Estimation of vehicle tyre model parameters from transient experimental data

Tires are one of the most important components of the vehicles due to their various functions such as supporting the vehicle load, transmitting the forces which drive, brake and guide the vehicle, acting like a secondary suspension for damping the effects of road irregularities without transmitting these to the vehicle suspension. Tires have a very complex structure including synthetic polymer layers, high-modulus cords, glass fiber and etc. materials. Due to being the only component to contact with road, almost all the forces and torques are transferred through the tires and to understand the vehicle dynamics it is crucial to know the forces and torques at the contact area. The forces which occur at the tires are the main factor which determine the behavior of tires. The difference between the direction that the body of the vehicle points and wheel's direction of movement is called 'slip angle' and the existing of this angle results in generating a 'lateral force'. Due to the tire deformations, reaction forces affect with a deflection which generates 'self aligning torque' that tends to steer the tire toward the direction in which it is travelling. At the fast maneuvers of tire, it is not assumed that the behavior of the tires are in steady state. If an instant change happens at slip angle, as carcass of tire moves away from the tire plane, the efficient slip angle between the tire and road changes from zero to the input value in a t time that different than zero. This efficient slip angle corresponds to a slip tension, which means the slip force of tire needs time to build up. This effect of the lag is called 'transient behavior'. To describe the effects of the tire on the vehicle, some fundamental elements must be considered. From the point of vehicle power, as it is called as traction in the terminology, to benefit the motor power in maximum, tires with strong holding ability must be chosen, otherwise the power will be lost by sliding and tires will be eroded quickly. Tires are produced according to the needs of the manufacturers to carry a certain weight, to be suitable for a significant maximum speed and to give a minimum brake distance. Comfort is more related to the rigidity of the tires and the height of its sidewall and for those reasons manufacturers usually demands ideal dimensions. For example, as the overall diameter stays the same, slick tires could be chosen; however, while this choice gives the advantage of from the point of actuation and maneuvering, it can create disadvantages related to comfort. Due to the desire in industry to manufacture the products on time with minimum cost, real world modeling and simulations are gone popular in recent years. In the automotive industry detailed multiple body simulations are widely used to understand the vehicle behaviors. Electronical technologies in Automotive area progress to upgrade comfort and safety factors. Control systems such as ABS, ESC exist for helping the driver and the amount of them increases day by day. Vehicle control algorithms provided huge steps in the area of developing the handling and driving behavior of vehicles. Today these algorithms are considered as a life saver technology and they are limited by the current knowledge of vehicle states. With the inexpensive and easy to apply measurements such as longitudinal velocity, accelerations, yaw rate; these systems continue to work but the main point which is the measurement of tire forces is more difficult due to technical and economical reasons. It is important to model the relation between tires and the road accurately as tires are efficient on properties of handling, braking and acceleration of vehicle and they are the only components have interaction with the road. The research of dynamic properties of cehicles has started at the 1920's. First researches about the effects of tire forces and torques were on the subject of finding solutions for yaw problems. Studies continued with analyzing diagonal movement and the effects of slip angle on lateral force and self aligning torque. To describe the tire behavior and tire structure, tire models such as brush, stretched-string or the combined 'Fiala' model have started to be developed and they provided a base for the current advanced tire models. In the mid-1980's Hans B Pacejka created a model similar to Fiala's and made researches with Bekker and Nyborg to calculate the lateral force, self aligning torque and the brake force. To calculate these forces and torques, they created a special formulation 'Magic Formula' and the semi-empirical model of them is named 'Magic Formula Tire Model' (Pacejka 1987). After creating the Magic Formula, Pacejka continued his studies with expanding the formulation to make the formulation fit to the tire characteristic as accurate as possible. The transient behavior of tires highly depends on tire velocity, relaxation length and frequency. In 1997, Pacejka added a relaxation length parameter to the formulation to define transient behavior of tires (Pacejka, 1997). Relaxation length is one of the properties of pneumatic tires that describes the delay between a slip angle is introduced and when the cornering force reaches the steady state (Pacejka, 2006). It is also described as the distance that a tire needs to roll before the lateral force builds up almost 63% of its steady state value (Cossalter, 2006). For the calculation of the relaxation length, Pacejka made researches and experiments with pendulum tests. Today in the softwares of multi body dynamics simulations old and new versions of Pacejka tire model are widely used along with the other tire models. The question 'which model to use in which situations' have been examined and still continues to examine with experiments of in and outdoor tests, handling, braking and driving manuevers to find the most accurate modelling of the real situations. For vehicle dynamics researches and simulations, 'Magic Formula Tire Model' is one of the most preferable models and estimating the coefficients of the Magic Formula is a very popular subject in the research area. For estimation, researchers used various methods such as IMMa algorithms, Kalman Filters, least square methods. In this study, diagonal movement characteristics obtained from a rotating drum test equipment with using a LASSA GT 70 175/70 SR 13 tire are adapted to Magic Formula Tire Model. To digitalize the data from the graphics, Get Data GraphDigitizer programme is used. The Magic Formula Tire Model is chosen to work with in this study due to its accuracy. Transient behavior of vehicles under the dynamic circumstances is examined depending on frequency and velocity. The estimation of the parameters of the Magic Formula tire model is aimed for lateral force and self aligning torque characteristics with constrained nonlinear optimization on Matlab software. As the optimization method, the Matlab software's 'fmincon' command is used due to its simplicity.