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Mechatronics 20 (2010) 402–414
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Cascade position control of a single pneumatic artificial muscle–mass system with hysteresis compensation Tri Vo Minh a,*, Tegoeh Tjahjowidodo b, Herman Ramon c, Hendrik Van Brussel a a
Mechanical Engineering Department, Division PMA, Celestijnenlaan 300B, B3001 Heverlee, Belgium School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore c Department of Biosystems, Kasteelpark Arenberg 30, B3001 Heverlee, Belgium b
a r t i c l e
i n f o
Article history: Received 1 May 2009 Accepted 3 March 2010
Keywords: Hysteresis modeling Pneumatic artificial muscle Tracking position control
a b s t r a c t The inherent hysteresis in a pneumatic artificial muscle (PAM) makes it difficult to control accurately the position of the PAM’s end effector. This hysteresis causes energy loss and the area of the hysteresis loop is dependent on the amplitude of the motion and on the underlying causes of the hysteresis phenomenon. This means that if the hysteresis energy loss is properly compensated, a more accurate positioning would be achieved. In this paper, the pressure/length hysteresis of a single PAM is modeled by using a Maxwellslip model. The obtained model is used in the feedforward path of a cascade position control scheme, in which the inner loop is designed to cope with the nonlinearity of the pressure buildup inside the PAM, whereas the outer loop is designed to cope with the nonlinearity of the PAM dynamics itself. The experimental results show that position control of a single PAM–mass system with hysteresis compensation (HC) is effectively improved compared to a control without HC, and the control system shows high robustness to load changes. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Recently, pneumatic artificial muscles (PAMs) have many desirable characteristics, such as high power-to-weight ratios, high power-to-volume ratios, and inherent compliance and have therefore been used in a wide variety of applications in humanoid robots, prostheses and orthoses. Most of the PAMs used today are based on the McKibben artificial muscle [4,7], which mainly consists of a rubber tube, a so-called bladder, and a surrounding sheath made of two helix inextensible fibers, a so-called braided shell. This cylindrically-shaped tube is closed at both ends with caps. When pressurized, the bladder tends to increase in volume (like a balloon), however, thanks to the non-extendable outside sheath, the bladder is restricted to expand in radius but shortens axially instead. If one end of the muscle (simply called a PAM in our paper) is fixed to a stationary support, the other end will exert a contracting force to the coupled load. The pneumatic energy in the compressed air is thus transformed into mechanical work. When depressurized, the muscle returns passively. Hence, similar to a human muscle, a PAM is a unidirectional actuator (Fig. 1). The contracting force of a muscle with a certain diameter is a nonlinear function of the input pressure and the length of the muscle. During a contraction/extension cycle hysteresis inherently occurs.
* Corresponding author. Tel.: +32 (016) 32 14 44. E-mail address:
[email protected] (T.V. Minh). 0957-4158/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2010.03.001
The high nonlinearity of the contracting force model and the complex hysteresis complicate the control problem. Two approaches to deal with this problem can be observed in literature; one using simplified models, the other using advanced nonlinear control algorithms. 1.1. Model accuracy The contracting force model of the PAM is difficult to obtain precisely. Most of the developed models are only approximations of its real behavior. There are three models that have received recognition in the PAM research community. The first model, developed by Inoue, is based on an empirical approach [1]. The second one was developed by Caldwell et al., and is based on the theorem of virtual work [2]. The last one was developed by Chou and Hannaford, and is based on geometric analysis [3]. With some restrictions, the three models are quite related to each other. In order to reduce the discrepancy between the output contracting force prediction and the measurement, many investigators have been working on the extensive assessment of the model accuracy. In papers [4–7], the authors took into account the effect of the nonzero wall thickness of the PAM, the non-cylindrical shape of the end parts of the PAM, and the inherent hysteresis. The model improvements are valuable to get an insight to PAM designers, but they increase the complexity in terms of control, since the modifications involve many unmeasurable parameters. Measuring and implementing
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uncertainties in the PAM control system. To increase control robustness, many different control methods have been tried: from conventional to nonlinear PID controllers, from learning controllers to robust control algorithms [10–28]. To get an insight, these reference papers can be divided into two groups: one group is related to the control of a single muscle [10–16], while the second group is directly related to the control of an antagonistic system, in which two muscles are coupled to make up a rotary joint [17–28]. Carefully considering the control objectives of these two groups, one can see that there are some similarities between them. The pressure/length relationship in the tracking position control of the single muscle is somehow similar to the pressure difference/angular displacement control of the antagonistic system. The torque/angular displacement control of the antagonistic system is comparable to the load disturbance control of the single muscle system. The algorithms developed for controlling the single muscle are applicable to controlling the antagonistic system. In fact, the control algorithms for a single PAM developed in [10,16] were applied in [17,27] respectively, dealing with antagonistic systems. Obviously, each of the two muscles contributes hysteresis to the antagonistic system. In literature, this hysteresis was almost always un-modeled and considered as part of the uncertainties of the contracting force model. This shortcoming was proposed to be overcome by using pole placement methods [10,11,17], neural networks control [22], variable structure control or sliding mode control [23–25], or using a combination of more advanced techniques such as nonlinear PID control [14], adaptive fuzzy modelbased control [15], adaptive fuzzy logic siding mode control [26].
