analytical modelling of the effects of manufacturing errors on

Analytical modelling of the eVects of manufacturingerrors on the accuracy for a TRR±XY hybridparallel link machine tool

S-L Chen1*, T-H Chang2 and M-H Hsei11Institute of Manufacturing Engineering, National Cheng-Kung University, Taiwan2Department of Mechanical Engineering, Keio University, Japan

Abstract: Using a parallel link mechanism as the basic structure of the parallel link machine tool is anew design concept and has become one of the most important research ®elds and attracts manyprevious researchers. However, the application of these machines as a machine tool has not provenitself in terms of accuracy enhancement over traditional machine tools. In general, the geometry ofthe machine tool (moving platform size relative to base size) has signi®cant eVects on the level ofaccuracy achievable. Therefore, an error model analysis for the parallel link machine tool to takeadvantage of the geometry to minimize the error and to increase the accuracy is very important andinteresting and is focused on in this research. A TRR±XY hybrid ®ve-degree-of-freedom parallellink machine tool is built for this research to investigate the error model theory. The errors fromthe component machining and assembly are de®ned and considered in the inverse kinematicsolution. The eVects of the manufacturing errors on the accuracy of the machine tool are shown inthis research.

Keywords: parallel link, machine, manufacturing errors, accuracy

NOTATION

hs thickness of the sliderL length of the linksr circle radius of tool platform around which the

three ball joints are evenly locatedR circle radius on upper base frame around which

the three driving axes are evenly locatedSi position of the slider on the driving axis for chain it length of cutter

¬i angular errors along X axis based on coordinate 1

i angular errors along Y axis based on coordinate 1

®i angular errors along Z axis based on coordinate 1

¯FXi position errors along X axis based on coordinate 1

¯FYi position errors along Y axis based on coordinate 1

¯JXi position errors along X axis based on coordinate 6¯JZi position errors along Z axis based on coordinate 6

¯SXi position errors along X axis based on coordinate 7

¯SYi position errors along Y axis based on coordinate 7

¶i angular errors along X axis based on coordinate 2

¿i angular errors along Y axis based on coordinate 2

’ tool axis orientation angle (see Fig. 8)

Subscripts

F errors de®ned in the [Frame] matrixi chain ˆ A, B, CJ errors de®ned in the [Ball] matrixS errors de®ned in the [Spindle] matrixX , Y , Z based on the X , Y or Z axis direction

1 INTRODUCTION

Using a parallel link mechanism as the basic structure ofthe parallel link machine tool is a new design concept andis one of the the most important research ®elds andattracts many previous researchers [1±6]. However, theapplication of these machines as a machine tool hasnot proven itself in terms of accuracy enhancementover traditional machine tools. In general, the geometryof the machine tool (moving platform size relative tobase size) has signi®cant eVects on the level of accuracyachievable. Therefore, an error model analysis for the

1203

B04400 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B

The MS was received on 6 June 2000 and was accepted after revision forpublication on 9 January 2001.*Corresponding author: Institute of Manufacturing Engineering,National Cheng-Kung University, Taiwan 701.

parallel link machine tool to take advantage of the geo-metry to minimize the error and to increase the accuracyis essential and interesting and is focused on in thisresearch. A TRR±XY hybrid ®ve-degree-of-freedom (5DOF) parallel link machine tool is built for this researchto investigate the error model theory (see Fig. 1). Thehybrid means that the machine tool is composed of a3-DOF parallel link mechanism and a 2-DOF serial-type XY table. Here, `TRR’ represents the tool frameindependent motion DOF of the upper parallel linkmechanism, where `T’ denotes the translational DOFand `R’ the rotational DOF. From a mechanism view-point, this system may also be referred to as a PRS±PPsystem, where `P’ denotes the prismatic motion pair,`R’ the rotational motion pair and `S’ the sphericaljoint. Because the XY table is commonly used in industryand the technology is well developed, this researchfocuses on the error model analysis of the 3-DOF parallellink mechanism. Several errors are usually encounteredin the machining and assembly of this machine tool.The base frame errors, the pin joint errors, ball jointassembly errors and tool frame assembly errors of thismachine tool are considered in this research and includedin the inverse kinematic solution derivation. Several tensof types of error will be included in the above-mentionederrors and make the theoretical analysis for completelyunderstanding the eVects of each error very di� cult.

From the practical viewpoint, only 11 manufacturingerrors are selected in this research. Some other factorsmay also in¯uence the machine accuracy [7±9]. Most ofthe 11 errors are component machining errors andassembly errors and data on the errors can be measuredby using a common measuring method. The componentmachining errors and assembly errors are considered asmanufacturing errors in this research. The simulationresults show the eVects of the manufacturing errors onthe accuracy of the machine tool. This is very usefulfor developing an eVective compensation theory toincrease the accuracy for this machine tool.

