INTRODUCTION
Friction is an unavoidable force which exists in all mechanical systems
involving moving parts; so it is constantly a hot topic in control community.
In certain systems such as braking system it is a desirable property,
whereas, as shown in Fig. 1, for servomechanism systems
the effects of the friction must be compensated in order to get a good
performance.
Although by using lubricating materials the effect of friction can be reduced,
this remedy is not generally feasible and practical. Consequently, different
nonlinear and adaptive control strategies have been used in the past decade
for friction compensation. Friction models may be categorized into two main
classes, namely, static and dynamic models. From a practical point of view,
dynamical effects of friction are often small and difficult to measure (Friedland
and Mentzelopoulou, 1993). On the other hand, static models with a nonlinear
map from velocity to frictional force play an important role in control system
analysis and design. Once the effects of Coulomb, viscous and other types of
friction are considered, one may develop different nonlinear maps from the velocity
variable to the frictional force.
The problem of friction modeling and estimating has been considered in the
past in many research works. Since the friction compensation concept was introduced
by Haessig and Friedland (1991), several adaptive friction
compensation controllers have been designed in many works (Friedland
and Park, 1991; Friedland and Mentzelopoulou, 1992;
Yazdizadeh and Khorasani, 1996a, b;
Liao and Chien, 2000; Kelly et al., 2004;
Suraneni et al., 2005; HuaXia
et al., 2006; Shang et al., 2008).
After the pioneering works by Friedland and Park (1991)
and Friedland and Mentzelopoulou (1992), several modifications
were introduced by Yazdizadeh and Khorasani (1996a, b)
and Liao and Chien (2000). They have focused on designing
new adaptation laws by proposing new forms of the tuning function g(v) where
v is the velocity. Mainly, their considerations were to design more stable nonlinear
adaptive controller. In the work by Friedland and Park (1991),
the asymptotically stable controller with restrictive conditions was designed.
The contribution in the work by Yazdizadeh and Khorasani
(1996a, b) which are more elaborated in this paper
was to relax those restrictive conditions imposed in the study by
Friedland and Park (1991).
Due to the recent advances in intelligent control, research in friction compensation
using an intelligent control scheme has appeared in the literature. Suraneni
et al. (2005) proposed an adaptive tracking control scheme based
on dynamic fuzzy logic system. The proposed method is an online identification
and indirect adaptive control, in which the control input is adjusted adaptively
to compensate the effects of the nonlinearities.