Fig. 1. (a) Structural layout of a PAM. (b) Description of the PAM working principle.
parameters such as the contacting surface of braiding fibers or their friction coefficient to the control scheme in [4] is an example. Consequentially, simple models, as presented in [1–3], are more frequently used in PAM control systems. 1.2. Hysteresis in PAMs It is indicated above that hysteresis is inherently present in a PAM and that it is complicated to model. The cause leading to hysteresis is assumed to be the friction, which exists between the braided shell and the bladder, between the cords of the braided shell, and the inherent hysteresis of the bladder. A few studies in literature have dealt with PAM hysteresis modeling. Tondu and Lopez [4] added the hysteresis to the contracting force model as Coulomb dry friction in order to capture not only the static but also the dynamics of the output force prediction. Due to the increased complexity of the contracting force model by accounting for hysteresis, the implementation of this model is of limited use for control purposes. Davis and Caldwell [7], tried to interpret in detail the contacting surface of braided cords which characterizes the hysteresis. Their assessment showed a good agreement between the measurement and the prediction of the static force, but also increased the complexity of the model. This work is valuable to provide an insight to PAM designers. Although Chou and Hannaford [3] did not model the hysteresis, they generalized and tested some interesting characteristics of the hysteresis such as quasi rate-independency and dependence on history (memory). 1.3. Control of PAMs The model error of the contracting force, due to the complex hysteresis and the effect of the rubber relaxation [22], introduces
Fig. 2. The single muscle–mass system (SMu).
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lengths of the PAM is found and shown in Fig. 3b. The force–pressure–length relationship resulting from this experiment is called the ‘‘isometric model”. The second setup is the ‘‘isobaric setup” as shown in Fig. 4a. The modeled muscle is stretched by another stronger muscle. During testing, the pressure in the test muscle is regulated at a constant value and different constant pressures
Fig. 3. (a) Photo of the isometric setup. (b) The contracting force is linearly proportional to the pressure activation at different contraction ratios.
Most of these studies are applied to tracking position problems. A few authors [12,16,20] consider loading effects. Hildebrandt et al. introduced the hysteresis model developed in [4] in an inverse model-based control algorithm [16], while Balasubramanian and Rattan suggested using an offset for compensating the hysteresis [15]. In this paper we introduce a new deterministic approach: the hysteresis is characterized and modeled carefully, the associated control is therefore implemented as a conventional PID controller with feedforward hysteresis compensation. This study focuses only on the single muscle (SMu) system, in which the muscle drives a vertically suspended mass, as shown in Fig. 2. This paper is organized as follows. Section 2 derives the novel contracting force model. Section 3 addresses the pressure/length hysteresis modeling. Section 4 presents the design of the cascade tracking position control. Section 5 gives experimental results and discussions. Section 6 brings the conclusion and future work. 2. The contracting force modeling In order to derive the contracting force model in a straightforward way, we need to establish two experimental setups. The first setup is the isometric setup as shown in Fig. 3a. The modeled muscle is put in the test-bed where the two ends of the muscle are fastened in two stationary supports. From this experiment, the contracting force/pressure activation relationship at different
Fig. 4. (a) Photo of the isobaric setup. (b) The contracting force is nonlinear to the contraction ratio at different pressures inside the muscle and appears to have different values during contraction or extension (hysteresis).
