An Efficient Identification Method for Friction in Single-DOF Motion ...
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 4, JULY 2004
555
An Efficient Identification Method for Friction in Single-DOF Motion Control Systems Seung-Jean Kim, Sung-Yeol Kim, and In-Joong Ha
Abstract—This brief describes a new identification method for friction in single-degree-of-freedom (DOF) motion control systems, in which the friction model is not necessarily linear in parameters. The proposed method works well with any measurement data of velocity and input control force, as long as the initial and final velocities are identical. Most importantly, the proposed method requires neither the information of acceleration nor the accurate information of mass for its implementation, in contrast with the previously known offline identification methods. This feature is due to the orthogonality property between acceleration and any function of velocity, which is a unique property of single-DOF motion control systems. The performance of the proposed method is demonstrated through some simulation and experimental results. Index Terms—Friction, mechanical systems, motion control, parameter estimation, parameter identification, time domain analysis.
I. INTRODUCTION
I
N motion control systems, friction is a primary source of disturbance, which can degrade control performance significantly. In particular, motion control systems usually exhibit relatively large steady-state tracking errors or even oscillations if the controllers are designed without considering friction. In this context, a great deal of research works have been done on finding effective compensation methods for friction in motion control systems [7]–[16]. Also, see the survey paper [1] and the vast literature therein. One such control method is the model-based friction compensation technique. It is well known that its performance is highly dependent on the accuracy of the friction model used in the friction compensation loop. For this reason, friction identification has been one of the most important issues in the design of high-performance motion control systems [12]–[16]. Until now, a lot of attention has been paid to mathematical modeling for friction. Recently, some researchers have developed some dynamic models which can better predict friction phenomenon [2]–[4]. However, these dynamic friction models require the estimation of certain fictitious state variables. Moreover, estimating the parameters of the dynamic models requires solving a complicated nonlinear-optimization problem. As a matter of fact, static-friction models have been widely used in control and identification of most motion control systems, since the friction can be well approximated as a function of velocity [8]–[15].
Manuscript received July 18, 2001. Manuscript received in final form June 16, 2003. Recommended by Associate Editor D. Dawson. This work was supported by the SNU BK21-IT Program. The authors are with ACRC/ASRI, School of Electrical Engineering, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2004.825145
In general, friction identification methods can be classified into two categories, online and offline methods [1]. The most well-known online identification methods are the least mean squares (LMS) methods and recursive least squares (RLS) methods. Unfortunately, however, these two methods cannot be applied directly to such empirical parametric models of friction as Gaussian model and Lorentzian model [1], which are not linear in parameters. On the other hand, friction identification using offline methods is carried out after data-acquisition experiments. As a result, offline methods can deal with more general parametric models of friction such as Gaussian model and Lorentzian model [1]. In this brief, we develop a new offline identification method for friction in single-degree-of-freedom (DOF) motion control systems using static-friction models, which are not necessarily linear in parameters. The proposed method works well with any measurement data of velocity and input-control force, as long as the initial and final velocities are identical. Most importantly, the proposed method dose not rely on the information of acceleration and mass, in contrast with the previously known offline identification methods. This feature is due to the orthogonality property between acceleration and any function of velocity, which is a unique property of single-DOF motion control systems. The performance of the proposed method is demonstrated through some simulation and experimental results. This brief is organized as follows. In Section II, we formulate the friction identification problem in single-DOF motion control systems. In Section III, we show that it can be solved without the information of acceleration and the prior knowledge of mass when the initial and final velocities are identical. In Section IV, we demonstrate the good performance of the proposed method through various simulation and experimental results. Finally, the conclusions are summarized in Section V. II. PROBLEM STATEMENT One-dimensional motion of a mass which moves on a surface with friction is governed by (1) where , , , and represent the mass, the velocity of the mass, the friction force, and the control force applied to the mass, respectively. The friction force in (1) is modeled as (2) Here, the function represents the slip-friction force at nonzero velocity. Hence, it is natural to assume that
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In the low-velocity region, the friction force decreases with increasing velocity, as shown in Fig. 1. This phenomenon is widely known as Stribeck effect. Many empirical parametric models such as Gaussian model and Lorentzian model have been proposed to describe the Stribeck effect [1]. On the other represents the friction force at zero hand, the stick friction velocity. The mathematical model is given by if if
(4)
where if if if
. Fig. 1.
Let us suppose that the slip friction the following parametric model with rameters :
Stribeck effect.
can be described by unknown friction pa(5)
represents the error inevitably involved Here, the function in parameterizing the slip friction . Then, the friction model in (2) can be written as (6) where
(7) Now the system in (1) is excited to acquire the data used to identify the friction parameter vector . The data acquisition can be carried out in an open-loop setting or in a closed-loop setting. Let us suppose that the signals and in (6) are measured over an interval , . Then, we can find the optimal estimate of the parameter vector by solving the following optimization problem:
III. IDENTIFICATION METHOD In our development, we make the following assumptions: , is piecewise continuous and (A.1) For each (9) (A.2) is finite. As a matter of fact, all the previously known parametric models of static friction, including Gaussian model and Lorentzian model [1] satisfy the above assumptions. Furthermore, all those . In particular, in view of (3), it is models satisfy natural to assume (9). such that We now arrange the friction parameters (10) where is the vector consisting of the parameters to is the vector consisting of the which is affine, and other parameters to which is not affine. As a result, we can write the parametric model in the following form: (11)
(8) where is the admissible parameter-vector set. Here and in what follows, indicates that minimizes the cost function over , namely, . We instead write , in particular when is known to be unique. Observe that solving the optimization problem (P) involves acquiring the acceleration numerically due to the presence of the term in . However, this can cause large parameter identification error. Furthermore, the identification accuracy of the parameter vector is directly affected by the information accuracy of . In the next section, we show that the optimization problem (P) can be solved without the information of the acceleration and the mass , provided that the initial and final velocities are identical, i.e., . Hence, we can solve the optimization problem (P) by measuring only the signals and , which are readily accessible in practical applications.
