Modeling Torque-Angle Hysteresis in A Pneumatic Muscle Manipulator

Transcription

1 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Montréal, Canada, July -9, Modeling Torque-Angle Hysteresis in A Pneumatic Muscle Manipulator Tri Vo Minh, Student Member, IEEE, Bram Kamers, Tegoeh Tjahjowidodo 3, Herman Ramon, Hendrik Van Brussel, Fellow IEEE Abstract Hysteresis is inherently present in Pneumatic Artificial Muscle (PAM) Actuators. Our new observation shows that the hysteresis in a PAM is characterized by quasirate independency and history dependency, and is completely described by the Maxwell-slip (MS) model. In this paper, we explain how to model the existing hysteresis in an antagonistic PAM configuration, using the models derived for the individual PAMs. Experimental results show that the pneumatic manipulator hysteresis has the same behaviors as those found in a single PAM. The proposed model is used to predict the output torque of the manipulator joint not only for any arbitrary angular displacements but also for any arbitrary pressure difference. Key words: McKibben Artificial Muscle, Hysteresis Modeling, Pneumatic Muscle Manipulator. P I. INTRODUCTION NEUMATIC muscle manipulators have recently become the alternative actuators for developing humanoid robots due to their soft characteristic. An antagonistic arrangement actuated by a pair of pneumatic artificial muscles can be ready for a -dof soft manipulator, which possesses the same advantageous characteristics as the single PAM unit, such as adjustable compliance, high power-to-weight ratio, high volume-to-weight ratio, etc. However, hysteresis is a drawback that limits the widespread use of this kind of actuators. This paper presents a new approach to capture the inherent hysteresis in a basic antagonistic muscle manipulator unit constructed by a pair of FESTO fluidic muscles. A PAM is a flexible actuator similar to human muscle, i.e. the shorter in length the muscle, the smaller the contracting force []. Most of the PAMs used today are based on the McKibben muscle []. During the history of their development, PAMs have received several names, such as Rubbertuator [3], braided pneumatic muscle [], McKibben artificial muscle [], McKibben pneumatic artificial muscle [5], fluidic muscle (Festo AG, Germany), or simply (pneumatic) muscle [5]. 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 [3], [5], [], and [7]. () Mechanical Engineering Department, Division PMA, Celestijnenlaan Basically, a McKibben based artificial muscle is made up of an inner rubber tube, the so-called bladder, and a nonextendable double-helix woven sheath, the so-called braided shell. The basic working principle of a pneumatic muscle is as follows: when the rubber tube is inflated, the braided shell experiences lateral expansion, resulting in axial contracting force and the movement of the end point position of the pneumatic muscle. Due to the lateral motions, hysteresis occurs which acts as a limitation of its widespread use. 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 [], shape memory alloy [9], piezoelectric [] actuators. In a PAM, the hysteresis can be determined in an isobaric test and in an isotonic contraction test [],[5], referring to force/length hysteresis and pressure/length hysteresis respectively. The PAM hysteretic phenomenon was 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 the braided cords and the rubber bladder, and the friction between cords themselves. These causes are also listed in [], [], and [5], amongst them the friction between cords being the most significant. Inherent hysteresis in actuators increases the level of system nonlinearity, and leads to the complexity of the associated control system. Many efforts have been paid for modeling actuator hysteresis and using the developed models to compensate for the nonlinear effects, for instance see [], []. For PAM hysteresis modeling, Tondu et al. [] 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. Davis et al. [3] 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. [] described hysteresis in pleated PAM by using a Preisach modeling approach. However, pleated PAMs are different from McKibben-based PAM family. Meanwhile, Chou and Hannaford [5] gave an insight into hysteretic characteristics of a PAM, such as quasi rate independency, history dependency, but they did not really go further into modeling that hysteresis. Tri et al. [5], [] brought Chou s and Hannaford s observations to a new model of PAM hysteresis by using the Maxwell-slip approach. 3B, B3 Heverlee, Belgium. Even though several studies focused on modeling hysteresis in a single PAM, such as [], [3], and [] that were found in literature, while only [] showed the capability of capturing hysteresis for any arbitrary motion of () Department of Biosystems, Kasteelpark Arenberg 3, B3 Heverlee, Belgium. (3) School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore //$. IEEE

