Department of Precision and Microsystems Engineering Estimation of Vehicle Handling States Shelav Jain Report no Coach Professor Specialisation Type of report Date : : : : : : AUT 2014.027 Mr. Anil Kunnappillil Madhusudhanan prof. dr. ir. E. G. M. Holweg Automotive M.Sc. Thesis 28 October 2014 Estimation of Vehicle Handling States Using Tire Force Measurements From Load Sensing Bearings Master of Science Thesis For the degree of Master of Science in Mechanical Engineering at Delft University of Technology Shelav Jain October 28, 2014 Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of Technology c Precision and Microsystems Engineering (PME) Copyright All rights reserved. Delft University of Technology Department of Precision and Microsystems Engineering (PME) The undersigned hereby certify that they have read and recommend to the Faculty of Mechanical, Maritime and Materials Engineering (3mE) for acceptance a thesis entitled Estimation of Vehicle Handling States by Shelav Jain in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering Dated: October 28, 2014 Supervisor(s): prof.dr.ir.E.G.M.(Edward) Holweg Mr. Anil Kunnappillil Madhusudhanan Reader(s): dr.ir.A.L.(Arend)Schwab dr.Barys Shyrokau Abstract With the ever increasing demand of cars, safety is of prime concern to automobile manufactures across the world. Companies strive for the vision of "Zero accidents" through high quality and innovative products that reduce the frequency of accidents as well as their consequences. The auto industry has been exerting a myriad of state-of-the-art technologies to make automotive safety systems that reduce driver’s strain and fatigue and assist safe driving. Advanced Driver Assistance Systems (ADAS) are technologies that capture the vehicle’s surrounding environment and assist the driver by keeping him informed about the current vehicle state, and if necessary intervene to prevent an impending danger, while the driver is in control of the vehicle at all times. Vehicle Dynamics Control (VDC) systems enhance the handling and safety of the car by assisting the driver in maintaining control of the vehicle. However, these features are limited by lack of knowledge of the vehicle states. Some of the vehicle states like sideslip angle are not measurable due to technical or economic reasons. Therefore, these must be estimated by using the available measurements. Most of the existing sideslip angle estimators are based on lateral acceleration, but these estimators have estimation errors arising from roll and pitch dynamics. The purpose of this study is to develop an algorithm that estimates the vehicle sideslip angle and tire cornering stiffnesses using tire force measurements. This is achieved by implementing a model based deterministic approach. Kalman filters and their extensions used for state and parameter estimation are investigated. Finally, the developed system is implemented using a multibody vehicle simulator and tested for different maneuvers. Keywords: Vehicle lateral dynamics, sideslip angle, cornering stiffness, Unscented Kalman filter Master of Science Thesis Shelav Jain ii Shelav Jain Master of Science Thesis Table of Contents Acknowledgements ix 1 Introduction 1-1 Lateral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 1-2 Automobile Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1-3 Active control of Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 7 1-3-1 Electronic Stability Control . . . . . . . . . . . . . . . . . . . . . . . . . 8 1-4 Application of State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1-4-1 Estimation of Sideslip Angle . . . . . . . . . . . . . . . . . . . . . . . . 16 1-4-2 Estimation of Tire Cornering Stiffness . . . . . . . . . . . . . . . . . . . 17 1-5 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1-6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Physical Modeling 21 2-1 Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1-1 Four-wheel vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1-2 The Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 22 24 2-2 Tire Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2-2-1 Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2-2-2 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2-3 Transient tire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 29 29 3 State Estimation and Filtering 31 3-1 Modeling of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3-2 Observability of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3-3 The Luenberger Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Master of Science Thesis Shelav Jain iv Table of Contents 3-4 Linear Kalman Filter . . . . . . . 3-5 Non-linear Kalman Filter . . . . 3-5-1 Unscented Kalman Filter 3-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Estimation using Bicycle Model 33 35 36 40 41 4-1 Nonlinear Observer: State Space Algorithm . . . . . . . . . . . . . . . . . . . . 41 4-2 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4-3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3-1 Tuning Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 4-3-2 Observer Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3-3 Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 Nonlinear observer with sideslip angle as measurement . . . . . . . . . . . . . . 44 50 50 4-5 Analysis and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 Estimation using Four Wheel Vehicle Model 53 5-1 Nonlinear Observer Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 55 5-3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3-1 Tuning Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 5-3-2 Observer Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3-3 Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Nonlinear observer with lateral velocity as measurement . . . . . . . . . . . . . . 56 61 64 5-5 Analysis and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Conclusions and Recommendations 6-1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 67 68 Bibliography 69 Glossary 75 List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shelav Jain 75 Master of Science Thesis List of Figures 1-1 Forces acting on a vehicle. Source: [1] . . . . . . . . . . . . . . . . . . . . . . . 2 1-2 Components of tire forces. Source: [1] . . . . . . . . . . . . . . . . . . . . . . . 3 1-3 Tire slip angle. Source: [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1-4 The lateral tire force versus slip angle. . . . . . . . . . . . . . . . . . . . . . . . 4 1-5 The friction circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6 The lateral tire force versus slip angle at different normal loads. . . . . . . . . . 5 6 1-7 Yaw stability control system for vehicle lateral dynamics. Source: [3] . . . . . . . 8 1-8 Sideslip angle of a vehicle. Source: [2] . . . . . . . . . . . . . . . . . . . . . . . 9 1-9 ESP control loop in a vehicle. Source: [1] . . . . . . . . . . . . . . . . . . . . . 10 1-10 Active chassis control. Source: [3] . . . . . . . . . . . . . . . . . . . . . . . . . 11 1-11 Yaw moment as function of sideslip angle for different steering angles. Source: [3] 12 1-12 Objective of yaw stability control system. Source: [3] . . . . . . . . . . . . . . . 12 1-13 Block diagram for integrated control objective. Source: [4] . . . . . . . . . . . . 13 1-14 The principle layout of a control loop using a state estimator. . . . . . . . . . . . 15 2-1 Four-wheel vehicle model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 The bicycle model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 24 2-3 Tire dynamics variables. Source: [5] . . . . . . . . . . . . . . . . . . . . . . . . 25 2-4 Four categories of possible types of approach to develop a tire model. Source: [6] 26 2-5 The Magic Formula parameters. Source: [6] . . . . . . . . . . . . . . . . . . . . 28 3-1 The principle of the Unscented Transformation. Source: [7] . . . . . . . . . . . . 36 3-2 Flowchart of the Unscented Kalman Filter (UKF) algorithm. . . . . . . . . . . . 38 4-1 Steering profile and lateral acceleration for sine steer at Vx of 80km/h. . . . . . 45 Master of Science Thesis Shelav Jain vi List of Figures 4-2 Sideslip angle for sine steer at Vx of 80km/h. . . . . . . . . . . . . . . . . . . . 45 4-3 Cornering stiffness for sine steer at Vx of 80km/h. . . . . . . . . . . . . . . . . 46 4-4 Steering profile and lateral acceleration for double lane change at Vx of 75km/h. 46 4-5 Sideslip angle for double lane change at Vx of 75km/h. . . . . . . . . . . . . . . 47 4-6 Cornering stiffness for double lane change at Vx of 75km/h. . . . . . . . . . . . 47 4-7 Steering profile and lateral acceleration for double lane change at Vx of 85km/h and µ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4-8 Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5. . . . . . . . 49 4-9 Cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. . . . . 49 4-10 Lateral acceleration and sideslip angle for steady state cornering for UKF without sideslip angle measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4-11 Steering profile and lateral acceleration for steady state cornering at Vx of 90km/h and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12 Cornering stiffness for steady state cornering at Vx of 90km/h and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 52 5-1 Steering profile and lateral acceleration for sine steer at Vx of 80km/h. . . . . . 57 5-2 Sideslip angle for sine steer at Vx of 80km/h. . . . . . . . . . . . . . . . . . . . 57 5-3 Front tires cornering stiffness for sine steer at Vx of 80km/h. . . . . . . . . . . . 58 5-4 Rear tires cornering stiffness for sine steer at Vx of 80km/h. . . . . . . . . . . . 58 5-5 Steering profile and lateral acceleration for double lane change at Vx of 80km/h. 59 5-6 Sideslip angle for double lane change at Vx of 80km/h. . . . . . . . . . . . . . . 59 5-7 Front tires cornering stiffness for double lane change at Vx of 80km/h. . . . . . 60 5-8 Rear tires cornering stiffness for double lane change at Vx of 80km/h. . . . . . . 60 5-9 Steering profile and lateral acceleration for double lane change at Vx of 85km/h and µ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5-10 Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5. . . . . . . . 62 5-11 Front tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. 62 5-12 Rear tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. 63 5-13 Lateral acceleration and sideslip angle for steady state maneuver for UKF without lateral velocity as measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14 Steering profile and lateral acceleration for steady state cornering at Vx of 95km/h and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Front tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16 Rear tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m radius circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shelav Jain 63 64 65 65 Master of Science Thesis List of Tables 4-1 D class vehicle parameters values. . . . . . . . . . . . . . . . . . . . . . . . . . 42 4-2 UKF tuning parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4-3 RMS error values for UKF with bicycle model. . . . . . . . . . . . . . . . . . . . 48 5-1 Root mean squared (RMS) error values for UKF with Four-Wheel vehicle model (FWVM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Master of Science Thesis Shelav Jain viii Shelav Jain List of Tables Master of Science Thesis Acknowledgements I would like to thank my supervisors prof.dr.ir.Edward Holweg and Mr.Anil Kunnappillil Madhusudhanan for their helpful suggestions and comments during discussions about the topic. I would also like to thank dr.Mustafa Ali Arat for guidance and direction in successful completion of this project. Furthermore, my sincere thank to dr.ir.A.L.(Arend)Schwab and dr.Barys Shyrokau for accepting to be on my M.Sc. defense committee. Delft, University of Technology October 28, 2014 Master of Science Thesis Shelav Jain Shelav Jain x Shelav Jain Acknowledgements Master of Science Thesis “The Future depends on what we do in the present” — Mahatama Gandhi Chapter 1 Introduction ”It has often said that the primary forces by which a high-speed motor vehicle is controlled are developed in four patches-each the size of man’s hand-where the tires contact the road. This is indeed the case. A knowledge of forces and moments generated by pneumatic (rubber) tires at the ground is essential to understanding highway vehicle dynamics.-Thomas D. Gillespie” From Newton’s laws of motion, it is very well understood that inertia is the property possessed by all bodies and by virtue of this it will remain in its state of motion or rest unless an external force is applied to it, to bring about a change to that status. On an automobile, various forces act upon it regardless of its state of motion, as shown in Figure 1-1. Vehicle dynamics is the branch of engineering which relates tire and aerodynamic forces to overall vehicle acceleration, velocities and motions using Newton’s Laws of Motion. It encompasses the behavior of the vehicle as affected by driveline, tires, aerodynamics and chassis characteristics. It is the study of forces which affect wheeled vehicles in motion and of the vehicle’s responses, either natural or driver induced. Vehicle Dynamics can be broken down into the following branches representing the degrees of freedom of a vehicle: • Longitudinal Dynamics - The ability of a vehicle to accelerate and decelerate comes under this branch of vehicle dynamics. Basic governing factors are: vehicle weight, net power available at the wheels (from engine during acceleration and from braking system during deceleration), tractive capacity of the driving tires etc. • Lateral Dynamics - This branch of vehicle dynamics deals with cornering behavior of a vehicle. The way in which vehicles perform traverse to their direction of motion, particularly during cornering and swerving, describes their handling performance. According to [8], there is a subtle difference between handling and cornering ability of a vehicle. Master of Science Thesis Shelav Jain 2 Introduction Figure 1-1: Forces acting on a vehicle. Source: [1] Cornering refers to objective properties of the vehicle when changing direction and sustaining lateral acceleration in the process, it may be quantified by the level of lateral acceleration that can be sustained in a stable condition. On the other hand, handling improves the vehicle’s quality by giving feedback to the driver, affecting the ease of the driving task or affecting driver’s ability to maintain control. Handling implies, not only to the vehicle’s explicit capabilities but also its contributions to the performance of the driver-vehicle combination. • Vertical Dynamics - It is also referred as the ride quality of a vehicle. While driving on an uneven road surface, the vehicle body moves upward and downward. According to [8], ride is a subjective perception, normally associated with the level of comfort experienced when traveling in a vehicle. The focus of this study is lateral dynamics. Therefore, longitudinal and vertical dynamics will not be treated hereafter. 