# 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
Using the expansion of the angular velocity tensor \(\Omega^{TI}=\Omega^{TB}+\Omega^{BI}\) and performing the Euler transformation results in  
\[D^{I}\big{(}\Omega^{TI}s_{ij}\big{)}=D^{T}\Omega^{TB}s_{ij}+\Omega^{TI}\Omega ^{TB}s_{ij}+D^{B}(\Omega^{BI}s_{ij})+\Omega^{BI}\Omega^{BI}s_{ij} \tag{38}\]  
\(\Omega^{TI}\) in the second term of RHS the above equation should be substituted using \(\Omega^{TI}=\Omega^{TB}+\Omega^{BI}\). For the third term on the RHS of the above equation, we take the rotational time derivative of the term, apply the Euler transformation, \(D^{B}s_{ij}=D^{T}s_{ij}+\Omega^{TB}D^{B}s_{ij}\) and consider that \(D^{T}s_{ij}=0\). These simplify the third term to  
\[D^{B}(\Omega^{BI}s_{ij})=D^{B}(\Omega^{BI}s_{ij})+\Omega^{TI}\Omega^{TB}s_{ij} \tag{39}\]  
Substituting Equation (39) into Equation (38) and then substituting Equations (37) and (38) into Equation (36) gives  
\[m^{T}S_{tl}D^{I}D^{I}s_{tl}=m^{T}S_{tl}\bigg{[} \begin{array}{c}D^{T}\Omega^{ TB}s_{ij}+\Omega^{TB}\Omega^{TB}s_{ij}\\ +D^{B}\Omega^{BI}s_{ij}+2\Omega^{BI}\Omega^{TB}s_{ij}+\Omega^{BI}\Omega^{BI}s_{ ij}\\ +D^{B}\Omega^{BI}s_{jb}+\Omega^{BI}\Omega^{BI}s_{jb}+D^{B}V^{I}_{b}+\Omega^{BI}V^{I}_{b} \end{array} \bigg{]} \tag{40}\]Finally, substituting Equations (35) and (40) into Equation (34) and further substituting Equations (32) and (34) into Equation (29), we obtain  
\[ \begin{array}{l}{{{{D}^{T}}{{\Omega}^{T{B}}}_{{{s_{ij}}}}+{{\Omega}^{T{B}}}{{ \Omega}^{T{B}}}{{s_{ij}}}}}\\ {{{+{D}^{B}}{{\Omega}^{B{I}}}{{s_{ij}}}+2{{\Omega}^{B{I}}}{{\Omega}^{T{B}}}{{s _{ij}}}+{{\Omega}^{B{I}}}{{\Omega}^{B{I}}}{{S_{{s_{ij}}}}}}}\\ {{{+{D}^{B}}{{\Omega}^{B{I}}}{{s_{j}}}+{{\Omega}^{B{I}}}{{\Omega}^{B{I}}}{{s _{j_{p}}}}+{{D}^{B}}{V_{b}^{I}}+{{\Omega}^{B{I}}}{V_{b}^{I}}}}\\ \end{array} \Bigg{]}\\ +{{{I}_{t}^{T}}}\bigg{(}{{{D}^{T}}{{\omega}^{T{B}}}+{{D}^{B}}{{\omega}^{B{I}} }-{{\Omega}^{T{B}}}{{\omega}^{B{I}}}}\bigg{)}+\ \bigg{(}{{({{\Omega}^{T{B}}}+{{\Omega}^{B{I}}})}{{I}_{t}^{T}}({{\omega}^{T{B} }}+{{\omega}^{B{I}}})}\bigg{)}\\ +{{{I}_{b}^{B}}}{{D}^{B}}{{\omega}^{B{I}}}+{{\Omega}^{B{I}}}{{I}_{b}^{B}}{{ \omega}^{B{I}}}+{{m}^{B}}{{S_{{{bl}}}}({{D}^{B}}{V_{b}^{I}}+{{\Omega}^{B{I}}} {V_{b}^{I}})}=\sum\limits_{k}{{M}_{k}}+\sum\limits_{k}{{S_{{{B}_{k}}}}{{I}_{k}}} \end{array} \tag{41}\]  
The second term on the RHS of this equation can be written as the sum of contributions from individual parts  
\[ \begin{array}{l}{\sum\limits_{k}{{S_{{{b}_{k}}}}{{I}_{k}}}{{F}_{k}}={{S_{{ {bl}}}}{F}_{B}}+\big{(}{{{S_{{{T}}}}}+{{S_{{{bl}}}}}}\big{)}{{F}_{T}}}&{{={S_{ {{bl}}}}}\sum\limits_{k}{{F}_{k}}+{{S_{{{bl}}}}}{{F}_{T}}}\\ \end{array}  \tag{42}\]  
The expression for \(\sum\limits_{k}{{F}_{k}}\) can be obtained from Equation (27). Multiplying both sides of Equation (27) by \({{S_{{{bl}}}}}\) and inserting into Equation (34) results in corresponding terms being canceled from both sides. Therefore, the rotational dynamic equations of motion in an invariant tensor form are  
\[\begin{array}{l}{{{{D}^{T}}{{\Omega}^{T{B}}}{{s_{ij}}}+{{\Omega}^{T{B}}}{{ \Omega}^{T{B}}}{{s_{ij}}}}}\\ {{{+{D}^{B}}{{\Omega}^{B{I}}}{{({s_{ij}}}+{s_{j_{p}}})}+{{\Omega}^{B{I}}}{{ \Omega}^{B{I}}}{{({s_  
where \(I_{y}^{T}\) is the mass moment of inertia of the abdomen about \(y\)-axis at the joint \(j\), and \(\kappa\) is the damping constant. \(\tau_{ly}\) is the load torque due to gravity and \(\tau_{ay}\) is the torque applied at the joint to actuate the abdomen.