s25061872.pdf

Academic Editor: Zheng Chen Received: 11 February 2025 Revised: 4 March 2025 Accepted: 13 March 2025 Published: 18 March 2025 Citation: Ji, Y .; Y u, J. T ime-Impact Optimal T rajectory Planning for W afer-Handling Robot…

Academic Editor: Zheng Chen

Received: 11 February 2025

Revised: 4 March 2025

Accepted: 13 March 2025

Published: 18 March 2025

Citation: Ji, Y.; Yu, J. Time-Impact

Optimal Trajectory Planning for

Wafer-Handling Robotic Arms Based

on the Improved Snake Optimization

Algorithm. Sensors 2025, 25, 1872.

https://doi.org/10.3390/s25061872

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

(https://creativecommons.org/

licenses/by/4.0/).

Article

Time-Impact Optimal Trajectory Planning for Wafer-Handling

Robotic Arms Based on the Improved Snake

Optimization Algorithm

Yujie Ji * and Jiale Yu

School of Mechanical Engineering, Shenyang Ligong University, Shenyang 110159, China; yjl2801036037@163.com

* Correspondence: jyj19700521@163.com

Abstract: To enhance the working efﬁciency of a wafer-handling robotic arm and simulta-

neously alleviate the impact and vibration during the motion process, a trajectory planning

approach based on an improved snake optimization (ISO) algorithm is proposed. The

following improvements have been made to the snake optimization (SO) algorithm: the in-

troduction of a Chaotic Tent Map for initializing the swarm, the use of randomly perturbed

dynamic learning factors to replace ﬁxed values, the application of a cosine annealing

learning rate for self-adaptively updating individual positions, and the incorporation of

Bayesian optimization for parameterization and ﬁne-tuning of the system’s selection pro-

cess. Furthermore, the ISO algorithm is applied in the Cartesian space of the robotic arm

to effectively address the trajectory planning challenge of the single-segment start–stop

S-shaped

speed curve with arc transitions. The simulation results indicate that the im-

proved S-shaped speed curve has increased by 24.1% compared with the original plan,

and the mean and variance rankings of ISO algorithm have, respectively, improved by

60.8% and 63.4%, compared with the SO algorithm. Meanwhile, this study has successfully

achieved the Pareto optimal solution with time and impact as the targets based on the

established MATLAB experimental simulation platform.

Keywords: ISO algorithm; S-shaped speed curve; trajectory planning; time-impact optimal

1. Introduction

With the rapid advancement of the semiconductor industry, the optimal trajectory

planning problem for manipulator arms has become a critical research focus [

–

]. It is

a fundamental and critical challenge in the ﬁeld of robotic manipulators. At its core, the

goal of optimal trajectory planning is to determine a motion path for the manipulator

that meets speciﬁc requirements such as minimizing operation time, energy consumption,

and mechanical stress, while ensuring the accuracy of reaching the target position and

orientation. To achieve the overarching objective, the following three critical aspects

should be thoroughly considered: the selection of an appropriate operational space for the

robotic arm, the design of suitable trajectory curves, and the implementation of an efﬁcient

optimization algorithm.

When deliberating the trajectory planning of the manipulator arm, a choice must

be made between the planning schemes in Cartesian space and Joint space, which are

typically determined by speciﬁc task requirements, characteristics of the manipulator arm,

and tracking accuracy, among other factors. For instance, trajectory planning in Cartesian

space enables direct control of the position and orientation of the manipulator end effec-

tor, which is relatively more intuitive than that in Joint space, and can more effectively

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Sensors 2025, 25, 1872

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plan smooth motion trajectories [

–

]. From the perspective of advanced computational

methods, Xiao et al. [

] successfully mitigated the shock and vibration experienced by a

robot during its movement from the starting position to the target position in Cartesian

space, thereby ensuring the accuracy and safety of point-to-point movements. In contrast,

Zhang et al. [

] successfully planned two sets of trajectories using polynomial interpolation

in Joint space and veriﬁed the model by using SimMechanics. However, owing to the

insufﬁcient robustness of the PID controller against external environmental factors, the sim-

pliﬁcation of the model signiﬁcantly affected the accuracy of the experiment. Additionally,

the improved quantum particle swarm optimization algorithm proposed by Luo et al. [

]

performed outstandingly in the trajectory planning of the robot joint space, effectively

enhancing the efﬁciency and accuracy of trajectory planning. Regarding the S-shaped

speed curve, Wu et al. [

] investigated how to plan trajectories in Joint space of industrial

robots that satisfy kinematic constraints and have continuous jerk. Nevertheless, trajectory

planning in Joint space requires complex inverse kinematic solutions. Due to joint coupling

effects, trajectories converted to Cartesian space may exhibit distortions or deviations from

the desired path. This constitutes a potential issue for manipulator arms employed in

wafer-handling scenarios.

