dimensional inspection of translating cams by cnc coordinate

DIMENSIONAL INSPECTION OF TRANSLATING CAMS BY CNC COORDINATEMEASURING MACHINE

Jung-Fa HsiehDepartment of Mechanical Engineering, Far East University, Tainan, Taiwan

E-mail: [email protected]

Received April 2012, Accepted September 2012No. 12-CSME-39, E.I.C. Accession 3359

ABSTRACTIn this paper, a simple and analytical method is proposed for accomplishing the inspection procedure.

In the proposed approach, a kinematic model of the cam profile is derived using homogenous coordinatetransformation theory. The normal vectors and principal curvature of the designed translating cam are thenderived from the analytical expression for the cam profile. Based on the coordinates and normal vector ofeach specified point on the cam profile, the NC data required to move the CMM inspection probe to thesurface of the cam are then obtained. A least-square fitting method is proposed to minimize the inspectionerror caused by a misalignment of the actual evaluation frame relative to the ideal frame.

Keywords: translating cam; conjugate surface theory; principal curvature.

INSPECTION DIMENSIONNELLE DES CAMES DE TRANLATION PAR MACHINE DEMESURE PAR COORDONNÉES CNC

RÉSUMÉDans cet article nous proposons une méthode simple et analytique pour effectuer la procédure d’ins-

pection. Dans l’approche proposée, un modèle cinématique du profil de la came est obtenu, en utilisant lathéorie de transformation des coordonnées homogènes. Les vecteurs normaux et la courbure principale dela came de translation sont ensuite dérivés de l’expression analytique du profil de la came. En se basant surles coordonnées et sur le vecteur normal de chaque point spécifique sur le profil de la came, les données NCrequises pour déplacer la sonde d’inspection CMM à la surface de la came sont alors obtenues. La méthodedes moindres carrés est proposée pour minimiser les erreurs d’inspection causées par le désalignement ducadre d’évaluation réelle par rapport au cadre d’évaluation idéale.

Mots-clés : came de translation ; théorie des surfaces conjuguées ; courbure principale.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 3, 2012 259

NOMENCLATURE

(xyz)0 coordinate frame (xyz)0 built in translating cam0S translating cam surfacerS surface equation of meshing element of roller0n unit outward normal of translating cama1 length of float linka2 length of crank linka3 distance between z2 and z3 axesb4 distance between x3 and x4 axesb4a displacement of roller-followers displacement of translating camNi ith point on nomminal surfacePi ith actual measured pointSi ith nearest point to nominal surfaceX ,Y,Z required NC data values for CMMλ distance between the design point and the set point along the normal vectorδ width of rollerθ3 rotation angle of crank

1. INTRODUCTION

In any manufacturing process, the finished component inevitably deviates from the nominal design dueto cutting force effects, thermal expansion of the machine tool, and so forth. As a result, some form ofinspection process is required to ensure that the component dimensions fall within tolerable limits. Of thevarious off-line inspection methods available, Coordinate Measuring Machines (CMMs) have emerged asthe method of choice for evaluating the dimensional quality of manufactured components since they enablea highly accurate and rapid characterization of complex surface profiles [1–8].

When inspecting the dimensional accuracy of a cam using a CMM, the coordinate data of multiple dis-crete points on the nominal cam surface are derived using some form of interpolation algorithm [2–7] andthe cam profile error is then evaluated by comparing the nominal coordinate data with the actual measuredcoordinates, However, modeling the cam contour using interpolation curves involves the use of cumbersomeand complex equations. Accordingly, Chang et al. [8] presented a method for evaluating the profile devi-ation of cams based on the correlation between the radial-dimension error and the normal-direction errorof the cam surface. However, although the proposed method is simpler than existing interpolation-basedtechniques, it is only suitable for disc cams.

