Improving Measurement and Quality of Products: A Case Study

This assignment discusses the ways to improve measurement and quality of products in a company. It includes recommendations for using advanced co-ordinate measuring machines and selecting appropriate probes. It also presents a case study on developing a least square estimated model in MATLAB for a ring gauge measurement. The assignment provides insights into the use of emerging technologies and techniques for quality improvement.

Running head: ASSIGNMENT
ASSIGNMENT
Name of the Student
Name of the University
Author Note

1ASSIGNMENT
Introduction:
In this particular assignment two tasks are completed to for a company one of which
is providing suitable recommendation for improving measurement and quality of product of
the company. The emerging technologies that are used as measuring techniques are discussed
in the first task with detail description. In the second task a least square estimated model is
developed in MATLAB to find the best fit to a sample ring gauge co-ordinates measurement
data. The Newton’s method of non-linear least square regression is used with standard
circular model function. The parameters of the fitted circle is obtained from the Newton
method of non-linear least square fitting and then from those parameters a fitted circle is
obtained compared with the test points of ring gauge by a plot.
Task 1:
Given, that the maximum level of dimension of any product produced by the company
is 400 mm. The primary products of the company are turned and milled components having
substantial features with different batch sizes of over thousands of varieties. The accepted
tolerance for the dimensions is 50 μm but for some specific high precision products the
tolerance level is as low as 5 microns. The range of the surface roughness is between the
range 0.025 to 12.5 μm.
The measuring quality can be improved by employing most advanced co-ordinate measuring
machines or CMMs based on the area of appliance. The bridge type CMM is best for
measuring medium size components that require high accuracy. The cantilever CMM is best
for the small component measurement where highest accuracy is required. The horizontal
arm CMM is useful for the measurement of large component machines where the accuracy
tolerance is high enough (Hocken and Pereira 2016). The Gantry type CMMs are useful for

2ASSIGNMENT
large component measurement where the tolerance is low or the high accuracy measurement
is very much required.
Furthermore, the measurement and the quality of the product can be improved by selecting
appropriate probe for the CMMs. The probes can be explicitly divided by contact probes
which measure by touching the products or components and non-contact probes that measure
the jobs or product by laser technology or by using computer vision. There exist some multi-
sensor probes which mainly works by touching and by the method of optical scanning. The
CMM contact probes have two types namely touch trigger probes and Analog scanning touch
probe. In the touch trigger probes inside the probe housing a stylus is attached to bearing
plate with which some pressure sensors are attached. An electrical signal is generated every
time the contact is made with the job. The probe can be manually rotated or automatically
rotated by the moving axis of the CMM and many of the stylus tips and attachments are
accommodated with it. Touch probes are used in many industries for its versatile uses and the
flexibility. The piezoelectric sensors installed in the probes eliminated the error introduced
for bending of stylus and with improvement in the strain gauge technology it is ensured that
the triggering of probes occur at a constant force irrespective of the angle of contact with the
work-pieces. Thus the directional sensitivity is eliminated and accuracy in terms of microns is
achieved. The Analog scanning probes are mainly used for measuring contoured surfaces like
assemblies of metal sheets (Li et al. 2016). Instead of touching the individual points the
probes are remained at contact with the work-piece as it dragged thorough and thus Analog
measurement can be employed. The data acquisition is improved in the Analog measurement
probes and in some continuous Analog scanning probes the data acquisition is continuous
which better that the point to point data acquisition. This types of probes are employed with
CMMs to measure complex structures like turbine blades, crankshafts, bodies of automobiles.
The continuous Analog scanning probes can be of two types namely closed loop and open

3ASSIGNMENT
loop system. In closed loop systems the changes in the direction of surface of the job is being
automatically detected by the probe and the probe adjusts itself for maintaining contact (Li et
al. 2016). In case of open loop system the probe is controlled by an input data file which
contains information about the dimensions of the job. The closed loop probes are needed for
unknown complex structures whereas the open loop systems are useful for jobs with regular
shapes. The operation time of open loop system is faster than the closed loop systems. The
non-contact probes are particularly categorized as laser probes and the machine vision
probes. The working of the laser probe is similar to the working of the touch-trigger probe,
however, the only difference is that laser probes use a concentrated laser beam in place of
stylus. When the laser beam is incident on a part the position of the part can be read through
triangulation in lens inside the receptor of probe. The technique is similar to the positions
finding of surveyors with the help of bearings where the distance between two fixed points is
known. In case of the computer vision based probes an archetype is digitized electronically
for generating exact dimensions of future jobs (Sładek 2016). An HD camera is then
employed for generating large amount of measurement points inside a frame and then the
features are compared with the electronic model by counting down the pixel values. In
machine vision the greatest advantage is that only one calibration is required for the lens
whereas in other types of probes many recalibrations are required.
In recent days for improved measurement and qualities the companies are employing CMMs
which can be controlled by software. This software can provide CAD/CAM or Solidworks
models of work-pieces and CMMs can measure work-pieces according to those model.
Finding the uncertainty with the measurement is one of the vital rule for quality improvement
as it is practically impossible to know true exact dimensions of a part. The ways for
measurement of uncertainty involves taking measurement of machine several times and then
applying statistical modelling techniques like Monte-Carlo simulation.

