IN THE PAST two decades, pneumatic artificial muscles

Transcription

1 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY A New Approach to Modeling Hysteresis in a Pneumatic Artificial Muscle Using The Maxwell-Slip Model Tri Vo-Minh, Student Member, IEEE, Tegoeh Tjahjowidodo, Herman Ramon, and Hendrik Van Brussel Abstract Two main challenges in using a pneumatic artificial muscle (PAM) actuator are the nonlinearity of pneumatic system and the nonlinearity of the PAM dynamics. The latter is complicated to characterize. In this paper, a Maxwell-slip model used as a lumped-parametric quasi-static model is proposed to capture the force/length hysteresis of a PAM. The intuitive selection of elements in this model interprets the unclear, but blended contributing causes of the hysteresis very well, which are assumed to originate from the dry friction of the double helix weaving of the PAM braided shell, the friction of the weaving and the bladder, the elasticity of the bladder and/or the deformation of the conical parts of a PAM close to the end caps. The obtained model is simple, but physically meaningful and easy to handle in terms of control. Index Terms Hysteresis, modeling, nonlinear systems, pneumatic systems. I. INTRODUCTION IN THE PAST two decades, pneumatic artificial muscles (PAMs) have raised the interest of scientists and researchers around the world, even though they were originally invented already in the 1950s by the American physician J. L. McKibben. The interest diminished gradually after first introduction because of the difficulty of air storage, inadequate control algorithms, and the predominant usage of electric motors for servo control [1], [8]. The increasing popularity of PAM usage nowadays is due to its clear advantages, such as inherent compliance, compactness, structural flexibility, and very low weight compared to other kinds of artificial muscle [2], [3], [18], [21], [31], and [32]. These advantageous properties of PAMs meet the emerging demand of state-of-art actuators for developing humanoids and anthropomorphic devices [1] [5]. Most of the PAMs used today are based on the McKibben muscle, which by times had other names, such as Rubbertuator Manuscript received January 16, 2009; revised July 4, 2009 and November 5, 2009; accepted November 20, Date of publication January 15, 2010; date of current version January 12, Recommended by Technical Editor V. Narayan. This work was supported by the Directorate-General for Development Cooperation of the Belgium Government, which also awarded a scholarship to T. Vo-Minh. T. Vo-Minh and H. Van Brussel are with the Department of Mechanical Engineering, Division of Production Engineering, Machine Design, and Automation, Katholieke Universiteit Leuven, B-3001 Heverlee, Belgium ( tri.vominh@student.kuleuven.be; hendrik.vanbrussel@mech.kuleuven.ac.be). T. Tjahjowidodo is with the Flanders Mechatronics Technology Center, B-3001 Heverlee, Belgium ( tegoeh.tjahjowidodo@mech.kuleuven.be). H. Ramon is with the Department of Biosystems, Division of Mechatronics, Biosensors and Statistics, B-3001 Heverlee, Belgium ( herman. ramon@biw.kuleuven.be). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMECH Fig. 1. (a) McKibben artificial muscle design and (b) working principle. [2], [3], braided pneumatic muscle [4], McKibben artificial muscle [5], McKibben pneumatic artificial muscle [21], and fluidic muscle (Festo AG, In this paper, a PAM is simply referred as a muscle. Basically, a McKibben-based artificial muscle consists of an inner rubber tube, which functions as an air enclosure, also called a bladder. This tube is tightened at the outside by a fiber layer of nonextendable double helix weaving, which functions as an antirupture layer and for the transmission of work, the so-called braided shell. The entire tube is closed by two end caps, of which one has a hose for connecting the air supply and the other is available for connecting to the mechanical load. The design of McKibben-based PAMs can be slightly different from producer to producer, but their working principle is unchanged [see Fig. 1(a)]. The PAM is used to convert pneumatic power into mechanical power. When it is being inflated the muscle is enlarged in radial and shortened in longitudinal direction [see Fig. 1(b)], /$ IEEE

