Quality Control (Exam 2001, Question 1)

Quality Control (Exam 2001, Question 1)

Usage notes

a) i)

The following part is manufactured.

The part can be assumed to be a square with side length of 10 cm with an angle cut off at 45 degrees, distant 3 cm from the upper right corner. All these values are exact!

The hole is manufactured using a perfectly circular drill. The position (x,y) of its center is not known. Also the radius r of the drill is not known (since it wears out with use).

A tool can measure the thickness of the part at the endpoints of a, b, c, d and e approximately. This tool always measures the thinnest thickness. Assume that the measurement errors are all normally distributed with a standard deviation of

= 0.1 mm.

The objective of this exercise is to write the problem as an equation system, to solve it in the least squares sense and to do the confidence analysis.

How many measured values are there in the problem?
___________________

Which are the measured values?
___________________

How many unknowns are there in the problem?
___________________

Which are the unknowns?
___________________

To how many equations does the problem lead?
___________________

Now we will derive the equations from the geometry of the problem. Some of them can be written down directly. Give these simple equations. Use '=', even though the identities will be only approximate.
There will be 5 equations at the end, but for now we only consider the 4 'simple' equations.
___________________
___________________
___________________
___________________

There is a special form of a straight line equation which is suited to find the distance between a straight line and a point. Give the name of this form.
___________________

Give the general formula for the distance between a point (x,y) and a straight line in the above mentioned form.
distance = ___________________

This will now be applied to our problem. Give the equation of the straight line through (10,7) and (7,10) (i.e. the oblique edge) in the above form. Again use (u,v) as coordinates, as (x,y) are already used for the position of the drill.
___________________ = 0

Give the distance between the position (x,y) of the drill and the oblique edge.
___________________

Now give the remaining equation. Again use = even though the identities will be approximate.
___________________

b)

Which of the following properties does the equation system fullfil:

nonlinear

overdetermined

underdetermined

differential

well-determined

linear

Give now the matrix A and the vector c in Maple5 syntax ( Matrix(2,2,[[1,2],[3,4]]) and Vector([1,2,3,4]) )
A = ___________________
c = ___________________

Set up the equation system (in matrix form) which takes into account the measurement error. Use

Let us call S the sum of squares. Give an expression for S.

What is the appropriate method to compute an estimate for the center of the hole and its radius?

Brent's golden section search

Newton's method

Normal equations

Spectral method

Give an equivalent but simpler equation system.

c)

How is S (the sum of squares) distributed? Write the distribution in Maple form. To get a choice type "?distribution" in Maple.
___________________

We set the confidence level to 98%. Explain with a formula what this means. Use P[...] to denote the probability. Do not use the character "%".
___________________

Change the formula using the concrete distribution of S. Use Maple notation and assume that the package "stats" is already available. From now on use S_max as upper bound.
___________________

Give the Maple command to compute S_max.
___________________

The result is S_max = 13.388...

For a given measurement, consider the set of all values satisfying this upper bound. It is called "confidence region". This solid is an ...

exact 3-dimensional ellipse

approximate 3-dimensional ellipsoid

approximate 3-dimensional ellipse

exact 2-dimensional ellipse

approximate 5-dimensional ellipsoid

exact 3-dimensional ellipsoid

exact 5-dimensional ellipsoid

Bonus Level:

To illustrate what we did, you may now compute a concrete numerical example of your choice. Enter the measured values in cm (A typical measurement would be a=2.49, b=2.51, c= 2.45, d=2.49 and e=2.51.):

a = ___________________ b = ___________________ c = ___________________ d = ___________________ e = ___________________

The values are:
x_star = y_star = r_star = S_star =

The Maple code used to compute this was:

with(LinearAlgebra):
qc:=proc(a,b,c,d,e)
   local AA,cc,zz,sigma,pp_star,S_star;
   AA:=<<1|0|-1>,<0|-1|-1>,<-1/sqrt(2)|-1/sqrt(2)|-1>,<-1|0|-1>,<0|1|-1>>:
   cc:=<0,10,17/sqrt(2),10,0>:
   zz:=:
   sigma:=0.01;
   pp_star:=LeastSquares(AA,zz-cc);
   S_star:=evalf(Norm((AA.pp_star)+cc-zz),2)/sigma)^2);
   RETURN(pp_star,S_star);
end:

The picture below shows the 98% confidence region:

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