where and
:
,
, : (12)
Finally, we make the following assumption on the admissible parameter-vector set : (A.3) The vector takes its value in a compact subset of , that is, . In what follows, we use the following notation:
(13) Here, .
is symmetric and positive semidefinite, for all
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We are now ready to state our main result. Theorem 1: Suppose that during the identification period , the trajectory of velocity satisfies the following two conditions: (14) (15)
where
and
is positive - definite for all Moreover, if problem:
. Then
Fig. 2.
The closed-loop system.
Then, it is easy to see that the unique solution of the optimization problem (P) can now be written as
(16)
is a solution of the following optimization
(17)
then ( , ) is a solution of the optimization problem (P). Conversely, if ( , ) is the solution of the . optimization problem (P), then The proof is given in Appendix A. It should be noted that in contrast with (P), the cost function of (L) depends on neither the acceleration , nor the mass . Consequently, by solving the reduced optimization problem (L), we can identify the parameter vector without the information of the acceleration and the mass . In fact, the first condition in (14) simply requires that the initial and final values of the velocity are identical. By a proper data-acquisition experiment, it can be easily met in practical applications, as will be seen in the next section. On the other hand, the second condition in (15) merely states that the excitation of the system in (1) in the data acquisition experiment should be sufficient for successful identification of the parameter vector. , this assumption is For a parametric model with , i.e., the mass should equivalent to requiring that not move at a constant velocity. Generally speaking, the solution of the reduced optimization problem (L) cannot be found in closed form due to the nonlinear nature of its cost function. Instead, it should be sought via some numerical search methods. However, if the parametric model in (5) is linear with respect to all the friction parameters, then its optimal friction parameters can be found in closed form. To show this, observe first that if the parametric model in (5) is , then linear with respect to all the friction parameters such that there exist basis functions
(18)
(19) The overall identification method for the slip friction is described below. ): Select an appropriate parametric model of the slip i) friction. ): Determine the corresponding parameter vectors ii) and in (10), and compute . ): Conduct a data acquisition experiment such that iii) the measured data of velocity satisfies two conditions stated in (14) and (15). ): Find by solving the optimization problem (L) in iv) (17) with the measured data of velocity . ): Construct the identified slip friction model as v) follows:
From now on, we illustrate the application of the above )– ) to the identification of the slip friction, method using two parametric models. First, we consider asymmetric Gaussian model defined by
(20) where if if
Here, , , , , , , , and are the friction parameters to be identified. Obviously, it is affine , , , , , and , but not to and to . Then, it is straightforward to see that . Note that it takes the form in (11) with
(21)
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Fig. 3. Time responses of the closed-loop system (simulation).
where
where
It is known that the typical values of and range from to , depending on systems [1]. Then, the admissible parameter vector set for asymmetric Gaussian model in (20) can be written as (22)
We can therefore identify the friction parameters , , , , , , , and of asymmetric Gaussian model in (20) by solving the following two separate one-dimensional optimization problems:
where
, , . The corresponding optimization problem (L) then becomes
where
and
(24a)
(23a)
(24b)
(23b)
The above two optimization problems can be solved via exhaustive search methods without resorting to sophisticated numerical search methods, as will be seen in the next section. As a second example, we use asymmetric Coulomb viscous friction model, which is defined by
are defined in (13). Using the fact that , , we can further simplify the expression of the cost function of the optimization problem in (23) as follows:
(25) Here, , , , and are the friction parameters to be identified. In other words, the above parametric model corre-
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TABLE I TRUE VALUES OF MODEL PARAMETERS ASSUMED IN SIMULATION
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TABLE II IDENTIFIED VALUES OF MODEL PARAMETERS IN SIMULATION. (A) ASYMMETRIC COULOMB VISOUS FRICTION MODEL; (B) ASYMMETRIC LORENTZIAN MODEL; (C) ASYMMETRIC GAUSSIAN MODEL
+
sponds to the case when the Stribeck effect is ignored in identification process. Obviously, the friction model in (25) takes the form in (18) with
The corresponding optimal estimate (19). Using the fact that as follows: the optimal estimate
is then determined by , , we can find
(26) We conclude this section by addressing the uniqueness of the solution of the optimization problem (P). In fact, unless some additional assumptions are imposed on the parametric model , the admissible parameter vector set , and the velocity trajectory used in identification process, the solution is not necessarily unique due to the presence of the parameterization error . However, in the absence of parameterization error, i.e., , the solution of the optimization problem (P) is always unique. The uniqueness result is summarized in the following theorem. Theorem 2: Suppose that there exists a parameter vector such that (27) Then, the solution of the optimization problem (P) is uniquely given by . The proof is given in Appendix B.
