DSmT Decision-Making Algorithms for Finding Grasping Configurations of Robot Dexterous Hands

2018·3 MB·

by  Ionel-Alexandru Gal

SS
symmetry
Article
DSmT Decision-Making Algorithms for Finding
Grasping Configurations of Robot Dexterous Hands
Ionel-AlexandruGal,DanutBucurandLuigeVladareanu*
InstituteofSolidMechanicsoftheRomanianAcademy,010141BucharestS1,Romania;
[email protected](I.-A.G.);[email protected](D.B.)
* Correspondence:[email protected];Tel.:[phone]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:14May2018;Accepted:29May2018;Published:1June2018
Abstract: Inthispaper,wepresentadecidingtechniqueforroboticdexteroushandconfigurations.
This algorithm can be used to decide on how to configure a robotic hand so it can grasp objects
indifferentscenarios. Receivingasinput, severalsensorsignalsthatprovideinformationonthe
object’sshape,theDSmTdecision-makingalgorithmpassestheinformationthroughseveralsteps
beforedecidingwhathandconfigurationshouldbeusedforacertainobjectandtask. Theproposed
decision-makingmethodforrealtimecontrolwilldecreasethefeedbacktimebetweenthecommand
andgraspedobject,andcanbesuccessfullyappliedonrobotdexteroushands. Forthis,wehave
used the Dezert–Smarandache theory which can provide information even on contradictory or
uncertainsystems.
Keywords: neutrosophy; DSmT; decision-making algorithms; robotic dexterous hands; grasping
configurations;grasptype
1. Introduction
The purpose of autonomous robotics is to build systems that can fulfill all kinds of tasks
without human intervention, in different environments which were not specially build for robot
interaction. Amajorchallengeforthisautonomousroboticsfieldcomesfromhighuncertaintywithin
real environments. This is because the robot designer cannot know all the details regarding the
environment. Mostoftheenvironmentparametersareunknown,thepositionofhumansandobjects
cannotbepreviouslyanticipatedandthemotionpathmightbeblocked. Besidethese,theaccumulated
sensorinformationcanbeuncertainanderrorprone. Thequalityofthisinformationisinfluencedby
noise,visualfieldlimitations,observationconditions,andthecomplexityofinterpretationtechnique.
Theartificialintelligenceandtheheuristictechniqueswereusedbymanyscientistsinthefieldof
robotcontrol[1]andmotionplanning.Regardinggraspingandobjectmanipulations,themainresearch
activitiesweretodesignamechanismforhand[2–4]anddexterousfingermotion[5],whichareahigh
complexityresearchtasksincontrollingrobotichands.
Currently, in the research area of robotics, there is a desire to develop robotic systems with
applications in dynamic and unknown environments, in which human lives would be at risk,
likenaturalornucleardisasterareas,andalsoindifferentfieldsofwork,rangingfromhousechores
oragriculturetomilitaryapplications. Inanyoftheseresearchareas,theroboticsystemmustfulfill
aseriesoftaskswhichimpliesobjectmanipulationandtransportation,orusingequipmentandtools.
Fromherearisesthenecessityofdevelopmentgraspingsystems[6]toreproduce,aswellaspossible
humanhandmotion[7–9].
Toachieveanaccurategraspingsystem,agrasptaxonomyofthehumanhandwasanalyzedby
Feixetal.[10]whofound33differentgrasptypes,sortedbyoppositiontype,virtualfingerassignments;
type in terms of power, precision, or intermediate grasp; and the position of the thumb. While
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Alvarezetal.[11]researchedhumangraspstrategieswithingrasptypes,Fermulleretal.[12]focused
onmanipulationactionforhumanhandondifferentobjecttypesincludinghandpre-configuration.
Tsai et al. [13] found that classifying objects into primitive shapes can provide a way to select
the best grasping posture, but a general approach can also be used for hand–object geometry
fitting[14].Thisclassificationworkswellforgraspingproblemsinconstrainedworkspaceusingvisual
datacombinedwithforcesensors[15]andalsoforunder-actuatedgraspingwhichusesrotational
stiffness[16]. However,forunknownobjects,scientistsfounddifferentapproachestosolvethehand
graspingproblem.Choietal.[17]usedtwodifferentneuralnetworksanddatafusiontoclassifyobjects,
Seredynskietal.[18]achievedfastgrasplearningwithprobabilisticmodels,whileSongetal.[19]used
atactile-basedblindgraspingalongwithadiscrete-timecontroller. Thesameapproachisusedby
Guetal.[20]whichproposedablindhapticexplorationofunknownobjectsforgraspplanningof
dexterousrobotichand. Usinggraspingmethods,Yamakawaetal.[21]developedarobotichandfor
knotmanipulation,whileNacyetal.[22]usedartificialneuralnetworkalgorithmsforslipprevention
andZaidietal.[23]usedamulti-fingeredrobothandtograsp3Ddeformableobjects,applyingthe
methodonspheresandcubes.
