Friction compensation for an industrial hydraulic robot
ABSTRACT A model based friction compensation scheme using a novel dynamical
friction model was implemented on an industrial Schilling Titan II
hydraulic robot. Off-line estimation of parameters was carried out,
using the results of two kinds of experiments. These experiments were
done independently at each joint. A nonlinear PI type controller was
used in the inner torque loop to improve its performance. The complete
control scheme has shown to substantially improve the position precision
in regulation and tracking. Higher precision applications can be
performed by the hydraulic robot with this controller
- SourceAvailable from: lth.se[show abstract] [hide abstract]
ABSTRACT: In this paper we propose a new dynamic model for friction. The model captures most of the friction behavior that has been observed experimentally. This includes the Stribeck effect, hysteresis, spring-like characteristics for stiction, and varying break-away force. Properties of the model that are relevant to control design are investigated by analysis and simulation. New control strategies, including a friction observer, are explored, and stability results are presentedIEEE Transactions on Automatic Control 04/1995; · 2.72 Impact Factor
Conference Proceeding: Adaptive friction compensation in DC motor drives[show abstract] [hide abstract]
ABSTRACT: This paper proposes a control scheme where the nonlinear effects of friction are compensated adaptively. When the friction is compensated the motor drive can approximately be described by a constant coefficient linear model. Standard methods can be applied to design a regulator for such a model. This results in a control law which is a combination of a fixed linear controller and an adaptive part which compensates for nonlinear friction effects. Experiments have clearly shown that both static and dynamic friction have nonsymmetric characteristic. They depend on the direction of motion. This is considered in the design of the adaptive friction compensation. The proposed scheme has been implemented and tested on a laboratory prototype with good results. The control low is implemented on an IBM-PC. The paper describes the ideas, the algorithm and the experimental results. The results are relevant for many precision drives like those found in industrial robots.Robotics and Automation. Proceedings. 1986 IEEE International Conference on; 05/1986
- Journal of Dynamic Systems Measurement and Control-transactions of The Asme - J DYN SYST MEAS CONTR. 01/1981; 103(1).
Friction Compensation for an
Industrial Hydraulic Robot
P. Lischinsky, C. Canudas-de-Wit, and G. Morel
oint friction is one of the major limitations in performing high
J precision manipulation tasks. It affects both static and dy-
namic performances, and may cause instability when coupled to
position or force feedback control. Thus, compensating for joint
friction has been one of the main research issues in robot design
and control over the years. The aim of this paper is to show how
friction compensation based on the LuGre (for Lund and Greno-
ble) dynamic model, , which was applied to an electric actua-
tor in , can be successfully used for an hydraulically actuated
manipulator. Fig. 1 is a photograph of the Schilling Titan I1 ma-
nipulator used in this research.
Friction compensation is particularly important for hydraulic
manipulators. First, due to high supply pressure, tight sealing is
required to prevent the actuators from significant internal leaks.
This generates very high joint friction. As an example, the joint
friction of the Schilling Titan I1 manipulator can reach 30% of
the nominal actuator torque. Secondly, nonlinear Stribeck fric-
tion, a well known source of stick-and-slip oscillations, has a
particular importance in hydraulic systems [ 151. (Fig. 4 illus-
trates the Stribeck effect for the first joint of the Titan I1 robot. A
25% drop between the static friction torque and the minimum
torque is observed.)
While compensating for friction is specially important for hy-
draulic devices, it is also particularly difficult. There are several
ways to compensate for friction, basically divided into non-
model based and model based compensation [ 11. Among the first
type there is, for example, classical dither noise, which consists
in adding a high frequency signal to the control signal. Another
possibility is to design a joint torque feedback, requiring extra
torque sensors to be mounted on the robot or on its base; see for
example . In most cases (including our industrial application),
none of those sensors are available. Model-based friction com-
pensation uses on-line friction torque estimation . The esti-
mated friction torque is added to the torque reference generated
by the position controller and gravity compensation. This kind of
compensation assumes that the actuator has a fast and accurate
torque response. This is generally verified with electric actua-
tors. Nevertheless, most servovalves, which are the control de-
vices of the hydraulic actuators, do not provide a sufficiently fast
and efficient torque response. Then an inner torque loop has to be
f! Lischinsky (firstname.lastname@example.org) is with the Department of Auto-
matic Control at the Systems Engineering School, Univer;vidad de
Los Andes, Mirida, Venezuela. C. Canudas-de- W i t is with the De-
partment of Automatic Control, Polytechnic Institute of Crenoble,
St-Martin-d’Ht.re.7 Cedex, France. G. Morel is with the University of
Strasbourg/ENSPS LSIIT/GRAVIR, Illkirch, France.