4ASSIGNMENT
Task 2:
A least square algorithm is developed in MATLAB which calculates the best fitted
model of the sample XY data of measurements of a ring gauge. The measurement results are
shown in the following table.
X(in mm) Y(in mm)
84.047 5.038
73.303 45.008
44.041 74.326
4.015 85.021
-35.996 74.319
-65.25 45.033
-75.972 5.038
-65.241 -34.997
-35.979 -64.263
4.025 -74.969
44.029 -64.241
73.330 -34.975
84.006 5.006
The ring gauge is circular and hence the nonlinear function is assumed to be a function of
circle which is given by,

5ASSIGNMENT
(X – a)^2 + (Y – b)^2 = c^2. (1)
The best matching a, b and c are required to be found which minimizes the least square
between the predict model values of y and the actual model values of y.
In Newton method the zero of the fitted objective function is found starting from an initial
guess. The iteration equation of the Newton method is given below (Bethea 2018).
x(k+1) = x(k) - f(x(k))/f’(x(k))
Hence, k is the iteration number and f(x) is the objective function. Here, the objective
function is the sum of square of error and this is minimized using Newton’s method with
finite number of iterations and the values of the parameters for three variables namely circle
radius, circle x co-ordinate and circle y-coordinate is obtained (Xing et al. 2018).
The uncertainty associated with the x and y co-ordinates is 30 μm. The measurement
uncertainties about the radius of the circle will be
Uncertainty (radius) = sqrt(30^2 + 30^2) = 42.4264 μm.
Uncertainty with the centre positions will be 30 μm for each as the distance is linear from the
locus points of circle to centre.
MATLAB code for Non-linear least square model fit:
xdata = [84.047 73.303 44.041 4.015 -35.996 -65.250 -75.972 -65.241 -35.979 4.025 44.029
73.330 84.006];
ydata = [5.038 45.008 74.326 85.021 74.319 45.003 5.038 -34.997 -64.263 -74.969 -64.241 -
34.975 5.006];
XY = [xdata' ydata'];
Par = CircleFitting(XY);

6ASSIGNMENT
sprintf('The center co-ordinate of the fitted circle with least sum of square is [%.4f,%.4f] and
the radius of the fitted circle is %.4f',Par(1),Par(2),Par(3))
theta = 0:0.1:2*pi;
xfit = Par(3).*cos(theta) - Par(1);
yfit = Par(3).*sin(theta) - Par(2);
plot(xdata,ydata,'ro',xfit,yfit,'bo-')
title('Sample points and fitted points')
legend('Oiginal gauge co-ordinates','fitted co-ordinates by Nonlinear least
square','Location','best')
xlabel('x co-ordinates')
ylabel('y co-orninates')
function Params = CircleFitting(XY)
points = size(XY,1); % number of data points
average = mean(XY); % calaculating average of data
% moment computation
Matxx=0; Matyy=0; Matxy=0; Matxz=0; Matyz=0; Matzz=0; % initializing with zeros
for i=1:points
Xi = XY(i,1) - average(1);
Yi = XY(i,2) - average(2);

7ASSIGNMENT
Zi = Xi*Xi + Yi*Yi;
Matxy = Matxy + Xi*Yi;
Matxx = Matxx + Xi*Xi;
Matyy = Matyy + Yi*Yi;
Matxz = Matxz + Xi*Zi;
Matyz = Matyz + Yi*Zi;
Matzz = Matzz + Zi*Zi;
end
Matxx = Matxx/points;
Matyy = Matyy/points;
Matxy = Matxy/points;
Matxz = Matxz/points;
Matyz = Matyz/points;
Matzz = Matzz/points;
% fitting polynomial coeffcient calculation
Mz = Matxx + Matyy;
Cov_xy = Matxx*Matyy - Matxy*Matxy;
Mxz2 = Matxz*Matxz;
Myz2 = Matyz*Matyz;

