A continuous version of the LuGre friction model applied to the adaptive control of a pneumatic servo system

Author’s Accepted Manuscript A Continuous Version of the LuGre Friction Model Applied to the Adaptive Control of a Pneumatic Servo System M.R. Sobczyk, V.I. Gervini, E.A. Perondi, M.A.B. Cunha www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)30182-X http://dx.doi.org/10.1016/j.jfranklin.2016.06.003 FI2621

To appear in: Journal of the Franklin Institute Received date: 30 October 2015 Revised date: 28 March 2016 Accepted date: 1 June 2016 Cite this article as: M.R. Sobczyk, V.I. Gervini, E.A. Perondi and M.A.B. Cunha, A Continuous Version of the LuGre Friction Model Applied to the Adaptive Control of a Pneumatic Servo System, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.06.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Continuous Version of the LuGre Friction Model Applied to the Adaptive Control of a Pneumatic Servo System M. R. Sobczyka,∗, V. I. Gervinib , E. A. Perondia , M. A. B. Cunhac , a

Departamento de Engenharia Mecˆ anica, Universidade Federal do Rio Grande do Sul, Rua Sarmento Leite, 425, Centro. Porto Alegre - RS, Brazil b Centro de Ciˆencias Computacionais, Universidade Federal do Rio Grande, Av. Italia, km. 8, Campus Carreiros. Rio Grande - RS, Brazil c Instituto Federal de Educa¸c˜ ao, Ciˆencia e Tecnologia Sul-rio-grandense, Pra¸ca Vinte de Setembro, 455, Pelotas, RS, Brazil

Abstract This work presents the application of a friction compensation scheme to the trajectory-tracking control of a pneumatic servo actuator. Such scheme is based on a continuous approximation of the LuGre friction model, developed so as to allow complete Lyapunov stability analyses without resorting to assumptions that are difficult to satisfy in practice due to their physical meaning. By using adaptive estimation, extensive identification procedures are also avoided for determining friction parameters. Experimental results illustrate the most significant advantages and potential limitations of the proposed scheme in real applications. Keywords: friction, motion control, hydraulics, pneumatics

∗

Corresponding author Email addresses: [email protected] (M. R. Sobczyk), [email protected] (V. I. Gervini), [email protected] (E. A. Perondi), [email protected] (M. A. B. Cunha)

Preprint submitted to Journal of The Franklin Institute

June 4, 2016

1. Introduction Pneumatic actuators are an interesting choice for many industrial applications because, while presenting good ratios between size/weight and available forces/torques, these systems are usually cheap and easy to assemble. Also, they are easy to integrate in industrial facilities (which usually have compressed-air supply lines as a standard feature) and environmentfriendly, because air leakages do not pose serious pollution hazards. On the other hand, as their dynamic behavior is highly nonlinear, they are poorly suited for performing high-precision tasks. As a result, since their application advantages are significant, much effort has been spent in finding ways to monitor these systems effectively [1–9]. Friction is one of the most difficult phenomena to deal with in highprecision applications [10–12], because it is the source of strongly nonlinear effects such as pre-sliding displacement, break-away force variations, and stick-slip motion [13]. Due to its relevance, this is a widely addressed problem [2, 6, 14–23], with model-based compensation techniques appearing as a frequent approach [2, 6, 15, 18, 20–23]. Several models for this purpose can be found and there is no consensus regarding which is the most suitable one [24], but the LuGre model [13] seems to be the most frequent choice (see [2, 6, 18, 22, 25–29], among many other examples), being even considered as “the reference model” for representing dry-friction forces [16]. In spite of its usefulness in many applications, the LuGre model has been criticized for flaws both in its theoretical properties and in its practical usage [9, 16, 24, 30–32]. More specifically, there are two points of interest in this work: (i) the experimental identification of the parameters employed by the 2

model is a difficult task [14, 16, 22, 26, 32]; (ii) the model depends on discontinuous mathematical terms, which limit its application for certain classes of systems [6, 22, 33, 34]. In the first case, direct measurement of the necessary variables requires sophisticated, high-resolution equipment [26, 32, 35], and measurements can be affected by variables that are not accounted for in the structure of the model, such as temperature and loading conditions [19]. Therefore, many works rely on time-consuming, indirect identification approaches [9, 36–38] and/or combined simulation-experiment procedures, using parameter values ranging within the same orders of magnitude that are usually found in works based on directly measured data [13, 22, 39]. Regarding the second problem, the discontinuous terms represent a major hindrance when it is explicitly required to estimate the time derivative of the friction force, a situation that is common to fluid-driven servo actuators [2, 6, 22, 33, 40, 41] and may also occur in systems whose actuators’ internal dynamics are significant [17, 18]. This difficulty affects directly the stability guarantees that can be provided for the closed-loop systems when modelbased controllers are employed, leading many authors to provide analysis only for restricted cases [6, 33, 42, 43] or to apply regularization methods that alter basic properties of the LuGre friction model, such as boundedness and passivity [12, 22, 33, 44]. In some cases, these measures lead to assumptions that are difficult to satisfy in the practical implementation of the proposed algorithms. In [6, 33], for instance, the time derivative of the friction force is neglected, which is not a valid hypothesis for low velocities. The control method discussed in [42] uses a discontinuous and potentially unbounded term for representing friction effects in its Lyapunov-based sta-