Fig. 1: 
A single mass servomechanism system 
A robust adaptive compensation
technique for tracking issue in a 1DOF mechanical system with stick–slip
friction is proposed by Lee and Kim (1995). However,
most of the conventional adaptive control technique relies on linear parameterizations
that usually tend to be restrictive.
In all above mentioned analytical efforts in attacking friction effects, the
Coulomb friction coefficient is assumed to be constant. In other words, the
existing stability analysis is all restricted to the time invariant friction
coefficient. But in reality, the magnitude of friction coefficient may depend
on velocity which practically is not a constant. Having considered this fact,
Ahn and Chen (2004) focused on the applications, where
the reference position and reference velocity are periodically time varying
and so the friction is also a periodic variable. An adaptive friction compensation
controller with a time varying friction coefficient was designed. In another
study by Ahn and Chen (2005), the situation in which
the friction force be related to the state is considered. The past information
of the trajectory along the state axis was used to update the current adaptation
since the friction is stateperiodic.
In this study, we first introduce and more elaborate on the original method
proposed by Yazdizadeh and Khorasani (1996a, b)
that relaxes the constraint imposed by previous works and then a situation
in which the frictional force is considered as a disturbance that may depend
on velocity, position, state or an external source is investigated. The variation
of the friction is not assumed to be restricted to periodic cases. The proposed
approach can be used to adaptively compensate the frictional force in many practical
applications including a mobile robot moving on a floor composed of different
materials with different friction coefficients that experiences different friction
forces depending on position. It can be seen from the simulation results that
the proposed approach makes more precision as well as higher speed of the convergence.
MATERIALS AND METHODS
The original lyapunovbased strategy for friction compensation: Generally,
if the nonlinearities in a system are known precisely, a control law based on
feedback linearization method can be constructed (under certain conditions)
and by proper selection of the controller gains, stability and desired performance
of the closedloop system can be ensured. In the case where for example friction
is unknown, the parameters are first estimated and based on the estimates control
command can be constructed and applied to the system. Recursive leastsquares
(RLS) and least meansquares (LMS) methods for parameter estimation for friction
compensation have been reported in the study by Canudas and Praly (2000). Basically,
the RLS and LMS algorithms are first used to estimate unknown parameters such
as viscous and Coulomb coefficients and subsequently the control command is
constructed based on the estimates. The other common approach reported in the
literature is based on Model Reference Adaptive Control (MRAC) strategy (Gilbert
and Winstone, 1974) In this approach the objective is to minimize the error
between the states of the system and the model states whose trajectories characterize
the desired signals to be followed.
Among different parameter estimation methods, the Lyapunovbased method has
shown promising results (Friedland and Park, 1992).
In the Lyapunovbased method the control command is designed (or constructed)
such that the derivative of a Lyapunov function candidate along the trajectories
of the system enjoys certain properties. For the sake of clarity and illustration,
the steps that are involved in the design of an adaptive friction compensation
for a singlemass system are worked out in detail.
Dynamic equation of a singlemass system (Fig. 1) is
given by:
Where:
f(v, k_{c}) = k_{c}
sgn(v) 
(2) 
is the Coulomb friction model, u is the effect of all forces except the
friction, a is the acceleration of the mass and m is the mass. Without
loss of generality and for the sake of simplicity we assume that m = 1,
so, the dynamic equation is written as:
or equivalently in the state space representation as:
where, x_{1} = x is the position of the mass and x_{2}
= x is its velocity. A linearizing feedback control for this system is
given by:
u = k_{c} sgn (x_{2})+g_{1}x_{1}+g_{2}x_{2} 
(5) 
that results in the closed loop system as:
where, g_{1} and g_{2} are chosen such that the system
is asymptotically stable and desired performance specifications are satisfied.
The main problem that we are faced with is that k_{c} is unknown.
Therefore, the above controller is rendered not implementable. Due to
this difficulty, it is proposed that the estimated value of k_{c}
should be used instead based on the following nonlinear expression:
where, z is an intermediate variable whose dynamics is to be specified
shortly based on a Lyapunov function candidate.
Defining the estimation error by ,
the Lyapunov function candidate is now chosen as:
Consequently, the time derivative of V along the trajectories of the
system becomes:
The term ż is at our disposal and is now selected in such a way
as to make
negative definite. By a simple manipulation, one may show that if ż
is governed by:
then we get, ė = ġ(x_{2})e and
= ġ(x_{2})e^{2}. Clearly,
is negative definite if and only if 0 < ġ(x_{2})
< K_{max}. Therefore, the conditions for asymptotic stability
of the error system may now be written as follows:
• 
ġ(x_{2}) is positive definite and g(x_{2})
is monotonically increasing 
• 
ġ(x_{2}) is bounded 
The nonlinear function g(x_{2}) is selected based
on the above two criteria. There are many functions that will satisfy
these conditions. As an example, let:
a monotonically increasing function in x_{2} with
a derivative:
bounded by 0.5, so, the nonlinear estimator is constructed according
to:
Using the above update rule, the following equations are obtained for
ė and
Implying that the error dynamics is asymptotically stable (Note that
e = 0 is an equilibrium point, so, e → 0 as t → ∞). In
