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Sequential identification of coulomb and viscous friction in robot drives
Aumnatica.
Pergamon
0 1997Elsevier
PII: solJo5-1098(%)00183-5
Vol. 33, No. 3, pp. 393-401, 1997 Science Ltd. All rights reserved Printed in Great Britain ooo54c98/97 s17.00 + 0.00
Brief Paper
Sequential Identification of Coulomb and Viscous Friction in Robot Drives* M. R. ELHAMI?
Key Words-Identification
and D. J. BROOKFIELDt
algorithms; friction; robot control; Coulomb friction; viscous friction.
Abstract-Modem robot control methods require knowledge of the form and coefficients of the joint friction. The main feature of the present work is the practical application of the nonlinear filtering approach of Detchmendy and Sridhar, on both computer simulation and experimental data, in frictional identification. The results confirm the feasibility of the proposed estimation approach. In addition, different models of friction have been examined for the estimation of Coulomb and viscous coefficients. In these models, asymmetry of the parameters, as an essential assumption, has been investigated and separate values determined for positive and negative directions of rotation. The experimental results justify the introduction of asymmetry, particularly for the Coulomb coefficient. It is shown that an asymmetric Coulomb and viscous frictional model is appropriate for the DC servo motor investigated. The quality of the fit of the coefficients was measured in terms of the RMS torque error, and low torque errors has been obtained. 0 1997 Elsevier Science Ltd.
and observations are obtained by linear measurement devices. No statistical assumptions were required concerning the nature of unknown inputs to the system or the measurement errors on the output. In recent years many authors have extended and developed this method both theoretically and for engineering applications. Among these authors are Kalaba and Tesfatsion (1981), who developed and implemented this method on discrete-time (that is, sampled data) systems, and Kalaba et al. (1981), who described a numerical implementation of the method and compared it to other techniques. Stanway et al. (1987a,b) used this approach to estimate coefficients in a squeeze-film vibration isolator and in an electro-rheological vibration damper. The basic problem in all of this work was the analysis of a system having dynamic behaviour described by the nonlinear differential equation
1. Introduction Control systems for industrial robots have conventionally been based on individual joint controllers that made no attempt to compensate for coupled torques or forces from other joints. Recently controllers have been studied that use model reference strategies where terms are included explicitly in each joint controller to compensate for these coupled torques or forces. These strategies have also included compensation for changing inertia and for friction. Compensation of friction requires a knowledge of the law describing the frictional behaviour of the joint and the associated coefficients. This paper describes an identification technique for the determination of frictional laws and coefficients in robot drives. This identification technique has been developed on the basis of the nonlinear filtering theory proposed by Detchmendy and Sridhar (1966). 1.1. Nonlinear filtering. In considering nonlinear problems, Detchmendy and Sridhar (1966) and Kagiwada et nf. (1969) derived filtering algorithms similar to the previous first-order filters using the least-squares error criterion and invariant imbedding techniques. The problem they considered was the sequential estimation of states and parameters in continuous time for noisy nonlinear dynamical systems. The class of systems considered were those in which the dynamical behaviour is described by an ordinary differential equation
Observations of the process are obtained in the form
i(t) = g@(t)) + E(t), t E [O, T].
y(t) = h@(t)) + v(t).
t E [O, Tl,
(1)
(2)
where r(t) is the state of the system, y(t) is the observed value, g@(t)) and h@(t)) are continuous functions in x, E(t) represents unknown inputs and q(t) represents the noise in the observations. The problem is defined for all times t from zero to the present time T. Using the usual least-squares sense, the problem of estimating a state x(T) at the present time T is then based on minimising a cost function (CF), which is the integral of the sum of the weighted squared residual errors.
CF = :[$(t) !
+ w(t)eZ(t)] dt,
(3)
where w(t) is a positive weighting factor. Invariant imbedding and linear approximation techniques were used by Detchmendy and Sridhar (1966) to derive a sequential least-squares estimator by minimising (3). The resulting estimator equations are analogous to the wellknown first-order filter or extended Kalman filter equations, which are described in vector form in later sections.
* Received 2 May 1995; revised 13 October 1995; received in final form 22 March 19%. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Bo Wahlberg under the direction of Editor Torsten Soderstrom. Corresponding author Dr David J. Brookfield. Tel. +44 51 794 4831; Fax +44 51 794 4848; E-mail [email protected]. ac.uk. t Department of Mechanical Engineering, University of Liverpool, Liverpool L69 3BX, U.K.
2. Theoretical background 2.1. Frictional model. A comprehensive survey of frictional behaviour was made by Armstrong-Helouvry et al. (1994). To represent the frictional behaviour of a typical robot drive component, the DC servo motor, it was decided to model the friction as stick/slip, with an exponential transition from stick to slip. This is a general model, which, with appropriate parameters, can represent other simpler frictional models. A 393
Brief Papers typical of simpler models are those used by SKF bearings. in which the viscous frictional torque is proportional to the two-thirds root of the shaft velocity and Canudas de Wit et al. (1987). who suggested a model where the viscous torque is proportional to the square root of the velocity. However Canudas de Wit et al. argue for the use of a square-root model as computationally easier than an exponential model, rather than because of a better fit to experimental data. The exponential model has therefore been used here. Figure 1 shows the exponential frictional model. In this figure K,, and Kv2 are the viscous friction coefficients in the positive and negative directions respectively. K,-, and K, 1 are the Coulomb coefficients for the stick or static state, and C, and Cz are the corresponding coefficients for the slip or dynamic state. The parameters B, and B, represent the transition from stick to slip behaviour for positive and negative directions of rotation. This model has been discussed by many authors, including Armstrong (1988). Canudas de Wit et al. (1987), Johnson and Lorenz (1991). Leonard and Krishnaprasad (1992) and Phillips and Ballou (1993). A number of these authors considered frictional models where the coefficients were different for the two directions of rotation. Nevertheless, we believe it is an essential task to include an asymmetric characteristic in the frictional model to achieve a more perfect model of friction as shown in Fig. 1 such that, generally, K,., Z K,.z. C, f C2. B, # B2 and Kv, f Kvz. The initial exponential model of friction modified to include an asymmetric characteristic and the frictional torque rr at angular velocity $ is then given by rF = (K,,
161+ C, + (K,., - C,)e
of sgn l(b) and sgn 2(h) are defined sgnl (e)=
sgn2 (&) =
1
is described
+ [Kv, 101+ C, + (Kc-, - C,)e
dt
g
+ [K,, t [K,,
‘2”‘) sgn 2(e) = K,I sin wr.
/V/ + C, + (K,., - C,)e IV1 + c
+ (G
- G)e
“I’“‘] sgnl (V) “““1 sgn2 (V) = K, sin r,
w&re r = wt, V =&. K,,., = Kvl,Z/Jo, Kc,., = K,,,,/Jw’, C,,, = C,,2/Jw’, B,,z = B,,*w and K, = K,I/Jw’. After normalisation, it is then only necessary to estimatm numerical values for the eight frictional parameters Kv,,Z, Kc-,.,. Cl,z and B,,,. The estimates obtained are converted into non-normalised form before being displayed by the identification program by applying the inverse of the definitions given above. 2.2. Estimarion procedure. Based on the discussion in the previous sections, a nonlinear filtering approach was chosen for estimating parameters from data. Using the method of Detchmendy and Sridhar (1966). a nonlinear filter is constructed based on the augmented state equations that describe the parameters of the identification problem as state variables. Therefore the process of estimating the viscous and Coulomb friction parameters in the exponential model becomes the estimation of both physical states and parameter states. In this procedure an augmented state vector is defined that includes the physical states as well as the parameter states.
as follows:
(eP0).
Identification of the frictional parameters requires the application of a known driving torque and the measurement of the resulting angular velocity. A sinusoidal torque is allowing parameters for both positive and convenient, negative velocities to be identified. When such a torque is
KC
Cl
Angular
Velocity
(rad s
c2
exponential
“1 ‘“‘1sgnl (8)
(6)
(8(O).
