Friction may well be nature s most useful phenomenon.

Transcription

1 Modeling and Measuring Friction Effects AVRAHAM HARNOY, BERNARD FRIEDLAND, and SIMON COHN PHYSICS, APPARATUS, AND EXPERIMENTS Friction may well be nature s most useful phenomenon. Without friction we would hae no belt dries, no clutches, no wheels, and no brakes. Walking, and een standing upright on a moderately inclined surface, would be impossible. In machinery in which it is not the driing force, howeer, friction is an undesirable parasitic phenomenon, generating heat and wasting energy. Large sums are spent each year on lubricants to eliminate as much friction as possible in mechanical deices. On the other hand, when friction is the source of traction and braking, it is important to keep friction at a high leel. To achiee this end, much effort and funding are expended on projects such as improing tires and antilock brakes. In traction applications, the process starts with the ehicle at rest, whereas in braking applications, the process ends with the ehicle at rest. In both applications, the behaior of friction when the elocity of the ehicle crosses zero is of little interest. In motion-control applications, howeer, the elocity of the controlled object typically crosses zero, often seeral times, during operation. Hence, in such applications, it is necessary to understand the behaior of friction in the icinity of zero elocity. Experiments [1] show that friction in the icinity of zero elocity is a dynamic phenomenon that static models fail to adequately describe. The goals of this article are first to reiew some of the physical characteristics of friction, especially the need for state-space dynamic models; then to describe apparatus for measuring friction effects and Digital Object Identifier 1.119/MCS to present the results achieed using this apparatus; and, finally, to discuss some of the issues relating to measurement of friction effects. PHYSICAL MODELS OF FRICTION PHOTODISC Static Friction Models Friction effects can be imagined as resulting from two mountainous surfaces, one inerted aboe the other, as 82 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER X/8/$25. 28IEEE

2 illustrated in Figure 1. The motion of one surface relatie to the other causes the friction force. The height and sharpness of the mountains (asperities) represent the roughness of the surfaces. The lower mountains support the normal force (load) pressing down on the upper surface. The larger the load, the more deeply the upper surface penetrates into the lower, making it more difficult for the surfaces to moe apart. This effect helps explain why the friction force depends on the load F. The simplest friction model is the Coulomb model, in which the force of friction is gien by f = μfsgn(), (1) where F is the normal force, μ <1 is the coefficient of friction, and is the elocity of one surface relatie to the other. Note that the Coulomb model (1) has a discontinuity at zero relatie elocity. This discontinuity implies that a friction force f, with f μf, can be present een when there is no relatie motion. This phenomenon, which is sometimes called starting friction, helps explain why an object can remain motionless on an inclined plane. Static friction also explains the idealization of rolling without slipping in which the translational elocity of the center of the wheel equals the angular elocity of the wheel multiplied by the wheel radius. The discontinuity at zero elocity presents mathematical and computational difficulties [2]. These difficulties may possibly be alleiated by smoothing the discontinuity, thus replacing the model of (1) by In many situations the leel of starting friction is greater than the leel of friction at nonzero elocity, and there is a rapid transition from starting friction to sliding friction, as shown in Figure 2. When linear iscous damping, that is, a force proportional to elocity, is included in the friction model, the cure may increase with increasing positie elocity as shown by the broken line. The phenomenon of friction decreasing and then increasing with elocity is known as the Stribeck effect. The Coulomb friction model, including the Stribeck effect shown in Figure 2, is often adequate for representing the effects of friction in practical systems. In some applications, howeer, the Coulomb model fails to proide a satisfactory description of system behaior. In one such application, a precision line-of-sight stabilization system [1], a limit cycle predicted by the Coulomb model is absent in the test results. Dynamic Friction Models The upper object in Figure 1 can t moe unless the asperities either break off (the phenomenon of polishing) or bend. Bending of the asperities implies energy storage, which introduces dynamics into the model of friction. Friction Force f Normal Force F FIGURE 1 Rubbing surfaces without lubrication. The mountains, which are called asperities, impede motion and result in friction force. The normal force F is supported by the asperities. The drawing is similar to a figure in [1] Velocity f = φ() = μnsgn()s(, a), (2) where the smoothing function s(, a) is a continuous, een function of, but, depending on a, rapidly rises to one as departs from zero. Candidates for the smoothing function include f μf and s(, a) = 1 e 2 /a 2 s(, a) = e a2 / 2. f μf FIGURE 2 Conentional friction representations. The idealized Coulomb friction cure shows larger starting friction than moing friction. The friction force is represented as a smooth function for >. The broken line shows the Stribeck effect when linear damping is included in the friction model. DECEMBER 28 «IEEE CONTROL SYSTEMS MAGAZINE 83