order to have control over the rate of parameter convergence, the estimator
structure is now modified to:
where, the parameters k and μ are design variables that are selected
to achieve proper transient and convergence performances.
For the sake of comparison, note that the estimator by Friedland
and Park (1991) may be considered as a special case of our systematic and
general strategy in which
The above choice results in conditional asymptotic stability of the error dynamics.
In other words, the error dynamics is asymptotically stable provided that velocity
is always maintained to be bounded away from zero (Friedland
and Park, 1992). It is precisely this specific condition that is now relaxed
by our proposed approach.
It is worth noting that the principal behind our approach, Friedland and Park’s,
RLS, LMS and MRAC methods are all the same. In all the above cases the parameter
estimation is proportional to the integral of the acceleration error. For example,
in the Lyapunovbased approach,
is proportional to z and ż is proportional to
which implies that
is proportional to the integral of .
This is also the case for the RLS, LMS and MRAC (ArmstrongHelouvry
et al., 1994) methods and is an important property that one should
consider for designing other nonlinear estimators.
It is also worth noting that in all the above cases k_{c} is
assumed to be an unknown parameter which is constant. But as mentioned
before it is known that in many applications this parameter is not constant
and may vary by variation of the system position or velocity. In the next
section we propose a new method in which the constant constraint on k_{c}
is also relaxed.
The new proposed controller for time varying friction coefficient:
Similar to the previous case, we consider a singlemass system with Coulomb
friction in the state space representation which is given by:
where, x_{1} is the position of the mass and x_{2} is
the velocity, f (x_{2}) is the frictional force and may be considered
as:
f (x_{2}) = k_{c} (t) sgn (x_{2}) 
where, k_{c} (t0 is Coulomb friction coefficient. Again u is
the control input representing the effect of all forces except the friction.
Using feedback u, it is possible to make the closed loop control system
asymptotically stable. In the new proposed controller for time varying
cases, we consider the feedback control input as follow:
where, a combinational definition of error is given by:
And the error on position and velocity are given by:
e_{x} (t) = x (t) x_{d}
(t) 
(23) 
where, α>0 and λ>0 are constant gains, e_{x}
(t) and e_{v} (t) are position and velocity errors in which x_{d}
is the desired position trajectory. Again
is an estimation of the Coulomb friction coefficient. In the proposed
new method, the adaptation law is designed as follows:
where, P is a positive design parameter.
To prove asymptotic stability, the Lyapunov stability theorem and LaSalle
theorem are adopted herein. If the Lyapunov function candidate is chosen
as:
Where:
Then the time variation of the V(t) on a period of time along the trajectory
of the closedloop system leads to:
It can be easily shown that if α+P/2>0, then ΔV ≤ 0.
Using the Invariant set theorem, it can be seen that the control law (Eq.
21) and the periodic adaptation law (Eq. 25) guarantee
the asymptotically stability of the equilibrium points as t → ∞.
RESULTS AND DISCUSSION
Simulation results: Here, the original proposed method is first
applied to a simple singlemass system in and then applied to a more complicated
system such as a twolink robot manipulator in, to compensate for friction.
For the sake of comparison between the Friedland and Park’s scheme
and our proposed scheme and in addition to see how the selection of g
(x_{2}) affects the results, singlemass system will
be simulated with different desired trajectories (different amplitudes)
and different Coulomb friction coefficients. It will be shown that friction
compensation by our original method is robust to amplitude variation of
the desired trajectory and to the variation of Coulomb friction coefficients
(different environments and surfaces), whereas the Friedland and Park’s
method is not robust to amplitude variation of the desired trajectory
and to the variation of Coulomb friction coefficients. The capabilities
of the new proposed controller for time varying cases are also shown by
performing some simulation. The simulation results in this part are compared
with the recent works for periodic time varying case presented by Ahn
and Chen (2004, 2005).
A singlemass system with the original proposed method: The dynamic
nonlinear equations for this system and the linearizing feedback control
law are as follows:
with
where, g_{1} and g_{2} are chosen so that the damping
ratio and the natural frequency of the closedloop system are given by
ξ = 0.707 and ω = 10 rad sec^{1}. This results in g_{1}
= 200 and g_{2} = 20. Also k_{c} is selected as k_{c}
= 50.
To see the advantage of the original proposed scheme, the closedloop
system is first simulated with no compensation for the cases with and
without friction. Figure 2 shows the result. It is clear
that without friction the output x_{1} follows the desired trajectory
quite satisfactorily, whereas when friction is present the same control
results in an unacceptable output tracking due to the presence of large
error. In other words, there is not only a steadystate tracking error
but also a considerable time delay. Our design goal is to guarantee that
the output tracks the desired trajectory even when the friction is present.
Figures 3a, b depict the simulation results for the
case when the proposed friction compensation is implemented.
As shown almost perfect tracking is achieved. The nonlinear function
g (x_{2}) = ke^{μx2} that
satisfies the two aforementioned criteria has been used for these simulations.
The desired trajectory is a square wave lower bounded by 1 and upper
bounded by 1 with a period of 2 sec.
It is possible to adjust the parameters k and μ so, that almost
the same results are achieved by the Friedland and Park’s method.
Figure 4a, b show the simulation results for the same
system using Friedland and Park’s method.

