Stability Analysis during Active Tire Excitation for Friction
Estimation
RUDRENDU SHEKHAR
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
Road accidents have been a persistent cause of death worldwide, and claim millions of lives every year. Recent developments in the active safety systems like Electronic Stability Control (ESC) have helped in reducing these numbers quite signi cantly over the years. However, a major challenge for these systems is to know the friction coe cient between the tire and the road, as this value limits the amount of force the tires can generate. Knowledge of the coe cient of friction can be used to adapt the driving style, thereby avoiding interventions by stability control at the limit, making vehicles safer. However, it is a major challenge within the automotive industry to estimate the coe cient of friction accurately, and with su cient availability, as that requires high levels of tire utilization, such that the tire is forced to reach the non-linear range of operation. Such events are very rare in everyday driving, and requires a system induced active excitation of the tires. One such method that has been proposed earlier, to carry out an active tire excitation, is by using a simultaneous propulsive and brake force on front and the rear the axles. However, applying an equal magnitude of propulsive and brake force results in a force neutral situation at the vehicle level, which forces the velocity to be constant, overriding driver acceleration requests. Thus, an active tire excitation method was proposed by Volvo Cars, which is able to apply an unequal propulsive and brake force to the front and the rear axle, such that the driver's acceleration demand can be met, during friction estimation. However, such an excitation can be dangerous to carry out, if it leads to instability of the vehicle.
Several methods have been developed to analyze and quantify stability of a vehicle, but detailed analysis about the stability under forced excitation, for friction estimation, is very rare. This thesis work investigates the lateral stability of a vehicle undergoing an active tire excitation for friction estimation. The objective is to understand which vehicle and tire models can be used to quantify the lateral stability of a vehicle under forced excitation, and how phase portrait methods can be used to develop a stability monitor that is able to indicate the lateral stability of the vehicle under a forced excitation.
The results of using a stability monitor during active tire excitation clearly show that it is able to indicate when the vehicle becomes unstable and looses control. It also shows that for slow speed steady-state maneuvers and straight line maneuvers, the stability monitor does not indicate instabil- ity. A comparison between phase portrait based and conventional side-slip based stability monitors show the eectiveness and generality of the phase portrait based monitor, which is able to detect instability earlier than the conventional side-slip based method.
This thesis work has been carried out in cooperation with Volvo Car Corporation, who have been responsible for the thesis, and the patent to which this thesis work is related to. I would like to extend my sincerest thanks to Volvo Cars for this opportunity and support.
I would especially like to convey my gratitude towards my main supervisors Mats Jonasson and An- ton Albinsson, who have guided me throughout the thesis. The weekly meetings have been invaluable for the progress of this thesis and the conceptual discussions during these meetings have been a great learning experience for me. I would also like to thank my examiner Lars Drugge, who's guidance and discussions have helped me shape my thesis work. Thank you for all your support and guidance.
Thanks to the Anders Peterson, Derong Yang, Mikael Thor, Rickard Nilsson, Tomas Halleröd, Niklas Ohlsson, and Alexander Varghese at Vehicle Dynamics and Motion Control department for helping me out in case of questions and for the valuable inputs. Also, thanks to Matthijs Klomp and Angelis Stavros from the Vehicle Dynamics CAE team for their support with the vehicle models that I used in my thesis.
Last but not the least, I would like to thank my girlfriend Padma Nayagam for her continued sup- port and motivation throughout the thesis work, and my friends and parents for their support and encouragement, without which this thesis would have been incomplete.
1 Introduction 1
1.1 Background and Motivation . . . 1
1.1.1 The Problem of Accidents . . . 1
1.1.2 The Role of Active Safety and Friction . . . 3
1.1.3 Challenges of Friction Estimation . . . 5
1.1.4 Probe-Friction - Volvo Cars . . . 7
1.1.5 Vehicle Lateral Stability . . . 8
1.2 Objective of the Thesis . . . 9
1.3 Limitations of the Thesis . . . 9
1.4 Outline of the Thesis . . . 10
2 Theory 11 2.1 Coordinate System . . . 11
2.2 Tire Modeling . . . 11
2.2.1 The Linear Tire Model . . . 14
2.2.2 The Brush Model . . . 15
2.2.3 The Dugo Tire Model . . . 17
2.2.4 The Modi ed Dugo Tire Model . . . 19
2.2.5 The Magic Formula Tire Model . . . 21
2.3 The Bicycle Model . . . 23
2.3.1 Bicycle Model with 2 DoF . . . 24
2.3.2 Bicycle Model with 5 DoF . . . 25
2.3.2.1 Wheel Dynamics . . . 26
2.4 Lateral Stability of the Bicycle Model . . . 27
2.4.1 Steady-State Stability in the Linear Region with 2 DoF . . . 27
2.4.2 The Handling Diagram . . . 29
2.4.3 The Phase Portrait Method . . . 31
2.5 Tire Force Estimation . . . 32
3 Tools and Methods 34 3.1 Test Vehicle . . . 34
3.2 Simulink Implementation . . . 35
4 Lateral Stability with 2 DoF 38 4.1 Cornering Stiness of the Linear Tire model . . . 38
4.2 Combined Slip Operation - Friction Circle . . . 39
4.2.1 Steady-State Stability Analysis - Friction Circle . . . 41
4.3 Combined Slip Operation - Brush Model . . . 42
4.3.1 Steady-State Stability Analysis - Brush Model . . . 44
5.3 Stability Analysis with Handling Diagram . . . 52
5.4 Stability Analysis using Phase Portraits . . . 54
5.4.1 Factors Aecting Stability Regions . . . 56
6 Stability Monitoring for Active Tire Excitation 59 6.1 Stability Region Lookup based Monitor . . . 59
6.1.1 Sensors . . . 60
6.1.2 Propulsion Torque Estimation . . . 60
6.1.3 Force Estimation . . . 61
6.1.4 Brake Torque Estimation . . . 61
6.1.5 Utilization Monitor . . . 61
6.1.6 Indexing Strategy . . . 62
6.1.7 Stability Region Lookup . . . 63
6.1.8 Stability Monitor . . . 63
6.2 Rear Axle Side-Slip Limit based Monitor . . . 64
6.3 Real-time Testing and Comparison . . . 65
6.3.1 Straight Ahead at 30 km/h . . . 67
6.3.2 Constant Radius Cornering at 30 km/h . . . 69
6.3.3 Constant Radius Cornering at 55 km/h . . . 70
6.3.4 High Speed Step Steer at 65 km/h . . . 72
7 Discussion and Conclusion 76
8 Future Work 78
Bibliography 79