2 178 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011 Fig. 2. Nonlinear relation between the contracting force, internal pressure, and length [or contraction ratio h (in percentage)] of a 20 mm nominal diameter muscle (source: fluidic muscle datasheet, whereas when it is being deflated, the muscle turns back to its original form. In the contracting stage, the PAM can exert a force to a connected load. This tensile or contracting force of the PAM is unidirectional, and its value depends on the internal pressure and the actual length for a certain designed diameter. Due to the nonlinear relation of the contracting force to its internal pressure and length or contraction ratio (see Fig. 2), this artificial muscle has a highly inherent compliance, which is similar to a skeleton muscle [8]; the shorter in length the muscle, the smaller the contracting force. Moreover, the direct coupling to the load makes it competitive to electric actuators in terms of structural optimization and power/weight ratio. Besides the aforementioned advantages, there exist, however, two main drawbacks that limit a PAM application, namely the nonlinearity of the pressure buildup and the hysteresis that is due to its geometric construction. These disadvantages cause difficulties in designing controllers for high-performance positioning systems. This paper is mainly dedicated to the latter problem: characterization of the inherent hysteresis, which is complicated to understand and handle. Section II gives an overview of the contracting force model development. Section III discusses hysteretic phenomena in a PAM and subsequently introduces a new method to model the force/length hysteresis. Section IV describes the experimental setup. The experimental results are presented in Section V, followed by some intensive discussions. Section VI gives the conclusions and opens some discussion points for future work. II. STATIC CONTRACTING FORCE OF A PAM Now that PAMs have been coming back with high research interest in scientific communities, the demand for a good model for this type of muscle is rising. There is a need for different types of model: a simple but physically meaningful model that is suitable for control purposes, and a complicated one that takes into account many aspects, making it suitable for model validation and improvements in the design stage. Lack of knowledge on forming the model in the former case will leave a difficulty for the control system afterward. Therefore, the more accurate but controllable the model is, the less effort is required for designing a control strategy. In order to increase the degree of understanding and the accuracy of the model, historical reviews of previous PAM modeling efforts in the past two decades should be performed carefully. Modeling of the static contracting force is still on-going to find a precise mathematic description. As stated in [3], it is nearly impossible to create a precise mathematic model. This is true up to now because most of the proposed models are approximations. Tondu et al. developed the static force based on the empirical model given in [3], which includes a correction factor due to the effect of the end caps, causing the discrepancy at highcontraction ratios [5], [6]. This model was afterward modified with a Coulomb dry-friction force model in order to accommodate the inherent hysteresis [7], [8]. The elastic force against the contracting force relationship was also mentioned in [9] when accounting for the effect of the bladder thickness. This effect was carefully investigated in [13] and [23]. Caldwell et al. stated that the static force is just the product of the internal pressure and the internal area of the muscle [10], [12]. As a rule of thumb, a conversion factor is later introduced to reduce the deviation between the measurement and the proposed model. In a straightforward approach, Chou and Hannaford derived a static model based on trigonometric analysis, but problems occurred in the validation phase, since the model neglected some aspects, such as noncylindrical shape and nonzero wall thickness [21], [22]. Davis et al. spent more effort on the model validation aspect by taking into account the wall thickness and the stress effect [16]. However, the models by Tondu and by Caldwell, being obtained by different approaches (empirical and energy conservation, respectively), can be converted to the Chou model, which is based on geometrical relationships only [8], [14]. In general, all available models are approximations, and therefore, no model has achieved a satisfactory output force prediction. There are many parameters needed, such as the braid angle, the braid length, braid contacting surface area, etc., to obtain an accurate model. One of simplified models can also be found in [22], which shows that the static force consists of an active term, which depends on the pressure; a passive term, which depends on the muscle length; and a nonlinear term, which depends on the muscle extreme length. The simulation of this simple model without considering hysteresis and the nonlinear term, however, shows a good resemblance to the measurements. In this paper, the static force is derived in a more straightforward way as follows: F isob = F const + F hys (1) where F isob is the measured static contracting force obtained from an isobaric experiment; F const is the static force obtained from an experiment with constrained PAM (see Section IV); F hys is the hysteresis force. Hysteresis force F hys can then be extracted as follows: F hys = F isob F const. (2) F hys, the so-called extracted hysteresis, exhibits the same behavior as in the presliding regime of mechanical friction, and will be thoroughly studied in the next section.