8ASSIGNMENT
A2 = 4*Cov_xy - 3*Mz*Mz - Matzz;
A1 = Matzz*Mz + 4*Cov_xy*Mz - Mxz2 - Myz2 - Mz*Mz*Mz;
A0 = Mxz2*Matyy + Myz2*Matxx - Matzz*Cov_xy - 2*Matxz*Matyz*Matxy +
Mz*Mz*Cov_xy;
A22 = A2 + A2;
epsilon=1e-12;
ynew=1e+20;
IterMax=30;
xnew = 0;
% Newton's method starting at x=0
for iter=1:IterMax
yold = ynew;
ynew = A0 + xnew*(A1 + xnew*(A2 + 4.*xnew*xnew));
if (abs(ynew)>abs(yold))
disp('Newton algorithm goes wrong direction: |ynew| > |yold|');
xnew = 0;
break;
end
Dy = A1 + xnew*(A22 + 16*xnew*xnew);
xold = xnew;

9ASSIGNMENT
xnew = xold - ynew/Dy;
if (abs((xnew-xold)/xnew) < epsilon), break, end
if (iter >= IterMax)
disp('Newton method does not converge');
xnew = 0;
end
if (xnew<0.)
fprintf(1,'Newton method gives negative root: x=%f\n',xnew);
xnew = 0;
end
end
% computing the circle parameters
DET = xnew*xnew - xnew*Mz + Cov_xy;
Center = [Matxz*(Matyy-xnew)-Matyz*Matxy ,
Matyz*(Matxx-xnew)-Matxz*Matxy]/DET/2;
Params = [Center+average , sqrt(Center*Center'+Mz+2*xnew)]; % obtaining the paramters
of fitted circle
end
Output:

10ASSIGNMENT
'The center co-ordinate of the fitted circle with least sum of square is [4.0280,5.0287] and
the radius of the fitted circle is 80.0012'
-100 -80 -60 -40 -20 0 20 40 60 80 100
x co-ordinates
-100
-80
-60
-40
-20
0
20
40
60
80
100
y co-orninates
Sample points and fitted points
Oiginal gauge co-ordinates
fitted co-ordinates by Nonlinear least square
Discussion:
Thus from the CMMs listed in task 1 it is recommended to use the Gantry type CMMs for
measurement of large turned and milled components of dimension close to 400 mm as with
this type of CMMs very accurate measurement with low tolerance level close to 50 μm to 5
microns can be achieved. It is also recommended to use closed loop Analog scanning probes
to measure surfaces having roughness in the range 0.025 to 12.5 μm as this probe is very
good for measuring contoured surfaces with unusual roughness. The closed loop system will
automatically adjust the probe according to dimension of the surface. Now, the Non-linear
least square technique which is employed in the task 2 for fitting a least square model to the

11ASSIGNMENT
test measurement data of X, Y co-ordinates of the ring gauge is based on Newton algorithm
for minimization or root finding of a function. The initial guesses of the parameters of non-
linear function (here is circle) plays an important role in terms of accuracy of the parameters
and the convergence time of the solution (Zhou, Xu and Gu 2018). The initial guesses for the
circle parameters of the circle are assumed to be zero here, other intuitive guesses can
produce more accurate solution parameters with less convergence time. The plot of sample
points and the fitted circle shows that the ring gauge is not exactly circular shape as the fitted
circle is somehow deviated from test points. The deviation is minimum in terms of non-linear
least square fitting with maximum of 30 iterations.
Conclusion:
In conclusion it can be stated that in this assignment the different types of equipment
and software that are used by industries for improvement of the quality and measurement of
products are listed and the most suitable equipment for the given company is recommended
for different dimensions of parts produced by the company. In the second task a non-linear
least square model is appropriately fitted with the provided sample data of ring gauge using
MATLAB. The objective function assumed here for data is to be circular from the shape of
sample data in XY plane, however, other complex objective function like ellipse or any other
type of improved algorithm like Gauss-Newton method, Levenberg-Marquardt or trust-
region-reflective methods can provide more accurate fitting to the sample data points.

12ASSIGNMENT
References:
Bethea, R.M., 2018. Statistical methods for engineers and scientists. Routledge.
Zhou, D., Xu, P. and Gu, Q., 2018. Stochastic variance-reduced cubic regularized newton
method. arXiv preprint arXiv:1802.04796.
Xing, L., Xiong, Z., Liu, J.Y., Luo, W. and Yue, Y.Z., 2018. Offline calibration for MEMS
gyroscope G-sensitivity error coefficients based on the newton iteration and least square
methods. The Journal of Navigation, 71(2), pp.352-370.
Hocken, R.J. and Pereira, P.H., 2016. Coordinate measuring machines and systems. CRC
press.
Li, R.J., Fan, K.C., Huang, Q.X., Zhou, H., Gong, E.M. and Xiang, M., 2016. A long-stroke
3D contact scanning probe for micro/nano coordinate measuring machine. Precision
Engineering, 43, pp.220-229.
Sładek, J.A., 2016. Coordinate Metrology. Berlin, Heidelberg: Springer Berlin Heidelberg.