3 One conception of energy storage is the electrical analogy shown in Figure 3, with oltage analogous to elocity and current analogous to friction force. In this circuit, the pair of back-to-back Zener diodes represents static Coulomb friction. The storage of energy in this analogy is represented by a capacitor in parallel with the diodes accounts for energy storage. Owing to the presence of this capacitor, the circuit is now a firstorder dynamic system, goerned by d/dt = (1/C)(i ψ()), (3) where ψ is the inerse of the nonlinear function φ defined in (2). Another conception of the effect of flexible asperities is the bristle model, in which the lower surface of the moing object is regarded as a brush moing oer an inerted brush that represents the fixed surface. The relatie motion of the two surfaces causes the bristles of the brushes to bend and thereby store energy [1]. A more detailed ersion of the bristle model is gien in [3]. The bristle model accounts for the physics of dynamic friction, but the number of bristles required for erisimilitude may preclude efficient simulation. The LuGre model [5], [6], widely used in the control system literature, captures the effect of the bristles as the output of a system goerned by a first-order system in which the friction force is gien by with where f = σ z + σ 1 ż + α 2, (4) ż = σ z, (5) g() g() = α + α 1 e (/ ) 2. As shown in [7], the LuGre model, with the appropriate selection of the parameters α i,σ i, subsumes seeral of the earlier dynamic friction models. A more general first-order model represents the friction force f as the output of a dynamic system in which the input is the relatie elocity. This model takes the form f = f (z, ), (6) ż = g(z, ). (7) The nature of the functions f and g determines the behaior of the friction force [4] Hydrodynamic Model In dry (unlubricated) friction, the asperities of the rubbing surfaces are in direct contact. As shown in Figure 4, the effect of introducing a lubricant between the surfaces is to create a band of separation between the upper surface and the lower one. This band reduces the depth of penetration of the upper surface into the lower one, thereby reducing the coefficient of friction. The LuGre model can cope with the effects of lubrication but is not based on the hydrodynamics of lubrication. Under the assumption that the mass of the lubricant is negligible, hydrodynamic theory [8] leads to a second-order model for the friction force in a lubricated journal bearing. In the resulting model, the friction force is gien by c 2 f = c 1 κ(ε)(ε ε T ) sgn() +, (8) (1 ε 2 ) 1 2 with J 11 (ε)ε ϕ + J 12 ε = c 3 Fsin ϕ + c 4 J 11 (ε), (9) J 12 (ε)ε ϕ + J 22 ε = c 3 Fcos ϕ + c 4 J 12 (ε), (1) in which ε and ϕ are dimensionless ariables proportional to the eccentricity of the shaft and the angular position of i i + 1/c 1/s Normal Force F ϕ () C Friction Force f Velocity FIGURE 3 An electric circuit that proides an analogue of friction. The back-to-back Zener diodes in parallel with a small capacitor comprise a possible electrical analogue of friction. The capacitor stores energy, which leads to a dynamic model. This block diagram represents the circuit of. FIGURE 4 Rubbing surfaces with lubrication. The lubrication has the effect of separating the surfaces. Hence the asperities are farther apart and impede motion less, resulting in a lower friction force than in Figure IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER 28

4 the point on the shaft nearest to the bearing wall, respectiely; J ij are integrals relating to the ariable film thickness around the bearing; and c i are constants related to the geometry of the shaft and journal bearing. The quantity ε T is the eccentricity at which the friction achiees its minimum alue, and { 1, for ε>ε = T,, for ε ε T, by a calibrated, full strain-gauge bridge bonded to the elastic ring. The total friction torque of all four bearings is measured by a calibrated rigid piezoelectric load cell, which preents rotation of the outer bearing housing K. This torque is and κ(ε) is an asperity stiffness function gien, for example, by κ(ε) = κ (ε ε ) n. The first term in (8) represents the friction component due to the