1 Car Modeling with Applications to Model-based Anti-spin Examensarbete utfört i Reglerteknik vid Tekniska Högskolan i Linköping av Erik Wernholt Reg nr: LiTH-ISY-EX-3111
3 Car Modeling with Applications to Model-based Anti-spin Examensarbete utfört i Reglerteknik vid Tekniska Högskolan i Linköping av Erik Wernholt Reg nr: LiTH-ISY-EX-3111 Supervisor: Urban Forssell, Anders Stenman Johan Löfberg Examiner: Fredrik Gustafsson Linköping, 26th January 21.
5 Abstract i Abstract Anti-spin systems for cars are becoming more and more common. Today's anti-spin control systems typically use a PID controller to limit each individual wheel's spin. The controller's parameters depend on different operating conditions and its implementation on new platforms has proven to be timeconsuming and costly. The trend in the automotive industry to solve many of these problems is to use model-based control systems. In this thesis, different aspects of model-based anti-spin have been investigated and it appeared to involve quite a few difficult problems. To be able to design and evaluate different anti-spin approaches, the car needs to be modeled. Two different models are constructed, one advanced simulation model for validating the complete system and one simpler model used for control purposes. The simulation model appears to work properly for all velocities of interest and while starting from rest. For the anti-spin system, the main contributions are an algorithm for estimation of tire/road friction and tire properties, a strategy for the calculation of a reference slip value, and design of a controller. The complete anti-spin system is tested on the simulation model and the results seem promising for possible improvements in future systems. Further validation and development of the system are needed, but perhaps the design of future anti-spin systems could be made more methodically when using the ideas presented in this thesis. Key Words: MODEL-BASED CONTROL, ANTI-SPIN, CAR MODEL- ING, FRICTION ESTIMATION.
6 ii Abstract
7 Abstract iii Acknowledgments First of all I would like to thank my supervisors at NIRA, Urban Forssell and Anders Stenman, for all inspiration and guidance throughout my work and for patiently reading and commenting the manuscript. I would also like to thank Mats Löfgren for introducing me to the important parts of an anti-spin system and for his descriptions of NSC (NIRA Spin Control). My supervisor at the University, Johan Löfberg, thanks for the new ideas about the slip controller. Sorry, I ran out of time and was unable to test all the ideas. Finally, many thanks to Martin Enqvist, Peter Hall and Martin Samuelsson for all help and fruitful discussions and to all the people at NIRA in Linköping for making my time so enjoyable.
8 iv Abstract
9 Notation v Notation Since many variables and parameters are used in the car model and in the anti-spin system, only the most essential and frequently used symbols are explained here. Extra indices are sometimes used on the symbols, e.g. fl (front left), fr (front right), rl (rear left), rr (rear right), f (front) and r (rear) for the wheels. Symbols i t k μ m b m w m tot r r nom r s s ref v b v x v y v z B F air F aux F d F lead F lag F s F sus Transmission ratio (function of gear) Tire parameter, slope in the μ x s curve Vehicle body mass Wheel mass Vehicle mass Wheel radius Nominal wheel radius Radius of an unloaded wheel Wheel slip Reference slip Vehicle body velocity (mass center) Longitudinal velocity (in some coordinate system) Lateral velocity (in some coordinate system) Vertical velocity (in some coordinate system) Track width Air resistance Force from anti-roll bar Damper force Phase lead controller Phase lag controller Spring force Suspension force, sum of F s, F d and F aux
10 vi Notation F x F y F z G s I b I e I t I w I w;eff K L L f L r L s L ff M b M b;driver M c M d M e M e;driver M r M s M t M w R T T e T s ff fi Longitudinal tire force Lateral tire force Vertical tire force Transfer function from M c to s Moment of inertia matrix of the vehicle body Engine moment of inertia Transmission moment of inertia Wheel moment of inertia Effective wheel moment of inertia Combined slip and slip angle Distance between front and rear wheels' axes Distance between front wheel axis and mass center Distance between rear wheel axis and mass center Slip relaxation length Slip angle relaxation length Wheel braking torque or Sum of moments acting on vehicle body Braking torque from driver Output signal from spin controller Output torque from driveline Engine torque (output) Engine torque request from driver Engine torque request (input to engine) Output torque from drive shaft Output torque from transmission Wheel torque, sum of M b and M d Curve radius Transformation matrix Engine time delay Sample period Wheel slip angle Vehicle side slip angle
11 Notation vii fl Camber angle (inclination of wheel) ffi w Road wheel angle ffi S Steering wheel angle Pitch angle, rotation around the lateral axis par Parameter vector used in the bicycle model μ Normalized total tire force μ max Tire/road friction coefficient μ x Normalized longitudinal tire force μ y Normalized lateral force fi s Drive shaft time constant ffi Roll angle, rotation around the longitudinal axis ψ Yaw angle, rotation around the vertical axis _ψ Yaw rate, speed of rotation around the vertical axis! Wheel speed of rotation! b Body speed of rotation vector! e Engine speed of rotation! t Transmission speed of rotation Abbreviations ABS CoG DSA MASC NSC In Un W Anti-lock braking system Center of gravity coordinate system Dynamic stability assistance Model-based anti-spin controller NIRA spin control Inertial coordinate system Undercarriage coordinate system Wheel coordinate system
12 viii Notation
13 Contents ix Contents 1 Introduction Vehicle Stability Systems Model-based Anti-spin Systems NIRA Automotive AB Objectives Limitations Thesis Outline Vehicle Stability Systems Introduction Block Structure Braking Torque Engine Torque Reduction Sensor Signals Existing Systems Car Modeling Introduction Coordinate Systems Suspension Wheel Body Driveline Braking system Steering System Implementation Validation of the Model Model-based Anti-spin Controller Simplified Vehicle Models Overall Structure Estimation of Tire Forces Estimation of Longitudinal Velocity, Slip and Slip Angle Estimation of Lateral Velocity and Yaw Rate Estimation of Tire/Road Friction and Tire Properties Reference Slip Calculation Slip Controller Test of Complete Anti-spin System
14 x Contents 5 Discussion Conclusions Extensions References 69 Appendix A: Test of Car Model: Varying Driving 71 A.1 Acceleration A.2 Cornering A.3 Braking Appendix B: Block Diagram of MASC 79 Appendix C: Test of MASC: Straight Ahead 81 Appendix D: Test of MASC: Cornering 85
15 1 Introduction 1 1 Introduction In today's automotive industry, much research and development aim at making the future vehicles safer. An anti-lock braking system (ABS) is often standard and airbags, seat-belt pretensioners and inflatable curtains are some other examples of new safety products. Other products that are becoming more and more common are different kinds of Vehicle Stability Systems", often called, e.g., anti-spin, anti-skid or traction control. These kinds of stability systems are the topic of this thesis. 1.1 Vehicle Stability Systems The name Vehicle Stability Systems reveals that there sometimes is some kind of vehicle instability that needs to be stabilized. Here this instability is divided into the quantities spin, slide and skid. A wheel spin often occurs during acceleration on slippery roads, like during snow or rain, causing the vehicle to lose steering ability. As a measure of the wheel spin, the concept slip is defined and often used instead of spin throughout the thesis. The most common types of slides are called understeer and oversteer, which mean that the vehicle turns less or more than some nominal value. The term skid is here used for the unstable state when the tires are unable to hold the car on track and the whole car slides to one side. There exist many stability systems on the market, but the manufacturers use different names for similar systems, leading to confusion. Three major groups can be identified, here denoted as ffl Traction Control (e.g. Volvo s TRACS) ffl Stability Assistance and Traction Control (e.g. Volvo s DSA or STC) ffl Stability and Traction Control (e.g. Volvo s DSTC) The differences between them are both sensors, actuators and strategy. The Traction Control system only controls the spin. To be able to stabilize the vehicle during a slide or a skid, the instability both needs to be detected and corrected. The Stability and Traction Control system performs that task. The Stability Assistance and Traction Control system reduces the risk for instability in curves by decreasing the allowed wheel slip, but is unable to actively correct instability when it occurs. Despite the differences, they all have in common that they try to control the tire/road contact forces by controlling the wheels' spin. Therefore the term anti-spin" is often used.
