## 3 Linear Motorcycle Model
 \tag{7}\]  
Further, the kinematic steering angle can be defined by the following equation:  
\[\delta_{R}=\tan^{-1}\left(\frac{\cos(\epsilon)\sin(\delta)}{\cos(\phi)\cos( \delta)-\sin(\phi)\sin(\epsilon)\sin(\delta)}\right). \tag{8}\]  
Equation (8) can be approximated in linear form as follows:  
\[\delta_{R}\approx\delta\cos(\epsilon). \tag{9}\]  
Substituting the value of kinematic steering angle from Equation (9) in Equation (7) gives the following expression:  
\[ \begin{array}{l}(I_{g}+Mh^{2})\ddot{\phi}-Mgh\phi=\\ -M\left(\frac{v^{2}}{l}\delta+v\frac{l_{r}}{l}\dot{\delta}\right)\cos( \epsilon)h-\frac{v^{2}}{l}\left(\frac{I_{fw}}{r_{f}}+\frac{I_{rw}}{r_{r}} \right)\cos(\epsilon)\delta-\frac{v}{r_{f}}I_{fw}\cos(\epsilon)\dot{\delta}-M _{f}dh_{f}\ddot{\delta}+Mg\frac{l_{r}}{l}d_{1}\delta.\end{array}  \tag{10}\]  
The open-loop transfer function for the low-speed stability of the motorcycle system can be described from Equation (10) by the following expression:  
Open-loop transfer function =  
\[\frac{\phi(s)}{\delta(s)}=\frac{-M\left(\frac{v^{2}}{l}+v\frac{l_{r}}{l}s \right)\cos(\epsilon)h-\frac{v^{2}}{l}\left(\frac{I_{fw}}{r_{f}}+\frac{I_{rw}} {r_{r}}\right)\cos(\epsilon)-\frac{v}{r_{f}}I_{fw}\cos(\epsilon)s-M_{f}dh_{f}s ^{2}+Mg\frac{l_{r}}{l}d_{1}}{(I_{g}+Mh^{2})s^{2}-Mgh}. \tag{11}\]  
The transfer function defines the stability of the motorcycle at all the speeds. Root-locus plot for Equation (11) is shown in Figure 4. The figure shows that one of the poles of the system is always positive at speeds 3, 5 and 10 km/h. Hence, the low-speed stability for the open-loop system of the motorcycle cannot be achieved. Further, the closed-loop feedback of the motorcycle can be defined using the following relationship between the steering angle (\(\delta\)) and roll angle (\(\phi\)):  
\[\delta(t)=a.\phi(t-\tau), \tag{12}\]  
where \(a\) is roll angle gain and \(\tau\) is a lead time for roll angle with respect to the steering angle. Equation (12) can be further simplified for \(\tau<1\) as follows:  
\[\delta=a.\phi-a\tau\phi. \tag{13}\]  
Substituting the value from Equation (13) in the Equation (10) following equation is obtained:  
\[C_{1}\ddot{\phi}+C_{2}\ddot{\phi}+C_{3}\dot{\phi}+C_{4}\phi=0, \tag{14}\]  
where,  
\[C_{1}=-M_{f}h_{f}da\tau \tag{15}\]  
\[C_{2}=Mh^{2}-\frac{Mhl_{r}v\cos(\epsilon)a\tau}{l}+I_{g}+M_{f}h_{f}da-\frac{I_ {fw}v\cos(\epsilon)a\tau}{r_{f}} \tag{16}\]  
\[C_{3}=Mhv\cos(\epsilon)a\left(\frac{l_{r}}{l}-\frac{\tau\tau}{l}\right)+\frac {I_{fw}v\cos(\epsilon)a}{r_{f}}-\frac{v^{2}\cos(\epsilon)a\tau}{l}\left( \frac{I_{fw}}{r_{f}}+\frac{I_{rw}}{r_{r}}\right)+\frac{Mgl_{r}d_{1}a\tau}{l} \tag{17}\]  
\[C_{4}=\frac{v^{2}\cos(\epsilon)a}{l}\left(\frac{I_{f}w}{r_{f}}+\frac{I_{r}w}{ r_{r}}\right)-Mgh-Ma\left(\frac{gdl_{r}+hv^{2}\cos(\epsilon)}{l}\right). \tag{18}\]  
The constants of the closed-loop characteristic Equation (14): \(C_{1},C_{2},C_{3}\) and \(C_{4}\) are shown in Equations (15)-(18). The eigenvalues of the characteristic equation are calculated for different values of the roll angle gain \(a\) and the lead time \(\tau\). The motorcycle is stable at a point where all the real parts of the eigenvalues are negative for a particular value of \(a\) and \(\tau\). The shaded zones in Figure 5a,b show regions for the stable motorcycle corresponding to the values of \(a\) and \(\tau\) at speeds 3 and 5 km/h  
Figure 4: Root-locus plot for open-loop motorcycle system. respectively. The rider must be operating inside the shaded zone shown in Figure 5 to achieve the low-speed stability. In the next section, the theoretical results are validated from experimental results.