3 VO-MINH et al.: NEW APPROACH TO MODELING HYSTERESIS IN A PNEUMATIC ARTIFICIAL MUSCLE USING THE MAXWELL-SLIP MODEL 179 Fig. 4. Sketch of the hysteresis function with nonlocal memory. Fig. 3. Possible causes of hysteresis in a PAM. (a) Cords friction. (b) Cords and bladder friction. (c) Concical deformation. (d) Bladder stretching due to volume increase. III. MODELING OF HYSTERESIS IN A PAM A. Reviews of Hysteresis in a PAM Hysteresis is defined as the lagging of an effect behind its cause as stated in Oxford dictionary. Hysteresis is a complex nonlinearity with memory that complicates tracking control and appears intrinsically in several types of actuators, such as electromagnetic [33], shape memory alloy [34], and piezoelectric [35] actuators. In a PAM, the hysteresis can be determined in an isobaric test and isotonic contraction test [8], [22], which refer to force/length hysteresis and pressure/length hysteresis, respectively. The PAM hysteretic phenomenon is reported earlier in [3], where it is indicated that this kind of hysteresis is caused by the inherent hysteresis of the elastic bladder, the friction between braided cords and rubber bladder, and the friction between cords themselves. These causes are also listed in [8], [9], [21], and [22], amongst them the friction between cords being the most significant. However, the effect of each cause separately has not been proved in previous studies. In our lumped parametric approach, not only the mentioned causes, but also many others, such as the deformation of the noncylindrical shape close to the end caps and the high nonlinear effect out of the operating range are also responsible for the resulting hysteresis. Some contributing causes of hysteresis are illustrated in Fig. 3. Hysteresis causes energy loss and increases the complexity of the control system [17]. Literature shows that it is difficult to control a PAM in which hysteresis is left unmodeled. Tondu et al. [7], [8] added a dry-friction model to the contracting force to increase the accuracy of the static and dynamic response, but this study is limited to the assumption that hysteresis is due to only the thread-on-thread dry friction between the cords [see Fig. 3(a)]. Davis and Caldwell [17] tried to statically capture hysteresis when braids friction is carefully considered. However, these studies give rise to complex problems in terms of control, since there are many parameters that are difficult to quantify. Van Damme et al. [27] described hysteresis in pleated PAM [28] by using a Preisach model. However, this modeling approach cannot reflect the elastoplastic property of hysteretic causes [24], [33], and the construction of a pleated PAM is different from the McKibben-based PAM family. Meanwhile, Chou and Hannaford gave an insight into hysteretic characteristics in a PAM, such as quasi-rate independency, history dependency, but they did not really go further into modeling that hysteresis [21], [22]. This paper picks up these interesting points mentioned by Chou and clarifies them by developing an analogy of hysteresis in a PAM to the behavior in the presliding regime of mechanical friction. B. Presliding Regime of Mechanical Friction Swevers et al. studied the microscopic phenomena of dry friction by modeling the complex interaction between the surface and the near-surface regions of two interacting materials, in order to construct a more comprehensive prediction model that accounts for the various aspects of the phenomenon [25]. In dry friction between two contacting surfaces, two regimes can be distinguished: presliding and sliding. In the presliding regime, the adhesive forces are dominant such that the friction force (at asperity contacts) appears to be a function of displacement rather than the velocity (rate independency). The sliding regime happens when the displacement increases until the asperity contacts are broken away. In the presliding regime, the asperity junctions deform elastoplastically, resulting in a nonlinear-spring behavior. Friction in this regime can be described by a hysteresis function, which consists of transition curves. The friction force will follow a certain function of displacement, the so-called virgin curve y(x) (see Fig. 4). Assume that the displacement is toward x+, and if, say at x 1 the motion is reversed, the friction force will follow the flipped-and-double-stretched virgin curve as represented by the lower-half loop in Fig. 4. Since this trajectory is fully characterized by the virgin curve, representation of the trajectory obviously requires the information of the reversal point 1. Therefore, at this stage, the system has to memorize point 1. Subsequently, if the motion is reversed again after it arrives at x 2, the trajectory will follow the double-stretched virgin curve through point 2. Thus, at this stage, the system has to memorize both points 1 and 2.