16 2 1 Introduction The type of system considered here is the Stability Assistance and Traction Control, but the others will be briefly discussed as well. 1.2 Model-based Anti-spin Systems Today's anti-spin control systems typically use a PID controller to limit each individual wheel's slip. The controller's parameters depend on operating conditions such as velocity and friction, and their implementation in new platforms has proven to be time-consuming and costly. After each change of car specific parameters, e.g. for a new version of a car, the system also must be retuned since the influence is unknown. The trend in the automotive industry to solve many of these problems is to use a model-based control system, like e.g. the Bosch VDC system. Some remarks about why to use a model-based anti-spin controller (MASC) are: ffl MASC has the potential to reduce implementation time and cost. ffl The model used in the controller includes the body, wheel and driveline, a friction estimator and a model for safety margins. Hence, MASC takes the whole state of the car into consideration, not only the wheel spins, which may improve certain safety margins. ffl The ideas presented here are applicable to vehicles such as cars, motorcycles, trucks, and trains. To be able to construct and evaluate several anti-spin approaches, the car needs to be modeled. At least two types of models are needed, one advanced simulation model for testing the system and one simpler model used for the control purposes. The minimum requirement of the simulation model is that the slip is modeled and that the normal forces change due to suspension dynamics. The model must also have similar behavior as a real car. The car modeling is one of the two parts in the thesis. The second part is a survey of the main parts in an anti-spin system. The main contributions in this part are an algorithm for estimation of tire/road friction and tire properties, a strategy for the calculation of a reference slip value, and a derivation of a controller. These parts are combined into an anti-spin system, which is tested on the simulation model. 1.3 NIRA Automotive AB This master thesis project has been carried out at NIRA Automotive AB. The company was established in 1985 and has 25 employees in three cities:
17 1.4 Objectives 3 Stockholm, Gothenburg and Linköping. The company is specialized in the design and implementation of reliable, time-critical electronic control systems for automotive, marine, and industrial applications. On three occasions NIRA Automotive has received the prestigious Volvo Technology Award for its innovative products. The Linköping office belongs to the business unit NIRA Dynamics and works in the areas of signal processing and control. The idea is to use information from existing or low-cost add-on sensors to compute high-precision virtual sensor signals. Two examples are virtual tire pressure and road friction sensors where wheel speed signals from the anti-lock braking system are the principal information sources. NIRA Dynamics is also responsible for the development of NIRA Spin Control (NSC), used for cars and motorcycles. More information about NIRA and their products can be found on the web page Objectives The objectives of this thesis can be summarized in the following items. ffl Learn more about the areas of car modeling and model-based antispin. ffl Construct a car simulation model that can be used for the development and testing of anti-spin systems and hopefully also in other NIRA projects. ffl Test some new ideas (from NIRA) about the design of a model-based anti-spin system. If possible, the following items will be considered as well. ffl Test the system in a real car for some simple driving cases. ffl Compare the performance with the NIRA Spin Control system (NSC). 1.5 Limitations The task of constructing both a simulation model and an anti-spin system in a master thesis project implies that some simplifications are needed. These are summarized in the following items. ffl The ground is assumed to be flat (no slopes) and the wheel suspension is simplified to always be perpendicular to the ground.
18 4 1 Introduction ffl A simple driveline model (engine, clutch, transmission) is used, which limits the design and testing of a high performance spin controller. ffl Validation of the simulation model is limited to the qualitative behavior. ffl The derived anti-spin system is a simplified version compared to a production system. In order to get a working product, much work is needed, including testing and tuning of parameters. ffl Since the NSC involves fuel blocking and this is a well working solution, the focus here is to improve the calculation of the slip reference value. ffl Sensor signals are used without offsets or noise. ffl Yaw rate (rotation around the vertical axis) is assumed to be known. ffl Parameter deviations in e.g. wheel radii, masses or time constants are not tested. 1.6 Thesis Outline Chapter 2 gives a more thorough introduction to the basic ideas of Vehicle Stability Systems". In Chapter 3 the car modeling is described and Chapter 4 deals with the model-based anti-spin system. The results are summarized in Chapter 5.
19 2 Vehicle Stability Systems 5 2 Vehicle Stability Systems 2.1 Introduction The only way to control the behavior of the car while driving is to change the forces acting on the tires. This is done by the driver using the engine, the brakes and the steering wheel. The relations between the driver inputs and the tire forces are important to understand in order to construct a well working stability system. Coordinate System Before we start describing the forces we need to define the vehicle geometry. According to ISO standard 1, a coordinate system is attached in the center of gravity of the car, with the x-axis pointing forward, the y-axis to the left and the z-axis upwards (see Figure 2.1). These directions are often referred to as longitudinal (x), lateral (y) and vertical (z). Rotations around these axes are described using the angles roll ffi, pitch and yaw ψ. This is further described in Section 3.2. Figure 2.1. Car with attached coordinate system. Tire Forces The forces acting at the tire/road contact point are the vertical force F z, the lateral force F y and the longitudinal force F x (see Figure 2.2). F z is the normal force which is varying depending on load and driving conditions. 1 ISO 8855, see e.g. . SAE standard is z-axis pointing downwards and y-axis right.
20 6 2 Vehicle Stability Systems This is further described in Section 4.3. F y is the force present during cornering and F x is the force due to braking and accelerations. F x is sometimes referred to as the traction force or just traction. From above From the side x z v α ω y F y x v x r F x F z Figure 2.2. A wheel seen from above and from the side. The tire/road contact can be modeled as a spring and in order to transfer a force, it needs to be stretched. In the longitudinal direction this is achieved if the wheel rotates faster than the corresponding longitudinal velocity. A so called tire slip s is then generated, defined as 2 s = r! v x v x (2.1) where! is the angular velocity of the wheel, r is the wheel radius and v x is the longitudinal velocity of the wheel (see Figure 2.2). The longitudinal force F x can then be seen as a function of the slip. The slip is often given in percent. To be noted is that the slip is not limited to 1 %, although this is used in most of the figures in this report. If there is a lateral force present then the tire will also undergo an elastic deformation in the lateral direction. This deformation gives rise to a slip angle ff defined (according to ISO and SAE) as ff = arctan v y jv x j 2 SAE J67e, see e.g. . ISO standard is s = r! vx r!. (2.2)
21 2.1 Introduction 7 where v x is the longitudinal velocity and v y is the lateral velocity of the wheel. As can be seen in Figure 2.2, ff is the angle between the wheel x-axis and the direction of movement. F y can be seen as a function of the slip angle. F x and F y also depend on the normal force F z and of course the tire/road friction coefficient μ max. Under normal driving conditions F x and F y can be seen as proportional to the normal force and the normalized forces μ x = F x =F z and μ y = F y =F z are defined. Examples of the normalized longitudinal force μ x are shown in Figure 2.3 as function of slip for different surfaces. μ max is then the peak value of the μ x s curve. 1.9 Asphalt, µ max = Wet asphalt, µ max =.7 µ x Snow, µ max = Ice, µ max = Slip (%) Figure 2.3. Normalized longitudinal force μ x for different slip and surfaces. During combined lateral and longitudinal forces, the forces depend on both slip and slip angles. This can be seen in Figures 2.4 and 2.5 for μ max = 1. Figures 2.3 to 2.5 are generated using a tire model described in Section 3.4. To be noted is that, according to the directions in Figure 2.2, a positive slip angle gives a negative lateral force. For simplicity, the absolute values are used throughout the report (e.g. in Figure 2.5) when the direction is not explicitly noted.
