Nauka
Identification of an electrically driven manipulator
using the differential filters
Leszek Cedro
Faculty of Mechatronics and Machinery Design, Kielce University of Technology
Abstract: The paper presents an example of solving the parameter identification problem in case of robot with three degrees of freedom has been also presented. The identification has been performed with the use of elaborated differential filters. The applied identifica-tion method does not require differential equaidentifica-tions solving but only determining the appropriate derivatives. Identification method and its generalizations using the object inverse model require informa-tion on time derivatives of the input and output signals [1, 2]. The required derivative order depends on the order of differential equ-ations describing the object.
Keywords: differential filters, identification
1. Introduction
The rapid developments in computer hardware and softwa-re and, consequently, the common use of computers to con-trol processes have aroused wide interest in mathematical modeling, control processes and, accordingly, control sys-tem identification.
The method of identification applied in the analysis in-volves fine-tuning of the inverse model. The method can be used only for such values of the input signals that are deter-mined from the measurement data. Identifying a dynamic system by means of the input error method (fig. 1) requires looking for a model that generates the same input as the object. Only in the case of model reversibility is such a pro-cedure possible. This reversibility is true for linear minimum-phase models and a certain class of non-linear models whe-re the input is determined basing on the output data [1, 2]. Let us assume, for instance, that the object is described by means of a differential equation:
( ) ( 1)
( n, n ,..., , )
f ϕ ϕ − ϕ θ =τ (1)
where f is a certain known function. Thus, the identification error is defined as:
ˆ e= −τ τ
,
( ) ( 1) ˆ
ˆ ( ,n n ,..., , )
x f y= y − y θ . (2)
A drawback of this method is that derivative estimates need to be determined. An advantage, on the other hand, is that it is not necessary to solve the differential equations describing the model at each step of iteration.
The fundamental problem related to the implementation of the input error method and its generalization is the
ne-cessity to determine the estimates of signal derivatives. This is achieved by applying differential filters [3].
2. Differential filters
Let us assume that the differential filter of the k-th order is a series connection of a low-pass filter with boundary frequ-ency Ωg and a difference quotient of the k-th order (fig. 2). Fig. 1. A block diagram of the process of estimation of the
inver-se model parameters
Rys. 1. Schemat blokowy procesu estymacji parametrów modelu odwrotnego
Fig. 2. Block diagram of the differential filter Rys. 2. Schemat blokowy filtru różniczkującego
The low-pass filter will be responsible firstly for reducing the signal spectrum and secondly for correcting the characteri-stics of the difference quotient in the range of low frequen-cies. Thus, the filter will be called a low-pass correction fil-ter. The desired transfer function of the low-pass filter is:
kor ( ) ( ) for ( ) 0 for . k k g k g H H H Ω ∇ Ω Ω ≤ Ω Ω = Ω > Ω (3)
As a result, the transfer function of the series connection of the difference quotient and the low-pass filter in the ran-ge of low frequencies will be equal to the transfer function of an ideal differential filter.
kor 2 3 ( ) ( )/ ( ) /sin 1 /2(1 cos ) 2 /( 2sin sin(2 )) 3. k k k H H H k k k ∇ Ω = Ω Ω = Ω Ω = = Ω − Ω = Ω − Ω + Ω = (4)
The filter impulse response is the inverse Fourier trans-form of its frequency characteristic, thus:
kor( ) 21 kor( ) g g j n k k h n H e d π Ω Ω −Ω =
∫
Ω Ω. (5)Unfortunately, integral (5) cannot be expressed by me-ans of the analytic functions. It needs to be determined using some approximation. By expanding function Hkork( )Ω
into a Taylor series around the value Ω =0, we obtain:
2 4 2 4 kor 2 4 1 ( ) 1 6 ( ) 1 ( ) 2 12 1 ( ) 3. 4 k O k H O k O k +Ω + Ω = Ω Ω = + + Ω = +Ω + Ω = (6)
The four-term approximation of the expansion appe-ars to be fairly sufficient. The inverse Fourier transform of the function obtained by rejecting the terms of the hi-gher orders is equal to:
3 2 2 2 3 kor 2 2 2 3 2 2 2 12 cos( ) 6 (6 1 (2 cos( ) 12 ( ) (12 1 (2 cos( ) 4 (4 g g g g g k g g g g g g n n n n n n h n n n n n n n n π π π Ω Ω + Ω Ω + = Ω Ω + (7)
Assume that the impulse response of the low-pass dif-ferential filter is:
kor Harris 1 ( ) ( ) ( ) dk k k h n h n W n χ = , (8)
where WHarris( )n is Harris window described by the
follo-wing equation: Harris( ) 0,36 0,49cos( / ) 0,14cos(2 / ) 0,01cos(3 / ). W n n M n M n M π π π = + + + + (9)
The parameter
χ
k should be selected in sum a way that the slope of the characteristic of the filter being designed at point Ω=0 is the same as that of the ideal differential equation, thus:3. A mathematical model of a robot
manipulator
In the next sections, the following problems will be solved: first, we will derive the equations for the DC motors, then, we will define the kinetic and potential energy of the sys-tem, and finally, we will symbolically derive the robot dynamic equations, using the second order Lagrange equ-ations.
