# Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot
## 6 Simulation Results
The maximum absolute values of \(e_{x}\), \(e_{y}\), \(e_{z}\), \(a_{x}\), \(a_{y}\), \(a_{z}\), \(\tau_{1}\), \(\tau_{2}\), and \(\tau_{3}\) are collected in Table 2.  
Figure 14: Comparison of motion profiles: actuation torque \(\tau_{3}\) [Nm]. Figure 12: Comparison of motion profiles: actuation torque \(\tau_{1}\) [Nm]. Figure 8:0 Comparison of motion profiles: actuation torque \(\tau_{2}\) [Nm]. In all the figures starting from Figure 8:1  
* continuous lines represent the results obtained with the KD controller, and dashed lines the results obtained with the KDHD controller;
* the colors of the graphs indicate the motion profile, with the same coding used in Figure 8:2  
From the analysis of Figures 6-14 and of Table 2 it is possible to outline the following'conclusions:  
* once a motion profile is selected, the performance differences between the KD and KDHD controllers are not remarkable: as a matter of fact, all the graphs with continuous lines (KD) are qualitatively similar to the graphs with dashed lines (KDHD) of the corresponding color; as it will be discussed in the following, the benefits of the FO controller are higher for faster motions (see Figures 15-18);
* as in SISO systems [10], the behaviors of the robot controlled by using the EJ, SJ and MS] profiles are quite similar; nevertheless, observing the values of Table 2, it is possible to notice that the EJ profile performs slightly better, having lower maximum absolute values of the errors \(e_{x}\), \(e_{y}\), and \(e_{z}\) than the SJ and MS] profiles, but with similar values of the actuation torques \(\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\); also the acceleration values are slightly lower adopting the EJ profile rather than the SJ or the MS] law;
* the TV, TA and CY laws allow to obtain lower errors; nevertheless, these profiles are characterized by jerk discontinuities [10], and this may cause vibrational phenomena which are not evidenced by the considered rigid body model; in particular, the TV profile has higher order discontinuities, since it has discontinuous acceleration and infinite jerk, while the TA and CY profiles have discontinuous but not infinite jerk; this causes oscillations of the actuation torques and of the end-effector accelerations with the TV profile (Figures 9-14). Figure 8:3 Maximum values of end-effector accelerations \(a_{x}\), \(a_{y}\), \(a_{z}\) with different motion profiles (percentage ratios with respect to the elliptic jerk profile case). Figure 8:4 Maximum values of the actuation torques \(\tau_{1}\), \(\tau_{2}\), and \(\tau_{3}\) with different motion profiles (percentage ratios with respect to the elliptic jerk profile case). Figure 8:5 Maximum values of the \(x\), \(y\) and \(z\) end-effector errors with different motion profiles (percentage ratios with respect to the elliptic jerk profile case). In order to investigate the effects of the motion speed, different simulations have been carried out with the same control parameters and with the same trajectory parameters of the results reported in Figures 9-14 and in Table 2, but performing the motion at different speeds. This has been obtained by varying the motion duration \(T\), which in the previous simulations was equal to 0.2 s, in the range between 0.1 s and 0.5 s, while maintaining the ratio \(T_{\text{stop}}/T=3/5\). Considering these assumptions, Figures 15-17 represent the maximum absolute values of Table 2 (external coordinate errors, end-effector accelerations and actuation torques) as functions of the motion duration \(T\). Moreover, Figure 18 shows, as overall indices of the error and of the actuation effort, the Integral Square Error of the end-effector translational coordinates (\(ISE\)) and the Integral Control Effort (\(ICE\)), defined according to the following expressions:  
Figure 8:6  
where \(T_{sim}\) is the simulation time, sufficient to evaluate adequately the residual vibrations.