research article screw theory based singularity...

Research ArticleScrew Theory Based Singularity Analysis ofLower-Mobility Parallel Robots considering the MotionForceTransmissibility and Constrainability

Xiang Chen12 Xin-Jun Liu12 and Fugui Xie12

1 State Key Laboratory of Tribology and Institute of Manufacturing Engineering Department of Mechanical EngineeringTsinghua University Beijing 100084 China

2 Beijing Key Lab of PrecisionUltra Precision Manufacturing Equipments and Control Tsinghua UniversityBeijing 100084 China

Correspondence should be addressed to Xin-Jun Liu xinjunliumailtsinghuaeducn and Fugui Xie xiefgmailtsinghuaeducn

Received 14 June 2014 Accepted 10 September 2014

Academic Editor Carlo Cosentino

Copyright copy 2015 Xiang Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Singularity is an inherent characteristic of parallel robots and is also a typical mathematical problem in engineering applicationIn general to identify singularity configuration the singular solution in mathematics should be derived This work introduces analternative approach to the singularity identification of lower-mobility parallel robots considering themotionforce transmissibilityand constrainabilityThe theory of screws is used as themathematic tool to define the transmission and constraint indices of parallelrobots The singularity is hereby classified into four types concerning both input and output members of a parallel robot thatis input transmission singularity output transmission singularity input constraint singularity and output constraint singularityFurthermore we take several typical parallel robots as examples to illustrate the process of singularity analysis Particularly theinput and output constraint singularities which are firstly proposed in this work are depicted in detail The results demonstratethat the method can not only identify all possible singular configurations but also explain their physical meanings Therefore theproposed approach is proved to be comprehensible and effective in solving singularity problems in parallel mechanisms

1 Introduction

In theory parallel robots compared with their counterpartshave the potential to answer the increasing need for highstiffness compactness load-to-weight ratio and so forthWith this in mind extensive attention has been focused onparallel robots Particularly most of the parallel robots usedsuccessfully in engineering are those with lower mobilitySuch engineering applications are nowadays with a rapidrate utilized in precise manufacturing medical science spaceequipment and others [1]

Before the usage of parallel robots in engineering singu-larity as one significant issue must be investigated Singu-larity is one of the inherent characteristics of parallel robotswhich influences a considerable number of performancesincluding the workspace dexterity stiffness and load capac-ity Normally at singular configurations a parallel robot loses

control over degrees of freedom (DOFs) Either of two effectsgenerally occurs (1) the robot gains one or more unexpectedDOFs thereby degrading natural stiffness and diminishingload capacity in the direction of the additional DOF or (2) therobot loses one or more DOFs and lies at a dead point whereit is uncontrollable Therefore the determination of singularconfigurations which is critical to understanding a robotrsquoskinematics should be identified and avoided if possible

Basically the singularity is also a typical mathematicalproblem The singularity that happened in parallel robots iscorresponding to the singular solution in mathematics Inthe past three decades a number ofmathematical approacheshave been devoted to analyzing the singularity problemsin parallel robots The methods could be typically dividedinto two categories analytical and geometric methods Inthe analytical methods the Jacobian matrix of a parallelrobot was always derived to identify singularity that is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 487956 11 pageshttpdxdoiorg1011552015487956

2 Mathematical Problems in Engineering

Gosselin and Angeles [2] identified singularity in closed-loopchains mechanism by analyzing the determinant of Jacobianmatrix The rank deficiency of Jacobian matrix leads to thesingularity of a parallel robot [3] In the geometric methodsscrew theory and line geometry were commonly used tosolve singularity problems [4] Park and Kim [5] proposeda coordinate-invariant differential geometric approach toanalyzing the singularity in a parallel mechanism Hunt [6]laid down a general framework for using screw theory inanalyzing singularity in parallel robots Hao and McCarthy[7] concluded the conditions for line-based singularitiesvia the theory of screws Merlet [8] noticed that it is thepossibility to identify all singular configurations in a parallelrobot via line geometric method Ben-Horin Shoham [9]analyzed the singularity of a class of 6-DOF parallel robotsusing the superbracket and the Grassmann-Cayley operatorsMore recent approaches to singularity analysis were discussedby Davidson and Hunt [10] Ma and Angeles [11] Bonev et al[12] and Tsai [13]

The other significant issue in singularity analysis isthe classification of singularities On the basis of singularproperties of Jacobian matrices Gosselin and Angeles [2]classified the singularity into three main groups With thesame definitions Tsai [13] termed these three kinds of singu-larities as the inverse forward and combined singularitiesGogu [14] distinguished two types of singular configurationsstructural and kinematic singularities Park and Kim [5]classified closed-chain singularities into three basic typesconfiguration space singularity actuator singularity and end-effector singularity In addition Liu et al [15] combined thethree groups of singularities proposed by Park to come upwith the first-order singularity and established two-ordersingularities as degenerate and nondegenerate types Fangand Tsai [16] divided the singularity cases as limb singularitymanipulator singularity and actuation singularity Further-more Zlatanov et al [17] considered both the input andoutput of a parallel robot and concluded that the singularityshould always happen for the combination cases among sixclassified types redundant input impossible input redun-dant output impossible output redundant passive motionand increased instantaneous mobility Note that the singular-ity is an inherent characteristic of parallel robots and not allsingularities could be avoided in practice The elimination ofdifferent singularities is another important issue in practicewhich attracted many researchers For example Li and Xu[18] discussed this problem from the aspect of optimalmechanism design

However the predecessorsrsquo contributions mainly focusedon finding new mathematical tools to search and dividethe solutions of singularities This work on the other handdiscusses the singularity problem from the view of itspractical physical meanings We put forward an alternativeviewpoint to the singularity in terms of the motionforcetransmission and constraint properties of lower-mobilityparallel robots As is well known the key function of aparallel robot is to transmit and constrain the underlyingmotion and forces between inputs and outputThereby in thefunctional process of a parallel robot any motion and forcewhich cannot be transmitted or constrained as expected

could lead to the singularities Thus by considering thetransmission and constraint properties we can evaluate thesingularities of lower-mobility parallel robots It should benoted that this work is an intuitive following study with theknowledge of the motionforce transmission and constraintanalysis of parallel robots proposed by our research team[19 20] The motionforce transmission concept and itsindices were proposed by Wang et al [19] in 2010 andthen the approach to motionforce constraint performanceanalysis was introduced in 2014 [20] In both literaturesthe indices used to evaluate the transmission and constraintperformance of parallel robots have the potential to indicatethe worst transmissibility and constrainability when they areequal to zero In such cases the mechanism cannot transmitor constrain the underlying motions and forces therebyresulting in singularities which are discussed in this work