3

bility analysis. The analysis given in [22] relies on the existence of an upper bound to the amplitude of the internal state that is employed for estimating the friction force, which is not true for the smooth approximation of the LuGre model that is used to that end. In [41], the terms related to viscous friction effects are taken to be null, which contrasts with the nature of the hydraulic fluid that is usually employed in this type of actuator. Finally, the analysis presented in [2] is built on the assumption that the friction force estimated by the algorithm actually converges to its corresponding value in the experimental system, without presenting any theoretical or experimental arguments to back up such claim. In [40], we introduced a continuous approximation of the Lugre friction model. Such approximation differs from other ones in two critical points, which were proven in that work: (i) its internal state z (t) is bounded, (ii) with an appropriate choice of its parameter values the input-output mapping defined by such approximation is passive. Here, we extend that work by employing such approximation with an adaptive control algorithm to be applied to trajectory-tracking problems in a pneumatic servo positioner. The main contribution presented in this paper lies in allowing the use of the LuGre model with fluid power actuators without relying on questionable assumptions or affecting any aspect of a formal, complete stability analysis of the closed-loop system. Also, since it is an adaptive controller, we avoid extensive efforts to identify the parameters of the friction model. Finally, experimental results are used to evaluate the main advantages and limitations of the proposed strategy in real applications. To the best of our knowledge, this is the first work to address these three features simultaneously.

4

This work is organized as follows. Initially, in Sec. 2, the mathematical model of the system is presented. The proposed control algorithm is discussed in Sec. 3, whereas the stability analysis of the closed loop system is the focus of Sec. 4. Sec. 5 is dedicated to presenting and discussing experimental results. Finally, the main conclusions are given in Sec. 6. 2. System model A typical pneumatic servo actuator is depicted schematically in Fig. 1. Its pneumatic source is assumed to provide a constant pressure Ps to a symmetric actuating cylinder with utile stroke L and piston area A. An electric servovalve regulates the flow rates of compressed air moving into and out of the chambers of the cylinder. The air passage area of the valve is directly proportional to its electric control voltage u. The dynamic model for this type of system is discussed at length in many works (see, for instance, [6] or [9]), and the details of the specific model that is used in this work can be found in [22] and [45]. The operation of this system can be understood as two interconnected subsystems: (i) mechanical, and (ii) pneumatic. The first one is regarded as a mass M subject to a friction force f and driven by a pneumatic force due to the differences in the pressures p1 and p2 in the chambers of the cylinder. Thus, considering that M encompasses the mass of the entire piston-load assembly, we apply Newton’s Second Law to obtain

M y¨ + f = A(p1 − p2 )

(1)

where y¨ is the second derivative with respect to time of the piston position y, 5

Figure 1: Pneumatic positioning system

i.e., the acceleration. Accordingly, the piston velocity is defined as y, ˙ and the time derivative of the acceleration, commonly referred to as jerk, is denoted ... as y . The friction force f will be discussed separately in Sec. 2.1. The second subsystem is composed of the equations that govern the dynamics of the chamber pressures p1 and p2 . These equations can be written with basis on energy conservation arguments and are given by:

p˙1 = − p˙2 =

Ary˙ rRT p1 + qm1 V10 + Ay V10 + Ay

Ary˙ rRT p2 + qm2 V20 + A(L − y) V20 + A(L − y)

(2) (3)

where r is the ratio between the specific heat values of the air, R is the gas constant, T is the air supply temperature, y is the piston position, V10 and V20 are the dead volumes of air at the extremities of both chambers, and qm1 and qm2 are the mass flow rates of air into or out of each chamber. These rates are nonlinear functions of the pressures in the chambers and of

6

the voltage signal u, whose effects are modeled by empirically-determined polinomial approximations, as described in Appendix A. As the piston moves due to the differences of pressure between the two chambers, it is convenient to rewrite equations (1), (2), and (3) in terms of a differential pressure p∆ = p1 − p2 . We accomplish this by subtracting equations (2) and (3) and rearranging terms, obtaining: ˆ 1 , p2 , y, y) p˙∆ = h(p ˙ + uˆ(p1 , p2 , u, y)

(4)

ˆ 1 , p2 , y, y) where h(p ˙ and uˆ(p1 , p2 , u, y) are grouped terms that are functions only of the states of the system and those that are affected by the input voltage u, respectively. Such terms are defined as: 

 qm1 qm2 uˆ(p1 , p2 , u, y) = RrT − V10 + Ay V20 + A(L − y)   p p 1 2 ˆ 1 , p2 , y, y) h(p ˙ = −rAy˙ − V10 + Ay V20 + A(L − y)

(5) (6)

Remark #1: from Eq. 4, it is clear that the control signal applied to the system (u) does not affect directly the force applied to the piston-load assembly. Instead, it affects the dynamics of the pressures in the chambers, i.e., the time derivative of the applied pneumatic force. Therefore, the proposed feedback-linearization control algorithm requires an explicit estimate of the time derivative of the friction force, because this is the quantity that is directly related to the control input. This topic is discussed with more detail in Section 2.1.