1.1 Top ten causes of death worldwide in the 15-29 age group, 2012 [14] . . . 1
1.2 Deaths due to tra c accidents worldwide since 2001. [14] . . . 2
1.3 Major causes of road accidents. [16] . . . 2
1.4 Role of ESC in reducing single vehicle crashes in USA for 5 states between 1997-2002 [17] . . . 4
1.5 Correlation between the number of accidents and the coe cient of friction for 10 million vehicle kilometers. [18] . . . 5
1.6 Diagram showing force characteristics at low and high friction with everyday driving. 6 1.7 The concept of simultaneous propulsive and brake torque. . . 6
1.8 Overview of the active tire excitation logic. A = Wheel-speeds and tire forces; C,E,F = Front and rear axle torque requests; D = Estimated friction coe cient; G = Other vehicle states; H = Activation or deactivation [25]. . . 7
2.1 Vehicle axis system de ned by ISO 8855:2011 [1] . . . 11
2.2 Slip angle in a tire . . . 12
2.3 Lateral and Longitudinal Force Characteristics of a Typical Tire at dierent Normal Loads . . . 13
2.4 Combined lateral and longitudinal characteristics for a typical tire [4] . . . 13
2.5 Lateral force vs slip angle for a tire showing the linear range of operation [3] . . . 14
2.6 Concept of brush model of a tire [4] . . . 15
2.7 Characteristic curves for the brush model in pure lateral, longitudinal and combined slip . . . 16
2.8 Tire-road contact geometry - Dugo model [10] . . . 17
2.9 Lateral, longitudinal and combined slip characteristics for Dugo tire model . . . 19
2.10 Lateral, longitudinal and combined slip characteristics for modi ed Dugo tire model 21 2.11 Lateral, longitudinal and combined slip characteristics for magic formula tire model . 22 2.12 Planar bicycle model [2] . . . 23
2.13 Forces and torques acting on the wheel . . . 26
2.14 Schematics for the drive-line . . . 27
2.15 Normalized lateral force characteristics of the front and the rear axle [2] . . . 30
2.16 Examples of handling diagrams showing normalized axle characteristics to the left and the corresponding handling diagram to the right [4] . . . 31
2.17 Phase portrait diagram showing the evolution of the yaw rate and side-slip angle for a bicycle model . . . 32
2.18 Structure of the tire force estimator showing three sub-estimators; vertical, longitu- dinal and lateral tire force estimators [7] . . . 33
3.1 Simulink implementation of the bicycle model with 2 degrees of freedom . . . 36
3.2 Simulink implementation of the bicycle model with 2 degrees of freedom . . . 36
4.1 Constant radius maneuver with gradually increasing speed at low lateral acceleration 38 4.2 Fy Comparison between Bicycle Model with Linear Tires and Measured Data for a constant radius maneuver with gradually increasing speed . . . 39
4.6 Eect of longitudinal slip on the cornering stiness of the brush model . . . 43
4.7 Longitudinal Force Comparison from Brush Model with Measured Data . . . 44
4.8 Slip Ratio vs Longitudinal Force for Brush Model . . . 45
4.9 Inverse Brush Model for Longitudinal Force . . . 46
4.10 Kuscs vs Fx for dierent µ - Brush Model . . . 46
5.1 Comaprison between the calculated wheel speed from the wheel dynamics model, and test data together with the gear shift . . . 49
5.2 Inputs of steering wheel angle, front propulsive torque and rear brake torque to bicycle model . . . 50
5.3 Front and Rear axle lateral force comparison between Dugo, Modi ed Dugo and Magic Formula with test data . . . 51
5.4 Comparison of vehicle ay, ωz, β and trajectory from bicycle model with test data . . 52
5.5 Handling diagram from the 5 degree of freedom bicycle model with Magic Formula tires with active excitation on dierent axles . . . 53
5.6 Handling diagram for 5 DoF bicycle model with Magic Formula tire model showing handling and axle characteristics with increasing excitation torque . . . 54
5.7 Phase portrait simulation for µ = 0.5, vx = 55 km/h, δsw = 0 deg, T = 400 Nm, and biasf = 0.5, showing the stable region . . . 56
5.8 Comparison of the relative eects of the input variables on the size of the stability region . . . 57
5.9 Stable regions for increasing friction coe cient . . . 58
6.1 Overall structure of the stability region lookup based monitor. . . 60
6.2 Diagram showing lower and upper bounds of the discretized variable between which the measured value x lies. . . 62
6.3 Working of the stability region based monitor showing stable and unstable conditions. 64 6.4 Structure of the rear axle side-slip based monitor. . . 65
6.5 Working of the rear axle side-slip limit based stability monitor with stable and un- stable operating points. . . 65
6.6 Straight ahead maneuver at 30 km/h. . . 66
6.7 Constant radius corner at 30 km/h and 70 km/h. . . 67
6.8 High speed step steer maneuver at 65 km/h. . . 67
6.9 Active tire excitation stability monitoring variables plotted against time for a straight ahead maneuver at 30 km/h. . . 68
6.10 Active tire excitation stability monitoring variables plotted against time for a constant radius cornering maneuver at 30 km/h. . . 69
6.11 Active tire excitation stability monitoring variables plotted against time for a constant radius cornering maneuver at 55 km/h. . . 70
6.12 Instantaneous states leave the stability region for a constant radius cornering maneu- ver at 55 km/h. . . 71
6.13 Comparison of the stability signals from both stability monitors wrt wheel speeds . . 71
6.14 Active tire excitation stability monitoring variables plotted against time for a high speed step steer maneuver at 65 km/h. . . 73
6.15 Instantaneous states leave the stability region for a high speed step steer maneuver at 65 km/h. . . 74
6.16 Comparison of the stability signals from both stability monitors wrt wheel speeds . . 74
2.1 Friction coe cients for modi ed Dugo model . . . 20
3.1 Vehicle Parameters . . . 34
3.2 Drive-line Parameters . . . 35
3.3 Sensor signals used for state estimation and stability monitoring . . . 35
4.1 Cornering stiness values for the bicycle model . . . 39
4.2 Brush Model Slip Stiness . . . 45
5.1 Range and discretization of input conditions for phase portrait generation . . . 55
5.2 Range and discretization of initial conditions for phase portrait generation . . . 55
5.3 Input variables with constant values for observing the eect on the stability region with change in each of them . . . 56
6.1 List of maneuvers for stability monitor validation. . . 66
This thesis work has been carried out in cooperation with the department of Vehicle Dynamics and Motion Control at Volvo Car Corporation, with an aim to understand the problem imposed by vehi- cle stability under forced excitation of the tires with a simultaneous propulsive and brake torque, in order to estimate the friction coe cient between the tire and the road. This chapter describes the main motivation behind the thesis work, its scope and limitations, and the organization of this report.