2 the muscle end point position and any internal muscle pressure. In a muscle pair configuration, the individual hystereses do not cancel out each other, and it apparently occurs as we can see in the pressure difference/joint angle relation [7], [] or in the torque/joint angle relation [9]. It is hard to find a reference on hysteresis modeling for a pneumatic muscle manipulator, except for [] where pressure difference/joint angle hysteresis was simulated with the assumption that hysteresis comes from the muscle diameter variation. This paper extends the work in [] to a further step to model the torque/angle hysteresis in a pneumatic muscle antagonistic system or a pneumatic muscle manipulator. The paper is organized as follows. Section II describes the experimental apparatus. Section III addresses how to extract and model the torque/angle hysteresis. Section IV presents the experimental results with extensive discussion. Section V closes with the conclusions and future work. II. EXPERIMENTAL APPARATUS MYPE-5-M5-B) was used to control the air mass flow charging and discharging the muscles. Two similar pressure sensors (SENSORTECHNICS type of PTE55DA) were used to measure the pressure in the muscles. In order to make a rotary joint, one muscle was connected to the other via a timing belt that covered a timing gear, which was mounted on an axis with two supporting bearings. Two coupling adaptors were designed to connect each end of the two muscles to the load cells (type DBBP- from BONGSHIN), which were fixed to the support. The joint axis and the load cell support were fixed to a rigid frame. The joint rotation was measured by using the incremental rotary encoder (PANASONIC type of EC-CWZC-M). An arm or a lever, which can be used to exert an external torque to the joint, was fixed to the joint axis. All I/O information from/to the plant setup was processed by a -bit data acquisition card DAQmx NI-9 from National Instruments, which was embedded in a realtime 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. At the initial position where the two pressures are equal and without external torque, the distance from the axis to the load cell support is designed such that the length of each muscle is equal to 5% of its working length (about 75 mm). Such a design allows each muscle to contract over its full operation length, i.e. from % to 5% contraction ratio or expressed in absolute length of each muscle: from 5 mm to mm. As specified in the datasheet, the muscle can operate with a working pressure range of to bar, so a mean pressure of 3 bar was selected to apply to each muscle at the initial position. In order to move the arm around, a pressure difference needs to be generated as the control input of the antagonistic system; the outputs can be the joint angle and/or the joint torque. A. Constraint torque model III. HYSTERESIS MODELING Static constraint torque (Nm) - - =- degree = degree stepwise=+5 =+ degree Fig.. Photograph of the antagonistic test setup - - stepwise=-5 = dregee Figure is the photograph of our designed apparatus; a pair of FESTO fluidic muscles (type MAS--N) was used to make up the antagonistic configuration. Even though one 5-way/3-position pneumatic proportional directional control valve can be used to control the pressure difference of the antagonistic system [], in our modeling approach, we need to control exactly the pressure in each muscle, therefore a pair of FESTO directional proportional valves (type Pressure difference (bar) Fig.. Static torque against pressure difference at different joint angles In order to find the hysteresis loop, we first introduced a new experiment setup, called the constraint setup because the movement of the arm was blocked during the measurement (see Fig. ). This measurement is used to derive the static torque model in a straightforward way. The arm 3

