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Sensors 2026 , 26 , 829 17 of 21 output module is constructed to generate the r eference pr oﬁles of the joint angle, angular velocity , and angular acceleration. Meanwhile, a random disturbance module is connected exter…

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Figure 16. Acceleration–time proﬁles for different time weight.

Table 7 shows that the sensitivity of both the total operation time and the vibration

energy index to w1 is relatively low, and the current optimization results therefore lie in an

approximately ﬂat Pareto region. Overall, although increasing the time weight reduces the

operation time, it may induce higher vibration energy, implying greater instability and/or

energy consumption.

6. Simulation Analysis Based on Simulink

In this study, a Simulink-based trajectory simulation platform was Established. The

optimal trajectory parameters obtained from the optimization algorithm were imported

into the platform to simulate and evaluate the manipulator’s stability under external

disturbances. The external disturbance shown in Figure 17 is modeled as a bounded random

torque applied to the joint actuator to emulate unmodeled dynamics and environmental

perturbations in wafer-handling operations. The disturbance is deﬁned as:

(t) = τ

max

ξ(t), ξ(t) ∈ [−0.1, 0.1] (20)

where

(t) is a zero-mean bounded random process and

max

denotes the disturbance

bound. In the Simulink implementation,

(t) is generated using a random source (e.g., a

band-limited noise or uniform random sequence) and then constrained by saturation to

ensure |

(t)|

≤ τ

max

. This bounded setting provides a physically meaningful robustness

test while avoiding unrealistically large excitation.

Figure 17. Robotic Arm Simulation System Flowchart.

This represents unmodeled dynamics and external perturbations typically encountered

in industrial wafer-handling environments.

To control the robot motion, the Simulink control program ﬁrst initializes the system

by setting the manipulator coordinates to the prescribed initial state [

]. Based on the

planned trajectory parameters obtained from the preceding simulations, a smooth-function

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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-

https://doi.org/10.3390/s26030829