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are selected. The proper pressure in the stretching muscle is preselected such that the test muscle with a given inside pressure is pulled and relieved in its permissible length or contraction ratio; at the smallest length the contracting force does not become negative and at the largest length, the highest force does not rupture the muscle. The experimental results of the isobaric setup are shown in Fig. 4b. In these two setups, the test muscle is a FESTO Fluidic muscle (type MAS-20-200N) with an internal diameter of 20 mm and a length of 200 mm. The stretching muscle is a MAS-40-300N type with an internal diameter of 40 mm and a length of 300 mm. The datasheet shows that these muscles have a maximum contraction ratio of 25%. The PTE5151D1A pressure sensor from SENSORTECHNICS 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 PD6506002 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. From the isometric and isobaric test results as shown in Figs. 3 and 4b respectively, one can observe that the contracting forces in the isometric test curves are linearly proportional to the pressure activation. The slopes and intercepts of these straight lines are changing for different contraction ratios. The contracting force curves are nonlinear function of the contraction ratio at different given pressures in the isobaric test, and the hysteretic loops are clearly visible. This brings us to the following model of the contracting force (similar work is found in [8,9]):
F isob ¼ F isom þ F hys
ð1Þ
where Fisob is the measured static contracting force from the isobaric experiment as shown in Fig. 4b, Fisom is the static force from the isometric model, and Fhys is the extracted force/length hysteresis, resulting from the subtraction of Fisob from Fisom. The isometric model of the contracting force is governed by the following equation:
F isom ¼ aðeÞP þ bðeÞ
ð2Þ
Experimental results show that a(e) and b(e) take the form:
aðeÞ ¼
2 X
C i ei and bðeÞ ¼
i¼0
3 X
C j ej
2 X i¼0
C i ei þ
3 X
C j ej
F hys ¼ Fðf ðeÞ; F hys ðen ÞÞ
ð5Þ
Substituting (4) and (5) into (1), one obtains the new form of the contracting force model (6):
F isob ¼ P
2 X
C i ei þ
i¼0
3 X
C j ej þ Fðf ðeÞ; F hys ðen ÞÞ
ð6Þ
j¼0
In the SMu system, we try to control the position with respect to the pressure activation, while the force is dynamically changing due to the acceleration effect on the load mass. In other words, the control input is the input pressure to the PAM which is inferred from the desired position, and the control output is the muscle length. The complete model derived in (6) contains the quasi-dynamic term Fhys, which can be used to compensate for the hysteresis in the position tracking control. However, written in such form, the pressure is vanishing in Fhys. In fact, zero pressure activation will lead to zero motion, and as a result there is no hysteresis. In order to show the pressure explicitly in Fhys, model (6) can be rewritten in the following form:
Pisot ¼ Pisom þ Phys
ð7Þ
where Pisot is the measured pressure in the isotonic test (see Section P3 ðF isom C j ej Þ P2 j¼0i the internal pressure P taken from the iso3), P isom ¼ Ce i¼0 i
metric model (4), and Phys = F1(f(e), Fhys(en)) is the extracted pressure/length hysteresis, experimentally resulting from the difference between Pisot and Pisom. Eq. (7) is extensively rewritten as:
Pisot
P F isom 3j¼0 C j ej ¼ þ F 1 ðf ðeÞ; F hys ðen ÞÞ P2 i C e i i¼0
ð8Þ
Comparing (6) with (8), one can infer that the force/length hysteresis would be transformed to the pressure/length hysteresis. Some questions may arise here, such as: Does the pressure/length hysteresis have a similar behavior as the force/length hysteresis? Is it possible to do the conversion mathematically between them? If they cannot be converted directly, can they be modeled or not? The answers are given in the following section.
