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Sensors 2026 , 26 , 829 14 of 21 Figure 11. V elocity-time proﬁles for dif ferent start acceleration. Figure 12. Displacement–time proﬁles for dif ferent start acceleration. Figure 13. Acceleration–time proﬁles for dif f…

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Displacement/deg

Figure 9. Displacement-time proﬁles for different planned displacements.

Figure 10. Acceleration-time proﬁles for different planned displacements.

As indicated in Table 5, the residual vibration energy is strongly inﬂuenced by the total

angular displacement. When the angular change increases from 60

◦

to 90

◦

, the residual

vibration energy rises by 103.28%. Therefore, constraining the forearm angular variation to

within 60

◦

can signiﬁcantly suppress vibrations induced by the ﬂexible joint.

(2) To investigate the inﬂuence of the start-up acceleration on the motion performance

and vibration characteristics of the wafer-transfer manipulator, comparative experiments

were conducted by varying the start-up acceleration a while keeping the other control

parameters constant. The results are listed in Table 6 and Figures 11–13.

Table 6. Variation in Objective Function Values of Wafer Transfer Robot Arm at Different Startup Acceleration.

Startup Acceleration Runtime Vibration Energy Index E

vib

−5 12.6952 32.8125

10 13.9619 131.2500

20 8.2476 525.0000

40 5.6347 2149.8891

80 2.0144 9029.1039

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Figure 11. Velocity-time proﬁles for different start acceleration.

Figure 12. Displacement–time proﬁles for different start acceleration.

Figure 13. Acceleration–time proﬁles for different start acceleration.

As shown in Table 6, as a increases from 5 to 80, the overall operation time exhibits

a decreasing trend, whereas the vibration energy index increases rapidly in a nonlinear

manner. These results demonstrate that a larger start-up acceleration is beneﬁcial for

reducing the operation time and improving transfer efﬁciency. However, excessive start-up

acceleration introduces stronger impact loads and excites high-frequency modes of the

manipulator, leading to a substantial increase in vibration. Hence, a typical trade-off exists

between efﬁciency and vibration suppression. Considering both efﬁciency and stability,

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a start-up acceleration of approximately a

≈

20 deg/s

can be regarded as a compromise

range that balances operation time and vibration mitigation, and this range is advantageous

for subsequent optimization and analysis.

(3) With the other control parameters unchanged, comparative experiments were

performed by varying the time-weighting factor w1, the results are shown in Table 7

and Figures 14–16. The experimental data indicate that, under the given constraints and

operating conditions, increasing w1 gradually reduces the total time obtained from the

objective function. When w1 > 0.9, the trajectory duration decreases markedly, while the

vibration energy index increases sharply.

Table 7. Variation in Objective Function Values of Wafer Transfer Robot Arm at Different Time Weight.

Time Weight Runtime Vibration Energy Index E

vib

0.1 22.0667 165.5172

0.3 22.0666 165.5172

0.5 21.4938 170.589

0.7 19.583 190.022

0.9 16.1414 239.1657

1.0 4.1635 69,522.553

Figure 14. Velocity-time proﬁles for different time weight.

Displacement/deg

Figure 15. Displacement–time proﬁles for different start acceleration time weight.

https://doi.org/10.3390/s26030829