2 THEORY

Figure 1 shows the hybrid 5-DOF parallel link machinetool that was built for this research. The upper part ofthis machine tool is a 3-DOF parallel link mechanism.The parallel mechanism has three links. Each link has aball joint on one end and a pin joint on the other end.The ball joints are connected to the tool frame and thepin joints are connected to the slider. The three verticalsliding axes are evenly arranged on a circle with aradius R. There are three ball joints on the tool frame.The ball joints are also evenly arranged on a circle witha radius r. The Denavit±Hartenberg notation method

Fig. 1 Schematic diagram showing the hybrid parallel link machine tool developed for this research

1204 S-L CHEN, T-H CHANG AND M-H HSEI

Proc Instn Mech Engrs Vol 215 Part B B04400 # IMechE 2001

(D±H method) is adopted in this research to derive theinverse kinematic solution [10, 11]. The D±H method isused to describe the geometrical relationship betweentwo coordinate systems. In general, the coordinatetransformation relationship between two coordinates…X ; Y ; Z†i and …X ; Y ; Z†i ¡ 1 can be obtained by thefollowing steps (see Fig. 2):

1. Move an oVset di along the Zi ¡ 1 axis from…X ; Y ; Z†i ¡ 1 to …X ; Y ; Z†0

i.2. Rotate a joint angle ³i about the Z0

i axis from…X ; Y ; Z†0

i to …X ; Y ; Z†00i .

3. Move a link length ai along the Xi axis from…X ; Y ; Z†00

i to …X ; Y ; Z†000i .

4. Rotate a twist angle ¬i about the X 0i axis from

…X ; Y ; Z†000i to …X ; Y ; Z†i.

The homogeneous transformation matrix (HTM)between coordinates …X ; Y ; Z†i and …X ; Y ; Z†i ¡ 1 iscommonly represented as

i ¡ 1Ai ˆ Trans…0; 0; di† Rot…Zi; ³i†£ Trans…ai; 0; 0† Rot…Xi; ¬i† …1†

D±H notation parameters for the hybrid parallellink machine tool are summarized in Table 1. Eight

coordinate systems are de®ned ‰…X ; Y ; Z†0±…X ; Y ; Z†8Šfor the 3-DOF parallel link mechanism. In this machinetool shown in Fig. 1, SX and SY are the displacements ofthe driving axes of the XY table and SA, SB and SC arethe displacements of the sliders of the driving axes A, Band C of the 3-DOF parallel link mechanism.

3 ERROR DEFINITIONS FOR THE MACHINETOOL

Two translation errors and three orientation errors areconsidered as the assembly errors of the machine toolstructure. The assembly errors of the B axis are selectedas an example to explain the errors considered in thisresearch and shown in Fig. 3. For the assembly errorsof the B axis, the translation errors are ¯FXB and ¯FYB

and the orientation errors are ¬B, B and ®B. Theassembly errors can be described by an HTM in theD±H notation method. The manufacturing errors of amachine tool are not so large with a normal machinetool assembly process. Therefore, the numerical valuesof the angle errors should be very small. Then, Taylorseries can be used to expand the error terms. Thehigher-order terms are truncated. For example, the

Fig. 2 Geometrical relationship between two coordinate systems using the D±H notation method

Table 1 D±H notation parameters for the hybrid parallel link machine tool

A chain B chain C chain

Link d ³ a ¬ d ³ a ¬ d ³ a ¬

1 0 ¡1508 R 1808 0 908 R 1808 0 ¡308 R 18082 SA 08 ¡hs 908 SB 08 ¡hs 908 SC 08 ¡hs 9083 0 ³A3 L 08 0 ³B3 L 08 0 ³C3 L 084 0 ³A4 0 908 0 ³B4 0 908 0 ³C4 0 9085 0 ³A5 0 908 0 ³B5 0 908 0 ³C5 0 9086 0 ³A6 0 08 0 ³B6 0 08 0 ³C6 0 087 0 08 r 908 0 08 r 908 0 08 r 9088 ¡t ¡308 0 08 ¡t 908 0 08 ¡t ¡1508 0 08

ANALYTICAL MODELLING OF THE EFFECTS OF MANUFACTURING ERRORS 1205


HTM for the B chain is displayed as [Frame]B, which isgiven by

‰FrameŠB ˆ

1 ¡®B B ¯FXB

®B 1 ¡¬B ¯FYB

¡ B ¬B 1 0

0 0 0 1

0

BBB@

1

CCCA …2†

Two orientation errors are considered as the assemblyerrors of the pin joints. The assembly errors of the pinjoint on the B axis are selected as an example to explainthe errors considered in this research and shown in Fig. 4.For the assembly errors of the pin joint on the B-chain,

the orientation errors are ¿B and ¶B. The assemblyerrors of the pin joint can be described by an HTM[Pin]B, which is given by

‰PinŠB ˆ

1 0 ¶B 0

0 1 ¡¿B 0

¡¶B ¿B 1 0

0 0 0 1

0

BBB@

1

CCCA …3†

For the ball joints, they are considered to have anoVset manufacturing error related to the ideal local coor-dinate system. The oVset manufacturing error is de®nedby two parameters (Fig. 5). An HTM [Ball] is used to

Fig. 3 De®nition of coordinate transformation for frame errors

Fig. 4 Geometric de®nition for pin joint manufacturing errors



include these two parameters. For example, the HTM forthe B chain is [Ball]B, which is given by

‰BallŠB ˆ

1 0 0 ¯JXB

0 1 0 0

0 0 1 ¯JZB

0 0 0 1

0

BBB@

1

CCCA …4†

The spindle shaft is considered to have an oVset manu-facturing error related to the ideal local coordinatesystem. The oVset manufacturing error is de®ned bytwo parameters (Fig. 6). An HTM [Spindle] is used toinclude these two parameters. For example, the HTMfor the B chain is [Spindle]B, which is given by

‰SpindleŠB ˆ

1 0 0 ¯SXB

0 1 0 ¯SYB

0 0 1 0

0 0 0 1

0

BBB@

1

CCCA …5†

The coordinate systems de®ned for the D±H coordinatetransformation are shown in Fig. 7.