Fig. I. Asymmetric
differential
where / is the polar moment of inertia (Nm s2), K, is the motor torque constant (Nm A--‘). I is the driving current amplitude (A) and w is the driving frequency (rad s -I). Prior to any estimation process, it is generally desirable to normalise the system differential equation into a nondimensional form. This normalisation avoids dimensional complexity in the equations, and is frequently essential to avoid numerical overflow in processing. For the DC servo motor considered here, normalisation of (5) gives the following equation, and it is the parameters in this equation that are estimated:
(4)
’ 82
by the first-order
(5)
(ecu). 0
dg
the motor
+ [Kvz 161+ Cz + (Kc2 - C,)e
(e>(l),
0
( -1
/-
‘I’“‘] sgnl (6)
+ [Kvz lb/ + Cz + (Kc.? - C2)emAzisf]sgn2 (e) The functions
applied. equation
Kc2
Coulomb
and viscous
friction.
’)
Brief Papers Consequently the estimation of parameters becomes the estimation of this augmented state vector. For the normalised equations describing the DC servo motor, normalised angular velocity V is a suitable choice for the physical state, and the normalised parameters above represent the parameter states. x7 X8 &IT ------= F’ Kv, Kvz Kc, Ka c, G B* &IT.
x= [x,
x2 x3 xq x5 X6
Over the short period of a test, it normalised parameters as constant time derivative of these is thus definitions and the parameter states differential equation can be derived g(f,r)=$=
[.a, 0 0
(7)
is sensible to model the and time-invariant. The zero. With the above as given in (7), a vector as follows:
0 0
0 0
0
O]T,
(8)
where i1 is given by ii = K, sin 7 - [x2 lx11+x6 + (x4 - x6)e-XHIXII] sgnl (xi) - [x3 lx,1 +x7 + (x5 - x,)e-+II]
sgn2 (xi).
(9)
The observation vector is y(7) + W,
1) + q(7).
(10)
Here h(Z, 7) =fl, since the only observation is Vobsr the observed velocity from the experiment. Hence it follows that ~(7) = y(7) - h(f, 7) = V,, - 9r.
(11)
The sequential estimator equations, which are of predictorcorrector type, are then
The set of equations (14)-(20) show that P is a function of sequential state estimate. It is clear from the analysis that the matrix P(n x n) is symmetric and thus only fn(n + 1) of the components require solution. In the identification procedure it is necessary to solve nine differential equations in the vector form (15) for the components of the derivative matrix and, in principle, a further 81 differential equations for the elements of the matrix P. However, as noted above, the matrix P is symmetric, and so only 45 out of 81 equations require solution. The technique described is an updating method, and so the x state vector requires to be set to initial values. Zero initial values have been assumed in the present work. Similarly, the matrix P also requires initial values. The diagonal elements of P were set equal to 4. Greater values than this led to numerical instability and overflow. Off-diagonal elements are set equal to unity. With any least-squares method, there is some risk of false minima being found. The RMS error (defined later in Section 2.3) is a measure of the quality of the fit of the coefficients. If this error is high, a search technique, such as described by Armstrong-Helouvry (1991), may be applied to ensure that a true minimum has been found. 2.3. RMS torque error. In order to determine the appropriate frictional model to represent a data set, a measure of the quality of the fit was necessary. In the present case this measure was the RMS torque error, that is, the RMS value of the difference between the actual motor torque and the motor torque predicted by the system differential equation (6) using the estimated values for the frictional parameters as T, = KJ sin wt -J$ (
+ (K,,
$ = g@, 7) + 2PHQ[y( 7) - h(f, r)],
(12)
where H is defined by
- 1=
II = ah@, 7) T [ &
[l
0
0 0
0
0
0 0
O]T. (13)
Here g(f, 7) and (y(7) - h(%, 7)} are obtained from (8) and (11). The observation error term is weighed by a matrix product of three terms: a continuously updated weighting matrix P(9 x 9); the matrix H and the scaling matrix Q, which is usually set equal to I and for the case of a scalar observation becomes unity. The elements of the weighing matrix P are updated in each recursive step using the equation dP %I(% 7) p + p %IT(%7) al X7a% a - 2P 7 {HQb( 7) - I@, ax
7)]}P + I.
(14)
The terms in (14) are defined by
ag(f, 7) -= a%
0 . I ::
Lo
0 . 0
.
0 ..f’
...
I
(15)
OJ
K, = -[x2 -xX(x4 - x6)e-XxIXrI]sgnl (x1) - [x3 - xy(xs - x7)e-J~IXIl]sgn2 (x,), KZ = -Ix,1 sgnl (xl), K4 = -e-“sb11 sgnl (x1), K, = -(l
(16)
K3 = -Ix,1 sgn2(xi),
(17)
KS = -e-x”lXll sgn2 (x,),
(18)
1
- [Kvl jbj+ Cl
- C,)eeBI ‘$1 sgnl(8)
- [KV2 @I+ C2 + (KC2 - C2)emB21bI]sgn2 (8).
(21)
The RMS torque error was determined from data for the last two cycles of the forcing function KJ sin or, i.e. after the estimates of the coefficients are likely to have converged. This error is then described as a percentage of the input torque amplitude KJ as follows:
&MS(%) =
%=“=I Tt)_ N
KiI >
x
lmo/
0,
(22)
where N is the number of observation in the last two cycles. Figure 2 shows the procedure followed in identifying parameters and determining the RMS torque error and its percentage. Determination of the RMS error required knowledge of the angular acceleration of the motor. In the absence of a accelerometer, this is found by digitally differentiating the velocity data. With sampled and quantised data, such differentiation requires pre-filtering of the velocity signal, and a symmetric Lanczos window filter was used for this purpose. It is important to note that in the process of finding the RMS torque error the inertial contribution to the total torque is required. The first expression inside the large parentheses on the right-hand side of (21) shows the subtraction of the inertial term J df3fdr from the input motor torque, KJ sin wr. This resulting torque is considered to be the frictional torque, which is then compared with the modelled or estimated frictional torque. 3. Simulation Initial tests on the least-squares identification method were made using simulated data. These data were generated by using the trapezium rule to numerically integrate (23) to give the velocity:
- eeXnlX1l)sgnl (xi), K7 = -(l
- e-XYlrll)sgn2 (xi),
(19)
K,, = jxl[ (x5 - x7)e-z’klI sgn2 (x1).
(20)
K, = /x1/ (x4 - x6)e-XfilX~Isgnl (x,),
$ = J-‘{[KJ
sin wt - [Kvl @I+ Cl
+ (Kc, - C& -h ‘“11sgnl(8)
+ [K,, lb/ + C2 + (KC2 - C,)e-a21il] sgn2 (6)).
