Eindhoven University of Technology

MASTER

High performance servo control for systems subjected to friction

Tebbens, H.G.L.

Award date:2000

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High performance servo control for systems subjected to friction

Rick Tebbens

Report No. 2000.36

Eindhoven, November 13,2000

Coaches: Dr. it-. M.J.G. van de Molengraft Ir. R.H.A. Hensen

Summary

Friction exists in all mechanisms to some extent. In many motion control applications, friction is a dominant factor that limits the performance of the system. However, because of its nonlinear nature, friction is often neglected or inadequately compensated by conventional controllers.

Classic linear controller design that has already proven to be successful when it comes to tackling friction difficulties, i.e., the classical PID or leadlag controller, will be applied to a rotating arm system subjected to friction. In spite of the many advantages of this controller, it has its drawbacks: the integral action can cause limit cycling to occur and errors after the desired setpoint are very slowly cnmpxsated fnr. -4 c!assica! PID cn?ltm!!p,r, extended with high nder dynamics c~~pcnsa t inn , high- frequency roll-off, is tuned in the frequency domain for the system by using (i) Frequency Response measurements and (ii) classical tools like the Nyquist diagram and Bode plots. Subsequently, various point to point movements are evaluated in order to obtain the best performance for the tuned PID controller plus mass feedforward.

An overview of the various friction compensation techniques, mentioned in the literature, is presented. Two clear distinctions can be observed with respect to the discussed techniques, i.e., (i) a model-based or a non model-based compensation techniques and (ii) a feedback or a feedforward friction compensation configuration. Due to the vast number of publications on various techniques, which are described for very specific practical applications, only the most promising and most recently proposed techniques will be addressed. Especially, the following friction compensation techniques for regulator or point to point movement will be discussed in more detail:

Stiff PD control, i.c., a gain-scheduling PD-controller. Fixed compensation based on a causal friction model, in a feedforward as well as a feedback configuration. Fixed compensation based on the dynamic LuGre friction model, again exploited in both a feedforward and feedback manner.

The addressed and promising friction compensation techniques are compared to the filtered leadlag controller with respect to their performance for the various regulator tasks. It can be concluded from the experimental work that:

A dynamic friction model, i.c., the LuGre friction model, has the best overall performance in a feedback compensation configuration. The dynamic LuGre friction model as well as the gain-scheduled PD-controller in a feedback configuration excel regarding their settling performance.

Moreover, all the fixed compensation feedback techniques give an improvement in overall performance in contrast with the fixed compensation feedforward techniques, which show large variations on the average performance.

S amenvatting

Wrijving is in alle mechanismen tot op zekere hoogte aanwezig. In veel servo toepassingen is wrijving een dominante factor die de performance van het systeem beperkt. Echter, vanwege haar niet lineaire aard, wordt wijving vaak vergeten of op ontoereikende wijze gecompenseerd door de conventionele regelaars.

Klassieke lineaire regelstrategien, zoals bijvoorbeeld de klassieke PID of leadlag regelaar, hebben zich in het verleden a1 bewezen met betrekking tot het oplossen van complicaties die voortvloeien uit wrijvingsfenomenen. Ceze P D of leadlag regelaar zal worden gei'mplemenieerd op eeri expex-imentele opstelling, te wden een roterende L- die onderhevig is am wrijving. Ondanks de vde voordelen van deze regelax, kleven er ook enkele nadelen aan: als gevolg van de btegrale actie kan limit cycling optreden en de fouten die optreden aan het einde van een setpoint worden relatief langzaam weggewerkt. In dit verslag wordt een klassieke PID regelaar, uitgebreidt met hogere orde dynamica compensatie en laag-doorlaat filtering, getuned in het frequentie domein voor het betreffende systeem door gebruikmaking van: (i) Frequentie Responsie metingen en (ii) klassieke hulpmiddelen als het Nyquist diagram en Bode plots. Vervolgens zijn verschillende opzetbewegingen onderzocht om zo tot een optimale performance van de getunede PID regelaar plus massa feedforward te komen.

Er wordt een overzicht van de verschillende wrijvingscompensatie technieken, zoals ze in de literatuur worden vermeld, gegeven. De besproken technieken kunnen op twee manieren worden ingedeeld, namelijk: (i) compensatie technieken die gebaseerd zijn op wrijvingsmodellen versus compensatie technieken die niet gebaseerd zijn op wrijvingsmodellen en (ii) configuraties gebaseerd op wrijvingsfeedback versus configuraties gebaseerd op wrijvingsfeedfonvard. Als gevolg van het onmetelijke aantal publikaties over de verschillende compensatie technieken, die zijn beschreven voor zeer specifieke praktische applicaties, zijn alleen de meest veelbelovende en de meest recente technieken besproken. De volgende wrijvingscompensatie technieken ten behoeve van punt naar punt beweging zijn in meer detail besproken:

Stijve PD regelaars, i.c., een "gain-scheduling" PD regelaar. Vaste wrijvingscompensatie gebaseerd op een causaal wrijvingsmodel, zowel in een feedback als een feedforward configuratie. Vaste wrijvingscompensatie gebaseerd op het dynamische LuGre model, zowel in een feedback als een feedforward configuratie.

De besproken wrijvingscompensatie technieken zijn vergeleken met de gefilterde PID regelaar met betrekking tot de performance voor de verschillende opzetbewegingen. Uit het experimentele werk kan geconcludeerd worden dat:

Het dynamische wrijvingsmodel, i.c., het LuGre model, de beste totale performance heeft in een feedbzck config-izik ien oopzicke vzn de zndere wfijvingscompeiisaiie iecfiiekeii. Zowel het LuGre wrijvingsmodel in een feedback configuratie als de gain-scheduled PD-regelaar vertonen uitstekend settling gedrag.

Bovendien kan opgemerkt worden dat alle vaste wijvingscompensatie technieken in een feedback configuratie, een verbetering opleveren in de totale performance in tegenstelling tot dezelfde technieken in een feedforward configuratie. In het laatste geval treden namelijk erg grote varianties op in de gemiddelde performance.

Contents

1 Introduction 2

2 Modeling and controller design 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Frequency Domain Modeling 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Controller Design 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Setpoints 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Implementation examples 16

3 Friction compensation 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Survey on friction compensation 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Friction compensation techniques 22

3.2.1 Non model-based friction compensation techniques . . . . . . . . . . . . . . . . . . . . . . . 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Model-based friction compensation techniques 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Friction compensation techniques for regulator tasks 27

. . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Stiff PD control: Gain-scheduled PD-controller 27 . . . . . . . . . . . . . . . . . . 3.3.2 Fixed friction compensation: causal model with stick region 30

. . . . . . . . . . . . . . . . . . . . 3.3.3 Fixed compensation: the Lund-Grenoble (LuGre) model 34 3.3.4 Adaptive control: adaptive friction compensation with partial knowledge based on LuGre

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . model 38

4 Experimental results 40 4.1 Integral control: filtered lead-lag controller plus mass feedforward . . . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fixed compensation: parameter estimation 43 4.3 Performance obtained with various friction compensation techniques . . . . . . . . . . . . . . . . . 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions 46

5 Conclusions and recommendations 53

A Velocity estimation 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . l Simulations 57

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Experimental results 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Conclusions 61

B Stability of the LuGre friction model 62

Chapter 1

Friction is present in all machines incorporating parts with relative motion. Although friction may be a desirable property, as it is for brakes, it is generally an impediment for servo control. Problems arising from friction in servo controlled systems can be divided into two groups, namely: (i) control errors occurring in positioning or pointing systems (application examples include pick and place units, ASML wafer stages, disc drives and robots) and (ii) control errors appearing in tracking systems. Each group is characterized by it's dominant friction effects. Systems from the first group will encounter control errors such as steady-state errors and hunting (limit cycle around a fixed point). Systems from the second group will encounter problems such as stand still, lost motion, stick slip and large tracking errors. Although a brief overview for both tasks will be given in this thesis, it will mainly discuss the problems involving positioning control (for references see article of B. Armstrong-HClouvry, P. Dupont and C. Canudas de Wit, "A survey of models, analysis and compensation methods for the control of machines with friction" [a]).

Problems induced by friction can be overcome, t o some extent, by linear controllers. A linear controller which has already proven t o be successful when it comes t o tackling friction difficulties, is the classical PID controller which can be extended with notches and low-pass filters to attenuate high-order dynamics of the controlled system. Frequency Response measurements play an important role in the tuning process which is carried out with classical tools such as Nyquist diagrams and bode plots. Further improvements can b e made by adding a mass feed-forward t o the controlled system. An estimation of the mass can also be derived from the Frequency Response Function (FRF) .

Although PID control can cope with friction difficulties very well, it has it's drawbacks: the integral action can cause limit cycling t o occur and errors after the desired setpoint are very slowly compensated for. Beside this drawback the cardinal question will be if the performance can be increased. Therefore various friction com- pensation techniques are examined with respect t o their performance, which, in case of point t o point movement, depend on settling time and position errors a t the end of the setpoint. These techniques can be divided into two groups: (i) non-model-based compensation, e.g., stiff PD control, dither, implusive control, and (ii) model-based compensation, e.g., fixed compensation such as feedback/feedforward of Coulomb/static friction or the more so- phisticated the Lund Grenoble (LuGre) model. The models that are used in the fixed compensation techniques are tuned in the time domain partially by evaluating controller outputs. In this thesis, the proposed techniques and models are applied on a rotating arm subjected to friction, which is driven by an asynchronous or induc- tion motor and the angular displacement is measured by a high resolution sin/cos encoder. Frequency Response Functions are measured with a SigLab system.

The outline of this thesis is as follows. In Chapter 2 a classical PID or lead/lag controller (extended with high order dynamics compensation and high-frequency roll-off) is tuned in the frequency domain by using Frequency Response measurements and classical tools like the Nyquist diagram and Bode plots. Subsequently, various point t o point movements are evaluated in order t o obtain the best performance for the tuned PID controller. The first part of Chapter 3 gives a global survey of friction compensation techniques. Some promising techniques mentioned in this chapter are discussed in the second part of Chapter 3 in more detail. In Chapter 4, a comparison is made between the techniques presented in the second part of Chapter 3 and a classically tuned PID controller on a

rotating arm subjected to friction. This thesis will be concluded in Chapter 5 and future prospects will be given. Finally, Appendix A gives a method for phase-free velocity reconstruction, which will be used for the friction

compensation techniques that depend upon the velocity.

Chapter 2

In linear control system design, Frequency Response Functions (FRF's) can play an important role in system modeling and controller design. In this chapter a rotating arm system will be investigated on in the frequency domain. Frequency response measurements will reveal all system dynamics over a certain frequency domain. This analysis will be performed in Section 2.1 and the obtained process Frequency Response Function (FRF), i.e., P( jw), will be used t o design a controller in Section 2.2. In the last section of this chapter setpoints are defined, which are used as a reference for the different friction compensation techniques described in the latter part of this thesis.

2.1 Frequency Domain Modeling

In this section, a brief strategy is given to measure the process FRF, P(jw), and subsequently using this to come to an estimation of a fourth order linear model. The system under consideration is a rotating arm subjected t o friction, which is driven by an asynchronous or induction motor as shown in Fig. 2.1. The angular displacement is measured by a high resolution sin/cos encoder.

Figure 2.1: Rotating arm

In order to obtain a FRF of the rotating arm, noise is injected in a closed-loop system, as depicted in Fig. 2.2. The controller C(s) can be any controller as long as the closed-loop dynamics are stable, where P(s ) is the transfer function of the process. From Fig. 2.2 the following expression can be obtained:

MI@) + C(s) . R(s) M2(S) = I+ L(s) MI (s) (1 + L(s))

where the loop gain L(s) = P(s) - C(s). As we are only interested in evaluating the linear dynamics of the system, only movements in one direction are considered: otherwise the non-linear Coulomb friction (at velocity

noise ( M I (s))

Figure 2.2: "Noise injection" in closed loop

reversals) and the non-linear static friction (at v = 0 [mls]) will play a significant role. A suitable reference signal in this case is a constant velocity trajectory, where the assumption is used that R(s) and Ml (s) are not correlated and therefore:

where S(s) is called the sensitivity function, which is a measure for disturbance attenuation. Disturbances entering the closed-loop system a t the output denoted with V(s) in Fig 2.2 wili be suppressed a t the output z( t ) according t o this sensitivity function. From Eq. 2.1 we are able t o determine L(s) (= P(s) . C(s)) :

Subsequently, because the transfer function of C(s) is known, an estimation of the plant transfer function P(s ) can be made:

The reconstructed process FRF P ( j w ) , obtained from the measured sensitivity FRF S(jw), is presented in Fig. 2.3. As we can see, resonance peaks appear at frequencies above 100 [Hz] . These dynamics can be explained

Figure 2.3: Frequency response function P

by evaluating the following simple fourth order model in Fig 2.4. In this case the rotating arm is reduced t o a

mass, spring and damper system (we could also evaluate higher order systems, but for the benefit of simplicity we evaluate a fourth order system). Because the motor and the (high resolution sin/cos) encoder both are located a t the lower end of the vertical shaft (see Fig. 2.1), the stiffness of the various parts will play a significant role in the total behavior of the rotating arm. The fourth order transfer function of the system is given by:

Figure 2.4: A simple model of the rotating arm

Fig. 2.5 shows the estimated transfer function of this theoretical system: a straight line with negative slope and typical higher order dynamics in a certain frequency region. If we had expanded the model with one or more masses, a more accurate model can be obtained. It is questionable if this extension is useful; it is not likely that such high frequencies will ever be excited. However, if these dynamics are in the used frequency range, they will influence the accuracy of the movements of the arm in a negative way and are therefore unwanted and must be eliminated. The controller design for these undesirable dynamics will be addressed in the next section, where these phenomena will be surpressed by two notches. When the measured FRF, Fig. 2.3, is compared with the

-2001 I 1 B l d la? ld

frequency [Hz]

Figure 2.5: Estimated FRF of system.

estimated FRF, Fig. 2.5, we notice that the phase of the experimental FRF considerably differs from that of the estimated transfer function. This distinction is caused by a time delay produced by a relative low sampling rate of the control loop of the induction motor. Fig. 2.6 shows an example of a Zero Order Hold (ZOH) sampled single-mass system. It can be seen that a decreasing sampling-frequency results in an increasing phase-lag. It can also be seen that there exists a linear relationship between the frequency and the phase-lag (look a t the linear representation of the phase-lag in Fig. 2.6). Therefore, the phase-lag that is introduced by the sampling can also be looked upon as a pure time delay. This can be explained as follows: because the transfer function of a pure time delay is represented in the s-domain by Hdelay(s) = e-At's (where At is the time delay), it would produce a

phase-lag of At [radls] added t o the phase-lag of the considered system. Because the phase-lag of a single-mass system is expected t o be constant (-180") over the entire frequency domain, At equals the slope of the linear representation of the phase-lag against the frequency (in [rad] and [radls] respectively) of the total (sampled) system:

where, cr is the slope. We would expect the time delay t o be correlated t o the sampling-time. When a system is

Phase-lag linear scale

--- 500 [Hz)

-

-

-

-

1 2 3 4 Fieq [ rndlS]

Figure 2.6: Phase-lag resulting from ZOH sampling of a single-mass system.

sampled (ZOH), the input or output is held constant over T [sec], which can be seen as a mean time delay of T/2 [sec] (Fig. 2.7). From Fig. 2.6 it can be seen that the slope (a ) is indeed equal t o the half of the sampling-time, and therefore:

1 At = -T

2 (2.3)

Figure 2.7: Time delay as a result of the sampling frequency.

