The Structure and Dimensional Design of a Reconfigurable PKMdsun/pubs/The... · International...

International Journal of Advanced Robotic Systems

The Structure and Dimensional Design of a Reconfigurable PKM Regular Paper

Xiaoqiang Tang1,*, Dengfeng Sun2 and Zhufeng Shao1 1 The State Key Laboratory of Tribology, Department of Precision Instruments and Mechanology, Tsinghua University,Beijing, P.R. of China 2 School of Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, IN,USA * Corresponding author E-mail:[email protected] Received 4Oct 2012; Accepted 26Oct 2012 DOI: 10.5772/54696 © 2013 Tanget al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Parallel Kinematic Machines (PKMs) have many advantages and have been widely used in the machine industry. Benefitting from its modular structure, a PKM is more reconfigurable than traditional serial machines. In this paper, a new type of driving strut module and innovative joints are designed for the Reconfigurable Parallel Kinematic Machine (RPKM). The new driving strut module can be changed from linear drive mode to telescopic drive mode easily, and the new spherical joint and universal joint are designed to achieve a large rotation angle. The inverse kinematics problems in relation to the 6‐DOF RPKM are analysed, and the Workspace Volume Index (WVI) and the Global ConditionIndex (GCI) are adopted to design the RPKM. According to the WVI and GCI analysis of the selected parameters for two types of 6‐DOF PKM, the dimensional parameters of the RPKM are designed. In the end, the new type of RPKM prototype is built, with which a wax pattern is machined. Keywords Structure Design, Dimensional Design, Reconfigurable PKM

1. Introduction

In recent years, the ever‐changing manufacturing environment caused by intensive competition on a global

scale and rapid changes in process technology has led to the requirement for an increasingly efficient production system. In order to achieve this, a reconfigurable manufacturing system characterized by modularity, integrability, customization, convertibility and diagnosability, which can easily be reconfigured and respond to unpredictable market changes cost‐effectively, is highly preferred. The earliest large‐scale research on reconfigurable manufacture systems reported in the literature is the Flexible Manufacture Cell (FMC) project launched in 1977 by MITI in Japan. Other reconfiguration projects, such as MOSYN, KERNEL and MOTION, focused on modular machine tools. After the foundation of the Engineering Research Center of Reconfigurable Machining Systems (ERC/RMS) at the University of Michigan in 1996, researchers began to develop the theory and technology of reconfigurable manufacturing systems systematically[1‐3]. A Parallel Kinematic Machine (PKM) has many advantages and has been widely used in machine industry. Thanks to its modular structure, a PKM is more reconfigurable than the traditional serial machine. With the modular design, several modules of PKMs can be integrated into a module set so that the modules can be reassembled rapidly for different applications. As a result, the recent trend in reconfigurable manufacturing

1Xiaoqiang Tang, Dengfeng Sun and Zhufeng Shao: The Structure and Dimensional Design of a Reconfigurable PKM

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systems has promoted the research and development of a novel type of machine tool called a Reconfigurable Parallel Kinematic Machine (RPKM) [4‐5], which is based on the parallel manipulator. More recently, the research on RPKMs has made some substantive progress, but some problems that should be solved for its further application still exist [6‐7]. Firstly, according to drive mode, the PKM can be mainly classified into two kinds: the telescopic drive PKM, whose struts are extendable with one end mounted on framework or base platform and the other on the moving platform; the linear drive PKM, whose struts have constant length with one end mounted on a slider and the other on the moving platform. Because both these PKMs have their pros and cons, they are designed and applied in respective cases: the linear drive PKM is inclined to be adopted in machining long‐shaped parts along its dominant movement direction, and the telescopic drive PKM is suitable to machine more complicated shapes other than those of a longshape [8]. It would therefore be ideal if the RPKM can realize these two types of PKMs with the same modular set. Moreover, these two types of PKMs need to use the universal joints and spherical joints. A universal joint can work with 2‐DOFs rotation, and a spherical joint with 3‐DOFs rotation. With the classic manufacturing process, it is difficult to fabricate a high precision spherical joint with a large rotation angle. To solve this problem, an integrated joint with either two DOFs or three DOFs should be designed. The dimensional design is another important and challenging problem for RPKMs. Two major issues need to be addressed in the optimal design of a PKM: performance evaluation and synthesis. After the design of a RPKM, it is necessary to evaluate its main characteristics. The second problem is to determine the dimensions (link lengths) of the RPKM, to make it most suitable for the specific task. The latter problem is one of the most difficult issues to solve in the field of RPKM. As is well known, the performances of parallel kinematic machines mainly depend upon the dimensions. Therefore, the process of mechanism dimensional design for a manipulator is largely based on the usage of the criteria, such as workspace, dexterity, payload, conditioning index and stiffness [9‐11]. Approaches in previous work dealing with the dimensional design of PKMs can be classified into two categories. One category refers to the local optimal design [11‐13], focusing on finding a family of dimensional parameters that generate an isotropy configuration, where the platform presents the desirable characteristics in terms of kinematics accuracy, noise rejection and singularity avoidance. However, due to the possible strut interference, this family of design parameters would not be useful in the design where the desirable orientation capability of the moving platform is also required. The other category