1-1 Lateral Dynamics As mentioned before, lateral dynamics involves stability and handling of vehicle during a cornering maneuver. Lateral dynamics is governed by the lateral tire forces. Lateral tire force (also known as side or cornering force) is the force necessary to sustain a vehicle through a turn. It is generated by the lateral tire deformation in the contact patch as shown in Figure 12. In the Figure 1-2, Fz is the vertical load, Fy is the lateral force and Fx is the longitudinal force. Shelav Jain Master of Science Thesis 1-1 Lateral Dynamics 3 Figure 1-2: Components of tire forces. Source: [1] The steady-state force generation of the tire is highly nonlinear with respect to tire slip angle. The slip angle of the tire α is the angle between the velocity vector Vf of the tire and the orientation of the tire, as shown in Figure 1-3. Figure 1-4 shows typical lateral tire force under pure lateral slip condition i.e. without any longitudinal tire force (no braking or acceleration). For small values of slip, the force generation is approximately proportional to slip. As the slip increases the force reaches a point of saturation after which it declines. The slope of the curve in the linear region of force generation is denoted as the linear cornering stiffness Cα . The Instantaneous Cornering Stiffness (ICS) Cins is defined as the slope of this curve at current tire slip angle. For small tire slip angles it is equivalent to the linear cornering stiffness. For higher tire slip angles it becomes smaller and becomes 0 when lateral tire force saturation is reached and even become negative beyond that [9]. The ICS can be given as: Cins (α) = dFy (α) . dα (1-1) The lateral friction coefficient µy is the relationship between the lateral force Fy and vertical load Fz . It can be given as: µy = Fy /Fz . (1-2) The lateral force generation is also influenced by the longitudinal slip and vice-versa. This effect can be seen by examining combined slip which depicts longitudinal and lateral forces for Master of Science Thesis Shelav Jain 4 Introduction Figure 1-3: Tire slip angle. Source: [2] Figure 1-4: The lateral tire force versus slip angle. Shelav Jain Master of Science Thesis 1-2 Automobile Safety 5 Figure 1-5: The friction circle. simultaneous longitudinal and lateral slip. The lateral force is maximum when no longitudinal slip is present and with increasing longitudinal slip, either due to brake or drive slip, the lateral force declines. This is shown with the help of Friction circle in Figure 1-5. The Friction circle combines the longitudinal and lateral forces and it represents the limits of the tire for a given set of operating conditions (load, temperature etc)[10]. As shown in Figure 1-5, the friction circle has lateral force Fy on the x-axis and longitudinal force Fx on the y-axis. In general, the use of a circle is a simplification, because the force generation in longitudinal direction is usually higher than in lateral direction. Therefore, the enveloping curve will be elliptical. The other important aspect is the influence of wheel load on tire forces. From the basics of coulomb friction it is evident that the maximum friction force Ff ric is proportional to the value of friction µ and the normal load Fz . Figure 1-6 shows lateral force Fy as a function of tire slip angle for different normal loads Fz . For different wheel loads, the curves follow the same pattern but for lower Fz it saturates at lower value of Fy . It can also be seen from this figure that the value of cornering stiffness changes for different normal loads. In [8], author represents the sensitivity of the cornering stiffness to the normal load by a second order polynomial given as: Cα = aFz − bFz2 , (1-3) where a and b are empirically determined and depends on the tire and road conditions. 1-2 Automobile Safety Nowadays cars have become an indispensable part of our life. We use them daily to commute to our work and for pleasure. Since the inception of the Ford Model T in the 1900s, automotive industry has undergone myriad change in terms of design and safety features of cars. Modern day cars are equipped with technologically advanced safety features that can takeover the Master of Science Thesis Shelav Jain 6 Introduction Lateral Force Characteristics 12000 Lateral tire force [N] 10000 8000 6000 Fz = 1594N Fz = 3187N Fz = 4781N Fz = 6374N Fz = 7968N Fz = 9561N Fz = 11155N Fz = 12749N 4000 2000 0 0 5 10 15 20 Slip angle [deg] 25 30 Figure 1-6: The lateral tire force versus slip angle at different normal loads. control of the car from the driver in case of emergency situations. In the automotive industry, safety features of a car are divided into two categories: Passive Safety: It is referred to the components present in a car that helps to protect the passengers during an accident. These primarily include airbags, seat belts and the physical structure of the car. These safety features mitigate the effect of the crash. Active Safety: These are the safety features present on a car that assist in preventing a crash. These features actively control the vehicle dynamics and assist the driver in case of an emergency situation. A variety of driver assistance systems are being developed by automotive manufacturers to automate mundane driving operations, reduce driving burden and reduce accidents. Examples of such driver assistance systems are: 1. Collision Avoidance System, 2. Anti-Lock Braking System (ABS), 3. Adaptive Cruise Control (ACC), 4. Traction Control System (TCS), 5. Electronic Stability Control, 6. Lane Departure Warning System, 7. Lane Keeping System, 8. Night Vision System, Shelav Jain Master of Science Thesis 1-3 Active control of Vehicle Dynamics 7 9. Driver Condition Monitoring System. These technologies help in reducing driver burden and make them less likely to be involved in accidents [11]. Vehicle stability control systems prevent a vehicle from drifting out of the desired path [12]. Such systems are referred by different names such as yaw stability control systems, electronic stability control systems or direct yaw moment control systems. To control the yaw rate, which is defined as the rate of change of its heading angle, three types of systems have been proposed widely in the literature: 1. Differential braking system that uses ABS that applies asymmetric braking between the left and the right wheels to control yaw moment. 2. Active Steering System that modifies the driver’s steering angle input and applies a correction to the steering angle. 3. Active torque distribution that uses active differential to control the drive torque distributed to each wheel. The focus of this study is to understand the functioning of active safety features especially those related to the control of lateral dynamics. Therefore, passive safety features are not discussed hereafter. 1-3 Active control of Vehicle Dynamics Vehicle handling behavior is not consistent in every situation mainly because of the nonlinearity in the tire characteristics. When driving in the linear range, the tire slip angle is small and the behavior of the vehicle can be easily predicted. While driving in the non-linear range i.e with larger slip angles, vehicle handling behavior is much harder to predict. It is widely known from empirical studies that the vehicle becomes unstable as it is near the limit of lateral acceleration during cornering. During cornering, acceleration increases the understeer tendency while deceleration during cornering decreases this trend. Understeer and oversteer are the undesired movements of a vehicle. When the vehicle goes beyond the steering input of the driver, it is said to be oversteered. When the vehicle does not wholly follow the drivers steering input, it is said to be understeered. This implies that acceleration or deceleration affects vehicle behavior significantly, especially near the limit of lateral acceleration during cornering. This is the major cause of vehicle instability [13]. Vehicle Dynamics Control (VDC) system is a closed-loop system designed to improve driveability through programmed intervention in the brake system, drivetrain, steering system, suspension system or a combination of any of these. Longitudinal dynamics control of road vehicles has been studied for quite some time. This led to the development of systems like ABS and TCS. These important longitudinal dynamics control systems can be described as given in [14]: Anti-lock Braking system - The loss of yaw response of the vehicle to steering inputs during full braking while the wheels are locked has lead to very early investigations to prevent wheel lock. ABS system detects wheel lock and manipulates the brake pressure to maintain high Master of Science Thesis Shelav Jain 8 Introduction Figure 1-7: Yaw stability control system for vehicle lateral dynamics. Source: [3] level of handling performance during full braking. It prevents the wheels from locking and the vehicle remains steerable. It was first introduced on a volume-production vehicle in 1978 [1]. Traction Control System - If the driven wheels spin with excess engine torque, handling becomes difficult, particularly if the driven and steered wheels are identical. A TCS preserves high level of handling performance during driving with excess engine torque. In addition to this safety relevant task of ensuring stability and steerability of the vehicle when accelerating, TCS also improves the traction of the vehicle by regulating the optimum slip. The upper limit here is, of course, set by the traction requirement stipulated by the driver [1]. It regulates the slip of driven wheels to the optimum level as soon as possible. TCS is not a braking system but it makes use of and actively operates the braking system to prevent a wheel from spinning. TCS is designed to prevent loss of traction of driven wheels. By extending the TCS with additional sensors: steering angle sensor, brake pressure, yaw rate and lateral acceleration, feedback control of the vehicle motion is possible. The important aspect of a VDC is to control the lateral dynamics. The most prominent approach in controlling the lateral vehicle dynamics is a yaw stability control system. These yaw stability control systems generate the yaw moment around the vehicle’s vertical axis to improve the lateral vehicle dynamics. As explained in [3], the elements of yaw stability control system are as shown in Figure 1-7. This study is focused around lateral dynamics of a road vehicle. Therefore, lateral dynamics control systems are discussed in detail in the subsequent sections. 1-3-1 Electronic Stability Control Skidding or lack of stability is one of the main causes of major traffic accidents and it has motivated engineers to develop Electronic Stability Control (ESC). There is a tremendous increase of interest in advanced safety features in automobiles. These systems have proven to be helpful in reducing vehicle accidents. The goal of active control systems is to warn the driver and/or induce some actions directly on the actuators. ESC or Electronic Stability Program (ESP) is an active safety feature for motor vehicles which aims at improving driving dynamics and at preventing crashes that result from loss of control. Shelav Jain Master of Science Thesis 1-3 Active control of Vehicle Dynamics 9 Figure 1-8: Sideslip angle of a vehicle. Source: [2] Since 2006, ESC has become more common and more vehicles that are not luxury vehicles have become equipped with ESC. ESC was first introduced as optional safety equipment in passenger cars on the European market in 1995, and was increasingly installed in passenger cars from 1998 [15]. ESC affects crash risk by enhancing the steerability and stability of vehicles. Crash types that are typically associated with ESC are crashes that are caused by high speed cornering maneuvers, collision avoidance maneuvers, low friction conditions etc. The types of crashes that are typically affected by ESC are often more serious than other crashes. The results indicate that ESC prevents about 40% of all crashes involving loss of control [15]. The greatest reductions were found for rollover crashes (50%), followed by run-off-road (40%) and single vehicle crashes (25%) [15]. Traditionally, understanding the steering response of a car in different operating conditions is a task of the driver. The driver should be able to judge the physical limits of various handling maneuvers. However, most drivers are not able to detect these limits until the vehicle reaches its physical handling limits. To improve the handling performance, depending on the steering wheel angle the yaw moment on the car needs to be controlled. ESC limits the slip angle β of the vehicle in order to prevent vehicle spin. The sideslip angle β of a vehicle is the angle between its velocity vector at the Centre of Gravity (COG) and with the longitudinal axis of the vehicle, as shown in Figure 1-8. There exists a large variety of ESC systems. They have in common that they enhance the controllability of vehicles and can prevent skidding and loss of control in cases of oversteering or understeering. ESC systems differ with respect to how they regulate driving parameters (yaw rate or sideslip) and how they counteract deviations (e.g. by braking individual wheels and reducing engine power) [16]. Master of Science Thesis Shelav Jain 10 Introduction Figure 1-9: ESP control loop in a vehicle. Source: [1] Figure 1-9 shows the architecture of ESP control provided by BOSCH. It shows the sensors required to determine the controller input parameters. The sensors used are: (1) yaw rate sensor and lateral acceleration sensor, (2) steering wheel angle sensor, (3) brake-pressure sensor and (4) wheel-speed sensor. The figure also shows: (5) ESP control unit, featuring a high-level vehicle dynamics controller and the low-level slip controller. The actuators used to fulfill the controller’s demand are also shown. The choice of actuator depends on the type of lateral dynamics control system used. The possible actuating principles are: (7) brakes (9) fuel injection system (Diesel Powertrain) (10) sprak plug and (11) throttle valve. Apart from these actuators, other possible actuators are active front and rear steering, active differential and active suspension. In ESC with asymmetric braking, braking is activated on individual wheels in a targeted manner, such as, on the inner rear wheel to counter understeer, or on the outer front wheel during oversteer. It helps to keep the vehicle’s course stable under all driving conditions. ESC can also accelerate the driven wheels by specific engine-control interventions to ensure the stability of the vehicle. As steering and braking actuators are part of the vehicle chassis, the active control of yaw stability control system can be achieved through active chassis control. The active chassis control using active braking or active differential as actuator is known as Direct Yaw Moment Control (DYC) and using steering is called as active steering control as shown in Figure 1-10. In active steering control, the control strategy can be implemented either using Active Front Steering (AFS) or Active Rear Steering (ARS) or Four-Wheel Active Steering (4WAS). Selection of the stability control system depends on the control objective and on the driving situation. In small lateral acceleration conditions, vehicle sideslip angle is small and tires are in linear operating region. Thus, good handling