22 8 2 Vehicle Stability Systems µ x slip angle (deg) slip (%) Figure 2.4. Normalized longitudinal force μ x for different slip and slip angles µ y slip (%) slip angle (deg) 15 Figure 2.5. Normalized lateral force μ y for different slip and slip angles.
23 2.2 Block Structure 9 Vehicle Instability As mentioned in Chapter 1, there exist three kinds of vehicle instabilities that need to be stabilized. These are spin, slide and skid which will be further described here. During acceleration on slippery roads (e.g. snow or rain) there is a risk for unwanted wheel spin. This gives less traction and is often called lowtraction. As can be seen in Figure 2.5, increased spin also leads to less lateral force and hence a risk of losing the lateral stability. This means that during cornering, or on a banked road, the car might slide off the road. The most common types of slides are referred to as understeer and oversteer. In an understeer situation, the front of the car plows toward the outside of a turn without following the curve of the turn. In an oversteer situation, on the other hand, the rear of the car slides toward the outside of a turn, and the car is turning too much. The term skid is here used for the unstable state where the car has got a significant lateral velocity. During normal driving the lateral velocity is almost zero (small slip angles), but in a skid the lateral tire forces are too small to keep the car on track and the lateral velocity increases. To detect slides and skid, two variables are used as a measure of the behavior. These are the vehicle side slip angle fi and the yaw rate ψ. _ The vehicle side slip angle is defined as fi = arctan v y jv x j (2.3) and is the angle between vehicle lateral and vehicle longitudinal velocity. Large side slip angles cause an uncomfortable driving feeling and are hard to manage by untrained drivers. Yaw rate is the vehicle speed of rotation around the vertical axis. During a slide, the yaw rate is too small (understeer) or too large (oversteer) relative to some nominal value. During a skid the vehicle side slip angle is large. 2.2 Block Structure Most anti-spin systems can be described by the block scheme in Figure 2.6. The two blocks ffl Reference Slip Calculation ffl Slip Controller are described next.
24 1 2 Vehicle Stability Systems Figure 2.6. Block scheme describing anti-spin systems. Reference Slip Calculation Depending on the complexity of the system, the slip reference value s ref is calculated in different ways. For some systems only a fix value is used. In other systems the curve radius R is estimated and the slip s ref is decreased in curves in order to enhance vehicle stability. In the most advanced systems (see e.g. ) the driver inputs such as steering angle, brake pressure and accelerator are used to calculate the desired vehicle performance. This is compared with the actual performance and from this, different s ref values for all four wheels are calculated and given to the slip controller. Slip Controller The slip s can be controlled by applying a braking torque to the spinning wheels and/or by changing the engine torque. It is important to note that the controller can only keep or reduce the engine torque requested by the driver. This is a safety aspect similar to the restriction that the steering angle is not allowed to be changed by any controller. To control each wheel slip individually, the brakes must be used. Therefore the brakes are used as actuators in the most advanced Vehicle Stability Systems. However, in future electric vehicles there might be a separate motor for each wheel, which would enable individual wheel slip control without using the brakes. 2.3 Braking Torque The braking torque can either be applied to each wheel individually or using some differential brake or lock. For example, a differential lock forces the driving wheels to rotate with the same speed and is used in many off-road vehicles to prevent one-wheel spinning. Using the brakes to control the slip is a fast method. However there are many drawbacks included if the brakes are used without reducing the engine torque as well. Some of the drawbacks
25 2.4 Engine Torque Reduction 11 are limited working time due to heat, extra wear of the brakes, noise, and of course energy loss. 2.4 Engine Torque Reduction In today's systems there are mainly three different ways of reducing the engine torque ffl Changing the Ignition Angle ffl Using a Throttle Servo ffl Fuel Blocking To understand torque reduction, a short introduction to the combustion engine is necessary. This is a summary from , see Figure 2.7 for the notation. Figure 2.7. Combustion engine. From . A common restriction for all the reduction methods are that the emissions stay low during the reduction. To keep the emissions low and use the fuel efficiently, the air/fuel ratio (A=F ) needs to be equal to the stoichiometric ratio (A=F ) s. The air/fuel equivalence ratio = (A=F ) (A=F ) s (2.4)
26 12 2 Vehicle Stability Systems is often used and should be kept equal to one. >1 leads to increased oxides of nitrogen and <1 leads to increased carbon monoxide and hydrocarbons. For a four stroke gasoline engine the crank shaft makes two revolutions for each operating cycle. The piston moves down-up-down-up during these four strokes called intake, compression, expansion and exhaust. During intake the intake valve is open and the piston moves downward and fresh air and fuel are filled into the cylinder. Then the piston moves upward and the mixture is compressed to a higher temperature and pressure. Before the top position is reached, the mixture is ignited by a spark which initiates the combustion. The piston then moves downward and an engine torque is generated during this expansion. When the bottom is reached, the exhaust valve opens and while the piston moves upwards the fluid in the combustion chamber is pushed out into the exhaust system. Then a new operating cycle starts. Normally an engine has four or more cylinders and the operating cycle for each cylinder is scheduled so that the expansions are evenly distributed over the engine cycle. This gives an ignition every half engine revolution for a four-cylinder engine. Changing the Ignition Angle Normally the ignition takes place before the piston reaches the upper position. The ignition angle 3 has a direct influence on the output torque and emissions and hence the output torque can be reduced by changing the ignition angle. However changing the angle too much increases the risk for not igniting the fuel, which can damage the catalyst and enlarge the emissions. By using this method the output torque can only be partly reduced. The advantage, however, is the reaction time, since the ignition angle can be changed after the fuel is injected and compressed. Using a Throttle Servo By using a throttle servo, the throttle angle, and therefore the airflow into the cylinder, can be changed. The whole torque can be reduced but the drawback is the reaction time. It takes time to move the throttle, but worse is that the manifold contains about as much air as the cylinder volume. 3 The ignition angle is the angle of the crank shaft relative to the upper position. It depends on engine design, operating conditions etc, but is normally -35 ffi. See  for details.