4 ERROR MODEL ANALYSIS

Considering the error de®nitions that are given in theabove section, the HTM between the coordinate system…X ; Y ; Z†8 (cutter tip) and the coordinate system…X ; Y ; Z†0 (upper base frame) can be written as follows.For the B chain,

…0A8†B ˆ 0A1‰FrameŠB 1A2‰PinŠB 2A33A4

4A55A6‰BallŠB

£ 6A7‰SpindleŠB 7A8 …6†

In the above equation,

0A1 ˆ

0 0 ¡1 0

1 0 0 R

0 ¡1 0 0

0 0 0 1

0

BBB@

1

CCCA …7a†

1A2 ˆ

1 0 0 ¡hs

0 0 ¡1 0

0 1 0 SB

0 0 0 1

0

BBB@

1

CCCA …7b†

2A3 ˆ

cos…³3B† ¡ sin…³3B† 0 L cos…³3B†sin…³3B† cos…³3B† 0 L sin…³3B†

0 0 1 0

0 0 0 1

0

BBB@

1

CCCA …7c†

3A4 ˆ

cos…³4B† 0 sin…³4B† 0

sin…³4B† 0 ¡ cos…³4B† 0

0 1 0 0

0 0 0 1

0

BBB@

1

CCCA …7d†

Fig. 5 Geometric de®nition for ball joint manufacturingerrors

Fig. 6 Geometric de®nition for spindle location manufacturing errors



4A5 ˆ

cos…³5B† 0 sin…³5B† 0

sin…³5B† 0 ¡ cos…³5B† 0

0 1 0 0

0 0 0 1

0

BBB@

1

CCCA …7e†

5A6 ˆ

cos…³6B† ¡ sin…³6B† 0 0

sin…³6B† cos…³6B† 0 0

0 0 1 0

0 0 0 1

0

BBB@

1

CCCA …7f†

6A7 ˆ

1 0 0 r

0 0 ¡1 0

0 1 0 0

0 0 0 1

0

BBB@

1

CCCA …7g†

7A8 ˆ

0 ¡1 0 0

1 0 0 0

0 0 1 ¡t

0 0 0 1

0

BBB@

1

CCCA …7h†

Fig. 7 Coordinate system de®nition based on the D±H notation method with manufacturingerrors considered



Rearranging equation (6) gives

…0A8†B ˆ 0A1‰FrameŠB 1AP2A3

3A44A5

5A8 …8†

In the above equation,

1AP ˆ 1A2‰PinŠB …9†

In equation (8), 5A8 is de®ned as

5A8 ˆ 5A6‰BallŠB 6A7‰SpindleŠB 7A8 …10†

On rearranging equation (8), an HTM T is used torepresent the product [Frame]¡1

B …0A1†¡1…0A8†B. Thecomponents of the HTM T are de®ned as follows:

T ˆ

N1 T1 B1 D1

N2 T2 B2 D2

N3 T3 B3 D3

0 0 0 1

0

BBB@

1

CCCA …11†

Substituting equations (9) to (11) into equation (8) gives

T ˆ 1AP2A3

3A44A5

5A8 …12†

Rearranging equation (12) gives

…1AP†¡1T…5A8†¡1 ˆ 2A33A4

4A5 …13†

The translation components of both sides of equation(13) are extracted and compared; the control positionSB of the B chain can then be obtained. The translationcomponent of the left-hand side of equation (13) is givenas follows:

‰I3 0Š…1AP†¡1T…5A8†¡1‰0 0 0 1ŠT

ˆ

FB ‡ ¶BGB ‡ hs

¡¿BGB ‡ HB ¡ SB

¶BFB ¡ GB ¡ ¿BHB ‡ ¿BSB ‡ ¶Bhs

0

BB@

1

CCA …14†

In the above equation, FB, GB and HB are de®ned as

FB

GB

HB

1

0BBBBB@

1CCCCCA

ˆ

N1 T1 B1 D1

N2 T2 B2 D2

N3 T3 B3 D3

0 0 0 1

0BBBBB@

1CCCCCA

£

0 0 1 ¡¯SYB ¡ ¯JZB

¡ cos…³6B† ¡ sin…³6B† 0 ¯JXB ‡ ¯SXB ‡ r

sin…³6B† ¡ cos…³6B† 0 t

0 0 0 1

0BBBBB@

1CCCCCA

£

0

0

0

1

0BBBBB@

1CCCCCA

…15†

The translation component of the right-hand side ofequation (13) is given as follows:

‰I3 0Š 2A33A4

4A5‰0 0 0 1ŠT ˆL cos…³3B†L sin…³3B†

0

0

B@

1

CA

…16†

By using the relationship that the results of equation (14)are equal to the results of equation (16), the inversekinematic solution of the machine tool can be obtainedas follows:

FB ‡ ¶BGB ‡ hs

¡¿BGB ‡ HB ¡ SB

¶BFB ¡ GB ¡ ¿BHB ‡ ¿BSB ‡ ¶Bhs

0

BB@

1

CCA

ˆ

L cos…³3B†

L sin…³3B†

0

0

BB@

1

CCA …17†

SB ˆ …¡¿BGB ‡ HB† ¡��L2 ¡ …FB ‡ ¶BGB ‡ hs†2

q

…18†

The solution procedures for the A and C chains are verysimilar to that for the B chain and the details are notincluded here. However, the results are summarized asfollows:

SA ˆ …¡¿AGA ‡ HA† ¡��L2 ¡ …FA ‡ ¶AGA ‡ hs†2

q

…19†

SC ˆ …¡¿CGC ‡ HC† ¡��L2 ¡ …FC ‡ ¶CGC ‡ hs†2

q

…20†

5 ®, ¯, ° DERIVATION WITH THE ERRORSCONSIDERED

One of the coordinate system relationships between…X ; Y ; Z†8 and …X ; Y ; Z†0 is given by equation (6). Thecoordinate system relationship …0A8†B between…X ; Y ; Z†8 and …X ; Y ; Z†0 can also be directly repre-sented by one translational HTM and three rotationalHTMs [3]. The translation displacements are PX , PY

and PZ and the rotation angles are ¬, and ®. …0A8†B

can be written as follows:

…0A8†B ˆ Trans…PX ; PY ; PZ†RZYX…®; ; ¬† …21†

In a real machining process, a neutral cutter location ®le(CL ®le) for the part surface is usually generated bycommercial software [12, 13]. The basic format for aCL ®le is (X Y Z i j k). (X Y Z) are the coordinate



data of the cutting point and (i j k) is the unit vectorof the tool axis. To simplify the derivation of the theory,the orientation of the tool axis is considered to coincidewith the normal direction of the part surface at thecutting point. Then

…i j k†T ˆ RZYX…®; ; ¬†‰0 0 1ŠT …22†

From equation (22), (i j k) can be obtained as

…i j k†T ˆsin… †

¡ sin…¬† cos… †cos…¬† cos… †

0

B@

1

CA …23†

From equation (23), the rotation angles ¬ and can beseparately given as follows:

ˆ sin¡1…i† …24†

¬ ˆ arctan 2…sin ¬; cos ¬† …25†

In equation (25),

sin ¬ ˆ¡j

cos …26†

cos ¬ ˆ kcos

…27†

In the ideal condition (no manufacturing errors in themachine tool), the rotation angle ® of this developedsystem can be obtained by using the geometric con-straints from the pin joints. The pin joints constrainthe three ball joints to move only on three planesseparately. The geometric relationships of the threeplanes are

X ˆ 0; B chain constraint plane

Y ˆ X��3

p ; A chain constraint plane

Y ˆ ¡ X��3

p ; C chain constraint plane

…28†

Using the geometric relationship in equations (28), therotation angle ® can be obtained as

® ˆ tan¡1

³¡ sin…¬† sin… †

cos…¬† ‡ cos… †

´…29†

With consideration of the manufacturing errors, it is veryobvious that the ® relationship given in equation (29) willnot be retained. In the derivation of the new ® relation-ship with the manufacturing errors de®ned in the pre-vious section considered, the three constraint planesthat are generated by the pin joints are still used. Thegeometric relationship [equations (28)] of the threeplanes is changed with the errors included. The deriva-tion of the equations for the three constraint planes isrequired to obtain the ® relationship.

Assume that P0 …X0; Y0; Z0† and P …X ; Y ; Z† are two(non-zero) points located on the constraint plane. Thenormal unit vector of the constraint plane is written as

N ˆ Ai ‡ Bj ‡ Ck …30†

From the geometric relationship, the equation of theconstraint plane is

A…X ¡ X0† ‡ B…Y ¡ Y0† ‡ C…Z ¡ Z0† ˆ 0 …31†

In each A chain, B chain or C chain, the centre point ofthe pin joint [origin point of …X2; Y2; Z2†e] and the centrepoint of the ball joint are separately located on theconstraint plane. Assume that the pin joint axis isalways perpendicular to the constraint plane. Therefore,the unit vector of the pin joint axis is selected as thenormal vector of the constraint plane. The two pointsgiven in equation (31) together with the normal vector(pin joint axis) can be used to solve the equation of theconstraint plane. The solution procedures for the Bchain constraint plane are given in the following as anexample. The constraint plane unit normal vector ofthe B chain can be written as

NB ˆ 0A1‰FrameŠB 1A2‰PinŠB‰0 0 1 0ŠT …32†

The origin point of the pin joint (coordinate 2) with theerrors considered can be written as

PB ˆ …X2; Y2; Z2†e

ˆ 0A1‰FrameŠB 1A2‰PinŠB‰0 0 0 1ŠT …33†

The coordinate data of the ball joint centre point can bewritten as

B1 ˆ …0A8†B‰0 r t 1ŠT …34†

On substituting equations (32) to (34) in equation (31)and rearranging the equation, the constraint planeequation can be rewritten as

B1PX ‡ B2PY ‡ BS sin…®† ‡ BC cos…®† ˆ BP …35†

In the above equation, B1, B2, BS, BC and BP are given inAppendix 1. The constraint plane equations of the A andC chains can be solved using the procedures discussedabove. The constraint plane equation for the A chaincan be obtained as