(23)
396
Brief Papers
square
N
Sum&Mean
r
w
R.U.S Error
square
*
SYSTEM
RMS Error (%) w
Parameters Identif~ation Method
:
Torque
Error
KiI
V&CitJ
T,=T-T
RMS
Root
I , , c I I
T
I
z=
I I
Fig. 2. Identification
We considered a number of simulation techniques, including those offered by standard packages such as ‘Simulink’ and techniques implemented in Pascal. The combination of the Karnopp (1985) method and the trapezium rule provided good velocity data as demonstrated by low RMS torque error for zero-noise contamination shown in Table 1. Each simulation test comprised 2000 samples: 200 per simulated torque cycle. To allow comparison with cxperimental results, the simulation program was thus designed to contaminate the velocity data with noise of a specified percentage. Simulation tests were undertaken with noise percentages of 0%. 5%, 10% and 20%. Figure 3 shows the frictional torque for an asymmetric Coulomb and viscous model. The solid line indicates the simulated frictional torque and the chain line indicates the estimate of this torque from the identification procedure. No noise contamination of the velocity signal was used. It can be seen that estimated torque converges to the simulated frictional torque within a few cycles. The estimated Coulomb and viscous coefficients converged torwards the values used in the simulation, and Table 1 shows that, for this case of zero-noise contamination, the largest error in estimated coefficient was less than 2%. A more genera1 frictional model is asymmetric exponential Coulomb and viscous. Figure 4 shows as a solid line the frictional torque and as a chain line the estimated frictional Table (a) Simulated K “I Wms) 1.0x10
1. Error
in estimated
frictional
coefficients K “1 Wms)
1
6.6
(b) B,, Bz, C, and friction model Noise level (%) 0 5 10 20
x
Error
in
torque for such a frictional model. In the same way as was seen in Fig. 3. the estimated torque converges to the simulated value within a few cycles. Figures 3 and 4 thus demonstrate the efficiency of the sequential least-squares method in estimating frictional coefficients. In Section 1 in (3) observation noise n(t) was included explicitly in the cost function that is minimised to obtain the parameter estimates. The sequential least-squares method thus acts as a filter and should be insensitive to observation noise. Table I shows a comparison of parameter estimates made for simulated asymmetric Coulomb and viscous friction with various noise-contamination percentages. It can be seen that the parameter estimates are not significantly biased by the presence of up to 20% noise contamination. For each level of contamination. the RMS torque error reflects the contamination of the velocity signal. 4. Practical experirnenl 4. I. i3,verimenfal procedure. The experimental arrangement is shown in Fig. 5. This arrangement was used to identify the frictional coefficients for a typical DC a MicroswitchTM DC Control Motor type servo-motor, 33VM82-020-11 designed for operation on a maximum of 24V DC and fitted with a tacho-generator to record the angular velocity. The motor current was supplied by a closed-loop controlled current source driven from an IBM
coefficients
lo-’
CZ selected
procedure
due to observation
noise
K, i (N m)
(k,
1.0x to give
Kv, (%)
Error in K,, (X,)
+91 -1.07 ~1.27 -1.54
-0.36 PO.30 0. I H +.01
10 ’
an asymmetric
Error
in
6.6 x IO Coulomb
Error
in
and
q viscous
;;>;
;;(
RMS error (%)
I.96 2.26
1.47 2.10 2.46 3.45
3.61 4.38 5.90 9.92
2.68 3.16
397
Brief Papers
Angular Velocity (red s.’ )
Fig. 3. Simulated and estimated frictional torques: asymmetric Coulomb and viscous model.
386 clone computer via a digital-to-analog, (D-A) converter. The tacho-generator output was recorded via an analog-todigital (A-D) converter. A program written in Pascal drove the D-A converter with a sinusoidal input at a frequency o (see (5)) and recorded the angular velocity via the A-D converter. The closed-loop current source was intended to maintain the required motor current despite the opposing motor back EMF. A high loop gain (I$ = 200) was used, and initial tests on the current source confirmed that the motor current was controlled to within 1% of the value programmed by the computer over the full range of angular velocities (i.e. the full range of motor back-EMFs). Each test comprised 10 cycles of sine-wave current input, and the velocity data were collected as 2000 samples, i.e. 200 samples per cycle. To ensure that the number of samples in each cycle was constant for each test, the normalised
frequency r= o At (where & is the sample interval) remained constant throughout the running tests. Therefore, for different frequencies, the sample interval Af was changed so that 7 always remain constant at 0.01~ For the test frequency of w = 39.54 rad s-l, the sample interval At was 0.8 ms. In identifying frictional coefficients from this data, the motor manufacturers stated inertia of 4.87 X 10m6kg m2 was used. 4.2. Experimental results. Figure 6 shows results from an
initial test on the motor at a frequency of w = 39.54 rad s-i. In this figure the motor driving torque and the resulting angular velocity are plotted. It can be seen that the angular velocity lags the driving torque, as is expected in a system with an integrating term (i.e. inertia). Furthermore, the velocity is asymmetric, with a maximum positive value of approximately 78rads-’ and a maximum negative value of -92 rad s-i.
25c
20-
15-
3
lo-
E $
5-
I-” ! =
o-
-5 -
-lO-
-15-
Fig. 4. Simulated and estimated frictional torques: asymmetric exponential Coulomb and viscous model.
398
Brief Papers
IBM 386 / 25 Pascal Identification
Fig. S. Motor identification system The sequential least-squares identification method was applied to these data. Nine states were estimated representing the velocity and the eight frictional parameters given in (7). Figure 7 shows the parameters K,,, Kvz (Fig. 7a), Kc,. KC2 (Fig. 7b). Ci. Cz (Fig. 7c) and B,. B2 (Fig. 7d) plotted as functions of the sample number. It can be seen that the parameters converge rapidly to stable values. Table 2 lists the final estimates of these parameters. Figure 8(a) shows the motor angular velocity (solid line) and the estimated angular velocity xi (chain line) plotted as functions of the sample number. The estimate clearly matches the observed velocity within one cycle. In Fig. S(b) the frictional torque determined as the forcing torque minus the inertial torque (solid line) and the frictional torque estimated from the coefficients (chain line) are shown plotted as functions of the sample number. These two curves are also coincident, indicating that the coefficients represent the frictional behaviour well. Figure 8(c) shows how the estimated frictional torque plotted against angular velocity develops as the data are processed. The converged frictional model for the motor determined from the final estimate of the coefficients in Table 2 is shown in Fig. 8(d), from which it is clear that the frictional behaviour of the test motor is asymmetric Coulomb and viscous. The test frequency w = 39.54 rads ’ was selected to ensure that a sufficient motor angular velocity was reached to allow convenient A-D conversion of the tacho-generattor voltage. However, we were concerned that the coefficients
obtained may have been influenced by the test frequency. To investigate this effect, a further 15 tests were undertaken over a total 1:6 frequency range from o = 6.18 rad s ~I to w = 39.54 rad ss’. In each case the least-squares identification procedure was applied to the data and the coefficients estimated. Figure 9(a) shows the viscous coefficients Kv, (solid line) and K,, (chain line), and Fig. 9(b) shows the Coulomb coefficients Kc, (solid line) and KC, (chain line) plotted as functions of the test frequency. Little frequency dependence. of the coefficients estimated is apparent. 5. Discussion Least-squares methods have been proposed by many authors. For example, Canudas de Wit ef 01. (1987) described an adaptive control strategy using a weighted RLS identification (described in further detail in Canudas de Wit and Carrillo, 1988. However, the general Detchmendy and Sridhar technique used in the present work does not appear to have been used previously for frictional identification, and differs significantly from the weighted RLS method reported by Canudas de Wit et al. It is a benefit that the Detchmendy and Sridhar least-squares method applied in the present work is also able to identify frictional behaviour described by simpler laws such as Coulomb and viscous and asymmetric Coulomb and viscous. The results of simulation data generated on the basis of two types frictional model are shown in Figs 3 and 4 with close and quick convergence. Table I shows that all of the
80 60
G
40
2
20 ,. .t: 0 8 t -20 > $ =-4O 2 Q-40
Veloeit$
-100 0
1 0.2
V
V
” 0.4
1 0.6
I 0.8
1 1
1 1.2
!,I/
-0.04
1.4
-0.05 1.6
Time (s) Fig. 6. Motor driving torque and angular velocity: w = 39.54 rad s ‘.
399
Brief Papers
v H
.I -lo
500
>
2000
;---,---
B
3
7
2)
;
11
E
8
1500
la)
LY
‘G Q
1000
’
i
:
Oi 0
v
Cl
500
1000
1500
500
2000
W
Pa
Fig. 7. Convergence of estimated coefficients plotted against sample number.
estimates of coefficients were within 2% of the given values, and the RMS torque error percentgaes were less than 3.7% for the asymmetric Coulomb and viscous model and 1.6% for the asymmetric exponential model. It can also be seen from Table 1 that parameter estimates are not biased by the introduction of up to 20% simulated white noise over the
velocity, the largest error in an estimated parameter being less than 3.5%. The results from physical experiments are shown in Figs 7-10. Figures 7(a-d) obtained at a test frequency of 39.54rad s-i show that the estimates converge rapidly to reasonably stable values. The coefficients Kv, and K,,
Table 2. Identified frictional coefficients Kv2
N m s)
(10-d Nms)
1.356
1.251
0
500
($& Nm)
$2 Nm)
($3 s-1)
$3 SK’)
Nm)
Nm)
1.06
0.82
2.14
1.91
2.64
3.57
1000 (a)
1500 2ooo Semple No.
z
0
z. 0.02 I E 0.01
0.02
a 6 0 '5 '2 -0.01
0
PE
'5 w"
1000 (W
1500 2000 Sample No
0.01
-0.01
-0.02 L3.7
-0.02 -100
500
-50
0
50
100
-100
-50
(c) AngularVelocity (reds-')
Fig. 8. Identified motor frictional behaviour.
