook by crossing the drawbars to improve low-speed maneuverability. Although increasing the effective drawbar length is successful at decreasing rearward amplification, the resulting dolly has decreased offtracking performance. Similady, the crossed drawbar solution improved low-speed offtracking, but at the expense of increased rearward amplification. In response to these problems, a "roller cam" idea has been suggested which shifts the IC far ahead of the pinde hook during maneuvers with small articulation angles, but allows the IC to move behind the pintie hook during large amplitude motions. 2.10.3 Forced-Steer and Skid-Steer Dollies Forced steer dollies are simply A-dollies which have been modified to have mechanical steering. A n additional rod is connected to the leading unit; the resulting steer angle is simply the articulation angle between the units multiplied by a gain value. The forced steering has negligible effect on the rearward amplification performance of the vehicle, but does improve low-speed maneuverability through the creation of Ackerman steering geometry [21]. Simulation reveals rearward amplification performance equal to, or slighdy worse than that of the standard A-dolly. A similar system is termed "skid-steer" dollies. In this configuration, the dolly is rigidly connected to the semitrailer through the fifth wheel, fonriing in essence a full trailer with no steering. This system has been utilized in Saskatchewan with limited success [21]. excessive tire scuffing and frame stresses, a self-steering axle can be added. To negate Generally, the dynamic performance of the skid-steer configuration is highly dependant on tire properties, Chapter 2. Review ofPrevious Research loading, and road conditions. In Winkler et aL [21], the skid-steer dollies performed less effectively than A-dollies for low-frequency steering inputs with results for higher-frequency inputs approaching that o f C-dollies. 2.10.4 Linked Articulation Dollies Dollies in this class remove a rotational degree o f freedom between the rear o f the lead trailer and the front o f the second trailer. This dolly creates an angular dependence between the trailers such that the kingpin o f the second trailer always remains on a projection o f the centedine o f the first trailer [21]. The change in dynamic performance is mixed; at low frequency the performance approximates that o f an A-dolly while for higher frequencies, rearward amplification performance is improved. 2.10.5 Roll Stiffened Pintle Hook Assembly C-dollies and B-trains improve vehicle dynamic performance in part due to the added roll coupling between trailer units they provide (in dynamic maneuvers when the motion o f the trailers is "out o f phase", roll coupling serves to add a counteracting moment to resist the roll). The standard pinde hook-eye assembly has no such roll coupling. Although the Roll-Stiffened Pinde Hook was created by U M T R I primarily to improve roll stability, the resulting reduction in dynamic loading can influence rearward amplification [21]. Simulation by Winkler et aL [21] reveals that this effect is negligible; the dynamic performance o f the roll-stiffened pinde hook was approximately the same as the standard A-dolly. 2.10.6 Locking A-Dolly A n alternate dolly has been created by V B G in Sweden [21]. Figure 2-8 below diagrams the device. 48 49 Chapter 2. Review ofPrevious Research (P) Figure 2-8 — Locking A-Dolly: (a) Locked, (b) Unlocked The dolly consists of two drawbars which may be locked (in four-bar linkage parlance, "toggled") or unlocked depending on driving conditions. At low velocities, the dolly is unlocked and the mechanism functions identically to a standard A-dolly. Likewise, the drawbars are locked at high velocity and the dolly performs dynamically as a C-dolly. Chapter 2. Review ofPrevious Research 2.10.7 Steerable C-Dollies To negate many of the difficulties in standard C-dollies, a self-steering axle can been added to the mechanism [21]. The steerable axle reduces low-speed offtracking, tire scuffing, and problems related to excessive frame stress. Two main steering types are used: (a) Automotive-type steering — The steering degree of freedom is provided by a kingpin and steering knuckle arrangement similar to the steering system on the tractor. (b) Tumtable-type steering — Dolly wheels are attached to a rigid axle which pivots relative to the dolly centerline about a single kingpin. In both variants of the mechanism, resistance to steering forces must be set high enough such that large steering angles are not permitted in highway conditions, but steering is freely allowed during large amplitude, low velocity maneuvers (such as city driving) [13]. The performance of the self-steering axle depends on the compromise made by the designer over the former requirements. Unfortunately, steerable C-dollies suffer from a unique problem in braking conditions. Since C-dollies steer in response to torque about the steering pivot, an additional steering force may be created by a differential in braking force on each side of the dolly. Variation of up to 20 percent is not uncommon due to differences in braking properties [21]; for different tire and road conditions the problem may be aggravated. The problem is most prevalent for tumtabletype steering, as the sensitivity to the differential in braking forces depends on the kingpin offset dimension (for turntable steering, this value is half the dolly track). 2.10.8 Reduction of Rearward Amplification Through Active Yaw Control El-Gindy et aL [19] and Palkovics [32] proposed several methods for the control of rearward amplification using modem control theory and components present aboard trucks having antilock braking systems. Although the analysis was based on a truck and full trailer configuration, the results are also valid for A-train trucks. 50 Chapter 2. Review ofPrevious Research 51 T o control and modify the dynamic performance of the test vehicle, El-Gindy proposed three control strategies: (a) Active yaw control of the truck center of gravity. (b) Active yaw control of the dolly center of gravity. (c) Active yaw control of the trailer center of gravity. The former control strategies were based on the "Active Unilateral Braking" controller detailed by Palkovics [32]. In the case of a simple tractor-semitrailer vehicle, Palkovics also compared active steering of both the tractor rear wheels, active steering of the trailer rear wheels, and active torque control at the fifth wheel hitch (fifth wheel braking) in addition to the above. The controller was based on a linear two-dimensional vehicle model, and took advantage of the truck's anti-lock braking components to provide a torque at the truck, dolly, or trailer center of gravity. The controller was also constrained to improve the roll stability of the vehicle without changes to the control input required by the driver or the trajectory to be followed. A Linear Quadratic Regulator (LQR) was selected to provide a control signal based on the minimization of a weighted combination of the system states [32]. Palkovics found that the most effective use of control involved active control of the vehicle's center of gravity. Additionally, El-Gindy et al demonstrated that although active yaw control of the tractor or the trailer yielded promising reductions in rearward amplification, their implementation was not desirable due to the strong effect they had on vehicle trajectory and driver control. Thus active control of the dolly center of gravity was chosen as the best compromise for the reduction of rearward amplification and the maintenance of fidelity in trajectory following. However, Palkovics [32] expressed reservations in using a standard L Q R controller in this application, as the robustness of the derived L Q R controller under parameter Chapter 2. Review ofPrevious Research and modeling uncertainties (non-linearities and modeling errors) was shown as less than optimal. 2.10.9 Reduction of Rearward Amplification using Externally-Mounted Dampers Vazquez et tfZ[33] proposed a unique method for improvement o f A-train dynamic performance. External dampers were mounted between a standard A-dolly and the leading semitrailer. The results o f the study show that damping the articulation angle rate between the dolly and the lead trailer can enhance vehicle lateral performance in terms o f peak lateral acceleration, yaw rate, and rearward amplification levels. 52 Chapter 3 Modeling Multiply-Articulated Vehicles The goal of this thesis is to derive an intelligent A-dolly which can reduce the practical occurrence of both rearward amplification and offtracking in multiply-articulated vehicles. This control will be manifested through the automatic steering of the wheels of the A-dolly. The present chapter details the derivation of a complex system modeL and the subsequent verification of its characteristics. The outputs of the model are used as "sensor inputs" to the control strategy for steering the intelligent A-dolly, and are also used to validate the operational characteristics of the control model. 3.1 Modeling Assumptions The model is a non-linear two-dimensional simulation, with provisions made to model the effects of pitch and roll as changes in the tire vertical loading. This approach has been shown as valid in the high-speed, low steering and articulation angle amplitude of operation in which this paper is focused [12]. As with any model of a real dynamic system, assumptions must be made to allow formulation of the equations of motion. The assumptions made were carefully chosen to display the 53 54 Chapter3. Modeling Multipfy-Articulated Vehicles essential s y s t e m characteristics (as n o t e d i n t h e literature r e v i e w ) w h i l e m a i n t a i n i n g e n o u g h simplicity for solution u s i n g standard methods. T h e key assumptions include: (a) S i g n i f i c a n t a n g u l a r m o t i o n o c c u r s i n the y a w p l a n e o n l y . T h e effects o f r o l l a n d p i t c h m a y b e a c c o u n t e d f o r u s i n g the accelerations a c t i n g o n t h e v e h i c l e c e n t e r o f g r a v i t y t o create semi-static l o a d transfer d u r i n g t h e m a n e u v e r (see S e c t i o n 3.4). (b) C r o s s - c o u p l i n g b e t w e e n the y a w , r o l l , a n d p i t c h m o d e s o f t h e a s s e m b l e d v e h i c l e are not significant (c) S y s t e m p a r a m e t e r s ( s u c h as t h e r o t a t i o n a l m o m e n t o f i n e r t i a / a n d the v e h i c l e m a s s m) are c o n s t a n t . (d) S t e e r i n g i n p u t s are a p p l i e d t o the v e h i c l e t h r o u g h t h e t r a c t o r f r o n t tires a n d the d o l l y tires o n l y . (e) S i m p l i f i e d b e a m - a x l e type s u s p e n s i o n is a s s u m e d f o r a l l axles; its effects are m a n i f e s t e d t h r o u g h l o a d transfer. (f) T h e s u s p e n s i o n s p r i n g c o e f f i c i e n t is m o d e l e d as a s i m p l e l i n e a r f u n c t i o n , a n d d o e s n o t i n c l u d e s u c h effects as b a c k l a s h o r hysteresis. (g) N o s t e e r i n g c o m p l i a n c e is m o d e l e d ; s t e e r i n g f o r t h e t r a c t o r is g i v e n t h r o u g h d i r e c t s p e c i f i c a t i o n o f the r e q u i r e d steer angle. (h) A l l v e h i c l e s are a s s u m e d t o b e r i g i d b o d i e s a n d h a v e n o c o m p l i a n c e i n r e s p o n s e t o t h e forces a p p l i e d t o t h e m (such as t w i s t i n g i n t o r s i o n , o r b e n d i n g a l o n g t h e vehicle's length). Q The tires are m o d e l e d o n l y a c c o r d i n g t o t h e i r lateral f o r c e and restoring m o m e n t characteristics; effects o f static d e f l e c t i o n , b o u n c e , r o l l steer, caster angle, o r k i n g p i n i n c l i n a t i o n are n o t p e r m i t t e d . 0 G y r o s c o p i c forces a r i s i n g f r o m r o t a t i n g e l e m e n t s are n e g l e c t e d . (k) B o t h t h e fifth w h e e l a n d p i n d e h o o k h i t c h i n g assemblies a l l o w u n c o n s t r a i n e d m o t i o n i n the y a w d i r e c t i o n , a n d d o n o t v a r y i n t h e i r p r o p e r t i e s t h r o u g h o u t the vehicle's motion. (1) T h e s i m u l a t i o n c o n d i t i o n s are a s s u m e d u n i f o r m ; this i m p l i e s a flat, s m o o t h r u n n i n g surface a n d the n e g l e c t o f a e r o d y n a m i c effects ( b o t h d u e t o v e h i c l e shape a n d w i n d loading). Chapter3. ModeHngMultipiy-Articulated Vehicles 3.2 55 Coordinate Systems The large number of degrees of freedom in the tractor-semitrailer vehicle suggests that the choice of coordinate systems is of utmost importance to facilitate the writing and manipulation of the system equations of motion. The approach taken in this model mirrors those taken by Mikulcik [27] and Eshleman etal [12] [29]. Each body in the A-train is assigned two coordinate frames according to the ISO standard. The first is an inertial (fixed) reference frame coinciding with the position of the tractor, first semitrailer, dolly, and second semitrailer directly in line with all deviations from static equilibrium zero. See Figure 3-1. Figure 3-1 — Inertial Reference Frames (black) and Tire Forces (red) and Hitch Forces (blue) for A-Train 56 Chapter 3. Modeling Multiply-Articulated Vehicles A second coordinate frame is attached to the center o f mass o f each vehicle and allowed to both translate and rotate with that body. The rotation between the local and inertial frames is related by Euler rotation angles utilized in a transformation matrix. In the general case, this matrix is composed of the three discrete components o f yaw, pitch, and roll. However, since the present analysis o f the vehicle is limited to the yaw plane, the required transformation matrix is reduced to the formulation o f Equation 3-1. cos(^) sin($) i -sin($) cos(^) y, x (3-1) Where: [^,Y3 refer to inertial coordinates, [x yJ refer to local coordinates, and $ refers to T T iJ the yaw angle (measured with respect to the inertial coordinates) of the rth unit Equation 3-1 can be used to relate any local position, velocityj or acceleration measurements from the local to inertial reference frames (and visa-versa). A version o f Equation 3-1 will be used in subsequent transformation of properties between units in the multi-articulated vehicle. 3.3 System Equations of Motion The Newtonian equations o f motion are derived for each unit in the combination vehicle separately, then related through the constraint equations at the fifth wheel and pintie hook hitches. Forces and couples are transmitted to the sprung masses through the suspension and the fifth wheel or pinde hook. It will be assumed that the suspension is attached to the tractor in four positions only; one location at each of the four tires. Similarly, the suspensions for both semitrailers as well as for the A-dolly are connected to their respective frames at two locations: the left and right rear. Although practical suspensions generally involve in-depth attachment geometry and multiply attachment points, the former simplification may be justified by imagining that the complex suspension assembly is itself a lightweight (essentially Chapter 3. Modeling Multiply-Articulated Vehicles 57 massless) member that connects to the frame at a single point. This formulation allows freedom to derive the general equations of motion without prior knowledge o f the specifics of the suspension geometry of the vehicle. Forces and moments are termed using an alpha-numeric description. Odd-numbered forces exist on the driver's side of the vehicle; even-numbered forces occur on the passenger side. Numbering starts from the tractor and moves backwards. When present, the x, y, or £ subscript on the force denotes its direction with respect to the local ISO reference frame. Table 3-1 details the naming nomenclature for forces and moments: Table 3-1 — Force and Moment Naming Nomenclature Number Location (Unit, Position) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Tractor, left front Tractor, right front Tractor, left rear Tractor, right rear Lead Semitrailer, left rear Lead Semitrailer, right rear A-dolly, left A-dolly, right Second Semitrailer, left rear Second Semitrailer, right rear Tractor, fifth wheel Lead Semitrailer, kingpin Lead Semitrailer, pintie hook Dolly, pinde hook Dolly, fifth wheel Second Semitrailer, kingpin The following subsections detail the equations of motion in the yaw plane, as well as those equations providing semi-static load transfer due to pitch and roll. Vehicle parameter values are listed in Appendix A : Reference A-train Data. 58 Chapter 3. Modeling Multiply-Articulated Vehicles 3.3.1 Yaw Equations of Motion The yaw equations of motion are at the heart of the non-linear A-train simulation. Each equation is derived using Newton's second law in concert with the ISO standard local coordinate axis; these axis are noted on each free body diagram (FBD). 3.3.1.1 Tractor Yaw Equations Figure 3-2 below denotes the FBD for the tractor. Figure 3-2 - Tractor FBD Using the set of axis attached to the tractor CG, the forward and lateral equations of motion are derived, along with a moment balance. ZF = m (V - V co ) = F x x xX yX x 4x + F + F - (F + F ) sin(£) ix Ux x £ X = m {V +V co ) = F + F +F +(F x ZM CG yX =I cb =-(F xX x x 3y x 4y 3y 2 x +F)cos(<?) X 4y 2 Uy 9 l Xz 2z l 3 z (3-3) 2 + F )b -F b +b cos(S)(F +F ) + d sin(£)(F -F ) + M + M + M 6 lly (3-2) 2 + d (F -F ) 2 7 3x 4x (3-4) +M +M 4z xu Note that Af^ (not pictured) is the restoring moment arising from the tires located at the indicated position, and Sis the applied steering angle. Chapter 3. Modeling Multiply-Articulated Vehicles 3.3.1.2 59 Lead Trailer Yaw Equations Figure 3-3 below denotes the F B D for the lead semitrailer. Figure 3-3 — Lead Semitrailer F B D In a method analogous to that of the tractor, the equations of motion are derived. Z* m V Y Fy= 2 m ( y2 C G V )= F F y+ 6 =I d> =F b -(F +F )b -F b +d (F -F )+M 3.3.1.3 2 2 l2y 3 5y 6y 4 l3y l0 & 5x 6x (3-5) + 6x + ^ 1 2 , + ^ 1 3 , S + *2<»2 V d £ M )'= f x = 2 (?