A comprehensive experimental setup for identification of friction model parameters

Mechanism and Machine Theory 100 (2016) 338–357

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

A comprehensive experimental setup for identification of friction model parameters Yun-Hsiang Sun a, * , Tao Chen b , Christine Qiong Wu a , Cyrus Shafai b a b

Department of Mechanical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada

A R T I C L E

I N F O

Article history: Received 20 October 2015 Received in revised form 23 February 2016 Accepted 24 February 2016 Available online xxxx Keywords: Friction models Parameter identification Capacitance displacement sensor Presliding

A B S T R A C T Parameter identification for advanced friction models is the most demanding task and involves a wide range of experimental dynamic analysis. The identification results directly affect the quality of the friction model and significantly depend on the setup. While many setups have been proposed to investigate the various characteristics of friction, rather less attention has been paid to designing one that is practical for parameter identification dedicated to certain common models. An experimental setup featuring simplicity, flexibility and good versatility is proposed here. The setup consists of a mass block connected to a motor-powered ball screw mechanism. Such a block slides on a metal base and the friction at the contact junction is varied accordingly. The materials and lubricants in the contact surface can be easily altered if needed. A capacitance displacement sensor is designed and installed at where the precise presliding profiles can be captured. The experimental setup, the calibration procedure for the displacement sensor and the experimental results are presented. It is shown that the experimental data produced by our setup leads to the success in parameter identification procedures of the selected friction models. Some limitations of the friction models, from the perspective of parameter identification, are discussed. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Friction appears in all mechanical systems incorporating parts in contact and with relative motion. Although friction can be desirable in certain applications, such as for brakes, in general, it has negative influences on the efficiency and motions of the mechanical systems. Control strategies developed based on an accurate friction model can eliminate the effect of friction and improve the system performance[1]. In the past decades, research in the control field has capitalized more on the advanced friction modeling based on the experimental observation and theoretical work of tribology [2–4]. However, the major disadvantage of these advanced models is that their parameters need to be identified through dynamic behaviors of friction such as friction hysteresis cycle [3,5,6] and presliding [4,7,8]. Demonstration of these dynamic behaviors of friction needs a setup featuring good versatility and precise displacement/force measurements in both presliding and gross-sliding regime. A setup absent from these characteristics may give rise to the inaccurate model parameters and hinder the further analyses. Therefore,

* Corresponding author. E-mail address: [email protected] (Y.H. Sun).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.02.013 0094-114X/© 2016 Elsevier Ltd. All rights reserved.

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designing a proper experimental setup that carries out the parameter identification is essential to ensure that the models are coupled with the accurate parameters. The setups to date have tend to demonstrate a limited number of dynamic behaviors of friction on one setup, which significantly reduce the practicability if more dynamic behaviors are expected in the identification procedures. A typical rig used to explore the friction behaviors based on translational motion is friction oscillator [9-11]. Such a rig normally consists of a vibrating base, several dampers, coil springs and frictional surfaces. During the experiments, the frictional surfaces are oscillating back and forth around the starting point by the excitation applied to the vibrating base, and the effect of the oscillation amplitudes/frequencies on the friction behaviors can be studied. Nevertheless, the friction oscillator usually incorporates many mechanical components as shown in Ref. [11], which makes it time-consuming to perform the assembly and the calibration. In addition, the oscillator only allows narrow velocity window near zero-velocity point to be applied on the frictional surfaces, a number of friction behaviors arisen from fast motion cannot be demonstrated as a result. Self-excited systems are frequently used to study a friction-induced phenomenon named as stick–slip oscillation [12-15]. Such an oscillation is responsible for unpredictable system output in many industrial applications [14] and indeed worth a full understanding. The earliest version of self-excited system usually incorporates a spring–mass system sliding on a belt [12] subject to the constant velocities. The transition in friction from presliding to gross-sliding regime will oscillate the mass block periodically around a point. Major modifications have been carried out to rectify some deficiencies within the original design, more information can be found in Refs. [14,15]. Although this setup is well-established by the later improvement, none of the improved variations has been reported able to control the force loaded on the mass block. As a result, the investigation of some friction behaviors induced by controlled loading force cannot be carried out. In addition, the micrometer-level displacement due to the plastic deformation of asperities under the load in the presliding regime [16,17] cannot be measured using this setup because of the lack of the proper displacement sensors. DC motors are one of the most widely used prime movers in various industry applications nowadays. Much research has been devoted to study the friction in the DC motors during the past decades [7,8,18,19]. Reliable measurements of displacement/velocity/loading force can be made through the armature current feedback and the optical encoder. By properly designing the controllers, most of the known dynamic behaviors of friction can be experimentally produced by the DC motors as shown in Ref. [20]. However, friction is generally a combined result of various factors such as material of frictional surfaces and type of lubricant added into the moving surfaces. The frictional surfaces of the DC motors are usually in a sealed environment, which does not allow the factors influencing friction to be easily altered. This limitation makes the DC motors less useful if changeable contact conditions are of great importance. Accurate measurement of presliding displacement is mandatory for the parameter identification of a few advanced friction models such as the LuGre model [4]. The presliding displacement is arisen from the deformable asperities at the contact junction where the loading forces are applied on. Such a displacement is experimentally difficult to measure since it is normally in the scale of a few micrometers [20]. Experimental setups proposed in Refs. [21-23] are adequate for the precise presliding measurement, whereas their high cost can be prohibitive. Therefore, it is highly desirable to have a sensor that is less expensive but does not compromise the accuracy. The present paper proposes an experimental setup that incorporates a mass block moved by a motor-powered ball screw mechanism. Its structure is simpler than most of the existing setups in which the harassment arisen from the complicated designs can be avoided. Our setup is designed to be able to alter the loads, materials and lubricants in the contact surface, which gives researchers a chance to investigate the friction behaviors subject to a wide range of contact conditions. By taking the advantage of a DC motor, different types of control tasks can be implemented on our setup in order to demonstrate most of the known friction features. In addition, a capacitance displacement sensor consisting of a pair of self-made conducting plates and the dedicated data acquisition system is built at affordable price with good resolution for precise presiding measurement. Parameter identification procedures of a group of friction models, namely the classical model, the Dahl model, the Bliman and Sorine model (B.S. model), and the LuGre model, are experimentally performed by our setup. The model selection is by no means exhaustive but ranges from the most basic model to advanced dynamic models. Experimental results reveal that the proposed setup is adequate to parameterize the selected friction models, more importantly the capacitance sensor built in-house can sufficiently capture the presliding. The identification procedures carried out by our setup demand a wide range of control tasks and the reliable measurements of friction behaviors. The proposed setup is assumed to be applicable to most of the existing friction models. This paper is structured as follows. In Section 2, the selected friction models are briefly introduced. Section 3 concerns the design of the mechanical testing structure and the sensing system of our setup. The experimental procedure is also discussed here, which is followed by the experimental results and the identified parameters. Section 5 contains the conclusions.