Fig. 3. An electrically-driven manipulator Rys. 3. Manipulator z napędem elektrycznym
Let ϕ=[ϕ ϕ ϕ1 2 3] denote the vector of joint variables
acting as generalized coordinates, mj – the mass, lj – the
arm length, lcj – the distance from the centre of gravity and Sj – the motor of the link j.
Using typical equivalent diagrams of DC motors ava-ilable in the literature, e.g. Ref. [4], and the second Kir-choff law, we can write the following electrical equation of the DC motor:
j j j j
z R L e
U =U +U +E , for j =1,2,3 (11)
where Uzjis the voltage supplied to the rotor.
Since an open-loop system may be difficult to control, it is essential that the identification be performed for a clo-sed-loop system with properly selected PD controllers. Let us assume that the equations of the controllers have the following form:
( ( ) ( )) ( )
j j j j
z p z j d j
U =K ϕ t −ϕ t −K ϕ t , (12) where: Kpj, Kdj – the parameters of the controllers, ϕzj( )t – the control signals, ϕj( )t – the variables describing
the position of the manipulator arms.
The voltage drops across the rotor winding resistance and inductance are:
Nauka ( ) j j j R w w U =R i t , (13) and wj( ) L j j di t U L dt = , (14)
where Rwj is the equivalent rotor winding resistance, Lj
is the equivalent rotor winding inductance, and iwj is the
current flowing through the rotor windings. The electromotive inductance force is
( )
j j
e e j
E =k ϕ t , (15)
where kej is an electromotive constant.
Substituting the subsequent components to eq. (11), we obtain: ( ) ( ) ( ) [ ( ) ( )] ( ) j j j j j j j w j w w e j p z j d j di t L R i t k t dt K t t K t ϕ ϕ ϕ ϕ + + = = − − , for j =1,2,3 (16)
The rotor torque is:
( )
j j j
s m w
M =k i t , (17)
where kmj is a mechanical constant.
Let us define the manipulator kinetic and potential ener-gy. The following geometrical relations take place:
2 2 cos( ( ))cos( ( ))2 1 c c x =l ϕ t ϕ t 3 3 2 2 1 2 3 1
cos( ( ))cos( ( )) cos( ( ) ( ))cos( ( )) c c x l t t l t t t ϕ ϕ ϕ ϕ ϕ = + + + 2 2 cos( ( ))sin( ( ))2 1 c c y =l ϕ t ϕ t 3 3 2 2 1 2 3 1
cos( ( ))sin( ( )) cos( ( ) ( ))sin( ( )) c c y l t t l t t t ϕ ϕ ϕ ϕ ϕ = + + + (18) 2 2 1 sin( ( ))2 c c z = +l l ϕ t 3 3 1 2sin( ( ))2 sin( ( )2 3( )) c c z = +l l ϕ t +l ϕ t +ϕ t
The velocity of the centre of gravity of the second arm of the manipulator is:
, 2 2 2 2 2 2 2 xc yc zc v = + + . 2 3 2 3 2 3 3 xc yc zc v = + + (19)
Thus, the kinetic energy of the system is:
1 2 3 E E= +E +E , (20) 2 1 1 1 Jc 2( )t E = ϕ , 2 2 () 2 2 2 2 2 2 2 mv J t E = + c
ϕ
2 2 3 3 3 3 3 m v2 Jc 2( )t E = + ϕ , 2 12 j j cj m l J = , 2 j j c l l = , where Jcj are moments of inertia of the robot armsassu-med for a uniform beam.
The potential energy of the system is:
1 2 3 U U U= + +U , (21) 1 1 1 c U =m gl , U2=m g l2 (1+lc2sin( ( )))ϕ2t , 3 3 3 (1 2sin( ( ))2 c sin( ( )2 3( )) U =m g l +l ϕ t +l ϕ t +ϕ t
Where g is the acceleration of gravity.
