# Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot
## 6 Simulation Results  
The set-point motion is composed of six phases. The start position \(\mathbf{x}_{ref}\) corresponds to the robot arm semi-bent with internal coordinates \(\theta_{ref}\) = [\(-\)45\({}^{\circ}\), 90\({}^{\circ}\), 0\({}^{\circ}\), \(-\)45\({}^{\circ}\)]\({}^{T}\). In each phase: the end-effector moves in a time \(T\) from \(\mathbf{x}_{ref}\) along a straight line with length \(d\); it stops at \(\mathbf{x}_{ref}\) + \(\Delta\mathbf{x}\) for \(T_{\mathrm{stop}}\); it returns to \(\mathbf{x}_{ref}\) along the same straight line; finally, it stops at \(\mathbf{x}_{ref}\) for \(T_{\mathrm{stop}}\); then, the next phase begins. In the six phases, the displacements \(\Delta\mathbf{x}\) are: (\(d\), 0, 0); (0, 0, \(d\)); (\(-\)\(d\), 0, 0); (0, \(-\)\(d\), 0); (0, \(0\), \(-\)\(d\)); consequently, all the linear displacements are along one axis of O(\(x\),\(y\),\(z\)). In each phase, there are two linear motions with opposite directions, characterized by the same duration \(T\) and displacement \(d\). According to the definition of the dimensionless position and dimensionless time discussed in Section 4, these two parameters are used to scale in time and position the dimensionless profiles of Figure 5.  
The end-effector rotation is maintained constant, since in robots with Schoenflies' motion, the vertical rotation is dynamically decoupled from the translations, and in the following, the analysis is focused on the first three external coordinates. Figures 6-14 show the comparison of the motion profiles for the discussed six-phase motion with \(d\) = 0.15 m, \(T\) = 0.25 s, and \(T_{\mathrm{stop}}\) = 3/5T = 0.15 s. In particular:  
* Figures 6-8 represent, respectively, the \(x\), \(y\) and \(z\) end-effector errors (\(e_{x}=x_{d}-x\), \(e_{y}=y_{d}-y\), \(e_{z}=z_{d}-z\));
* Figures 9-11 show, respectively, the \(x\), \(y\) and \(z\) end-effector accelerations \(a_{x}\), \(a_{y}\), and \(a_{z}\);
* Figures 12-14 represent, respectively, the torques \(\tau_{1}\), \(\tau_{2}\), and \(\tau_{3}\). \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Symbol** & **Parameter** & **Value** & **Unit** \\ \hline \(l_{0}\) & link 0 length & 900 & mm \\ \(l_{1}\) & link 1 length & 330 & mm \\ \(l_{3}\) & links 3–4 length & 330 & mm \\ \(l_{6}\) & distance along \(z\) between point E and link 1 & 387 & mm \\ \(l_{G1}\) & C.O.M. position of link 1 (Figure 2) & 165 & mm \\ \(l_{G3}\) & C.O.M. position of link 3 (Figure 2) & 165 & mm \\ \(l_{G4}\) & C.O.M. position of link 4 (Figure 2) & 165 & mm \\ \(m_{1}\) & link 1 mass & 10 & kg \\ \(m_{2}\) & link 2 mass & 5 & kg \\ \(m_{3}\) & link 3 mass & 5 & kg \\ \(m_{4}\) & link 4 mass & 5 & kg \\ \(m_{5}\) & link 5 mass & 12 & kg \\ \(m_{6}\) & link 6 mass & 3 & kg \\ \(\Theta_{ref}\) & internal coordinates of the reference position & [\(-\)45, 90, 0, \(-\)45] & \({}^{\circ}\) \\ \(\mathbf{x}_{ref}\) & external coordinates of the reference position & [467, 0, 513, 0] & mm, \({}^{\circ}\) \\ \(\theta_{3p}\) & neutral \(\theta_{3}\) angle of the balancing spring & \(-\)15 & \({}^{\circ}\) \\ \(k_{3}\) & stiffness coefficient of the balancing spring & 247.3 & Nm/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters and mass properties considered in the simulations. Figure 8: Comparison of motion profiles: external coordinate error \(e_{z}\) [m]. Figure 6: Comparison of motion profiles: external coordinate error \(e_{x}\) [m]. Figure 7: Comparison of motion profiles: external coordinate error \(e_{y}\) [m]. Figure 11: Comparison of motion profiles: end-effector acceleration \(a_{z}\) [m/s\({}^{2}\)]. Figure 10: Comparison of motion profiles: end-effector acceleration \(a_{y}\) [m/s\({}^{2}\)]. Figure 9: Comparison of motion profiles: end-effector acceleration \(a_{x}\) [m/s\({}^{2}\)]. The torque \(\tau_{4}\) is not represented since it is null in the absence of friction and with constant end-effector orientation \(\theta\), as in the considered motion.