Error Averaging Effect in Parallel Mechanism ...

applied sciences Article

Error Averaging Effect in Parallel Mechanism Coordinate Measuring Machine Peng-Hao Hu *, Chang-Wei Yu, Kuang-Chao Fan, Xue-Ming Dang and Rui-Jun Li School of Instrument Science and Opto-electronic Engineering, Hefei University of Technology, Hefei 230009, China; [email protected] (C.-W.Y.); [email protected] (K.-C.F.); [email protected] (X.-M.D.); [email protected] (R.-J.L.) * Correspondence: [email protected]; Tel./Fax: +86-551-62902797 Academic Editor: Richard Leach Received: 15 September 2016; Accepted: 18 November 2016; Published: 25 November 2016

Abstract: Error averaging effect is one of the advantages of a parallel mechanism when individual errors are relatively large. However, further investigation is necessary to clarify the evidence with mathematical analysis and experiment. In the developed parallel coordinate measuring machine (PCMM), which is based on three pairs of prismatic-universal-universal joints (3-PUU), error averaging mechanism was investigated and is analyzed in this report. Firstly, the error transfer coefficients of various errors in the PCMM were studied based on the established error transfer model. It can be shown how the various original errors in the parallel mechanism are averaged and reduced. Secondly, experimental measurements were carried out, including angular errors and straightness errors of three moving sliders. Lastly, solving the inverse kinematics by numerical method of iteration, it can be seen that the final measuring errors of the moving platform of PCMM can be reduced by the error averaging effect in comparison with the attributed geometric errors of three moving slides. This study reveals the significance of the error averaging effect for a PCMM. Keywords: error averaging effect; parallel mechanism; parameter error; coordinate measuring machine

1. Introduction Error averaging effect has always existed in precision measurement, which is an important factor in improving accuracy. Sensors such as capacitance sensor, linear grating, and inductosyn have averaging effects in reading data. For instance, if a rotary encoder is installed with two reading heads with a 180◦ separation, the eccentricity error of the grating disk with respect to the shaft can be eliminated [1]. Another example can be seen on an end-tooth precision indexing table: several teeth of the upper gear and the lower gear are engaged at the same time during lapping or indexing, and individual tooth errors are averaged and reduced [2]. Hydrostatic bearing or air-bearing spindles and stages also perform with high accuracy using error averaging effect [3]. Considerable researches on the error averaging effect in precision engineering have been conducted to reduce overall errors in the past. In the serial-type machine or machine tool, the volumetric errors are cumulated from individual geometric errors. Although, the parallel-type mechanism has the error averaging effect, it has been rarely studied [4]. Generally speaking, the error sources in the parallel mechanism can be classified into two categories: one error source always exists in common precision machines due to assembly and applied load, such as geometric errors in each axis, spindle motion error, and deformation error on account of heat and force; the other error originates from the parallel mechanism itself, such as input motion error, structure parameter error, component assembled error, location error of parts, and clearance error of joints. All these errors contribute to the positional error of the end functional point of the moving platform. When the parallel mechanism works, however, all these errors will have influence on one Appl. Sci. 2016, 6, 383; doi:10.3390/app6120383

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and clearance error of joints. All these errors contribute to the positional error of the end functional  2 of 12 point of the moving platform. When the parallel mechanism works, however, all these errors will  have influence on one another when transferring to the moving platform (i.e., the error averaging  effect). In recent years, we focused on the development of a three‐pair prismatic‐universal‐universal  another when transferring to the moving platform (i.e., the error averaging effect). In recent years, joint  (3‐PUU)  parallel  coordinate  machine  (PCMM).  Some  phenomena  about  error  we focused on the development of a measuring  three-pair prismatic-universal-universal joint (3-PUU) parallel averaging effect have been observed due to parameter errors. This study aims to reveal the essence  coordinate measuring machine (PCMM). Some phenomena about error averaging effect have been of error averaging effect due to geometric errors.  observed due to parameter errors. This study aims to reveal the essence of error averaging effect due to geometric errors. 2. Measuring Principle of 3‐PUU PCMM  2. Measuring Principle of 3-PUU PCMM The structure of the developed PCMM is shown in Figure 1. Three sliders are connected to the  The the developed PCMM shown in Figure 1. Three sliders aresliders  connected to the moving  structure platform of through  universal  joints isand  three  pairs  of  link  rods.  The  share  one  moving platform joints separately  and three pairs of driven  link rods. sliders one horizontal horizontal  linear through guider  universal and  can  move  while  by The steel  belts  share and  stepper  motors,  linear guider and can move separately while driven by steel belts and stepper motors, respectively. respectively. The linear grating ruler is fastened to one side of the linear guide. Three reading heads  The linear grating ruler is fastened to one side of the linear guide. Three reading heads are installed are installed with respect to each slider. The moving platform is connected to the end of each link  with respect to each slider. The moving platform is connected to the end of each link rod through rod through universal joints, and the moving platform translates in the space parallel to the three  universal joints, and the moving platform translates in the space parallel to the three sliders without sliders  without  rotation.  The  trigger  probe  assembled  on  the  moving  platform  triggers  three  rotation. The trigger probe assembled on the moving platform triggers three reading heads to collect reading  heads  to  collect  the  instantaneous  positions  synchronously,  when  the  probe  touches  the instantaneous positions synchronously, whencan  thethen  probe touches workpiece The coordinate of workpiece  [5,6].  The  coordinate  of  the  probe  be  calculated  based [5,6]. on  a  measuring  model  the probe can then be calculated based on a measuring model [7,8]. [7,8].  Appl. Sci. 2016, 6, 383

(a)

(b) Figure  1.  3‐PPU  (three  pairs  of  prismatic‐universal‐universal  joints)  PCMM  (parallel  coordinate  Figure 1. 3-PPU (three pairs of prismatic-universal-universal joints) PCMM (parallel coordinate measuring  machine).  (1)  Base,  (2)  table,  (3)  steel  belt,  (4)  reducer,  (5)  motor,  (6)  linear  grating,  (7)  measuring machine). (1) Base, (2) table, (3) steel belt, (4) reducer, (5) motor, (6) linear grating, (7) reading reading  head,  (8)  beam,  (9)  linear  guider,  (10)  rod,  (11)  universal  joints,  (12)  slider,  (13)  moving  head, (8) beam, (9) linear guider, (10) rod, (11) universal joints, (12) slider, (13) moving platform, platform, (14) measuring head, and (15) limit switch. (a) front view; (b) upward view.  (14) measuring head, and (15) limit switch. (a) front view; (b) upward view.