0 (4
50 100 Angular Velocity (reds-')
400
Brief Papers
Test Frequency (rad s ’ Fig. 9. Variation
of parameter
estimates
)
with test frequency.
Kc I. K,.Z, Kv, and Kv2, as listed in Table 2. The remaining coefficients simply take low values, indicating the Coulomb and viscous character. It has been shown above that the motor is modelled well by asymmetric Coulomb and viscous frictional behaviour, and so plots of coefficients B,, B,, C, and Cz have not been included. In particular. B, and B, have very low values. Since in the tests reported above the torque amplitude was high (approximately three times the static Coulomb friction), most of the data collected were in the linear viscous region, and therefore there were little data in the exponential transition region that contributed to the estimation of B, and Bz. In an attempt to determine the coefficients better a
associated with viscous behaviour in Fig. 7(a) were symmetric, having approximately equal values for the two directions of rotation. The Coulomb coefficients Kc-, and Kc.L in Fig. 7(b) were asymmetric, with Kc, being approximately 30% greater than Kcz. It is clear from Figs 8(a,b) that both the velocity and frictional torque estimates agreed well with the experimental data. Furthermore, the RMS torque error was less than 8.8%. indicating that the coefficients represented a good fit to the data. The form of the motor friction is not immediately apparent from the coefficient estimates in Fig. 7. However, Fig. g(d) shows that the friction was clearly asymmetric Coulomb and viscous. Hence the significant coefficients for this motor are
(a)
0901
0.012 I
0.014 I
0.016
0.016 I
0.02
0.022 I
0.024
0.026 I
0.026 1
0.03
(b)
i-~~
I:-;;:;
0901 0.012
0.014
0.016
0.016
0.02
Torque Amplitude Fig. 10. Variation
1
--;;
of identitied
coefficients
0.022
0.024
0.026
0.026
torque
amplitudes.
Nm
against
0.03
401
Brief Papers using torque amplitudes at the lower part of this range, it was not possible to estimate coefficients other than K-i and KC2, since the torque only exceeded the static Coulomb friction for a small part of each cycle, and thus the motor spent most further series of 16 tests were conducted, with torque amplitudes ranging from 1.1&i to 2.8Kc1. Unfortunately, of its time stationary. In the remaining tests at higher torque amplitudes the RMS error decreased as the torque amplitude increased, and the best coefficient estimates were obtained at 2.t3Kc,. In each test the coefficients B,, &, Ci and C2 were less than 10m3, and thus the motor behaviour was identified as asymmetric Coulomb and viscous, as suggested by the earlier data. Figures lO(a, b) show the variation of the identified values of Kv,, KV2, K,-, and KC2 with torque amplitude. Figure 9 shows that the estimates of the coefficients do not vary greatly with test frequency over a wide frequency range. The viscous coefficients Kv, and KV2 remain approximately equal, and the Coulomb coefficients K,, and K, maintain an approximate 30% difference. For comparison purposes, approximate values of the motor Coulomb and viscous coefficients were determined using the Kubo et al. (1986) method modified to measure armature current, rather than voltage. The Coulomb coefficient found by this method was Kc=O.O1l Nm and the viscous coefficient was Kv = 9.4 X lo-’ N m s. Using the sequential least-squares method, the mean estimates of the Coulomb coefficients over the range of test frequencies shown in Fig. 9 were K,, =O.O121Nm and K,=O.O0882Nm. The mean Coulomb coefficient from least-squares identification was thus 0.0105 N m, which compares well with the approximate value found using the modified Kubo et al. method. The mean of the viscous coefficients shown in Fig. 9 was 1.05 X lo-“ N ms, which is also in reasonable agreement with the Kubo et al. value. The Coulomb coefficients estimated were within 15% of those determined by Brookfield (1994) for the present motor. The asymmetric exponential model used in deriving the sequential least-squares estimator is of general applicability to rolling friction at low velocities. Although friction in different drive systems may not show a exponential stick/slip characteristic or asymmetry of coefficients, the sequential least-squares identification method will determine the frictional coeffiicients properly. 6. Conclusions This paper has described the practical application of nonlinear filtering in friction identification with a general model of asymmetric exponential stick/slip Coulomb and viscous friction in robot drives. The sequential least-squares estimator has been used to determine eight frictional coefficients. It has been shown through computer simulation that the sequential leastsquares method correctly identifies both the form of the frictional behaviour and the associated coefficients. The coefficient estimates have been shown to be reasonably insensitive to observation noise, and the RMS torque error percentage has been used as a measure of the quality of the fit. Acknowledgements-We
are grateful to Dr Roger Stanway for his initial suggestion of the use of a sequential method for identifying friction in robots. One of us (M.R.E.) has been supported by a scholarship from the government of the
Islamic Republic acknowledged.
of Iran, and this support
is gratefully
References
Armstrong, B. (1988). Friction: experimental determination, modelling and compensation. In Proc. IEEE International Cot@ on Robotics and Automation, pp. 1422-1427. Armstrong-Helouvry, B. (1991). Control of Machines with Friction: Kluwer, i\lorweil, MA. Armstrone-Helouvrv. B.. P. Duoont and C. Canudas De Wit (1994). ‘k survey
Brookfield, D. J. (1994). Identification of Coulomb friction in robot drives and other mechanical systems through observation of third harmonic generation. Proc. IMechE, Pt C., 208,329-336.
Canudas de Wit, C. and J. Carillo (1988). A weighted RLS algorithm for systems with bounded distrubances. In Proc. IFAC Conf. on Identification and System Identification, Beijing, PRC, pp. 879-884.
Parameter
Canudas De Wit, C., K. J. Astrom and K. Braun (1987). Adaptive friction compensation in DC motor drives. IEEE J. Robotics Automatioi, RA-3,681-685. Detchmendv. D. M. and R. Sridhar (1966). Seauential estimation of states and parameters in noisy nonlinear dynamic systems. Trans. ASME J. Basic Engng, 362-368. Johnson, C. T. and R. D. Lorenz (1991). Experimental identification of friction and its compensation in precise, position controlled mechanisms. In Proc. IEEE Industrial Applications Society Annual Meeting, Vol. 1-2, pp. 1400-1406. Kagiwada, H. H., R. E. Kalaba, A. Schumitzky and R. Sridhar (1%9). Invariant imbedding and sequential interpolating filters for non-linear processes. Trans. ,I
ASME:
\
J. Basic Engng, 91,195-200.
Kalaba, R. and L. Tesfatsion (1981). An exact sequential solution procedure for a class of discrete-time non-linear estimation problems. IEEE Trans. Autom. Control, AC-26,1144-1149. Kalaba, R., K. Springarn and L. Tesfatsion (1981). A sequential method for non-linear filtering: numerical implementation and comparisons. J. Optim. Applies, 34, 541-559.
Kamopp, D. (1985). Computer simulation of slip-stick friction in mechanical dynamic systems. Trans. ASME: J. Dyn. Syst., Measurement,
Control,
107,100-103.
Kubo, T., G. Anwar and M. Tomizuka (1986). Application of nonlinear friction compensation to robot-arm control. In Proc. IEEE International Automation, pp. 722-727.
Conf
on
Robotics
and
Leonard, N. E. and P. S. Krishnaprasad (1992). Adaptive friction compensation for bi-directional low velocity position tracking. In Proc. 31st IEEE Conf. on Decision and Control, Tuscan, AZ, pp. 267-273. Phillips, S. M. and K. R. Ballou (1993). Friction modelling and compensation for an industrial robot. J. Robotic Syst.,
10,947-971. Stanway, R., R. Firoozian and J. E. Mottershead (1987a). Estimation of the linearised damping coefficients of a squeeze-film vibration isolator. Proc. IMechE, Pt C, 201, 181-191. Stanway, R., J. L. Sproston and N. G. Stevens (1987b). Non-linear modelling of an electro-rheological vibration damper. J. Electrostatics, 20, 167-184.