4 180 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011 Consider a case when the motion is reversed half-way after it arrived at the reversal point 1. If the motion is then reversed after it arrives at point 3, the trajectory will follow the doublestretched virgin curve, similar to the previous example except that it will be drawn from point 3. Therefore, beyond this point, the system has to memorize points 1 and 3. The trajectory will behave in the same way if, after it arrives at point 3, we reverse the motion again. Hence beyond point 3, the trajectory will follow the flipped-double-stretched virgin curve and the system memorizes three reversal points (1, 3, and 3 ). Unique behavior is markedly seen when we keep pushing the system to the left after it arrives again at point 3. Instead of following the dashed curve as a continuation of the flipped-double-stretched virgin curve from 3 to 3, it will follow the previous lower loop trajectory from point 3 to 2. Therefore, it can be concluded that if the loop is closed (as it is formed by loop ), the previous reversal points that formed the loop can be removed from the memory. Strictly speaking, at present, beyond point 3, the system will only memorize points 1 and 3. Such hysteresis behavior is usually referred to as a nonlocal memory hysteresis. Mathematically, the friction force can be formulated by using state equation (3), which is based on a virgin curve equation ( ) x x1 F hys out = F hys 1 +2f 2 where f(x) =y(x) if x 0; f(x) = y( x) if x 0; F hys out is the friction force output; F hys 1 = f(x 1 ) is the friction force value at reversal point x 1. C. Maxwell-Slip Model is Well-Suited for Modeling PAM Hysteresis The virgin curve, characterizing the friction in the presliding regime, can be identified with a piecewise-linearization approximation using the Maxwell-slip model [26]. Iwan [24] developed a discrete mathematical element for hysteresis modeling in which each contributing element is referred to as a Maxwellslip element (see Fig. 5). The element is simply determined by two parameters, namely a stiffness k and a saturation force w. When the element starts to displace, it will stick and behave like a linear spring with a certain stiffness. If the element is continuously displaced, it will reach a maximum force, which is referred to as the saturation force, and beyond this state, the elementary force equals the saturation force. By putting several elements (with different parameters of stiffness and saturation force ) in parallel, the complex behavior of nonlocal memory hysteresis can be modeled discretely and the output force at any instant can be calculated by using (4). For a typical design, such as the Festo fluidic muscle in which a braided shell is incorporated into the rubber bladder, or for all muscles in which braiding strands are not relatively moved during extension/contraction and the friction resembles the presliding regime of mechanical friction between two contacting (3) Fig. 5. surfaces Modeling hysteresis using the Maxwell-slip model. F hys out = n F i. (4) If the virgin curve of PAM hysteresis is determined, any state of the output hysteretic force with respect to the muscle length can be interpolated by using representative elements. In other words, if the representative element parameters are known, the hysteretic force with nonlocal memory can be simulated for any length change. The virgin curve is experimentally obtained by exciting the muscle to the extremum length. This virgin curve contains information of all Maxwell-slip elements, which are contributing to the hysteresis loop. The number of representing elements in a virgin curve can be intuitively selected. IV. EXPERIMENTAL SETUP It is clear that the methodology for hysteresis modeling based on Maxwell-slip elements can be an alternative approach to capturing hysteresis in a PAM. It is necessary to find the hysteresis loop and to observe whether this hysteresis exhibits properties of the hysteresis function with nonlocal memory or not. Two experimental setups were built. The first is the constrained setup as can be seen in Fig. 6(a). The aim of this experiment is to find the empirical model of the contracting force. The test muscle is constrained at its both end caps, and one is fixed via the force sensor. After being constrained at a certain length, the pressure inside the muscle is increased, and as a result, the contracting force is increased. The model in this case is called constrained model in order to differ it from the free-moving model, which is obtained in the second setup, a so-called isobaric setup. In the second setup, one end cap is free to move but connected via a force sensor to another freely moving end cap of a stronger muscle, called the stretching muscle [see Fig. 6(b)]. The stretching muscle is used to actuate the test muscle while the inside pressure of the test muscle is regulated at a desired value. The 1