*2 - y2®2 F 5z Dolly Yaw Equations Figure 3-4 below denotes the F B D for the A-Dolly. 6y F + Uy F + (3-6) ^3y +M 6z +M l2z + M l3z (3-7) Chapter 3. Modeling Multiply-Articulated Vehicles 60 Figure 3-4 - A-Dolly FBD The equations of motion are as follows: ZF = m (y, -V co ) = F +F -sin( x 3 3 y3 3 Ux ] T F , = m (V +V a> ) = F 3 ^ M C G -F b l5y n = y3 ~(i F i4y 5 F x3 b +M l4z 3 l4y l5x +F l5y 6 l5z +M lz (3-8) +cos(/)(F + F ) (3-9) 8 7 7 +M %Z Note that /is the applied steering angle to the dolly wheels. 3.3.1.4 7 8 + d9 sin(/)(F - F ) + ^)cos(r)b +M )(F + F ) r Second Trailer Yaw Equations Figure 3-5 below denotes the FBD for the second semitrailer: 8 (3-10) Chapter 3. Modeling Multiply-Articulated Vehicles 61 Figure 3-5 — Second Semitrailer FBD The equations of motion are as follows: Z* = F m Z^ £ M C 3.3.1.5 G = m ^ -^) 4 4 x xA 10x l6x 4 ( ^ 4 + V a> ) = F =I d> ^M -F )-b,(F +F ) A - F F 9y i0y A l6y +F +F 9x (3-H) l0x + F +F 9y (3-12) l0y + F b M +M +M l6y 1+ l6! gz l02 (3-13) Articulation Angles and Force Constraints Having derived the individual equations of motion, the results must now be tied together to yield a model for the vehicle as a whole. The lateral and forward velocity, as well as the lateral and forward acceleration can all be resolved into functions of the behavior of the tractor. To facilitate this process, angular transformations and constraints pertaining to the hitching forces are determined. Angular transformations allow information expressed in one local reference frame to be converted into another local frame. In the present simulation, this entails conversion between adjacent units. Articulation angles between the tractor and first trailer, first trailer and dolly, and the dolly and second trailer are defined in Equations 3-14,3-15, and 3-16, respectively. 62 Chapter 3. Modeling Multiply-Articulated Vehicles (3-14) Ti =<f>2-<f>\ r F (3-15) 2 = h~ t>2 < (3-16) 3 =</>4-<f>3 Using the ideas o f Equation 3-1, the transformation matrix from the truck frame to the lead trailer frame is expressed in Equation 3-17. cos(r,) (3-17) sir^r,) - sin(r!) cos(r,) The fifth wheel forces between the lead semitrailer and the tractor may now be related: (3-18) F,12* -T, F I2y Uy 1 Similarly, the transformation matrices between the first trailer and the dolly T and the dolly 2 and the second trailer T are given below: . } i cos(T ) sin(T ) -sin(r ) cos(T ) cos(T ) sin(T ) 2 T = 2 2 2 2 3 T = (3-19) 3 (3-20) -sin(r )' cos(r ) 3 3 The forces between the lead semitrailer and the dolly, as well as between the dolly and second semitrailer may now be determined: 1 _ F 14* i<<y_ ~F = -T, 2 1 13* (3-21) 63 Chapter 3. Modeling Multiply-Articulated Vehicles 1 I6x = -T* F F _ (3-22) 15* 1 y. _\5y_ l6 The moment constraints at the hitching locations are presumed to be equal and opposite. ^ l l z (3-23) = - ^ 1 2 z (3-24) (3-25) M =-M i5z 3.3.1.6 l6z Velocity and Acceleration Constraints The formulation of the relative velocities and accelerations between the units in the A-train are formulated according to relative motion theory in rigid body dynamics [34] [35]. Results were obtained by expressing the velocity and acceleration o f the hitch point with respect to the local reference frame for both the leading and trailing units, then rotating the results into the same coordinate frame and equating the results. The relative velocities and accelerations between the tractor and the lead semitrailer are given in Equations 3-26 and 3-27, respectively. xl V. y2 •b (o ~v ~ = 9 Vxi-<»iV +(oX x2 yX i T (3-26) V. x2 Ki+0)^-0)^ + x -b a) 3 2 (3-27) -(o\b -(o V 3 2 y2 (b b +co V 2 3 2 x2 Note that the value o f dT,/dt is neglected; this is permissible since the value of drjdt is very small. Similady, the relative velocities for the lead semitrailer and the A-dolly, as well as the A-dolly and the second semitrailer are expresses in Equations 3-28 — 3-31. Chapter 3. Modeling Multiply-Articulated Vehicles *3 I (3-28) V. x2 = T, •b a> y2 73 w 2 'V 2 -O) V +d) b =T 2 X 2 y2 2 2 ~y x2 2 w V. = T5 Vy4_ 3.3.1.7 = T, 3 3 0 3 5 ^ 5 (3-29) + y3 ^ 3 (3-30) 7 v -co V +o) b 3 3 -b 0) 3 x3 5 0 y3 x3 (3-31) 2 u 4 y* + G) V -d) b _ 3 A -co b -a> V 2 'Vx4~ -b co -o) b -co V l0 + o) V -d) b _ * x4 JV y4 _ 64 3 n 7 4 y4 _ d) b +a> V 4 7 4 x4 _ Tire Forces and Aligning Moments Tire forces and aligning moments are modeled as nonlinear functions / of slip angle a and vertical loading F^ F = f(a,F ),i=1-10 (3-32) M f(a,F ),i=1-10 (3-33) ty ty = iz iz The function / is extremely difficult to find via analytical means, and is thus determined through experimental tests on the tire itself. The method for relating the slip angle and the vertical loading to the lateral tire force and the aligning moment is described in Section 3.5. Additionally, values o f the vertical tire loading F^ are determined using a linear combination of static loading, and semi-static load transfer arising from pitch and roll, are explained in Section 3.4. The slip angle relation for any tire has been given in Equation 2.1 and is reproduced below. a. = tan X*. ,1=1-10 (3-34) The expressions for the lateral and forward velocities at the tire/road interface is determined again using simple relative motion theory. Equations 3-35 to 3-38 give the derived results. 65 Chapter 3. Modeling Multiply-Articulated Vehicles Note that the terms in the denominators are added for wheels 1, 3, 5, 7, and 9 and subtracted for tires 2,4,6,8, and 10. (3-35) oc = tan V ±d co l2 xX 6 x (3-36) « = tan" 34 a = tan" 56 (3-37) Py2-M>2 ,^2±<V»2. (3-38) or = tan 7g (3-39) tan a, 9,10 3.3.1.8 ±J, « 0 4 Manipulation into State Space Form Equations 3-2 to 3-13 are manipulated and the coupling forces and moments are eliminated between the units, resulting in six independent nonlinear equations of motion. T o simulate the system of equations, it is required to have the equations in the form y = f(y) (3-40) Where: j is the state vector and y is the derivative of the state vector. T o achieve the form of Equation 3-40, four additional "intermediate" equations are introduced. < ,l=1-4 A — A </,.=6) d at i ( " 3 41) The result is ten equations composed of a nonlinear combination of the elements of the state vector and its derivative. Chapter 3. Modeling Multiply-Articulated Vehicles 66 vxl » v CO, CO, fa y= fa fa CO, (3-42) y CO, (3-43) co A CO, CO, co CO, co co 3 A 3.4 A Load Transfer The roll and pitch variables were excluded from the former equations o f motion. The effect of the roll and pitch motions are instead introduced into the analysis through the mechanism of load transfer. Load transfer, in turn, affects the available cornering force and restoring moment from a given tire. A t each tire, the instantaneous vertical loading is composed of two primary components: (a) Static loading (b) Dynamic loading due to acceleration and deceleration of the vehicle sprung masses The total loading for a given tire at a given time is expressed as the algebraic sum of effects (a) and (b). The simulation terminates i f any normal loading on a tire becomes less than zero; the previously-derived equations o f motion are no longer valid in their current form, and the vehicle is on the verge of rollover. 3.4.1 Static Loading The static loading for the reference A-train is based on the geometry o f the vehicle as well as the mass and its distribution throughout the vehicle. For the purposes o f this calculation, the 67 Chapter3. Modeling Multiply-Articulated Vehicles vehicle is assumed at rest on a level surface, with equal loadings on both sides. Figure 3-6 is a schematic diagram of the A-train used to find static loading. 10 J ™38 'llzs Fl5zs| | '9zs P Fi6 1 I ^Vzs f i ZS 10zs 1 1 I m F 4 z s L i7S lg Flzs F2zs ^3zs 12zs 4zs r 6zs L 8ZS r zStL — b — x Figure 3-6 - Static Loading, Standard A-train A simple Newtonian force balance is utilized, assuming static conditions. The subscript '\s" refers to static vertical loading in Equations 3-44 to 3-48; the calculated loading is the total value per side of the vehicle. F lzs r F 3is r =F =Z1 2zs r 2 b, — bb m g b +b [b + b 4 n b +b x 2 x 2 2 3 4 b, +b Km g -F - Z i b^g , b +b b +b L^3+6 ~ 4zs " 2 2 a r x 2 x bjn g =F =Z1 5zs 625 2 b +b 2 3 [ b^-b^m^g b (b -b )(b -b )m g +• (b +b )(b +b ) (b +b )(b +b )(b +b ) 3 | 4 5 g 6 b^b.-b^mjg 3 4 5 3 | 4 & 6 4 X0 5 6 xx 6 | 4 (3-44) 4 1 i xo (3-45) 4 4 5 & 6 xx 6 7 8 (3-46) 4 6 (3-47) [b +b 5 6 {b +b )(b,+b ) F -F - Z i •'to MO» ~ 2 _ n b (b -b ){b -b )m g I F.Izs 6 ^ b, +b, bfJM-jg b (b -b )m g b +b (b.+Wb.+b,)^ 5 4 f (Z>+6)(Z>+*) (*3+6 )(* +6 )(* +* ) 4 2 F | 1 % (i (3-48) Z>+6 7 g Chapter3. ModelingMultiply-Articulated Vehicles 3.4.2 Dynamic Loading Due to Roll Changes in loading due to the motion of the vehicle are modeled through two primary modes: roll which exhibits the largest response, and pitch. In the roll mode, moments are summed about a projection of the vehicle centerline on the ground plane. It should be noted that the pinde hook-eye hitch assembly between the A-dolly tongue and the lead semitrailer is not capable of transmitting moments along the vehicle. For this reason, the roll behavior of the vehicle may be broken into two units: the A-dolly-second semitrailer and the tractor-lead semitrailer. In both units, the procedure for analysis is the same; the moments in each unit are summed equal to zero (permissible in this case due to the relative speed of the roll mode compared to that of yaw [12]). The product of the lateral acceleration and the mass of the unit acts as a lateral force through the mass center. It is this force which is resisted by vertical loading in the tires; that is, the rolling of the sprung mass will cause the wheels on one side of the unit to be more heavily loaded than the other. To yield a solution, the individual equations from A-dollysecond semitrailer or tractor-lead semitrailer are combined by eliminating the moments through the fifth wheel. The left andrighttires on a given axle carry equal dynamic loadings with opposite signs. Figure 3-7 shows the roll-plane FBD for the A-dolly-second semitrailer unit. 68 Figure 3-7 - Load Transfer Due to Roll for the A-dolly-Second Semitrailer Unit (a) A-dolly, (b) Second Semitrailer 70 Chapter3. ModelingMultiplyArticulated Vehicles Summing moments about the ground point A for the A-dolly and the second semitrailer yields the following relations: M M x + ™i *K ~ 2 ^ 9 #3 = a y ~ Mi <A Me* + m*ayAi 2 2 (3-49) 0 = (3-50) 0 Note that 6 is the roll angle o f the rth vehicle. In Equations 3-49 and 3-50, the lateral motion { of the center o f gravity due to roll is neglected; this is again an accurate approximation due to the small angles involved. A force balance in the ^-direction yields Equations 3-51 and 3-52. The subscript "%f' refers to loading due to rolL %zr ~ F \0zr F i z r F ~ = F — (3-51) kd0 4 9 3 (3-52) ~ ~ ^5^10^4 9 z r Due to the roll coupling through the fifth wheel, Mikulcik [27] and Eshleman et al [12] proposed the following constraints: (3-53) 6> = # C O S ( ^ - ^ ) 3 4 4 3 (3-54) The resulting dynamic loading is obtained through combination o f Equations 3-49 to 3-54; the final expression is given in Equations 3-49 and 3-50. (3-55) F %zr F =-F 10zr 1 = ~ F l z r 1 = =-kd 9zr ~ 2(/r d '5 l0 n k A u 5 COS(^ 4 fa) 2 1 0 + k4d9) cos(fa - fa) 2(k d + k d ) 2 5 ( 2(k d s 2 x 0 + k dg) c o s ( ^ - 0 ) 4 4 3 x 0 4 9 ^ ( ^ 4 + ^ 4 ) 2(k d + k d ) 2 5 { 0 Similarly, Figure 3-8 shows the roll-plane F B D for the tractor-lead semitrailer unit. 4 9 (3-56) (b) Figure 3-8 - Load Transfer Due to Roll for the Tractor-Lead Semitrailer Unit (a) Tractor, (b) Lead Semitrailer Chapter 3. Modeling Multiply-Articulated Vehicles 72 As for the A-dolly/second semitrailer assembly, moment and force balances are created for the lead trailer and the tractor: M 2 y2 5 3 2 M +m a h -2k d 0 -2k d 0 2 XXx (3-57) + m a }\ - 2k d\0 = 0 X2x x yX F u 6 z r ~ r x l (3-58) =0 2 2 l (3-59) Szr (3-60) F = ~F = -k d 0 F = -F = 2 3zr 4zr 2zr n x , (3-61) l2r Expressions for the roll coupling in the tractorfifthwheel are given in expressions 3-62 and 363. (3-62) (3-63) 0, = f ? c o s ( & - ^ ) 2 M =-M cos(<f> -<j> ) nx X2x 2 x Combination of Equations 3-57 to 6-63 yields the resulting dynamic loadings due to roll. (3-64) F2 — F ZT -+- lzr 2{k d\ + k d + k dl) cos(& - $ ) 3 F, = ~F zr 3zr 2 2 n 2 k d% + k d + k d 2 5zr 2 2 3 x x 2 2 x 6 (3-65) k d\ + k d + k d 2 3 2 2 x (3-66) -F = -hd % 2(k d + k d + k d\) cos(& - <p\) *3 d + k d] + k d] 2 3 3.4.3 x = -k d cos(& - 4)1 2(k d + k d + k d\) cos(& - <j> ) 3 6zr 7 2 2 2 x 2 x Dynamic Loading Due to Pitch The dynamic loading due to the pitch mode is determined in the same fashion as the roll mode. The product of the fore-aft acceleration and the vehicle's mass were assumed to create a load acting through the mass center. In the side view (looking towards the positive ydirection), a steady-state moment balance was taken about the contact point of the rear tires as Chapter 3. Modeling Multiply-Articulated Vehicles well as the forward tow point (kingpin or pinde hook). 73 Substitution o f one of the former equations into the other resulted in an explicit equation for the change in loading in pitch. Figure 3-9 displays the pitch F B D for each unit in the vehicle. z II16zp 7 7 7 7 7 7 7 T '7zp (a) 7 7 T ~~r~r 74 Chapter 3. Modeling Multiply-Articulated Vehicles pitch -vllzp 7 7" 7 7 7 7 / / h lzp 32p F F 4zp 2, (d) Figure 3-9 - Load Transfer Due to Pitch (a) Second Semitrailer, (b) A-Dolly, (c) Lead Semitrailer, (d) Tractor Starting from the second semitrailer, the pitch loadings are determined. The subscript refers to the pitch mode. • ^ 4 ( ^ 4 - ^ 4 ) ^ 2 9zp F ~ F\o — • zp 2(A + \ ) (3-67) Chapter 3. Modeling Multiply-Articulated Vehicles 75 m ^ - V ^ a o ^ F\(,zp — - Fizp = F %zp l r = . [ w i ^ F^zp = -Fnzp = b +b ^=^ F F 5 = ^ =^ F p =F 3z 4zp 3 - ^ ) (3-69) +W ^ i T^hiKi-Vy&^o+F^ib,-b y] Lkfe-W+W io+^)] i 2(b +b ) = 7~T"[^i9 = , * y (3-70) u 6 4 Fnzp = (3-68) F\5zp ~ (3 " 71) 3 m 2(7,2 ~ ^ 2 ® 2 ) + F (b -b )] l3zp l0 , k ( ^ i - ^ ^ « + ^ . ^ - ^ ) ] = + ^) + ^(^ - ^^«] " (3 4 ( 3 ' ?2) 7 3 ) " (3 ?4) Its should be noted that the effect o f pitch during high-speed, constant or nearly-constant velocity maneuvers is quite small. The effect would become more prominent in tests requiring substantial forward acceleration or deceleration. 3.5 Tire Model Implementation Accurate modeling o f the mechanical properties o f tires is critical in the creation o f a simulation which reflects reality well. Unfortunately, there are substantial variations in the behavior o f individual tires. The lateral cornering force and aligning moment versus slip angle plots for four typical heavy truck tires are depicted in Figure 3-10 for a normal loading o f 26.7 kN. Note that even for tires with like cornering properties, there is significant spread in the aligning moment characteristics. Chapter 3. Modeling Multiply-Articulated Vehicles (b) Figure 3-10 - Comparison of Tire Properties: (a) Lateral Force vs. Slip Angle, (b) Aligning Moment vs. Slip Angle To account for this, it is necessary to implement a tire modeling strategy that allows: (a) Variation in normal loading due to dynamic effects. (b) Relative ease of implementation from experimentally-derived data. (c) Switching between different tires easily. A n approach following that of the U M T R I Phase models has been chosen to represent the lateral force and aligning moments o f a given tire. In addition, the effects of dual tires are also included, where appropriate. 77 Chapter3. Modeling Multiply-Articulated Vehicles 78 In general, tire data is available in tabulated form since tire test programs involve setting a slip angle, and testing the lateral force generated for a range of normal loads [22]. This test is repeated throughout the operating range of the tire. Although the resulting data exists for test combinations of slip angle and normal loading, a method must be devised to interpolate the data for conditions lying between known data points. The former is achieved by a two-dimensional linear interpolation scheme. The process for determining tire properties for intermediate points involve four steps. (a) Assume the desired slip angle and normal tire loading are known. Using the tabulated tire data, determine the lateral force for all vertical loadings corresponding to slip angles above and below the actual slip angle value. Note the range between the upper and lower bounding slip angles. (b) Using the reduced data set from (a), eliminate lateral forces attributed to all normal loadings except those immediately above and below the actual normal force. (c) Determine the location of the actual slip angle in the bounding range. Use this result to determine expected lateral force values at the corresponding normal forces above and below the actual value. (d) Using the range from (c), interpolate the location of the actual vertical loading and determine the expected lateral force for input slip angle and normal loading Although the process has been explained in terms of the lateral force, the procedure for the aligning moment is an exact analog. In axles having dual tires, an additional effect occurs as the vehicle comers. Where dual tires are mounted on an axle there exists a longitudinal force at each contact patch because the tires are forced to rotate at a common angular velocity while traveling on paths of different curvature [13]. These forces form a moment about the center of the wheel set. The moment can be expressed in terms of the vehicle state variables by Equation 3-75 [19]. 