Position control of a single pneumatic artificial muscle with hysteresis compensation based on modified Prandtl

Abstract

Background:

High-performance position control of pneumatic artificial muscles is limited by their inherent nonlinearity and hysteresis.

Objective:

This study aims to model the length/pressure hysteresis of a single pneumatic artificial muscle and to realize its accurate position tracking control with forward hysteresis compensation.

Methods:

The classical Prandtl–Ishlinskii model is widely used in hysteresis modelling and compensation. But it is only effective for symmetric hysteresis. Therefore, a modified Prandtl–Ishlinskii model is built to characterize the asymmetric length/pressure hysteresis of a single pneumatic artificial muscle, by replacing the classical play operators with two more flexible elementary operators to independently describe the ascending branch and descending branch of hysteresis loops. On the basis, a position tracking controller, which is composed of cascade forward hysteresis compensation and simple proportional pressure controller, is designed for the pneumatic artificial muscle.

Results:

Experiment results show that the MPI model can reproduce the length/pressure hysteresis of the pneumatic artificial muscle, and the proposed controller for the pneumatic artificial muscle can track the reference position signals with high accuracy.

Conclusion:

By modelling the length/pressure hysteresis with the modified Prandtl–Ishlinskii model and using its inversion for compensation, precise position control of a single pneumatic artificial muscle is achieved.

Keywords

Pneumatic artificial muscle position control hysteresis compensation modified Prandtl–Ishlinskii model

1. Introduction

Pneumatic artificial muscle (PAM), also referred to as rubbertuator [1], McKibben artificial muscle [2], braided pneumatic actuator [3], artificial muscle actuator [4], or other synonymous names in the literature, is a relatively new type of compliant pneumatic actuator [5]. Originally invented by the American physician Joseph L. McKibben in the 1950s, PAM has aroused extensive interest and attention of researchers [6].

In contrast to traditional actuators including motors, hydraulic actuators and pneumatic cylinders, PAM possesses many desirable and unique advantages, such as high power-to-weight ratio, low cost, and inherent compliance [7,8]. These characteristics allow PAM to better simulate the behavior of biological muscles, resulting prosperous applications in bionic robots [9–12], rehabilitation robots [13,14], prostheses and orthoses [15,16], and some other areas where smooth and natural movement, and safe human-machine interaction are required. However, PAM exhibits strong non-linear and hysteretic behavior, which increases the difficulties of achieving high-performance position control. To overcome these challenges, two approaches are adopted mostly in existing literature: one is model-based advanced control algorithms, and the other is forward hysteresis compensation.

In the first approach, simplified mathematical models are built to describe the static and/or dynamic characteristics of PAM [17]. The most popular used models of PAM include geometrical models [18], phenomenological models [19] and empirical models [20]. Based on the analytic models, different position tracking controllers are applied on PAM: from nonlinear PID controller [21] to fuzzy controller [22], from neural network controller [23] to adaptive robust controller [24], from sliding mode controller [25,26] to hybrid controller [27–29]. These intelligent controllers can compensate the model errors and uncertainties, and can improve position tracking performance. However, they have a drawback of dependence on model accuracy. What’s more, their applications in practice are affected by the high complexity and computational load.

In the second approach, the hysteretic behavior of PAM is measured through experiments and characterized by a hysteresis model. Then the inverse hysteresis model is applied as a forward compensation term of the position controller for PAM. Up to present, there are only a few implementations of hysteresis models in the position control of PAM. Specifically, Tri Vo-Minh et al. modeled the pressure/length hysteresis of a single PAM using the Maxwell-slip model [30,31]. The classical Preisach model [32–34] and the stop model [35] were adopted by researchers to describe the static hysteresis behavior of a manipulator driven by antagonistic PAMs. With feedforward hysteresis compensation, the PAM was controlled by conventional PID controller. But the obtained results only met with limited success because the above hysteresis models could not reproduce the hysteresis with sufficient accuracy.

The objective of this research is twofold: to model the length/pressure hysteresis of PAM with a more accurate model, and furthermore, to realize position tracking control of a single PAM with forward hysteresis compensation based on the hysteresis model. In order to achieve this, first of all, the length/pressure hysteresis of PAM is measured by an isotonic test, which is found to be asymmetric. The classical Prandtl–Ishlinskii model is a powerful tool in hysteresis modelling, but only works for symmetric hysteresis. Thus a modified Prandtl–Ishlinskii (MPI) model is then proposed to characterize the asymmetric length/pressure hysteresis. This model utilizes two asymmetric operators to describe the ascending branch and descending branch of hysteresis loops independently. And finally an open-loop position controller composed of cascade hysteresis compensator and pressure controller is designed for a single PAM. Comparisons between the model simulated and experimentally measured hysteresis loops show that the MPI model can accurately reproduce the length/pressure hysteresis of the PAM. And owing to the high accuracy of the MPI model, the hysteresis of the PAM can be well compensated, enabling the proposed control scheme to achieve quite effective position tracking, which is verified by experiment results.

