# Modeling and Control of an Articulated Multibody Aircraft
## 2 Development of the Multibody Equations of Motion
### Multibody Equations of Motion
#### 2.6.2 Rotational Dynamics  
For a system of \(k\) clustered rigid bodies, the rotational dynamic equations are developed as the sum of their individual contributions based on [PERSON]'s law, as follows [125; 38]:  
\[\underbrace{\sum_{k}D^{I}\bigg{(}I_{B_{k}}^{B_{k}}\omega^{B_{k}I}\bigg{)}}_{ \text{Term I}}+\underbrace{\sum_{k}D^{I}\bigg{(}m^{B_{k}}S_{B_{k}I}D^{I}s_{B_{ k}I}\bigg{)}}_{\text{Term II}}=\sum_{k}M_{k}+\sum_{k}S_{B_{k}I}F_{k} \tag{29}\]  
where the expression \(\sum_{k}M_{k}+\sum_{k}S_{B_{k}I}F_{k}\) on the right-hand side (RHS) of Equation (29) is the contributing sum of all the moments acting on the aircraft from each rigid body, which will be discussed in Section 2.7. For the system being considered in this study, the tail and body contributions are separated in _Term I_ in Equation (29) as  
\[\sum_{k}D^{I}\bigg{(}I_{B_{k}}^{B_{k}}\omega^{B_{k}I}\bigg{)}=D^{I}\bigg{(}I_ {B}^{B}\omega^{BI}\bigg{)}+D^{I}\bigg{(}I_{T}^{T}\omega^{T}\bigg{)} \tag{30}\]  
Applying the chain rule and Euler transformation to each term on the RHS of Equation (30) leads to  
\[ \begin{split}\sum_{k}D^{I}\bigg{(}I_{B_{k}}^{B_{k}}\omega^{B_{k }I}\bigg{)}&=\omega^{BI}D^{B}I_{b}^{B}+I_{b}^{B}D^{B}\omega^{BI} +\Omega^{BI}I_{b}^{B}\omega^{BI}\\ &+\omega^{TI}D^{T}I_{t}^{T}+I_{t}^{T}D^{T}\omega^{TI}+\Omega^{ TI}I_{t}^{T}\omega^{TI}\end{split}  \tag{31}\]  
Since the moment of inertia of the body and wing are constant in their corresponding frame, the first and fourth terms on the RHS of Equation (31) vanish. Additionally, considering \(D^{T}\omega^{TI}=D^{T}\omega^{TB}+D^{T}\omega^{BI}\) and \(\Omega^{BT}=-\Omega^{TB}\), and performing the Euler transformation to establish \(D^{T}\omega^{BI}\) in the body frame results in  
\[ \begin{split}\sum_{k}D^{I}\bigg{(}I_{B_{k}}^{B_{k}}\omega^{B_{k}I }\bigg{)}&=I_{b}^{B}D^{B}\omega^{BI}+\Omega^{BI}I_{b}^{B}\omega^{ BI}\\ &+I_{t}^{T}\bigg{(}D^{T}\omega^{TB}+D^{B}\omega^{BI}-\Omega^{TB} \omega^{BI}\bigg{)}\\ &+\;\bigg{(}(\Omega^{TB}+\Omega^{BI})I_{t}^{T}(\omega^{TB}+\omega ^{BI})\bigg{)}\end{split}  \tag{32}\]Applying the chain rule to _Term 11_ in Equation (29) yields  
\[\sum_{k}D^{I}\bigg{(}m^{B_{k}}S_{B_{k^{2}}}D^{I}s_{B_{k^{I}}}\bigg{)}=\sum_{k}m^{ B_{k}}\bigg{(}D^{I}S_{B_{k^{I}}}D^{I}s_{B_{k^{I}}}+S_{B_{k^{I}}}D^{I}D^{I}s_{B_{k^{I}}} \bigg{)} \tag{33}\]  
The cross-product of the vector derivatives is zero; therefore, the first term on the RHS of Equation (33) is zero. Additionally, the body and abdominal contributions are separated, leading to  
\[\sum_{k}D^{I}\bigg{(}m^{B_{k}}S_{B_{k^{I}}}D^{I}s_{B_{k^{I}}}\bigg{)}=m^{B}S_{ B_{l}}D^{I}D^{I}s_{B_{l}}+m^{T}S_{tl}D^{I}D^{I}s_{tl} \tag{34}\]  
Using Euler transformation, the first term on the RHS of Equation (34) becomes  
\[m^{B}S_{bl}D^{I}D^{I}s_{bl}=m^{B}S_{bl}(D^{B}V^{I}_{b}+\Omega^{BI}V^{I}_{b}) \tag{35}\]  
The second term on the RHS of Equation (34) could be developed using the vector expansion \(s_{tl}=s_{ij}+s_{jb}+s_{bl}\) and performing Euler transformation, results in  
\[m^{T}S_{tl}D^{I}D^{I}s_{tl}=m^{T}S_{tl}D^{I}\bigg{(}D^{T}s_{ij}+\Omega^{TI}s_ {ij}+D^{B}s_{jb}+\Omega^{BI}s_{jb}+V^{I}_{b}\bigg{)} \tag{36}\]  
As \(s_{ij}\) and \(s_{jb}\) are constant in their corresponding frames, the first and third terms of the equation above vanish. Additionally, the rotational time derivatives of the last two terms can be transformed to the body frame as follows  
\[D^{I}(\Omega^{BI}s_{jb})=D^{B}\Omega^{BI}s_{jb}+\Omega^{BI}\Omega^{BI}s_{jb} \quad\text{and}\quad D^{I}V^{I}_{b}=D^{B}V^{I}_{b}+\Omega^{BI}V^{I}_{b} \tag{37}\]  
Only the rotational derivative of the second term on the RHS of Equation (36) needs to be calculated.