## 3 FPTLIR/PTL Dynamics Modelling
### FPTLIR/PTL Contact Formulation
If the centroid line of the PTL is above one of the dashed lines (\(\delta>0\)), the normal pressure is proportional to the penetration depth. Conversely, if the centroid line of the PTL is below one of the dashed lines (\(\delta<0\)), the normal force is zero. Due to the depth limitations of the groove on the travelling wheel, the contact force ceases when the modulus of \(\delta\) exceeds a certain value. The angle \(\theta_{XY}\) is defined as the angle between the positive direction of \(\mathbf{s}_{rel}\) along the z-axis and the yz plane. Since the geometric dimensions of the beam element are much larger than those of the travelling wheel, the influence of the FPTLIR's contact process on the tilt angle of the beam element can be considered to be negligible. Consequently, \(\theta_{XY}\) can be approximated as follows:  
\[\theta_{XY}=\frac{3}{2}\pi-\theta(x_{c},t) \tag{35}\]  
Figure 7: The contact cross-section between the travelling wheel and the PTL. Here, \(\theta(x_{c},t)\) is the angle of the contact beam element in yz plane relative to the positive direction of the z-axis, which can be written as:  
\[\theta(x_{c},t)=\arccos\left(\frac{\frac{\partial\tau_{2}}{\partial x}}{\sqrt{ \left(\frac{\partial\tau_{2}}{\partial x}\right)^{2}+\left(\frac{\partial\tau _{2}}{\partial x}\right)^{2}}}\right) \tag{36}\]  
The normal vector of the contact plane in the inertial frame can be expressed by:  
\[\mathbf{u}_{N}=C_{\mathrm{I}}^{\mathrm{B}}\mathbf{R}_{y}(\theta_{YZ})^{-1} \mathbf{V} \tag{37}\]  
where:  
\[\mathbf{R}_{y}(\theta_{YZ})= \begin{bmatrix}\cos\theta_{YZ}&0&\sin\theta_{YZ} \\ 0&1&0\\ -\sin\theta_{YZ}&0&\cos\theta_{YZ}\end{bmatrix}  \tag{38}\]  
In the absence of external forces, the initial configuration of the PTL is determined by gravity alone, resulting in a catenary shape. As the FPTLIR descends, it forms an angle with the PTL. If this angle exceeds a critical threshold, the FPTLIR may slide downward along the PTL. Considering only the tangential \(t\) interaction force between the FPTLIR and the PTL within the yz plane, the unit tangential vector at the contact point is \(\mathbf{V}_{t}\), which is perpendicular to the vector \(\mathbf{s}_{rel}\). The generalized contact force can be calculated as:  
\[\mathbf{Q}_{N}^{\mathrm{z}}=s^{\mathrm{z}}(\eta)\mathbf{F}_{c} \tag{39}\]  
Given that the minimum bending radius of the PTL is considerably larger than that of the travelling wheel, it is reasonable to assume that the contact length, represented by the variable \(l_{0}\), remains constant over time. The contact force in the inertial coordinate system \(\mathbb{I}\) is expressed as:  
\[\mathbf{F}_{c}=\int_{0}^{l_{0}}\left(\mathbf{\ell}_{N}^{\mathrm{1}}+\mathbf{f}_{N}^{ \mathrm{2}}+\mathbf{f}_{t}\right)dx \tag{40}\]  
The tangential friction force is formulated as:  
\[\mathbf{f}_{t}=\mathbf{V}_{t}\mu\left(\left|\mathbf{\ell}_{N}^{\mathrm{1}}\right|+\left| \mathbf{\ell}_{N}^{\mathrm{2}}\right|\right) \tag{41}\]  
where:  
\[\eta=\frac{x_{c}-x_{k-1}}{l_{\mathrm{e}}} \tag{42}\]  
During the attachment phase of the FPTLIR's flight, the travelling wheels are in a braking state, preventing any relative rotation between the wheels and the FPTLIR. Consequently, in the absence of relative sliding between the travelling wheels and the PTL, the arc length coordinate \(x_{c}\) remains constant over time. Within the global coordinate system, as the FPTLIR stabilizes during its descent, the contact points remain aligned on the contact plane, ensuring no penetration or separation between the FPTLIR and the PTL. The kinematic constraints imposed by the contact can be written as:  
\[\mathbf{V}_{t}^{\mathrm{T}}(r^{\mathrm{zw}}-r^{\prime})=0 \tag{43}\]  
The global position vectors of the contact point on the PTL and the travelling wheel are respectively denoted by \(r^{\mathrm{zw}}\). It is assumed that the coordinates of the contact point on the PTL and the travelling wheel are known.