2. Review of friction models In this section, we will give a brief introduction of the candidate models. The classical model has the longest history and is the only static friction model among our selection. The rest of our selection are dynamic friction models which in general capture more frictional features observed in laboratory experiments than static models.

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2.1. Classical model The classical model [8] consists of three components, the Coulomb friction [24], static friction and viscous friction all having the form of ⎧ ⎪ ⎪ ⎨Fe F = Fc sgn (v) + s2 v ⎪ ⎪ ⎩F sgn (F ) s

e

if

v=0

if

v = 0

if

v=0

and

  Fe  < Fs

and

  Fe  ≥ Fs

(1)

where F is the friction, v is the velocity, Fe is the external force, Fs is the break-away force, Fc is the Coulomb friction and s 2 is the viscous friction coefficient. The static friction describes the friction counteracting the external input Fe when objects are at rest. Thus, during the static stage the friction is equal to the external force. Once the break-away force Fs is reached, the object under observation begins to slide. The friction is estimated by adding the Coulomb friction and the viscous friction together while the observed object is in motion. The basic idea of the Coulomb friction, Fc , is that the friction opposes motion arising from external loading, and its magnitude is independent of contact area and velocity, but proportional to the normal force. The viscous friction accounts for the lubricant effect on the friction. It is velocity-dependent and varies linearly. Fig. 1 is used to demonstrate the classical model. The break-away force, Fs , does not account for the friction at null velocity; on the contrary, it can take on any value between ±Fs to balance the external force. Once the velocity is not null, the friction estimation becomes the superposition of the Coulomb friction and the viscous friction. 2.2. Dahl model Dahl has completed a series of papers [2,25,26] since 1968 which details the fundamental differences between sliding friction and rolling friction. The interface bond material properties dictate the former, while the latter is mainly governed by tensile and compressive properties of the object. A conclusion was made that the static friction is not distinctly identifiable in rolling friction. A dynamic friction model, named as Dahl model, was developed to simulate the friction in a bearing as detailed below  a dF F = s 1 − sgn (v) dx Fc

(2)

where F is the friction, x is the displacement, s is the slope of the friction curve at F = 0, Fc is the Coulomb friction, and a determines the shape of the friction curve. Fig. 2 graphically illustrates how the friction varies with respect to the displacement. Note that the friction is only a function of the sign of the velocity and the displacement(x). The friction will never go beyond ±Fc if its initial value |F(0)| < Fc . The time domain model is needed and can be obtained by relating dF and dt using the chain rule  a dF dx dF F dF = = v = s 1 − sgn (v) v. dt dx dt dx Fc

(3)

The Dahl model avoids the discontinuity in the model and is capable of capturing friction hysteresis and presliding. However, the ability to predict basic friction features such as sliding friction and break-away force is beyond its scope, leaving the motivation for extensions of the Dahl model.

F

s

F

Slope=σ2

Friction

c

Velocity Fig. 1. The velocity–friction relation of the classical model.

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V reversal point

F

Friction

c

341

Rest slope σ

V>0 V<0

0 x

−F

c

Displacement Fig. 2. Dahl model implies that the friction is position dependent.

