Vehicle Dynamics Modeling and Control of the TowPlow, A Steerable Articulated

Vehicle Dynamics Modeling and Control of the TowPlow,

A Steerable Articulated Snow Plowing Vehicle System

By

Jae Young Kang

B.S. (Sungkyunkwan University, Korea) 2009

M.S. (Sungkyunkwan University, Korea) 2011

DISSERTATION

Submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in

Mechanical and Aerospace Engineering

in the

OFFICE OF GRADUATE STUDIES

of the

UNIVERSITY OF CALIFORNIA

DAVIS

Approved:

____________________________

Steven A. Velinsky, Chair

____________________________

Donald L. Margolis

___________________________

Sanjay S. Joshi

Committee in Charge

2014

UMI Number: 3646318

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Abstract

The TowPlow is a novel type of snowplow that consists of a conventional snowplow vehicle and a steerable, plow-mounted trailer. The trailer is equipped with hydraulic-powered steerable axles so that it can be steered up to 30 degrees with respect to the tractor. The combination of the front plow of the towing snowplow and the trailer-equipped plow is able to clear a path up to approximately 24-ft wide, which is the width of two typical traffic lanes. While the TowPlow may increase the efficiency and performance of the snow removal operation, the stability of the system under the harsh winter conditions may be compromised by implementation of the steerable trailer, and safety of the system must be ensured. The kinematic characteristics of the TowPlow are derived using instantaneous centers of velocity. Based on the derived equations, the relation between the radius of curvature and the trailer wheel steering angle that allows the tractor-trailer to maintain its initial articulation angle is defined to be used in a kinematics-based control of the TowPlow. Then, a linear dynamic model is developed in order to investigate the dynamic behavior of this system and its stability limits. Dynamic simulations of various maneuvers are performed, and kinematics-based control is implemented to investigate performance of the trailer’s corrective steering. The goal is to ensure that the trailer does not intrude into adjacent lanes during plowing operations while also ensuring that both lanes are sufficiently cleared. Even though the control input is obtained from kinematic analysis, which does not take forces and inertia into account, the simulation results clearly show that the corrective steering helps the TowPlow meet its performance goals.

Also, a nonlinear dynamic model of the TowPlow for longitudinal, lateral, and yaw motions is developed with the state variables of longitudinal velocity, lateral velocity and yaw rate of the towing unit, yaw rate of the trailer unit, and the articulation angle between the two units. The model includes a modified Dugoff’s tire friction model, tire rotation dynamics and the load transfer effect. The model is validated through full-scale experiments of the TowPlow under both steady-state and transient conditions. For completion of the nonlinear dynamic model, a snow resistance model is developed to estimate the snow resistant forces on each plow of the TowPlow. Dynamic simulations of the nonlinear TowPlow model including the snow resistance are performed without any controller. The effect of the snow resistance on the dynamics and stability of the TowPlow is discussed for various maneuvers such as cornering, slalom, up and down hill, and split friction coefficient braking.

Finally, an active steering control of the trailer axle is introduced to prevent the TowPlow from intruding into the adjacent lane and also from missing certain portions of the lane during its snow removal operation. The linear quadratic regulator (LQR) based closed-loop controller is developed utilizing the linear TowPlow model. Performance of the LQR controller is compared to that of a simple PI controller. Dynamic simulations of the TowPlow with the trailer active steering control are performed for the same maneuvers simulated with the uncontrolled system. The comparison of the simulation results between the controlled system and the uncontrolled system clearly demonstrates that the implementation of active steering control for the trailer axle will improve safety and efficiency of the TowPlow. Such control keeps the TowPlow from either intruding into the adjacent lane or missing large portions of the lane by maintaining its total articulation angle in its snow removal operation.

Acknowledgements

First, I would like to thank my advisor, Professor Steven A. Velinsky for his immeasurable support and supervision. I deeply appreciate his technical insight and management of the AHMCT. I learned many things from him. It was great opportunity for me to work with him at AHMCT.

I would like to also thank my dissertation committee, Professor Donald L. Margolis and Professor Sanjay S. Joshi, for their time, consideration, and encouragement.

I would also like to thank Duane Bennett, George Burkett, Kin Yen and Victor Reveles for their time and assistance on preparation and execution of the TowPlow experiment. With their assistance, this dissertation was brought up to a whole new level.