As  shown  in  Figure  1,  three  sliders  are  coplanar,  thus,  the  centers  of  six  universal  joints  on  As shown in Figure 1, three sliders are coplanar, thus, the centers of six universal joints on three three sliders also share one common plane, which is plane OCDB in Figure 2. The angle between the  sliders also share one common plane, which is plane OCDB in Figure 2. The angle between the OCDB plane and the horizontal plane of OHPB is 45◦ . B1 , B2 are centers of two joints on the left slider; B3 ,

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B4 are centers of the middle slider; and B5 and B6 are centers of the right slider. A fixed coordinate system O–XYZ is established based on the plane OCDB, and every center of universal joints Bi can be expressed as: B1 (x1 , 0, 0), B2 (x1 , c, 0), B3 (x2 , b + c, 0), B4 (x2 + a, b + c, 0), B5 (x3 , c, 0), B6 (x3 , 0, 0). Here, x1 , x2 , x3 indicate the positions of three reading heads, respectively, and a, b, c are position parameters between joints on three sliders. The moving platform is in the form of an uneven six-sided polygon on which ei is the center of universal joint i on the moving platform. Let point e1 be the unknown position in the coordinate system of O–XYZ and set it to (x, y, z). Thus, e2 –e6 coordinates can be determined with position parameters (e, f, g) between joints. According to the spatial two-point formula, a set of equations expressing all link lengths can be established as below [9].   ( x − x 1 ) 2 + y 2 + z 2 = l1 2     ( x − x 1 ) 2 + ( y + g − c ) 2 + z 2 = l2 2     ( x + e − x )2 + ( y + g + f − b − c )2 + z2 = l 2 2 3  ( x + d + e − a − x 2 ) 2 + ( y + g + f − b − c ) 2 + z 2 = l4 2      ( x + d + 2e − x3 )2 + (y + g − c)2 + z2 = l5 2    ( x + d + 2e − x3 )2 + y2 + z2 = l6 2

(1)

The group consists of six equations and only three unknowns. If two rods have the same length on each slider (i.e., l1 = l2 , l3 = l4 , l5 = l6 ), the unknown position of point e1 can be solved as:  l 2 −l 2  x = x1 + x32−2e−d + 2e+d5− x 1+ x   3 1  l32 −l12 +( x − x1 )2 −( x +e− x2 )2 y= + 2( f − b )  q    z = l 2 − ( x − x )2 − y2 1

b− f 2

(2)

1

The probing position in space can be obtained from point e1 with a transformation matrix containing sliders’ positions (x1 , x2 , x3 ) and structural parameters of link li (i = 1–6), joints positions (a, b, c, d, e, f, g), and the stylus length, if the probe-mounting position on the moving platform is known. In this study, we only focus on the motion errors of sliders that would affect the positional errors of the probe and, therefore, assume all structural parameters to be constant. In such a case, the probing position in space (xp , yp , zp ) can be simplified to the function of sliders positions as below.    xp = f 1 ( x1 , x2 , x3 ) y p = f 2 ( x1 , x2 , x3 )   z = f (x , x , x ) p 3 1 2 3

(3)

Equations (2) and (3) are measuring models that are also called positive solution in parallel mechanism. The positive solution is the key in analyzing the working space of the probe and establishing the error measuring model. The working space refers to the set of points in space wherein the trigger probe can reach. The measuring accuracy is closely related to the position of the probe tip within the working space. For a specified measuring accuracy, however, only within a certain range of working space can the accuracy of the PCMM achieve the required level. This range is called the measurement space. The measurement space is expected to be as large as possible and is largely dependent upon the accuracy of each slider’s motion and each structural parameter. Based on the required working space of parallel mechanism and the accuracy index as target, the optimization design with respect to all structural parameters of the PCMM was conducted [10,11]. In this study, the motion errors of each slider were measured, and the error averaging effect to the probing position is analyzed in the following sections.

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Based on the required working space of parallel mechanism and the accuracy index as target, the  optimization design with respect to all structural parameters of the PCMM was conducted [10,11].  In this study, the motion errors of each slider were measured, and the error averaging effect to the  Appl. Sci. 2016, 6, 383 4 of 12 probing position is analyzed in the following sections. 

  Figure 2. Geometrical model of PCMM.  Figure 2. Geometrical model of PCMM.