5 VO-MINH et al.: NEW APPROACH TO MODELING HYSTERESIS IN A PNEUMATIC ARTIFICIAL MUSCLE USING THE MAXWELL-SLIP MODEL 181 Fig. 7. lengths. Contracting force via the internal pressure at different constrained contraction ratio of 25%. The PTE5151D1A pressure sensor from SENORTECHNICS was employed to measure the pressure. The air was supplied via a pneumatic 5/3 directional proportional valve (Festo type of MYPE-5-M5-010B). The length of the muscle was measured by a laser displacement sensor type PD from BAUMER ELECTRIC, while the contracting force was measured by using a load cell type DBBP-200 from BONGSHIN. All I/O information from/to the plant setup was processed by a 16-bit data acquisition card DAQmx NI-6229 from National Instrument, which was embedded in a real-time desktop PC. The control and measurement algorithms were developed based on LabView Professional Development System for Windows with the add-on LabView Real-Time Module. Fig. 6. (a) Photograph of the constrained setup. (b) Photo of the isobaric setup. V. RESULTS AND DISCUSSION From the constrained setup, all measurement data of the contracting force against the internal pressure of the muscle at different constrained lengths were plotted as shown in Fig. 7. These lines look very straight, but the slope and the intercept of each line with the ordinate axis are different to one another. This brings us to formulate the constrained model of the contracting force F const as a linear function of the internal pressure as follows: F const = a(x)p + b(x) (5) aim of this setup is to measure the contracting force/length relation while the pressure is kept constant. If there is hysteresis, it will apparently appear in the data plot. The data are called free-moving data. The test muscle is a Festo fluidic muscle (type MAS N) with an internal diameter of 20 mm and a length of 200 mm. The stretching muscle is a MAS N type with an internal diameter of 40 mm and a length of 300 mm. The datasheet shows that these muscles have a maximum where a(x), b(x) are the slopes and the intercepts of the linear lines, respectively; P is the internal pressure; x is the length of the muscle. By fitting these lines with a linear equation, the slopes and the intercepts are obtained as functions of the muscle length. These functions are then again fitted by using the least squares curve fitting tools in MATLAB. The slopes and the intercepts

6 182 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011 Fig. 8. Contracting force via muscle length at different internal pressures. of the test muscle as functions of muscle length are given in (6) a(x) = x x b(x) = x x x (6) Substituting functions (6) into (5), one can obtain the model of the contracting force. The obtained model (6) is comparable to the work of [30], however our final contracting force model, expressed in (1), has a supplementary part, i.e., the extracted hysteresis. The advantage of forming the model in such a way is that it is not affected by the friction that is due to the motion and by stretching of elastic bladder, due to the volume change. These effects appear later in the free-moving model. From the isobaric setup, all measurement data of the contracting force against the length of the muscle at different constant pressures inside the muscle are plotted as shown in Fig. 8. Hysteresis, which is referred to as the force/length hysteresis apparently occurs. Observing the plot of these hysteresis loops, one can find out that the higher the given pressure in the test muscle, the longer the length interval the test muscle can be stretched or pulled. At the same given pressure, a larger length will give a higher contracting force, where the reconstructed data, which come from the constrained model are also plotted and displayed as a single line corresponding to each constant pressure. These lines show that there is no hysteresis comparable to that in the isobaric test. The isobaric tests have to be executed carefully to avoid stretching the muscle beyond the extremum at each internal pressure. The test muscle with a certain inside pressure will have a certain initial length, where the contracting force is equal to zero. The stretchable range of the test muscle with a given internal pressure is equal to its maximum length minus the initial length. This interval determines approximately the pressure interval of the stretching muscle in order to actuate the test muscle. This pressure interval determines the amplitude of a sinusoidal excitation for stretching the test muscle forward and backward. Fig. 9. (a) Stretching trajectory applied to the test muscle. (b) History dependency of hysteresis during extension/contraction. The maximum amplitude or full stroke of excitation will give the largest hysteresis loop. However, the test muscle with a highly internal pressure cannot be stretched to the maximum length due to rupture. That is why the 4-bar internal pressure was selected for the test muscle in order to determine the full stroke of hysteresis loop. In order to see the nonlocal memory behavior of this hysteresis loop, the pressure interval of the stretching muscle can be recalculated such that the test muscle is driven several times close to both ends of the full stroke with smaller amplitude [see Fig. 9(a)]. Fig. 9(b) shows that the small hysteresis loops are stuck at the upper side of the big loop in the extension phase and at the lower side of the big loop in the contraction phase. This is comparable to the behavior of the hysteresis function with nonlocal memory as described earlier. The quasi-rate independency characteristic of the hysteresis was examined by changing the stretching rate. In Fig. 10, the dotted line hysteresis loop was taken at a rate of 0.8 Hz, which