I want to especially thank my wife, Hyelyun Lee, and my son, Jeremy E. Kang, for their love, continuous support and encouragement. Special thanks to my parents for their unceasing prayers.

Contents

Abstract

Acknowledgements

1 Introduction

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 Kinematics of the articulated vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.2 Dynamics of the articulated vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.3 Snow resistance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.4 Stability control of the articulated vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Kinematics of the TowPlow

2.1 Kinematic model – instantaneous centers of velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Derivation of kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Defining steering inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Simulation of constant radius turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iiiv1122691112 141416172022

3 Linear vehicle dynamics and stability of the TowPlow

3.1 Linear planar model of the TowPlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Stability and controllability of the TowPlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Dynamics and open-loop control of the TowPlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Nonlinear vehicle dynamics of the TowPlow

4.1 Equations of motion for the TowPlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Modified Dugoff’s tire friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Tire rotation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Load transfer effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.1 Experimental configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.2 Steady-state circular test – constant speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.3 Transient maneuver test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Snow resistance model and dynamic simulation of the TowPlow

5.1 Snow resistance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Application of the snow resistance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Dynamic simulation of the TowPlow without control of the trailer axle . . . . . . . . . . . . . . . .

5.3.1 Driver model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2 Deploying trailer plow and cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.3 Slalom, up and down hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.4 Split friction coefficient braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Control of the TowPlow for the snow removal operation

6.1 Optimal controller design – LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 PI controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Dynamic simulation of the TowPlow with PI control of the trailer axle . . . . . . . . . . . . . . . . .

6.3.1 Slalom, up and down hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.2 Split friction coefficient braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2424323740 44444856575960616566 686874777880818589 909095959696100

7 Conclusions and recommendations

7.1 Conclusions and contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References 105101101 7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

List of Figures

Figure 1.1 TowPlow [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.2 Jindra’s tractrix integrator [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.3 Pretty’s tractrix from steering in circle [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.4 Instantaneous centers of logging trucks by Erkert et al [9] . . . . . . . . . . . . . . . . . . . . . . Figure 1.5 Cornering of tractor-trailer combination by Chen and Velinsky [10] . . . . . . . . . . . . . . . Figure 1.6 Manesis’ sliding kingpin mechanism [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.7 Typical unstable states of articulated vehicles by Vlk [14] . . . . . . . . . . . . . . . . . . . . . . . Figure 1.8 Coordinate system for the articulated vehicle by Chen and Tomizuka [19] . . . . . . . . . . Figure 1.9 Concept of friction between bristles for the LuGre model [20] . . . . . . . . . . . . . . . . . . . . Figure 1.10 Friction circle concept for the Dugoff’s tire friction model by Guntur and Sankar [26] . Figure 1.11 Mellor’s wedge plow model [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 1.12 Kaku’s snow flow assumption in snow resistance model [30] . . . . . . . . . . . . . . . . . . . . Figure 1.13 Control volume in front of the plow by Ravani et al. [31] . . . . . . . . . . . . . . . . . . . . . . . Figure 1.14 Control scheme of active braking control in [36] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.15 Control scheme of active all-wheel steering control in [44] . . . . . . . . . . . . . . . . . . . . . .

Figure 2.1 Schematic of the TowPlow system and associated notations . . . . . . . . . . . . . . . . . . . . . Figure 2.2 Radius of curvature of the road vs. Trailer wheel steering angle for constant total

articulation angle θt =30º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.3 Tractor steering angle vs. Trailer wheel steering angle for constant total articulation

angle θt =30º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.4 Simulation results of the constant radius turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23455677891011121215192022

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7 Forces at the hitch points and the tongue assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Locus of the eigenvalues of the matrix M-1A with varying longitudinal velocity . . . . . . 34Locus of the eigenvalues of the matrix M-1A with varying inertias . . . . . . . . . . . . . . . . . 35Locus of the eigenvalues of the matrix M-1A with varying inertias . . . . . . . . . . . . . . . . . 36Scheme of the uncontrolled system simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Scheme of the open-loop controlled system simulation . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.8