1.1 Background and Motivation
1.1.1 The Problem of Accidents
The modern day road car is one of the most valuable manifestation of science and technology that has empowered mankind with freedom of movement like no other. Cars have become an integral part of modern day lifestyle and today it is almost impossible to imagine a world without it. However, with all these perks, the motor vehicle has also aected the global environment with huge amounts of carbon-di-oxide and other pollutants through the burning of fossil fuel, and has been a leading cause of accidents worldwide, that has claimed uncountable lives and injured many more. Tra c accident is the most common cause of death worldwide among the youth aging between 15 and 29 years that claims more than 300,000 lives every year as can be seen in the Figure 1.1 [14].
Figure 1.1: Top ten causes of death worldwide in the 15-29 age group, 2012 [14]
Claiming over 1.25 million lives worldwide in 2013, tra c accidents have been the eighth leading cause of deaths worldwide [15], and this number has been growing since the beginning of the millen-
nium, as seen in Figure 1.2. However, increased awareness and steps taken to make road transport safer has shown that the number of deaths has stayed fairly constant since 2007, certifying the ef- fectiveness of the implemented safety measures as seen in Figure 1.2 [14].
Figure 1.2: Deaths due to tra c accidents worldwide since 2001. [14]
Despite the decrease in the rate of increase of deaths due to tra c accidents over the years, the absolute number is alarming, and to address the same, the United Nations proposed the sustainable development goal (SDG) of reducing the number of deaths due to tra c accidents by half within 2020 [14]. This is an massive target that puts enormous demands on tra c safety as a whole, in- volving infrastructure, vehicles and the legal systems associated with it.
Before it is possible to respond to this demand, it is important to understand the major causes behind these road accidents. A research on the causes of road accidents by Volvo Trucks reveals that the reason for tra c accidents can be broadly grouped into three major areas as shown in the Figure 1.3.
Figure 1.3: Major causes of road accidents. [16]
1. Environment: About 30% of road accidents can be attributed to the environment and an interaction between the environment, driver and the vehicle itself. These include external conditions that make the road tra c unsafe, like poor visibility due to darkness, rain, fog or snow, slippery conditions, and poor condition of the road and associated infrastructure. A combination of the eects of the environment, driver and vehicle can be a misjudgment on the part of the driver about the slipperiness of the road and the inability of the active safety systems to stabilize the vehicle in the event of loss of grip in such conditions.
2. Vehicle: The condition of the vehicle itself also plays an important part in road safety.
However, the probability of an accident caused by the fault of the vehicle alone is relatively low, and the combined eects of faulty vehicle, environment and the driver together make up for about 10% of the causes of tra c accidents. These reasons could be explosion of tires, blind spots and other mechanical or electrical failure in the vehicle.
3. Driver: The role of the driver in a tra c accident is the most common and can be accounted to contribute to 90% of the causes, which includes some combined eect of the driver and the environment and the vehicle. The involvement of the driver with the tra c plays a crucial role and a majority of the accidents are caused due to distraction and tiredness or under the in uence of alcohol.
1.1.2 The Role of Active Safety and Friction
To tackle these main causes of road accidents, simultaneous eort is being put on to improve the infrastructure, and the vehicle, to be able to reduce the eect of the environment and the vehicle on road accidents. Also, a major trend within the automotive industry now is to shift towards autonomous driving, so that the in uence of driver carelessness, tiredness and intoxication on road accidents can be minimized. Major steps within improvement of infrastructure include raising the safety standard of roads, improving maintenance during slippery conditions and improving the light- ing in darkness [16]. To be able to make vehicles safer, a lot of research is being carried out within the domains of passive and active safety, which have improved tremendously in the last decade. Passive safety deals with the protection of the occupants of a vehicle in case of an accident. These primarily include seat belts, airbags, crumple zones and the side impact protection. However, it is also im- portant to focus on making vehicles safer in a way that accidents do not happen in the rst place.
Here, active safety systems like Electronic Stability Control (ESC) play a major role in ensuring the safety of a vehicle under dangerous driving conditions like slippery road, evasive maneuvers, etc.
Apart from ensuring stability, the active safety systems like the traction control and the Anti-lock Braking System (ABS) provide adequate traction for accelerating and braking respectively, without compromising the steer-ability of the vehicle, by preventing wheel lock or wheel-spin.
Active safety systems have been quite crucial in preventing road accidents since their advent in the late 90's, and this was evaluated in the Unites States between the years 1997 and 2002. Single vehicle crashes were monitored during that period from 5 dierent states for speci c cars with ESC, and were compared to the earlier versions of the same car without any ESC, and the results can be seen in Figure 1.4a and 1.4b. It shows the percentage reduction of single vehicle crashes, and fatal single vehicle crashes, both in passenger cars and SUVs.
(a) Single vehicle crashes (b) Fatal single vehicle crashes
Figure 1.4: Role of ESC in reducing single vehicle crashes in USA for 5 states between 1997-2002 [17]
From the gure above, it can be clearly seen that ESC systems have been very eective in reducing the number of accidents by a substantially signi cant amount. This justi es the motivation behind improvement of active safety systems to make vehicles safer, and thus to reduce the number of ac- cidents.
Active safety systems are mainly control functions that need a lot vehicle state estimates to be able to calculate the required level of control intervention to stabilize the vehicle. As drivers, we are sometimes able to adapt our driving style when the road feels slippery, and help the active safety systems. However, with the industrial trend towards autonomous drive, it becomes even more essen- tial to know the friction level. This is because as drivers, we rely on our senses to judge the current road condition and take corrective actions accordingly, but with autonomously driven cars, it will become very important to know the friction coe cient with substantial con dence to be able to help the active safety systems to take corrective actions, to avoid accidents. Then, to be able to calculate the vehicle states accurately it is quite important to know what is the friction level between the tire and the road, as a lot of states depend on it. Furthermore, the coe cient of friction µ limits the amount of force a tire can generate, which can aect both state estimates and control functions quite signi cantly. Driving on slippery surfaces with low µ increases the probability of an accident as the tires are incapable of producing enough forces to be able to stabilize the vehicle in time. A study at the Swedish national road and transport research institute shows the correlation between the friction coe cient and the number of accidents as seen in Figure 1.5. This shows how the chance of an accident increases with decrease in the friction coe cient. This places higher demands on the active safety controllers, as the tires are not able to respond to the control inputs, reducing their eectiveness. Thus, if the friction can be estimated, control actions can be taken properly by autonomously driven cars in low µ conditions, to be able stabilize the vehicle before it is too late.