The remainder of this paper is organized as follows Basedon the theory of screws Section 2 proposes a singularityidentification approach in terms of the transmission andconstraint analysis of lower-mobility parallel robots Severalmodified indices are introduced to identify the singularitiesand the classification of singularity is discussed based onthe indices In Section 3 several typical parallel robots aretaken as examples to demonstrate the usage of the proposedapproach and indices Finally Section 4 follows with a con-clusion of this work

2 Approach to Analyzing Singularityand the Classification

There is no doubt that screw theory is an efficient and pro-found mathematical tool for studying parallel mechanismsand it has been used in evaluating the kinematic perfor-mance type synthesis and singularity analysis [21ndash25] Weintroduce our analytical approach to elucidating singularityand its classifications in which we consider motionforcetransmissibility and constrainability in parallel robots on thebasis of screw theory It should be noted that this work onlyconsiders the nonredundant parallel robots with fewer thansix DOFs wherein there exist the constraint motions andforces

21 Mathematical Foundation In general a screw can berepresented in dual-vector form as follows [26]

$ = 119904+ isin 1199040 (1)

where real part 119904 is a unit vector in the direction of thescrew axis dual part 119904

0denotes the moment of the screw

with respect to the origin and isin refers to dual unity whichis defined such that

isin = 0 isin2= 0 (2)

Considering one arbitrary point 119875 of position vector pon the screw axis we have

s0= p times s + ℎs (3)

Mathematical Problems in Engineering 3

Twist screw system

Linearlyindependent

Linearlyindependent

Restricted twist screw(RTS)

Dual

Dual

Reciprocal

Wrench screw system

Transmission wrench screw(TWS)

Constraint wrench screw(CWS)

Permitted twist screw(PTS)

$P

$R

$T

$C

Figure 1 Interrelationship among four twist and wrench screwsystems

where ℎ is defined as the pitch of the screw Here we get thefollowing relationship

ℎ = s119879s0 (4)

Given two screws $1and $2 the inner product between $

1

and $2is given as

$1sdot $2= 11990411199042+ isin (119904

111990402

+ 119904011199042) (5)

Hereby the dual part in (5) is defined as the reciprocalproduct labeled as ∘ between two screws

$1∘ $2= 119904111990402

+ 119904011199042 (6)

Considering a wrench represented by $1= 119891(f+ isin 120591) and

a twist denoted by $2= 119908(w+ isin v) their reciprocal product

$1∘ $2 indicates the power generated by the wrench on the

body that moves with the foregoing twist

$1∘ $2= 119891119908 (f sdot v + 120591 sdot w) (7)

where 119891 and 119908 are the amplitudes of the wrench screw andtwist screw respectively

The reciprocal product is a value indicating the instanta-neous power between thewrench and twist Particularly if thevalue equals zero $

1∘$2= 0 itmeans that the relevantwrench

applies nowork on the rigid body in the direction of the twistThis is a singular solution in mathematics corresponding toa singularity configuration in mechanism

22 Approach to Singularity Identification In a parallel robottwo important screw systems are involved that is the twistscrew system and wrench screw system The twists typicallyinclude permitted twist screws (PTSs denoted by $

119875) and

restricted twist screws (RTSs denoted by $119877) Meanwhile

the wrenches (forces and moments) are composed of trans-mission wrench screws (TWSs) and constraint wrench screws(CWSs) in a lower-mobility parallel robotTheTWS denotedby $119879 plays the role of transmitting the PTS and the

CWS denoted by $119862 is supposed to constrain the RTS in a

mechanism Figure 1 shows the interrelationship among thefour subscrew systems spanned by PTSs RTSs TWSs andCWSs respectively The details about the calculation processhave been depicted in [26 27]

As is well known the essential functions of a lower-mobility parallel robot contain two sides of a coin transmit-ting the expected motionforce from its input members tothe outputmember and constraining unwantedmotionforcebetween inputs and output of the parallel robot Recalling(5) one can see that if the wrench $

1= $119879 indicates

the transmission wrench exerted on the rigid body whilethe twist $

2= $119875 indicates the underlying permitted

motion their reciprocal product signifies their instantaneouspower If the product is equal to zero it means that theTWS applies no work in the direction of the PTS at thisconfiguration of the parallel robot Thus the TWS loses theability of transmitting the PTS stemming from a transmissionsingularity In a similar way if we set the wrench screwas $1

= $119862 the constraint wrench existed in the rigid

body and the twist screw as $2

= $119877 the restricted

motion their reciprocal product could be used to identifyconstraint singularity If the product between the CWS andRTS is zero it means the RTS is not constrained by theCWS At this configuration the parallel robot adds one DOFresulting in a constraint singularity Hereby the singularity ofa parallel robot can be identified as transmission singularityand constraint singularity

In addition a parallel robot as previously mentionedcontains the inputs and output performing input twistscrews and output twist screws respectively Normally theinput members are the actuators and the output memberis the moving platform Inspired from the classificationidea proposed by Zlatanov et al [17] we can divide thetransmission singularity into input transmission singularity(ITS) and output transmission singularity (OTS) Similarlywe consider the input and output constraint singularities (ICSand OCS) correspondingly The four indices of singularitiescan be defined as

ITS =1003816100381610038161003816$119879 ∘ $119875119868

1003816100381610038161003816 (8)

OTS =1003816100381610038161003816$119879 ∘ $119875119874

1003816100381610038161003816 (9)

ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 (10)

OCS =1003816100381610038161003816$119862 ∘ $119877119874

1003816100381610038161003816 (11)where $

119875119868is the selected input PTS in each limb $

119875119874is the

output twist of the robot related to the analyzed transmissionwrench $

119877119868is the RTS in the limb and $

119877119874is the output twist

related to the constraint wrench of the robotHere we demonstrate the calculation process of $

119875119874and

$119877119874

of the output using the tricks of ldquolockingrdquo and ldquoreleasingrdquoin screw theory In an 119899-DOF (0 lt 119899 lt 6) nonredundantparallel robot the wrench screw system comprises 119901 TWSsand 119902 CWSs where 119901 = 119899 and 119901 + 119902 = 6 are alwaysestablished In order to calculate the underlying $

119875119874119894related

to the 119894th TWS we can lock all other (119901 minus 1) TWSs therebyyielding five constraint wrenches including (119901minus1) TWSs and119902 CWSs Then using reciprocal product in screw theory onecan calculate the related $