3. Analysis of Grating Reading Error  3. Analysis of Grating Reading Error It  It can  can be  be seen  seen from  from Equation  Equation (3)  (3) that  that the  the measuring  measuring result  result depends  depends on  on the  the positions  positions of  of three  three reading heads. The reading error of the grating scale, which is often called “input motion error” in  reading heads. The reading error of the grating scale, which is often called “input motion error” in the the parallel mechanism, affects the measuring error. The sensitivity of reading error to measuring  parallel mechanism, affects the measuring error. The sensitivity of reading error to measuring error, error,  uncertainty  analysis,  can  be  from derived  from  Equation  (3)  with  differential  coefficient  called called  uncertainty analysis, can be derived Equation (3) with differential coefficient theory. theory.     ∂ f1 ∂ f1 ∂ f1    δ1 x dx ∂x ∂x ∂x 1 2 3 1    ∂f1f2 ∂f1f2 ∂f1f2    (4)  δ1 y  =   ∂x1 ∂x2 ∂x3   dx2  δ1 zδ x   ∂x1f3 x∂ 2f3 ∂xf33  dx dx3 1 1 ∂x1 ∂x2 ∂x3 δ y    f 2 f 2 f 2   dx    (4)  1   x x x   T2  where, [δxp , δyp , δzp ]T are measuring errors while [dx21 , dx23, dx ] are reading errors. The middle 3 × 3 1  3   δ1 z   dx3  matrix is called the error transfer coefficient  matrix. f3 f3 f3    Putting practical values of the PCMM structure into Equations (2) and (3), we can x1 x2 parameters x3  derive the error transfer coefficient matrix of every point in the measuring space. By conducting substantialp, δy simulation calculation, we found that the PCMM averaged the original reading head errors p, δzp]T are measuring errors while [dx 1, dx2, dx 3]T are reading errors. The middle 3 × 3  where, [δx through link rods. Table 1 lists the error transfer coefficients of two randomly selected points in the matrix is called the error transfer coefficient matrix.  measuring space. Other points generally have the same characteristics. Putting practical values of the PCMM structure parameters into Equations (2) and (3), we can  derive  the  error  transfer  coefficient  matrix  of  every  point  in  the  measuring  space.  By  conducting  Table 1. Error transfer coefficient of two points. substantial  simulation  calculation,  we  found  that  the  PCMM  averaged  the  original  reading  head  errors through link rods. Table 1 lists the error transfer coefficients of two randomly selected points  Serial (∂fi /∂x1 )/mm (∂fi /∂x2 )/mm (∂fi /∂x3 )/mm in the measuring space. Other points generally have the same characteristics.  Number (x, y, z) = (600, 420, 240) Table 1. Error transfer coefficient of two points.  i=1 0.5 0 0.5 Serial  i=2 0.13 0.16 − 0.43 ( fi / x1 )/mm  ( fi / x2 )/mm  ( fi / x3 )/mm  i=3 −0.11 −1.24 1.21 Number (x, y, z) = (600, 420, 240)  (x, y, z) = (600, 360, 290) i = 1  0.5  0  0.5  i=1 0.5 0 0.5 i = 2  0.13  0.16  −0.43  i=2 −1.13 1.26 0.13 i = 3  −0.11  1.71 −1.24  1.21  i=3 −0.86 −0.23 (x, y, z) = (600, 360, 290)  i = 1  0.5  0  0.5  As given in Equation (2), the−1.13  x coordinate of point e1.26  by the first and the third 1 is determined only0.13  i = 2  reading head data. The value of the second reading head is not related. The error i = 3  1.71  −0.86  −0.23  transfer coefficient

of the first and the third reading head are the same at 0.5. The error transfer coefficient of the second reading head is zero. In other words, if the length of the first and the third link rod are equal (i.e., l1 = l2 = l5 = l6 ), then x coordinate of point e1 will be equal to (x1 + x3 − 2e − d)/2. This phenomenon is very close to what we conduct in angle measurement, with two reading heads installed 180◦ apart on the circular grating. In the Y and Z directions, positive and negative error transfer coefficients

As  given  in  Equation  (2),  the  x  coordinate  of  point  e1  is  determined  only  by  the  first  and  the  third  reading  head  data.  The  value  of  the  second  reading  head  is  not  related.  The  error  transfer  coefficient of the first and the third reading head are the same at 0.5. The error transfer coefficient of  the second reading head is zero. In other words, if the length of the first and the third link rod are  equal  (i.e.,  l1  =  l2  =  l5  =  l6),  then  x  coordinate  of  point  e1  will  be  equal  to  (x1  +  x3  −  2e  −  d)/2. 5This  Appl. Sci. 2016, 6, 383 of 12 phenomenon  is  very  close  to  what  we  conduct  in  angle  measurement,  with  two  reading  heads  installed  180°  apart  on  the  circular  grating.  In  the  Y  and  Z  directions,  positive  and  negative  error  alternately appear, and the sum tends to zero. Therefore, we can conclude that three grating reading transfer coefficients alternately appear, and the sum tends to zero. Therefore, we can conclude that  head errors have been averaged in the PCMM. other words, thePCMM.  error average effect of thethe  parallel three  grating  reading  head  errors  have  been In averaged  in  the  In  other  words,  error  mechanism is embodied in the motion input errors. average effect of the parallel mechanism is embodied in the motion input errors.  Using we have studied thethe  length manufacturing error of six Using the the same same total total differential differential method, method,  we  have  studied  length  manufacturing  error  of  link rods and its influence on the measuring error. A similar conclusion was determined. For the X six link rods and its influence on the measuring error. A similar conclusion was determined. For the  direction, the error transfer coefficient could realize compensation by itself. In Y and Z direction, the X direction, the error transfer coefficient could realize compensation by itself. In Y and Z direction,  error averaging effect still exists, but is weaker than that in the X direction [11]. the error averaging effect still exists, but is weaker than that in the X direction [11].  4. Property of Slider Motion Errors in PCMM 4. Property of Slider Motion Errors in PCMM  Linear motion stage is always employed in precision engineering. The stage provides straight Linear motion stage is always employed in precision engineering. The stage provides straight  line motion reference, which is is  expected to travel along a straight line. line.  However, in practice, the actual line  motion  reference,  which  expected  to  travel  along  a  straight  However,  in  practice,  the  path and orientation of the stage deviate from the straight line of motion because of its own geometric actual path and orientation of the stage deviate from the straight line of motion because of its own  errors and errors  assembling errors. As shown 3, in  five unexpected straightness geometric  and  assembling  errors.  in As Figure shown  Figure  3,  five motions—two unexpected  motions—two  motion errors and three angular motion errors—occur in five degrees of freedom (5-DOF). Thefreedom  motion straightness  motion  errors  and  three  angular  motion  errors—occur  in  five  degrees  of  straightness errors include two linear errors Y and Z direction, respectively. The angular motion (5‐DOF).  The  motion  straightness  errors along include  two  linear  errors  along  Y  and  Z  direction,  errors around each axis are called pitch, yaw, and roll. respectively. The angular motion errors around each axis are called pitch, yaw, and roll. 