A1PX ‡ A2PY ‡ AS sin…®† ‡ AC cos…®† ˆ AP …36†

In the above equation, A1, A2, AS, AC and AP are given inAppendix 2. For the C chain, the constraint plane equa-tion can be written as

C1PX ‡ C2PY ‡ CS sin…®† ‡ CC cos…®† ˆ CP …37†

In the above equation, C1, C2, CS, CC and CP are givenin Appendix 3, in which above equation, NC is the



constraint plane unit vector of the C chain. Both sin…®†and cos…®† appear in the constraint plane equations(35) to (37). The following relationship between sin…®†and cos…®† is required to solve the constraint planeequation:

‰sin…®†Š2 ‡ ‰cos…®†Š2 ˆ 1 …38†

Solving equations (35) to (38) simultaneously, therelationship between ® and ¬, can be obtained asfollows:

sin…®†cos…®†

ˆ

B1 B2 BP BC

A1 A2 AP AC

C1 C2 CP CC

0 0 1 cos…®†

0

BBB@

1

CCCA

B1 B2 BS BP

A1 A2 AS AP

C1 C2 CS CP

0 0 sin…®† 1

0

BBB@

1

CCCA

…39†

It is very clear that the angle ® is a dependent variableand ¬, are independent variables. Equation (39) can beused to solve for the rotation angle ®. However, thecontrol positions of the three-motion chain …SA; SB; SC†appear in the coe� cients of equations (47), (52) and(57). This makes the solution procedure of equation(39) very complicated. The angle ® can be expressed as

® ˆ f …error parameter; SA; SB; SC† …40†

A ®nite diVerence concept is adopted to solve the aboveequation for the rotation angle ® and SA, SB, SC. In thenormal situation, the manufacturing errors appearing inmachine tools are small. The diVerences in SA, SB and SC

between an ideal machine tool and the real machine tool(with errors considered) should be not very large. There-fore, the SA, SB and SC data for the ideal machine toolare adopted as the initial guesses for the ®nite diVerencesolution scheme to solve for SA, SB and SC for themachine tool with errors considered.The displacementsfor the centre point of tool frame (parallel link mechan-ism) in the X and Y direction are PX and PY . It is worthmentioning that PX and PY are not independent vari-ables. Solving equations (35) to (38) simultaneously,PX and PY can also be obtained as

PX ˆ

BP B2 BS BC

AP A2 AS AC

CP C2 CS CC

1 0 sin…®† cos…®†

0

BBB@

1

CCCA

B1 B2 BS BC

A1 A2 AS AC

C1 C2 CS CC

0 0 sin…®† cos…®†

0

BBB@

1

CCCA

…41†

PY ˆ

B1 BP BS BC

A1 AP AS AC

C1 CP CS CC

0 1 sin…®† cos…®†

0

BBB@

1

CCCA

B1 B2 BS BC

A1 A2 AS AC

C1 C2 CS CC

0 0 sin…®† cos…®†

0

BBB@

1

CCCA

…42†

6 EFFECTS OF MANUFACTURING ERRORS ONTHE POSITION ACCURACY

After the inverse kinematic solution has been obtained inthe above sections, a very interesting question is `How dothe manufacturing errors aVect the position accuracy ofthe machine tool?’ There are four diVerent types ofmanufacturing errors included in this research: frameerrors, pin joint errors, ball joint errors and tool frame(location of spindle shaft) errors. To ®nd the errorsthat strongly aVect the position accuracy will be veryhelpful for engineers to keep their attention on improv-ing the accuracy of these key components. This isobviously helpful for improving the position accuracyof the machine tool. Finding the errors from somecomponents that have little eVect on the accuracy isalso bene®cial for reducing costs. These componentscan be designed with lower precision requirements.

The machine tool developed in this research has ®veDOF. Three DOF come from the upper parallel linkmechanism of the machine tool. Two DOF comes fromthe lower XY table. The XY table is widely acceptedand used by industry. It will not be discussed in thisresearch. The upper parallel link mechanism will befocused on and investigated. Therefore, the tool pathsfor analysing the eVects of manufacturing errors aredesigned for the upper parallel link mechanism. Figure8 shows the de®nition of the tool paths for the toolframe. i, j and k give the components of the cutter orien-tation. The i and j components vary with varying ’ withthe k component ®xed. A circular tool path will begenerated with a ®xed k value (Fig. 8). The inclinationangle of the tool axis is represented by ³. The eVects ofthe manufacturing errors on the position accuracy ofthe parallel link machine tool are separately discussedin the following:

1. Separately analyse and compare the eVects of theerrors from the A, B and C chains. If the eVects ofdiVerent chains are similar, the analysis and discus-sion will be focused on only one chain.

2. Analyse the eVects from the 11 errors (in each motionchain). Find which one most signi®cantly aVects theposition accuracy.

3. Investigate the eVects of errors with diVerent toolpath (k is varied). In this research, the k component



is set as follows: 0.2 (³ ˆ 78:48), 0.5 (³ ˆ 608), 0.8(³ ˆ 378).