Now, we will return t o our application. Because we want t o estimate the transfer function without the phase-shift due t o t he sampling rate of the control-loop of the motor, we have to compensate this time delay by multiplying the FRF with an 'lead phase' of the following form:

where P(jw) is the estimated FRF without the time delay as a result of the sampling. At can be estimated by determining the slope of the phase in the linear representation of the phase in Fig. 2.3, as shown in Fig. 2.8, where we also could determine the sample-frequency, which would be 2 - At, but it has no further practical implication. Fig. 2.9 (the solid line) shows the result of this compensation when At is set t o & [see]. In this figure an estimation of the former theoretical fourth order transfer function, Eq. 2.2, is made using the phase-compensated FRF. Equation 2.4 shows the estimated transfer function after it has been modified so it will suit Eq. 2.2.

2.2 Controller Design

Next, the measured process FRF will be used t o tune a PID controller where various tools from the classical as well as the more recent linear control theory will be used. At first we will examine the possibilities of a commonly used PID controller. As the controller will be tuned in the frequency domain, the PID controller has the following form:

where 7. 2 - - ' 2 ~ f , and 7 d = &. At frequencies higher than f, the integral action looses its dominant share in the total action of the controller. Above fd the differential action starts t o gain a dominant role. Fig. 2.10 gives a typical bode plot of a PID controller: the integral action manifests a t low frequencies where the gain approximates infinity and decreases as the frequency increases. The corresponding phase lag approaches -90'. In the high frequency region, the differential action will cause the gain to increase again as the frequency increases. This time the phase lead approaches +90°. The region in between is shaped by the P-action, i.c., an increasing proportional gain (K) will shift the gain function t o higher values.

The tuning of the PID-controller involves the shaping of the curves presented in Fig. 2.10 i.e. the three regions (or K, fi and fd in Fig. 2.11) have t o be shifted in such a manner that the following frequency response design goals are satisfied:

The closed-loop stability. By the Nyquist stability criterion, for closed-loop stability the loop gain should be shaped such that the Nyquist plot of L(jw) does not encircle the point -1 (assuming that the open-loop system is stable). In order t o keep the closed-loop system stable under perturbations of the loop gain L(s), several stability margins can be evaluated. In Fig. 2.12 only the modulus margin [7] is regarded; it is represented by the vector m. Because l&l = / I + HoL(jw)l, the modulus margin can also be found in the sensitivity plot:

If the modulus margin must be larger than 0.5 (circle in Fig. 2.12) or I m l > 4 then, IS(jw)l < 2 (E 6 d B ) . In Fig. 2.13 the experimental data has been used t o tune the controller with respect t o closed-loop stability; the modulus margin was set t o 0.5. The Nyquist plot starts at the lower left (low frequencies) and evaluates into the right part for higher frequencies. The Nyquist plot tells us that a proportional controller is not sufficient to guarantee closed-loop stability. If a D-action is added t o the controller, it can be seen that it is possible t o stabilize the system within the demanded modulus margin of 0.5. The D-action causes stability because it rotates a certain part of the curve over +90°. Which part will be rotated, depends on the choice of fd: if fd is increased only the points of high frequency will be rotated over +SO0 and if f d

decreases more points will be influenced. Thus from the Nyquist criterion follows that a PD-controller is sufficient t o guarantee stability, but this controller is not capable of eliminating errors due to friction. In this case an integral action is needed. This b r i n e us t o the next point:

Figure 2.8: Estimation of At

frequency [Hz]

-20

-40

- -60 m D -80 5 g, -100

-120

-140

-1 60

frequency [Hz]

Figure 2.9: Phase-compensated FRF and estimated transfer function

Compensated frf - K~ Estmated transf fun - 1 -

- -

- -

- -r -

- -I

-

I I

1 no 10' 1 o2 1 o3

PID contmiler

Figure 2.10: The bode plot of a PID-controller

Frequency (Hz)

Figure 2.11: Characteristic parameters of a PID controller

Satisfactory closed-loop command response. The response to the command signal R(s) (Fig. 2.2, MI (s) = 0, V ( s ) = 0) follows from the signal balance equation z = P(s)C(s)(-Z(s) + R(s)). Solution for z results in

with P(s)C(s) = L(s) the loop gain and T(s ) the complementary sensitivity function.

Adequate loop shaping ideally results in a complementary sensitivity function T ( s ) that is close t o 1 up t o the bandwidth, and transits smoothly to zero above this frequency. For large values of the loop gain L(s), the sensitivity function approaches 1. From Fig. 2.10 and 2.11 follows that an integral action induces very large values for the loop gain (at low frequencies) and will therefore be suitable for adequate loop shaping a t low frequencies. Fig. 2.13 shows the Nyquist plot of the system that is controlled by a PID controller. Further improvement of the complementary sensitivity by increasing fi can not be done without adverse effects: if fi is increased the curve will be rotated over -90'. This results in a curve that passes through the forbidden area or worse: it intersects the horizontal real axis line to the left of -1, which means that the controlled system is unstable. Another negative effect of making the loop gain L(jw) large over a large frequency band is that this easily results in error signals e and resulting plant inputs u that are larger than the plant can absorb. Therefore, L(jw) can only be made large over a limited frequency band. This is usually a low-pass band, that is, a band that ranges from frequency zero up to a maximal frequency wb.

The number of wb is called the bandwidth of the feedback loop (Fig. 2.14).

I

1 6 db line in the 1 sensitivity plot

,,.----.)7j

Figure 2.12: Nyquist plot

. , . . . . , ?"loop) . .

. .

ler I /': '. :

real part

Figure 2.13: Nyquist plot of the experiment

Figure 2.14: Example: open loop L(s) and bandwith.

The complementary sensitivity function is directly correlated with the sensitivity function S , therefore adequate loop shaping also involves keeping an open mind for disturbance attenuation.

Disturbance attenuation. To study disturbance attenuation, consider the block diagram of Fig. 2.2 (Ml(s) = 0), where V ( s ) represents the equivalent disturbance a t the output of the plant. In terms of Laplace transforms, the signal balance equation may be written as Z(s ) = V ( s ) - L(s}Z(s). Solution for Z(s) results in

The smaller IS (jw)l is, with w real, the more the disturbances are attenuated a t the angular frequency w. IS(jw)l is small if the magnitude of the loop gain L(jw) is large. Hence, for disturbance attenuation it is necessary t o shape the loop gain such that it is large over those frequencies where disturbance attenuation is needed. As we saw that L(jw) can only be made large over a band that ranges from frequency zero up to maximal frequency wb, effective disturbance attenuation is only achieved up t o the frequency wb.

For measurement noise reduction and high-frequency robustness we also provide high-frequency roll-off by including additional lag compensation of the form

Fig. 2.15 shows the sensitivity function S(jw) and the complementary function T( jw) of the tuned PID controller. The higher order dynamics can be found a t the frequencies of approximately 150 and 230 [Hz] . It is possible t o reduce these peaks by adding two of the following second order transfer function to the controller

where wl and w2 are the peak frequencies and PI and P2 are the damping coefficients.

Finally the controller will be extended with a second order low pass filter (tuned in DIET [13]). In DIET the total controller is implemented as follows:

Compl. sensitivily

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 ' 10 frequency [Hz]

frequency [Hz]

frequency [Hz]

frequency [Hz]

Figure 2.15: The sensitivity and the complimentary sensitivity function

where: Value Dimension Note

K 12 v - r a d K is the "gain"

1 7 . - - 1 - - 2Tf i 2~ 7 [.I fi is the "zero" of the integrator, dimension: [Hz}

7 d = - 1 - ""fd 2n 8 bl fd is the "zero" of the lead/lag, dimension: [Hz]

1 7~ = a.rrf, 2 ~ 2 0 0 bl f p is the "pole" of the lead/lag, dimension: [Hz]

wl, = 27rf1, 2x200 1 fl, is the "pole" of the low-pass filter, dimension: [Hz] & " A

PI , 0.3 [-I pin is the "damping" of the low pass filter "t' . .

wznl = 2xfZnl 2x149 [e] f2nl is the "pole" of notch 1, dimension: P2nl 0.024 [-I P2,, is the "damping pole" of notch 1 wl,l = 2xf ln l 2x153 [e] flnl is the "zero" of notch 1, dimension: Plnl 0.02 [-I Plnl is the "damping zero" of notch 1 ~ 2 ~ 2 = 2xf2n2 2x230 [ ] f2n2 is the "pole" of notch 2, dimension: P2n2 0.045 [-I P2n2 is the "damping pole" of notch 2 wln2 = 2x f ln2 2 ~ 2 2 8 [*] fln2 is the "zero" of notch 2, dimension: P1n2 0.015 [-I Pln2 is the "damping zero" of notch 2

Here, the integral action is increased in order t o eliminate the error a t the end of the setpoint as fast as possible. Fig. 2.17 shows the Nyquist-plot of the final controller. It can be seen that the modulus margin of 6 [dB] is slightly violated (from Fig. 2.18 follows that the modulus margin is 7.7 [dB]). The bandwidth of this new controller is 25.7 [Hz]. A large part of the higher order dynamics is attenuated by the notches.

When a controller is implemented in a real-time environment, it isn't always possible to maintain a high sampling rate. Before we go on t o the next chapter, we therefore want t o investigate first what influence a (relative) low sampling rate has on the stability of the former controller. Fig. 2.19 shows the impact of various sampling rates on the Nyquist plot. The continuous line represents the system when it is sampled with a frequency of 500 [Hz]. Up t o this frequency, the system has a modulus margin larger than 0.5. Only for sampling frequencies lower than 18 [Hz], the Nyquist plot intersects the horizontal real axis line to the left of -1 (dash/dot line in Fig. 2.19).

2.3 Setpoints

In this section we will evaluate the possibilities regarding point t o point movement. First we will consider the essential demands we require when it comes to point t o point movement: (i) the trajectory has t o be smooth, (ii)

Controller

U) '6/ 40 , ,~ yd 20 1 is2 10' 18 lo' 18 ld

frequency [Hz]

Figure 2.16: Shape of the final controller in the frequency domain.

Figure 2.17: Nyquist-plot of the final controIler

the system must be able t o follow the given trajectory with sufficient accuracy and (iii) the limits of the system (e.g. maximum velocity, acceleration, jerk) should not be exceeded while following the given trajectory. In this chapter we will investigate various feed functions when we use the PID controller tuned in the former chapter.

In order t o obtain smooth movements, i.e., no discontinuities in the acceleration, setpoints of a t least order three have to he ~ s d . !%poifits of a !ewer order will cause unacceptable high accelerations. Our choice of feed functions is restricted t o functions of no higher order than three. This can be explained by evaluating the steady state error E [7]:

~ e ) = lim 1

(2.10) .LO sn (1 + L (s))

With: L(s) = P(s) . C(s), where P is the plant transfer function represented as a simple one-mass system and C the controller transfer function as seen in Eq. 2.5; n is the order of the feed function.

where J is the mass of the system.

Sensitivity . . . . . . . . . . .

: I : : : : : :

: I : : : : : : . . . . . . . . . . . . . : 8 : ' . . . . . " ' . ,

10 ' wb 10 frequency [Hz]

Compl. sensitivity

. . . . . -25 ' ' . . '

10 ' wb 10 frequency [Hz]

200

150

100

50

0

-50

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . - . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . 10 ' 10

frequency [Hz]

-

-

-

-

Figure 2.18: The sensitivity and the complimentary sensitivity function of the closed-loop system

10 ' 10 frequency [Hz]

When a feed function of order three is used, relation 2.10 gives a nonzero finite steady-state error. Feed functions of a higher order will result in an infinite error and feed functions of order less than three will result in a zero steady-state error. Fig. 2.20 shows us the same results, when a simulation with three different feed functions is carried out.

Generating an optimal setpoint means compromising between the accuracy and the smoothness of the move- ment: a setpoint that is based on low order feedfunctions will lead t o excessive large acceleration and jerk (resulting in plant inputs that are larger than the plant can absorb). High order feed functions, however, will result in errors. Fig. 2.21 shows a common used setup function that is composed of various feed functions (highest order is three), which has no discontinuities in the acceleration. For these third order setup functions a number of parameters can be set a t a desired value:

hm= stroke [rad]

v,= maximum velocity [rad/s]

a,= maximum acceleration [rad/s2]

0 rm= jerk [rad/s3]

t,= feed time [s]

which car, be lumped in$= ttm paramters (see \ vim. - b 2.21): (i) constant acceleration and (ii) Cv which represents the fraction of

Ca represer?t,ing the fraction of feed time with feed time with constant velocity.

Next, various third order setup functions, with different values for Ca and Cv, are tested for on the controlled system. Before we do this, we have t o make sure that the generated feed functions will not exceed the limitations of the rotating arm concerning the velocity and acceleration. In the area where the velocity exceeds 13 [rad/s], the induced torque is not proportional t o the input signal anymore. Therefore the velocity should not exceed 13 [rad/s] and the maximum acceleration is restricted by constructive properties, which is set t o 250 [rad/s2].

Finally, a feedforward of the mass has been added t o the controller in order to reduce the error (see Fig. 2.22). In the previous section, an estimation of the mass has been made. For low frequencies (up to about 200 Hz) the system behavas as a single mass without stiffness or damping. In this case the mass equals ml+m2= 0.0229

[ I : it is assumed that the system will mainly excite the lower frequencies in order t o realize the imposed setpoint. For a single-mass system we know that F(s ) = ms2x(.s), where F is the force, m is the mass and X is the displacement. So, the mass feedforward in Fig. 2.22 will be: Ff (s) = (ml + m2)s2xTe (s) = M (s)XTe (s). Fig. 2.23 shows us that this gives a significant reduction of the position error as well as a reduction of the input.

