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Sensors 2026 , 26 , 829 3 of 21 the inﬂuence of random disturbances [ 15 – 17 ]. Researchers have proposed various new optimization algorithms and intr oduced differ ent solution strategies to address the short- comings …

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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 [

]. 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 [

]. 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 [

]. 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 [

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 [

]. 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 [

]. 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 [

]. 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 [

]. 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 [

]. 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 [

–

]. 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 [

However, the gray wolf population may converge around suboptimal individuals, leading

to premature convergence issues [

]. Pu Q et al. introduced an improved wild dog

optimization algorithm for robotic arm time-impact optimization [

], 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 [

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 [

]. 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 [

]. 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 [

]. However, the performance of this algorithm signif-

icantly deteriorates in a multi-constraint environment and requires further integration with

various improvement strategies [

]. The Graylag Goose Optimization (GGO) algorithm

introduced a group migration strategy [

], 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 [

]. 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

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deﬁned by the DH parameters, whereas

represents the elastic deformation angle of the

ﬂexible transmission element.

Figure 1. Motion Model of Wafer Transport Robotic Arm.

Table 1 shows the simpliﬁed model of the wafer transport robot, which consists of the

rigid link I (upper arm) and the ﬂexible link II (lower arm). The world coordinate system

is established with the base as the origin, the coordinate system X

is set at the

end-point of link I (upper arm), and the coordinate system X

is established at the

end-point of link II (lower arm). C

and C

represent the center of mass of the two links.

is the rotation angle of the primary arm, q

is the rotation angle of the secondary arm,

and

is the deformation of the ﬂexible lower arm. The structural parameters of the wafer

transfer robotic arm in this study are shown in Table 2.

Table 1. D-H Parameter Table of the Wafer Handling Robot Arm.

Arm/li q

◦

/mm l

/mm α

◦

0–120 0 240 0

Table 2. Parameters of Components of the Wafer Transport Robotic Arm.

Structure Value

Primary arm length/L

240 mm

Secondary arm length/L

240 mm

Mass of Primary arm/m

2.757 kg

Mass of secondary arm/m

2.165 kg

The center of gravity of the primary arm/Lc

0.4 L

The center of gravity of the secondary arm/Lc1 0.5 L

First, based on the Lagrange method, the dynamic equations of a planar two-degree-of-

freedom rigid robot are established. First, the robot’s potential energy equation is analyzed.

The robot’s forearm can be regarded as a ﬂexible arm rod, and its elastic potential energy

can be expressed as

k(q

− θ

) (1)

Here, the

refers to the ideal angle of the forearm when no deformation occurs. Since

the robot prototype moves parallel to the ground throughout, gravitational potential energy

is not taken into account. The kinetic energy expression includes both the rotational kinetic

energy of each link and the translational kinetic energy of the link centers of mass.

https://doi.org/10.3390/s26030829