refers to the global optimal design [14,15], which attempts to find a set of design parameters to minimize a weighted cost function in terms of the specified dexterous performance and workspace volume. The reason for implementing global optimal design would be that kinematics performance could not be guaranteed simply by conducting local optimal design. However, the above two dimensional design methods are only used for the telescopic drive PKM or the linear drive PKM respectively. In this paper, we will try to use the global optimal design method to designthe RPKM. In this paper, a new type of driving struts moduleand joints are designed for the RPKM. The inverse kinematics problems with regard to the two types of 6‐DOF parallel manipulators are described and a global optimal design index is used to design the RPKM, which can satisfy the requirement of the RPKM and realize workspace and dexterity capability. In the end, we show two wax patterns milled by the RPKM.

2. Structure design

The structure of reconfigurable PKMs is similar to that of some ordinary PKMs. They consist of common parts like: a base platform, a moving platform on which ending executor is installed, and some parallel‐actuated extendible legs or movable legs between the mobile and base platforms. Universal joints and spherical joints are mounted on either the moving platform or the base platform to connect platforms and legs, and to provide multi‐position and versatile attitude. The key modular designs of reconfigurable PKMs include strut modules and joint modules.

2.1 Strut modules design [16]

According to driving mode, PKMs can mainly be catalogued into two kinds: telescopic drive PKMs, whose struts are extendable with one end mounted on the framework or base platform and the other on the moving platform; linear drive PKMs, whose struts have constant length with one end mounted on a slider and the other on the moving platform. In figure 1, the telescopic drive mode is represented: a driving motor (1) is mounted at one end of the telescopic drive module. The motor drives the ball screw (2), and the work‐piece module (3) moves along the axis of the ball screw (2). A rod‐fix module (4) is connected to the work‐piece module along the direction of the screw in the linear drive module, then a constant‐length rod module (5) and a spherical joint (6) are mounted in turn. The length of the rod module (5) is represented by kl , and the maximal move distance of the telescopic drive mode is denoted by l .

2 Int J Adv Robotic Sy, 2013, Vol. 10, 267:2013 www.intechopen.com

In figure 2, lidriving motomodule, thwork‐piece mscrew(2). A upiece modulthe constant‐(6) are mounrepresented represented b According tochanged intconstant‐lengwork‐piece measy and can

Figure 1.Telesc

Figure 2.Linea

2.2 Joint modu

In a RPKM,spherical jomanufacturinprecision sphangle of the 20 . To res

either two DOfigure 3. Therotation axesthree DOFs.

inear drive mor(1) is mountehe motor drivmodule(3) mouniversal joine, then a rod‐‐length rod mnted in turn. Tby kl , and thby l .

o the design, thto linear drigth rod modumodule (3). Tn be finished in

copic drive mod

ar drive mode.

ule design [17]

, a universaloint has thrng process, itherical joint. Mtraditional s

solve this issuOFs or three De new type os, which are CAs shown i

ode is represeed at one end ves the ball soves along thnt (7) is moun‐fix module (4

module (5) andThe length of he maximal

he telescopic dive mode bule (5)at anotThe assemblinn a short perio

de

l joint has twree DOFs. Wt is difficult toMoreover, thepherical jointue, integratedDOFs are desif spherical joC‐ axis, A‐axiin figure 4, w

ented as followof the linear dscrew(2), andhe axis of thented on the w4) is connected a spherical rod module (move distanc

drive mode caby mounting ther place onng process is od of time.

wo DOFs, anWith the clo fabricate a e general rotat is no more d joints possesigned, as showoint includes ts and C‐axis whenwe use

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work‐ed to joint (5) is ce is

an be the

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2.3 T

Bothhavdesidrivshaptelescomshowwith

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C-axis

ure 3. Spherical j

rotation angle of A‐axis isnew type sphe

C-axi

ure 4. Universal

Two types of 6‐

h the linear de their pros aigned to be ve PKM is incped parts alonscopic drive

mplicated shapwn in figures h the same mo

y, the sphericaith two DOFs.