Using the expressions for the kinetic and potential ener-gy, we obtain two second-order Lagrange equations:
j s j j j d E E U M dt ϕ ϕ ϕ ∂ − ∂ + ∂ = ∂ ∂ ∂ , for j =1, 2,3. (22)
After substitution and simplification of all the variables, we have a system of three equations (where:
( ) j j t ϕ =ϕ ,
ϕ
j
=
ϕ
j
(t
)
, ϕ j =ϕ j( )t , ϕ j =ϕ j( )t ): 1 2 2 2 2 1 1 1 3 3 2 2 3 2 2 3 2 3 3 2 2 2 3 2 3 2 3 1 2 2 3 2 3 3 3 2 3 2 2 3 2 1 3 3 2 2 3 1 (( (2 3( ( 4 )) 3 (4 )cos(2 ) 3 (8 cos( )cos( ) cos(2( )))) 12 ( ( 4 )sin(2 ) ( sin(2( )) 4 sin(2 ))) 24 (2 cos( ) cos( m R l m l m l m m l m k m l m l l L l m m l m l l L l m l l ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + + + + + + + + + − + + + + + + + − + 1 1 2 3 2 2 3 3 1 1 1 2 2 3 2 3 3 3 2 3 2 2 2 3 2 3 3 1 2 2 3 2 3 2 3 3 1 3 3 2 2 2 3 3 2 3 )sin( ) ) 6 (4 2 ( (
4 )cos(2 ) ( cos(2( )) 4 cos(2 ))) 2 (2 cos( ) cos(
)sin( ) 2 (2 cos( )cos( ) cos(2( ))) e m k k L l m m l m l l l m R l l L l m l l ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + − + + + + + + + − + + + + − + + + + 2 2 3 2 1 2 2 3 2 3 3 3 2 3 2 2 3 1 3 3 3 2 3 2 2 2 3 3 1 2 2 3 2 3 3 3 2 3 2 2 3 2 1 3 3 2 ( ( ( 4 )sin(2 ) ( sin(2( )) 4 sin(2
))) 4 ( cos(2( )) 2 cos(2 )) ) ( ( 4 )sin(2 ) ( sin(2( )) 4 sin(2 ))) 2 (2 cos( R l m m l m l l L l m l l L l m m l m l l L l m l ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + − + + + + + + + − + + + + − + + + + + + + − 1 2 3 2 3 2 2 2 2 3 3 1 1 1 2 2 3 2 2 2 3 3 2 2 3 3 3 2 2 3 1 ) cos( ))sin( ) ) 2 ( 3 ( 4 )cos( ) 12 cos( )cos( ) 3 cos(
) ) ) z l L l m l m m l l m l m U ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + + + + + + + + + =
2 3 2 2 3 2 2 2 3 2 3 3 3 2 3 2 2 3 2 2 3 3 3 2 3 2 2 3 3 2 2 2 2 3 )) 4 sin(2 ))) 2 ( ( 4 )cos(2 ) ( cos(2( )) 4 cos(2 ))) 2 ( cos(2( )) 2 cos(2 )) 6 ( ( 4 )s l L l m m l m l l L l m l l L l m m ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + + + + + + + + + + + + + + + 2 3 3 3 2 3 2 2 3 1 1 2 2 2 2 3 2 3 3 2 3 2 2 2 2 2 2 2 2 3 2 2 3 3 2 2 3 3 2 3 3 3 3 3 2 2 3 2 2 2 2 2 in(2 ) ( sin(2( )) 4 sin(2 ))) 2(6 ( ( 2 )cos( ) cos( )) 4 12 3 3 6 sin( ) ( ( cos( ) 4 3 l m l l gR l m m l m l m R l m R l m R l m R l m l R L g l l ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + + + + + + + + + + + + − + + + + + 3 2 2 2 2 2 2 3 2 2 3 3 2 3 2 3 3 2 3 3 2 2 3 2 3 2 3 2 2 2 2 2 2 3 2 2 2 2 3 3 2 2 3 3 3 3 3 3 3 2 2 3 2 2 3 )) 6 (2 ( ( 2 )sin( ) sin( ) 2 ( cos( ) sin( )( ))) 4 12 3 3 6 cos( )( sin( ) e m k k gL l m m l m l l m L R L l L m l L m L l m L l m l m gL l L ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + − + + − + − + + + + + + + + + + − − 2 2 2 2 2 3 2 2 2 3 (2 2 )))) z l R R L L U ϕ ϕ ϕ ϕ + + + + + = (23) 3 3 3 3 3 3 2 3 3 2 3 3 2 2 3 3 3 3 3 2 1 3 2 2 3 2 3 2 3 3 3 2 3 2 2 3 2 3 2 2 2 3 3 2 3 1 (6( cos( ) ) (6 ( sin( ) cos( ) )
3 ( (2 cos( ) cos( ))sin( ) ( cos(2( )) 2 cos(2 )) (2 cos( )cos( ) cos(2( )))
e m m gl m R k k k l m l R L R l l L l l L l l ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + + + + + + + + + + + + + + + + + 3 3 3 2 2 3 2 3 2 3 1 1 3 3 2 3 2 3 2 2 3 2 3 2 3 3 3 3 3 2 3 2 3 2 3 2 3 2 3 3 ) 6 (2 cos( ) cos( ))sin( ) 3 ( 2 cos( )) 6 ( sin( ) 2 sin( ) ) 4 ( 6 ( sin( ) sin( ) ) 3( 2 cos( )) 4 ))) z L l l R l l L g l l R L g l l l l U ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + + + + + + + + − + − + + + − + + + + + =
4. Simulation of the manipulator model