Figure 1.5: Correlation between the number of accidents and the coe cient of friction for 10 million vehicle kilometers. [18]
1.1.3 Challenges of Friction Estimation
Realizing the importance of friction, various methods have been developed to estimate friction, and it is one of the most researched topic in the automotive industry. The methodologies can however be divided into two broad categories, namely:
1. Non-contact based method: The idea behind the non-contact based methods is a predictive approach to understand the road surface, to determine the coe cient of friction. These meth- ods rely on the use of camera and microphones to detect the road texture through luminance and using auto-correlation techniques to predict the friction level [19]. Although the method is predictive, but the involvement of the tire itself in the estimation process is absent. This might cause errors in the estimation, as the friction coe cient not only depends on the road texture, but also equally on the tire properties. Moreover, the estimate is dependent on a lot of additional sensors that might be unavailable in a production car due to economic reasons.
2. Sliding contact based method: To involve the tire in the estimation process, it is important to determine the forces acting on the tire. Majority of these methods involve estimation of the tire force and slip values, and then using a tire model to t the estimated force and the slip to calculate the maximum friction coe cient [20] [21]. However, the friction level can only be estimated with certainty if there is some sliding in the contact patch of the tire. With increased sliding, the estimation of friction becomes better. This is because, when the tires are operating in their linear range of operation, it is di cult to dierentiate whether the surface is slippery or not, since the force build up is very similar for lower values of slip. It is only when the tire contact patch starts to slide at higher values of slip, that the force starts to saturate earlier for low friction surfaces, and thus the estimation of the friction coe cient becomes reliable [5].
To make the tires reach their non-linear range of operation, for friction estimation, it is important that they must be excited up to a level where there is some sliding in the contact patch, to be able to get a reliable friction estimate. This can be done by an active tire excitation, where the tires are purposefully excited to the non-linear range of operation. Such active excitations can be induced by the driver while hard braking, accelerating or cornering. However, the possibility of a driver intentionally doing so it very rare and during normal driving tires are hardly utilized above their linear range as shown in Figure 1.6. The advantage with such an excitation is that the driver himself
is aware of it and is mentally prepared to encounter some behavioral changes in the vehicle during the excitation [22].
Figure 1.6: Diagram showing force characteristics at low and high friction with everyday driving.
Since the probability of a normal driver carrying out an active tire excitation during everyday driving is extremely rare, the possibility of getting a reliable friction estimate through this method is slim.
Although, there are situations in which the initial slope of the force-slip curve (slip stiness) varies for dierent surfaces, however, it is di cult to determine how much and what other factors in uence it apart from the surface itself. Thus, some sort of system induced tire excitation is needed to ensure that a reliable friction estimation can be carried out whenever required. The disadvantage of this method, however, is that the driver is not involved in carrying out the excitation and thus might be mentally unprepared for behavioral changes in the vehicle during the excitation [22].
Several active tire excitations have been investigated, which mostly involve longitudinal tire excita- tion through the power-train and brake control [23] [24]. The common idea is to add a propulsive force to one of the axles and to add an equal amount of brake force to the other axle to cancel out the net force on the vehicle and keep the longitudinal motion undisturbed, as seen in Figure 1.7.
However, the results from these are mostly based on simulations and an implementation of these have not been carried out on a real vehicle. Moreover, the possibility to accelerate, decelerate and turn is not investigated when the active excitation is being carried out, as the longitudinal excitation forces cancel out each other.
Figure 1.7: The concept of simultaneous propulsive and brake torque.
1.1.4 Probe-Friction - Volvo Cars
To address these challenges, a patent was led by Volvo Cars [25] in 2014, to devise a way of carrying out an active tire excitation for friction estimation, such that the driver demanded acceleration of the vehicle is unaected. The idea is to add a propulsive torque to the front axle, and the braking torque to the rear axle is controlled in a way that the driver demanded acceleration is achieved. An overview of the process is shown diagrammatically in Figure 1.8. In this gure, vehicle states like yaw rate, side-slip, steering angle etc. are represented by "G", which are used by the activation logic, together with the estimated friction coe cient "D" to check whether the vehicle is stable before and during the excitation. The activation signal "H" activates the "Vehicle controller 1"
which ramps up the propulsion torque, while maintaining the driver requested acceleration by con- trolling brake torque through a PID controller. These torque requests are represented by "C" and are sent to the "Vehicle." The estimated forces and the wheel speeds represented by "A" are used by the "Friction estimator." The estimated friction is then used by the "Auxiliary friction receivers"
and the "Vehicle controller 2", which represents all the other propulsion and brake controllers like ESP, ABS etc, that also request wheel torques through "E." The activation logic decides whether vehicle controller 1 or 2 will be sending the torque requests, depending on the stability of the vehicle.
Figure 1.8: Overview of the active tire excitation logic. A = Wheel-speeds and tire forces; C,E,F = Front and rear axle torque requests; D = Estimated friction coe cient; G = Other vehicle states; H
= Activation or deactivation [25].
This method of active tire excitation allows to maintain the drive-ability of the vehicle, while in- creasing the tire utilization for friction estimation. However, there are several challenges associated with this method which need special consideration:
1. Stability: The rst and foremost challenge is to ensure that the vehicle is stable during the active tire excitation. This is very important from a safety perspective, and any accidents due to active excitation can not be tolerated. It can be very dangerous if the amount of excitation is too large, which leads to a lock-up of the rear axle due to the brake torque, and thus induces lateral instability in the vehicle. This eect can be ampli ed a lot if the friction is too low.
2. Comfort: Having a forced excitation during normal driving may not be a very comfortable experience for the occupants of the car. The change of vehicle behavior due to the excitation can be totally unexpected by the driver which might lead to a loss of feeling of the car. Moreover the sound and vibration created by the actuators can be unpleasant for the occupants.
3. Fuel consumption: Adding a propulsion torque to the front axle needs excitation of the engine. This might cause the fuel consumption to increase signi cantly if the excitations are
repeated often. Thus, it is important to carry out the forced excitations in a way that the fuel consumption is minimized.
4. Component wear: The forced excitations may cause the components in the power-train and the brake system to be over-stressed due to the application of large amounts of propulsive and brake torque. Moreover the components in these systems are not designed for a loading scenario like this, and hence an active excitation can lead to a heavy wear and tear of the components.