While other scientists developed grasping strategies for different robotic hands [21–23],
ananthropomorphicrobotichandhasthepotentialtograspregularobjectsofdifferentshapesand
sizes[24,25],butselectingthegraspingmethodforacertainobjectisadifficultproblem. Aseriesof
papershaveapproachedthisproblembydevelopingalgorithmsforclassifyingthegraspingbythe
contactpoints[26,27].Thesealgorithmsarefocusedonfindingafixnumberofcontactareaswithout
takingintoconsiderationthehandgeometry. Othermethodsdevelopedgraspingsystemsforacertain
robotic hand architecture, scaling down the problem to finding a grasping method with the tip of
thefingers[27]. Thesemethodsareusefulincertainobjectmanipulation,butcannotbeappliedfor
a wide range of objects because it does not provide a stable grasping due to the face that it is not
used,thefinger’sinteriorsurfaceorthepalmofthehand. Amethodforfilteringthehighnumber
ofhandconfigurationsistousepredefinedgraspinghandconfigurations. Beforegraspinganobject,
humans,unconsciouslysimplifythegraspingaction,choosingoneofthefewhandpositionswhich
matchtheobject’sshapeandthetasktoaccomplish. Inthescientificliteraturetherearepaperswhich
havetriedtologinthepositioningforgraspingandtaxonomy,andoneofthemostknownpapers
is[28]. CutkoskyandWeight[29]haveextendedNapier’s[28]classificationbyaddingtherequired
taxonomyintheproductionenvironment,bystudyingthewayinwhichtheweightandgeometry
oftheobjectaffectschoosingthegraspingpositioning. Iberall[30]hasanalyzeddifferentgrasping
taxonomies and generalized them by using the virtual finger concept. Stransfield [31] has chosen
asimplerclassificationandbuiltasystembasedonruleswhichprovidedagraspingpositioningset,
startingfromasimplifieddescriptionoftheobjectgainedfromavideosystem.
The developed algorithm presented in this paper has the purpose to determine the grasping
positionaccordingtotheobject’sshape. Toprovethealgorithm’sefficiencywehavechosenthreetypes
ofobjectsforgrasping: cylindrical,spherical,andprismatic. Forthis,westartfromthehypothesis
thattheenvironmentdataarecapturedthroughastereovisionsystem[32]andaKinectsensor[33].
Onthisdata,whichthetwosystemobserversprovide,weapplyatemplatematchingalgorithm[34].
Thisalgorithmwillprovideamatchingpercentageoftheobjectthatneedstobegraspedwithatemplate
object. Thus,eachofthetwosourceswillprovidethreematchingvalues,foreachofthethreegrasping
types. Thesevaluesrepresenttheinputforourdetectionalgorithm,basedofDezert–Smarandache
Theory(DSmT)[35]fordatafusion. Thisalgorithmhasasinputdatafromtwoormultipleobservers
andinthefirstphasetheyareprocessedthroughaprocessofneutrosoficationwhichissimilarwith
thefuzzificationprocess. Then,theneutrosophicobservers’dataarepassedthroughanalgorithm
whichappliestheclassicDSmtheory[35]inordertoobtainasingledatasetonthesystem’sstates,
bycombiningtheobservers’neutrosophicvalues. Onthisobtaineddataset,weapplythedeveloped
DSmTdecision-makingalgorithmthatdecidesonthecategoryfromwhichthetargetobjectispartof.
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Thisdecisionfacilitatesthedetection–recognition–graspingprocesswhicharobotichandmustfollow,
obtainingintheendareal-timedecisionthatdoesnotstopordelaytherobot’stask.
Inrecentyears,usingmoresensorsforacertainapplicationsandthenusingdatafusionisbecoming
morecommoninthemilitaryandnonmilitaryresearchfields.Thedatafusiontechniquescombine
theinformationreceivedfromdifferentsensorswiththepurposeofeliminatingdisturbancesandto
improveprecisioncomparedtothesituationswhenasinglesensorisused[36,37].Thistechniqueworks
onthesameprincipleusedbyhumanstofeeltheenvironment.Forexample,ahumanbeingcannot
see over the corner or through vegetation, but with his hearing he can detect certain surrounding
dangers. Besidethestatisticaladvantagebuildfromcombiningthedetailsforacertainobject(through
redundantobservations),usingmoretypesofsensorsincreasestheprecisionwithwhichanobjectcan
beobservedandcharacterized. Forexample,anultrasonicsensorcandetectthedistancetoanobject,
butavideosensorcanestimateitsshape,andcombiningthesetwoinformationsourceswillprovide
twodistinctdataonthesameobject.
The evolution on the new sensors, the hardware’s processing techniques and capacity
improvementsfacilitatemoreandmoretherealtimedatafusion. Thelatestprogressweremadeinthe
areaofcomputationalanddetectionsystems,andprovidetheabilitytoreproduce,inhardwareand
software,thedatafusioncapacityofhumansandanimals. Thedatafusionsystemsareusedfortarget
tracking[38],automatictargetidentification[39],andautomatedreasoningapplications[40]. Thedata
fusionapplicationsarewidespread,rangingfromthemilitary[41]applications(targetrecognition,
autonomousmovingvehicles,distancedetection,battlefieldsurveillance,automaticdangerdetection)
tocivilianapplication(monitoringtheproductionprocesses,complextoolsmaintenancebasedon
certainconditions,robotics[42],andmedicalapplications[43]). Thedatafusiontechniquesundertake
classicelementslikedigitalsignalprocessing,statisticalestimation,controltheory,artificialintelligence,
andnumericmethods[44].
Combineddatainterpretationrequiresautomatedreasoningtechniquestakenfromtheareaof
artificialintelligence.Thepurposeofdevelopingtherecognitionbasedsystems,wastoanalyzeissues
likethedatagatheringcontext, therelationshipbetweenobservedentities, hierarchicalgroupingof
targets or objects and to predict future actions of these targets or entities. This kind of reasoning
is encountered in humans, but the automated reasoning techniques can only closely reproduce it.
Regardlessoftheusedtechnique,foraknowledgebasedsystem,threeelementsarerequired:oneormore
reasoningdiagrams,anautomatedevaluationprocessandacontroldiagram.Thereasoningdiagramsare
techniquesoffactsrepresentation,logicalrelations,proceduralknowledge,anduncertainty. Forthese
techniques, uncertainty from the observed data and from the logical relations can be represented
usingprobabilities,fuzzytheory[45,46],Dempster–Shafer[47]evidenceintervalsorothermethods.