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2.1. Friction model In this work, we use a continuous approximation of the LuGre model to account for the friction forces in the system. The original LuGre model was proposed by Canudas de Wit et al. in [13], being well known and widely used in control applications because of its ability to represent most friction effects with satisfactory accuracy. The model relies on estimating an average deflection z of the microscopic asperities (bristles) of two contacting surfaces, which is accomplished by means of the following nonlinear state observer:

z˙ = y˙ −

˙ |y| z g (y) ˙

(7)

where g (y) ˙ is a positive function defined as [13]:

g (y) ˙ =

i 1 h ˙ y˙ S )2 FC + (FS − FC ) e−(y/ σ0

(8)

where FC is the Coulomb friction force, FS is the static friction force, and y˙ S is the Stribeck velocity. The meaning of each of these quantities is illustrated in Fig. 2, which depicts the friction force as a static function of velocity.

Figure 2: Friction force f as a function of piston velocity v

8

Based on the state z, the friction force f acting between two bodies with relative velocity y˙ is represented as:

f = σ0 z + σ1 z˙ + σ2 y˙

(9)

where σ0 is a stiffness coefficient, σ1 is a damping coefficient related to z, ˙ and σ2 is a viscous friction coefficient, which is also illustrated in Fig. 2. As discussed in further detail in [40], due to the presence of the term |y˙ (t)|, whose time derivative is discontinuous when y˙ (t) = 0, application of the LuGre model becomes seriously hindered in control tasks involving actuators whose internal dynamics cannot be neglected, i. e., the control input u affects the time derivative of the force instead of the force itself, which is the case for fluid-driven servo actuators. This occurs because, for such systems, it is necessary to estimate the time derivative of the friction force, which cannot be done for all operation conditions due to the aforementioned discontinuity. In many works, this problem is avoided by replacing the term |y˙ (t)| by “intuitive” continuous approximations, such as 2yarctan ˙ (kv y) ˙ /π or ytanh ˙ (kv y) ˙ [12, 22, 33], where kv is a positive constant in both cases. However, this procedure causes the loss of two of the most important properties of the LuGre model, namely, boundedness of its internal state z(t) and passivity. So, in order to avoid these problems in applying the LuGre model to the control of a pneumatic actuator, we use the following continuous approximation:

z˙ = yS ˙ 1 (y) ˙ −

S2 (y) ˙ z g (y) ˙

where S1 (y) ˙ and S2 (y) ˙ are respectively defined as:

9

(10)

  S (y) ˙ 2 1 ˙ = [S0 (y)]  S (y) ˙ = yS ˙ (y) ˙ 2

(11)

0

and S0 (y) ˙ is any odd, continuous and monotonically increasing function whose absolute value converges asymptotically to 1 as |y| ˙ → ∞. Here, we use S0 (y) ˙ = 2arctan (kv y) ˙ /π. Even though this version of the LuGre model differs from other ones in minor details, it presents an important advantage since it can be proven to maintain the properties of boundedness and passivity. For the complete proofs of these two properties, please refer to [40]. The calculation of the friction force itself is not modified, i.e., Eq. (9) is still employed in its original form. As for its corresponding time derivative f˙, it can be readily determined as: f˙ = σ0 z˙ + σ1 z¨ + σ2 y¨

(12)

where the term z¨ can be calculated by taking the time derivative of Eq.(10): ˙ + Γz) z¨ = S˙ 1 y˙ + S1 y¨ − (Γz ˙

(13)

with Γ (y) ˙ = S2 (y) ˙ /g (y). ˙ The derivatives of the intermediate terms involved in this equation are given by the following expressions: S˙ 1 = 2S0 S˙ 0

S˙ 0 =

2kv y¨ π [1 + (kv y) ˙ 2]

10

(14)

(15)

g˙ =

S˙ 2 g − S2 g˙ Γ˙ = g 2 (y) ˙

(16)

S˙ 2 = S˙0 y˙ + S0 y¨

(17)

−2 (Fs − Fc ) −(y/ ˙ y˙ s )2 y ˙ y ¨ e σ0 y˙ s2

(18)

