A Polynomial-Based Framework for Understanding Hysteresis in Pneumatic Muscle Actuators

Muna Mohammed and Alaa Al-Ibadi
Department of Computer Engineering, Engineering College, University of Basrah, Basrah, Iraq
Correspondence to: Muna Mohammed, pgs.muna.mohammed@uobasrah.edu.iq

DOI: https://doi.org/10.70389/PJS.100147

Additional information

• Ethical approval: N/a
• Consent: N/a
• Funding: No industry funding
• Conflicts of interest: N/a
• Author contribution: Muna Mohammed and Alaa Al-Ibadi – Conceptualization, Writing – original draft, review and editing
• Guarantor: Muna Mohammed
• Provenance and peer-review: Unsolicited and externally peer-reviewed
• Data availability statement: N/a

Keywords: Pneumatic muscle actuator hysteresis, Fourth-degree polynomial modeling, Pressure–length characterization, Arduino-based experimental setup, Mckibben actuator control.

Peer Review
Received: 15 August 2025
Last revised: 26 September 2025
Accepted: 2 October 2025
Version accepted: 4
Published: 27 October 2025

Plain Language Summary Infographic

Introduction

In recent years, machines with autonomous controls—like human-coexisting robots—have started to function in settings that are near or include human interaction. Human safety is the primary concern in these settings, hence hardware components like actuators and mechanical structures must be lightweight and soft.^1,2 Compactness, cheap cost, high power-to-weight ratio, insensitivity to a dirty work environment, lack of stick-slip effects, and remarkably similar compliance to muscles are only a few of the special benefits that the pneumatic muscle actuator (PMA) offers. These features contribute to its widespread use.^3–6

They consist of a deformable, closed, strengthened, elastic membrane fastened to the ends.⁷ The PMA produces displacement and contraction force when a specific internal pressure is applied to it; under these circumstances, the displacement and contraction force are intimately connected to the internal pressure. However, the length/pressure hysteresis characteristic refers to the fact that the displacement of the PMA during inflation and deflation is not entirely consistent with the internal pressure. The hysteretic features of the PMA-driven system are one of the causes for its high nonlinearity, which makes it more challenging to precisely regulate the system’s trajectory or force tracking.^8–10 The internal bladder’s viscoelastic characteristics, air pressure, and the intricate structure and behavior of the outer braided sleeve covering the PMA are the causes of the nonlinear behavior, whereas the internal bladder is the cause of the hysteretic behavior, which causes the PMA to perform differently under various pressure conditions.^11,12

 One of the main drawbacks of pneumatic muscles is the occurrence of hysteresis, which can lead to inaccuracies that are challenging to regulate in a system that demands precise placement. The deformation of the flexible tube and the internal friction between each aramid fiber and the surrounding elastic material are the two main causes of pneumatic muscle hysteresis.^13,14 The parametric uncertainty and nonlinearity in PAM systems pose various limits to the mathematical modeling technique. Analytical modeling-based controllers are therefore ineffective and might not provide reliable results. Consequently, experimental modeling techniques that faithfully represent the intricate behavior of PAM systems are the focus of a lot of research. Polynomial regression is used in its modeling.^15,16 The most popular technique for estimating the relationship between muscle pressures and length during muscular contraction is polynomial regression¹⁷ the model for PAM that is based on polynomials. It has drawn a lot of interest from subject-matter specialists.^18–20 In this research, a fourth degree polynomial is used to characterize the setup’s deflated dynamics.²¹ The fourth-degree polynomial is made to be neutralized when multiplied by itself, allowing complexity to be removed while term degrees are decreased.²²

The Structure of the Pneumatic Muscle Actuator

The PMAs’ behavior as an contractor actuator is shaped by their construction. Figure 1 depicts the fundamental architecture of the pneumatic muscle actuator. L and D stand for the actuator’s length and diameter, respectively, while L0 and D0 are the initial values, which change depending on the size of the inner tube and the braided sleeve. The actuator’s ability to function as an extension or contraction muscle is determined by its braiding angle when it is relaxed (unpressurized). A contractor PMA is produced when the braided angle is less than 54.7, but an extensor PMA is produced by the actuator when the initial braided angle is larger than 54.7.^23–27

Muscle Materials and Manufacturing

This muscle consists of a rubber tube and a braided sleeve, each of which is 33.5 cm long and 2.5 cm in diameter. In addition to the solid caps made of Teflon, where the length of the first cap is 2.5 cm and its diameter is 1.8 cm, the length of the second cap is 3.6 cm, and its diameter is 1.8 cm, but it has a small hole where air is pumped through using the fitting. Finally, Adhesive tape and cable ties connected the braided sleeve, rubber tube, and Teflon caps together. The muscle weighs 70 grams. Figure 2 shows a picture of each of them, Then, it shows the muscle manufactured with a length of 30 cm and a diameter of 1.8 cm.

