# Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot
## 6 Simulation Results
In Figures 15-18, all the performance parameters are expressed as percentage ratios of one parameter obtained with a specific motion law with respect to the same parameter obtained with the EJ profile. From the observation of these graphs, it is possible to outline the following conclusions:  
* the benefits of the KDHD controller over the KD controller are greater for faster motions, with lower motion duration \(T\), and using motion laws with higher discontinuities, in particular, TV and CY; for \(T=0.1\) s the percentage reduction in the maximum \(x\), \(y\) and \(z\) end-effector errors using the FO controller are, respectively, \(-5.1\%\), \(-7.8\%\) and \(-5.7\%\) with the TV profile, and \(-0.9\%\), \(-3.4\%\) and \(-5.8\%\) with the CY profile (Figure 15);
* also considering the \(ISE\), for \(T=0.1\) s the percentage reduction with the KDHD controller is \(-3.4\%\) with the TV profile and \(-1.5\%\) with the CY profile (Figure 18, left);
* this better accuracy is obtained even with a lower overall control effort: for \(T=0.1\) s the \(ICE\) reduction with the KDHD controller is \(-6.7\%\) for the TV profile and \(-2.1\%\) for the CY profile (Figure 18, right);
* on the other hand, if \(T\) increases (slower motions) the advantage of the KDHD controller over the KD in terms of \(ICE\) decreases gradually (Figure 18, right); for \(T=0.5\) s  
Figure 8:7 Integral Square Error and Integral Control Effort with different motion profiles (percentage ratios with respect to the elliptic jerk profile case). the _ICE_ is almost equal and the differences in terms of maximum absolute and integral square errors are below 1%, with a slight advantage for the IO controller;
* focusing on the comparison among the motion profiles, it is possible to see that, with the considered rigid body modeling, the profiles with higher discontinuities (TV, TA, CY) have in general lower error, both in terms of maximum absolute values (Figure 15) and of _ISE_ (Figure 18, left), independently of the motion speed, but also higher torque peaks for \(\tau_{2}\), up to +18%, and for \(\tau_{3}\), up to +47% (Figure 17);
* limiting the comparison to the smoother profiles (EJ, SJ, MSJ), more suitable for real systems, the EJ profile performs better than the other two, in particular with respect to the SJ: the increase in maximum absolute error with the SJ profile with respect to the EJ is up to +3.3% for all the three coordinates (Figure 15), while the increase in _ISE_ is +3.8% (Figure 18, left); the error increase with the MSJ profile with respect to the EJ is lower, up to +0.8% for the maximum absolute errors and up to +0.7% for the _ISE_;
* it is interesting that the EJ profile allows obtaining lower error with lower control effort: the maximum _ICE_ increase with respect to EJ is +8.2% with the SJ law for \(T\) = 0.1, and +1.3% with the MSJ law, for \(T\) = 0.16 (Figure 18, right); as regards the maximum values of the actuation torques (Figure 17), they are higher for the SJ and MSJ profiles with respect to the EJ in almost all the range of motion duration \(T\).