In sum, the friction forces acting on the piston of the actuating cylinder are calculated by means of eqs. (9), (10) and (12), with the aid of their corresponding auxiliary terms. This completes the open-loop model. 3. Control algorithm The proposed controller is based on the cascade control methodology, which consists in interpreting the whole system as two interconnected subsystems, mechanical and pneumatic, in which the output of one subsystem (the differential pressure resulting from the pneumatic dynamics) is regarded as an intermediate input that is applied to the other. Thus, regarding first the mechanical subsystem, we develop a control law that generates a desired differential pressure p∆d that leads the system to track a given desired trajectory yd . Then, we choose another control law to be applied to the pneumatic subsystem, which calculates the values of the servovalve control voltage u

11

that cause the actual pressure difference p∆ to track p∆d as closely as possible. The advantage of this procedure lies in the fact that the two control laws are not directly tied to each other, so that the control choices for one subsystem can be made without addressing the other one explicitly. The proposed control law to be applied to the mechanical subsystem is a version of the well known one by Slotine and Li [46], which was augmented by Xie [21] to include an adaptive friction-compensating term. In order to describe the proposed control algorithm, we now define a number of auxiliary quantities. First, the trajectory tracking errors of the piston-load assembly in terms of position, velocity, acceleration and jerk are given by, respectively: ... ... ... y˜ = y − yd , y˜˙ = y˙ − y˙ d , y¨˜ = y¨ − y¨d , y˜ = y − y d

(19)

... where yd , y˙ d , y¨d , and y d are the desired values for their corresponding variables. Additionally, the auxiliary error measure s is established as

s = y˜˙ + λ˜ y

(20)

With the aid of these terms, the control laws applied to the mechanical and pneumatic subsystems are, respectively:
Ap∆d = −KD s + M y¨d − λy˜˙ + fˆ

(21)

ˆ (p1 , p2 , y, y) uˆ = −h ˙ − As − KP p˜∆ + p˙∆d

(22)

where KD and KP are positive constants, p˜∆ = p∆ − p∆d is the error in the ˆ 1 , p2 , y, y) differential pressure, h(p ˙ is the part of the pneumatic subsystem 12

dynamics that is independent of the control signal u(t) in (4), and fˆ is the observed value of the friction force, i.e.: fˆ = σ ˆ0 zˆ + σ ˆ1 zˆ˙ + σ ˆ2 y˙

(23)

In Eq. (23), all terms marked with a wedge (∧) are regarded as the estimated counterparts of their corresponding quantities in the real system, as modeled in Section 2.1. The dynamics of the estimated friction state zˆ is: S2 (y) ˙ zˆ˙ = S1 (y) ˙ y˙ − zˆ − µ0 s fs (y) ˙

(24)

Calculation of friction forces with the LuGre model requires six parameters: σ0 , σ1 , σ2 , FC , FS , and vs . The first two are called dynamic parameters, and their experimental identification is a rather difficult task [14, 16, 22, 26, 32]. The remainder are called static, because they can be easily identified in a static friction map such as the one illustrated in Fig. 2, which can be drawn from data obtained in relatively simple experiments as described in Section 5. Here, in order to avoid the experimental identification of the dynamic friction parameters of the model, we employ adaptive control, i.e., these parameters are updated on line as the task is performed. Adaptive control is a widely addressed research topic, with numerous examples of different techniques and applications (see, for instance, [47–51]). In our case, the adaptation scheme is based on the one presented in [21], which estimates the dynamic parameters of the LuGre model in the case of a DC servomotor. According to such scheme, the update rules for the estimated parameters are:

σ ˆ˙ 0 = −γ zˆs, σ ˆ˙ 1 = −β zˆ˙ s, σ ˆ˙ 2 = −χ ys ˙ 13

(25)

where µ0 , γ, β, and χ are positive constants. The complete procedure performed by the proposed algorithm is summarized in Fig. 3: based on the measurement of the mechanical variables of the system (position, velocity and acceleration), the control law given in Eq. (21) determines an appropriate value for the desired differential pressure p∆d . Then, a second control law (Eq. (22)) is calculated, seeking to force the actual pressure of the system to track such desired value. Finally, as such control law uˆ does not express the actual control voltage u applied to the servovalve, a nonlinear inversion procedure is used to calculate this value. This inversion consists in solving Eq. (5) for u, using the desired value of uˆ and the polynomials given in Appendix A that represent the air mass flow rates going through the servovalve ports.

Figure 3: Proposed control scheme

4. Stability analysis In this section, we analyze the convergence properties of the closed-loop tracking errors in the system with basis on Lyapuonv’s direct method. Prior to the analysis itself, in order to clarify its development, we rearrange terms in some of the relations presented in the previous sections. First, from the way 14

that all tracking errors in the system are defined, we can write p∆ = p˜∆ +p∆d . Thus, substitution of expressions (9), (21) and (23) into Eq. (1) yields:

  M s˙ = KD s + σ ˆ0 zˆ + σ ˆ1 zˆ˙ + σ ˆ2 y˙ − (σ0 z + σ1 z˙ + σ2 y) ˙ + A˜ p∆

(26)