Figure 3.9

Figure 3.10

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 5.1

Figure 5.2

Figure 5.3

Simulation results of the TowPlow comparing uncontrolled and controlled system for the step input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the TowPlow comparing uncontrolled and controlled system for the pulse input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the TowPlow comparing uncontrolled and controlled system for the sine input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scheme of the tractor unit and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Scheme of the trailer unit and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Scheme of the tongue assembly and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow chart of the tire force calculation [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computed longitudinal and lateral tire forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carpet plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Load factors in relation with (a) friction coefficient and (b) lateral stiffness . . . . . . . . . Carpet plots considering load change effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free body diagram for a driving wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Side and (b) rear views of the tractor unit and applied forces . . . . . . . . . . . . . . . . . . Layout of sensors and microcontrollers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test procedure of the steady-state test for a speed and direction . . . . . . . . . . . . . . . . . . Steady-state test results compared with simulation results . . . . . . . . . . . . . . . . . . . . . . . Test procedure of the transient maneuver test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient test inputs for the experiment and simulation . . . . . . . . . . . . . . . . . . . . . . . . . Transient test results compared with simulation results . . . . . . . . . . . . . . . . . . . . . . . . . Components of the snow resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scheme of the snow resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of resistance ratios for longitudinal snow resistance . . . . . . . . . . . . . . . . . . 4142434851525355565758616264656667696973

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9 Schemes of the snow plows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Longitudinal snow resistant forces of the plows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Lateral snow resistant forces of the plows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Driver model – control scheme of the driving/braking torque . . . . . . . . . . . . . . . . . . . . 79Driver model – control scheme of the tractor steering angle . . . . . . . . . . . . . . . . . . . . . . 79Figure 5.10 Simulation results of the TowPlow running straight with and without driver model . . . 82Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Simulation results of deploying trailer plow and cornering . . . . . . . . . . . . . . . . . . . . . . . Simulation results of slalom, up and down hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of braking on a snow packed road (μ0 = 0.4) . . . . . . . . . . . . . . . . . . . Simulation results of split friction coefficient braking – tractor on a wet road (μ0 = 0.6) and trailer on a snow packed road (μ0 = 0.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of split friction coefficient braking – tractor on a snow packed road (μ0 = 0.4) and trailer on a yet road (μ0 = 0.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Locus of the eigenvalues of the controlled system with varying longitudinal velocity . . LQR control scheme for the active steering of the trailer axle . . . . . . . . . . . . . . . . . . . . Cornering simulation results of the active trailer steering control . . . . . . . . . . . . . . . . . . PI control scheme for the active steering of the trailer axle . . . . . . . . . . . . . . . . . . . . . . PI control scheme for the active steering of the trailer axle . . . . . . . . . . . . . . . . . . . . . . Slalom, up and down hill simulation results of the active trailer steering control . . . . . . Split friction coefficient simulation results of the active trailer steering control - tractor on a wet road (μ0 = 0.6) and trailer on a snow packed road (μ0 = 0.4) . . . . . . . . . . . . . . Split friction coefficient simulation results of the active trailer steering control - tractor on a snow packed road (μ0 = 0.4) and trailer on a wet road (μ0 = 0.6) . . . . . . . . . . . . . . 83848687889293949596979899

List of Tables

Table 2.1 Vehicle parameters for kinematic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.1 Vehicle parameters for stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.1 Parameters for tire friction calculation [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.2 Vehicle parameters for model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 5.1 Plow parameters for the snow resistance calculation [30, 31] . . . . . . . . . . . . . . . . . . . . . Table 5.2 Vehicle parameters for dynamic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193351637278

Chapter 1

Introduction

1.1 Background

The snow removal operation is an important highway maintenance operation during the winter season. It is also a hazardous operation that requires significant budget and labor. To clear the whole road width for multi-lane roadways, typically, the operation has been accomplished by several snowplows forming a trapezoidal formation, called ‘gang plowing’.

A novel type of snowplow, the TowPlow (Figure 1), has been invented lately to allow highway agencies to reduce their budget for the winter operations by enhancing the clearing capacity of a single snowplow vehicle. The TowPlow consists of a conventional snowplow vehicle, referred to as the tractor henceforth, which tows a steerable trailer with a moldboard. The trailer is equipped with steerable axles so that the trailer can be steered up to 30 degrees with respect to the tractor. The hydraulic rams connected to the tractor’s hitch also assist the control of the trailer. The combination of the front plow of the towing snowplow and the trailer-equipped plow is able to clear a path up to approximately 24-ft wide, which is the width of two typical traffic lanes.

