Actuators are devices that are responsible for moving and/or controlling a machine or a system, converting input energy (photons, thermal energy, magnetism, and/or electric energy) into mechanical energy under the action of control signals.[1,2] Since humans began using tools, mechanical energy has always been a form of energy that can directly alter the world, playing a pivotal role in the evolution of human society and technological advancement. All previous industrial transformations have been facilitated by mechanical energy conversions devices such as the steam engine, internal combustion engine, and motor. In the context of the fourth industrial revolution, robotics, wearable devices, bionics, and other technologies have an increasing demand for actuators. Frequent and intimate human–computer interaction requires equipment that is soft, compact, lightweight, and energy efficient.[3] Compared to conventional rigid actuators, soft actuators exhibit remarkable advantages in terms of biocompatibility, portability, and power efficiency, and they provide more precise than hydraulic and pneumatic actuators at subcentimeter scales.[4] Due to the flexibility and compliance of soft actuators, they are particularly beneficial for physical interaction with fragile objects or living organisms.[5] Ionic polymer–metal composite (IPMCs) are types of electrically stimulated soft actuators that have lower operating voltage than the safe voltage of the human body, low power density, and relatively high efficiency compared to other actuators (muscles, shape memory alloy, piezoelectric polymer, and hydraulic).[6,7] It is noiseless, has a high strain capacity, can operate in a humid environment, and can bend, sway, and axial movement.[8–14] Due to their superior performance characteristics, IPMC actuators have been created as micropumps, soft grippers, drug release devices, biomimetic fishes, etc.[15–18]
IPMCs are a novel class of soft electroactive materials that are micrometer-scale layered devices.[8,19] The IPMC is a sandwich-like structure with metal electrodes on both sides, and its interior (called a membrane) consists of negatively charged polymers immersed in a liquid phase consisting of solvents and positively charged mobile ions (called counter ions).[7,12,20] IPMCs can function as both an actuator (electromechanical transduction) and a sensor (mechanoelectrical transduction).[21] The deformation mechanism of the IPMC actuator is that under the external electric field, the counter ions migrate to the cathode,[22,23] resulting in uneven charge distribution, osmotic pressure, and electrostatic effect, and the IPMC bends to the anode.[23–25] In order to realize the practical application of IPMC actuators, the researchers improved their performance through a series of chemical methods.[26–28] However, the physical basis of IPMC actuator's bending behavior is not fully understood, which will severely limit its real-world applications. The powerful role of the model for IPMC-based device development is often overlooked. In order to quantify its electromechanical behavior to promote the application of IPMC, the electromechanical transduction characteristics of IPMC are incorporated into the mathematical model framework, that is, the displacement and/or blocking force of IPMC actuators are simulated.
In recent years, there have been several reviews related to IPMC actuators. Ref. [29] reviewed the optimization of IPMC drive performance from the perspective of chemical preparation. Ref. [30] introduced the modeling of IPMC sensing as a sensor and self-sensing as an actuator, as well as the implementation and characteristics of different IPMC sensing methods. The preparation process and material progress of exchange membrane and electrode materials for IPMC actuators were reviewed in ref. [7]. Ref. [31] reviewed the effects of different preparation processes on the properties of IPMC. The IPMC model was reviewed in refs. [6,32]. There are obvious differences between this article and the refs. [7,29–31], and there are similarities with refs. [6,32]. However, 1) Compared with this article, refs. [6,32] lacked a review of 'multisegment model’ and the latest 'thermodynamically consistent continuum (TCC) models '. Ref. [32] did not involve neural network models. 2) Refs.[6,32] divided the model into black box, gray box, and white box according to its visibility into the system interior. In this article, according to the modeling principle of the model, the model is divided into nonphysical identification models, partial-physical models, and physical-based models. 3) Refs. [6,32] did not summarize the specific functions of the model to promote application. From the perspective of model function, this article reviews not only the model improvement methods but also the specific performance of the model function to promote the application of IPMC. 4) Ref. [32] did not propose the future trend of the model. Ref. [6] gave the future development trend of model development technology and control. Based on the concept of theory guiding application (i.e., IPMC model–model function–IPMC application), this article presents the characteristics of future models and the model–based development of IPMC-based devices. In summary, according to the modeling principle of IPMC actuator model, this article proposes a classification method different from the traditional classification methods (black box, grey box, and white box), which divides the model into nonphysical identification models, partial-physical models, and physical-based models. In this article, according to the characteristics of different types of models, they are subdivided into small classes. Compared with the traditional classification method, this classification method can be well combined with the function of the model and is more suitable for the development concept of 'IPMC model–model function–IPMC application'. The functions (prediction, optimization, control, and development) of the model are introduced in the process of summarizing the ideas for improving the model. The IPMC actuator models promote the application of IPMC, and the status quo of IPMC application is also reviewed. Finally, this article gives the characteristics of the future model and the future model-based development of IPMC-based devices.