Dezert–Smarandachetheory[35]comestoextendthesemethods,providingadvancedtechniquesof
uncertaintymanipulation. Theautomatedreasoningsystem’sdevelopingpurposeistoreproducethe
humancapabilityofreasoninganddecisionmaking,byspecifyingrulesandframesthatdefinethe
studiedsituation. Havingathandaninformationdatabase,anevaluationprocessisrequiredsothis
informationcanbeused. Forthis,thereareformaldiagramsdevelopedontheformallogic,fuzzylogic,
probabilisticreasoning,templatebasedmethods,casebasedreasoning,andmanyothers. Eachofthese
reasoningdiagramshasaconsistentinternalformalismwhichdescribeshowtousetheknowledge
databaseforobtainingthefinalconclusion.Anautomatedreasoningsystemneedsacontroldiagramto
fulfillthethinkingprocess. Theusedtechniquesincludesearchingmethods,systemsformaintaining
the truth based on assumptions and justifications, hierarchical decomposition, control theory, etc.
Eachofthesemethodshasthepurposeofcontrollingthereasoningevolutionprocess.
Theresultspresentedinthispaper,wereobtainedusingtheclassicDezert–Smarandachetheory
(DSmT)tocombineinputsfromtwodifferentobserversthatwanttoclassifyobjectsintothreecategories:
sphere,parallelepiped,andcylinder. Thesecategorieswerechosentoincludemostoftheobjectsthat
amanipulatorcangrasp. Thealgorithm’sinputsweretransformedintobeliefvaluesofcertainty,falsity,
uncertainty,andcontradictionvalues. Usingthesefourvaluesandtheircombinationsaccordingto
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DSmT,weappliedPetrinetdiagramlogicfortakingdecisionsontheshapetypeoftheanalyzedobjects.
Thistypeofalgorithmhasneverbeenusedbeforeforrealtimedecisiononhandgraspingtaxonomy.
Comparedtootheralgorithms[13–15]andmethods[16–18],ourshastheadvantagetodetecthigh
uncertaintiesandcontradictionswhichinpracticehasaverylowencounterratebutcanhavedrastic
effectsonthedecisiontypeorrobot,becauseiftheobject’sshapeisnotdetectedproperly,thenthe
robot might not be able to grasp it, which can lead to serious consequences. In deciding how to
graspobjects,researchershaveuseddifferentmethodstochoosethegraspingtaxonomyusingablind
hapticexploration[20]orindifferentapplicationsfortyingknots[21]orgraspdeformableobjects[23].
Becausetheproposedalgorithmcandetectanomaliesofcontradictinganduncertaininputvalues,
wecansaythattheproposedmethodtransformsthedecidingprocessintoalessdifficultproblemof
graspingmethod[24,25].
2. ObjectsGraspingandItsClassification
Mechanicalhandshavebeendevelopedtoprovidetherobotswiththeabilityofgraspingobjects
with different geometrical and physical properties [48]. To make an anthropomorphic hand seem
natural,itsmovementandthegraspingtypemustmatchthehumanhand.
Inthisregard,graspingpositiontaxonomyforhumanhandshasbeenlongstudiedandapplied
forrobotichands. Seventeendifferentcategoriesofhumanhandsgraspingpositionswerestudied.
However,wemustconsidertwoimportantthings: thefirstthingisthatthesecategoriesarederived
fromhumanhandstudies,whichprovesthattheyaremoreflexibleandabletoperformamultitude
ofmovementsthananyotherrobotichand, sothatthegraspingtaxonomyforrobothandscanbe
onlyasimplesubsetofthehumanhand. Thesecondisthatthehumanbehaviorstudiesofrealobject
graspinghaveshownsomedifferencesbetweentherealobservationsandtheclassifiedproperties[49].
In conclusion, any proposed taxonomy is only a reference point which the robot hand must
attain. Belowthemostusedgraspingpositionsaredescribed(extractedfrom[50]),whichshouldbe
consideredwhendevelopinganablerobotichand:
1. Powergrasping: Thecontactwiththeobjectsismadeonlargesurfacesofthehand,including
handphalangesandthepalmofthehand. Forthiskindofgrasping,highforcescanbeexerted
ontheobject.
• Sphericalgrasping: usedtograspsphericalobjects;
• Cylindricalgrasping: usedtograsplongobjectswhichcannotbecompletelysurroundedby
thehand;
• Lateralgrasping: thethumbexertsaforcetowardsthelateralsideoftheindexfinger.
2. Precisiongrasping: thecontactismadeonlywiththetipofthefingers.
• Prismaticgrasping(pinch): usedtograsplongobjects(withsmalldiameter)orverysmall.
Canbeachievedwithtwotofivefingers.
• Circulargrasping(tripod): usedingraspingcircularorroundobjects. Canbeachievedwith
three,four,orfivefingers.
3. Nograsping:
• Hook: thehandformsahookontheobjectandthehandforceisexertedagainstanexternal
force,usuallygravity.
• Buttonpressingorpointing
• Pushingwithopenhand.
IntheTable1,manipulationactivitiesthattherobotichandcanachieveareshown,correlated
withtherequiredactivitygraspingpositions[51].
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Table1.Graspingpositionforcertaintasks.