asperities; the second term represents the contribution due to the hydrodynamics [8]. APPARATUS FOR MEASURING FRICTION COEFFICIENTS Friction in a Journal Bearing When the purpose of measuring the force of friction is simply to estimate the energy loss due to friction at a constant elocity, only a rudimentary apparatus is required for measuring the static friction characteristic. Measuring the friction force when the relatie elocity is time arying and crosses through zero, howeer, is more complicated. First, it is necessary to generate and apply periodic or nonperiodic elocities of the magnitudes necessary to elicit the friction effect that is being inestigated. Second, it is necessary to isolate the friction force from all other forces in the system. (Since forces other than the friction force may be present in the system, these forces must not be allowed to corrupt the measurement.) Finally, it is necessary to measure the elocity of one of the rubbing surfaces relatie to the other. To achiee these requirements, the apparatus shown in Figure 5 is designed to measure the friction force in a journal bearing. This apparatus comprises an actuated shaft that can oscillate within a journal bearing housed in a structure designed to measure the friction force between the shaft and the bearing. The components of the apparatus are identified in Table 1. The apparatus measures the aerage dynamic friction force of four identical sleee bearings in isolation from all other sources of friction in the system, for example, friction in the ball bearings supporting the shaft. The apparatus is rigid enough to minimize errors [9]. The design concept is based on applying an internal load (action and reaction) between the inner housing N and the outer housing K by tightening the nut P on the bolt R, and preloading the elastic steel ring E. The apparatus contains the four sleee bearings H, with two bearings inside each of the inner and outer housings. All four test bearings thus hae equal radial load, but in opposite direction for each pair of bearings, due to the preload in the elastic ring. The load on the bearings is measured D A B C K FIGURE 5 Apparatus for measuring friction effect in a journal bearing. This apparatus is designed to minimize all forces on the shaft except the force due to friction. The photograph shows the apparatus in use. The cross-sectional iew shows the basic components as described in Table 1 and the text. TABLE 1 Components of the friction-measuring apparatus. The deice is illustrated in Figure 5. Component Description/function A Ball-bearing support for rotating shaft B Apparatus frame C Rotating shaft D Belt drie pulley E Elastic steel ring F, K Outer housing G Oil retainer disk H Sleee bearings (four) N Inner housing P Tightening nut R Tightening bolt P R N E F G H DECEMBER 28 «IEEE CONTROL SYSTEMS MAGAZINE 85

5 transferred to the load cell by a radial arm attached to the external housing as shown in Figure 5. Thus the measured friction torque of the four bearings is isolated from all other sources of friction. Oil is fed into the four bearings through four segments of flexible tubing and is drained from the bearings through a hole in the external housing into a collecting essel. The shaft is actuated by a position sero designed to track a ariety of reference signals. The apparatus is designed so that it can operate dry or with arious lubricants. I P Friction in a Line Contact The apparatus shown in Figure 6 is designed to measure the friction force in a sliding line contact at ery low elocity. This apparatus comprises a linear motion sliding table, drien by a seromotor and a ball screw drie. The apparatus is designed on the concept of a ball-screw-drien linear positioning table in which backlash is eliminated by preloading the screw drie. Low elocity is achieed by speed reduction of a screw drie. In addition, the speed of the motor is reduced by a set of pulleys and a timing belt. Closed-loop controlled motion is generated by a computer-controlled dc seromotor. The line contact is created between a short, finely ground, cylindrical shaft K and the flat friction surface N. The shaft K is clamped in the housing assembly I, J, and H, which is designed to hold arious shaft diameters. The normal load, which is centered aboe the line contact, is supplied by a rod P, which has weights attached to it that are not shown in the figure. When the friction test surface moes, the friction force is transmitted through the housing assembly to a piezoelectric load cell. The load cell generates a oltage signal, proportional to the friction force magnitude, which is fed to a dataacquisition system. Another concept of measurement of effects of friction at low elocities is presented in Another Concept. N L M O FIGURE 6 Apparatus for measuring friction force in a line contact. This diagram shows the completed apparatus. The cross-sectional iew shows the basic components described in