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Sensors 2025 , 25 , 1872 22 of 24 Sensors 2025 , 25 , x FOR PE ER REV I E W 23 of 25 Figure 14. Moti on trajectorie s in a two − dimensional pl a n e . 4. Conclusions This …
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(a) (b)
(c) (d)
Figure13.Kinematicimage:(a)DisplacementcurveinCartesianspace;(b)speedcurveinCartes ian
space;(c)accelerationcurveinCartesianspace;(d)jerkcurveinCartesianspace.
Inaddition,thisstudymeticulouslyrecordsthekeyparameters,particularlythejoint
angles,duringtheS‐shapedspeedcurvemovement.Twosetsofjointanglesconsistingof
severalinterpolationpointsaresubstitutedintotheforwardkinematicsformula,asshown
inEquat ion(30).XandYrepr es ent thecoordi natevaluesinthe
Cartesiancoo r dinat e
system,respectively.
112 1 2
112 1 2
cos cos cos
sin sin sin
Xlink link
Y link link
(30)
TheactualmotiontrajectorydepictedintheplanarspaceisshowninFigure14.By
comparingFigures4and14,thehighconsistencyofthetwotrajectoriescanbecl earl y
verified.ThiscomparisonclearlyvalidatestheaccuracyandreliabilityoftheISO. This
planningmethodnotonlyelevatestheaccuracy
oftrajectoryplanning,butalsoensures
thesafetyandreliabilityofthemotion.
Figure 13. Kinematic image: (a) Displacement curve in Cartesian space; (b) speed curve in Cartesian
space; (c) acceleration curve in Cartesian space; (d) jerk curve in Cartesian space.
In addition, this study meticulously records the key parameters, particularly the joint
angles, during the S-shaped speed curve movement. Two sets of joint angles consisting of
several interpolation points are substituted into the forward kinematics formula, as shown
in Equation (30). X and Y represent the coordinate values in the Cartesian coordinate
system, respectively.
(
X = link
1
×cos θ
1
+ link
2
×
(
cos θ
1
+ cos θ
2
)
Y = link
1
×sin θ
1
+ link
2
×
(
sin θ
1
+ sin θ
2
)
(30)
The actual motion trajectory depicted in the planar space is shown in Figure 14. By
comparing Figures 4 and 14, the high consistency of the two trajectories can be clearly
verified. This comparison clearly validates the accuracy and reliability of the ISO. This
planning method not only elevates the accuracy of trajectory planning, but also ensures the
safety and reliability of the motion.
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Figure14.Motiontrajectoriesinatwo−dimensionalplane.
4.Conclusions
Thisstudyproposesanimprovedsnakeoptimization(ISO)algorithm.Meanwhile,a
single‐segmentstart–stopS‐shapedspeedcurvewitharctransitionisconstructedbased
onISO,whichisa methodspecificallydesignedfortherapidandstableoperationrequire‐
mentsof thewafer‐handling roboticarm.Thismethod extensively investigates
and
optimizesonthekeyissuesoftimeandimpactwithinthecontextofroboticarmtrajectory
planning.ThroughacomparativeanalysiswiththeoriginalS‐shapedspeedcurvescheme
underthe same para met e r sett ings, the new method improvesthe time efficiencyby
24.1%,therebysignificantlyreducingthetime
requiredfortheroboticarmtocomplete
thetask.Furthermore,theperformanceoftheISOiscomparedwiththatofSO,SA,PSO,
GWO,andWOAonaseriesofbenchmarktestfunctions.Theresultsindicatethat,interms
ofthetwo key performance indicators,averagevalueandvariance,the ISO
algorithm
outperformsthesecondrankedGWOalgorithmbyamarginof38.9%.Theseimproved
simulationsfullydemonstratetheeffectivenessandsuperiorityoftheproposed method
inaddressingthetrajectoryplanning problemforwafer‐handlingroboticarms.
Inthisstudy,theshortcomingsoftheSOalgorithmhavebeenextensivelystudied
andimproved.Whiletheproposedenhancementsimproveglobalsearchcapabilities,the
riskoflocaloptimaconvergencepersists.Additionall y,althoughBayesianopt imization
improvesthesearchefficiency,itconsumesmorecomputerresources.Therefore,future
researchwil l prioritizerefining thesea rchmechanismtoaddresstheseli mitationsofthe
SOalgorithm.Simultaneously,weplanto
explorethecombinationofBayesianoptimi‐
zationandbandit‐basedmethods,aimingtoreducecomputationalcostswhileimprov‐
ingtheoptimizationefficiency.
AuthorContributions:Conceptualization,Y.J.andJ.Y.;methodology,J.Y.;software,J.Y.;validation,
J.Y.;formalanalysis,J.Y.;investigation,Y.J.andJ.Y.Allauthorshavereadandagreedtothepub‐
lishedversionofthemanuscript.
Funding:Thisresearchwasfunded bytheBasicScientificResearchProjectofHigherEducation
InstitutionsofLiaoning
Province(grantnumberJYTQN2023064).
InstitutionalReviewBoardStatement:Notapplicable.
InformedConsentStatement:Notapplicable.
DataAvailabilityStatement:Datasharingnotapplicable.Nonewdatawerecreatedoranalyzed
inthisstudy.Datasharingisnotapplicabletothisarticle.
Figure 14. Motion trajectories in a two−dimensional plane.
4. Conclusions
This study proposes an improved snake optimization (ISO) algorithm. Meanwhile, a
single-segment start–stop S-shaped speed curve with arc transition is constructed based on
ISO, which is a method specifically designed for the rapid and stable operation requirements
of the wafer-handling robotic arm. This method extensively investigates and optimizes on
the key issues of time and impact within the context of robotic arm trajectory planning.
Through a comparative analysis with the original S-shaped speed curve scheme under
the same parameter settings, the new method improves the time efficiency by 24.1%,
thereby significantly reducing the time required for the robotic arm to complete the task.
Furthermore, the performance of the ISO is compared with that of SO, SA, PSO, GWO, and
WOA on a series of benchmark test functions. The results indicate that, in terms of the two
key performance indicators, average value and variance, the ISO algorithm outperforms
the second ranked GWO algorithm by a margin of 38.9%. These improved simulations
fully demonstrate the effectiveness and superiority of the proposed method in addressing
the trajectory planning problem for wafer-handling robotic arms.
In this study, the shortcomings of the SO algorithm have been extensively studied
and improved. While the proposed enhancements improve global search capabilities, the
risk of local optima convergence persists. Additionally, although Bayesian optimization
improves the search efficiency, it consumes more computer resources. Therefore, future
research will prioritize refining the search mechanism to address these limitations of the SO
algorithm. Simultaneously, we plan to explore the combination of Bayesian optimization
and bandit-based methods, aiming to reduce computational costs while improving the
optimization efficiency.
Author Contributions: Conceptualization, Y.J. and J.Y.; methodology, J.Y.; software, J.Y.; validation,
J.Y.; formal analysis, J.Y.; investigation, Y.J. and J.Y. 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 Education
Institutions of Liaoning Province (grant number JYTQN2023064).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data sharing not applicable. No new data were created or analyzed in
this study. Data sharing is not applicable to this article.
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Conflicts of Interest: The authors declare no conflicts of interest.
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