Object Activity GraspingPosition
Bottles,cups,andmugs Transport,pouring/filling Force:Cylindricalgrasping(fromthesideorthetop)
Force:Lateralgrasping
Cups(usinghandles) Pouring/filling
Precision:Prismaticgrasping
Power:Lateralgrasping
Transport
Plates/trays Precision:Prismaticgrasping
Receivingfromhumans
Nograsp:pushing(openhand)
Pens,cutlery Transport Precision:Prismaticgrasping
Force:Cylindricalgrasping
Doorhandle Open/Close
Nograsp:Hook
Power:Sphericalgrasping
Smallobjects Transport
Precision:Circulargrasping(tripod)
Switches,buttons Pushing Nograsp:Buttonpressing
Force:Lateralgrasping
Roundswitches,bottlecaps Rotation
Precision:Circulargrasping(tripod)
3. ObjectDetectionUsingStereo-VisionandKinectSensor
Objectrecognitioninartificialsightrepresentsthetaskofsearchingacertainobjectinapicture
oravideosequence. Thisproblemcanbeapproachedasalearningproblem. Atfirst,thesystemis
trainedwithsampleimageswhichbelongtothetargetgroup,thesystembeingtaughttospotthese
amongotherpictures. Thus,whenthesystemreceivesnewimages,itcan‘feel’thepresenceofthe
searchedobject/sample/template.
Template matching is a techniques used to sort objects in an image. A model is an image
region, and the goal is to find instances of this model in a larger picture. The template matching
techniquesrepresentaclassicapproachforlocalizationproblemsandobjectrecognitioninapicture.
Thesemethodsareusedinapplicationslikeobjecttracking,imagecompression,stereograms,image
segmentation[52],andotherspecificproblemsofartificialvision[53].
Objectrecognitionisveryimportantforarobotthatmustfulfillacertaintask.Tocompleteitstask,
therobotmustavoidobstacles,obtainthesizeoftheobject,manipulateit,etc. Forthecaseofdetected
objectmanipulation,therobotmustdetecttheobject’sshape,size,andpositionintheenvironment.
The main methods for achieving the depth information use stereoscopic cameras, laser scanners,
anddepthcameras.
To achieve the proposed decision-making algorithm, we assumed that the environment
informationiscapturedwithastereoscopicsystemandaKinectsensor.
Stereovision systems [32] represents a passive technique of achieving a virtual 3D image of
the environment in which the robot moves, by matching the common features of an image set of
the same scene. Because this method works with images, it needs a high computational power.
Thedepthinformationcanbenoisyincertaincases,becausethemethoddependsonthetextureofthe
environmentobjectsandontheambientlight.
Kinect[33]isafairlyeasytoobtainplatform,whichmakesitwidespread. Itusesadepthsensor
basedonstructuredlight. ByusinganASICboard,theKinectsensorgeneratesadepthmapon11bits
witharesolutionof640×480pixels,at30Hz. Giventhepriceofthedevice,theinformationqualityis
prettygood,butithasbothadvantagesanddisadvantages,meaningthatthedepthimagescontain
areaswherethedepthreadingcouldnotbeachieved. Thisproblemappearsfromthefactthatsome
materialsdonotreflectinfraredlight. Whenthedeviceismovedreallyfast,likeanyothercamera,
itrecordsblurrypictures,whichalsoleadstomissinginformationfromtheacquiredpicture.
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4. NeutrosophicLogicandDSmTheory
4.1. NeutrosophicLogic
The neutrosophic triplet (truth, falsity, and uncertainty) idea appeared in 1764 when J.H.
Lambert investigates a witness credibility which was affected by the testimony of another person.
HegeneralizedHooper’sruleofsamplecombination(1680),whichwasanon-Bayesianapproachfor
findingaprobabilisticmodel. Koopmanin1940introducesthelowandhighprobability,followedby
GoodandDempster(1967)whogaveacombinationruleoftwoarguments. Shafer(1976)extended
thisruletoDempster–ShaferTheoryfortrustfunctionsbydefiningthetrustandplausibilityfunctions
and using the inference rules of Dempster for combining two samples from two different sources.
The trust function is a connection between the fuzzy reasoning and probability. Dempster–Shafer
theoryfortrustfunctionsisageneralizationofBayesianprobability(Bayes1760,Laplace1780). Ituses
themathematicalprobabilityinamoregeneralwayanditisbasedontheprobabilisticcombinationof
artificialintelligencesamples.
Lambertonesaidthat“thereisachancepthatthewitnesscanbetrustworthyandfair,achanceq
thathewillbedeceivingandachance1-p-qthathewillbeindifferent”. ThisideawastakenbyShafer
in1986andlater,usedbySmarandachetofurtherdeveloptheneutrosophiclogic[54,55].
4.1.1. NeutrosophicLogicDefinition
A logic in which each proposition has its percentage of truth in a subset T, its percentage of
uncertaintyinasubsetI,anditspercentageoffalsityinasubsetFiscalledneutrosophiclogic[54,55].
Thispaperextendsthegeneralstructureoftheneutrophicrobotcontrol(RNC),knownasthe
Vladareanu–Smarandache method [55–57] for the robot hybrid force-position control in a virtual
platform[58,59],whichappliesneutrosophicsciencetoroboticsusingtheneutrosophiclogicandset
operators. Thus,usingtwoobservers,astereovisionsystemandaKinectsensor,willprovidethree
matchingvaluesforDSmTdecision-makingalgorithms. Asubsetoftruth,uncertaintyandfalsityis
usedinsteadofasinglenumberbecauseinmanycasesonecannotknowwithprecisionthepercentage
oftruthorfalsity,butthesecanbeapproximated. Forexample,asuppositioncanbe30%to40%true
and60%to70%false[60].