This paper proposes a simple and direct analytical method for evaluating the dimensional accuracy ofa 2-D translating cam. In the proposed approach, the coordinates and normal vectors of the designatedinspection points are derived from an analytical model of the cam profile and are used to compute theNC commands required to drive the inspection probe of the CMM. The probe is moved along the normalvector until it contacts the cam surface, and the coordinates of the stylus ball center of the probe are thenrecorded. Finally, the measured coordinates of the inspection point are compensated by the stylus ball radiusalong the normal direction and are compared with the designed coordinate values in order to determine thedimensional accuracy of the cam profile. The accuracy of the evaluation results is enhanced by means ofan iterative scheme designed to minimize the total errors [9] caused by a misalignment between the actualevaluation frame and the ideal frame. Several algorithms including least-square fitting were used to evaluatethe form error, roundness [10].

The mathematical modeling performed in this study is based on the coordinate transformation methodoutlined in [11]. The proposed methodology comprises four basic steps: (1) designing the cam profile

260 Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 3, 2012

using conjugate surface theory, (2) determining the normal vectors of the translating cam, (3) calculatingthe NC data required to drive the 3-axis CMM during the inspection process, and (4) minimizing the sumof the squared errors by establishing the best-fit coordinate frame. The validity of the proposed approachis demonstrated by inspecting a machined cam on a CMM using the NC data derived by the developedalgorithm.

2. PROFILE DESIGN OF TRANSLATING CAM

In conventional cam systems, the cam is installed directly on the main shaft. However, in the case ofspace constraints, the cam mechanism is driven remotely by a rotating shaft and crank mechanism. Figure 1presents a schematic illustration of the offset slider-crank and translating roller-follower considered in thepresent study.

Fig. 1. Geometry of offset slider-crank mechanism with translating roller-follower.

To inspect the profile of a manufactured cam on a CMM, it is first necessary to determine the profile ofthe designed cam mathematically in accordance with conjugate surface theory [12]. To synthesize the camprofile, the links of the cam mechanism are numbered sequentially, starting with the slider cam (markedas "0" in Fig. 1) and finishing with the translating roller-follower (marked as "4" in Fig. 1). Once frame(xyz)i(i = 0, .,4) has been assigned to link i in accordance with the Denavit-Hartenberg (D-H) notation [13],the kinematic parameters of the slider-crank/roller-follower mechanism can be tabulated, as shown in Ta-ble 1.

Link bi θi ai αi

1 0 θ1 a1 00 2 0 −θ2 −a2 00 3 0 −θ3 a3 −900 4 b4 0 0 900

Table 1

Table 1. Kinematic parameters of slider crank mechanism with meshing translating roller-followers.


The configuration of frame (xyz)4 with respect to frame (xyz)0 is given by:

0A4 =4

∏i=1

i−1Ai =

1 0 0 m0 1 0 b4− e0 0 1 00 0 0 1

. (1)

Note that b4 = b4(θ3) specifies the input-output relationship of the mechanism. In Fig. 1, parameter b40denotes the initial position (i.e. the low dwell position) of the roller-follower, while b4a is the displacementof the roller-follower in the vertical direction. The configuration of frame (xyz)r (embedded in the roller)with respect to frame (xyz)4 can be expressed as:

4Ar =

1 0 0 00 1 0 00 0 1 00 0 0 1

. (2)

Meanwhile, the configuration of frame (xyz)r with respect to frame (xyz)0 can be expressed as:

0Ar =0A4

4Ar =

1 0 0 m0 1 0 b4− e0 0 1 00 0 0 1

. (3)

Fig. 2. Conical roller location with respect to translating roller-follower.

