Research Article Recognition of Kinematic Joints of 3D Assembly …downloads.hindawi.com/journals/mpe/2016/1761968.pdf · 2019-07-30 · resultant moment of the force system about

Research ArticleRecognition of Kinematic Joints of 3D Assembly ModelsBased on Reciprocal Screw Theory

Tao Xiong1 Liping Chen1 Jianwan Ding1 Yizhong Wu1 and Wenjie Hou2

1National Engineering Research Center of CAD Software Information Technology School of Mechanical Science andEngineering Huazhong University of Science and Technology Wuhan 430074 China2Eaton (China) Investments Co Ltd Shanghai 200335 China

Correspondence should be addressed to Jianwan Ding dingjwhusteducn

Received 17 February 2016 Accepted 17 March 2016

Academic Editor Chaudry Masood Khalique

Copyright copy 2016 Tao Xiong 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

Reciprocal screw theory is used to recognize the kinematic joints of assemblies restricted by arbitrary combinations of geometryconstraints Kinematic analysis is common for reaching a satisfactory design If a machine is large and the incidence of redesignfrequent is high then it becomes imperative to have fast analysis-redesign-reanalysis cycles This work addresses this problem byproviding recognition technology for converting a 3D assembly model into a kinematic joint model which is represented by agraph of parts with kinematic joints among themThe three basic components of the geometric constraints are described in termsof wrench and it is thus easy to model each common assembly constraint At the same time several different types of kinematicjoints in practice are presented in terms of twist For the reciprocal product of a twist and wrench which is equal to zero thegeometry constraints can be converted into the corresponding kinematic joints as a result To eliminate completely the redundantcomponents of different geometry constraints that act upon the same part the specific operation of a matrix space is applied Thisability is useful in supporting the kinematic design of properly constrained assemblies in CAD systems

1 Introduction

Currently we can design a machine in terms of a top-downapproach or a bottom-up approach in a CAD system Allparts of an assembly model are positioned using variousgeometry constraints so that they are in the correct relativepositions and orientations The 3D assembly models recordthe design parameters and some information that can beused for concept-level and detailed-level design Its benefitsare usually recognized only by those who design precisionmachines [1ndash3]

Any machine that consists of moving parts has to bedesigned to properly perform its kinematic functions Toanalyze the kinematic property of assemblies because of thelack of a kinematic description many researchers have usedscrew theory Applying screw theory to kinematics was firstperformed by Ball [4] Hunt [5] identified 22 cases of Ballrsquosreciprocal screw systems References [6ndash8] applied screwtheory to determine the degrees of freedom of any body

in a mechanism Eddie Baker [9] analyzed many types ofcomplex mechanisms based on [6] Mason and Salisbury[10] used screw theory to characterize the nature of contactsbetween robot gripper hands and objects To analyze roboticmultiple peg-in-hole Fei and Zhao [11] described the contactforces by using the screw theory in three dimensions Leeand Saitou [12] determined the dimensional integrity ofrobustness based on screw theory Kim et al [13] establisheda framework to estimate the reaction of constraints about aknee

Assembly features encode geometric and functional rela-tionships between parts that are widely used Thus manyresearchers perform kinematic analysis by using the assemblyfeatures Konkar and Cutkosky [14] created screw systemrepresentations of assembly mating features Adams et al[15ndash17] used Konkarrsquos algorithm and extended their workby defining extensible screw representations of some typesof assembly features Gerbino et al [18 19] proposed analgorithm to decompose the liaison diagram representing an

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1761968 13 pageshttpdxdoiorg10115520161761968

2 Mathematical Problems in Engineering

assembly into simple paths to which the application of thetwist union or intersection algorithms may be easily appliedThey also highlighted the complexity behind the analysis ofoverconstraints with respect to the motion analysis Celen-tano [20] improved the algorithm presented in [18] anddeveloped a tool to analyze the motions and constraints ofCAD mechanisms Su et al [21] presented a theory-basedapproach for the analysis and synthesis of flexible jointsDixon and Shah [22] provided a mechanism for user-definedcustom assembly features by using an interactive systemAfter that twist and wrench matrices could be extractedfrom face pair relations and used in kinematic and structuralanalysis

Geometry constraints between two parts in an assemblymay be arbitrary combinations References [23ndash25] analyzedthe state of a constraint of an assembly mechanism or fixtureto see if it is properly constrained or whether it containsunwanted overconstraints Rusli et al [26 27] addressedattachment-level design in which decisions are made toestablish the types locations and orientations of assemblyfeatures Su et al [28] analyzed the mobility of overcon-strained linkages and compliant mechanisms To providea desired constrained motion for synthesizing a pattern offlexures a new screw theory that addresses ldquoline screwsrdquo andldquoline screw systemsrdquo was presented by Su and Tari [29]

According to [4ndash29] if the screw representations betweenany two parts of the assemblymodel are known we can easilyanalyze the modelrsquos kinematic functions References [4ndash1323ndash29] manually model the corresponding screw systemThus it is time consuming when the machine is large andcomplex On the other hand to determine the screw systemby using the methods proposed in [14ndash22] all of the screwrepresentations of different features that may be used firsthave to bemodeledThis is awkward and resembles geometricreasoning [30] If the incidence of redesign frequency is highthe screw system has to be rebuilt again and again which hasa great influence on the efficiency of the analysis procedure

We hope to automatically and easily obtain the screwsystem of any assembly model Thus when we modify theassembly model during analysis-redesign-reanalysis cyclesthe kinematic functions can be directly validated For con-venience we adopt a graph of parts with twist amongthem to express the screw system A screw system can beeasily used for kinematic analysis such as determining thedegrees of freedom estimating the reaction of constraintsand characterizing the nature of contacts between any twocomponents

In this paper we propose a new method to automaticallyconvert a 3D assembly model into a screw system based onreciprocal screw theoryThe two fundamental concepts in theconstraint-based approach that is freedom and constraintcan be represented mathematically by a twist and a wrenchBoth the twist and kinematic joint represent motion boththe wrench and geometry constraint represent constraintWhen a twist is given we can obtain the correspondingwrench as the reciprocal product of a twist and wrenchwhich is equal to zero Conversely if we know a wrench thenthe corresponding twist could be deduced as well To avoidgeometric reasoningwemodel the three basic components of

z

x

yo

A

Instantaneousscrew axis (ISA)

120596A

120596A

o

Figure 1 Twist

geometry constraints in terms of wrench Thus all commonassembly constraints can be automatically deduced For theredundant property of geometry constraints the specificoperation of a matrix space is applied Finally we achieve thegraph of parts with kinematic joints among themwith respectto an assembly model

11 Organization of the Paper Section 2 reviews the twist rep-resentations of common kinematic joints Section 3 analyzesthe basic components of geometry constraints and modelsthese components in terms of wrench Section 4 shows howto recognize the corresponding kinematic joints of assembliesrestricted by arbitrary combinations of geometry constraintsSection 5 contains several examples Section 6 presents ourconclusions

2 Screw Models of Kinematic Joints

To recognize the corresponding kinematic joints from geom-etry constraints a mathematical model must be developedthat can sufficiently and simultaneously describe the proper-ties of kinematic joints and geometry constraintsThis sectionshows how to define kinematic joints using twist

21 Twist A twist [32 33] takes the form

T = [120596119909120596119910120596119911 120592119909120592119910120592119911] (1)

A twist is a screw that describes the instantaneousmotionto the first order of a rigid body This means that the motiondescribed by a twist is allowed The first triplet representsthe angular velocity of the body with respect to a globalreference frameThe second triplet represents the velocity inthe global reference frame of that point on the body or itsextension which is instantaneously located at the origin ofthe global frame Therefore the unique line associated withthe first triplet is the axis of rotation and the unique pointassociated with the second triplet is the point on the body orits extension which is at that instant located at the origin ofthe global reference frame See Figure 1

The ISA in Figure 1 is the axis introduced in ChaslesThe-orem Both the twist and kinematic joint represent motion

Mathematical Problems in Engineering 3

ISA

Pv

Figure 2 Prismatic joint

Therefore we can use a twist to characterize a componentof the kinematic joint between two parts Afterwards thekinematic joint can be transformed into a correspondingtwist matrix

22 Twist Representations of Kinematic Joints There are6 types of kinematic joints in common prismatic jointrevolute joint helical joint cylindrical joint spherical jointand planar joint In this section we use s to characterizethe allowed motions of each kinematic joint and show theircorresponding twist matrices

221 Prismatic Joint The degrees of freedom (dof) of a pris-matic joint (119875) is one We use the unit vector 119875v = (V