A-axs

joint

gle of C‐axis s 60 , so theerical joint and

is

joint

‐DOF PKMs

rive PKM andand cons. Theapplied in reclined to be adng its linear mPKM is su

pes other than5 and 6, we asodule set.

al joint will be

xis C-axi

is 90 ande general rotad universal jo

A-axis

d the telescoperefore, they espective casedopted in ma

movement direuitable to mn those of a lssemble two ty

e changed into

is

d the rotationation angle ofint is 60 .

pic drive PKMare generallyes: the linearachining long‐ction, and the

machine moreongshape. Asypes of PKMs

o

n f

M y r ‐e e s s


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Figure 5.The te

Figure 6. The l

3. Dimension

As soon as wmost importRPKM. It is design methothe criteria athe kinematicindexes are results throu(WVI) and th

3.1 Kinematic

Kinematics developed amoving plat( i 1, 2, 3,...,denoted as system : o

1 6B ...B with y axis in coiof points Bframe, definithe centre perpendiculato the vecto

1 6A ...A is de( i 1, 2, 3,...,denoted as lstructure

A 2 / 3

elescopic drive P

linear drive PKM

nal design

we have the strtant step is toof primary imod that can reand link dimecs model of thformulated.

ugh analysinghe Global Cond

s model

models of thas shown in tform are de6 ), and the

iB ( i 1, 2, 3,o xyz is locathe z axis vencidence with

1 6...B is denoting the top fraof the circlar to the movior 1OA . The enoted as AR6 ), and the l , where i iA Bis symm

A .

PKM

M

ructure designo decide the dmportance to eveal the relatensions of thehe RPKM is buThen we get

g the Workspaditioning Inde

hese two typfigure 7. Th

enoted as plavertices of th,...,6 ). A fixedated at the ceertical to the bh 1OB . The cited as BR . Aame : o ʹ xle 1 6A ...A . ng platform acircumcircle . Vector iA Blink length l , ( i 1, 2,

metrical, B 2

n of the RPKMdimensions ofdevelop a utionships betwe machine. Firuilt and the crit the dimensiace Volume Inex (GCI) [14].

pes of PKMhe vertices ofatform jointshe base plated global referentre of the cbase plate andircumcircle raAnother referx y z , is locateThe z axi

and y axis parradius of po

i is denoted afor each le

...,6 ). Because

B2 / 3

M, the f the seful ween rstly, iteria ional ndex

are f the iA e are rence circle d the adius rence ed at is is rallel oints as iL eg is e the and

Figuman

Thedefiin thordeof tposiorie

Theexpr

The c

a

ure 7. The geomnipulators.

objective of ine a mappinghe Cartesian ser to achieve the moving pition is givenentation is give

rotation of coressed by the

c cs cs

R

vectors i Ban be written

a. The telescopic

b. The linear d

etric architectur

the inverse g from the pospace to the sethe aim pose.platform is asn by the positen by a matrix

ʹ x

O

oordinate systerotation matri

c s s s c s s s c c c s

in the base pas:

c drive PKM

drive PKM

re of the 6‐DOF

kinematics sose of the movet of actuated For this analssumed to betion vector OxR . Then it is

Tx y z

em ʹ relativix

c c s c sc s s c c c c

platform coord

parallel

solution is toving platformd leg inputs inlysis, the posee known. TheOʹ

and the

shown

(1)

e to can be

s sc s

(2)

dinate system

o m n e e e

)

e

)

m


i B B iR cos((i 1) / 3),R sin((i 1) / 3),z ,(i 1,3,5)

B

B Bi

B B i

R cos((i 2) / 3 ),R sin((i 2) / 3 ),z

(i 2,4,6)

B

(3)

where iz 0,i 1...6 , for PKM with telescopic drives. The vectors i A in the moving platform coordinate system can be written as:

i A AR cos((i 1) / 3),R sin((i 1) / 3),0 ,(i 1,3,5)

A

A A

iA A

R cos((i 2) / 3 ),,

R sin((i 2) / 3 ),0(i 2,4,6)