ð3Þ
j¼0
3. Pressure/length hysteresis modeling
where a(e) and b(e) are the slopes and the intercepts respectively of the straight lines in Fig. 3b, Ci, Cj are the coefficients of the polynomial function, P the internal pressure of the muscle, and e is the contraction ratio defined as the ratio of the difference between the maximum length lmax and the actual length l to the maximum l length of the muscle: e ¼ lmax lmax Substituting (3) into (2), the isometric model of the contracting force can be rewritten as follows:
F isom ¼ P
a PAM behaves like the friction in a two-object surface contact in the presliding regime. This hysteresis is well described by a hysteresis function with non-local memory. This type of hysteresis has the properties of quasi rate-independency and history dependency. The general form of this hysteresis, which is a nonlinear function F() of the virgin curve f(e) and the hysteretic force Fhys(en) at the last reversal extremum en, can be written as follows:
ð4Þ
j¼0
The isometric model described in Eq. (4) is quite similar to the one found in [27], but for us it is just an isometric part in our novel model as shown in Eq. (1). The remaining part in this model is the extracted hysteretic force, which can be mathematically described as we have shown in [9]. This study indicated that the hysteresis in
3.1. From the force/length hysteresis to the pressure/length hysteresis As discussed earlier, the pressure activation is vanishing in the hysteresis term in Eq. (6), therefore the inverse hysteresis term in Eq. (8) cannot be directly derived from (6). The pressure/length hysteresis term in (7) and (8) obviously occurs in an isotonic test. The isotonic test setup uses the same configuration as the SMu system depicted in Fig. 2. The muscle carries a certain mass and the manipulating pressure will activate the mass up and down. Different masses are tested and the set of test results for different loads is shown in Fig. 5b, which are similar to the test results in [3]. This figure shows that during moving up and down the pressure/length hysteresis inherently appears. The pressure activation in this setup, denoted as the isotonic pressure Pisot (8), not only contains the direct power conversion term Pisom but also the extra amount of pressure needed to overcome the friction during movement. This
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Fig. 5. The force/length hysteresis (in an isobaric test) leads to the pressure/length hysteresis (in an isotonic test).
Fig. 6. The pressure/length hysteresis exhibits a non-local memory characteristic.
additional pressure is the compensation term Phys and exists in the form of extracted pressure/length hysteresis. The pressure/length hysteresis therefore originates from the force/length hysteresis. This resulting hysteresis has also been explained in [27]. However, the pressure/length hysteresis can be visually explained based on the isobaric test results shown in Fig. 5a. If we draw a horizontal line crossing the force/length hysteresis loops, the intersecting points can be used to rebuild the pressure/length hysteresis loops in the isotonic test (Fig. 5b) while noting that the horizontal crossing line in the isobaric test implies a certain mass in the isotonic test . The force/length behavior has been characterized in [8,9]. Preliminary tests showing the non-local memory and the quasi rateindependency characteristics of the pressure/length hysteresis are treated in this paper. A clue for the non-local memory behavior of the pressure/length hysteresis is shown in Fig. 6. This figure shows the hysteresis loop 1-2-3-4-1 as the muscle carries 6 kg mass (equivalent to a force of 60N approximately). In order to obtain this loop, the activation pressure is selected such that during moving up and down the muscle length interval is in the permissible range, to avoid the muscle being directly stretched, and that
Fig. 7. The pressure/length characteristic.
hysteresis
exhibits
a
quasi
rate-independent
it is gradually changing so that the measurement is quasi static, to avoid the effect of mass acceleration. Following the upper half loop 1-2-3 corresponds to moving up the mass, if the tracking pressure is reversed at point 2, the system will memorize point 2 and when it goes to point 20 , meaning that the mass is moving down, the trace goes from 2 to 20 . If the tracking pressure is again reversed at point 20 , the system will memorize points 2 and 20 and when it goes back to point 2, meaning that the mass is moving up again, the trace goes from 20 back to 2. Reaching point 2, if the tracking pressure is rising, the system will follow the upper half loop 1-2-3 instead of following the dotted curve 20 -2-200 . It means that when the tracking pressure finishes 2-20 -2 loop, the system will forget memory point 2. This behavior is called ‘‘non-local memory”. The same behavior is found when following the lower half loop 3-4-1 corresponding to moving down the mass. The rate independent characteristic of the hysteresis loop is shown in Fig. 7, in which the 0.2 Hz loop is compared with the 0.05 Hz loop. Higher loop rates are difficult to test due to the acceleration effect on the carrying mass. The two main characteristics of the pressure/length hysteresis resulting from the
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above analysis bring us to the conclusion that this kind of hysteresis can be completely described by the hysteresis function with non-local memory, which is the same solution as explained in [9,30,32]. Consequently, the force/length hysteresis term in Eq. (6) is hard to convert to the pressure/length hysteresis term in Eq. (8), but it can be rewritten in another form such that it can be mathematically modeled as:
407
– Extracting the pressure/length hysteresis loop experimentally. – Shrinking the upper (or lower) half of extracted hysteresis loop to get the virgin curve. – Picking up intuitively the segments which represent the Maxwell-slip elements, a kind of piecewise linearization of the virgin curve. – Identifying the representing parameters for those selected elements.