119909 V119910V119911)

to characterize the allowed translation See Figure 2Thus 119875s has the form

119875s = 119875v (2)

The twist matrix representation of 119875 is given by

T119875 = [0 119875s] (3)

We use 0 instead of [0 0 0] unless otherwise stated

222 Revolute Joint The dof of a revolute joint (119877) is oneWeuse 119877L to characterize the axis of 119877 The unit vector of 119877Lis characterized by 119877v = (V

119909 V119910 V119911) 119877P = (119875

119909 119875119910 119875119911) is a

single point on the 119877L See Figure 3Thus 119877L has the form

119877L = 119877P + 119886 sdot 119877v (4)

where 119886 is a real number 119877s has the form

119877s = 119877P times 119877v (5)

Then the twist matrix representation of 119877 is given by

T119877 = [119877v 119877s] (6)

ISA

RP

Pv

RL

Figure 3 Revolute joint

ISA

HL

HP

Hv

Hp

Figure 4 Helical joint

223 Helical Joint The dof of a helical joint (H) is one Weuse119867L to characterize the axis of119867 The unit vector of119867Lis characterized by 119867v = (V

119909 V119910 V119911) 119867P = (119875

119909 119875119910 119875119911) is a

single point on the119867L See Figure 4Thus119867L has the form

119867L = 119867P + 119886 sdot 119867v (7)

where 119886 is a real number119867s has the form

119867s = 119867P times 119867v + 119867119901 sdot 119867v (8)

where119867119901 is the pitch of119867 Then the twist matrix represen-tation of119867 is given by

T119867 = [119867v 119867s] (9)

224 Cylindrical Joint The dof of a cylindrical joint (119862) istwo We use 119862L to characterize the axis of 119862 The unit vectorof119862L is characterized by119862v = (V

119909 V119910 V119911)119862P = (119875

119909 119875119910 119875119911)

is a single point on the 119862L See Figure 5Thus 119862L has the form

119862L = 119862P + 119886 sdot 119862v (10)


ISA

CL

CP

Cv

Figure 5 Cylindrical joint

SP

ISA

v1

v2

v3

Figure 6 Spherical joint

where 119886 is a real number 119862s has two components as shownin

119862s1= 119862P times 119862v

119862s2= 119862v

(11)

Therefore the twist matrix representation of119862 is given by

T119862 = [119862v 119862s

1

0 119862s2

] (12)

225 Spherical Joint The dof of a spherical joint (119878) is threeWe use 119878P = (119875

119909 119875119910 119875119911) to characterize the central point of

119878 and define v1= (1 0 0) v

2= (0 1 0) and v

3= (0 0 1) See

Figure 6Thus 119878s has three components as shown in

119878s1= 119878P times v

1= (0 119875

119911 minus119875119910)

119878s2= 119878P times v

2= (minus119875

119911 0 119875119909)

119878s3= 119878P times v

3= (119875119910 minus119875119909 0)

(13)

EP

ISA

En

Ev1

Ev2

Figure 7 Planar joint


T119878 = [[[

v1 119878s1

v2 119878s2

v3 119878s3

]

]

]

(14)

226 Planar Joint The dof of a planar joint (119864) is three Thevectors 119864v

1= (v1199091 v1199101 v1199111) and 119864v

2= (v1199092 v1199102 v1199112) are

orthogonal in the planar surface and the unit vector 119864n isthe outer normal vector 119864P = (119875

119909 119875119910 119875119911) is a single point

on the surface See Figure 7Thus 119864s has three components as shown in

119864s1= 119864v1

119864s2= 119864v2

119864s3= 119864P times 119864n

(15)


T119864 = [[[

0 119864s1

0 119864s2

119864n 119864s3

]

]

]

(16)

3 Reciprocal Screw Models ofGeometric Constraints

To transformarbitrary combinations of geometric constraintsbetween two parts into the corresponding kinematic jointwe should use the same mathematical model with kinematicjoints or a model that can be easily transformed into a twistThis section shows how to define geometry constraints usingwrench

31 Wrench A wrench [32 33] takes the form

W = [119891119909119891119910119891119911 119898119909119898119910119898119911] (17)

A wrench is also a screw and describes the resultant forceand moment of a force system acting on a rigid body Anyset of forces and couples applied to a body can be reduced toa set comprising a single force acting along a specific line inspace and a pure couple acting in a plane perpendicular tothat line This means that the motion described by a wrench


z

x

yo

M

Force system

Instantaneous screw axis (ISA)

Fi

F1 F2

F3

Fr

Mr

ri

Line of action of Fr

Fr

Figure 8 Wrench

is forbidden The first triplet describes the resultant force ina global reference frame The second triplet represents theresultant moment of the force system about the origin of theglobal frame See Figure 8

Both a wrench and geometry constraint are structuralconstraints Therefore we can use a wrench to characterizea component of the geometry constraints between two partsAfterwards the geometry constraints can be transformedinto a corresponding wrench matrix

32 Basic Components of Geometric Constraints When wedescribe a part in space the orientation and position shouldbe specified [34] The orientation cannot be described inabsolute terms An orientation is given by a rotation fromsome known reference orientationThe amount of rotation isknown as an angular displacement In other words describ-ing an orientation is mathematically equivalent to describingan angular displacement In the same way when we specifythe position of a part we cannot do so in absolute terms wemust always do so within the context of a specific referenceframeTherefore specifying a position is actually the same asspecifying an amount of translation from a given referencepoint (usually the origin of some coordinate system)

In practice there are many geometry constraints suchas align and point on line We can use one or arbitrarycombinations of geometric constraints to restrict the relativeorientation and position between two parts As we knowcommon geometry constraints such as coaxial constraintsand spherical constraints usually have a high degree ofconstraint (doc) However these geometry constraints are allcombined by the three different types of basic constraintsdot-1 constraint dot-2 constraint and the distance constraint[35] which can be divided into two categories the orientationpart and the position part

Consider a pair of rigid bodies denoted as bodies 119894 and119895 in Figure 9 Reference points P

119894and P

119895 reference frames

f119894-g119894-h119894and f

119895-g119895-h119895 and nonzero unit vectors a

119894and a

119895

are fixed in bodies 119894 and 119895 respectively Vector d119894119895connects

z

x

y

i

j

ai

Pihi

gi

f i

ri

sPidij

fj

hjgj

sPj

Pj aj

rj

Figure 9 Vectors fixed in and between bodies

P119894and P

119895between bodies The basic constraints are often

characterized by such elements

321 Dot-1 Constraint Dot-1 constraint restricts the relativeorientation of a pair of bodies that is

Φ

1198891

(a119894 a119895 1198621) equiv a119894

Ta119895minus 1198621= 0 119862

1isin (minus1 1) (18)

where doc(Φ1198891) = 1 and dot-1 constraint is symmetric withregard to bodies 119894 and 119895 Note that dot-1 constraint would notbe a basic constraint if119862

1= plusmn1Thismeans that the vectors a

119894

and a119895are parallel in the same or opposite directions In this

case doc(Φ1198891(a119894 a119895 plusmn1)) = 2

Writing the vectors a119894and a119895in terms of their respective

body reference transformationmatricesA119894andA

119895and body-

fixed constant vectors a1015840119894and a1015840

119895 where a

119894= A119894a1015840119894and a

119895=

A119895a1015840119895 (18) has the form

Φ

1198891

(a119894 a119895 1198621) equiv a1015840119894

TA119894

TA119895a1015840119895minus 1198621= 0

1198621isin (minus1 1)

(19)

where the transformation matrices A119894and A

119895in (19) depend

on the orientation generalized coordinates of bodies 119894 and 119895respectively

As a result the wrench representation for dot-1 constraintshould also freeze the relative orientation of the vectors a

119894and

a119895 Therefore the wrench matrix is given by

W1198891 = (0 a119894times a119895) (20)

If we use the body reference transformationmatrices (20)has the form

W1198891 = (0 (a1015840119894

TA119894

T) times (A

119895a1015840119895)) (21)

322 Dot-2 Constraint Dot-2 constraint restricts the relativeposition of a pair of bodies that is

Φ

1198892

(a119894 d119894119895 1198622) equiv a119894

Td119894119895minus 1198622= 0 d

119894119895= 0 (22)


where doc(Φ1198892) = 1 the same as that with dot-1 constraintalthough dot-2 constraint is not symmetric with regard tobodies 119894 and 119895 Analytically it is important to recall that dot-2constraint breaks down if d

119894119895= 0 Write the vector d

119894119895as

d119894119895= r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P (23)