27 2.5 Sensor Signals 13 Even if the throttle is fully closed it will take a couple of engine cycles before the engine torque is sufficiently reduced. Fuel Blocking For a fuel injection engine, the amount of fuel injected into each specific cylinder can easily be changed. Since the air/fuel ratio needs to be constant, the only alternative to the other methods is to fully block the fuel injected. This means that the air/fuel mixture in that specific cylinder becomes very lean (almost no fuel) and the emissions are insignificantly affected. This method is faster than the throttle servo but slower than changing the ignition angle. The engine torque can be totally reduced with a mean reaction time of 5/4 engine revolutions. This can be understood since the blocking takes place during the intake and the torque is generated during the expansion one revolution later. Furthermore, from the blocking signal there is a mean reaction time of 1/4 revolution until any intake takes place. One drawback with the fuel blocking is that the torque reduction is discrete. However, by blocking a different number of cylinders in one or several engine cycles the mean value of the engine torque can be changed in several steps. 2.5 Sensor Signals The performance of a stability system highly depends on how accurate the vehicle behavior can be determined. To measure the signals of interest different sensors are used (see e.g. ). The number of measured signals and their quality affect how accurate the vehicle behavior can be determined. The sensor signals considered here are ffl Wheel speed ffl Engine speed ffl Engine torque ffl Yaw rate ffl Lateral acceleration ffl Steering angle ffl Brake pressure
28 14 2 Vehicle Stability Systems ffl Accelerator pedal position The minimum requirement is that the wheel speeds are measured since these are used in the calculation of wheel slip. To estimate the important tire/road friction, the engine speed and engine torque and/or brake pressure are often used. The other signals are used in the most advanced systems and will be further described in Section Existing Systems In Chapter 1, three groups of existing systems were identified: ffl Traction Control ffl Stability Assistance and Traction Control ffl Stability and Traction Control These groups will be further described throughout this section, mainly using Volvo s stability systems as examples. The information about their systems is gathered from their homepage Traction Control One traction control system is the Volvo TRACS. It is used only during low velocities (less than 4 km/h) and is a kind of start-off system which prevents wheel spin. In order to get maximum traction force on all driven wheels, the brakes are used as actuators. The advantage is that different torques can be applied on the driven wheels, which provides additional traction when starting off on "μ-split" surfaces 4. This system is good for off-road vehicles and is used, e.g., in the Volvo Cross Country. A constant slip reference value is used with no consideration of cornering. Stability Assistance and Traction Control One such system is the Volvo DSA (Dynamic Stability Assistance) used in the Volvo S4/V4. In Volvo S6, S8 and V7 another system called STC (Stability and Traction Control) is used. STC works in all velocities. In low speed the brakes are used actuators and in high speed engine torque reduction is adopted. Here the DSA system will be used as an example. 4 A μ-split surface means different friction coefficients on the driving wheels.
29 2.6 Existing Systems 15 DSA functions at all speeds above 5 km/h. This system is fast enough to control the slip during all velocities which gives stability. For example, when driving on icy surfaces at 16 km/h, the system initiates a torque reduction within a distance of 25 cm! DSA is a system which only uses engine torque reduction and hence the start-off performance is not as good as for the TRACS system on μ-split surfaces. However, the drawbacks in using only the brakes are avoided. The difference between this system and the Bosch VDC described next, is that DSA does not control the vehicle side slip angle and yaw rate actively by applying a yaw moment 5. DSA tries to avoid sliding and skidding by reducing the slip in curves, but is unable both to detect the unstable state and generate a yaw moment. DSA is developed by NIRA Automotive AB. Stability and Traction Control There exist many systems which control both the vehicle stability and traction, see e.g.  for an updated list. These systems are called Stability and Traction Control throughout this report. Volvo uses the name DSTC (Dynamic Stability and Traction Control) in the models S6 and S8 for their system and Audi and Mercedes-Benz use the name ESP (Electronic Stabilization Program). The Bosch VDC (Vehicle Dynamics Control), described in , will be used here as an example. VDC uses both brakes and engine torque reduction to control the slip of each wheel individually. Controlling each wheel slip makes it possible to generate a yaw moment if the vehicle behavior differs from the wanted. This system measures wheel speeds, brake pressure, steering angle, accelerator, yaw rate and lateral acceleration. The system then estimates tire/road friction coefficients and the controlled state variables (vehicle side slip angle and yaw rate). From friction coefficients, vehicle speed and driver inputs the desired vehicle behavior is calculated. By comparing the desired behavior with the vehicle states, instability is detected. The VDC controller then calculates the necessary tire slip to generate yaw moment and traction/braking. The necessary tire slip is sent to the brake slip controller (ABS) and the traction controller, which are using both brakes and engine torque reduction. When the system detects understeer, it applies light brake pressure to the inside rear wheel. This increases the yaw rate and helps the car back onto the intended line. In an oversteer situation, the system applies braking 5 Ayaw moment is often generated by braking an individual wheel, causing the vehicle to rotate around the vertical axis.
30 16 2 Vehicle Stability Systems to the outside front wheel, decreasing the yaw rate and bringing the rear end back in line.
31 3 Car Modeling 17 3 Car Modeling 3.1 Introduction A car simulation model is needed in order to design and test an anti-spin system. A simpler model, used for the control purposes, is needed as well. For an anti-spin system, it is important to accurately model the slip dynamics. As mentioned in Chapter 2, the normal force affects the relation between slip and longitudinal force. Hence, varying normal forces due to suspension dynamics are modeled. In Chapter 1 some limitations were mentioned. As a first approximation, it is assumed that the ground is flat (no slopes) and that the wheels and suspension are always perpendicular to the ground. This is an adequate compromise between complexity of the model and describing the factors affecting the anti-spin system. Slopes have a minor effect on the anti-spin system. However, if sensors such as accelerometers are used, slopes influence their measurements since the gravitation is measured as well. Slopes can be introduced quite easily when needed or when the anti-spin system is working on flat ground. The suspension will be somewhat non-realistic during cornering. In reality both the wheel motion relative to the vehicle body and the way the tire/road contact forces are transferred to the body are complex (see e.g. ). Hence, to get a more accurate model during cornering, the suspension model needs to be further refined as well. In order to reduce complexity of both the model and the task of constructing an anti-spin system, this is left for future work. We have chosen to divide the car model into the following blocks ffl suspension ffl wheel ffl body ffl driveline ffl brakes ffl steering described later in this chapter. The model is a state space model and the states are mostly positions/angles and velocities/angular velocities. Forces and torques are calculated from these states and signals from the driver. The states' derivatives are then calculated from the states, forces and torques.
32 18 3 Car Modeling The simulation model is validated somewhat in Section 3.1 against real world data and theory, but further validation is needed later. To be able to describe the motions of the vehicle, different coordinate systems are needed and the transformation between these systems must also be investigated. 3.2 Coordinate Systems According to Kiencke/Nielsen , four different coordinate systems are needed to describe the motions (see Figure 3.1): Figure 3.1. Different coordinate systems used to describe the motions of the car. Fixed Inertial Coordinate System (In): Attached to the earth with the z-axis pointing upwards. Center of Gravity Coordinate System (CoG): The origin is attached to the center of gravity (cog) of the car. The x-axis pointing forward, the y-axis to the left and the z-axis up in the roof. Undercarriage System (Un): The origin lies at road-level in the perpendicular projection of cog. Same yaw angle as in CoG system.
33 3.2 Coordinate Systems 19 Wheel System (W): Used in the calculation of tire forces. The system is attached to the wheel center and is rotated with the steering angle ffi w in respect to the Un system. To distinguish between all coordinate systems, In, CoG, Un and W are used as indices for the different coordinate systems (when not obvious). Since there are four wheels, there actually exist four W systems. The indices ij are used for the different wheels, where i 2 ff;rg (front, rear) and j 2fl; rg (left, right). Euler angles are used to describe how the CoG system is oriented with respect to the In system: ffl ψ specifies rotation around the z-axis and is called yaw. ffl specifies rotation around the y-axis and is called pitch. ffl ffi specifies rotation around the x-axis and is called roll. The order of which the rotations are performed affects the result. The standard is yaw-pitch-roll (from In to CoG). See Figure 3.2 where these rotations are carried out separately. To describe each rotation, transformation matrices can be used. ffl Rotation around z: ffl Rotation around y: ffl Rotation around x: T RotZ = T RotY = T RotX = cos ψ sin ψ sin ψ cos ψ 1 cos sin 1 sin cos cos ffi sin ffi sin ffi cos ffi 3 5 (3.1) 3 5 (3.2) 3 5 (3.3) A rotation around several axes corresponds to a multiplication of the matrices.