4.1.2. NeutrosophicComponentsDefinition
LetT,I,Fbethreestandardornon-standardsubsetsof]−0,1+[with
supT = t infT = t
sup inf
supI =i infI =i
sup inf
supF = f infF = f
sup inf
and
n = t +i + f
sup sup sup sup
n = t +i + f
inf inf inf inf
TheT,I,andFsetsarenotalwaysintervals,butcanbesubsets:discreteorcontinuum;withasingle
element;finiteorinfinite(theelementsarecountableoruncountable);subsetsunionorintersection.
Also,thesesubsetscanoverlap,andtherealsubsetsrepresenttherelativeerrorsinfindingthet,i,
andf values(whentheT,I,andFsubsetsarereducedtosinglepoints).
T,I,andFarecalledtheneutrosophiccomponentsandrepresentthetruth,uncertainty,andfalsity
values,whenreferringtoneutrosophy,neutrosophiclogic,neutrosophicsets,neutrosophicprobability,
orneutrosophicstatistics.
This representation is closer to the human reasoning and defines knowledge imprecision or
linguisticinaccuracyreceivedfromdifferentobservers(thisiswhyT,I,andFaresubsetsandcanbe
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morethatasetofpoints),theuncertaintygivenbyincompleteknowledgeordataacquisitionerrors
(forthiswehavethesetI)andthevaguenesscausedbymissingedgesorlimits.
Afterdefiningthesets,weneedtospecifytheirsuperior(x )andinferior(x )limitsbecause
sup inf
inmostofthecasestheywillbeneeded[61,62].
4.2. Dezert–SmarandacheTheory(DSmT)
Todevelopartificialcognitivesystemsagoodmanagementofsensorinformationisrequired.
Whentheinputdataaregatheredbydifferentsensors,accordingtotheenvironmentcertainsituations
mayappearwhenoneofthesensorscannotgivecorrectinformationortheinformationiscontradictory
betweensensors. Toresolvethisissueastrongmathematicalmodelisrequired,especiallywhenthe
informationisinaccurateoruncertain.
The Dezert–Smarandache Theory (DSmT) [53,54,60] can be considered an extension of
Dempster–Shafertheory(DST)[46]. DSmTallowsinformationcombining,gatheredfromdifferent
andindependentsourcesastrustfunctions. DSmTcanbeusedforsolvinginformationfusiononstatic
ordynamiccomplexproblems,especiallywhentheinformationdifferencesbetweentheobserversare
veryhigh.
DSmTstartsbydefiningthenotionofaDSmfreemodel,denotedbyMf(Θ)andsaysthatΘis
asetofexhaustiveelementsθ,i =1,...,nwhichcannotoverlap. Thismodelisfreebecausethereare
i
noothersuppositionsoverthehypothesis. AslongastheDSmfreemodelisfulfilled,wecanapply
theassociativeandcommutativeDSmruleofcombination.
DSm theory [62] is based on defining the Dedekind lattice, known as the hyper power set of
frame Θ. In DSmT, Θ is considered a set {θ ,...,θ } of n exhaustive elements, without adding
1 n
otherconstraints.
DSmTcantackleinformationsamples,gatheredfromdifferentinformationsourceswhichdonot
allowthesameinterpretationofthesetΘelements. LetΘ = θ ,θ bethesimplecase,madeoftwo
1 2
assumption,then[54]:
• theprobabilitytheoryworks(assumingexclusivityandcompletenessassumptions)withbasic
probabilityassignments(bpa)m(.) ∈ [0,1]suchthat
m(θ )+m(θ ) =1
1 2
• theDempster–Shafertheoryworks,(assumingexclusivityandcompletenessassumptions)with
basicbeliefassignments(bba)m(.) ∈ [0,1]suchthat
m(θ )+m(θ )+m(θ ∪θ ) =1
1 2 1 2
• theDSmtheoryworks(assumingexclusivityandcompletenessassumptions)withbasicbelief
assignment(bba)m(.) ∈ [0,1]suchthat
m(θ )+m(θ )+m(θ ∪θ )+m(θ ∩θ ) =1
1 2 1 2 1 2
Θ
4.2.1. TheD HyperpowerSetNotion
OneofthebaseelementsofDSmtheoryisthenotionofhyperpowerset. LetΘ = {θ ,...,θ }be
1 n
Θ
afiniteset(calledframe)withnexhaustiveelements. TheDedekindlattice,calledhyperpowersetD
withinDSmTframe,isdefinedasthesetofallbuiltstatementsfromtheelementsofsetΘwiththe∪
and∩operatorssuchthat:
1. ∅,θ ,...,θ ∈ DΘ
1 n
2. If A, B ∈ DΘ,then A∩B ∈ DΘ and A∪B ∈ DΘ.
Θ
3. NootherelementisincludedinD withtheexceptionofthosementionedat1and2.
Symmetry2018,10,198 8of26
DΘ duals(obtainedbychangingwithinexpressionstheoperator∩withtheoperator∪)isDΘ.
InDΘ thereareelementsthataredualwiththemselves. ThecardinalityofDΘ increaseswith2nwhen
thecardinalityofΘisn. GeneratingtheDΘ hyperpowersetiscloseconnectedwiththeDedekind[54,
55]knownproblemofisotoneBooleanfunctionset. BecauseforanyfinitesetΘ, |DΘ| ≥ |2Θ|,wecall
DΘ thehyperpowersetofΘ.
Theθ,i =1,...,nelementsfromΘformthefinitesetofsuppositions/conceptsthatcharacterize
i
thefusionproblem. DΘ representsthefreemodelofDSmMf(Θ)andallowsworkingwithfuzzy
conceptsthatdescribetheintrinsicandrelativecharacter. Thiskindofconceptcannotbeaccurately
distinguishedwithinanabsoluteinterpretationbecauseoftheunapproachableuniversaltruth.