119875119874119894as

$119875119874119894

∘ $119879119895

= 0 119895 = 1 2 119901 minus 1

$119875119874119894

∘ $119862119896

= 0 119896 = 1 2 119902(12)


Limited-DOFparallel

manipulator

Wrench analysis of themanipulator

Wrench screw subsystemsTWS and CSW

p + q = 6

Motionforce constraint analysis Motionforce transmission analysis

ICS = 0 OCS = 0 ITS = 0 OTS = 0

Input constraintsingularity

Output constraintsingularity

Input transmissionsingularity

Output transmissionsingularity

$T1 $T2 $Tp $C1 $C2 $Cq

Figure 2 Process of singularity analysis in a lower-mobility parallel robot

On the other hand if we release the constraint of the119894th CWS while locking all the TWSs the parallel robot isassumed to be a single-DOF one The only imagined twistmotion $

119877119874119894 is considered to be the output RTS with regard

to the 119894th CWS Hereby we can achieve five constraintwrenches as well that is 119901 TWSs and (119902 minus 1) CWSs Therelated $

119877119874119894could be calculated from

$119877119874119894

∘ $119879119895

= 0 119895 = 1 2 119901

$119877119874119894

∘ $119862119896

= 0 119896 = 1 2 119902 minus 1(13)

To sum up the entire process of the singularity analysisof a lower-mobility parallel robot is depicted in Figure 2Using (8)ndash(11) we can evaluate four classified types ofsingularities that is input constraint singularity output con-straint singularity input transmission singularity and outputtransmission singularity Expressions (8)ndash(11) are nothingbut the numerators of the existing performance indicesproposed in [18 19] The ranges of modified indices arezero to infinity Clearly the purpose here is to identify thesingularity of a parallel robot thus only the indices beingzero aremeaningful Four indices defined from the reciprocalproduct can be used not only to identify the singularitiesbut also to explain their physical meanings from the view ofmotionforce transmission and constraint performance

3 Examples

We now turn towards illustrating how the classified sin-gularities happen and function by analyzing some typical

lower-mobility parallel robots To realize this analysis weshould firstly study kinematics of the parallel mechanismsand describe their twists and wrenches via screw theory

31 Input Constraint Singularity Input constraint singularityis a new defined type of singularity which was never involvedin the literatures It is the singularity caused by the constraintwrench in the limb of a parallel robot At this singularconfiguration the constraint force in input limbs cannotconstrain unwanted motions as expected We say that theconstrainability of the limb becomes the worst resulting inan input constraint singularity Hence the parallel robotadds a DOF and loses control when its limbs lose theconstrainability Here we take 3-US robot (Figure 3) a purerotational mechanism as an example to show details of theusage of ICS index

Figure 3 is a 3-US parallel robot with three identical US-type limbs (U denotes universal joint S refers to sphericaljoint) Normally U joint is a kinematic equivalent with tworevolute joints with perpendicular axes and S joint can becomposed of three orthogonal revolute joints Clearly eachUS-type limb has five DOFs yielding one constraint forceThe mechanism has three US-type limbs achieving threeconstraint forces together Thus the mechanism performsthreeDOFs that is three rotationalDOFs In order to controlit three actuators should be settled for this mechanism Forthe sake of simplicity the revolute joints connected to thefixed base platform (in the U joint) are chosen to be actuated

In order to calculate the ICS of themechanism we shouldfirstly analyze the input US-type limbs via screw theory


Fixed Actuated

Uz

S

x

yO

Figure 3 A 3-US parallel robot

z

x

y

O$P1 $P2

$P5$P3

$P4

120572

Figure 4 A US-type limb

(Figure 4) According to the relations depicted in Figure 1 itis intuitive to represent permitted twists constraint wrenchesand restricted twists in this limb

Figure 4 is a US-type limb where a local coordinatesystem 119874-119909119910119911 is established in the center point of U jointUnder the coordinate we can achieve five PTSs as

$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 0 0)

$1198753

= (cos120572 0 sin120572) + isin (0 0 0)

$1198754

= (0 1 0) + isin (minus119897 sin120572 0 119897 cos120572)

$1198755

= (minus sin120572 0 cos120572) + isin (0 119897 0)

(14)

Then using reciprocal and dual relations among thetwists and wrenches we can achieve a CWS and an input RTSas

$119862= (cos120572 0 sin120572) + isin (0 0 0)

$119877119868

= (0 0 0) + isin (0 0 1) (15)

1

09

08

07

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180

120572 (deg)

ICS

Figure 5 The relationship between the ICS and the angle 120572

Recalling the defined ICS index (see (8)) we can calculatethe ICS of the US-type limb as

ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 = |sin120572| (16)

The relationship between ICS index and the angle 120572 isshown in Figure 5 From this figure we can see that whenthe angle equals 0 or 120587 the ICS equals zero indicting that theinput constraint singularity happens at this configuration

One thing should be noted the singularity is only relatedto a whole parallel robot rather than a serial limb If we con-sider that a robot suffers from a singularity the instantaneousDOF of the mechanism varies rather than the motilities ofthe limb Hence we illustrate the singular configuration of3-US parallel robot in Figure 6 caused by the zero indexICS = 0 Figure 6 shows an input constraint singularityconfiguration of 3-US parallel robot At this configurationthe robot cannot constrain the translational motion along thedirection of the 119911-axis thereby adding a translational DOFalong the 119911-axis In other words the robot cannot resist anyexternal force along the direction of the 119911-axis suffering fromthe poorest stiffness

32 Output Constraint Singularity Figure 7 shows a typical 3-RPS parallel robot [28]This robot has been analyzed by someresearchers including its singularity property that is Briotand Bonev [29] discussed its singularity loci by calculatingthe Jacobian matrix However one type of singularity wasmissed in [29] that turns out the so-called output constraintsingularity in this paperTherebywe take 3-RPS parallel robotas an example to illustrate the usage of the introduced indexOCS

The mechanism has a moving platform that connects thebase platform by three identical RPS limbs Each RPS limbcontains a revolute joint prismatic joint and spherical jointin turn Three P joints are actuated Considering that therobot has zero-torsion performance we apply the tilt-and-torsion (TampT) angles (120593 120579 120590) to describe the orientation ofthe moving platform where 120593 120579 and 120590 are referred to as


Fixed

Uz

S

xy

Figure 6 A singular configuration of the 3-US parallel robot

z

xy

O

Figure 7 A 3-RPS parallel robot

azimuth tilt and torsion angles respectively [30] Here welet 120590 be equal to 0 indicating zero torsion angle

