Robots are increasingly assisting humans, especially the elderly and disabled. However, building rigid robotic joints becomes more complex when trying to match the softness of human arms and flexible joints, which are moved by many muscles. Researchers inspired by the need for human-like actuator systems developed the pneumatic muscle actuator (PMA). The mechanism moves as a result of the PMA’s conversion of pneumatic power into pulling force. The performance characteristics of these systems are comparable to and more robust than those of human muscle.^1,2,3 Soft robots are robots composed of elastic, moldable materials like soft plastic, silicone, and rubber. These robots imitate living things like octopuses, worms, fish, and even certain plants using designs influenced by nature. Soft robots can bend, stretch, and compress without suffering damage, making them versatile and able to adapt to diverse settings.^4,5,6,7

Soft robots possess desired traits Because of their consistently deformable and compliant structures, which can tolerate large deformations during normal operation these features include like conformability to their surroundings embedded in their mechanical constructions, but their flexibility and compliance make accurate, high-bandwidth control more difficult than their rigid counterparts. Therefore, researchers are increasingly developing soft robots for applications in crowded and unstructured environments, including manipulation, search and rescue operations, and human-centered jobs.^8,9,10,11

PMA are more secure than other devices that use heat, electricity, or chemically active materials because they can be made from inexpensive materials that are readily available, have high degrees of freedom due to the soft actuator’s ability to bend, are inexpensive because they can be made from readily available compressed air, have a lifting force of 6000 N despite only weighing about 800 grams, and have high strength relative to volume that is five times greater than that provided by electric motors.^12,13

And having very comparable compliance to muscles. Because of these features It is commonly utilized in arm/ankle orthotics, elbow exoskeleton lower-limb rehabilitation, and medical nursing.¹⁴ Although PMAs offer many benefits, they also have a number of drawbacks that might restrict their use. For the most part, these PMAs exhibit hysteretic behavior due to the friction between the braided sleeves’ wires, and because there is inherent friction, the muscle’s temperature rises over time during operation, significantly affecting its dynamic properties. A rise in temperature can accelerate material fatigue, modify the inner tube’s viscoelastic response, and affect the air’s compressibility. These changes can lead to performance drift and less repeatability in the actuator’s behavior. Additionally, they exhibit nonlinear behavior due to the viscoelastic properties of the inner tubes, the compressibility of the air inside, and the geometrically complex behaviors of the PMA outer covering, and hysteresis causes a change in the way the muscle reacts to pressure changes, which means that depending on whether the muscle is in the process of inflating or deflating, the force it exerts for a given pressure may vary. Because of this, it is more difficult to attain accurate control, particularly in applications that need precise force or location. However, building a precise dynamic model of PMA may successfully resolve the aforementioned issues.^15,16,17

Currently, modeling and hysteresis compensation are gradually emerging as important study topics. From a mathematical perspective, we can easily divide the PMA’s dynamic models into two categories: physics-based models and phenomenon-based models. The physics-based model mostly figures out how the PMA’s shape and material properties relate to the thickness of the rubber tubes so that a static model of the PMA can be made using the force balance method. Physics-based models include many model parameters and complex mathematical derivations, which make controller design very challenging and raise processing costs significantly. Alternatively, we create phenomenon-based models using reduced model descriptions and experimental data.^18,19

The contraction-type PMA is designed from a rubber tube surrounded by a braided sleeve that has two ends made of plastic (or any solid material). Figure 1 shows the structure of the contractor muscle. The braided mesh represents the length of the PMA, which is similar to the rubber tube length. The PMA produces its contraction performance when the angle θ is less than 54.7°. Although it varies from muscle to muscle, the contraction ratio never exceed 30%.²⁰

The intended pneumatic muscle for contraction consists of a braided sleeve of 35.5 cm in length and 1.6 cm in diameter. The inner tube is rubber used by bicycles. It is simple to use, and its material structure allows it to expand and contract. The tube length is 35.5 cm and 1.6 cm in diameter, and the muscle length is 30 cm and 1.8 in diameter.

Two cylinders of Teflon caps were applied to either end of the muscle. The first one is 3.5 cm long and has a 2 cm diameter with an air-passing hole inside. The second one is 2.5 cm long and has a comparable diameter. Which are tightly held using cable ties.