Withallofthis,therearecertainparticularfusionproblemsthatimplydiscreteconcepts,wherethe
θ elementsareexclusivelytrue. Inthiscase,alltheexclusivityconstraintsofθ,i =1,...,nmustbe
i i
includedinthepreviousmodeltoproperlycharacterizethetruthinesscharacterofthefusionproblem
Θ Θ
andtomatchreality. Forthis,thehyperpowersetD isdecreasedtotheclassicpowerset2 ,forming
thesmallesthybridDSmmodel,notedwithM0(Θ),andcoincideswithShafer’smodel.
Besides the problem types that correspond with the Shaffer’s model M0(Θ) and those that
correspond with the DSm free model Mf(Θ), there is an extensive class of fusion problems that
includeinΘstates,continuousfuzzyconceptsanddiscretehypothesis. Inthiscasewemusttakeinto
considerationcertainexclusivityconstraintsandsomenon-existentialconstraints. Eachfusionhybrid
problemisdescribedbyaDSmhybridmodelM(Θ)withM(Θ) (cid:54)= Mf(Θ)andM(Θ) (cid:54)= M0(Θ).
4.2.2. GeneralizedBeliefFunctions
Starting from a general frame Θ, we define a DΘ → [0,1] transformation associated with
aninformationsourceBlike[54]
m(∅) =0and∑ m(A) =1. (1)
A∈DΘ
Them(A)valueiscalledgeneralizedbasicbeliefassignmentofA.
ThegeneralizedtrustandplausibilityaredefinedinthesamewayasinDempster–Shafertheory[47]
∑
Bel(A) = m(B), (2)
B ⊆ A
B ∈ DΘ
∑
Pl(A) = B∩A (cid:54)=∅ m(B). (3)
B ∈ DΘ
These definitions are compatible with the classic trust function definition from the
Θ Θ
Dempster–Shafer theory when D is reduced to 2 for fusion problems where the Shafer model
M0(Θ)canbeapplied. Westillhave∀ A ∈ DΘ, Bel(A) ≤Pl(A). Tonoticethatwhenweworkwith
thefreeDSmMf(Θ)model,wewillalwayshavePl(A) =1, ∀ A (cid:54)=∅∈ DΘ,whichisnormal[54].
4.2.3. DSmClassicRuleofCombination
When the DSm free model Mf(Θ) can be applied, the combination rule mMf(Θ) ≡ m(.) (cid:44)
[m ⊗m ](.)oftwoindependentsourcesB andB thatprovideinformationonthesameframeΘ
1 2 1 2
with the belief functions Bel (.) and Bel (.) associated to gbba m (.) and m (.) correspond to the
1 2 1 2
conjunctiveconsensusofsources. Datacombinationsaredonebyusingtheformula[54]
Θ ∑
∀C ∈ D , mMf(Θ)(C) ≡ m(C) = A,B ∈ DΘ m1(A)m2(B). (4)
A∩B =C
Symmetry2018,10,198 9of26
BecauseDΘ isclosedunder∩and∪operators,thisnewcombinationruleguaranteesthatm(.)Is
ageneralizedtrustvalue,meaningthat m(.) : DΘ → [0,1]. Thisruleofcombinationiscommutative
andassociativeandcanbeusedallthetimeforsourcesfusionwhichimpliesfuzzyconcepts. Thisrule
canbeextendedwitheaseforcombiningk>2independentinformationsources[55,56].
BecauseofthehighnumberofelementsinDΘ,whenthecardinalityofΘincreases,theneedof
computationalresourcesalsoincreasesforprocessingtheDSmcombinationrule. Thisobservationis
trueonlyifthecore(thesetofgeneralizedbasicbeliefassignmentfortheneededelements)K (m )
1 1
andK (m )coincidewithDΘ,meaningthatwhenm (A) >0andm (A) >0forany A (cid:54)=∅∈ DΘ.
2 2 1 2
For most practical applications, the K1(m1) and K2(m2) dimensions are much smaller than (cid:12)(cid:12)DΘ(cid:12)(cid:12)
becausSeymtmheetryi n20f1o8r, m10,a xt FioOnR PsEoEuRr RcEeVsIEpWro videmostofthetimethegeneralizedbasicbeliefassi9g nofm 26e ntfor
onlyonesubsetofhyperpowerset. ThisfacilitatestheDSmclassicruleimplementation.
assignment for only one subset of hyper power set. This facilitates the DSm classic rule
Figure 1 presents the DSm combination rule architecture. The first layer is formed by all the
implementation.
generalizedbasicbeliefassignmentvaluesoftheneededelementsA, i =1,...,nofm (.). Thesecond
Figure 1 presents the DSm combination rule architecture. The ifirst layer is formed1 by all the
layer gisenmeraadlizeedo ubtasoifc ablellitehf easgseignnemraelnitz evdalubeass iocf btheeli enfeeadsesdig nelmemeennttsv a(cid:1827)l(cid:3036)u,(cid:1861)e=s 1B,i…, ,i(cid:1866) =of 1(cid:1865),(cid:2869).(...),. kTohfe m2(.).
