Instrumentation & Measurement Magazine 26-1 - 43

and deviates from target values. Therefore, in dealing with
nonlinear problems, it is difficult for the PCM to obtain a good
modeling accuracy even by increasing the model order. This is
a significant drawback of the PCM.
Unconstrained Bezier Calibration Method
(UBCM)
As depicted in Fig. 2a, take into consideration the measurement
range into consideration, the unconstrained Bezier
calibrating model could be established as:
 C wp1 11
n1
i1
f 
Fig. 1. Runge phenomenon of the PCM.
where n, pi
Polynomial Calibration Method (PCM)
According to [20], the polynomial calibration model is described
as:
f x px

n 1
pi
i
 
1
where n and pi denote the model order and unknown parameters.
pi could be obtained by minimizing ep2
as:
Oe
ppmin

where ep=f-fp
2

(2)
(x) is the residual error of the PCM. Currently,
PCM is the most commonly used method, since it has a concise
form, as shown in (1). In dealing with strongly nonlinear
modeling problems, the PCM needs to increase its model order
n to achieve a better approximation. However, Runge
phenomenon will occur in the higher-ordered polynomial calibration
model. As depicted in Fig. 1, due to the increment of
the model order, the polynomial calibration model fluctuates
Bezier Calibration Method (BCM)
As depicted in Fig. 2b, the UBCM is uncoordinated with the
calibrated data at its boundary. Under some circumstances,
sensor calibration may require an accurate approximation at
the two bounds of the calibrated data x. To achieve an accurate
calibration at the boundary of x, the unknown parameters
of the 1st and n-th handle points could be constrained as p1
pn=uf, w1=1 and wn
=lf
ni 1

(1)
where p  1 andwub  ww  are unknown displacement
and weight coefficient vectors.
fx
ub ,,ub ub
pw
T
ub pp,
n


T
1 , n

and wi
rameters of the UBCM. lx, ux and rx=ux-lx
denote the model order and unknown padenote
the lower
bound, upper bound and measurement range of x. Equation
(3) could be rewritten as:
(4)
ub x 
 Cw   rr xx 

n1
i xx
i ni
1
1 11

xl 




xl

  
1


 
1
i xx
n ii
 
 
 rr xx 

  

xl 





1

xl

i11
ni

(3)
i11
ni

,
=1, respectively. In this case, the UBCM becomes
a constrained form, known as the Bezier Calibrating
Method (BCM), as depicted in Fig. 2b. The BCM could be obtained
as:
Fig. 2. (a) Illustration of the UBCM. (b) Illustration of the BCM.
February 2023
IEEE Instrumentation & Measurement Magazine
43

Instrumentation & Measurement Magazine 26-1

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