performance can be obtained by the Shelav Jain Master of Science Thesis 1-3 Active control of Vehicle Dynamics 11 Figure 1-10: Active chassis control. Source: [3] active steering system alone. In emergency cases however, when the lateral acceleration and sideslip angle are significant, stability can not be guaranteed only by steering control, in this case direct yaw moment control by differential braking can enhance the performance of the vehicle. Most of the stability control systems rely on the yaw stability controller to force the vehicle to follow a desired path. These controllers give an output to the specific actuator to generate the required yaw moment. The yaw moment can be described as the moment around the vehicle’s vertical axis. This is caused by different longitudinal forces acting on the left and right sides of the vehicle and different lateral forces acting at the front and rear axles. Yaw moments are required to turn the vehicle when cornering. However, it is well known that at large vehicle sideslip angles, changing the steering angle produces very little change in the yaw rate of the vehicle. In designing a good stability controller, yaw-rate control and sideslip control should be considered together. As mentioned in [17], it is necessary to control the sideslip angle along with the yaw motion in order to maintain the stability of a vehicle. The instability of vehicle at its limit is brought about by decreased control of the yaw moment at a large sideslip angle. Sensitivity of the yaw moment on the vehicle w.r.t changes in steering angle decreases rapidly as the sideslip angle of the vehicle increases, as shown in Figure 1-11 [13]. From this curved line it can be seen that, when the sideslip angle is small, the yaw moment required is proportional to the slip angle. But for the larger sideslip angle, yaw moment saturates and starts to decrease at a certain point. This means that at the larger slip angles, the required yaw moment can not be generated using steering as an actuator for active chassis control. Therefore, to improve the vehicle handling and stability performance, it is essential to control both yaw rate and sideslip responses [18]. Clearly in lateral vehicle dynamics control both yaw rate and sideslip angle play an important role. As explained in [19], control objective of yaw stability control system may be classified into three categories; yaw rate control, sideslip control and the combination of yaw rate and sideslip control as given in Figure 1-12. Master of Science Thesis Shelav Jain 12 Introduction Figure 1-11: Yaw moment as function of sideslip angle for different steering angles. Source: [3] Figure 1-12: Objective of yaw stability control system. Source: [3] Shelav Jain Master of Science Thesis 1-3 Active control of Vehicle Dynamics 13 Figure 1-13: Block diagram for integrated control objective. Source: [4] The objective of yaw rate control system is to control the actual yaw rate close to the desired yaw rate which is generated by the reference model. This will improve the handling or maneuverability of the vehicle. In the steady state condition, the desired yaw rate response γd can be obtained by [3]: γd = Vx δf , (lf + lr ) + kus Vx 2 (1-4) where Vx is the longitudinal velocity, lf and lr are the distance from COG to the front and rear axle respectively, δf is the steering angle of the front wheels and kus is the stability factor also known as the understeer gradient and is defined as: kus = m(lr Cr − lf Cf ) , (lf + lr )Cf Cr (1-5) where m is the vehicle mass, Cf and Cr are the front and rear tire cornering stiffnesses. From equation (1-5) the term (lr Cr − lf Cf ) plays an important part in cross coupling of sideslip and yaw motions. This term is the yaw moment caused by unit tire slip angle. When this value is positive, side slip motion yields a restoring yaw moment and reduces the slip angle, but when it is negative the yaw moment results in an increase in the slip angle. The former is termed as understeer and the later is oversteer [20]. The sideslip angle based lateral dynamics control systems control the lateral stability of the system by keeping the desired sideslip βd as zero in steady state condition. To control these two control objectives effectively, two different control mechanisms are required. The block diagram for such a control system is shown in Figure 1-13. The vehicle yaw moment controller gives the desired wheel slip λd as the output and wheel slip controller applies the required braking torque on the wheels to generate the desired yaw rate and sideslip angle. Any chassis control system that aims to improve the vehicle handling performance must rely on the tire lateral force and longitudinal force. These two forces rely upon the vertical load and are interdependent. One of the major advantages of direct yaw moment control method Master of Science Thesis Shelav Jain 14 Introduction is that the tire longitudinal force has no feedback from the vehicle lateral motion as long as it is within the limit of the tire capacity with respect to the vertical load [20]. As explained earlier, the direct yaw moment control is one of the most effective methods of active chassis control. It must be noted that direct yaw moment control can become highly important and crucial for Electric Vehicle (EV) when compared with the Internal Combustion Engine (ICE) type vehicles. In an EV with in-wheeled electric motors, it is easier to control the yaw moment. Most of the yaw moment based control systems applied to improve the vehicle lateral dynamics are model-based. This means that in order to decide the control output the mathematical model of the vehicle is used. Most of the yaw stability control system architectures use linear vehicle models to simplify the control law. But, in the real world environment, the roadvehicle dynamics is highly nonlinear and uncertain. The main problem of yaw rate and sideslip tracking control systems are uncertainties caused from variations of dynamic parameters such as road surface adhesion coefficients, tire cornering stiffness, vehicle mass, vehicle speed and moment of inertia. Therefore, the control system designed should be robust to overcome the effect of these uncertainties, as given in [21], or these dynamic parameters should be available for real time model update either by the use of suitable sensors or estimation algorithms. As shown in [20], a robust control system is designed for lateral vehicle control to compensate for change in the dynamic parameters. It is useful when the change in tire cornering stiffness is not significant. It is conceivable that there may be a running condition in which robust control theory cannot be sufficient to absorb conditions encountered when the tire cornering stiffness is drastically reduced. In particular, when the tire force exceeds the maximum frictional force and the cornering stiffness indicates a negative value, it is conceivable that the control system will aggravate the instability. In addition the controller may be conservative. In designing any lateral dynamics control law, determination of lateral force saturation or nonlinear operating region of tires is very important. The lateral force saturation can be because of heavy steering command and/or low friction road. The lateral tire force saturation can be detected by estimation of ICS, as shown in Figure 1-4. In [22], [23], [24], [25], authors have proposed a yaw moment control system for EV based on adaptive DYC with cornering stiffness estimation. The adaptive DYC can be explained using the following equations. The yaw motion of an EV can be given as: I dγ = Nz − Nt − Nd , dt (1-6) where I is the vehicle inertia, γ is the yaw rate, Nz is the control yaw moment, Nt is the yaw moment generated by tire and road contact and Nd is the disturbance yaw moment. For the linear vehicle model and linear tire model the yaw moment Nt can be written as: Nt = 2Cr lf lr γ − β lr − 2Cf δf − γ − β lf . Vx Vx (1-7) This yaw-moment can be estimated by the yaw moment observer. Then the cornering stiffnesses can be identified using a recursive least square algorithm. Alternatively, if real-time cornering stiffnesses values are known from some other estimation process then the adaptive DYC can be updated directly. Shelav Jain Master of Science Thesis 1-4 Application of State Estimation 15 Figure 1-14: The principle layout of a control loop using a state estimator. 1-4 Application of State Estimation In most feedback control systems in the automotive field, the control action depends on some important variables like vehicle lateral velocity, sideslip angle, tire-road forces etc, which are often not measured directly because of technical and/or economic reasons. Therefore, it is required to use estimators or virtual sensors for estimation of these important variables. In order to implement any of the active control strategies discussed above, the vehicle’s lateral dynamics states have to be estimated from the available sensor signals. Effective operation of each of these systems depends on an accurate knowledge of the vehicle states, such as velocity, lateral acceleration, yaw rate, as well as vehicle and tire side slip. To accurately estimate the state values, mathematical model of the physical system used in the estimation algorithm must be as precise as possible [26]. Estimation means the extraction of information of any physical variable not available directly from sensors, by using available information. An observer is an algorithm that computes at every instant the values of variable of interest, not directly measured. The objective of the observer is to estimate sequentially the state of the vehicle dynamic system using a sequence of noisy available measurements made on the system. For dynamic state estimation, the discrete time approach is convenient for real-time application using on-board systems. The idea of a state estimator is to implement a model of the real system in an on-board computer in parallel with the system itself, as shown in Figure 1-14. The model of the real system can be a full nonlinear model or a simplified linearlized model. For a linear model, dynamics of the plant are represented by a state-space description where matrix A is the state matrix, B and G are input vectors for the controller input u and plant noise w respectively, and matrices C and D form the output y. The discrete time representation of A and B are denoted by φ and Γ. In Figure 1-14, a Kalman filter is used as the stochastic state estimator. It is a set of mathematical equations that provides an effective computational Master of Science Thesis Shelav Jain 16 Introduction means to estimate the state of a process, in a way that minimizes the mean of the squared error [27]. The prediction and correction steps of Kalman filter state estimation algorithm are discussed in detail in Chapter 3. The plant to be observed in Figure 1-14 is excited by noise w that is characterized by stochastic quantities and that the sensors used are also corrupted by stochastic noise v as well. Essentially, the state estimator is driven by the same inputs as the plant with exception of the process noise. The principle of estimating the system states x is based on the comparison of measured outputs y and estimated outputs yˆ. With a good state estimator, the difference e which is fed back into the Kalman filter and the estimated states x ˆ will follow the plant states x. A necessary condition for estimating the state variables is that the plant is observable. This means that all major plant dynamics affect output y. For a more accurate representation of system dynamics, a complex nonlinear model is required. The Kalman filter explained above needs to be extended to accommodate the nonlinearities of the model. Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are the most commonly used state estimators for nonlinear systems. Although, both are computationally expensive, UKF is more accurate than EKF. Their differences are treated in detail in Chapter 3. 1-4-1 Estimation of Sideslip Angle As discussed extensively in the previous section, tracking of sideslip angle is also required along with tracking of yaw rate for satisfactory lateral dynamics response. But, measurement of sideslip angle is not possible commercially. To date, no vehicle sideslip angle sensor is available that is accurate or economical enough to be implemented on the current vehicles. The estimation of sideslip angle has been a widely studied topic. Several strategies were proposed, mainly based on state observers; the procedures rely on tire models and evaluation of its parameters. These approaches can lead to good estimation if the tire parameters are correctly identified, but if changes occur in tires’ cornering stiffness due to different friction conditions or to the tire-wear, the estimation might be affected by errors. Most of the estimation techniques employs lateral acceleration sensor to obtain the sideslip angle. This method is not very accurate because road bank angle and roll angle of the vehicle acts as a bias element on the real lateral acceleration. The conventional estimation methods of sideslip angle are based on model-based observer design and direct sensor integration [2]. The model based observer method has higher accuracy in the linear tire region and it is robust against sensor bias. But, the estimation depends on vehicle parameter like vehicle mass, inertia and tire parameters like cornering stiffness. It is difficult to identify these parameters in real-time, therefore a model-based estimation algorithm cannot provide reliable results over all driving situations. The direct sensor integration method is a kinematic based approach, a differential relation between the sideslip angle and vehicle’s measurable dynamic parameters can be easily obtained. Since, the relation is differential, its application leads to a progressive drift during the integration process. In [28], [2], authors have proposed a new methodology by combining a vehicle model based method and a kinematics-based method. The kinematic based model provides reliable results in transient condition while vehicle model based approach is used to obtain sideslip angle in Shelav Jain Master of Science Thesis 1-4 Application of State Estimation 17 steady state condition. The experimental results indicate that the algorithm produces robust estimation of sideslip angle. In [29], [30], authors have proposed a method to estimate lateral velocity using lateral tire force sensor for EV. But, authors have assumed that the tire cornering stiffness for left and right tire is equal, this assumption is applicable only for EV because of lower position of COG, hence less weight transfer. So, this method cannot be extended to ICE vehicles. In [31], authors have presented a vehicle nonlinear model based estimation of sideslip angle using EKF. They have used the Dugoff tire model, which incorporates tire-road friction and tire cornering stiffness. But, in estimating sideslip angle they have used constant tire cornering stiffness. Therefore, applicability of the results cannot be verified. In [32], authors have proposed an adaptive approach (considering tire-road friction adaptation) for real-time estimation of sideslip angle. They have estimated the tire cornering stiffnesses and further used them in estimating the sideslip angle. But, the proposed algorithm does not give accurate results for the vehicle running on slippery roads e.g. snowy or icy roads. Authors in [33], have proposed a kinematic model based approach of estimation lateral velocity. The model relates longitudinal velocity, lateral velocity, longitudinal acceleration and yaw rate. The observer based on this method is not sensitive to change in vehicle parameters, but it produces noisy estimates. In addition, the estimates drift when yaw rate is zero because the observer becomes unobservable. In [34], authors have proposed an UKF based sideslip angle estimation algorithm. Since, tire-road friction plays an important role in estimation of vehicle states. This algorithm also estimates tire-road friction parameter as an augmented state. This estimation method using nonlinear models shows the practical potential for calculating lateral tire forces and sideslip angle. In [35], two block strategy to estimate tire-road forces, sideslip angle and cornering stiffness is given. Sliding mode observer is used to estimate tire-road forces and then these forces are used in EKF algorithm to estimate the sideslip angle and cornering stiffness. The proposed observer gives reliable results for lateral acceleration less than 0.6g. However, for lateral acceleration greater than 0.6g, results are not sufficiently accurate. This method is complicated to apply in the field vehicle. 