COORDINATE METROLOGY
Contents
1. compute 5 ten Al shank ER Figure 2 Touch Trigger Probe Components Pa a Pa JO l ye pS es a e a gt N C3 I Ro N 7 Y Ne ON e o wg oe Ya Sen ay C IN 9 J Figure 3 Orthogonal Least Squares Planar Circle Parameters page 7 of 7 Manufacturing Engineering 3M022. y 1 740 x 9 400 y 3 420 x 9 850 y 1 740 x 9 400 y 3 420 1 Use these points with the supplied cir2dm m function file to fit the circle centre and radius Include the results in your submitted laboratory report Comment Are the fitted circle parameters correct For cases in which the sample data is not evenly distributed around the circle circumference an alternative initial parameter estimate can be obtained by using the linear approximation 1 E amp RN o 5 E OIT 03 r 12 z e 2r Manufacturing Engineering 3M02 page 4 of 7 Coordinate Metrology for small form errors The original and the linearly approximated problem have nearly identical solutions for x y 7 From 12 the sum of squares is written as E y El i 1 n SG y 02 x2 y2 2x 4 29 y 13 i l n Es w 2x x 2y y i l where z x y and w r x y The linear approximation minimum is found by simultaneously solving 0 0 0 14 an 14 In matrix form n n n is ae ti ini i int i n 2 n A D x ii Lo BEZA 15 2 n n ai goes ii c 0 ay ii After the value for w has been determined the initial radius is calculated from 1 2 of ca Yeo w 16 As before iterative improvement using the cir2dm m function is then carried out to minimize the original problem of minimizing the sum of squares Use this modified approach to recalculate the circle centre position and radius
3. for the points listed in Include the results in your submitted laboratory report Comment The OLS best fit sphere used for probe calibration is determined by extending the above concepts to three dimensions CMM Scale Uncertainty As stated earlier the resolution of the teaching CMM used in this experiment is 0 010 mm Hence even under perfect conditions sample points are reported as X 0 005 Y 0 005 mm Use this information to calculate the uncertainty of the circle centre position and radius for the evenly distributed data points that you collected Repeat for the points listed in Include your calculations in your submitted laboratory report page 5 of 7 Manufacturing Engineering 3M02 Coordinate Metrology In real applications thermal expansion axis lack of squareness and other geometrical errors will further increase the sample point uncertainty References 1 G T Anthony and M G Cox Reliable Algorithms for Roundness Assessment to BS3730 M G Cox and G N Peggs editors Software for Coordinate Measuring Machines pp 30 37 National Physics Laboratory Teddington UK 1986 2 The Omni Tech Group OMNI 3D User s Manual Fenton ML USA 1998 z E i Ne C Figure 1 Coordinate Measuring Machine Components Legend 1 touch trigger probe 2 motor and scale wiring 3 DCC teach pendant 4 operator interface computer Manufacturing Engineering 3M02 page 6 of 7 Coordinate Metrology
4. Manufacturing Engineering 3M02 Laboratory Coordinate Metrology Introduction For interchangeable assembly each component part must satisfy dimensional tolerance requirements Because of its versatility the Coordinate Measuring Machine CMM is widely used to inspect parts A number of discrete dimensional data points are collected for each feature plane cylinder sphere circle etc on the part Mathematical algorithms are then used to calculate the substitute feature parameters that best fit the collected points This subject is referred to as coordinate metrology In this laboratory you will be exposed to basic CMM operation and to the coordinate metrology mathematics used for planar circle orthogonal least squares data fitting CMM Basics A typical CMM consists of three mutually orthogonal axes a dimensional data point collection probe and a computer based operator interface Figure 1 Each axis is equipped with an electronic scale system that reports the axis position back to the computer For Direct Computer Controlled DCC CMMs motors are used to move the axes The CMM used for this laboratory is moved manually by the operator The axis scale resolution is 0 010 mm Most commonly a touch trigger probe is used for dimensional data point collection This device is a normally closed electronic switch that opens when its stylus tip contacts the part feature surface The switch state is monitored by the computer and when it opens the