A

(4)

Therefore, the vectors i A can be expressed in the base platform coordinate system as:

i i RA A O , (i 1,2,...,6) (5)

The relationship between vectors i A , iB and strut vectors has the following form according to the geometric constraint of the architecture:

i i i il B A L ， (i 1,2,...,6)

(6)

where i kl l ,i 1,2,...,6 if the PKM is equipped with linear drives. The inverse kinematics problem of the machine can be solved by Eq. (6), where the unknown items are either il ( i 1,2,...,6 ) for the PKM with telescopic drives, or iz ( i 1,2,...,6 ) for the PKM with linear drives. Hence, for a given mechanism and prescribed position and orientation of the moving platform, the required actuator inputs can be directly computed from Eq. (6). Eq. (6) can be differentiated with respect to time to obtain the velocity equation. This leads to an equation as:

p qJ J p q (9)

where q is the velocity vector of the moving platform defined in the Cartesian coordinate systemas

T

x,y,z, , , q (10)

and p is the vector of input velocities. As mentioned earlier, there are two different types of drive modes. When the PKM is equipped with telescoped legs, p can be expressed as

T1 2 3 4 5 6[l ,l ,l ,l ,l ,l ] p (11)

while when the PKM is actuated with slider legs, its expression is

T

1 2 3 4 5 6[z ,z ,z ,z ,z ,z ] p (12)

Matrices pJ and qJ are the 6 6 forward and inverse Jacobian matrices of the mechanism, and can be expressed as

p1

p2

p3p

p4

p5

p6 6 6

J

J 0

J

J

0 J

J

J (13)

1 1 1

2 2 2

3 3 3q

4 4 4

5 5 5

6 6 6 6 6

( )( )( )( )( )( )

w v ww v ww v ww v ww v ww v w

J (14)

where iw is the unit vector of iL and i 1 i ʹ v R A .

q J I , qi 1,(i=1,2,...,6)J ; When the PKM is actuated with telescoped legs,

qi 1,(i 1,2,...,6) J and q J I ; if the PKM is equipped with slider legs, qi i z( )J w , where i z( )w is the third element of vector iw with respect to the z axis coordinate. If the q 0J , the equation can be written as:

1

p q p q q J J J (15)

3.2 Workspace and index

The workspace of a PKM is the work region which the tool can reach, and is an important indication of the PKM’s performance. In this section, the Workspace Volume Index (WVI) is used to represent the workspace volume of the RPKM. A numeric method is applied to compute the PKM’s reachable points to integrate the workspace surface. According to the earlier research of Huang [13], the same optimal design result for workspace can be obtained by setting the rotation angles of the moving platform to zero or other values. To simplify the analysis, the three rotation angles of the moving platform are all set to zero in this calculation. If we are given the initial value as shown in Table 1, the workspaces of the two kinds of PKMs are shown in figure 8 and their WVIs are 1.5x108 mm3 and 1.2x107 mm3

respectively.


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DimensionaVariable

AR (mm)

BR (mm)

l (mm)

A (o)

B (o)

minl (mm)

maxl = minl +(mm)

minz (mm)

maxz = minzl (mm)

kl (mm)

Table 1. The d

Figure 8. RPKM

3.3 Conditiona

The conditioto evaluate expression is

al D

Radius

Radius o

The maxi

The strutmoving The strut

base plPKM with

) Minimal l

l Maximal

PKM wi

) Maximapi

+ Minimapi

Length

dimensional vari

a. The teles

b. The lin

M workspace su

al number and

on number of the dexterity

s written as:

Description

of base platform

of moving platfo

imal move rangstrut

t mounted angleplatform(figuret mounted anglelatform(figure 7 telescopic legs length of telesco

legs

length of telescolegs

th linear legs al height of woriece module

al height of woriece module

h of rod module

iables

scopic drive PKM

near drive PKM

urface

GCI index

Jacobian maty [18]. The c

InitialValue

m 200

orm 400

ge of 200

e on e 7)

60( / 3

e on 7)

30( / 6

opic 500

opic 700

rk‐200

rk‐400

e 600

M

trix J can be condition num

l e

3 )

6 )

used mber

wheJaco Accvalu

The

Noweval

It hindiplatvarican

Figu

Howplan

ere i (i=1,2,obian matrix J

ording to theue

singularity va

w, the Local luating the pe

has been acknicates a betttform. As shoiables in Tablbe shownin o

a

ure 9.LCI of RPK

wever, the LCne forthe PK

1 1k

…,6) are the J .