3 was fastened in a stationary support at different joint angles, for instance in steps of 5 degrees from the center (lever vertically downwards as in Fig ) in both directions, clockwise and counter-clockwise. At one joint angle, the pressure difference was gradually changed from the minimum to the maximum values and the other way around for several times. The maximum and the minimum of pressure difference were determined such that the contracting force in the individual muscle could not become negative. This is just because we want to obtain the hysteresis contributed by both muscles, not a single one, and to avoid a practical situation where only one muscle is active. The static torque against pressure difference relationship at different constant angles is plotted in Fig.. This relationship is quite linear so that we can adopt following linear model: Tconst a( ) p b( ) () where T const is the constraint torque, defined as the product of the timing gear radius times the force difference between F and F ; p is the pressure difference, defined as the difference between the internal pressures p and p of the two muscles (subscripts and refer to the right hand side muscle and the left hand side muscle respectively); a( ) and b( ) are the slopes and the intercepts of the straight lines at different joint angles respectively. After fitting these straight lines by using curve fitting toolbox in Matlab, a( ) and b( ) take following forms: i a( ) ci i () b( ) d Substituting () into (), we obtain : T ( i const ci ) p d (3) i where ci ( i,..,) are the coefficients of the quadratic function of the slope approximation, and d is the slope of the straight line of the intercept approximation. B. Extracted hysteresis The static torque model (3) obtained in the constraint setup already somehow looks like the models shown as Eq. in [9] or Eq. in []. However, these models have an extension, which used to describe the stick behavior of the manipulator. In our observation, the static torque against the angular displacement relationship in an isobaric condition (pressure difference stays unchanged) revealed the occurrence of hysteresis with nonlocal memory. Since the static torque model obtained in the condition of no relative motion of the muscle sheath (muscle length is blocked), no hysteresis occurred. This convinces us to formulate the static torque in the isobaric condition in the following form: T T T () isob const hys where T isob is the static torque measured in the isobaric condition and is the so-called extracted hysteresis, T hys because it can be extracted by subtracting Tisob from Tconst as follows: Thys Tisob Tconst (5) Fig. 3 shows the isobaric test results (loop type curves) together with the reconstructed static torque/joint angle relationship from the constraint model (single lines) with the same constant pressure differences, e.g. p,, bar. Applying equation (5), the extracted hysteresis is obtained and shown in Fig.. Static torque (Nm) p=- p= p= Fig. 3. Isobaric torque and constraint torque at the same pressure difference Extracted hysteresis torque (Nm) Fig.. Extracted hysteresis As briefly described in Appendix A or referring to [5][], the virgin curve can be shrunk from the extracted hysteresis loop. The extracted hysteresis (Fig.) is imaginarily bounded by a double-stretched virgin curve and a flipped-double-stretched virgin curve. This is known as an intuitive way to select representative elements for MS modeling approach. Here we intuitively chose four MSelements, and they are sufficient to reconstruct the extracted hysteresis with acceptable accuracy. Their stiffness and saturation parameters are shown in table I. Table I_Identified parameters of four representative Maxwell-slip elements. Element 3 k w

4 IV. EXPERIMENTAL RESULTS AND DISCUSSION A. Modeling achievements (c) The identified parameters of the -element MS model as. shown in table (I) were entered in the algorithm developed. based on the modeling approach shown in Fig. to simulate online the extracted hysteresis. Even though the model. parameters were determined based on the full hysteresis loop (see again Fig. ), the proposed model can capture accurately the hysteresis at different isobaric conditions, for -. instance at,, and - bar, as shown in Fig.5a. The observed small mismatch could be foreseen when realizing -. that a compromise was made to intuitively select the segments for MS modeling approach. However, the discrepancy is significant in the upper right corner and lower left corner, while the fit is good at the center. In practice, the mismatch areas occur when one muscle has a very high inside pressure while the other has a very low inside pressure so that both of them are working in a very high nonlinear region as discussed in []. However, the output joint torque predictions (Eq.) corresponding to the extracted hysteresis shown in Fig.5a are very well captured with an error (±.Nm) of about 5% to the maximum joint torque (±Nm) as shown in Fig.5c. The