To evaluate output constraint performance of the mech-anism we should firstly analyze its twists and wrenchesFigure 8 shows one of the three identical RPS limbs Screwtheory is used to analyze the limb wherein the S joint can beregarded as a combination of three R joints with orthogonalaxes In the local coordinate frame attached to R joints inFigure 8 five PTSs can be easily achieved as

$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 cos120573 sin120573)

$1198753

= (1 0 0) + isin (0 119897 sin120573 119897 cos120573)

$1198754

= (0 1 0) + isin (minus119897 sin120573 0 0)

$1198755

= (0 0 1) + isin (119897 cos120573 0 0)

(17)

$P1

$P2

$P5

$P3

$T$C

$P4

z998400

x998400

y998400

120573

Figure 8 A RPS limb in 3-RPS mechanism

where 120573 is the angle between the leg and 1199101015840-axis and 119897 is the

length of telescopic limbThen the CWS and TWS of the limb can be correspond-

ingly calculated by reciprocal and dual properties as

$119862= (1 0 0) + isin (0 119897 sin120573 minus119897 cos120573)

$119879= (0 119897 cos120573 119897 sin120573) + isin (0 0 0)

(18)

We can find that the CWS ($119862) indicates a pure force in

the direction of the 1199091015840-axis passing through the center of Sjoint and the TWS ($

119879) is a pure force in the direction of the

legFrom the analysis of the RPS limb we can conclude that

3-RPS robot contains three CWSs and three TWSs in sumThen according to (13) in Section 2 we can achieve the119894th RTS $

119877119874119894 by releasing the 119894th CWS at one time Then

three CWSs and their relevant RTSs can be calculated athand Recalling (11) we can identify the output constraintsingularity by using OCS index

Figure 9 shows the relationship between OCS andazimuth angle 120593 by fixing the tilt angle 120579 = 0 and Figure 10presents the relationship between OCS and tilt angle 120579

by fixing the azimuth angle 120593 = 0 We can find thatthe CTP value keeps constant with the change of angle 120593while differing with the change of angle 120579 When the tiltangle 120579 = 180

∘ CTP equals zero indicating the constraintsingularity That is to say a 3-RPS parallel robot suffers froman output constraint singularity when its moving platformis upside down (Figure 11) We now turn towards explainingthe physical meaning of this type of singularity from theview of motionforce constrainability At this configurationthree CWSs (Figure 11) intersect at one point (119862 point)the robot cannot counterbalance any external torque on theintersection point behaving the poorest stiffness around therotational axis Thus the mechanism adds one rotationalDOF resulting in an output constraint singularity


15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS

Figure 9 The relationship between the value of OCS and azimuthangle 120593 by fixing tilt angle 120579 = 0

1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)

Figure 10 The relationship between the value of OCS and tilt angle120579 by fixing azimuth angle 120593 = 0

33 Input Transmission Singularity Input transmission sin-gularity indicates that the motionforce cannot be transmit-ted out from inputs This concept has been proposed in ourprevious research [25] wherein the index is used to analyzethe closeness between a pose and the singularity Howeverthe aim of this work is to identify all possible singularities ina parallel robot thus we modified the OCS index as (9)

Figure 12 is the kinematic schematic of a planar 3-RRRparallel robot The mobile platform is connected to the baseby three identical RRR limbs each with three revolute jointsand two bars Figure 13 shows a planar RRR limb where thejoint connected to the base is actuated With respect to theglobal coordinate system 119874-119909119910 three PTSs can be expressedas

$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O

Figure 11 Output constraint singular configuration of 3-RPS robot

A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400

ActuatorPassive joint

Figure 12 A planar 3-RRR parallel robot

O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti

Figure 13 One RRR leg of the parallel robot


ActuatorPassive jointActuatorPassive joint

(a)


(b)

Figure 14 Input transmission singularity (a) when 1205791= minus90∘ and 120595

1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘

where 119886119898 119887119898 119898 = 1 2 3 are constants related to the

instantaneous position of every kinematic pairSince the first joint connected to the base is actuated

the corresponding twist screw is an input PTS which can beexpressed as

$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)

Clearly this planar RRR limb contains three twists yield-ing no CWS in the plane Hereby we conclude that the 3-RRRparallel robot does not suffer from any constraint singularityEach limb has its own TWS transmitting the motionforcebetween input members and the output platform which canbe calculated as

$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)

where 119891119894is a pure force vector moving along the direction

of the centers of the two passive joints and 119904119894represents

a position vector at the original point of the proposedcoordinate system (Figure 13)

According to (8) we can calculate the ITS index toelucidate the input transmission singularity

ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)

where 1199031denotes the length of the leg AB120595 is the angle of leg

BC and the 119909-axis and 120579 is the angle of leg AB and the 119909-axis(Figure 13)

From (20) one can find that ITS equals zero if and only if120595 = 120579+119894120587 (119894 = 0 1 2 )That is to say this singularity occurswhen the active legAB and passive leg BC are aligned Indeedone of the three legs is fully stretched-out or folded-back atthe input singular configurations (Figures 14(a) and 14(b))An infinitesimal rotation on the actuator of the link results

15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0

Figure 15 Input transmission singularity loci of a 3-RRR robot withrotational angle 120593 equal to 0

∘ 30∘ 60∘ 90∘ and 120∘

in no output motion in the mobile platform That means themotion cannot be transmitted from the inputs to the outputmember at such input transmission singular configurationsThemobile platformdegenerates the translational DOF alongdirection of the limb

Singularity loci comprise a series of continuous singular-ity points Accordingly input transmission singularity loci aremade up of all points with ITS = 0 The workspace enclosedby the singularity loci is always called the usable workspaceFigure 15 presents input transmission singularity loci of


1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05

0minus55 minus48 minus40 minus30 110 120 132 140

Figure 16 Relationship between OTS index and rotational angle 120593when the mobile platform is located at the original point

3-RRR planar robot with the rotational angles 120593 equal to 0∘30∘ 60∘ 90∘ and 120∘ respectively The usable workspacevaries with the change of the rotational angle

34 Output Transmission Singularity The key to analyzingthe output transmission singularity of a parallel robot iscalculating the output PTS in relation to each TWS Wealso take the planar 3-RRR parallel robot as an example toillustrate the usage of the defined index OTS