Fig. 1. Schilling Titan II manipulator during an assembly task in
Another difficulty in applying friction compensation to hy-
draulic systems arises from the variation of operating conditions.
In particular, oil temperature affects its viscosity, bulk modulus
and compressibility. Actuator wear also influences friction. Fur-
thermore, the nonlinear dynamics of these actuators are position
dependent (see (1) below). These variations significantly affect
the joint friction and the torque inner-loop characteristics during
a manipulation task. Adaptation is then required.
The LuGre friction model used in this paper fits the require-
ments for friction compensation of hydraulic systems because it
can describe complex friction behavior, such as stick-slip mo-
tion, presliding displacement, Dah1 and Stribeck effects and fric-
tional lag. In addition, it can be coupled with a simple and robust
adaptation algorithm, as shown in . This model has two kinds
of parameters. A first set of four static parameters are used to
characterize the steady state static map between velocity and
friction force or torque. This includes static, Coulomb, Stribeck
and viscous frictional effects. A second set of two dynamic para-
meters affects the dynamic friction response.
A two step off-line identification methodology of the LuGre
parameters has been proposed in . First, closed-loop constant
velocity experiments are performed to identify the static velocity
to friction map. In a second step, stick slip motions are per-
formed. For this kind of motions, which emphasizes the dynamic
friction effect, a simplified friction model is used in order to
identify the two dynamic parameters.
Finally, to cope with changes in friction characteristics, a sin-
gle parameter adaptive scheme can be combined with the LuGre
observer-based friction compensation scheme.
Februav 1999 0272- 1708/99/$10.00019991EEE
In order to compensate for friction torque, it is required to
provide the manipulator with the ability of accurately applying
the desired torque. This is generally not achieved in hydraulic
systems, as the servovalve (electro-hydraulic valve) controls the
flow rather than the pressure into the actuator. To provide the ro-
bot with torque control, two pressure sensors were installed at
each joint, one for each chamber of the actuator. Through differ-
ential pressure measurement it was possible to estimate the ac-
tual hydraulic torque applied by the actuators. Then, a torque
controller was designed based on the hydraulic actuators model.
In the context of the overall control problem, this torque control-
ler constitutes an inner loop for the outer position controller.
Modeling a hydraulic actuator is quite complex, , . As
indicated in [ 1 11, in principle a hydraulic actuator consists of two
oil compartments or chambers, separated by a movable part. In
linear movement actuators, this part is a piston fixed to a shaft; in
the case of the rotary actuator this part is the vane, which is con-
nected to the output shaft. The oil flows into and out of the cham-
bers are provided by the servovalve. There are three main stages.
1. The input current u drives the position of the servovalve
spool with a fast second order linear transfer functionG,7(s). Be-
cause the bandwidth of the servovalve is well beyond that of the
actuator, these dynamics were neglected.
2. The servovalve spool position controls the oil flow into the
chambers of the actuator with a nonlinear map b(u, AP), which
depends on complex internal piping and geometry of the servo-
3. This flow, combined with disturbing flows due to both ac-
tuator motion and internal leaks, controls the variation of the dif-
ferential pressure between the two compartments M = P, -P2
with nonlinear coupled dynamics. AP depends on the actuator
geometry and its position q.
This can be formally written using the continuity equations as
where B is the oil bulk modulus parameter,
ternal leakage flow between the two compartments, V , 4 models
the disturbing flow due to the actuator motion, Vr being the actua-
tor displacement corresponding to piston area in linear actuators.