The braided sleeve fully encased the rubber tube, and the covers are fastened to the tube with tape and a water cap to prevent them from moving when air was pumped into the tube. show in Figure 2.

When compressed air is pumped into the rubber tube’s interior, it expands radially and contracts in length. The inner netting serves as a spring to return the tube’s natural shape as air leaves it show Figure 3.²¹

It’s been completed conducted several experiments to measure the length of the muscle. Changes in muscle length were observed with a change in pressure applied to it. The aim of these experiments was to understand the behavior of the muscle during contraction and relaxation. The experiments are performed as follows:

See Figure 4 for the equipment used in the experiment.

The experiment is carried out as follows: The experiment begins with the muscle at its natural length of 30 cm. Next, we apply air pressure ranging from (0–500) kPa. Pressures above 500 kPa may cause harm to humans. Furthermore, muscle movement is relatively stable between (450 and 500) kPa, so there is no need for higher pressures. This pressure causes the muscle to contract, causing its length to decrease until it reaches 24 cm at a pressure of 500 kPa. In the second stage, we decrease the air pressure from (500–0) kPa, allowing the muscle to return to its natural size. Sensor calibration is rest at each experiment set to zero. By analyzing the collected data, Figure 5 illustrates the pressure and length curves, as well as the muscle’s behavior and response to pressure changes. The numerical results are displayed in the Table 1, which are an average of several conducted experiments.

The blue curve illustrates the muscle’s length change behavior under increasing pressure, while the red curve illustrates the muscle’s behavior under decreasing pressure.

Observe that at the beginning, when the pressure is between 0 and 50, the resistance of the materials that make up the muscle, such as the rubber tube and sleeves, prevents any change in the muscle’s length because the applied pressure is insufficient.

Since the muscle’s materials are not elastic, it requires a certain amount of pressure to expand and contract, so length doesn’t change.

Then, when the pressure increases, the muscle undergoes a significant change in length because it has reached a sufficient pressure to resist the flexibility and friction of the materials that make up the muscle. Moreover, the muscle tends to exhibit exponential behavior. And once the pressure reaches 400, the length stabilizes as the sleeve’s diameter reaches its maximum value.

It can be observed that the muscle length shows a different behavior when the pressure increases compared to when the pressure decreases. During the extension and contraction cycles, there is a noticeable slowdown, which shows that the muscle does not respond in a straight line. This is because of the hysteresis phenomenon, which is caused by the interaction of several factors, including the internal friction of the fibers and the rubber tube and the variation between the pressure inside the muscle and the air pressure outside.

To determine the relationship between the length and pressure of the pneumatic muscle, several authors have tried to create a mathematical model that depicts the behavior of the actuator.²¹ An accurate and important mathematical model is created based on how muscle length changes under pressure, which resembles the shape of the sigmoid function, so the sigmoid function was chosen to represent the mathematical model. Researcher Alaa Al-Abbadi employs this method.²³ General Equation for sigmoid. Where the length of the pneumatic muscle actuator (PMA) as a function of applied pressure is modeled using the following equation:

Where:

To find the best fit between the model and experimental data, the exponent (6.214) was determined by iteratively adjusting its value and performing curve fitting to experimental data. Due to the highly nonlinear behavior of pneumatic muscles, analytical determination was not feasible, so trial-and-error was used to minimize fitting error and accurately match the real muscle response.

As a result, a collection of equations was established. used the fitting curve approach to model the system in (2). This method is among the simplest ways to identify a system, whether it be dynamic or static.²⁴

Used MATLAB’s lsqcurvefit function to fit the parameters a and b of the model. The fitting procedure uses the experimental data of muscle length in a relation of pressure. The function minimizes the difference between the experimental data and the theoretical model by adjusting the parameters a and b. The curve fitting function has the following form:

Where:

In (3) To account for the hysteresis, we introduce a second relationship for the length of the muscle when pressure is decreasing. This model incorporates a hysteresis term, which is the difference between the muscle length at increasing and decreasing pressures.

The model for the decreasing pressure can be expressed as:

Where:

In this section, the results of the mathematical model presented in the modeling section will be compared with experimental data in order to confirm the validity of the model. The following diagrams will illustrate the relationship between the specified factors, such as length and pressure.

Figure 6 shows the connection between the pneumatic muscle’s length and the pressure applied, discussed earlier, to illustrate the experimental data during pressure increases and decreases. A curve fitting technique was used to improve the agreement between the experimental data and the theoretical data.