Experiment of PMA

An experiment took place conducted to study the hysteresis of Artificial pneumatic contractile muscle. After that, the Arduino Mega 2560, the pressure sensor and the ultrasonic sensor are used and connected in a circuit through connecting wires. Figure 3 shows a picture of each of them.

The manufactured air muscle is suspended to the frame by fixing it with screws; then, the ultrasonic sensor is placed on the muscle from below. The compressor is connected to the muscle; between them, there is a valve to control the amount of air entering the muscle. Then, the Arduino is connected to the computer, and the code is uploaded to it, as shown in Figure 4, which shows a picture of the air-controlled valve.

The compressor is turned on, and the air is pumped into the muscle, as shown in Figure 5, a picture of the muscle at rest and a picture at Pressurized. We continue to pump air at different rates ranging from (0–500) kPa, where each pressure rate has a different distance (the ultrasonic sensor measures the distance). In the end, these results are obtained in Table 1 below.

Table 1: Exprement result of length and pressure.
Length with Decreased Pressure(cm)	Length with Increased Pressure (cm)	Pressure (KPa)
30.06	30.06	0
30.05	30.06	50
28.54	29.14	100
27.46	27.94	150
26.38	26.74	200
25.64	26.006	250
25.22	25.47	300
25.08	25.32	350
24.59	24.82	400
24.39	24.4	450
24.2	24.2	500

The Table 1 shows that when the pressure value is zero, the length values ​​when increasing and decreasing are equal in value, and after that the length values ​​begin to decrease with increasing pressure until they stabilize at a pressure of 400, where the muscle reached its maximum length of 24.2.

Figure 6 shows the variation of the PMA length with different applied air pressure from (0–500 Kpa). The length remains constant when the pressure is from 0 to 50, and this is due to the resistance of the rubber tube. The connection between the rubber tube and the braided sleeve is not achieved, so no change in length occurs. We notice after that the length changes significantly until it reaches 400, and then it remains constant again because the diameter of the braided sleeve has reached its maximum.

Modeling of the Hysteretic

The proposed model of the hysteretic equation of the pneumatic artificial muscles aims to reduce the error between the increasing and decreasing pressure values as much as possible. The Hysteretic is modeled using fourth-degree polynomials. In contrast to lower-degree polynomials, fourth-degree polynomials are better able to capture complicated behaviors, depending on the material’s flexibility, amount of compressed air, and kind. They have the ability to identify various trends and curvatures in the data. according to

(1)

(2)

a_i is increase factor that has four values. The polynomial coefficients for the increasing phase are listed in Table 2.

Table 2: Values of increase factor.
ai1	ai2	ai3	ai4
0.0064937	–0.00022509	7.001e–07	–6.4747e–10

a_d is decrease factor that has four values. The polynomial coefficients for the decreasing phase are summarized in Table 3.

Table 3: Values of decrease factor.
ad1	ad2	ad3	ad4
–3.9456e–07	–0.00019412	6.4832e–07	6.1569e–10

(3)

After calculating the values of the lengths when increasing and decreasing, one calculates the error according to:

(4)

(5)

Then Calculate Mean Squared Error (MSE)

(6)

The notation and symbols used in this study are described in Table 4.

Table 4: Nomenclature.
Symbol	Description	Unit
L0	Initial muscle length	cm
p	Applied pressure	kPa
Lincm	Modeled length during increasing pressure	cm
Ldecm	Modeled length during decreasing pressure	cm
ai1 … ai4	Polynomial coefficients (increasing)	–
ad1 …ad4	Polynomial coefficients (decreasing)	–
Einc (pi)	Error during increasing	cm
Edec (pd)	Error during decreasing	cm
E	Error (difference between experimental and modeled length)	cm
H	Hysteresis (length difference)	cm
MSE	Mean Squared Error	cm²
N	Number of samples	–

Validation of Modeling

Different lengths resulted when performing the hysteresis experiment and pumping different pressures, as shown in Table 1. After obtaining the results, we deduced equations and drew them using the MATLAB program. These results appeared as shown in Figure 7, where the graph obtained during the experiment is very similar to the graph when modeling. The hysteresis in the graph is the difference between the values ​​when the pressure is increased compared to the values ​​when the pressure is decreased, which indicates the presence of non-linear effects in the muscle response to pressure. When the pressure is increased, the length of the muscle decreases, and when the pressure is decreased, the length remains slightly higher, which highlights the behavior of aerobic muscles under different pressures.