the text. TABLE 2 Conditions for friction measurement experiments. The apparatus shown in Figure 5 is used in the experiment. Bearing diameter Bearing length Bearing material Clearance between bearing and shaft Journal mass Lubricant 2.54 cm 1.9 cm Brass.5 mm 2.27 kg SAE 1W-4 oil RESULTS OF FRICTION- COEFFICIENT MEASUREMENT in a Lubricated Journal Bearing The apparatus of Figure 5 is deployed in a series of measurements to examine the alidity of the hydrodynamic model (8) (1). The experimental conditions gien in Table 2 are established. All the experiments are performed with the shaft subjected to a controlled sinusoidal elocity with frequency ω rad/s, calibrated to impart a tangential elocity (t) of the shaft surface gien by (t) = r θ =.127 sin ωt m/s (11) where r is the shaft radius, and θ is the shaft angular elocity. 86 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER 28

6 Static In principle, the static friction-coefficient cure of Figure 2 can be obtained by establishing a series of constant elocity settings of the shaft relatie to the bearing and, at each setting, measuring the friction force. To expedite the experiment, howeer, the apparatus is run at an extremely low oscillation frequency ω in (11), namely,.55 rad/s (around 2 cycles/min). The results of this measurement are gien in Figure 7, which reeals a pronounced Stribeck effect. Also notice that the measured friction cure for increasing elocity is not identical to that for decreasing elocity, in the range of.2.5 m/s. This result may be due to not using a sufficiently low input frequency or to a hysteresis effect, as discussed in [1]. Dynamic To inestigate the dynamic effects of friction in the lubricated journal bearing, the apparatus is operated at f FIGURE 7 Measured static friction in a lubricated journal bearing. The input elocity is sinusoidal at the ery low frequency of.55 rad/s (about 2 cycles/min). The load is 14 N; the shaft is steel, 2.5 cm in diameter; the sleee is brass; the lubricant is SAE 1W-4 automotie oil. The Stribeck effect is eident..3 Another Concept An apparatus for measuring friction effects often comprises a massie fixed base and a relatiely light object, such as a machine shaft, that is actuated to moe at a prescribed elocity relatie to the base. This configuration is necessary for friction measurements at substantial elocities, since it may be impractical to moe the base adequately. But keeping the base fixed leads to a problem associated with measuring dynamic friction, namely ensuring that the friction is the only force on the object subjected to friction. In control applications, low elocities and elocity reersals can occur. For measuring friction effects in such applications, an alternatie configuration, shown schematically in Figure S1, might be more appropriate. In this configuration, the object that is actuated is the base. A relatiely light test object of mass m rests on the base. The only force that makes the test object stick to the base is friction, since, if friction were absent, the test mass would remain stationary in inertial space while the base would moe under it. It can be shown that the friction acceleration f/m on the test mass is equal to μg, where μ is the coefficient of friction. For the Coulomb friction model, the test mass remains stationary in inertial space if the base acceleration is greater than μg but remains fixed to the base if the acceleration is less than μg. (Remember the familiar tablecloth trick: if the acceleration of the tablecloth is large enough, the objects on it remain stationary as the tablecloth is pulled from under them.) We are interested in determining what happens when the relatie elocity is close to zero, which occurs when the frictional acceleration, which is the only acceleration on the test mass, is around μg, and μ is the quantity to be determined. Since friction is the only horizontal force on the test mass, it can be measured by means of an accelerometer mounted as shown in Figure S1. To measure the elocity of the test mass relatie to the moing base, a noncontacting sensor is used. The sensor can be optical, acoustic, inductie, capacitie, or based on the Hall effect. Motion can be imparted to the base by means of a sero actuator designed to track the reference elocity. The same principle can be used to measure rotational friction. The base is a hollow bearing housing actuated to proide the desired motion. The shaft within the bearing, being completely free, moes only because of friction between it and the housing. The elocity of the shaft relatie the housing can be measured without contact by means of a shaft encoder. Instruments for direct measurement of angular acceleration are uncommon, but the inertial elocity of the shaft is readily measured by means of a gyro. If the noise in the gyro is low enough, it might be feasible to differentiate the inertial angular elocity signal to estimate the inertial angular acceleration. Noncontacting Velocity Sensor Test Mass m Reciprocating Base (t ) Accelerometer FIGURE S1 Conceptual representation of an apparatus for measuring translational friction at low elocity. Friction is the only force acting on the upper object. This force is measured by the accelerometer. f (Friction Force) Actuation DECEMBER 28 «IEEE CONTROL SYSTEMS MAGAZINE 87