Figure 1.1. TowPlow [1]

While the TowPlow may increase the efficiency and performance of the snow removal operation, the stability of the system under the harsh winter conditions may be compromised by implementation of the steerable trailer, and stability of the system must be ensured in terms of the lateral and yaw dynamics, load transfer, hill climbing, and low friction road conditions. This dissertation examines the stability of the TowPlow through both kinematic analysis and detailed dynamic modeling considering snow resistance, load transfer and gradability. The addition of control to the TowPlow to enhance its operational performance and stability, and broaden its applicability in the challenging winter operational conditions is also studied.

1.2 Literature survey

For the study of dynamic modeling and control of the TowPlow, related literature is reviewed in the following categories: kinematics, dynamics, snow resistance, and stability control. To constrain the scope of the review, it is confined to work related to articulated vehicles since the TowPlow is a unique type of this vehicle.

1.2.1

Kinematics of the articulated vehicle Kinematics of articulated vehicles has been studied mainly to investigate off-tracking, which

describes the difference in path radii between the front axle of the towing unit and the rear axle of the

trailer unit, and to generate trajectories of an autonomous vehicle or a mobile robot system. The design of articulated vehicles, highway exit ramps, and parking lots is affected by kinematics of the articulated vehicle due to its geometry [2, 3]. Jindra (1963) published his work on the tracking of a tractor-trailer combination in a steady turn. He developed equations that determine the kinematic path of a single-unit vehicle using general tractrix, and applied the results to the tractor-trailer combination. He also developed the tractrix integrator instrument, shown in Figure 1.2, which can trace maneuvering patterns of any trailer combination with two-wheel steering [4].

Figure 1.2. Jindra’s tractrix integrator [4]

Pretty (1964) provided a full evaluation of off-tracking paths for large vehicle combinations considering the basic geometry of steering and tracking in a circular curve and a straight line. Figure 1.3 shows the tractrix generated from steering in a circle and by the rear of the trailer when the towing pintle follows the curve [5].

The Western Highway Institute (1970) performed a set of comprehensive analyses measuring off-tracking of vehicles and vehicle combinations using the following methods: (1) the use of models (i.e. the general tractrix), (2) the graphical method, and (3) the mathematical method. They concluded that there are no significant differences in measuring off-tracking for the same equipment whichever methods are applied, and that the amount of off-tracking is most likely dependent on the components of the

wheelbases such as the distance between each axle and the articulation point [6].

Figure 1.3. Pretty’s tractrix from steering in circle [5]

Off-tracking has been a significant issue causing disruption to traffic flow by large trucks and tractor-trailer combinations intruding into adjacent lanes. Saito (1979) associated articulation angle and forward velocity of the semi-trailer with rear-wheel steering to reduce the off-tracking [7]. Alexander and Maddocks (1988) derived equations that relate the centers of curvature of the wheels to the center of rotation of the vehicle, and utilized the results for problems of off-tracking and optimal steering [8]. Erkert et al. (1989) investigated off-tracking of logging trucks for road design in forests utilizing the method of general tractrix and instantaneous centers of rotation (Figure 1.4), and the results compared favorably with experimental data [9].

Chen and Velinsky (1992) suggested a kinematic design methodology to optimize the geometry of the vehicles and the roadways for low-speed maneuverability. Also, they ascertained that the low-speed maneuverability of an articulated vehicle can be improved through steering of trailer axles as a linear function of the articulation angle and front-wheel-steer angle, as shown in Figure 1.5 [10].

Figure 1.4. Instantaneous centers of logging trucks by Erkert et al [9]

(a)(b)

Figure 1.5. Cornering of tractor-trailer combination: (a) without trailer steering, (b) with trailer

steering by Chen and Velinsky [10]

Manesis (1998) introduced a sliding kingpin mechanism, shown in Figure 1.6, to eliminate the off-tracking of heavy duty trucks with semi-trailers, and also designed various types of sliding control

[11-13].