79 Chapter 3. Modeling Multiply-Articulated Vehicles M,dualz _pX a> = ! L (3-75) Where D is the dual tire spacing and C is the tire circumferential stiffness (experimentally ; d measured tire property) for the /th tire. The total moments on axles having dual tires is the sum of the aligning moments of the individual tires and the moment arising from the dual tire effect The tire data used in this study is presented in Appendix B: Tire Data. 3.6 Driver Model The simulation proposed in this chapter must function in two respects: (a) Open loop steering — the user inputs direcdy the steering angle as a function of time. (b) Closed loop steering — the tractor must follow a prescribed trajectory by generating the required steering inputs. The closed loop driver model is based on controllers derived in Park et al [36] and Yang et al [37]. Both controllers were originally created as automatic steering for cars in highway conditions. The controller proposed here is a simplification of the solution derived by Park et al and is valid for maneuvers having small steering amplitudes only (such as highway lanechange). To facilitate creation of the controller, tractor velocities and accelerations are converted from the local coordinate frame to the inertial frame. Only inertial positions and orientations o f the tractor are considered; the controller accounts for the vehicle dynamics indirectiy through the tractor's response to steering inputs. 80 Chapter3. Modeling Multiply-Articulated Vehicles The user supplies a trajectory for the tractor center of gravity to follow as well as a preview time T. The combination o f the preview time and the vehicle forward velocity V XimlkU to generate the look-ahead distance d ateaf is used Refer to Figure 3-11 below. Figure 3-11 — Schematic of the Look-Ahead Driver Model The heading vector is perpendicular to the front axle of the tractor, and forms the orientation angle 0ft) with the inertial x-axis; the current position o f the tractor center of gravity (X (t), f Y (t)) with respect to the inertial coordinate frame is assumed known. p Errors between the desired and estimated position and orientation of the vehicle at the lookhead distance drive the controller. F (t+T) refers to the desired lateral position o f the tractor N at time (t+T), given vehicle information available at time t. The look-ahead angle fajj) is given by Equation 3-76: F (t + N T)-Y(t) ahead (3-76) Chapter 3. Modeling Multiply-Articulated Vehicles ; 81 T o extrapolate the actual lateral position o f the tractor to a point d^^ a Taylor series expansion is used and truncated at the second derivative. The predicted lateral position is a function of the current inertial position, velocity, and acceleration. 1 •• , Y (t + T)*Y (t) + Y (t)T + -Y (t)T p p p 2 (3-77) p The path deviation errors may now be expressed: e (t) = F (t + T)-Y (t lal N p + T) ^ ( 0 =***(0-4(0 (3-78) (- ) 3 79 The tractor steering angle is taken to be a linear combination of Equations 3-65 and 3-66. 8{j) = k e mk e (t) l lat 2 cmg (3-80) The values o f the gains k, and k are tuned via trial and error to yield the desired path tracking 2 of the vehicle. 3.7 Simulation Protocols The behavior o f any simulation is sensitive to the magnitude and frequency o f the tractor steering inputs. As such, standardization o f the methods by which rearward amplification and offtracking data is obtained has been presented to allow for uniform comparison o f results between researchers. There are two general methods for testing the high-speed dynamic performance o f a heavy vehicle: (a) Frequency Analysis (b) S A E Recommended Practice S A E J2179 82 Chapter 3. Modeling Multiply-Articulated Vehicles Frequency analysis methods predate the S A E standard method, and were utilized largely by the U M T R I researchers Mallikarjunareo [20], Fancher [16] [17], Ervin et al [20] and Winkler et al [21]. The vehicle test velocity is set at 24.6 m/s (55 M P H ) , the standard speed limit for heavy trucks on interstate highways in the United States. The driver inputs a sinusoidal steering input over the a range of typical frequencies (typically between 0 and 3.25 H z , although computerized simulations usually raise the upper bound to 4 Hz). The lateral accelerations of the tractor and the second trailer are compared to yield the rearward amplification. Unfortunately, the magnitude of the observed rearward amplification is dependant on the magnitude o f the steering input; increasing the maximum steering amplitude increases load transfer and thus, rearward amplification. It is desirable to test near to the maximum steer amplitude before rollover to yield a "worst case" result. For this reason, full-scale tests performed using this protocol are outfitted with outrigger assemblies to prevent the destruction o f the vehicle [21]. A n alternate, more rigid testing protocol was introduced in 1993 as S A E Standard J2179 [38]. In S A E J2179, the vehicle passes through a rigidly-defined single lane-change path at 24.6 + 0.44 m/s. Additionally, S A E J2179 makes provisions for measuring high-speed offtracking during the test. The standard test maneuver is detailed in Figure 3-12. RA Observation Region 1.31 m 1.46 m Inrun 61 m Figure 3-12 - S A E J2179 Dynamic Stability Test Maneuver Chapter 3. Modeling Multiply-Articulated Vehicles 83 T o encourage standardization o f the results, many stipulations are placed on the vehicle, driver, and test conditions. These conditions include: (a) Thetiresin new, unworn condition, inflated to the manufacturer's rated pressure. (b) The vehicle must be loaded uniformly, with load locations and center of gravity heights must be measured, if possible. (c) The period of the test maneuver must be between 2 and 3 seconds. (d) The deviation from 24.6 m/s throughout the test must be less than 0.83 m/s The test vehicle is fully instrumented using accelerometers on both the tractor and the second trailer, as well as a water jet device to aid in determining the high-speed offtracking behavior of the last trailer. The standard also provides a recommended statistical method for accurately processing the experimental data. The majority o f the ideas o f S A E J2179 can be carried over into the simulation environment, the key difficulty being the selection o f an appropriate lane-change trajectory (expressed in inertial coordinates). Sledge, et al [39] investigated a variety o f candidate trajectories, and concluded that the best performance was yielded by so-called "3-4-5" or "4-5-6-7" polynomials . 3 The lane change tests performed in this report will involve the "3-4-5" polynomial trajectory. Refer to Equation 3-81 for the formulation. 3 Y laleral (X) = W 10 ~X~ L -15 'X' L 4 5~ 'X~ +6 (3-81) L Where: X is the absolute longitudinal position of the tractor, W is the width o f the lane change, and L is the length required to complete the lane change. 3 These polynomials are derived from the mechanical theory of cams, and have the properties of continuous position, velocity, and acceleration profiles. 84 Chapter 3. Modeling Multiply-Articulated Vehicles This report will investigate both simulation protocols as required to compare the derived model with results obtained in the literature. 3.8 Method Of Solution Once assembled, the equations detailed in the preceding sections form the basis for the dynamic simulation o f the A-train heavy truck. T o facilitate a solution, the equations of motion are manipulated into the form of Equation 3-82. Once so arranged, the resulting equation can be solved numerically using the O D E suite built into The Math Works MatLAB 5.3. T o ensure a solution, the mass matrix M(ty) is assumed to be non-singular. Using the function ode45, Equation 3-82 may be solved. Ode45 is based on the Runge-Kutta solution algorithm, specifically that of the Dormand-Prince pair. The solver is a one-step process, meaning that in computing the current value of the state matrixy(tj, only the solution at the previous time step j(t„.,) is required. Although ode45 allows adaptive step size in computing the solution, this feature has been disabled to allow output of the previous value of the state vector at each time step. This modification allows numeric estimation of the lateral and forward acceleration of each unit, perniitting the estimation of semi-static dynamic loading throughout the test maneuver. Figure 3-13 details the variety of files and their interactions required to achieve the simulation. Chapter 3. Modeling Multiply-Articulated Vehicles 85 (jStart Simulation Driver Program (Prcproccss) ODE Control Program Write State Values from Last Loop Execution to Global Variable ODE 45 Mass Matrix M(t,y) Driver Program (Postprocess) End Simulation v_ (a) . Chapter 3. Modeling Multiply-Articulated Vehicles Obtain Previous State Values 86 Numerically Estimate Vehicle Accelerations Manual Operation Y Closed-loop Driver Determine Current Tractor Inertial Position/Orientation yUser Programmed. \ Steer Angle / Determine Desired Future Tractor Inertial Position/Orientation Find Quasi-static Load Transfer Using Accelerations Calculate Steer Angle Rollover (End Simulation) Set Steer Angle With State Feedback Controller Find Tire Lateral Force Find Tire Aligning Moment Find F(t,y) using Equations of Motion Return F(t,y) •<2> (b) Figure 3-13 - Solution Method Flow Chart (a) Overall, (b) State Equations The simulation is started by running the driver program. This program "oversees" the entire simulation, and serves to initialize the system variables, and provide the global values of the A train parameters. The driver program calls the O D E controller file which sets the error tolerance and accuracy of the simulation as well as providing ode45 the locations where the Chapter 3. Modeling Multiply-Articulated Vehicles 87 mass matrix M(tji) and the state transition m a t r i x / ^ reside on the computer system. The O D E controller file then calls ode45. Ode45 proceeds through its algorithm, calling the m-files for the mass matrix and the state equations as required. The execution o f the mass matrix m-file is straightforward; however, the function o f the state transition involves calculation or specification o f the tractor steer angle, determining the semi-static load transfer, accounting for intelligent dolly steering (if enabled), finding the required tire properties, and finally, computation o f F(tj) based on the former information. The results are then passed back to the O D E controller file. The handoff of information between the O D E controller file, ode45, the mass matrix m-file, and the state equation m-file continues until the end o f the specified simulation time domain. A t this point, the computed results for the state vector at each time step are passed back to the driver program. The driver program completes the simulation by post-processing the state data, and plots the required results. 3.9 Simulation Results The standard A-train combination vehicle has been simulated using the analysis prescribed in this chapter. Parameter data referring to vehicle geometry, mechanical properties, and tire properties has been derived from Winkler et aL [21]. The following sections detail the verification o f the model, and the dynamic behavior o f the vehicle with respect to both rearward amplification and high-speed offtracking. 3.9.1 Verification of Results The simulation was first run through a test conforming to S A E J2179. See Figure 3-14 below. Chapter 3. Modeling Multiply-Articulated Vehicles 88 Figure 3-14 - Vehicle Lateral Accelerations for S A E J2179 In Figure 3-14, each plot represents the lateral acceleration of a unit in the A-train with respect to its local coordinate frame, as would be measured in the full-scale test (differences between the local and inertial accelerations are negligible due to the small attitude angles). The tractor steering controller maintains the position of the tractor's center of gravity over the desired trajectory. Immediately apparent is the increase in lateral acceleration along the units in the vehicle. One also notes that the peak lateral acceleration o f the dolly is slighdy larger than that of the last trailer, due to its geometry, small mass, and low rotational moment o f inertia. The dolly and second trailer, in particular, are seen to behave in a lightiy-damped manner, with significant overshoot and settling times after the completion of the maneuver. The observed rearward amplification for this test was 1.60. Chapter 3. Modeling Multiply-Articulated Vehicles 89 The current model is verified for reasonable rearward amplification behavior against full-scale test results published by Winkler et aL in 1986 [21]. In their tests, low-level tractor lateral accelerations were employed. The test maneuvers consisted o f open-loop sinusoidal steering inputs provided by a driver for a range of frequencies. Although standard test procedure dictates testing at 24.6 m/s (55 M P H ) , Winkler et aL were confined by the layout of the test facility to vehicle velocities between 20.1 m/s and 21.5 m/s. The observed rearward amplification readings were then "corrected" to 24.6 m/s using a linear interpolator whose constants were derived using the U M T R I Phase I V model. Refer to Equation 3-83 below. = RAy + (C, + C F )(55 - V) 2 Where: RA (3-83) r is the adjusted rearward amplification level at 55 M P H , RA SS V is the observed rearward amplification in the test, C, and C are "velocity sensitivity constants" (configuration2 dependant), F i s the frequency of the maneuver, and T^is the test vehicle velocity in M P H . r The vehicle tested by Winkler et aL had significant differences from the reference A-train. Both trailer vans were shorter than the reference vehicle; the wheelbases in the test vehicle were reduced by 0.61 m (7.92 m vans versus 8.53 m vans). The trailers were also modified to include outrigger assemblies, affecting the rotational moments of inertia of the units. Finally, the A-dolly drawbar length was also reduced compared to the reference A-train. Figure 3-15 compares the behavior of the modified vehicle in the present simulation against the corrected rearward amplification data presented in [21]. Chapter 3. Modeling Multiply-Articulated Vehicles 90 Figure 3-15 — Comparison o f Simulation and U M T R I Test Results The test data and the simulation results are seen to exhibit a reasonable fit at the upper test frequencies. However, there is some discrepancy at input frequencies between approximately 1.5 and 2.5 H z . In this range, the U M T R I tests predict a much higher level of rearward amplification than does the simulation. This apparent difference was also noted by Winkler et al and is at odds against other simulation data in this regime [28]. It is possible that this result may be attributed to the interpolation methods utilized by Winkler et aL to obtain the "corrected" rearward amplification readings. Conversely, these points may simply be outliers from the test data. The differences between the reference and test A-train vehicles are shown in Figure 3-16. Chapter 3. Modeling Multiply-Articulated Vehicles 91 Input Frequency (Hz) Figure 3-16 - Comparison of Reference and U M T R I Test A-Trains The U M T R I test vehicle has higher rearward amplification levels throughout the frequency range. The peak value o f rearward amplification is also moved to higher frequencies. Differences in vehicle geometry account for the majority of differences seen in Figure 3-16. 3.9.2 Rearward Amplification Sensitivity Factors Within a given multiply-articulated vehicle geometry, there are several factors which lend themselves to increasing the observed rearward amplification. forward velocity; the effects of velocity are shown in Figure 3-17. The largest factor is vehicle Chapter 3. Modeling Multiply-Articulated Vehicles Figure 3-17 — 92 Effect of Velocity on Rearward Amplification Figure 3-17 shows representative frequency responses for the reference A-train vehicle at nominal velocity (24.6 m/s) as well as 25 percent above (30.7 m/s) and below (18.4 m/s) that value. Most prominent is the marked increase in rearward amplification as the velocity increases; the peak value o f rearward amplification at 30.7 m/s is 1.6 times that seen at the nominal velocity. One also observes that at low velocity, there is a slight deamplification behavior at high frequency. The frequency at which the peak value of rearward amplification occurs also moves to higher values as the velocity increases. Tire size and construction also play a significant roll in the level o f observed rearward amplification. See Figure 3-18. Chapter 3. Modeling Multiply-Articulated Vehicks 2.25 93 r Figure 3-18 — Effect of Tire Type on Rearward Amplification The tires tested in Figure 3-18 include both bias-ply and radial constructions. Although similar in dimension, the differing cornering and aligning moment behaviors lead to differences in peak results which differ by up to 11% given the same magnitude o f sinusoidal input. Figure 3-19 investigates the effects of load transfer on rearward amplification. Simulations were conducted for sinusoidal inputs of 2.0 H z and 71 H z at differing values o f peak steer angle. Chapter 3. Modeling Multiply-Articulated Vehicles 94 1.9 1.85 1.8 3.14 H z 1.75 c o u — 2.00 Hz Input ----- 3.14 Hz Input 1.65 1.6 1.55 0.3 0.4 0.5 Maximum Tractor Steer Angle (deg) 0.7 Figure 3-19 — Effect of Frequency and Load Transfer on Rearward Amplification As the maximum magnitude of the input is increased, so also is the load transfer. This causes rearward amplification levels to rise, as the available lateral force from the tires is less for one heavily-loaded tire and one lighdy-loaded tire than that obtainable from two equally-loaded tires (see Figures 2-3 and 2-4) . The load transfer effect is neady the same for the two input 4 frequencies tested (the increase in rearward amplification due to load transfer for the 2.0 H z steering input was 6 percent while the result for the n H z steering input was 7 percent). 