A fully developed model of the nonlinear terms b(u,AP) and
@(q) for the Schilling Titan I1 actuators based on [1 11 can be
found in . A simplified form of this model is given by (1) with
represents the in-
b( U, AP) = Cd-
whereC is the servovalve constant gain, Py is the supply pressure,
and 1 IS a constant depending on actuator geometry, with
P, > sign(u) AP and12 > q2.
By using this model, a torque controller has been designed in
[lo]. Both the servovalve input nonlinearity b(u, AP) and the dis-
turbing flow V7q are compensated by this torque controller, while
the unknown disturbances, such as the leak flow (Dl, are
supposed to be rejected by a linear compensator H,(s).
The torque control law is then given by
( 4 )
where k' = APc - AP is the pressure error. AP, is the desired
value for the differential pressure that is directly computed from
the desired torque rc as A P ' = (1 I V , ) r,.
good rejection of the disturbances, including static and dynamic
leaks, the linear pressure controller H,(s) has been designed to be
a Proportional plus Integral regulator. From (1) and (4) the closed
loop dynamics are given by
In order to achieve a
where the leak flow (Dl has been modeled as (Dl = KfAP, K, be-
ing a constant parameter. As shown by ( S ) , the resulting closed
loop dynamics of the pressure subsystem does not depend any-
more on the joint velocity 4. However, it still depends on the joint
position q. This is due to the non-compensated nonlinear term
qq) in (5). A stability analysis of the torque loop has shown that
the worst position for the joint was the one which maximizes Nq)
and thus, the bandwidth of the closed loop (5). In practice, the PI
gains of H,(s) are tuned in this configuration, which is the me-
chanical limit of the actuator. The gains are adjusted in order to
provide a high inner loop bandwidth compared to the outer posi-
tion loop. Experimental results presented later show that robust-
ness of the nonlinear PI controller is good enough to provide
accurate torque control in other joint configurations. Also non-
linear PI control exhibits better closed loop response than a sim-
ple linear PI controller.
Friction Modeling and Compensation
Friction is usually modeled as a discontinuous static map be-
tween velocity and friction torque which depends on the veloc-
ity's sign. It is often restricted to Coulomb and viscous friction
components. A more complete static model is shown in Fig. 2.
However, there are several interesting properties observed in sys-
tems with friction which cannot be explained only by static mod-
els. This is basically due to the fact that friction does not have an
t Friction [Nm]
Constant Velocity [radk]
Static Friction Level
Fig. 2. Friction-(constant) velocity description.
IEEE Control Systems
instantaneous response on a change of velocity, i.e., it has inter-
nal dynamics. Examples of these dynamic properties [ 11,  are:
stick-slip motion which consists of limit cycle oscillation at
low velocities, caused by the fact that friction is larger at
rest than during motion,
presliding displacement which shows that friction behaves
like a spring when the applied force is less than the static
friction break-away force,
frictional lag which means that there is an hysteresis in the
relationship between friction and velocity.
All these static and dynamic characteristics of friction are
captured by the dynamical and analytical model proposed in ,
called LuGre, which is suitable for the design of model-based
friction compensation schemes. Experimental results on a D.C.
motor with constant parameters and adaptive compensation
schemes based on this model were reported in . The LuGre
model is given by
( 7 )
where q[rad I sec] is the angular velocity, and F[Nm] is the fric-
tion torque. Equation (6) represents the dynamics of the friction
internal state z, which describes the average relative deflection of
the contact surfaces during the stiction phases. This state is not
measurable. The function (CO > a, + a, 2 g(4) L a, > 0) de-
scribes part of the "steady-state'' characteristics of the model for
constant velocity motions: v, [rad I sec] is the Stribeck velocity,
(a, + a,)
[Nm] is the static friction, and a, is the Coulomb fric-
tion. The steady-state friction characteristics (when the velocity
q is constant and dz I dt = 0) are given by
where a2 [Nmsecl rad] represents the viscous friction. Thus,
the complete friction model is characterized by four static pa-
rameters a,, a,, a2 andv,, and two dynamic parameters on, 0,.