When using MATLAB’s lsqcurvefit function, the code modifies a and b to make the predicted length function as close to the real length data as possible. To decrease the error (difference) between the experimental and anticipated values.

Where:

Fitted value for a= 0.19073.

Fitted value for b= 0.011333.

The blue dotted curve represents curve fitting when pressure increases, while the red dotted curve represents curve fitting when pressure decreases. The gap between them represents hysteresis, the non-linear behavior of a pneumatic muscle.

The analysis of the proposed model showed that it provides excellent accuracy, as there is a clear match between the experimental and theoretical values at most points with a slight deviation. calculated the mean square error (MSE) and the correlation coefficient Using MATLAB, finding that the MSE was 0.15776, indicating a low level of error. found the correlation coefficient when increased pressure to be 0. 99603.The correlation coefficient when decreased pressure is 0.99598. These numbers indicate a high strength of agreement between the experimental data and the mathematical model.

The Mean Absolute Error (MAE) was 0.16642 for both increasing and decreasing pressure. The coefficient of determination (R²) was 0.99115 for increasing pressure and 0.991 for decreasing pressure, demonstrating the high accuracy of the proposed model.

Figure 7 shows the error difference between the mathematical model data and the experimental data, where (A) represents the error in the case of increasing pressure, while (B) represents the error in the case of decreasing pressure. can analyze the results in the following way: The model accurately represents the experimental values when the error is close to zero. However, when the error goes beyond zero towards positive values, it signifies that the experimental values surpass the theoretical values. In other words, the model assumes that the muscle length is shorter than its actual length. And when the error value is negative, the model assumes that the muscle length is longer than its original length. This error is because of the behavior of the muscle material.

The Sigma method and the Bouc-Wen model were compared using the same experimental dataset to evaluate the performance of the proposed models. The coefficient of determination (R²), mean square error (MSE), and mean absolute error (MAE) were used as the main statistical indicators for comparison. Table 2 below illustrates these differences.

The sigmoid model, which more smoothly and simply captured the nonlinear muscle action, generally demonstrated a closer alignment with the experimental data. Although the Bouc–Wen model was effective at depicting hysteresis in general, it tended to overstate the loop area and produced a less accurate fit. The sigmoid method turned out to be the more accurate and trustworthy way to depict the behavior of the actuator in this situation.

Figure 8 plots the experimental findings against the two models to provide a visual comparison. The Bouc–Wen curve exhibits pronounced deviations and a greater hysteresis loop area, whereas the sigmoid curve more closely reflects the experimental trend. The statistical results shown in Table 3 are supported by this visual evidence.

This study investigated the contractile behavior of pneumatic muscle actuators (PMAs) using both mathematical modeling and experimental methods. Muscle length was measured in laboratory tests under different applied pressures.

One of the most important findings was the presence of hysteresis, which had a significant effect on muscle length as pressure increased and decreased. Different contraction and relaxation trajectories illustrate the nonlinear behavior of the hysteresis effect in PMA. On the other hand, used a curve-fitting technique to calculate muscle length/pressure using a sigmoid function, which improved the accuracy and consistency of the theoretical model equation for muscle length with the experimental data.

The mean square error (MSE), which showed a high degree of accuracy in the model’s predictions, validated the agreement between the experimental data and the suggested mathematical model. This validation demonstrates the dependability of the sigmoid function in simulating PMA contraction dynamics and Several research studies have put forth models, with differing degrees of accuracy, to forecast the behavior of pneumatic muscle actuators (PMAs). For instance, a model displayed MSE values ranging from 7.23 to 22.68, demonstrating a substantial discrepancy between theoretical and experimental outcomes.¹³

Conversely, the model in this paper shows enhanced accuracy with a significantly lower MSE of 0.15776. The study’s conclusions advance the use of PMAs in domains where precise and flexible actuation is essential, such as industrial automation, robotics, and prosthetics. Further research could build on this foundation to explore real-time control techniques and improve PMA performance in dynamic operating environments. For future research, this model can be used for control systems.

I would like to express my thanks and appreciation to the Department of Computer Engineering at the University of Basrah for providing a laboratory to work in. This acknowledgement highlights the significance of institutional support in facilitating academic research. Such support not only offers monetary aid but also produces an atmosphere that is suitable to academic growth.