4 For Michelin 10.00x20 tires, with even 13.3 kN normal loading per side and a slip angle of 1 degree, the available force and aligning moment are 4804 N and 149 Nm, respectively. For the same slip angle, and vertical loading of 26.3 kN on one side and 0 N on the other, the force and aligning moment are 3640 N (-24 %) and 176 N m (+15 %). The aligning moment has the effect of softening the effective cornering coefficient of a tire. Chapter 3. Modeling Multiply-Articulated Vehicles 3.9.3 95 Offtracking Results Testing according to S A E J2179 allows estimation of vehicle offtracking. The offtracking corresponding to the test in Figure 3-14 is shown in Figure 3-20. Figure 3-20 - Offtracking During S A E J2179 Test Protocol Figure 3-20 depicts the motion of the center of gravity of each unit in the A-Train in the yaw plane. One observes the moderate amount of offtracking in the first trailer, but a large overshoot (13 percent) in the response of both the A-dolly and the second semitrailer. The A dolly and second semitrailer oscillate an additional two times before coming to steady-state. The effect o f velocity is pronounced in offtracking behavior. The S A E J2179 test trajectory was repeated for vehicle velocities 25 percent above (30.7 m/s) and below (18.4 m/s) the nominal velocity (26.4 m/s). Figure 3-21 denotes the results. Chapter 3. Modeling Multiply-Articulated Vehicles 1.6 96 -1 1.5 - 0 10 20 30 40 50 70 60 80 90 100 110 120 130 140 X-Position (m) Figure 3-21 - Effect of Speed on Offtracking The offtracking observed at the nominal velocity was much greater than that seen at 18.4 m/s. Conversely, the test was not able to proceed for the 30.7 m/s velocity as the wheels of both the lead and second semitrailers lifted off the ground during the simulation, indicating immanent rollover. Both of the former results coincide with the findings of Bernard et al [18]; that is, there exists a velocity at which no offtracking will occur, and above that, offtracking will increase in proportion to the increase in velocity. Finally, a simulation was performed to investigate vehicle performance through a long-radius, sweeping comer, such as would be found on an interstate-type highway. The simulation was performed at the nominal 24.6 m/s velocity. After effects due to entering the comer, a maximum offtracking of 0.15 m was noted. See Figure 3-22. Chapter 3. Modeling Multiply-Articulated Vehicles 97 -0.25 _Q g I 0 i i 10 20 i 30 X Position (m) i 40 i 50 I 60 Figure 3-22 — Steady-State Cornering Offtracking Performance 3.10 Discussion The formulation for a non-linear, two-dimensional yaw plane model with semi-static load transfer has been given. An appropriate system of reference frames was selected. A modeling concept was chosen in which the effects of pitch and roll would be decoupled from those in the yaw plane. The effects of pitch and roll would instead be modeled as changes in the normal loading of the A-train tires. The yaw plane equations of motion were derived separately for each unit in the reference Atrain. Projections of the vehicle in the roll and pitch frames were utilized to derive the 98 Chapter3. ModelingMultipfy-Articulated Vehicles mathematics of semi-static load transfer. Forward and lateral accelerations acting upon the mass center of each unit as utilized as inputs to the load transfer equations. A two-dimensional interpolator was proposed for accurate tire modeling using available experimental data. In addition to cornering force and aligning moment, the effects of tandem tires was also accounted for. To accurately track an intended trajectory in the yaw plane, a driver model was developed to control the simulation. Two valid test protocols were presented. The first, S A E Recommended Practice J2179, was intended to simultaneously test rearward amplification and offtracking. The second, sinusoidal steering inputs of frequencies throughout the normal domain of operation is a simpler testing procedure meant only to characterize rearward amplification behavior. The driver controller was intended for use in simulations under S A E Recommended Practice J2179; frequency response testing is performed in open-loop. The process by which the equations of motion can be combined into a complete simulation in MarLAB is also provided. The applicability of the non-linear two-dimensional vehicle model in simulating both rearward amplification and offtracking events was given. Simulation results compared favorably to those generated by U M T R I through full-scale testing, verifying the soundness of the model formulation and assumptions made during its derivation. As a result of the A-train modeling, vehicle velocity, tire characteristics and steering input degree and frequency were verified as influential factors in the incidence of rearward amplification and offtracking. Chapter 4 Control System Design As demonstrated in the Section 3, the dynamic performance of multiply articulated vehicles are compromised by offtracking and rearward amplification at high velocities. This chapter addresses this deficiency by automatically steering the wheels located on the A-dolly axle. A linear model of the A-dolly — second semitrailer is derived, and compared with the results of the preceding complex analysis. The linear model is then utilized to formulate the control strategy based on eigenvalue (pole-placement) techniques. 4.1 Linear Model Derivation The primary step in devising a control strategy is simplification of the modeling of the plant Since the pinde hook hitch between the lead semitrailer and the A-dolly cannot transmit moments, the A-dolly and the second semitrailer are largely independent of the lead semitrailer and tractor in their behavior (in particular, the lead units are vital only as inputs to the position of the A-dolly; the lateral force in the pinde hook between the A-dolly and the leading semitrailer has been proved by Fancher [16] to be "small" compared to the other forces on the system). One can then thus model the full trailer as a separate unit having the position and magnitude of the velocity vector of the A-dolly drawbar as an input Additionally, because the intended control strategy is to depend only on steering the A-dolly to stabilize the motion of the second semitrailer (that is, we do not propose to control the lead semitrailer in any way), 99 Chapter 4. Control System Design 100 moving the performance of the full trailer assembly near some critical value represents the objective of the control. The model chosen for control is a linear yaw plane analysis. As such, it carries with it the stipulations of all linearized vehicle models (see Section 2.4). The major assumptions are repeated in the specific context of the control model are: a) The vehicle velocity is constant, thus mamtaining linear slip angle formulations for the tires. b) The full trailer is modeled using the "bicycle" premise; that is, the width of the vehicle is neglected and tires on the left andrightside of the vehicle are lumped together into an effective tire. A single tire coefficient is introduced to model the lateral force, aligning moment, and dual tire behavior of the effective tire. c) Small angles are assumed throughout for all rotational variables. d) In expressions for relative velocity and acceleration between the A-dolly and the second semitrailer, terms that are the product of two variables having "small" values (such as the articulation angle and the angular velocity of one of the units) are taken to be very small, and are thus neglected. e) The fifth wheel of the A-dolly is assumed to lie directiy over the center of gravity of the unit (a reasonable assumption as the A-dolly drawbar is usually balanced when the full trailer assembly is created). f) Resisting yaw moments occurring at thefifthwheel hitch are assumed negligible. 4.1.1 Preface to Modeling The process by which the equations of motion are generated follows the same process as in Section 3 — Newtonian methods tofindthe basic equations,rigidbody dynamics theory to determine the velocity and acceleration constraints, and elimination of the hitch forces using the constraints to generate the final equation set. The reference frames utilized are also identical for the A-dolly and the second semitrailer. There are, however, several important deviations. 101 Chapter 4. Control System Design There are two system inputs. T h e first is the applied steer angle o n the dolly; this is the variable to be controlled. T h e second is the angular velocity o f the velocity vector U relative to the A - d o l l y reference frame QJ is a vector representing the velocity o f the pinde hook). A l t h o u g h the magnitude o f the velocity vector is. constrained to remain constant, the rate o f change o f its direction w i l l be directiy responsible for the dynamic behavior o f the resulting full trailer simulation. T h e system outputs are chosen as variables that are easily measured by devices located o n the dolly, and yield results that are expected to be made relative to the dolly frame o f reference. D o l l y angular velocity C0 and the "articulation angle" between the velocity vector U and the 3 dolly centedine £ were chosen for this reason. Estimation o f the remaining parameters and states are explored i n Section 5. 4.1.2 Derivation of the Linear Equations of Motion A s i n Section 3, the equations o f motion are derived i n the yaw plane. T h e modeling approach has its roots in the analysis performed by Bernard et aL [18] i n 1980, but is modified in terms o f application, number o f units modeled, system states chosen, and state variable formulation i n the current research. Figure 4-1 denotes the dimensions and variables for the A-dolly. U Figure 4-1 - F B D F o r Linear A - D o l l y C Chapter 4. Control System Design 102 A s the f o r w a r d v e l o c i t y is t a k e n t o b e c o n s t a n t , n o s u m m a t i o n o f forces a l o n g t h e c e n t e r l i n e o f the A - d o l l y (x-direction) is necessary. T h e r e s u l t i n g e q u a t i o n s o f m o t i o n are g i v e n b e l o w . = » » 3 « , 3 = l4y + l5y + U F F F M (4-2) yh YJ CG= h<»3= l4 F (4-1) S i m i l a d y , F i g u r e 4-2 displays the F D B f o r the l i n e a r i z e d s e c o n d semitrailer. 910 16y '8 "7 1 Figure 4-2 — F B D F o r L i n e a r Semitrailer T h e e q u a t i o n s o f m o t i o n are g i v e n i n E q u a t i o n s 4-3 a n d 4-4. ^, y F ~ 4 y4 m a ~ l6y F + (4-3) 910 F (4-4) 4.1.3 Constraint Equations T h e s a m e i d e a o f r e l a t i n g terms f r o m adjacent S e c t i o n 3 is s i m p l i f i e d a n d repeated here. frames u s i n g a t r a n s f o r m a t i o n m a t r i x f r o m T h e " a r t i c u l a t i o n a n g l e " 6 f o r m e d b e t w e e n the d o l l y centerline a n d the i n p u t v e l o c i t y v e c t o r U is g i v e n b y E q u a t i o n 4-5 = fc-$rdt W h e r e : r is the angular v e l o c i t y o f the i n p u t v e c t o r U . A - d o l l y a n d the s e c o n d trailer is g i v e n b y E q u a t i o n 3-16. (4-5) T h e a r t i c u l a t i o n angle 7^ b e t w e e n the Chapter 4. Control System Design 103 With r defined, the transformation matrix can be found, assuming small angles. } (4-6) -r, l Applying the results of Equation (4-6) to Equation 3-22 yields the small angle equivalence between the lateral fifth wheel force in the A-dolly and second semitrailer reference frames. F, = -F, 6J (4-7) SJ The lateral velocity of the A-dolly is determined using the velocity vector U and rigid body dynamics (assuming small angles). The negative sign in Equation 4-8 is a result o f the sign convention assumed in Figure 4-1. (4-8) ' U ' >3," _-sU_ The formulation for A-dolly lateral acceleration is expressed in Equation 4-9. V =Ur-b cb y3 5 (4-9) 3 Using the linearized transformation matrix 4-6 and the relative velocity and acceleration results from Equations 3-30 and 3-31, respectively, the approximate formulations are given for the second semitrailer. x~ = T U 3 + - sU - b co 5 b o) 3 (4-10) 0 -b co _ 7 4 (4-11) 2 =T 5 + 3 Ur - b d> 5 3 -b cb _ 7 4 Chapter 4. Control System Design 104 Note that after expansion of Equation 4-11, the term h has been neglected as the 5 product of the terms 0)f and / ] are small compared to the rest of the formulation. 4.1.4 Slip Angles and Tire Modeling The behavior of the tires was simplified into a single equivalent tire whose properties can be deduced from the simple linear expression found in Equation 4-12. F 1 =C lateral,! a (4-12) ^effective *i 1 v ' Ctfaut is a simplified representation of the interaction between various nonlinear tire effects. Refer to Section 5 for derivation of C^ during operation of the proposed control system. Drawing upon the velocity Equations 4-8 and 4-10, expressions for the slip angles of the A dolly and second semitrailer tires are found. b 0) 5 «9io = Jj[~ W-eU- (4-13) 3 (4-14) b co - (b + b )co1 ] 5 3 7 % A Note that tan(^) = £ w h e n small angle approximations are evoked. 4.2 State Space Formulation of the Equation of Motion To facilitate modem control analysis of the system, the equations of motion described in the former section are manipulated into state space form. T o do so, the tire forces, and the velocity and acceleration constraints are substituted into expressions 4-1 to 4-4. The forces F H F t6 and are then eliminated through substitution of Equations 4-1 and 4-3 into Equations 4-2 and 4-4. The result is two linear equations of motion involving four states. 105 Chapter 4. Control System Design To complete the state transition matrix, an additional two equations involving the derivative of the acceleration angles are determined. (4-15) f3 = (4-16) 60 - C0 4 3 The state space formulation of the system may now be created: 3 CO, (4-17) ~co ~cb ~ 3 CO* = A r s s (4-18) co 3 Where: M and A are square 4x4 matrices, B and B are 4x1 column vectors, and C is a 4x2 t 2 matrix mapping the system states to the measured output. The mass matrix in Equation 4-17 is of full rank and is easily invertible symbolically using computational tools available in MatLAB 5.3. This allows formulation of Equation 4-17 in the standard state space form. ~cb CD~ 3 C0 4 co = A p r 4 n Where: A „ = M ' A , B 1 > P = M% 3 £ £ and B 2 I P = M^B,. (4-19) Chapter 4. Control System Design 106 The dynamics of the full trailer system are now uniquely defined by the elements of the Ap matrix in Equation 4-19. Similarly, matrices B 1>p and map the effects of the dolly steer angle y and the angular velocity of the vector U to the system states. The value of r is regarded as a disturbance input and cannot be used in the control strategy. Thus, the method by which the dynamic effects of the plant are to be modified are created solely from the inputs provided by the controller to the dolly steer angle y. 4.2.1 Controllability and Observability To ensure the feasibility of the proposed state space formulation given in Equation 4-19 for control design, the concepts of controllability and observability must be verified. Controllability refers to the freedom the designer has in moving the plant matrix eigenvalues to more desirable locations to improve system performance [40]. A system is said to be controllable if the matrix proposed in Equation 4-20 is of full rank; that is, the rank of the matrix Cont is equal to the number of states. If the rank of Cont is less than the number of system states, the plant eigenvalues cannot be moved to arbitrary locations using the current formulation. 5 Cont = [B lp Note that the matrix B 1>p AB p A B, 2 lp p p A% \ p (4-20) is utilized here, as y is the only available controllable input. For values associated with "typical" A-dolly/semitrailer combinations, Cont is of full rank. Observability refers to the ability to construct a system's state vector completely from measurements made at the output y. If the Obs matrix formulated in Equation 4-21 is of full rank, the system is completely observable (that is, the value of all the system states may be 5 For more information, refer to The Art of Control Enrineerine by Dutton et al [40]. 107 Chapter 4. Control System Design determined from the system output). If Obs is not of full rank, some (but not all) of the system's state information may be found using output measurements [40]. (4-21) c Obs CA CA CA Assuming the measured values are co and £, Obs is of full rank for typical full trailer data } This topic will be investigated more fully in Section 5. 