Figure 2.19: Nyquist plots for various sampling- frequencies

2.4 Implementation examples

Throughout this report, five different setpoints will be used t o evaluate the various compensation techniques for large as well as small movements. In this subsection we will use these setpoints as examples how the various parameters can be set. As we already mentioned above, the velocity should not exceed 13 [rad/s] and the maximum acceIeration is restricted by constructive properties and should not exceed 250 [rad/s2]. vm and a, are therefore set to 11.5 [rad/s] and 200 [rad/s2] respectively or 2.88 [m/s] and 50 [m/s2] a t the end of the arm. The maximum jerk is set to 4000 [rad/s3] or 1000 [m/s3] a t the end of the arm. The five different setpoints are given below (the final values of the various parameters are summarized in the table below):

h,= .s? [rad]. If the values of c, and c, are chosen correctly it is possible to reach the maximum acceleration as well as the maximum velocity.

hm= in- [rad]. In this case the stroke is too short for the acceleration t o reach its maximum value with the given maximum jerk. The jerk is the slope of the acceleration in the first part of the setpoint; if the slope is too small, the maximum acceleration is not reached a t the point where 25% of the feed time has passed. At this point the system has t o deccelerate in order t o reach the endpoint a t t,. If the maximual allowable jerk had been higher the maximal acceleration would have been reached and the feed time tm would have been shorter. From the table below can be observed that this setpoint has reached the system limits by evaluating c, and c,: both c, and c, are zero.

1 1 hmZ3r, h m = r n ~ , h m = & j ~ [rad]. For very small movements the maximum acceleration, and conse- quently t he maximum velocity, will not be reached. Because the stroke is decreasing, the time for the acceleration t o increase, will be shorter. Therefore the real maximal acceleration will also decrease, as well as the real maximal velocity.

Time [sec]

." f 10 ci

2 4 6 Time [sec]

400 fourth order feed function

LT

OO 2 4 6 Time [sec]

0 , , 1 OO 2 4 6

10 -4 Time [sec]

-1 1 I 0 2 4 6

1 0 -5 Time [sec] - 8

- f w :TI 3

0 ' 0 2 4 6

Time [sec]

Figure 2.20: Comparison various feed functions

Figure 2.21: Definition third order setup function

Setpoint hm [rad] c, cu rm [rad/s3] a, [rad/s" vm [rad/s] t, [s] 1 IT 0.0197 0.4353 4000 200.0000 11.5000 0.3807

2 i 7r 0 0 4w0 i84.4000 8.5i26 0.i845

3 -T 0 0 4000 135.9213 4.6213 0.1360 4 I,? 0 0 4000 63.0147 0.9956 0.0631 - 5 Yo IT 0 0 4000 29.2000 0.2145 0.0293 -

i nnn Final Settings for five setpoints.

Figure 2.22: Mass feedforward

8 I - leadllag + Mass ff - - leadllag . . . Mass ff

Figure 2.23: Response to third order feed function with and without feed forward

Chapter 3

Friction compensation

In this chapter, different friction compensation techniques will be discussed for various controller tasks. Depending on the specific controller task, different dominant frictional phenomena are essential and therefore limit the performance. The different important frictional characteristics and their specific influence on performance are described in Section 3.1. Various friction compensation techniques for the different controller tasks are presented in Section 3.2. Due t o the diversity of control problems and the large number of possible compensation techniques, the focus in this thesis will only be on the regulator task for simple mass systems subjected t o friction. In Section 3.3, recently proposed and most promising compensation techniques for the regulator task will be discussed in more detail. These methods will be applied and tested for on the rotating arm in Chapter 4.

3.1 Survey on friction compensation

In this section an overview on the various controller tasks together with different compensation techniques is given, for a complete survey on friction compensation see 181. Before this classification can be given, it is essential that all frictional phenomena are discussed. Figure 3.1 represents the Stribeck friction curve together with its four distinct friction regimes illustrating the various dominant frictional behaviours. A brief description of each regime can be given as follows:

The first regime: static friction and presliding displacement. In this regime contact occurs a t the asperity junctions of the surfaces that are in contact. From the standpoint of control, these junctions have two important behaviors: (i) they deform elastically, giving rise t o presliding displacement (ii) and both the boundary film and asperities deform plastically, giving rise t o static friction.

The second regime: boundary lubrication. In the second regime -that of very low velocity sliding- fluid lubrication is not important, the velocity is not adequate t o build a fluid film between the surfaces; the boundary layer serves t o provide lubrication. It must be solid so that it will be maintained under the contact stress, but of low shear strength to reduce friction.

The third regime: partial fluid lubrication. Lubricant is brought into the load bearing region through motion, either by sliding or rolling. Some is expelled by pressure arising from the load, but viscosity prevents all of the lubricant from escaping thus a film is formed. The greater viscosity or motion velocity, the thicker the fluid film will be. When the film is not thicker than the height of the asperities, some solid-to-solid contact will result and there will be partial fluid lubrication. When the film is sufficiently thick, seperation is complete and the load is fully supported by the fluid.

The fourth regime: full fluid lubrication. In full fluid lubrication, solid t o solid contact is eliminated and the contact area is fully supported by the fluid lubricant.

For different controller tasks, one or more of the described phenomena are present and dominant. Hence, for the various compensation tasks a classification on the dominant friction contributor and its dominant controller

Sliding velocity

Figure 3.1: Steady-state friction-velocity curve

error can be given as presented in the table below. From the four different controller tasks, one is the regulator and the remaining three are various versions of a tracking problem.

Compensation task Control error Dominant friction contributor Regulator (pointing Steady state error, Stiction. or position control) hunting (limit cycle

around fixed point) Tracking with velocity Stand still, Stiction. reversal lost motion Tracking a t low velocities Stick-slip Negative-sloped Stribeck

curve: stiction. Tracking a t high velocities Large tracking errors Viscous behavior of lubricant

A description of each controller task can be given as follows:

Task I , the regulator, is encountered with positioning and pointing systems. Application examples include telescopes, antennas, machine tools, disk drives and robots. In this case, a system spends most of its time either near or within the stiction regime. When the friction-velocity curve of a system is negatively sloped a t the origin, the equilibria of a P D position regulator consists of an interval on the position axis in the phase space. By adding integral control, the equilibria set consists only of points with the desired position (and velocity); however, this set can be unstable with nearby trajectories diverging away t o a limit cycle. This integral-induced stick-slip oscillation about the goal position is referred t o as "hunting".

As to its frictional cause, task 2, tracking with velocity reversals, is closely related with task I. Due t o a higher static ievei of friction, motion through zero velocity is not smooth. A system inay pa-ise a t zero velocity until sufficient force is applied to exceed the maximum stiction level. This task is encountered with machine tools, tracking mechanisms and robots under position or force control. An important example of the effect of friction on this task occurs for machine tool slideways, where it is known as stand still or quadrant glitch. In multiple degree of freedom motion, the joint undergoing velocity reversals pauses while the others continue freely. The resulting motion manifests itself as a defect in the workpiece contour.

Task 3, tracking a t low velocities, differs from task 2 in that the desired motion is of constant direction and perhaps constant velocity. This task also arises for machine tools, tracking mechanisms and robots under position or force control. It is the task most often associated with stick-slip. The common pictorial representations include the pin on flat apparatus and its kinematic inversion depicted in Fig. 3.2.

The potential for stick-slip limit cycling exists when the operating point, vo lies on a negatively sloped portion of the steady-state friction-velocity curve such as regime 3 of Fig. 3.1. This task is often studied by addressing one of two criteria for smooth motion:

Figure 3.2: 2acking a t low velocities. The free end of the spring and damper are moved a t constant velocity, vo.

- Does a stick-slip limit cycle exist and for what system parameter values will it be stable?

- For what system parameter values is the equilibrium point x=vo stable?

Task 4, tracking at high velocities, arises for machine tools, position-controlled robots and tracking mecha- nisms. High-speed operation not only increases productivity, it may actually be necessary t o meet process constraints. For example, in high speed machining, a critical cutting velocity must be exceeded in order t o avoid the excessive tool temperatures which lead to premature failure.

This task is significantly different from the previous three tasks because high-velocity friction is dominated by viscous effects. The friction-velocity curve is positively sloped and stability is usually not a problem. Instead, tracking error is observed t o increase as a function of velocity.

Often, machines performing high-velocity tracking must also cope with velocity reversals. Due to the nonlinearity of friction, a linear fixed-gain controller that is tuned for low velocities may perform poorly a t high velocities and vice versa. This suggests the need for nonlinear compensation.

In literature [8] a clear distinction can be found with respect t o the friction compensation techniques for the various controller tasks, i.e., (i) model-based friction compensation and (ii) non model-based friction compensa- tion. In Fig. 3.3 an overview on the available friction compensation techniques for the different controller tasks is given. From this overview another important distinction can be seen, i.e., the difference in feedforward and

- Stiff position control X X X 1

Figure 3.3: Tasks and their associated compensation techniques.

U

Integral controVDeadband a c -

Joint torque control

impulsive control 2 8

Dither

Coulomb friction feedfonvardhedback 8 General friction feedfonvardhedback

5 Adaptive feedfonvardhbedback

X

X

X

X

X

X

X

X

X

X X X X X

X

X

X

feedback control. The effectiveness of a particular compensation technique depends strongly on the task and therefore on the dominant friction phenomenon. A division in compensation methods, that are model-based or non model-based, will be made in the folowing two sections.

3.2 Friction cornpensat ion techniques

In this section, several friction compensation techniques, both model-based and non model-based, are discussed for the various controller tasks. A complete survey on these techniques and discussion on the applicability can be found in [8]. Detailed information on the literature references on the compensation techniques presented in this section can be found in this survey.

3.2.1 Non model-based friction compensation techniques

Stiff PD control

While the regulator problem is stable under P D control, the tracking problem does exhibit stick slip a t low velocities. For many years, it has been known that by increasing the damping or the stiffness of a system, stick- slip can be eliminated. In a control context, this can be accomplished by increasing the P D gains. The success of stiff P D control can only be fully understood by considering frictional memory. The importance of a delay between a change in sliding velocity and the corresponding change in friction lies in its role in determining the presence or absence of stick-slip in the force control of a stiff contact. By applying dimensional analysis t o a model with non-linear, low-velocity friction, but with no frictional lag, it has been shown that in the absence of frictional lag, increasing stiffness will not eliminate stick-slip. However, in [12], by applying dimensional and perturbation analysis to a friction model with frictional lag, the elimination of stick-slip by increasing stiffness is explained.

In summary, stick-slip can be eliminated through either high derivative (velocity) feedback or high proportional (position) feedback. They are best used together as they are complementary.

Integral control

While stiff P D control can be used t o achieve stable tracking, integral control of position or velocity is almost always introduced t o minimize steady-state errors. Using integral action, systems are found to limit cycle when tracking at low or zero velocities. Integral action, and the limit cycling it induces, are rarely discussed in the literature except as a motivation for more complex control methods. This is in direct contrast with its widespread use and with the variety of techniques developed t o circumvent its shortcomings.

To overcome limit cycling, one standard technique is to employ a deadband as the input t o the integrator block. This, of course, imposes its own steady-state error which hopefully is less than the error before the integral action was added.

In addition t o friction inducing Iimit cycles, integral control can be ineffective and even detoriating a t velocity reversals. Integral windup from prior motion can actually inhibit breakaway. To prevent this, the integral term is typically reset a t velocity reversals. While this eliminates the windup problem, the ensuing integral action produces minimal effect when needed the most t o overcome stiction.

Dither

Dither is a high frequency signal introduced into a system to modify it's behavior. For machines with friction, control engineers focus on the capability of dither t o smooth the discontinuity of friction a t low velocity.

How smoothing arises with dither can be seen in an example: the relationship

is discontinuous. However, when a dither of amplitude CY and frequency w is added t o the input, the averaged

output becomes:

Through averaging, g(t) can be a continuous function of u(t).

a Tangential and normal dither

The analysis presented in the control literature focus on a dither signal added to the command input,, which, for the configuration shown in Fig. 3.4 will give rise t o vibrations that are tangential t o the sliding contact. In tribology literature, on the other hand, the impact of vibrations normal to the contact have been considered. The distinction between normal and tangential dither in a friction contact is a considerable one: the effect of tangential dither is t o modify the influence of friction (by averaging the nonlinearity); the effect of vibrations normal t o the contact is to modify the friction by reducing the friction coefficient.

Tangential vibration Modifies influence of friction (e.g. control input dither)

e--+ Normal vibration Modifies friction (e.g. external vibrator)

~rict i& interface

Figure 3.4: Direction and effect of dither

Working with a simple Coulomb + viscous friction model one would not expect friction to be reduced by normal vibrations, so long as the contact is not broken; but when contact compliance (the origin of presliding displacement) and asperity contacts are considered, more sliding is seen to occur during periods of reduced loading and less during periods of during periods of increased loading. This arises because the mechanical bandwidth of individual asperities may be orders of magnitude higher than the bandwidth of the macroscopic mechanical elements. Godfrey (1967) reports a reduction of coefficient of friction from 0.15 to 0.06 in a lubricated steel contact with addition of 1000 [Hz] normal vibrations.

For control engineers, the importance of this distinction lies in how dither is t o be applied. The original applications of dither involved external mechanical vibrators. On gun mounts and other large pointing systems, such vibrators, sometimes called "dipplers" are still being used. While any form of dither will result in both tangential and normal forces through the coupling arising out of asperity contacts, more freedom exists for orienting the dither when an external vibrator is used.

a Depth of discontinuity and dither in hydraulic servos

When dither is applied, filtering exists between the source of the vibration and the point where it is have to have its effect, as shown by the transfer function Gl(s) in Fig. 3.5. In all cases, but particularly when dither is applied a t the control input, this filtering is important. Cebuhar (1988) evaluates the filtering in terms of the "depth" of the discontinuity, and defines a formal measure of this depth relating t o the number of integrators in Gl(s). He finds that when the depth is great, the designer is more restricted in the application of a dither. The ratio

plays an important role in the effectiveness of the dither and Cebuhar proposes that wd, the dither frequency, should be chosen to maximize a slightly modified form of Eq. 3.1.

Figure 3.5: A transfer function between the input and the nonlinearity where dither is to have its influence.

Based on input from engineers in industry, it seems that dither is only occasionally applied tc motor servos, but is often app:id and wkh g ra t effect t o spee! ~a lves in hydrru!ic servos. The !arger Gl(iw),/Gz(iw) achievable in the hydraulic servo may account fcr the greater success of dither in these systems.