2.3. Bliman and Sorine model This model was developed by Bliman and Sorine in a series of papers [3,5,6]. The authors emphasized that the friction is not explicitly velocity-dependent, instead it depends on the distance traveled after a zero velocity crossing. The model was developed to exhibit following properties: Firstly, it reproduces the basic features of friction such as break-away force, Coulomb friction, friction hysteresis cycle and presliding. Secondly, it is a dissipative model since the friction naturally dissipates energy at all times. Thirdly, the identification of the parameters is straightforward once the experimental friction hysteresis cycles are available. The Bliman and Sorine model has the form of dxs1 −xs1 f1 v = |v| + dt gef gef −xs2 f2 v dxs2 = |v| − dt ef ef F = xs1 + xs2

(4)

where the xs1 and xs2 are two states added to produce the friction F. f1 , f2 , g, and ef do not embrace any physical meanings but are determined through two reference points selected on the friction hysteresis cycle, as can be seen in Fig. 3. The first reference point (xe , Fs ) is chosen based on the presence of break-away force Fs and its corresponding displacement xe . The friction approaches the Coulomb friction Fc for the displacement grater than xp . This information is used to form the second reference point (xp , Fc ). Note that the second reference point does not have to lie on the hysteresis cycle. The four parameters, f1 , f2 , ef

Friction F

Point 1

V reversal point

V>0

s

F

c

0

Point 2

x

e

x

x

p

V<0

Displacement Fig. 3. Friction hysteresis cycle. xe is the displacement where the break-away force Fs is experienced. Friction converges to Coulomb friction Fc for relative |F (xp )−Fc | displacement greater than xp . xp is the displacement which makes the equal to 5% [6]. 2 Fc

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and g, can be determined with the known coordinates of two reference points using the following map [Fs Fc xe xp ] → [f1 f2 ef g] from [6,20]: ⎧ m m2 +2)p f1 = ( 21(p−1 ⎪ ⎪ ) Fc ⎪ ⎪ ⎨f = m1 m2 p+2 F c 2 2(p−1) xp ⎪ e = ⎪ f ⎪ 3 ⎪ ⎩ m m +2 g = m 1m 2p+2 1

(5)

2

with m1 and m2 defined as m1 =

Fs − F c , m2 = e3xe /xp Fc

(6)

and p is the solution of m1 m2 + 2 ln p = (p − 1) ln m2 m1 m2

p>1

and

(7)

where the solution exists and is unique if and only if ln m2 <

m1 m2 + 2 m1 m2

(8)

for example when 3xe < xp , as suggested in Refs. [6,20]. This strong constraint on the coordinates of the two experimental reference points requires a relatively large xp to obtain a set of valid parameters of the model. A sharp drop in the experimental friction profile will give rise to a unsolvable p value. 2.4. LuGre model Friction models developed after the 80’s have become more sophisticated since the qualitative mechanisms of friction were fairly well understood by taking advantage of tribology. When examined on a microscopic scale, the contact surface is extremely irregular due to the presence of a number of asperities, and it was proved that the junctions deform elastically before the lubricant enters, completely separating two solid bodies [16]. The authors of the LuGre model [4] visualized that two rigid bodies were in contact through elastic bristles, giving rise to the friction force when a tangential force is exerted on the mass. While the tangential force is high enough such that the bristles becomes over-deflected, the stationary mass then becomes a slider. Fig. 4 graphically conveys this notion. The LuGre model regarding the aggregate behavior of the bristles has the form of F = s 0 z + s1

dz + s2 v dt

(9)

s0 |v| dz =v− z dt g (v ) g (v) = Fc + (Fs − Fc ) e−(v/vs )

(10) 2

(11)

where F is the friction, s 0 is the stiffness of the bristles, s 1 is the damping of the bristles, s 2 indicates the viscous friction coefficient, v is the relative velocity between two surfaces, z is the average deflection of the bristle, Fc is the Coulomb friction, Fs is the break-away force, and vs is the Stribeck velocity. This model provides a good approximation on friction that involves in a wide range of the engineering applications with dissimilar operating environments. It benefits from the assumption that the true contact occurs at points where the elastic bristles of the contact surface come together. Such an assumption agrees with the observations found in tribology of engineering surfaces. For the object experiencing constant velocities, the steady-state friction denoted by Fss associated with the velocities is given by Fss (v) = g (v) sgn (v) + s2 v 2

= Fc sgn (v) + (Fs − Fc ) e−(v/vs ) sgn (v) + s2 v.

(12)

Eq. (12) characterizes the famous Stribeck curves [27,28] which models friction as a function of velocity, see Fig. 5. The Stribeck curve provides us an insight into the smooth transition between static friction and sliding friction instead of having a discontinuous drop once the object is at the onset of motion. The Stribeck velocity vs in Eq. (11) determines the slope of the friction curve during the transition period.

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Fig. 4. (a) Contact area between two solid bodies is thought of as a contact between bristles. (b) The average deflection of the bristles between the contact area is represented by a single bristle with the average deflection z. Using the spring and the damper is essential to avoid the oscillations in the average deflection.

3. Experimental approach An experimental approach is developed to identify the parameters for the friction models discussed in Section 2. This section presents the design of the setup and its sensing system. The experimental procedure is discussed as well. 3.1. Design of experiment table The experiment table shown in Fig. 6 mainly consists of a ball screw, a motor and a brass block. The ball screw is coupled with the motor shaft by a beam coupler. One optical encoder is used to provide information about the motion of the shaft. A thin metal plate is bolted to the ball screw nut to transfer the longitudinal motion to the brass block. The load cell provides a rigid connection between the thin metal plate and the brass block to ensure no bending will take place during the experiments. The micrometer head is used to calibrate the dedicated high-sensitivity sensor which is required in the identification of parameters in the LuGre model. Detailed information on the design, working principle, installation, sensitivity and calibration of such a sensor are addressed in Section 3.2.

Friction

Smooth transient

0

Smooth transient

Velocity Fig. 5. The Stribeck curve characterizes friction as a function of velocity.

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Fig. 6. Photo of the sliding table.