The remaining sections are organized as follows. As shown in Figure 1, Section 2 covers classical and intelligent identification models. Section 3 includes equivalent circuit models, equivalent beam models, and multisegment models. Section 4 classifies the physical-based models. It is worth noting that this classification method can also distinguish the functions of different types of models, as shown in Figure 1. Section 5 introduces the application of IPMC actuators. Section 5 does not exist in isolation and has been facilitated by the functions of the models in Sections 2, 3, and 4. Then, the modeling methods and applications of IPMC actuators, the characteristics of future models, and the future model-based development of IPMC-based devices are discussed. Finally, the full text is summarized in Section 7.
The nonphysical identification model is a method of obtaining a mathematical model for the system using the IPMC actuator displacement data. It is unnecessary to understand the system and its components nor derive parameter values from separate tests when using this modeling technique. This article divides nonphysical identification models into classical identification models and intelligent identification models based on distinct modeling methods. Considering that the precision of the IPMC actuation displacement data influences model development, Section 2.2 concludes with a summary of some data measurements.
Classical Identification ModelsBhat et al.[33] described the displacement of the IPMC tip using an algebraic polynomial function with time exponents. Experiments were performed to determine the output displacement under different voltages, and a MATLAB toolbox for nonlinear curve fitting was used to determine the displacement expression of IPMCs, solving the problem of nonlinear curve fitting in the sense of least squares. Yun et al.[34] used the Box–Jenkins (BJ) method to identify the dynamic behavior of IPMC actuators, handled the inherent nonlinear behavior of IPMCs under the linear model with skill, and provided a system identification method to enhance the accuracy and reliability of IPMC actuators and sensors. Notably, the identification model derived from simple experimental identification is extensively used in developing IPMC controllers,[35] but not all controls are based on the model.[36] Open-loop control typically relies on the precise model of IPMCs. However, temperature influences the dynamic characteristics of IPMCs. Dong et al.[37] proposed a new temperature-based method for modeling IPMC actuation dynamics. The IPMC actuator is modeled using a transfer function derived from experience, where the zero and pole points are temperature dependent. This nonlinear dependence is approximated by a polynomial function under the assumption that each pole and zero point of the transfer function is nonlinearly dependent on temperature. The established temperature-dependent model is able to approximate the direct-fitting model and predict the frequency response at new temperatures. However, none of the aforementioned methodologies account for the hysteresis and creep behavior of the IPMC actuator as a piezoelectric actuator. Ref. [38] proposes a comprehensive model to characterize the hysteresis and creep of piezoelectric actuators (Figure 2a). The hybrid model based on the weighted superposition of the phenomenology model, Preisach operator, and creep operator is applied to the modeling of IPMC actuators. Hao et al.[39] analyzed hysteresis using the P–I model (Figure 2b), obtained the creep model of IPMCs by modifying the creep model of piezoelectric materials, and proposed an inverse model for hysteresis. Chen et al.[40] noted that the P–I model could not adequately characterize the IPMC actuator's hysteresis behavior. In order to soften the sharpness of the P–I model, a filter is added to the output of the P–I hysteresis, and a new IPMC actuator model is proposed. Wang et al.[41] developed a Hammerstein model for IPMCs in a creep operator superposition model that characterizes the static nonlinear component model, and the dynamic transfer function is used to compensate for the error in the static nonlinear component. The dynamic linear portion is described by an autoregressive model of the second order.
