# Measurement and Modeling of a Cargo Bicycle Tire for Vehicle Dynamics Simulation
## 2 Materials and Methods
### Tire Model  
In order to use a vehicle dynamics simulation for the design and parameterization of an ECU controller for electric cargo trailers, real-time capability is required. This criterion also applies to the tire model used in the simulation. To integrate the tire model developed in the course of this research into a real-time simulation model, a semi-empirical Magic Formula (MF) model was created according to [16]. The model uses the previously measured force  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test No.**} & **Tire** & **Tire** & \multirow{2}{*}{\(F_{N,z}\) in N} & \multirow{2}{*}{\(\mathbf{v}_{v,x}\) in m/s} & **Range \(\mathbf{\alpha}\)** & \multirow{2}{*}{\(\mathbf{\gamma}\) in \(\circ\)} \\  & **Pressure \(\mathbf{p}\)** & **Pressure \(\mathbf{p}\)** & & & & **in \(\circ\)** \\  & **in psi** & **in bar** & & & & \\ \hline
1.1 & 43.511 & 3.0 & 625 & 5.556 & \(-2\)–\(18\) & 0 \\
1.2 & 43.511 & 3.0 & 765 & 5.556 & \(-2\)–\(18\) & 0 \\
2.1 & 50.763 & 3.5 & 625 & 5.556 & \(-2\)–\(18\) & 0 \\
2.2 & 50.763 & 3.5 & 765 & 5.55 and torque curves of a tire and can fit the model curve to the real data by adjusting a specified set of coefficients. The combination of nested arctangent and sinusoidal functions in the formula allows for both the longitudinal and lateral forces \(F_{t,x}\) to be described as a function of \(\kappa\), and \(F_{t,y}\) as a function of \(\alpha\), as well as the aligning torque \(T_{t,z}\) as a function of \(\alpha\), with one equation by adjusting the coefficients. As a result, the name of the formula is derived from the empirical development of the basic equation, which nevertheless allows for the model to be fit to a wide range of tires. Since the MF is in constant development, there are different models and stages of development. While [19] shows the early stages of the model, newer versions such as [20] also take tire pressure into account. Ref. [21] provides a good overview of the different iterations and changes that were made with respect to the generic MF model. The model used in the context of this research is given in Equation (4) and refers to the general model of the MF according to [16]. This form is chosen to investigate whether a realistic model of the measurement results of bicycle tires for load transportation can be represented at all with the help of the general MF representation. In further steps, modeling with one of the other MF forms can be investigated.  
\[y=D\cdot sin(C\cdot\arctan(B\cdot x-E\cdot(B\cdot x-\arctan(B\cdot x)))) \tag{4}\]  
with the vertical shift \(S_{V}\) and horizontal shift \(S_{H}\) going into the equation as:  
\[Y(X)=y(x)+S_{V} \tag{5}\]  
\[x=X+S_{H} \tag{6}\]  
The variables of Equation (4) are defined as follows:  
* Y as the output, which can be \(F_{t,x},F_{t,y}\) or \(T_{t,z}\);
* x as the input, which can be \(\kappa\) or \(\alpha\) depending on the input;
* B as the stiffness factor, which affects the slope;
* C as the shape factor, which affects the peak and horizontal asymptote;
* D as the peak value respective to the central x-axis, as well as for \(C\geq 1\);
* E as the curvature factor, which affects the curvature at the peak point as well as the horizontal position of the peak.  
Fitting the model to the measured data by optimizing the so-called Pacejka or MF coefficient's B, C, D and E is carried out in MATLAB with the help of the nonlinear least squares method and the trust region algorithm.