The rotary motion of the shaft is transferred to the ball screw shaft via the beam coupler. The resultant rotary motion of the shaft moves the ball screw nut linearly along the longitudinal axis. The high stiffness of the metal plate and the load cell ensures that one dimensional sliding motion is accurately reproduced on the brass block. The load cell is used to continuously measure either the tension or the compression force applied on the brass block. An optical encoder is used to measure the linear displacement of the mass, which is used in the upcoming analysis as discussed in Section 3.3. Such an encoder generates up/down pulses with a resolution of 512 count per revolution. Table 1 summarizes the electrical/mechanical parameters of the developed experiment table. Fig. 7 demonstrates how the setup interacts with the PC, the servo motor amplifier and the Quanser data acquisition (DAQ). The encoder measurements and the load cell data are collected by the Quanser DAQ and transmitted into the PC at 200 Hz. The PC is manufactured by Dell, which features a dual-core processor with a clock speed of 2.40 GHz and 4 GB of Ram. The controllers programmed within the Matlab Simulink environment generate the control voltage to the motor based on the tracking error. The amplifier, Maxon ADS/50-5, bridges the controller voltage commands sent from the PC to the motor installed on the experimental setup. Measurements of the microfine displacement of the block occurring before the gross sliding have posed a well-known challenge to any encoder-based displacement sensing system. Such a displacement, alternatively named presliding, is essential to the identification procedure of the LuGre model and is typically on the scale of a few micrometers. To record the presliding of

Table 1 Parameters of the motor, ball screw and load cell. Symbol and name

Nominal value

Unit

Equipment

mmotor , weight of motor R, terminal resistance L, terminal inductance Km , torque constant Kv , speed constant Jm , rotor inertia Vnominal , nominal voltage P, assigned output power gm , max. efficiency at nominal voltage Screw shaft diameter Lead Pitch Thread direction Screw thread length Total length Excitation Output Capacity Resolution Safe overload Ultimate overload Weight

71 21.4 0.953 37.4 255.0 4.2 36.0 11.0 83 10 1 1 Right 262 310 5 2 44.5 0.089 66.75 133.5 1

g Y mH mN m/A rpm/V g cm2 V W % mm mm rev/mm

Motor

mm mm V mV/V N N N N kg

Ball screw

Load cell

Brass block

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Fig. 7. The main control loop, the power source and the sensing system.

the block, a capacitance displacement sensor featuring high sensitivity is designed and built. The detailed information of the displacement sensor is addressed in Section 3.2. 3.2. Design of capacitance displacement sensor Measuring the presliding necessitates a capacitance displacement sensor, which is introduced in this section. The measurements are used to determine s 0 and s 1 presented in Eq. (9). Fig. 8a shows all components included in the sensor, which consists of two conducting plates. Each conducting plate is fabricated by depositing a thin layer of aluminum on one piece of a glass slide using the sputtering system. The stencil mask helps define the pattern of the conducting plate. Placing two conducting plates in parallel forms a capacitor. The resultant capacitance varies with the separation between two plates if the overlap area and the background capacitance remain unchanged. Fig. 8b shows how the capacitance measurements are transmitted to the Simulink environment. The capacitance arisen from the two conducting plates is converted to the digital signal and sent to the Matlab Simulink. Fig. 9a depicts the installation of two conducting plates. Conducting plate 1 is attached on supporting plate 1 bolted to the the spindle. Conducting plate 2 is attached on supporting plate 2 bolted to the brass block. The two conducting plates are positioned and aligned in a way that its separation gives measurable capacitance, and the edges of the two square plates are precisely aligned to form an overlap area. The displacement measured by the sensor consists of two parts. The first part is the true presliding occurring at the contact junction, and the second part comes from the elastic deformation of the material of the brass block due to the applied force. To reduce the displacement of the second part, the capacitance displacement sensor is installed at the bottom of the brass block with a clearance of 2mm with respect to the table, as shown in Fig. 9b. In this case, the presliding will dominate the sensor reading since it is measured where close to the frictional surface. Fig. 9c depicts a typical configuration of the sensor. The capacitance caused by such a parallel-plate capacitor has the form of C=

eA x

(13)

where e is the permittivity of air, normally 8.854 × 10 −12 F/m is used, A is the overlap area of the two plates, and x is their separation. The sensitivity, S, of this capacitance displacement sensor is defined as the ratio of the change in capacitance with respect to the displacement as S=

dC eA =− 2 dx x

(14)

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Fig. 8. (a) Tools to build the sensor. (b) How the measurements transmit to the Simulink environment. AD7747 helps convert the analog capacitance signal to digital, and MX7-ck board transmits the digital signal to Matlab.

where the sensitivity increases for low x that consists of the initial separation and the displacement. This favorable feature makes the capacitance displacement sensor popular in the application of precise position sensing. Fig. 10 shows typical calibration results on the capacitance sensor used in this paper. All the capacitance measurements are sampled at various displacements having the equal interval of 10 lm in between. A quadratic polynomial, rather than Eq. (13), is used to relate the displacements to the corresponding average capacitances as the sensor output may drift slightly from the ideal response. Such a calibration needs to be carried out for each trial since the initial separation varies from trial to trial. The calibration can be done as follows: Capacitance displacement sensor calibration: 1. Bring the brass block to a position close to the spindle of the micrometer head. Turn on the capacitance displacement sensor. Start the data acquisition system. 2. Fine-tune the position of the spindle to ensure that the initial separation of two plates is minimized. Confirm that the two conducting plates are precisely overlapped. Once the initial separation is determined, the corresponding capacitance is recorded as Cini .