The classical identification model reveals the physical characteristics of the IPMC actuator to a low degree, and the output displacement and input voltage are the physical quantities directly modeled. These models can predict the displacement of IPMC actuator well,[33–41] but only a few models can characterize the hysteresis or creep of IPMC.[38–41] Based on this model, a controller can be developed to achieve more accurate IPMC control.[33–35,37–41]
Intelligent Identification ModelsIn the classical recognition model, the method to solve the hysteresis problem proposed by Chen et al.[38] can describe the bending behavior of IPMC, but its accuracy is unknown. Anh et al.[42] created a nonlinear autoregressive exogenous (NARX) model based on fuzzy algorithms for identifying pneumatic artificial muscle actuators. Nam et al.[43] cascaded the Preisach operator with the fuzzy NARX model. They used the particle swarm optimization (PSO) algorithm in the identification process to produce accurate nonlinear black box models (NBBMs) of IPMCs. However, none of the aforementioned models accounted for the large bending behavior of IPMC actuators when identifying their nonlinear behavior. To identify the large deformation behavior of IPMCs, Annabestani et al.[44] proposed an adaptive neuro fuzzy inference system (ANFIS)-NARX model. In actuality, the metal electrodes of IPMCs are not ideal conductors, and the electric field applied to IPMCs decreases longitudinally, resulting in nonuniform deformation of IPMCs.[45] However, research on the nonuniform deformation behavior of IPMCs is scarce. The team then utilized the model proposed in another study[44] to investigate the nonuniform deformation and curvature trajectory of IPMCs. Zamyad et al.[46] proposed an improved ANFIS-NARX model that extends the input space by substituting a hysteresis operator for the previous output samples. In order to overcome the IPMC hysteresis phenomenon, Huang et al.[47] used the least squares support vector machine (LSSVM) NARX model to model the tip displacement of IPMC actuators and proposed an artificial honeycomb algorithm to optimize the LSSVM NARX model's parameters. A related study has shown that there is a definite relationship between the electrical characteristics and shape of IPMCs, especially between the resistance and shape of IPMCs.[48] Truong et al.[49] proposed an NBBM model based on a recurrent multilayer perceptron neural network (RMLPNN) and optimized by advanced training methods of self-adjustable learning mechanisms (SALM) (Figure 3a). The team then designed a combined a general multilayer perceptron neural nerwork (GMLPNN) and a SALM model[50] with the same maximum accuracy as another study.[49] A novel NBBM constructed by GMLPNN with integrated smart learning mechanism (SLM) is proposed, which is based on an extended Kalman filter with self-decoupling ability (SDEKF).[51] Using SDEKF-based SLM can optimize the parameters of GMLPNN with less computational effort and improve modeling accuracy. Compared to using SALM, the number of training iterations for SDEKF is significantly reduced. When the neural network model is available, system identification can develop a nonlinear controller to adjust the performance of the actuator.[52]
The autoregressive model, such as models,[33,34,36,38,48,52] uses previous samples from the system output during its recognition process, constituting a fundamental constraint on the practical applications of IPMCs. However, acquiring information about the output is not feasible in dominant applications of IPMCs, especially in biomedical applications.[53]
Annabestani et al.[53] proposed a Volterra-based IPMC nonautoregressive nonlinear identification technique. Although the nonautoregressive identification of IPMCs is feasible, it is not as accurate as the autoregressive model. The accurate nonautoregressive model of Laguerre multilayer perceptron (MLP) network is used to predict the displacement of the tip point of the IPMC actuator, taking into account the influence of ambient humidity and temperature on the actuator characteristics.[54] Zamyad et al.[55] proposed a hybrid model of parallel nonautoregressive recurrent networks with internal memory units, and the accuracy of the proposed recurrent network is comparable to that of the autoregressive neural network model (Figure 3c)[54] and the ANFIS-NARX model.[46] However, compared with the two nonautoregressive MLP models based on Volterra[53] and Laguerre,[54] the model in another study[55] has significant advantages over the Volterra model, and there is no significant difference in performance between the model and the Laguerre MLP model.[54] Annabestani et al.[56] proposed a hybrid model combining resistor–capacitor (RC) distributed model and a deep MLP (D-MLP) neural network model (Figure 3b). The RC model with low fidelity is used to convert the input voltage into some information data, which can be combined with the model based on machine learning (ML) to identify the behavior of IPMC with high accuracy. The hybrid model uses a nonautoregressive structure, showing a high fidelity to the IPMC physical model, and is more accurate than other autoregressive models.[43,44,50]
Interestingly, Annabestani et al.[57] proposed a model for describing large deformations of IPMCs, which can better represent the true displacement of IPMCs. However, the traditional method of using laser displacement sensors to measure the displacement of the IPMC tip can only measure the small deformation displacement of the IPMC actuator (Figure 4a). Laser displacement sensors cannot accurately map the complex bending exhibited by IPMCs.[58] The laser displacement device is also affected by changes in beam refraction, making it unsuitable for use underwater or under humid conditions and on highly reflective objects.[58] Some studies use digital cameras to record the bending deformation of IPMCs to overcome the shortcomings of laser displacement sensors.[53–56] The digital camera is usually placed on one side of the IPMC actuator (Figure 4b), and the tip displacement of IPMCs can be measured using a scale to obtain a grid paper. The charge-coupled device (CCD) video camera is also important for displacement measurement in IPMC. In order to improve the measurement efficiency of IPMC actuator displacement, Tsiakmakis et al.[59] developed a visual measurement method based on CCD cameras. Table 1 summaries the characteristics of the intelligent identification model.