Fig. 4. Identification result using ACVM (simulation). Solid line: true slip friction curve; dotted line: identification using ACVM.
over the interval and use them to identify the friction curve. In our simulation work, we assume that the true slip fricin (2) can be described by the following asymmetric tion Lorentzian model: (29) where
IV. SIMULATION AND EXPERIMENTAL RESULTS In this section, we present some simulation and experimental results to illuminate further the practical use of the proposed identification method )– ). In our simulation and experimental works, we carry out the data acquisition in a closed-loop is taken as the setting, as shown in Fig. 2. The controller with proportional-derivative (PD) controller and . the control gains Then, the following periodic function is chosen as an excitation input to the corresponding closed-loop system: (28) The time responses of the closed-loop system to the excitation input in (28) are shown in Fig. 3. As can be seen from Fig. 3, the steady oscillation is established after two periods. In our simulation work, we measure the velocity and the control force
The true values of the mass and the friction coefficients, which are assumed in our simulation work, are given in Table I. by using three We identify the slip friction function parametric models of friction, asymmetric Coulomb + viscous friction model (ACVM), asymmetric Gaussian friction model (AGM), and asymmetric Lorentzian model (ALM). To identify the friction parameters of ACVM, it suffices to calculate (26) using the measured signals and . The identification results are given in Table II. The identified friction model is plotted graphically in Fig. 4. As can be seen from Fig. 4, it describes the actual slip friction quite well except at low velocity where the Stribeck effect is noticeable.
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Fig. 6. Experimental setup.
Fig. 5. Identification result using AGM (simulation). Solid line: true slip friction curve; dotted line: identification using AGM.
In order to identify accurately the shape of the slip friction at low velocity, we have to use a parametric model taking the Stribeck effect into consideration. The second parametric model is AGM in (20). To identify the friction parameters of AGM, it suffices to solve the optimization problems in (24). To do this, over the we first choose N equally-spaced data points . In all simulation and experiment works, the value range of is chosen to be 100. Next, we solve numerically the optimization problems in (24) The identification results are given in Table II. As can be seen from Fig. 5, using AGM provides better identification accuracy, although computationally more complex, than using ACVM. The third parametric model is ALM in (29), which takes the form in (11) with
Here, the typical values of and range from to , depending on systems [1]. Then, the admissible , parameter vector set for ALM in (29) is given by , , and where .
Through some arguments similar to those used to derive (24), , , , , we can show that the friction parameters , , , and of ALM in (29) can be identified by solving the following two separate one-dimensional optimization problems:
(30a)
(30b) where
The above optimization problems are solved via exhaustive grid points uniformly sampled over the search with . The identification results are given in Table II. interval As can be seen from Table II, the proposed method can identify consistently the friction parameters of ALM used in simulation, since there exists no parameterization error. Next, we demonstrate the practical use of the proposed iden-potification method through some experiment using an sitioning system. The schematic diagram of our experimental setup is shown in Fig. 6. For our experimental results, we activated only its axis. The linear resolution of the optical encoder . The velocity information is obtained using is the well-known M/T method based on the encoder signal. The excitation input and the gains of the PD controller used in our experiment were chosen to be the same as those used in
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Fig. 7. Time responses of the closed-loop system during steady oscillation (experiment).
TABLE III IDENTIFIED VALUES OF MODEL PARAMETERS IN EXPERIMENT
the simulation. In our experimental work, we measured the ve, as locity and the control force over the interval done in the simulation. The measured data are plotted graphically in Fig. 7. In our experimental work, we have used AGM. The identification results are summarized in Table III. Note that the true slip friction is not known. In this context, we evaluate the accuracy of the experimental identification results in an indirect way. This evaluation method is based on the so-called constant velocity test [1], in which the mass is forced to move at a constant velocity . Then, the control is equal to the friction force . In Fig. 8, the force identification results obtained from this procedure are marked by , while the curve of AGM with the identified values of model parameters given in Table III is plotted in solid line. This figure directly demonstrates that the proposed method can accurately at the velocity region identify the slip friction where the Stribeck effect is negligible. However, the accuracy of the proposed identification method in the region of very low velocity cannot be evaluated, since it was impossible to control the velocity of the mass accurately in that region. The main reason lies in the performance limitation of the servo pack and the low resolution of the optical encoder used in our experimental setup. V. CONCLUSION In this brief, we have proposed a novel identification method for friction in motion control systems and have demonstrated its
Fig. 8. Direct evaluation of identification accuracy (experiment). Solid line: AGM with the identified values of model parameters in Table III; +: identification results obtained through constant-velocity tests.