Also, as in the case of the previously defined variables, we denote the estimation error of the friction state z as z˜ = zˆ − z. Thus, by subtracting (10) from (23), the time derivative of such error can be shown to be: S2 (y) ˙ z˜ − µ0 s z˜˙ = − fs (y) ˙

(27)

Finally, we apply the control law given in Eq. (22) to the dynamics of the pneumatic subsystem (Eq. (4)), obtaining

p˜˙∆ = −As − KP p∆

(28)

It is now possible to analyze the convergence properties of the trajectorytracking errors of the closed-loop system, which can be stated by means of Proposition #1. Proposition #1 Consider the pneumatic positioning system described  T in Section 2. For this system, we define ρt = y˜, y˜˙ , p˜∆ as the trajectorytracking error vector, ρe = [˜ z, σ ˜0 , σ ˜1 , σ ˜2 ]T as the estimation error vector, and  T ρg = ρt T , ρe T as the global error vector. Assuming that the desired values for the position, velocity, acceleration and jerk of the actuating piston are all bounded, when this system is controlled in closed-loop form by means of the algorithm proposed in Section 3, it is possible to choose the parameter values for such controller so as to ensure that ρg = 0 is a stable equilibrium. 15

Furthermore, such choice of parameters leads to kρt k → 0 as t → ∞, i.e., the trajectory-tracking errors of the closed-loop system converge asymptotically to zero. Proof Consider the following non-negative function: 1 V (t) = 2

  1 2 1 2 1 2 σ0 2 2 2 Ms + σ ˜ + σ ˜ + σ ˜ + z˜ + p˜∆ γ 0 β 1 χ 2 µ0

(29)

by differentiating Eq. (29) and using (25) to (28), we can show that

σ0 S2 (y) ˙ ˙ 2 σ1 S2 (y) V˙ (t) = − (KD + σ1 µ0 ) s2 − KP p˜2∆ − z˜ − z˜s µ0 g(y) ˙ g(y) ˙ Now, applying the inequality ab ≤

1 2

(30)

(a2 + b2 ) to the term that depends

on the product z˜s and rearranging terms, we obtain: V˙ (t) ≤ −βs s2 − KP p˜2∆ − βz z˜2 where βs and βz are, respectively:   βs = KD + σ1 µ0 − σ1 S2 (y) ˙ / [2g (y)] ˙   ˙ σ0  βz = S2 (y) − σ21 fs (y) ˙ µ0

(31)

(32)

Thus, V˙ (t) can be made non-positive if   µ0 < 2σ0 /σ1 h i ˙  KD > σ1 S2 (y) − µ 0 2g(y) ˙

(33)

Since S2 (y) ˙ and g(y) ˙ are bounded, the aforementioned values of µ0 and KD exist. Thus, V˙ (t) ≤ 0 and this equilibrium is stable, i.e., kρt k and kρe k are both limited. Moreover, as the dependence on s and p˜∆ is negative for any 16

nonzero combination of these quantities, both must tend to zero as t → ∞. However, this is not true for z˜, as its multiplying factor in Eq. (30) depends on y, ˙ nor the remainder elements of ρe , because they do not appear in V˙ (t). As s = y˜˙ + λ˜ y can be interpreted as a first order filter applied to y˜, and since s converges asymptotically to zero, it follows that both y˜˙ and y˜ must also converge asymptotically to zero, completing the proof. Remark #2: The second condition stated in Eq. 33 can be difficult to attain in practice because, even for moderate velocities (about 0.1 m/s), the lower bound required for KD is roughly equal to σ0 σ1 /100, which can often reach orders of magnitude of 103 or higher [15, 22, 37, 42, 43], implying very large feedback gains in the practical implementation of the controlled system. Even if the control apparatus is able to apply such values in practice, it is undesirable to do so because that can render the system to be very sensitive to measurement noise or unmodeled dynamic effects. This particular topic will be discussed with more detail in Section 5. 5. Experimental evaluation The experiments were performed on a pneumatic workbench comprising a rodless, double-acting cylinder (DGPL-1000 FESTO) and a 5/3 servovalve (MPYE-5-1/8 FESTO). A variable-resistance sensor (MLO - POT - 1000 TLF FESTO) was used for measuring piston position, whereas velocity and acceleration were estimated by numerical differentiation. Chamber pressures were monitored by membrane-type transducers (510 - Huba Control). All signals were fed to a real-time control electronic board (DS1104 - dSPACE GmbH), hosted by a standard PC, running the algorithm described in Section 17