Pergamon
0 1997Elsevier
PII: solJo5-1098(%)00183-5
Vol. 33, No. 3, pp. 393-401, 1997 Science Ltd. All rights reserved Printed in Great Britain ooo54c98/97 s17.00 + 0.00
Brief Paper
Sequential Identification of Coulomb and Viscous Friction in Robot Drives* M. R. ELHAMI?
Key Words-Identification
and D. J. BROOKFIELDt
algorithms; friction; robot control; Coulomb friction; viscous friction.
Abstract-Modem robot control methods require knowledge of the form and coefficients of the joint friction. The main feature of the present work is the practical application of the nonlinear filtering approach of Detchmendy and Sridhar, on both computer simulation and experimental data, in frictional identification. The results confirm the feasibility of the proposed estimation approach. In addition, different models of friction have been examined for the estimation of Coulomb and viscous coefficients. In these models, asymmetry of the parameters, as an essential assumption, has been investigated and separate values determined for positive and negative directions of rotation. The experimental results justify the introduction of asymmetry, particularly for the Coulomb coefficient. It is shown that an asymmetric Coulomb and viscous frictional model is appropriate for the DC servo motor investigated. The quality of the fit of the coefficients was measured in terms of the RMS torque error, and low torque errors has been obtained. 0 1997 Elsevier Science Ltd.
and observations are obtained by linear measurement devices. No statistical assumptions were required concerning the nature of unknown inputs to the system or the measurement errors on the output. In recent years many authors have extended and developed this method both theoretically and for engineering applications. Among these authors are Kalaba and Tesfatsion (1981), who developed and implemented this method on discrete-time (that is, sampled data) systems, and Kalaba et al. (1981), who described a numerical implementation of the method and compared it to other techniques. Stanway et al. (1987a,b) used this approach to estimate coefficients in a squeeze-film vibration isolator and in an electro-rheological vibration damper. The basic problem in all of this work was the analysis of a system having dynamic behaviour described by the nonlinear differential equation
1. Introduction Control systems for industrial robots have conventionally been based on individual joint controllers that made no attempt to compensate for coupled torques or forces from other joints. Recently controllers have been studied that use model reference strategies where terms are included explicitly in each joint controller to compensate for these coupled torques or forces. These strategies have also included compensation for changing inertia and for friction. Compensation of friction requires a knowledge of the law describing the frictional behaviour of the joint and the associated coefficients. This paper describes an identification technique for the determination of frictional laws and coefficients in robot drives. This identification technique has been developed on the basis of the nonlinear filtering theory proposed by Detchmendy and Sridhar (1966). 1.1. Nonlinear filtering. In considering nonlinear problems, Detchmendy and Sridhar (1966) and Kagiwada et nf. (1969) derived filtering algorithms similar to the previous first-order filters using the least-squares error criterion and invariant imbedding techniques. The problem they considered was the sequential estimation of states and parameters in continuous time for noisy nonlinear dynamical systems. The class of systems considered were those in which the dynamical behaviour is described by an ordinary differential equation
Observations of the process are obtained in the form
i(t) = g@(t)) + E(t), t E [O, T].
y(t) = h@(t)) + v(t).
t E [O, Tl,
(1)
(2)
where r(t) is the state of the system, y(t) is the observed value, g@(t)) and h@(t)) are continuous functions in x, E(t) represents unknown inputs and q(t) represents the noise in the observations. The problem is defined for all times t from zero to the present time T. Using the usual least-squares sense, the problem of estimating a state x(T) at the present time T is then based on minimising a cost function (CF), which is the integral of the sum of the weighted squared residual errors.
CF = :[$(t) !
+ w(t)eZ(t)] dt,
(3)
where w(t) is a positive weighting factor. Invariant imbedding and linear approximation techniques were used by Detchmendy and Sridhar (1966) to derive a sequential least-squares estimator by minimising (3). The resulting estimator equations are analogous to the wellknown first-order filter or extended Kalman filter equations, which are described in vector form in later sections.
* Received 2 May 1995; revised 13 October 1995; received in final form 22 March 19%. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Bo Wahlberg under the direction of Editor Torsten Soderstrom. Corresponding author Dr David J. Brookfield. Tel. +44 51 794 4831; Fax +44 51 794 4848; E-mail [email protected]. ac.uk. t Department of Mechanical Engineering, University of Liverpool, Liverpool L69 3BX, U.K.
2. Theoretical background 2.1. Frictional model. A comprehensive survey of frictional behaviour was made by Armstrong-Helouvry et al. (1994). To represent the frictional behaviour of a typical robot drive component, the DC servo motor, it was decided to model the friction as stick/slip, with an exponential transition from stick to slip. This is a general model, which, with appropriate parameters, can represent other simpler frictional models. A 393
Brief Papers typical of simpler models are those used by SKF bearings. in which the viscous frictional torque is proportional to the two-thirds root of the shaft velocity and Canudas de Wit et al. (1987). who suggested a model where the viscous torque is proportional to the square root of the velocity. However Canudas de Wit et al. argue for the use of a square-root model as computationally easier than an exponential model, rather than because of a better fit to experimental data. The exponential model has therefore been used here. Figure 1 shows the exponential frictional model. In this figure K,, and Kv2 are the viscous friction coefficients in the positive and negative directions respectively. K,-, and K, 1 are the Coulomb coefficients for the stick or static state, and C, and Cz are the corresponding coefficients for the slip or dynamic state. The parameters B, and B, represent the transition from stick to slip behaviour for positive and negative directions of rotation. This model has been discussed by many authors, including Armstrong (1988). Canudas de Wit et al. (1987), Johnson and Lorenz (1991). Leonard and Krishnaprasad (1992) and Phillips and Ballou (1993). A number of these authors considered frictional models where the coefficients were different for the two directions of rotation. Nevertheless, we believe it is an essential task to include an asymmetric characteristic in the frictional model to achieve a more perfect model of friction as shown in Fig. 1 such that, generally, K,., Z K,.z. C, f C2. B, # B2 and Kv, f Kvz. The initial exponential model of friction modified to include an asymmetric characteristic and the frictional torque rr at angular velocity $ is then given by rF = (K,,
161+ C, + (K,., - C,)e
of sgn l(b) and sgn 2(h) are defined sgnl (e)=
sgn2 (&) =
1
is described
+ [Kv, 101+ C, + (Kc-, - C,)e
dt
g
+ [K,, t [K,,
‘2”‘) sgn 2(e) = K,I sin wr.
/V/ + C, + (K,., - C,)e IV1 + c
+ (G
- G)e
“I’“‘] sgnl (V) “““1 sgn2 (V) = K, sin r,
w&re r = wt, V =&. K,,., = Kvl,Z/Jo, Kc,., = K,,,,/Jw’, C,,, = C,,2/Jw’, B,,z = B,,*w and K, = K,I/Jw’. After normalisation, it is then only necessary to estimatm numerical values for the eight frictional parameters Kv,,Z, Kc-,.,. Cl,z and B,,,. The estimates obtained are converted into non-normalised form before being displayed by the identification program by applying the inverse of the definitions given above. 2.2. Estimarion procedure. Based on the discussion in the previous sections, a nonlinear filtering approach was chosen for estimating parameters from data. Using the method of Detchmendy and Sridhar (1966). a nonlinear filter is constructed based on the augmented state equations that describe the parameters of the identification problem as state variables. Therefore the process of estimating the viscous and Coulomb friction parameters in the exponential model becomes the estimation of both physical states and parameter states. In this procedure an augmented state vector is defined that includes the physical states as well as the parameter states.
as follows:
(eP0).
Identification of the frictional parameters requires the application of a known driving torque and the measurement of the resulting angular velocity. A sinusoidal torque is allowing parameters for both positive and convenient, negative velocities to be identified. When such a torque is
KC
Cl
Angular
Velocity
(rad s
c2
exponential
“1 ‘“‘1sgnl (8)
(6)
(8(O).