generality and practical use through various simulation and experimental results. In contrast with the previously known offline identification methods, the proposed method does not require the information of acceleration, which is not practically available without large error, since it can exploit the orthogonality property between acceleration and any function of velocity. In this context, we believe that the proposed method should find a wide range of applications in industry. Nonetheless, some important issues remain untouched. First, the issue of model selection is important. In fact, the identification accuracy depends on the parametric model used for identifying the slip friction function. Thus, it would be important to find a good parametric model of the slip friction from the viewpoint of identification. Second, when the parameters are varying with time because of changes in operating conditions, aging of
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equipment and so forth, offline identification methods become ineffective. In this context, it would also be an interesting topic of further research to develop an online identification method for friction based on the orthogonality property. APPENDIX A PROOF OF THEOREM 1
(36) Now, suppose that (17). Then,
is a solution of the optimization (L) in
We first state some useful properties of the functions , and below which can be easily deduced from (A.1)–(A.3) and the condition in (15). and for each For all are piecewise continuous are linearly independent on
(31) (32) (33)
To prove Theorem 1, we need the following lemma. Lemma 1 (Orthogonality Property): Suppose that the velocity signal satisfies the condition in (14). Then, for any piecewise continuous
(37) This along with (35) and (36) implies that
Proof: Observe from the piecewise continuity of that the function is bounded over the the function . Then, there exists a sequence of interval such that continuous functions . Then, the well-known Lebesgue Convergence theorem [17] along with the condition in (14) implies that
where . Proof of Theorem 1: By way of contradiction, we first show is singular, that the assertion in (16) is true. Suppose that . Then, there exists a nonzero vector for some such that . This along , a.e. on , with (31) implies that which is contradictory to (33). Consider the following optimization problem depending on : (34) As will be seen soon, the solution of the optimization problem in (34) is uniquely given by
(38) In other words, ( , ) is the solution of the optimization problem (P) in (8). ) is a solution of Conversely, suppose that ( , the optimization problem (P). Then, it is obvious that . Thus, this along with (35) . shows that We complete the proof by showing (35) and (36). It is easy to see that the optimization problem in (34) has the unique solution given by (39)
(35) such that
It follows from (32) and Lemma 1 that: (40)
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Further, it follows from (4) that:
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Finally, (36) is the direct consequence of (44) and (47). (41)
and, hence, that
APPENDIX B PROOF OF THEOREM 2 Note from (27) that and hence that
(42)
(48)
As the consequence of (7), (40), and (42), we obtain that
(43) This along with (39) leads to the fact stated in (35). On the other hand, we can see through some direct calculations that
(44) Here, it follows from Lemma 1 that (45) Through some arguments to used to derive (42), we can show that (46) This along with (45) implies that
(47)
such Suppose that there exists a parameter vector . Note that that . Hence, it holds that . This along with (48) implies that . This, in turn, implies that , a.e. on and hence that , a.e. on . Finally, It follows from the condition in (15) that . REFERENCES [1] B. Armstrong-Hélouvry, P. Dupont, and C.Canudas de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138, 1994. [2] B. Armstrong-Hélouvry, “Stick slip and control in low-speed motion,” IEEE Trans. Automat. Contr., vol. 38, pp. 1483–1496, Oct. 1993. [3] P. Dahl, “A Solid Friction Model,” Aerosp. Corp., El Segundo, CA, Technical report TOR-0158(3107-18)-1, 1968. [4] C.Canudas de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, pp. 419–425, Mar. 1995. [5] D. P. Hess and A. Soom, “Friction at a lubricated line contact operating at oscillating sliding velocities,” J. Tribology, vol. 112, no. 1, pp. 147–152, 1990. [6] L. C. Bo and D. Pavelescu, “The friction-speed relation and its influence on the critical velocity of the stick-slip friction,” Wear, vol. 82, no. 3, pp. 277–289, 1982. [7] S. I. Cho and I. J. Ha, “A learning approach to tracking in mechanical systems with friction,” IEEE Trans. Automat. Contr., vol. 45, pp. 111–116, Jan. 2000. [8] C. Canudas de Wit, “Robust control for servo-mechanisms under inexact fricton compensation,” Automatica, vol. 29, no. 3, pp. 757–761, 1993. [9] P. Vedagarbda, D. M. Dawson, and M. Feemster, “Tracking control of mechanical systems in the presence of nonlinear dynamic friction effects,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 446–456, July 1999. [10] R. M. Hirschorn and G. Miller, “Control of nonlinear systems with friction,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 588–595, May 1999. [11] M. R. Popovic, D. M. Gorinevsky, and A. A. Goldenberg, “High-precision positioning of a mechanism with nonlinear friction using a fuzzy logic pulse controller,” IEEE Trans. Contr. Syst. Technol., vol. 8, pp. 151–158, Jan. 2000. [12] M. R. Elhami and D. J. Brookfield, “Sequential identification of Coulomb and viscous friction in robot drives,” Automatica, vol. 33, no. 3, pp. 393–401, 1997. [13] B. Friedland and Y. J. Park, “On adaptive friction compensation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1609–1612, Oct. 1992. [14] C.Canudas de Wit, K. J. Åström, and K. Braun, “Adaptive friction compensation in DC motor drives,” IEEE J. Roboti. Automat., vol. RA-3, Dec. 1987. [15] H. S. Lee and M. Tomizuka, “Robust motion controller design for highaccuracy positioning systems,” IEEE Trans. Ind. Electron., vol. 43, pp. 48–55, Feb. 1996. [16] S. J. Kim and I. J. Ha, “A frequency-domain approach to the identification of mechanical systems with friction,” IEEE Trans. Automat. Contr., vol. 46, pp. 888–893, June 2001. [17] E. Hewitt and K. Stromberg, Real and Abstract Analysis. New York: Springer-Verlag, 1967.
