How to Build a Magic Formula Tire Modelling Using MATLAB

Aidan Mercer

United States

Matlab

Aidan Mercer has over 7 years of experience in vehicle dynamics and simulation using MATLAB. He completed his Master’s degree from Clarkson University, USA.

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FSAE student teams can access detailed tire test data through the Tire Test Consortium, offering a solid foundation for modeling. This data allows you to create highly realistic, empirical models that can be simulated in MATLAB/Simulink or exported to other tools like ADAMS or Siemens NX.

Among the various empirical approaches available, the Magic Formula model stands out for its balance between complexity and computational efficiency. This model enables both fast real-time simulations and accurate behavior prediction across various operating conditions.

Introduction to the Magic Formula Tire Model

At its core, the Magic Formula expresses tire behavior as a non-linear equation that can be tuned with just a few parameters:

y = D * sin(C * atan(B * x - E * (B * x - atan(B * x))))

Here:

• y is the output variable (e.g., lateral force)
• x is the input (e.g., slip angle)
• B, C, D, and E are parameters that define the curve shape

A more sophisticated, semi-empirical form of this model contains over 100 parameters and includes insights into real tire physics. That’s why it's widely used in motorsport and automotive simulation.

The semi-empirical model offers two major advantages:

1. It allows extrapolation of data into unmeasured areas using similarity rules.
2. It is accurate yet computationally efficient, suitable for real-time control and simulation.

Creating the Tool

To simplify the modeling process, an open-source tool was created as a graphical user interface (GUI) wrapped around a command-line MATLAB library. This GUI tool lets students:

• Import and process TTC tire test data
• Fit multiple tire force models (Fx, Fy, Mz, etc.)
• Export the final model in formats compatible with MATLAB and external simulators

It builds on core MATLAB functions but adds ease of use through visual configuration and analysis.

Understanding the Methodology

The tool uses the MATLAB Optimization Toolbox for curve fitting, replacing lsqcurvefit with fmincon to allow nonlinear constraints. This lets users maintain parameter realism. For instance:

• Cx > 0
• Dx > 0
• Ex ≤ 1

These constraints ensure physically realistic tire model parameters during the optimization process.

The fitting order follows dependencies within the model equations. Pure slip forces (Fx0, Fy0) are fitted first, followed by combined slip equations like Fx, Fy, and Mz.

Implementing the Model in MATLAB

For implementation, students can refer to the tire modeling literature such as "Tire and Vehicle Dynamics" by Pacejka. Each component of the model (like Fx0, which calculates longitudinal force) is translated into MATLAB functions.

Here’s a simplified version of a function that calculates Fx0:

Parameter naming and formulas closely follow the MF-Tyre/MF-Swift manual and literature conventions.

Building and Using the GUI

The GUI was developed using MATLAB’s custom UI component framework (not AppDesigner), allowing modular development. Although AppDesigner is used for layout mockups, the final application avoids its limitations.

The GUI supports isolated testing of each component and includes:

• A Tire Data tab for importing TTC data
• A Tyre Model tab for creating and saving models
• A Tyre Analysis tab for plotting and validating the fit

Workflow Demonstration

1. Install the App
   Download the app as a MATLAB toolbox file (.mltbx). Once installed, it appears in your App catalog.
2. Import Tire Data
   Data should be in MAT format with SI units. The toolbox includes example datasets if you don’t have access to the TTC yet. Two types of data are essential:
   • Drive/Brake: For pure longitudinal slip
   • Cornering: For pure lateral slip
   Both are required to build a complete model.
3. Configure and Run the Fitter
   Create a new tire model file and configure fit-modes (e.g., Fx0, Fy0). Fit these first to ensure dependencies are resolved correctly.
   Once satisfied, append the fitted parameters to your model or manually adjust them through the GUI interface.

Analyzing and Validating the Fit

Use the Tyre Analysis tab to plot your model against measured data. You can select steady-state conditions and visualize how well the model predicts tire forces.

If needed, manually tune parameters with real-time plot updates. This helps students understand how each parameter affects model behavior.

Exporting the Final Model

After successful fitting and validation, export your model in two formats:

• TIR-file: Compatible with commercial software
• MAT-file: Saves the model as a struct for direct use in MATLAB

Here’s an example of the TIR output:

To use it in MATLAB:

[Fx, Fy] = mftyre.v62.eval(p, slipangl, longslip, inclangl, inflpres, FZW, tyreSide);

For external simulators, the TIR file integrates seamlessly with tools like Simulink, ADAMS, or Simpack.

Manual Tuning and Visualization

The GUI provides auto-refresh capabilities, enabling students to see instant changes as they tweak parameters. This interactive process is ideal for educational purposes and helps achieve a perfect data fit.

Results

Even skipping a few optional steps, the entire modeling process—from importing data to exporting the model—can be completed in under 5 minutes. The tool allows exporting either to struct (for MATLAB) or TIR (for commercial use). The evaluation function is fast enough to be used in real-time simulations.

However, some outputs like aligning moments (Mz0, Mz) are not yet supported. You can use other projects such as mfeval for full evaluation capabilities.

What’s Next

Future improvements for the tool are in the works:

• Compatibility with older MF versions
• Kamm-circle analysis
• Better UI/UX features
• Custom weighting during fitting

This roadmap ensures that the tool will evolve with student needs.

Final Thoughts

This tool simplifies the complex task of tire modeling and makes it accessible to university students working on MATLAB assignments or participating in Formula Student. It bridges the gap between theoretical knowledge and real-world application, especially when paired with FSAE tire data.

Students who explore this tool will not only be able to model tires effectively but also gain a deeper understanding of vehicle dynamics modeling in MATLAB.