Equation (22) becomes

Φ

1198892

(a119894 d119894119895 1198622) equiv a1015840119894

TA119894

T(r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P)

minus 1198622= 0

(24)

As a result the wrench representation for dot-2 constraintmust freeze the relative translation along the vector a

119894

Therefore the wrench matrix is given by

W1198892 = (a119894 0) (25)

Equation (25) also has the form

W1198892 = (A119894a1015840119894 0) (26)

323 Distance Constraint In practice we often require a pairof points on two bodies to coincide or maintain a determineddistance that is

Φ

119904

(119875119894 119875119895 1198623) equiv d119894119895

Td119894119895minus 1198623sdot 1198623= 0 (27)

where doc(Φ119904) would be different depending on whether 1198623

is zero Therefore we have

doc (Φ119904) = 1 if 1198623

= 0

doc (Φ119904) = 3 if 1198623= 0

(28)

As a result the wrench representation for the distanceconstraint should have two types of expressions as shown in(29) and (30)

W119904 = [d119894119895 0] if 119862

3= 0 (29)

where the wrench representation for the distance constraintshould freeze the relative translation along the vector d

119894119895if

1198623

= 0 However if 1198623= 0 the wrench representation has

the form

W119904 = [[[

1 0 0 00 1 0 00 0 1 0

]

]

]

if 1198623= 0 (30)

where the wrench which is illustrated in (30) freezes all therelative translations along the axes 119909 119910 and 119911

33 Wrench Representations of Geometry Constraints Tointroduce geometry constraints we give Figure 10

In Figure 10 reference points P1and P

2 reference frames

f1-g1-h1and f2-g2-h2 and central axes L

1and L

2are fixed in

cylinders 1198611and 119861

2 respectively Vector d

12connects P

1and

P2between the cylinders

L1

B1

h1

P1

f1

F1 g1

L2

B2

h2

P2

f2F2

g2

d12

Figure 10 Instance of geometry constraints

The reference point P1has the form

P1= (1198751199091 1198751199101 1198751199111) (31)

The central axis L1has the form

L1= P1+ 119886 sdot h

1 (32)

where 119886 is a real number The planar surface F1has the form

F1= P1+ 119887 sdot f

1+ 119888 sdot g

1 (33)

where 119887 and 119888 are both real numbers At the same timethe elements P

2 L2 and F

2have the same forms The

following geometry constraints are often characterized bysuch elements

We can divide all different types of geometry constraintswhich have more than one doc into six categories coaxialconstraint coplanar constraint spherical constraint align con-straint distance between line and surface and point on lineWe have modeled the three basic components of geometryconstraints in terms of wrench Thus the wrench models ofthese six types of geometry constraints can be automaticallydeduced

331 Coaxial Constraint Weuse119862119900119894119871119871(L1 L2) to denote the

coaxial constraint of axes L1and L

2 as shown in Figure 11

The doc of 119862119900119894119871119871(L1 L2) is four which restricts two

relative orientations and two relative translations In otherwords it comprises two dot-1 constraints and two dot-2constraints as shown in

Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(f1 d12 0)

Φ4

1198892

(g1 d12 0)

(34)

Then the wrench matrix of 119862119900119894119871119871(L1 L2) is given by

W119862119900119894119871119871(L

1L2)=

[

[

[

[

[

[

0 h1times f2

0 h1times g2

f1 0

g1 0

]

]

]

]

]

]

(35)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)

Figure 11 Coaxial constraint

f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12

Figure 12 Coplanar constraint

The corresponding kinematic joint of coaxial constraint iscylindrical joint

332 Coplanar Constraint We use 119862119900119894119865119865(F1 F2) to denote

the coplanar constraint of planar surfaces F1and F

2 as shown

in Figure 12The doc of 119862119900119894119865119865(F

1 F2) is three which restricts two

relative orientations and one relative translation In otherwords it comprises two dot-1 constraints and one dot-2constraint as shown in

Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)

Then the wrench matrix of 119862119900119894119865119865(F1 F2) is given by

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)

The corresponding kinematic joint of coplanar constraintis planar joint

333 Spherical Constraint We use 119862119900119894119875119875(P1P2) to denote

the spherical constraint of two points P1and P

2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)

Figure 13 Spherical constraint

f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)

Figure 14 Align constraint

The doc of 119862119900119894119875119875(P1P2) is three which restricts three

relative translations in three unrelated directions In otherwords it comprises one distance constraint which has 119862

3= 0

in Section 323 as shown in

Φ

119904

(119875119894 119875119895 0) (38)

Then the wrench matrix of 119862119900119894119875119875(P1P2) is given by

W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)

The corresponding kinematic joint of spherical constraintis spherical joint

334 Align Constraint Align constraint (119875119886119903119865119865) is used tomake two planar surfaces F

1and F

2parallel as shown in

Figure 14The doc of 119875119886119903119865119865(F

1 F2) is two which restricts two

relative orientations In other words it comprises two dot-1constraints as shown in

Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)

Figure 15 Distance between line and surface


W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)

We cannot find a corresponding kinematic joint for alignconstraint among the common kinematic joints

335 Distance between Line and Surface We use 119863119894119904119871119865(L1

F2 119863) to denote the axis L

1and the surface F

2that maintain

a certain distance119863 as shown in Figure 15The doc of 119863119894119904119871119865(L

1 F2 119863) is two which restricts one

relative orientation and one relative translation In otherwords it comprises one dot-1 constraint and one dot-2constraint as shown in

Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)

Then the wrench matrix of119863119894119904119871119865(L1 F2 119863) is given by

W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)

We also cannot find a corresponding kinematic jointfor distance between line and surface among the commonkinematic joints

336 Point on Line Weuse119862119900119894119875119871(P1 L2) to denote the point

P1on line L


The doc of 119862119900119894119875119871(P1 L2) is two which restricts two

relative translations In other words it comprises two dot-2constraints as shown in

Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)

Then the wrench matrix of 119862119900119894119875119871(P1 L2) is given by

W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)

We still cannot find a corresponding kinematic joint forpoint on line among the common kinematic joints

f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)

Figure 16 Point on line

4 Recognition of Kinematic Joints

The key to recognizing kinematic joints in terms of geometryconstraints is the reciprocity relation We show how to definegeometry constraints using wrench The space of screws thatdefines the range of wrenches that can be exerted upon abody and the space of screws that defines the range of twiststhat the body can perform without breaking any contacts inthe joints of the mechanism are mutually reciprocal screwspaces When we know the wrenches acting upon a body wecan determine the motions it can perform Conversely if weknow the motions a body is capable of performing then theconstraints that are exerted upon it can also be deduced

41 Algorithm We assume that 119860 is a part in a 3D assemblymodel restricted by parts 119861

1 1198612 119861

119899 The twist space of 119860

is the space of screws reciprocal to the wrenches of contactacting upon it The space of wrenches acting upon 119860 is theunion of all the individual wrenches acting upon it fromthe parts 119861

119894 These individual wrenches are to be considered

as acting alone Algorithm 1 provides the twist space of 119860denoted by T119860

The 119877119890119888119894119901119903119900119888119886119897(P) in Algorithm 1 means to exchangethe first triplet and the second triplet of P For exampleif P = [119886 119887 119888 119889 119890 119891] then we have 119877119890119888119894119901119903119900119888119886119897(P) =

[119889 119890 119891 119886 119887 119888] The recursive nature of the procedurerequires the computation of the wrench space and twistspace of each component This proves to be computationallyexpensive because wrench space and twist space are definedin terms of the reciprocal of one another The operation ofderiving reciprocal screws involves computing the null spaceof matrices The speed of the algorithm is linearly related tothe number of 119861

119894 That is the algorithm is of the order 119874(119899)

42 Reciprocal Product According to reciprocal screw the-ory two screws (a twist and a wrench) are said to be mutuallyreciprocal if their reciprocal product is zero The reciprocal


BEGINFOR each part Bi which restrict AConvert all the geometry constraints between A and Bi into basic components 119862119895

119860 119895 = 1 119897

Determine the wrenchesW119896119860 119896= 1 119898 acting upon A through C119895

119860 We havem not less than l

Collect all the wrenchesW119896119860into a matrixW

END FORCompute the orthogonal matrixQ in terms ofW by Gram-Schmidt orthogonalization [31]IF Rank(Q) = 6 THENReturn ldquoIMMOBILErdquo

ELSECompute the orthogonal complement matrix P of Q

END IFTA =Reciprocal(P)

END

Algorithm 1 Function twist space

f2

A2

h2F2

g2

B2

f1A1

F1

g1B1

Figure 17 Redundant constraint

product of a twist T1and a wrenchW

1thatisequal to zero is

given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)