In Fig. 2, the surface equation, rS, and unit outward normal, rn, of the meshing element of the roller canbe expressed with respect to frame(xyx)r as follows:

rS =[

rCθ rSθ u 1]T

(−δ/2≤ u≤ δ

/2,0≤ θ ≤ 2π) , (4)

rn =[

Cθ Sθ 0 0]T

, (5)

where θ is the polar angle , r is the radius and δ is the width of the roller.Once the input-output relation of the mechanism is given, the conjugate points and complete cam profile

can be determined from the meshing element of the roller in accordance with:

0nT • d0Sdt

= (0Arrn)T d(0Ar

rS)dt

= 0 , (6)


where 0n and 0S are the unit outward normal and surface equation with respect to frame (xyz)0, respectively.Imposing the condition implied in Eq. (6) on Eqs. (4) and (5), the conjugate points (denoted as θ) can be

expressed as:

θ =− tan−1(dm/

dtdb4/

dt) , (7)

where dmdt = (a2Sθ3 +

(a2Sθ3−e)a2Cθ3√a2

1−(a2Sθ3−e)2)w3, and db4

dt = db4adt is the velocity of the roller-follower.

The cam profile and its normal vectors can be obtained by substituting Eq. (7) into Eqs. (4) and (5), andthen transforming rS and rn to frame (xyz)0, i.e.:

0S =[

0Sx0Sy

0Sz 1]T

=[

rCθ +m rSθ +b4− e u 1]T

, (8)

0n =[

0nx0ny

0nz 0]T

=[

Cθ Sθ 0 0]T

. (9)

To determine the conjugate points, it is necessary to derive the kinematic model of the translating cam.Readers who are interested in the methodology for the kinematic model are referred to Hsieh [14] for detaileddescriptions.

3. NC VALUES OF PROBE LOCATIONS

CMM machines are off-line metrology instruments. In other words, the cam has to be physically movedfrom the machine tool to the CMM in order to evaluate its dimensional accuracy. Consequently, frame(xyz)0, originally defined on the machine tool, disappears and must be re-determined on the CMM. Havingmounted the cam on the CMM, the original cam frame, (xyz)0, can be determined from surface A andcircle B (see Fig. 3). A minimum of two hits as far apart as possible are required on surface A in order toobtain a datum line for the x-axis. Meanwhile, a minimum of 3 hits on the circle B are required to establishthe datum point of the cam.

Figure 3 presents a schematic illustration showing the profile measurement of a translating cam usinga touch-trigger probe. When inspecting the cam profile using a CMM device, three probe positions arerequired, namely the preparatory point, the approach point and the target point, where all three points liealong the normal vector to the measurement point of interest, Ni. The probe is moved along the normal vectoruntil it contacts the cam surface at point Ni. At the moment the probe is triggered, the CMM controllerrecords the NC values corresponding to the position of the probe center (denoted as X , Y and Z). In theinspection process, the NC values required to drive the probe to the required measurement position can beobtained from the design point coordinates and a knowledge of the normal distance between the design pointand the set point. In other words, the required probe coordinates can be expressed as:

X = rCθ +m+λ0nx , (10)

Y = rSθ +b4− e+λ0ny , (11)

where λ is the distance between the design point and the set point along the normal vector.To obtain the measured coordinate, Pi, of the machined cam profile, it is necessary to compensate for the

probe radius. In other words, the coordinates of the measured point with respect to cam frame (xyz)0′ areobtained as:

X = X−R0nx , (12)

Y = Y −R0ny , (13)

where R is the radius of the probe.


Fig. 3. Measurement of translating cam profile with touch-trigger probe.

The dimensional errors of the machined cam can be evaluated by comparing the measured coordinate, Pi,with the designed coordinates given by Eq. (8) provided that the evaluation frame (xyz)0′ is aligned with theideal cam frame (xyz)0. To minimize the effects of any misalignment between the two frames, the best-fitprocess described in Section 4 is proposed.