Eachnseocdoendfr olamyert hies fimrasdtel aoyuetr oifs aclol ntnhee cgteendewraliitzhede abcahsinc obdeleieof fatshsiegnsemceonnt dvalaluyeesr .(cid:1828)T,h(cid:1861)e=o1u,…tp,u(cid:1863) tolfa yeris
(cid:3036)
create(cid:1865)db(y.).c oEmachb innoindge ftrhoemg tehnee friraslti zlaeyderb aiss icconbneelicetefda swsiitghn emacehn ntovdael uoef sthoef saelclotnhde lpaoyesrs.i bTlheei onutetprsuet ctions
(cid:2870)
A ∩Bla,yeir= is1 c,r.e.a.t,edn bayn dcojm=bin1i,n.g. .t,hek .gIefnwerealwizeodu lbdashica vbeelaieft haisrsdigsnomuerncte vtaoluperso vofi daell gthene eproaslsiizbelde basic
i j
beliefianstseirgsnecmtioennst v(cid:1827)a(cid:3036)l∩ue(cid:1828)s(cid:3037),m(cid:1861)=(.1),,…th,i(cid:1866)s wanodu ld(cid:1862)=ha1v,…e,b(cid:1863)e.e nIf cowme bwinoeudldb yhapvlea cain tghiirtdb estowureceen ttoh eporouvtipduet layer
3
generalized basic belief assignment values (cid:1865) (.), this would have been combined by placing it
andthesecondonethatprovidesthegeneraliz(cid:2871)edbasicbeliefassignmentvaluesm (.). Duetothe
2
between the output layer and the second one that provides the generalized basic belief assignment
commutativeandassociativepropertiesofDSmclassicruleofcombination,indevelopingtheDSm
values (cid:1865) (.). Due to the commutative and associative properties of DSm classic rule of
(cid:2870)
network,aparticularorderoflayersisnotrequired[54].
combination, in developing the DSm network, a particular order of layers is not required [54].

Figure 1. Graphical representation of DSm classic rule of combination for ℳ(cid:3033)(Θ) [35]
Figure1.GraphicalrepresentationofDSmclassicruleofcombinationforMf(Θ)[35].
5. Decision-Making Algorithm
5. Decision-MakingAlgorithm
As observed in this paper, according to the object shape and assigned task, grasping is divided
Ainstoob esiegrhvte cdatiengtohriisesp [a6p3e]:r ,sapchceorricdailn ggratosptihneg,o bcyjelicntdsrhicaapl egaransdpiansgs,i glanteerdalt agsrka,spg,r apsrpisimngatiisc dgirvaisdpe, dinto
eightccairtceuglaorr igersas[p6i3n]g:,s hpohoekr gicraaslpginrags, pbuinttgo,nc pyrleinssdinrgic, aalngdr paussphiinngg,. lFartoemra tlhgesrea sgpra,spprinisgm tyaptiecs,g trhaes mpo,scti rcular
used ones are cylindrical and prismatic grasping (see Table 1). These can be used in almost any
grasping,hookgrasping,buttonpressing,andpushing. Fromthesegraspingtypes,themostused
situation and we can say that spherical grasping is a particular grasping of these two. The spherical
onesarecylindricalandprismaticgrasping(seeTable1). Thesecanbeusedinalmostanysituation
grasp is used for power grasping, when the contact with the object is achieved with all the fingers’
andwecansaythatsphericalgraspingisaparticulargraspingofthesetwo. Thesphericalgraspis
phalanges and the hand’s palm. This is why a requirement for classification by the shape of the
usedforpowergrasping,whenthecontactwiththeobjectisachievedwithallthefingers’phalanges
object is needed. Due to the fact that these types are more often encountered, they were taken into
andthceonhsaindedr’astipoanl fmor. tThhe isstuisdiwedh yfuasiorenq puriorbelmemesn. tforclassificationbytheshapeoftheobjectisneeded.
DuetotheTfhaec tfuthsiaotnt hperosbeletymp eaismasr etom aochreievoeft ea ncleanssciofiucanttioenre, db,yt shheaypew oefr eobtjaekctesn toin gtroascpo,n ssoi dthearat ttihoensef orthe
studiecdanfu msiaotcnhp wroitbh ltehme so.ther three types of grasping studied. The target objects are classified into three
Tchaetefguosriieosn: spprhoebrel,e pmaraalilmelesptiopeadc,h ainedv ecyalincdlaesrs. iFfiocra etaicohn c,abteygsohrya,p ae gorafsopibnjge cttyspeto isg arsassigpn,esdo [5th6]a. tthese
Following the presented theory in Section 4, the information is provided by two independent
canmatchwiththeotherthreetypesofgraspingstudied. Thetargetobjectsareclassifiedintothree
sources (observers): a stereovision system and a Kinect sensor. The observers are presented in
categories: sphere,parallelepiped,andcylinder. Foreachcategory,agraspingtypeisassigned[56].
Section 3, and are used to scan the robot’s work environment. By using the information provided by
FollowingthepresentedtheoryinSection4,theinformationisprovidedbytwoindependent
the two observers, a 3D virtual image of the environment is achieved, from which the human
sourceosp(eorbatsoerr vcheross)e:s athsete orbejoevcti stioo nbes gyrsatespmeda,n tdhuas Kdienfiencitnsge tnhseo gr.raTshpeinogb tsaesrkv tehrast amreusptr bees eanctheideviend Sbeyc tion3,
andarteheu sreodbotto. sTchaen 3thDe rimobaoget’ sowf othrke eonbvjeicrto, nimsoelantet.d Bfyroumsi ntghet hseceinnef,o rims actoimonpaprerdo viwdiethd bthyreteh etwo
templates—formed by similar methods—which represent a sphere, a parallelepiped, and a cylinder.
Afterwards, a template matching algorithm is applied to place the object in one of the three
categories, with a certain matching percentage. This percentage can vary according to the conditions
in which the images are obtained (weak light, object from which the light is reflected, etc.). The data
taken from each sensor are then individual processed with a neutrosophication algorithm, with the
purpose of obtaining the generalized basic belief assignment values for each hypothesis that can
characterize the system. In the next step, having the basic belief assignment values, we combine the
Symmetry2018,10,198 10of26
observers,a3Dvirtualimageoftheenvironmentisachieved,fromwhichthehumanoperatorchoses
theobjecttobegrasped,thusdefiningthegraspingtaskthatmustbeachievedbytherobot. The3D
imageoftheobject,isolatedfromthescene,iscomparedwiththreetemplates—formedbysimilar
methods—whichrepresentasphere,aparallelepiped,andacylinder. Afterwards,atemplatematching
algorithmisappliedtoplacetheobjectinoneofthethreecategories,withacertainmatchingpercentage.
