# A Screw-Axis Approach to the Stability of Two-Wheeled Vehicles
## 2 Multibody Model  
The multibody model is built in Adams, which is one of the most used environments for vehicle dynamic analysis. The motorcycle model is standard, including the eleven degrees of freedom associated to the gross motion (chassis position and orientation, steering rotation, suspension travels and wheels rotation). The modelling of the structural compliance is limited to the lateral bending and torsion of the chassis (one additional degree of freedom), which is well known to be essential for a proper modelling of the vibration modes [5; 7; 8]. The tyres are modelled using the Magic Formula approach [5], which is included in Adams by default. The inertial properties of the bike were measured in the laboratory of the University of Padova, while the rider inertial properties were estimated using the Zatsiorsky-DeLeva dataset, which is one the most used [9]. The hands are assumed to be off the handlebar [16]. The vibration properties are usually extracted in two alternative ways: Using a time domain simulation or a frequency domain simulation. The former case resembles the approach employed also during experimental tests, where the system is excited using an impulse (e.g., on the handlebar) and the resulting oscillation is analysed in order to extract its modal component. The latter approach--frequency domain--is the preferred approach in this work, because it allows to compute the vibration properties more efficiently. In practice, all of the vibration modes are extracted simultaneously without applying any impulse on the system. The following procedure is carried out at all the speeds of interest. A time domain simulation is started at the speed of interest using a proportional-derivative-integral (PID) engine torque controller to keep the vehicle at constant speed (the observed variable is the speed error): Indeed, at least the aerodynamic drag and the rolling resistance need to be balanced. The simulation is stopped when a steady-state condition is reached, i.e., when the transient oscillations have disappeared (which usually happen after 3-5 s). At this stage the PID is disengaged and the plant linearised in order to obtain the standard state space formulation  
\[\hat{x} = Ax+Bu \tag{1}\] \[y = Cx+Du \tag{2}\]  
where \(A\) is the state matrix, \(B\) is the input matrix, \(C\) is the output matrix and \(D\) is the feed-through matrix; \(y\) denotes the observed variables, \(u\) are the inputs (steering torque and engine torque) and \(x\) are the model states:  
\[x = [\dot{X},X,\dot{Y},Y,\dot{Z},Z,\dot{\phi},\dot{\theta},\theta, \dot{\psi},\dot{\phi},\dot{\phi},\dot{\xi},\dot{\tau},\ddot{\xi}_{f},\ddot{ \xi}_{f},\ddot{\beta}_{r},\dot{\beta}_{r},\dot{\beta}_{f},\dot{\gamma}_{r}, \sigma_{1},\cdots,\sigma_{9}]^{T}. \tag{3}\]  
The model has 31 states: The first 24 states in (3) consists of the 12 user selected position states (which are manually defined through the PSTATE command in Adams) and their derivatives, while the latter seven states are automatically generated by Adams and are likely associated to the (transient) Magic Formula tyre model (e.g., longitudinal and lateral relaxation equations of the front and rear tyres) and other internal variables not directly controlled by the user. In more detail, in the user-selected states \(X\), \(Y\) and \(Z\) are the longitudinal, lateral and vertical displacements of the saddle position (however any other point of the chassis works equally fine), \(\phi\), \(\theta\) and \(\psi\) are the roll, pitch and yaw angles of the saddle reference marker (i.e., of the chassis, since it is assumed rigid), \(\delta\) is the steering angle, \(\xi_{r}\) and \(\xi_{f}\) are the rear and front suspension deformations, \(\beta_{r}\) and \(\beta_{f}\) are the rear and front wheel spin angles and \(\gamma\) is the bending deflection of the front frame [7; 8], while \(\sigma_{1},\cdots,\sigma_{9}\) are the states automatically generated by [PERSON].