The parameter B, [Nm I rad] can be understood as a stiffness co-
efficient of the microscopic deformations of z [rad] during the
presliding displacement, while (T, [Nmsecl rad] is a damping
coefficient associated with dz I dt.
The dynamic equations of the robot without centripetal and
Coriolis terms are:
where H(q) is the inertia matrix, G(q) is the gravity torque vec-
tor, F is the friction torque vector and l - is the applied hydraulic
torque. For each axis i, the locally decoupled dynamics become
whereh,,(q) is theithdiagonal term ofH(q),G,(q)is theithcom-
ponent of G, rt is the torque applied to the ith joint and F, is the ith
joint friction torque given by (6) and (7).
Friction parameters are difficult to estimate since they appear
nonlinearly in the model, and the internal friction state is not
The four static parameters can be estimated by the construc-
tion of the friction-velocity map measured during constant ve-
locity motions. For constant velocity experiments, from (8) and
(10) and assuming exact gravity compensation, we have
In this work, closed-loop experiments under position PD control
with gravity compensation were performed over the six joints of
the robot. The friction-velocity data values are then obtained by
averaging the measured velocity and the input torque values.
Nonlinear optimization algorithms were used to fit the experi-
mental data to equation (1 1) with the static parameters .
The estimation of the two dynamic parameters B,, and oIt is
not possible using linear estimation techniques due to the nonlin-
ear dependence of friction with these two parameters, and to the
fact that the internal state z, is not measurable.
Nevertheless, an approximated estimation can be done. In or-
der to estimate oat, a small magnitude, slowly varying torque in-
put r, is applied in open loop. Assuming that it remains smaller
than the break-away torque, the system exhibits presliding mi-
cro-displacements. In this case, it can be assumed that ij, = 0,
4, = 0 and 2, is constant. From (10) and (7)
Thus, from equations (6) and (12), we get
This equation can be explicitly integrated to obtain z,(t) by using
an input ramp function S(t) = ct, c > 0. Assuming that the initial
configuration is ~ ~ ( 0 )
= 0, and also 4, > 0, it yields:
Therefore, from the actual measurement of q,(t) and the previ-
ously estimated values for a,, and a L, z, (t) can be computed dur-
ing a time interval (0, T). An estimation for ( T , ! can be obtained
by fitting this data in the approximate linear relationship
S(t) = B",Z,(t).
Another way of estimating ( T , , is to directly use the measures
of the micro pre-sliding displacement Aqc:
..; ........... ; ........... :
_. ......... i ........... I ........... i ...........
_ .......... i ........... j ........... j ........... i ........... k ........... ; ........... + .......... ~
15 20 25
15 20 25 30
....... j ........... j ........... i ....................... i ........... i .......... :
- .......... i ........... i .......................
_ .......... j ........... j ........... j ........... i ........... i ........... j ............ i ........
- ...................... i ........... ........................................................... .
; ........... ; ........... ; ........... :
....... i . 0 ......... j ........... i..
Fig. 3. Experimental comparison of torque controllers on joint 1:
position reference, torque reference, and torque erro'oI:
Static Friction Estimation : Axe 1
Fig. 4. Experimental friction-(constant) velocity curve
corresponding to joint 1.
-0.15 -0.1 -0.05
Static Velocity [radisec]
0.1 0.15 0.2
where Art = $( T ) - r,(O) is the differential torque value re-
quired to obtain a presliding differential micro-displacement
Aqz = q,(T) -4,(O).
Finally, in order to estimate o,,,
scription of the system (6)-(7) in the stiction phase (4, =O,
z, = 0) given by
we used the linearized de-
4, 4, +(Oh + a,, 1 4, + 0 0 , 4, = l - ; '
Here, o , , is determined such that this system has near critical damp-
ing, i.e., 0.8 < 6 < 1, using the expression o,,
This condition was found to be necessary to achieve a damped
transient response of the friction observer used for compensation
=2 6 d c
The estimation of friction parameters allows fixed and adap-
tive friction compensation of the robot, see the complete analysis
in , , and . Other friction compensation schemes based
on the LuGre model were presented in [ 121 and [ 131.