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Fig. 9. (a) Installation of two conducting plates. (b) The vertical clearance between the table surface and the conducting plate edge. (c) Left: schematic and notations for a parallel-plate capacitor. Right: two square conducting plates with a square overlap area are used to form the capacitor.

3. Rotate the thimble to move conducting plate 1, shown in Fig. 9a, away from conducting plate 2 by 10 lm increments until 70 lm of accumulated displacement is achieved, and the capacitance in response to each displacement is recorded. 4. Establish the relationship between the displacement and the capacitance. A quadratic polynomial is recommended to best fit to all recorded points plotted on the displacement–capacitance map. Once the calibration is properly performed, we move the spindle back to the initial position where the Cini is observed. The quadratic polynomial obtained in the calibration process is valid for the trial having the same Cini value. 10.5

Calibration curve Experimental result Capacitance (pF)

10

9.5

9

8.5

8

0

10

20

30

40

50

Displacement (μM) Fig. 10. Calibration curve versus experimental measurements.

60

70

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3.3. Experimental procedure The aim of the experiments is to identify the parameters of the selected friction models. The reference inputs, controlled variables and the post-processing procedures of the trials differ from model to model. The details are addressed in the following subsections. For each experimental trial, the procedures outlined below are strictly followed to ensure the friction to be accurately estimated. All the trials in this study are performed from rest, i.e., the force applied on the block and the displacement are null before we launch each trial. The following procedure ensures that null force is achievable. The brass block is positioned where the ball screw nut has equal travel to both ends of the shaft. This adjustment ensures that the back and forth motion required for some identification processes can be achieved. After the ball screw nut is in position, we manually zero the force applied on the block by rotating the ball screw shaft. The incremental encoder used in this study provides the relative angle between the origin and the current position of the motor shaft. The origin is the position of the motor shaft when the power is turned on. The control tasks are implemented once null force and displacement are secured. Newton’s second law is used to estimate the friction as follows: Fext − friction = mx¨

(15)

where the Fext is the input force, m is the mass of the block, and x is the linear displacement of the block converted from the encoder reading. The input force sensed by the load cell and the linear displacement is filtered by a second order low-pass filter. Friction estimation at each time instance is achievable using Eq. (15). The following subsections summarize the reference inputs, controlled variables and the post-processing procedures needed to identify the parameters for the models. 3.3.1. Classical model Three parameters, Fs , Fc , and s 2 , included in Eq. (1) are determined through the Stribeck curve [27,28]. The first step is to estimate the friction during constant velocity motion. An experimental test is performed for 15 trials at dissimilar desired velocities ranging from v = ±2.0mm/s to ±5.5mm/s with the equal interval of 0.5mm/s. A PID controller regulates the velocity output and operates on tracking error. During the steady state the friction is therefore equal to the force applied to the block due to the null acceleration. The mean values of the friction profiles, measured for various velocities, are plotted on a friction–velocity map which partly constructs the Stribeck curve. The second step is to explore the highest load before the dynamic motion of the block is triggered. Thirteen trials are performed by assigning the desired input force to a closed-loop force PID controller. Desired forces for the trials are listed in Table 2. Plotting the experimental points on the friction–velocity map completes the the Stribeck curve and the function Eq. (12) is used to perform the curve fitting. The curve fitting toolbox in Matlab is used to fit the data for Eq. (12) in which the sign function will be replaced by either 1 or −1, depending on which velocity region to be identified. The selected method and algorithm are NonlinearLeastSquares and Trust-Region, respectively. Other options are left with the default settings. The identified parameters for positive and negative velocity regions are averaged out to obtain the nominal parameters. This post-processing ensures the model predictions to be symmetrical in both regions of velocities. Note that the parameter Vs in Eq. (12) is not applicable to the classical model. 3.3.2. Dahl model Two parameters, s and Fc in Eq. (2), are identified through sixteen trials. The desired sinusoidal velocity trajectories and durations of conducted trials are listed in Table 3. A PID controller regulates the velocity output and operates on tracking error. The frequency and duration for each trial are chosen to slide the block with either positive or negative velocity. The friction–displacement map resulting from each trial contributes to one set of parameters. The absolute peak of 1mm/s is adopted throughout all the trials. Slow motion and the sinusoidal reference input were also used by Dahl [2] to perform the identification process, whereas in his study the friction behavior in rotary motion was of interest.

Table 2 Trials conducted with static motion (Force control/step input). Trial No.

Desired force [N]

Trial No.

Desired force [N]

1 2 3 4 5 6 7

0.3 0.6 1.0 1.5 1.8 2.0 2.1

8 9 10 11 12 13

2.2 −0.3 −0.6 −2.1 −2.3 −2.4

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Table 3 Desired velocity trajectories of the trials conducted for identification process. Trial No.

