s26030829.pdf
Academic Editor: Zheng Chen Received: 19 December 2025 Revised: 21 January 2026 Accepted: 22 January 2026 Published: 27 January 2026 Copyright: © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an…
Academic Editor: Zheng Chen
Received: 19 December 2025
Revised: 21 January 2026
Accepted: 22 January 2026
Published: 27 January 2026
Copyright: © 2026 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license.
Article
Wafer Handing Robotic Arm Vibration Trajectory Planning
Based on Graylag Goose Optimization
Yujie Ji * and Peiyan Hu
School of Mechanical Engineering, Shenyang Ligong University, Shenyang 110159, China
* Correspondence: jyj19700521@163.com
Abstract
In contemporary semiconductor manufacturing, wafer-handling robots are essential for
achieving high-speed and high-precision wafer transportation. However, the demand for
rapid motion and lightweight design introduces flexible transmission components that are
prone to residual vibrations, which degrade positioning accuracy and system stability. To
address this challenge, this paper proposes a vibration-suppression trajectory planning
method based on the Gray Goose Optimization (GGO) algorithm. The proposed algorithm
integrates grouped global search with local optimization capabilities, making it well suited
for solving multi-objective optimization problems. Comparative tests conducted on eight
randomly selected multimodal benchmark functions from the CEC2013 test suite verify
the effectiveness and robustness of the GGO algorithm. Establishing a multi-objective
function that considers both motion time and vibration energy enables the GGO algorithm
to determine the switching time points of an S-shaped velocity profile, thereby generating
smooth trajectories with continuous velocity and acceleration. By varying different initial
conditions, the trade-off between motion time and vibration energy is systematically ana-
lyzed with respect to angular displacement, initial acceleration, and time-weighting factors.
Simulation results indicate that the planned trajectories exhibit negligible displacement vari-
ation under zero-mean disturbances. The velocity error remains within 0.1 deg
·
s
−1
, and the
acceleration error is confined within 0.2 deg
·
s
−2
. Consequently, Pareto-optimal solutions
are successfully obtained with respect to both motion time and residual vibration energy.
Keywords: wafer handing robotic arm; trajectory planning; Graylag Goose Optimization
1. Introduction
The wafer transport robotic arm is an important actuator in semiconductor manufac-
turing equipment, and its performance directly affects production throughput and yield.
As semiconductor process nodes continue to shrink, stringent requirements are placed
on the robotic arm’s positioning accuracy, stability, and operational efficiency [
1
–
3
]. Tra-
ditional time-optimal trajectory planning methods often focus on achieving the fastest
point-to-point motion of the robotic arm’s end effector while neglecting the issue of residual
vibrations at the end effector caused by the flexible characteristics of the transmission sys-
tem. Such vibrations can lead to a decrease in the positioning accuracy of the robotic arm’s
end effector, and in severe cases, may cause wafer displacement or breakage, resulting in
reduced productivity and significant economic losses [
4
,
5
]. Therefore, efficient and precise
vibration-suppression trajectory planning has become one of the core technologies in the
development of advanced wafer robots.
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The vibration suppression methods for robotic arms can be categorized into passive
and active vibration suppression. Passive vibration suppression primarily involves op-
timizing the topology to improve mass distribution, using lightweight, stiff materials to
enhance the equivalent stiffness, and incorporating damping devices and energy-absorbing
components to suppress vibrations through physical mechanisms. These passive vibration
suppression schemes do not require external energy input, are cost-effective, but have
limited adaptability to time-varying operating conditions, making them difficult to meet
the requirements of wafer transport robotic arms [
6
]. Active vibration suppression, on the
other hand, reduces the vibrations of the robot by improving control strategies, thereby en-
hancing motion precision and stability. PID control, with its simple and intuitive structure
and low implementation difficulty, is widely used in the motion control of industrial robots
to suppress elastic vibrations in robotic arms. Yang C et al. employed PID fuzzy control to
suppress elastic vibrations in flexible robotic arms, improving the positioning and control
accuracy of piezoelectric flexible arm ends [
7
]. Guo et al. proposed a tracking control sys-
tem integrating an incremental PID controller and a sliding mode robust controller, which
demonstrated stronger accuracy, stability, and anti-interference capabilities compared to
traditional PID control [
8
]. In addressing similar robotic arm control problems, Wazzan
utilized PID sliding mode control combined with a genetic algorithm to tune controller
parameters, effectively improving the performance of the robotic arm’s PID controller [
9
].
Despite the simplicity and intuitiveness of these control methods, these controllers are
often optimized based on specific models, and their control parameters are difficult to meet
the requirements for high precision and stability, especially when dealing with complex
robotic arm structures or variable working environments.
Recently, researchers have commonly employed vibration suppression trajectory op-
timization to address the limitations of both passive and active vibration suppression,
providing a more cost-effective and universally applicable solution for robotic arm vibra-
tion control. This approach has become one of the key research focuses in the field of
robotic arm control [
10
]. These methods formulate trajectory planning as a multi-objective
or constrained optimization problem by simultaneously considering system dynamics,
motion time, energy consumption, joint smoothness, and end-effector vibration, and they
commonly introduce metaheuristic optimization algorithms to search for optimal solutions.