Table 1 The properties of intelligent identification models
| Pulication, Year, Reference | The structure of model | Data measurement mode | Displace-ment identifica-tion features | Accuracy of the model | Model Contribution |
| Truong et al., 2010,[49] | RMLPNN | CCD laser sensor | Tip displacement | The highest accuracy is 91.5% | Self-estimated based on the applied power without any measuring sensor |
| SALM | Self-sensing based on the surface resistance | ||||
| Truong et al., 2011,[50] | GMLPNN | CCD laser sensor | Tip displacement | The highest accuracy is 91.5% | It reduces the cost and complexity of the system while improving system reliability |
| SALM | Self-sensing based on the surface resistance | ||||
| Nam et al., 2012,[43] | Novel Preisach type fuzzy NARX | Laser sensor | Tip displacement | Excellent approximating | |
| Modified PSO | capability | ||||
| Truong et al., 2014,[51] | GMLPNN | CCD laser sensor | Tip displacement | The highest accuracy is 93.87% | Contributing to optimizing the GMLPNN decisive parameters. |
| SLM based on the SDEKF | Self-sensing based on the surface resistance | ||||
| Annabestani et al., 2014,[44] | ANFIS | Camera | Tip displacement | The smallest NMSE is 5.86e-5 | Interpretability, Readability |
| NARX | Better estimation accuracy | ||||
| Annabestani et al., 2014,[57] | ANFIS | CCD laser sensor | Large Non-uniform deformation | The smallest NMSE is 1.1098e-04 | The nonuniform deformation of IPMC can be predicted |
| NARX | |||||
| Annabestani et al., 2016,[53] | Nonautoregressive nonlinear identification using Generalized Volterra-Based Approach | Camera | Large Tip displacement | The smallest NMSE is 0.0692 | Nonautoregressive identification |
| Nonautogressive identification | This model can be used in situations where the output information is not easily obtained | ||||
| Nakshatharan et al., 2017,[52] | Neural network modeling of patterned IPMC | Laser displacement sensor | Tip displacement | System identification based on neural network model | |
| Self-sensing | Model-based control | ||||
| Zamyad et al., 2018,[54] | Laguerre-MLP network | Camera | Tip displacement | The smallest NMSE is 3.28e-06 | Consideration of Ambient Temperature and Humidity Effects on ipmcs’ |
| Nonautogressive identification | Performance | ||||
| Zamyad et al., 2020,[55] | RNN-based model | Camera | Bending and Tip displace-ment in large deformat-ion | The smallest NMSE is 4.67e-05 | Achieve high-precision output of non-autoregressive models |
| A hybrid model of parallel non-autoregressive recurrent network with internal memory cells | Nonautogressive identification | ||||
| Annabestani et al., 2022,[56] | Physical-Data driven machine learning based model | Camera | Tip displacement | The smallest NMSE is 9.5781e-04 | Hybrid model |
| Hybrid model | Nonautogressive identification | The high fidelity model is obtained |
The intelligent identification model takes the input voltage and output displacement of the IPMC actuator as the input and output of the neural network. Such models can predict the displacement of IPMC actuators with high accuracy,[43,44,46,47,49–51,53–56] and can characterize the hysteresis or creep of IPMC.[43,44,46,47,49,51] The prediction of the entire deformation shape of the IPMC[44,46,53] and the position prediction without external data can be achieved.[53–56] Some models have been developed for higher precision controllers.[47]
Partial-Physical ModelsA partial-physical model is created by combining existing physical laws and identifying parameters and material properties by experiments, and its modeling concepts are primarily derived from analogy. Different models are established according to different modeling principles, including equivalent circuit models, equivalent beam models, and multisegment models.