1-4-2 Estimation of Tire Cornering Stiffness As discussed in earlier sections, a tire cornering stiffness is an important dynamic parameter. It plays an important in designing an ESC system, estimation of vehicle states and determination of lateral tire force saturation. In the determination of control law to enhance the handling of road vehicles, most of the VDC systems use constant cornering stiffness as input to the system. But, in real working situations, cornering stiffness vary due to change in tire-road friction and tire wear. Therefore, it is important to obtain these dynamic parameters for robust working of ESC systems. In order to minimize the effect of change in tire cornering stiffnesses, several researchers have proposed estimation for cornering stiffness. But to estimate the stiffnesses, present values of either tire-road friction or vehicle sideslip angle are required. Authors in [36] have proposed a recursive parameter estimation to estimate the cornering stiffness, but the proposed method depends on the measurement of sideslip angle and yaw rate. However, measurement of sideslip angle using sensors is difficult and expensive. Master of Science Thesis Shelav Jain 18 Introduction In [37], authors have presented a comprehensive summary of cornering stiffness estimation based on one-track vehicle model. Estimation methods can be divided into to categories: timedomain methods and transfer-function methods. The time-domain methods use the dynamic equations of the vehicle motion, and the underlying equations are correct even when the vehicle state is time-varying. On the other hand, the transfer-function approach is accurate only when the vehicle is time-invariant, and the effects of transient dynamics are ignored. The time-domain approach presented, requires real-time information of vehicle sideslip angle to estimate the cornering stiffness. They have also proposed a ‘beta-less’ method to estimate the front and rear tire cornering stiffness, but the system is under-determinate. To solve the under-determinate issue, information from multiple sensing points needs to be used and the information has to persistently exciting for the estimation to converge to the true values. Another way to solve the under-determinate issue is to assume the ratio of front and rear cornering stiffness as constant. The assumption of constant ratio of stiffnesses seems promising for real-time implementation, as it resolves the under-determinate issue. But, after various simulations it has been confirmed that in actual application of the vehicle, this ratio is not constant. Hence, the estimation of parameters diverge from its true values. According to [38], assumption of fixed front and rear cornering stiffness ratio (as described in [37]) has no strong foundation, especially when road condition changes. So, they proposed a new Beta-less estimation of cornering stiffness. But, this approach does not give accurate results for higher lateral acceleration because of the assumption that the left and right tire cornering stiffness are almost the same, as explained earlier the value of cornering stiffness depends on vertical load Fz on the tire. This assumption is valid only for lateral acceleration below 0.2g and for EV because of their low COG, hence less weight transfer. In [39], a transfer function approach to estimate the cornering stiffness has been given. From the bicycle model, the transfer function from steering angle to yaw rate was obtained and using parameter estimation approach unknown process parameters were calculated. The equation was numerically solved using Newton-Raphson method, which is cumbersome and time consuming and therefore not suitable for real time application. Recently, with the availability of cheap Global Positioning System (GPS) sensors, many researchers have shifted their focus to estimation techniques using them. In one such approach by authors in [40], they have used vehicle heading information from GPS along with yaw rate to obtain reliable estimate of the sideslip angle. Then they have used the estimated sideslip angle to obtain the tire cornering stiffnesses. The estimated tire cornering stiffness can then be used in the model based estimator in order to provide a more accurate model of the vehicle. In one of the research works at DLR, German Aerospace Research Center, researchers have demonstrated the application of tire cornering stiffness to active car steering. In [9], researchers have first estimated the tire cornering stiffness and then incorporated it in designing the control law for active steering. Basic function of cornering stiffness is to detect the saturation of lateral force, so that the controller can stabilize it. But, the estimation of cornering stiffness requires derivatives of steering angle and lateral acceleration. These measured signals are noisy and requires good filtering techniques. The cornering stiffness estimation is given as: Shelav Jain Master of Science Thesis 1-5 Thesis Objective 19 Cf = m a˙ y,f lr , lf + lr δ˙f − ay + γf Vx (1-8) where m is the mass of the front axle (according to the vehicle weight distribution), lf and lr are the distance of front and rear axle from COG respectively, Vx is the longitudinal velocity, ay,f is the lateral acceleration of front axle and γf is the yaw rate of front axle. In [41], authors have proposed a betaless estimation of ICS using lateral tire force measurements. This estimation is only available if the tire slip angle is profound enough and it is not available during constant cornering. Moreover, the proposed scheme requires derivative of measured forces to calculate the ICS. Special signal processing techniques are required to overcome the inaccuracy due to the sensor noise. 1-5 Thesis Objective Significant amount of research has been conducted in the field of state estimation but future active safety systems need more accurate information about the state of a vehicle. The main disadvantage of most of the vehicle state estimation approaches is the requirement of prior knowledge of tire parameters such as cornering stiffness and friction coefficient. However, the tire models can be used to estimate these parameters but it makes the system too complex for real-time application [42], [43]. Neglecting the aerodynamic effects, the motion of a vehicle is governed by the tire-road forces in each direction [5]. The load sensing bearing technology from SKF [44] and NSK [30], [45] presents an alternative approach towards vehicle state and parameter estimation. Control of vehicle dynamics using these sensors is a new topic of research. The application of tire force sensors has been proven successful for lateral tire force control of EV in [29], [45]. Another group of researchers [44], [46], [47], have worked on the application of load sensing in the area of hybrid ABS control. They have proven that the use of load sensing information along with wheel acceleration measurement makes the algorithm simpler and robust. Along with longitudinal dynamics control, lateral vehicle dynamics control using tire forces have also been studied in [48], [49]. The measured tire forces give an operating state of each tire and they are the most important variables as they are the only point of interaction of a vehicle with the road. However, most of the approaches discussed in the literature to estimate vehicle lateral dynamics states, use lateral acceleration measurement as one of the important input signals to the estimator. But the roll angle φ and road bank angle λ introduce an offset in the lateral acceleration measurement due to the gravity component. In [42], the author gives an expression to compensate the measured lateral acceleration for the road bank angle and roll angle: ay,global = ay,measured − g sin(φ − λ) . cos(φ − λ) (1-9) where ay,global is the actual lateral acceleration, ay,measured is the measured lateral acceleration and g is the acceleration due to gravity. Master of Science Thesis Shelav Jain 20 Introduction The use of measured tire forces for estimation eliminates the computationally expensive algorithms, which are otherwise necessary to estimate the road bank angle and roll angle. As mentioned earlier, sideslip control along with yaw rate control is required for satisfactory steerability and stability of a vehicle. But, measuring vehicle sideslip requires expensive sensors. Therefore, the objective of this study is to estimate the vehicle lateral dynamics states based on measured tire forces. The UKF algorithm is used to estimate the vehicle sideslip angle and tire cornering stiffnesses. The vehicle model used to evaluate the performance of the estimator is a multibody mechanical system for a four-wheeled vehicle with 15 mechanical Degree of Freedom (DOF) from CarSim [50]. CarSim is a vehicle dynamics simulation software developed by Mechanical Simulation Corporation in Ann Arbor, USA. It is a parametric modeling software widely used in research and industry to simulate and analyze vehicle dynamic behavior. 1-6 Thesis Outline This section is intended to give a brief overview of the contents of this thesis report. Chapter 1 has provided a detailed description of the VDC system. The main focus of this chapter was to understand the concept of Active Chassis Control. It further discussed the importance of sideslip angle tracking, which is the basis of this work. A brief description of previous work related to sideslip angle and cornering stiffness estimation was also given. In Chapter 2, physical modeling of the vehicle considered for the project is explained. In addition to that it gives an overview of the important tire models. Chapter 3 is devoted to the theory of estimation algorithms. It gives a detailed description starting from most basic observers to the nonlinear estimation algorithms. The two extensions namely, EKF and UKF are explained. Chapter 4 and Chapter 5 discuss the results of this study for different vehicle maneuvers. These chapters describe the estimation process of sideslip angle and cornering stiffness for the bicycle model and Four-Wheel vehicle model (FWVM). The estimation algorithm, vehicle model, observability analysis and observer error analysis constitute this chapter. The tuning parameters for the UKF algorithm are also given. Finally, Chapter 6 contains the conclusions and future recommendations. Shelav Jain Master of Science Thesis Chapter 2 Physical Modeling The estimator use physical model of the system to estimate the system’s states. With respect to the automotive systems, accuracy of the estimation process mainly depends on the modeling accuracy of the two primary subsystems i.e. vehicle model and tire model. Based on the level of complexity, various vehicle and tire models are available in the literature. The degree of complexity of the model also depends on the desired objective. For example, in simulator design, it is necessary to reproduce the behavior of each individual component. But, in the real time application, a compromise has to made due to the limited calculation capacity. 2-1 Vehicle Model The vehicle model should represent all dynamics of interest as simply as possible. In this analysis, the vehicle is modeled as a rigid body and is studied in a body-fixed coordinate system with the origin located at the Centre of Gravity (COG). Complete dynamic behavior of the vehicle can be described by three individual model or combination of them, namely, 1. Vertical dynamics model- This model is used for analyzing ride of the vehicle and it incorporates the suspension dynamics. 2. Planar dynamics model- This model is also known as yaw plane model. It is used to understand the lateral and longitudinal dynamics of the vehicle, which is the topic of interest for this work. There are two commonly used vehicle models: (a) Four-Wheel vehicle model (FWVM) or 2 track model. (b) Single track model or Bicycle model. 3. Roll Plane model- It is used to describe the roll dynamics. The focus of this work is to understand the lateral dynamics, which can be accurately modeled by yaw plane model. Both the planar dynamics model i.e. the 2-track model and bicycle model are treated in detail in the following sections. Master of Science Thesis Shelav Jain 22 Physical Modeling Figure 2-1: Four-wheel vehicle model. 2-1-1 Four-wheel vehicle model It is commonly called as two track model and is widely used to analyze and control the longitudinal and transverse vehicle dynamics. Figure 2-1 shows the schematic of a 3 Degree of Freedom (DOF) vehicle model that represents the lateral, longitudinal and yaw motions. The front and rear track widths E are assumed to be equal. The distances from the vehicle’s COG to the front and rear axles are lf and lr respectively. Here δij is the steering angle of the ij tire, Fyij is the lateral tire force of ij tire and Fxij is the longitudinal tire force of ij tire. The side-slip at the vehicle’s COG (β) is the angle between the velocity vector (V ) and the ˙ is the angular velocity about the vertical axis true heading of the vehicle. The yaw rate (ψ) with Iz being the yaw moment of inertia. The longitudinal and lateral velocities are Vx and Vy , respectively. The longitudinal and lateral forces (Fx,y,i,j ) acting during the movement, are shown for front and rear tires of the vehicles. Describing the inter-relationship between different vehicle dynamics parameters using Newton’s second law gives: Shelav Jain Master of Science Thesis 2-1 Vehicle Model 23 lf [FyF L cos δF L + FyF R cos δF R + FxF L sin δF L + FxF R sin δF R ] ψ¨ = Iz lr [FyRL cos δRL + FyRR cos δRR + FxRL sin δRL + FxRR sin δRR ] − Iz E [FyF L sin δF L − FyF R sin δF R + FxF R sin δF R − FxF L cos δF L + 2Iz −FxRL cos δRL + FyRL sin δRL + FxRR cos δRR − FyRR sin δRR ], ay = ax = (2-1) 1 [FyF L cos δF L + FyF R cos δF R + FxF L sin δF L + FxF R sin δF R + FyRL cos δRL mv +FyRR cos δRR + FxRL sin δRL + FxRR sin δRR ], (2-2) 1 [−FyF L sin δF L − FyF R sin δF R + FxF L cos δF L + FxF R cos δF R + FxRL cos δRL mv +FxRR cos δRR − FyRL sin δRL − FyRR sin δRR ], (2-3) V˙x = Vy ψ˙ + ax , (2-4) V˙y = −Vx ψ˙ + ay , (2-5) where mv is mass of the vehicle, ax and ay are the longitudinal and lateral acceleration respectively. The sideslip angle can be calculated from lateral and longitudinal velocities: ! Vy . Vx β = arctan (2-6) The longitudinal and lateral velocities, the steer angle of the front wheels and the yaw rate are then used as a basis for the calculation of the tire slip angles αij : αF L Vy + lf ψ˙ = δF L − arctan , ˙ Vx − E ψ/2 (2-7) αF R Vy + lf ψ˙ = δF R − arctan , ˙ Vx + E ψ/2 (2-8) αRL Vy − lr ψ˙ = δRL − arctan , ˙ Vx − E ψ/2 (2-9) αRR Vy − lr ψ˙ . = δRR − arctan ˙ Vx + E ψ/2 (2-10) " " " " Master of Science Thesis # # # # Shelav Jain 24 Physical Modeling Figure 2-2: The bicycle model. 