Fig. 2: 
Performance of the closedloop system with () and
without (— —) friction compensation 
Now to demonstrate the robustness of our proposed method subject to variation
in the friction coefficient as well as different desired trajectories,
the adaptive system is first simulated for a new desired trajectory, namely,
the same square wave but with amplitude of 0.1. Figure
3c, d depict the simulation results for this case. As shown almost
perfect tracking is achieved in this case too. However, when Friedland
and Park’s method is used the output does not track the desired trajectory
perfectly. Figure 4c, d show that the output is almost
zero for the first 5 sec and the parameter estimate converges to its true
value very slowly.
The second advantage of our proposed method is its robustness to variation
in the Coulomb friction coefficient. Figure 3e, f depict
the simulation results for the case when the friction coefficient is 500.
As shown almost perfect tracking is achieved by our method confirming
its robustness to Coulomb friction coefficient variation. However, as
shown in Fig 4e and f, the tracking result by the Friedland
and Park’s method is not satisfactory.
A twolink planar robot manipulator with the original proposed method:
The dynamic equations of a twolink rigid manipulator are given by Craige
(1989) as:
where, τ is the applied torque, θ is the position vector, that
is:



Fig. 3: 
The results for the original proposed scheme. (a, c,
e) Depict the performance of the closedloop system and (b, d, f)
Depict the parameter estimate for (k_{c} = 50, x_{d}
= 1), (k_{c} = 50, x_{d} = 0.1) and (k_{c}
= 500, x_{d} = 1) respectively. In all cases k = 135 and μ
= 1 
M (θ) is the symmetric positive definite mass matrix:
is the centrifugal and Coriolis forces:
G (θ) is the term due to gravity:
and
is the coulomb friction that is modeled by:
By using partitioned controller design, τ is chosen as:
with
and
where,
and
and θ_{d} are desired trajectories with θ_{1d}
= sin (πt) and θ_{2d} = 5 sin (πt). The gains k_{v}
and k_{p} are chosen such that desired performance specifications
are satisfied. Since,
is assumed to be unknown, the estimates of k_{c1} and k_{c2}
based on the results of the previous section are used in the controller.
In other words, the modified controller is given by:
By simple manipulations, the output error equations are now derived as:



Fig. 4: 
The results for the Friedland and Park’s scheme, Figure a, c and e
depict the performance of the closedloop system and Figure
b, d and f depict
the parameter estimate for (k _{c} = 50, x _{d} = 1),
(k _{c} = 50, x _{d} = 0.1) and (k _{c} = 500,
x _{d} = 1) respectively. In all cases k = 10 and μ =
2 

Fig. 5: 
Performance of the closedloop system with (  ) and
without (..) friction compensation. The solid line is the desired
trajectory. Figure a shows the first joint position
θ and Figure b shows the second joint position
θ _{2} with k _{1} = 10, μ _{1} = 1,
k _{2} = 100 and μ _{2} = 1 

Fig. 6: 
Performance of the new proposed controller for position
tracking error where, 

Fig. 7: 
Performance of the proposed method by Ahn
and Chen (2004) for position tracking error, where, 
Clearly, if
and ,
then it follows that e_{1} → 0 and e_{2} → 0
as t → ∞. Simulation results are performed using the following
parameters: m_{1} = 2, m_{2} = 1, l_{1} = 2, l_{2}
= 1, k_{v1} = k_{v2} = 20, k_{p2} = 200, k_{ci}
= 500 and k_{c2} = 100.
The estimator is constructed by taking g (x2) = ke^{μx2}.
Figures 5a, b show the tracking performance of the system
with and without friction compensation. Despite the large magnitude of
the friction in the first link, the adaptive estimator successfully compensates
for the friction.
A singlemass system with the new proposed controller for time varying case:
In this section the new proposed controller for time varying case is applied
to a simple singlemass system. The simulation results of the proposed method
as compared to the methods by Ahn and Chen (2004, 2005)
are presented in Fig. 69 for two different
cases, namely, k_{c} (t) as a periodic signal and k_{c} (t)
as a state depended parameter.

Fig. 8: 
Performance of the new proposed controller for position
tracking error where, k_{c} (t) = [50+5sin(2πx (t))+2sin(4πX(t))
+2sin(4πX(t))+sin(6πX(t))] sgn (v) 

Fig. 9: 
Performance of the proposed method by Ahn
and Chen (2005) for position tracking error where, k _{c} (t)
= [50+5sin(2πx (t))+2sin (4πX(t)) +2sin (4πX(t)) +sin (6πX(t))]
sgn (v) 
Desired trajectory of the position is d_{x} (t) = sin πt and α, λ are selected as α = 10 and α
= 10.
In the first case, the coulomb friction coefficient k_{c} (t) is considered
as a periodic signal. The position tracking error signal achieved by the proposed
new method is shown in Fig. 6 and the result of the proposed
method by Ahn and Chen (2004) is given in Fig.
7.
In the second case, the coulomb friction coefficient k_{c} (t) is considered
as a state depended parameter. The position tracking error signal achieved by
the new proposed controller is shown in Fig. 8 and the result
of the proposed method by Ahn and Chen (2005) is given
in Fig. 9. As can be seen from the Fig. 9,
the proposed approach has more precision and higher rate of convergence.
CONCLUSION
The new nonlinear adaptive scheme based on Lyapunov analysis developed in the
work by Yazdizadeh and Khorasani (1996a, b)
to estimate and to compensate for friction are more elaborated in this paper.
The method is applied to a simple mass system and a twolink rigid robot manipulator.
The proposed strategy adaptively compensates for unknown static Coulomb friction
nonlinearities. The method is general and systematic in construction. It was
shown analytically that the estimation error dynamics is asymptotically stable
without requiring a constraint on the velocity. Simulation results confirm the
robustness and the advantages of the proposed scheme compared to the other similar
works in the literature. The shortage and drawback of the proposed method is
in its assumption on k_{c}. The proposed original method, similar to
many other references, assumes the friction coefficient as a constant. To remove
this assumption another new controller is proposed in this study. Simulation
results of the new proposed method confirm the advantages of the proposed scheme
compared to the other main similar works in the literature.