Figure 1.6. Manesis’ sliding kingpin mechanism [12]

1.2.2

Dynamics of the articulated vehicle From the 1930s, a substantial amount of work has been performed concerning the directional dynamics of articulated vehicles. Vlk (1985) comprehensively reviewed and summarized studies on handling performance of truck-trailer vehicles. According to his review, the early theoretical works of articulated vehicles are limited to only unstable states of the trailer until Schmid (1964) and Jindra (1965) introducing the interdependence between truck and trailer motions [14]. In the 1960s, Jindra (1965) and Bundorf (1967) developed linear differential equations for the simplified mechanical model of a tractor double trailer combination and an automobile-trailer combination, respectively, and examined the directional instability and steady-state turning performance through steady-state and transient responses to steering inputs [15, 16]. Ellis (1969) developed both linear and simplified nonlinear models for the planar motion of articulated vehicles neglecting the roll motion of the vehicles, and analyzed dynamic responses to show how instability of the trailer occurs [17]. Segal and Ervin (1981) classified handling instability of

articulated vehicles into: (1) jack-knifing – occurring when the tractor oversteers and the trailer

understeers or slightly oversteers above a critical speed; (2) trailer swing – occurring when the tractor oversteers and the trailer oversteers strongly above a critical speed [18]. Vlk (1985) also characterized three typical directional unstable states of articulated vehicles: (1) snaking – trailer yaw oscillation that occurs at high speed; (2) jack-knifing – instability of tractor yaw motion; and (3) trailer swing – instability of trailer yaw motion [14].

Figure 1.7. Typical unstable states of articulated vehicles by Vlk [14]

Later, a more complex nonlinear model of articulated vehicles considering the lateral, yaw and roll motions together was developed by Chen and Tomizuka (1995) [19].

Figure 1.8. Coordinate system for the articulated vehicle by Chen and Tomizuka [19]

Analysis on non-linear dynamics of the vehicle had been enhanced through development of non-linear tire friction models because forces and moments generated by the friction between tire and road surface influence vehicle dynamics significantly. The tire models that have been used commonly for vehicle dynamics are the LuGre model, Pacejka’s model, and Dugoff’s model. The LuGre friction model is originally suggested by Canudas de Wit et al. [20]. It describes the mechanism of friction as contact of two rigid bodies through elastic bristles. When one body travels on the other, the bristles randomly deflect like springs, and the bending of the bristles generates the friction force. Initially, the LuGre model was only used for the longitudinal friction force. However, it was extended to the combined longitudinal and lateral motion [21].

Figure 1.9. Concept of friction between bristles for the LuGre model [20]

Pacejka’s model, also known as the Magic Formula tire model, is mathematical equations composed of several tunable coefficients to accurately describe the measured data of the longitudinal and lateral tire force [22]. The coefficients in the model may not have physical interpretation.

Dugoff’s model is a derivative of the freely rolling tire by Fiala [23]. Dugoff extended the previous work to general tire-road interaction either for pure-slip or combined-slip condition [24]. A simplified Dugoff’s model assuming that both longitudinal and lateral forces are linearly dependent on the normal force of the tire is developed by Krauter [25]. In addition to the simplified model, Guntur and Sankar implemented the friction circle concept to Dugoff’s model; i.e., if the desired friction is less than

or equal to half of the available friction, described by inside of the circle in Figure 1.10, the longitudinal

and lateral tire forces have linear relationship with the slip ratio and slip angle, respectively; however, if the desired friction outside of the circle, the tire forces attenuate nonlinearly. They also presented a procedure of the tire forces calculation for vehicle simulation [26].

Figure 1.10. Friction circle concept for the Dugoff’s tire friction model by Guntur and Sankar [26]

1.2.3

Snow resistance model The snow resistance model is significant in modeling of the TowPlow because forces on the plows affect the dynamics of the TowPlow, and may cause instability. There has been an effort to estimate forces on the plow during the snow removal operation. Some of the models found in the literature are based on Croce’s model, which is a simple Bernoulli fluid flow model under the assumption that the velocity of the snow is constant throughout the entire process. The model approximates the snow resistance force more closely at higher plowing speed [27]. Mellor (1965), for modeling of wedge shaped plow (Figure 1.11), modified Croce’s model through introducing a coefficient that compensates the velocity change of the snow due to compression of the snow [28]. Zhou et al. (2000) modified Mellor’s model to be used in vehicle dynamic modeling [29]. However, the model still fails to consider the compression of the snow accurately.

Kaku’s model (1979, Figure 1.12), based on the theory of conservation of momentum, considers the velocity change of the snow due to its compressibility [30]. Kempainen et al. (1998) presented a

complex snow resistance model that includes compressibility of the snow, shear and turbulent zones in

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