2-1-2 The Bicycle Model Vehicle model can be considerably simplified by using a classical bicycle model as shown in Figure 2-2. It is also known as the single track model. It is mostly used to describe the lateral dynamic behavior, especially for evaluation of sideslip angle [5]. It is simplified from the nonlinear FWVM based on the following assumptions: 1. Tire forces operate in the linear region. 2. The two left and right front wheels are represented by one single wheel. Similarly, the rear wheels are represented by one central rear wheel. 3. Longitudinal speed is constant i.e. the longitudinal acceleration is zero. 4. In this model, vertical movements are ignored, roll motion is not taken into account. 5. Both front wheels have the same steering angle and also the rear wheels. 6. No braking is applied at all wheels. The simplified bicycle model is formulated by the following relationship: 1 ψ¨ = [lf [Fxf sin δf + Fyf cos δf ] − lr [Fxr sin δr + Fyr cos δr ]], Iz β˙ = 1 ˙ [Fyf cos δf + Fxf sin δf + Fyr cos δr + Fxr sin δr ] − ψ, mv Vx (2-11) (2-12) where, • ψ˙ is the yaw rate. • β is the vehicle sideslip angle. • Iz is the yaw moment of inertia. • mv is the mass of the vehicle. Shelav Jain Master of Science Thesis 2-2 Tire Dynamics 25 Figure 2-3: Tire dynamics variables. Source: [5] • lf is the distance between front axle and COG. • lr is the distance between rear axle and COG. • Fyf and Fyr are the front and rear lateral forces respectively. • Fxf and Fxr are the front and rear longitudinal forces respectively. • Vx is the longitudinal velocity of the vehicle. • δf and δr are the front and rear steering angles respectively. Assuming small angles, front (αf ) and rear (αr ) tire slip angles are calculated using kinematic relations with respect to the vehicle’s speed and yaw rate: 2-2 αf = δf − β − lf ψ˙ , Vx (2-13) αr = δr − β + lr ψ˙ . Vx (2-14) Tire Dynamics Tires are the main vehicle components generating external forces that can be effectively manipulated to affect vehicle motions. The developed forces are the function of tire properties Master of Science Thesis Shelav Jain 26 Physical Modeling Figure 2-4: Four categories of possible types of approach to develop a tire model. Source: [6] (material, tread pattern, tread depth, profile, etc.), the normal load on the tire, road friction and the velocities experienced by the tire. Tires deform due to the vertical load, and makes contact with the road surface over a nonzero footprint area called the contact patch. The force that a tire receives from the road is assumed to be at the center of the contact patch and can be decomposed along the three wheel axes. The lateral force, Fy , is the force along the Y axis, the longitudinal force, Fx , is the force along the X axis and the normal or vertical force, Fz , is the force along the Z axis. Figure 2-3 shows the tire dynamics variables. In Figure 2-3, the moment along the Z axis, Mz , is the aligning moment, moment along the X axis Mx , is the overturning moment and the moment along the Y axis, My , is the rolling moment. All the vehicle states depend on the accurate measurement of vehicle accelerations, which in turn depends on the accurate knowledge of tire forces, as can be seen in equations (2-2) and (2-3). Until very recently, the actual tire forces cannot be measured, they need to be calculated based on the estimated and measured states. Several types of tire models have been developed during the past half century; each type for a specific purpose. Different levels of accuracy and complexity may be introduced in the various categories of utilization. This often involves entirely different approaches. Figure 2-4 illustrates the effect of different modeling approaches i.e. from empirical to theoretical model. Shelav Jain Master of Science Thesis 2-2 Tire Dynamics 27 From left to right in Figure 2-4 the model is less experimental and more theoretical. In the middle, the model will be simpler but possibly less accurate while at the far right the description becomes complex and less suitable for application in the simulation of vehicle motions and may be more appropriate for the analysis of detailed tire performance in relation to its construction [6]. The focus of this study is to understand the lateral dynamics of a vehicle. So the lateral force models used to understand this important characteristic are discussed further. A vehicle can turn because of the applied lateral tire forces. While negotiating a turn, lateral force originates at the center of the contact patch which lies in the horizontal plane and is perpendicular to the direction in which the wheel is headed if no camber exists. To transmit these lateral forces, the tire must turn laterally. In turn, the direction of motion of the tire deviates from the wheel plane. The purpose of a tire model is to obtain a structure in which the measurement data can be fitted with the use of suitable parameters. From the point of simplicity and real-time implementation of this model-based estimation technique, linear tire model has been used in this study. But, for the sake of completeness most commonly used nonlinear tire model is also briefed in this work, for detailed description interested readers may follow [6]. 2-2-1 Magic Formula According to Pacejka [6], the tire model should be: • able to describe all steady-state tire characteristics, • easily obtainable from measured data, • physically meaningful; its parameters should characterize in some way the typifying quantities of the tire, • compact and easy to use, • able to contribute to better understanding of tire behavior. The basic formula for this model is [6]: y = Dsin[Carctan(Bx − E(Bx − arctanBx))], (2-15) with Y (x) = y(x) + SV and x = X + SH . In these formulas, Y is the output variable, which stands for longitudinal force Fx or lateral force Fy or aligning moment Mz . X is the input variable, which stands for lateral slip angle αy or longitudinal slip αx . The parameters B, C, D, E, SV and SH of these formulas as shown in Figure 2-5 are defined as follows: • D: the peak value. • C: the shape factor that controls the limits of the range of the sine function and thereby determines the slope of the resulting curve. Master of Science Thesis Shelav Jain 28 Physical Modeling Figure 2-5: The Magic Formula parameters. Source: [6] • B: the stiffness factor. This factor determines the slope at the origin and is also called the stiffness factor. • E: the curvature factor; it controls the value of the slip at which the peak of the curve occurs. • BCD: this product corresponds to the slope at the origin (x = y = 0). For lateral force, this factor corresponds to the cornering stiffness. The Magic Formula typically produces a curve that passes through the origin. To allow the curve to have an offset with respect to the origin, shifts SH and SV have been introduced. 2-2-2 Linear model In normal driving situations, the tires are well under their saturation limit and have small tire slip angles. For these conditions, the lateral tire force is a linear function of the slip angle with the slope equal to the linear cornering stiffness. Lateral forces of the front and rear tires (Bicycle model) can be written using front equation (2-13) and rear slip angle equation (2-14) as: ψ˙ δf − β − lf , Vx (2-16) ψ˙ δr − β + lr . Vx (2-17) Fyf = Cαf αf = Cαf Fyr = Cαr αr = Cαr Shelav Jain Master of Science Thesis 2-3 Summary 2-2-3 29 Transient tire model As tires have a complex structure, tire forces are not developed instantaneously at the maneuvering action. Owing to the flexible structure of the tire, generation of forces require some rolling distance. In turn the force and moment response is delayed in response to the external input namely steering angle. As explained in [11], the dynamic lateral tire force model can be given as: τ F˙y + Fy = F¯y , (2-18) where τ is the relaxation time constant, Fy is the dynamic lateral force and F¯y is the static lateral force calculated from one of the tire models either discussed earlier or from the literature. The relaxation length behavior of tires is not analyzed in this study but interested readers may follow [51], [52], [53]. 2-3 Summary This chapter explained the physical models of a vehicle and tire. Different types of vehicle models are described and classified according to the complexity of implementation and accuracy of the results. The FWVM is more accurate than the bicycle model as it captures the load transfer from left to right wheels or vice versa during cornering maneuvers. But, implementation of the FWVM is difficult. Next, tire dynamics is described and different models are studied. Static tire models like the Magic tire formula and a linear model are presented. The transient model on the basis of relaxation tire behavior is briefly touched upon. The dynamic modeling of the vehicle-tire system forms the backbone of the estimation problem. The estimation algorithms which use these physical models are be explained in the next chapter. Master of Science Thesis Shelav Jain 30 Shelav Jain Physical Modeling Master of Science Thesis Chapter 3 State Estimation and Filtering In this chapter, the state estimators used in this study are discussed. First, a short introduction is given to modeling of a general system, then its observability conditions are discussed and then the estimator algorithm is explained. In many dynamic mechanical systems, it is often impractical to assume that all states describing the system’s response can be measured. Sometimes, they cannot be measured by physical means and sometimes it is not economical. Also, resources can be saved when the states may be estimated using information from other signals and physical models. Whenever the state of a system is to be estimated from noisy sensor information, some kind of state observer is implemented to combine the information from different sensors to produce an accurate estimate of the state. State reconstruction is a two-step procedure: in the first step a physical model is identified and then an estimator is designed using suitable observation techniques. Vehicle and tire model used in this study are already discussed in the chapter 2. 3-1 Modeling of the system The design of observers is based on the system dynamics. The dynamics of a system under study can be described by a non-linear function f of the state variables x, the system inputs u and a noise signal v known as process noise. The system outputs y can be described as continuous nonlinear function of x, u and a noise signal w, known as measurement noise. The system can be shown as: x˙ = f (x, u, v), y = h(x, u, w). (3-1) In general the functions f and h are nonlinear but in many cases, it is assumed that f and h are linear, functions of x, u and the noise. The system can be described as: Master of Science Thesis Shelav Jain 32 State Estimation and Filtering x˙ = Ax + Bu + v, y = Cx + Du + w, (3-2) where A is the state evolution matrix, B is the input matrix, C is the output matrix, and D is the feed through matrix. Assumption of linearity makes the estimation problem of easier to solve. In reality, however linear systems are very rare and models are never free of uncertainties. 3-2 Observability of the system As explained in [54], A system is observable if for any t > 0 it is possible to determine the state of the system x(t) through measurements y(t) and inputs u(t) in the interval [0, t]. If a system is observable, then there is no "hidden" dynamics inside it; we can understand everything that is going on by observing the inputs and outputs. The observability of the system is of practical interest because it determines if a set of sensors is sufficient for controlling a system. The linear system given by equation (3-2) is observable if the observability matrix O given by equation (3-3) has full rank n (i.e. n linearly independent rows) [54]: O = [C CA CA2 ....CAn−1 ]T . (3-3) The question for nonlinear system is the same, whether it is possible to reconstruct the states from measurement of outputs and inputs. The estimation problem observability of nonlinear systems is investigated by [55], the authors have presented two sufficient conditions of global observability. But, in this study, the observability studied for non-linear system is local, as discussed in [56], [57]. The system is locally observable at x0 if there exists a neighborhood of x0 such that every x in that neighborhood other than x0 is distinguishable from x0 . The local observability analysis uses Lie derivatives of hi function. The Lie derivative at (r + 1) order is defined as: ∂Lf r hi (x) Lf r+1 hi (x) = f (x, u), (3-4) ∂x with ∂hi (x) Lf 1 hi (x) = f (x, u), (3-5) ∂x where i = 1, ..., p and p is the number of measurements, x is the state and u is the input. The observability function oi corresponding to the measurement function hi is defined as: dhi (x) dL 1 h (x) i f oi = , .... dLf n−1 hi (x) (3-6) where d is the operator: dhi = Shelav Jain ∂hi (x) ∂hi (x) , ...., . ∂x1 ∂xn (3-7) Master of Science Thesis 3-3 The Luenberger Observer 33 The observability function of the system is calculated as: o1 O = .... . (3-8) op The system given by equation (3-1) is locally observable if the observability matrix O given by equation (3-8) has full rank n, otherwise the system is unobservable. 