4.3 Comparison of Linear and Non-Linear Models The results of the preceding linear analysis was compared to the results of the complex nonlinear simulation of Section 3. An effective cornering coefficient C^ = 7.2(10) N/rad was 5 assumed for thetires;the lane-change maneuver simulated was identical to that in Section 3. Figures 4-3 and 4-4 compare the angular velocity of the A-dolly and the second trailer, and the two articulation angles, respectively, for the A-dolly and the second semitrailer for both models. Chapter 4. Control System Design Figure 4-3 - Comparison of Linear and Non-linear Results: Angular Velocity 108 109 Chapter 4. Control System Design Figure 4-4 - Comparison of Linear and Non-linear Results: Articulation Angles In both cases, the agreement between the linear and non-linear models is quite close. The linear model does well to correctiy predict the motion o f the units under small amplitude maneuvers. Such maneuvers correspond to the primary operational conditions for typical A - train vehicles. As the amplitude of the motion increases, the deviation between the two models increases. In particular, motion at or near the extremities o f each state demonstrate the importance o f non-linearities. escalating Large amplitudes of motion generally indicate very severe or abrupt operation of the vehicle. In this case, the bicycle model of the vehicle in the yaw plane and the simplification of the tire properties into a single effective cornering constant begin to break down; the effects of load transfer on the observed behavior of the A-train increases in 110 Chapter 4. Control System Design prominence. None-the-less, it is clear that for high-speed, highway operation at or slighdy exceeding the normal operational domain, the proposed linear control model adequately describes the dynamic behavior of the system. 4.4 System Eigenvalues Mallikarjunarao [20] and Fancher [17] demonstrated the importance of eigenvalues (poles) in characterizing the dynamic behavior of a particular plant. Investigation of the structure of the eigenvalues can yield information such as effective damping ratios of each mode, a qualitative estimate of the system rate o f convergence, and a means to determine the relative stability of a given linear system. The eigenvalues for the full trailer system consist of two complex conjugate pairs. Each pair is representative of the behavior of a particular mode o f operation. As such, the eigenvalues are determined using Equation 4-22. tyi-A plant l (4-22) = 0 Each individual eigenvalue may be represented in terms of real and imaginary numbers in the form of Equation 4-23 [41]. (4-23) Where: 0)^ is the undamped natural frequency and ^ is the corresponding damping ratio of the rth mode. Equations 4-22 and 4-23 are used in conjunction with the system plant matrix A pl3nt to determine the eigenvalues for the A-dolly/semitrailer vehicle. Figure 4-5 displays the behavior Chapter 4. Control System Design 111 of the positive half o f the complex conjugate pairs over velocities ranging from 18 m/s to 32 m/s in 2 m/s increments for the reference A-train full trailer assembly. -10 - 9 - 8 - 7 - 6 - 5 -4 ^ ( r a d / s e c ) [Real] - 3 - 2 - 1 0 Figure 4-5 - Sensitivity of Eigenvalues to Velocity The horizontal axis of Figure 4-5 denotes the real coefficient o f the eigenvalue (C,co^; the vertical axis represents the damped natural frequency or imaginary part of the eigenvalue (ooJI-C^f*). Lines o f constant damping ratio are teal and noted with their respective values. Conversely, curves of constant natural frequency are represented as arcs with their origins on the real axis. Investigation of Figure 4-5 reveals some reasoning behind the observed rearward amplification and offtracking phenomenon. As the vehicle velocity increases, the effective damping ratio of Chapter 4. Control System Design both plant modes decrease. 112 The eigenvalues also tend towards relatively "slower" locations (recall that the speed o f response o f a system can be reasonably estimated by the magnitude o f the negative real part o f the eigenvalue; the greater the magnitude o f the real part, the faster the resulting system response will be). The damping ratio o f the individual modes are not the same at like velocities; the higher-frequency mode always exhibits a more lighdy-damped response than does the lower-frequency mode. Decreasing the effective damping ratios increases the tracking error during transient vehicle maneuvers. A s a corollary response, the rearward amplification performance is degraded as well. Thus, the proposed control system must move the plant eigenvalues to locations which specify increased damping for each o f the two modes. The movement o f the eigenvalues must be balanced against the effort required; the control input necessary to move the eigenvalues to their new locations must be reasonable in light o f sensor, actuator and other implementation issues. 4.5 State Variable Feedback Control By feeding back all o f the states in a completely controllable system, it becomes possible to locate the closed-loop eigenvalues at any location in the complex plane. Any desired dynamic response of the system is thus possible, given the correct controller. The feedback controller is designed assuming all states are measured and available. However, as explained in Section 4.1.1, the angular velocity o f the second semitrailer C0 and the 4 articulation angle between the A-dolly and the second semitrailer r } measured, but instead will be estimated from available data. will not be directiy Fortunately, the design o f the state-feedback controller and the method for estimation can be designed independendy [42] . 6 6 This is called the "separation principle" in controls literature. For a complete proof, see DeRusso et al State Variables for Enmneers [42]. Chapter 4. Control System Design 113 The topic of this section is the selection and performance of the state variable feedback controller. The procedure for estimation is investigated in Section 5. 4.5.1 State Feedback and Ackerman's Formula The term "state feedback control" implies a control law for the system of the form in Equation 4-24. (4-24) co. y = -K co A Where: Kis a 1x4 matrix of feedback gains which moves the closed-loop plant eigenvalues to their intended positions. Two primary methods of determining and placing system poles are available. The first method, optimal control, is developed by balancing system response and control effort. System eigenvalues are not direcdy specified in this instance and the feedback gains are generated automatically as part of the design process. The second method, pole placement, involves specification of the characteristic equation of the closed loop system according to some criteria, and subsequent calculation of the feedback gains required to achieve it. A n algorithm known as Ackermann's Formula is utilized in this regard [41]. Ackerman's formula is an extension to the controllability theory presented in Equation 4-20. The design characteristic equation is determined using the desired eigenvalues and expanded to yield Equation 4-25. X + ax?i + a A + a A + a = 0 A 2 2 3 A (4-25) Chapter 4. Control System Design 114 A relationship between the coefficients a o f Equation 4-25 and the plant matrix A { plaat is achieved by the Cayley-Hamilton theorem, which states that a matrix satisfies its own characteristic polynomial [41]. Through extensive manipulation, the required system feedback gains are computed by Equations 4-27 and 4-28 . 7 b Cont = [0 0 r 0 lf (4" ) 27 K = b\(A ) (4-28) planl The algorithm proceeds as follows. The desired system characteristic equation is determined, allowing for the computation o f Equation 4-26. Coefficients o f b are determined using the T controllability matrix Cont and Equation 4-27; Gaussian pivot and elimination matrix operations are utilized in this regard (Ackermann's Formula can also be solved using explicit inversion o f the controllability matrix; however, since this technique is numeric, the computational burden and resulting numerical inaccuracies make this option less desirable). Equation 4-28 then allows for computation of the required feedback matrix K 4.5.2 Screening of Candidate Control Strategies To investigate the behavior o f candidate A-dolly steering strategies, a simple screening simulation was undertaken using the linear full-trailer model derived in this section. The input value r was selected such that the pinde hook o f the A-dolly underwent a sinusoidal lanechange type maneuver of period 2.5 seconds, maximum amplitude o f 0.15 g, and forward velocity of 24.6 m/s. The lateral acceleration behavior of the uncontrolled full trailer assembly is depicted in Figure 4-6. 7 For a complete proof of Ackermann's Formula, refer to Franklin etaL Feedback. Controloj'Dynamic Systems [41]. Chapter 4. Control System Design 115 Figure 4-6 - Uncontrolled Full-Trailer Behavior Note that the increase in lateral acceleration seen in Figure 4-6 is between the pinde hook and the second semitrailer center of gravity only; this is not the same as the comparison between tractor and second semitrailer lateral accelerations in the definition of rearward amplification. However, reduction o f lateral acceleration o f the semitrailer in the linear model will correspond to a reduction in the overall rearward amplification o f the entire A-train vehicle. The uncontrolled peak lateral acceleration of the second semitrailer was noted to be 0.17 g The following subsections will detail plausible solutions to move the system eigenvalues to locations yielding improved dynamic behavior. Chapter 4. Control System Design 116 4.5.3 The Linear Quadratic Regulator Linear quadratic regulators (LQR) have their roots in optimal control theory, in which the control objectives must be established in a specific function format, known as a performance index. Such controllers derive their name from the maximum (quadratic) order o f the performance index. The goal o f the controller is to minimize the value contained within the performance index. The selection of an appropriate performance index is contingent on two conflicting ideas [43]: (a) The index must adequately reflect the designer's concept of "good" performance. (b) The index must be o f a form such that a solution is tractable using optimal control theory. Using L Q R control to run various active yaw torque schemes for reducing rearward amplification was proposed by El-Gindy et al [19] and Palkovics et aL [32]. In the present context, the L Q R controller is derived to provide the dolly steer angle y. The standard performance index for L Q R control in reproduced in Equation 4-30 [19]. 1 T ( LQR=l^j] J . (4-30) Q y y) x x+ R dt o Where: A , Q, y, and R represent the state vector, the state-weighting matrix, the steering input, and the input-weighting matrix, respectively. The relative magnitude of the elements o f Q and R determine the effort by the controller to minimize the corresponding state and control values. The minimization o f Equation 4-30 can be found by solving the standard Algebraic Riccati Equation for X^g. Chapter 4. Control System Design ^LQ A R +A X T planl plant 117 - X B RT B X x LQR LQR lp Xp LQR +Q =0 (4-31) The result from Equation 4-31 is then used to compute the required state feedback gain matrix K K = R-^Bl X p («2) LQR Equations 4-31 and 4-32 were implemented using the Iqr function in Madab 5.3's Control Toolbox. After much trial and error, values of Q — diag[l 0,100,10,10] and R = 1 was found to yield the best results with the observed second semitrailer peak lateral acceleration of 0.130,g. Implementation o f the L Q R method is limited in practice by the need to "tune" the values o f Q and R for each value of the plant matrix. In the present case, the plant matrix may change quite substantially with differing vehicle velocities, loadings, geometry, and tire types. The L Q R controller also requires a very accurate plant model for best results, and demands greater computational requirements from the system than do some of the other strategies. Furthermore, there is a general "lack o f feel" to the results; the designer relinquishes direct control of the eigenvalue locations, and is instead confounded with abstract weighting factors to arrive at a solution. 4.5.4 Prototype Control A n alternate method of selecting the closed-loop plant poles is that of prototype control, proposed by Franklin et al [41]. The desired eigenvalues are set by selecting a function exhibiting desirable transient behavior. purpose, many functions have been This function is known as the prototype; for this cataloged according to the particular transient characteristics at which they excel. The most common prototype designs for control are the Bessel and I T A E functions. The eigenvalues proposed by the Bessel and I T A E functions are applied to the system using state variable feedback and Ackermann's formula to determine the required gains. 118 Chapter 4. Control System Design Bessel functions are used frequentiy in Electrical Engineeringforfilter design. They offer the advantage of a transient response neady devoid of overshoot. The prototype characteristic equation for a plant of order n is given by an «th degree Bessel polynomial. In the present case, Equation 4-33 denotes the required 4 degree Bessel prototype. th 00 • + 0.6573 ± y0.8302 X + 0.9047 ± y0.2711 (4-33) n Where: oo is the cutoff frequency of the desired response. 0 The other proposed set of prototype responses involve minimization of the integral of the time multiplied by the absolute values of the error (ITAE). As compared to the Bessel family of polynomials, the ITAE functions exhibit greater overshoot, but with the advantage of decreased rise time and reduced sensitivity to noise at higher frequencies [41]. The formulation is given in Equation 4-34. + 0.4240 ± yi.2630 oo. • + 0.6260± y0.4141 (4-34) oo n When applied to the screening test, a cutoff frequency of 00 = 7.5 rad/s was applied to the 0 desired characteristic responses in Equations 4-33 and 4-34. The observed lateral accelerations were 0.133 g and 0.137 £ for the Bessel and ITAE prototypes, respectively. As expected, the tracking response of the Bessel test exhibited negligible overshoot following the maneuver, and the ITAErisetimedecreased over the uncontrolled situation. 4.5.5 Critical Damping Control of the A-Dolly Investigation of the lateral acceleration plots for the entire A-train vehicle in Figure 3-14 reveals that the majority of amplification of acceleration occurs between the lead semitrailer and the A-dolly. A simple controller which ensured critical or nearly-critical motion for the 119 Chapter 4. Control System Design dolly could thus be expected to have a favorable impact on the lateral acceleration observed at the second semitrailer center of gravity. The derivation o f the controller follows the work o f Ackermann et al [44]; however, the system and control objectives are different Where Ackermann et al sought to find an automatic control law to limit the understeer or oversteer behavior o f a typical automobile using critical yaw damping applied to rear wheel steering, this paper desires to use critical yaw damping to reduce the acceleration o f the second semitrailer. The A-dolly is modeled linearly in the yaw plane in which the dolly mass plus the proportion of the trailer mass m acting on the dolly is assumed to be a point mass located at the A-dolly 4 axle. See Figure 4-1. Only the tire lateral force is considered to be acting on the system; the lateral pinde hook and fifth wheel forces are neglected. A simple moment balance about the pinde hook yields Equation 4-35 in terms o f the angle between the velocity vector and the dolly centerline #and its derivatives. Q j ^effective Q Um ^ ^-effective Q_ mL ^effective ' (4"35) mL Where: Cg^ is the effective cornering coefficient, U is the magnitude of the forward velocity, and L is the drawbar length. Comparison of Equation 4-35 with the "standard" form for a second-order formulation allows determination of the undamped natural frequency co and the damping ratio £ n _ J effective ~i~mir n ^effective ''effective 2Umco n (4-37) Chapter 4. 120 Controt System Design Equation 4-37 can be re-arranged to yield the critical velocity, assuming the value of the damping coefficient is 1.0 (below the critical velocity rearward amplification is not significant). ^crit fc—T i ~ii -' 21 V ^•effective m (- ) 4 38 1 Using the results from Equations 4-35 and 4-38 and assuming a feedback control law of the form y = -NO, the control formulation may be determined. (4-39) r= u \L9 Although the controller o f Equation 4-39 exhibited good tracking performance in the screening test, the lateral acceleration of the second semitrailer was found to be 0.141 g. The present control law has the advantage of simplicity and ease of implementation in a physical system. Unfortunately, the estimation of the effective tire cornering coefficient is difficult to perform accurately with an over-simplified plant model and the improvement in dynamic performance is less pronounced than the other options tested. 4.5.6 Critical Damping of Each Mode Section 4.5.5 introduced the idea of critical damping as a method by which the system eigenvalues may be set. Here, we extend this problem to the linear formulation presented at the onset of this section. Figure 4-5 clearly depicts the lightiy-damped nature of the system eigenvalues at moderate to high velocities. The goal of the control law is to increase the effective damping coefficient of each o f the two modes while mamtaining the same value o f the undamped natural frequency. Once the eigenvalues have been selected, Ackermann's formula is utilized to determine the appropriate state variable feedback gains. Chapter 4. Control System Design 121 Each eigenvalue o f the plant matrix may be expressed in the form of Equation 4-40. l=<T,±j<o -(4-40) d<l With reference to Figure 4-5, one can deduce that the undamped natural frequency co is equal n to the hypotenuse o f a right triangle with cr on the horizontal axis and 0)^ on the vertical axis. TZ^T (4-41) co„, - Jcr, + co j d Similarly, by comparing Equations 4-23and 4-40, the effective damping coefficient may be deduced. — T ~ 2 . (4-42) 2 The plant eigenvalues from their initial uncontrolled locations (damping values £ - 0.47 and = 0.54) to the values corresponding to critical damping of each mode. The screening simulation was then conducted; the peak value of lateral acceleration was noted to be 0.127 g. Due to the manageable size o f the plant (4x4), symbolic expressions for the uncontrolled eigenvalues were derived using the Symbolic Toolbox in MatLab 5.3. The equations deterrnining the critical damping o f each mode are flexible, with the same calculations required for any value o f the system parameters. This fact, along with a symbolic realization o f Ackermann's formula makes the present control law easy to implement in a computing system without the requirement o f matrix capabilities. The sum o f the former, in addition to the superior attenuation o f the second trailer's lateral acceleration, made the critical damping o f each mode the control strategy of choice. 