7 VO-MINH et al.: NEW APPROACH TO MODELING HYSTERESIS IN A PNEUMATIC ARTIFICIAL MUSCLE USING THE MAXWELL-SLIP MODEL 183 Fig. 10. Quasi-rate independency of hysteresis during extension /contraction. Fig. 12. Calculating the Maxwell-slip element parameters directly on the coordinate analysis. Fig. 11. Extracted hysteresis with 4-bar internal pressure in the test muscle. is 16 times faster than the solid line loop, which was obtained at 0.05 Hz. We can see that the size and the slope of these two loops are almost unchanged. The waviness in the higher rate loop resulted from the oscillation of the pressure regulation in the stretching muscle, while the internal pressure of the test muscle was shown to be well-regulated for all stretching rates [29]. The force/length hysteresis shown in Figs. 8 and 9(b) is tested under isobaric conditions. This hysteresis is assumed to be a combination of the constrained model and the hysteresis contributed by Maxwell-slip elements, referred to as the extracted hysteresis F hys in (1). Therefore, the extracted hysteresis can be obtained by subtracting the free-moving measured data to the reconstructed force/length data from the constrained model. Fig. 11 shows the extracted hysteresis at 4-bar internal pressure. The full stretchable stroke is about 40 mm, starting at the minimum length of 153 mm to the maximum length of 193 mm (contraction ratio of 20%). Having the hysteresis loop, which exhibits the behavior of nonlocal memory, one can easily identify the representative Maxwell-slip elements in order to model and simulate that hysteresis. At first the virgin curve must be drawn by shrinking the upper side of the big loop, and then, asymptotes are drawn to approximate the virgin curve. However, selection of the number of elements that sufficiently capture the behavior of hysteresis is quite intuitive. Although the hysteretic causes in a PAM cannot be exactly quantified, four representative Maxwell-slip elements are visibly sufficient to fit the virgin curve. Eight parameters qualifying these four elements have to be identified (see Fig. 12), which consist of four stiffness parameters k 1, k 2, k 3, and k 4 and four saturation values w 1, w 2, w 3, and w 4.The system of eight equations is thus established based on the coordinate analysis (7). The values a 1, a 2, a 3, a 4 and b 1, b 2, b 3, b 4 are directly measured in the plot and used to calculate the slope of four segments K a, K b, K c, and K d. Solving these equations yields eight representative parameters as shown in Table I. These parameters are introduced in the model in order to be simulated and validated with the measurement data k 1 + k 2 + k 3 + k 4 = K a = a 1 b 1 k 2 + k 3 + k 4 = K b = a 2 b 2 b 1 k 3 + k 4 = K c = a 3 b 3 b 2

8 184 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011 TABLE I IDENTIFIED PARAMETERS OF FOUR REPRESENTATIVE MAXWELL-SLIP ELEMENTS Fig. 14. Total contracting force, combining the constrained model and the simulated hysteresis force based on Maxwell-slip model, compared to the freemoving measured data at different given pressures. Fig. 13. Extracted hysteresis force is captured by the Maxwell-slip model not only with a full stroke excitation, but also with different excitation intervals. k 4 = K d = a 4 b 4 b 3 w 1 k 1 = b 1 w 2 k 2 = b 2 w 3 k 3 = b 3 w 4 = b 4. k 4 (7) Fig. 13 shows the online simulation of the extracted hysteresis model and the measured data at different motion intervals. Fig. 14 shows the online simulation of the combination of the constrained model and the extracted hysteresis model compared to the free-moving measured data at different given pressures. These figures indicate a good agreement between the model and the measured data. However, there is a small discrepancy at the extremes of the curve, which can be explained due to the contracting force becoming highly nonlinear and the effect of the rubber bladder. For the high nonlinearity of the contracting force at the extreme lengths, we can see that close to the maximum length zone the fibers tend to be directly stretched. As a result, the contracting force is rapidly increasing. In fact this zone starts from 193 mm, which corresponds to the length at rest of the muscle, and goes to the maximum length. On the other hand, close to the minimum length zone, the contracting force tends to have a negative value due to the pillow-like reaction. This Fig. 15. points. Creep resulting from a long-run cyclic excitation around two reversal zone depends on the internal pressure; it is about 150 mm at 5 bars. In practice, these extreme zones are not expected to occur. In addition, out of these zones the behavior of the hysteresis is beyond the characterization of the model. For the effect of the rubber bladder, i.e., the rubber relaxation, Fig. 15 depicts the hysteresis loop with creep, which is described in [12], [20] as a time-varying system or temperature effect. This loop was obtained by cyclic stretching of the test muscle for a large number of cycles (about 50 cycles with a period of 20 s each). The first few cycles indicated in this figure refer to the beginning of cyclic stretching. Gradually, the muscle was warmed up due to dissipation of the energy loss caused by friction. The minimum length of the muscle after a long run is increasing a bit, and the contracting force is slightly increasing. As long as the test muscle is stretched, the final cycles loops will remain constant. The loop shifting is more predominant in