This section discusses the results of a simulation of clo-sed-loop equations including a robot model with PD con-trollers (fig. 4). The collected data will then be used in the identification algorithm.
First, the pre-determined signal was defined:
]
[
ϕ
z1ϕ
z2ϕ
z3 . The signal was assumed to be a properly de-layed step function (each arm with a different delay) pas-sing through an additional low-pass filter with a bounda-ry frequency Ωg =0.025rad/s. The filtering wasresponsi-ble for limiting the signal spectrum.
The responses are not satisfactory from the point of view of regulation. The aim of the study was to generate signals to be used in the identification process. It is advisa-ble that the pre-determined signals and the controller
pa-rameters be carefully selected so that the signals provide sufficient information about the object dynamics.
5. Identification
Let us recall that the robot mass and arm length are the unk-nown parameters denoted as
θ
=
[
m
1,
m
2,
m
3,
l
1,
l
2,
l
3]
.The method used for the parameter identification is repre-sented graphically in Figure 1. It is assumed that the measurement data concerning the trajectories of the gene-ralized variables and the necessary input signals are ava-ilable. The estimate of the input signals,
τ
ˆf, isdetermi-ned basing on the current estimates of the object para-meters θˆ=[ , , , , , ]m m m l l lˆ ˆ ˆ1 2 3 1 2 3ˆ ˆ ˆ . These equations have the
same structure as eq. (23); yet, the unknown parameters
θ
, are replaced by the estimates θˆ, the generalized varia-bles are replaced by variavaria-bles filtered through a low-pass filter, and their derivatives (which are not measured) are replaced by their estimates obtained by using relevant dif-ferentiating filters. Let us assume that the boundary fre-quency of the differentiating filters is: Wg = 0.2 rad/s. Theidentification requires determining the estimates of the parameters responsible for the quality factor minimization.
2 0 1 ˆ ˆ ( ) T( f f) J dt T θ =
∫
τ −τ , (24)where τf is an input signal filtered with a low-pass filter.
The identification procedure is commenced for the follo-wing initial values: θˆ=[99.2, 151.7, 49.9, 0.49, 1.01, 0.75]. The final values of the parameters are determined after 24 iterations of the minimization algorithm. The estima-tes θˆ=[108.462, 150.44, 49.6122, 0.491266, 1.00157, 0.70392] slightly depart from the real values of the para-meters, θ =[100, 150, 50, 0.5, 1, 0.7].
6. Identification and measurement
noise
In this point we will examine how far the elaborated fil-ters eliminate the measurement and quantization noise [5, 6]. We will also examine the influence of the measu-rement and quantization noise on the result of
identifica-Fig. 4. Responses of ϕ1,ϕ2,ϕ3
Nauka
tion process with the use of finite elements differentiation method and elaborated filters.
The signal processing theory comprises activities aimed on selection of substantial information on the examined phenomena and elimination of redundant information. It is commonly known that the measured signals contain com-ponents resulting from the disturbances. In our case the qu-antization noise value is connected directly with the num-ber of bits of the n-bit A/D converter [7, 8].
Using the same identification method and elaborated filters following parameters have been obtained for the no-isy signal (n=20) θˆ=[108.827, 150.446, 49.6164, 0.490419, 1.00154, 0.703882], (n=16) θˆ=[112.219, 150.266, 49.8673, 0.480005, 1.0005, 0.701312], (n=14) θˆ=[122.175, 147.789, 52.3693, 0.435243, 0.991148, 0.676069].