1.1.5 Vehicle Lateral Stability
The most critical challenge in carrying out an active tire excitation lies in making it safe. Thus a good understanding of vehicle stability is imperative to the development of such a method for friction estimation. The lateral stability of a vehicle is best described by the yaw rate and the side-slip angle of a vehicle. When the driver turns the steering wheel, lateral forces are generated at the front axle, due to a slip angle at the front, which creates a yaw rate and a side-slip angle of the vehicle. This is then translated to the rear axle, which then develops a corresponding slip angle, generating some lateral force. The yaw moment generated by the front axle is then balanced by the rear axle, and the vehicle reaches a steady-state cornering situation. However, if the front axle looses grip before the rear, further turning of the steering wheel does not increase the yaw rate, and the vehicle tends to continue straight ahead. Such a situation is called under-steer. On the contrary, if the rear axle looses grip before the front axle, then the balancing yaw moment from the rear axle is not enough to counteract the yaw moment from the front axle, and thus the yaw rate starts to increase. In such a situation, the vehicle seems to turn more than the driver intends, and is called over-steer. While, if both the front and the rear axle loose grip simultaneously, then the side-slip angle of the vehicle tends to grow, but not the yaw rate. Then the heading direction of the vehicle stays as intended by the driver, and such a situation is called neutral-steer. The most dangerous of these cases is the over-steer case, in which the yaw rate can increase signi cantly which leads to the vehicle spinning about the z axis. The driver looses complete control of the vehicle and might get into a side-way collision, which is not so well protected for crash as the front.
The under and over-steer behavior of a vehicle is also dependent on a lot of factors like suspen- sion setup, mass distribution between the front and the rear axle, the coe cient of friction, and the amount of longitudinal force on the axle. The most important of these, during an active tire excitation is the longitudinal force. This is because tires have a limited capacity to generate forces, which depends on the friction coe cient and the normal load on the tire. If the tire is utilized fully in the longitudinal direction, its ability to generate forces in the lateral direction is severely reduced.
If this happens to the rear axle during an active excitation, where the rear axle is braked too much that it is close to getting locked up, then the lateral forces at the rear axle almost vanishes, leading the vehicle into an unstable over-steer condition.
Many stability control systems for vehicles have been developed, which are commercially available as the ESC systems. These are typically used to control the under and over-steer of a vehicle by dierential braking on one or more wheels. The main control logic for these systems are either side-slip control or yaw rate control, where the controller acts upon a deviation of these states from a model-predicted value. However, these controllers usually do not have an estimate of the fric- tion coe cient, which makes it di cult for them to be equally eective across dierent surfaces.
Friction estimation on the other hand, can be used to overcome this shortcoming. By using the information about the vehicle states and the friction coe cient, it is possible to create boundaries for stable vehicle handling. Knowing the limits of stability, it is then possible to take control actions such that the vehicle is never able to cross these boundaries. This could provide a clear advantage
by taking preventive action based on the knowledge of the handling limits, over conventional ESC systems, which react when the vehicle crosses its handling limit. Such a limit based predictive control system can be compared to the envelope protection systems [26] found in commercial air- crafts, where the system restricts excitations that can lead to instability. Control methodologies on similar principles have been discussed, where the yaw rate and the side-slip phase plane diagrams were used to identify bifurcation points that indicates vehicle instability [27]. The Milliken Moment Method also describes the handling limits of a vehicle using lateral acceleration and normalized yaw moment over a broad range of steering and side-slip angles [28]. Phase portrait diagrams of yaw rate and side-slip angle was also used in [29] and [30] to de ne boundaries between stable and unstable regions, and to design a controller to keep the vehicle within these boundaries. However, these techniques primarily use a simple vehicle model, focusing only on the lateral dynamics during a cornering maneuver. No discussion of vehicle stability under forced excitation both at the front and the rear axle, for friction estimation, has been carried out, where the longitudinal dynamics also becomes important, as the vehicle might need to accelerate and decelerate during friction estimation.
1.2 Objective of the Thesis
The scope of this thesis is to dive deeper into the understanding of vehicle lateral stability during forced excitation for friction estimation. The aim is to focus only on the safety aspect of friction estimation using active tire excitation, from a stability point of view. In general, this thesis will try to answer the following research questions:
1. What types of vehicle and tire models can be used to understand the lateral stability of a vehicle under forced excitation of both the axles, and which of them represent the actual vehicle behavior more accurately under such a scenario.
2. What methods are suitable to evaluate and quantify lateral stability when longitudinal dy- namics is aected by a forced excitation at the axles.
3. How to understand when it is safe enough to carry out an active tire excitation, and how to decide when to stop, depending on the vehicle's stability during the forced excitation.
The expected outcome from this thesis would be a stability monitor, that is able to decide whether it is safe to carry out an active tire excitation, and indicate to stop the excitation if the vehicle tends to become unstable.
1.3 Limitations of the Thesis
The scope of this thesis is limited to gain an understanding of vehicle lateral stability under forced excitation, and to use that knowledge in building a stability monitor that indicates whether the vehicle is stable before and during an excitation. This thesis work does not discuss control methods to stabilize the vehicle if it tends to become unstable.
The challenges involved in realizing active tire excitations are many, as discussed in Section 1.1.4, but this thesis will not try to discuss all the challenges except the stability aspect.
The most detailed vehicle model used in this thesis work is limited to the bicycle model with 5 degrees of freedom, and more complicated vehicle models like the two-track model etc. are not used due to higher computational demands. 4 dierent tire models are discussed in this thesis in order to compare which one represents the actual tire in the closest possible way, without increasing compu- tational need and mathematical complexity. However, there might be more tire models that can be used and compared, which give a better compromise between mathematical complexity and accuracy.
1.4 Outline of the Thesis
This thesis is organized into 8 chapters and a brief overview of each chapter is given below:
Chapter 1: This chapter forms as an introduction to the thesis, discussing the background, objectives and the limitations of the thesis.
Chapter 2: This chapter describes the theory about the vehicle and tire models, tire force estimation, and stability analysis that is used throughout the thesis.
Chapter 3: This chapter describes the simulation tools and the test vehicle that is used in the thesis for simulation and implementation of the work.
Chapter 4: This chapter discusses the lateral stability of a bicycle model with 2 degrees of freedom, under a forced excitation.
Chapter 5: The use of an extension of the bicycle model into 5 degrees of freedom, together with dierent tire models, to understand the stability of the vehicle using phase portrait methods is presented in this chapter.
Chapter 6: This chapter describes the use of the knowledge from Chapter 5, about vehicle stability using phase portrait methods, to develop a stability monitor, which is able to indicate the stability of the vehicle in the event of a forced excitation.
Chapter 7: This chapter contains a general discussion of the results and its applicability in a broader picture. It also discusses the applicability of the proposed method for stability monitoring, its advantages, limitations and the assumptions
Chapter 8: This chapter discusses the future prospects of the thesis, and how it can be ex- tended.
2.1 Coordinate System
The coordinate system used throughout this thesis is based on the ISO 8855:2011 [1] and can be seen in Figure 2.1. It is a vehicle xed coordinate system, with its origin on the x − y or the road plane, exactly below the CoG, and moves along with the vehicle. Thus it is a non-inertial frame of reference.