In a 3-RRR robot as mentioned there exist three TWSsin sum We isolate the permitted output twist by lockingany two TWSs while leaving one analyzed TWS free At thismoment the 3-RRR robot could be regarded as a single-DOFmechanism It is intuitive to see that the output PTS is onlycontributed by the analyzed TWS $

119879119894 whereas the other two

locked TWSs apply no work in the direction of the twistThen from (12) we can derive this twist $

119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)

The singularity depends not only on the pose of themobile platform but also on the positionThere are uncount-able configurable points in the usable workspace For the sakeof brevity we here choose original point (119909 119910) = (0 0) toevaluate the output transmission singularity From (9) we canidentify this type of singularity using the index OTS

OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)

When the mobile platform of 3-RRR parallel robot islocated at the original point (119909 119910) = (0 0) the OTSindex varies with the rotational angle 120593 of mobile platformFigure 16 illustrates the relation between OTS index and therotational angle 120593 varying in (minus180∘ 180∘] From Figure 16we can see that the OTS index equals zero when 120593

1= minus48∘

and 1205932

= 132∘ with regard to two output transmission

singular configurations respectively Figures 17(a) and 17(b)show the output transmission singular configurations

We now turn towards explaining the physical meaningin view of motionforce transmissibility At the output trans-mission singularities three TWSs converge to one pointresulting in the fact that any external torque along the axisthrough the convergence point cannot be counterbalancedby the TWSs That is to say the mechanism behaves thepoorest stiffness along the direction of the normal axis Atsuch configurations the mobile platform adds a rotationalDOF when all the actuators are locked

It should be noted that there exist some configurationsthat happen to suffer from more than one type of singularityThat is to say at some configurations both input and outputtransmission or constraint performances are poor yieldingdifferent kinds of singularities at the same time We candemonstrate that from the 3-RRR case with other parametersthat is two legs in each chain are equal 119903

1= 1199032 and the

radii of the base and mobile platforms are equal to eachother 119903

3= 1199034 In such architecture the base platform has

the same size as the mobile platform Using the proposedapproach presented in Section 2 we can calculate the ITS andOTS of this manipulator Figure 18 shows the relationshipsamong the two indices and the rotational angle of the mobileplatform in which the ITS and OTS values are both equalto zero when the rotational angle 120593 equals zero In otherwords the 3-RRR parallel manipulator suffers from bothinput and output transmission singularities (Figure 19) atone configuration in which the mobile and base platformscoincide with each other At this transmission singular con-figuration the transmission wrenches cannot transmit anymotion or force between the input members and the outputmemberThus the actuators could no longer drive themobileplatform

4 Conclusion

In this work we propose a new approach to identifying thesingularities of lower-mobility parallel robots which is animportant issue in engineering Although certain existingapproaches have been introduced to analyze singularitiesfromdifferent aspects this work provides an alternative view-point and approach to singularity identification and classifi-cationThe theory of screws is used as themathematical foun-dation to identify singularities considering the motionforcetransmissibility and constrainability Meanwhile we classifythe singularities into four types based on the transmissionand constraint performance in both the input and outputmembers Using screw theory four generalized indices ICSOCS ITS and OTS are proposed to identify input constraintsingularity output constraint singularity input transmissionsingularity and output transmission singularity respectivelyThe approach stands out with some merits including that itcan identify all possible singularities as well as explainingtheir physical meanings from the view of essential propertiesof parallel robots Furthermore based on the proposedapproach and indices we analyze the singularities of 3-US


y

xO


(a)


y

xO

(b)

Figure 17 Output transmission singularity configurations with rotational angles (a) 1205931= minus48

∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e

minus150 minus100 minus50 0 50 100 150

120593 (deg)

ITSOTS

Figure 18 Relationships among ITS and OTS values and therotational angle 120593

3-RPS and 3-RRR parallel robots All the singularity resultsaremeaningful to facilitate better understanding and usage ofthe mechanisms in engineering In addition the eliminationof these four classified types of singularities will be investi-gated in our further research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

O x

y


Figure 19 Input and output transmission singular configuration ofthe 3-RRR manipulator with parameters 119903

1= 1199032and 1199033= 1199034

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51375251 and 51135008)

References

[1] Y Patel and P George ldquoParallel manipulators applicationsmdashasurveyrdquo Modern Mechanical Engineering vol 2 no 3 pp 57ndash64 2012

[2] C Gosselin and J Angeles ldquoSingularity analysis of closed-loop kinematic chainsrdquo IEEE Transactions on Robotics andAutomation vol 6 no 3 pp 281ndash290 1990


[3] Q Xu and Y Li ldquoDesign and analysis of a new singularity-free three-prismatic-revolute-cylindrical translational parallelmanipulatorrdquo Proceedings of the Institution of Mechanical Engi-neers Part C Journal ofMechanical Engineering Science vol 221no 5 pp 565ndash577 2007

[4] N Shvalb M Shoham H Bamberger and D Blanc ldquoTopolog-ical and kinematic singularities for a class of parallel mecha-nismsrdquoMathematical Problems in Engineering vol 2009 ArticleID 249349 12 pages 2009

[5] F C Park and J W Kim ldquoSingularity analysis of closedkinematic chainsrdquo Journal of Mechanical Design Transactionsof the ASME vol 121 no 1 pp 32ndash38 1999

[6] K H Hunt Kinematic Geometry of Mechanisms Oxford Uni-versity Press London UK 1978

[7] F Hao and J M McCarthy ldquoConditions for line-based sin-gularities in spatial platform manipulatorsrdquo Journal of RoboticSystems vol 15 no 1 pp 43ndash55 1998

[8] J-P Merlet ldquoSingular configurations of parallel manipulatorsand Grassmann geometryrdquo International Journal of RoboticsResearch vol 8 no 5 pp 45ndash56 1989

[9] P Ben-Horin and M Shoham ldquoSingularity condition ofsix-degree-of-freedom three-legged parallel robots based onGrassmann-Cayley Algebrardquo IEEE Transactions on Roboticsvol 22 no 4 pp 577ndash590 2006

[10] J K Davidson and K H Hunt Robots and Screw TheoryApplications of Kinematics and Statics to Robotics OxfordUniversity Press London UK 2004

[11] O Ma and J Angeles ldquoArchitecture singularities of platformmanipulatorsrdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation pp 1542ndash1547 SacramentoCalif USA April 1991

[12] I A Bonev D Zlatanov and C M Gosselin ldquoSingularityanalysis of 3-DOFplanar parallelmechanisms via screw theoryrdquoJournal ofMechanicalDesign Transactions of theASME vol 125no 3 pp 573ndash581 2003