Impulsive control

A number of investigators have devised controllers which achieve precise motions in the presence of friction by applying a series of small impacts, the so-called "impulsive controllers". Impulsive control is distinguished from dither in that it is the impulses themselves which are t o carry out the desired motion. The impulses are not zero mean and must be calibrated t o produce the desired result. Impulsive control is also distinct from standard pulse width modulation (PWM) controllers, where voltage pulses are applied t o a motor. In PWM controllers, the motor inductance averages the relatively high frequency (perhaps 20 kHz) voltage pulses to produce a nearly constant motor current, and therefore nearly constant torque.

In literature (Yang and Tomizuka (1988), Suzuki and Tomizuka (1991), Armstrong (1988), Armstrong- HBlouvry (1991), Hojjat and Higuchi (1991)), the impulses are applied when the system is a t rest, i.e., in the stick phase. The effect of the impulse is a small displacement or a controlled breakaway, leading t o transition t o another controller which regulates macroscopic movements. By making the impulses of great magnitude but short duration, the static friction is overcome and sensitivity t o the details of friction is reduced. A typical behavior is shown in Fig. 3.6.Hojjat and Higuchi (1991) present an apparatus designed especially t o demonstrate impulsive

400 400

Impulsive force I ,

Displacement

i Friction force

0 10

Time (ms)

Figure 3.6: Behaviour of motion under impulsive control.

control. They reliably achieved a remarkable 10 nm per impulse motion and speculate that repeatable 1 nm per impulse motions may be possible. Their mechanism was not unlike a machine slideway; the slider measured about 3 cm on a side and weighted 155 g. Hojjat and Higuchi (1991) control the amplitude of their impulses, typically applying a force about 10 times the static friction for about 1 ms, and show that displacement is given by the square of the amplitude times an empirical constant.

Joint torque control

Joint torque control is a sensor-based technique which encloses the actuator-transmission subsystem in feed- back loop t o make it behave more nearly as an ideal torque source. Disturbances due to undesirable actuator

characteristics, e.g., friction, ripple, or transmission behaviors, e.g., friction, flexibility, inhomogeneities, can be significantly reduced by sensing and high gain feedback. The basic structure is shown in Fig. 3.7; an inner torque loop functions to make the applied torque, T, follow the commanded torque, T,.

Actuator currcnt Torquc applicd to command the mechanism

xd & TC Tt T, Motor, T, X ---+ Transmission Mechanism & their friction

! Scnscci torquc (high bandwidth inna loop) I (lower bandwidth outer loop)

Figure 3.7: Block diagram of a joint torque control (JTC) system.

Joint torque control has been implemented as a means of compensating for actuator and transmission friction, as a means of compensating or more precisely controlling transmission ffexibilities, and as a means of sensing and compensating for the nonlinear rigid body dynamics and gravitational loads experienced in robotics.

Implementation of joint torque control requires torque or force sensing as near as practical t o the output element of the system so that all or nearly all of the actuator and transmission friction will be enclosed in the joint torque feedback loop.

As shown in Fig. 3.7, the controller design is a nested one, with an inner loop which maintains applied torque and compensates for friction, and an outer loop which governs the execution of the mechanism task. The multi-loop structure would in general require a full multi-loop analysis; most of the research in this field, however, is focussed on the inner torque loop, and has required that the frequency domain separation between the inner and the outer loops be sufficient t o protect against dynamic interactions. Eismann (1992), who has substantial experience with the commercial Robotics Research arm, suggests that a 4:l ratio is required between the cross-over frequency of the inner joint torque control loop and that of the outer task control loop.

Dual mode control

High precision applications, such as semiconductor manufacturing and diamond turning of optical elements, require nanometer position accuracy over millimeters of motion range. The standard technology for nanometer positioning involves two stage mechanisms. The coarse positioning stage might comprise a piezoelectric actuator. The liabilities of the two stage mechanisms are weight, size and complexity; two actuators and two controllers are required per degree of freedom.

By capitalizing on presliding displacement, referred to as microdynamics in the nanotechnology literature, it is possible t o achieve two modes of control in a single mechanism: gross motion in the standard way, and fine motion in the presliding displacement. Presliding displacement is motion that occurs by the deformation of asperities in the sliding interface. In this motion, position is a function of the applied force; the junction appears t o be a stiff spring rather than a sliding (or rolling) bearing. The result is a two markedly different mechanism dynamics: "macrodynamics", the ordinary dynamics of the mechanism, and "microdynamics", which governs motions that depend upon elastic deformation in the frictional contact. Because the dynamics are drastically different, two different controller structures are required, thus dual mode control.

3.2.2 Model-based friction compensation techniques

Fixed compensation

When a model is available, it is possible t o compensate for friction by applying a force/torque command equal and opposite t o the instantaneous friction force. The basic construction of model-based compensation, which is applied on our rotating arm, is shown in Fig. 3.8.

Friction Friction Velocity, sensed, compcnsation Predictor estimated or desired

Figure 3.8: Block diagram: model-based friction compensation for the rotating arm

The resulting differential equation for the system sybjected to friction and friction compensation based on a fixed model reads:

A

where u is the induction motor input voltage, C, is the motor gain, Ff,,, is the estimated friction torque, Ff,,, is the actual friction torque, q is the position and M is the effective inertia of the motor-transmission-rotating arm combination.

The model-based schemes can be classified according t o what estimate of the velocity (sensed, estimated or desired) is used t o evaluate the friction model and what portions of the friction model are applied (i.e. Coulomb, Coulomb f static, Coulomb+static+viscous or more sophisticated models that include frictional memory). Since only the position is measured on the rotating arm, the model-based schemes will be classified into two groups, regarding the velocity estimate: (i) model-based schemes based on the estimated velocity, also referred t o as fixed model feedback, and (ii) model-based schemes based on the desired velocity, also referred to as fixed model feedforward.

One of the major difficulties in performing friction compensation is the difficulty in modeling friction at very low velocities. Several practical problems can appear as a consequence of doing friction compensation on the basis of a discontinuous model. The discontinuity a t zero speed allows the friction t o take on an infinite number of values. This problem appears as errors or instabilities in algorithms that depend on "true" zero velocity t o correctly compensate for friction. Systems that differentiate a quantized position signal to estimate vebcity are especially vulnerable since a calculated zero speed may not occur during velocity reversals.Conceptually, state variable models are better adapted to describe and hence t o compensate for friction at very small velocities. They better reflect the fact that friction is a continuous function of time. Examples of such dynamic friction models are (i) the Dahl model [14], (ii) the second order Blimann-Sorine model [15], and (iii) the LuGre model [2] which is a modii5cation of t he Dah! mode!. The latter f r k t i m m d e l , which nazx is ar? ~bbrevatim ef L ~ n d and Grenobk, exhibits a large number of practically observed friction phenomena. Hence, this model is beneficial from a control engineering point of view. However, when these dynamic models are used as a basis for friction compensation, one major difficulty arises because the internal state of these dynamic models is not measurable. However, internal state observers can be designed and stability of the observer-based control schemes can be shown. In connection with control design, these models feature the input-output property of passivity, or more precisely, dissipativity. These properties can be explicitly exploited during control design leading t o an explicit determination of the class of compensators that render the feedback loop stable.

Adaptive control

Friction may change as a function of the normal forces in contact, temperature variations, position etc. For the systems that exhibit such time dependent frictional behaviour, adaptation mechanisms can be designed that compensate for the varying friction forces. The challenges t o adaptive control of a machine with friction are not

unlike the general challenges of adaptive control: problems of stability, the need for persistent (indeed, sufficient) excitation, difficulties that arise when the true model is not in the model set, the need for methods for setting rate parameters and other parameters in the adaptive algorithm, etc. With friction, the motivation for adaptive control is also unlike the general motivation: friction will, in many cases, be a variable quantity which the controller must track. A substantial number of papers concerning adaptive control in robotics and elsewhere have touched on friction: [8] has focused on several papers where friction has been a major concern in the design of an adaptive control algorithm. All of the papers surveyed employ a model-based friction compensation and adaptively update the model parameters. The adaptive algorithms, however, span a great range including the Recursive-Least-Square (RLS) and Least-Mean-Square (LMS) algorithms, the model reference adaptive control (MRAC) algorithms, or the Lyapunov function based algorithms.

3.3 Friction cornpensat ion techniques for regulator tasks

This section will evaluate those friction compensation techniques that can be applied for regulator tasks or point t o point movement. The various friction compensation techniques available in literature can be found in the table presented in Fig. 3.3. However, due to the vast number of publications on these techniques presented in literature, which are described for very specific practical applications, only the most promising and most recently proposed techniques will be addressed in the sequel. The following friction compensation strategies are considered and applied t o the rotating arm in Chapter 4:

S t 8 PD control: Gain-scheduling PD-controller

o Fixed compensation based on a causal friction model

Fixed compensation based on the dynamic LuGre Friction model

Adaptive compensation based on the dynamic LuGre friction model

3.3.1 St iff PD control: Gain-scheduled PD-controller

As seen in Section 3.2.1., limit cycling can for regulator tasks be avoided by choosing a stiff PD-controller. In [12], a gain scheduled PD-controller is derived in such a way that for low velocities the PD-gain can be increased and closed-loop stability (and performance) is guaranteed. Two desirable properties of the proposed controller design are: (i) a high gain controller where it is needed, i.e. for low velocities where the friction is dominant and strongly non-linear, and (ii) the stability of the closed-loop system is guaranteed. Due t o the absence of an integral action in the control loop, friction induced limit cycling for the point t o point movements is avoided. To obtain this controller a promising technique of non-linear modeling, i.e., Polytopic Linear Models [17], is used and a controller synthesis based on this model is performed resulting in the desired favourable properties. The unified idea of the Polytopic Linear Modeling technique is to represent a nonlinear dynamic system by a 'global' model which is the result of convex combinations of locally valid affine linear models. Two desirable properties are: (i) the richness of this mode! dass t o apprsximate a large chss of nonlinear systeris arbitrarily close and (ii) the fact that the model structure is suitable for system analysis and controller synthesis based on linear matrix inequalities (LMI's) .

Polytopic Linear Model

The polytopic model is composed of several locally valid linear models. The structure of each model is in correspondence with the topology of a linearized mechanical system, i.e., x = Aix + Ci + Biu with x = [q 4.IT. With each local model a model validity function pi : R -+ [O, 11 is associated which, by definition, is close to 1 for those regions in the input and state space where the corresponding local linear model is valid. Here, the partitioning only depends on the angular velocity q due t o the choice of the nonlinear friction torque as a function of q. A typical choice for the validity function pi is the Gaussian function

where di is the center and ai is the variance of the Gaussian function. Now a set of normalized validity functions wi : B -+ [O, 11 can be defined

where N is the number of local models used to compose one global model. This definition implies that for all Q: EL, wi(Q, 8) = 1. The polytopic friction model becomes

where a;(q-di)+bi is the affine mcdei of the friction locally valid arcund angular velocity di. For the identification of the PLM, the centers di, slopes ai and offsets bi of the linear models and the variance ai of the Gaussian validity functions have t o be estimated.

One way t o construct an odd function with the PLM is the following; Choose an even number of local models N which are divided in pairs of two. The centers are chosen opposite d; = -dzi, the variances are chosen equal ai = a 2 i as well as the slopes a; = azi and the offsets are again chosen opposite bi = -bZi with i = 1,. . . , +. An advantage of this construction is a reduction of the number of parameters by a factor 2. These conditions ensure an odd friction function and the state space representation of the PLM becomes

For the PLM, the model parameters become

So, the modeling problem is reduced t o two sub-modeling problems: (i) divide the operating space of the system into a set of operating regimes and (ii) identify with every operating regime a locally valid mechanical model together with a corresponding validation function. The PLM can be represented as

N

x = C wi(& 8) {Ai(8)x + Ci(Q)} + Bu (3.4) i=l

where the input matrix B is considered t o be known exactly, since c, = 16 [N.m/V] and J = 0.03 [kg.m2] were identified in Hensen (2000).

Controller synthesis

The controller design consists of two parts, i.e. (i) feedforward control on the basis of estimated PLM parameters and (ii) a nonlinear optimal controller for the remaining error dynamics. The optimal controller design is derived by choosing a proper Lyapunov function assuming that the real system is within the PLM set.

The feedforward control part which is based on the model estimates of Eq. 3.4 can be written as

where 5 d are the desired state trajectories and (BTB)-' gT is the (pseudo) inverse of B. With u = u j f +ufeedbaCk the remaining error dynamics become

where the tracking error e = x - x d .

In contrast t o chapter 2, the remaining error is controlled by a nonlinear (feedback) controller which is optimized in time domain by minimization of the following integral criterion

The optimal control law uOptimal can be computed by solving the Hamilton-Jacobi-Bellman equation

by choosing the optimal control law Ufee&ack as

U f eedback = Uoptimal

and the structure of the costfunction L(e,ufeedback) as

where Qij > 0 V(i . j ) . Subsequently, the following conditions for optimality can be found

where Qij and Rj are symmetric weight matrices that balances the accumulated deviation of the state error e from zero and the accumulated amplitude of the feedback control input ufeedback in the cost function L(e, u eedback).

So, if there exists a P > 0 satisfying the following set of Algebraic Riccati Equations (ARES)

then the error feedback control law with ufeedback = uoptimal minimizes the the costfunction J = J(e,ufeedback) for the system described by Eq. 3.4. Furthermore, V = e T p e serves as a Lyapunov function for the closed-loop system (details can be found in [dl).

Optimal controller design

One way of designing an optimal controller is to find one state feedback which is independent of the validity of any local model. This can be done by setting all Rj equal t o R which results in

which is simular t o a PD control law of the following form

If we reconsider the controller tuned in Section 2.2 again, K was set t o 12 [s] and fd was set t o 8 [Hz] . In order to make a comparison between this controller and the formally tuned PID controller, we have chosen the parameters fl and f 2 equal to PC and D, respectively. By solving [ fl f2 ] = R - ~ B ~ P (with R=lO), P will have the following form

Pll can be found by solving the the AREs (Eq. 3.6)

so that P > 0 and &; > 0. Since the friction function is odd, the nuiilber of regimes wi!i be reduced from 4 regimes t o 2 (in the friction model 4 local models are used). Therefore, only 2 AREs have t o be solved (i = 1,2). P, Q1 and Qz will have the following values:

Next, a different input weight is used for regime I (this is the low velocity regime). The following value for R1 is proposed

R1 = a:Rz, with 0 <a: 6 1.