Mathematically because reciprocity imposes the relationT1∘W1 we can compute one of these quantities if the other is

known This allows us to compute the twists that a body mayexecute while in contact with other bodies without breakingthe contacts

43 Redundant Constraint When more than one geometryconstraint acts upon the same part because some compo-nents of these constraints may be redundant the resultantmotion is the logical intersection of the individual wrenchmatrix that defines each constraint for example when wemodel a revolute joint 119877

12with two links 119861

1and 119861

2 The

constraints used are 119862119900119894119871119871(A1A2) and 119862119900119894119865119865(F

1 F2) See

Figure 17Also dof(119877

12) = 1Thismeans that doc(119861

1 1198612) = 5 How-

ever we have doc(119862119900119894119871119871(A1A2)) = 4 and doc(119862119900119894119865119865(F

1

F2)) = 3 In other words the dimension of the orthog-

onal wrench space which comprises W119862119900119894119871119871(A

1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1

Figure 18 Recognition of planar joint

W119862119900119894119865119865(F

1F2) is 5 Therefore the motions that a body is

capable of performing while under the action of multiplewrenches are the logical intersection of the motions it canperform under the action of each wrench acting alone Thuswe can eliminate the redundant components of differentgeometry constraints that act upon the same part by usingGram-Schmidt orthogonalization in Algorithm 1

5 Examples

In each of the examples below different types of kinematicjoints are recognized

Example 1 (planar joint) Figure 18 shows a planar jointconsisting of two parts 119861

1and 119861

2 and a coplanar constraint

119862119900119894119865119865(F1 F2)

Following the process outlined in Section 332 we findbased on the definition of the coplanar constraint that thewrench matrix is

W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001

Figure 19 Recognition of prismatic joint

whereW119862119900119894119865119865(F

1F2)is already an orthogonalmatrixThen the

orthogonal complement matrix P119862119900119894119865119865(F

1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)

Therefore the twist matrixT12between 119861

1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12

represents the motion of a planar joint which isillustrated in Section 226 Moreover the recognition processof a cylindrical joint or spherical joint is approximately thesame as that of the planar joint

Example 2 (prismatic joint) Figure 19 shows a prismaticjoint consisting of two parts 119861

1and 119861

2 and two coplanar

constraints 119862119900119894119865119865(F1 F2) and 119862119900119894119865119865(F1015840

1 F10158402)

Following the process outlined in Example 1 we find thatthe wrench matrices between 119861

1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y

Figure 20 Hydraulic grab

where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840

1F10158402) The orthogonal

matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)

The orthogonal complement matrix P12ofQ12is

P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12

represents the motion of a prismatic joint whichis shown in Section 221 Moreover the recognition processof a revolute joint is approximately the same as that of theprismatic joint

Example 3 (hydraulic grab) Here we present a projectexample A hydraulic grab with three degrees of freedom inCartesian coordinates is illustrated in Figure 20

The hydraulic grab comprises 12 parts 1198610 1198611 1198612 1198613

1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861

11 12 coplanar constraints

119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and

119862119900119894119865119865(11986110 11986111) and 15 coaxial constraints 119862119900119894119871119871(119861

0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See

Figure 21Here we present the processes of recognition among 119861

1

1198612 and 119861

3 as shown in Figure 22 The geometry constraint

between 1198612and 119861

3is 119862119900119894119871119871(119861

2 1198613) Following the process


CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

Figure 21 The constraint graph of a hydraulic grab

h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861

3in a hydraulic grab

outlined in Section 331 we find based on the definition ofthe coaxial constraint that the wrench matrix is

W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)

The orthogonal complement matrix is

P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)

Then the twist matrix T23


3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)

The geometry constraints between 1198611

and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861

1 1198613) Thus the wrench matrices


3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861

11198613) The orthogonal

matrixQ13ofW13is given by

Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)

Then the orthogonal complement matrix P13ofQ13is

P13= [0 h1015840

3] (59)

Therefore the twist matrix T13between 119861

1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)

In this paper we propose a new method to automaticallyconvert a 3D assembly model into a screw system basedon reciprocal screw theory According to the geometryconstraints between any two parts in the hydraulic graband Algorithm 1 we obtain the screw system as shown inFigure 23

As mentioned in [4ndash29] we can perform some differenttypes of kinematic analyses using the screw system such asdetermining the degrees of freedom of 119861

1and 119861

4 that is

T1198611

= T10cap (T20cup T32cup T31)

T1198614

= (T1198611

cup T41) cap (T

1198611

cup T51cup T65cup T64)

(61)

where the resultant of the motions that the end-effector ofa purely serial chain may perform due to the action of eachjoint acting alone is the union of that set and the resultant ofthe set of twists allowed separately by multiple contacts upona body is the intersection of the individual twists Thus thedegrees of freedom of 119861

1are equal to the dimension of the

orthogonal twist space of1198611 denoted byT

1198611

In the samewaywe can obtain the degrees of freedomof119861

4and any part in the

hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107

Figure 23 The screw system of a hydraulic grab

R revolute jointC cylindrical joint

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65

Figure 24 The kinematic joint model of a hydraulic grab

According to Sections 222 and 224 the kinematic jointsbetween 119861

2and 119861

3and 119861

1and 119861

3are the cylindrical joint and

revolute joint As a result we can obtain the kinematic jointmodel See Figure 24 If the corresponding kinematic jointbetween any two parts cannot be recognized we can easilyanalyze the modelrsquos geometry constraints

6 Conclusions

This paper investigates a method for determining the kine-matic joints of assemblies restricted by arbitrary combina-tions of geometry constraints using reciprocal screw theoryWe achieve the process of recognition in terms of thebasic components of geometry constraints and kinematicjoints We describe the three basic components of geometricconstraints in terms of wrench At the same time the specificoperation of a matrix space is applied to eliminate the redun-dant components between different geometry constraintsthat act upon the same part A user of this method can auto-matically and easily convert any assembly model into a screwsystem in any CAD system Afterwards the screw system canbe easily used for validating kinematic functions analyzingover- or underconstrained assembly configurations and soon

Additional Points

(i) We provide a new method for converting any 3Dassembly model into a screw system based on recip-rocal screw theory

(ii) We describe the three basic components of geometricconstraints in terms of wrench

(iii) We eliminate the redundant components of differentgeometry constraints that act upon the same part bythe specific operation of a matrix space

(iv) We establish the maps between geometry constraintsand kinematic joints as the reciprocal product of atwist and wrench which is equal to zero

(v) We present several application examples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to thank the Ministry of Scienceand Technology of the Peoplersquos Republic of China (nos2011ZX02403-005 2011CB706502 and 2013AA041301) Thiswork is also supported by the National Natural ScienceFoundation of China (nos 61370182 and 51175198)

References

[1] A H Slocum Precision Machine Design Prentice Hall NewYork NY USA 1991

[2] M Rajh S Glodez J Flasker K Gotlih and T KostanjevecldquoDesign and analysis of an fMRI compatible haptic robotrdquoRobotics and Computer-Integrated Manufacturing vol 27 no 2pp 267ndash275 2011

[3] S Awtar and G Parmar ldquoDesign of a large range XY nanoposi-tioning systemrdquo Journal of Mechanisms and Robotics vol 5 no2 Article ID 021008 2013

[4] R S Ball A Treatise on the Theory of Screws CambridgeUniversity Press Cambridge UK 1998

[5] K H Hunt Kinematic Geometry of Mechanisms ClarendonPress Oxford UK 1978

[6] K J Waldron ldquoThe constraint analysis of mechanismsrdquo Journalof Mechanisms vol 1 no 2 pp 101ndash114 1966

[7] R Konkar Incremental kinematic analysis and symbolic syn-thesis of mechanisms [PhD dissertation] Stanford UniversityStanford Calif USA 1993

[8] J-S Zhao Z-J Feng and J-X Dong ldquoComputation of theconfiguration degree of freedomof a spatial parallel mechanismby using reciprocal screw theoryrdquo Mechanism and MachineTheory vol 41 no 12 pp 1486ndash1504 2006

[9] J Eddie Baker ldquoScrew system algebra applied to special linkageconfigurationsrdquoMechanism and Machine Theory vol 15 no 4pp 255ndash265 1980

[10] M TMason and J K Salisbury Robot Hands and theMechanicsof Manipulation MIT Press Cambridge Mass USA 1985

[11] Y Fei and X Zhao ldquoAn assembly process modeling and analysisfor robotic multiple peg-in-holerdquo Journal of Intelligent andRobotic Systems vol 36 no 2 pp 175ndash189 2003