Fixed friction compensation. Fixed friction compensation
consists of designing a friction observer for the system for each
joint i. Here, for the sake of readability, the subscript i is omitted.
From the friction model given by equations (lo), (6)-(7) the ob-
z =q-o, 7
1 4 1
z-k e , k > 0
and adding it to the position controller we obtain
where e = q -q7 is the tracking error, H(s) is the position con-
troller, ? and F are the estimated internal state z and friction
torque F, respectively; k is the observer gain and h the linearized
joint inertia. The torque reference r, is the input to the inner
torque loop presented previously. The closed loop error dynam-
ics are given by the loop interconnection of two systems
e = (-F)=
h(s2 + H(s))
0, s + 0,
(-Z) = G(s) (-Z)
whereF = F-Fand? =z -z^.IfH(s)ischosensuchth_atG(s)is
strictly positive real (SPR) then the observer error, F - F, and the
position error, e, will asymptotically converge to zero.
To prove it let's introduce a state space representation of G(s)
= A( + B(-z")
and a Lyapunov function
v = ('P<+ -. k
Since G(s) is SPR, it follows from the Kalman-Yakubovich
Lemma, , thatthereexistmatricesp =PT > OandQ =QT > 0
Now evaluating the time-derivative of V along the solutions
of the closed-loop system
IEEE Control Systems
1 4 1 ? + ke)
= -c7 Q< - 2ez" + -?(
2 - 7
where the last inequality comes from the fact that g(4) > 0. The
radial unboundedness of V together with the semi-definiteness of
dV / dt implies that the states are bounded. In the regulation case
(9, constant), we can apply LaSalle's theorem-to prove that
5 -+ 0 and z + 0 which means that both e and F converge to-
wards zero and the global asymptotic stability is proven. See [ 141
for the stability proof of the general tracking case.
This result can also be understood from the fact that the ob-
server error dynamics (20) correspond to a dissipative map from
e to Z (see  for details). By adding the friction estimate to the
control signal, the position error will be the output of a linear sys-
tem operating on 2. This means that we have an interconnection
of a dissipative system with a linear SPR system. Such a system
is known to be asymptotically stable.
The SPR condition on G(s) excludes the use of a pure PID
compensator for H( s ) . If H( s) is chosen as a PD controller, then
the second order closed loop function is G(s) = I ( 6:zi+K,,
where KP and Kd are the proportional and derivative gains, re-
spectively. Thus, the SPR condition on G(s) yields:
0 Kd >=.
In practice, this condition was found to be too restrictive due
to the high frequency zero of (19). The estimated values for
CT,, / (J, are between 100 and 2000 [rads] for the different robot
joints, which results in high gains. In  it was experimentally
shown that the SPR requirements are too restrictive and do not
give rise to a sharp stability condition.
Adaptive friction compensation. Friction can change for
different reasons, such as oil temperature variations and actuator
wear. Two adaptive schemes were presented in  based on the
adaptation of only one parameter, and using the previously esti-
mated nominal friction model. The first of them is given by
where the nominal value of the parameter 0 = 1.
Here, we assume that the six nominal friction parameters and
the linearized joint inertia h are known, the dynamic friction pa-
rameters are invariant, and the variation of the static friction pa-
rameters are captured by the model (21)-(22), where 0 is
assumed to be unknown and bounded as 0 < 0 4 m. Then the fol-
lowing adaptive controller is used:
where, z,:= us -af with the following filtered signals of the ve-
locity and the applied torque
f'- 0,s + 0"
5 10 15
30 35 40 45
Position of Axe 1
5 10 15
30 35 40 45 50
Fig. 5. Presliding micro-displacement on joint 1 for estimation of 0,.
Fixed Friction ComPensation
Fig. 6. Tracking error without and with friction compensation on