=Paper=
{{Paper
|id=Vol-2711/paper9
|storemode=property
|title=The Phase Response Values Application of Second-Order Digital Frequency-Dependent Components for Calculating Transfer Function Coefficients in Robotic Systems
|pdfUrl=https://ceur-ws.org/Vol-2711/paper9.pdf
|volume=Vol-2711
|authors=Hanna Ukhina,Ivan Afanasyev,Valerii Sytnikov,Oleg Streltsov,Pavel Stupen
|dblpUrl=https://dblp.org/rec/conf/icst2/UkhinaASSS20
}}
==The Phase Response Values Application of Second-Order Digital Frequency-Dependent Components for Calculating Transfer Function Coefficients in Robotic Systems==
https://ceur-ws.org/Vol-2711/paper9.pdf
The Phase Response Values Application of Second-Order
Digital Frequency-Dependent Components for
Calculating Transfer Function Coefficients in Robotic
Systems
Hanna Ukhina[0000-0003-3797-1460], Ivan Afanasyev[0000-0002-3207-5741X],
Valerii Sytnikov[0000-0003-3229-5096], Oleg Streltsov[0000-0002-4691-5703],
Pavel Stupen[0000-0003-1952-6144]
Odessa National Polytechnic University, Odessa, Ukraine
ukhinanna@gmail.com, nekaktotak2@gmail.com,
sitnvs@gmail.com, ovstreltsov@gmail.com,
stek2000@gmail.com
Abstract. This study deals with the influence of the second order tunable digital
frequency-dependent components' transfer function coefficients on the phase
response during those components tuning. The obtained approximations of
phase frequency responce dependence on the ripple level allowed us to find the
digital filter transfer function of the denominator coefficients values. It is worth
noting that this expression works at ripple level up to 40 with a standard error of
0,06% for Chebyshev filter-1 and 0.28% for Chebyshev filter-2. Also in these
work we fix the ripple level. With this value, for each cutoff level and ripple
level we determine the phase value and to obtain the three-dimensional graph
approximation. Theoretical studies have been verified experimentally
Keywords: Industry 4.0, Internet of Things, Robotic Platforms, Frequency
Characteristics
1 Introduction
At modern epoque the development of information management systems and robotic
systems takes place in accordance with the Industry 4.0 concept. In the framework of
this direction, there is a need to improve existing and develop new approaches to the
components development of such systems with the criteria of mobility, flexibility, the
ability to quickly restructure and adaptability to their operating conditions [1].
Exiting of the Internet of things (IoT) technology implies the information ex-
change not only between people and "things", but also between "things", i.e. comput-
erized machines, devices, components, and sensors.
On the one hand, "things" equipped with sensors can exchange data and process
information without human engaging to that process, and on the other hand, a human
Copyright © 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). ICST-2020
can actively participate in the process, for example, when it comes to "smart devices"
or "smart enterprise " concepts.
A smart enterprise is a subcategory of IoT, embodying as its kind, the industrial
Internet of things (IIoT). Therefore, today the technological equipment and facilities
components are equipped with multifunctional sensors, actuators and controllers.
Using of Internet technologies and wireless networks to them facilitates the data
rapid collection, their primary processing and presentation to the operator in a con-
venient form for prompt and timely decision-making. However, increasing the auto-
mation level in all areas of the enterprise allows you to organize production without
the people participation. In this regard, robotic industries are in the first place, as are
robotic platforms.
The personnel role in this case is reduced to the control of the technological pro-
cess and response to emergency situations [1-6]. This task is especially acute for ro-
botic platforms and critical application systems that must operate in difficult condi-
tions without human access [7-10].
Then, there arises a problem of building the robotic computer systems capable for
a comprehensive rearrangement of their characteristics by soft- and hardware tools,
depending on the operating conditions to increase the system efficiency is relevant. It
should be noted that the signal processing path of such systems uses frequency-
dependent components (FDC), representing a combination of hard- and software ar-
ranged in a complex architecture and including a variety of linkages [11-14].
Well-known is that the FDC responses do uniquely depend on the transfer function
coefficients [7].
Then, while calculating the coefficients, we can take into account the peculiar
characteristics of both the amplitude-frequency response (AFR) and phase-frequency
(PFR) response.
Let's consider the influence of the second order tunable digital frequency-
dependent components' phase-frequency response values when a smooth amplitude-
frequency response while calculating the transfer function coefficients [15].
2 The influence of the frequency-dependent components
transfer function coefficients on the its characteristics
Here we shall consider the widely used typical digital filters as a component of signal
processing and filtering path. Well known is that for ease of tuning the high-order
components are built on the basis of first and second orders frequency-dependent
components (FDC) [7].
