# Aerial Physical Interaction in Grabbing Conditions with Lightweight and Compliant Dual Arms
## 3 Modelling
### Dynamics  
The dynamic model of the compliant joint dual arm aerial manipulation robot is derived from the Lagrangian and the generalized equations of the forces and torques:  
\[L=K-V \tag{5}\]  
\[\frac{d}{dt}\left\{\frac{\partial L}{\partial\mathbf{\xi}}\right\}-\left\{\frac{ \partial L}{\partial\mathbf{\xi}}\right\}=\mathbf{\Gamma}+\mathbf{\Gamma}_{ext} \tag{6}\]  
where \(L\) is the Lagrangian; \(K\) and \(V\) are the kinetic and potential energies, respectively; \(\mathbf{\xi}\) is the vector of generalized coordinates; and \(\mathbf{\Gamma}\) and \(\mathbf{\Gamma}_{ext}\) respectively represent the generated and external wrenches acting on the aerial robot. The vector of generalized coordinates includes the multirotor position and orientation, as well as the servo shaft and output link angular position vectors, \(\mathbf{\theta}^{i}=\left[ \begin{array}{ccc}\theta_{1}^{i}&\theta_{2}^{i}& \theta_{3}^{i}\end{array} \right]\) and \(\mathbf{q}^{i}=\left[ \begin{array}{ccc}q_{1}^{i}&q_{2}^{i}&q_{3}^{i}\end{array} \right]\), respectively, and thus \(\mathbf{\xi}\) is defined as follows. \[\mathbf{\xi}=\left[ \begin{array}{ccc}\mathbf{\ {}^{E}}_{\mathbf{P}}&\mathbf{\ {}^{E}}_{\mathbf{\eta}} \mathbf{B}&\mathbf{\theta}^{1}&\mathbf{q}^{1}&\mathbf{\theta}^{2}&\mathbf{q}^{2}\end{array} \right]^ {T}\in\Re^{18} \tag{7}\]  
Analogously, the vector of generalized forces comprises the forces and torques acting over the multirotor and the joints of the manipulator. \[\mathbf{\Gamma}=\left[ \begin{array}{ccc}\mathbf{\ {}^{F}}_{\mathbf{B}}&\mathbf{\ {}^{\tau}}_{\mathbf{B}}&\mathbf{ \ {}^{1}_{m}}&\mathbf{\ {}^{\tau}}^{1}&\mathbf{\ {}^{\tau}_{m}}&\mathbf{\ {}^{\tau}}^{2}\end{array} \right]^{T}\in\Re^{18} \tag{8}\]  
The vector of external forces \(\mathbf{\Gamma}_{ext}\) models the disturbance wrenches exerted on the multirotor base [18; 19; 20], the contact forces at the end effector [4; 6; 17], as well as the aerodynamic forces raised when the thrust of the rotors is affected by close surfaces in the environment [7; 26]. The kinetic energy of the aerial manipulator can be expressed as the sum of the kinetic energy of the aerial platform and the kinetic energy of the robotic arms:  
\[K=K_{UAAV}+K_{arms} \tag{9}\]  
Each of these components comprises two terms corresponding to the translation and rotation of the masses with respect to the inertial frame \(\{\mathbf{E}\}\):  
\[K_{UAAV}=\frac{1}{2}m_{UAAV}\|\mathbf{\dot{F}^{j}}\mathbf{\dot{B}}^{2}+\frac{1}{2}\bm {\dot{F}}_{\mathbf{\omega}}\mathbf{\dot{T}}_{\mathbf{I}UAV}\mathbf{\dot{F}}_{\mathbf{\omega}B} \tag{10}\]  
\[K_{arms}=\sum_{i=1}^{2}\sum_{j=1}^{4}\Bigl{(}\frac{1}{2}m_{j}^{i}\|\mathbf{\dot{F}^ {j}}_{i}\|^{2}+\frac{1}{2}\mathbf{\dot{\alpha}}_{j}^{i,T}\mathbf{\dot{r}}_{j}^{i,E}\bm {\omega}_{j}^{i}\Bigr{)} \tag{11}\]  
where \(m_{UAAV}\) and \(\mathbf{I}_{UAAV}\) are the mass and inertia tensor of the aerial platform, respectively, whereas \(m_{j}^{i}\) and \(\mathbf{\dot{F}^{i}_{j}}\) are mass and inertia of the \(j\)-th joint of the \(i\)-th arm, respectively. The potential energy of the aerial manipulator also includes two terms, the gravity and elastic potential of the compliant joints:  
\[V=g\biggl{(}m_{UAAV}z_{UAV}+\sum_{i=1}^{2}\sum_{j=1}^{4}m_{j}^{i}z_{j}^{i} \biggr{)}+\sum_{i=1}^{2}\sum_{j=1}^{4}k_{j}^{i}\Bigl{(}\theta_{j}^{i}-q_{j}^{ i}\Bigr{)}^{2} \tag{12}\]  
Here, \(g\) is the gravity constant, and \(k_{j}^{i}\) is the corresponding joint stiffness.