4. Investigate the eVects of errors with diVerent machinetool sizes. In this research, the dimensionless param-eters r=R and L=R are selected to represent the dimen-sions of the machine tool. The machine tool size is setas follows: r=R ˆ 0:4; r=R ˆ 0:6; r=R ˆ 0:8;L=R ˆ 5; L=R ˆ 2:4; L=R ˆ 1. L is the link length,R is the radius of the upper base frame and r is thetool frame radius.

7 RESULTS AND DISCUSSION

There are 11 errors included in this research. It isessential to understand the eVects of diVerent errors onthe accuracy of the machine tool. However, the eVectsof errors vary with diVerent tool paths (tool axis orienta-tion and cutting point location). To display the analyticalresults with diVerent tool paths, the variation of tool axisorientation (angle ’ or i, j, k) is used as the horizontalaxis. The diVerences (dSA, dSB and dSC) of the controlpositions SA, SB and SC with and without manufacturingerrors are used to display the eVects.

1. Figures 9, 10 and 11 separately show the controlposition variations dSA, dSB and dSC for the 11manufacturing errors considered for only B, A andC chains. From the results in Figs 9 to 11, it isfound that the 11 manufacturing errors generatedfrom diVerent kinematic chains (A, B and C chains)have nearly the same eVects on the position accuracyof the hybrid parallel link machine tool. This resultseems very reasonable because the three movingaxes (with an actuator for each axis) are evenlyarranged on a circle to form an equilateral triangle.The circle is located on the X ±Y plane of the…X ; Y ; Z†0 coordinate system. From the geometricrelationship, it is obvious that the arrangement hasa symmetric structure. Therefore, only the eVectsfrom the 11 manufacturing errors that appeared onthe B-chain will be discussed in the following. Thesymmetric situation can also be understood by shift-ing forward 1208 the results in Fig. 10 and shiftingforward 2408 the results in Fig. 11. The shifting

Fig. 8 Tool path planning for the error model analysis. i, j, kare the components representing the orientation of thetool axis

Fig. 9 The eVects of B chain manufacturing errors on the variations dSA, dSB and dSC of the control position

(k ˆ 0:8). (a) [Frame error]B: ¬B, B, ®B ˆ 18, ¯FXB ˆ ¯FYB ˆ 1 mm. (b) [Pin error]B: ¿B ˆ 18, ¶B ˆ 18.(c) [Ball error]B: ¯JXB ˆ 1 mm, ¯JZB ˆ 1 mm. (d) [Spindle error]A: ¯SXB ˆ 1 mm, ¯SYB ˆ 1 mm



angle 1208 is the angle between the B and A chainsand 2408 is the angle between the B and C chains. Itis found that the results of Figs 10 and 11 are thesame as the results of Fig. 9 after the shifts.

2. Figure 12 shows the eVects of the 11 manufacturingerrors in the B chain on dSA, dSB and dSC of themachine tool …k ˆ 0:8†. The eVects of ¬B ˆ 18, B ˆ 18 and ¿B ˆ 18 are shown in Fig. 12a. TheeVects of ®B ˆ 18, ¶B ˆ 18 and ¯FXB ˆ 1 mm are

shown in Fig. 12b. The eVects of ¯FYB ˆ 1 mm,

¯JXB ˆ 1 mm, ¯JZB ˆ 1 mm, ¯SXB ˆ 1 mm and¯SYB ˆ 1 mm are shown in Fig. 12c. From the resultsin Fig. 12, it is clearly seen that dSA, dSB and dSC

vary not only with the error type but also with thetool position ’. The magnitudes of dSA, dSB anddSC are varied using diVerent ’ values. On compar-ing the analytical results about the eVects of errorson the control position variation dSA, dSB and dSC

Fig. 10 The eVects of A chain manufacturing errors on the variations dSA, dSB and dSC of the control posi-tion (k ˆ 0:8). (a) [Frame error]A: ¬A, A, ®A ˆ 18, ¯FXA ˆ ¯FYA ˆ 1 mm. (b) [Pin error]A: ¿A ˆ 18,¶A ˆ 18. (c) [Ball error]A: ¯JXA ˆ 1 mm, ¯JZA ˆ 1 mm. (d) [Spindle error]A: ¯SXA ˆ 1 mm,

¯SYA ˆ 1 mm

Fig. 11 The eVects of C chain manufacturing errors on the variations dSA, dSB and dSC of the control posi-tion (k ˆ 0:8). (a) [Frame error]C: ¬C, C, ®C ˆ 18, ¯FXC ˆ ¯FYC ˆ 1 mm. (b) [Pin error]C: ¿C ˆ 18,¶C ˆ 18. (c) [Ball error]C: ¯JXC ˆ 1 mm, ¯JZC ˆ 1 mm. (d) [Spindle error]C: ¯SXC ˆ 1 mm,

¯SYC ˆ 1 mm



in Fig. 12, however, it is found that ¬B ˆ 18 is themost signi®cant and ¿B ˆ 18 is the second mostsigni®cant. The eVects of ®B ˆ 18, ¶B ˆ 18, ¯FXB ˆ1 mm, ¯FYB ˆ 1 mm, ¯JXB ˆ 1 mm, ¯JZB ˆ 1 mm,¯SXB ˆ 1 mm and ¯SYB ˆ 1 mm are relatively small.

