## 3 Modelling
### Motorcycle Model  
The model used in this work is based on [10], where a model for BMW sport motorcycles from the 1990s was developed and validated with several experimental data, especially regarding the weave and wobble modes. The kinematic structure of the rear suspension and the dataset in the present paper are based on [12]. This reference uses a model with 13 DoF:  
* Six DoF related to the main frame rigid body motion: translation about \(x,y,z\) axes, roll (\(\rho\)), pitch (\(\phi\)) and yaw (\(\psi\)). * Two wheel spins (\(\theta_{\text{fw}},\theta_{\text{rw}}\)). * Two suspension travels (\(z_{\text{f}},z_{\text{r}}\)). * Steer rotation (\(\delta\)). * Frame torsion (\(\alpha_{\text{fr}}\)) at the steering head joint, simulated with a revolute joint with rotation axis perpendicular to the steering axis. * Rider lean (\(\alpha_{\text{ri}}\)) simulated with a revolute joint between the rider's lower and upper body with rotation axis parallel to the main frame x axis. In the present work some additional DoF are added, which are shown in Figure 1.
* Three compliances of the front fork, simulated with three revolute joints placed approximately at the half of fork length:
* Bending about the x axis (\(\alpha_{\text{ff}}\)); the rotation axis of the revolute joint is perpendicular to the steering axis. This compliance is particularly important for wobble, as explained in Section 2. It allows a lateral motion of the wheel along the y axis. * Bending about the y axis (\(\beta_{\text{ff}}\)); the rotation axis of the revolute joint is parallel to the front wheel axis. This flexibility allows a translation of the wheel along the x axis. * Torsion (\(\gamma_{\text{ff}}\)); the rotation axis of the revolute joint is parallel to the steering axis. * Lateral motion of the rider's lower body with respect to the saddle (\(y_{\text{ri}}\)), simulated with a linear joint with axis parallel to the main frame y axis. * Rider yaw rotation (\(\gamma_{\text{ri}}\)) simulated with a revolute joint between the rider's lower and upper body with rotation axis parallel to the main frame z axis. Figure 1: Motorcycle sketch with the DoF associated with the stiffnesses. The small rectangles indicate the related joints. The red points represent the bodies’ centre of mass. * Swingarm torsion (\(\alpha_{\rm sw}\)), simulated with a revolute joint at the swingarm anchor point with rotation axis parallel to the main frame x axis. The motorcycle parameters in [12] were obtained through experimental measurements or through estimates. However, some parameter variations have been used in the present work:  
* The data for the rider's upper body mass and inertia tensor were taken from [12]. In [12] the mass and inertia tensor of the rider's lower body and of the motorcycle's main frame were fused together. In the present work, with the aim of allowing lateral translation of the rider's hip (considered as lower body in the present work), the mass and inertia tensor of the main frame in [12] were split between hip and main frame, while ascertaining that the combined inertia tensor and the overall centre of gravity remain unchanged. The data for hip's mass and inertia tensor were taken from [38]. According to these data, the mass and inertia tensor of the main frame were properly modified. In particular, the hip mass was subtracted from the main frame mass in [12]. Moreover, as explained above, the inertia tensor of the main frame was adapted so that the whole inertia tensor of the lower body and main frame together was equal to the main frame inertia in [12]. The rider's legs were considered fixed to the main frame, so their mass and inertia were included in it. As reported in [12] (p. 252), the rider's total mass was 72 kg. * In the year of publication of [12] the experimental data of the front fork were not completely available. Some of them were judged by the authors of the present work as not realistic (for example, the mass of the lower fork and its lacking inertia tensor).