3. The processing time for each control loop was 1.0 ms. According to its manufacturer’s catalog, the nominal parameters values for this workbench are: A = 4.19 · 10−4 m2 , L = 1 m, V10 = 1.96 · 10−6 m3 , and V20 = 4.91 · 10−6 m3 . The parameters for compressed air were assumed as r = 1.4, R = 286.9 Jkg/K, T = 293.15 K, as commonly found in other works on the subject [22, 9, 8]. The payload mass was M = 3.66 kg. As for the admissible trajectory requirements, the maximum stroke and velocity for this pneumatic cylinder are 0.9 m and 1.0 m/s, respectively. Also, due to the continuity conditions implied in the stability analysis presented in Section 4 [52], all required position trajectories and its derivatives up to the 3rd order must be bounded continuous functions of time. So, if a discontinuous trajectory (e.g., a step input) is required, it is necessary to filter its corresponding signal before applying it to the system. As already mentioned in Section 3, LuGre model parameters can be classified in two categories: static and dynamic. Those in the first group can be experimentally identified by the following procedure: using a simple algorithm such as P- or PI-controller, the actuator is led to track a constant velocity trajectory while the pressure difference applied to the piston is measured. Since the velocity is constant, this pressure difference multiplied by the piston area gives the friction force. Repeating the test for several speed values leads to a static map such as the one depicted in Fig. 2, from which the following parameters were determined: σ ˆ2 = 89.9 Ns/m, FC = 64.3 N, FS = 75 N, vs = 0.02 m/s. As the parameters in the second group are very difficult to determine experimentally, their values were assumed as unknown, being estimated on line as described in Section 3. The workbench used here

18

is of the same manufacturer and model as the one employed in [22], which deals with friction compensation in a pneumatic actuator using the LuGre model with fixed parameters. Thus, the initial values for our dynamic friction parameters were the same as those used in that work: σ ˆ0 = 4500 N/m, σ ˆ1 = 93.1 Ns/m. Due to its cascaded structure, the proposed controller allows a clear separation of the effects of each control law: while Eq.(21) relates an intended position/velocity trajectory to its corresponding desired pressure p∆d , Eq.(22) is concerned with causing the actual differential pressure p∆ to match such desired value. Likewise, the controller gains KD and λ have a direct effect on the tracking of the mechanical variables (y and y) ˙ of the system, whereas KP affects mostly the applied pressure p∆ . Since our focus is on the effect of the proposed friction compensation scheme on the mechanical trajectory performed by the system, only the gains referring to the mechanical subsystem were varied, whereas the pneumatic subsystem gain was set as KP = 150 s−1 . The tests were repeated for five different sets of controller gains, as informed in Table 1. These sets were adjusted by means of approximate transfer function considerations combined with direct experiments, and the chosen values reflect an increasing degree of required control effort.

Table 1: Gain sets employed during the experimental tests

set #

1

2

3

4

5

λ (s−1 )

50 60 70 80 90

KD (Ns/m)

10 15 20 25 30

19

In each test, the system was required to track a sinusoidal trajectory with amplitude 0.4 m and frequency 2.0 rad/s. Because the proposed friction compensating scheme is adaptive, in order to highlight its specific effects, the time span for each test was 5 min (300 s). Initially, the friction compensating term of the control law (Eq. (23)) was turned off. Then, this term was turned on, with fixed dynamic parameters. Finally, parameter values were allowed to adapt with time as determined in Eq. (25). The results from the experiments are illustrated in figures 4, 5, and 6. Due to the relatively long test period when compared to that of each movement cycle, to facilitate the visualization of the results, Fig. 4 depicts only a few cycles of interest at the beginning and at the end of one of the tests, whereas the complete picture is given in the other two graphs. Also for the sake of facilitating their understanding, the results in these figures are summarized in periods of 15 seconds. As seen in figures 4 and 5, the compensation term with nominal (initial) parameters caused overcompensation of the friction forces and the amplitudes of the tracking errors were increased. However, as the adaptation scheme acted, such errors were reduced until reaching approximately constant amplitudes, the smallest of which was 2.9 cm, representing 7.3% of the amplitude of the desired trajectory. Moreover, due to the adaptation of the friction parameters, the performed trajectory became significantly closer to its reference, a fact confirmed by the expressive reduction of the rms tracking errors. When compared to traditional control approaches, the proposed control scheme led to a clear improvement in the trajectory tracking performance of the controlled system. With linear state feedback control, for instance, the obtained tracking error amplitude for the same trajectory was about 6 cm, or

20

Figure 4: Trajectory tracking performance of the closed-loop system - general view: (a) initial stages, (b) final stages. For 0 ≤ t < 15s, without friction compensation, stiction always caused the piston to stop before reaching the end of its desired trajectory. As the compensator was turned on with nominal parameters (t = 15 s), the piston was able to reach the entire trajectory, but at the price of overcompensation. As the parameters were adapted, such overcompensation was progressively reduced and the overall tracking performance was enhanced.