2. Methods

2.1. Measurement of length-pressure hysteresis of PAM

PAM is an elastic actuator with similar characteristics to human muscles, which can convert pneumatic power to mechanical power. To date, commercially available PAMs come from three dominant providers: Bridgestone located in Japan, Shadow Robot Company in UK and Festo Corporation in Germany. The specific construction of PAMs may be slightly different from producer to producer, but their basic structure and working principle are alike.

Basically, a PAM consists of two main components: a silicon rubber tube which functions as a bladder, and a braided fiber mesh sleeve which surrounds the bladder. The inner bladder is sealed at both ends in such a way that compressed air can be supplied to or exhausted from the bladder via a gas inlet/outlet at one end. When the bladder is inflated, its volume increases. Consequently, the surrounding sleeve generates contraction in the axial direction because it cannot be extended laterally. Whereas, the bladder restores to its original state when deflated. The value of axial contraction is dependent on the internal pressure and the load applied on the muscle. An example of PAMs manufactured by Festo is shown in Fig. 1.

Fig. 1.

Pneumatic artificial muscle: (a) Initial condition; (b) Pressurized condition.

Due to the elasticity of the muscle materials and the frictional effects between the inner tube and outer sleeve, the contraction of PAM in the pressurizing and depressurizing processes exhibits a hysteretic behavior. This hysteretic behavior, termed as length/pressure hysteresis, increases system nonlinearity and prohibits precise position tracking control of a single PAM.

The length/pressure relationship of PAM can be measured through an isotonic test by applying a certain quasi-static pressure signal on the muscle and recording its contraction. Figure 2 illustrates the schematic experimental apparatus. The PAM is fixed on the frame at the upper end, with a certain load hanging at the other end. The pressure inside the PAM is adjusted by a proportional pressure regulator. A pressure transducer and a linear displacement transducer are used to measure the internal pressure and the contraction of the PAM respectively. Considering that the maximum operating pressure of the PAM is 0.6 MPa, a triangle waveform pressure signal with varying amplitudes, i.e. 0.5 MPa, 0.45 MPa, 0.4 MPa, 0.35 MPa, and 0.3 MPa is supplied to the PAM. Plotting contraction versus pressure, the length/pressure hysteresis loops including both major loop and minor loops are readily available, as presented in Fig. 3.

Fig. 2.

Schematic experimental apparatus.

Fig. 3.

Measured length/pressure hysteresis loops.

2.2. Modelling length-pressure hysteresis with modified Prandtl–Ishlinskii model

The classical Prandtl–Ishlinskii hysteresis model is widely used for hysteresis modelling and compensation. It uses the summation of a finite number of weighted backlash operators with different thresholds to characterize hysteresis behavior in the continuous space [36,37]. But it only works well for symmetric hysteresis, unable to model the length/pressure hysteresis of PAM which is shown to be asymmetric. Therefore, a MPI model is proposed based on the classical Prandtl–Ishlinskii model. This model utilizes two modified asymmetric play operators to describe the ascending branch and descending branch of the hysteresis loops respectively.

Analytically, suppose $C_{m}$ represents the space of the piecewise monotone and continuous functions, and $0 = t_{0} < t_{1} < \dots < t_{N} = t_{E}$ are subintervals in $[0, t_{E}]$ . For any piecewise monotone input function $u (t) \in C_{m} [0, t_{E}] \to [0, 1]$ , and a given threshold value $r_{i}$ which satisfies $0 ⩽ r_{i} < 1$ , the elementary play operator of the ascending branch is defined as: $\begin{matrix} (1) & H_{r_{i}}^{a} [u, w_{0}] (t) = \{\begin{matrix} w_{i - 1} & Δ u ⩾ 0, u_{i} ⩽ r_{i} \\ \frac{1 + r_{i}}{1 - r_{i}} (u_{i} - r_{i}) & Δ u ⩾ 0, r_{i} < u_{i} ⩽ \frac{1 + r_{i}}{2} \\ u_{i} & Δ u ⩾ 0, u_{i} > \frac{1 + r_{i}}{2} \\ u_{i} & Δ u < 0 \end{matrix} \end{matrix}$