4. BEST-FIT PROCESS

In practice a certain degree of misalignment inevitably exists between the ideal coordinate system (xyz)0and the actual evaluation frame (xyz)0 when the cam is mounted on the CMM for inspection purposes (seeFig. 4). To account for the effects of this misalignment when evaluating the dimensional accuracy of thecam profile, it is necessary to determine the relative configuration of the two frames. In accordance withcoordinate transformation, the configuration of frame (xyz)0 relative to frame (xyz)0 is given by:

0A0 =

Cβ −Sβ txSβ Cβ ty0 0 1

. (14)

Denoting the ideal cam surface as S, the relationship between the actual measured points and the idealsurface is expressed as:

Pi = Si + εi , (15)

where Pi represents the ith measured point, Si is the nearest point on S to Pi, and εi is the geometric error ofthe feature evaluated in the surface normal direction at point Si. The least-squares best fit of the measureddata can then be obtained by minimizing the sum of the squared errors of the measured coordinates from theideal surface with respect to the feature parameters, i.e.:

F =j

∑i=0

ε2i =

j

∑i=0|Pi−Si|2 , (16)

where F is the objective function to be minimized, j is the total number of measured points. When perform-ing computer-aided measurement, a nominal geometry is available from either a CAD model or a blueprint.


Fig. 4. Evaluation frame (xyz)0 and cam frame (xyz)0.

In measuring the cam profile on the CMM, the relationship between the nominal surface and the ideal surfacecan be expressed as:

Ni = ASi , (17)

where Ni is the position vector on the nominal surface as viewed from the local coordinate frame of thenominal geometry and Si is the corresponding point as seen from the machine coordinate frame.

In practice, it is meaningless to evaluate the dimensional error of the cam profile by comparing the coor-dinates of target point Ni with those of measured point Pi since the error between the two sets of coordinatesis dependent to a large extent on the misalignment between the two coordinate frames. A better evaluationof the dimensional error can be obtained by comparing the coordinates of measured point Pi with those ofthe nearest point Si located on the cam profile. In this case, the geometric error can be defined as:

εi = Pi−A−1Ni . (18)

Therefore, the objective function which is to be minimized becomes:

F =j

∑i=0

ε2i =

j

∑i=0

∣∣Pi−A−1Ni∣∣2 . (19)

Rather than transforming the nominal surface to obtain the best fit to the measurement data, the measurementdata can be inversely transformed to obtain the best fit to the nominal surface. As a result, the least-squaresbest fit is obtained by minimizing the sum of the squared errors between the measured coordinates and theideal surface with respect to parameters tx, ty and β , i.e.:

F(tx, ty,β ) =j

∑i=0

ε2i =

j

∑i=0|APi−Ni|2 . (20)

In order to determine the optimal solution to Eq. (20), the gradients of the objective function F with respectto parameters tx, ty, and β are set to zero. In minimizing Eq. (20), the nearest point Si is determined bysearching the neighboring region of target point Ni. Consequently, solving Eq. (20) results in three nonlinearequations which need to be solved via an iterative scheme. In each iteration of the solution procedure, thedistance from each measured point to the nominal surface is reduced by determining the nearest point, and


the squared sum of the distances between corresponding points in the two point sets is minimized by best-fitting the two-point set. As a result, the solution procedure guarantees convergence toward the optimalsolution.

5. IMPLEMENTATION

To verify the methodology developed in this study, a translating cam with parameters a1 = 100 mm,a2 = 60 mm, a3 = 180 mm, e = 15 mm, b40 = 58 mm, p2 = 300, p3 = 1500, h = b4a = 50 mm, δ = 10 mmand r = 8 mm was machined on a 3-axis vertical machine tool. The input-output relation of the rotatingcrank/sliding cam was defined in accordance with the following modified sine acceleration motion curve:

b4a(θ3) =

h[ π

4+π

θ3−θdτ− 1

4(4+π)S(4πθ3−θd

τ)],0≤ θ3−θd ≤ τ

8

h[ 24+π

+ π

4+π

θ3−θdτ− 9

4(4+π)S(4π

3θ3−θd

τ+ π

3 )],τ

8 ≤ θ3−θd ≤ 7τ

8

h[ 44+π

+ π

4+π

θ3−θdτ− 1

4(4+π)S(4πθ3−θd

τ)], 7τ

8 ≤ θ3−θd ≤ τ

. (21)