9 VO-MINH et al.: NEW APPROACH TO MODELING HYSTERESIS IN A PNEUMATIC ARTIFICIAL MUSCLE USING THE MAXWELL-SLIP MODEL 185 Fig. 16. Volume against length of the test muscle. the region from about 175 mm up to the maximum length, which has a logical connection to the volume/length curve shown in Fig. 16. The volume changes rapidly in the mentioned region. This means that the bladder is significantly expanded, and as a result the rubber structure has crept. The hysteresis capture therefore shows a mismatch and is very dependent on the intuitive selection of the elements of the modeled loop. However, the discrepancy is small (about ±3 N approximately) compared to the total hysteresis, and becomes negligible at higher internal pressures (i.e. 5 and 6 bar) and in the smaller length region (see Fig. 14). VI. CONCLUSION Using the Maxwell-slip model in capturing the complex hysteresis in PAMs has shown to be a very distinguished approach. The PAM hysteresis behaves similar to the presliding regime in mechanical friction. The lumped-parametric modeling approach gives an adequate interpretation to the blended contributing causes of the PAM complex hysteresis. The proposed model based on the hysteresis function with nonlocal memory describes very well the force/length hysteresis not only at different excitation intervals, but also with different internal pressures. This is encouraging for our future work, intending to use the model to improve the control of the PAM. REFERENCES [1] V. L. Nickel, J. Perry, and A. L. Garrett, Development of useful function in the severely paralysed hand, J. Bone Joint Surg., vol. 45-A, no. 5, pp , Jul [2] EPW, Rubber muscles take robotics one step further Rubber Develop., vol. 37, no. 4, pp , [3] K. Inoue, Rubbertuators and applications for robots, in Proc. 4th Symp. Robot. Res., Tokyo, 1987, pp [4] J. M. Winters, Braided artificial muscles: Mechanical properties and future uses in prosthetics/orthetics, in Proc. RESNA 13th Conf., Washington, DC, 1990, pp [5] B. Tondu, V. Boitier, and P. Lopez, Naturally compliant robot-arms actuated by McKibben artificial muscles, in Proc. IEEE-SMC Conf., San Antonio, TX, 1994, pp [6] B. Tondu and P. Lopez, Theory of an artificial pneumatic muscle and application to the modelling of McKibben artificial muscle, C.R.A.S., French Nat. Acad. Sci., vol. 320, Series IIb, pp , 1995 (in French with an abridged English version). [7] B. Tondu, Analysis and modeling of the dynamic behavior of the McKibben artificial muscle, in Proc. IFAC Robot Control, Nantes, France, 1997, pp [8] B. Tondu and P. Lopez, Modeling and control of McKibben artificial muscle robot actuators, IEEE Control Syst. Mag.,vol.20,no.2,pp.15 38, Apr [9] B. Tondu, S. Ippolito, J. Guiochet, and A. Daidie, A seven-degrees-offreedom robotarm driven by pneumatic artificial muscles for humanoid robots, Int. J. Robot. Res., vol. 24, no. 4, pp , [10] D. G. Caldwell, A. Razak, and M. Goodwin, Braided artificial muscle actuators, in Proc. IFAC Intell. Auton. Veh., Southampton, U.K., 1993, pp [11] D. G. Caldwell, G. A. Medrano-Cerda, and M. Goodwin, Characteristics and adaptive control of pneumatic muscle actuators for a robotic elbow, in Proc. Int. Conf. Robot. Autom., San Diego, CA, 1994, vol. 4, pp [12] D. G. Caldwell, G. A. Medrano-Cerda, and M. Goodwin, Control of pneumatic muscle actuators, IEEE Control Syst. Mag., vol. 15, no. 1, pp , Feb [13] N. Tsagarakis and D. G. Caldwell, Improved modeling and assessment of pneumatic muscle actuators, in Proc. IEEE Int. Conf. Robot. Autom., San Francisco, CA, 2000, vol. 4, pp [14] D. G. Caldwell, N. Tsagarakis, and G. A. Medrano-Cerda, Biomimetic actuators: Polymeric pseudo muscular actuators and pneumatic muscle actuators for biological emulation, Mechatronics, vol. 10, pp , [15] D. G. Caldwell and N. Tsagarakis, Biomimetic actuators in prosthetic and rehabilitation applications, Technol. Health Care, vol. 10, pp , [16] S. Davis, N. Tsagarakis, J. Canderle, and D. G. Caldwell, Enhanced modeling and performance in braided pneumatic muscle actuators, Int. J. Robot. Res., vol. 22, no. 3 4, pp , [17] S. Davis and D. G. Caldwell, Braid effects on contractile range and friction modeling in pneumatic muscle actuators, Int. J. Robot. Res., vol. 25, no. 4, pp , [18] D. W. Repperger, C. A. Phillips, D. C. Johnson, R. D. Harmon, and K. Johnson, A study of pneumatic muscle technology for possible assistance in mobility, in Proc. 19th Annu. Int. Conf. IEEE Eng. Med. Biol. Soc., Chicago, IL, 1997, pp [19] D. B. Reynolds, D. W. Repperger, C. A. Phillips, and G. Bandry, Dynamic characteristics of pneumatic muscle, Ann. Biomed. Eng., vol. 31, no. 3, pp , [20] J. H. Lilly and L. Yang, Sliding mode tracking for pneumatic muscle actuators in opposing pair configuration, IEEE Trans. Control Syst. Technol., vol. 13, no. 4, pp , Jul [21] C. P. Chou and B. Hannaford, Static and dynamic characteristics of McKibben pneumatic artificial muscles, in Proc. IEEE Int. Conf. Robot. Autom., San Diego, CA, May 1994, vol. 1, pp [22] C. P. Chou and B. Hannaford, Measurement and modeling of McKibben pneumatic artificial muscles, IEEE Trans. Robot. Autom., vol. 12, no. 1, pp , Feb [23] G. K. Klute and B. Hannaford, Accounting for elastic energy storage in McKibben artificial muscle actuators, Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 122, pp , [24] W. D. Iwan, A distributed-element model for hysteresis and its steadystate dynamic response, J. Appl. Mech., vol. 33, pp , [25] J. Swevers, F. Al-Bender, C. Ganseman, and T. Prajogo, An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Trans. Autom. Control, vol. 45, no. 4, pp , Apr [26] V. Lampaert, J. Swevers, and F. Al-Bender, Modifications of the leuven integrated friction model structure, IEEE Trans. Autom. Control,vol.47, no. 4, pp , Apr [27] M. Van Damme, P. Beyl, B. Vanderborght, R. Van Ham, I. Vanderniepen, R. Versluys, F. Daerden, and D. Lefeber, Modeling hysteresis in pleated pneumatic artificial muscles, in Proc. IEEE Int. Conf. Robot., Autom. Mechatronics, 2008, pp [28] F. Daerden and D. Lefeber, The concept and design of pleated pneumatic artificial muscles, Int. J. Fluid Power, vol. 2, no. 3, pp , [29] T. Vo Minh, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, Nonlocal memory hysteresis in a pneumatic artificial muscle, in Proc. 17th

10 186 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011 Mediterranean Conf. Control Autom., Thessaloniki, Greece, Jun. 2009, pp [30] A. Hildebrandt, O. Sawodny, R. Neumann, and A. Hartmann, Cascaded control concept of a robot with two degrees of freedom driven by four artificial pneumatic muscle actuators, in Proc. Amer. Control Conf.,2005, pp [31] D. H. Plettenburg, Pneumatic actuators: A comparison of energy-to-mass ratio s, in Proc. IEEE Int. Conf. Rehabil. Robot., Chicago, IL, 2005, pp [32] X. C. Zhu, G. L. Tao, B. Yao, and J. Cao, Adaptive robust posture control of parallel manipulator driven by pneumatic artificial muscles with redundancy, IEEE/ASME Trans. Mechatronics, vol. 13, no. 4, pp , Aug [33] S. Mittal and C.-H. Menq, Hysteresis compensation in electromagnetic actuators through preisach model inversion, IEEE/ASME Trans. Mechatronics, vol. 5, no. 4, pp , Dec [34] S. M. Dutta and F. H. Ghorbel, Differential hysteresis modeling of a shape memory alloy wire actuator, IEEE/ASME Trans. Mechatronics, vol. 10, no. 2, pp , Apr [35] U.-X. Tan, W. T. Latt, C. Y. Shee, C. N. Riviere, and W. T. Ang, Feedforward controller of Ill-conditioned hysteresis using singularity-free Prandtl- Ishlinskii model, IEEE/ASME Trans. Mechatronics, vol. 14, no. 5, pp , Oct Herman Ramon was born in Oostende, Belgium, in He received the M.Eng. degree in bioscience from the University of Gent, Flanders, Belgium, in 1982, and received the Ph.D. degree in applied biological sciences from the Katholieke Universiteit Leuven, Heverlee, Belgium, in He is currently a Full Professor in the Department of Biosystems, Division of Mechatronics, Biosensors and Statistics, Katholieke Universiteit Leuven, where he is involved in teaching courses on system dynamics, identification, and control of biosystems, biomechatronics, precision agriculture, and ecosystems modeling. His research interests include the development of precision mechanisms for improved control of processes occuring in the food chain, from the field to the fork taking into account the human factor in the process chain. This encompasses better understanding of the process dynamics using advanced modeling techniques (discrete- and finite-element modeling, multibody modeling, neurofuzzy modeling, etc.), advanced control systems engineering (robust and model predictive control, neurofuzzy control, etc.), and mechatronic design. of actuators. Tri Vo-Minh (S 09) was born in Bentre, Vietnam, in He received the B.Eng. degree in mechanical engineering from Cantho University, Cantho, Vietnam, and the M.Sc. degree in mechanical engineering from National University, Ho Chi Minh City, Vietnam, in 1993 and 1998, respectively. He is currently working toward the graduate degree at Katholieke Universiteit Leuven, Heverlee, Belgium. His current research interests include developing a soft arm actuated by pneumatic artificial muscles and modeling the intrinsic hysteresis in these types Hendrik Van Brussel was born in He received the B.Sc.M.E. (Technisch Ingenieur) degree from Hoger Technisch Instituut, Oostende, Belgium, in 1965, and the M.Sc.E.E. (Burgerlijk Ingenieur) and Ph.D. degrees from the Katholieke Universiteit Leuven, Heverlee, Belgium, in 1968 and 1971, respectively. He is currently a Full Professor of mechatronics and automation in the Department of Mechanical Engineering, Katholieke Universiteit Leuven. He was a pioneer in robotics research in Europe and an active promoter of the mechatronics idea as a new paradigm in machine design. He has authored or coauthored on different aspects of robotics, mechatronics, and flexible automation. His current research interests include mechatronics, medical and behavior-based robotics, holonic manufacturing systems, and precision engineering. Dr. Brussel is the past President of the International Academy for Production Engineering and the President of the European Society for Precision Engineering and Nanotechnology. in machinery. Tegoeh Tjahjowidodo received the M.Eng. degree from the Institut Teknologi Bandung, Bandung, Indonesia, in 1999, and the Ph.D. degree from the Katholieke Universiteit Leuven, Heverlee, Belgium, in He is currently a Senior Researcher at the Flanders Mechatronics Technology Center, Heverlee, a research center in the mechatronics area, since 2006, where he was involved in several projects, mainly in developing a model-based diagnosis methodology of mechatronic systems and noise reduction techniques