7 seeral angular elocities and at two leels of normal force. Figure 8 shows the friction coefficient f/f ersus elocity for three frequencies with a normal force of (c) FIGURE 8 Measured dynamic friction in a lubricated journal bearing. The shaft is steel, and the sleee is brass. These results show that the friction coefficient depends on frequency. The normal force is 14 N; the input elocity is sinusoidal. Three frequencies are used for the input elocity: ω =.1 rad/s, ω =.25 rad/s, and (c) ω = 1 rad/s. The solid cures are experimental data; the dashed cures are fit to the hydrodynamic model. N. Figure 9 shows the behaior at three frequencies for a normal force of 84 N. The solid lines show the experimental results, and the dashed lines show the results predicted by the hydrodynamic model of (8) (1). In this model, the parameters c 1, c 2, and ε T are fitted to the static friction cure Figure 7, and the other coefficients are determined theoretically from the bearing dimensions and the iscosity of the lubricant (c) FIGURE 9 Measured dynamic friction in a lubricated journal bearing. The shaft is steel, and the sleee is brass. These results show that the friction coefficient depends on the frequency of oscillation of the input elocity. The normal force is 84 N; the input elocity is sinusoidal at three frequencies: ω =.45 rad/s, ω =.25 rad/s, and (c) ω =.5 rad/s. The solid cures are experimental data; the dashed cures are fit to the hydrodynamic model. 88 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER 28

8 In machinery in which it is not the driing force, friction is an undesirable parasitic phenomenon, generating heat and wasting energy. Both sets of cures form a loop, enclosing a nonzero area, typical of dynamic friction. The loops enclose more area as the frequency increases, as would be expected with a dynamic model such as discussed in (6) and (7). The upper portion of the cure shows the behaior for increasing elocity, while the lower portion shows the behaior for decreasing elocity. This phenomenon is often erroneously referred to as hysteresis, since the phenomenon may be a consequence of the dynamics of the process rather than of the nonlinearity. Een in a linear dynamic system, the cure is a loop; in fact, for sinusoidal inputs, it is an ellipse, which is a special case of a Lissajous figure. Note also that the coefficient of friction, namely, the ratio of the friction force to the normal force, depends somewhat on the magnitude of the normal force. The Coulomb model does not predict this dependence. Another property of these characteristics is that they appear to lie entirely in the first and third quadrants, implying that energy is dissipated by friction, not only on the aerage but at eery instant of time. Careful examination of the experimental cures, howeer, reeals a slight penetration into the second and fourth quadrants, as is the case with a stable linear electric circuit. (In an electric circuit, the ellipse of oltage ersus current, for sinusoidal input current, has its major axis in the first and third quadrant, but with the instantaneous alue sometimes in the second and fourth quadrants, representing time instants in which the circuit is returning stored energy to the external system.) Operation of the experiment at lower maximum elocity would likely reeal this characteristic in more detail. The experimental cures exhibit the characteristic loops and approximately the same maximum and minimum alues as the theoretical cures; they also show similar frequency dependence. But the experimental cures are not as smooth and not always symmetric. These discrepancies could be due to effects not included in the model or to imperfections in the apparatus. The need for more inestigation is eident. Dry Measurements The coefficient of friction for an unlubricated (dry) bearing is studied using the same apparatus. Figures 1 and 11 show the results for two different sleee treatments. The cure of the friction coefficient for an uncoated steel shaft 2.5 cm in diameter is shown in Figure 1. The cure of the same sleee, coated with ultra-high molecular FIGURE 1 Measured static friction in a dry journal bearing with a steel, uncoated shaft, 2.5 cm in diameter. The input elocity is sinusoidal at a frequency of.5 rad/s. The load is 14 N. The sleee is uncoated brass FIGURE 11 Measured friction in a dry journal bearing with a steel sleee coated with ultra-high molecular weight polyethylene, 2.5 cm in diameter. The input elocity is sinusoidal with a frequency of.25 rad/s. The load is 14 N; the shaft is steel. Note that the lower and upper cures are separated and that the coefficient of friction increases with elocity..5 DECEMBER 28 «IEEE CONTROL SYSTEMS MAGAZINE 89