  Figure 3. Degrees of freedom (DOFs) of the stage.  Figure 3. Degrees of freedom (DOFs) of the stage.

Over  the  years,  numerous  scholars  have  intensively  studied  the  characteristics  of  the  Over the years, numerous scholars have intensively studied the valuable  characteristics of the aforementioned  geometric  errors  on  precision  machines  and  made  many  contributions,  aforementioned geometric errors on precision machines and made many valuable contributions, especially in analysis and discussion of the relationship between the motion straightness errors and  especially in analysis and discussion of the relationship between the motion straightness errors and angular motion errors. Their conclusion can be summarized as follows [12–17]:  angular motion errors. Their conclusion can be summarized as follows [12–17]: (1)  The straightness motion error of the linear stage is closely related to the straightness error of  (1) the  Theguide  straightness motion of the lineardifferent in  stage is closely to the straightness error of the rail  itself,  but  error these errors are  that related the  guide  rail straightness  error is a  guide rail itself, but these errors are different in that the guide rail straightness error is a “cause”, “cause”, whereas the straightness motion error is a “result”;  whereas the straightness motion is astraightness  “result”; (2)  The  angular  motion  error  and  error motion  error  are  also  related,  but  no  definite  (2) function relation exists between them;  The angular motion error and motion straightness error are also related, but no definite function (3)  When  a exists different  point them; on  the  stage  is  measured,  its  value  of  straightness  motion  error  may  relation between (3) differ.  When a different point on the stage is measured, its value of straightness motion error may differ. Our study showed that the motion straightness error of the slider is the key error source that  Our study showed that the motion straightness error of the slider is the key error source that influences and even determines the accuracy and performance of the PCMM. This is because since  influences and even determines the accuracy and performance of the PCMM. This is because since the the three sliders are shared with only one linear guider rail, their motion errors will have an impact  three sliders are shared with only one linear guider rail, their motion errors will have an impact on on the moving platform through six link rods. Figure 4 illustrates one slider working on the guide  the moving platform through six link rods. Figure 4 illustrates one slider working on the guide rail. A detailed discussion about the slider angular motion errors and motion straightness errors will be given, respectively, in the following sections.

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rail.  A  detailed  Appl. Sci. 2016, 6, 383discussion  about  the  slider  angular  motion  errors  and  motion  straightness  errors  6 of 12 will be given, respectively, in the following sections. 

  Figure 4. Slider motion in PCMM.  Figure 4. Slider motion in PCMM.

4.1. Abbe Error and Angular Motion  4.1. Abbe Error and Angular Motion The Abbe principle, which was proposed in 1890, is often referred to as the first principle for  The Abbe principle, which was proposed in 1890, is often referred to as the first principle for instrument  design.  This  principle  could  avoid  or  eliminate  the  first‐order  measuring  error  caused  instrument design. This principle could avoid or eliminate the first-order measuring error caused by by angular motion error. The Abbe error also plays a key role in 3‐PUU PCMM. As shown in Figure  angular motion error. The Abbe error also plays a key role in 3-PUU PCMM. As shown in Figure 4, 4, point P is the center of one joint on the slider, while point A is the center of grating reading head.  point P is the center of one joint on the slider, while point A is the center of grating reading head. Two points are located at the different corners of the cuboids diagonal. Distance H and S are Abbe  Two points are located at the different corners of the cuboids diagonal. Distance H and S are Abbe offsets,  which  are  perpendicular  to  each  other  in  Y  and  Z  directions,  respectively.  Pitch  β(x)  and  offsets, which are perpendicular to each other in Y and Z directions, respectively. Pitch β(x) and yaw yaw γ(x) of the slider produce Abbe error of point P with respect to point A. When the slider moves  γ(x) of the slider produce Abbe error of point P with respect to point A. When the slider moves to to a point xi, its yaw angle is γ(xi). This means that the slider will rotate a small angle around the Z  a point xi , its yaw angle is γ(xi ). This means that the slider will rotate a small angle around the Z axis, axis,  which  changes  the  position  of  reading  head  to  linear  grating;  this  is  one  kind  of  Abbe  error  which changes the position of reading head to linear grating; this is one kind of Abbe error which is which  is  equal  to  Abbe  offset  H  times  γ(xi).  The  pitch  β(xi)  also  produces  similar  results  with  equal to Abbe offset H times γ(xi ). The pitch β(xi ) also produces similar results with another Abbe another  Abbe  offset  S.  Thus,  total  Abbe  error  can  be  expressed  as  Equation  (5).  This  issue  can  be  offset S. Thus, total Abbe error can be expressed as Equation (5). This issue can be analyzed as an input analyzed as an input error. The method is similar to the analysis of the sensor reading error in part  error. The method is similar to the analysis of the sensor reading error in part 3 [18–20]. 3 [18–20].  We substitute the grating reading error in Equation (4) with Abbe error. Here, dx can be We  substitute  the  grating  reading  error  in  Equation  (4)  with  Abbe  error.  Here,  dxii  can  be  expressed by: expressed by:  dxi = − Hi tanγ( xi ) − Si tanβ( xi ) (5)

dxi small, Hi tan γ( xi)  S(5) i tanβ( i )   simplified as: As the pitch and yaw angle is very Equation canxbe dxi = − Hi γ( xi ) − Si β( xi ) As the pitch and yaw angle is very small, Equation (5) can be simplified as: 

