ComputerAided_Design_Engineering_amp_Manufactur.pdf

deviation between the nominal CAD data and the converted measurement data. The measured data
�1
can be converted to the nominal CAD coordinate system, by multiplying T1 to the measured data,
MM. Thus, the relationship can be formed as,
(4.21)
The T2 matrix of 4�4 can be determined by the least squares technique, minimizing the sum of
�1 squares of distance between the converted measurement data ( T1 MM)
and the nominal CAD data (D
� rN). The sum, E, is
E � S|T1 (4.22)
The variational principle can be applied to solve the components of T 2 matrix. A convenient method
for the evaluation of T 2 matrix is to use the pseudo-inverse technique for computational efficiency.
Therefore, T 2 matrix can be calculated as follows
T 2
�
(4.23)
where MM is the CMM measurement data for the surface, and ( )T indicates the transposed matrix. The
calculated T 2 matrix is now used to generate the updated CNC codes downloaded into the CMM, and
the second measurement data, MM(2), are obtained. When the generated measurement path still fails
to measure the thin section of parts, further iterative procedures can be introduced.
Based on the second measurement data MM(2), the second iterative T2(2) matrix can be evaluated
in that case. Prior to going to further iterations, ERR, the sum of squares of distance can be used as a
criterion. That is,
where ERR � sum of squares of distance
�1 T1 MM � T2 D�rN �1 MM T2 D�rN 1
T1 ( )
� ( )| 2 for all measurement points
�
MM( D � rN )T[ ( D�rN) ( D�rN)T] � 1
�
� |T 1
ERR £ TOL
�1 ( k)
MM
�
( k)
T2 D � rN
( )|
(4.24)
(4.25)
and is the kth iterative measurement data, and is the kth iterative T2 matrix. When the
criterion is not met, further iterations can be performed, and finer alignment procedures can be followed.
Figure 4.5 shows the flowchart for a typical alignment system using the rough- and fine-phase alignments.
The illustrated alignment technique can be applied to practical manufacturing of turbine blades, for
example. A turbine blade is modeled into four features such as two chord-length-based cubic spline
curves and two very small edge circles of around 0.15 mm radius. After locating on a CMM, the turbine
blade is probed around the reference block by the six points based on the rough alignment procedure.
The transformation matrix T1 of the rough phase alignment is calculated, and initial measurement
procedures are performed on the pressure/suction surfaces. Based on the measurement data, the T2 transformation matrix is calculated using the least squares technique. The sum of squares of distance
are calculated as a criterion for further iterative measurement procedures. Figure 4.6 shows a typical
distance deviation between the nominal CAD data points and the measured points. It is noteworthy
that, in this figure, the sum of squares of distance is greatly reduced at the first iteration, and further
iterations are not needed. Figure 4.7 shows a typical measurement path before and after the fine phase
alignment; the measurement path is slightly changed so as to measure the thin edges after the fine
alignment procedure. 12
MM k ( )
T2 ( k)
2