other advantage of - this modeling approach is that the hysteresis can be captured for any arbitrary trajectory as shown in Fig.. - (a) Model error (Nm) Fig. 5. Capture of hysteresis at arbitrary pressure differences: capture extracted hysteresis (a), prediction of joint torque with hysteresis (b), model error. Static torque (Nm) - error measured modeled Extracted hysteresis torque (Nm) Static torque (Nm) Extracted hys. loop at p=- Extracted hys. loop at p= modeled measured Extracted hys. loop at p= (b) measured at p= measured at p= measured at p=- modeled at p= modeled at p= measured at p= Fig.. Capture of hysteresis at arbitrary angular displacements B. Check for hysteresis with nonlocal memory and quasirate independency A simple test was conducted to see the nonlocal memory behavior in the manipulator hysteresis. The arm was driven with the trajectory as shown at the top of Fig.7. As we can see, on the way going around to close a big loop (see the bottom of Fig.7), the arm was driven backwards and forwards several times to create some smaller loops Following the lower half of the big loop for example, if a small loop is finished, e.g. --, then the output hysteresis continues the lower half of the big loop. This is called a nonlocal memory behavior (see also [5]). time (xms) Extracted hysteresis torque (Nm) Fig. 7. Nonlocal memory behavior of torque/angle hysteresis

5 Static torque (Nm) Fig.. Quasi-rate independency of torque/angle hysteresis The quasi rate independency characteristic of the manipulator was also tested by changing the excitation rate. Fig. shows that the hysteresis loop closed at a frequency of.9 Hz stays the same as the one closed at a rate of.9 Hz. Higher rates could be applied to compare. However, a higher rate needs a smaller excitation amplitude to ensure the pressure regulation loop remains within its bandwidth. Fig. 9 shows that for an isobaric test, for instance with p bar (the center position), during excitation of the arm the pressure difference is almost kept constant. ACKNOWLEDGMENT rate at.9 Hz rate at.9 Hz The author gratefully acknowledges the DGDC (Directorate-General for Development Cooperation), of the Belgium Government, for awarding a scholarship to the first author and for funding this research. APPENDIX A The Maxwell-slip hysteresis modeling approach is an alternative method to capture hysteresis with nonlocal memory. This method assumes that the output hysteresis is a sum of the individual hystereses of several Maxwell-slip elements, which are connected in parallel. Each Maxwellslip element has its own stiffness and saturation level (see Fig. ). The key feature of this approach is that these elements contribute to the virgin curve. Stated in other words, if the virgin curve is found, the piecewise linearization of this curve will turn out the elements of the MS-model. The following steps briefly describe how to calculate the stiffness and saturation levels of the representative Maxwell-slip elements: Maxwell-slip element H ( w, k ) Pressure difference (bar) p desired p response Maxwell-slip element Maxwell-slip element n H ( w, k) H hys _ out H n( wn, kn) time (xms) Fig. 9. Response of pressure difference during an isobaric test V. CONCLUSION The inherent torque/angle hysteresis in a pneumatic muscle manipulator has been characterized and shown to be similar to the hysteresis contributed by the individual muscles, i.e. hysteresis with nonlocal memory and quasi rate independency. This manipulator hysteresis is well captured by applying the MS modeling approach, not only at different angular displacements but also at arbitrary pressure differences. A new model of the static torque-pressure difference-joint angle relationship, the so-called constraint model was proposed and it was shown that it plays a key role for extracting the hysteresis behavior. The overall model showed a significant performance in predicting the joint torque for arbitrary arm motions. It will be very important in future control applications, for instance for using this inverse model to control the stiffness of the arm. where the i-th element is: w i H i Fig.. Maxwell-slip modeling approach. - Obtain the hysteresis loop experimentally - Shrink the upper (or lower) half of hysteresis loop to get the virgin curve. - Pick up intuitively the segments which are asymptotes to the virgin curve. - Calculate the stiffness and saturation level for each element by establishing and solving a system of equations that is directly formed on the plot coordinates. Fig. shows an example for calculating stiffness and saturation level based on a virgin curve with representative k i w i x