Fig. I. Asymmetric
differential
where / is the polar moment of inertia (Nm s2), K, is the motor torque constant (Nm A--‘). I is the driving current amplitude (A) and w is the driving frequency (rad s -I). Prior to any estimation process, it is generally desirable to normalise the system differential equation into a nondimensional form. This normalisation avoids dimensional complexity in the equations, and is frequently essential to avoid numerical overflow in processing. For the DC servo motor considered here, normalisation of (5) gives the following equation, and it is the parameters in this equation that are estimated:
(4)
’ 82
by the first-order
(5)
(ecu). 0
dg
the motor
+ [Kvz 161+ Cz + (Kc2 - C,)e
(e>(l),
0
( -1
/-
‘I’“‘] sgnl (6)
+ [Kvz lb/ + Cz + (Kc.? - C2)emAzisf]sgn2 (e) The functions
applied. equation
Kc2
Coulomb
and viscous
friction.
’)
Brief Papers Consequently the estimation of parameters becomes the estimation of this augmented state vector. For the normalised equations describing the DC servo motor, normalised angular velocity V is a suitable choice for the physical state, and the normalised parameters above represent the parameter states. x7 X8 &IT ------= F’ Kv, Kvz Kc, Ka c, G B* &IT.
x= [x,
x2 x3 xq x5 X6
Over the short period of a test, it normalised parameters as constant time derivative of these is thus definitions and the parameter states differential equation can be derived g(f,r)=$=
[.a, 0 0
(7)
is sensible to model the and time-invariant. The zero. With the above as given in (7), a vector as follows:
0 0
0 0
0
O]T,
(8)
where i1 is given by ii = K, sin 7 - [x2 lx11+x6 + (x4 - x6)e-XHIXII] sgnl (xi) - [x3 lx,1 +x7 + (x5 - x,)e-+II]
sgn2 (xi).
(9)
The observation vector is y(7) + W,
1) + q(7).
(10)
Here h(Z, 7) =fl, since the only observation is Vobsr the observed velocity from the experiment. Hence it follows that ~(7) = y(7) - h(f, 7) = V,, - 9r.
(11)
The sequential estimator equations, which are of predictorcorrector type, are then
The set of equations (14)-(20) show that P is a function of sequential state estimate. It is clear from the analysis that the matrix P(n x n) is symmetric and thus only fn(n + 1) of the components require solution. In the identification procedure it is necessary to solve nine differential equations in the vector form (15) for the components of the derivative matrix and, in principle, a further 81 differential equations for the elements of the matrix P. However, as noted above, the matrix P is symmetric, and so only 45 out of 81 equations require solution. The technique described is an updating method, and so the x state vector requires to be set to initial values. Zero initial values have been assumed in the present work. Similarly, the matrix P also requires initial values. The diagonal elements of P were set equal to 4. Greater values than this led to numerical instability and overflow. Off-diagonal elements are set equal to unity. With any least-squares method, there is some risk of false minima being found. The RMS error (defined later in Section 2.3) is a measure of the quality of the fit of the coefficients. If this error is high, a search technique, such as described by Armstrong-Helouvry (1991), may be applied to ensure that a true minimum has been found. 2.3. RMS torque error. In order to determine the appropriate frictional model to represent a data set, a measure of the quality of the fit was necessary. In the present case this measure was the RMS torque error, that is, the RMS value of the difference between the actual motor torque and the motor torque predicted by the system differential equation (6) using the estimated values for the frictional parameters as T, = KJ sin wt -J$ (
+ (K,,
$ = g@, 7) + 2PHQ[y( 7) - h(f, r)],
(12)
where H is defined by
- 1=
II = ah@, 7) T [ &
[l
0
0 0
0
0
0 0
O]T. (13)
Here g(f, 7) and (y(7) - h(%, 7)} are obtained from (8) and (11). The observation error term is weighed by a matrix product of three terms: a continuously updated weighting matrix P(9 x 9); the matrix H and the scaling matrix Q, which is usually set equal to I and for the case of a scalar observation becomes unity. The elements of the weighing matrix P are updated in each recursive step using the equation dP %I(% 7) p + p %IT(%7) al X7a% a - 2P 7 {HQb( 7) - I@, ax
7)]}P + I.
(14)
The terms in (14) are defined by
ag(f, 7) -= a%
0 . I ::
Lo
0 . 0
.
0 ..f’
...
I
(15)
OJ
K, = -[x2 -xX(x4 - x6)e-XxIXrI]sgnl (x1) - [x3 - xy(xs - x7)e-J~IXIl]sgn2 (x,), KZ = -Ix,1 sgnl (xl), K4 = -e-“sb11 sgnl (x1), K, = -(l
(16)
K3 = -Ix,1 sgn2(xi),
(17)
KS = -e-x”lXll sgn2 (x,),
(18)
1
- [Kvl jbj+ Cl
- C,)eeBI ‘$1 sgnl(8)
- [KV2 @I+ C2 + (KC2 - C2)emB21bI]sgn2 (8).
(21)
The RMS torque error was determined from data for the last two cycles of the forcing function KJ sin or, i.e. after the estimates of the coefficients are likely to have converged. This error is then described as a percentage of the input torque amplitude KJ as follows:
&MS(%) =
%=“=I Tt)_ N
KiI >
x
lmo/
0,
(22)
where N is the number of observation in the last two cycles. Figure 2 shows the procedure followed in identifying parameters and determining the RMS torque error and its percentage. Determination of the RMS error required knowledge of the angular acceleration of the motor. In the absence of a accelerometer, this is found by digitally differentiating the velocity data. With sampled and quantised data, such differentiation requires pre-filtering of the velocity signal, and a symmetric Lanczos window filter was used for this purpose. It is important to note that in the process of finding the RMS torque error the inertial contribution to the total torque is required. The first expression inside the large parentheses on the right-hand side of (21) shows the subtraction of the inertial term J df3fdr from the input motor torque, KJ sin wr. This resulting torque is considered to be the frictional torque, which is then compared with the modelled or estimated frictional torque. 3. Simulation Initial tests on the least-squares identification method were made using simulated data. These data were generated by using the trapezium rule to numerically integrate (23) to give the velocity:
- eeXnlX1l)sgnl (xi), K7 = -(l
- e-XYlrll)sgn2 (xi),
(19)
K,, = jxl[ (x5 - x7)e-z’klI sgn2 (x1).
(20)
K, = /x1/ (x4 - x6)e-XfilX~Isgnl (x,),
$ = J-‘{[KJ
sin wt - [Kvl @I+ Cl
+ (Kc, - C& -h ‘“11sgnl(8)
+ [K,, lb/ + C2 + (KC2 - C,)e-a21il] sgn2 (6)).