Therefore, the analysis of the digital filter transfer function coefficients' influence
on the digital filter responses properties is carried out using the second-order transfer
function, which has the form
a0 + a1 z −1 + a2 z −2
H ( z) =
1 + b1 z −1 + b2 z −2
(1)
where a 0 , a1 , a 2 , b1 , b2 are the real coefficients of the numerator and denominator.
or using Euler's formula z = cos( ) − j sin ( ) ,
−1
When substituting z −1 = e − j
f
2
, 0, , f , f d are
fd
where is the normalized angular frequency, =
the linear frequency and the sampling frequency, respectively, we get a complex
transfer coefficient, and based on this coefficient, the amplitude-frequency response
(AFR) and phase-frequency response (PFR): at a 0 = − a 2 [13]
2
a a
2 + 1 + 4 1 cos ( ) + 2 cos ( 2 )
H ( ) = a02 0
a a0 , (2)
1 + b1 + b2 + 2b1 (1 + b2 ) cos ( ) + 2b2 cos ( 2 )
2 2
( b2 − 1) sin ( )
( ) = arctg (3)
( b2 + 1) cos ( ) + b1
It should be noted that similar equations are used to describe the digital filters, and
typical second order links at automatic control systems. Therefore, we consider the
influence of the transfer function coefficients onto the frequency response of second
order low-pass (LP) digital filters, as the most used frequency-dependent component
used at computer systems and control systems.
For polynomial filters, the relation a1 = 2a0 , which makes it possible to get the ampli-
tude frequency response square from the formula (2) upon certain transformations
carried out
4a02 (1 + cos ( ) )
2
H ( ) =
2
(4)
(1 − b2 ) + ( b1 + 2cos ( 2 ) ) ( b1 + 2b2 cos ( ) )
2
From equation (3) for the Butterworth filter and with respect to PFR type (Fig.1), it
can be specified that the phase value at cutoff frequency ( c ) = − . Then the PFR
2
denominator will appear like
( b2 + 1) cos ( ) + b1 = 0
We define b1 as
b1 = − ( b2 + 1) cos ( ) (5)
Fig. 1. Frequency and phase response of the second order Butterworth filter
For the Chebyshev filter-1, this solution is not possible, because here the PFR has
different phase values at frequency the cutoff level c set by the ripple value in the
bandwidth . It should be noted that in the most mathematical packages, the ripple
level is set in RP decibels further recalculated to c as follows
1
c=
100.1RP
For the Butterworth digital filter, the RP value is constantly -3 dB ( c = 0.707 ), and
for other filters it has different values, but the value c is in the range , refer
to Fig. 2 and 3.
Fig. 2. Amplitude frequency response and frequency response of the second order Chebyshev
filter-1 at RP= -3 dB
Fig. 3. Amplitude frequency response and frequency response of the second order Chebyshev
filter -1, at RP= - 7dB
Studies of PFR have shown that one of its main features is not a dependence on the
cutoff frequency, but the dependence on the ripple level. This phenomenon is mani-
fested differently in different filters.
After all transformations based on equations (2) and (3), the coefficients b1 and b2
for the Chebyshev filter -1 will have the form
1 − b2
b1 = − (1 + b2 ) cos ( c ) + sin ( c )
2 1 + + (1 + cos ( ) ) − sin ( )
2
b2 =
c c
2 1 + 2 + (1 + cos ( ) ) + sin ( )
c
c
where = tg ( ) from formula (3).
As evidently seen, to calculate the coefficients b1 and b2 it is necessary to know the
cutoff frequency PFR values at different ripple level values in the bandwidth.
It follows from Figures 2 and 3 that the phase values at the cutoff frequency depend
only on the ripple level in the bandwidth, Fig.4 [12].
Fig. 4. Dependence of Chebyshev's LP-1 FDC PFR at cutoff frequency ω ̀=0.5 on the ripple
level
To be noted is that if the ripple level is constant and the cutoff frequency varies, this
value does never change. The approximation of this dependence has the form
x (1 + 0.1666 x 2 ) + 0.008 x 4 (1 + 0.0241x 2 )
= − thx = −
(1 + 0.5 x 2 ) + 0.0417 x 4 (1 + 0.0335 x 2 )
It is worth noting that this expression works at RP up to 40 with a standard error of
0,06%. The resulting transformation is shown in Figure 5.
Fig. 5. Tabulated values (1) and approximating function (2) graphs
If we consider the Chebyshev filter-1 phase value as the limit value equal to -π, then
the expansion into series allows using it only up to the bandwidth ripple level value
equal to 35.
Applying this approach to finding the non-polynomial Chebyshev filter-2 coefficients
has its own peculiarities. For the Chebyshev filter-2, this solution is not possible,
because here the PFR has different phase values at the cutoff frequency level cs set
by the ripple value in the bandwidth . The ripple level is set in RP decibels further
recalculated to cs the same as c in the most mathematical packages.
For the Chebyshev filter-2, the phase also has a different value depending on the cs
value, Fig. 6 and 7.
