Instrumentation & Measurement Magazine 26-1 - 44

f pw x
b
b
where p , and w , 2 are unknownbn1ww 
T
bn
pp 
21
displacement and weight coefficient vectors. The main difference
between (4) and (5) is are the dimensions of displacement
and weight coefficient vectors. The unknown displacement
and weight coefficient vectors pub
and wub, pb and wb of the
UBCM and BCM, respectively, could be obtained by numerical
optimization as:
 
 
 
 
P21
n




p ub
wub
ub 
O min eub2
 P n2 41   lu

p b 




wb 
 bbO mine 2

where Oub and Ob are objective functions of the UBCM and
BCM optimization model, respectively. (lub,uub)and (lb
note the design space of design variables P2nx1
respectively. eub=fub
and P(2n-4)x1
-f and eb
discrete form of fub(x), fb

=fb
-f where fub
(lub,uub) and (lb,ub) are defined as:
 

lub   sn 

  




uub
 sn


  


lb

  


ub
a Amp
a Amp
 
 n
a Amp
a Amp
s
s
a Amp
a Amp
s
s
fI 


 
 
 
 
 n
 n
fI
fI
fI 




21
21
21
21
respectively. Inx1 and I(n-2)x1 are column vector, of which all
the elements are 1. Amp(f)=max(f)-min(f) denotes the absolute
amplitude of f. as
is a scaling factor of the design space.
Generally, the design space will expand as as increases and
increase the computing cost of the optimization. When as
is
set as 10, we found that we could achieve a balance between
the computing cost and versatility. A graphic parameter optimization
process of i-th handle point is displayed in Fig.
3. By optimizing the position of pi
, the curve near the i-th
handle point will move towards the target value. Further optimizing
the value of wi
, the radius of the curve near the i-th
handle point will be locally adjusted to approximate the target
curve. In the actual optimization, all design variables will
be optimized simultaneously, achieving a global approximation
of the target data.
44


a Amp
a Amp
sn
sn



fI
fI


fI 


1
1
1
1

 
 

 
 n
(9)
fI
(8)
, fb
,ub) de,
and
f denote the
(x) and f, respectively. The design space


lu,ub ub
 
bb,
(6)
Fig. 3. Schematic diagram of the parameter optimization process.
(7)
Sensor Inherent Nonlinearity
Calibration
Experimental Setup
Experimental platform of the sensor calibration is depicted
in Fig. 4. The calibration object is a micro linear potentiometer
with an inherent nonlinearity. The nominal measurement
range of the potentiometer is 60 mm. A laser displacement sensor
HG-C1200-P is utilized to measure the nonlinear response
of the micro linear potentiometer. The nominal measurement
range, repeatability and linearity of the laser displacement
sensor are 160 mm, 0.2 mm and ±0.2% full scale, respectively.
As the probe of the micro linear potentiometer moves
within the measurement range, it will produce a displacement
response x. Meanwhile, the displacement of the probe is measured
by the laser displacement sensor as f. In the experiment,
x and f are sampled by an ADS1115 module, which is a 16-bit
Analog-to-Digital Converter (ADC).
Sensor Calibration Results
Fig. 5 illustrates the sensor nonlinearity calibrations by the
PCM, UBCM and BCM. The measured x, f and the corresponding
calibration results are displayed in Fig. 5c. Fig. 5a and Fig.
T
b ,, 
(5)
Fig. 4. Linear potentiometer calibration experiments.
IEEE Instrumentation & Measurement Magazine
February 2023
Instrumentation & Measurement Magazine 26-1

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