Desired velocity trajectory [m/s]

Duration [Seconds]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 × 10 −3 sin(t) 1 × 10 −3 sin(t) 1 × 10 −3 sin(t/2) 1 × 10 −3 sin(t/2) 1 × 10 −3 sin(t/4) 1 × 10 −3 sin(t/4) 1 × 10 −3 sin(t/6) 1 × 10 −3 sin(t/6) −1 × 10 −3 sin(t) −1 × 10 −3 sin(t) −1 × 10 −3 sin(t/2) −1 × 10 −3 sin(t/2) −1 × 10 −3 sin(t/4) −1 × 10 −3 sin(t/4) −1 × 10 −3 sin(t/6) −1 × 10 −3 sin(t/6)

3.15 3.15 6.3 6.3 12.6 12.6 19 19 3.15 3.15 6.3 6.3 12.6 12.6 19 19

To obtain the friction model which is used to carry out the curve fitting, the Dahl model shown in Eq. (2) with sgn(v) = 1 and a = 1 [8, 26]–29-31] is integrated using the initial condition F(x = 0) = 0. It has the form of −s

F (x) = −Fc e Fc

x

+ Fc .

(16)

Furthermore, sgn(v) = −1 gives s

x

F (x) = Fc e Fc − Fc

(17)

where F and x are friction force and displacement, respectively. Parameters, Fc and s, become identifiable with the friction– displacement maps experimentally recorded. Selecting Eq. (16) or (17) to perform the curve fitting depends on the sign of the velocity of the conducted trial. Trials 1–8 correspond to Eq. (16) and trials 9–16 use Eq. (17). The least squares method is programmed in a Matlab m-file for the purpose of finding the best fit to the experimental measurements. The output error cost function is defined as E=

n

[F (k) − F (k, s, Fc )]

2

(18)

k=1

where F(k) is the kth sampled friction measurement and F(k, s, Fc ) is the kth value of model prediction given by either Eq. (16) or (17). The parameters identified within the positive and negative velocity regions are averaged out to obtain the nominal parameters. 3.3.3. Bliman and Sorine model Four parameters, f1 , f2 , ef and g in Eq. (4) are identified through nine trials. The desired sinusoidal velocity trajectories and durations of the conducted trials are listed in Table 4. A PID controller regulates the velocity output and operates on tracking error. The frequency and duration in each trial are chosen to slide the brass block for two complete cycles. Only the second

Table 4 Desired velocity trajectories of the trials conducted for identification process. Trial No. 1 2 3 4 5 6 7 8 9

Desired velocity trajectory [m/s] −3

2 × 10 sin(2t/3) 2 × 10 −3 sin(4t/5) 2 × 10 −3 sin(t) 3 × 10 −3 sin(2t/3) 4 × 10 −3 sin(2t/3) 5 × 10 −3 sin(2t/3) 6 × 10 −3 sin(2t/3) 6 × 10 −3 sin(4t/5) 6 × 10 −3 sin(t)

Duration [Second] 18.85 15.7 12.57 18.85 18.85 18.85 18.85 15.7 12.57

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hysteresis cycle is adopted to perform the identification process. The friction–displacement map resulting from each trial contributes to one set of parameters. 3.3.4. LuGre model Six parameters within the LuGre model, Fc , Fs , Vs , s 2 , s 0 , and s 1 shown in Eqs. (11) and (9) fall into two categories. The first category, containing the first 4 parameters, is named static parameters and can be identified through the Stribeck curve obtained in Section 3.3.1. The remaining two parameters, s 0 and s 1 named dynamic parameters, allow the model to predict the microfine motion within the presliding region. These two dynamic parameters are identified through six trials associated with force control. The desired force of the conducted trials is listed in Table 5. For each trial, A PID controller regulates the force applied on the block and operates on tracking error. The time series of the microfine displacement resulting from each trial contributes to one set of dynamic parameters. To identify the dynamic parameters, the first step is to relate the microfine displacement sensed by the capacitance displacement sensor to the force applied on the mass block. In the presliding region, there is no gross motion and the displacement is caused by the junctional deformation, in which z = x is satisfied [7,8,20,32–36], and Eq. (9) is revised using the notion z = x F = s 0 x + s1

dx + s2 x˙ dt

(19)

substitute the above equation into Eq. (15) and keep the F at the left hand side, we have F = mx¨ + (s1 + s2 ) x˙ + s0 x.

(20)

This second-order system can be further reduced to F = s0 x

(21)

if the input force has a step profile and the system state x reaches its steady state in the presliding region. Eq. (21) can be used to parameterize the s 0 , and the identified values are averaged out to obtain the nominal value. Once the nominal s 0 is determined, the last coefficient, s 1 , is ready to be identified. Eq. (20) is equivalent to the following, given m = 1 F = x¨ + 2fyn x˙ + y2n x

(22)

where yn is the natural frequency and f represents the damping ration. Comparing Eq. (22) with Eq. (20), we obtain yn = and also √ 2f s0 = s1 + s2

√

s0

(23)

where f is determined as 1 as suggested in Ref. [37] to properly damp the system state x while the velocity approaches to zero [20]. Having f = 1 leads to the equation √ 2 s0 = s 1 + s 2

(24)

where s 1 can be identified with known s 1 and s 2 values. 4. Results The procedures presented above have allowed the parameters for four chosen models to be successfully identified. Each subsection below covers the experimental results obtained from the trials along with the parameters. Furthermore, descriptive error bars for the identified parameters of the Dahl model and the LuGre model are shown and discussed as well. Table 5 Trials conducted for identification process. Trial No.

Desired force [N]

Trial No.

Desired force [N]

1 2 3

0.5 0.5 1.0

4 5 6

1.0 1.5 1.5

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351

Fig. 11. Friction–velocity map: experimental findings and classical model output.