34 2 3 Car Modeling a) z b) In z Un y In y Un x Un ψ x In c) d) θ z Temp z CoG y CoG y Temp φ x Temp x CoG Figure 3.2. Different coordinate systems, a) In system, b) rotate ψ about z In gives Un system, c) rotate about y Un gives Temp system and d) rotate ffi about x T emp gives CoG system. According to  a base is defined as e = e 1 e 2 e 3 where ei are the base vectors. Avector v is then expressed like v = x y z 1 A = e1 e 2 e x y z 1 A = e1 x + e 2 y + e 3 z (3.4) Bases for the systems in Figure 3.2 can then be constructed as e = ^x In ^y In ^z In f = ^x Un ^y Un ^z Un g = ^x T emp ^y T emp ^z T emp h = ^x CoG ^y CoG ^z CoG (3.5a) (3.5b) (3.5c) (3.5d)
35 3.3 Suspension 21 According to linear algebra  their relations are f = et RotZ (3.6) g = ft RotY = et RotZ T RotY (3.7) h = gt RotX = et RotZ T RotY T RotX = et RotZYX (3.8) A vector expressed in the CoG system is then transformed to the In system x In y In z In 1 A = x CoG y CoG z CoG 1 A = x CoG y CoG z CoG 1 A (3.9) Let T RotW denote the rotation around the z W -axis 6. Similar expressions can then be derived for the transformation between all systems like 3.3 Suspension T CoGIn = T RotZ T RotY T RotX (3.1a) T InCoG = T T CoGIn (3.1b) T CoGUn = T RotY T RotX (3.1c) T UnCoG = T T CoGUn (3.1d) T UnIn = T RotZ (3.1e) T InUn = T T UnIn (3.1f) T WUn = T RotW (3.1g) T UnW = T T WUn (3.1h) The suspension is approximated to always be perpendicular to the ground. This means that roll and pitch angles only affect the height of the suspension, not the orientation. Lateral and longitudinal forces from the wheels are transferred to the chassis through linkages not described here. The vertical forces are transferred through the spring, damper and anti-roll bar (see Figure 3.3) described next. Spring The spring is modeled as a linear spring with coefficient k s. The spring force is then F s = k s (l sus; l sus ) (3.11) 6 The only difference between T RotW and T RotZ is that ψ is replaced by ffi w.
36 22 3 Car Modeling Figure 3.3. Suspension and wheels of a car, seen from above. where l sus; is the uncompressed spring length and l sus is the actual spring length. The spring force F s is defined to be positive when the spring is compressed. Damper The damper force is proportional to the suspension velocity v sus = d dt (l sus), but the coefficient is different for bump and rebound. According to [16, page 8] the wheel velocities in the upward direction (bump) are generally considerably higher (about a factor 2) than in the downward (rebound) direction. To keep the damper forces symmetric, the damper coefficient for rebound is made twice as large as for bump. The damper force F d then is F d = ρ kd;rebound v sus if v sus k d;bump v sus if v sus < (3.12) where k d;rebound and k d;bump are the damper coefficients for rebound and bump respectively. F d is defined to be positive during bump. Anti-roll Bar During cornering, when the vehicle begins to lean or roll to one side, the wheels are also forced to lean or roll to one side. This leads to a so called camber angle fl, which is the angle between z Un and the wheel plane (defined by x W and z W ). The tire that originally enjoyed a complete and flat contact patch prior to body roll must operate on only the tire edge during body roll. The resulting loss of traction leads to larger slip angles and a risk for totally losing the grip and slide. Large roll angles are also uncomfortable for the
37 3.3 Suspension 23 passengers if driving on curved roads. In order to get small roll angles, the so called roll stiffness must be high. According to the approximation about vertical suspensions and wheels, the camber angle is zero in this car model and the tire enjoys complete road contact. Nevertheless, in order to get a correct roll angle during cornering the roll stiffness must be realistic. The effects of camber angles for the wheel forces are further described in Section 3.4. To increase the roll stiffness, stiffer springs can be used. A stiffer spring will compress less than a softer spring when subject to an equal amount of force, leading to less roll. However, stiffer springs have an immediate and substantial effect on ride quality. So, even though the roll stiffness is increased, this is not a good solution. Instead, an anti-roll bar can be used. As described in , an anti-roll bar is a U-shaped metal bar that links both wheels on the same axle to the car body (see Figure 3.3). Essentially, the ends of the bar are connected to the suspension while the center of the bar is connected to the body of the car. In order for body roll to occur, the suspension on the outside edge of the car must compress while the suspension on the inside edge simultaneously extends. However, since the anti-roll bar is attached to both wheels and the car body as well, such movement is only possible if the metal bar is allowed to twist. The antiroll bar hence works as an extra spring during roll, leading to increased roll stiffness. Since this extra spring" is only active during cornering, the negative effect on ride quality isavoided. The anti-roll bar can be described as generating auxiliary spring forces proportional to the the roll angle like F aux;fl = k aux;f ffi (3.13a) F aux;fr = k aux;f ffi (3.13b) F aux;rl = k aux;r ffi (3.13c) F aux;rr = k aux;r ffi (3.13d) where k aux;f describes the stiffness on the front axis and k aux;r the stiffness on the rear axis. F aux acts upon the wheel and car body in the same way as the spring force F s. Dynamics The suspension is assumed to have no mass, which means that the dynamics of the suspension is considered in the wheel and car body. The length l sus
38 24 3 Car Modeling and velocity v sus is calculated from wheel and car body states. The total suspension force F sus = F s + F d + F aux (3.14) is used in the dynamics of the wheel and car body. 3.4 Wheel The wheel is modeled as a rigid body with mass and moment of inertia. The wheel also consists of a tire which is generating forces and torques. As an approximation, torque generated by the tire around the z-axis is neglected. Another approximation is that the wheel is always perpendicular to the ground, leading to zero camber angles fl. The effects on the tire forces are similar as for slip angles, but approximately ten times smaller. The camber angles are also relatively small which make zero camber angles a fairly good approximation. For example during heavy cornering on asphalt (worst case for the relation fl=ff) the roll angle is less than 5 ffi for a normal car, according to [16, page 584]. The camber angle is often smaller than the roll angle due to suspension geometry. The slip angle is also about 5 ffi for a normal tire (P195/7R-14 described in [16, chapter 14]), leading to less than 1 % error. To be noted is that all calculations in this section are carried out in the wheel coordinate system, and indices are with respect to this system. Tire Model In the vertical direction, the tire is modeled as a linear spring with coefficient k tire and the normal force is F z = max (;k tire (r r)) (3.15) where r is the nominal wheel radius and r is the wheel radius. In the longitudinal and lateral directions the tire model is an empirical model which is based on measured data for a specific tire, described in . Apart from the notation, this section is a summary from . The raw data is normalized so that the dependences on friction and load are eliminated. A normalized curve is then obtained describing the relationship between normalized forces and normalized slip and slip angle. This curve depends only on the tire and the inflation pressure. The slip s and slip angle ff are combined and normalized to a variable K as K = 1 q (k x s) μ 2 +(k y tan ff) 2 (3.16) max
39 3.4 Wheel 25 where k x is the longitudinal stiffness and k y is the cornering stiffness. These are defined as the initial slopes of the μ x s and μ y tan ff curves respectively. Their values depend on the specific tire and inflation pressure. The normalized resultant" force μ is defined as μ = 1 q μ μ 2 x + μ 2 y (3.17) max and is found to be a function of K independent of load. The so called magic formula", developed by Pecejka et al. (see e.g. ), describes this function: EB μ(k) =D sin C arctan B (1 E)K + arctan(bk) (3.18) where B, C, D and E are parameters describing a specific tire and inflation pressure. Another variable is defined to make the equations hold for both small and large slips and slip angles: 1 (K) =( 2 1+ ky k x 1 ky k x cos( K 2 ) if jkj»2ß (3.19) 1 if jkj > 2ß The normalized longitudinal and lateral forces can then be calculated as s μ x = sgn(v x )μ max μp (3.2) s2 +( tan ff) 2 tan ff μ y = μ max μp (3.21) s2 +( tan ff) 2 where sgn(v x ) makes the equation hold also for reverse drive. Dynamic Slip and Slip Angle Since the longitudinal velocity appears in the denominator of both slip and slip angle in equations (2.1) and (2.2), the calculations deteriorate for low velocities. Furthermore, the transient behavior needs to be modeled in some way. For example, a step steer input would lead to a step in the lateral force, but in reality the tire needs to be stretched in order to transfer the force, leading to a lag. The solution is to use dynamic slip and slip angle as described in . The basic idea is to view a hypothetical element of the tire in the contact patch and define slip and slip angle using distances instead of velocities.