When drawing a number of horizontal lines from the bottom to the top of Fig. 5 and aligning vertically the starting point and the final point when these lines are crossing through the force/length loops, one can obtain the displacement intervals at different loads. The heavier the loads the shorter the intervals and this agrees with the experimental results shown in Fig. 8. In this figure, the looptype curves are plotted from the isotonic experiment, but the single-line curves are rebuilt from the isometric model. At the same load a loop-type curve can be subtracted from a single-line curve in order to obtain the pressure/length hysteresis loop, namely extracted hysteresis. In order to reach the extreme lengths for modeling the pressure/length hysteresis, one can chose the isotonic loop corresponding to the lighter mass since the interval in such selection should spread out over the working length of the muscle. However, in this way the extracted hysteresis cannot accommodate the effect of the full-ranged loads, which are in forms of the forces that are vanishing in the model as discussed earlier. Bearing this in mind, a number of different-load hysteresis extractions is carried out and shown in Fig. 9. We have just distinguished how to obtain the extracted pressure/length hysteresis loops which accommodate the load effect. The remaining steps in the modeling procedure described above are intensively exploited based on what has been seen in Fig. 9 and shown in Fig. 10. The pressure/length hysteresis loops at different carrying loads are plotted in Fig. 10. These loops are bounded by the outer dottedline loop. This bound is assumed to be a symmetric loop of the double-stretched virgin curve and the flipped-double-stretched virgin curve [8,9]. When the pressure–velocity is positive, the virgin curve can be shrunk from the double-stretched virgin curve, while the virgin curve can be shrunk from the flipped-double-stretched virgin curve when the pressure velocity is negative. After the virgin curve is extracted (the dashed line), the piecewise linearization is applied by intuitively picking up the representative Maxwell elements [9,29,31]. As presented in [32], four representative Maxwell elements are sufficient to capture and simulate the hysteresis behavior. These elements are shown as asymptotic segments to the virgin curve. Based on the coordinates of these segments, one can calculate the representative parameters of each element by solving Eq. (11):
Fig. 8. The isotonic test (loop-type curve) at different loads compared to the isometric model (single-line curve).
Fig. 9. The extracted pressure/length hysteresis at different loads.
Phys ¼ PðpðeÞ; Phys ðen ÞÞ
ð9Þ
where P() is the nonlinear function of pressure/length hysteresis prediction, p(e) is pressure hysteretic function, namely the virgin curve, and Phys(en) is the hysteretic pressure at the last reversal extremum en. Eq. (8) now takes the new form:
Pisot
P F isom 3j¼0 C j ej ¼ þ PðpðeÞ; P hys ðen ÞÞ P2 i i¼0 C i e
ð10Þ
Comparing (8) and (10), one obtains:
Phys ¼ PðpðeÞ; P hys ðen ÞÞ ¼ F 1 ðf ðeÞ; F hys ðen ÞÞ Hence there are two forms of the pressure/length hysteresis, and expressed in the form P(p(e), Phys(en)) this hysteresis can be modeled. However, the hysteresis force is in turn vanishing in this new form. 3.2. Modeling the pressure/length hysteresis The modeling work of the pressure/length hysteresis in this paper resembles the work of [9,31]. It is briefly described in the following steps:
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Fig. 10. Modeling of the extracted hysteresis based on the virgin curve of the bounded loop.
Table 1 The identified parameters of the four representative Maxwell-slip elements. Element
1
2
3
4
k w
10 0.05
7.5 0.075
1.6 0.056
0.9 0.081
k1 þ k2 þ k3 þ k4 ¼ K a ¼ a1 =b1 k2 þ k3 þ k4 ¼ K b ¼ a2 =ðb2 b1 Þ k3 þ k4 ¼ K c ¼ a3 =ðb3 b2 Þ k4 ¼ K d ¼ a4 =ðb4 b3 Þ w1 =k1 ¼ b1
ð11Þ
w2 =k2 ¼ b2 w3 =k3 ¼ b3 w4 =k4 ¼ b4 where ki, wi (i, j = 1, . . . , 4) are the stiffness and the saturation values of the elements respectively and ai, bj (i, j = 1, . . . , 4) are the coordinates directly measured on the graph. The identified parameters are given in Table 1 and the scheme of the prediction of the output hysteretic pressure is shown in Fig 11. This hysteresis model is applied in the feedforward path of the control scheme that will be analyzed in the next section.