Trajectory planning curves are vital for motion control strategies. The common types

include linear interpolation, B-spline curves, polynomial curves, T-shaped speed curves,

and S-shaped speed curves. Polynomial interpolation is frequently employed for this

purpose. By selecting the appropriate polynomial order, its adaptability can be ﬂexibly

adjusted to match different numbers of data points and smooth the curve, thus effectively

reducing the vibration and shock of the robotic arm during movement [

–

]. Although

high-order polynomial interpolation can provide a smoother curve ﬁt, it may lead to the

Runge phenomenon with an excessive number of data points. In addition, they are highly

sensitive to minor data point variations, potentially leading to numerical instability. Fur-

thermore, Cong et al. [

] proposed the use of ﬁfth-order Non-uniform B-spline curves in

the trajectory planning and crafted a multi-objective trajectory optimization model employ-

ing the Non-dominated Sorting Genetic Algorithm-II (NSGA2). Regarding the T-shaped

speed curve, Ayazbay et al. [

] applied a mini-mum time-impact planning algorithm to

optimize the curve, targeting an enhanced dynamic performance and reduced mechanical

wear. However, this curve lacked smoothness at the endpoints, necessitating additional

processing to ensure the overall continuity and smoothness of the trajectory. In response to

these limitations, researchers have proposed S-shaped speed curves

[17,18]

as trajectory

planning curves, which are distinguished by smooth transitions at the velocity inﬂection

points. This results in less impact force and noise compared to the T-shaped accelera-

tion and deceleration phases. S-shaped speed curves are commonly used in applications

requiring precise control and minimization of mechanical vibration.

Heuristic algorithms are increasingly employed in trajectory planning and typically

address complex optimization problems when it is difﬁcult to ﬁnd an exact solution or when

the cost of computing an exact solution is too high. Intelligent algorithms, such as genetic

algorithms (GA), particle swarm optimization (PSO) algorithms, annealing algorithms (SA),

whale optimization algorithms (WOA), and grey wolf optimization (GWO) algorithms

provide novel solutions for addressing intricate trajectory planning problems. They have

the characteristics of ﬂexibility, robustness, and diversity, which can help users ﬁnd the

global optimal solutions. Among them, the PSO algorithm [

], which simulates the social

behavior of bird ﬂocks to achieve optimization, is widely used to optimize the use of robotic

arms owing to its simplicity, ease of implementation, and strong ability for multi-modal

optimization. However, the PSO algorithm cannot avoid problems of parameter sensitivity

and convergence acceleration. Therefore, research teams have improved to varying degrees

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to overcome these shortcomings [

–

]. Lu and Wang [

] built an optimal trajectory

planning method based on a B-spline and a WOA to improve the work efﬁciency of a

robotic arm. However, they did not fully compare the performance differences between the

WOA and other existing algorithms. By combining the back propagation neural network

(BPNN) with the GA, the overall performance of the upper limb rehabilitation robotic arm

trajectory planning was improved to some extent. However, the parameter adjustment

after the combination was very complex and required a large number of experiments and

professional knowledge for correction [

]. The SO algorithm excels outstanding in the

realm of optimization owing to its exceptional ﬂexibility and robustness. It is not only easy

to integrate chaos theory and learning mechanisms to bolster search efﬁciency but also

to maintain efﬁcient exploration ability under poor initial conditions. In addition, it can

rapidly adapt to the changes in dynamic environments and optimize search strategies in

real time, thereby safeguarding the continuity and effectiveness of the optimization process.

Li et al. used a SO algorithm to ﬁnd optimal paths in dynamic environments. Although its

performance has been improved compared with other intelligent algorithms, the improved

algorithm still requires careful parameter tuning [

]. To address the shortcomings of the

SO algorithm, recent literature [

] begun to focus on solving its core mechanism to

raise the overall performance of optimization problems. The ISO has also been gradually

applied to path planning and to solve general constrained optimization problems.

Therefore, considering the shortcomings of the afore-mentioned studies and the main

factors that should be prioritized during the actual operation of the wafer manipulator,

this study proposes an improved control strategy for a manipulator that combines the

single-segment start–stop S-shaped speed of the circular arc transition, an ISO algorithm,

and a time-impact optimization method of trajectory planning. It is worth noting that

it has more advantages in terms of the stability of the control effect or the convergence

ability of the algorithm compared with other types of methods. And it can adapt to

parameter optimization in different scenarios. In summary, the contributions of this study

are as follows.

(1)

In the area of intelligent optimization methods, the following improvements have

been made to the SO algorithm.

•

The Tent Chaotic Map initializes populations, instead of randomly generating

populations with ‘rand’ functions.

•

Dynamic learning factors, integrated with random perturbations, replace static

learning factors.

•

A cosine annealing learning rate scheme is employed for the adaptive updating

of individual positions.

•

The Bayesian method is utilized to optimize thresholds, surpassing the reliance

on empirical thresholds.

(2)

In the area of trajectory planning, the following improvements have been made to the

S-shaped speed curve.

• The interpolated arc transition supplants the abrupt right-angle transition.

•

The continuous single-segment start–stop mechanism supersedes the segmented

start–stop mechanism.

2. Materials and Methods

2.1. The Structure Parameters of the Wafer-Handling Robotic Arms

Assuming that the task requirements can be realized in a plane, the system model

is simpliﬁed when the robotic arm transports the wafer, as shown in Figure 1. Before

planning the trajectory, it is necessary to establish the link coordinate system correctly. In