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Sensors 2026 , 26 , 829 18 of 21 Figure 20. Robotic Arm Acceleration Curve. As indicated by the simulation results in Figure 18 , the planned joint angular displace- ment proﬁle of the forearm joint is in close agreement…

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output module is constructed to generate the reference proﬁles of the joint angle, angular

velocity, and angular acceleration. Meanwhile, a random disturbance module is connected

externally to provide an excitation torque, which is applied as the external input torque

the dynamic planning module. A closed-loop control function is then established according

to the governing dynamic equations. The output angular acceleration is integrated twice to

obtain the simulated angular velocity and angular displacement, and the resulting data

are exported to spreadsheets. Furthermore, inverse kinematics is employed to inversely

infer the effect of the external disturbance and back-calculate the corresponding driving

torque, thereby completing the closed-loop simulation test module [

]. In this study,

angular displacement, angular velocity, angular acceleration, and joint torque are adopted

as evaluation metrics to assess the time-planning performance and vibration-suppression

capability of the wafer-transfer robot using the Simulink test platform. The integrated

simulation platform is executed, and the Scope outputs are exported to the workspace via

the To Workspace function, yielding the simulation results shown in Figures 18–20.

0 5 10 15 20 25 30 35

Time (s)

100

120

140

Displacement/deg

displacement curv

Simulated angular displacement

Planned angular displacement

Figure 18. Robotic Arm Displacement Curve.

Figure 19. Robotic Arm Velocity Curve.

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Figure 20. Robotic Arm Acceleration Curve.

As indicated by the simulation results in Figure 18, the planned joint angular displace-

ment proﬁle of the forearm joint is in close agreement with the simulated tracking angular

displacement over the entire motion interval. The two curves almost overlap in the initial

stage, the acceleration and constant-velocity transition stage, and the ﬁnal deceleration

stage, exhibiting an overall smooth S-shaped trajectory. In terms of tracking performance,

the tracking angular displacement shows no evident overshoot or oscillation throughout the

motion, and the steady-state error at the end is virtually negligible, demonstrating that the

system achieves favorable dynamic response characteristics and high steady-state accuracy.

As shown by the simulation results in Figure 19, the planned angular velocity of

the forearm joint agrees well with the simulated tracking angular velocity over the entire

motion interval, with only a negligible deviation observed in the vicinity of the peak

velocity. No evident overshoot or oscillation is observed, indicating that the controller can

accurately track the desired angular-velocity command and that the system exhibits high

velocity-tracking accuracy and stability. Such a smooth velocity proﬁle is beneﬁcial for

reducing vibration levels in the joint and links, and it also prevents inertial impacts induced

by abrupt acceleration/deceleration, which is crucial for maintaining posture stability and

operational safety during wafer handling.

Furthermore, the planned and tracked angular accelerations of the forearm joint

remain highly consistent throughout the motion, as shown in Figure 19. Only minor local

high-frequency ﬂuctuations are present, and no abrupt spikes or severe reverse jumps

are detected. Therefore, the proposed planning method can fundamentally suppress

structural vibrations of the manipulator, thereby supporting the stable operation of the

wafer-transfer robot.

Based on the simulation results shown in Figures 18–20, the vibration suppression per-

formance of the proposed trajectory planning method is further validated. Compared with

conventional polynomial-based trajectory planning approaches, the planned joint angular

displacement, velocity, and acceleration proﬁles exhibit smooth transitions without evident

oscillations or abrupt ﬂuctuations. In particular, the angular-displacement ﬂuctuation is

constrained within 0.01 deg/s, quantitatively indicating effective disturbance rejection and

enhanced dynamic smoothness, thereby supporting the vibration-suppression effect of the

proposed jerk-minimization trajectory planning strategy.

Moreover, by introducing the proposed improved multi-objective Gray Goose Opti-

mization (MOGGO) algorithm, the solution quality of the dual-objective vibration suppres-

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sion trajectory optimization problem is signiﬁcantly enhanced, beneﬁting from improved

population diversity and global search capability.

To further evaluate the robustness of the proposed method, random external torque

disturbances are introduced into the system. Under such disturbances, the tracking re-

sponses remain highly consistent with the planned trajectories, and no signiﬁcant overshoot

or residual oscillation is observed during acceleration, constant-velocity, or deceleration

phases. In particular, the angular velocity ﬂuctuation is effectively constrained within a

small range, indicating that high-frequency vibration components are signiﬁcantly sup-

pressed. These results conﬁrm that minimizing the jerk-based vibration energy index leads

to improved dynamic smoothness and enhanced vibration suppression capability of the

wafer-handling robotic arm.

7. Conclusions

This paper addresses the micro-vibration problem of wafer transfer robotic arms and

proposes and validates a vibration suppression trajectory optimization method based on the

Graylag Goose Optimization (GGO) algorithm. After theoretical derivation and simulation

tests, the following conclusions are drawn:

(1) This study established and analyzed the dynamic model of the wafer transfer

robotic arm based on the Lagrange method, providing a theoretical foundation for vi-

bration suppression trajectory planning. The model fully considers the ﬂexible dynamic

characteristics of the robotic arm and accurately describes the system’s dynamic behavior.

(2) By combining a multi-segment polynomial trajectory planning method with a

residual vibration energy dissipation equation, a multi-objective optimization function

targeting both time-optimal performance and vibration energy was constructed. By rea-

sonably setting weight coefﬁcients according to working conditions, a multi-objective

coordinated optimization was achieved to ensure trajectory smoothness and minimal

residual vibration energy.

(3) Simulation results show that when solving the dual-objective vibration suppression

trajectory optimization, introducing the proposed improved multi-objective Gray Goose

Optimization (MOGGO) algorithm effectively solves the dual-objective problem. The

results indicate that the derived trajectory, when subjected to random external disturbances,

maintains speed ﬂuctuations basically within 0.01 deg/s, demonstrating good vibration

suppression performance.

However, this study only performed Pareto optimization for two objectives and has not

been extended to higher-dimensional, multi-objective optimization problems. Moreover, it

is limited to SIMULINK and MATLAB simulation environments. Future work will expand

to validation on a PLC platform and further explore the algorithm’s applicability under

more complex operating conditions and with additional optimization objectives.

Author Contributions: Conceptualization, Y.J.; Software, P.H.; Formal analysis, P.H.; Writing—

original draft, P.H.; Writing—review & editing, P.H.; Supervision, Y.J. All authors have read and

agreed to the published version of the manuscript.

Funding: This research was funded by the Basic Scientiﬁc Research Project of Higher EducationInsti-

tutions of Liaoning Province (grant number JYTQN2023064).

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The original contributions presented in this study are included in the

article. Further inquiries can be directed to the corresponding author.

Conﬂicts of Interest: The authors declare no conﬂicts of interest.

https://doi.org/10.3390/s26030829