Fig. 6. Amplitude frequency response and phase frequency response of the second order Che-
byshev filter-2 at RP= -3
Fig. 7. Amplitude frequency response and phase frequency response of the second order Che-
byshev filter-2, at RP= - 7dB
After all transformations based on equations (2) and (3), the coefficients b1 and b2
for the Chebyshev filter -2 will have the form
(1 − b2 ) sin ( c ) + (1 + b2 ) cos ( c )
b1 = −
2cs 1 + 2 − (1 + cs ) − (1 − cs ) cos ( ) + (1 − cs ) sin ( )
b2 =
c c
2cs 1 + 2 − (1 + cs ) − (1 − cs ) cos ( ) − (1 − cs ) sin ( )
c
c
where = tg ( ) from formula (3).
As evidently seen, to calculate the coefficients b1 and b2 it is necessary to know the
cutoff frequency PFR values at different ripple level values in the bandstop. If the
ripple level is constant and the cutoff frequency varies, this value does never change.
From figures 6 and 7 we can conclude that the phase values at the cutoff frequency
depend only on the level of ripples in the bandstop, Fig. 8 [15].
Fig. 8. Tabulated values (1) and approximating function (2) graphs
The first approximations gave us the dependence of the kind shown in Fig.9.
x (1 + 0.1666 x ) + 0.008 x (1 + 0.0241x )
2 4 2
= − thx = −
(1 + 0.5 x ) + 0.0417 x (1 + 0.0335 x )
2 4 2
The mean square error in this case is 0.28%.
Transitions to the non-polynomial Cauer filter (elliptic) after all transformations based
on equations (2) and (3), the coefficients b1 and b2 will have the form
(1 − b2 ) sin ( c ) + (1 + b2 ) cos ( c )
b1 = −
2c 1 + + ( c + cs ) + ( c − cs ) cos ( ) − ( c − cs ) sin ( )
2
b2 =
c c
2c 1 + + ( c + cs ) + ( c − cs ) cos ( ) + (1 − cs ) sin ( )
2
c
c
where = tg ( ) from formula (3).
Fig. 9. Tabulated values (1) and approximating function (2) graphs
As evidently seen, to calculate the coefficients b1 and b2 it is necessary to know the
cutoff frequency PFR values at different ripple level values c and cs . If the ripple
level is constant c and cs , this value does never change, Fig. 10.
Fig. 10. Tabulated values graph
To obtain the three-dimensional graph approximation, we go the following way. We
fix the ripple level cs or RS.
With this value, for each c or RP we determine the phase value. Thus, having drawn
sections for a three-dimensional graph, Fig. 10, one can find the approximating func-
tion description = f ( RP ) , for a given RS. Such a function will be a quadratic
equation and have the form
= − ( aRP 2 + bRP + c )
The coefficient values a, b and c have found for different RS values, table 1.
Table 1. The coefficient values a, b, and c for different RS values
RS a b c
5 -0.0097 0.1368 0.2685
7 -0.0063 0.1236 0,3051
10 -0,0040 0,1110 0,3489
15 -0,0022 0,0986 0,4032
20 -0,0015 0,0910 0,4421
30 -0,0008 0,0820 0,4923
40 -0,0005 0,0765 0,5232
50 -0,0003 0,0726 0,5451
60 -0,0002 0,0698 0,5623
Based on these data, the dependences of graphs a = f ( RS ) , b = f ( RS ) , c = f ( RS )
are plotted, respectively Fig. 11-13.
Fig. 11. Tabulated values (1) and approximating function a = f ( RS ) (2) graphs
Fig. 12. Tabulated values (1) and approximating function b = f ( RS ) (2) graphs
Fig.13. Tabulated values (1) and approximating function c = f ( RS ) (2) graphs
The square error in each case was 0.016%, 0.038%, and 3.14%, respectively, and the
approximating functions have the form
a = −0.1305038 RS −1.533 ;
b = 0.2084 RS −0.271 ;
c = 0.1732 RS 0.2986 .
Then, the phase value will be described by an approximating function
= − ( −0.1305RS −1.533 ) RP 2 + ( 0.2084 RS −0.271 ) RP + ( 0.1732 RS 0.2986 )
3 Experimental verification
To verify experimentally the exposed theoretical provisions, we used an ultrasonic
rangefinder, installed on the robotic system for assessing the distance to the obstacle.
The rangefinder improved accuracy has been achieved by tuning the radiation fre-
quency and, accordingly, rearrangement of channel for reflected signal processing. In
the reflected signal processing path, we used a digital refrangible filter. Refer to Fig.
14 and 15 for details of filter operation.
Fig.14. Filter input signal and signal spectrum
Fig.15. Filter output signal and signal spectrum
4 Conclusion
The obtained approximations of the PFR dependence on the ripple level and equations
for finding the denominator coefficients of the digital filters transfer contributes to a
more rapid using them in robotic systems. However, the resulting implementations are
not quite simple. Therefore, expedient shall be continuing the search for a more con-
venient implementation on the microprocessor technology.
It should be borne in mind that the values of RP and RS do not change in large ranges.
Usually they vary in narrow ranges and rarely. In this case, you can pre-calculate and
use some pre-prepared values stored in memory. In the case of a sharp change in
working conditions, these formulas can be useful for operational calculation, in order
to maintain the best robotic system functioning.
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