If P, Q1 and R are chosen so that Eq. 3.6 is satisfied (for regime I) , it is always possible t o find a Q1 > 0 when P is unchanged and R1 = a:R (0 < a: 6 1). With a small we obtain a high gain controller where a is limited due t o the fact that high gain control will amplify measurement noise and the torque control u is limited in practice. In this state space regime a stiff PD-controller is required t o reduce steady state errors due t o static friction.

3.3.2 Fixed friction compensation: causal model with stick region

In this section two primary objectives are dealt with. The first is t o present methodologies developed for ex- perimentally determining accurate models for nonlinear friction. The second objective is t o present alternative closed-loop controller strategies for decoupling the effect of friction in order to improve positioning accuracy. The identification methodology is novel in the manner in which it extracts the nonlinear friction properties from the closed-loop errors via an iterative signal processing technique.

A fundamental problem with classical friction models is that they are not causal, that is, the discontinuity a t zero speed allows the friction t o take on an i n f i ~ t e mmber ef values. Thk pr&!em appears 2s errms er instabilities in algorithms that depend on "true" zero velocity t o correctly compensate for friction. Systems that differentiate a quantized position signal t o estimate velocity are especially vulnerable since a calculated zero speed may not occur during velocity reversals.

A solution to this problem is proposed in [ 5 ] . This solution circumvents the discontinuity by defining a small "stick" region a t a minute speed in which the velocity is defined t o be zero, as shown in Fig. 3.9. In this region the friction force balances the net force acting on the rest of the system. Equilibrium is maintained until the breakaway force is exceeded, and the system moves into the slip region.

[5] Proposes a method that extracts the friction characteristic from the loop errors of a state feedback motion controller. The controller is initially designed by incorporating all available system knowledge into the model. Fig. 3.10 shows a block diagram of such a controller. The feedforward controller provides accurate command tracking whereas the feedback controller attempts to reject disturbances in the form of external loads or unmodeled dynamics, such as friction. In doing so, it acts as a disturbance estimator. Thus, the poles of the controller should be placed t o obtain disturbance rejection properties (we will use the controller as derived

A- Velocity

Figure 3.9: Causal model with stick region

Figure 3.10: Block diagram of a motion controlled system with feedforward and linear state feedback control

in Section 2.2. for this application), whereas the model-based feedforward controller should be independently designed for command following. By using signal processing analysis techniques t o isolate the errors as functions of the states, e.g., velocity or position, the physical relationship between friction and the spatial states become readily apparent. This observed relationship is used t o formulate an appropriate model structure. Then, these same torque errors are used t o identify the model parameters. This model is then used t o implement a nonlinear friction compensation in a feedforward or feedback format. By applying this method in an iterative scheme, the systems's performance can be continually improved by progressively extracting and compensating the dominant, unmodeled terms until the tracking errors are reduced to an acceptable level.

An iterative identification methodology

Suppose the feedforward in Fig. 3.10, where j represents the estimated system mass-inertia and FfTiC(w) rep- - resents the estimated velocity dependent friction model, is ideal (FfTiC(w) = Ffric(w) and J = J). Then the system will track the commands perfectly with zero state errors. However, if the feedforward model does not accurately represent the physical system, tracking errors will result. As previously stated, unmodeled dynamics of the system will appear as feedback torque commands Tfb.

As a systems test procedure, assume that the system can be driven over trajectories that includes a velocity reversal. If the feedback torque error Tfb is measured during the trajectory and plotted versus velocity w, the result may look similar t o one of the cases shown in Fig. 3.12. Here, feedforward has cancelled all dynamics except for the system friction, and torque error is entirely caused by the physical friction. However it is more

Figure 3.11: Reference acceleration (or wref) versus time.

likely that the feedforward compensation is not perfectiy accurate, and errors due t o inertia and damping wiii also be present. Fig. 3.12(a) represents a worst case since no feedforward compensation is used, which implies

A

J = 0, b = 0. The basic friction characteristic is present, but it is contaminated by other errors. For example, the hysteresis loop of Fig. 3.12(a) is caused by inertial errors and will be present whenever the system is accelerating. The size of the hysteresis loop can be used to estimate the inertia of the system

where ATfb is the width of the loop (Gref is a step function, as depicted in Fig. 3.11, and therefore the trajectory is a second order feed function).

After a good estimate of inertia is made, it will be added t o the feedforward path t o reduce or eliminate state errors while accelerating. With the improved tracking control, the system is again exercised over the same trajectory. The new T f b versus w relationship has no hysteresis loop if the inertial compensation was successful, as observed in Fig. 3.12(b).

The sloped line in the Tfb that is characteristic of Fig. 3.12(b) suggests that a viscous damping estimate is

also present. The slope of this line allows the damping coefficient b t o be estimated directly by standard least squares curve fitting.

At this point, the feedforward terms include accurate inertial and viscous damping terms. Repeating the command trajectory, the friction characteristic of Fig. 3.12(c) is observed. The two parameters that define the friction model Tstzck = Fs and Tslzp = Fc are obtained by inspection. Incorporating these parameters in the feedforward model yields complete friction compensation, with only residual torque feedback noise remaining, as shown in Fig. 3.12(d).

Figure 3.12: Feedback torques versus velocity with progressively improved feedforward model: (a) No feedforward; (b) inertia compensated; (c) viscous damping compensated; (d) stiction and sliding friction compensated.

The use of spatial synchronous averaging

A potential problem with any identification method is the contamination of data by uncorrelated signals inherent in the data. Most methods attempt t o reduce the effects of such unwanted signals by the selection of robust signal processing techniques. One signal processing technique (synchronous averaging) has proven particularly effective in attenuating such noise during the experimental stage.

Most averaging methods are based on attenuating signals that are not correlated with the common reference variable (usually time). White noise is an example of such a signal that is asynchronous and uncorrelated in time. However, other types of noise can still taint the results due t o their correlation to the common reference. Examples include other physical characteristics that vary with position, acceleration, or current (e.g., gravitational loads, torque ripple, spatially varying loads, Coriolis effects, etc.). These unknown physical effects are unmodeled so they will also be contained ir? the Tfb signal, along with the frictionai properties of interest as shown by (3.8).

Thus, t o improve the friction model, it is desirable t o attenuate the components of Tfb that are asynchronous (uncorrelated) with the spatial variable, velocity (i.e., Tother). This will leave only the desired signal that is synchronous with velocity (i.e., TfTictio,). By proper design of the experiment, a deterministic effort is made t o average out the signals that are asynchronous with the spatial variables, most especially with velocity.

This concept suggests that the data be averaged in the velocity (not time domain). A common velocity distribution should be used in all experiments, but all other motion variables are varied in order t o average out effects not included in the velocity-dependent friction model. When repetitive setpoints are used it can be better not t o change the other motion variables so that acceleration dependent behavior (for instance) can be captured in velocity dependent friction model (i.e., we know that static friction is higher for slow acceleration than for fast acceleration). A number of velocity trajectories can be generated, where each has the same shape and size, but the corresponding position and acceleration trajectories can have different magnitude, direction, and initial positions.

Fig. 3.13 shows the result

0 5 1 5 2 ;5 3 5 b 4 5 ; 5!5 Velocity [radls]

0.5 I 5 5 b i .5 5 s'5 Velocity [radls]

Figure 3.13: Experimental controller torque errors (a: low acceleration, b: high acceleration).

Friction compensation: nonlinear state feedback

Feedback compensation implements the causal friction model described in Fig. 3.9 with stick region width Aw:

I w I > Aw (system is in the "slip7' region)

I w ] < Aw (system is in the "stick" region)

where TC is the estimation of the Coulomb friction and the min function here selects the number closest to zero and limits the friction torque to the breakaway value T, in the stick region. T ~ ~ ~ ~ ~ ; ~ ~ is the calculated friction torque used for the compensation, and T * is the estimated torque based on the known torques, i.e., Tf + Tfb in a feedback configuration and T f f in a feedforward configfiration. As depicted in the bkock diagram in Fig. 3.14, L 1 m e sum of feedback and feedforward torque corrections, is a good estimate of T*

The quality of the T* estimate is good if the assumption of a high bandwidth motor torque control loop is satisfied and if the motor torque is accurately known. A velocity estimate is also required for the digital controller and is obtained by a direct difference approximation of the encoder signal

where T is the sampling period. The differential estimate proved t o be fairly smooth in this application because the encoders resolution was high relative t o the sampling frequency.

Friction model Velocity estimator

Figure 3.14: Nonlinear decoupling state feedback friction compensation technique

Altough it is unlikely that the closed-loop system will be unstable because the feedback is restricted to T,, no stability-proof is given in [5].

Friction compensation: nonlinear state forward

The feedforward controller uses a friction model similar t o Eq. 3.9 and 3.10 but with some modifications. By definition, commanded states are used in place of the actual states. In addition, since no estimate of the feedback Tfb is available, the T* estimate must be derived completely from feedforward (i.e., T* = cmTff). Fig. 3.15 shows the controller topology.

3.3.3 Fixed compensation: the Lund-Grenoble (LuGre) model

In [2] a new dynamic model for friction is given. This model captures most of the frictional behaviour that has been observed experimentally.

The qualitative mechanism of friction are fairly well understood. Surfaces are very irregular a t the microscopic level and two surfaces therefore make contact a t a number of asperities. We visualize this as two rigid bodies that

Figure 3.15: Nonlinear decoupling state feedforward friction compensation technique

Figure 3.16: The friction interface between two surfaces.

make contact trough elastic bristles. When a tangential force is applied, the bristles will deflect like springs which gives rise to the friction force, as depicted in Fig. 3.16. The friction interface between two surfaces is thought of as a contact between bristles. For simplicity, the bristles on the lower part are shown as being rigid.

If the force is sufficiently large some of the bristles deflect so much that they will slip. The phenomenon is highly random due t o the irregular forms of the surfaces. Therefore, a model is proposed that is based on the average behavior of the bristles. The average deflection of the bristles is denoted by z and is modeled by

where v is the relative velocity between the two surfaces. The first term gives a deflection that is proportional t o the integral of the relative velocity. The second term asserts that the deflection z approaches the value

is steady state i.e., when v is constant. The function g is positive and depends on many factors such as material properties, !ubricat.tion, temperature. It need not t o be symmetrical. Direction dependent behavior can therefore be captured. For typical bearing friction, g(v) will decrease monotonically from g(0) when v increases. This corresponds to the Stribeck effect. The friction force generated from the bending of the bristles is described as

where a0 is the stiffness and a1 a damping coefficient. A term proportional t o the relative velocity could be added to the friction force t o account for viscous friction so that

The model given by (3.11) and (3.13) is characterized by the function g and the parameters oo, ol and 02. A parameterization of g that has been proposed to describe the Stribeck effect is

aog(u) = Fc + (F, - E ) e - ( ~ 1 7 ~ ~ ) ~ (3.14)

where F, is the Coulomb friction level, F, is the level of the stiction force, and v, is the Stribeck velocity, as plotted in Fig. 3.17. With this description the model is characterized by six parameters 0 0 , al, oz, F,, F,, and

Figure 3.17: Stribeck curve

v,. It follows from (3.12)-(3.14) that for steady state motion the relation between velocity and friction force is given by

F,,(v) = oog(v)sign(v) + o2v = ~ , s ign (v ) f (F, - ~ , ) e - ( ~ ~ ~ ~ ) ~ s i ~ n ( v ) + ozu

Note however, that when velocity is not constant, the dynamics of the model will be very important and give rise t o different types of phenomena, as shown in Fig. 3.18.The low velocity friction characteristics are particularly

Figure 3.18.: Friction force for various accelerations (dotted line acceleration approximates zero)

important for high performance pointing and tracking. The model can describe arbitrary steady-state friction characteristics. It supports hysteretic behavior due t o frictional lag, spring-like behavior in stiction and gives a varying break-away force depending on the rate of change of the applied force. All these phenomena are unified into a first order nonlinear differential equation.

Position control with a friction observer using the LuGre friction model

Consider the problem of position tracking for a simple mass system subjected t o friction

(where d x l d t = v is the velocity and F the friction force given by 3.13). Assume that the parameters oo, 81, a 2

and the function g in the friction model are known. The state z is, however, not measurable and hence has to be

observed to estimate the friction force. For this we use a nonlinear friction observer [2] given by

Stability conditions follow from Eq. 3.19.

and the following control law

A dZxd u = -C(s)e + F f m-

dt"

where e = x - xd is the position error and xd is the desired reference which is assumed to be twice differentiable. The term ke in the observer is a correction term from the position error. The closed-loop system is represented by the block diagram in Fig. 3.19. With the observer based friction compensation, we achieve position tracking as shown in the following theorem. Theorem: Consider system (3.15) together with the friction model (3.11) and

Figure 3.13: Fricti~r, c~mpensatior, using the LuGre friction mode!

(3.13), friction observer (3.16) and (3.17), and control law (3.18). If C(s) is chosen such that

A

is strictly positive real (SPR) [20] then the observer error, i? = F- F, and the position error, e, will asymptotically go to zero.

The proof of this theorem is given in appendix B. The theorem can also be understood from the following observations. By introducing the observer we get a dissipative map from e to Z (Z = z - 2 ) and by adding the friction estimate to the control signal, the position error will be the output of a linear system operating on Z. This means that we have an interconnection of a dissipative system and a linear SPR system as seen in Fig. 3.20. Such a system is known to be asymptotically stable [20].

Figure 3.20: Block diagram of a linear and nonlinear block.

When the SPR condition is checked for our system (rotating arm with !ead/!ag cor?tro!ler and second order low pass filter), it follows that the SPR condition is violated for frequencies higher than 31 [rad/s]. The SPR condition could be satisfied when for instance, a very stiff PD controller with a high gain and low fd is used. Unfortunately, such a stiff controller will cause the total system t o become unstable, so we can conclude that in this case it is impossible to satisfy the SPR condition. However, since C. Canudas the Wit and P. Lischinsky [I] showed that the SPR condition is not so restrictive and that other analyis tools may be used to find sharper conditions, the LuGre model will still be implemented in combination with the derived (lead/lag) controller.

3.3.4 Adaptive control: adaptive friction compensation with partial knowledge based on LuGre model

Friction may vary as a function of the normal forces in contact, temperature changes, position, etc. A variation in one of these factors may affect the six friction parameters in a rather complicated way. It will thus be necessary to design adaptive mechanisms that can cope with variations of the full set of parameters. However, owing to the fact that the friction parameters appear highly non linear in the model and the state z is not measurable, this task appears very difficult, if not impossible.

It is thus important t o establish the physical cause of the friction variations and try to explicitly relate this cause to the parameters of the friction model. Disturbances acting on the friction model can thus be "structured" with a minimum number of unknown parameters. We will consider the following two cases.