[13] L W Tsai Robot Analysis The Mechanics of Serial and ParallelManipulators Wiley-Interscience New York NY USA 1999

[14] G Gogu ldquoFamilies of 6R orthogonal robotic manipulators withonly isolated and pseudo-isolated singularitiesrdquoMechanism andMachine Theory vol 37 no 11 pp 1347ndash1375 2002

[15] G Liu Y Lou and Z Li ldquoSingularities of parallel manipulatorsa geometric treatmentrdquo IEEE Transactions on Robotics andAutomation vol 19 no 4 pp 579ndash594 2003

[16] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoFparallelmanipulatorswith identical limb structuresrdquoThe International Journal of Robotics Research vol 21 no 9 pp799ndash810 2002

[17] D Zlatanov R G Fenton and B Benhabib ldquoIdentification andclassification of the singular configurations of mechanismsrdquoMechanism and Machine Theory vol 33 no 6 pp 743ndash7601998

[18] Y Li and Q Xu ldquoKinematic analysis and design of a new 3-DOF translational parallel manipulatorrdquo Journal of MechanicalDesign Transactions of the ASME vol 128 no 4 pp 729ndash7372006

[19] J Wang C Wu and X-J Liu ldquoPerformance evaluation ofparallel manipulators motionforce transmissibility and itsindexrdquo Mechanism and Machine Theory vol 45 no 10 pp1462ndash1476 2010

[20] X-J Liu X Chen andM Nahon ldquoMotionforce constrainabil-ity analysis of lower-mobility parallel manipulatorsrdquo Journal ofMechanisms and Robotics vol 6 no 3 Article ID 031006 2014

[21] Y F Fang and L W Tsai ldquoEnumeration of a class of overcon-strained mechanisms using the theory of reciprocal screwsrdquoMechanism and Machine Theory vol 39 no 11 pp 1175ndash11872004

[22] H R Fang Y F Fang and K T Zhang ldquoReciprocal screwtheory based singularity analysis of a novel 3-DOF parallelmanipulatorrdquo Chinese Journal of Mechanical Engineering vol25 no 4 pp 647ndash653 2012

[23] X W Kong and C M Gosselin ldquo Type synthesis of 3-DOFtranslational parallel m anipulators based on screw theoryrdquoJournal of Mechanical Design Transactions of the ASME vol126 no 1 pp 83ndash92 2004

[24] Q Xu and Y Li ldquoAn investigation on mobility and stiffness ofa 3-DOF translational parallel manipulator via screw theoryrdquoRobotics and Computer-IntegratedManufacturing vol 24 no 3pp 402ndash414 2008

[25] X-J Liu C Wu and J Wang ldquoA new approach for singularityanalysis and closeness measurement to singularities of parallelmanipulatorsrdquo ASME Journal of Mechanical and Robotics vol4 no 4 Article ID 041001 2012

[26] G R Veldkamp ldquoOn the use of dual numbers vectors andmatrices in instantaneous spatial kinematicsrdquo Mechanism andMachine Theory vol 11 no 2 pp 141ndash156 1976

[27] T Huang H T Liu and D G Chetwynd ldquoGeneralizedJacobian analysis of lower mobility manipulatorsrdquo Mechanismand Machine Theory vol 46 no 6 pp 831ndash844 2011

[28] C-C Kao and T-S Zhan ldquoModified PSO method for robustcontrol of 3RPS parallel manipulatorsrdquoMathematical Problemsin Engineering vol 2010 Article ID 302430 25 pages 2010

[29] S Briot and I A Bonev ldquoSingularity analysis of zero-torsionparallel mechanismsrdquo in Proceedings of the IEEERSJ Interna-tional Conference on Intelligent Robots and Systems (IROS rsquo08)pp 1952ndash1957 Nice France September 2008

[30] X-J Liu and I A Bonev ldquoOrientation capability error analysisand dimensional optimization of two articulated tool headswith parallel kinematicsrdquo Journal of Manufacturing Science andEngineering vol 130 no 1 pp 0110151ndash0110159 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Gosselin and Angeles [2] identified singularity in closed-loopchains mechanism by analyzing the determinant of Jacobianmatrix The rank deficiency of Jacobian matrix leads to thesingularity of a parallel robot [3] In the geometric methodsscrew theory and line geometry were commonly used tosolve singularity problems [4] Park and Kim [5] proposeda coordinate-invariant differential geometric approach toanalyzing the singularity in a parallel mechanism Hunt [6]laid down a general framework for using screw theory inanalyzing singularity in parallel robots Hao and McCarthy[7] concluded the conditions for line-based singularitiesvia the theory of screws Merlet [8] noticed that it is thepossibility to identify all singular configurations in a parallelrobot via line geometric method Ben-Horin Shoham [9]analyzed the singularity of a class of 6-DOF parallel robotsusing the superbracket and the Grassmann-Cayley operatorsMore recent approaches to singularity analysis were discussedby Davidson and Hunt [10] Ma and Angeles [11] Bonev et al[12] and Tsai [13]

The other significant issue in singularity analysis isthe classification of singularities On the basis of singularproperties of Jacobian matrices Gosselin and Angeles [2]classified the singularity into three main groups With thesame definitions Tsai [13] termed these three kinds of singu-larities as the inverse forward and combined singularitiesGogu [14] distinguished two types of singular configurationsstructural and kinematic singularities Park and Kim [5]classified closed-chain singularities into three basic typesconfiguration space singularity actuator singularity and end-effector singularity In addition Liu et al [15] combined thethree groups of singularities proposed by Park to come upwith the first-order singularity and established two-ordersingularities as degenerate and nondegenerate types Fangand Tsai [16] divided the singularity cases as limb singularitymanipulator singularity and actuation singularity Further-more Zlatanov et al [17] considered both the input andoutput of a parallel robot and concluded that the singularityshould always happen for the combination cases among sixclassified types redundant input impossible input redun-dant output impossible output redundant passive motionand increased instantaneous mobility Note that the singular-ity is an inherent characteristic of parallel robots and not allsingularities could be avoided in practice The elimination ofdifferent singularities is another important issue in practicewhich attracted many researchers For example Li and Xu[18] discussed this problem from the aspect of optimalmechanism design