3-3 The Luenberger Observer The Luenberger observer, on which the majority of the later developments in the field of state estimation are based, uses a mathematical model of the system to simulate it in real-time parallel to the system, using the inputs as well as the outputs to estimate the state. It assumes a system of the form in equation (3-2) but without considering the noise. The variables of a state observers are commonly denoted by ‘hat’: x ˆ and yˆ to distinguish them from the variables of the equations describing by the physical system. The equations describing the Luenberger observer are: x ˆ˙ = Aˆ x + L[y − yˆ] + Bu, (3-9) In the above equation L is a feedback gain. As can be seen, the observer is simply a copy of the system, augmented by a feedback term to correct estimation errors due to model uncertainty and process noise v. It is not always possible to choose L such that the state can be obtained. This depends on the so called observability of the system. The pair [A, C] should be observable. If this condition is met and the system is noise free, the observer gain L can be chosen such that the estimation error converges arbitrarily fast. In fact, the system describing the error dynamics will become: e˙ = (A − LC)e. (3-10) If there is noise on the system, a Luenberger observer in general cannot guarantee convergence of the error to zero. The presence of noise on the state variables as well as on the output measurements results in a dilemma. On one hand, it is desirable to choose L large, to quickly compensate for errors due to the process noise. On the other hand, this will make the estimation more sensitive to measurement noise, which makes it more attractive to choose a smaller L. 3-4 Linear Kalman Filter The Kalman filter is a linear observer with the observer gain L chosen such that the estimation error is least-square optimal under the conditions that: Master of Science Thesis Shelav Jain 34 State Estimation and Filtering • The system is linear and the system matrices are known. The stochastic discrete-time state-representation of a linear time-invariant system is given by: Xk = AXk−1 + BUk + vk , (3-11) Yk = CXk + wk . (3-12) • The process noise v and the measurement noise w are temporarily uncorrelated, zero mean, Gaussian white noise signals: vk = N (0, Qk ), wk = N (0, Rk ), (3-13) where Q and R are the covariance matrices describing the second-order properties of the state and measurement noise. The optimal estimation problem of Xk based on input-output data and knowledge of the model can be solved by minimizing the loss function: ˆ k/k ) = E{(X ˆ k/k−1 − Xk )2 }, ∀k, J(X (3-14) ˆ k/k−1 and X ˆ k/k are, respectively, the prediction and the prior estimate of Xk . where X A recursive estimation of Xk can be expressed in the form: ˆ k/k = X ˆ k/k−1 + Lk (Yk − Yˆk/k−1 ), X (3-15) where Yˆk/k−1 is the prediction of Yk and Lk is the Kalman gain. The difference between Yˆk/k−1 ˆ k−1/k−1 and and Yk is called the filter innovation at instant k. Assuming the prior estimate X the current observation Yk to be Gaussian random variables, the optimal solution is given by the following equations: • Initialization: The initial state and the initial covariance are determined by: ˆ 0 = E[X0 ], P0 = E[(X0 − X ˆ 0 )(X0 − X ˆ 0 )T ]. X (3-16) • Time update: – The prediction of the state is given by: ˆ k|k−1 = AX ˆ k−1|k−1 + BUk . X (3-17) – The predicted covariance is computed as: Pk|k−1 = APk−1|k−1 AT + R. (3-18) • Measurement update: – The filter gain is calculated by: Lk = Pk|k−1 C T [HPk|k−1 C T + Q]−1 . Shelav Jain (3-19) Master of Science Thesis 3-5 Non-linear Kalman Filter 35 – The state estimation is determined by: ˆ k|k = X ˆ k|k−1 + Lk [Yk − C X ˆ k|k−1 ]. X (3-20) – The estimated covariance is: Pk|k = [I − Lk C]Pk|k−1 . (3-21) The first step consists of initializing the filter by choosing a starting estimate for the state and its variance. In general, the effect of these initial estimates diminishes with time and they do not affect the steady-state performance of a filter. The second step introduces equations to estimate the state vector. These equations are divided into time and measurement equations. The time update projects the current state estimate ahead in time whereas the measurement update adjusts the projected estimate by an actual measurement at that time [5]. 3-5 Non-linear Kalman Filter Kalman Filter is an optimal filter for estimating a linear system states. Due to its simplicity, it has found its usage in numerous practical applications. However, most of the real-world systems are non-linear in nature. Estimation of these non-linear systems is extremely important because all practical systems are non-linear for example target tracking, vehicle navigation, plant control etc. In order to deal with the non-linearity, many extensions have been developed over the Kalman filter. Two of the main extensions are: 1. Extended Kalman Filter (EKF) where non-linearities are accounted by linearising the system about its last-known best estimate with the assumption that the error incurred by neglecting the higher-order terms is small in comparison to the first-order terms. 2. Unscented Kalman Filter (UKF) which implements the unscented transformation. In this work, UKF is used and its properties and advantages are discussed in detail in the next section. EKF is widely used in the industry for estimation of nonlinear systems. In this study, it is not implemented but interested readers can follow [27]. Although the EKF is conceptually simple, it has three well-known drawbacks, as given in [58]: 1. Linearisation can produce unstable filter performance if the time-step intervals are not sufficiently small. 2. Derivation of Jacobians are non-trivial in most applications and often lead to significant implementation difficulties. 3. Sufficiently small time-step intervals usually imply high computational overhead as the number of calculations demanded for the generation of the Jacobian and the prediction of state estimate and covariance is large. Master of Science Thesis Shelav Jain 36 State Estimation and Filtering Figure 3-1: The principle of the Unscented Transformation. Source: [7] 3-5-1 Unscented Kalman Filter In Kalman filter a Gaussian Random Variable (GRV) is propagated through the system dynamics. In the EKF, the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearisation of the non-linear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but it is now represented using a minimal set of carefully chosen sample points known as sigma points. These sigma points completely capture the true mean and covariance of the GRV, and when propagated through the true non-linear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any non-linearity [59]. The EKF, in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF [59]. Since UKF algorithm is based on the Unscented Transformation (UT), it is explained in the next subsection. The Unscented Transformation The UT is a method of calculating the statistics of a random variable which undergoes a nonlinear transformation. It is based on the intuition that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary non-linear function or transformation [60]. As shown in Figure 3-1, a set of sigma points are chosen so that their mean and sample ¯ and Pxx . Then the non-linear function is applied at each point to get a covariance are x ¯ and Pyy . cloud of transformed points, their mean and covariance are y ¯ and covariance Pxx is approximated by The L-dimensional random variable x with mean x 2L + 1 sigma points with corresponding weights Wi , according to the following: Shelav Jain Master of Science Thesis 3-5 Non-linear Kalman Filter 37 χ0 = x ¯, q χi = x ¯ + ( (L + λ)Pxx )i , i = 1, ...., L, q χi = x ¯ − ( (L + λ)Pxx )i−L , i = L + 1, ...., 2L, W0 (m) = λ\(L + λ), W0 (c) = λ\(L + λ) + (1 + α2 + β), Wi (m) = Wi (c) = 1\[2(L + λ)], i = 1, ...., 2L, (3-22) where λ = α2 (L + κ) − L is a scaling parameter. α determines the spread of the sigma points ¯ and is usually set to a small positive value. κ is a secondary scaling parameter around x which is usually set to 0 and β is used to incorporate prior knowledge of the distribution of x. p The term ( (L + λ)Pxx )i is the ith row of the matrix square root. In [61], it is given that the sigma points capture the same mean and covariance irrespective of the choice of linear algebra method used to obtain matrix square root. To obtain the matrix square root in this study, Cholseky decomposition is used, which is a numerically efficient and stable method. The term p ( (L + λ)Pxx )i is obtained form lower triangular matrix of the Cholseky factorization. For detailed explanation about matrix square root methods, [62] can be followed. These sigma points are then propagated through the non-linear output functions. The transformation procedure is as follows as given in [61]: 1. Instantiate each point through the function to yield the set of transformed sigma points, Yi = h(χi ), i = 0, ..., 2L. (3-23) ¯ is given by the weighted average of the transformed points, 2. The mean y ¯= y 2L X Wi (m) Yi . (3-24) i=0 3. The covariance Pyy is the weighted outer product of the transformed points, Pyy = 2L X ¯}{Yi − y ¯ }T . Wi (c) {Yi − y (3-25) i=0 The Unscented Filter The Unscented Filter is an extension of the UT. The flowchart of the algorithm is given in Figure 3-2. Master of Science Thesis Shelav Jain 38 State Estimation and Filtering Figure 3-2: Flowchart of the UKF algorithm. The transformation process which occur in an UKF consists of the following steps, as given in [59]: 1. Initialize the filter with the initial value of state and its covariance: x ˆ0 = E[x0 ], P0 = E[(x0 − x ˆ0 )(x0 − x ˆ0 )T ]. (3-26) 2. Calculate sigma points using this initial estimate of state and covariance as given in equation (3-22): q q χk−1 = [ˆ xk−1 x ˆk−1 + ( (L + λ)Pxx )i=1,..,L x ˆk−1 − ( (L + λ)Pxx )i=L+1,..,2L ]. (3-27) 3. Time Update: The predicted mean and covariance are computed by instantiating each point through the process model: χk|k−1 = f [χk−1 , u(k)]. Shelav Jain (3-28) Master of Science Thesis 3-5 Non-linear Kalman Filter 39 (a) The predicted mean is computed as: x ˆ− k = 2L X Wi (m) χi,k|k−1 . (3-29) i=0 (b) The predicted covariance is computed as: Pk − = 2L X T Wi (c) [(χi,k|k−1 − x ˆ− ˆ− k )(χi,k|k−1 − x k ) ] + Qk , (3-30) i=0 where Qk is the process noise covariance matrix. (c) Again obtain the sigma points using the predicted mean and covariance: χk−1 = [ˆ x− ˆ− k x k ± q (L + λ)Pk − ]. (3-31) (d) Instantiate each of the sigma points through the observation model: Yk|k−1 = h(χk−1 , u(k)). (3-32) i. The predicted observation is calculated as: yˆk− = 2L X Wi (m) Yi,k|k−1 . (3-33) i=0 ii. The innovation covariance is: Pyk yk = 2L X Wi (c) [(Yk|k−1 − yˆk− )(Yk|k−1 − yˆk− )T ] + Rk , (3-34) i=0 where Rk is the measurement noise covariance matrix. (e) Finally, predict the cross correlation as: P xk yk = 2L X Wi (c) [(χk|k−1 − x ˆ− ˆk− )T ]. k )(Yk|k−1 − y (3-35) i=0 4. The filter gain can be calculated as: Kk = Pxk yk Pyk yk −1 . (3-36) 5. Measurement Update: (a) The estimated state is given as: x ˆk = x ˆ− ˆk− ). k + Kk (yk − y (3-37) Pk = Pk − − Kk Pyk yk Kk T . (3-38) (b) The estimated covariance is: Master of Science Thesis Shelav Jain 40 3-6 State Estimation and Filtering Summary This chapter discussed the approaches for linear and non-linear Kalman filtering. The observability conditions for linear and non-linear systems are discussed. Then the formulation of state observer has been given. Next, the linear and non-linear Kalman filters are described, which constitute the backbone of this study. The filters work under the assumption that the noise has Gaussian distribution. For nonlinear filtering, the UKF algorithm is discussed in detail and a brief description of EKF is also given. In chapter 4 and chapter 5, the results obtained using the UKF algorithm described in this chapter are discussed. Shelav Jain Master of Science Thesis Chapter 4 Estimation using Bicycle Model The focus of this chapter is to analyze the accuracy of the nonlinear filter designed to estimate the sideslip angle and tire cornering stiffness. Two different configurations of the filter are studied using different vehicle maneuvers. First Unscented Kalman Filter (UKF) algorithm is without the sideslip angle as measurement signal and second UKF has sideslip angle as measurement signal. The measurement of sideslip angle is taken from the estimation algorithm designed in [63] using tire force measurements. 4-1 Nonlinear Observer: State Space Algorithm To estimate the sideslip angle and tire cornering stiffness, the vehicle’s equations of motion are considered to model these variables as function of other vehicle parameters. The nonlinear vehicle model used is described in section 2-1-2 and linear tire force model is described in section 2-2-2. To build a model based UKF, the nonlinear bicycle model equations (2-11) - (2-12) and linear tire model equations (2-16) - (2-17) are converted to discrete form by first-order Euler method as: xk = fk−1 (xk , uk ) + vk , yk = h(xk , uk ) + wk . (4-1) The state vector xk , at each time instant k, comprises of sideslip angle, yaw rate, front tire cornering stiffness, rear tire cornering stiffness: xk = [βk , ψ˙ k , Cαf,k , Cαr,k ]T . (4-2) The input vector uk comprises the front and rear wheels steering angles and front and rear wheels longitudinal forces: uk = [δf,k , δr,k , Fxf,k , Fxr,k ]T . (4-3) Master of Science Thesis Shelav Jain 42 Estimation using Bicycle Model The measurement vector comprises of yaw rate, front and rear wheels lateral forces: yk = [ψ˙ k , Fyf,k , Fyr,k ]T . (4-4) The nonlinear function f (·) that relates the states at time k to the states at time k − 1 and to the inputs uk is given as: T ψ˙ k−1 f1 = βk−1 + Cαf,k−1 δf,k − βk−1 − lf cos δf,k + Fxf,k sin δf,k mv Vx Vx ψ˙ k−1 + Cαr,k−1 δr,k − βk−1 + lr cos δr,k + Fxr,k sin δr,k − ψ˙ k−1 , Vx T ψ˙ k−1 ˙ lf Cαf,k−1 δf,k − βk−1 − lf cos δf,k + Fxf,k sin u1,k f2 = ψk−1 + Izz Vx ψ˙ k−1 − lr Cαr,k−1 δr,k − βk−1 + lr cos δr,k + Fxr,k sin δr,k , Vx f3 = Cαf,k−1 , f4 = Cαr,k−1 . (4-5) The measurement equations are given as: h1 = ψ˙ k , ψ˙ k h2 = Cαf,k δf,k − βk − lf cos δf,k , Vx ψ˙ k h3 = Cαr,k δr,k − βk + lr cos δr,k . Vx (4-6) The vehicle parameters are taken from the CarSim multi-body simulation software. A D-Class sedan vehicle is chosen for the simulation. The parameters values are given in table 4-1. S.No 1 2 3 4 5 Parameter Mass of the vehicle Yaw moment of inertia Distance from Centre of Gravity (COG) to front axle Distance from COG to rear axle Sampling Time Symbol mv Izz lf lr T Unit kg kgm2 m m s Value 1530 2315.3 1.110 1.670 0.001 Table 4-1: D class vehicle parameters values. 4-2 Observability Analysis As discussed in section 3-2, observability is the measure of how well the system states are related to its inputs and measurements. The observability study for the nonlinear system is local and uses the Lie derivatives as given in equation (3-4). For the discrete time state Shelav Jain Master of Science Thesis 4-3 Simulation Results 43 space system described above, observability matrix is evaluated using MATLAB symbolic environment and it is given as: O = [dhi (x) dL1f hi (x) dL2f hi (x) dL3f hi (x)]T . (4-7) For initial condition of non zero cornering stiffness the system is locally observable. The dimension of the observability matrix is 12 × 4 and the rank of the matrix O is 4. Therefore, the nonlinear system described by equations 4-5 and 4-6 is locally observable. 4-3 Simulation Results In this section, the algorithm developed in the previous section is validated and simulation results are presented for various experiments. The algorithm is simulated in CarSim and Simulink. The measurement and input signals are taken from the CarSim environment and then post-processed in Simulink. The vehicle model in CarSim simulation package has 15 mechanical Degree of Freedom (DOF) [50]. The CarSim package uses built-in nonlinear tire models with dependency on slip, load and camber. 