Instrumentation & Measurement Magazine 25-5 - 19

where C wp w p ,
i ,1 ,1
i
,
i  [ ,, ]iiin i n

T
is the vector composed of the
control points of the i-level NURBS curve.
According to the principle of wavelet decomposition, φi
can be decomposed into φi-1
pressed as:
and ψi-1
  ii 11i
PQ
 is the nodal interpolation matrix, and it can be
calculated by Oslo method [11].
connects the function in the space Vi
where P
nn ii 1

Q nnni () 1ii
 
function in Wi-1
by Wang [12].
expressed as:
     1, ,n k nn
i
11i
,
0 ,
j
i1 , 1, , ii1
  
(5)
Therefore, the i-level NURBS curve can be decomposed
into the sum of i-1 level NURBS curve and B-spline wavelet
curve belonging to space i−1. It can be expressed as:
r u uC u P uQ Ci
i() () ( ()

  () )
T
T TT T
i iii 11

(6)
For some complex surfaces, the distance and arc length of
sampling points are directly related to the characterization
of the surface complexity. The curve is parameterized, and
the complexity of sampling points is comprehensively evaluated
by calculating the arc length and B-wavelet parameters
of the curve. The calculation formula of parameter curve in
11
t tt 
, n
 is:
l t  r u du( )
()

t 1
t
(7)
The approximation difference between continuous curves
can be measured by the value of B-wavelet curve. In this paper,
we use the square of B-wavelet basis function in [9] to measure
the complexity of shape. Since this value is a function on
B-spline curve r(t), Ei
is a matrix that
with the B-spline wavelet
, which can be solved by the method proposed
Using the orthogonal relation between φi-1 and ψi-1
, Q can be
(4)
. This process can be exwhere
l(tn+1
of l(t) and ()
definition curve based on the mixture of these two quantities
is:
c t w wE t
lt 
() 
2 ()


lt
()
 1
le
n
1
()
(10)
) is the sum of arc length of curve, and the values
Et are coded in the range of (0,1) by normalization
algorithm. wl and we are the weights of arc length and energy
function, and the value in this paper is 1.
For the sampling process of a complex curve, the spacing
and density of initial sampling points can be rezoned by fixing
the value of c(t). The complexity of sampling points is positively
correlated with ()
curve is large, the value of ()
Et . When the local complexity of the
Et increases. In (10), the sampling
spacing is adjusted by fixing the coefficient on the left side of
the equation. When the value of l(t) decreases, the sampling
spacing becomes smaller. On the contrary, the distance between
sampling points becomes larger.
Experiment and Results
(t) is defined as the cumulative integral of
arc length and shape measure, which represents the energy of
B-wavelet basis function along the curve. The energy formula
of the curve is:
nn
E t
ijt
j
()    ( ) ( )
t ii
1
 

1
2 t r u du

1
The mean value method is used to express the numerical
relationship between the local energy of the curve and the
overall energy of the curve. The energy mean function is:
n
Et
ET  nE t
()
i 1
i()
in1
()
(9)
where Ei(tn+1) is the sum of all the values of the ith B wavelet
curve.
In conclusion, based on [9], the cumulative arc length and
the function based on B-wavelet can be calculated through
(7)-(9). The complexity relationship of the parameterized
August 2022
(8)
Based on the section line method, data sampling experiments
are carried out on the spherical crowns. The spherical crowns
usually have the characteristics of semi ellipsoid shape. According
to the structural characteristics of the end surface of
the parts to be tested, the parabolic curve is used to approximate
the longitudinal section of the spherical crown. Based
on the abscissa of the parabola, the radius of the measurement
track circle in the sampling plane is planned. The workpiece
to be measured is an equal scaled sample of a large ellipsoidal
workpiece in aerospace fields, and the diameter of the
workpiece is 400 mm. The sampling strategy proposed in this
paper is used to plan the distribution of sampling points on
the parabola.
The non-contact measurement system is adopted for the
data acquisition experiment of the workpiece to be measured.
The profile measurement system mainly includes the serial
manipulator, measurement sensor, sensor installation device
and high-speed analog input module (DMP) of a CNC system,
which is mainly used to complete the data measurement of
each profile. The profile measurement system, the workpiece
and the sampling results are shown in Fig. 2. The non-contact
laser displacement sensor is fixed at the end of the manipulator
and used to measure the profile data. The position of the laser
sensor can be changed by controlling the manipulator, and the
whole surface of the workpiece can be measured by planning
the measurement trajectory.
The measurement trajectory is planned in the reference
plane by using the sampling point planning results in Fig. 2,
and the data simulation is carried out by MATLAB 2018b. The
measuring track of the measuring device consists of k concentric
circles. According to the results in Fig. 2, the number
of concentric circles is set to 20. Taking the uniform sampling
method as the control group, the number of concentric circles
is also set to 20, and the equal radian method is adopted for
the sampling points in each circle of sampling track, which is
120 sampling points, as shown in Fig. 3. At the same time, the
IEEE Instrumentation & Measurement Magazine
19

Instrumentation & Measurement Magazine 25-5

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 25-5

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