the position in case of controlling an antagonistic system [19]. This control scheme is suitable for decreasing only the steady state error, but not sufficient for handling the oscillation in the transient response, because the desired position is obtained after the controlled torque is attenuated to zero. In case of controlling the SMu system, Hildebrandt et al. proposed to use the cascaded scheme with the inverse model-based control, in which the inner loop is designed to control the force and the outer loop is designed to control the position [16]. This study also shows an improvement of the steady state error, but oscillations occurred during a step change of the load. The inverse model-based control, however, requires precise models of the system elements such as the control valve, the muscle, etc., which are difficult to obtain due to difficulty in air leakage modeling, the asymmetry in the charging/discharging air flow, and particularly the hysteresis in the PAM. A combination of a feedforward controller, which is used to linearize the nonlinearity of hysteresis behavior, and a PID controller resulted in a significant improvement in tracking control of some other actuators [33,34], but it was rarely implemented in tracking control of PAM actuators. In this paper we propose a novel control scheme to deal with the tracking position control problem of the SMu system as shown in Fig. 12. In this control scheme, the inner loop is designed to cope with the building-up pressure problem and the outer loop with the feedforward path of hysteresis compensation is designed to cope with the highly nonlinear dynamics of the muscle.
4. Tracking position control of the SMu system 4.1. The inner loop Cascade control is a traditional control strategy to be applied to the PAM system. Tsagarakis and Caldwell designed the inner loop to guarantee the torque while the outer loop is used to control
Assume that during charging and discharging of the air, the process is in-between the isothermal and adiabatic states. The
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Fig. 11. Description of the predicted pressure output: four Maxwell-slip elements are intuitively selected, the output prediction of the extracted hysteresis pressure Phys is the sum of the individually contributing outputs Phys1,. . .,4 of these elements.
Fig. 12. The scheme of the cascade position control of the SMu system.
relationship of the mass of air, the muscle volume and the internal pressure is governed by the polytropic gas law:
c V P ¼ const: m
ð12Þ
Taking the total differential of Eq. (12), one can get:
C n en
ð15Þ
n¼0
where Cn are the coefficients of the polynomial function. The mass flow rate through a non-ideal nozzle is governed by the following equation:
Pd 6 b ðsonic flowÞ Pu
ð16aÞ
and
For an ideal gas, we have also:
ð13Þ
Substituting (13) into (12), the building-up pressure can be described by the following equation:
c _ _ P_ ¼ ðmRT PVÞ V
3 X
sffiffiffiffiffi T0 _ ¼ P u C q0 if m T1
_ _ þ cPV_ ¼ c m PV PV m
PV ¼ mRT
V¼
ð14Þ
where P is the pressure inside the muscle, V the volume of the muscle, m is the mass of air inside the muscle, c the polytropic exponent, R the universal gas constant, and T is the air temperature inside the muscle. The muscle volume varies during contraction/extension [27], thus one can obtain the volume as a function of the contraction ratio as follows:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiv !2 u Pd b T0u t _ ¼ P u C q0 1 Pu if m 1b T1
Pd b ðsubsonic flowÞ Pu ð16bÞ
where Pu, Pd are the upstream and downstream pressure respec_ is mass flow rate through the tively, b is critical pressure ratio, m nozzle, C the flow conductance, q0, T0 the air density and air temperature respectively at standard conditions, and T1 is the air supply temperature. Eq. (16a) and (16b) can be rewritten in short form as follows:
: 1 Pd m ¼ Pu C pffiffiffiffiffi w Pu T1
ð17Þ
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where w PPud is called the flow function, which is dependent only on the pressure ratio. In order to regulate and manipulate the pressure inside the muscle by using the proportional directional control valve, the flow function is frequently switching between charging and discharging and the response of the mass flow rates are totally different between these two processes. That is because during charging Pu = Ps and Pd = P, whereas during discharging Pu = P and Pd = P0 (Ps, P0 are the supply pressure and the atmospheric pressure respectively). A schematic drawing of the building-up pressure is depicted in Fig. 13 in order to explain graphically the relationships in Eqs. (14), (15), and (17). The building-up pressure is not only dependent on the nonlinearity and switching of the flow function, but also on the muscle volume variation, the air temperature, the polytropic exponent and the leakage through the control valve as well. These dependencies are difficult to handle, the ‘PI’ controller is therefore introduced to close the loop. The ‘P’ control action is sufficient for
controlling such a first order system (Eq. (14)). The ‘I’ control action is however needed to eliminate the steady state error due to the leakage. The ‘P’ and ‘I’ control gains are obtained by trial and error.