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An Efficient Identification Method for Friction in Single-DOF Motion Control Systems Seung-Jean Kim, Sung-Yeol Kim, and In-Joong Ha
Abstract—This brief describes a new identification method for friction in single-degree-of-freedom (DOF) motion control systems, in which the friction model is not necessarily linear in parameters. The proposed method works well with any measurement data of velocity and input control force, as long as the initial and final velocities are identical. Most importantly, the proposed method requires neither the information of acceleration nor the accurate information of mass for its implementation, in contrast with the previously known offline identification methods. This feature is due to the orthogonality property between acceleration and any function of velocity, which is a unique property of single-DOF motion control systems. The performance of the proposed method is demonstrated through some simulation and experimental results. Index Terms—Friction, mechanical systems, motion control, parameter estimation, parameter identification, time domain analysis.
I. INTRODUCTION
I
N motion control systems, friction is a primary source of disturbance, which can degrade control performance significantly. In particular, motion control systems usually exhibit relatively large steady-state tracking errors or even oscillations if the controllers are designed without considering friction. In this context, a great deal of research works have been done on finding effective compensation methods for friction in motion control systems [7]–[16]. Also, see the survey paper [1] and the vast literature therein. One such control method is the model-based friction compensation technique. It is well known that its performance is highly dependent on the accuracy of the friction model used in the friction compensation loop. For this reason, friction identification has been one of the most important issues in the design of high-performance motion control systems [12]–[16]. Until now, a lot of attention has been paid to mathematical modeling for friction. Recently, some researchers have developed some dynamic models which can better predict friction phenomenon [2]–[4]. However, these dynamic friction models require the estimation of certain fictitious state variables. Moreover, estimating the parameters of the dynamic models requires solving a complicated nonlinear-optimization problem. As a matter of fact, static-friction models have been widely used in control and identification of most motion control systems, since the friction can be well approximated as a function of velocity [8]–[15].
Manuscript received July 18, 2001. Manuscript received in final form June 16, 2003. Recommended by Associate Editor D. Dawson. This work was supported by the SNU BK21-IT Program. The authors are with ACRC/ASRI, School of Electrical Engineering, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2004.825145
In general, friction identification methods can be classified into two categories, online and offline methods [1]. The most well-known online identification methods are the least mean squares (LMS) methods and recursive least squares (RLS) methods. Unfortunately, however, these two methods cannot be applied directly to such empirical parametric models of friction as Gaussian model and Lorentzian model [1], which are not linear in parameters. On the other hand, friction identification using offline methods is carried out after data-acquisition experiments. As a result, offline methods can deal with more general parametric models of friction such as Gaussian model and Lorentzian model [1]. In this brief, we develop a new offline identification method for friction in single-degree-of-freedom (DOF) motion control systems using static-friction models, which are not necessarily linear in parameters. The proposed method works well with any measurement data of velocity and input-control force, as long as the initial and final velocities are identical. Most importantly, the proposed method dose not rely on the information of acceleration and mass, in contrast with the previously known offline identification methods. This feature is due to the orthogonality property between acceleration and any function of velocity, which is a unique property of single-DOF motion control systems. The performance of the proposed method is demonstrated through some simulation and experimental results. This brief is organized as follows. In Section II, we formulate the friction identification problem in single-DOF motion control systems. In Section III, we show that it can be solved without the information of acceleration and the prior knowledge of mass when the initial and final velocities are identical. In Section IV, we demonstrate the good performance of the proposed method through various simulation and experimental results. Finally, the conclusions are summarized in Section V. II. PROBLEM STATEMENT One-dimensional motion of a mass which moves on a surface with friction is governed by (1) where , , , and represent the mass, the velocity of the mass, the friction force, and the control force applied to the mass, respectively. The friction force in (1) is modeled as (2) Here, the function represents the slip-friction force at nonzero velocity. Hence, it is natural to assume that
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In the low-velocity region, the friction force decreases with increasing velocity, as shown in Fig. 1. This phenomenon is widely known as Stribeck effect. Many empirical parametric models such as Gaussian model and Lorentzian model have been proposed to describe the Stribeck effect [1]. On the other represents the friction force at zero hand, the stick friction velocity. The mathematical model is given by if if
(4)
where if if if
. Fig. 1.
Let us suppose that the slip friction the following parametric model with rameters :
Stribeck effect.
can be described by unknown friction pa(5)
represents the error inevitably involved Here, the function in parameterizing the slip friction . Then, the friction model in (2) can be written as (6) where
(7) Now the system in (1) is excited to acquire the data used to identify the friction parameter vector . The data acquisition can be carried out in an open-loop setting or in a closed-loop setting. Let us suppose that the signals and in (6) are measured over an interval , . Then, we can find the optimal estimate of the parameter vector by solving the following optimization problem:
III. IDENTIFICATION METHOD In our development, we make the following assumptions: , is piecewise continuous and (A.1) For each (9) (A.2) is finite. As a matter of fact, all the previously known parametric models of static friction, including Gaussian model and Lorentzian model [1] satisfy the above assumptions. Furthermore, all those . In particular, in view of (3), it is models satisfy natural to assume (9). such that We now arrange the friction parameters (10) where is the vector consisting of the parameters to is the vector consisting of the which is affine, and other parameters to which is not affine. As a result, we can write the parametric model in the following form: (11)
(8) where is the admissible parameter-vector set. Here and in what follows, indicates that minimizes the cost function over , namely, . We instead write , in particular when is known to be unique. Observe that solving the optimization problem (P) involves acquiring the acceleration numerically due to the presence of the term in . However, this can cause large parameter identification error. Furthermore, the identification accuracy of the parameter vector is directly affected by the information accuracy of . In the next section, we show that the optimization problem (P) can be solved without the information of the acceleration and the mass , provided that the initial and final velocities are identical, i.e., . Hence, we can solve the optimization problem (P) by measuring only the signals and , which are readily accessible in practical applications.