122 Chapter 4. Control System Design 4.6 Discussion A linear model was developed for use in the design and execution of a control strategy to reduce the lateral acceleration levels of the second semitrailer in an A-train vehicle. The model assumed the A-dolly and second semitrailer to have negligible width effects (bicycle model), have a single "effective" tire located at each axle, and move at a constant velocity. Once derived, the linear equations of motion were compared to their counterparts, developed in Section 3. Good agreement was seen between the two models, although peak values of the measured states exhibited some divergence between the models as the linear assumptions broke down. The eigenvalues were determined and investigated for the linear control model. It was determined that the poor high-speed dynamic behavior could be attributed, in part, to light damping of the two system modes. To increase the damping at the system modes, the concept of state variable feedback was introduced. The model was proved both controllable and observable, thus allowing for arbitrary eigenvalue placement using Ackermann's Formula. The question of where to propedy locate the closed-loop eigenvalues was investigated through five candidate control laws: (a) Linear Quadratic Regulator (b) Bessel function prototype (c) ITAE function prototype (d) Critical damping of the dolly (e) Critical damping of each system mode 1 The results of the screening analysis are shown in Figure 4-7 below. Chapter 4. Control System Design 0.18 -I Uncontrolled Figure 4-7 — Effectiveness LQR Bessel ITAE Dolly Critical Damp. Mode Critical Damp. o f Candidate C o n t r o l Laws F o r reasons o f computational and ease o f implementation, as well as superior dynamic performance, critical damping o f each system mode was chosen as the best control realization. Chapter 5 State And Parameter Estimation In Section 4, it was assumed that measurements of each value in the state vector was possible, and available for use in feedback control. Unfortunately, determination o f the second semitrailer angular velocity co and the articulation angle between the A-dolly and the second 4 semitrailer F is not easily achieved using sensors mounted to the dolly only. In addition, due 3 to variability in tire properties with tire type, manufacturer, loading, and environmental conditions, the researcher has little a priori knowledge of the effective tire cornering coefficient r The former properties must thus be estimated using available output data measurements and the system control model. A n estimation algorithm known as the Extended Kalman Filter (EKF) will be employed in this regard to simultaneously estimate the unknown states and parameters. Methodology for determining properties o f the second semitrailer such as the rotational moment of inertia I and the location of the center of gravity are also investigated. 4 5.1 State Augmentation and the EKF The basic linear Kalman filter algorithm addresses the problem of estimating the state of a given process in the presence o f process and measurement noise. The Kalman filter utilizes a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of measurements (often corrupted by noise). As such, the equations for 124 Chapter 5. State and Parameter Estimation 125 the Kalman filter fall into two groups: time update equations and measurement update equations. The filter alternately predicts the value of the state, then makes corrections as more recent data is available. Computation is recursive; that is, current results depend on the preceding data and estimate of the state. The time update equations are responsible for projecting forward the current state and error covariance estimates to obtain the a priori estimates for the next time step. These equations may be thought o f as the predictive component of the filter. The measurement update equations are responsible for the feedback (i.e. for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate); such results constitute the corrector equations. The preceding argument is true for linear systems, such as the model derived in Section 4, and would be valid i f all terms in the plant matrix Ap were known with certainty. However, this is not the case; we wish to estimate the effective cornering coefficient Cg^ at the same time as the states co and F . This is achieved most easily by augmenting the state vector with } 3 0) 3 C^^ (5-1) CO, x= s C.effective By augmenting the state, the originally-linear plant matrix Ap becomes a nonlinear function of the system states f(x). As the plant is now non-linear, modifications to the basic Kalman filter scheme are required to compute a result The E K F approach is to apply standard Kalman filter methodology to nonlinear systems through linearization of the system about the current mean and covariance of the state estimate [44]. Mathematically, this implies a linear Taylor series approximation of the system Chapter 5. State and Parameter Estimation 126 equations of motion at the previous state estimate, and Taylor linearization o f the corrective equations at the corresponding predicted position. These linearizations are updated at each time step in the analysis. This method gives a simple and efficient algorithm by which to handle the given nonlinear model. 5.2 Discretization of the Equations of Motion The system equations derived in Section 4 were defined for continuous time. However, the E K F algorithm requires a formulation of the system with respect to discrete time (sampled) measurements. Each variable co , C0 , r e, y, and r is replaced by co (k), co (k), r/k), s(k), y(k), 3 4 p 3 4 and r(k), their equivalent values at the time step k, respectively. Assuming a sufficiendy small sample time h > 0, Equation 5-2 may be used to replace the derivative of the state variables [45]. • a>3,* i-*>3,* CO, = ( + 5 - 2 ) h 3 The results of Equation 5-2 are repeated for the derivatives of co^ 7^, and s\ the accuracy of the derivative approximation increases as the sample time decreases. The discretized values of the state and their derivatives are then combined with the plant matrix A p and the control matrices i? ,and B^to yield the required model. 4/ «3,* l + AdiPeffective) r ,*+i 3 (5-3) A * + [ \,p( effective) B C 2,p\ B 127 Chapter 5. State and Parameter Estimation 1 0 0 0 0 A*" 0 0 1 A/ =c (5-4) Note that the output equation 5-4 does not change, at the measurement matrix C does not depend on the unknown cornering coefficient C^^. Rearrangement of terms due to Equation 5-2 changes the elements o f the plant matrix A^; this difference is noted by a new symbol for the plant \(C^J. T o complete the formulation, an equation relating the future (k+1) and present (k) values of the effective cornering coefficient must be determined. Assuming that C^, holds a constant value, an expression can be created assuming a random walk (i.e. the output value varies only due to the observed noise sequence). C = C ^effective,k+\ ^effective.k (5-5) +E T Tk Where: E, is any zero-mean Gaussian white noise sequence. k Note that E, must have an assigned positive definite variance Varf^J = S, where S is constant. k Reasons for this stipulation are supplied in the following section on E K F convergence. It should be noted that for accurate estimation, all parameters and states should be of the same order of magnitude. In the particular case of this paper, C^, is of order 10 while 0J and 7 ] s 4 are of order Iff or Iff , depending on the maneuver. T o counteract this discrepancy, all terms 1 involving 2 are multiplied by a scaling factor of Iff ; estimated values of C^, 7 ofmeformC^=(C^/0 . 7 will thus be Chapter 5. State and Parameter Estimation 128 5.3 Formulation of the Extended Kalman Filter As stated, the E K F estimates the required system states by first predicting the result using the data from the previous iteration, then correcting that estimate as required using data from the current iteration. A complete deviation of the E K F algorithm may be found in Chui etal [45]. There exist two matrices which allow modeling of system noise. The first matrix Q 4x4 state error covariance while the second matrix R M is the is the measurement noise covariance. Since the noise generated by individual sensors are generally independent of each other, the form of Q and R is usually diagonal, with the expected noise covariance of each state (in the case o f Q) or measured output (in the case o f R) located at their respective locations on the matrix diagonals. Tuning of the Q and R matrices is commonly required to obtain the best system performance. The algorithm is initialized in Equations 5-6 and 5-7. ~E(x ) r 0 r p Where: Xg — \(O )0 C0 r 4fl = Var(x ) 0 0 S 0 (5-6) (5-7) £ ] , the initial values o f the system states and Var(x^ is the initial T )0 0 variance of the state estimates. In most instances, the diagonal values of Varfx^) are set to large positive values, indicating large uncertainties in the initial values in matrix of the estimated values. The matrix P is known as the error covariance 129 Chapter 5. State and Parameter Estimation The estimation process starts with the predictor formulation. Equation 5-8 represents the projection o f the state estimate ahead using prior data. Similarly, Equation 5-9 projects the error covariance ahead using the data from the previous sample interval. ^d,k-l (^eff.k-l Ad,k-l(C ff,k-l) P eff,k-l + dCeff e k\k-l 0 X c eff,k\k-l 1 (5-8) ) k-l B \ (5-9) Yk- 2p 'k-l -p-i 1 d Ad,k-\(Peff,k-\) 0 4i.t-i(e«.t-i)*k-i+k,(Q,*-i) dC.eff 2, \ B P 'A-l. + i ~TQT 0" 0 S T Note that the matrix T 4x4 in Equation 5-9 maps the process noise to the corresponding state. The estimates arising from Equations 5-8 and 5-9 are then corrected using the current measurement data. Equation 5-10 computed the Kalman gain matrix, Equation 5-11 updates the state and parameter estimates using the current measurements, and Equation 5-12 updates the error covariance matrix. ~ Pk\k-l " k (5-10) [c Qltc A (5-11) ~ X r yeff,k_ X peff,k\k-l k\k - F + k\k-l = 1/5*5 G [y -Cx _] k _ G [C k k m Oj (5-12) Recursive application of Equations 5-8 to 5-12 yields the desired state and parameter estimation. Equation 5-10 is known as the Kalman Gain of the estimator algorithm. 130 Chapter 5. State and Parameter Estimation 5.4 Estimating Performance of the EKF To verify the predictive performance of the E K F , the full nonlinear model of Section 3 was simulated in a standard lane change maneuver and the results compared. In this test, the actual magnitude of sensor noise was unknown and the results were therefore tested in the no-noise case. T o this end, the vectors Q, R, and Twere set to diagfO.001 0.001 0.001 0.001J, diag[0.001 0.001], and diag[1 111], respectively. The variance of the unknown cornering coefficient was set to S — 10 , corresponding to a variance on the estimated value o f ± 100 N/rad. s results are shown in Figures 5-1 and 5-2. Time (sec) Figure 5-1 - Comparison of Estimated and Actual Parameters: Trailer Angular Velocity The Chapter 5. State and Parameter Estimation 131 Figure 5-2 - Comparison of Estimated and Actual Parameters: Trailer/A-dolly Articulation Angle The estimated values of both co and T] show an excellent match to the actual states, 4 particularly at the peak values of the parameters where linear assumptions began to break down. This is due in part to the negative feedback between the measured results from the nonlinear model and the linear model states forcing the behavior of the estimated parameters to better match the actual values. The state estimates also show no ill effect o f mismatch or abnormal behavior as the identification is started. The results o f Figures 5-1 and 5-2 represent the best possible conditions for the estimator to function within. However, results gained once the system noise can be accurately characterized should also bear like behavior. 132 Chapter 5. State and Parameter Estimation Figure 5-3 depicts the performance of the estimator in obtaining a value for C, Figure 5-3 - Performance of the Tire Coefficient Estimator The E K F estimate is shown to react quickly in light of the large initial diagonal values o f the covariance matrix P. The results settle at an equilibrium near the value o f 7.2(10 ) N / r a d ; the s estimated value of Cg fluctuates slighdy about this point. Small values o f the Kalman Gain matrix G at the end of the simulation indicate the settling of the algorithm. Chapter 5. State and Parameter 'Estimation 5.4 133 Convergence of the EKF Algorithm Although the E K F algorithm facilitates a relatively simple and straightforward method to estimate system states, several authors in the literature have expressed reservations about its use. The fundamental flaw of the E K F lies in the fact that the distributions o f the various random variables assumed in deriving the standard Kalman filter are not preserved exactiy [55]. Specifically, the assumed normal distributions only approximate the optimality of Bayes' rule by linearization. The key arguments arising from this fact are given below with discussion of the current problem in response. (a) The E K F will not converge reliably if the initial estimate of the unknown parameter is poor [46] [48] or if the sign is incorrect [47]. Reliable initial conditions are necessary to ensure accurate results [48]. In the linear full trailer model, both the general order of magnitude (C^is on the order of 10 N / r a d for the tested heavy truck tires) and the sign (the effective cornering coefficient is always positive) are known. The estimator is started during steady-state operation, thus ensuring that all states are at a known null " value. 5 (b) The convergence of the E K F is not guaranteed i f the magnitude of the disturbance inputs on the system are large enough such that the linearization about the current state estimate is inadequate to propedy describe the system [46]. The control model matches the behavior of the nonlinear vehicle very well in the operational domain of interest. Disturbances large enough to disrupt the general accuracy of this assumption would likely result in a vehicle accident event regardless of the control applied. (c) The plant must be observable [47]. This fact was shown in Equation 4-21. (d) The performance of some multi D O F systems may deteriorate under low sampling rates [48]. However, setting a reasonable sampling rate for the resulting physical system will alleviate this concern. (e) The estimated results may be biased according to the method by which system noise is modeled [49]; this bias is not caused by the structure of the E K F , but rather by assumptions made concerning noise corruption in the model. The simplicity of the full trailer model, and the well-known characteristics of the required sensors (accelerometers and rate gyroscopes) bode well in this regard. The characteristics of the system can thus be modeled with a fair degree of accuracy. 134 Chapter 5. State and Parameter Estimation (f) A lack o f coupling between the Kalman gain G and the estimated parameter C$ leads to divergence in systems where only coefficients of the plant matrix A/C^ are unknown [49]. In the present system, this concern is moot as both states and the cornering coefficient are to be estimated. (g) A non-zero value for the parameter noise coefficient S must be assumed [45]. If S is chosen to be zero, the E K F algorithm makes the estimate of Cg independent o f the input data, thus precluding identification. The current model assumes S = 1(10 ). 5 By adhering to the concerns in (a)-(g) above, the E K F can be expected to operate correcdy with the intended convergence properties on the full trailer model. 5.5 Evaluation of Miscellaneous Parameters Besides the effective tire cornering coefficient, there are additional parameters of the second semitrailer which must be determined in order to property apply control. Although the wheelbase L of the semitrailer (the distance between the kingpin and the rear axle) and both axle loadings are generally known at the time of creation of the A-train, the center of gravity position, trailer mass, and trailer rotational moment of inertia must be determined. We will assume that the mass m of the A-dolly and the corresponding rotational moment o f } inertia I } are known, as these parameters are constant with all loading and trailer configurations. Additionally, it is assumed that the fifth wheel of the A-dolly is coincident with its center of gravity; that is, the dolly is balanced. The location of the center of gravity and semitrailer mass is easily found using a twodimensional static model with respect to the pitch plane. Equations 5-13 to 5-15 give the results in terms of the variables defined in Section 3. 