(5) (6)

In an experiment, the pitch and yaw motion errors of three sliders (HIWIN Technologies dxangular i   Hiγ( xi )  Siβ( xi )   (6) Corp., Taichung, China) were measured by an autocollimator (Jingda Measurement Technology Co. Ltd., Jiujiang, China), as shown in Figure 5. The overall trends of the three sliders’ angle errors were In  an  experiment,  the  pitch  and  yaw  angular  motion  errors  of  three  sliders  (HIWIN  very similar. Figure 6 shows the angular motion data of the left slider, which contains 37 points. Technologies  Corp.,  Taichung,  China)  were  measured  by  an  autocollimator  (Jingda  Measurement  Angular data are collected once after the slider moves 50 mm. Thus, the serial number of the horizontal Technology Co. Ltd., Jiujiang, China), as shown in Figure 5. The overall trends of the three sliders’  axis in Figure 6 is corresponds to the different positions of the left slider. Several models of spline curve angle  errors  were  very  similar.  Figure  6  shows  the  angular  motion  data  of  the  left  slider,  which  interpolation for the angular motion error of three sliders were established after obtaining experimental contains 37 points. Angular data are collected once after the slider moves 50 mm. Thus, the serial  data, which was important for completing the subsequent error calculation. number of the horizontal axis in Figure 6 is corresponds to the different positions of the left slider.  As shown Table 2, 3curve  cross-sections (x = 750, 1250) inmotion  the measurement spacesliders  are selected Several  models inof  spline  interpolation  for 1000, the  angular  error  of  three  were  initially, and 9 points are selected as feature points in each section. Therefore, 27 points are distributed in established after obtaining experimental data, which was important for completing the subsequent  the measurement space. Corresponding to these points are each slider’s position coordinate, which can error calculation.    be determined through the inverse kinematic solution of PCMM using Equation (2). Thus, the pitch and yaw angle error value of every slider can be achieved through the angle error model. Putting measured angle errors of Figure 6 into in Equations (3) and (6), we can determine the contribution of Abbe errors of the three sliders to the probe’s positional errors in the measurement space as listed in Table 2.

Appl. Sci. 2016, 6, 383

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Appl. Sci. 2016, 6, 383  Appl. Sci. 2016, 6, 383 

7 of 12  7 of 12 

   

Figure 5. Autocollimator detection.  Figure 5. Autocollimator detection. Figure 5. Autocollimator detection. 

   

Figure 6. Pitch and yaw angle error of slider 1.  Figure 6. Pitch and yaw angle error of slider 1.  Figure 6. Pitch and yaw angle error of slider 1.

As  shown  in  Table  2,  3  cross‐sections  (x  =  750,  1000,  1250)  in  the  measurement  space  are  As  shown  in  Table Table 2,  3 2.cross‐sections  (x  by =  750,  1250)  the  measurement  space  are  Error contributed Abbe1000,  error of three in  sliders. selected initially, and 9 points are selected as feature points in each section. Therefore, 27 points are  selected initially, and 9 points are selected as feature points in each section. Therefore, 27 points are  distributed  in  the  measurement  space.  Corresponding  to  these  points  are  each  slider’s  position  distributed  the  measurement  space. through  Corresponding  to  these  points solution  are  each  slider’s  position  Selectedin  Points/mm Corresponding Slider the  Position/mm Error Value/µm coordinate,  which  can  be  determined  inverse  kinematic  of  PCMM  using  coordinate,  which  can  be  determined  through  the  inverse  kinematic  solution  of  PCMM  δyA δzA using  (X, Y, Z) x1 x2 x3 δxA Equation (2). Thus, the pitch and yaw angle error value of every slider can be achieved through the  Equation (2). Thus, the pitch and yaw angle error value of every slider can be achieved through the  750, 150, 200 193.58 1167.44 1319.58 −7.60 −21.96 −12.13 angle error model. Putting measured angle errors of Figure 6 into in Equations (3) and (6), we can  angle error model. Putting measured angle errors of Figure 6 into in Equations (3) and (6), we can  750, 150, 300 240.49 1047.23 1285.55 2.53 19.79 6.84 determine the contribution of Abbe errors of the three sliders to the probe’s positional errors in the  750, 150, 400 314.56 1074.49 1211.45 4.59 16.38 11.68 determine the contribution of Abbe errors of the three sliders to the probe’s positional errors in the  measurement space as listed in Table 2.  750, 300, 200 257.96 1357.98 1468.03 2.55 −15.23 5.77 measurement space as listed in Table 2.  750, 300, 300 750, 300, 400 750, 450, 200 Selected  750, 450, 300 Selected  750, 450, 400 Points/mm  1000, 150, 200 Points/mm  (X, Y, Z)  1000, 150, 300 (X, Y, Z)  750, 150, 200  ... 750, 150, 200  1250, 450, 200 750, 150, 300  1250, 450, 300 750, 150, 300  750, 150, 400  1250 ,450, 400

750, 150, 400  750, 300, 200 

311.70

1308.28

1414.32

5.90

17.06 15.68 −20.05 1.74 −18.34 −1.57 1.28 7.87 1.40 Error Value/μm  −5.36 14.45 1.97 Error Value/μm  −3.57 −14.40 4.15 δxA   δ− yA  δzA− 0.25 −3.52 11.86 δxA  δyA  δzA  −7.60  −21.96  −12.13  ... ... ... −7.60  −21.96  −12.13  3.11 17.34 4.51 2.53  19.79  6.84  2.62 17.07 3.33 2.53  19.79  6.84  4.59  16.38  11.68  2.27 −11.78 11.68  −2.10 4.59  16.38 

400.57 1228.27 1325.46 −5.69 Table 2. Error contributed by Abbe error of three sliders.  Table 2. Error contributed by Abbe error of three sliders.  390.01 1371.35 1486.67 −2.79