4.2. The outer loop The dynamics of the muscle are described by Eq. (10), which consists of the direct conversion part (static part) and the hysteresis part (quasi-dynamic part). Both are modeled, but the latter is dynamically changing according not only to the current position but also to the history of the last position reversal. If a certain amount of pressure needed to overcome the hysteresis friction is dynamically compensated, the position control of the SMu system will not experience the oscillation in the transient state since the position is very sensitive to the pressure (showing in Fig. 5b that there are many possible pressure levels at the same position).
Fig. 13. Illustration of the high nonlinearity of the building-up pressure, which is dependent on the switching and nonlinear flow function, volume variation, un-modeled air leakage and the surrounding environment conditions as well.
Fig. 14. The muscle dynamics are online compensated by computing the hysteresis and feeding forward in the outer loop.
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Fig. 15. A measured pressure/length hysteresis is captured by the developed model.
The pressure hysteresis compensation (HC) is therefore placed in the feedforward path to reduce the control effort of the PI controller, which is used to close the position loop (Fig. 14). The desired position or contraction ratio ed will be converted to the required pressure Preq in order to serve the inner loop. 5. Results and discussions Two main nonlinear problems in the SMu system have been analyzed: the nonlinearity of the building-up pressure and the nonlinearity of the muscle dynamics. For solving the former problem, a PI controller is sufficient to regulate and track the required pressure for controlling the muscle position, because (i) the building-up pressure is governed by a first-order behavior and (ii) the increasing/decreasing pressure corresponding respectively to the shortening/elongating muscle adapts the damping of the pressure control system thanks to the volume damper (Eq. (15)). By trial
Fig. 16. The effectiveness of the hysteresis compensation in the SMu position control.
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and error, the parameters of the proportional gain k = 3 and the integral time constant Ti = 0.003 s for this controller are selected and kept constant during the tests. For solving the latter problem, the position loop is closed and a similar PI controller is used. As usual, the ‘I’ action is used to eliminate the steady state error. The parameters of the proportional gain k = 0.03 and the integral time constant Ti = 0.0009 s for this controller are selected and also kept constant during the tests. However, due to the pressure/ length hysteresis, at the same position or contraction ratio there are many possible pressures to reach this position. Therefore, when applying a step change in the position, the SMu system easily oscillates around that new position. Fig. 15 shows the measured hysteresis pressure which is captured at different positions or contraction ratios by the developed model. The hysteresis pressure prediction against the position is fed forward to the outer loop of the SMu control system and gives a proper value to the total pressure needed to reach the new position. The controller thus needs less effort to correct for the hysteresis instead of using its effort to correct the model discrepancy. The effectiveness of the hysteresis compensation in the SMu position control can be seen in Fig. 16. The SMu system carries the 6 kg mass. A 0.2 Hz square wave is applied to manipulate the muscle around the equilibrium position of 185 mm length with an amplitude of 5 mm. When the hysteresis compensation is switched off (without HC) (the top), the transient response becomes oscillating due to the ‘sponginess’ [2] of the pressure, as can be seen in the pressure loop response (the bottom). When the hysteresis compensation is on (with HC), the new position is smoothly reached. An additional amount of extra pressure given by the online model-based computation can also be seen in the pressure control loop. According to Repperger et al. [12], there exist two different dynamics for contraction or inflation and extension or deflation in such a SMu system, which are dependent on the position. The next test therefore aims at investigating how the hysteresis model is adequately adapted to the different equilibrium positions along the working range of the muscle. In fact, the input of the position control system is kept the same as in the previous test with the 0.2 Hz square wave. Three different equilibrium positions 165 mm, 175 mm, and 185 mm are chosen. At each equilibrium position, four different excitation amplitudes 0.5 mm, 1 mm, 2 mm, and 5 mm are applied. The load is changed to 24 kg. The test results are plotted in a 4 3 array as shown in Fig. 17. At the bottom of the figure, the muscle shapes are put close to the equilibrium position in order to illustrate the volume change. At the right side of the figure, different amplitudes are indicated. For the first two waves approximately in each subplot the HC is switched off, and at rest the HC is active. With such an arrangement, it can be observed that: – The HC takes effective action at any equilibrium position and with any excitation amplitude. – At the same equilibrium position, the higher the excitation amplitude, the less oscillating the position response (follow each column, top to bottom). This is due to the volume damping and the flow friction. – At the same excitation amplitude, the higher the equilibrium position, the more oscillating the position response (follow each row, left to right). This is due to the volume damping against the equilibrium position. As shown at the bottom of the figure, the higher the equilibrium, the smaller the volume of the muscle. – At the highest equilibrium position, the excitation towards shortening of the muscle length, the position response is more oscillating than that of the excitation towards elongating of the muscle length. This is due to the asymmetry of the flow function, which is more visible when the volume damping is rather small (the most upper right subplot).