However the predecessorsrsquo contributions mainly focusedon finding new mathematical tools to search and dividethe solutions of singularities This work on the other handdiscusses the singularity problem from the view of itspractical physical meanings We put forward an alternativeviewpoint to the singularity in terms of the motionforcetransmission and constraint properties of lower-mobilityparallel robots As is well known the key function of aparallel robot is to transmit and constrain the underlyingmotion and forces between inputs and outputThereby in thefunctional process of a parallel robot any motion and forcewhich cannot be transmitted or constrained as expected

could lead to the singularities Thus by considering thetransmission and constraint properties we can evaluate thesingularities of lower-mobility parallel robots It should benoted that this work is an intuitive following study with theknowledge of the motionforce transmission and constraintanalysis of parallel robots proposed by our research team[19 20] The motionforce transmission concept and itsindices were proposed by Wang et al [19] in 2010 andthen the approach to motionforce constraint performanceanalysis was introduced in 2014 [20] In both literaturesthe indices used to evaluate the transmission and constraintperformance of parallel robots have the potential to indicatethe worst transmissibility and constrainability when they areequal to zero In such cases the mechanism cannot transmitor constrain the underlying motions and forces therebyresulting in singularities which are discussed in this work

The remainder of this paper is organized as follows Basedon the theory of screws Section 2 proposes a singularityidentification approach in terms of the transmission andconstraint analysis of lower-mobility parallel robots Severalmodified indices are introduced to identify the singularitiesand the classification of singularity is discussed based onthe indices In Section 3 several typical parallel robots aretaken as examples to demonstrate the usage of the proposedapproach and indices Finally Section 4 follows with a con-clusion of this work

2 Approach to Analyzing Singularityand the Classification

There is no doubt that screw theory is an efficient and pro-found mathematical tool for studying parallel mechanismsand it has been used in evaluating the kinematic perfor-mance type synthesis and singularity analysis [21ndash25] Weintroduce our analytical approach to elucidating singularityand its classifications in which we consider motionforcetransmissibility and constrainability in parallel robots on thebasis of screw theory It should be noted that this work onlyconsiders the nonredundant parallel robots with fewer thansix DOFs wherein there exist the constraint motions andforces

21 Mathematical Foundation In general a screw can berepresented in dual-vector form as follows [26]

$ = 119904+ isin 1199040 (1)

where real part 119904 is a unit vector in the direction of thescrew axis dual part 119904

0denotes the moment of the screw

with respect to the origin and isin refers to dual unity whichis defined such that

isin = 0 isin2= 0 (2)

Considering one arbitrary point 119875 of position vector pon the screw axis we have

s0= p times s + ℎs (3)


Twist screw system

Linearlyindependent

Linearlyindependent


Dual

Dual

Reciprocal

Wrench screw system




$P

$R

$T

$C



ℎ = s119879s0 (4)


1

and $2is given as

$1sdot $2= 11990411199042+ isin (119904

111990402

+ 119904011199042) (5)


$1∘ $2= 119904111990402

+ 119904011199042 (6)





$1∘ $2= 119891119908 (f sdot v + 120591 sdot w) (7)






119875) and















ITS =1003816100381610038161003816$119879 ∘ $119875119868

1003816100381610038161003816 (8)

OTS =1003816100381610038161003816$119879 ∘ $119875119874

1003816100381610038161003816 (9)

ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 (10)

OCS =1003816100381610038161003816$119862 ∘ $119877119874

1003816100381610038161003816 (11)where $


119875119874is the





119875119874and

$119877119874


119875119874119894related


119875119874119894as

$119875119874119894

∘ $119879119895

= 0 119895 = 1 2 119901 minus 1

$119875119874119894

∘ $119862119896

= 0 119896 = 1 2 119902(12)


Limited-DOFparallel

manipulator



p + q = 6


ICS = 0 OCS = 0 ITS = 0 OTS = 0





$T1 $T2 $Tp $C1 $C2 $Cq






$119877119874119894

∘ $119879119895

= 0 119895 = 1 2 119901

$119877119874119894

∘ $119862119896

= 0 119896 = 1 2 119902 minus 1(13)


3 Examples







Fixed Actuated

Uz

S

x

yO


z

x

y

O$P1 $P2

$P5$P3

$P4

120572




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 0 0)

$1198753

= (cos120572 0 sin120572) + isin (0 0 0)

$1198754


$1198755


(14)


$119862= (cos120572 0 sin120572) + isin (0 0 0)

$119877119868

= (0 0 0) + isin (0 0 1) (15)

1

09

08

07

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180

120572 (deg)

ICS



ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 = |sin120572| (16)






Fixed

Uz

S

xy


z

xy

O




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 cos120573 sin120573)

$1198753

= (1 0 0) + isin (0 119897 sin120573 119897 cos120573)

$1198754

= (0 1 0) + isin (minus119897 sin120573 0 0)

$1198755

= (0 0 1) + isin (119897 cos120573 0 0)

(17)

$P1

$P2

$P5

$P3

$T$C

$P4

z998400

x998400

y998400

120573






$119879= (0 119897 cos120573 119897 sin120573) + isin (0 0 0)

(18)












15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS


1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)




$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O


A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400



O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti




(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Twist screw system

Linearlyindependent

Linearlyindependent


Dual

Dual

Reciprocal

Wrench screw system




$P

$R

$T

$C



ℎ = s119879s0 (4)


1

and $2is given as

$1sdot $2= 11990411199042+ isin (119904

111990402

+ 119904011199042) (5)


$1∘ $2= 119904111990402

+ 119904011199042 (6)





$1∘ $2= 119891119908 (f sdot v + 120591 sdot w) (7)






119875) and















ITS =1003816100381610038161003816$119879 ∘ $119875119868

1003816100381610038161003816 (8)

OTS =1003816100381610038161003816$119879 ∘ $119875119874

1003816100381610038161003816 (9)

ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 (10)

OCS =1003816100381610038161003816$119862 ∘ $119877119874

1003816100381610038161003816 (11)where $


119875119874is the





119875119874and

$119877119874


119875119874119894related


119875119874119894as

$119875119874119894

∘ $119879119895

= 0 119895 = 1 2 119901 minus 1

$119875119874119894

∘ $119862119896

= 0 119896 = 1 2 119902(12)


Limited-DOFparallel

manipulator



p + q = 6


ICS = 0 OCS = 0 ITS = 0 OTS = 0





$T1 $T2 $Tp $C1 $C2 $Cq






$119877119874119894

∘ $119879119895

= 0 119895 = 1 2 119901

$119877119874119894

∘ $119862119896

= 0 119896 = 1 2 119902 minus 1(13)


3 Examples







Fixed Actuated

Uz

S

x

yO


z

x

y

O$P1 $P2

$P5$P3

$P4

120572




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 0 0)