Corresponding Slider  467.86 1310.48 1458.19 Corresponding Slider  652.02 1203.11 1374.04 Position/mm  443.58 1403.64 1582.45 Position/mm  x1  x2  x3  490.49 1397.26 1735.50 x1  x2  x3  193.58  1167.44  1319.58  ... ... ... 193.58  1167.44  1319.58  890.00 1671.33 1836.56 240.49  1047.23  1285.55  967.86 1610.46 1858.18 240.49  1047.23  1285.55  314.56  1074.49  1211.45  1152.0 1653.10 1874.07

314.56  257.96 

1074.49  1357.98 

1211.45  1468.03 

2.55 

−15.23 

5.77 

. . . 750, 300, 200  represents a few sets of 257.96  data was omitted, because there are 27 sets of data if all data are 5.77  shown, 1357.98  1468.03  2.55 in table,−15.23  750, 300, 300  311.70  1308.28  1414.32  5.90  17.06  15.68  the table will occupy too large columns.

750, 300, 300  311.70  1308.28  1414.32  5.90  17.06  15.68  750, 300, 400  400.57  1228.27  1325.46  −5.69  −20.05  1.74  750, 300, 400  400.57  1228.27  1325.46  −5.69  −20.05  1.74  750, 450, 200  390.01  1371.35  1486.67  −2.79  −18.34  −1.57  Based on the calculation results, for 27 selected feature points, it can be seen that the contribution 750, 450, 200  390.01  1371.35  1486.67  −2.79  −18.34  −1.57  750, 450, 300  1310.48  1458.19  1.28  7.87  1.40 smallest. of the angular motion errors467.86  to measuring accuracy of the probe in the X direction is the 750, 450, 300  467.86  1310.48  1458.19  1.28  7.87  1.40  750, 450, 400  652.02  1203.11  1374.04  −5.36  14.45  1.97  Error variation fluctuates from −7.6 to +5.9 µm. The error contribution to the Y direction is the largest, 750, 450, 400  652.02  1203.11  1374.04  −5.36  14.45  1.97  1000, 150, 200  443.58  1403.64  1582.45  −3.57  −14.40  4.15  1000, 150, 200  443.58  1403.64  1582.45  −3.57  −14.40  4.15  1000, 150, 300  490.49  1397.26  1735.50  −3.52  −11.86  −0.25  1000, 150, 300  490.49  1397.26  1735.50  −3.52  −11.86  −0.25 

1250 ,450, 400 

1152.0 

1653.10 

1874.07 

2.27 

−11.78 

−2.10 

… represents a few sets of data was omitted, because there are 27 sets of data in table, if all data are  shown, the table will occupy too large columns.   

Based  Appl. Sci. 2016, on  6, 383the 

calculation  results,  for  27  selected  feature  points,  it  can  be  seen  that  8 ofthe  12 contribution of the angular motion errors to measuring accuracy of the probe in the X direction is  the  smallest.  Error  variation  fluctuates  from  −7.6  to  +5.9  μm.  The  error  contribution  to  the  Y  ranging −21.9 to ~+22.3 µm.from  The −21.9  error to  variation observed in Figurecan  7. Compared with direction from is  the  largest,  ranging  ~+22.3  can μm. be The  error  variation  be  observed  in  the errors in Y and Z directions, the Abbe error has a minimal effect in the X direction because the Figure 7. Compared with the errors in Y and Z directions, the Abbe error has a minimal effect in the  error averaging effect in X direction is strongest. It is similar to the results of our previous reports on X direction because the error averaging effect in X direction is strongest. It is similar to the results of  reading head error analysis [21]. our previous reports on reading head error analysis [21]. 

  Figure 7. Variation of error caused by Abbe error.  Figure 7. Variation of error caused by Abbe error.

4.2. Straightness Motion Error  4.2. Straightness Motion Error As  shown  in  Figure  4,  three  sliders  also  generate  motion  straightness  errors  in  Y  and  Z  As shown in Figure 4, three sliders also generate motion straightness errors in Y and Z directions, directions, namely δY(x) and δZ(x), respectively [20,21]. Three sliders moving one after another not  namely δY (x) and δZ (x), respectively [20,21]. Three sliders moving one after another not only cause the only  cause  the  moving  platform  to  produce  positioning  errors,  but  also  result  in  straightness  and  moving platform to produce positioning errors, but also result in straightness and orientation changes. orientation  changes.  Thus,  the  change  of  the  probe  position  leads  to  measurement  error.  Thus, the change of the probe position leads to measurement error. Considering the influence of δY (x) Considering the influence of δY(x) and δZ(x), we can modify Equation (1) to:  and δZ (x), we can modify Equation (1) to:  2 2 z1)2)2 = l12 l 2  δδz 1)  ( y  δy12  ( x −(xx1 )2x+ (y − δy1 ) ) + ((zz − 1 1   2 2 2 2  2  g c c−δy l2 (xx1 )2x+ (+y g − δy11))22 +( (zz− δzδz 1) ( 1)   ( x − y  1 ) = l2    ( x +(ex− xe )2x 2+ 2 2 2 2 y g g+f f−bb− y 2)2 )  + c 2l)322 = l3 2 −δδy (zz− ) (y (+ ( z -δ 2) δz  2 2 2 2 )2 + ( z − 2δz )22 = l 2  ( x +(dx+ 4   ea −ax2 )x 2)+ (y( y+ gg+ f−  bb−c c− δyδy ( z  δz 2) 2l 4 2) 2    de −   2 2 2 2      ( x + d + 2e − x ) + ( y + g − c − δy ) + ( z − δz ) = l 2 2 2 2 3 3 3 5    ( x  d  2e  x3)  ( y  g  c  δy 3)  ( z  δz 3)  l 5 ( x + d + 2e − x3 )2 +2 (y − δy3 )22 + (z − δz23 )2 =2 l6 2 ( x  d  2e  x3)  ( y  δy 3)  ( z  δz 3)  l 6