1 ™4 =-l( g r F 9 : s + loJ F + ( 1zs F + F Szs) ~ ™gJ 3 l ( 5 4 3 ) Chapter 5. State and Parameter Estimation ( 9zs F i lzs F + F 135 (5-14) lOzs)L + F $zs) + ( 9zs + F F lOzs) ~ "hS (5-15) b= L— b g n The rotational moment o f inertia I is approximated quite accurately for van-type trailers by 4 assuming the trailer to be a rectangular prism. The result is given in Equation 5-16 [35]. (5-16) 7 =^-[L 4<] 2 4 5.6 + Discussion This section has presented the methodology for estimating the unmeasurable states C0 and r 4 } as well as the effective cornering coefficient C^ T o do so, the Extended Kalman Filter algorithm was introduced and the original four states o f the linear model were augmented with C^as an additional "state". The value o f C^was assumed to follow a random walk behavior in this regard. Comparison o f the estimated and actual state values revealed an excellent match throughout the test maneuver. N/rad. The cornering coefficient achieved a constant identified values at 7.2(1 (f) Convergence properties o f the E K F were verified against results from literature for the present A-dolly — semitrailer simulation and found to be acceptable. Formulas for determining the remainder o f the unknown physical parameters of the second semitrailer were provided using static loading and trailer wheelbase data. As a result, all physical parameters and states of the full trailer control model can be defined using sensor data located on the A dolly uniquely and static data available at the time of vehicle assembly. Chapter 6 Results The present chapter draws together the results derived in Sections 3 - 5 and investigates the resulting improvement in the dynamic performance of the reference A-train vehicle. The nonlinear two dimensional simulation serves as a virtual vehicle for the evaluation of the control law; control signals are extracted from the simulation and used to drive both the controller and the state/parameter E K F to steer the "smart" A-dolly. The controlled A-train was then evaluated for both the frequency response and S A E J2179 test protocols, and the results noted. 6.1 Tire Property Identification As mentioned in Section 3, accurate modeling of the vehicle's tires are paramount to achieving accurate control of the system. The nonlinear simulation was run with both sinusoidal and lane-change steering inputs for each of the four tires tested. A l l tests were conducted using identical maneuvers and assuming each tire on the vehicle was of the same type. See Table 61. 136 Table 6-1 — Identified Tire Parameters Tire Name Effective Cornering Coefficient C (N/rad) t0iahe Michelin 10.00x20 Firestone 10.00x22F Firestone 10.00x22RTB Freuhauf 10.00x20 7.2(10*) 1.2{\tf) (>.5{\tf) 63(10") The estimator was able to accurately distinguish between the behavior of the specific tires. In general, the E K F converged quickly on the final parameter value. N o significant differences in the identified value were noted according to the magnitude, frequency, or type of maneuver conducted. Additionally, in tests in which the tire effective cornering coefficient was reduced by 50 percent, the estimator again converged to the correct value. It should be noted that the parameters defined in Table 6-1 are accurate only within operating conditions in which the linear control model is valid; beyond this range, a single coefficient is insufficient to describe the tire parameters as nonlinearities increase in importance. 6.2 Dynamic Performance Improvement The eigenvalues of the control system are located to introduce criticality in damping for each mode. However, additional investigation revealed that locating all the closed loop eigenvalues at the critically-damped value for the "faster" mode produced an improvement in the attenuation of rearward amplification. This position represented good system response while maintaining the required steering inputs within a feasible range. To investigate the function of the controller, simulations were performed under both S A E J2179 and sinusoidal inputs of differing frequencies. The S A E J2179 tests provided both an indication of the improvement in rearward amplification performance, and well as a mechanism for the evaluation of the second trailer's transient offtracking behavior. The Chapter 6. Results 138 frequency response tests extended the rearward amplification results over the operational domain of the vehicle. 6.2.1 Testing Under SAE J2179 The required steering input is noted in Figure 6-1. Figure 6-1 — Required Steering Input The peak magnitude of the A-dolly steer angle is seen to be approximately 2.5 degrees during the lane change and generally follows the lateral acceleration of the second trailer. In this test, a controller steer input of 4 deg/sec is required. As expected, the dolly steer angle nulls out as the A-train returns to a straight line trajectory. Figure 6-2 compares the steering input to the tractor to that required by the A-dolly. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 S Time (sec) Figure 6-2 — Comparison Between Tractor and A-Dolly Steering Angles The dolly steer angle is seen to be out of phase with respect to the tractor steering angle. Note also that the peak magnitude of the dolly steer is approximately 250 percent o f that o f the tractor. As the radius of curvature of the intended tractor trajectory decreases, the required control effort is seen to increase (i.e. the controller effort is seen to be largest when the A dolly-semitrailer combination is at the portion of the tractor trajectory at which the largest rearward amplification effect occurs). The overall lateral acceleration performance of each vehicle in the reference A-train vehicle is given in Figure 6-3. Figure 6-4 compares the controlled and uncontrolled lateral accelerations of the second semitrailer to the lateral acceleration of the tractor. Chapter 6. Results Figure 6-3 - A-Train Lateral Acceleration Performance Under Control 140 Chapter 6. Results 141 0.2 0.15 j -0.15- V -0.2 -I r 0 1 r— 2 V / r^— 3 , 4 • , , 5 6 Time (sec) Figure 6-4 — Comparison Between Controlled and Uncontrolled Second Trailer Lateral Acceleration Inspection of the former figures reveals several trends. First, the peak value o f the second trailer lateral acceleration is reduced from 0.190 ^ to 0.157 g, the peak value occurs slightiy later than the uncontrolled case. The reduction in lateral acceleration translates to an improvement in rearward amplification performance from 1.58 in the uncontrolled case to 1.31 under control. Second, the magnitude of the acceleration overshoot after the maneuver is reduced. The lateral acceleration behavior of the second trailer more closely approximates that o f the lead semitrailer. Last, one notes that the control is acting smoothly on the vehicle, yielding a trajectory which is devoid of jerk or other acceleration anomalies. Similarly, Figures 6-5 and 6-6 display the displacement of each component in the vehicle, and a comparison between the trajectory following ability of the controlled and uncontrolled systems, respectively. .n 11_ i i i -20 0 20 40 l 1 60 80 X Position (m) 1 100 Figure 6-5 - A-train Displacement Performance Under Control 1 120 1 140 1 160 143 Chapter 6. Results -0.1 H 0 — , 1 0 , 20 1 30 1 40 1 50 1 60 1 —i 70 80 X-Posltion (m) < 90 100 110 120 130 140 Figure 6-6 - Comparison Between Controlled and Uncontrolled Second Trailer Trajectory Following Immediately evident is the decrease in transient offtracking for the controlled vehicle. With respect to Figure 6-5, one notes that the second trailer's trajectory is approaching that o f the lead semitrailer. Since the behavior of the lead semitrailer is not controlled (and is effectively isolated from the effects o f the A-dolly and second semitrailer due to the pinde hook connection), the controlled trajectory following is approaching the best results possible given the assumed control implementation. The path following ability of the second trailer is improved throughout the maneuver. The addition of control reduces the peak offtracking amount from 0.146 m to 0.073 m. The controlled A-dolly semitrailer unit also offtracks the lead semitrailer by a peak value o f only 0.02 m, compared to 0.093 m without control. Settling time is also greatiy reduced through the addition of control. Chapter 6. Results 6.2.2 144 Testing Using the Frequency Response Method As in Section 3, the frequency response behavior of the reference A-train is evaluated using sinusoidal inputs over the frequencies known to be within the operational domain of the vehicle. The most important results deal with inputs ranging in frequency from approximately 1.5 to n Hz. These frequencies are most representative of steering inputs during avoidance situations. Input frequencies above K H z are ergonomically difficult for the driver to obtain in practice, since the large gain between the actual wheel steer angle and the steering wheel angle in the tractor's cab (typical gains are 30 — 60 degrees steering wheel motion to 1 degree actual wheel steer angle [29]) require rapid, large motion inputs in this case. The uncontrolled frequency response plots are reproduced from Section 3. O n each plot, control was applied and the same input frequencies and magnitudes used to create the "controlled" plots. As before, the results are computed for nominal velocities (24.6 m/s) as well as ± 25% velocities (30.8 m/s and 18.5 m/s, respectively). Chapter 6. Results ••P"™" 11111111 145 1 mmmmmmm i mi mmmsgem Figure 6-7 - Controlled Versus Uncontrolled Response: V = 18J mls x3 146 Chapter 6. Results Figure 6-8 - Controlled Versus Uncontrolled Response: V - 24.6 mls x3 Chapter 6. Results 147 Figure 6-9 — Controlled Versus Uncontrolled Response: V = 30.8 mls x3 A l l three velocities exhibit an improvement in the observed rearward amplification levels over all frequencies. In the 24.6 m/s and 30.8 m/s tests, the improvement in dynamic performance increases as the input frequency increases. For the 18.5 m/s test however, the gap narrows somewhat at higher frequencies. This is due, in part, to the tractor steering input being too rapid to permit a full vehicle response. The peak reductions in rearward amplification and the frequencies they occur at are given in Table 6-2. Chapter 6. Results 148 Table 6-2 - Peak Reductions in Rearward AmpliflcatioQ Velocity (m/s) Percent of Uncontrolled RA Peak Reduction Frequency (Hz) 18.5 24.6 30.8 23 25 48 3 4 4 Note that the effectiveness of the controller diminishes somewhat as the vehicle velocity increases. This may be due to the fact that only one "end" of the trailer is actively being controlled; addition o f control to the rear axle of the second semitrailer could likely improve the observed results (but would violate the design constraint o f localizing all control equipment on the dolly only). The addition of control pushes the peak value of rearward amplification to slighdy lower frequencies. Comparison of Figures 6-8 and 6-9 indicates the stabilizing action of the A-dolly controller makes the controlled second semitrailer moving at 30.8 m/s behave much like the uncontrolled semitrailer moving at 24.6 m/s. In general, for a given threshold lateral acceleration of the second trailer (based on rollover stability), the controlled system allows for improved safety of operation at higher velocities. 6.3 Comparison of Proposed Controller With UMTRI Results Figure 6-10 contrasts both the controlled and the uncontrolled "shortened" A-train vehicle to the test results of Winkler et al [10]. The two mechanical units proposed by Winkler et al are both C-dollies; that is, they connect without articulation using two drawbars to the lead semitrailer. The first is a self-steer C-dolly, deriving its wheel steer angle from the caster angle of the wheel kingpins (a similar effect to a typical car's wheels straightening out automatically after a tight comer). The second design is a controlled-steer C-dolly, developed at U M T R I . 8 Values in the table represent the percentage the controlled rearward amplification is of the uncontrolled value. Smaller values thus indicate best performance. Note that the percentages are normalized relative to "zero" amplification, while occurs when the value of rearward amplification is 1.00. For example, if the uncontrolled and controlled rearward amplifications were 1.58 and 1.31, respectively, the listed result would be (1.31-1)7(1.58-1) = 53% This device is steered via a mechanical gain based on the articulation angle between the units. The objective o f each o f the U M T R I dollies is to minimize some o f the C-dolly faults while maintaining their high speed dynamic performance. 2.5 2.25 - —— —— x 0 Controlled A-Dolly Uncontrolled A-Dolly Self-Steer C-Dolly (UMTRI Test) Controlled-Steer C-Dolly (UMTRI Test) NJ Ul Rearward Amplification Uncontrolled O X A * Ul x 0 X x 0 yr Controlled X 0 > 1.25 1 \ I 0.5 1 1 1 1.5 1 i 2 2.5 Input Frequency (Hz) i i 3 3.5 4V Figure 6-10 — Comparative Performance: Controlled and Uncontrolled A-train, Self-steer Cdolly, and Controlled Steer C-Dolly The results o f Figure 6-10 are encouraging. One observes that the controlled A-dolly is as effective, or superior to both U M T R I designs over the entire frequency range. In particular, the performance in the 1.5 to n H z input range demonstrates a marked improvement in dynamic behavior. Both U M T R I C-dollies perform roughly the same. Note also that the data to the left (Le. at lower frequencies) o f the controlled A-dolly plot appear to be outliers; the same behavior was determined with respect to the Phase I V model by Winkler et al [10]. 150 Chapter 6. Results None-the-less, it appears as though the "smart" A-dolly concept is a viable alternative (at least in the vehicle dynamics sense) to the best conventional converter dollies. 6.4 Steady-State Offtracking Performance Figure 6-11 below denotes the offtracking behavior of the controlled A-dolly vehicle during steady-state cornering. Figure 6-11 - Steady-state Cornering Under Control Comparison with the results in Section 3 reveal that the steady-state offtracking has been reduced from 0.15 m to 0.09 m with the application of control. The controlled A-dolly - Chapter 6. Results 151 semitrailer unit demonstrates good trajectory following. Here again, the overall reduction of offtracking is limited by the dynamic behavior of the lead semitrailer. 6.5 Parameter Sensitivity To test the effect o f parameter misidentification, additional runs of the frequency response simulation were conducted. In these tests, the maneuver and vehicle properties o f the nonlinear model were held constant. linear model was varied between However, the effective cornering coefficient for the 75 and 125 percent of the nominal value (C^. = 7.2(10)). 5 The results are given in Figure 6-12. 0 0.5 1 1.5 2 2.5 Input Frequency (Hz) 3 Figure 6-12 - Effect of Parameter Error in Rearward Amplification Attenuation 3.5 4 Chapter 6. Results 152 The controlled A-dolly remains stable under all three tests. If the cornering coefficient is underestimated in the control model, the resulting dynamic behavior is slightiy better than a controller created with the "exact" tire properties. In effect, the system poles have been placed between the locations mandated by the critically-damped natural frequency o f the primary system modes. The penalty for the additional decrease in rearward amplification over the nominal condition controller is additional control effort; the controller permits larger steer angles for the A-dolly. Similady, i f the cornering coefficient is overestimated by the controller, the rearward amplification effects are worse than the nominal controller, as the magnitude o f the steer angle becomes less than the nominal case. Even so, the resulting controlled vehicle exhibits dynamic behavior which remains superior to the reference A-train. The former shows that while exact determination o f the tire cornering properties is not an absolute necessity in the sense o f stability, it does make for a better-performing controller. By placing the eigenvalues of both system modes on the critically-damped location o f the "faster" mode, the resulting controller is robust and makes leeway for error in parameter estimation. 6.6 A-Dolly Sensor and Actuator Requirements The required sensor inputs for the proposed system may be classified into three categories: (a) Direcdy Measured (b) Processed Measurements (c) Estimated Measurements The quantities which may be directly measured are the dolly forward velocity V , the angular x3 velocity and acceleration of the A-dolly co and dldt(co ), the steer angle of the dolly wheel y, 3 3 and the lateral acceleration of the center of gravity a . } It is assumed that the user enters the 153 Chapter 6. Results wheelbase and track of the second trailer, the axle loads, and the moment of inertia I . The 3 mass m and the vehicle geometry of the A-dolly are assumed constant and known. } Using on-line numeric integration, the lateral velocity V of the center of gravity may be } found. An expression for the lateral velocity and acceleration of the pintie hook is found using rigid body motion theory. (6-1) (6-2) Using the equations from Section 5, expressions for e and r may be determined, assuming small angles and the sign convention of Section 5. (6-3) (6-4) "-1 r = co? ayp v x3 The remaining unknown quantities are estimated using the techniques of Section 5. The sensors required to achieve the above raw inputs are listed in Table 6-3 along with their preliminary requirements. Table 6-3 — Sensor and Actuator Requirements Measurement Range Sensor Type 0-35 m/s Hall Effect Velocity Sensor Accelerometer 0-0.5,2 Rate gyro 0 ± 8 deg/s Steer Angle Potentiometer 0 ± 5 deg Steering Actuator 0 ± 5 deg (displacement) 0± 10 deg/s (rate) Resolution ± 0.1 m/s ± 0.01 ^ ±0.05 deg/s ±0.05 deg ±0.05 deg ±0.05 deg/s Chapter 6. Results 154 Note that the operational range of the steering actuator was chosen larger than required by the controller as a measure to prevent saturation of the actuator. 