Construction, modeling, and control of a three‐beam prototype using interactive fiber rubber composites

In this contribution, we discuss the construction, modeling, and control of a three‐beam prototype that utilizes interactive fiber rubber composites integrated with shape memory alloys (SMAs) as actuators. These actuators are integrated into a textile layer and covered with an elastomer layer, providing flexibility and protection to the prototype. A mathematical model was developed to describe the behavior of the prototype, employing system identification techniques. However, precise position control of systems actuated by SMAs is challenging due to their inherent nonlinearities and hysteretic behavior. To tackle this challenge, a proportional‐integral (PI) controller was designed based on robust stability conditions. The effectiveness of the designed PI controller was validated through both numerical simulations and real experiments.


INTRODUCTION
Interactive structures, which can modify their properties in response to external influences, have gained attention in various fields such as robotics, aerospace, and biomedicine [1].These structures rely on smart materials that act as sensors and/or actuators, with shape memory alloys (SMAs) being a prominent choice [2].SMAs have the ability to recover their predefined shape or size when subjected to thermal stimulus, making them suitable for actuation in morphing wings, catheters, flexible link robots, and spacecraft antennas [3][4][5][6].SMAs offer advantages such as sensing and actuation capabilities, flexibility, lightness, and ease of integration into structures [7].They can generate large displacements and higher force-to-weight ratios compared to other smart actuators, making them ideal for developing functionally integrated adaptive structures.Heating SMAs can be achieved by passing electrical current or exposing them to thermal radiation [8].
One approach to creating soft structures is the integration of textile actuators based on SMAs into fiber-reinforced elastomer composites [9,10].Fiber-reinforced polymers (FRPs) provide design freedom, allowing tailored combinations of matrix and reinforcement fibers for specific applications.Composite structures with highly stretchable rubber matrices are particularly promising due to their ability to undergo large deformations [11].Advancements in textile technology enable the customization of reinforcement textiles for highly deformable fiber-rubber composites (FRCs) and the direct integration of wire-shaped SMAs into the fabric, resulting in interactive FRCs (IFRCs) capable of active deformations.IFRCs possess excellent physical and mechanical characteristics, including stiffness, strength, fracture toughness, and damage resistance [12,13].These composites offer adjustable mechanical properties and show great potential for applications in soft robotics and human-machine interactions [14].
Mathematical modeling, experimental modeling, or system identification techniques can be employed to derive simplified equations for the system [15,16].In this study, we follow the approach developed in [16] to represent the relationship between the inputs and outputs of the system using a rational transfer function that incorporates the system's nonlinearities as unstructured uncertainty.
The focus of this study is the construction, modeling, and design of a three-beam prototype integrated with SMA actuators capable of controlling spatial deformation.The activation of the SMAs through applied voltage induces deformation in the prototype.While the prototype exhibits significant deformation in vertical movements, its performance in lateral movements is negligible.Therefore, the control is specifically focused on vertical movements when all three beams are activated.
This paper is organized into five sections.Section 2 describes the manufacturing.Section 3 is dedicated to the identification of the model.Section 4 focuses on the control strategy.Finally, in Section 5, a brief summary of the conclusions is provided.

PROTOTYPE MANUFACTURE
This prototype draws inspiration from cephalopods to enhance the movements of soft structures.The concept involves three beams that enable movement in multiple spatial directions.The initial concept sketch (Figure 1) shows a setup with three arms.The three beams of the prototype are fixed on one end of a plate and interconnected at the center of the plate with the other beams.It utilizes SMAs in the form of 0.3 mm thick wires as actuators.
F I G U R E 1 Conceptual model of the prototype [17].The textile fabrication process involves the use of the tailored fiber placement (TFP) technique.The 0.3-mm-thick NiTi shape memory wire (Memry GmbH, Weil, Germany) is covered with a sheath made up of glass and polypropylene fibers using a friction spinning machine.This cover improves the wire's sliding capability and allows greater deformations.
The beams are manufactured with a composite structure.A mold, as shown in Figure 2, with a height of 3 mm is used for fiber insertion in a planar plane.The fiberglass textile scrim with the stitched-on shape memory wire is placed inside the mold.Sylgard R 184, a resin mixed with a hardener and 0.5 wt.% ZnS white pigment, is poured into the mold, infiltrating the textile.The material is then cured for 72 h at room temperature.The elastomer provides flexibility and deflection to the structure without the need for hinges.
Finally, the structure is fixed on one end to an acrylic and aluminum plate for support as it is shown in Figure 3, whereas the other end is interconnected at the center of the plate with the other beams.This prototype allows for both vertical displacement and lateral movement, depending on the activated beams.When all three SMA wires are actuated, the main movement occurs in the vertical direction, achieving maximum displacement.Activating only two beams enables small lateral movements, although the vertical displacement is reduced compared to the previous scenario.Lastly, activating only one beam results in minimal vertical and lateral movement.