4-3-1 Tuning Parameters For satisfactory working of the UKF, it is important to tune the process noise covariance matrix Q and measurement noise covariance matrix R. The yaw rate and sideslip angle are modeled using system dynamic equations, therefore low uncertainty is assigned to them. However, the cornering stiffnesses are not modeled at all, hence, they are given high uncertainties. The uncertainties in the measurement noise covariance matrix are taken from the measurement signal. The uncertainty for yaw rate signal is taken from the available gyroscope sensor. While the uncertainties in the forces are taken from the measurement data obtained from a test vehicle mounted with SKF Load Sensing Bearings. The process and measurement noises are assumed to be constant and uncorrelated, therefore, the off-diagonal elements are assigned to 0. The measurement noise matrix R in terms of standard deviation of the measured signal is given as: σ ˙2 0 0 ψ R = 0 σFyf 2 0 . 0 0 σFyr 2 After a number of simulations and careful tuning, the following values of Q and R gave desirable results: 0.00001 0 0 0 0 15.3 0 0 Q = 10e − 08 , 0 0 15e11 0 0 0 0 11e11 Master of Science Thesis Shelav Jain 44 Estimation using Bicycle Model 0.001 0 0 6.3458e3 0 R= 0 . 0 0 6.3458e3 In addition to the Q and R matrices, there are some other parameters for tuning in UKF. These parameters determine the scaling of Unscented Transformation (UT), their significance is given in subsection 3-5-1. The values of these parameters used for this study are given in table 4-2. S.No 1 2 3 Parameter α β κ Value 1e-03 2 0 Table 4-2: UKF tuning parameters. 4-3-2 Observer Evaluation The algorithm is verified by comparing the outputs of the estimator with the measurements from CarSim for different maneuvers. The observer error is also calculated to analyze the accuracy of the estimation results. The measurement signals, i.e. the yaw rate and lateral tire forces are polluted with white noise having standard deviation as given in the R matrix. Figure 4-1 to Figure 4-9 describe the comparison between the estimated and measured sideslip angle and cornering stiffnesses. Sine-sweep maneuver - First, the model is simulated for sine steering input with vehicle speed of 80km/h. The applied steering profile and the lateral acceleration are shown in Figure 4-1. Figure 4-2 and Figure 4-3 compare the estimated sideslip angle and tire cornering stiffnesses with the output from CarSim. The estimated sideslip angle tracks the reference well. From the Figure 4-3, it can be seen that the algorithm captures the change in cornering stiffness. But at low steering angles the error between the estimated cornering stiffness and the reference is significant. It is due to the low magnitude of the lateral force close to the crossover region. Therefore, resulting in a low signal to noise ratio and thereby results in an estimate mainly driven by sensor noise. Double lane change maneuver- Next, the vehicle is simulated for double lane change maneuver. The steering profile and lateral acceleration are shown in Figure 4-4. As can be seen from the Figure 4-5 the estimated sideslip angle tracks the reference. In Figure 4-6 comparison between the estimated and reference cornering stiffness is given. The filter follows the reference but, in case of slow dynamics, i.e. for close to zero steering angles, the estimate is driven by noise in the lateral force signal. Hence, not resulting in an accurate estimate. Shelav Jain Master of Science Thesis 4-3 Simulation Results 45 Steering Angle 5 0 f δ [deg] Steer −5 0 5 10 15 Lateral Acceleration 1 a y a [g] y 0 −1 0 5 10 15 t [s] Figure 4-1: Steering profile and lateral acceleration for sine steer at Vx of 80km/h. Sideslip Angle Reference Estimated 3 Sideslip Angle [deg] 2 1 0 −1 −2 −3 0 5 10 15 t [s] Figure 4-2: Sideslip angle for sine steer at Vx of 80km/h. Master of Science Thesis Shelav Jain 46 Estimation using Bicycle Model Front Cornering Stiffness Reference Estimated Cαf [N/deg] 2500 2000 1500 0 5 10 15 Rear Cornering Stiffness Cαr [N/deg] 2000 1500 1000 0 5 10 15 t [s] Figure 4-3: Cornering stiffness for sine steer at Vx of 80km/h. Steering Angle 5 0 f δ [deg] Steer −5 0 2 4 6 8 10 Lateral Acceleration 1 y a [g] ay 0 −1 0 2 4 6 8 10 t [s] Figure 4-4: Steering profile and lateral acceleration for double lane change at Vx of 75km/h. Shelav Jain Master of Science Thesis 4-3 Simulation Results 47 Sideslip Angle Reference Estimated 3 Sideslip Angle [deg] 2 1 0 −1 −2 −3 0 2 4 6 8 10 t [s] Figure 4-5: Sideslip angle for double lane change at Vx of 75km/h. Front Cornering Stiffness Cαf [N/deg] 4000 Reference Estimated 3000 2000 1000 0 2 4 6 8 10 8 10 Rear Cornering Stiffness Cαr [N/deg] 2500 2000 1500 1000 0 2 4 6 t [s] Figure 4-6: Cornering stiffness for double lane change at Vx of 75km/h. Master of Science Thesis Shelav Jain 48 Estimation using Bicycle Model Steering Angle 5 0 f δ [deg] Steer −5 0 1 2 3 4 5 6 7 8 Lateral Acceleration 0.5 a ay [g] y 0 −0.5 0 1 2 3 4 5 6 7 8 t [s] Figure 4-7: Steering profile and lateral acceleration for double lane change at Vx of 85km/h and µ = 0.5. Low friction surface- To check the applicability of the developed algorithm on low friction surface which results in higher sideslip angles. The vehicle is simulated for double lane change on low friction surfaces with µ = 0.5. The steering profile and lateral acceleration are shown in Figure 4-7. The sideslip angle and cornering stiffness are given in Figure 4-8 and Figure 4-9 respectively. The cornering stiffness estimate in Figure 4-9 shows inaccurate results in the region of slow dynamics. The cornering stiffness values are not converging because of lack of system information. Hence, resulting in the estimate driven by sensor noise. To measure the performance of the state estimator, Root mean squared (RMS) values of estimation error for different maneuvers are evaluated. The formulation of RMS error is given below: v u N u1 X t RM S error = (ˆ xn − xn )2 , (4-8) N n=1 where x ˆn is the estimated state, xn is the measured state and N is the total number of samples. The RMS error values of sideslip angle estimation are shown in the table below. S.No 1 2 3 Maneuver Sine Steer at Vx of 80km/h Double Lane Change at Vx of 75km/h Double Lane Change at Vx of 85km/h and µ = 0.5 RMS error 0.2918 deg 0.2726 deg 0.4107 deg Table 4-3: RMS error values for UKF with bicycle model. Shelav Jain Master of Science Thesis 4-3 Simulation Results 49 Sideslip Angle 8 Reference Estimated 6 Sideslip Angle [deg] 4 2 0 −2 −4 −6 0 1 2 3 4 5 6 7 8 t [s] Figure 4-8: Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5. Front Cornering Stiffness Cαf [N/deg] 4000 Reference Estimated 2000 0 0 2 4 6 8 Rear Cornering Stiffness Cαr [N/deg] 3000 2000 1000 0 0 2 4 6 8 t [s] Figure 4-9: Cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. Master of Science Thesis Shelav Jain 50 Estimation using Bicycle Model Lateral Acceleration 1 ay ay [g] 0.5 0 −0.5 0 2 4 6 8 10 Sideslip angle [deg] Sideslip Angle 1 Reference Estimated 0 −1 −2 0 2 4 6 8 10 t [s] Figure 4-10: Lateral acceleration and sideslip angle for steady state cornering for UKF without sideslip angle measurement. 4-3-3 Limitation The UKF presented in the section above does not give satisfactory results in the steady state cornering maneuver. The estimates diverge from the actual measurements, as shown in Figure 4-10. The reason for the divergence in steady state maneuver needs to be investigated. Therefore, to estimate the cornering stiffness of a vehicle in steady state cornering maneuver, measurement of sideslip angle is incorporated in the algorithm. The sideslip angle is obtained from another estimator designed in [63]. 4-4 Nonlinear observer with sideslip angle as measurement The measurement vector given in (4-4) is extended with sideslip angle as measurement signal. The extended vector is: yk = [β, ψ˙ k , Fyf,k , Fyr,k ]T . (4-9) Then the output equation (4-6) is extended as: h1 = βk , h2 = ψ˙ k , ψ˙ k h3 = Cαf,k δf,k − βk − lf cos δf,k , Vx ψ˙ k h4 = Cαr,k δr,k − βk + lr cos δr,k . Vx Shelav Jain (4-10) Master of Science Thesis 4-5 Analysis and Observations 51 Steering Angle 5 δ [deg] Steer f 0 −5 0 2 4 6 8 10 Lateral Acceleration 1 a y ay [g] 0.5 0 −0.5 0 2 4 6 8 10 t [s] Figure 4-11: Steering profile and lateral acceleration for steady state cornering at Vx of 90km/h and 100m radius circle. Steady state maneuver - The vehicle is tested for steady state cornering maneuver on a track of constant radius 100m with vehicle speed of 90km/h. The steering input and lateral acceleration are given in Figure 4-11. With the sideslip angle information, filter tracks the cornering stiffness accurately, as can be seen in the Figure 4-12. For other maneuvers the UKF with sideslip angle as measurement shows the same performance as the one without the sideslip angle as measurement signal. 4-5 Analysis and Observations The developed algorithm tracks the reference well in transient maneuver. It is able to estimate vehicle sideslip angle in the lateral acceleration range {−0.8, 0.8} g. The model also gives accurate estimate of sideslip angle on low friction surface, as shown in the results for µ = 0.5. But, for steady state maneuver it is not able to estimate the sideslip angle accurately. The filter is more accurate for lower lateral acceleration values. This can be understood from the fact that the linear tire model is used in this study. From the cornering stiffness curves it can be seen that the changes in the cornering stiffness are well captured. 4-6 Summary This chapter presented the results of the developed sideslip and cornering stiffness estimation algorithm. First, the estimator algorithm is given for the bicycle model. The observability analysis results show that the system is locally observable. Next, the simulation results are given which constitutes the values of tuning parameters, observer evaluation on different maneuvers and its RMS error. This part also gives the Master of Science Thesis Shelav Jain 52 Estimation using Bicycle Model Front Cornering Stiffness Cαf [N/deg] 2100 Reference Estimated 2000 1900 1800 0 2 4 6 8 10 8 10 Cαr [N/deg] Rear Cornering Stiffness 1700 1600 1500 1400 0 2 4 6 t [s] Figure 4-12: Cornering stiffness for steady state cornering at Vx of 90km/h and 100m radius circle. limitation of the algorithm and presents a method to improve the developed algorithm. The performance of the filter in steady state maneuvers can be improved by incorporating sideslip angle as measurement signal in the filter. The results in this chapter are based on the bicycle model. In the next chapter the UKF is designed on the basis of Four-Wheel vehicle model (FWVM). Shelav Jain Master of Science Thesis Chapter 5 Estimation using Four Wheel Vehicle Model The algorithm designed in the last chapter is extended to accommodate the Four-Wheel vehicle model (FWVM). The FWVM is more accurate than the bicycle model, as it also captures the load transfer from inside to outside wheels while cornering. 5-1 Nonlinear Observer Formulation As can be seen from the equations (2-1) - (2-5), determination of lateral velocity requires information of the longitudinal forces. These forces should be taken into account to accurately estimate vehicle sideslip angle (or lateral velocity) during braking or acceleration in a turn. The state vector xk , at each time instant k, comprises of sideslip angle, yaw rate, front and rear tires lateral forces, front and rear tires cornering stiffness: xk = [Vx,k , Vy,k , ψ˙ k , FyF L,k , FyF R,k , FyRL,k , FyRR,k , CαF L,k , CαF R,k , CαRL,k , CαRR,k ]T . (5-1) Therefore, the vehicle sideslip angle can be calculated using: Vy,k , Vx,k x2,k . = arctan x1,k βk = arctan (5-2) The input vector uk comprises the front and rear wheels steering angles and front and rear wheels longitudinal forces: uk = [δF L,k , δF R,k , δRL,k , δRR,k , FxF L,k , FxF R,k , FxRL,k , FxRR,k ]T . Master of Science Thesis (5-3) Shelav Jain 54 Estimation using Four Wheel Vehicle Model The measurement vector comprises of vehicle’s longitudinal velocity, yaw rate, front and rear wheels lateral forces: yk = [Vx,k , ψ˙ k , FyF L,k , FyF R,k , FyRL,k , FyRR,k ]T . (5-4) The nonlinear function f (·) that relates the states at time k to the states at time k − 1 and to the inputs uk is given as: 1 FxF L,k cos δF L,k − FyF L,k−1 sin δF L,k + FxF R,k cos δF R,k mv − FyF R,k−1 sin δF R,k + FxRL,k cos δRL,k − FyRL,k−1 sin δRL,k f1 = Vx,k−1 + T Vy,k−1 ψ˙ k−1 + + FxRR,k cos δRR,k − FyRR,k−1 sin δRR,k , 1 FyF L,k−1 cos δF L,k + FxF L,k sin δF L,k + FyF R,k−1 cos δF R,k f2 = Vy,k−1 + T − Vx,k−1 ψ˙ k−1 + mv + FxF R,k sin δF R,k + FyRL,k−1 cos δRL,k + FxRL,k sin δRL,k + FyRR,k−1 cos δRR,k + FxRR,k sin δRR,k f3 = ψ˙ k−1 + , T lf FyF L,k−1 cos δF L,k + FxF L,k sin δF L,k + FyF R,k−1 cos δF R,k + FxF R,k sin δF R,k Izz − lr FyRL,k−1 cos δRL,k + FxRL,k sin δRL,k + FyRR,k−1 cos δRR,k + FxRR,k sin δRR,k E + FyF L,k−1 sin δF L,k − FxF L,k cos δF L,k + FxF R,k cos δF R,k − FyF R,k−1 sin δF R,k 2 − FxRL,k cos δRL,k + FyRL,k−1 sin δRL,k + FxRR,k cos δRR,k − FyRR,k−1 sin δRR,k f4 = CαF L,k−1 δF L,k − arctan Vy,k−1 + lf ψ˙ k−1 Vx,k−1 − E ψ˙ k−1 2 , , Vy,k−1 + lf ψ˙ k−1 f5 = CαF R,k−1 δF R,k − arctan , E ψ˙ Vx,k−1 + 2k−1 Vy,k−1 − lr ψ˙ k−1 f6 = CαRL,k−1 δRL,k − arctan , E ψ˙ Vx,k−1 − 2k−1 Vy,k−1 − lr ψ˙ k−1 , f7 = CαRR,k−1 δRR,k − arctan E ψ˙ Vx,k−1 + 2k−1 f8 = CαF L,k−1 , f9 = CαF R,k−1 , f10 = CαRL,k−1 , f11 = CαRR,k−1 . Shelav Jain (5-5) Master of Science Thesis 5-2 Observability Analysis 55 The observation system h(·) which gives the relation between the measurements at time k with the states and inputs is: h1 = Vx,k , h2 = ψ˙ k , h3 = FyF L,k , h4 = FyF R,k , h5 = FyRL,k , h6 = FyRR,k . (5-6) The vehicle parameters are taken from the CarSim simulation package. A D-Class sedan vehicle is chosen for the simulation. The parameters values are given in table 4-1 on page 42 and E is the track width which is equal to 1.550m. 5-2 Observability Analysis The observability study for the nonlinear system is local and uses the Lie derivatives as given in equation (3-4). For the discrete time state space system described above, observability matrix is evaluated using MATLAB symbolic environment and it is given as: O = [dhi (x) dL1f hi (x) dL2f hi (x) dL3f hi (x) dL4f hi (x) dL5f hi (x) dL6f hi (x) dL7f hi (x)]T . (5-7) For initial condition of non zero cornering stiffness the system is locally observable. The dimension of the observability matrix is 66 × 8 and the rank of the matrix O is 8. Therefore, the nonlinear system described by equations 5-5 and 5-6 is locally observable. 5-3 Simulation Results The efficiency of the estimation process to reconstruct the sideslip angle and cornering stiffness has been tested using CarSim simulation environment. The details of the CarSim model are given in section 4-3. This section gives the Unscented Kalman Filter (UKF) tuning parameters and simulation results for different maneuvers. 5-3-1 Tuning Parameters After a number of simulations and careful tuning, the following values of Q and R gave desirable results: Master of Science Thesis Shelav Jain 56 Estimation using Four Wheel Vehicle Model 55.5 0 0 0 0 0 0 0 0 0 0 1e − 7 0 0 0 0 0 0 0 0 0 0 0 0 15.3 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 , Q = 10e−07 0 0 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 0 0 85.3e09 0 0 0 0 0 0 0 0 0 0 0 85.3e09 0 0 0 0 0 0 0 0 0 0 85.3e09 0 0 0 0 0 0 0 0 0 0 0 0 85.3e09 0.001 0 0 0 0 0 0 0.001 0 0 0 0 0 0 6.3458e03 0 0 0 R= . 0 0 0 6.3458e03 0 0 0 0 0 0 6.3458e03 0 0 0 0 0 0 6.3458e03 In addition to Q and R matrices, there are some other parameters for tuning in UKF. These parameters are given in table 4-2. 