Thispercentagecanvaryaccordingtotheconditionsinwhichtheimagesareobtained(weaklight,
objectfromwhichthelightisreflected, etc.). Thedatatakenfromeachsensorarethenindividual
processedwithaneutrosophicationalgorithm,withthepurposeofobtainingthegeneralizedbasic
beliefassignmentvaluesforeachhypothesisthatcancharacterizethesystem. Inthenextstep,having
thebasicbSeymlimeeftray 2s0s1i8g, 1n0,m x FeOnRt PvEEaRl uREeVsI,EWw ecombinethedataprovidedbythetwoobserve1r0 sofb 2y6 usingthe
classicDSdmatar uplreovoidfecdo bmy tbhien tawtoio onb.seTrvheers nbye xutsisntge tphei sclatossiac pDpSmly rauled oefn ceoumtbrionsaotiopnh. iTchaet nioexnt satlegpo isr ittoh monthe
obtainedvaaplpuleys a, dtoenaecuhtrioesvoephthiceatdioenc ailsgioornithomn othn ethseh oabptaeinoefdt vhaeluoebs,j etoc tacbhyiepvela tchien dgeciitsiionnt oont htheet shhraepee categories
mentionedofa tbhoe voebj.ecTt hbey epnlatciirneg pitr ionctoe sthsei sthvreise ucaatlelgyorrieeps rmeesnetniotneedd ianboFvieg. uTrhee 2e.ntire process is visually
represented in Figure 2.

Figure 2. Diagram of the proposed algorithm.
Figure2.Diagramoftheproposedalgorithm.
5.1. Data Neutrosophication
5.1. DataNeutrEoascohp ohbisceartvieorn provides a truth percentage for each system’s state. The state set Θ=(cid:4668)(cid:2016) ,(cid:2016) ,(cid:2016) (cid:4669)
(cid:2869) (cid:2870) (cid:2871)
that characterizes the fusion problem is
Eachobserverprovidesatruthpercentageforeachsystem’sstate. ThestatesetΘ = {θ ,θ ,θ }
1 2 3
Θ=(cid:4668)(cid:1845)(cid:1868),(cid:1842)(cid:1853),(cid:1829)(cid:1877)(cid:4669), (5)
thatcharacterizesthefusionproblemis
where Sp = sphere, Pa = parallelepiped, and Cy = cylinder.
To compute the belief values for thΘe h=yp{erS ppo,wPear, sCety }(cid:1830),(cid:2944) elements we developed an algorithm (5)
based on the neutrosophic logic. The hyper power set (cid:1830)(cid:2944) is formed by using the method presented
in Section 4.2.1 and has the form
whereSp=sphere,Pa=parallelepiped,andCy=cylinder.
(cid:1830)(cid:3061)=(cid:4668)∅,(cid:1845)(cid:1868),(cid:1842)(cid:1853),(cid:1829)(cid:1877),(cid:1845)(cid:1868)∪(cid:1842)(cid:1853),(cid:1845)(cid:1868)∪(cid:1829)(cid:1877),(cid:1829)(cid:1877)∪(cid:1842)(cid:1853),(cid:1845)(cid:1868)∩Θ(cid:1842)(cid:1853),(cid:1845)(cid:1868)∩(cid:1829)(cid:1877),(cid:1829)(cid:1877)∩(cid:1842)(cid:1853),(cid:1845)(cid:1868)∩
TocomputethebeliefvaluesforthehyperpowersetD elementswedevelopeda(n6)a lgorithm
((cid:1829)(cid:1877)∪(cid:1842)(cid:1853)),(cid:1829)(cid:1877)∩((cid:1845)(cid:1868)∪(cid:1842)(cid:1853)),(cid:1842)(cid:1853)∩((cid:1829)(cid:1877)∪(cid:1845)(cid:1868)),(cid:1845)(cid:1868)∪(cid:1829)(cid:1877)∪(cid:1842)(cid:1853),(cid:1845)(cid:1868)∩(cid:1829)(cid:1877)∩(cid:1842)(cid:1853)(cid:4669) .
Θ
basedontheneutrosophiclogic. ThehyperpowersetD isformedbyusingthemethodpresentedin
The statements of each observer are handled in ways of truth (T), uncertainty (I), and falsity (F),
Section4.2.1andhastheform
specific to the neutrosophic logic. Due to the fact that (cid:1832)=1−(cid:1846)−(cid:1835), the statements of falsity are not
tDakΘen =int(cid:8)o ∅co,nSsipd,erPataio,nC. y,Sp∪Pa,Sp∪Cy,Cy∪Pa,Sp∩Pa,Sp∩Cy,Cy∩Pa,Sp∩
The neutrosophic algorithm has as input the certainty probabilities (truth) provided(cid:9) by the (6)
observ(eCrsy o∪n tPhae )s,yCstyem∩’s( sStapte∪s. TPhae)s,eP paro∩ba(bCiliytie∪s aSrpe )th,Senp p∪roCceyss∪edP uasi,nSgp th∩e dCeysc∩ribPeda ru.les in
Figure 3. If the difference between the certainties probabilities used at a certain point by the
processing algorithm is larger than a certain threshold found by trial and error, then we will
consider that the uncertainty percentage between the compared states is null, and the probability