## 3 FPTLIR/PTL Dynamics Modelling
### FPTLIR Subsystem Dynamics  
The moment of inertia of the FPTLIR in the body coordinate system is calculated using the parallel axis theorem as follows:  
\[\mathbf{I}_{GB}^{\mathbb{B}}=\mathbf{I}_{bB}^{\mathbb{B}}+m_{\mathbb{B}}\Big{[}\Big{(} \mathbf{r}_{BG}^{\mathbb{B}}\times\mathbf{r}_{BG}^{\mathbb{B}}\Big{)}\mathbf{I}_{3}-\mathbf{r}_ {BG}^{\mathbb{B}}\otimes\mathbf{r}_{BG}^{\mathbb{B}}\Big{]} \tag{44}\]  
In this context, \(\mathbf{r}_{BG}^{\mathbb{B}}\) represents the displacement vector from the FPTLIR's center of mass to its center of gravity. The symbol \(\otimes\) denotes the cross product, and \(\mathbf{I}_{bB}^{\mathbb{B}}\) is a diagonal matrix containing the principal moments of inertia \(\mathbf{I}_{x}\), \(\mathbf{I}_{y}\), and \(\mathbf{I}_{z}\):  
\[\mathbf{I}_{bB}^{\mathbb{B}}=\begin{bmatrix}\mathbf{I}_{x}&0&0\\ 0&\mathbf{I}_{y}&0\\ 0&0&\mathbf{I}_{z}\end{bmatrix} \tag{45}\]  
The angular velocity and linear velocity in Equation (44) can be expressed in terms of the FPTLIR's generalized velocity, as outlined in reference [32]:  
\[\frac{d}{dt}\Big{(}\mathbf{r}_{iB}^{\mathbb{B}}\Big{)}_{\mathbb{I}}=\mathbf{I}_{\mathbb{ B},D}^{\mathbb{B}}\dot{\mathbf{q}}_{r} \tag{46}\]  
\[\mathbf{\omega}_{\mathbb{IB}}^{\mathbb{B}}=\mathbf{I}_{\mathbb{B},R}^{\mathbb{B}}\dot {\mathbf{q}}_{r} \tag{47}\]  
where:  
\[\mathbf{J}_{ib,D}^{\mathbb{B}}=\begin{bmatrix}\mathbf{C}_{\mathbb{I}}^{\mathbb{B}}&0 _{3\times 3}\end{bmatrix} \tag{48}\]  
\[\mathbf{\omega}_{\mathbb{IB}}^{\mathbb{B}}=\begin{bmatrix}1&0&-\mathrm{s}(\theta )\\ 0&\mathrm{c}(\phi)&\mathrm{s}(\phi)\mathrm{c}(\theta)\\ 0&-\mathrm{s}(\phi)&\mathrm{c}(\theta)\mathrm{c}(\phi)\end{bmatrix}\begin{bmatrix} \dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\end{bmatrix}=\mathbf{C}_{\phi\theta}^{\mathbb{B}}\begin{bmatrix}\dot{ \phi}\\ \dot{\theta}\\ \dot{\theta}\end{bmatrix} \tag{49}\]  
\[\mathbf{J}_{\mathbb{IB},R}^{\mathbb{B}}=\begin{bmatrix}0_{3\times 3}&\mathbf{\omega}_{ \phi\phi}^{\mathbb{B}}\end{bmatrix} \tag{50}\]  
\[\frac{d}{dt}\Big{(}\mathbf{r}_{iG}^{\mathbb{B}}\Big{)}_{\mathbb{I}}=\Big{(}\mathbf{I}_ {ib,D}^{\mathbb{B}}-\Big{[}\mathbf{r}_{BG}^{\mathbb{B}}\Big{]}\times\mathbf{I}_{ \mathbb{IB},R}^{\mathbb{B}}\Big{)}\dot{\mathbf{q}}_{r}=\mathbf{I}_{iG,D}^{\mathbb{B}} \dot{\mathbf{q}}_{r} \tag{51}\]  
\[T_{B}=\frac{1}{2}\dot{\mathbf{q}}_{r}^{\,T}M_{B}\dot{\mathbf{q}}_{r} \tag{52}\]  
where \(M_{B}\) can be defined as:  
\[M_{B}=m_{\mathbb{B}}\mathbf{I}_{iG,D}^{\mathbb{B}}\mathbf{T}_{iG,D}^{\mathbb{B}}+\mathbf{I }_{\mathbb{B},R}^{\mathbb{B}}\mathbf{T}^{T}\Big{(}\mathbf{I}_{bB}^{\mathbb{B}}+m_{B} \Big{[}\mathbf{r}_{BG}^{\mathbb{B}}\Big{]}_{\mathbb{X}}^{2}\Big{)}\mathbf{I}_{ \mathbb{IB},R}^{\mathbb{B}} \tag{53}\]  
The control vector of the FPTLIR is written as:  
\[\mathbf{F}_{\mathbb{B}}=\Big{[}\begin{array}{cccc}0&0&F_{z}^{\mathbb{B}}&M_{x} ^{\mathbb{B}}&M_{y}^{\mathbb{B}}&M_{z}^{\mathbb{B}}\end{array}\Big{]}^{T} \tag{54}\]  
Therefore, the kinetic energy of the FPTLIR can be derived as:  
\[T_{B}=\frac{1}{2}\bigg{(}m_{B}\frac{d}{dt}\Big{(}\mathbf{r}_{iG}^{\mathbb{B}} \Big{)}_{\mathbb{I}}^{\mathbb{T}}\frac{d}{dt}\Big{(}\mathbf{r}_{iG}^{\mathbb{B}} \Big{)}_{\mathbb{I}}+\Big{(}\mathbf{\omega}_{\mathbb{IB}}^{\mathbb{B}}\Big{)}^{T} \mathbf{I}_{GB}^{\mathbb{B}}\mathbf{\omega}_{\mathbb{IB}}^{\mathbb{B}}\bigg{)} \tag{55}\]  
where velocity of the FPTLIR's center of mass is:  
\[\frac{d}{dt}\Big{(}\mathbf{r}_{iG}^{\mathbb{B}}\Big{)}_{\mathbb{I}}=\frac{d}{dt} \Big{(}\mathbf{r}_{ib}^{\mathbb{B}}\Big{)}_{\mathbb{I}}+\frac{d}{dt}\Big{(}\mathbf{r}_ {bG}^{\mathbb{B}}\Big{)}_{\mathbb{I}}=\frac{d}{dt}\Big{(}\mathbf{r}_{ib}^{\mathbb{ B}}\Big{)}_{\mathbb{I}}+\mathbf{\omega}_{\mathbb{IB}}^{\mathbb{B}}\times\mathbf{r}_{bG}^{ \mathbb{B}} \tag{56}\]