9 Friction effects can be imagined as resulting from two mountainous surfaces, one inerted aboe the other. Friction (kg-m Torque) Friction (kg-m Torque) Velocity (rad/s) FIGURE 12 Estimated and measured friction force in a friction-compensation experiment. The estimated friction force is determined by the friction-compensation algorithm. The measured friction force is determined using the experimental apparatus shown in Figure 5. Note that the measurements show a calibration bias. Neertheless, the estimate of friction force tracks the measurement. The largest discrepancy is near zero elocity. weight polyethylene, is shown in Figure 11. In the former case, the cure exhibits the typical Stribeck effect, namely the reduction of friction coefficient magnitude with increasing elocity. For the coated sleee, howeer, the cure exhibits a negatie Stribeck effect, in which the friction coefficient magnitude increases with elocity. Both results show a separation between the cure for increasing elocity and the cure for decreasing elocity, which suggests either that the frequency is not sufficiently low or that pure hysteresis is present. The angular elocity of the coated sleee is fie times higher than it is for the uncoated sleee; neertheless, the separation between the cures in this case is not significantly greater than it is for the uncoated sleee. The effect of the coating is eident. For the uncoated shaft, the starting friction coefficient is about.4, which decreases to about.2 as the elocity increases, owing to the Stribeck effect. For the coated sleee, the starting friction coefficient is about.2 and does not rise as high as.4 as the elocity is increased. All the cures in the figures 7 11 are obtained using a sinusoidal input elocity. It can be demonstrated by simulation that the cures corresponding to the force-ersuselocity cures for other input waeforms are generally not the same as those for sinusoidal inputs [1]. Online Friction Force Estimation If the source of friction in a system can be isolated for measurement, it may be possible to determine the parameters of a friction model with accuracy sufficient for control system design and performance simulation. If the source of friction cannot be isolated, howeer, an online friction estimation algorithm might be effectie for this purpose. The adaptie friction compensation algorithm described in [11] is an example of such an algorithm. Based on the static Coulomb friction model, this algorithm is designed to determine the maximum friction force leel μf and thereby deelop a control force to cancel it. Although the algorithm is based on the static Coulomb friction model (1), the friction force estimate that it produces is time arying and tracks the experimentally measured friction force, as shown in Figure 12. The experimental results [12], obtained using the apparatus of Figure 5, show that the online-estimated friction force and the directly measured friction force both exhibit the characteristic loops and are approximately the same magnitude. The discrepancy between the online estimate and the direct measurement is greatest in the lowelocity range, where the Coulomb friction model is least accurate, because the cures for increasing elocity and for decreasing elocity differ most at low elocities. Another experimental inestigation of this technique is reported in [13]. CONCLUSIONS Much progress has been made during the past two decades in understanding and modeling the physical characteristics of friction and its effects in motion-control systems. The recognition that friction is a dynamic phenomenon that can be represented by state-space models is a significant deelopment that is still ongoing. The nature of the elocity input is part of a larger question of the goal or goals of measuring dynamic friction effects. From the scientific iewpoint, the goal is to determine which of seeral friction models best represents the obsered measurements and to determine the parameters of the model that best fit the data. The choice of input elocity is part of the goal. In the case of a linear system, the only issue is the order of the model and the alues of the parameters therein. 9 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER 28