(7) (7)

The analytical solution of the above set of equations is difficult to find. Only a numerical The  can analytical  solution  the  above  set  of  equations  difficult  to  find.  Only  a  numerical  solution be obtained byof  numerical iteration analysis. is Other parallel mechanisms are also solution  can  be  obtained  iteration  analysis.  Other  parallel  mechanisms  also  generally encountered. Any by  typenumerical  of numerical algorithm in the calculation process consists of are  attempt, generally  encountered.  Any  and type convergence. of  numerical Inalgorithm  in  the  calculation  process  consists  of of  approximation, compromise, real parallel mechanism motion, the position attempt,  approximation,  compromise,  and  convergence.  parallel  mechanism  the  the moving platform is determined by the plurality of barsIn  in real  common. In this process, motion,  mechanical position of the moving platform is determined by the plurality of bars in common. In this process,  tensile, compression, torsion, and elastic deformation of involved parts occur. When a certain position mechanical tensile, compression, torsion, and elastic deformation of involved parts occur. When a  can achieve balance among these factors, this is the actual position that the moving platform reaches. certain position can achieve balance among these factors, this is the actual position that the moving  Error averaging effect happens during this moving process. Reflecting on mathematics, it is iteration platform  reaches. Thus, Error  happens  during  has this weak moving  on  and convergence. weaveraging  find that ifeffect  any iterative algorithm errorprocess.  averagingReflecting  effect, it will mathematics, it is iteration and convergence. Thus, we find that if any iterative algorithm has weak  be unsuitable for use in parallel mechanism calculation. error averaging effect, it will be unsuitable for use in parallel mechanism calculation.  As mentioned by Bryan in 1979 [12], the slider motion straightness errors are different when different points on the slider are measured. We are mainly interested in the error of the point where two universal joints center on the slider. Therefore, when a dual-frequency laser straightness interferometer was used to detect slider motion straightness error, the interference mirror (Wollaston prism) required adjustment to be as close as possible to the center point of two joints on the moving slider. The large reflection mirror was statically fixed to the guide support. The experimental setup is shown in Figure 8.

different points on the slider are measured. We are mainly interested in the error of the point where  two  universal  joints  center  on  the  slider.  Therefore,  when  a  dual‐frequency  laser  straightness  interferometer  was  used  to  detect  slider  motion  straightness  error,  the  interference  mirror  (Wollaston prism) required adjustment to be as close as possible to the center point of two joints on  the  moving  The  large  reflection  mirror  was  statically  fixed  to  the  guide  support. 9 of The  Appl. Sci. 2016, 6,slider.  383 12 experimental setup is shown in Figure 8. 

  Figure 8. Straightness motion error experiment.  Figure 8. Straightness motion error experiment.

Straightness errors in Y and Z directions were captured by the interferometer (Renishaw XL‐80,  Straightness errorsUK).  in Y and directions were captured bystraightness  the interferometer (Renishaw XL-80, Renishaw,  New  Mills,  The Zoverall  trend  of  three  slider  motion  errors  is  highly  Renishaw, New Mills, UK). the  The overall trend of three slider straightness motion highly similar. similar.  Figure  9  shows  straightness  motion  errors  of  first  slider  in errors two  is directions.  The  Figure 9 shows the straightness motion errors of first slider in two directions. The horizontal axis is the horizontal axis is the coordinate x1 of the first slider. The interferometer picks up δY(x) once after the  coordinate x of the first slider. The interferometer picks up δ (x) once after the slider moves 50 mm in 1 50  mm  in  one  direction,  thus,  the  “forward”  Y slider  moves  curve  can  be  established.  The  one direction, thus, thewas  “forward” curve canreverse  be established. The process was repeated measurement  process  repeated  in  the  direction  to measurement established  the  “reverse”  curve.  It  in the reverse direction to established the “reverse” curve. It can be seen that the two directional can be seen that the two directional measurements are quite consistent, and the average value of the  measurements are quite consistent, and the average value of the data is then used as the straightness data is then used as the straightness motion error of the slider in the Y direction, as shown in Figure  motion errorFor  of the in the Y motion  direction,error  as shown in slider  Figurein  9 (above). For the straightness motion 9  (above).  the slider straightness  of  the  the  Z  direction,  we  repeated  the  error of the slider in the Z direction, we repeated the detection process, but the interference mirror detection process, but the interference mirror and the reflection mirror needed to be rotated 90° in  and the reflection mirror needed to be rotated 90◦ Zin advance. Figure 9 (below) shows the variation advance. Figure 9 (below) shows the variation of δ (x).    ofAppl. Sci. 2016, 6, 383  δZThe  (x). value  difference  among  six  sets  of  data  of  straightness  motion  errors  is  extremely  small.  10 of 12  Thus, only one series of data on a slider was selected to establish the error model for the subsequent  error  calculation.  A  quadratic  function  was  used  to  establish  the  model  for  straightness  motion  error  in  the  Z  direction.  The  straightness  motion  error  in  the  Y  direction  is  relatively  complex,  where the error model needs to be constructed with the third‐order spline curve function.  As  shown  in  Table  3,  3  groups  (x  =  750,  1000,  1250)  in  the  measurement  space  were  selected  initially, and 9 points were selected as feature points in each group. Thus, a total of 27 points were  distributed in the measurement space. Corresponding to each point is a slider position coordinate  that  can  be  determined  through  the  inverse  kinematics  of  PCMM.  Putting  the  measured  straightness errors of the three sliders into Equation (7) and solving the actual probe position in the  measurement space by Equation (3), the contributions of straightness motion errors to the probe’s  positional error are listed in Table 3. 

  Figure 9. Straightness motion error of slider 1.  Figure 9. Straightness motion error of slider 1. Table 3. Error contributed by straightness motion errors. 