5-3-2 Observer Evaluation The observer is verified by comparing the outputs of the estimator with the measurements from CarSim. The observer error is also calculated to analyze the accuracy of the estimation results. The measurement signals i.e. the longitudinal velocity, yaw rate and tire forces are polluted with white noise having standard deviation as given in the R matrix. The Figure 5-1 to Figure 5-12 describe the comparison between the estimated and measured sideslip angle and tires cornering stiffnesses. Sine sweep maneuver- First, the vehicle is tested on sine steering input with longitudinal velocity of 80km/h. The steering profile and lateral acceleration are as shown in Figure 5-1. The estimated sideslip angle and tires cornering stiffness are compared with the measurements from CarSim as shown in Figure 5-2 - Figure 5-4. The estimates track the reference well. But close to the crossover region, i.e. close to zero steering angles, cornering stiffness estimates are driven by the sensor noise. Therefore, shows large deviation from the reference. Double lane change maneuver - To simulate the vehicle behavior in transient conditions, it is tested for Double Lane Change maneuver with vehicle velocity of 80km/h. The lateral acceleration and steering input is as shown in Figure 5-5. The sideslip angle and tires cornering stiffness are given in Figure 5-6 - Figure 5-8. The algorithm gives accurate results in region where sufficient system information is available. But in the region where steering angle is almost zero, the filter does not converge and estimates are driven by sensor noise. Shelav Jain Master of Science Thesis 5-3 Simulation Results 57 Steering Angle 5 0 f δ [deg] Steer −5 0 5 10 15 Lateral Acceleration 1 a y a [g] y 0 −1 0 5 10 15 t [s] Figure 5-1: Steering profile and lateral acceleration for sine steer at Vx of 80km/h. Sideslip Angle Reference Estimated 3 Sideslip Angle [deg] 2 1 0 −1 −2 −3 0 5 10 15 t [s] Figure 5-2: Sideslip angle for sine steer at Vx of 80km/h. Master of Science Thesis Shelav Jain 58 Estimation using Four Wheel Vehicle Model Front Left Tire Cornering Stiffness CαFL [N/deg] 1500 Reference Estimated 1000 500 0 0 5 10 15 Front Right Tire Cornering Stiffness CαFR [N/deg] 1500 1000 500 0 0 5 10 15 t [s] Figure 5-3: Front tires cornering stiffness for sine steer at Vx of 80km/h. Rear Left Tire Cornering Stiffness CαRL [N/deg] 1500 Reference Estimated 1000 500 0 0 5 10 15 Rear Right Tire Cornering Stiffness CαRR [N/deg] 1500 1000 500 0 0 5 10 15 t [s] Figure 5-4: Rear tires cornering stiffness for sine steer at Vx of 80km/h. Shelav Jain Master of Science Thesis 5-3 Simulation Results 59 Steering Angle Steer 0 f δ [deg] 5 −5 0 2 4 6 8 Lateral Acceleration 1 a y a [g] y 0 −1 0 2 4 6 8 t [s] Figure 5-5: Steering profile and lateral acceleration for double lane change at Vx of 80km/h. Sideslip Angle 4 Reference Estimated 3 Sideslip Angle [deg] 2 1 0 −1 −2 −3 −4 −5 0 2 4 6 8 t [s] Figure 5-6: Sideslip angle for double lane change at Vx of 80km/h. Master of Science Thesis Shelav Jain 60 Estimation using Four Wheel Vehicle Model Front Left Tire Cornering Stiffness CαFL [N/deg] 2000 Reference Estimated 1000 0 0 2 4 6 8 Front Right Tire Cornering Stiffness CαFR [N/deg] 2000 1000 0 0 2 4 6 8 t [s] Figure 5-7: Front tires cornering stiffness for double lane change at Vx of 80km/h. Rear Left Tire Cornering Stiffness CαRL [N/deg] 1500 Reference Estimated 1000 500 0 0 2 4 6 8 Rear Right Tire Cornering Stiffness CαRR [N/deg] 1500 1000 500 0 0 2 4 6 8 t [s] Figure 5-8: Rear tires cornering stiffness for double lane change at Vx of 80km/h. Shelav Jain Master of Science Thesis 5-3 Simulation Results 61 Steering Angle 5 0 f δ [deg] Steer −5 0 1 2 3 4 5 6 7 8 Lateral Acceleration 0.5 a ay [g] y 0 −0.5 0 1 2 3 4 5 6 7 8 t [s] Figure 5-9: Steering profile and lateral acceleration for double lane change at Vx of 85km/h and µ = 0.5. Low friction surface - Next, the vehicle is simulated on low friction surface. It is important for an estimator to accurately estimate the sideslip angle on low friction surface because skidding on low friction surfaces can cause accidents. To generate larger sideslip angle, the vehicle is tested for a longitudinal velocity of 85km/h at µ = 0.5. The developed lateral acceleration and steering input are as shown in Figure 5-9. The vehicle sideslip angle angle cornering stiffness are given in Figure 5-10 - Figure 5-12. The filter tracks the sideslip angle even at larger values. From the cornering stiffness curves vehicle’s handling limit can be determined. As the value of cornering stiffness becomes very low for high sideslip angle condition on low friction surfaces. To measure the performance of the state estimator, Root mean squared (RMS) values of estimation error for different maneuvers are also evaluated. The RMS error values of sideslip angle estimation are shown in the table below. S.No 1 2 3 Maneuver Sine Steer at Vx of 80km/h Double Lane Change at Vx of 80km/h Double Lane Change at Vx of 85km/h and µ = 0.5 RMS error 0.3635 deg 0.3043 deg 0.3454 deg Table 5-1: RMS error values for UKF with FWVM. 5-3-3 Limitation The estimation accuracy of the algorithm is only limited to fast and transient maneuvers. It does not give satisfactory results in the steady state cornering maneuver. The estimates diverge from the actual measurements, as shown in Figure 5-13. Therefore, to estimate the cornering stiffness of a vehicle in steady state cornering maneuver, measurement of lateral Master of Science Thesis Shelav Jain 62 Estimation using Four Wheel Vehicle Model Sideslip Angle 8 Reference Estimated Sideslip Angle [deg] 6 4 2 0 −2 −4 −6 0 1 2 3 4 5 6 7 8 t [s] Figure 5-10: Sideslip angle for double lane change at Vx of 85km/h and µ = 0.5. Front Left Tire Cornering Stiffness CαFL [N/deg] 2000 Reference Estimated 1000 0 0 2 4 6 8 Front Right Tire Cornering Stiffness CαFR [N/deg] 1500 1000 500 0 0 2 4 6 8 t [s] Figure 5-11: Front tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. Shelav Jain Master of Science Thesis 5-3 Simulation Results 63 Rear Left Tire Cornering Stiffness CαRL [N/deg] 1500 Reference Estimated 1000 500 0 0 2 4 6 8 Rear Right Tire Cornering Stiffness CαRR [N/deg] 1500 1000 500 0 0 2 4 6 8 t [s] Figure 5-12: Rear tires cornering stiffness for double lane change at Vx of 85km/h and µ = 0.5. velocity (related to sideslip angle by equation (5-2)) is incorporated in the algorithm. The lateral velocity is obtained from another estimator designed in [63]. Lateral Acceleration 1 a y ay [g] 0.5 0 −0.5 0 2 4 6 8 10 Sideslip angle [deg] Sideslip Angle 0 Reference Estimated −1 −2 −3 0 2 4 6 8 10 t [s] Figure 5-13: Lateral acceleration and sideslip angle for steady state maneuver for UKF without lateral velocity as measurement. Master of Science Thesis Shelav Jain 64 Estimation using Four Wheel Vehicle Model Steering Angle Steer δf [deg] 4 2 0 0 2 4 6 8 10 Lateral Acceleration 1 a ay [g] y 0.5 0 −0.5 0 2 4 6 8 10 t [s] Figure 5-14: Steering profile and lateral acceleration for steady state cornering at Vx of 95km/h and 100m radius circle. 5-4 Nonlinear observer with lateral velocity as measurement The vehicle lateral velocity is incorporated into the system given in equation (5-4). The measurement vector for the updated observer is: yk = [Vx,k , Vy,k , ψ˙ k , FyF L,k , FyF R,k , FyRL,k , FyRR,k ]T . (5-8) The output equation (5-6) is extended with lateral velocity as measurement. h1 = Vx,k , h2 = Vy,k , h3 = ψ˙ k , h4 = FyF L,k , h5 = FyF R,k , h6 = FyRL,k , h7 = FyRR,k . (5-9) Steady state maneuver- To understand the behavior of the vehicle during steady state cornering, it is tested on constant radius turn of 100m with vehicle velocity of 95km/h. The steering profile and lateral acceleration are given in Figure 5-14. The estimated cornering stiffness for front and rear tires are given in Figure 5-15 and Figure 5-16. It is evident for the results that the lateral velocity information improves the performance of the filter especially in steady state conditions. In all other maneuvers, filter with lateral velocity as measurement has similar performance as compared to the one without the lateral velocity as measurement. Shelav Jain Master of Science Thesis 5-4 Nonlinear observer with lateral velocity as measurement 65 [N/deg] Front Left Tire Cornering Stiffness Reference Estimated 600 C αFL 500 400 0 2 4 6 8 10 8 10 CαFR [N/deg] Front Right Tire Cornering Stiffness 1400 1200 1000 0 2 4 6 t [s] Figure 5-15: Front tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m radius circle. [N/deg] Rear Left Tire Cornering Stiffness Reference Estimated 400 C αRL 300 200 0 2 4 6 8 10 8 10 Rear Right Tire Cornering Stiffness CαRR [N/deg] 1400 1200 1000 800 0 2 4 6 t [s] Figure 5-16: Rear tires cornering stiffness for steady state cornering at Vx of 95km/h and 100m radius circle. Master of Science Thesis Shelav Jain 66 5-5 Estimation using Four Wheel Vehicle Model Analysis and Observations The model is able to follow the reference sideslip angle in the lateral acceleration range of {−0.8, 0.8}g. It is also able to track the reference in the case of low friction surface for example ice, snow, water etc. The developed algorithm also accurately tracks the cornering stiffness of all four tires. From the figures, it is evident that the cornering stiffness changes significantly during a turn especially due to the lateral load transfer. But the algorithm is not able to capture the sideslip angle correctly in the steady state maneuvers. Therefore, it is incorporated with the measurement of lateral velocity to give satisfactory results. 5-6 Summary This chapter discussed the results of the FWVM based UKF algorithm. First the algorithm was formulated and then the observability analysis has been performed. Next, the observer is evaluated on the basis of simulation results from different maneuvers along with the RMS error. The analysis of the results is given along with its limitation. The filter without lateral velocity as measurement signal does not give satisfactory results in steady state situations. Therefore, a different observer with lateral velocity as measurement is proposed to overcome this limitation. Shelav Jain Master of Science Thesis Chapter 6 Conclusions and Recommendations 6-1 Conclusions The thesis report gives an overview of the Vehicle Dynamics Control (VDC) systems. The objective of these systems can be either control of yaw rate or sideslip angle or both. The importance of controlling the sideslip angle for satisfactory handling characteristics has been discussed in detail. But effective measurement of sideslip angle is expensive. This thesis work developed a model-based algorithm to estimate the vehicle sideslip angle and tire cornering stiffness. The nonlinear model based Unscented Kalman Filter (UKF) accurately estimates the sideslip angle and cornering stiffness for different maneuvers. The UKF algorithm is designed for two types of vehicle models, the bicycle model and two-track model. The results of the two-track model are more accurate than the bicycle model, as it also captures the lateral load transfer during cornering. The developed model is tested using multibody simulation package CarSim. From the results of different maneuvers it has been shown that the algorithm is able to capture the dynamics of sideslip angle in the lateral acceleration range {−0.8, 0.8}g and on different friction surfaces. The model also accurately captures the changes in tire cornering stiffness during different maneuvers. In the filter design, cornering stiffness is not modeled with any dynamics and high uncertainty is assigned to it, so that the filter tracks the changes. But due to the lack of system information close to zero steering angles, cornering stiffness estimate is not accurate. In case of low lateral forces, the signal to noise ratio is low and the estimates are guided by noise rather than the actual signal. The filter also does not show adequate performance in steady state maneuvers. But the availability of lateral velocity or sideslip angle signal as measurement improves the performance of the filter. The accuracy of model-based estimation approach is limited by the knowledge of vehicle and tire parameters. The cornering stiffness obtained from this study can be used to update the vehicle and tire model. Hence, improving the accuracy of the estimation algorithm. VDC systems depend on vehicle parameters to obtain the control law. The results from this study can be used for real-time model update to improve the controller efficiency. Master of Science Thesis Shelav Jain 68 6-2 Conclusions and Recommendations Recommendations The following section presents recommendations that can be performed to extended the results obtained in this study. Improvement of cornering stiffness estimation As discussed, the developed algorithm does not give satisfactory results when lateral forces are low or when system information is not adequate. The accuracy of the results can be improved in these conditions. Also, the reason for divergence of the estimates in steady state maneuvers in UKF without sideslip angle as measurement needs to be investigated. Complexity of the vehicle and tire model In any model based estimation process accuracy of the algorithm depends on the accuracy of the system model. In this case, vehicle model can be extended to include the roll and pitch dynamics. To capture the dynamic behavior of the tire, the transient tire model as described in the chapter 2 can be implemented in the algorithm. As the tire forces are considered as states in the UKF designed with Four-Wheel vehicle model (FWVM), the transient tire model can be incorporated provided the relaxation time constant is known. Estimation of vehicle parameters Vehicle’s inertial parameters like vehicle mass and moment of inertia change often in the actual operating conditions. In controller design, accurate information of these vehicle parameters is important. The algorithm developed in this study can form the basis of estimation of these parameters. Determination of lateral force saturation For VDC systems, information of nonlinear operating region of tires is important. This information can be obtained using Instantaneous Cornering Stiffness (ICS), as explained in section 1-1. 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Master of Science Thesis Shelav Jain 74 Shelav Jain Bibliography Master of Science Thesis Glossary List of Acronyms ACC Adaptive Cruise Control ABS Anti-Lock Braking System TCS Traction Control System ESC Electronic Stability Control EKF Extended Kalman Filter COG Centre of Gravity DOF Degree of Freedom ADAS Advanced Driver Assistance Systems GRV Gaussian Random Variable UT Unscented Transformation UKF Unscented Kalman Filter FWVM Four-Wheel vehicle model EV Electric Vehicle VDC Vehicle Dynamics Control ESP Electronic Stability Program AFS Active Front Steering ARS Active Rear Steering 4WAS Four-Wheel Active Steering DYC Direct Yaw Moment Control Master of Science Thesis Shelav Jain 76 Glossary ICE Internal Combustion Engine ICS Instantaneous Cornering Stiffness GPS Global Positioning System RMS Root mean squared Shelav Jain Master of Science Thesis

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