# Cartesian Aerial Manipulator with Compliant Arm
## 3 Modelling
### Dynamic Model  
The equations of the dynamic model of the Cartesian aerial manipulator with compliant arm can be derived following the Euler-Lagrange formulation based on the Lagrangian and the generalized equation of the forces and torques:  
\[\frac{d}{dt}\left\{\frac{\partial L}{\partial\xi}\right\}-\frac{\partial L}{ \partial\xi}=\boldsymbol{\Gamma}+\boldsymbol{\Gamma}_{ext}\;;\;L=K-V \tag{10}\]  
Figure 8: Volume of operation in the Cartesian manipulator (**up**) and in a 3-degrees of freedom (3-DOF) revolute joint arm (**down**). Here \(L\) is the Lagrangian, which is the difference between the kinetic and potential energies, \(K\) and \(V\), respectively, \(\xi\) is the vector of generalized coordinates, whereas \(\mathbf{\varGamma}\) and \(\mathbf{\varGamma}_{ext}\) are the vectors of endogenous and exogenous (external) generalized forces, defined as follows. \[\mathbf{\zeta}=\left[ \begin{array}{cccccc}\mathbf{\ {}}^{F}_{\mathbf{B}}&\mathbf{\ {}}^{F}_{\mathbf{B}}^{T}&\mathbf{ \varGamma}_{\mathbf{B}}^{T}&d_{1}&d_{2}&q_{3}&\theta_{3}&\Delta l_{4}\end{array}  \right]^{T}\in\Re^{11} \tag{11}\]  
\[\mathbf{\varGamma}=\left[ \begin{array}{cccccc}\mathbf{\ {}}^{T}_{\mathbf{B}}&\mathbf{\ {}}^{T}_{\mathbf{B}}&\mathbf{ \ {}}^{T}_{\mathbf{B}}&F_{1}&F_{2}&\tau_{3}&\tau_{m,3}&F_{4}\end{array} \right]^{T}\in \Re^{11} \tag{12}\]  
where \(\mathbf{\ {}}^{F}_{\mathbf{B}}=\left[x,y,z\right]^{T}\) and \(\mathbf{\varGamma}_{\mathbf{B}}=\left[\phi,\theta,\psi\right]^{T}\) represent the position and orientation of the aerial platform in the Earth fixed frame, \(d_{1}\), \(d_{2}\), and \(q_{3}\) are the joint variables defined earlier, and \(\Delta l_{4}\) is the linear elongation of the end effector. The forces and torques acting over the hexa-rotor are denoted by \(\mathbf{F_{B}}\) and \(\tau_{\mathbf{B}}\), whereas \(F_{1}\) and \(F_{2}\) are the force exerted by the Cartesian manipulator in the X and Y axes, respectively, \(\tau_{3}\) is the torque applied over the compliant joint, and \(F_{4}\) is the force exerted or supported in the direction of the arm link. The kinetic energy is the sum of two terms corresponding to the aerial platform, considered as a rigid solid, and the masses of the manipulator:  
\[K=\frac{1}{2}m_{AP}\parallel\mathbf{\ {}}^{F}_{\mathbf{B}}\parallel^{2}+\frac{1}{2} \mathbf{\epsilon}\mathbf{\omega}_{\mathbf{B}}^{T}\mathbf{I}_{AP}\mathbf{\epsilon}\mathbf{\omega}_{B}+ \frac{1}{2}\sum_{i=1}^{4}\Bigl{(}m_{i}\parallel\mathbf{\ {}}^{F}_{i}\mathbf{\ {}}^{2}+\mathbf{ {}^{E}}\mathbf{\omega}_{i}^{T}\mathbf{I}_{i}\mathbf{{}^{E}}\mathbf{\omega}_{i}\Bigr{)} \tag{13}\]  
where \(m_{AP}\) and \(\mathbf{I}_{AP}\) are the mass and inertia tensor of the aerial platform (including the X-axis support structure), \(m_{i}\) and \(\mathbf{I}_{i}\) are the mass and inertia of the \(i\)-th link of the manipulator (Y-axis linear structure, servo structure, arm link and forearm, as indicated in Figure 2 and Table 2), \(\mathbf{{}^{E}}_{\mathbf{B}}\) and \(\mathbf{{}^{E}}_{\mathbf{A}}\mathbf{\omega}_{\mathbf{B}}\) are the translational and rotational speed of the aerial platform, whereas \(\mathbf{{}^{E}}_{\mathbf{f}}\) and \(\mathbf{{}^{E}}_{\mathbf{A}}\mathbf{\omega}_{i}\) are the speed of the \(i\)-th link of the manipulator.