¬B ˆ 18 is one of the frame errors and ¿B ˆ 18 isone of the pin joint errors. This result implies thata precision machining process and a very preciseassembly process are very important in the frameassembly compared with the manufacture of theother components. Pin joints are also consideredto be key components of the hybrid parallel linkmachine tool.

3. The tool path plans for the analysis were discussed inthe previous section. The smaller k value means thatthe inclination angle of tool axis is larger (Fig. 8).In Fig. 13, smaller k values are found to have largerdSA, dSB and dSC variations. In other words, largereVects of the manufacturing errors are found withthe larger tool axis inclination angle. Although onlythe errors in the B chain are considered, the sametendency of the results appeared in the A and Cchains. From a practical machining viewpoint, thisresult in Fig. 13 implies that a small tool axis inclina-tion angle is better and is suggested for the machinedesign.

4. The eVect of the dimensional variation of the machinetool is another very interesting topic for applying thedeveloped machine tool to real machining processes.In this research, two dimensionless variables (r=Rand L=R) are adopted to represent the dimensionvariation of the machine tool. Figure 14 shows theeVects of r=R on the variation of dSA, dSB and dSC

with (a) r=R ˆ 0:8, ¬B ˆ 18, (b) r=R ˆ 0:6, ¬B ˆ 18,and (c) r=R ˆ 0:4, ¬B ˆ 18. No variation of dSB is

Fig. 12 Comparison of the eVects of diVerent types of B chain

errors on the control position variations dSA, dSB

and dSC (k ˆ 0:8): (a) ¬B ˆ 18, B ˆ 18, ¿B ˆ 18; (b)

®B ˆ 18, ¶B ˆ 18, ¯FXB ˆ 1 mm; (c) ¯FYB ˆ 1 mm,

¯JXB ˆ 1 mm, ¯JZB ˆ 1 mm, ¯SXB ˆ 1 mm, ¯SYB ˆ1 mm

Fig. 13 Comparison of the eVects of various k components on

the control position variations dSA, dSB and dSC withB chain errors considered: (a) k ˆ 0:2, ¬B ˆ 10; (b)

k ˆ 0:5, ¬B ˆ 10; (c) k ˆ 0:8, ¬B ˆ 10



found with r=R values of 0.8, 0.6 and 0.4 in the wholeangle range. In the A and C chains, dSA and dSC

signi®cantly increase with r=R decreasing from 0.8to 0.6 to 0.4 in the range 08 4 ’ 4 2408. In contrast,dSA and dSC decrease when r=R decreases from 0.8to 0.6 to 0.4 in the range 2408 4 ’ 4 3608. It isworth mentioning that no manufacturing errorsfrom A and C chains are included in this analysis.Figure 15 shows the eVects of L=R on the variationof dSA, dSB and dSC with (a) L=R ˆ 5, ¬B ˆ 18, (b)L=R ˆ 2:4, ¬B ˆ 18 and (c) L=R ˆ 1:0, ¬B ˆ 18.

Very little variation of dSB is found for L=R valuesof 5, 2.4 and 1.0 in the whole range of angles ’. Inthe A and C chains, dSA and dSC decrease slightlywhen the L=R ratio increases from 1.0 to 2.4 to 5.

8 SUMMARY

The eVects of manufacturing errors on position accuracyfor the hybrid parallel link machine tool are investigatedin this research with advantage taken of the geometry tominimize the errors and to increase the accuracy. Elevenmanufacturing errors are considered in each kinematicchain. ¬B, B and ¿B are the most signi®cant errors inthe B chain that have large eVects on the positionaccuracy of the machine tool. As the machine tool hasa symmetric structure, ¬A, A and ¿A in the A chainand ¬C, C and ¿C in the C chain also very signi®cantlyaVect the position accuracy of the machine tool. There-fore, it is suggested that the engineer should pay moreattention on the tolerance and manufacturing errors ofthe frame structure in the design process. The smallertool axis inclination angle has better position accuracywith the same manufacturing errors. It is suggestedthat engineers should use a small tool inclination anglein real machining processes. The eVects of the r=R ratioon the position accuracy variation are found to bemore signi®cant than those of the L=R ratio.

ACKNOWLEDGEMENTS

Part of the research was funded by the ITRI of ROC andNSC of ROC under Grant NSC 89-2212-E-006-023.This®nancial support is gratefully acknowledged.

REFERENCES

1 Giddings and Lewis Variax. http://www.giddings.com/.2 Ingersoll. http://www.mel.nist.gov/gallery/hex/hexph.htm.

3 Toyoda. http://www.toyoda-ouki.co.jp/_pub_html/

sub_html/tmw/prodlines/paralink/paralink.jpg.4 Geodetic. http://www.hexapod.co.uk/g500.htm.

5 Path®nders. http://www.execpc.com/path®nders/

construct2.html.6 Honegger, M., Codourey, A. and Burdet, E. Adaptive

control of the hexaglide, a 6 DOF parallel manipulator.

In Proceedings of the 1997 IEEE International Conferenceon Robotics and Automation, Albuquerque, New Mexico,

1997, pp. 543±548.

7 Wiens, G. J. and Walker, C. W. Hexapod machine toolerror analysis with inclusion of system dynamics. In

Proceedings of the ASME, MED-Vol. 8, 1998, pp. 705±713.