$1198753

= (cos120572 0 sin120572) + isin (0 0 0)

$1198754


$1198755


(14)


$119862= (cos120572 0 sin120572) + isin (0 0 0)

$119877119868

= (0 0 0) + isin (0 0 1) (15)

1

09

08

07

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180

120572 (deg)

ICS



ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 = |sin120572| (16)






Fixed

Uz

S

xy


z

xy

O




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 cos120573 sin120573)

$1198753

= (1 0 0) + isin (0 119897 sin120573 119897 cos120573)

$1198754

= (0 1 0) + isin (minus119897 sin120573 0 0)

$1198755

= (0 0 1) + isin (119897 cos120573 0 0)

(17)

$P1

$P2

$P5

$P3

$T$C

$P4

z998400

x998400

y998400

120573






$119879= (0 119897 cos120573 119897 sin120573) + isin (0 0 0)

(18)












15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS


1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)




$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O


A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400



O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti




(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Limited-DOFparallel

manipulator



p + q = 6


ICS = 0 OCS = 0 ITS = 0 OTS = 0





$T1 $T2 $Tp $C1 $C2 $Cq






$119877119874119894

∘ $119879119895

= 0 119895 = 1 2 119901

$119877119874119894

∘ $119862119896

= 0 119896 = 1 2 119902 minus 1(13)


3 Examples







Fixed Actuated

Uz

S

x

yO


z

x

y

O$P1 $P2

$P5$P3

$P4

120572




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 0 0)

$1198753

= (cos120572 0 sin120572) + isin (0 0 0)

$1198754


$1198755


(14)


$119862= (cos120572 0 sin120572) + isin (0 0 0)

$119877119868

= (0 0 0) + isin (0 0 1) (15)

1

09

08

07

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180

120572 (deg)

ICS



ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 = |sin120572| (16)






Fixed

Uz

S

xy


z

xy

O




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 cos120573 sin120573)

$1198753

= (1 0 0) + isin (0 119897 sin120573 119897 cos120573)

$1198754

= (0 1 0) + isin (minus119897 sin120573 0 0)

$1198755

= (0 0 1) + isin (119897 cos120573 0 0)

(17)

$P1

$P2

$P5

$P3

$T$C

$P4

z998400

x998400

y998400

120573






$119879= (0 119897 cos120573 119897 sin120573) + isin (0 0 0)

(18)












15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS


1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)




$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O


A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400



O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti




(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fixed Actuated

Uz

S

x

yO


z

x

y

O$P1 $P2

$P5$P3

$P4

120572




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 0 0)

$1198753

= (cos120572 0 sin120572) + isin (0 0 0)

$1198754


$1198755


(14)


$119862= (cos120572 0 sin120572) + isin (0 0 0)

$119877119868

= (0 0 0) + isin (0 0 1) (15)

1

09

08

07

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180

120572 (deg)

ICS



ICS =1003816100381610038161003816$119862 ∘ $119877119868

1003816100381610038161003816 = |sin120572| (16)






Fixed

Uz

S

xy


z

xy

O




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 cos120573 sin120573)

$1198753

= (1 0 0) + isin (0 119897 sin120573 119897 cos120573)

$1198754

= (0 1 0) + isin (minus119897 sin120573 0 0)

$1198755

= (0 0 1) + isin (119897 cos120573 0 0)

(17)

$P1

$P2

$P5

$P3

$T$C

$P4

z998400

x998400

y998400

120573






$119879= (0 119897 cos120573 119897 sin120573) + isin (0 0 0)

(18)












15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS


1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)




$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O


A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400



O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti




(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fixed

Uz

S

xy


z

xy

O




$1198751

= (1 0 0) + isin (0 0 0)

$1198752

= (0 1 0) + isin (0 cos120573 sin120573)

$1198753

= (1 0 0) + isin (0 119897 sin120573 119897 cos120573)

$1198754

= (0 1 0) + isin (minus119897 sin120573 0 0)

$1198755

= (0 0 1) + isin (119897 cos120573 0 0)

(17)

$P1

$P2

$P5

$P3

$T$C

$P4

z998400

x998400

y998400

120573






$119879= (0 119897 cos120573 119897 sin120573) + isin (0 0 0)

(18)












15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS


1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)




$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O


A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400



O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti




(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15

1

05

00 50 100 150 200 250 300 350

120593 (deg)

OCS


1

09

08

07

06

05

04

03

02

01

00 50 100 150 200 250 300 350

OCS

5

25

0175 180 185

times10minus3

120579 (deg)




$1198751

= (0 0 1) + isin (1198871 minus1198861 0)

$1198752

= (0 0 1) + isin (1198872 minus1198862 0)

$1198753

= (0 0 1) + isin (1198873 minus1198863 0)

(19)

$C2

$C3

$C1

z

x y

C

O


A

B Leg 1

C

r4

yy998400

r3

x

x998400

F

Leg 3

Leg 2

H

r2G

I

r1 D

E

120593

OO998400



O

O998400

x

y

si

120601i

A

B

C

Leg i

120579i

120595i

$Ti




(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a)


(b)


1= minus90∘ (b) when 120579

1= 90∘ and 120595

1= 270∘




$119868= $1198751

= (0 0 1) + isin (1198871 minus1198861 0) (20)


$119879119894

= 119891119894+ isin (119904

119894times 119891119894) (21)





ITS = $119868∘ $119879119894

=10038161003816100381610038161199031 sin (120595 minus 120579)

1003816100381610038161003816 (22)




15

15

1

1

05

05

0

0

minus05

minus05

minus1

minus1minus15

minus15

y

x

120593 = 120∘

120593 = 90∘

120593 = 60∘

120593 = 30∘

120593 = 0


∘ 30∘ 60∘ 90∘ and 120∘




1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0

OTS

minus180

minus150

minus100

minus50 0

50

100

150

180

120593 (deg)

1

05

0

1

05








119875119874119894

$119879119895

∘ $119875119874119894

= 0 (119894 119895 = 1 2 3 119895 = 119894) (23)


OTS =1003816100381610038161003816$119879119894 ∘ $119875119874119894

1003816100381610038161003816 (24)


1= minus48∘

and 1205932





1= 1199032 and the




4 Conclusion



y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y

xO


(a)


y

xO

(b)


∘ and (b) 1205932= 132

∘

1

08

06

04

02

0

Valu

e


120593 (deg)

ITSOTS





O x

y



1= 1199032and 1199033= 1199034

Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article screw theory based singularity...

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