The cam was then moved off the machining center and mounted on a Mitutoyo BHN-710 CMM (see Fig. 5).The evaluation frame, (xyz)0, was determined manually according to the statement in Section 3. During theinspection process, the cam profile was evaluated at 21 pre-defined points along the cam profile. The normal

Fig. 5. Photograph showing cam profile inspection on CMM.

vectors of the 21 points were determined from Eq. (9) and are shown as Fig. 6. The NC data required by theCMM to evaluate the dimensional accuracy of the cam were obtained from Eqs.(12) and (13) and are shownas Fig. 7. As discussed in Section 3, the coordinates of the measured points were determined from Eqs. (14)and (15). The best-fit process was then applied to derive the dimensional error vector. The transformationmatrix 0A0 was found to be:

0A0 =

0.99999999 −0.00007495 0.002053630.00007495 0.99999999 −0.00081360

0 0 1

, (22)


while F reduced from 0.00296545 mm 2 to 0.00155287 mm2. Table 2 shows the design coordinates, mea-sured coordinates and nearest coordinates for the 21 pre-defined positions on the cam surface. The dimen-sional errors of the cam profile in the x- and y-directions are illustrated graphically in Fig. 8. The errorbetween the measured data and the design data is most likely the result of volumetric errors and vibration ofthe machine tool during the machining process.

Fig. 6. Normal vectors of translating cam.

Fig. 7. Probe path when measuring translating cam.


Designed coordinates Measured coordinates Nearest coordinates

xi (mm) yi (mm) xi (mm) yi (mm) xi (mm) yi (mm)

20.705 50 20.7056 50.007 20.7067 49.9993

22.105 50 22.1053 50.006 22.1067 49.9992

26.183 50 26.182 50.007 26.1847 49.9989

32.902 50.042 32.905 50.049 32.9037 50.0403

42.606 51.084 42.609 51.08 42.6078 51.0816

53.67 54.036 53.678 54.028 53.672 54.0328

65.377 58.76 65.386 58.755 65.3793 58.7559

77.224 64.904 77.23 64.883 77.2268 64.899

88.7 71.97 88.709 71.954 88.7033 71.9642

99.358 79.343 99.362 79.328 99.3619 79.3364

108.859 86.354 108.877 86.342 108.8634 86.3467

116.974 92.347 116.986 92.341 116.9789 92.339

123.522 96.752 123.533 96.742 123.5272 96.7436

128.244 99.205 128.246 99.197 128.2494 99.1962

131.012 99.932 131.014 99.924 131.0174 99.923

134.163 100 134.162 99.995 134.1684 99.9908

137.325 100 137.324 99.986 137.3304 99.9905

139.75 100 139.749 99.99 139.7554 99.9903

141.487 100 141.486 99.99 141.4924 99.9902

142.551 100 142.55 99.995 142.5564 99.9901

142.919 100 142.918 99.999 142.9244 99.9901

Table 2 Table 2. Comparison of designed, measured and nearest coordinates at designated points on cam surface.


Fig. 8. Dimensional errors of machined cam in x- and y-directions.

6. CONCLUSIONS

This study has presented a generalized kinematic model for generating the NC data required by a CMMto carry out the automatic measurement of a translating cam. In developing the model, three key issueshave been addressed, namely (1) developing an analytical expression for the cam profile; (2) determiningthe probe location matrix; and (3) accounting for the measurement error resulting from a misalignment ofthe evaluation frame relative to the ideal frame when mounting the cam on the CMM. The validity of theproposed approach has been demonstrated by inspecting a machined cam on a 3-axis CMM in accordancewith the NC data generated by the proposed algorithm. The method presented in this study integrates thecam design, manufacturing and inspection activities, and therefore makes possible a versatile, automatic,cost-efficient and controllable production process.

ACKNOWLEDGEMENTS

The author gratefully acknowledges the financial support provided to this study by the National ScienceCouncil of Taiwan under Grant No. NSC100-2221-E-269-009.

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dimensional inspection of translating cams by cnc coordinate

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