MODELING
The mathematical model provides a description of the dynamic behavior of a system, in either the time domain or frequency domain.Various approaches exist to obtain dynamical models [15,16].One such approach is experimental modeling, also known as system identification, which involves measuring multiple variables of the process and constructing a model that closely matches the measured data.For the model identification of our prototype, we conducted several actuation tests and observed that the horizontal displacement was negligible compared to the vertical displacement (the biggest one achieved is 50 mm).Therefore, we only considered the vertical displacement of the prototype when all the beams were activated for the model.We chose to use a system identification approach to obtain the mathematical model of the IFEC prototype, since the SMA wires exhibit a time lag in response to the applied voltage and do not exhibit overshoot, we aimed to identify a first-order linear time-invariant system that describes the system's essential behavior.
We use an Arduino to send the proper input using Pulse-width modulation (PWM) signals and a circuit base on power MOSFETs to switch the current for each SMA wire.To measure the system's output, the vertical displacement between the center of the beams and the center of the plate was measured using a SHARP sensor (GP2Y0A51SK0F), and we read the data sent by the sensor in the ADC of the Arduino and then transform it into millimeters after a previous calibration of the sensor.We also use Simulink that serves as a bridge between the real systems and the data acquisition, letting us to observe in real time and save the data of each experiment.In Figure 4, the input and the output of one test can be observed.
The input and output values from these tests were utilized to obtain the parameters of the first-order transfer function where  is the gain and  is the time constant, as shown in (1).We coded a MATLAB R script that offers a convenient and efficient method for determining the parameters of our model.
After performing the tests, we simulated our system to compare the actual output with the simulated output.This facilitated a comprehensive analysis and validation of the model against experimental data sets, ensuring accuracy and reliability.An example of such a comparison is illustrated in Figure 5.It is important to mention that we obtained a range of gains and time constants based on our experiments, we present each range with lower and upper values.
Finally, we calculated the nominal transfer function of the system G().In this case, the nominal gain and time constants were determined as K̃=0.0112 and T̃= 21.355, respectively.

F I G U R E 5
Comparison between the measured output and the simulated output.

CONTROL
A crucial step in designing a robust controller involves the consideration of the potential uncertainties that could affect the behavior of the prototype.These uncertainties may include unknown nonlinearities, parametric variations, additive or multiplicative uncertainties, nonmodel dynamics, and more.Based on [18] and the research conducted in [16], we use a robust stability approach to achieve robust performance for the prototype, even in the presence of uncertainties.In this case, we utilized additive and multiplicative uncertainties, assuming that the nominal transfer function of the plant is represented by G(s).Consequently, the perturbed transfer function G(s) can be described using in this case additive or multiplicative uncertainty.

Additive uncertainty Multiplicative uncertainty
where W(s) represents a proper and stable weight function that characterizes the uncertainty dynamics, whereas Δ() contains the uncertainty itself, which can be any stable transfer function that satisfies the following inequality: To satisfy these conditions, an appropriate weighting function () and Δ() for additive uncertainty was calculated and it is shown in (4).
The same for the multiplicative uncertainty shown in (5): Then, assuming an additive uncertainty, the closed-loop system is according to [18] robustly stable if and only if F I G U R E 6 Valid and invalid combinations for   and   to assure robust stability condition for additive and multiplicative uncertainty models.
And in the case of the multiplicative, the condition for robust stability is given by To control the position of the IFEC prototypes, we propose a proportional-integral (PI) controller, which is tuned based on the robust stability conditions where the proportional and integral actions are the coefficients   and   , respectively: The (nominal) open-loop transfer function of the system with the controller is expressed as The (nominal) corresponding complementary sensitivity functions are By substituting the above equations into the robust stability condition for each uncertainty model, we can calculate a range of valid combinations that guarantee the robust stability of the system.A plot shows that the set of valid combinations of   and   for both additive and multiplicative uncertainties is shown in Figure 6.
The objective of the controller in the prototype is to stabilize the vertical displacement of the system when all the SMA wires are activated.The figure demonstrates the deflection control of the system, with a reference of 30 mm and   = 50 and   = 0.75.As shown in Figure 7, the system is capable of reaching the desired reference.

CONCLUSIONS
The construction of the prototype and the development of a mathematical model for FRCs actuated by SMAs have provided valuable insights to further work.Through system identification techniques, we were able to gain a deeper understanding of the dynamics of the IFRCs actuated with SMAs and effectively control their actions.The mathematical model F I G U R E 7 Deflection of the controlled system for a reference of 30 mm using a robust PI controller.derived from the system identification process captures the complex characteristics of the prototype, including hysteresis and uncertainties.

A C K N O W L E D G M E N T S
This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), research project 380321452/GRK2430.Open access funding enabled and organized by Projekt DEAL.

R E F E R E N C E S
Draw of the tripod with dimensions and components.(B) Layered structure of the individual beams [17].F I G U R E 3 (A) Real prototype.(B) Detailed view of the integration of the active component.

F I G U R E 4
Input and different measured outputs in different experiments.