The positive x direction is towards the front of the vehicle. The positive y direction is towards the left of the vehicle and the positive z axis is vertically upwards. All rotations and angles around these axes follow the right hand convention for the positive direction. A rotation about the x axis is called the roll and the corresponding roll rate is denoted by ωx. Similarly, a rotational velocity about the yaxis is the pitch rate ωy and a rotational velocity about the z axis is the yaw rate ωz. The angular dierence between the heading direction and the pointing direction of the vehicle, or in other words, the angular dierence between the total velocity vector v and the x axis is called the side-slip angle β.
Figure 2.1: Vehicle axis system de ned by ISO 8855:2011 [1]
2.2 Tire Modeling
The importance of tires in a vehicle can not be overstated. They generate all the forces and moments that are required to propel, control and stabilize a vehicle. The force generation from a tire is usually
described as a function of the slip. The lateral force from a tire depends on the slip angle α, which is the angular dierence between the heading direction and the pointing direction of the tire, which can be seen in Figure 2.2. The longitudinal force, on the other hand, can be expressed as a function of the slip ratio s, which can be understood as a longitudinal deformation in the tire contact patch, and can be expressed as in Equation 2.1. Here, ω is the angular velocity of the tire, Reis the eective rolling radius, and vx is the longitudinal velocity of the tire. However, the de nition of slip ratio varies slightly with dierent tire modeling approaches and will be mentioned.
Figure 2.2: Slip angle in a tire
s =
ωRe−vx
ω if |ω| > |vx|
ωRe−vx
vx if |ω| < |vx|
(2.1)
Typical pure slip-force curves for a tire can be seen in Figure 2.3. The lateral force (Fy) vs slip angle curves for dierent normal load levels (Fz) are shown in Figure 2.3a. It can be observed that for lower slip angles, the lateral force is linearly proportional to the slip angle, but as the slip angle increases, this relation deviates from linearity, and achieves a peak value before it drops o for even higher slip angles. Similar to the lateral force characteristics, it can be seen in Figure 2.3b, that the longitudinal force also shows a similar behavior with slip ratio, where it is linear for small values of longitudinal slip, and then shows a non-linear behavior for higher slip ratios. Moreover, compared to the lateral force curve, it can be observed that the peak of the longitudinal force is rather sharp, and occurs at relatively low value of slip ratio. Also, like the lateral force curve, the longitudinal force also shows a declining trend for higher slip ratios, but tends to saturate to a constant value for very high slips. Another interesting aspect of the tires is that the grip in the longitudinal and lateral direction might be dierent. As seen from the gure, it can be noticed that the grip in the lateral direction is slightly more than the longitudinal direction, with a higher peak in the lateral force curve. However, this characteristic might show an opposite trend with the longitudinal grip being higher than the lateral grip. Moreover, these lateral and longitudinal force characteristics are also aected in combined slip operation, when lateral and longitudinal slips are present together.
(a) Lateral force characteristics (b) Longitudinal force characteristics
Figure 2.3: Lateral and Longitudinal Force Characteristics of a Typical Tire at dierent Normal Loads
Tires have a physical limit on the amount of total force that it can generate, which depends on the friction coe cient µ and the normal load on the tire Fz. The lateral and the longitudinal force gener- ation capability of a tire is thus limited, and dependent on each other. Figure 2.4 shows a typical tire characteristics in combined lateral and longitudinal slip. The lateral force Fy has been plotted for dierent levels of slip angle and increasing longitudinal braking force Fx. It can be seen very clearly that for a tire operating at a constant slip angle, an increase in the longitudinal force decreases the lateral force. This eect is less profound for lower slip angles and lower levels of longitudinal force, and increases substantially for higher slip angles and longitudinal forces. Another interesting observation from this gure is that, as the longitudinal force increases beyond a threshold, that is, when the slip ratio corresponding to the peak longitudinal force is crossed, both the longitudinal and lateral forces start decreasing, and this eect becomes more important when the wheel is close to locking, then, the tire looses substantial ability to generate both lateral and longitudinal forces.
Figure 2.4: Combined lateral and longitudinal characteristics for a typical tire [4]
It can be concluded from the above discussion that tires, although one of the most important com- ponents of the vehicle is highly non-linear, and di cult to express mathematically. Many modeling approaches have been tried till date, in order to express the force generation mathematically, and all of them depend on simpli ed assumptions regarding the physics of the tire. Each of them have their own advantages, disadvantages and area of applicability, depending on the purpose for which it is used. Some of these models relevant to this study are discussed subsequently in more detail,
with a focus on the assumptions and the limitations of each.
2.2.1 The Linear Tire Model
One of the fundamental tire models that can be used to describe the lateral force generation of a tire is the linear tire model, which expresses the lateral force generated by the tire, as a linear function of the slip angle. However, an inherent assumption in this case is that the magnitude of the slip angle is essentially small. This is because at higher slip angles the force generation is actually non-linear, and it can no longer be expressed as a linear function of the slip angle. A typical lateral force curve, as a function of the slip angle is shown in Figure 2.5.
Figure 2.5: Lateral force vs slip angle for a tire showing the linear range of operation [3]
Fy = −Cyα (2.2)
It can be observed from this gure that a linear relationship exists between the lateral force and the slip angle close to the origin, and the lateral force can be expressed as in Equation 2.2, where Cy is the cornering stiness. The cornering stiness Cy, as de ned, is the slope of the lateral force vs slip angle curve when the slip angle α −→ 0 and it depends on the normal load Fz of the tire.
An important thing to note here is that, due to the sign convention, as described in Section 2.1, a negative slip angle α generates a positive lateral force Fy, and thus the lateral force is expressed as a negative product of the slip angle and the cornering stiness. For a real tire, the rate of increase of the lateral force gradually decreases as the slip angle exceeds the linear range of operation, and thus exhibits a non-linear behavior. It is also worth nothing that the maximum force that the tire can generate is limited by the amount of normal load and the friction coe cient µ. Thus a linear tire model can be used under the assumption that the slip angles are small.
2.2.2 The Brush Model
One of the earliest and the most fundamental physical model to understand the force generation in a pneumatic tire is described by the Brush model [4]. It is based on a simpli ed assumption that the tire is in contact with the road through exible elements called bristles. The relative motion between the tire and the ground gives rise to a slip, that leads to a deformation of the bristles.
The bristles have a stiness in the lateral and longitudinal direction, and hence are able to produce forces as a result of these deformations. Thus, the forces can be expressed as a function of the tire slip. Figure 2.6 shows the tire depicted in the form of bristles around its circumference. It shows a combined slip operation, where the bristles in the contact patch C are deformed in both longitudinal and lateral direction, and are producing force in both directions (Fy and Fx). It is also assumed that the contact patch has a parabolic pressure distribution with maximum pressure at the middle of the contact patch which gradually drops down to zero at the edges. Here, ω is the rotational speed of the tire and α is the slip angle caused due to lateral deformation.
