# Towards Autonomous Bridge Inspection: Sensor Mounting Using Aerial Manipulators
## 3 Mathematical Model
### Dynamics  
The most complicated task of the aerial manipulator is attaching the sensor to a wall and maintaining the required force reference while the two-component adhesive hardens. Figure 3: The visual decomposition of dexterity \(\mathfrak{D}\), reach \(\mathfrak{R}\), limit \(\mathfrak{L}\) and the overall combined surface. This analysis is performed for \(\delta=0^{\circ}\): (**a**) the dexterity \(\mathfrak{D}\) surface shows the measure of how far the manipulator is from the null space. Values around zero are closer to the null space; (**b**) the reach \(\mathfrak{R}\) surface shows how far the end-effector can move in a certain configuration. This value tends towards zero as the arm approaches a folded configuration; (**c**) the limit \(\mathfrak{L}\) depicts how far a certain configuration is from the physical limits of the manipulator joints; and (**d**) the combined manifold \(\mathfrak{M}\) of the formerly described surfaces. Higher values offer better trade-off between dexterity, reach and limit, defining the optimal manipulator configuration \(\mathbf{q}_{\mathbf{M}}^{*}\). To successfully perform such a task, the coupled UAV-manipulator system dynamics have to be addressed for the precise end-effector configuration planning. Considering the UAV dynamics only, the derivative of the generalized coordinates can be defined as \(\dot{\mathbf{q}}_{\mathrm{B}}=\left[(\dot{\mathbf{p}}_{\mathrm{W}}^{\mathrm{B}})^ {T}\quad(\dot{\mathbf{o}}_{\mathrm{W}}^{\mathrm{B}})^{T}\right]^{T}\in\mathbb{R}^ {6}\). Here, the \((\dot{\mathbf{p}}_{\mathrm{W}}^{\mathrm{B}})^{T}\) is the linear velocity of the body of the UAV in the world frame and the \((\dot{\mathbf{o}}_{\mathrm{W}}^{\mathrm{B}})^{T}\) represents the angular velocity of the UAV in the world frame. The UAV's propulsion system consists of \(n_{p}\) propellers rigidly attached to the body. Each propeller produces force and torque along the \(z_{\mathrm{B}}\) axis. The vector of the propeller rotational velocities is simply defined as  
\[\mathbf{\Omega}_{\mathrm{UAV}}= \begin{bmatrix}\Omega_{1}&\dots&\Omega_{n_{p}} \end{bmatrix} ^{T}\in\mathbb{R}^{n_{p}}. \tag{7}\]  
Force and torque produced by each propeller are non-linear functions depending on the rotational velocity \(\mathbf{\Omega}_{\mathrm{UAV}}\). Rather than using the rotational velocities as control inputs, they can be mapped to a more convenient space. Namely, the mapped control input space can be written as  
\[\mathbf{u}_{\mathrm{UAV}}=\mathbf{K}\cdot diag(\mathbf{\Omega}_{\mathrm{UAV}} )\cdot\mathbf{\Omega}_{\mathrm{UAV}}, \tag{8}\]  
where \(\mathbf{K}\in\mathbb{R}^{4\times n_{p}}\) is the mapping matrix and \(\mathbf{u}_{\mathrm{UAV}}= \begin{bmatrix}u_{1}&u_{2}&u_{3}&u_{4}\end{bmatrix} ^ {T}\), where \(u_{4}\) represents the net thrust and \(u_{1}\), \(u_{2}\) and \(u_{3}\) are moments around the body frame axes. As stated earlier, the manipulator consists of three rotational DoFs. Therefore, the joint positions of the manipulator are defined as \(\mathbf{q}_{\mathrm{M}}= \begin{bmatrix}q_{1}&q_{2}&q_{3}\end{bmatrix} ^{T}\). The rotational velocity of each joint is a time derivative of joint positions \(\dot{\mathbf{q}}_{\mathrm{M}}=d\mathbf{q}_{\mathrm{M}}/dt\). The torque of each joint is considered the control input of the manipulator \(\mathbf{u}_{\mathrm{M}}= \begin{bmatrix}\tau_{1}&\tau_{2}&\tau_{3}\end{bmatrix} ^ {T}\). The resulting generalized coordinates of the aerial manipulator can be written as \(\mathbf{q}= \begin{bmatrix}\mathbf{q}_{\mathrm{UAV}}&\mathbf{q}_{\mathrm{M}} \end{bmatrix} ^{T}\in\mathbb{R}^{9}\), and the velocities can be obtained in the same manner as \(\dot{\mathbf{q}}= \begin{bmatrix}\dot{\mathbf{q}}_{\mathrm{UAV}}&\dot{\mathbf{ q}}_{\mathrm{M}}\end{bmatrix} ^{T}\in\mathbb{R}^{9}\). The resulting control inputs of the system can be expressed as \(\mathbf{u}= \begin{bmatrix}\mathbf{u}_{\mathrm{UAV}}&\mathbf{u}_{\mathrm{M}} \end{bmatrix} ^{T}\in\mathbb{R}^{7}\).