where and
:
,
, : (12)
Finally, we make the following assumption on the admissible parameter-vector set : (A.3) The vector takes its value in a compact subset of , that is, . In what follows, we use the following notation:
(13) Here, .
is symmetric and positive semidefinite, for all
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We are now ready to state our main result. Theorem 1: Suppose that during the identification period , the trajectory of velocity satisfies the following two conditions: (14) (15)
where
and
is positive - definite for all Moreover, if problem:
. Then
Fig. 2.
The closed-loop system.
Then, it is easy to see that the unique solution of the optimization problem (P) can now be written as
(16)
is a solution of the following optimization
(17)
then ( , ) is a solution of the optimization problem (P). Conversely, if ( , ) is the solution of the . optimization problem (P), then The proof is given in Appendix A. It should be noted that in contrast with (P), the cost function of (L) depends on neither the acceleration , nor the mass . Consequently, by solving the reduced optimization problem (L), we can identify the parameter vector without the information of the acceleration and the mass . In fact, the first condition in (14) simply requires that the initial and final values of the velocity are identical. By a proper data-acquisition experiment, it can be easily met in practical applications, as will be seen in the next section. On the other hand, the second condition in (15) merely states that the excitation of the system in (1) in the data acquisition experiment should be sufficient for successful identification of the parameter vector. , this assumption is For a parametric model with , i.e., the mass should equivalent to requiring that not move at a constant velocity. Generally speaking, the solution of the reduced optimization problem (L) cannot be found in closed form due to the nonlinear nature of its cost function. Instead, it should be sought via some numerical search methods. However, if the parametric model in (5) is linear with respect to all the friction parameters, then its optimal friction parameters can be found in closed form. To show this, observe first that if the parametric model in (5) is , then linear with respect to all the friction parameters such that there exist basis functions
(18)
(19) The overall identification method for the slip friction is described below. ): Select an appropriate parametric model of the slip i) friction. ): Determine the corresponding parameter vectors ii) and in (10), and compute . ): Conduct a data acquisition experiment such that iii) the measured data of velocity satisfies two conditions stated in (14) and (15). ): Find by solving the optimization problem (L) in iv) (17) with the measured data of velocity . ): Construct the identified slip friction model as v) follows:
From now on, we illustrate the application of the above )– ) to the identification of the slip friction, method using two parametric models. First, we consider asymmetric Gaussian model defined by
(20) where if if
Here, , , , , , , , and are the friction parameters to be identified. Obviously, it is affine , , , , , and , but not to and to . Then, it is straightforward to see that . Note that it takes the form in (11) with
(21)
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Fig. 3. Time responses of the closed-loop system (simulation).
where
where
It is known that the typical values of and range from to , depending on systems [1]. Then, the admissible parameter vector set for asymmetric Gaussian model in (20) can be written as (22)
We can therefore identify the friction parameters , , , , , , , and of asymmetric Gaussian model in (20) by solving the following two separate one-dimensional optimization problems:
where
, , . The corresponding optimization problem (L) then becomes
where
and
(24a)
(23a)
(24b)
(23b)
The above two optimization problems can be solved via exhaustive search methods without resorting to sophisticated numerical search methods, as will be seen in the next section. As a second example, we use asymmetric Coulomb viscous friction model, which is defined by
are defined in (13). Using the fact that , , we can further simplify the expression of the cost function of the optimization problem in (23) as follows:
(25) Here, , , , and are the friction parameters to be identified. In other words, the above parametric model corre-
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TABLE I TRUE VALUES OF MODEL PARAMETERS ASSUMED IN SIMULATION
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TABLE II IDENTIFIED VALUES OF MODEL PARAMETERS IN SIMULATION. (A) ASYMMETRIC COULOMB VISOUS FRICTION MODEL; (B) ASYMMETRIC LORENTZIAN MODEL; (C) ASYMMETRIC GAUSSIAN MODEL
+
sponds to the case when the Stribeck effect is ignored in identification process. Obviously, the friction model in (25) takes the form in (18) with
The corresponding optimal estimate (19). Using the fact that as follows: the optimal estimate
is then determined by , , we can find
(26) We conclude this section by addressing the uniqueness of the solution of the optimization problem (P). In fact, unless some additional assumptions are imposed on the parametric model , the admissible parameter vector set , and the velocity trajectory used in identification process, the solution is not necessarily unique due to the presence of the parameterization error . However, in the absence of parameterization error, i.e., , the solution of the optimization problem (P) is always unique. The uniqueness result is summarized in the following theorem. Theorem 2: Suppose that there exists a parameter vector such that (27) Then, the solution of the optimization problem (P) is uniquely given by . The proof is given in Appendix B.