Similarly, the elementary play operator of the descending branch is defined as: $\begin{matrix} (2) & H_{r i}^{d} [u, w_{0}] (t) = \{\begin{matrix} f (u_{m}) & Δ u < 0, u_{i} ⩾ 1 - r_{i} \\ \frac{1 + r_{i}}{1 - r_{i}} u_{i} - r_{i} & Δ u < 0, \frac{1 - r_{i}}{2} ⩽ u_{i} < 1 - r_{i} \\ u_{i} & Δ u < 0, u_{i} < \frac{1 - r_{i}}{2} \\ u_{i} & Δ u ⩾ 0 \end{matrix} \end{matrix}$

In the above equation, $f (u_{m})$ is the function of the dominant extrema $u_{m}$ of the input, which is expressed as a fifth degree polynomial with the following form: $\begin{matrix} (3) & f (u_{m}) = c_{0} \cdot u_{m}^{5} + c_{1} \cdot u_{m}^{4} + c_{2} \cdot u_{m}^{3} + c_{3} \cdot u_{m}^{2} + c_{4} \cdot u_{m} + c_{5} \end{matrix}$

The input-output relationships of the play operators of the ascending branch and of the descending branch are presented in Figs 4 and 5 respectively.

Fig. 4.

Play operator of the ascending branch.

Fig. 5.

Play operator of the descending branch.

Calculating the weighted sum of the two play operators, the output of the MPI hysteresis model can be written as: $\begin{matrix} (4) & y (t) = p u (t) + \sum_{i = 1}^{n} q_{i}^{a} H_{r i}^{a} [u, w_{0}] (t) + \sum_{i = 1}^{n} q_{i}^{d} H_{r i}^{d} [u, w_{0}] (t) \end{matrix}$ where p is a constant parameter, n is the number of play operators, $q_{i}^{a}$ is the density function of the play operator of the ascending branch, and $q_{i}^{d}$ is the density function of the play operator of the descending branch.

2.3. Position control scheme for the PAM based on MPI model

In this paper, in order to compensate the length/pressure hysteresis, a position controller with forward hysteresis compensation is designed for a single PAM based on the previously proposed MPI model. The basic idea of hysteresis compensation is simple, that is, putting the inverse hysteresis model in cascade with the physical hysteresis system. Thereupon, the system hysteresis and its inversion can cancel out each other, achieving a pseudo-linear system.

The block diagram of the proposed control system is shown in Fig. 6, which corresponds to a very simple open-loop controller. The control scheme works as follows. The desired contraction of the PAM denoted as $Δ L_{d}$ is input to the inverse hysteresis model. The inverse model will generate the desired control pressure signal $P_{d}$ which is then fed to the pressure controller. The pressure controller, which is simply set to be a proportional controller, regulates the pressure inside the PAM through a proportional pressure regulator by sending varying voltages. Theoretically, the output of the PAM, i.e. the measured contraction $Δ L_{m}$ should follow the reference input.

Fig. 6.

The position control scheme of a single pneumatic artificial muscle.

3. Results

3.1. Identification and verification of the MPI model

The parameters of the MPI model can be identified using recursive least mean square algorithm. Ensured by the iteration law, the parameters will converge to stable values within a few steps of iteration. In the MPI model, the number of play operators determines the accuracy of hysteresis modelling. More play operators will guarantee higher accuracy but demand heavier computation. Thus, an appropriate number of play operators is chosen to be 20. Table 1 presents the parameters of the MPI model including the predefined threshold values and the identified weightings of the play operators. Figure 7 shows the comparisons between the measured and simulated length/pressure hysteresis loops. It is fairly easy to see that the MPI model can predict not only the major hysteresis loop, but also minor loops very accurately.

3.2. Experiment results of position control for PAM

The proposed position control scheme is applied on the physical experimental apparatus previously depicted in Fig. 2 to evaluate its performance. In the experiments, two of the most commonly used waveforms are adopted as reference contraction signals: one is a triangle waveform signal and the other is a sinusoidal signal, illustrated in Fig. 8. What’s more, in order to better test the control performance over both major hysteresis loop and minor loops, the two signals have variable amplitudes, namely 40 mm, 35 mm and 30 mm. Figures 9 and 10 present the desired and measured contractions as well as the tracking errors in the cases of the triangle waveform input, and the sinusoidal input respectively. In the first case, the maximal tracking error is 1.29 mm. And in the second case, the maximal tracking error is 1.38 mm.