40 26 3 Car Modeling This corresponds to the explanation in Section 2.1 where the tire is seen as a spring in the contact area. The result needs to be modified slightly in order to work with the definitions used in this report. This gives the equation _fi = 1 ( jv x jfi + v y ) (3.22a) L ff fi = tan ff (3.22b) for the slip angle, where L ff is called lateral relaxation length. For the slip the equation is _s = 1 L s ( jv x js + r! sgn(v x ) jv x j) (3.23) where L s is called longitudinal relaxation length. When v x changes sign during a spin out maneuver or a stopping maneuver, the sign of s must be changed in the differential equation in order to get a correct value. Let ff and s denote the old definitions in equations (2.1) and (2.2). In steady state, the dynamic slip and slip angle are the same as before. The difference is that a good behavior for all velocities (including zero) is achieved and that a first order lag is introduced. If a step change in ff is introduced (for example by a step in the steering angle), then ff will rise to about 63 % of ff in L ff =v x seconds. A step change in s will in the same way make s rise to about 63 % of s in L s =v x seconds. Dynamics The dynamics of the wheel is limited to the vertical direction and the wheel spinning axis. In the lateral and longitudinal direction all wheels are lumped together with the car body and considered to be one single rigid body. The states are the height r and velocity v z of the wheel center and the speed of rotation! of the wheel. The dynamics can be described by the following equations _v z = 1 (F z m w g F sus ) m w (3.24) _r = v z (3.25) _! = 1 (M w rf x ) I w;eff (3.26) where m w is the mass of the wheel 7, I w;eff is the effective moment of inertia for the wheel, described later in equation (3.47) and M w is the sum of the 7 The weight of brakes and suspension is included in the wheel mass.
41 3.5 Body 27 torques from driveline and brakes like M w = M d M b sgn(!) (3.27) where M d is the torque from the driveline (described in Section 3.6) and M b is the torque generated by the brakes (described in Section 3.7). Furthermore, there are the dynamics of slip angle ff and slip s, described in equations (3.22) and (3.23), which gives two more states for each wheel. 3.5 Body The car body is modeled as a rigid body with mass m b and a moment of inertia matrix I b expressed in the CoG system. The forces acting on the car body are suspension forces, gravitation and air drag. The states needed to describe the motions are position and velocity of the center of gravity in the In system, the Euler angles and the angular velocity of the body. According to the approximation about vertical suspension, the dynamics in the lateral and longitudinal directions include the wheels as well. Hence, the car body mass is seen as the sum of the body and wheels, m tot, in the lateral and longitudinal directions. Furthermore, the forces acting on the car body can be seen as acting in the tire/road contact point. Dynamics Let r cog be the vector from center of gravity to the tire/road contact point. The forces and torques acting on the body can then be expressed as F ij F x;ij F y;ij F sus;ij F b = F air + F g + X i;j 1 A (3.28) F ij (3.29) M b = X i;j r cog;ij F ij (3.3) where ij are the wheel indices and denotes the cross-product defined in  as: Definition 3.1. The cross-product a b between the vectors a and b is a vector defined by 1. ja bj = jajjbj sin v where v is the angle between a and b
42 28 3 Car Modeling a b v a b Figure 3.4. The cross product a b. 2. a b? a, a b? b 3. a b is directed according to the so called right-hand rule (see Figure 3.4) The translatory motion, expressed in the In system, is described by _v m tot m tot m b 1 A = Fb (3.31) _r b = v b (3.32) where v b is the velocity of the center of gravity and r b is the vector from some fixed point in the In system to the center of gravity. The moment equation in the CoG system is, according to , M b = I b _! b +! b (I b! b ) (3.33) where! b is the angular velocity of the car body. Since the car body is rotating, the transformation matrices need to be updated. From equation (3.8), there is a relation h = et (3.34) T = t 1 t 2 t 3 (3.35) between the bases for the CoG and In systems. If the base vectors h i are differentiated, two different expressions can be achieved _h i =! b h i (3.36) _h i = _t i (3.37)
43 3.6 Driveline 29 Combining equation (3.36) and (3.37) gives the expression T _ =! b t 1! b t 2! b t 3 (3.38) for the derivative of the transformation matrix T. A simple solution would be to integrate (3.38). However, because of numerical problems, the matrix would soon be non-orthonormal, leading to strange results. Instead an expression for the derivative of the Euler angles is needed. T_ can be expressed as T _ _ _ (3.39) and combining (3.38) and (3.39) gives 9 equations and 3 unknown quantities. Solving for _ ffi, _ and _ψ give an expression for the derivative of the Euler angles, which can be integrated to get the Euler angles. Remark To beinvestigated is if yaw, pitch and roll could be treated separately in the moment equation (3.33) as in . Hence the term! b (I b! b ) is neglected, which is correct only if the the body is symmetric or the rotation is around one of the principal axes 8. The moment is expressed in the Un system and the roll and pitch axis are assumed to lie at road level. This I xx ffi I yy I zz ψ 1 A = Mb (3.4) where I xx, I yy and I zz are the diagonal elements of I b and M b is expressed in the Un system. Integrating (3.4) twice gives the Euler angles. This would be a simpler solution and maybe the error introduced is negligible. However, this is not satisfactory explained and needs to be further investigated. 3.6 Driveline A vehicular driveline usually consists of engine, clutch, transmission, shafts and wheels. Here, the dynamics of the wheels are treated separately. The driveline can be seen as describing the relation between requested engine torque M r and wheel torque M d. In Figure 3.5 there is a block diagram of the driveline. Here a very simple model is used. In order to get a fast and well working anti-spin system, further modeling is probably needed, but this is left for future work. 8 The moment of inertia matrix is diagonal if calculated in respect to the principal axes.