[12] B Lee andK Saitou ldquoThree-dimensional assembly synthesis forrobust dimensional integrity based on screw theoryrdquo Journal ofMechanical Design vol 128 no 1 pp 243ndash251 2006

[13] W Kim A P Veloso D Araujo V Vleck and F Joao ldquoAninformational framework to predict reaction of constraintsusing a reciprocally connected knee modelrdquo Computer Methodsin Biomechanics and Biomedical Engineering vol 18 no 1 pp78ndash89 2015

[14] R Konkar andM Cutkosky ldquoIncremental kinematic analysis ofmechanismsrdquo Journal of Mechanical Design Transactions of theASME vol 117 no 4 pp 589ndash596 1995

[15] J D Adams and D E Whitney ldquoApplication of screw theory toconstraint analysis ofmechanical assemblies joined by featuresrdquoJournal ofMechanicalDesign Transactions of theASME vol 123no 1 pp 26ndash32 2001

[16] J D Adams and S Gerbino ldquoApplication of screw theory tomotion analysis of assemblies of rigid partsrdquo in Proceedingsof the IEEE International Symposium on Assembly and TaskPlanning (ISATP rsquo99) pp 75ndash80 Porto Portugal July 1999

[17] J D Adams Feature-based analysis of selective limited motion inassemblies [MS thesis] Mechanical Engineering DepartmentMassachusetts Institute of Technology Cambridge Mass USA1998

[18] S Gerbino and F Arrichiello ldquoHow to investigate constraintsand motions in assemblies by Screw Theoryrdquo in Proceedingsof the CIRP International Seminar on Intelligent Computationin Manufacturing Engineering (CIRP ICME rsquo04) pp 331ndash336Sorrento Italy June-July 2004

[19] P Franciosa and S Gerbino ldquoA CAD-based methodology formotion and constraint analysis according to screw theoryrdquo inProceedings of the ASME International Mechanical EngineeringCongress and Exposition (IMECE rsquo09) Paper no IMECE2009-13159 pp 287ndash296 Lake Buena Vista Fla USA November2009

[20] M Celentano Development of a CAD-based graphic environ-ment for motion and constraint analysis according to ScrewTheory [MS thesis in Mechanical Engineering] University ofNaplesmdashFederico II Naples Italy 2007

[21] H J Su D V Dorozhkin and J M Vance ldquoA screw theoryapproach for the conceptual design of flexible joints for com-pliant mechanismsrdquo Journal of Mechanisms and Robotics vol 1no 4 Article ID 041009 pp 1ndash8 2009

[22] A Dixon and J J Shah ldquoAssembly feature tutor and recognitionalgorithms based onmating face PairsrdquoComputer-Aided Designand Applications vol 7 no 3 pp 319ndash333 2010

[23] D E Whitney R Mantripragada J D Adams and S J RheeldquoDesigning assembliesrdquo Research in Engineering Design vol 11no 4 pp 229ndash253 1999

[24] D K Smith Constraint analysis of assemblies using screw theoryand tolerance sensitivities [Master of Science] Department ofMechanical Engineering Brigham Young University 2001

[25] G Shukla and D E Whitney ldquoThe path method for analyzingmobility and constraint of mechanisms and assembliesrdquo IEEETransactions on Automation Science and Engineering vol 2 no2 pp 184ndash192 2005

[26] L Rusli A Luscher and J Schmiedeler ldquoAnalysis of constraintconfigurations in mechanical assembly via screw theoryrdquo Jour-nal of Mechanical Design vol 134 no 2 Article ID 021006 pp1ndash12 2012

[27] L Rusli A Luscher and J Schmiedeler ldquoOptimization ofconstraint location orientation and quantity in mechanical

assemblyrdquo Journal of Mechanical Design vol 135 no 7 ArticleID 071007 pp 1ndash11 2013

[28] H J Su L Zhou and Y Zhang ldquoMobility analysis and typesynthesis with screw theory from rigid body linkages tocompliant mechanismsrdquoAdvances in Mechanisms Robotics andDesign Education and Research vol 14 pp 67ndash81 2013

[29] H-J Su and H Tari ldquoOn line screw systems and their applica-tion to flexure synthesisrdquo Journal of Mechanisms and Roboticsvol 3 no 1 Article ID 011009 2011

[30] D C Anderson and T C Chang ldquoGeometric reasoning infeature-based design and process planningrdquo Computers andGraphics vol 14 no 2 pp 225ndash235 1990

[31] J Hefferon Linear Algebra 2014 httpjoshuasmcvtedulinea-ralgebra

[32] J Phillips Freedom in MachinerymdashIntroducing Screw Theoryvol 1 Cambridge University Press Cambridge UK 2007

[33] ZHuang Q Li andHDing ldquoBasics of screw theoryrdquo inTheoryof ParallelMechanisms vol 6 pp 1ndash16 Springer DordrechtTheNetherlands 2013

[34] F Dunn and I Parberry 3DMath Primer for Graphics andGameDevelopment CRC Press Boca Raton Fla USA 2nd edition2011

[35] E J HaugComputer Aided Kinematics andDynamics ofMecha-nical Systems vol 1 of Allyn and Bacon Series in EngineeringBasic Method Boston Mass USA 1989

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


assembly into simple paths to which the application of thetwist union or intersection algorithms may be easily appliedThey also highlighted the complexity behind the analysis ofoverconstraints with respect to the motion analysis Celen-tano [20] improved the algorithm presented in [18] anddeveloped a tool to analyze the motions and constraints ofCAD mechanisms Su et al [21] presented a theory-basedapproach for the analysis and synthesis of flexible jointsDixon and Shah [22] provided a mechanism for user-definedcustom assembly features by using an interactive systemAfter that twist and wrench matrices could be extractedfrom face pair relations and used in kinematic and structuralanalysis

Geometry constraints between two parts in an assemblymay be arbitrary combinations References [23ndash25] analyzedthe state of a constraint of an assembly mechanism or fixtureto see if it is properly constrained or whether it containsunwanted overconstraints Rusli et al [26 27] addressedattachment-level design in which decisions are made toestablish the types locations and orientations of assemblyfeatures Su et al [28] analyzed the mobility of overcon-strained linkages and compliant mechanisms To providea desired constrained motion for synthesizing a pattern offlexures a new screw theory that addresses ldquoline screwsrdquo andldquoline screw systemsrdquo was presented by Su and Tari [29]

According to [4ndash29] if the screw representations betweenany two parts of the assemblymodel are known we can easilyanalyze the modelrsquos kinematic functions References [4ndash1323ndash29] manually model the corresponding screw systemThus it is time consuming when the machine is large andcomplex On the other hand to determine the screw systemby using the methods proposed in [14ndash22] all of the screwrepresentations of different features that may be used firsthave to bemodeledThis is awkward and resembles geometricreasoning [30] If the incidence of redesign frequency is highthe screw system has to be rebuilt again and again which hasa great influence on the efficiency of the analysis procedure

We hope to automatically and easily obtain the screwsystem of any assembly model Thus when we modify theassembly model during analysis-redesign-reanalysis cyclesthe kinematic functions can be directly validated For con-venience we adopt a graph of parts with twist amongthem to express the screw system A screw system can beeasily used for kinematic analysis such as determining thedegrees of freedom estimating the reaction of constraintsand characterizing the nature of contacts between any twocomponents

In this paper we propose a new method to automaticallyconvert a 3D assembly model into a screw system based onreciprocal screw theoryThe two fundamental concepts in theconstraint-based approach that is freedom and constraintcan be represented mathematically by a twist and a wrenchBoth the twist and kinematic joint represent motion boththe wrench and geometry constraint represent constraintWhen a twist is given we can obtain the correspondingwrench as the reciprocal product of a twist and wrenchwhich is equal to zero Conversely if we know a wrench thenthe corresponding twist could be deduced as well To avoidgeometric reasoningwemodel the three basic components of

z

x

yo

A

Instantaneousscrew axis (ISA)

120596A

120596A

o

Figure 1 Twist

geometry constraints in terms of wrench Thus all commonassembly constraints can be automatically deduced For theredundant property of geometry constraints the specificoperation of a matrix space is applied Finally we achieve thegraph of parts with kinematic joints among themwith respectto an assembly model

11 Organization of the Paper Section 2 reviews the twist rep-resentations of common kinematic joints Section 3 analyzesthe basic components of geometry constraints and modelsthese components in terms of wrench Section 4 shows howto recognize the corresponding kinematic joints of assembliesrestricted by arbitrary combinations of geometry constraintsSection 5 contains several examples Section 6 presents ourconclusions