The static friction parameters (apart from viscous friction) are assumed to vary and the dynamic friction parameters are assumed t o be invariant, i.e.

where the friction changes dne t~ vari&icns ir? the normz! forces a,re captured by 8.

All static and dynamic parameters are assumed t o vary as

Here 8 may represent variations probably due to temperature changes.

The reasoning behind the two assumptions above is as follows. In the first case (eq. (3.20) and(3.21)), variations in the normal forces may have a substantial impact on the static parameters, in particular on the

function g(v) that includes Coulomb and break away friction. Besides, the parameters a 0 and al are considered t o be invariant, since the lubricant between the two surfaces in contact and the material of these surfaces do not depend on the normal forces. The possible dependence of the viscous friction 0 2 on 8, is assumed in this case t o be dealt with by the linear compensator C(s), and is thus not modeled in the above description. In the second case , Eqs. 3.22 and 3.23, variations in temperature and material wear are assumed t o change uniformly both static and dynamic parameters.

In this context, parameter uncertainties are captured by the single parameter 8, since the six nominal friction parameters are assumed t o be known. The problem now is t o modify the in Section 3.2.3 described observer-based friction compensation scheme t o cope with uncertainties and "slow" variations in 8 while preserving closed loop stability.

The first case: the six nominal friction parameters and the system inertia J are a priori known, the dynamic friction parameters are invariant and the variations in the static friction parameters are captured by the model (3.20), (3.21), where 8 is assumed t o be unknown and bounded as 0<8 < oo. Then the adaptive controller

where uf and af are the filtered signals

applied to a single-mass system: J$ = u - F with (3. l l ) , (3.13) and (3.14) yields asymptotic stability if a linear operator C(s) can be found so that the linear map G(s) in (3.19) satisfies the SPR condition, of which the proof can be found in [I].

Chapter 4

Exp 0 n ~ . ~ m n + v 1 1 1 + ~ c a I G S U A l t J r 3

In this chapter, the friction compensation techniques, as presented in Section 3.3, are implemented on the rotating arm in order to make a comparison between their effectiveness in dealing with friction for point to point movement. An attempt has been made to translate the results from the experiments into indices, which make it possible to make a qualitative comparison in the last chapter. The performance is measured by evaluating the settling time, the error a t the end of the setpoint and the error during the setpoint. Although, the latter property is more a measure for tracking performance as it is for point t o point movement, one can not obtain good performance in the sense of the first two points without optimizing the third. The settling time is defined as the time, which is needed after the setpoint, t o reduce the error of the system within a predefined error bound. The margins are set t o f 0.0001 * 7r [rad] which is 8.192 times the resolution of the angular displacement of the arm, which are referred t o as Dmin and Dmaa representing respectively the lower and the upper bound.

Semoint 5 5 .................................... Rotating arm

Friction I ' -0

compcnsatlon

Actual friction L 3

Figure 4.1: Block diagram of a compensated closed-loop system

All the implemented compensation techniques are tuned by minimizing the controiier output Tfb (Fig. 4.i) as well as the error a t the end of the setpoint: every compensation serves the purpose of reducing the error and consequently reducing the controller output. Suppose the total compensation (i.e., the mass feedforward plus the friction feedforward, -back) is ideal. Then, the system depicted in Fig. 4.1, will track the commands perfectly without state errors. However, if the compensation model does not represent the physical system exactly, tracking errors will result. Thus, unmodelled dynamics of the system will appear as feedback torque commands Tfb. Ideally, the controller output will be zero, while the compensation output takes care of the total movement. For the sake of clarity, the following sections will, besides the performance results, also contain the controller and compensation outputs.

The performance of the applied compensation techniques are compared with the results obtained with the conventional lead/lag controller, which is derived in Section 2.2. To this end all controllers are tested with the setpoints that are described in Section 2.3, consisting of long stroke movements and very short movements. The acceleration profiles corresponding t o these setpoints are shown in Fig. 4.2. During all experiments, the sample

frequency is set t o 1000 [Hz] and the discrete integration scheme used is ode5 [18]. The position is measured by a high resolution sin/cos encoder and the resolution of the arm movement is 2x1 (5OOO* 8.192 *4) = 3.835. [rad].

."

0 1 0 2 0 3 0 4 0 5 0 6 Time [secl

0

K i Time [sec]

Time [sec]

. -

0 1 0 2 0 3 0 4 0 5 0 6 T~me [sec]

0 1 0 2 0 3 0 4 0 5 0 6 Tlrne [sec]

a) Setpoint 1 b) Setpoint 2 c) Setpoint 3 d) Setpoint 4 e) Setpoint 5

Figure 4.2: Acceleration of the 5 setpoints.

During the experiments with the classical lead/lag controller with mass feedforward, a strange event occurred during the constant velocity period of the setpoint, the position error persistently remained negative (which means that the measured position of the arm is ahead of the setpoint), while the controller maintains this error by employing a force in the direction of the error, as depicted in Fig. 4.3. Because this error only occurs when a

PID with 1st order hold PID without 1st order hold

I , : 3 I ' , ; i

I /

-4 I j

0 0.1 0.2 03 0.4 0.5 06 0.7 0.8 0.9 1 Time [sec]

-2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time [sec]

Figure 4.3: Error with and without 1st order hold on output encoder.

higher order integration scheme is used, it is assumed that the error lies in the fact that these integration schemes use intermediate (and therefore obsolete) data points within the sampling period. We expect that the controller compensates the delay following from the Zero Order Hold (ZOH) and therefore causes the system t o be ahead of the setpoint, but further examination of this problem (especially the role of Simulink in combination with dSPACE) is recommended. As expected, a first order hold a t the output of the position measurement reduces this error significantly, also shown in Fig. 4.3. Therefore a first order hold is applied in all the experiments in the

following sections. In Section 4.1 the performance obtained with a classic filter lead/lag controller combined with mass feed-

forward, as described in Section 2.2, is presented and will be used as a reference performance for the other compensation techniques. In Section 4.3, the performances obtained from the various proposed friction compen- sation techniques, as presented in Section 3.3, are compared with respect to the performance of the filter lead/lag controller plus mass feedforward. The experimental results of the proposed friction compensation techniques are combined in the last part of this section.

4.1 Integral control: filtered lead-lag cod roller plus mass feedfor-

The experimental results for the 5 different setpoints, as given in Section 2.3 and shown in Fig. 4.2, are depicted in Fig. 4.4.

x 1 0 ' ~ I

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x lo* Time [sec] x 10" Time [secJ

I I 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x Time [sec] Time [sec] I I I

I 0 0.2 0.4 0.6 0.8 1

Time [sec]

Figure 4.4: Error of system with filtered lead/lag plus mass feedforward. Setpoint 1 (= fig. a) to setpoint 5 (= fig. e).

The performance of the friction compensation techniques that follow in the next sections, are all expressed in perce~tages ef the r s d t s ..rising from the experiments in this section. The table below shows the absolute performance of the controller expressed in three different criteria.

Setpoint Max. error during setpoint [rad] Max. error after setpoint [rad] Settling time [see]

1 -0.0021 -0.0021 0.1169 2 -0.0024 -0.0024 0.1049 3 0.0023 -0.0020 0.1169 4 0.0020 -0.0007 0.2233 5 0.0018 0.0017 0.0302

Results filtered lead/lag controller.

4.2 Fixed compensat ion: parameter estimation

In this section the estimation technique, as described in Section 3.2.2, is used t o make a global estimation of the parameters that will be used in the following sections, i.e., the sections that consider fixed compensation techniques. The fine tuning is done later on, by evaluating the controller output when the specific compensation technique is used.

-0.5~ I 0 2 4 6 8 1 0

Velocity [radlsec]

-0.5 t.....l 0 2 4 6 8 1 0

Velocity [radsec]

-0.5; 2 4 6 8 1 0

Velocity [radlsec]

-0.5; 2 4 6 8 1 0

Velocity [radsec]

Figure 4.5: Input versus velocity. a) No feedback b) Mass feedforward c) Mass + viscous friction feedforward d) Mass + viscous + Coulomb friction feedforward

Fig. 4.5 shows the output torque of the controller (input of the arm) versus the velocity for the imposed setpoint.. The setpoint has the following form: it starts with a constant positive acceIeration of 17.7 [rad/s2] during 0.6 seconds followed by a constant negative acceleration of 17.7 [rad/s2] during another 0.6 seconds. Fig. 4.5a) shows the results when no feedforward is used. Next, a mass feedforward (J = 0.028 [kgm2/rad]f is added t o the controlled system which leads t o the results depicted in Fig. 4.5b). In Fig. 4 . 5 ~ ) the feedforward is extended with viscous friction (b = 0.06 [Nmslrad]). Finally, Coulomb friction is added t o the feedforward in Fig. 4.5d) (F, = 0.5 [Nm]). From Fig. 4.5 also follows that, in this case F, -t- F, = 1 [Nm].

4.3 Performance obtained with various friction compensat ion tech- niques

Stiff PD control: gain-scheduled controller

The experimental results for the 5 different setpoints (see section 2.3) are depicted in Fig. 4.7, 4.8. The table below shows the relative performance of the gain-scheduled controller expressed in three different

criteria in comparison with the filtered lead/lag. The gain-scheduling parameter a is set to 0.2, which resulted in acceptable input torques and is the lowest value possible with respect t o noise amplification.

Max. error during setpoint Max. error after setpoint Setpoint % of settling time Leadllag

(% of error Lead/Iag) (% of error Leadllag) 1 157.1 23.4 9.7

5 27.8 28.8 1.6 Mean: 93.2 60.6 8.2 --

Results gain scheduled controller.

From the experiments follows that the gain-scheduled controller has excellent settling behavior and a relatively iow error after i, has been reached in comparison with the fi!tered leadjiag controiier. The maximum error between t , and t , however, is relativeiy large. This can be explained by the fact that there is no integral action (or high gain) t o reduce the error in the (constant) high velocity regime. Another possible drawback is the high torque that is used a t the moment that the controller switches t o the high gain feedback.

Fixed compensation: causal model with stick region

The experimental results of a feedforward for the 5 different setpoints (see section 2.3) are depicted in Fig. 4.9, 4.10.

The table below shows the relative performance of the feedforward compensation expressed in three different criteria in comparison with the filtered lead/lag.

Max. error during setpoint Max. error after setpoint Setpoint % of settling time Lead/lag

(% of error Lead/lag) (% of error Lead/lag) I 61.9 52.3 7.6 2 41.7 34.6 153.3 3 69.6 60.2 186.3 4 65.0 175.9 4.9 5 77.7 82.2 157.9

Mean: 63.2 81.0 102.0 Results causal model f eed forward .

From the experiments follows that the feedforward increases the performance regarding the maximum error during the setpoint. However, the settling time as well as the maximum error after setpoint, exhibit large variations on the average performance. Therefore, in spite of the coincidental performance improvements, this feedforward has little practical value for point t o point movement.

Next, we consider the results of experiments with a feedback of the causal model (Fig. 4.11, 4.12).The table below shows the relative performance of the feedback compensation expressed in three different criteria in comparison with the filtered leadliag.

Max. error during setpoint Max. error after setpoint Setpoint % of settling time Lead/lag (% of error Leadllag) (% of error Lead/iag)

1 85.5 36.0 3.1 2 40.5 17.3 92.3 3 54.2 35.0 76.0 4 85.1 55.7 0.4 5 59.4 58.0 58.6

Mean: 59.5 40.4 46.08 Results causal model f eedback .

From the figures can be concluded that there is a significant improvement on the last 2 points (i.e. max error after setpoint and settling time) when a feedback is used, compared to the feedforward. The maximum error during setpoint however, remains practically the same. Although the variance on the average relative settling time is still large, the performance stays under 100% for each setpoint and therefore the feedback contributes to an increase of the performance.

For the feedforward as well as the feedback the following parameter values are used (derived by minimizing the PID controller output as well as the error a t the end of the setpoint): J = 0.026 [kgm2/rad], b = 0.08 [Nmslrad], Fc = 0.4 [Nm], F, = 0.5 [Nm] and dw = 0.5 [radls]. It can be seen that the optimal values for compensation are not equal to the estimated parameters in Section 4.3, since we are not interested in an optimal estimation but in an optimal compensation.

Fixed compensation: Lund Grenoble model

When the LuGre model is implemented on the real system, some numerical problems will arise when trying to integrate the observer (3.16) using the first-order Euler approximation (or any other discrete version of (3.16)). The discrete version of (3.16) with constant 'high' velocity (v 2 us for which g(v) E 5) is as follows (using the Euier scheme) :

where T is the sample time and v(k) is the sampled velocity. The stability condition will then be:

where it is preferred that v(k) is significantly larger than v,. When v(k) exceeds 3, the observer becomes unstable, therefore the integration will be stopped and the steady-state value of 2 will be used:

Now, the adjusted LuGre model is used in a feedforward. The experimental results of this feedforward for the 5 different setpoints (see section 2.3) are depicted in Fig. 4.13, 4.14.

The table below shows the relative performance of the feedforward compensation expressed in three different criteria in comparison with the filtered leadllag.

Max. error during setpoint Max. error after setpoint Setpoint % of settling time Leadllag (% of error Lead/lag) (% of error Lead/lag)

1 90.5 28.8 70.8 2 58.3 39.4 107.5 3 60.9 54.4 82.0 4 80.0 53.5 63.4 5 44.4 15.4 0.0

Mean: 66.8 38.3 64.7 Results LuGre f e e d f o ~ w a r d .

From the experiments follows that the feedforward increases the overaii average performance. The settling behaviour however, exhibits relatively large variations on the average performance. Therefore, in spite of the improvement of the average behaviour, this feedforward has little practical value for point t o point movement.

Next, we consider the results of experiments with a feedback of the LuGre model (Fig. 4.15, 4.16).

Max. error during setpoint Max. error after setpoint Setpoint % of settling time Lead/lag (% of error Lead/Iag) (% of error Lead/lag)

1 71.4 14.4 0.0

Mean: 56.2 20.7 6.9 Resul ts LuGre feedback.

From the experiments can be concluded that this feedback will result in excellent overall performance and the average performance computed as the mean error over the five different setpoints as well as the corresponding variances are very good. Especially, the settling behaviour has improved significantly in comparison with the non compensated leadllag and the LuGre feedforward.