44 3 3 Car Modeling Driver M_e,driver > M_r Anti spin system M_c < Engine M_e M_t M_s Transmission Drive shaft Differential M_d,fl Front left wheel M_d,fr Front right wheel Figure 3.5. Block diagram of the driveline. The requested engine torque M r depends on the requested torque from the driver M e;driver (, e.g. throttle position) and the reduction torque from the anti-spin controller M c like M r = 8 < : if M c» M e;driver M e;driver + M c if M e;driver <M c < (3.41) M e;driver if» M c Hence, the controller is only able to reduce requested engine torque. The engine is modeled as a simple time delay from the requested engine torque M r to the actual engine torque M e, reflecting the delay of5=4 engine revolutions introduced while using fuel blocking. This delay T e varies with the engine speed! e like T e = 5 2ß 4! e which gives M e (t) =M r (t T e ) (3.42) The transmission ratio i t depends on the gear and transforms the engine speed! e and engine torque M e like! e = i t! t (3.43) M e i t = M t (3.44) where! t and M t are the speed and torque of the transmission respectively. No clutch is modeled which makes the gear shifting non-physical. The gear is instantaneously changed when the driver requires another gear. To keep the engine speed within reasonable limits the transmission has automatic gear shifting as well (active when the driver specifies gear = ). When a specific engine speed is obtained, the gear is changed. Hence, in order to get accurate results during gear shifting, further modeling is needed.
45 3.7 Braking system 31 The shafts are modeled as first order low-pass filters like _M s = 1 fi s (M t M s ) (3.45) where M s is the torque transfered by the drive shaft and fi s is the shaft time constant. The torque is then split evenly between left and right wheels in the differential (called open differential) like M d;fl = M d;fr = 1 2 M s (3.46) where M d;fl and M d;fr are the driveline torque for the left and right front wheels. The driveline inertia affects the acceleration of the wheels and since the driveline is seen as only generating wheel torques, the inertia must be taken care of in the wheels. The effective moment of inertia for the wheels I w;eff can then be seen as I w;eff = I w I d (3.47) I d = I t + i 2 t I e (3.48) where I w is the wheel moment of inertia, I e is the moment of inertia for the engine and I t is the moment of inertia of the transmission and shafts. I d is the effective driveline moment of inertia when taken care of in the wheels. The engine speed is used in the automatic gear shifting and is approximately the front wheel speed multiplied by the transmission ratio. 3.7 Braking system When the driver steps on the braking pedal, a braking torque M b;driver ( ) is generated, proportional to the brake pressure. This braking torque is divided 7/3 between front and rear wheels like M b;fl = M b;fr =:35M b;driver (3.49) M b;rl = M b;rr =:15M b;driver (3.5) No anti-lock braking system (ABS) is presently implemented. 3.8 Steering System The input to the system is the steering angle ffi S from the driver and the outputs are steering angles ffi w for all the wheels. The steering angles for
46 32 3 Car Modeling the rear wheels are zero and both the front wheels have the same steering angle, proportional to the input as where k steer is the gain ratio. 3.9 Implementation ffi w = k steer ffi S (3.51) The model is implemented in Simulink, which is an extension of MATLAB TM . Simulink is a tool for modeling, analyzing and simulating physical and mathematical systems, including those with nonlinear elements and those that make use of continuous and discrete time. The vehicle model is divided into blocks that are connected with wires as can be seen in the block scheme in Figure 3.6. The simulation model consists of the following blocks: ffl Body ffl Wheel ffl Driveline ffl Braking system ffl Steering system The suspension is included in the body block since the suspension has no extra states. No driver model exists yet. The signals from the driver are given "by hand". These are: ffl Engine torque, M e;driver ffl Braking torque, M b;driver ffl Steering angle, ffi S ffl Gear No roads are specified, different friction coefficients μ max are also given by hand". The continuous equations describing the vehicle dynamics, which are defined earlier in Chapter 3, are then implemented via s-functions, written in MATLAB TM code. These s-functions contain routines for update of discrete
47 3.9 Implementation 33 Inputs Accelerator Gear Steering angle Brake Tire/road friction Driveline Steering system Braking system Front left wheel Front right wheel Rear left wheel Rear right wheel Position and velocities Tire forces Tire forces Tire forces Tire forces Body Outputs Figure 3.6. Block scheme of the car simulation model.
48 34 3 Car Modeling states and calculation of the derivative of continuous states and the outputs of the block. To run a simulation, car specific parameters and driver inputs are needed. The parameters are specified in an m-file and the driver inputs are specified as matrices in the MATLAB TM workspace. The result from the simulation is returned to the workspace automatically. The s-functions are written in MATLAB TM code, which is a slow implementation. One way to make it approximately ten times faster would be to implement the s-functions in C-code. This is preferably done when the development of the model is finished. 3.1 Validation of the Model A thorough validation of a complete vehicle model against real world data involves too much work to be considered here. Furthermore, extra sensors are needed in order to measure e.g. the driveline properties. The validation of the model is then more qualitative than quantitative. Further validation of the model is needed in order to get same results from the simulation model as for a real car. Car specific parameters also need to be estimated. Example 1: Straight Ahead Real world data are collected while driving straight ahead in a Volvo V4 test car. These are used as inputs to the simulation model and the results are compared. Since no clutch is modeled, the start and gear shifting is different and the signals are modified slightly. In Figure 3.7, the engine torque and transmission ratio can be seen and in Figure 3.8 the engine speed and vehicle velocity are plotted. A constant engine torque is used until t =1:77 s in order to achieve the same engine speed in the simulation as in the real car. Between t = 4:3 s and t = 5:8 s the engine torque is set to zero since the gear is shifted (gear 1 to 2) and no torque is transferred while the clutch is used. The solid lines in Figure 3.7 together with μ max = 1 and zero steering angle are used as input to the simulation model. By comparing the overall behavior, like e.g. the acceleration, the model can be validated for straight ahead driving. As can be seen in Figure 3.8, the model gives about the same result. The small differences might depend on e.g. unmodeled friction in the driveline, rolling resistance, a small slope or parameter deviations in driveline and vehicle mass.
49 3.1 Validation of the Model 35 8 Engine torque 6 M e (Nm) Transmission ratio i t t (s) Figure 3.7. Measured engine torque and transmission ratio (dotted) and slightly modified input signals to the simulation model (solid). 3 Engine speed ω e (rpm) Velocity v x (km/h) t (s) Figure 3.8. Measured engine speed and vehicle velocity (dotted) and the output from the simulation model (solid). Example 2: Steady State Cornering In the test car, the steering angle cannot be measured. In order to validate the cornering behavior, the simulation results are compared with the theory.
50 36 3 Car Modeling From equation (4.12) in Section 4.3, the following relation is derived R = L tan ffi w (3.52) between steering angle ffi w, curve radius R and distance L between front and rear wheels. From Figure 3.9, the curve radius R ß 14 m is derived for a simulation with ffi w = ß=16. Comparing this with the calculated radius from equation (3.52), R = 2:7= tan(ß=16) ß 13:6 m, gives a close correspondence. Consequently the cornering behavior is satisfactory. 3 Simulation 25 Theory 2 14 m y (m) m 1 5 Start of steady state cornering x (m) Figure 3.9. Position of mass center while driving with constant velocity v x =8 m/s and steering angle ffi w = ß=16 rad. Example 3: Varying Driving In Appendix A the results from a varying drive is presented, including hard acceleration, varying steering angle and brake to stop. The plots are commented there, but to sum up one can say that the model seems to be working very well. The only problem is when the wheels are locked up during braking or when the vehicle comes to stop. The car does not come completely to rest but oscillate with a small amplitude and the simulation takes approximately ten times longer per second than while driving.