5. d the four form errors e in your submitted laboratory report Understanding Least Squares Data Fitting When the part surface is very smooth and form errors are small the orthogonal least squares OLS method provides a simple method to obtain substitute feature parameters To reduce floating point roundoff all coordinates are first transformed by their mean value That is if the n collected sample points are denoted by X i 1 n then the mean is calculated as Bee fae XY XY 1 i 1 The transformed points denoted by x y i 1 n are calculated as 4 59 XY X Y 2 Manufacturing Engineering 3M02 page 2 of 7 Coordinate Metrology Unless otherwise specified the transformed points are used hereafter For a planar circle the form error or residual for point 7 is defined as Figure 3 1 2 03 3 where x y is the substitute circle centre location and r is the substitute circle radius The OLS method seeks to determine values for x y and r that minimizes the sum of squares E Ye 4 Because of the square root term in the problem is non linear and must be solved iteratively from an initial approximation X 9 Y 9 If the sample points are evenly distributed around the circle circumference a practical approximation for the circle centre is the mean Xo X A n C 1 il 1 n 1S y 5 nj 1 Yeo gt Y gt loo So O y eal ik i l At
6. e errors the probe approach direction should be opposite to the outward pointing part feature surface normal vector Circular Part Inspection Using the calibrated probe a part feature can now be inspected Proceed as follows 1 Place the supplied wheel bearing outer cone on the CMM table fixture with the cone opening upwards in the Z direction For convenience it is suggested that you place the part near to the interface computer 2 Start the OMNI 3D operator interface software on the computer Move each axis to its extreme negative value and press lt Enter gt on the computer to re zero the axes scales 3 Calibrate the probe diameter 4 Using menu choice Measure Point collect four points on the outside circular ring of the bearing Position the centre of the stylus sphere approximately 3 mm below the top for best results Collect one point at each quadrant of the circular ring Approach along the surface normal in the X Y X and Y directions Check that the software displays the correct compensation direction for each collected point Record the four points in your submitted laboratory report Each student must obtain his her own data points for this portion of the laboratory Note These points have already been compensated for the probe diameter 5 Using menu choice Measure Circle collect four points again at to the best of your ability the same locations Record the displayed circle centre position diameter an
7. instantaneous axes positions are recorded The operator interface program allows substitute feature data fitting calculation of distance and orientation between features etc For DCC machines a CAD CAM system can be used to offline program the CMM for unattended operation Probe Calibration For touch trigger probes a stylus deflects to trigger the contact switch Figure 2 At the end of the stylus a spherical tip larger in diameter than the stylus shaft is attached This permits approach to the part feature surface from many directions with confidence that the stylus tip and not the shaft will contact the part first Because of differences in switch spring stiffness and cantilever bending of the stylus shaft the effective diameter of the tip must be experimentally determined Normally this is accomplished with the aid of a reference sphere with a known radius r Five points are collected on the sphere four equally spaced around the equator and one at the north pole Without applying any compensation a data fitting algorithm calculates the measured reference sphere radius r The effective probe radius is then calculated as r r r The TA will demonstrate this calibration procedure during the laboratory period page 1 of 7 Manufacturing Engineering 3M02 Coordinate Metrology After calibration the effective probe radius is added as a vector along the direction of probe approach to the part feature surface To minimize cosin
8. the minimum of JE JE JE 0 0 0 z dx dy dr n To improve the estimate at step k the second partial derivatives are also calculated and the matrix equation SE OE k ee OX DY y k 7 E E h ck EGON oy T i a is solved for CA Sa The improved centre position estimate is 8B WE Yerri Yek ck page 3 of 7 Manufacturing Engineering 3M02 Coordinate Metrology where 0 lt lt 1 is a scaling factor Start with 1 0 1 If the sum of squares E 4 lincreases reduce A until E decreases after the iteration The improved radius estimate is 1X 1 2 Foky 6 Xe O Aa 9 i l This iterative improvement procedure is repeated until a user defined convergence condition is satisfied For this application a reasonable condition is 0010 p gt 4 07 10 A MATLAB function file named cir2dm m that implements this algorithm will be provided by the TA Use it to calculate the fitted circle centre and radius for your point data Compare the MATLAB result with the OMNI 3D result and comment Include this information in your submitted laboratory report Unevenly Distributed Sample Points A full circle circumference is not always available or accessible for sample data point collection However it still may be desireable to attempt to fit a substitute feature For example consider a circle centred at x 0 y 0 with radius r 10 mm Four sample points are collected at the locations x 9 850
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