May 23, 2023
Tire dynamics model identification
Formula Student is a student engineering competition held annually all over the world. Student teams from around the world design, build, test, and race a small-scale formula-style racing cars, including the electric ones. The competition is run by the Institution of Mechanical Engineers and uses the same rules as the original Formula SAE with supplementary regulations. In this use case, pSeven was used to calibrate car’s tire dynamics model by one of the racing teams.
You can also explore another project for Formula Student when pSeven used for Bolide Suspension Optimization.
The tire dynamics of a racing car according to MF 5.2 model (so called Pacejka’s Magic Formula) is based on 16 longitudinal parameters describing the behavior of the tires during the longitudinal movement of the car and 6 correction factors (scale factors). The aim of this study was to identify the scale factors for calibration of the bolide tires behavior model. To obtain a reference design curve describing the behavior of the tires in different modes, full-scale experiments were carried out on the racing track, during which the longitudinal acceleration of the car and the speed of the car were determined (Fig. 1). This data was used as an input for Amesim simulation model of tire dynamics.
Figure 1. Input data for Amesim simulation model: longitudinal acceleration of the car (top), speed of the car (bottom)
After processing the experimental data and importing it into the simulation model, virtual tests are carried out in Amesim simulator, which makes it possible to obtain a design curve that describes the behavior of the tires in different modes (Fig. 2).
Figure 2. Importing data into the Amesim model
The calibration stage involves minimizing the difference between the longitudinal design force and the longitudinal analytical force in the contact patch by searching for appropriate parameters – scale factors.
The main challenge was that the bolide operation data was available only for low operating modes: the forces that arise in the contact patch of the tire with the track at the presented time intervals are in fairly low ranges: +/- 500 N (Fig. 3, a). On the other hand, the mathematical model describing the behavior of the tires at these small forces in the contact area is in a narrow range (Fig. 3, b - red square). In this area, it is very difficult to catch the change in the force and bind this change to a specific scale factor.
Figure 3. a) Forces in the tire contact patch analytical / model; b) Diagram showing tire test results (from the manufacturer)
To automate the calibration process of the tire dynamics model the simulation model performed in Amesim was integrated into pSeven workflow.
Figure 4. Integration with Amesim in pSeven
Using the Gradient-Based Optimization (GBO) algorithm implemented in pSeven the problem of single-criteria optimization for 6 variables was solved (Fig. 5).
Figure 5. Optimization loop in pSeven
The optimization criteria is the function of the area between the analytical and design curves calculated by formulas from the Table 1.
Table 1. Functions of longitudinal forces: Fx_analytical, Fx_model and area = aim_delta_Fx_2
As the result, the accuracy of calibration increased by 1.36%. The total identification time for the scale factors was exactly 300 seconds (5 min). When using the built-in optimization algorithms in Amesim (Genetic algorithm, NLPQL – gradient descent method), an adequate solution to the problem was not obtained.
The results allowed increasing the accuracy of the mathematical model of the longitudinal movement of the racing car, which even closer brought it to the real model of the car's behavior on the track.
The performed exploration also made it possible to analyze the influence of each scale factor on the ability of the tire to work properly, and to identify the requirements for the study of more severe operating modes of the racing bolide.