# Cartesian Aerial Manipulator with Compliant Arm
## 3 Modelling
### Positioning Accuracy and Workspace  
It is interesting to compare the workspace, joint limits and position errors of the proposed Cartesian manipulator with compliant joint arm with respect to other kinematic configurations, like the 3-DOF robotic arm described in [19], where the TCP position is given by:  
\[\mathbf{\mathsf{o}}_{\mathbf{r}TCP}^{*}=\left[ \begin{array}{c}r(q_{2},\ q_{3}) \cdot\cos(q_{1})\\ r(q_{2},\ q_{3})\cdot\sin(q_{1})\\ L_{1}\cos(q_{2})+L_{2}\cos(q_{2}+q_{3})\end{array} \right] \tag{4}\]  
Here \(L_{1}\) and \(L_{2}\) are the upper arm and forearm link lengths, respectively, and:  
\[r(q_{2},\ q_{3})=L_{1}\sin(q_{2})+L_{2}\sin(q_{2}+q_{3}) \tag{5}\]  
As stated in the introduction and in Section 2.1, the Cartesian base reduces the propagation of positioning errors in the TCP due to angular deviations (joint deflection) or manufacturing errors (misalignments, clearance). This can be quantitatively evaluated through the Jacobian matrix:  
\[\mathbf{\varepsilon}_{TCP}=\mathbf{J}(\mathbf{q})\cdot\mathbf{\varepsilon}_{\mathbf{q}} \tag{6}\]  
where \(\mathbf{\varepsilon}_{TCP}\) is the position deviation of the TCP in the XYZ axes due to a variation in the vector of joint variables \(\mathbf{\varepsilon}_{\mathbf{q}}\), which depends on the Jacobian of the arm particularized to the nominal pose, \(\mathbf{J}(\mathbf{q})\). Although it is omitted for space reasons, the Jacobian of the manipulator described by Equation (4) introduces coupled terms in the XYZ axes, i.e., each of the Cartesian coordinates depends on the three joint variables simultaneously, whereas in the kinematic configuration considered in this paper, the Jacobian matrix is significantly simplified, resulting in the following expression. As can be seen, the positioning accuracy in the X-axis depends on the accuracy of the X-axis actuator and on the compliant joint, whereas the accuracy on the YZ axes dep  
\[\mathbf{\varepsilon}_{TCP}=\left[ \begin{array}{c}\varepsilon_{Y}\\ \varepsilon_{Y}\\ \varepsilon_{Z}\end{array} \right]=\left[ \begin{array}{ccc}1&0&L\cdot\cos(q_{3 })\\ 0&1&0\\ 0&0&L\cdot\sin(q_{3})\end{array} \right]\cdot\left[ \begin{array}{c}\varepsilon _{d_{1}}\\ \varepsilon_{d_{2}}\\ \varepsilon_{q_{3}}\end{array} \right]=\left[ \begin{array}{c}\varepsilon_{d_ {1}}+L\cdot\cos(q_{3})\cdot\varepsilon_{d_{3}}\\ \varepsilon_{d_{2}}\\ L\cdot\sin(q_{3})\cdot\varepsilon_{q_{3}}\end{array} \right] \tag{7}\]  
ends on the accuracy of the \(d_{2}\) and \(q_{3}\) coordinates. The volume of operation of the Cartesian manipulator is a parallelepiped of length \(D_{1}=0.5\) [m], depth \(D_{2}=0.2\) [m], and height \(L=0.2\) [m], with two additional circular sections generated by the revolution of the compliant joint with maximum rotation. \(q_{3}^{max}\) is around 120\({}^{\circ}\):  
\[V=D_{1}D_{2}L+D_{2}L^{2}q_{3}^{max}=0.037\left[\text{m}^{3}\right] \tag{8}\]  
whereas the volume of operation of a 3-DOF robotic arm corresponding to a hollow semi-sphere with upper arm link length \(L_{1}\) = 0.2 [m] and forearm link length \(L_{2}\) = 0.2 [m] is given by:  
\[V^{*}=\frac{4}{3}\pi\left[(L_{1}+L_{2})^{3}-R_{min}^{3}\right]=0.234\left[ \text{m}^{3}\right] \tag{9}\]  
This is represented graphically in Figure 8, and compared with respect to the workspace of the 3-DOF manipulator described in [19]. The hollow area is determined by the maximum rotation angle of the elbow joint, \(q_{3}^{max}\), defining a minimum radius \(R_{min}\) from the origin of the manipulator frame to the end effector. Although the difference is significant, the dexterity and performance of configurations with revolute joints are typically reduced when integrated in multirotor platforms due to the motion constraints associated with the landing gear and the effect of the kinematic singularities when the arm tends to be stretched. The effective workspace of the Cartesian prototype described here can be extended by exploiting the rotation of the multirotor in the yaw angle, equal to \(V=0.13\)\(\left[\text{m}^{3}\right]\) for a full rotation.