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Fig. 17. Step responses of the SMu position control under a square wave of 0.2 Hz. The test set is done with the same load, but different excitation amplitudes and different equilibrium positions.
Fig. 18. Step responses of the SMu position control under a square wave of 0.2 Hz. The tests are performed with the different loads, but with the same excitation amplitude and equilibrium position. The bottom plots show the pressure loop responses corresponding to each load.
Consequentially, the highest equilibrium position is the most sensitive and the smaller the excitation amplitude, the more oscillating the position response will be. This forms the basis for performing the next test, which aims at investigating how the hysteresis model is adequately adapted to the different loads at the most sensitive equilibrium position and amplitude. In fact, three different loads 6 kg, 24 kg, and 50 kg are selected for the test. The equilibrium is at 185 mm and the excitation amplitude is 2 mm. The wave type and the wave frequency are kept unchanged. The test results are plotted and shown in Fig. 18, in which the first row shows the position response and the second row shows the response of the pressure loop. From the left to the right, one can see that the heavier the load, the less oscillat-
ing the position response. This is due to the decrease of the natural frequency of the mass–spring system. It is more convincing if we look at the pressure loop response. A heavier load needs a higher pressure. A higher pressure will not only increase flow damping due to flow friction but also the muscle stiffness due to the relationship in Eq. (6). The control system with activated HC performs an adequate action for different loads. However, the effect of the asymmetry of the flow function is not removed after the HC takes action. For example, in the case of the 6 kg load, the manipulating pressure interval of about 0.8 to 1.2 bar is needed to move around the equilibrium position. For moving up the mass, the charging rate of air in this pressure interval goes very fast because the flow is in the sonic state. Sequentially,
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than the charging process. The reason for this is related to the asymmetry of the flow function and its switching property. For the details, damping of the flow can be illustrated in Fig. 20; (a) the control signal is applied such that the valve is fully opened for charging and discharging, (b) the corresponding pressures are built up in the muscle, (c) the curves of the building-up pressure for charging and discharging are put at the same coordinate to compare, and (d) the rate of charging and discharging are shown a part form each other. The charging rate runs very high in a short time after which it saturates, while the discharging rate goes down gradually. In summary, we can state that for all changes of the load and equilibrium position, the SMu position control with the hysteresis compensation has shown an efficient performance. The effects of the asymmetric flow cannot be removed. However, these effects on the position control system are not significant, and they come from the inner pressure control loop, where the controller parameters are kept constant for all tests. Fig. 19. Two ‘sensitive’ equilibrium positions are selected to test for the asymmetric effect of the building-up pressure.
6. Conclusion the small overshoot in the position response still occurs when the muscle is shortened. These overshoots occur not only at a high rate of charging but also at a high rate of discharging. Fig. 19 shows the overshoot in both cases when two different loads are examined. The lighter load (6 kg) is used to examine the high rate of charging. The heavier load (50 kg) is used to examine the high rate of discharging. The high rate of discharging happens when the SMu system carries a heavy load and elongates from the shortest equilibrium position. This is why the equilibrium position of the heavy load is at 175 mm, whereas the equilibrium position of the light load is at 185 mm, which is based on the previous sensitivity analysis. The overshoots, when shortening and elongating the muscle corresponding to the lighter and heavier loads, are visibly seen in Fig 19, and indicated that the discharging process is more damped
This paper addressed the difficult problem of modeling hysteresis of PAMs and the use of the resulting models in accurate position control of single PAM systems. This hysteresis is the pressure/ length hysteresis that features a rate-independent and history dependent characteristics, and is well described by using the Maxwell-slip model. The developed model has been implemented into the cascade PI–PI control architecture. The overall control system with the feedforward hysteresis compensation has shown a consistent performance regardless of the choice of equilibrium position and load changes. By using conventional PI controllers only, together with the simple but deterministic hysteresis compensation, the tracking position control of a SMu system is effectively improved. These results inspire us to apply PAMs in the development of humanlike robots.
Fig. 20. The asymmetry of the building-up pressure during charge and discharge.
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Acknowledgment The author gratefully acknowledges the DGDC (DirectorateGeneral for Development Cooperation), of the Belgium Government, for awarding a scholarship to the first author and for funding this research.
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