# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 5 Experimental Results  
To further verify the practicality of the proposed method, experiments were conducted on a linear-motion stage manufactured by PARADOX Inc., which is commonly used in production equipment such as bonding machines, transfer robots, inspection instruments, and dispensers. A stainless-steel beam was attached to the base to imitate the flexible behavior of the tool side. During operation, high-speed movements of the stage base can induce vibrations in the beam along the motion direction that need extra standby time to dissipate at the end positions. Minimizing the induced vibrations is crucial for improving the positioning accuracy and efficiency. The motion stage was controlled by a host computer through a motion control board (GHN, Googoltech) connected via the glLink-II bus. The optimization algorithms were programmed and implemented on the host computer, and the resulting motion profiles were subsequently sent to the control board as the reference input to drive the motor. To measure the vibration accelerations at the beam tip, an accelerometer (1B103, DHTEST) was utilized, with a sampling frequency of 5.12 kHz. The experiment platform is depicted in Figure 21.  
In the experiment, the motion stage was required to travel a distance of 0.3 m, and the actuator limits were specified as follows: \(V_{\text{max}}=0.4\) m/s, \(A_{\text{max}}=2\) m/s\({}^{2}\), and \(J_{\text{max}}=20\) m/s\({}^{3}\). The beam was characterized by a natural frequency of 8.81 Hz and a damping ratio of 0.008. The desired positioning accuracy at the target point was set to \(\pm\) 0.2 m/s\({}^{2}\). The proposed optimized sinusoidal jerk profile, as well as other baseline methods in previous studies [16; 23], were tested. Figure 22 shows the actual position profiles of the base read from the linear encoder and the velocity profiles deduced from the derivation of the actual positions. It can be seen that the actual profiles of the base are basically consistent with the reference design and accurately reach the desired positions. Figure 23 illustrates the measured vibration accelerations corresponding to different profiles. An additional test was performed to verify the robustness of the profiles by attaching an extra mass at the beam tip, causing its natural frequency to change from 8.81 Hz to 7.55 Hz, which represents  
Figure 20: Magnitude of residual-vibration envelope for \(\pm\)10% variation in actual natural frequency with optimized sinusoidal jerk profiles. Figure 26: Power spectral density (PSD) curves of vibration signals induced by different motion profiles [16; 23]. Figure 24: Comparison of vibration accelerations induced by different motion profiles with modeling error [16; 23]. Figure 25: Comparison of vibration accelerations induced by optimized sinusoidal jerk profiles with different robustness degrees. From the analysis in Figures 22-26 and Table 9, it is possible to outline the following points:  
* Starting with the visual impression in Figure 22, the sinusoidal jerk profile guarantees good smoothness of movement. Due to its continuity up to the jerk level, the adverse influence of high-frequency harmonics can be eliminated;
* As shown in Figure 23, the classic trapezoidal velocity profile, characterized by acceleration discontinuities and infinite jerk, offers the minimum execution time. However, it induces the highest vibration level and, consequently, the longest settling time. By adjusting the acceleration time, the technique in [16] significantly reduced the vibration to 6.5% of the initial level. The proposed sinusoidal jerk profile yields the lowest residual-vibration amplitude, achieving improvements of 59.3% and 77.4% compared to the profiles in [16] and [23] when \(r=1\), respectively;
* The employed profile optimization introduces a slight delay in the reference motion completion time. However, this delay is more than compensated for by the reduced settling time, as the beam reaches a complete standstill state earlier, effectively increasing the operational productivity.