Fig. 4. Identification result using ACVM (simulation). Solid line: true slip friction curve; dotted line: identification using ACVM.
over the interval and use them to identify the friction curve. In our simulation work, we assume that the true slip fricin (2) can be described by the following asymmetric tion Lorentzian model: (29) where
IV. SIMULATION AND EXPERIMENTAL RESULTS In this section, we present some simulation and experimental results to illuminate further the practical use of the proposed identification method )– ). In our simulation and experimental works, we carry out the data acquisition in a closed-loop is taken as the setting, as shown in Fig. 2. The controller with proportional-derivative (PD) controller and . the control gains Then, the following periodic function is chosen as an excitation input to the corresponding closed-loop system: (28) The time responses of the closed-loop system to the excitation input in (28) are shown in Fig. 3. As can be seen from Fig. 3, the steady oscillation is established after two periods. In our simulation work, we measure the velocity and the control force
The true values of the mass and the friction coefficients, which are assumed in our simulation work, are given in Table I. by using three We identify the slip friction function parametric models of friction, asymmetric Coulomb + viscous friction model (ACVM), asymmetric Gaussian friction model (AGM), and asymmetric Lorentzian model (ALM). To identify the friction parameters of ACVM, it suffices to calculate (26) using the measured signals and . The identification results are given in Table II. The identified friction model is plotted graphically in Fig. 4. As can be seen from Fig. 4, it describes the actual slip friction quite well except at low velocity where the Stribeck effect is noticeable.
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Fig. 6. Experimental setup.
Fig. 5. Identification result using AGM (simulation). Solid line: true slip friction curve; dotted line: identification using AGM.
In order to identify accurately the shape of the slip friction at low velocity, we have to use a parametric model taking the Stribeck effect into consideration. The second parametric model is AGM in (20). To identify the friction parameters of AGM, it suffices to solve the optimization problems in (24). To do this, over the we first choose N equally-spaced data points . In all simulation and experiment works, the value range of is chosen to be 100. Next, we solve numerically the optimization problems in (24) The identification results are given in Table II. As can be seen from Fig. 5, using AGM provides better identification accuracy, although computationally more complex, than using ACVM. The third parametric model is ALM in (29), which takes the form in (11) with
Here, the typical values of and range from to , depending on systems [1]. Then, the admissible , parameter vector set for ALM in (29) is given by , , and where .
Through some arguments similar to those used to derive (24), , , , , we can show that the friction parameters , , , and of ALM in (29) can be identified by solving the following two separate one-dimensional optimization problems:
(30a)
(30b) where
The above optimization problems are solved via exhaustive grid points uniformly sampled over the search with . The identification results are given in Table II. interval As can be seen from Table II, the proposed method can identify consistently the friction parameters of ALM used in simulation, since there exists no parameterization error. Next, we demonstrate the practical use of the proposed iden-potification method through some experiment using an sitioning system. The schematic diagram of our experimental setup is shown in Fig. 6. For our experimental results, we activated only its axis. The linear resolution of the optical encoder . The velocity information is obtained using is the well-known M/T method based on the encoder signal. The excitation input and the gains of the PD controller used in our experiment were chosen to be the same as those used in
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Fig. 7. Time responses of the closed-loop system during steady oscillation (experiment).
TABLE III IDENTIFIED VALUES OF MODEL PARAMETERS IN EXPERIMENT
the simulation. In our experimental work, we measured the ve, as locity and the control force over the interval done in the simulation. The measured data are plotted graphically in Fig. 7. In our experimental work, we have used AGM. The identification results are summarized in Table III. Note that the true slip friction is not known. In this context, we evaluate the accuracy of the experimental identification results in an indirect way. This evaluation method is based on the so-called constant velocity test [1], in which the mass is forced to move at a constant velocity . Then, the control is equal to the friction force . In Fig. 8, the force identification results obtained from this procedure are marked by , while the curve of AGM with the identified values of model parameters given in Table III is plotted in solid line. This figure directly demonstrates that the proposed method can accurately at the velocity region identify the slip friction where the Stribeck effect is negligible. However, the accuracy of the proposed identification method in the region of very low velocity cannot be evaluated, since it was impossible to control the velocity of the mass accurately in that region. The main reason lies in the performance limitation of the servo pack and the low resolution of the optical encoder used in our experimental setup. V. CONCLUSION In this brief, we have proposed a novel identification method for friction in motion control systems and have demonstrated its
Fig. 8. Direct evaluation of identification accuracy (experiment). Solid line: AGM with the identified values of model parameters in Table III; +: identification results obtained through constant-velocity tests.