2 Screw Models of Kinematic Joints

To recognize the corresponding kinematic joints from geom-etry constraints a mathematical model must be developedthat can sufficiently and simultaneously describe the proper-ties of kinematic joints and geometry constraintsThis sectionshows how to define kinematic joints using twist

21 Twist A twist [32 33] takes the form

T = [120596119909120596119910120596119911 120592119909120592119910120592119911] (1)

A twist is a screw that describes the instantaneousmotionto the first order of a rigid body This means that the motiondescribed by a twist is allowed The first triplet representsthe angular velocity of the body with respect to a globalreference frameThe second triplet represents the velocity inthe global reference frame of that point on the body or itsextension which is instantaneously located at the origin ofthe global frame Therefore the unique line associated withthe first triplet is the axis of rotation and the unique pointassociated with the second triplet is the point on the body orits extension which is at that instant located at the origin ofthe global reference frame See Figure 1

The ISA in Figure 1 is the axis introduced in ChaslesThe-orem Both the twist and kinematic joint represent motion


ISA

Pv





119909 V119910V119911)


119875s = 119875v (2)


T119875 = [0 119875s] (3)



119909 V119910 V119911) 119877P = (119875

119909 119875119910 119875119911) is a


119877L = 119877P + 119886 sdot 119877v (4)


119877s = 119877P times 119877v (5)


T119877 = [119877v 119877s] (6)

ISA

RP

Pv

RL


ISA

HL

HP

Hv

Hp



119909 V119910 V119911) 119867P = (119875

119909 119875119910 119875119911) is a


119867L = 119867P + 119886 sdot 119867v (7)


119867s = 119867P times 119867v + 119867119901 sdot 119867v (8)


T119867 = [119867v 119867s] (9)


119909 V119910 V119911)119862P = (119875

119909 119875119910 119875119911)


119862L = 119862P + 119886 sdot 119862v (10)


ISA

CL

CP

Cv


SP

ISA

v1

v2

v3



119862s1= 119862P times 119862v

119862s2= 119862v

(11)


T119862 = [119862v 119862s

1

0 119862s2

] (12)



119878 and define v1= (1 0 0) v

2= (0 1 0) and v

3= (0 0 1) See


119878s1= 119878P times v

1= (0 119875

119911 minus119875119910)

119878s2= 119878P times v

2= (minus119875

119911 0 119875119909)

119878s3= 119878P times v

3= (119875119910 minus119875119909 0)

(13)

EP

ISA

En

Ev1

Ev2



T119878 = [[[

v1 119878s1

v2 119878s2

v3 119878s3

]

]

]

(14)


1= (v1199091 v1199101 v1199111) and 119864v

2= (v1199092 v1199102 v1199112) are


119909 119875119910 119875119911) is a single point


119864s1= 119864v1

119864s2= 119864v2

119864s3= 119864P times 119864n

(15)


T119864 = [[[

0 119864s1

0 119864s2

119864n 119864s3

]

]

]

(16)




W = [119891119909119891119910119891119911 119898119909119898119910119898119911] (17)



z

x

yo

M

Force system


Fi

F1 F2

F3

Fr

Mr

ri


Fr

Figure 8 Wrench






119894and P


f119894-g119894-h119894and f


119894and a

119895


z

x

y

i

j

ai

Pihi

gi

f i

ri

sPidij

fj

hjgj

sPj

Pj aj

rj


P119894and P




Φ

1198891

(a119894 a119895 1198621) equiv a119894

Ta119895minus 1198621= 0 119862




119894





119895and body-


119895 where a

119894= A119894a1015840119894and a

119895=

A119895a1015840119895 (18) has the form

Φ

1198891

(a119894 a119895 1198621) equiv a1015840119894

TA119894

TA119895a1015840119895minus 1198621= 0


(19)





119894and


W1198891 = (0 a119894times a119895) (20)


W1198891 = (0 (a1015840119894

TA119894

T) times (A

119895a1015840119895)) (21)


Φ

1198892

(a119894 d119894119895 1198622) equiv a119894

Td119894119895minus 1198622= 0 d

119894119895= 0 (22)




119894119895as

d119894119895= r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P (23)


Φ

1198892

(a119894 d119894119895 1198622) equiv a1015840119894

TA119894

T(r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P)

minus 1198622= 0

(24)


119894


W1198892 = (a119894 0) (25)


W1198892 = (A119894a1015840119894 0) (26)


Φ

119904

(119875119894 119875119895 1198623) equiv d119894119895

Td119894119895minus 1198623sdot 1198623= 0 (27)



doc (Φ119904) = 1 if 1198623

= 0

doc (Φ119904) = 3 if 1198623= 0

(28)


W119904 = [d119894119895 0] if 119862

3= 0 (29)


119894119895if

1198623


the form

W119904 = [[[

1 0 0 00 1 0 00 0 1 0

]

]

]

if 1198623= 0 (30)




2 reference frames


1and L

2are fixed in



12connects P

1and


L1

B1

h1

P1

f1

F1 g1

L2

B2

h2

P2

f2F2

g2

d12



P1= (1198751199091 1198751199101 1198751199111) (31)


L1= P1+ 119886 sdot h

1 (32)


F1= P1+ 119887 sdot f

1+ 119888 sdot g

1 (33)


2 L2 and F









Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(f1 d12 0)

Φ4

1198892

(g1 d12 0)

(34)


W119862119900119894119871119871(L

1L2)=

[

[

[

[

[

[

0 h1times f2

0 h1times g2

f1 0

g1 0

]

]

]

]

]

]

(35)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)


f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12





2 as shown




Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)


W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)




2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)




3= 0


Φ

119904

(119875119894 119875119895 0) (38)


W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)



1and F





Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ISA

Pv





119909 V119910V119911)


119875s = 119875v (2)


T119875 = [0 119875s] (3)



119909 V119910 V119911) 119877P = (119875

119909 119875119910 119875119911) is a


119877L = 119877P + 119886 sdot 119877v (4)


119877s = 119877P times 119877v (5)


T119877 = [119877v 119877s] (6)

ISA

RP

Pv

RL


ISA

HL

HP

Hv

Hp



119909 V119910 V119911) 119867P = (119875

119909 119875119910 119875119911) is a


119867L = 119867P + 119886 sdot 119867v (7)


119867s = 119867P times 119867v + 119867119901 sdot 119867v (8)


T119867 = [119867v 119867s] (9)


119909 V119910 V119911)119862P = (119875

119909 119875119910 119875119911)


119862L = 119862P + 119886 sdot 119862v (10)


ISA

CL

CP

Cv


SP

ISA

v1

v2

v3



119862s1= 119862P times 119862v

119862s2= 119862v

(11)


T119862 = [119862v 119862s

1

0 119862s2

] (12)



119878 and define v1= (1 0 0) v

2= (0 1 0) and v

3= (0 0 1) See


119878s1= 119878P times v

1= (0 119875

119911 minus119875119910)

119878s2= 119878P times v

2= (minus119875

119911 0 119875119909)

119878s3= 119878P times v

3= (119875119910 minus119875119909 0)

(13)

EP

ISA

En

Ev1

Ev2



T119878 = [[[

v1 119878s1

v2 119878s2

v3 119878s3

]

]

]

(14)


1= (v1199091 v1199101 v1199111) and 119864v

2= (v1199092 v1199102 v1199112) are


119909 119875119910 119875119911) is a single point


119864s1= 119864v1

119864s2= 119864v2

119864s3= 119864P times 119864n

(15)


T119864 = [[[

0 119864s1

0 119864s2

119864n 119864s3

]

]

]

(16)




W = [119891119909119891119910119891119911 119898119909119898119910119898119911] (17)



z

x

yo

M

Force system


Fi

F1 F2

F3

Fr

Mr

ri


Fr

Figure 8 Wrench






119894and P


f119894-g119894-h119894and f


119894and a

119895


z

x

y

i

j

ai

Pihi

gi

f i

ri

sPidij

fj

hjgj

sPj

Pj aj

rj


P119894and P




Φ

1198891

(a119894 a119895 1198621) equiv a119894

Ta119895minus 1198621= 0 119862




119894





119895and body-


119895 where a

119894= A119894a1015840119894and a

119895=

A119895a1015840119895 (18) has the form

Φ

1198891

(a119894 a119895 1198621) equiv a1015840119894

TA119894

TA119895a1015840119895minus 1198621= 0


(19)





119894and


W1198891 = (0 a119894times a119895) (20)


W1198891 = (0 (a1015840119894

TA119894

T) times (A

119895a1015840119895)) (21)


Φ

1198892

(a119894 d119894119895 1198622) equiv a119894

Td119894119895minus 1198622= 0 d

119894119895= 0 (22)