The value difference among six sets of data of straightness motion errors is extremely small. Thus, only one series of data on a sliderCorresponding Slider  was selected to establish the error model for the subsequent error Selected  Error Value/μm  calculation.Points/mm  A quadratic function wasPosition/mm  used to establish the model for straightness motion error in the Z direction. (X, Y, Z)  The straightness xmotion errorxin is relatively complex, where 1  2  the Y direction x3  δxS  δyS  δzS  the error model needs to be constructed with the third-order spline curve function. 750, 150, 200  193.58  1167.44  1319.58  3  8  −8  As shown in Table 3, 3 groups (x = 750, 1000, 1250) in the measurement space were 750, 150, 300  240.49  1047.23  1285.55  −7  5  17  selected initially, and 9 points were selected in each group.6 Thus, a total points were 750, 150, 400  314.56  as feature 1074.49 points1211.45  2  of 27 −9 

750, 300, 200  750, 300, 300  750, 300, 400  750, 450, 200  750, 450, 300 

257.96  311.70  400.57  390.01  467.86 

1357.98  1308.28  1228.27  1371.35  1310.48 

1468.03  1414.32  1325.46  1486.67  1458.19 

−3  3  −5  −1  −2 

−4  −1  −10  1  13 

4  9  −25  3  −13 

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distributed in the measurement space. Corresponding to each point is a slider position coordinate that can be determined through the inverse kinematics of PCMM. Putting the measured straightness errors of the three sliders into Equation (7) and solving the actual probe position in the measurement space by Equation (3), the contributions of straightness motion errors to the probe’s positional error are listed in Table 3. Table 3. Error contributed by straightness motion errors. Selected Points/mm

Corresponding Slider Position/mm

Error Value/µm

(X, Y, Z)

x1

x2

x3

δxS

δyS

δzS

750, 150, 200 750, 150, 300 750, 150, 400 750, 300, 200 750, 300, 300 750, 300, 400 750, 450, 200 750, 450, 300 750, 450, 400 1000, 150, 200 1000, 150, 300 ... 1250, 450, 200 1250, 450, 300 1250, 450, 400

193.58 240.49 314.56 257.96 311.70 400.57 390.01 467.86 652.02 443.58 490.49 ... 890.00 967.86 1152.0

1167.44 1047.23 1074.49 1357.98 1308.28 1228.27 1371.35 1310.48 1203.11 1403.64 1397.26 ... 1671.33 1610.46 1653.10

1319.58 1285.55 1211.45 1468.03 1414.32 1325.46 1486.67 1458.19 1374.04 1582.45 1735.50 ... 1836.56 1858.18 1874.07

3 −7 6 −3 3 −5 −1 −2 2 4 3 ... 1 3 3

8 5 2 −4 −1 −10 1 13 −8 8 −9 ... 10 −1 8

−8 17 −9 4 9 −25 3 −13 14 6 −22 ... 18 5 −8

. . . represents a few sets of data was omitted.

As shown in Table 3, the straightness motion errors of the sliders contribute the smallest measuring error in the X direction and produce the largest error in the Z direction. This means that, if the purpose is to promote the Z direction measuring accuracy, the slider straightness motion error needs to be improved significantly. The error curves in three directions are shown in Figure 10. The total variation in three directions ranges from −0.03 to ~0.02 mm. Compared with the original slider motion straightness error data, the final influence of the slider motion straightness error on the measurement accuracy is relatively decreased [9,22]. This also demonstrates the error averaging effect for the PCMM. Appl. Sci. 2016, 6, 383  11 of 12 

  Figure 10. Variation of error caused by straightness motion error.  Figure 10. Variation of error caused by straightness motion error.

5. Conclusions  5. Conclusions This study analyzed the error averaging effect on the parallel mechanism. Although this issue  This study analyzed the error averaging effect on the parallel mechanism. Although this issue had  not  been  fully  explored,  we  have  found  solid  evidence  from  theoretical  analysis  and  had not been fully explored, we have found solid evidence from theoretical analysis and experimental experimental  results  that  this  effect  is mainly  dependent  on  the  symmetry  of  structure.  The  input  results that this effect is mainly dependent on the symmetry of structure. The input motion error, motion error, angle errors, and straightness errors of each slider can be averaged and reduced at the  angle errors, and straightness errors of each slider can be averaged and reduced at the probe position probe position through parallel links rods. In the 3‐PUU PCMM, the precision linear guider, which  through parallel links rods. In the 3-PUU PCMM, the precision linear guider, which is shared by is  shared  by  three  sliders,  is  an  important  measuring  part.  Our  future  studies  on  error  averaging  effect  will  further  explore  the  correlation  between  different  error  sources  and  the  error  transfer  coefficient to establish an integrated error model to improve the accuracy of the developed PCMM.  Acknowledgments:  The  authors  are  grateful  for  the  financial  support  provided  by  the  Natural  Science  Foundation  of  China  (51475133,  51675157)  and  Natural  Science  Foundation  of  Anhui  Province 

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three sliders, is an important measuring part. Our future studies on error averaging effect will further explore the correlation between different error sources and the error transfer coefficient to establish an integrated error model to improve the accuracy of the developed PCMM. Acknowledgments: The authors are grateful for the financial support provided by the Natural Science Foundation of China (51475133, 51675157) and Natural Science Foundation of Anhui Province (1508085MF122). Author Contributions: Peng-Hao Hu and Chang-Wei Yu conceived, designed the instrument structure and wrote the paper; Kuang-Chao Fan guided the instrument design and provided new ideas about the instrument structure; Xue-Ming Dang designed and produced the motion control system; Rui-Jun Li finished the experiment and processed the obtained data. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviation The following abbreviations are used in this manuscript: PCMM 3-PUU

Parallel coordinate measuring machine 3 pairs of prismatic-universal-universal joints

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