## 3 Hardware System Design
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameter** & **Value** \\ \hline \multirow{3}{*}{Whole robot} & Width & 640 [mm] \\ \cline{2-3}  & Maximum height & 540 [mm] \\ \cline{2-3}  & Total weight & 3000 [g] \\ \cline{2-3}  & Rotors & 6 \\ \hline \multirow{3}{*}{Single manipulator} & DOF & 3 \\ \cline{2-3}  & Link lengths & 50 [mm], 40 [mm], 100 [mm], \\ \cline{2-3}  & (L1, L2, L3, L4, L5) & 95 [mm], 205 [mm] \\ \cline{2-3}  & Joint angle range & \(-100[^{\circ}]\) to \(100[^{\circ}]\), \(-190[^{\circ}]\) to \(90[^{\circ}]\), \\ \cline{2-3}  & (J1, J2, J3) & \(-60[^{\circ}]\) to \(60[^{\circ}]\) \\ \cline{2-3}  & Maximum width & 5 [mm] \\ \cline{2-3}  & Weight & 330 [g] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the whole robot and three-arm aerial manipulator system with parallel link mechanism. Figure 3: UAV with parallel link three-manipulator system. \[x=L_{1}c_{1}+L_{3}c_{1}c_{2}-L_{4}s_{2}c_{1}c_{3}-L_{4}s_{3}c_{1}c_{2}-L_{5}(s_{ 2}s_{3}c_{1}-c_{1}c_{2}c_{3})c_{3}\\ -L_{5}(-s_{2}c_{1}c_{3}-s_{3}c_{1}c_{2})s_{3}, \tag{1}\]  
\[y=L_{1}s_{1}+L_{3}s_{1}c_{2}-L_{4}s_{1}s_{2}c_{3}-L_{4}s_{1}s_{3}c_{2}-L_{5}(s_ {1}s_{2}s_{3}-s_{1}c_{2}c_{3})c_{3}\\ -L_{5}(-s_{1}s_{2}c_{3}-s_{1}s_{3}c_{2})s_{3}, \tag{2}\]  
\[z=-L_{2}+L_{3}s_{2}-L_{4}s_{2}s_{3}+L_{4}c_{2}c_{3}-L_{5}(-s_{2}s_ {3}+c_{2}c_{3})s_{3}\\ -L_{5}(-s_{2}c_{3}-s_{3}c_{2})c_{3} \tag{3}\]  
The inverse kinematics are derived from above equations using Jacobian and are written as:  
\[J_{v}= \begin{bmatrix}\delta_{1}&\delta_{2}&\delta_{3}\\ \delta_{4}&\delta_{5}&\delta_{6}\\ \delta_{7}&\delta_{8}&\delta_{9}\\ 0&0&1\\ 1&0&0\\ \end{bmatrix}  \tag{4}\]  
where  
\[\delta_{1} =(-L_{1}-L_{3}c_{2}+L_{4}s_{23}-L_{5}c_{2})s_{1}, \tag{5}\] \[\delta_{2} =(L_{1}+L_{3}c_{2}-L_{4}s_{23}+L_{5}c_{2})c_{1},\] \[\delta_{3} =0,\] \[\delta_{4} =-(L_{3}s_{2}+L_{4}c_{23}+L_{5}s_{2})c_{1},\] \[\delta_{5} =-(L_{3}s_{2}+L_{4}c_{23}+L_{5}s_{2})s_{1},\] \[\delta_{6} =L_{3}c_{2}-L_{4}s_{23}+L_{5}c_{2},\] \[\delta_{7} =-L_{4}c_{1}c_{23},\] \[\delta_{8} =-L_{4}s_{1}c_{23},\] \[\delta_{9} =-L_{4}s_{23}\]  
The springs attached at the link \(L4\) of the manipulator applies some torque on the Joint \(J3\). In other words, when the motor attached at the joint \(J3\) does not apply any torque at this joint, the torque produced by the springs will bring the links \(L3\) and \(L5\) closer to each other, resulting in an extending of the manipulator. The springs are attached to link \(L4\) as depicted in Figure 5 (marked in green). The increase in joint angle at the joint \(J3\) will result in stretching of the spring, which in turn will induce counter torque at joint \(J3\). The relation between the joint \(J3\) and spring length \(l_{s}\) is used to understand the spring stretching. In  
Figure 4: Link lengths and joints marked for the designed manipulator. Figure 5a, the set of joints marked in blue and in black are always parallel to each other. The blue dotted line is drawn parallel to the blue links from the spring intersection point at link \(P\)2. The spring is attached between links marked as \(P\)1 and \(P\)2. Using the triangle formed by the green, red and blue dotted lines is used to find the length \(l_{s}\) of the spring. \[\delta_{2} =(L_{1}+L_{3}c_{2}-L_{4}s_{23}+L_{5}c_{2})c_{1},\]0  
The Equation (6) produces a relationship between the angle of the joint \(J\)3 and spring length \(l_{s}\), which is shown in the form of a graph in Figure 5b. A hexarotor airframe is constructed using the above-described design, where the manipulators can reach below, sideways and above the airframe. Since most of the tasks such as landing and grasping requires the manipulators to be at the bottom part of the airframe, the base of the manipulators is attached at the side of the airframe, facing downwards. Although the manipulators can easily reach to the sides of the airframe from this position, to reach above it must pass between two arms of the airframe including the propeller span.