4.1. Classical model The experimental trials associated with static and dynamic motions are plotted on the friction–velocity map to create the Stribeck curve. As can be seen in Fig. 11, the points collected from the trials with static motions distribute vertically in the figure. It indicates that the brass block remains static if the applied force is in the interval of [−2.4, 2.2]N. In addition, the friction data collected from the trials with nonzero velocities show a rate-dependent behavior. Overall, the higher the velocity, the larger the friction. Following the procedure presented in Section 3.3.1, the parameters are identified and shown in Table 6. Such parameters are taken to reproduce the friction which is drawn in the solid curve in Fig. 11. The experimental measurements and the predictions produced by Eq. (1) are superimposed and agree well, indicating Eq. (1) realistically represents the experimental friction. Note that the parameter Vs is not applicable to the classical model. 4.2. Dahl model All the friction profiles estimated from the trials presented in Section 3.3.2 are plotted on the friction–displacement map to identify the parameters. Fig. 12a and b shows a typical set of the velocity and the corresponding friction profiles out of 16 trials. Fig. 12a shows that the velocity of the mass block follows the desired trajectory reasonably well. A minor fluctuation is found around the desired trajectory due to the tracking error occurring at each sampled time instant. It can be seen in Fig. 12b that the friction converges to a fixed value along with mild spikes once the block begins to move. Such spikes are attributed to varying velocities during operation. Following the procedure presented in Section 3.3.2, the parameters are identified for each trial and shown in Table 7. Please note that the Fc identified for the Dahl model is 2.0 which is significantly different from the Fc shown in Table 6. Such a discrepancy comes from the structures of the two models. In the classical model the friction data obtained from non-zero velocities are modeled by Fc + s 2 v, but the similar data are modeled by only one parameter Fc in the Dahl model. This is to say that, the Fc in the Dahl model is a more general description of friction than in the classical model. Therefore, it is expected to see the discrepancy in Fc values. Experimental results of Trial 7 are superimposed with the model prediction as shown in Fig. 12b. Overall, the model can capture the sliding friction except the spikes. Fig. 12c zooms in Area 1 marked by dashed line in Fig. 12b. As can be seen that the model prediction monotonically approaches to a fixed value once a critical displacement is reached, which indicates that the break-away force cannot be reproduced. According to Table 7, each identified Fc value does not have a significant variation with respect to the nominal Fc . The highest variation of approximately 13 % of the nominal value can be found in Trial 11. However, the identified s are less clustered around Table 6 Nominal model parameters for Stribeck curve. Velocity Symbol and name

V>0

V<0

Nominal value

Unit

Fc , Coulomb friction Fs , static friction Vs , Stribeck velocity s 2 , viscous friction coefficient

1.11 2.3 0.001943 193.4

1.458 2.4 0.0003365 152.6

1.284 2.35 0.00113975 173

N N m/s N − s/m

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Fig. 12. Experimental data and curve fitting of Trial No. 7.

the nominal value compared with the Fc . This is due to the fact that the s, coupled with an exponential function as shown in Eqs. (16) and (17), accounts for the change of the friction prediction with respect to the displacement during the transition period. It is numerically observed that a minor shift of the friction prediction during the transition period leads to a significant discrepancy in the s. Thus a high standard deviation of the experimental results is expected in this parameter. Fig. 13 shows the descriptive error bars verifying whether the parameters fit within the normal range. According to Ref. [38] each experimental point is considered normal if it falls within two standard deviation(SD) of the mean. Any point falling outside this 2 SD boundary is treated as an abnormal point, which does not represent the typical behavior of the object to be measured/estimated. The standard deviation of s and Fc , calculated based on Table 7, is 18,346.7 and 0.164, respectively. Each point shown in Fig. 13a indicates the s value independently identified in each trial and n refers to the number of conducted trials in the identification process. The bar on the left shows how the identified s varies. The bar on the right scales up to two standard deviation of the mean. Since all the points fall within the error bars covering the range of two standard deviation, they are considered to be normal. Similarly, the points displayed in Fig. 13b are fully encompassed within the two standard deviation band, but even closer to the mean.

4.3. Bliman and Sorine model All the hysteresis loops resulting from the trials presented in Section 3.3.3 are plotted on the friction–displacement map to identify the parameters. Fig. 14 shows the velocity profile of Trial 6 and the resultant hysteresis loop which is taken to determine the two reference points. Fig. 14a shows that overall the velocity of the brass block follows the desired trajectory reasonably well except for the block that is in the vicinity of null velocity. Such a tracking error is attributed to the time that the controller spends overcoming the break-away force. Second part of Fig. 14a shows two peaks observed after the velocity zero-crossing on the hysteresis loop. Such peaks account for the break-away force. Fig. 14b shows that the friction curve experiences a sharp drop after the break-away force is attained along with the coordinates of the two reference points.

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Fig. 13. How identified parameters spread out within two standard deviation band.