6.7 Discussion This section presented the dynamic performance of the controlled A-train vehicle. The behavior was investigated using both S A E J2179 and the frequency response testing protocols. A marked improvement in both the level of rearward amplification and the trajectoryfollowing ability of the second semitrailer was noted. The identified effective cornering coefficients for the test tires were also presented. The controlled A-train was compared to results for C-dollies published by Winkler et al [10]. It was found that the "smart" A-dolly performed as well or superior to the listed C-dollies, particulady in the input frequency range of 1.5 to nYLz. The system was shown to perform in a stable manner under variations of the estimated tire cornering properties by ± 25 percent. To implement the system, the required sensors, actuators, and conversion equations were given, along with approximate measurement ranges and resolutions. Chapter 7 Conclusions and Recommendations The occurrence o f dynamic instability in current A-train vehicles has been presented. The combination of a nonlinear dynamic simulation, simplified linear control model, eigenvalue placement state variable feedback controller, and E K F parameter and state estimation scheme have been combined to yield the final controlled vehicle. To best compare the dynamic performance of a typical A-train vehicle incorporating control and one without, a complex two-dimensional nonlinear yaw plane model was derived. The Newtonian equations of motion were determined with respect to local body-fixed reference frames. Effects o f roll and pitch were solved for the semi-static sense using the lateral and forward accelerations on the center of gravity of each unit in the vehicle; the results were applied to the simulation through dynamic load transfer to the tires. The tires themselves were modeled using empirical data for cornering force, aligning moment, and the moment couple effect with dual tires. A driver model was incorporated to move the simulated vehicle along an intended trajectory. Simulation using MatLab revealed a favorable comparison between the model results and data gathered from full-scale testing. A linear controller was applied to the system using a dedicated, simplified state-space formulation of the A-dolly and second semitrailer. Key assumptions included constant magnitude of the input velocity vector, linear tire modeling as a function of a single coefficient, 155 Chapter 7. Conclusions and Recommendations _56 and a reduction o f width to yield the "bicycle model". Output from the control model was proven to match the nonlinear simulation results well for high-speed, constant velocity, low amplitude maneuvers, but diverged in the case o f significant acceleration and large steering amplitudes as the effects o f nonlinearities became more prominent Investigation o f the model eigenvalues revealed two lighdy-damped, complex conjugate modes. A screening test of candidate control laws proved that a controller which moved the system eigenvalues to the critically-damped location o f the faster mode produced both good attenuation o f rearward amplification and offtracking, and feasible control demands. By constraining the control system to be localized on the A-dolly only, two states o f the control system {co and T]) were not able to be determined direcdy. In addition, variability in 4 performance properties. and environmental conditions necessitated estimation o f the tire cornering A n Extended Kalman Filter was employed in this regard to simultaneously and recursively find the required states and tire properties. Application o f the eigenvalue placement controller to the nonlinear simulation was met with favorable results. In all tests, the levels of rearward amplification and dynamic offtracking were reduced by the controlled A-dolly. Robustness was also shown with respect to the estimated cornering properties and the resulting control. The effectiveness o f the controller was shown to be greatest at or below 24.6 m/s, with the performance o f the controller diminishing for velocities beyond this mark. Comparison with U M T R I test results [10] revealed the performance o f the present system is as effective or better than the best Cdolly/B-train vehicles in common use today. 7.1 Contributions of the Present Work The present work has achieved the following: Chapter 7. Conclusions and Recommendations 157 (a) Vehicle Modeling (b) Simplified Control Model - A unique bicycle full trailer model was proposed and verified against the nonlinear simulation to be accurate in the case o f high-speed, low amplitude maneuvers. The variables and reference frames utilized in the formulation were chosen to yield a model measurable from the local A-dolly coordinate frame. (c) State Space Feedback Controller - A simple self-tuning controller based on critical damping o f the system modes and pole placement was given, and was shown to be effective in attenuating both rearward amplification and offtracking behavior. The controller was shown to be robust to errors in both state and parameter estimation, but decreased in effectiveness at very high vehicle velocities. (d) System Integration — The required components in the steering system are feasible to implement; the required sensors and actuator can be localized on the A-dolly only. 7.2 The standard A-train vehicle has been accurately modeled using a two-dimensional nonlinear yaw plane formulation. This model yields good results in the domain o f operation o f interest for this model. The formulation method using MatLAB in conjunction with semi-static load transfer was novel. — Recommendations for Future Work Constraints on time and resources precluded exploration o f all possible research inherent in the study and design o f the proposed "smart" A-dolly system. The following is a partial list of research areas which may be pursued further. (a) Signal filtering and Noise Modeling Sensor data arising from the actual system will undoubtedly be corrupted to some extent by noise. Appropriate filters must be constructed to attenuate this noise while preserving the signal o f interest for control. A filter must also be designed which is capable o f numeric integration o f the lateral acceleration signal o f the A-dolly to yield the corresponding lateral velocity (a successful method for this purpose has been proposed by Vagstedt in [50] and [51]). — In addition, the noise and interaction of the noise between sensors in the system must be determined with a moderate degree of accuracy for optimal function of the E K F . (b) Modification of the EKF to Account for Sudden Parameter Changes - In the current implementation, the E K F performs well to identify the tire cornering coefficient after startup. However, since the error covariance matrix P and the Kalman Gain matrix G both have small values after the E K F has run for a time, the algorithm is slow to react to sudden changes in tire properties, caused, for example, by changing pavement friction characteristics. See Figure 7-1 below. Chapter 7. Conclusions and Recommendations 158 0.08 O 0.02- ™ °0 5 Time (sec) 10 15 Figure 7-1 — Slow Subsequent Convergence o f the Identified Tire Property Using the E K F Investigation o f controls literature reveals several methods to improve the E K F performance [49] [52] [53]. A simple, yet promising technique has been proposed by Gustafsson [54]. This algorithm looks for trends (either upwards or downwards) in the identified value of the parameter. When the algorithm determines the value of the parameter is indeed changing, the element values of the error covariance matrix are gready increased, causing the E K F to react quickly in response. A similar technique based on a "forgetting factor" [40] (i.e., dividing the error covariance matrix by a number less than 1.0) may also be effective in preventing drift of the estimated parameter and ensuring rapid reaction to changes in its value. (c) Control Implementation Issues - Engineering problems in sensor mounting, calibration, power and computing sources, and user interface must be addressed. Some instrumentation issues have been investigated in Vagstedt in [50] [51]. (d) Suitability of System Inputs for Control - During operation under highway conditions, a typical A-train vehicle may travel for long periods o f time at steady-state. Such conditions are not conducive for identification. Additional research into Chapter 7. Conclusions and Recommendations 159 determining i f random road noise during operation w i l l be sufficient for identification, or, i f not, the magnitude and type o f a dithering-type signal must be conducted. Mechanical Design of the Dolly Steering System (e) - Design o f the steering system will be mandated by issues relating to performance, cost, packaging, maintenance, and reliability. Proposed methods include automobile-type steering (similar to the steer system i n the tractor), wagon-type steering (instead o f pivoting both wheels about individual kingpins, the entire axle is pivoted about the A - d o l l y centerline) o r movement o f the mounting point o f the A - d o l l y drawbar. T h e current analysis has been conducted assuming automobile-type steering; changes to b o t h the nonlinear simulation and the linear control m o d e l will be required i f a different steering strategy is utilized. (f) Prototype Verification - A s a final step i n its development, the "smart" A - d o l l y system w i l l require full-scale testing using the protocols laid out i n Section 3. 161 References [11] Wong, J.Y., Theory of Ground Vehicles (Second Edition), John Wiley and Sons, Inc., Toronto, 1993 [12] Echleman, R.L. et al Articulated Vehicle Handling, NTIS PB-211 201, 1972 [13] Ellis, J.R., Vehicle Handling Dynamics, Publications Limited, London, 1994 [14] Schmid, I., Engineering Approach to Truck and Tractor Train Stability, SAE Paper 670006,1967 [15] Jindra, F., Handling Characteristics o f Tractor-Trailer SAE Paper 650720,1965 [16] Fancher, P.S., The Transient Directional Response of Full Trailers, SAE Paper 821259,1982 [17] Fancher, P.S., Directional Dynamics Considerations for MultiArticulated, Multi-Axled Heavy Vehicles, SAE Paper 892499,1989 [18] Bernard, J.E., and Vanderploeg, M . , Static and Dynamic Offtracking of Articulated Vehicles, SAE Paper 800151,1980 [19] El-Gindy, M . , Kulakowski, B.T., Tong, X . , and Mrad, N . , Rearward Amplification Control of a Truck/Full Trailer, Advances in Automotive Control 1998, Elsevier Science Publishing Co., New York, 1998 [20] Mallikarjunarao, C. , and Fancher, P.S. Analysis o f the Directional Response Characteristics of Double Tankers, SAE Paper 781064,1978 [21] Winkler, C.B., et al Improving the Dynamic Performance of Multitrailer Vehicles: A Study o f Innovative Dollies. Volume II: Appendices, NTIS PB87-194031,1986 [22] MacAdam, C.C. et al A Computerized Model For Simulating the Braking and Steering Dynamics of Trucks, Tractor-Semitrailers, Doubles, and Triples Combinations, NTIS PB80-227994,1980 [23] Fancher, P.S. et aL Simulation of the Directional Response Characteristics of Tractor-Semitrailer Vehicles, NTIS PB80-189632, 1979 Mechanical Engineering Combinations, 162 References [24] Bernard, J.E. et al A Computer Based Mathematical Method for Predicting the Directional Response of Trucks and Tractor-Trailers, NTISPB-221 630,1973 [25] D'Souza, A . F . and Eshleman, R.L., Maneuverability Limits and Handling Criterion of Articulated Vehicles, Computational Methods in Ground Transportation Vehicles (AMD Vol 50), The American Society of Mechanical Engineers, New York, 1982 [26] Bakker, E . , Pacejka, H . B . , and Lidner, L . , A New Tire Model with an Application in Vehicle Dynamics Studies, SAE Paper 890087,1989 [27] Mikulcik, E.C., The Dynamics of Tractor-Semitrailer Vehicles: The Jackknifing Problem, SAE Paper 710045,1971 [28] Ervin, R.D., and MacAdam, C C , The Dynamic Response of MultiplyArticulated Truck Combinations to Steering Input, SAE Paper 820973, 1982 [29] Echleman, R.L. et aL Stability and Handling Criteria of Articulated Vehicles. Part I - Technical Report, NTIS PB-225 021/5,1973 [30] Day, T.D., Differences Between the E D V D S and the Phase 4, SAE Paper 1999-01-0103,1999 [31] Jindra, F, Maneuverability of Trailer Trains, SAE Paper 630491,1963 [32] Palkovics, L . , and El-Gindy, M . , Examination o f Different Control Strategies of Heavy-Vehicle Performance, Journal of Dynamic Systems, Measurement, and Control, vol.118 pp.489-497,1996 [33] Vazquez, V . Directional Performance Analysis of an A-train Double With Externally-Mounted Dampers, Heavy Vehicle Systems, vol.8 no.2 pp.155-176,2001 [34] Hibbeler, R.C., Engineering Mechanics - Statics and Dynamics (Sixth Edition), Maxwell Macmillan Canada, Toronto, 1992 [35] Sandor, B.L, Engineering Mechanics Dynamics (Second Edition), Prentice Hall International, Toronto, 1987 [36] Park, J. and Nikravesh, P.E., A Look-Ahead Driver Model Autonomous Cruising on Highways, SAE Paper 961686,1996 for Appendix A: Reference A-train Data This Appendix presents the geometry and physical data for the reference A-train vehicle [21]. Table A-l - Tractor Data Parameter Value Units i 5987 kg Ii 8474 kgm 0.61 m 2.44 m 2.217 m 1.016 m 0.914 m m b t b 2 b 9 d 7 2 K 21027'4 N/m V 1128400 N/m K 1.12 m 1.12 m h 18 165 Appendix A: Reference A-train Data Table A-2 — L e a d S e m i t r a i l e r D a t a Parameter Value Units m 2 \ kg 53037 kgm 3.56 m 3.15 m io 4.064 m d 8 0.914 m k 3 1564551 h 5 0.846 m 1 9 2.065 m 2.065 m b 3 b 4 b 14424 h K Table A-3 Parameter N/m A-dolly Data Value Units 1143 kg I3 289 kgm V 2.032 m b 6 0.0 m b u 0.0 m d, 0.914 m k. 1564551 h 0.0 m K 1.12 m K 1.12 m 8 2 N/m 2 166 Appendix A: Reference A-train Data Table A-4 — Second Semitrailer Data Parameter Value Units m 4 I 4 b 7 b 8 dio 14737 kg 55360 kgm 3.404 m 3.302 m 0.914 m 1564551 h 12 h 17 h 21 2 N/m 0.831 m 2.05 m 2.05 m Table A-5- Miscellaneous Data Parameter Value D 0.33 CS; 123392 Units m N m/rad 167 Appendix B: Tire Data This Appendix presents the tire cornering force and aligning moment plots for the Firestone 10.00x22F, Firestone 10.00x22RIB, and Freuhauf 10.00x20 heavy truck tires, respectively. Figure B - l - Cornering Force Versus Slip Angle, Firestone 10.00x22F 168 Appendix B: Tire Data 169 Slip Angle (rad) Figure B-2 - AUgning Moment Versus Slip Angle, Firestone 10.00x22F Appendix B: Tire Data 170 171 Appendix B: Tire Data Slip Angle (rad) Figure B-4 - Mgning Moment Versus Slip Angle, Firestone 10.00x22RIB Appendix B: Tire Data 172 Slip Angle (rad) Figure B-5 - Cornering Force Versus Slip Angle, Freuhauf 10.00x20 Appendix B: Tire Data Figure B - 6 — Aligning Moment Versus Slip Angle, Freuhauf 10.00x20 173
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Improving the dynamic performance of multiply-articulated vehicles Rempel, Michael R. 2002
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Title | Improving the dynamic performance of multiply-articulated vehicles |
Creator |
Rempel, Michael R. |
Date Issued | 2002 |
Description | Current heavy vehicles having two or more trailers suffer from unique dynamic characteristics which limit both their stability and maneuverability at speed. The control of these characteristics in A-train vehicles is the objective of this work; specifically, the attenuation of rearward amplification and high-speed offtracking. Control is attained is through automatic steering of the A-dolly axle; the control system is localized entirely on the A-dolly, creating a modular and easily-implemented unit. The equations of motion were derived for a reference A-train vehicle, and the results simulated and compared to literature. A nonlinear two-dimensional yaw plane model with semi-static load transfer in the pitch and roll modes was found to adequately display the intended system characteristics. To apply control, a second linear state space model was created, based on the behavior of the A-dolly and the second semitrailer only. For high-speed, low amplitude maneuvers under feasible input frequencies, the results corresponded to the nonlinear simulation. Control was achieved using a state variable feedback controller to steer the dolly wheels; the gains were determined by moving the plant eigenvalues via Ackermann's method to the critically-damped locations of the "faster" mode. The controller was shown as to be robust to parameter estimation errors and to balance performance and required control inputs well. An Extended Kalman Filter (EKF) was employed to estimate the unknown tire cornering properties and states not measurable direcdy from the A-dolly. Through simulation, it was found that the controller was effective in reducing both the observed rearward amplification and the dynamic offtracking, although the effectiveness decreased with increasing forward velocity. At the nominal velocity for interstate highways in the United States (24.6 m/s), the peak improvement in rearward amplification under control was reduced to a minimum of 25 percent of the uncontrolled value; the peak value of offtxacking was reduced up to 50 percent. |
Extent | 6629777 bytes |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-08-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0090274 |
URI | http://hdl.handle.net/2429/12208 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2002-05 |
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UBCV |
Scholarly Level | Graduate |
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