13.3% of relative error. Direct comparison with other works is more difficult because there is significant variation in experimental apparatuses. However, in those examples where experimental setups and trajectory requirements were similar to ours, the reported relative errors ranged from about 7.5% to 11% [6, 22, 53–55]. Therefore, although such comparison must be considered just in qualitative terms, it indicates that the proposed controller can be regarded as a suitable approach. Addressing specifically the largest observed errors, Fig.4 clearly shows 21

Figure 5: Position tracking errors in the controlled system along 300 s (20 periods of 15 s), for the 5 sets of controller gains given in Table 1: (a) peak errors, (b) rms errors. In the first period (0-15 s), no friction compensation was used. During the second period (15-30 s), friction compensation was used with nominal parameters. Starting from t = 30 s, friction parameters were adapted with time. After long periods of operation in adaptive mode, the peak errors obtained by using the friction compensation term were just slightly smaller than those achieved without it, but the rms errors were clearly reduced, thus improving the overall trajectory tracking performance.

that they occurred mostly at the moments of trajectory reversal, regardless of the use of friction compensation. This feature can be explained by stiction: when compensation was not used, at near-zero velocities, the sudden rise in friction force caused the piston to stop short of its desired position; with the aid of the compensation scheme, the controller was able to track friction disturbances more closely as velocity was reduced. However, after the piston stopped, it took relatively long for it to move once again. The main reason

22

for such delay lies in the structure of the model (see equations (10) and (11)): since y˙ = 0, the value of the estimated friction state z(t) remains fixed. Therefore, the estimated friction force that is used in the mechanical subsystem control law (Eq. (21)) cannot match the variations of the actual one unless the velocity is nonzero. This is a fundamental limitation not only of our version of the LuGre model, but of the original one as well. To overcome it, it is necessary to add an entirely new term to the structure of the model, allowing for the update of the estimated friction force while the body remains still. Here, this necessity is somewhat abated by estimating the time derivative of the friction force, which is part of the control law applied to the pneumatic subsystem (Eq.(22)) and depends on piston acceleration, but the problem clearly persists. The values of the estimated friction parameters over time during the tests are depicted in Fig. 6. In most cases, such estimates tended to stabilize with time, but two important exceptions must be pointed out. First, for higher gain values, the estimates for σ ˆ0 (t) and σ ˆ2 (t) behaved much differently from the cases with reduced gains. In particular, σ ˆ2 (t) for gain set #5 showed signs of a possibly unstable behavior, because its value decreased until it saturated at its inferior bound (all three σ ˆi are lower-bounded by zero because of their physical meaning). Second, σ ˆ1 (t) was monotonically increasing in all tests (even for gain set #5, although much more slowly), which is also an evidence of possible instability. From the analysis carried out in Section 4, this result may seem unexpected, because it was proven that all tracking errors and estimated parameters should be limited quantities. However, such proof relies on the existence of a lower bound for KD , which, for the values of the

23

Figure 6: Average values of the estimated friction parameters during the tests: (a) σ ˆ0 (t), (b) σ ˆ1 (t), (c) σ ˆ2 (t). The periods depicted here are the same shown in Fig. 5

parameters used in this work, should be of an order of magnitude of 103 (see Eq. (33) and Remark #2). In practice, however, this gain could be hardly set at values greater than about 100 Ns/m, because of the appearance of intense oscillations in the response of the controlled system. These oscillations may be related to unmodeled dynamics in the controlled plant, but there is no doubt that they are largely affected by measurement noise, which, in the case of the employed workbench, is seriously enhanced by the use of numeric 24

differentiation for obtaining the velocity and acceleration signals. Due to this problem, the gains of the controller had to be limited to the relatively narrow range given in Table 1, which reflects a compromise between avoiding excessive oscillations and obtaining acceptable tracking-error amplitudes. Noise is ubiquitous to any experimental apparatus, and even more so in common industrial environments, representing a problem that can only be mitigated at very large costs in terms of equipment, time and effort. Thus, even in those uncommon cases when the restrictions imposed by the control hardware can be neglected, limiting gains to avoid noise amplification is always necessary. This is where a major drawback in using the LuGre model appears: since it emulates the elastic behavior of microscopic contact elements, its dynamic parameters tend to present rather large values: in the closely related case of a hydraulic actuator, for instance, the elastic deformation of seals lead to pre-sliding displacements of a few tenths of millimeter [19], which are consistent with an order of magnitude of 105 N/m or higher for σ0 . In other works, typically found values are about 104 N/m or 105 N/m for σ0 and 102 Ns/m for σ1 [15, 22, 37, 42, 56]. Therefore, since the stability analyses for nonlinear systems are often based on inequalities that depend on the values of their parameters (see [9, 21, 22, 57–60], among many other examples), when the LuGre model is used, it can be very difficult to attain in practice the gain values required by such analyses. It must be stressed that Lyapunov stability methods are based on sufficient conditions, not on necessary ones. Thus, even if the usual values associated with the LuGre model cause such analyses to fail in guaranteeing the stability of the controlled system, it does not follow that the system is

