# Towards Autonomous Bridge Inspection: Sensor Mounting Using Aerial Manipulators
## 3 Mathematical Model
### Kinematics
When \(\beta=0\), the situation is reversed and the manipulator motion is used to move the end-effector. For every other \(\beta\), the end-effector motion is obtained in part by the UAV body and the manipulator arm motion. There are obvious advantages in combining the motion of the UAV and the manipulator arm. The UAV can move in 3D space beyond the reach of the arm; however, the motion of the UAV is not as precise and dynamically decoupled. The kinematics of the arm enable the end-effector to obtain the desired approach angle \(\mathbf{z}_{0}^{ee}=[\cos(\delta),\sin(\delta),0]^{\rm T}\), which under the hovering assumption, becomes equal to the global approach vector \(\mathbf{z}_{W}^{ee}\) pointing towards the contact point on the infrastructure. The straightforward mathematical manipulation of Equation (1) allows for writing the constraint equation:  
\[q_{3}=\delta-q_{1}-q_{2}, \tag{5}\]  
which ensures that the manipulator points in the right direction, where \(\delta\) is the desired manipulator inclination in the body \(x\)-\(z\) plane. Figure 2: Coordinate systems of the world, UAV and the 3-DoF manipulator. To find the optimal manipulator pose during contact, the dexterity \(\mathfrak{D}\) and the reach \(\mathfrak{R}\) of the pose were taken into account, while considering that the joints are as far as possible from their physical limits \(\mathfrak{L}\). Since the motion of the arm is constrained with its approach axis condition, a reduced form of a Jacobian matrix was used \(\mathbf{J}=\left[\frac{\delta p_{0}^{\alpha^{\prime}}}{\delta q_{1}},\frac{ \delta p_{0}^{\alpha^{\prime}}}{\delta q_{2}}\right]\) to derive the pose dexterity index \(\mathfrak{D}=\left|\mathbf{J}^{T}\cdot\mathbf{J}\right|\) and determine how far the current pose is from the null space of the manipulator [29]. The reach of the pose \(\mathfrak{R}=(\mathbf{p}_{0}^{\alpha^{\prime}})^{T}\cdot\mathbf{p}_{0}^{\alpha}\) was also taken into account, since the goal was to keep the end-effector and the contact point as far away from the UAV body. Finally, the following equation is defined:  
\[\mathfrak{L}=\frac{(q_{1}^{2}-Q_{1 max}{}^{2})(q_{2}^{2}-Q_{2 max}{}^{2})}{Q_{1 max}{}^{2}Q_{2 max}{}^{2}}, \tag{6}\]  
to measure how far away the given configuration is (i.e., \(q_{1},q_{2}\)) from the joint limits \(Q_{1 max},Q_{2 max}\). Normalizing \(\mathfrak{D},\mathfrak{R}\) and \(\mathfrak{L}\) enables combining the three conditions into a single manifold \(\mathfrak{M}=\mathfrak{D}\cdot\mathfrak{R}\cdot\mathfrak{L}\) and find the optimal configuration \(\mathbf{q}_{\mathbf{M}}^{*}=\left[q_{1}^{*}\quad q_{2}^{*}\quad q_{3}^{*} \right]^{T}\) for the desired approach angle \(\delta\). The described method is depicted in Figure 3 for the specific case of the approach angle \(\delta=0^{\circ}\), but can be extended to any value of the approach angle through Equation (5). As a side note, the manipulator attachment on the top of the UAV body was chosen to be able to reach surfaces underneath the bridge. Although this shifts the center of gravity upwards, the stability of the system is not compromised since the manipulator is constructed of lightweight materials.