## 3 FPTLIR/PTL Dynamics Modelling  
### PTL Modeling  
Initially, the PTL is assumed to be unaffected by gravity and concentrated forces. It is typically a multi-layer composite structure. To facilitate calculations, the unstressed PTL is simplified as an Euler-Bernoulli beam with a circular cross-section, neglecting the effects of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Value** & **Formula** & **Range** \\ \hline Roll angle & \((-\Theta_{\mathrm{c}}+\alpha(x),\Phi_{\mathrm{c}}+\alpha(x)\ )\) & \((-19.9,9.1)\) to \((-9.1,19.9)\) \\ \hline Pitch angle & \((\Theta_{\mathrm{c}},\Theta_{\mathrm{c}})\) & \((-63,63)\) \\ \hline Yaw angle & \((-\Psi_{\mathrm{c}},\Psi_{\mathrm{c}})\) & \((-16.34,16.34)\) \\ \hline Z-displacement & \((0,-Z_{c})\) & \((0,-1.25)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Workspace parameters of the FPTLIR. Figure 5: Schematic diagram of the operating zones in the Z direction for the FPTLIR. rotational moments of inertia and shear stress [30]. Given that the initial unstressed total length of the PTL is \(L\) and dividing the PTL into \(N\) beam units, then the stress-free length of a single element is \(l_{\rm e}=L/N\) and the arc coordinate lengths of the beam node of the unit are \(x_{j}=jl_{e}\ j=1,2,3,\cdots,N\ x_{j}\in[0,L]\). Point \(b\) is located at the center of the FPTLIR's rotor plane, which is the origin body reference frame \(\mathbb{B}\). The \(z\) axes of reference frame \(\mathbb{B}\) are orthogonal to the rotor plane. The center of mass of the FPTLIR is indicated by G, and the center of the robot's travelling wheel is denoted by \(w\). The contact point is marked by \(c\). Figure 6 depicts the \(k^{th}\) 12-degree element PTL. There are two node coordinates, \(\mathbf{r}^{k-1}\) and \(\mathbf{r}^{k}\), for a \(k^{th}\) beam element. The subscript \(x\) denotes the first-order derivative with respect to \(x\), such as \(\mathbf{r}^{k}_{x}=\partial\mathbf{r}^{k}/\partial x\) and \(\mathbf{r}^{k-1}_{x}=\partial\mathbf{r}^{j+1}/\partial x\). For subsequent coupling with the FPTLIR, a 6 DOF (degrees of freedom) node is used, where \(r_{1}\), \(r_{2}\), and \(r_{3}\) represent the coordinates of the element node in the X, Y, and Z directions. Within a single element, the global shape function is used to estimate the generalized coordinate vector \(\mathbf{r}\) at the arc-length coordinate \(x\):  
\[\mathbf{r}(x,t)=\sum_{\alpha=1}^{4}s^{\alpha}\mathbf{q}_{\alpha} \tag{6}\]  
where the global shape function is defined as follows, using the Hermite method:  
\[s^{1}=1-3\xi^{2}+2\xi^{3},s^{2}=l_{\rm e}\Big{(}\xi-2\xi^{2}+\xi^{3}\Big{)},s^ {3}=3\xi^{2}-2\xi^{3},s^{4}=l_{\rm e}\Big{(}\xi^{3}-\xi^{2}\Big{)} \tag{7}\]  
where \(\xi=(x-x_{k-1})/l_{  
\[\overline{\epsilon}=\frac{1}{2}\left(\mathbf{r}_{x}^{\intercal}\mathbf{r}_{x}- \overline{\mathbf{r}}_{x}^{\intercal}\overline{\mathbf{r}}_{x}\right) \tag{11}\]  
\[\kappa=\frac{\left\|\mathbf{r}_{x}\times\mathbf{r}_{xx}\right\|}{\left\| \mathbf{r}_{x}\right\|^{2}} \tag{12}\]  
where \(\overline{\mathbf{r}}\) represents the undeformed vector of the centroid line. When the element is in the initial unstressed configuration, there are \(\left\|\overline{\mathbf{r}}_{x}\right\|=1\).