# A Screw-Axis Approach to the Stability of Two-Wheeled Vehicles
## 3 Screw-Axis and Velocity Centres  
The screw axis of the chassis is computed from the eigenvectors of the state matrix \(A\) in (1), which can be 'animated' in the time domain through  
\[x(t)=x_{0}e^{-\omega_{n}\zeta t}\sin\left[\left(\sqrt{1-\zeta^{2}}\right)\omega _{n}t+\phi\right], \tag{4}\]  
where \(x_{0}\) is the magnitude of the eigenvector state (i.e., the length of the arrow of the compass plot), \(\omega_{n}\) is the (undamped) frequency of the corresponding eigenvalue, \(\zeta\) is the damping ratio of the corresponding eigenvalue, \(\phi\) is the phase (i.e., the angle of the arrow of the compass plot), and \(t\) is the time. Each point \(P_{0}\) of the chassis has a velocity \(v_{0}\) and angular velocity \(\omega\) during the oscillation--the latter is the same for each point of the chassis, which is here treated as a rigid body. The velocity \(v_{0}\) can be divided in two different components: \(v_{\parallel}\) parallel to \(\omega\) and \(v_{\perp}\) normal to \(\omega\), which are given by  
\[v_{\parallel} = \left(v_{0}\cdot\frac{\omega}{\|\omega\|}\right)\frac{\omega}{\| \omega\|} \tag{5}\] \[v_{\perp} = v_{0}-v_{\parallel} \tag{6}\]  
The normal component can also be written as  
\[v_{\perp}=\omega\times d, \tag{7}\]  
Figure 2: Weave compass diagrams at different speeds: \(y\) is the saddle point lateral displacement, \(\delta\) is the steer angle, \(\phi\) is the roll angle and \(\psi\) is the yaw angle. where \(d\) is the distance between the screw axis and \(P_{0}\). When multiplying both sides by \(\omega\) one obtains  
\[v_{\perp}\times\omega=(\omega\times d)\times\omega, \tag{8}\]  
which can be rewritten using triple-product identity as  
\[v_{\perp}\times\omega = (\omega\cdot\omega)d-(d\cdot\omega)\omega \tag{9}\] \[= \|\omega\|^{2}d. \tag{10}\]  
Therefore, the distance \(d\) can be computed at each instant from  
\[d = \frac{v_{\perp}\times\omega}{\|\omega\|^{2}}, \tag{11}\]  
while the equation of the screw axis is given by  
\[S_{a} = P_{0}-d+\mu\omega, \tag{12}\] \[ \begin{pmatrix}x_{a}\\ y_{a}\\ z_{a}\end{pmatrix}  = \begin{pmatrix}x_{0}\\ y_{0}\\ z_{0}\end{pmatrix} -\frac{1}{\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}} \begin{pmatrix}-v_{z}\omega_{y}+v_{y}\omega_{z}\\ -v_{x}\omega_{z}+v_{z}\omega_{x}\\ -v_{y}\omega_{x}+v_{x}\omega_{y}\end{pmatrix} +\mu \begin{pmatrix}\omega_{x}\\ \omega_{y}\\ \omega_{z}\end{pmatrix}  \tag{13}\]  
where \(\mu\) is a scalar associated with the infinite points of the screw axis line. The three-dimensional motion of the chassis can be projected on an \(x-y\) transverse plane (parallel to the road plane, i.e., top view), a \(y-z\) frontal plane (i.e., front view) or an \(x-z\) sagittal plane (i.e., side view). In this case the screw axis is replaced by the velocity centre, whose trajectory on the selected plane can be obtained by intersection between the screw line and the selected plane. The scalar \(\mu\) corresponding to the transverse plane is obtained when solving the third equation in (13) for \(z_{a}=z_{S}\), with \(z_{S}\) being the height of the selected plane:  
\[\mu=\frac{1}{\omega_{z}}\Bigg{(}z_{S}-z_{0}-\frac{v_{y}\omega_{x}-v_{x}\omega_ {y}}{\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}}\Bigg{)}. \tag{14}\]  
Similarly, the scalar \(\mu\) corresponding to the projection on the frontal plane is given by  
\[\mu=\frac{1}{\omega_{x}}\Bigg{(}x_{M}-x_{0}-\frac{v_{z}\omega_{y}-v_{y}\omega _{z}}{\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}}\Bigg{)}, \tag{15}\]  
which is obtained from the second of (13) in the case \(x_{a}=x_{S}\), with \(x_{S}\) being the coordinate of the selected plane. Finally, the intersection with the sagittal plane is not considered, since in straight motion the lateral modes are decoupled from the longitudinal modes: As a consequence \(\omega_{y}=0\) (pitch rate) during weave and wobble, and the screw axis remains parallel to the vehicle sagittal plane. In principle, the approach presented can also be applied to experimental data, i.e., logged from the IMU of the vehicle, in addition to numerical data.