8 Weck, M. and Dammer, M. Design, calculation and controlof machine tools based on parallel kinematics. In

Proceedings of the ASME, MED-Vol. 8, 1998, pp. 715±721.

Fig. 14 Comparison of the eVects of various r=R ratios on thecontrol position variations dSA, dSB and dSC with B

chain errors considered: (a) r=R ˆ 0:8, ¬B ˆ 18; (b)

r=R ˆ 0:6, ¬B ˆ 18; (c) r=R ˆ 0:4, ¬B ˆ 18

Fig. 15 Comparison of the eVects of various L=R ratios on

the control position variations dSA, dSB and dSC

with B chain errors considered: (a) L=R ˆ 0:8,¬B ˆ 18; (b) L=R ˆ 0:6, ¬B ˆ 18; (c) L=R ˆ 0:4,

¬B ˆ 18



http://www.execpc.com/pathfinders/construct2.html

http://www.toyoda-ouki.co.jp/_pub_html/sub_html/tmw/prodlines/paralink/paralink.jpg

http://www.giddings.com/

http://www.mel.nist.gov/gallery/hex/hexph.htm

http://www.toyoda-ouki.co.jp/_pub_html/sub_html/tmw/prodlines/paralink/paralink.jpg

http://www.hexapod.co.uk/g500.htm

http://www.execpc.com/pathfinders/construct2.html

9 Soons, J. A. On the geometric and thermal errors of ahexapod machine tool. In Parallel Kinematic Machines

(Eds C. R. Boer, L. Molinari-Tosatti and K. S. Smith),

1999, pp. 151±169 (Springer, London).10 Hartenberg, R. and Denavit, J. Kinematic Synthesis of

Linkages, 1964 (McGraw-Hill, New York).

11 Craig, J. J. Introduction to Robotics: Mechanics andControl, 1989 (Addison-Wesley, Canada).

12 Zeid, I. CAD/CAM Theory and Practice, 1991 (McGraw-

Hill, New York).13 Unigraphics Solutions. In UG CAD/CAM User Manual,

1999 (EDS, Maryland Heights).

APPENDIX 1

B1 ˆ NB…1; 1† …43†

B2 ˆ NB…2; 1† …44†

BS ˆ NB…1; 1†‰¡ cos… †rŠ

‡ NB…2; 1†‰¡r sin… † sin…¬†Š

‡ NB…3; 1†‰r cos…¬† sin… †Š …45†

BC ˆ NB…2; 1†‰r cos…¬†Š ‡ NB…3; 1†‰r sin…¬†Š …46†

BP ˆ PB…1; 1†NB…1; 1† ‡ PB…2; 1†NB…2; 1†

‡ PB…3; 1†NB…3; 1† ¡ NB…1; 1†‰sin… †tŠ

¡ NB…2; 1†‰¡ cos… † sin…¬†tŠ

¡ NB…3; 1†‰cos…¬† cos… †t ‡ PZŠ …47†

APPENDIX 2

A1 ˆ NA…1; 1† …48†

A2 ˆ NA…2; 1† …49†

AS ˆ NA…1; 1†‰12cos… †rŠ

‡ NA…2; 1†‰¡ 12

��3

pr cos…¬† ‡ 1

2r sin… † sin…¬†Š

‡ NA…3; 1†‰¡ 12

��3

pr sin…¬† ¡ 1

2r cos…¬† sin… †Š

…50†

AC ˆ NA…1; 1†‰¡ 12cos… †

��3

prŠ

‡ NA…2; 1†‰¡ 12

��3

pr sin…¬† sin… † ¡ 1

2r cos…¬†Š

‡ NA…3; 1†‰12

��3

pr cos…¬† sin… † ¡ 1

2r sin…¬†Š

…51†

AP ˆ PA…1; 1†NA…1; 1† ‡ PA…2; 1†NA…2; 1†

‡ PA…3; 1†NA…3; 1† ¡ NA…1; 1† sin… †t

¡ NA…2; 1†‰¡ cos… † sin…¬†tŠ

¡ NA…3; 1†‰cos…¬† cos… †t ‡ PZŠ …52†

APPENDIX 3

C1 ˆ NC…1; 1† …53†

C2 ˆ NC…2; 1† …54†

CS ˆ NC…1; 1†‰12cos… †rŠ

‡ NC…2; 1†‰12

��3

pr cos…¬† ‡ 1

2r sin… † sin…¬†Š

‡ NC…3; 1†‰12

��3

pr sin…¬† ¡ 1

2r cos…¬† sin… †Š

…55†

CC ˆ NC…1; 1†‰12cos… †

��3

prŠ

‡ NC…2; 1†‰12

��3

pr sin…¬† sin… † ¡ 1

2r cos…¬†Š

‡ NC…3; 1†‰¡ 12

��3

pr cos…¬† sin… † ¡ 1

2r sin…¬†Š

…56†

CP ˆ PC…1; 1†NC…1; 1† ‡ PC…2; 1†NC…2; 1†

‡ PC…3; 1†NC…3; 1† ¡ NC…1; 1† sin… †t

¡ NC…2; 1†‰¡ cos… † sin…¬†tŠ

¡ NC…3; 1†‰cos…¬† cos… †t ‡ PZŠ …57†



analytical modelling of the effects of manufacturing errors on

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