## 3 FPTLIR/PTL Dynamics Modelling
### FPTLIR/PTL Contact Formulation  
The generalized coordinates of the FPTLIR are denoted by \({q_{r}}^{T}=\left[u,v,w,\phi,\theta,\psi\right]\), and the generalized velocity of the FPTLIR is represented by \(\dot{\mathbf{q}}_{r}^{T}=\left[\dot{u},\dot{v},\dot{w},\dot{\phi},\dot{\theta},\dot{\psi}\right].\) The vector from point \(b\) to point \(i\) is denoted by \(u,v,w\). In the inertial coordinate system \(\mathbb{I}\), the Euler angles \(\phi,\theta,\psi\) represent the rotation of the body coordinate system \(\mathbb{B}\) relative to \(\mathbb{I}\). In particular, \(\psi\) represents the yaw angle, indicating a rotation about the z-axis, \(\phi\) denotes the roll angle, signifying a rotation about the x-axis, and \(\theta\) is the pitch angle, denoting a rotation about the y-axis. The rotation matrix from the body coordinate system \(\mathbb{B}\) to the inertial coordinate system \(\mathbb{I}\) is defined as:  
\[\mathbf{C}_{\mathbb{I}}^{\mathbb{B}}= \begin{bmatrix}c\psi c\theta-s\phi s\psi s \theta&-c\phi s\psi&c\psi s\theta+c\theta s\phi s\psi\\ c\theta s\psi+c\psi s\phi s\theta&c\psi c\psi&s\psi s\theta-c\psi c\theta s\phi \\ -c\phi s\theta&s\phi&c\psi c\theta\end{bmatrix}  \tag{28}\]  
As the FTPLIR declines, the gap between the travelling wheel and the PTL diminishes until contact is established between the two. During this interaction, the PTL undergoes deformation. The Hunt-Crossley contact model is used to characterize the contact interface as a nonlinear spring-damper system. The contact force per unit length is formulated as:  
\[\mathbf{f}_{N}=k_{N}\delta^{n}(1+D\dot{\delta})\mathbf{u}_{N} \tag{29}\]  
where \(\mathbf{u}_{N}\) is the orthogonal vector perpendicular to the contact surface of the travelling wheel, \(k_{N}\) is the contact stiffness, \(\delta\) is the relative penetration depth, \(D\) is the contact damping, and \(n\) is a positive integer, determined by the local geometric characteristics of the material and the contact surface; it is taken in the current analysis as \(n=2.5\) to describe similar constraints [25]. The contact point between the FPTLIR and the PTL is \(c\). The arc length coordinate of the corresponding central of the contact point is \(x_{c}\), where \(x_{k}\geq x_{c}\geq x_{k-1}\). The vector from the center of the travelling wheel \(c\) to the contact point on the PTL's centerline is given by:  
\[\mathbf{s}_{rel}=\mathbf{r}(x_{c},t)-\mathbf{s}_{w} \tag{30}\]  
where \(\mathbf{r}(x_{c},t)\) represents the position of the PTL centerline at the contact point in the inertial coordinate system, and \(\mathbf{s}_{w}\) is the position of the travelling wheel center in the inertial coordinate system. \[\mathbf{s}_{w}=C_{\mathbb{I}}^{\mathbb{B}}\left(\mathbf{r}_{ib}^{\mathbb{B}}+\mathbf{r}_{ bw}^{\mathbb{B}}\right)=\mathbf{r}_{ib}^{\mathbb{I}}+C_{\mathbb{I}}^{\mathbb{B}}\mathbf{r}_{ bw}^{\mathbb{B}} \tag{31}\]  
Then the relative penetration depth is given as:  
\[\delta=R+\frac{d}{2}-|\mathbf{s}_{rel}| \tag{32}\]  
To accurately describe the contact between the travelling wheel and the PTL, it is necessary to analyze the cross-section at the contact point. The groove surface of the travelling wheel consists of two straight lines. The contact force is applied to the centroid line of the PTL, and the contact area is represented by a dashed line, offset from the actual contact area which is represented by solid lines (Figure 7). The contact planes are denoted by \(s1\) and \(s2\), intersecting at a point designated as \(\mathbf{p}_{0}\). The respective normal vectors of these planes are \(\mathbf{V}_{n}^{s1}\) and \(\mathbf{V}_{n}^{s2}\). \[\mathbf{V}_{n}^{s1}=[0\ \cos(\theta_{w}/2)\ \sin(\theta_{w}/2)]^{T} \tag{33}\] \[\mathbf{V}_{n}^{s2}=[0\ -\cos(\theta_{w}/2)\ \sin(\theta_{w}/2)]^{T} \tag{34}\]  
The variable \(\theta_{w}\) represents the groove angle of the travelling wheel. The radius of the groove bottom surface of the travelling wheel is smaller than that of the PTL.