For the feedforward as well as the feedback, the following parameter values are used (derived by minimizing the PID controller output as well as the error a t the end of the setpoint): J = 0.025 [kgm2/rad], oo = 1.73 + lo3 [Nmlrad], ol = 3.02 - 0.08 [Nmslrad], b or 0 2 = 0.08 [Nms/rad], F, = 0.32 [Nm], F, = 0.5 [Nm], v, = 0.3 [radls] and v,,, = ;$30,001 0.35 [radls]. In this thesis the estimation of a0 and al is left outside of consideration; more information about this subject can be found in [3]. Again, it can be seen that the optimal values for compensation are not equal to the estimated parameters in Section 4.2, since we are not interested ir? an @ma! zstimatim but ir, ar, epti=d cnrr?psnsaticln,

Note: although the regarded performances of the different friction compensation techniques differ, the outputs of the various controllers are significantly reduced by the use of compensation techniques. This indicates that the used feedback, -forward models will to some extent, contribute t o an improvement of the tracking performance of the controlled system.

4.4 Conclusions In this last chapter the considered friction compensation methods from the previous sections are compared with respect to their performance. In order t o make a comparison between the various methods, the results of the experiments are translated into three indices (i.e. settling time, error after setpoint and error during setpoint) for each method. These indices are established as follows

(100 - RE) I =

10

where I is the index of the concerning performance and RE is the average relative error or settling time (compared to the filtered lead/lag controller plus mass feedforward that is derived in Section 2.2). Thus, an index of 10 is a perfect score, an index of 0 means that there is no improvement and a negative index indicates that there is a deterioration in performance in comparison with the filtered lead/Iag controller. Fig. 4.6 shows the results of the five friction compensation methods compared to the filtered leadllag controller.

Performance of compensation techniques in comparison with the filtered leadlag controller

Gain scheduled Causal ff Causal fb LuGre ff LuGre fb

Figure 4.6: The performance of the various compensation techniques expressed in indices.

From Fig. 4.6 follows that the filtered lead/lag controller plus mass feedforward extended with a LuGre feedback has the best overal performance. The gain-scheduled controller has (Iike the LuGre feedback) an excellent settling behaviour. The maximum error during the setpoint however, is relatively large, which can be explained by the fact that there is no integral action (or high gain) t o reduce the error in the (constant) high velocity regime. A notable fact is that if one should decide to use the LuGre feedback, it is possible that the estimated parameters can not be used in combination with the chosen sampling frequency in the experimental setup because of numerical stability problems when trying to integrate the observer, as described in Section 4.5. Problems of this kind will dissolve in the future when we have disposition of faster computers.

In the group of model based friction compensation techniques, a distinction between feedforward and feedback can be made. From the experiments follows that regulator (or pcsitioning) tasks that are carried out by a C I A ..,.A l...'.A / I n m ~ l l b e l ~ ~ lcau, la5 extended with a frictkr? feedfnrward, will not ensure an improvement in settling behaviour: both the cafisal mode! feedforward as well as the L u g e feedforward have a large variance on the average settling performance. It is a remarkabe fact that the poor settling behaviour is caused by a reduction of the error a t the end of the setpoint: the integral action needs (because of its cumulative nature) more time to deal with the decreased error than with the originally higher error. Therefore, these feedforwards have little practical value for point t o point movement. The feedback of the causal model however, shows significant improvement on all three performance criteria.

I 0 0.2 0.4 0.6 0.8 1 x Time [sec]

L 0 0.2 0.4 0.6 0.8 1 x 10" Time [sec]

I 0 0.2 0.4 0.6 0.8 1 x l o 3 Time [sec]

-1 I 0 0.2 0.4 0.6 0.8 1

Time [sec]

- Gain scheduled controller - - Leadlla~ controller

-1 0 0.2 0.4 0.6 0.8 1

Time [sec]

Figure 4.7: Error of system with gain scheduled controller plus PLM feedforward/fedback compared with error of system with filtered lead/lag plus mass feedforward. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

).2 0.3 0.4 0.5 0.6 Time [sec]

I .

0.2 0.3 0.4 0.5 0.6 Time [sec]

1.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Time [sec] Time [sec]

-5 I i

0.2 0.3 0.4 0.5 Time [sec]

Figure 4.8: Torque of various components of system with gain scheduled controller plus PLM feedforward/feedback compared with output filtered lead/lag plus mass feedforward. Setpoint 1 (= fig. a) to setpoint 5 (= fig e).

I Dmax u m $ O L - -

-2 - - Dmin I I

0 0.2 0.4 0.6 0.8 1 m3 Time [sec]

I I I

I 0 0.2 0.4 0.6 0.8 1 x Time [sec] I I I

I I

0 0.2 0.4 0.6 0.8 1 x lo3 Time [sec]

, 6

I 0 0.2 0.4 0.6 0.8 1

Time [sec]

- Friction feedforward - - No friction compensation

I 0 0.2 0.4 0.6 0.8 1

Time [sec]

Figure 4.9: Error of filtered lead/lag controlled system plus mass feedforward extended with causal model feed- forward compared with error filtered lead/lag plus mass feedforward without causal model feedforward. Setpoint 1 (= fig. a) to setpoint 5 (= fig. e).

5 - a ) 1 / \ I 1 1 \ I Z

P 2 I

-5 - I I

0.1 0.2 0.3 0.4 0.5 0.6 Time [sec]

O ! 0:; 0:3 0:4 0:s 0:s Time [sec]

-1 I 0.1 0.2 0.3 0.4 0.5 0.6

Time [sec]

-5 I I I,

0.1 0.2 0.3 0.4 0.5 0.6 Time [sec]

2 r . .

-20!1 Oi2 '0:3 0:4 0:5 0:6 Time [sec]

- Output controller system with feedforward) - - Mass feedforward - Friction feedforward

Output controller (system without feedforward)

Figure 4.10: Torque of various components of filtered lead/lag controlled system plus mass feedforward extended with causal model feedforward compared with output filtered lead/lag plus mass feedforward without causal model feedforward. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

I Dmax - 2

f -2

I I

0 0.2 0.4 0.6 0.8 1 Time [sec]

A ,

. . . . -4 1 . . ,

0 0.2 0.4 0.6 0.8 1

x lo3 Time [sec] 3 r I '

-1 1 0 0.2 0.4 0.6 0.8 1

Time [sec]

-2 - I i-' I I

-4 - I I

-6 I I.

0 0.2 0.4 0.6 0.8 1 Time [sec]

3 r

-1 I / I . 0 0.2 0.4 0.6 0.8 1

Time [sec]

- Friction feedback - - No friction compensation

Figure 4.11: Error of filtered lead/lag controlled system plus mass feedforward extended with Causal model feedback compared with error filtered lead/lag plus mass feedforward without causal model feedback. Setpoint 1 (= fig. a) to setpoint 5 (= fig. e).

ts

-5 ' - .

0.2 0.3 0.4 0.5 0.6 Time fsecl

' :

-4 IOi2 0:3 64 0:s Time [sec]

- l ' , i 0.2 0.3 0.4 0.5

Time [sec]

-5 !

0.2 0.3 0.4 0.5 0.6 Time [sec]

B -1

-2 0.2 0.3 0.4 0.5

Time [sec]

- Output controller system with feedback) - - Mass feedforward - LuGre feedback

Output controller (system without feedback)

Figure 4.12: Torque of various components of filtered lead/lag controlled system plus mass feedforward extended with causal model feedback compared with output filtered lead/lag plus mass feedforward without causal model feedback. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

0 0.2 0.4 0.6 0.8 1 x 10" Time [sec]

- 0 0.2 0.4 0.6 0.8 1 x 10" Time [sec] I / I

i I " E 2 j e ) ill, - I ill,

-2

0 0.2 0.4 0.6 0.8 1 x 10" Time [sec]

-1 ( . / I ,

0 0.2 0.4 0.6 0.8 1 Time [secl

- LuGre friction feedforward - - No friction compensation

i I -1

0 0.2 0.4 0.6 0.8 1 Time [sec]

Figure 4.13: Error of filtered lead/lag controlled system plus mass feedforward extended with LuGre feedforward compared with error filtered lead/lag plus mass feedforward without LuGre feedforward. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

ts

\I I

I . 0.2 0.4 0.6

Time [sec]

I I .

3.2 0.4 0.8 Time [sec]

I I

p I - e) I jj.

-1 0.2 0.4 0.6

Time [sec]

-2 1 I

0.2 0.4 O.6 Time [sec]

- Output controller (system with LuGre feedforward) - - Mass feedforward - LuGre feedforward

. . . . Output controller (system without LuGre feedforward)

Figure 4.14: Torque of various components of filtered lead/lag controlled system plus mass feedforward extended with LuGre feedforward compared with output filtered lead/lag plus mass feedforward without LuGre feedforward. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

0 0.2 0.4 0.6 0.8 1 Time [sec]

-2 I IL<

0 0.2 0.4 0.6 0.8 1 x 10" Time [sec]

-2 I

0 0.2 0.4 0.6 0.8 1 x Time [sec]

-1 I I, i

0 0.2 0.4 0.6 0.8 Time [secf

- LuGre friction feedback - - No friction compensation

-1 I 0 0.2 0.4 0.6 0.8 1

Time [sec] t

Figure 4.15: Error of filtered lead/lag controlled system plus mass feedforward extended with LuGre feedback compared with error filtered lead/lag plus mass feedforward without LuGre feedback. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

-4 I '. I 0.2 0.3 0.4 0.5 0.6

Time [sec] I I 1

-2 1 ' , 0.2 0.3 0.4 0.5 0.6

Time [sec]

-1 I , . 0.2 0.3 0.4 0.5 0.6

Time [sec]

- Output controller (system with LuGre feed-back) - - Mass feed-forward - LuGre feed-back

- - Output controller (system without LuGre feed-back)

Figure 4.16: Torque of various components of filtered lead/lag controlled system plus mass fedforward extended with LuGre feedback compared with output filtered lead/lag plus mass feedforward without LuGre feedback. Setpoint 1 (= fig. a) t o setpoint 5 (= fig. e).

Chapter 5

The Frequency Respons Functions (FRF's) have proven t o be very useful when it comes t o frequency domain identification and controller tuning. In case of point t o point movement, consideration should be paid to the shape of the setup function in order to avoid unnecessary errors and unacceptable high torques and velocities.

Though often used, the performance of the classical PID controller is still capable of improvement, with respect t o point t o point movement, by the use of friction compensation techniques. From the experimental work can be concluded that:

A dynamic friction model, i.c., the LuGre friction model has the best overall performance in a feedback compensation configuration.

The dynamic LuGre friction model as well as the gain-scheduled PD-controller in a feedback configuration excel regarding their settling performance.

The performance of the feedback compensation configurations is superior t o the performance of the feed- forward compensation configurations.

Although this thesis is restricted to point to point movement, it can be seen that the considered friction compensation techniques, with the exception of the gain-scheduled controller, also have excellent prospects for tracking purposes.

The performance of the various friction compensation techniques has only been examined for short periods of time. Since the friction may vary as a function of the normal forces in contact, temperature changes, position, etc., long duration experiments should be carried out in order t o evaluate the real impact of the former parameters on the performance. Subsequently, the adaptive friction compensation technique as described in Section 3.3.4., can be implemented.

Finally, this thesis has only discussed a few of the most promising friction compensation techniques as presented in Chapter 3. However, further research on other techniques, e.g., dithering, impulsive control and other friction models, remains recommendable. Since the LuGre friction model resulted in an excellent performance, it is recommendable t o examine other dynamic friction models first.

Bibliography

[I] C. Canudas de Wit and P. lischinsky, "Adaptive friction compensation with partially known dynamic friction model", International journal of adaptive control and signal processing, vol. 11, 65-80 (1997).

[2] C. Canudas de Wit, H. Olsson, K.J. k t r 6 m and P. Lischinsky, "A new model for control of systems with friction", IEEE Transactions on automatic control, vol 40, no. 3 (1995).

[3] R.H.A. Hensen, M.J.G. van de MolengraR, "Frequency domain identification of dynamic friction model parameters", accepted for publication in IEEE Transactions on Control Systems Technology.

[4] R.H.A. Hensen, G.Z. Angelis, M.J.G. van de Molengraft, "Adaptive optimal friction control based on a polytopic linear model", MTNS 2000, Perpignan, Proceedings CD-ROM.

[5] C.T. Johnson, R.D. Lorenz, "Experimental identification of friction and its compensation in precise, position controlled mechanisms", IEEE Transactions on industry applications, vol. 28, no. 6 (1992).

[6] M.R.L. Kuijpers, "Modelling and identification of a one-mass system with friction", (October 22 1999).

[7] O.H. Bosgra, H. Kwakernaak, "Design methods for control systems", Notes for a course of the Dutch Institute of Systems and Control (1997-1998).

[8] B. Armstrong-Hhlouvry, P. Dupont and C. Canudas de Wit, "A survey of models, analysis and compensation methods for the control of machines with friction", Automatica, 30, 1083-1138 (1994).

[9] ir. N.B. Meijerman, "Leren programmeren in C f t " , l e druk (1995).

[lo] G. Strang, "Linear Algebra and its applications", third edition (1986).

[ll] B. Kolman, "Linear Algebra with applications", sixth edition (1997).

[12] "Friction-induced vibration, chatter, squeal, and chaos", The winter annual meeting of The American society of mechanical engineers Anaheim, California, November 8-13 (1992).

[13] Philips, "The Do It Easy Tool", a program which facilitates the design and tuning of controllers, (1998).

[I41 P. Dahl, "A solid friction model", Aerospace Corp., El Segundo, CA. Tech. Rep. TOR-0158(3107-18)-1, (1968).

[15] P.-A. Bliman and M. Sorine, "Friction modelling by hysteresis operators. Application to Dahl, Sticktion, and Stribeck effects", in Proc. Conf. Models of Hysteresis, Trento, Italy, (1991).

[I61 Gene F. Franklin, J . David Powell, Abbas Emami-Naeini, "Feedback control of dynamic systems", (1986).

[17] Hensen R.H.A., Angelis G.Z., et al.,"Grey-box modeling of friction: An experimental case study", in Pro- ceedings of the European Control Conference, (September 1999).

[18] Simulink version 3.0 for Matlab version 5.3.0.10183 (R11) (1999), (01-Sep-1998).

[19] Cambridge University Press, "Numerical recipes in C: the art of scientific computing", (1988-1992).

[20] J.-J.E. Slotine, Weiping LI, "Applied nonlinear control", (1991).

Appendix A

As friction depends upon the velocity, it is essential t o be able to make an accurate estimation of the velocity on behalf of friction compensation. A method which is often used for on-line velocity estimation is filtering and subsequently estimating the velocity by means of an Euler scheme. The main disadvantage of this method is that filtering (and also the use of a Euler scheme) will produce a significant phase lag. The following solution is proposed: the idea is t o use past information t o make an estimation of the current position in a so called "prediction filter".