The findings plotted the error equation at increasing and decreasing pressures, as shown in Figure 8. The blue line (increasing pressure) shows how the errors change as the pressure increases, with some negative and positive values indicating that the model may be more accurate at some points than others. The red line (decreasing pressure) also shows fluctuations in errors, but the negative values ​​can be more pronounced at some points, indicating that the model may be less accurate as the pressure decreases. The differences between the two lines indicate that the behavior of the aerobic muscle is different when the pressure increases compared to when it decreases, reflecting the hysteresis phenomenon mentioned in the previous Figure 7. These differences could indicate that the model must be improved or modified when dealing with increasing or decreasing pressures to improve accuracy.

The mean square error (MSE) of 0.05201 indicates that there is some error in the predictions, but it is not very large. The correlation coefficient for increasing pressure is 0.9978, and the correlation coefficient for decreasing pressure is 0.9964, indicating a solid relationship between pressure and muscle length when pressure increases and when pressure decreases. Despite the presence of hysteresis, the high correlation coefficient values ​​indicate that the model is still able to predict the expected values ​​with high accuracy. The Root mean squared Error (RMSE) is 0.23 and the Mean absolute error is 0.127809258. The report mean ± SD of length at each pressure show in Table 5.

Table 5: Mean and standard deviation of the length.
Pressure (KPa)	Mean (cm)	SD (cm)
0	30.06	0
50	30.06	0.00471
100	28.84333	0.41955
150	27.70333	0.33941
200	26.56	0.25456
250	25.825	0.25692
300	25.34833	0.17206
350	25.20333	0.16971
400	24.70667	0.16028
450	24.395	0.00707
500	24.2	0

Error metrics were assessed during the entire pressure in order to test the fidelity of the suggested Preisach-type model against the experimental results. The model captures both the general trend and the hysteresis loop, closely matching the actual muscle-length–pressure trajectories as show Figure 9. In terms of numbers, it obtained a Mean Absolute Error (MAE) of 0.2384, a Mean Squared Error (MSE) of 0.0968, and a Root Mean Squared Error (RMSE) of 0.3111. These numbers show sub-millimetric average deviations from the working range, confirming that the Preisach formulation is appropriate for simulating the hysteretic response of the actuator.

Conclusion

In this study, the behavior of pneumatic muscle actuators (PMAs) is investigated, with a focus on the hysteresis effect between displacement and pressure during operation. comprehend and forecast the actuator’s response to changing pressures, the data offered important insights into the hysteresis loop. The behavior of the actuator is described using a polynomial equation that is developed and put into practice in order to overcome the difficulties associated with hysteresis modeling. MATLAB simulations served to validate the model, and graphical analyses of the predicted and experimental data took place. Mean Squared Error (MSE) and correlation coefficient functioned as evaluation measures that demonstrated the model’s efficacy. The model’s capacity to reduce prediction errors is validated by the low MSE value, and there is a significant correlation between the simulated and experimental measurements.

As evidenced by the high correlation coefficient.

According to the investigation, the suggested polynomial model provides a trustworthy PMA control and system design tool by accurately capturing the hysteresis phenomena. In applications requiring delicate actuator control, the model’s reduced error guarantees increased precision and performance. These findings open the door for PMA systems to be further optimized so they may be incorporated into cutting-edge automation and robotics solutions. Future research could build on this work by investigating different modeling techniques or putting real-time control strategies into practice to improve the accuracy and responsiveness of the system.

Acknowledgements

I would like to sincerely thank the University of Basrah Department of Computer Engineering for providing me with a good laboratory, an appropriate workspace, and materials for my research projects.

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Cite this article as:
Mohammed M and Al-Ibadi A. A Polynomial-Based Framework for Understanding Hysteresis in Pneumatic Muscle Actuators. Premier Journal of Science 2025;15:100147

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