## 3 Linear Motorcycle Model  
A linear mathematical model for the motorcycle is described in this section. An equation of motion for the stability of the motorcycle is determined in the roll direction. Open-loop and closed-loop systems for the motorcycle are defined using the same equation. In this model, it is assumed that the tire thickness is zero, and all the angles are small. The layout of a typical motorcycle is shown in Figure 2. The descriptions of the symbols used in the figure are given in Table 1 from Section 2. The points A and O are contact points of the front and rear wheels with ground respectively, and B is the point where the steering axis intersects the ground. Figure 3 depicts the schematic of the motorcycle, which shows its different states while balancing. The symbols used in the figure are listed in Table 2. The figure shows that the motorcycle-rider system (mass \(M\)) first steered by an angle (\(\delta\)) as shown by legend 2, and then rotated about the \(x\)-axis by a roll angle (\(\phi\)) as shown by legend 3. These steering angle and roll angle are generated due to disturbances to the motorcycle. Figure 3a,b show rear-view and top-view of the motorcycle respectively. Various forces acting on the motorcycle are due to weight (\(Mg\)), normal reaction (\(N\)) and lateral force (\(F_{y}\)) as shown in the Figure 3a. The Figure 3b shows the effect of steering angle (\(\delta\)) and roll angle (\(\phi\)) on the motorcycle state. Marginal changes in \(\delta\) and \(\phi\), which result in yaw angle (\(\psi\)), make the motorcycle to follow a circle of radius R. The significance of the steering mechanism on the stability of the motorcycle is described in this section. Figure 1: Block diagram for the low-speed stability of a motorcycle and rider system. The equation of motion for the stability of the motorcycle in roll direction about the \(x\)-axis (i.e., the axis intersecting vehicle plane and ground plane at point O) is defined as follows:  
\[I_{o}\ddot{\phi}+M\dot{y}_{o}h=Mgh\phi+M_{gyro}+M_{stearing}+M_{front\;normal\;reaction}, \tag{1}\]  
where: \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Symbols** \\ \hline Roll angle & \(\phi\) \\ Steering angle & \(\delta\) \\ Kinematic steering angle & \(\delta_{R}\) \\ Yaw angle & \(\psi\) \\ Lateral displacement at O & \(y_{o}\) \\ Speed & \(v\) \\ Instantaneous turning circle radius & \(R\) \\ \hline \end{tabular}
\end{table} 
Table 2: Input and output parameters used in linear model. Figure 3: Schematic of linear model of motorcycle showing its different states. Figure 2: A typical layout of a motorcycle. The inertia of the motorcycle with respect to point O is as follows:  
\[I_{o}=I_{g}+Mh^{2}. \tag{2}\]  
The lateral acceleration of the motorcycle \(\ddot{y}\) about point O is a function of the centrifugal force and lateral velocity. These parameters are derived from motorcycle speed \(v\) and kinematic steering angle \(\delta_{R}\) as shown below:  
\[\ddot{y}_{o}=\frac{v^{2}}{l}\delta_{R}+v\frac{l}{l}\delta_{R}. \tag{3}\]  
Gyroscopic moments by both front and rear wheels on the motorcycle, and by the front wheel on the front steering system can be described as:  
\[M_{gyro}=-\frac{v^{2}}{l}\left(\frac{I_{fw}}{r_{f}}+\frac{I_{rw}}{r_{r}} \right)\delta_{R}-\frac{v}{r_{f}}I_{fw}\dot{\delta}_{R}. \tag{4}\]  
Moment due to the front steering system vibration is as follows:  
\[M_{steering}=-M_{f}dh_{f}\delta. \tag{5}\]  
Moment due to the front wheel normal reaction force is as follows:  
\[M_{front\;normal\;reaction}=Mg\frac{l_{r}}{l}d_{1}\delta. \tag{6}\]  
The following equation is achieved by substituting the values from Equations (2)-(6) in Equation (1). \[ \begin{array}{l}(I_{g}+Mh^{2})\ddot{\phi}+M\left(\frac{v^{2}}{l}\delta_{R} +v\frac{l_{r}}{l}\delta_{R}\right)h=\\ Mgh\phi-\frac{v^{2}}{l}\left(\frac{I_{fw}}{r_{f}}+\frac{I_{rw}}{r_{r}}\right) \delta_{R}-\frac{v}{r_{f}}I_{fw}\delta_{R}-M_{f}dh_{f}\ddot{\delta}+Mg\frac{l _{r}}{l}d_{1}\delta.\end{array}