10 The bristle model accounts for the physics of dynamic friction, but the number of bristles required for erisimilitude may preclude efficient simulation. Identification techniques are aailable to determine the appropriate order of the model and the parameter alues. For nonlinear systems, howeer, similar identification techniques are not readily aailable. The proper inputs to use and the means to discriminate between contending models remain as open questions. The deelopment of effectie means for compensating friction effects in control systems is another research area that continues to receie attention. Experimental inestigations lag behind the deelopment of theory. In particular, control algorithms for systems in which friction is important are typically demonstrated by simulations in which the friction model, not experimentally alidated, is the same as the model used in the deelopment of the algorithm. Notwithstanding the alue of such simulations, it must be recognized that these are only first steps toward practical designs. To merit serious consideration in practice, the algorithms must ultimately be demonstrated experimentally. REFERENCES [1] D.A. Haessig, Jr., and B. Friedland, On the modeling and simulation of friction, Trans. ASME, J. Dyn. Syst. Meas. Contr., ol. 113, pp , Sept [2] J. Cortes, Discontinuous dynamic systems, IEEE Control Syst. Mag., ol. 28, no. 3, pp , June 28. [3] Y. Guo, Y. Braiman, Z. Zhang, and J. Barben, Nanotribology and nanoscale friction control, IEEE Control Syst. Mag., ol. 28, no. 6, pp. 92 1, 28. [4] B.S.R. Amstrong and Q. Chen, The Z-Properties chart, IEEE Control Syst. Mag., ol. 28, no. 5, pp , 28. [5] C. Canudas de Wit, H. Olsson, K.J. Åstrom, and P. Lischinsky, A new model for control systems with friction, IEEE Trans. Automa. Contr., ol 4, no. 3, pp , Mar [6] H. Olsson, K.J. Åstrom, C. Canudas de Wit, M. Gäfert, and P. Lischinsky, Friction models and friction compensation, Euro. J. Contr., ol. 4, pp , [7] H. Olsson, Control systems with friction, Ph.D. dissertation, Dept. Automatic Control, Uniersity of Lund, Sweden, [8] A. Harnoy and B. Friedland, Dynamic friction model of lubricatioed surfaces for precise motion control, Tribology Trans., ol. 37, no. 3, pp , [9] A. Harnoy, B. Friedland, R. Semenock, H. Rachoor, and A. Aly, Apparatus for empirical determination of dynamic friction, in Proc. American Control Conf., Baltimore, MD, 1994, pp [1] A.K. Padthe, J. Oh, D.S. Bernstein, D.D. Rizos, and S.D. Fassois, Duhem modeling of friction-induced hysteresis, IEEE Control Syst. Mag., ol. 28, no. 5, pp. 9 17, 28. [11] B. Friedland and Y.-J. Park, On adaptie friction compensation, IEEE Trans. Automat. Contr., ol. 37, no. 1, pp , Oct [12] J. Amin, B. Friedland, and A. Harnoy, Implementation of a friction estimation and compensation technique, IEEE Control Syst. Mag., ol. 17, no. 4, pp , Aug [13] S. Tafazoli, C.W. de Sila, and P.D. Lawrence, Tracking control of an electrohydraulic manipulator in the presence of friction, IEEE Trans. Contr. Syst. Technol., ol. 6, no. 3, pp , May AUTHOR INFORMATION Araham Harnoy is a professor in the Department of Mechanical Engineering at New Jersey Institute of Technology where he has taught since He has been engaged in teaching and directing the mechanical laboratories and has been inoled in research in tribology, bearings and lubrication, fluid mechanics, heat transfer, and rheology. He has many years of diersified industrial and academic experience in seeral countries. He holds the B.S. and M.S. degrees in mechanical engineering and a doctor of science in mechanics, all from the Technion Israel Institute of Technology. He is the author of the textbook, Bearing Design in Machinery, Engineering Tribology and Lubrication (Marcel Dekker, 23). Bernard Friedland (bf@njit.edu) is a distinguished professor in the Department of Electrical and Computer Engineering at the New Jersey Institute of Technology (NJIT), which he joined in 199. He was a Lady Dais isiting professor at the Technion Israel Institute of Technology and has held appointments as an adjunct professor of electrical engineering at the Polytechnic Uniersity, New York Uniersity, and Columbia Uniersity. He was educated in New York City and receied the B.S., M.S., and Ph.D. degrees from Columbia Uniersity. He is the author of two textbooks on automatic control and coauthor of two other textbooks, one on circuit theory and the other on linear systems theory. He is the author or coauthor of oer 1 technical papers on control theory and its applications. For 27 years prior to joining NJIT, he was manager of systems research in the Kearfott Guidance and Naigation Corporation. While at Kearfott, he was awarded 12 patents in the field of naigation, instrumentation, and control systems. He is a fellow of the ASME, which awarded him the 1982 Oldenberger Medal. He is a Fellow of the IEEE and has receied the the IEEE Third Millennium Medal and the Control Systems Society s Distinguished Member Award. He can be contacted at the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 712. Simon Cohn receied the B.S. degree in mechanical engineering from Boston Uniersity and the M.S. degree from the New Jersey Institute of Technology with a thesis on the measurement of dynamic friction. He is a principal engineer with Ethicon, Inc., a diision of Johnson and Johnson, deeloping surgical instruments. He holds ten patents in surgical instrumentation. DECEMBER 28 «IEEE CONTROL SYSTEMS MAGAZINE 91