Next, an example is given in which the method is explained. In this example the current data is predicted with a second order poIynomia1. We therefore need a t least 3 data points to construct a second order polynomial. These data points are stored in two history columns = [t,-z, tnPl, t,] and cpH = [P,-~, (P,-~, pn] for time and position history respectively. This fictive example is depicted in Fig. A.1. In the figure all points that are used for estimating the polynomial (i.e. the values that a stored in and E) are marked with a circle, the measured data points are marked with a cross.

Every time that the position changes one ore more increments cpH is updated with the expected position of a 't 't+(Pt-ht half sample time earlier: we assume that the position on t = t - $ would have been pt-6t/2 = 7. Next, the

estimated polynomial is extrapolated to the current time in order to check whether it stays within the boundaries

of Pt f encoder resolution

3 . If it stays within the boundaries, the estimated polynomial is considered t o be correct. - Otherwise, the last point (on t = t - %) is temporarily rejected and a new (temporary) point is chosen namely ptem, = p, (this is the current position). Now, a new polynomial is constructed trough the last three points (the last point will only be used for this calculation and not for following calculations).

When the position remains unchanged, the former estimated polynomial is extrapolated t o the current time and again is checked if it exceeds the boundaries. If it exceeds the boundaries, the history is updated with the estimated position on t = t - % and a new polynomial is constructed which will subsequently be checked if it exceeds the boundaries on the current time. If it doesn't stay within the boundaries, the same procedure a s before is used: the last point is temporarily rejected and replaced by ptemp = p,. Finally, a polynomial is constructed trough the last three points.

Wher, the p&f,ior, remiics c n n s t s ~ t for a longer period of time, it is assumed that the setpoint is completed and we will therefore reset the history of the filter. This will prevent the filter from using obsolete data points when the movements has already finished, which would lead t o a delay in the velocity estimation.

Figure A. l shows how the data is processed in the prediction filter (for this example):

a) The position remains unchanged, therefore the former estimated polynomial will be extrapolated t o the current time. As can be seen, the estimated polynomial exceeds the boundaries and therefore has t o be adjusted.

b) The history columns will be updated with the estimated position on t = t - 9. The calculated polynomial trough these last three points doesn't exceed the boundaries on the current time and will therefore be considered t o be sufficient accurate.

c) Again, the position remains unchanged. St Seconds after the last measurement, the former estimated polyno- mial still suffices and thus will be maintained. However, after 2St seconds the position still hasn't changed but the polynomial exceeds the boundaries.

Time [sec]

A "1

*-2, flc.-l) ?P)

Time [sec]

Figure A.l: "Data fitting" with the prediction filter

d) The last values in the history columns are updated with the estimated position on t = t- 9 and the calculated polynomial is extrapolated t o the current time; it can be seen that the polynomial suffices as it doesn't exceed the boundaries.

e ) Finally, the measured position changes one increment. The history columns will therefore be updated with the estimated position on t = t - %. Subsequently, a new polynomial is estimated trough the last three points. This polynomial also suffices.

The velocity (and higher derivatives like acceleration) can be estimated by differentiating the various polyno- mials at the considered point in time.In general the prediction filter is tuned by adjusting several parameters:

Number of data points through which the various polynomials are constructed. An advantage of using a large number of data points is that the filter will give better predictions for smooth (relatively slow) movements. On the other hand, the filter will not be able to follow fast movements. This slow behavior of the filter will also result in an increasing initial error. If the number of data points is larger than the order of the polynomial + 1, it is not guaranteed that the estimated position stays within the boundaries. For this reason an extra option is build in the filter to guarantee that the estimated points do not exceed the boundaries. However, this option increases the time needed for calculating the individual polynomials

and it is questionable if this option will increase the accuracy of the speed estimation. The latter will be investigated in the next section.

The order of the polynomials. In order to estimate the velocity, a polynomial of a t least order one should be used. Likewise, if we want t o estimate the acceleration we should use polynomials of a t least order two. Our goal is t o estimate the velocity for third order setpoints and therefore it is expected that we should use polynomials of order three.

The reset period. If the period over which no increment variation occurs is larger than the reset period, the position history column will be filled with the last measured value of the position. The duration of the reset period is mainly determined by the maximum velocity of the setpoint. For setpoints with high maximum velocities, the reset period can be short. For setpoints with a low maximum velocity, the reset period shoiild be longer.

The resolution of the incremental encoder. This parameter isn't really a tuning parameter but it can be increased if a smoother (but less accurate) estimation is required.

A. 1 Simulations

In this section the previous filter is tested for several sampled movements (sample frequency= 10 [kHz], encoder resolution= 5000E92 [rad]). These movements are almost similar t o the five setpoints given in chapter 2 (regarding the maximum velocity and acceleration) with the exception that in this case we will regard setpoints of order infinity. This difference can be seen as a disturbance on the original third order setpoints. The formula that describes the movement of this setpoint is given below:

where h, is the stroke and t, is the feed time. We will compare three different methods of velocity reconstruction namely:

1. Zero-phase forward and reverse digital filtering (with Butterworth filter) combined with centered difference approximation of the velocity.

The filter works as follows: after filtering in the forward direction, the filtered sequence is then reversed and run back through the filter. The result has precisely zero phase distortion and magnitude modified by the square of the filter's magnitude response. Care is taken to minimize start-up and ending transients by matching initial conditions.

Next, the velocity is established using the centered difference formula:

which also doesn't induce phase lag of the velocity. The main shortcoming of this somewhat ideal method of filtering and velocity estimation is that it can only be used off-line.

2. Ordinary digital filtering (with Butterworth filter) combined with Euler approximation of the velocity.

Although this way of filtering leads to phase lag, it can be used on-line. This together with the fact that i t is easy to use, is probably the main reason for it's frequent use.

3. Prediction filtering which leads directly t o velocity estimates by differentiating the estimated polynomials.

The results of the various simulations are depicted in Fig. A.2: Upper-left: setpoint 1, hm= T [rad], tm= 0.3807 [s], vm,= 16.5 [rad/s], a,,= 136.20 [rad/s2]. Upper-right: setpoint 2, h,= 2 [rad], t,= 0.1845 [s], vm,,= 8.51 [rad/s}, am,= 144.97 [rad/s2]. Middle-left: setpoint 3, h,= 6 [rad], t,= 0.1360 [s], vm,= 4.62 [rad/s], am,= 106.72 [rad/s2]. Middle-right: setpoint 4, hm= 6 [rad], t,= 0.0631 [s], vm,= 1.00 [rad/s], %,= 49.58 [rad/s2].

Time [s]

P u

Prediction filt. Zero phase filt.

0.2 0 0.1

> -0.2 0 0 0.01 0.02 0.03

Time [s]

-d" 0 w.., - ,I" \. - ./' \I.

Figure A.2: Velocity error for three different methods of velocity estimation tested on five setpoints (sample frequency= 10 kHz).

r z -0.2 - '...! > . P'

- 0 0.1 0.2 0.3 0.4 0.5 9 Time [s]

Lower-left: setpoint 5, h,= & [rad], t,= 0.0293 [s], vm,= 0.21 [rad/s], am,= 22.99 [rad/s2]. Below, the various parameters of the prediction filter are given for the five setpoints.

Setpoint Number of data points Order of polynomial Reset period [number of intervals] I 125 3 20 2 100 3 20 3 100 3 20 4 50 3 20 5 30 2 20

In all cases there was little or no improvement when the last point of estimated polynomials was forced into the boundaries. It can be seen that for slow setpoints the prediction filter is superior t o the normal filter which introduces a significant phase lag. For fast setpoints however (setpoint 5), there is no improvement when the prediction filter is used. Figure A.3 shows a part of the data set which was filtered with the prediction filter. This shows us that the prediction filter is capable of estimating constant velocity in spite of t he suddenly changing position.

Figure A.4 shows the relative error for various velocities during a setpoint. Again, the following setpoint is used: h,= z? [rad], tm= 0.3807 [s], v,,= 16.5 [rad/s], am,= 136.20 [rad/s2]. It can be seen that the prediction filter has a small relative error for low velocities compared t o the ordinary filter. Therefore the prediction filter is very suitable for low velocity applications (note that tm must be sufficiently long).

A-E irr?portact diffi_cu!ty that hasn't been investigated in the former simulations is the chattering of the encoder in between the increments. This problem arises a t the end of the setpoint when the velocity should be zero. In the next section we will therefore evaluate the various filtering techniques on real data.

A.2 Experimental results

When the prediction filter is implanted in a real-time environment we have t o make sure that all the calculations on behalf of the filter, can be made in the prescribed sample time. The main problem with respect to the calculation time is the estimation of the polynomial. In our filter the polynomial is calculated with a least squares method which is solved by using the QR algorithm with preparatory Householder transformations [lo]. This method has the advantage that it is faster than the Gram-Schmidt process and more accurate than the Gaus- Jordan elimination method. In the simulation environment we choose a sample rate of 10000 [Hz]; in the real time environment however, we were only able t o maintain a sample rate of 1000 [Hz] with a maximum of 12 data

0.158 0 1585 0.159 0.1595 0.16 0.1605 0.161 Time [sec]

Figure A.3: Measurements versus prediction filter

Velocity [radlsec]

'5: , I I I

Figure A.4: The relative error (percentage of the velocity) versus the velocity

t I t ; i '!

- Prediction filt. Zero phase filt.

- -- - Ordinary filt.

points or 100 [Hz] with a maximum of 26 data points. Therefore we repeated the simulations of the previous section with the exception that in this case the sample rate is set t o 1000 [Hz] and number of data points is 10. The results of these simulations are depicted in Fig. A.5. In contrast with the results of the previous simulations (on 10000 [Hz]) the velocity estimation became more smooth without filtering. Unlike the simulations on 10000 [Hz] it is now possible to show the unfiltered Euler approximation of the velocity in the same figure as the filtered estimations (the errors due to discretization have significantly decreased and will therefore not ruin the figure of the filtered estimation) . However, the phase delay due t o Euler approximation of the velocity, has increased.

I

0 0.1 0.2 0.3 0.4 0.5 Time Is1

V) a 9 0.2 - -

0 0.05 0.1 0.15 0.2 - Time [s]

I I

0 0.05 0.1 0.15 - Time [s]

Prediction filt. Zero phase filt.

- Ordinary filt. No filtering

Time [s]

Figure A.5: Velocity error for three different methods of velocity estimation tested on five setpoints (sample frequency= 1 kHz).

Next, our approach is somewhat identical t o the last section: three different filtering and velocity estimation techniques are compared with respect t o their accuracy for five different setpoints (see Section 2.4.). The various techniques are tested on a trajectory generated by the controller tuned in Section 2.2. (sample frequency: 1000 [Hz] and number of data points is set t o 10 for every setpoint) . In this case however, it is impossible to compare the results of these techniques with the exact velocity. In the former section we saw that filter method 1 (Zero- phase forward and revers2 digital filtering c~rnbixd with centered difference appr~ximit ion of the ve!ocity) gave a rather good estimation of the velocity. We will therefore use the results of this technique a s a reference for the other two estimation techniques.Because the accuracy depends upon the performance of the zero phase estimation technique, it is not possible t o get accurate quantitative information from the results depicted in A.6. Despite of this we may still conclude from Fig. A.6 that the prediction filter remains superior t o t he ordinary filter for high velocities and it deals with chattering in a satisfactory manner.

Next, we want t o evaluate an alternative filtering method: each sample is reconstructed by determining a polynomial fit through the current point and a number of previous neighboring samples. This method is different than the previous method (Fig. A.l) with respect to manner in which the data is used: the first filter does not always use neighboring samples for estimation of the polynomials but for slow velocities it uses samples that are more than one sample-time separated. This means that off-line calculation of parameters is impossible, as the Vandermonde matrix isn't constant. For the alternative filtering method we want to use the Singular Value Decomposition (SVD) method for calculating the various polynomials [lo]. Although this method can be

Time [s]

I

0 0.2 0.4 0.6 - U) Time [s] 3

- Prediction filt. . - . Ordinary filt.

_m - ,- 0.2 e

Time [s]

- , / -\

.,I ..

Figure A.6: Velocity error for two different methods of velocity estimation tested on five setpoints.

2 -9.2 - '>

8 *_

0 0.1 0.2 0.3 - Time [s]

significantly slower than solving the normal equations, its great advantage is that it (theoretically) cannot fail plus the fact that it also fixes the roundoff problem [19]. In this case however, we don't care about the speed of the SVD, as it is calculated in advance (off-line). Although, the principle of the SVD prediction filter slightly differs from the QR prediction filter, we expect that there will be little or no difference between the results of these two: we already saw that the QR prediction filter uses a large amount of data-points in order t o achieve sufficient accuracy. This means that the method depicted in Fig. A.1 looses its strength.

Finally, we tested the SVD method analog to former tests on the QR prediction filter. From the experi- ments followed that there was indeed no considerable difference between the two methods regarding the accuracy (with the exception that the SVD filter can maintain the accuracy for high sampling frequencies in a real time environment).

A.3 Conclusions

For fast setpoicts the prediction filter looses its superior position. In this case there is not enough data available for the filter to make an accurate prediction. For low velocity long setpoints however (for instance certain tracking trajectory), it remains superior. In spite of the phase-lag improvement, the noise on the velocity estimation remained an impediment when the prediction filter was used in the sxperimer,ts in Chapter 4. Fer this r e s o n the filtering technique has not been used in chapter 4. However, it remains recommendable to investigate if the results can be improved when for instance, the prediction filter is combined with other filtering techniques.

In the previous section we saw that in a real-time environment the QR prediction filter could not be set to its most optimal parameter values. It is therefore recommendable t o use the more faster and stable SVD method.

Appendix B

Theorem: Consider system (B. 1) together with the friction model (3.1 1) and (3.13), friction observer (3.16) and (3.17), and control law (3.18).

If C(s) is chosen such that

A

is strictly positive real (SPR) then the observer error, F - F, and the position error, e, will asymptotically go to zero.

Proof: The control law yields the following equations

- where F = F - F and I = z - 2. Now introduce

as a Lyapunov function and

which is a state-space representation of G(s). Since G(s) is SPR (see H.K. Khalil, nonlinear systems. New York: Macmillian, 1992) i t follows from the Kalman-Yakubovitch Lemma [20] that there excist matrices P = pT > 0 and Q = QT > 0 such that

Now

The radial unboundedness of V together with the semi-definiteness of implies that the states are bounded.

We can now apply LaSalle's theorem to see that t -+ 0 and ? --+ 0 which means that both e and I: tends to zero and the theorem is proven.