e characterist

6det( E

alue is

Ti i( ) J J

Conditioningrformance of

1LCI 1 / k

nowledged thter motion down in figurele1, the changone X‐Y plane

a. The telescopic

b. The linear d

KM

CI can only evKM workspac

6/

singularity v

tic equation o

T ) J J =0

) , (i=1…6)

g Index (LCIthe RPKM, wh

1 6 1/

hat a larger dexterity of e 9, given thege trends of tof the worksp

c drive PKM

drive PKM

valuate the dexce, and we a

(16)

values of the

of singularity

(17)

(18)

I) is used tohich is:

(19)

value of LCIthe moving

e dimensionalthe LCI valuepace.

xterity of X‐Yadopt Global

)

e

y

)

)

o

)

I g l e

Y l


Condition Index (GCI) [14] to evaluate the dexterity of whole workspace, a promising approach defined as

1W WGCI 1 / k dW dW (20)

where dW is a differential workspace volume, and 1k is the condition number of the Jacobian matrix at a given moving platform pose, which is defined as the ratio of maximum and minimum singularity values of the manipulator Jacobian matrix. It is well known that a larger value of GCI reveals a better motion dexterity of the moving platform.

3.4 Workspace and GCI analysis

From Table 1, we can know that ten dimensional variables should be designed. It will be very difficult to analyse all the variables at one time. So the discussed design variables are defined as 1 A ,

2 B , 3 AR / l , 4 BR / l , 5 minl / l and

6 kl / l , where 3 , 4 , 5 and 6 are all dimensionless variables. Based on the workspace and formula of GCI, the effects of the design variables on the dexterity and workspace can be discussed and evaluated. The ball screw length limitation givesl 200mm . Other design parameters that are not

discussed below are the same as specified in Table 1.

3.4.1 Workspace effect analysis

Figure 10 shows the WVI curve when 1 and 2 are given different values. It indicates that the workspace expands as 1 increases. When 2 30 , the maximal workspace volume can be obtained. Because the proposed RPKM structures are symmetrical, the value of 1 should not be larger than 60 , so we will let 1 60 and 2 30 for the following analysis.

a. The telescopic drive PKM

b. The linear drive PKM

Figure 10. Workspace volumes of RPKM with different 1 & 2

As shown in figure 11, the workspace volumes increase with 3 increasing and 4 decreasing. If we only want to get the largest workspace volume, selecting the largest value of 3 and the smallest value of 4 will be a good choice.



Figure 11. Workspace volumes of the RPKM with different 3and 4


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As shown in figure 12, the design variable 5 only has impact on the workspace volumes of the telescopic drive PKM, while the design variable 6 only affects the workspace volumes of the linear drive PKM. Workspace increases with 5 increasing and 4 decreasing. Workspace volumes increase as 6 increases and 4decreases. If we only want to get the largest workspace, selecting the largest values of 5 and 6 will be a good choice. However, the increasing values of 5 and 6 will decrease the stiffness of the machine.



Figure12. Workspace volumes of the RPKM with different 4 ,

5 and 6

3.4.2 GCI effect analysis

As shown in figure 13, we can get the maximal GCI as long as the value of 3 is close to one, and the values of GCI increase as 4 decreases. As shown in figure 14, the design variable of 5 only affects the GCI of the telescopic drive PKM, while the design variable of 6 only affects the linear drive PKM. The maximal value of GCI can be obtained as the value of

5 and 6 are near 2.5. The values of GCI increase when

4 increases.



Figure 13.GCI of the RPKM with different 3 and 4



Figure14.GCI of the RPKM with different 4 , 5 and 6


3.5 Dimension

The dimensishown in Tab

Dimensionvariables

Getting larWVI

Getting largeDimensionvariables

Getting larWVI

Getting large

Table 2. The d

According toof WVI and Ga better motianalyse the selected as and 6 4 . PKM’s perfoand then eTherefore, acdesign variaband 6 2.5 when 3 1.2 will get a largThen the dimratio of desdimensional shown in Tab Dimensional

Valu

Dimensional

Valu

Table 3.The di

Based on Tmanufacture16(a), the telpattern, andprototype wproved the fe

4. Conclusion

In this paperassembled inlinear drive Pcan be drawn

nal variables de

ional variableble 2.