These optimization algorithms, inspired by natural phenomena and biological behaviors,
offer new approaches for solving trajectory planning and constrained optimization prob-
lems of this nature [
11
]. The development of metaheuristic optimization algorithms can be
broadly divided into two directions: global search and local optimization. Among them,
the Genetic Algorithm (GA) is widely used in various global search optimization problems
due to its strong capability in handling complex constraints, combinatorial optimization,
and global exploration. Xu Qiang et al. proposed a genetic algorithm combined with
a simulated annealing mechanism, which further improved the optimization accuracy,
shortened the robotic arm’s motion time, and enhanced its work efficiency [
12
]. Meanwhile,
Kheshti et al. applied genetic algorithms to adjust the gain optimization of sliding mode
controllers, reducing the time required for a manipulator arm to complete a predefined mo-
tion trajectory [
13
]. However, the computational model of genetic algorithms is relatively
complex, and its use has certain limitations. In contrast, Particle Swarm Optimization (PSO)
has a simpler algorithm structure and stronger local optimization capability. In this context,
Wu N et al. proposed an improved particle swarm optimization algorithm (RTPPSO) to
optimize the robotic arm’s joint angles or path. By introducing an adaptive weight strategy,
they significantly improved the robotic arm’s work efficiency [
14
]. However, when solving
complex problems with multiple dimensions, strong constraints, and strong coupling,
PSO is prone to accuracy degradation and insufficient global search capability due to
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the influence of random disturbances [
15
–
17
]. Researchers have proposed various new
optimization algorithms and introduced different solution strategies to address the short-
comings of these two algorithms. Yue et al. applied a gray wolf optimization algorithm with
a probabilistic disturbance strategy to solve the time-optimal problem. This optimization
strategy effectively balanced the weight between local optimization and global search [
18
].
However, the gray wolf population may converge around suboptimal individuals, leading
to premature convergence issues [
19
,
20
]. Pu Q et al. introduced an improved wild dog
optimization algorithm for robotic arm time-impact optimization [
21
,
22
], but they did not
perform tests for solving high-dimensional problems using the wild dog optimization
algorithm. Additionally, to improve global search capability and address the issue of search
direction disturbance, Lv Yun peng et al. proposed an inverse kinematics solver based on
a multi-strategy improved dung beetle optimization algorithm. However, this improved
algorithm has not been applied to the multi-objective vibration optimization direction [
23
].
Widyianto Agus et al. addressed the issue of the shipboard two-degree-of-freedom robotic
arm being affected by environmental factors and proposed a real-time tuning of PD con-
trol parameters based on an ant colony algorithm, significantly improving the stability of
shipboard equipment [
24
]. Abro et al. combined ant colony optimization’s pheromone
learning with the Denavit–Hartenberg (DH) method to develop a robotic arm error model,
further enhancing the accuracy of global search and improving the reliability and accuracy
of industrial robotic systems [
25
]. However, the ant colony algorithm is heavily depen-
dent on the initial model, making it difficult to avoid local optimal solutions. The Whale
Optimization Algorithm (WOA) introduced a random spiral search step, eliminating the
need for crossover and mutation operators, with a simpler program structure. Wang Fang
et al. applied a hybrid whale optimization algorithm to optimize the time-impact optimal
trajectory of industrial robotic arms, using fifth-order B-splines for interpolation planning
of motion trajectories in joint space [
26
]. However, the performance of this algorithm signif-
icantly deteriorates in a multi-constraint environment and requires further integration with
various improvement strategies [
27
]. The Graylag Goose Optimization (GGO) algorithm
introduced a group migration strategy [
28
], demonstrating strong search capabilities in
multi-objective problems and efficient local optimization when facing suboptimal solu-
tions. Ashish Sharma et al. conducted eight multi-objective benchmark tests, with the
results showing that the Multi-Objective Graylag Goose Optimization (MOGGO) algorithm
exhibited excellent performance in terms of convergence [29].
In summary, this paper introduces the GGO algorithm into the vibration suppression
trajectory optimization problem of wafer transport robotic arms, providing a new feasible
solution for multi-objective trajectory optimization. This study constructs the end dynamics
equation of the flexible lower arm and integrates a multi-objective optimization model
involving motion time, energy, and impact. Based on the multi-objective GGO algorithm,
the trajectory parameters are solved, aiming to generate a set of Pareto-optimal trajectories
with superior overall performance, offering a novel and efficient solution to improve the
performance of the wafer transport system.
2. Establishment of the Dynamics Model for Wafer Transport
Robotic Arm
In this study, the wafer transport robotic arm mainly consists of two links [
30
,
31
]. The
upper (first) link is rigidly connected to the motor, while the lower (second) link is driven
by another motor via a flexible steel belt inside the upper link, which can be regarded as a
flexible motion joint. During motion, only the bending stress of the steel belt is considered.
The end-effector is coupled to the posture of the upper link through the steel belt embedded
inside the lower link. As shown in Figure 1, q1 and q2 denote the rotational joint angles
https://doi.org/10.3390/s26030829