generality and practical use through various simulation and experimental results. In contrast with the previously known offline identification methods, the proposed method does not require the information of acceleration, which is not practically available without large error, since it can exploit the orthogonality property between acceleration and any function of velocity. In this context, we believe that the proposed method should find a wide range of applications in industry. Nonetheless, some important issues remain untouched. First, the issue of model selection is important. In fact, the identification accuracy depends on the parametric model used for identifying the slip friction function. Thus, it would be important to find a good parametric model of the slip friction from the viewpoint of identification. Second, when the parameters are varying with time because of changes in operating conditions, aging of
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equipment and so forth, offline identification methods become ineffective. In this context, it would also be an interesting topic of further research to develop an online identification method for friction based on the orthogonality property. APPENDIX A PROOF OF THEOREM 1
(36) Now, suppose that (17). Then,
is a solution of the optimization (L) in
We first state some useful properties of the functions , and below which can be easily deduced from (A.1)–(A.3) and the condition in (15). and for each For all are piecewise continuous are linearly independent on
(31) (32) (33)
To prove Theorem 1, we need the following lemma. Lemma 1 (Orthogonality Property): Suppose that the velocity signal satisfies the condition in (14). Then, for any piecewise continuous
(37) This along with (35) and (36) implies that
Proof: Observe from the piecewise continuity of that the function is bounded over the the function . Then, there exists a sequence of interval such that continuous functions . Then, the well-known Lebesgue Convergence theorem [17] along with the condition in (14) implies that
where . Proof of Theorem 1: By way of contradiction, we first show is singular, that the assertion in (16) is true. Suppose that . Then, there exists a nonzero vector for some such that . This along , a.e. on , with (31) implies that which is contradictory to (33). Consider the following optimization problem depending on : (34) As will be seen soon, the solution of the optimization problem in (34) is uniquely given by
(38) In other words, ( , ) is the solution of the optimization problem (P) in (8). ) is a solution of Conversely, suppose that ( , the optimization problem (P). Then, it is obvious that . Thus, this along with (35) . shows that We complete the proof by showing (35) and (36). It is easy to see that the optimization problem in (34) has the unique solution given by (39)
(35) such that
It follows from (32) and Lemma 1 that: (40)
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Further, it follows from (4) that:
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Finally, (36) is the direct consequence of (44) and (47). (41)
and, hence, that
APPENDIX B PROOF OF THEOREM 2 Note from (27) that and hence that
(42)
(48)
As the consequence of (7), (40), and (42), we obtain that
(43) This along with (39) leads to the fact stated in (35). On the other hand, we can see through some direct calculations that
(44) Here, it follows from Lemma 1 that (45) Through some arguments to used to derive (42), we can show that (46) This along with (45) implies that
(47)
such Suppose that there exists a parameter vector . Note that that . Hence, it holds that . This along with (48) implies that . This, in turn, implies that , a.e. on and hence that , a.e. on . Finally, It follows from the condition in (15) that . REFERENCES [1] B. Armstrong-Hélouvry, P. Dupont, and C.Canudas de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138, 1994. [2] B. Armstrong-Hélouvry, “Stick slip and control in low-speed motion,” IEEE Trans. Automat. Contr., vol. 38, pp. 1483–1496, Oct. 1993. [3] P. Dahl, “A Solid Friction Model,” Aerosp. Corp., El Segundo, CA, Technical report TOR-0158(3107-18)-1, 1968. [4] C.Canudas de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, pp. 419–425, Mar. 1995. [5] D. P. Hess and A. Soom, “Friction at a lubricated line contact operating at oscillating sliding velocities,” J. Tribology, vol. 112, no. 1, pp. 147–152, 1990. [6] L. C. Bo and D. Pavelescu, “The friction-speed relation and its influence on the critical velocity of the stick-slip friction,” Wear, vol. 82, no. 3, pp. 277–289, 1982. [7] S. I. Cho and I. J. Ha, “A learning approach to tracking in mechanical systems with friction,” IEEE Trans. Automat. Contr., vol. 45, pp. 111–116, Jan. 2000. [8] C. Canudas de Wit, “Robust control for servo-mechanisms under inexact fricton compensation,” Automatica, vol. 29, no. 3, pp. 757–761, 1993. [9] P. Vedagarbda, D. M. Dawson, and M. Feemster, “Tracking control of mechanical systems in the presence of nonlinear dynamic friction effects,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 446–456, July 1999. [10] R. M. Hirschorn and G. Miller, “Control of nonlinear systems with friction,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 588–595, May 1999. [11] M. R. Popovic, D. M. Gorinevsky, and A. A. Goldenberg, “High-precision positioning of a mechanism with nonlinear friction using a fuzzy logic pulse controller,” IEEE Trans. Contr. Syst. Technol., vol. 8, pp. 151–158, Jan. 2000. [12] M. R. Elhami and D. J. Brookfield, “Sequential identification of Coulomb and viscous friction in robot drives,” Automatica, vol. 33, no. 3, pp. 393–401, 1997. [13] B. Friedland and Y. J. Park, “On adaptive friction compensation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1609–1612, Oct. 1992. [14] C.Canudas de Wit, K. J. Åström, and K. Braun, “Adaptive friction compensation in DC motor drives,” IEEE J. Roboti. Automat., vol. RA-3, Dec. 1987. [15] H. S. Lee and M. Tomizuka, “Robust motion controller design for highaccuracy positioning systems,” IEEE Trans. Ind. Electron., vol. 43, pp. 48–55, Feb. 1996. [16] S. J. Kim and I. J. Ha, “A frequency-domain approach to the identification of mechanical systems with friction,” IEEE Trans. Automat. Contr., vol. 46, pp. 888–893, June 2001. [17] E. Hewitt and K. Stromberg, Real and Abstract Analysis. New York: Springer-Verlag, 1967.