119894119895as

d119894119895= r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P (23)


Φ

1198892

(a119894 d119894119895 1198622) equiv a1015840119894

TA119894

T(r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P)

minus 1198622= 0

(24)


119894


W1198892 = (a119894 0) (25)


W1198892 = (A119894a1015840119894 0) (26)


Φ

119904

(119875119894 119875119895 1198623) equiv d119894119895

Td119894119895minus 1198623sdot 1198623= 0 (27)



doc (Φ119904) = 1 if 1198623

= 0

doc (Φ119904) = 3 if 1198623= 0

(28)


W119904 = [d119894119895 0] if 119862

3= 0 (29)


119894119895if

1198623


the form

W119904 = [[[

1 0 0 00 1 0 00 0 1 0

]

]

]

if 1198623= 0 (30)




2 reference frames


1and L

2are fixed in



12connects P

1and


L1

B1

h1

P1

f1

F1 g1

L2

B2

h2

P2

f2F2

g2

d12



P1= (1198751199091 1198751199101 1198751199111) (31)


L1= P1+ 119886 sdot h

1 (32)


F1= P1+ 119887 sdot f

1+ 119888 sdot g

1 (33)


2 L2 and F









Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(f1 d12 0)

Φ4

1198892

(g1 d12 0)

(34)


W119862119900119894119871119871(L

1L2)=

[

[

[

[

[

[

0 h1times f2

0 h1times g2

f1 0

g1 0

]

]

]

]

]

]

(35)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)


f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12





2 as shown




Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)


W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)




2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)




3= 0


Φ

119904

(119875119894 119875119895 0) (38)


W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)



1and F





Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ISA

CL

CP

Cv


SP

ISA

v1

v2

v3



119862s1= 119862P times 119862v

119862s2= 119862v

(11)


T119862 = [119862v 119862s

1

0 119862s2

] (12)



119878 and define v1= (1 0 0) v

2= (0 1 0) and v

3= (0 0 1) See


119878s1= 119878P times v

1= (0 119875

119911 minus119875119910)

119878s2= 119878P times v

2= (minus119875

119911 0 119875119909)

119878s3= 119878P times v

3= (119875119910 minus119875119909 0)

(13)

EP

ISA

En

Ev1

Ev2



T119878 = [[[

v1 119878s1

v2 119878s2

v3 119878s3

]

]

]

(14)


1= (v1199091 v1199101 v1199111) and 119864v

2= (v1199092 v1199102 v1199112) are


119909 119875119910 119875119911) is a single point


119864s1= 119864v1

119864s2= 119864v2

119864s3= 119864P times 119864n

(15)


T119864 = [[[

0 119864s1

0 119864s2

119864n 119864s3

]

]

]

(16)




W = [119891119909119891119910119891119911 119898119909119898119910119898119911] (17)



z

x

yo

M

Force system


Fi

F1 F2

F3

Fr

Mr

ri


Fr

Figure 8 Wrench






119894and P


f119894-g119894-h119894and f


119894and a

119895


z

x

y

i

j

ai

Pihi

gi

f i

ri

sPidij

fj

hjgj

sPj

Pj aj

rj


P119894and P




Φ

1198891

(a119894 a119895 1198621) equiv a119894

Ta119895minus 1198621= 0 119862




119894





119895and body-


119895 where a

119894= A119894a1015840119894and a

119895=

A119895a1015840119895 (18) has the form

Φ

1198891

(a119894 a119895 1198621) equiv a1015840119894

TA119894

TA119895a1015840119895minus 1198621= 0


(19)





119894and


W1198891 = (0 a119894times a119895) (20)


W1198891 = (0 (a1015840119894

TA119894

T) times (A

119895a1015840119895)) (21)


Φ

1198892

(a119894 d119894119895 1198622) equiv a119894

Td119894119895minus 1198622= 0 d

119894119895= 0 (22)




119894119895as

d119894119895= r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P (23)


Φ

1198892

(a119894 d119894119895 1198622) equiv a1015840119894

TA119894

T(r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P)

minus 1198622= 0

(24)


119894


W1198892 = (a119894 0) (25)


W1198892 = (A119894a1015840119894 0) (26)


Φ

119904

(119875119894 119875119895 1198623) equiv d119894119895

Td119894119895minus 1198623sdot 1198623= 0 (27)



doc (Φ119904) = 1 if 1198623

= 0

doc (Φ119904) = 3 if 1198623= 0

(28)


W119904 = [d119894119895 0] if 119862

3= 0 (29)


119894119895if

1198623


the form

W119904 = [[[

1 0 0 00 1 0 00 0 1 0

]

]

]

if 1198623= 0 (30)




2 reference frames


1and L

2are fixed in



12connects P

1and


L1

B1

h1

P1

f1

F1 g1

L2

B2

h2

P2

f2F2

g2

d12



P1= (1198751199091 1198751199101 1198751199111) (31)


L1= P1+ 119886 sdot h

1 (32)


F1= P1+ 119887 sdot f

1+ 119888 sdot g

1 (33)


2 L2 and F









Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(f1 d12 0)

Φ4

1198892

(g1 d12 0)

(34)


W119862119900119894119871119871(L

1L2)=

[

[

[

[

[

[

0 h1times f2

0 h1times g2

f1 0

g1 0

]

]

]

]

]

]

(35)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)


f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12





2 as shown




Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)


W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)




2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)




3= 0


Φ

119904

(119875119894 119875119895 0) (38)


W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)



1and F





Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

x

yo

M

Force system


Fi

F1 F2

F3

Fr

Mr

ri


Fr

Figure 8 Wrench






119894and P


f119894-g119894-h119894and f


119894and a

119895


z

x

y

i

j

ai

Pihi

gi

f i

ri

sPidij

fj

hjgj

sPj

Pj aj

rj


P119894and P




Φ

1198891

(a119894 a119895 1198621) equiv a119894

Ta119895minus 1198621= 0 119862




119894





119895and body-


119895 where a

119894= A119894a1015840119894and a

119895=

A119895a1015840119895 (18) has the form

Φ

1198891

(a119894 a119895 1198621) equiv a1015840119894

TA119894

TA119895a1015840119895minus 1198621= 0


(19)





119894and


W1198891 = (0 a119894times a119895) (20)


W1198891 = (0 (a1015840119894

TA119894

T) times (A

119895a1015840119895)) (21)


Φ

1198892

(a119894 d119894119895 1198622) equiv a119894

Td119894119895minus 1198622= 0 d

119894119895= 0 (22)




119894119895as

d119894119895= r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P (23)


Φ

1198892

(a119894 d119894119895 1198622) equiv a1015840119894

TA119894

T(r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P)

minus 1198622= 0

(24)


119894


W1198892 = (a119894 0) (25)


W1198892 = (A119894a1015840119894 0) (26)


Φ

119904

(119875119894 119875119895 1198623) equiv d119894119895

Td119894119895minus 1198623sdot 1198623= 0 (27)



doc (Φ119904) = 1 if 1198623

= 0

doc (Φ119904) = 3 if 1198623= 0

(28)


W119904 = [d119894119895 0] if 119862

3= 0 (29)


119894119895if

1198623


the form

W119904 = [[[

1 0 0 00 1 0 00 0 1 0

]

]

]

if 1198623= 0 (30)




2 reference frames


1and L

2are fixed in



12connects P

1and


L1

B1

h1

P1

f1

F1 g1

L2

B2

h2

P2

f2F2

g2

d12



P1= (1198751199091 1198751199101 1198751199111) (31)


L1= P1+ 119886 sdot h

1 (32)


F1= P1+ 119887 sdot f

1+ 119888 sdot g

1 (33)


2 L2 and F









Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(f1 d12 0)

Φ4

1198892

(g1 d12 0)

(34)


W119862119900119894119871119871(L

1L2)=

[

[

[

[

[

[

0 h1times f2

0 h1times g2

f1 0

g1 0

]

]

]

]

]

]

(35)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)


f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12





2 as shown




Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)


W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)




2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)




3= 0


Φ

119904

(119875119894 119875119895 0) (38)


W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)



1and F





Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894119895as

d119894119895= r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P (23)


Φ

1198892

(a119894 d119894119895 1198622) equiv a1015840119894

TA119894

T(r119895+ A119895s1015840119895

Pminus r119894minus A119894s1015840119894

P)

minus 1198622= 0

(24)


119894


W1198892 = (a119894 0) (25)


W1198892 = (A119894a1015840119894 0) (26)