(23)
396
Brief Papers
square
N
Sum&Mean
r
w
R.U.S Error
square
*
SYSTEM
RMS Error (%) w
Parameters Identif~ation Method
:
Torque
Error
KiI
V&CitJ
T,=T-T
RMS
Root
I , , c I I
T
I
z=
I I
Fig. 2. Identification
We considered a number of simulation techniques, including those offered by standard packages such as ‘Simulink’ and techniques implemented in Pascal. The combination of the Karnopp (1985) method and the trapezium rule provided good velocity data as demonstrated by low RMS torque error for zero-noise contamination shown in Table 1. Each simulation test comprised 2000 samples: 200 per simulated torque cycle. To allow comparison with cxperimental results, the simulation program was thus designed to contaminate the velocity data with noise of a specified percentage. Simulation tests were undertaken with noise percentages of 0%. 5%, 10% and 20%. Figure 3 shows the frictional torque for an asymmetric Coulomb and viscous model. The solid line indicates the simulated frictional torque and the chain line indicates the estimate of this torque from the identification procedure. No noise contamination of the velocity signal was used. It can be seen that estimated torque converges to the simulated frictional torque within a few cycles. The estimated Coulomb and viscous coefficients converged torwards the values used in the simulation, and Table 1 shows that, for this case of zero-noise contamination, the largest error in estimated coefficient was less than 2%. A more genera1 frictional model is asymmetric exponential Coulomb and viscous. Figure 4 shows as a solid line the frictional torque and as a chain line the estimated frictional Table (a) Simulated K “I Wms) 1.0x10
1. Error
in estimated
frictional
coefficients K “1 Wms)
1
6.6
(b) B,, Bz, C, and friction model Noise level (%) 0 5 10 20
x
Error
in
torque for such a frictional model. In the same way as was seen in Fig. 3. the estimated torque converges to the simulated value within a few cycles. Figures 3 and 4 thus demonstrate the efficiency of the sequential least-squares method in estimating frictional coefficients. In Section 1 in (3) observation noise n(t) was included explicitly in the cost function that is minimised to obtain the parameter estimates. The sequential least-squares method thus acts as a filter and should be insensitive to observation noise. Table I shows a comparison of parameter estimates made for simulated asymmetric Coulomb and viscous friction with various noise-contamination percentages. It can be seen that the parameter estimates are not significantly biased by the presence of up to 20% noise contamination. For each level of contamination. the RMS torque error reflects the contamination of the velocity signal. 4. Practical experirnenl 4. I. i3,verimenfal procedure. The experimental arrangement is shown in Fig. 5. This arrangement was used to identify the frictional coefficients for a typical DC a MicroswitchTM DC Control Motor type servo-motor, 33VM82-020-11 designed for operation on a maximum of 24V DC and fitted with a tacho-generator to record the angular velocity. The motor current was supplied by a closed-loop controlled current source driven from an IBM
coefficients
lo-’
CZ selected
procedure
due to observation
noise
K, i (N m)
(k,
1.0x to give
Kv, (%)
Error in K,, (X,)
+91 -1.07 ~1.27 -1.54
-0.36 PO.30 0. I H +.01
10 ’
an asymmetric
Error
in
6.6 x IO Coulomb
Error
in
and
q viscous
;;>;
;;(
RMS error (%)
I.96 2.26
1.47 2.10 2.46 3.45
3.61 4.38 5.90 9.92
2.68 3.16
397
Brief Papers
Angular Velocity (red s.’ )
Fig. 3. Simulated and estimated frictional torques: asymmetric Coulomb and viscous model.
386 clone computer via a digital-to-analog, (D-A) converter. The tacho-generator output was recorded via an analog-todigital (A-D) converter. A program written in Pascal drove the D-A converter with a sinusoidal input at a frequency o (see (5)) and recorded the angular velocity via the A-D converter. The closed-loop current source was intended to maintain the required motor current despite the opposing motor back EMF. A high loop gain (I$ = 200) was used, and initial tests on the current source confirmed that the motor current was controlled to within 1% of the value programmed by the computer over the full range of angular velocities (i.e. the full range of motor back-EMFs). Each test comprised 10 cycles of sine-wave current input, and the velocity data were collected as 2000 samples, i.e. 200 samples per cycle. To ensure that the number of samples in each cycle was constant for each test, the normalised
frequency r= o At (where & is the sample interval) remained constant throughout the running tests. Therefore, for different frequencies, the sample interval Af was changed so that 7 always remain constant at 0.01~ For the test frequency of w = 39.54 rad s-l, the sample interval At was 0.8 ms. In identifying frictional coefficients from this data, the motor manufacturers stated inertia of 4.87 X 10m6kg m2 was used. 4.2. Experimental results. Figure 6 shows results from an
initial test on the motor at a frequency of w = 39.54 rad s-i. In this figure the motor driving torque and the resulting angular velocity are plotted. It can be seen that the angular velocity lags the driving torque, as is expected in a system with an integrating term (i.e. inertia). Furthermore, the velocity is asymmetric, with a maximum positive value of approximately 78rads-’ and a maximum negative value of -92 rad s-i.
25c
20-
15-
3
lo-
E $
5-
I-” ! =
o-
-5 -
-lO-
-15-
Fig. 4. Simulated and estimated frictional torques: asymmetric exponential Coulomb and viscous model.
398
Brief Papers
IBM 386 / 25 Pascal Identification
Fig. S. Motor identification system The sequential least-squares identification method was applied to these data. Nine states were estimated representing the velocity and the eight frictional parameters given in (7). Figure 7 shows the parameters K,,, Kvz (Fig. 7a), Kc,. KC2 (Fig. 7b). Ci. Cz (Fig. 7c) and B,. B2 (Fig. 7d) plotted as functions of the sample number. It can be seen that the parameters converge rapidly to stable values. Table 2 lists the final estimates of these parameters. Figure 8(a) shows the motor angular velocity (solid line) and the estimated angular velocity xi (chain line) plotted as functions of the sample number. The estimate clearly matches the observed velocity within one cycle. In Fig. S(b) the frictional torque determined as the forcing torque minus the inertial torque (solid line) and the frictional torque estimated from the coefficients (chain line) are shown plotted as functions of the sample number. These two curves are also coincident, indicating that the coefficients represent the frictional behaviour well. Figure 8(c) shows how the estimated frictional torque plotted against angular velocity develops as the data are processed. The converged frictional model for the motor determined from the final estimate of the coefficients in Table 2 is shown in Fig. 8(d), from which it is clear that the frictional behaviour of the test motor is asymmetric Coulomb and viscous. The test frequency w = 39.54 rads ’ was selected to ensure that a sufficient motor angular velocity was reached to allow convenient A-D conversion of the tacho-generattor voltage. However, we were concerned that the coefficients
obtained may have been influenced by the test frequency. To investigate this effect, a further 15 tests were undertaken over a total 1:6 frequency range from o = 6.18 rad s ~I to w = 39.54 rad ss’. In each case the least-squares identification procedure was applied to the data and the coefficients estimated. Figure 9(a) shows the viscous coefficients Kv, (solid line) and K,, (chain line), and Fig. 9(b) shows the Coulomb coefficients Kc, (solid line) and KC, (chain line) plotted as functions of the test frequency. Little frequency dependence. of the coefficients estimated is apparent. 5. Discussion Least-squares methods have been proposed by many authors. For example, Canudas de Wit ef 01. (1987) described an adaptive control strategy using a weighted RLS identification (described in further detail in Canudas de Wit and Carrillo, 1988. However, the general Detchmendy and Sridhar technique used in the present work does not appear to have been used previously for frictional identification, and differs significantly from the weighted RLS method reported by Canudas de Wit et al. It is a benefit that the Detchmendy and Sridhar least-squares method applied in the present work is also able to identify frictional behaviour described by simpler laws such as Coulomb and viscous and asymmetric Coulomb and viscous. The results of simulation data generated on the basis of two types frictional model are shown in Figs 3 and 4 with close and quick convergence. Table I shows that all of the
80 60
G
40
2
20 ,. .t: 0 8 t -20 > $ =-4O 2 Q-40
Veloeit$
-100 0
1 0.2
V
V
” 0.4
1 0.6
I 0.8
1 1
1 1.2
!,I/
-0.04
1.4
-0.05 1.6
Time (s) Fig. 6. Motor driving torque and angular velocity: w = 39.54 rad s ‘.
399
Brief Papers
v H
.I -lo
500
>
2000
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B
3
7
2)
;
11
E
8
1500
la)
LY
‘G Q
1000
’
i
:
Oi 0
v
Cl
500
1000
1500
500
2000
W
Pa
Fig. 7. Convergence of estimated coefficients plotted against sample number.
estimates of coefficients were within 2% of the given values, and the RMS torque error percentgaes were less than 3.7% for the asymmetric Coulomb and viscous model and 1.6% for the asymmetric exponential model. It can also be seen from Table 1 that parameter estimates are not biased by the introduction of up to 20% simulated white noise over the
velocity, the largest error in an estimated parameter being less than 3.5%. The results from physical experiments are shown in Figs 7-10. Figures 7(a-d) obtained at a test frequency of 39.54rad s-i show that the estimates converge rapidly to reasonably stable values. The coefficients Kv, and K,,
Table 2. Identified frictional coefficients Kv2
N m s)
(10-d Nms)
1.356
1.251
0
500
($& Nm)
$2 Nm)
($3 s-1)
$3 SK’)
Nm)
Nm)
1.06
0.82
2.14
1.91
2.64
3.57
1000 (a)
1500 2ooo Semple No.
z
0
z. 0.02 I E 0.01
0.02
a 6 0 '5 '2 -0.01
0
PE
'5 w"
1000 (W
1500 2000 Sample No
0.01
-0.01
-0.02 L3.7
-0.02 -100
500
-50
0
50
100
-100
-50
(c) AngularVelocity (reds-')
Fig. 8. Identified motor frictional behaviour.