nal s 1

ger 60

r GCI ‐‐ nal s 4

ger The smalvalue

r GCIThe larg

value

dimensional vari

o the definitionGCI indicate aion dexterity oworkspace,

1 60 , 2 3 However, thermance, so wendeavour toccording to thebles are select5 . According 25 , 4 2 , ger workspacemensional varsign parametvariables of tble 3.

l variables ARue 250

l variables maxlue 800

imensional para

Table 3, a 6d in 2010, aescopic drive d in figure was machiningeasibility of th

n

r, a 6‐DOF RPnto either thePKM with then as follows:

esign

es effects on

2

30

‐‐

5

llest e

The largvalue

gest e

Near 2

iables effects on

n of WVI and a larger worksof moving pladesign vari

30 , 3 1.5 ,e GCI is verye should cons obtain a lae GCI effect anted as 3 1 , to figures 1

5 3 and e and reasonariables can be ters i ,i 1,2,. the RPKM can

BR l 400 200

minz maxz

200 400

ameter of the RP

6‐DOF RPKMs shown infiPKM was m16(b), the lig another wahe 6‐DOF RPK

PKM is propoe telescopic de same modul

WVI and GC

3

The largvalueNear

6

gest e

The largvalue

2.5 Near 2

n WVI and GCI

GCI, larger vaspace volumeatform. If we iables should 4 1.5 , 5 y important insider the GCI arger workspnalysis results

4 2.5 , 5 1, 12, 13 and

6 3 , the RPable GCI relativcalculated by...,6 . Finally, n be obtained

A (o) B (o) l60 30 6

kl A (o) 600 60

PKM prototype

M prototype igure 15.In fi

machining onenear drive Pax pattern, wKM prototype.

sed, which cadrive PKM orle set. Conclus

CIare

gest e 1

gest e

2.5

alues e and only d be 3.5

n the first, pace. s, the 2.5

d 14, PKM vely. y the the

d and

minl600

B (o)

90

was igure wax PKM which

an be r the sions

(1)

(2)

(3)

Howpapotheinde

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A new typetelescopic drmodules, whchanged intoperiod of timA compounddesigned to universal jorotation is lothe general spherical joinand can reachThe indices odesign the vato WVI and Gare obtained,experiments implement th

wever, only ter. In the futer RPKMswithex.

a

ure 15. The 6‐DO

e of strut is rive or linearhere the teleso linear drive me. d joint with tbe a sphericaint can alsoocked. Comparotation ang

nt or universah 60 . of WVI and GCariables of theGCI analysis, t, and then theproved that the machining

wo 6‐DOF PKture work, weh the same d

a. The telescopic

b. The linear d

OF RPKM proto

designed to r drive withoscopic drive mmode easily a

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CI are used toe 6‐DOF RPKMthe dimensione prototype is the new type function.

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drive PKM

otype

realize eitherout any othermode can beand in a short

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o evaluate andM. Accordingnless variablesbuilt. Millingof RPKM can

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r r e t

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Figure 16. Mill

5. Acknowled

This researcScience Founand the Na2011ZX04015

6. References

[1] Y. Koren,Systems,ReconfigReport #Ml, 1997

[2] Y. KorenSystems,

[3] Farhad, LockableRobotics,

[4] F. JovaneDesign Parallel Germany

[5] V. Angeloa reconf

Milling by tel

Milling by l

ling wax pattern

dgments

ch is sponsorndation of Chational S&T 5‐011).

s

, A. G. Ulsoy, , Engineerigurable Ma#1, The Unive7. n, Computer, McGraw‐HilP. Kourosh,

e Cylindrical vol. 25,2009, pe, S. P. NegriIssues for RKinematics My, 2002, pp. 69o, L. Giovannifigurable redu

lescopic drive P

linear drive PKM

n

red by the hina (nos. 111Major Projec

Reconfigurabing Researcachining Syersity of Mich

r Control oll, New York, 1A Reconfigul Joints, IEEEpp. 785‐797.. i, I. Fassi, L. ReconfigurableMachines Intern9‐82.. i, et al., Perforundant paralle

PKM

M

National Na178012, 50975ct of China

ble Manufactuch Center ystems(ERC/Rhigan, Ann A

of Manufactu1983. rable Robot WE Transactions

Molinari Toe PKMs, in national Confer

rmance analysel manipulato

tural 149), (no.

uring for

RMS), rbor,

uring

With s on

satti, 2002 rence,

sis of or, in

[6] X

[7] G

[8] X

[9] K

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