25

unstable. Moreover, the experimental results presented here indicate that, with smaller feedback gains and an appropriate set of friction parameters, the proposed friction compensation scheme tends to improve the performance of the controlled system. As for the potential instability, since the only indication for this possibility lies in the behavior of the estimated parameters, this difficulty can be overcome by employing other adaptation rules, or setting strict limits for the variation range of their values. Finally, as already pointed out in Section 1, even though the stability condition found here could not be implemented in practice, we must emphasize that the use of the proposed approximation allowed the corresponding analysis to be carried out in a complete and formal manner, without need to assume invalid or excessively restricted operation hypotheses. In contrast, all of the aforementioned works with similar experimental results [6, 22, 53–55] suffer from more severe restrictions regarding this topic: the first two examples rely on stability analyses based on physically unrealistic hypotheses, as already discussed in Section 1, whereas the analysis presented in the third one does not take dryfriction effects explicitly into account, and the last two discuss stability only in the context of linear approximations. This reinforces the notion that just the possibility of performing a more complete stability analysis of the system is already an important advance by itself. Therefore, although there are some important issues to overcome, the overall performance of the proposed control approach can be classified as promising.

26

6. CONCLUSIONS The primary objective of this work was to propose a friction compensation scheme that allowed the application of the LuGre friction model to fluid power-based actuators and still retain the ability to produce a complete, formal stability analysis for the controlled system without resorting to questionable assumptions regarding its operation conditions. Additionally, it was intended to develop this compensation scheme in an adaptive manner, to avoid the experimental determination of parameters that are difficult to measure. From a theoretical perspective, these objectives were fulfilled. However, with respect to practical usefulness, even if some results show potentially significant improvements, there are still some difficulties to overcome before this algorithm may be considered as a fully attractive solution. Future works will focus on avoiding such difficulties. Besides altering the adaptation rule employed in the proposed control scheme, such works may also include further modifications in the structure of the LuGre model, or the use of alternative models that are not as prone to the same hindrances. Appendix A. Modeling of air flow rates In this work, the air mass flow functions are modeled directly from experimental data, using polynomial functions to represent the effects of the chamber pressures, p1 and p2 , and of the cylinder control voltage u. Such functions are normalized, 3rd order polynomials, whose coefficients are adjusted so as to present only one real root within the interval [0,1]. The objective is to facilitate the calculation of the inverse function u (ˆ u), because

27

such root can be easily determined from well-known analytic relations. The general structure of these functions is given by:

proc qm (pu , pd , u) = [qm ]max fp (pu , pd )fu (u)

(A.1)

where pu and pd are, respectively, the upstream and downstream pressures with respect to the valve orifice, [qm ]max is the maximum air mass flow value identified during the experiment, fp (pu , pd ) is the polynomial function that models the effect of the pressures, and fu (u) is the polynomial that represents the influence of the control input u. The index proc refers to the process being carried out in the chamber of interest: filling (fil ) or exhausting (exh). For instance, for a positive control input, chamber 1 is filled while chamber 2 is exhausted. Under such conditions, for the valve orifice that regulates the flow into chamber 1, the upstream pressure is equal to the supply pressure Ps , whereas the downstream pressure is p1 . Likewise, for chamber 2, the corresponding values for these pressures are p2 and the atmospheric pressure P0 . As Ps and P0 are constant, they do not need to be mentioned explicitly proc1 proc2 in the equations. Thus, the expressions for qm1 and qm2 are, respectively:

f il f il f il qm1 (p1 , u) = [qm1 ]max fpf il (p1 )fu1 (u)

(A.2)

exh exh exh ]max fpexh (p2 )fu2 (u) qm2 (p2 , u) = [qm2

(A.3)

Following this notation, the complete list of experimentally determined polynomials is:

fpf1il (p1 ) = −1, 888p31 + 1, 157p21 − 0, 191p1 + 1, 007 28

(A.4)

fpexh (p1 ) = 0, 346p31 − 1, 747p21 + 2, 312p1 + 0, 09 1

(A.5)

fpf2il (p2 ) = −2, 395p32 + 1, 851p22 − 0, 362p2 + 1, 014

(A.6)

fpexh (p2 ) = −0, 029p32 − 0, 857p22 + 1, 88p2 + 0, 109 2

(A.7)

fuf1il (u) = 1, 2u3 − 3, 5u2 + 3, 58u − 0, 3

(A.8)

fuexh (u) = 1, 265u3 − 3, 282u2 + 3, 148u − 0, 112 1

(A.9)

fuf2il (u) = 1, 198u3 − 3, 446u2 + 3, 42u − 0, 161

(A.10)

fuexh (u) = 1, 2u3 − 3, 5u2 + 3, 6u − 0, 3 2

(A.11)

All these functions employ normalized values for its corresponding variables, i.e., u = u/umax , where umax is the maximum input voltage that can be applied to the servovalve, and pi = (pi − P0 )/(Ps − P0 ). This procedure facilitates the calculation of these functions for cases with different values of supply pressure or servovalves with other supply voltage definitions. With these functions, the entire inversion procedure is carried out as follows. First, with the desired value for the control law uˆ and using Eq. (5), we calculate a weighted sum of the air mass flow rates qm1 and qm2 . Then, we substitute the polynomial functions that correspond to the process determined by the desired control law, which result in a new 3rd order polynomial 29

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