0 (4
50 100 Angular Velocity (reds-')
400
Brief Papers
Test Frequency (rad s ’ Fig. 9. Variation
of parameter
estimates
)
with test frequency.
Kc I. K,.Z, Kv, and Kv2, as listed in Table 2. The remaining coefficients simply take low values, indicating the Coulomb and viscous character. It has been shown above that the motor is modelled well by asymmetric Coulomb and viscous frictional behaviour, and so plots of coefficients B,, B,, C, and Cz have not been included. In particular. B, and B, have very low values. Since in the tests reported above the torque amplitude was high (approximately three times the static Coulomb friction), most of the data collected were in the linear viscous region, and therefore there were little data in the exponential transition region that contributed to the estimation of B, and Bz. In an attempt to determine the coefficients better a
associated with viscous behaviour in Fig. 7(a) were symmetric, having approximately equal values for the two directions of rotation. The Coulomb coefficients Kc-, and Kc.L in Fig. 7(b) were asymmetric, with Kc, being approximately 30% greater than Kcz. It is clear from Figs 8(a,b) that both the velocity and frictional torque estimates agreed well with the experimental data. Furthermore, the RMS torque error was less than 8.8%. indicating that the coefficients represented a good fit to the data. The form of the motor friction is not immediately apparent from the coefficient estimates in Fig. 7. However, Fig. g(d) shows that the friction was clearly asymmetric Coulomb and viscous. Hence the significant coefficients for this motor are
(a)
0901
0.012 I
0.014 I
0.016
0.016 I
0.02
0.022 I
0.024
0.026 I
0.026 1
0.03
(b)
i-~~
I:-;;:;
0901 0.012
0.014
0.016
0.016
0.02
Torque Amplitude Fig. 10. Variation
1
--;;
of identitied
coefficients
0.022
0.024
0.026
0.026
torque
amplitudes.
Nm
against
0.03
401
Brief Papers using torque amplitudes at the lower part of this range, it was not possible to estimate coefficients other than K-i and KC2, since the torque only exceeded the static Coulomb friction for a small part of each cycle, and thus the motor spent most further series of 16 tests were conducted, with torque amplitudes ranging from 1.1&i to 2.8Kc1. Unfortunately, of its time stationary. In the remaining tests at higher torque amplitudes the RMS error decreased as the torque amplitude increased, and the best coefficient estimates were obtained at 2.t3Kc,. In each test the coefficients B,, &, Ci and C2 were less than 10m3, and thus the motor behaviour was identified as asymmetric Coulomb and viscous, as suggested by the earlier data. Figures lO(a, b) show the variation of the identified values of Kv,, KV2, K,-, and KC2 with torque amplitude. Figure 9 shows that the estimates of the coefficients do not vary greatly with test frequency over a wide frequency range. The viscous coefficients Kv, and KV2 remain approximately equal, and the Coulomb coefficients K,, and K, maintain an approximate 30% difference. For comparison purposes, approximate values of the motor Coulomb and viscous coefficients were determined using the Kubo et al. (1986) method modified to measure armature current, rather than voltage. The Coulomb coefficient found by this method was Kc=O.O1l Nm and the viscous coefficient was Kv = 9.4 X lo-’ N m s. Using the sequential least-squares method, the mean estimates of the Coulomb coefficients over the range of test frequencies shown in Fig. 9 were K,, =O.O121Nm and K,=O.O0882Nm. The mean Coulomb coefficient from least-squares identification was thus 0.0105 N m, which compares well with the approximate value found using the modified Kubo et al. method. The mean of the viscous coefficients shown in Fig. 9 was 1.05 X lo-“ N ms, which is also in reasonable agreement with the Kubo et al. value. The Coulomb coefficients estimated were within 15% of those determined by Brookfield (1994) for the present motor. The asymmetric exponential model used in deriving the sequential least-squares estimator is of general applicability to rolling friction at low velocities. Although friction in different drive systems may not show a exponential stick/slip characteristic or asymmetry of coefficients, the sequential least-squares identification method will determine the frictional coeffiicients properly. 6. Conclusions This paper has described the practical application of nonlinear filtering in friction identification with a general model of asymmetric exponential stick/slip Coulomb and viscous friction in robot drives. The sequential least-squares estimator has been used to determine eight frictional coefficients. It has been shown through computer simulation that the sequential leastsquares method correctly identifies both the form of the frictional behaviour and the associated coefficients. The coefficient estimates have been shown to be reasonably insensitive to observation noise, and the RMS torque error percentage has been used as a measure of the quality of the fit. Acknowledgements-We
are grateful to Dr Roger Stanway for his initial suggestion of the use of a sequential method for identifying friction in robots. One of us (M.R.E.) has been supported by a scholarship from the government of the
Islamic Republic acknowledged.
of Iran, and this support
is gratefully
References
Armstrong, B. (1988). Friction: experimental determination, modelling and compensation. In Proc. IEEE International Cot@ on Robotics and Automation, pp. 1422-1427. Armstrong-Helouvry, B. (1991). Control of Machines with Friction: Kluwer, i\lorweil, MA. Armstrone-Helouvrv. B.. P. Duoont and C. Canudas De Wit (1994). ‘k survey
Brookfield, D. J. (1994). Identification of Coulomb friction in robot drives and other mechanical systems through observation of third harmonic generation. Proc. IMechE, Pt C., 208,329-336.
Canudas de Wit, C. and J. Carillo (1988). A weighted RLS algorithm for systems with bounded distrubances. In Proc. IFAC Conf. on Identification and System Identification, Beijing, PRC, pp. 879-884.
Parameter
Canudas De Wit, C., K. J. Astrom and K. Braun (1987). Adaptive friction compensation in DC motor drives. IEEE J. Robotics Automatioi, RA-3,681-685. Detchmendv. D. M. and R. Sridhar (1966). Seauential estimation of states and parameters in noisy nonlinear dynamic systems. Trans. ASME J. Basic Engng, 362-368. Johnson, C. T. and R. D. Lorenz (1991). Experimental identification of friction and its compensation in precise, position controlled mechanisms. In Proc. IEEE Industrial Applications Society Annual Meeting, Vol. 1-2, pp. 1400-1406. Kagiwada, H. H., R. E. Kalaba, A. Schumitzky and R. Sridhar (1%9). Invariant imbedding and sequential interpolating filters for non-linear processes. Trans. ,I
ASME:
\
J. Basic Engng, 91,195-200.
Kalaba, R. and L. Tesfatsion (1981). An exact sequential solution procedure for a class of discrete-time non-linear estimation problems. IEEE Trans. Autom. Control, AC-26,1144-1149. Kalaba, R., K. Springarn and L. Tesfatsion (1981). A sequential method for non-linear filtering: numerical implementation and comparisons. J. Optim. Applies, 34, 541-559.
Kamopp, D. (1985). Computer simulation of slip-stick friction in mechanical dynamic systems. Trans. ASME: J. Dyn. Syst., Measurement,
Control,
107,100-103.
Kubo, T., G. Anwar and M. Tomizuka (1986). Application of nonlinear friction compensation to robot-arm control. In Proc. IEEE International Automation, pp. 722-727.
Conf
on
Robotics
and
Leonard, N. E. and P. S. Krishnaprasad (1992). Adaptive friction compensation for bi-directional low velocity position tracking. In Proc. 31st IEEE Conf. on Decision and Control, Tuscan, AZ, pp. 267-273. Phillips, S. M. and K. R. Ballou (1993). Friction modelling and compensation for an industrial robot. J. Robotic Syst.,
10,947-971. Stanway, R., R. Firoozian and J. E. Mottershead (1987a). Estimation of the linearised damping coefficients of a squeeze-film vibration isolator. Proc. IMechE, Pt C, 201, 181-191. Stanway, R., J. L. Sproston and N. G. Stevens (1987b). Non-linear modelling of an electro-rheological vibration damper. J. Electrostatics, 20, 167-184.