Φ

119904

(119875119894 119875119895 1198623) equiv d119894119895

Td119894119895minus 1198623sdot 1198623= 0 (27)



doc (Φ119904) = 1 if 1198623

= 0

doc (Φ119904) = 3 if 1198623= 0

(28)


W119904 = [d119894119895 0] if 119862

3= 0 (29)


119894119895if

1198623


the form

W119904 = [[[

1 0 0 00 1 0 00 0 1 0

]

]

]

if 1198623= 0 (30)




2 reference frames


1and L

2are fixed in



12connects P

1and


L1

B1

h1

P1

f1

F1 g1

L2

B2

h2

P2

f2F2

g2

d12



P1= (1198751199091 1198751199101 1198751199111) (31)


L1= P1+ 119886 sdot h

1 (32)


F1= P1+ 119887 sdot f

1+ 119888 sdot g

1 (33)


2 L2 and F









Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(f1 d12 0)

Φ4

1198892

(g1 d12 0)

(34)


W119862119900119894119871119871(L

1L2)=

[

[

[

[

[

[

0 h1times f2

0 h1times g2

f1 0

g1 0

]

]

]

]

]

]

(35)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)


f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12





2 as shown




Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)


W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)




2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)




3= 0


Φ

119904

(119875119894 119875119895 0) (38)


W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)



1and F





Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

CoiLL(L1 L2)


f2

P2

h2

F2

g2

L2

B2

f1P1

h1

F1

g1

L1

B1

CoiFF(F1 F2)

d12





2 as shown




Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) Φ

3

1198892

(h1 d12 0) (36)


W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

(37)




2 as shown in

Figure 13

f2

P2

h2

F2 g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPP(P1P2)


f2 P2 h2

F2g2

L2

B2

f1 P1 h1

F1g1

L1

B1

ParFF(F1 F2)




3= 0


Φ

119904

(119875119894 119875119895 0) (38)


W119862119900119894119875119875(P

1P2)=

[

[

[

1 0 0 00 1 0 00 0 1 0

]

]

]

(39)



1and F





Φ1

1198891

(h1 f2 0) Φ

2

1198891

(h1 g2 0) (40)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

DisLF(L1 F2 D)



W119875119886119903119865119865(F

1F2)= [

0 h1times f2

0 h1times g2

] (41)




1and the surface F

2that maintain




Φ1

1198891

(h1 h2 0) Φ

2

1198892

(h2 d12 119863) (42)


W119863119894119904119871119865(L

1F2119863)

= [

0 h1times h2

h2 0

] (43)



P1on line L




Φ1

1198892

(f2 d12 0) Φ

2

1198892

(g2 d12 0) (44)


W119863119894119904119871119865(L

1F2119863)

= [

f2 0

g2 0

] (45)


f2 P2 h2

F2g2

L2

B2

f1

P1

h1

F1

g1

L1

B1

CoiPL(P1 L2)





1 1198612 119861











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860 119895 = 1 119897







END


f2

A2

h2F2

g2

B2

f1A1

F1

g1B1




given by

T1∘W1= T11minus3

sdotW24minus6

+ T14minus6

sdotW21minus3

= [0 0] (46)





1and 119861

2 The


1 F2) See



1 1198612) = 5 How-


1



1A2)and

f2

h2

F2g2

B2

f1h1

g1

B1

F1


W119862119900119894119865119865(F



5 Examples



1and 119861


119862119900119894119865119865(F1 F2)


W119862119900119894119865119865(F

1F2)=

[

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

]

(47)


f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










f2h2

F2

g2

B2

f1

h1 g1

B1

F1

f 9984002

h9984002g9984002

F9984002

f 9984001

h9984001

g9984001

F9984001


whereW119862119900119894119865119865(F



1F2)is

P119862119900119894119865119865(F

1F2)=

[

[

[

f1 0

g1 0

0 h1

]

]

]

(48)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119865119865(F1F2)) =

[

[

[

0 f1

0 g1

h1 0

]

]

]

(49)

where T12



1and 119861

2 and two coplanar


1 F10158402)


1and 119861

2are

W119862119900119894119865119865(F

1F2)=

[

[

[

0 h1times f2

0 h1times g2

h1 0

]

]

]

W119862119900119894119865119865(F1015840

1F10158402)=

[

[

[

[

0 h10158401times f10158402

0 h10158401times g10158402

h10158401 0

]

]

]

]

(50)

B0

B1

B2

B3

B4

B5B6

B7

B8

B9

B10

B11

x

y


where W12

= W119862119900119894119865119865(F

1F2)cup W119862119900119894119865119865(F1015840


matrixQ12ofW12is

Q12=

[

[

[

[

[

[

[

[

[

[

[

0 h1times f2

0 h1times g2

0 h10158401times g10158402

h1 0

h10158401 0

]

]

]

]

]

]

]

]

]

]

]

(51)


P12= [g1 0] (52)


1and 119861

2has the

following form

T12= 119877119890119888119894119901119903119900119888119886119897 (P

12) = [0 g

1] (53)

where T12




1198614 1198615 1198616 1198617 1198618 1198619 11986110 and 119861


119862119900119894119865119865(1198610 1198612)119862119900119894119865119865(119861

0 1198611)119862119900119894119865119865(119861

1 1198613)119862119900119894119865119865(119861

1 1198615)

119862119900119894119865119865(1198611 1198614) 119862119900119894119865119865(119861

4 1198616) 119862119900119894119865119865(119861

4 11986111) 119862119900119894119865119865(119861

4

1198617) 119862119900119894119865119865(119861

4 1198618) 119862119900119894119865119865(119861

9 11986110) 119862119900119894119865119865(119861

7 11986110) and


0 1198612)

119862119900119894119871119871(1198610 1198611) 119862119900119894119871119871(119861

1 1198613) 119862119900119894119871119871(119861

2 1198613) 119862119900119894119871119871(119861

1 1198615)

119862119900119894119871119871(1198611 1198614) 119862119900119894119871119871(119861

4 1198616) 119862119900119894119871119871(119861

5 1198616) 119862119900119894119871119871(119861

4

11986111) 119862119900119894119871119871(119861

4 1198617) 119862119900119894119871119871(119861

4 1198618) 119862119900119894119871119871(119861

8 1198619)

119862119900119894119871119871(1198619 11986110) 119862119900119894119871119871(119861

7 11986110) and 119862119900119894119871119871(119861

10 11986111) See


1

1198612 and 119861



3is 119862119900119894119871119871(119861



CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










CoiLL

CoiFF

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11


h2 g2

f 9984003

g9984003

B2

B1

B0

g3h3

g9984001

f 9984001

B3

x

y

Figure 22 Parts 1198611 1198612 and 119861



W119862119900119894119871119871(119861

21198613)=

[

[

[

[

[

[

0 h2times f3

0 h2times g3

f2 0

g2 0

]

]

]

]

]

]

(54)


P119862119900119894119871119871(119861

21198613)= [

0 h2

h2 0

] (55)



3has the

following form

T23= 119877119890119888119894119901119903119900119888119886119897 (P

119862119900119894119871119871(11986121198613)) = [

h2 0

0 h2

] (56)


and 1198613

are119862119900119894119871119871(119861

1 1198613) and 119862119900119894119865119865(119861



3are

W119862119900119894119871119871(119861

11198613)=

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

]

]

]

]

]

]

]

W119862119900119894119865119865(119861

11198613)=

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

h10158403 0

]

]

]

]

(57)

where W13

= W119862119900119894119871119871(119861

11198613)cup W119862119900119894119865119865(119861



Q13=

[

[

[

[

[

[

[

[

[

[

0 h10158403times f10158401

0 h10158403times g10158401

f10158403 0

g10158403 0

h10158403 0

]

]

]

]

]

]

]

]

]

]

(58)


P13= [0 h1015840

3] (59)


1and 119861

3is

T13= 119877119890119888119894119901119903119900119888119886119897 (P

13) = [h1015840

3 0] (60)



1and 119861

4 that is

T1198611


T1198614

= (T1198611

cup T41) cap (T

1198611


(61)




1198611



hydraulic grab


T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










T twist

B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

T32

T31

T10

T20

T109

T1110

T114

T98

T74 T8

4

T41

T64

T65

T51

T107



B0B1

B2B3

B4

B5B6

B7 B8

B9B10

B11

R31

R10

R20

1110

R114

C98

R74 R8

4

R41

R64 R5

1

R107

R

R109 C3

2

C65



2and 119861

3and 119861

1and 119861



6 Conclusions


Additional Points






Competing Interests


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Recognition of Kinematic Joints of 3D Assembly …downloads.hindawi.com/journals/mpe/2016/1761968.pdf · 2019-07-30 · resultant moment of the force system about