Figure 2.6: Concept of brush model of a tire [4]
The lateral and longitudinal force generation from the brush model in combined slip conditions can be described by the Equations 2.3 - 2.11. Here, a is the length of the contact patch and cp is the stiness per unit length. Assuming that the tire is isotropic with equal stiness in the lateral and longitudinal direction (cpx= cpy= cp), cp can be derived from the cornering stiness Cy. σx and σy
are the longitudinal and lateral slips respectively, derived from the longitudinal velocity vx, wheel rotational velocity ω and the eective rolling radius Re. σ is the total slip, and is dependent on the longitudinal and the lateral slip quantities. F is the total force, with longitudinal and lateral com- ponents Fx and Fy respectively. Since this formulation assumes a parabolic pressure distribution, there is always a sliding region in the contact patch, and σm describes the value of slip at which the sliding starts. It also assumed that the lateral and longitudinal coe cient of friction are equal and constant (µx = µy = µ).
cp= Cy
2a2 (2.3)
θ = 2cpa2
3µFz (2.4)
σm = 1
θ (2.5)
σx= ωRe− vx
ωRe (2.6)
σy = vxtan α
ωRe (2.7)
σ = q
σ2x+ σy2 (2.8)
F =
(µFzθσ(3 − 3|θσ| + (θσ)2) if |σ| < σm
µFz if |σ| ≥ σm
(2.9)
Fx= F σx
σ (2.10)
Fy = F σy
σ (2.11)
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−1.5
−1
−0.5 0 0.5 1 1.5
Normalized Lateral Force Fy/Fz
Slip Angle [rad]
Lateral Force vs Slip Angle for a Brush Model
µ = 1 µ = 0.5 µ = 0.3
(a) Pure lateral slip characteristics
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1.5
−1
−0.5 0 0.5 1 1.5
Normalized Longitudinal Force Fx/Fz
Slip Ratio
Longitudinal Force vs Slip Ratio for Brush Model
Traction →
← Braking
Traction →
← Braking
Traction →
← Braking
µ = 1 µ = 0.5 µ = 0.3
(b) Pure longitudinal slip characteristics
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Combined Slip Operation in Brush Model
Fx/Fz Fy/Fz
α = 2 deg α = 5 deg α = 10 deg
(c) Combined slip characteristics at dierent slip angles
Figure 2.7: Characteristic curves for the brush model in pure lateral, longitudinal and combined slip
Figure 2.7 shows some of the characteristic curves of the brush model in pure lateral, pure longi- tudinal and combined slip. All of these plots are normalized with the normal load Fz. It can be seen clearly that the brush model tends to capture the non-linear behavior of the tires. Both the lateral and the longitudinal forces behave linearly for lower slips, and then saturate for higher slips as seen in Figure 2.7a and 2.7b. This saturation limit is given by µFz. However, it does not have a well de ned peak force and neither the drop in force for higher slips. This is why the combined
slip characteristics does not totally resemble the real tire, as it can be seen in Figure 2.7c. Here, the normalized lateral force is plotted against the normalized longitudinal force for dierent levels of constant slip angles, and it can be seen clearly that unlike the real tire shown in Figure 2.4, the lon- gitudinal force does not decrease abruptly at high longitudinal slips, when the tire is close to locking.
2.2.3 The Dugo Tire Model
The Dugo tire model [10] is based on the Fiala model [11] with some assumptions for mathematical simpli cation. It is based on an idealized tire-road contact geometry as seen in Figure 2.8 under a uniform pressure distribution over the contact patch. The model can be understood as a simpli ed version of the Brush model, by using a uniform pressure distribution in the contact patch. The camber angle of the tire is considered to be zero, and the eects of camber (if any) are included using an equivalent slip angle.
Figure 2.8: Tire-road contact geometry - Dugo model [10]
Here, the ground surface is denoted by the − η plane. 3-4 represents the carcass centerline, 0-1-2 represents the tread centerline and α is the slip angle. 0 is the point where the tread enters the con- tact patch and leaves from point 2. Point 1 indicates the initiation of the sliding region. The carcass is assumed to be elastically connected to the tread through orthogonal springs, which generate forces independently in the η and directions, depending on the individual stinesses and the deformation.
The tread is assumed to start sliding from point 1 where the stress due to deformation of the tread exceeds the tire-road shear stress limit. Using the above concept for the tire-road contact geometry, Equation 2.12 - 2.17 describe the force generation of the Dugo tire model.
¯
s = sCx
µFz(1 − s) (2.12)
¯
α = Cytan α
µFz(1 − s) (2.13)
¯ sr=p
¯
s2+ ¯α2 (2.14)
Fr
µFz
=
¯
sr if ¯sr< 0.5 1 −4 ¯1s
r if ¯sr≥ 0.5
(2.15)
Fx = ¯s
¯ sr
Fr (2.16)
Fy = − ¯α
¯ sr
Fr (2.17)
Here, ¯s is the normalized longitudinal slip vector derived from s, the slip ratio given by the Equation 2.43, Cx is the longitudinal stiness, µ is the friction coe cient and Fz is the normal load on the axle. Similarly, ¯α is the normalized lateral slip vector derived from the slip angle α and the corner- ing stiness Cy. The normalized longitudinal and the lateral slip vectors are used to calculate the resultant normalized slip vector given by ¯sr. This is then used to calculate a normalized resultant force µFFrz, which constitutes the normalized longitudinal and lateral force components µFFxz and µFFyz. Figure 2.9a and 2.9b shows the normalized lateral and longitudinal force characteristics at pure slip for a Dugo model at dierent friction levels, at a constant normal load Fz. These gures clearly show that the forces from the Dugo tire model tends to saturate at higher slips. In contrast to the linear model, this is more realistic as and it is able to capture the limitation of the force generation capability of the tires at higher slips. However, when compared to the Brush model, the forces asymptotically approach the friction limit (µFz) for higher slips in the Dugo model as opposed to a constant value in the Brush model. This is dierence is because of the assumption that the pressure distribution is constant in the contact patch in the Dugo model as opposed to a parabolic pressure distribution in the Brush model.