Table 1
Parameters of the modified Prandtl–Ishlinskii hysteresis model

$r_{i}$ $q_{i}^{a}$ $q_{i}^{d}$ $r_{i}$ $q_{i}^{a}$ $q_{i}^{d}$ $r_{i}$ $q_{i}^{a}$ $q_{i}^{d}$ $r_{i}$ $q_{i}^{a}$ $q_{i}^{d}$

0 −0.0008 −0.0008 0.25 0.1038 0.0468 0.5 −0.0262 0.0521 0.75 −0.0162 0.0650

0.05 0.6265 0.0429 0.3 −0.0639 0.0180 0.55 0.0069 0.0577 0.8 −0.0056 0.0210

0.1 0.2471 0.0335 0.35 −0.1425 0.0517 0.6 0.0100 0.0673 0.85 −0.0458 0.0109

0.15 0.2716 0.0267 0.4 −0.0693 0.0364 0.65 −0.0230 0.0736 0.9 −0.0118 0.0190

0.2 0.2020 0.0346 0.45 −0.1309 0.0517 0.7 −0.0927 0.0677 0.95 −0.0440 −0.0152

$r_{i}$	$q_{i}^{a}$	$q_{i}^{d}$	$r_{i}$	$q_{i}^{a}$	$q_{i}^{d}$	$r_{i}$	$q_{i}^{a}$	$q_{i}^{d}$	$r_{i}$	$q_{i}^{a}$	$q_{i}^{d}$
0	−0.0008	−0.0008	0.25	0.1038	0.0468	0.5	−0.0262	0.0521	0.75	−0.0162	0.0650
0.05	0.6265	0.0429	0.3	−0.0639	0.0180	0.55	0.0069	0.0577	0.8	−0.0056	0.0210
0.1	0.2471	0.0335	0.35	−0.1425	0.0517	0.6	0.0100	0.0673	0.85	−0.0458	0.0109
0.15	0.2716	0.0267	0.4	−0.0693	0.0364	0.65	−0.0230	0.0736	0.9	−0.0118	0.0190
0.2	0.2020	0.0346	0.45	−0.1309	0.0517	0.7	−0.0927	0.0677	0.95	−0.0440	−0.0152

Fig. 7.

Modelling result of the MPI model.

Fig. 8.

Desired contraction signals: (a) Triangle waveform signal; (b) Sinusoidal signal.

Fig. 9.

Experiment results of position control in case of triangle waveform signal: (a) Comparison between desired contraction and measured contraction; (b) Tracking error.

Fig. 10.

Experiment results of position control in case of sinusoidal signal: (a) Comparison between desired contraction and measured contraction; (b) Tracking error.

4. Discussion and conclusion

Modelling and compensation of hysteresis behavior has always been a complex problem. The inherent length/pressure hysteresis in the PAM makes it difficult to achieve precise position control. If the hysteresis can be accurately modeled and properly compensated, more precise positioning will be achieved. In this paper, an effective solution to the problem is provided. To begin with, the length/pressure hysteresis of a single PAM is measured and then modeled using a MPI model, by adopting two independent operators to describe the ascending branch and descending branch of hysteresis loops. After that, the inversion of the hysteresis model is used for forward hysteresis compensation in a cascade position controller for a single PAM. Although the controller simply acts as an open-loop proportional algorithm, it exhibits sufficient accuracy in position tracking in the experiments.

The performance of the proposed control scheme is compared with those from other researches. Choi controlled the PAM according to the relation between contraction length and inside pressure with maximum 3 mm errors [38]. Liu adopted discrete sliding mode algorithm for PAM to track a sinusoidal signal, and the biggest error was 2.25 mm [26]. Fan designed a BP neural network tuned PID controller for position tracking of PAM, and the maximal error for a sinusoidal wave was 1.6 mm [29]. On one hand, these comparisons show that the proposed control scheme has benefits in simpler form and higher accuracy. On the other hand, the MPI model is validated to be effective in hysteresis modelling and compensation. Since the MPI model has a general form, it suggests that the effectiveness of the MPI model is not confined to PAMs. We believe it can also play some roles in other objects with hysteresis characteristics. And this will be further studied in the future.

Footnotes

Acknowledgement

This research is funded and supported by the National Natural Science Foundation of China (Grant No. 51675116).

Conflict of interest

The authors have no conflict of interest to report.

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Position control of a single pneumatic artificial muscle with hysteresis compensation based on modified Prandtl–Ishlinskii model

Abstract

Background:

Objective:

Methods:

Results:

Conclusion:

Keywords

1. Introduction

2. Methods

2.1. Measurement of length-pressure hysteresis of PAM

3.1. Identification and verification of the MPI model

3.2. Experiment results of position control for PAM

Footnotes

Acknowledgement

Conflict of interest

References