Following the procedure presented in Section 3.3.3 the parameters are identified and shown in Table 8. Experimental results of Trial 6 are superimposed with the model prediction as shown in Fig. 14c. One can see that the model highly agrees with the experimental friction except for the sharp drops occurring before the friction converges to the steady-state value. It is found that the experimental friction drops more dramatic than the model prediction. According to Table 8, only two valid p values are found over the nine conducted trials. The issue of having unsolvable p values has been brought up in Refs. [8,20] and is attributed to the sharp drop of the experimental friction after the break-away force is reached. Such a sharp drop makes the two reference points too close to guarantee the strong condition expressed by Eq. (8). This constraint limits the availability of the parameters for this model. As a matter of fact, frictional surface featuring low stiffness of the asperities in contact will lead to a relatively large value of xe compared with those with high stiffness of the asperities. Such a large xe simply makes the constraint 3xe < xp almost impossible to achieve. Furthermore, it has been found that the loading rate of input force during sticking affects the coordinate of first reference point (xe , Fs ) as well. A lower force loading rate gives higher value of Fs and xe in general. Overall, the parameter identification of the B.S. model is partially determined by a reference point (xe , Fs ) that is highly sensitive to the factors varying from system to system, which makes the model less useful in practice. Ref. [20] proposed a solution to deal with this constraint by manually adjusting the coordinates of the two reference points, whereas the rational and the procedure of the adjustment have not been discussed in the work. 4.4. LuGre model The time series data of the microfine displacements obtained from the trials presented in Section 3.3.4 are taken to identify the dynamic parameters. The static parameters have been identified in Section 4.1. Fig. 15 shows the input force and the displacement profiles obtained from Trial 4 which is considered as a representative trial. It can be seen in Fig. 15a that the force takes approximately 4 s to reach the steady state. The controller gains are carefully tuned to ensure that the block is loaded in

Table 7 Nominal model parameters for Dahl model. Trial No.

Fc [N]

s

Trial No.

Fc [N]

s

1 2 3 4 5 6 7 8 Fc nominal

1.94 1.91 2.25 2.02 1.95 2.24 2.2 2.1 2.0

55,000 73,000 82,000 64,000 89,000 73,000 112,000 79,000 s nominal

9 10 11 12 13 14 15 16 87,750

1.99 1.85 2.27 1.79 1.76 2.0 1.93 1.87

84,000 70,000 87,000 107,000 103,000 105,000 108,000 113,000

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a)

b) 2.6

(x ,F )=(2.4366e−4,2.502) e

s

Friction (N)

2.4 2.2

(x ,F )=(3.642e−4,2.0531)

2

p

k

1.8 1.6 0

c)

5

Displacement (m)

10

15 −4

x 10

Fig. 14. (a) The input profile giving the block cyclic motion and its resultant hysteresis loop. (b) The selected two reference points plotted on the break-away region in zoom. (c) Model estimation vs. realistic friction profile.

the presliding region. Fig. 15b shows the variation in the capacitance measured by the AD7747 chip. The capacitance decreases due to the increasing separation between two conducting plates. The variation in capacitance is converted to the displacement of the block using the conversion obtained in the calibration process. Such a displacement is treated as presliding as discussed in Section 3.2 and converges to approximately 15 lm after the applied force is stabilized. Following the procedure presented in Section 3.3.4 the dynamic parameter s 0 is identified for each trial and shown in Table 9. The coverage of the two standard deviation band of the s 0 shown in Fig. 16 indicates that each identified s 0 is considered to be normal. The nominal value of s 0 is used to determine the s 1 using Eq. (24).

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Table 8 Identified parameters for B.S. model. Trial No.

p

(f1 , f2 , ef , g)

1 2 3 4 5 6 7 8 9

No solution No solution No solution No solution No solution 1.2305 No solution No solution 1.1613

N/A N/A N/A N/A N/A (19.875, 17.8227, 0.000121, 0.9063) N/A N/A (9.4050, 7.3367, 0.000188, 0.7931)

Fig. 15. Experimental results obtained from the representative trial.

5. Conclusions Friction is present in a variety of engineering applications and has a strong influence on the performances of the control systems. Development of friction compensation schemes based on advanced friction models can largely improve the system performance. However, the parameter identification is perhaps the most demanding task before the models are coupled with the realistic parameters. It requires a setup featuring the successful demonstration of various types of control tasks and the reliable sensing systems. This paper presents an experimental setup which is valid, but not limited to, for parameter identification procedures of the selected friction models. The main advantage of this setup is the simplicity, flexibility and versatility. The velocity and the loading force of the block are well-controlled with this setup. They can be either a constant value or varying in a specific fashion. The development of the capacitance displacement sensor in this setup allows it to have the capability to capture the microscope Table 9 Identified parameters for LuGre model. Trial No.

s 0 [N/m]

s 0 nominal

s 1 nominal

1 2 3 4 5 6

65,669.888 68,265.363 67,188.616 66,851.513 63,950.064 69,877.942

66,967.231

344.56

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Fig. 16. How identified parameters spread out within two standard deviation band.

displacement occurring in the presliding regime. The location where the capacitance displacement sensor is installed ensures the accuracy of the presliding measurements. With this setup, the behaviors of friction in either presliding or gross-sliding regime can be accurately captured. It has been shown that the parameter identification procedures of the selected friction models can be satisfactorily performed by the proposed setup, though for the B.S. model the unsolvable p values are observed in some of the conducted trials. Such a limitation in parameter identification for the B.S. model has been pointed out by other researchers in this field and is attributed to the strong constraints imposed by the model designers. The rectification dealing with this limitation is beyond the scope of this paper. While friction does vary from system to system, its basic behaviors are the same in many mechanical systems. The experimental friction behaviors produced by the proposed setup not only complete the selected friction models but also can be used for development of new advanced friction models. It is a very effective and valuable setup for friction-related studies.

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