51 4 Model-based Anti-spin Controller 37 4 Model-based Anti-spin Controller As pointed out earlier in Section 2.2, the optimal slip value for the system, the slip reference value s ref, changes depending on e.g. the tire/road friction and cornering. Hence, in order to get a high-performance system, the friction coefficient μ max and required lateral force during cornering need to be determined. Other factors are the wheels' slip angles and the vehicle side slip angle and yaw rate and hence, these states also need to be estimated. Since the relations between slip, slip angle and forces are through normalized forces, a correct estimation of the normal forces are important as well. As an approximation, the driven whels are treated as a single wheel and slip, slip angles and forces are the mean values of the left and right wheels. Estimation of lateral velocity and yaw rate are briefly analyzed with respect to known strategies, but the implementation of the state estimator is left for future work. Since these states are needed by other parts in the anti-spin system, they are sometimes assumed to be known. The main contributions in this chapter are an algorithm for estimation of tire/road friction and tire properties, a strategy for the calculation of s ref and a derivation of a controller. These parts are combined into an anti-spin system, which is tested on the simulation model. 4.1 Simplified Vehicle Models Different simplified models can be used to describe the vehicle dynamics. For the slip controller, the driveline and the slip dynamics for the driven wheels are the most important things to model. Hence, a complete vehicle model is not needed in order to get a working slip controller. It is sufficient to use a quarter vehicle model. For the lateral velocity and yaw rate some of the steering dynamics needs to be modeled as well and a bicycle model is used. Quarter Vehicle Model For the design of the slip controller and the estimation of the longitudinal tire force, the most important parts to model are the driveline and the slip dynamics. A quarter vehicle model describes these properties. The driven wheels are lumped together to a single wheel and only the longitudinal dynamics of the vehicle body is modeled. No braking torque is considered. The equations (3.23), (3.26) and (3.31) from the simulation model are then
52 38 4 Model-based Anti-spin Controller slightly modified for the quarter vehicle like _v x = _! f = 1 (F x;f F air;x ) m tot (4.1) 1 (M w;f rf x;f ) 2I w;eff (4.2) _s f = 1 L s ( v x s f + r! f v x ) (4.3) where v x is the velocity of the body (and wheel), index f denotes front wheels, F x;f = F x;fl + F x;fr and M w;f = M w;fl + M w;fr. The air resistance F air;x can be described by F air;x = k air;x v 2 x. The dynamics from engine torque M e to wheel torque M w;f is described by equations (3.27), (3.44), (3.45) and (3.46), giving (when no braking torque is considered) _M w;f = 1 fi s (i t M e M w;f ) (4.4) where fi s is the time constant for the drive shaft. The model is further described and used in Sections 4.3 and 4.8. Bicycle Model A bicycle model describes the major properties of a vehicle during cornering. The model can then be used e.g. in the estimation of lateral velocity v y and yaw rate ψ, _ described in Section 4.5. If the steering angle ffi w is small, then the lateral dynamics can approximately be described as (see Figure 4.1 for notation) _x = f(x; u; par ) (4.5) where x = f(x; u; par )= _ ψ v y, u = ffi w, par = v x s f s r F z;f F z;r μ max and ψ 1 I zz (L f F y;f (x; u; par ) L r F y;r (x; u; par )) 1 m tot (F y;f (x; u; par )+F y;r (x; u; par )) v x _ψ! (4.6) I zz is the moment of inertia of the vehicle body around the z-axis (element 3; 3 in I b from Section 3.5) and v x is the longitudinal velocity of the mass center (v b;x ). The front and rear lateral forces F y;f and F y;r are functions of slip, slip angle, normal force and friction as described in Section 3.4. The model is further described and used in Sections 4.3 and 4.5.
53 4.2 Overall Structure 39 α f δ w v f F y,f v b β ψ. L f mass center α r L r v r F y,r Figure 4.1. Bicycle model. 4.2 Overall Structure The overall structure of the anti-spin system considered here can be seen in Figure 4.2. Six different blocks are used, these are ffl Estimation of tire forces ffl Estimation of longitudinal velocity, slip and slip angle ffl Estimation of lateral velocity and yaw rate ffl Estimation of tire/road friction and tire properties ffl Reference slip calculation ffl Slip controller
54 4 4 Model-based Anti-spin Controller Sensors Estimation of tire forces Estimation of longitudinal velocity, slip and slip angle Estimation of lateral velocity and yaw rate Estimation of tire/road friction and tire properties Reference slip calculation Gain scheduling (velocity and gear) Slip Slip controller Torque reduction Car Figure 4.2. Block diagram of complete anti-spin system.
55 4.3 Estimation of Tire Forces 41 and are further described in the following six sections. See Appendix B for a more detailed block diagram with all inputs and outputs specified for each block. 4.3 Estimation of Tire Forces One way of estimating the longitudinal tire force F x;f is to use the measurements of engine torque M e, engine speed! e and front wheel speeds! fl and! fr. From these, the wheel torque M w;f and front wheel angular acceleration _! f are calculated as! f =! fl +! fr (4.7) 2 i t =! e (4.8)! f M w;f ß i t M e (4.9) _! f ß! f! f;old T s (4.1) where T s is the sample time and! f;old is the front wheel speed from time t T s. F x;f can then be calculated from equation (4.2). The lateral force during cornering can be estimated from the bicycle model in Section 4.1 and a steady state assumption f(x; u) = (equation (4.6)). The front wheel lateral force is then F y;f = L r L m totv x _ψ (4.11) Furthermore, at low speeds, the vehicle will follow the direction of the front wheel, specified by the angle ffi w. This means driving in a circle with radius R defined by tan ffi w = L=R (4.12) or, for small angles, ffi w = L=R. The curve radius R can be expressed in v x and _ ψ as v x = R _ ψ (4.13) Combining equations (4.12) and (4.13) gives _ψ = v x tan ffi w L (4.14) which can be used as an alternative to measured yaw rate in equation (4.11). During cornering with higher speeds, there will be slip angles present which
56 42 4 Model-based Anti-spin Controller means that equation (4.14) does not necessarily hold. However, the steering angle indicates how much lateral force the driver wants, in contrast to measured yaw rate, which indicates how much lateral force is presently used. Hence, the steering angle can be used in the anti-spin system. This will be further discussed in Section 4.7. Correct values of the normal forces F z;ij are important since they affect both the lateral and longitudinal normalized forces. As described in , the normal forces change drastically during cornering and acceleration/deceleration. Since the driven wheels are treated as a single wheel, the effects during cornering are neglected. Only the changes from a nominal value in the normal forces are treated. Figure 4.3. Car seen from the side. Let F x;tot denote the total longitudinal force acting on the car during acceleration. For a front wheel driven car F x;tot ß F x;f, if air resistance and slopes are neglected. The steady state equations during acceleration are (See Figure 4.3) F z;f + F z;r = m tot g (4.15) L f F z;f + L r F z;r = hf x;tot (4.16) If these equations are combined, the forces can be calculated as F z;f = L rm tot g hf x;tot L f + L r (4.17) F z;r = L fm tot g + hf x;tot L f + L r (4.18)