6 elements. In this figure, four segments with slopes K a, K b, K c, and Kd are visibly drawn such that they are asymptotes to the virgin curve. The corresponding coordinates are then determined, i.e. a, a, a3, and a in y- axis and b, b, b3, and b in x-axis respectively. The elemental stiffness k, k, k3, and k and elemental saturation level w, w, w3, and w respectively for four elements are obtained by solving equation (A.). a a 3 a a Output hysteresis K a b k K b b k Linearizing segments K c b 3 K d Fig.. Calculating stiffness and saturation based on a virgin curve with representative elements. k 3 b w Virgin curve k k k k K a / b 3 a k k3 k Kb a/( b b) k3 k Kc a3/( b3 b) k Kd a/( b b3) w/ k b w/ k b w3/ k3 b3 w / k b displacement k w w 3 w (A.) REFERENCES [] B. Tondu, P. Lopez, Modeling and Control of McKibben Artificial Muscle Robot Actuators, IEEE Control Systems Magazine, Vol., No.,, pp [] V. L. Nickel, J. Perry and A. L. Garrett, Development of Useful Function in the Severely Paralysed Hand, The Journal of Bone and Joint Surgery, Vol.5-A, No.5, July 93, pp [3] K. Inoue, Rubbertuators and applications for robots, Proceedings of the th Symposium on Robotics Research, Tokyo, 97, pp [] J. M. Winters, Braided Artificial Muscles: Mechanical Properties and Future Uses in Prosthetics/Orthetics, RESNA 3 th Conference, Washington, USA, 99, pp [5] C. P. Chou, & B. Hannaford, Measurement and modeling of McKibben pneumatic artificial muscles, IEEE Transactions on Robotics and Automation, 99, Vol., pp. 9-. [] Dick H. Plettenburg, Pneumatic Actuators: a Comparison of Energyto-Mass Ratio s, IEEE International Conference on Rehabilitation Robotics, Chicago, IL, 5, pp [7] 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 Transactions on Mechatronics,, Vol. 3, pp. -5. [] Samir Mittal and Chia-Hsiang Menq, Hysteresis Compensation in Electromagnetic Actuators Through Preisach Model Inversion, IEEE Transactions on Mechatronics,, Vol. 5, pp [9] Sushant M. Dutta and Fathi H. Ghorbel, Differential Hysteresis Modeling of a Shape Memory Alloy Wire Actuator, IEEE Transactions on Mechatronics, 5, Vol., pp [] U-Xuan Tan, Win Tun Latt, Cheng Yap Shee, Cameron N. Riviere, and Wei Tech Ang, Feedforward Controller of Ill-Conditioned Hysteresis Using Singularity-Free Prandtl-Ishlinskii Model, IEEE Transactions on Mechatronics, 9, Vol., pp [] B. Tondu, S. Ippolito, J. Guiochet, A. Daidie, A Seven-degrees-offreedom Robotarm Driven by Pneumatic Artificial Muscles for Humanoid Robots, The International Journal of Robotics Research, Vol., No., 5, pp [] Ming-Jyi Jang, Chieh-LiChen, Jie-RenLee, Modeling and control of a piezoelectric actuator driven system with asymmetric hysteresis, Journal of the Franklin Institute,Vol. 3, 9, pp [3] S. Davis, D. G. Caldwell, Braid Effects on Contractile Range and Friction Modeling in Pneumatic Muscle Actuators, The International Journal of Robotics Research, Vol. 5, No.,, pp [] Van Damme Michael, Beyl Pieter, Vanderborght Bram, Van Ham Ronald, Vanderniepen Innes, Versluys Rino, Daerden Frank, Lefeber Dirk, Modeling Hysteresis in Pleated Pneumatic Artificial Muscles, IEEE International Conference on Robotics, Automation & Mechatronics,, pp [5] Tri Vo Minh, Tegoeh Tjahjowidodo, Herman Ramon, and Hendrik Van Brussel, Non-local Memory Hysteresis in A Pneumatic Artificial Muscle, Proceedings of the 7 th Mediterranean Conf. on Control and Automation, Thessaloniki, Greece, June -, 9, pp, -5. [] Tri Vo Minh, Tegoeh Tjahjowidodo, Herman Ramon, and Hendrik Van Brussel, A new approach to modeling hysteresis in a pneumatic artificial muscle using the Maxwell-slip model, IEEE/ASME Transaction on Mechatronics, Vol. PP (99),, pp.-. [7] B. Tondu, V. Boitier, P. Lopez, Naturally compliant robot-arms actuated by McKibben artificial muscles, Proceedings of 9 IEEE- SMC Conference, San Antonio, TX, 99, pp [] P. van der Smagt, F. Groen, K. Schulten, Analysis and control of a rubbertuator arm, Biol. Cybern, Vol. 75, 99, pp. 33. [9] J. Schröder, D. Erol, K. Kawamura, and R. Dillmann, Dynamic pneumatic actuator model for a model-based torque controller, IEEE Int. Symp. On Computational Intelligence in Robotics and Automation, 3, pp [] TU Diep Cong Thanh, Kyoung Kwan Ahn, Nonlinear PID control to improve the control performance of axes pneumatic artificial muscle manipulator using neural network, Mechatronics, Vol.,, pp [] D. Cai, H. Yamaura, A robust controller for manipulator driven by artificial muscle actuator, Proceedings of the 99 IEEE international conference on control applications, 99, pp