Using the finite elements method following parameters
have been obtained for the noisy signal (n=20) θˆ=
[2.22592×106, –609 462, 313 468, –0.000121342, 0.0149072,
0.000171592].
Comparing the obtained results we can state that the differential filters eliminate the measurement noise in a ma-jor degree and the parameters determined in the identifica-tion process are close to the actual ones. Tradiidentifica-tional diffe-rentiation does not ensure noise elimination and the iden-tified parameters differ significantly from the actual ones. Using the elaborated filters in identification methods we obtain well determined parameters in case of quantization on the level of 16-bit cards.
7. Conclusions
In contrast to the conventional output error method, which involves comparing and estimating input signals, the input error method is considerably faster. The identification pro-cedure does not require solving a series of differential equ-ations in each iteration of the algorithm minimizing the quality factor.
It should be noted that the spectrum of the pre-deter-mined signals is limited. In spite of the fact that the robot system is a non-linear system, the following relationship is obtained for the filtered signals: τˆf ≅τf, if
θ
ˆ
=
θ
. As theslight differences are due to the system non-linearity and quantization errors, the equation can be solved approxi-mately.
Elaborated differential filters have low-pass character. This feature enables removing of high-frequency compo-nents of the signal, for example the noise. Differential fil-ters ensure determining of appropriate derivatives of signal with errors far more less than simple differentiation me-thods, what plays particularly important role in the iden-tification process. In various calculations which have been performed, proper operation of the method for more com-plicated mechanical systems and for systems of greater number of identified parameters has been stated.
Bibliography
1. Cedro L., Janecki D., Model parameter identification with nonlinear parameterization applied to a manipula-tor model, Monographic series of publications –
“Com-puter science in the age of XXI century”, Radom 2011, ISBN 978-83-7789-006-6, ISBN 978-83-7351-324-2. 2. Cedro L., Janecki D., Differential filters and the
iden-tification of a manipulator using Mathematica softwa-re, XXXIV. Seminar ASR ‘2009 Instruments and Con-trol, Ostrava, ISBN 978-80-248-1953-2.
3. Janecki D., Cedro L., Differential Filters With
Applica-tion To System IdentificaApplica-tion, 7th European
Conferen-ce of Young Research and ScienConferen-ce Workers in Trans-port and Telecommunications TRANSCOM 2007, Żili-na, Slovakia, 115.
4. Kowal J., Fundamentals of control engineering, Vol. II, 2004, UWND, Kraków.
5. Mocak J., Janiga I.I, Rievaj M., Bustin D., The Use of Fractional Differentiation or Integration for Signal Improvement, “Measurement Science Review”, 2007, Vol. 7, Section 1, No. 5.
6. Rabiner L.R., Gold B., Theory and Application of Digi-tal Signal Processing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975.
7. Lyons R.G., An introduction to digital signal proces-sing (in Polish), WKiŁ, Warsaw 1999.
8. Pintelon R., Schoukens J., Real-Time Integration and Differentiation of Analog Signals by Means of Digital Filtering, “IEEE Transactions on Instrumentation and Measurement”, Vol. 39, No. 6. December 1990.
Identyfikacja manipulatora
z napędem elektrycznym z użyciem
filtrów różniczkujących
Streszczenie: Artykuł przedstawia przykład identyfikacji para-metrów manipulatora o trzech stopniach swobody. W identyfika-cji wykorzystano opracowane filtry różniczkujące. Zastosowano metodę identyfikacji, która nie wymaga rozwiązywania układu równań różniczkowych tylko użycia zróżniczkowanych sygna-łów. Metoda identyfikacji wykorzystuje model odwrotny oraz sy-gnały wejściowe i wyjściowe. Wymagany rząd użytych sygna-łów zależy od równań różniczkowych opisujących obiekt. Słowa kluczowe: filtry różniczkujące, identyfikacja
Leszek Cedro, PhD
A graduate of the Kielce University of Technology (Faculty of Mechatro-nics and Machine Building). Rece-ived his doctor’s degree in 2007 pre-senting a thesis on Identification of hydraulic drive systems using diffe-rentiating filters. Now employed as an assistant professor at the Centre for Laser Technologies of Metals, which is a joint unit of the Kielce University of Technology and the Polish Aca-demy of Sciences. His research inte-rests include control theory and iden-tification methods.