Figure 2.9c shows the normalized combined slip characteristics of the Dugo tire model at a constant normal load Fz. Here, the normalized lateral force Fy/Fz is plotted against the normalized longi- tudinal force Fx/Fz, at dierent slip angles, for increasing slip ratios. It can be noticed from these
gures that as the slip ratio is increased, maintaining a constant slip angle (moving horizontally along a particular curve), the lateral force drops o quite signi cantly at higher slip ratios. This eect is more pronounced at higher slip angles, and the drop in lateral force with longitudinal slip is quite signi cant. The Dugo model shows similar trend like the Brush model due to their modeling similarities. However, they both have the same limitation while capturing the real behavior of the tires at high longitudinal slips when the tire is close to locking. This is because both these models do not have a drop in longitudinal forces for high slip ratios, and thus are unable to capture the eect of locked wheels on lateral force generation capability.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Normalized Longitudinal Force Fx/Fz
Slip Ratio
Longitudinal Force vs Slip Ratio for Dugoff Model
µ = 0.3 µ = 0.5 µ = 1
(a) Pure longitudinal slip characteristics
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Normalized Lateral Force Fy/Fz
Slip Angle [rad]
Lateral Force vs Slip Angle for Dugoff Model
µ = 0.3 µ = 0.5 µ = 1
(b) Pure lateral slip characteristics
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fx/Fz Fy/Fz
Combined Slip Operation for Dugoff Model
α = 2 deg α = 5 deg α = 10 deg
(c) Combined slip characteristics at dierent slip angles
Figure 2.9: Lateral, longitudinal and combined slip characteristics for Dugo tire model
2.2.4 The Modi ed Dugo Tire Model
To overcome the limitations of the Dugo tire model as mentioned in Section 2.2.3, a modi ed Dugo tire model was proposed [12]. The objective of this model was to capture the characteristics of a real tire as closely as possible, without too many parameters and thus reduce computational de- mand. The main reason why the Dugo tire model does not show a de nite peak, both in the lateral and longitudinal force characteristics, is because the friction coe cient is assumed to be constant throughout the contact patch. However, there is always dierence between the static and the sliding friction coe cient, where the static coe cient is higher than the sliding one. The modi ed Dugo
tire model implements this concept to express the friction coe cient as a linear function of slip, where the friction coe cient reduces for higher slips. Equations 2.18 to 2.21 describe the variation of the friction coe cient with slip in the modi ed Dugo model.
µx = µmaxx − (µmaxx − µminx )ss (2.18)
µy = µmaxy − (µmaxy − µminy )sα (2.19)
ss= |s| (2.20)
sα= min(1,
vxsin α max(vxcos α, ωRe)
) (2.21)
Here, the longitudinal and lateral coe cients of friction (µx and µy), are expressed as a function of the absolute longitudinal slip ratio ss, and absolute lateral slip ratio sα respectively. µmaxx is the peak longitudinal coe cient of friction, µminx is the longitudinal sliding coe cient of friction, µmaxy
is the maximum lateral friction coe cient, and µminy is the lateral sliding friction coe cient. These can be derived from the coe cients of the Magic Formula tire model as given in Equation 2.22 and 2.23. In these equations, Dx,y is the peak longitudinal or lateral force at pure slip condition, and F0x,y|s=1,α=π
2 is the value of the longitudinal force at s = 1 or the value of lateral force at α = π2 in pure slip conditions. IFx and IFy are increase factors for scaling the longitudinal and lateral coe cient of friction respectively, and usually varies between 10% − 20% [12]. Table 2.1 enlists the maximum and minimum friction coe cients in the longitudinal and lateral direction, used in the modi ed Dugo model, for increase factors IFx= 12.5%and IFy = 15%for a nominal friction value of µ = 1.
µmaxx,y = Dx,y
Fz IFx (2.22)
µminx,y = F0x,y|s=1,α=π
2
Fz IFy (2.23)
Table 2.1: Friction coe cients for modi ed Dugo model
Parameter Symbol Value Unit
Maximum Longitudinal Friction Coe cient µmaxx 1.215 - Minimum Longitudinal Friction Coe cient µminx 0.8 - Maximum Lateral Friction Coe cient µmaxy 1.265 - Minimum Lateral Friction Coe cient µminy 0.13 -
The longitudinal and lateral coe cients of friction from Equations 2.18 and 2.19 are used along with Equations 2.12 to 2.17 to obtain the lateral and longitudinal force characteristics of the modi-
ed Dugo tire model. Figure 2.10a and 2.10b shows the normalized longitudinal and lateral force characteristics for the modi ed Dugo model for dierent levels of friction, for a constant normal load Fz. From these gures, it can be seen clearly that, due to the introduction of a slip dependent friction, both the lateral and longitudinal force characteristics now have a distinct peak. The forces then decrease linearly after the peak up to a point where µ = µmin. This de nitely addresses the limitation posed by the Dugo tire model, where the forces asymptotically approached the friction limit. However, the decrease of forces beyond the peak is usually non-linear for a real tire, which might lead to an overestimation of the force drop by the modi ed Dugo model for higher slips.
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−1.5
−1
−0.5 0 0.5 1 1.5
Normalized Longitudinal Force Fx/Fz
Slip Ratio
Normalized Longitudinal Force characteristics for Modified Dugoff model
µ = 0.3 µ = 0.5 µ = 1
(a) Pure longitudinal slip characteristics
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Normalized Lateral Force Fy/Fz
Slip Angle [rad]
Normalized Lateral Force characteristics for Modified Dugoff model µ = 0.3 µ = 0.5 µ = 1
(b) Pure lateral slip characteristics
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Combined Slip Operation for Modified Dugoff Model
Fx/Fz Fy/Fz
α = 2 deg α = 5 deg α = 10 deg
(c) Combined slip characteristics at dierent slip angles
Figure 2.10: Lateral, longitudinal and combined slip characteristics for modi ed Dugo tire model
Figure 2.10c shows the normalized combined slip operation of the modi ed Dugo model. In com- parison to the Dugo model, it can be clearly seen that as the longitudinal slip is increased for a constant lateral slip, the lateral force starts decreasing, and there comes a point, after which, increas- ing the longitudinal slip further, reduces both the lateral and the longitudinal forces abruptly. This is more realistic as compared to the Dugo model, where this simultaneous reduction of longitudinal and lateral forces is absent for higher slip ratios. This phenomenon is captured with the modi ed Dugo model, as the coe cient of friction is not constant and decreases as a linear function of the slip. This makes the friction coe cient reduce for higher slip ratios, where most of the contact patch is sliding, and thus, both the lateral and the longitudinal forces reduce suddenly.
2.2.5 The Magic Formula Tire Model
The Magic Formula is one of the most widely used tire models in the automotive industry due to its robustness at high slips. It is a semi-empirical tire model, the coe cients of which, are derived using a curve tting technique to measured test data. This model was jointly developed by TU Delft and Volvo Cars, and several versions of the same have been published [13] [?]. The modi ed Dugo
model is able to represent the behavior of a real tire to quite a large extent, however, as seen in Sec- tion 2.2.4, the modi ed Dugo model is based on the Magic Formula tire model for the calculation of its friction coe cients. This makes the model dependent, on the Magic Formula. Equations 2.24 to 2.26 describe the force generation using the Magic Formula tire model.