Equivalent Circuit ModelsAmong the early IPMC electrical models, the distributed circuit model proposed by Shahinpoor et al.[60] and Bao et al.[61] are representative models. Given the presence of the polymer membrane on the actuator, the preceding models considered only the capacitance of the actuator (Figure 5a). Moreover, the linear models considered in the preceding studies are incapable of describing the nonlinear actuation behaviors of IPMCs.[62] Bonomo et al.[62] used a novel nonlinear lumped parameter equivalent circuit model (Figure 5b) that can describe the linear and nonlinear phenomena observed when current flows through the material, but this model cannot describe the relationship between current and external macroscopic deformation of IPMC. Another study[63] further considered the dependence of IPMC absorption current on the applied voltage on its thickness, proposed an improved nonlinear circuit model to describe the relationship between the current and voltage in IPMCs, and transformed the absorbed current into the response of IPMC machinery for modeling. On the basis of the model,[63] the response of IPMC actuators to temperature and relative humidity is analyzed, and it is discovered that the IPMC actuator is highly dependent on ambient humidity level but has no significant dependence on ambient temperature function.[64] Vahabi et al.[65] utilized the same equivalent circuit model as ref. [63] and identified model parameters using linear and nonlinear least squares methods. McDaid et al.[66] proposed a novel equivalent circuit model that partitions the equivalent circuit into clamped and free parts. The electromechanical model is divided into three processes: the nonlinear current model, the electromechanical coupling term, and the segmented mechanical beam model. Diab et al.[67] also proposed a comparable electromechanical model composed of an equivalent circuit model and beam model. Compared with the previous equivalent circuit model,[60–62,64] the models[66,67] can not only accurately simulate the deformation of the actuator but also predict the blocking force. Moeinkhah et al.[68] provided a distributed equivalent circuit model (Figure 5c). Based on the distributed equivalent circuit model, it was determined that charge density, current, and voltage are functions of actuator length. Using the Golubev method, an infinite-dimensional impedance model was developed and replaced with a simple second-order electromechanical model. Caponetto et al.[69] used an improved equivalent circuit model to identify the electrical model and electromechanical model using the objective optimization algorithm of the Nelder–Mead simplex method. Chang et al.[70] introduced a new linear time-varying model that can demonstrate the hysteresis phenomenon of IPMC actuating process. Taking into account the IPMC as a physical cantilever with force distribution on the upper surface, the team believes the model can be more precise and require less time due to its more realistic properties. Considering that the surface resistance of IPMC may change during the deformation process,[48] but the research on the self-sensing of IPMCs has not provided an accurate mathematical model, Nam et al.[71] developed an accurate mathematical model for the self-sensing of IPMC actuators and analyzed the impact of surface resistance changes on IPMCs using mathematical methods. In order to avoid more complex modeling and accurately represent the degradation of IPMC materials over time due to repeated use, Carrico et al.[72] used a control method based on feedforward learning, Bayesian optimization, to implement motion control of IPMC equipment. The IPMC electromechanical model (including electrical model and mechanical model) was used for gait optimization of the ML algorithm. Improved actuator performance and improved robot motion speed can be achieved through ML, and proof-of-concept can be utilized to further advance more complex IPMC-based devices.
The equivalent circuit model equates the IPMC actuator as a circuit and incorporates the operating current of IPMC into the model. The modeling principle is that when the IPMC actuator is working, the current flows through the actuator, and the output displacement of the actuator is related to the current. Such models can predict actuator displacement,[60–68,70–72] develop controllers,[72] optimize actuator performance,[63,64,71] and develop IPMC-based devices.[72]
Equivalent Beam ModelsThe elastic modulus of IPMCs contributes to its bending stiffness, but due to its complex deformation under an electric field, its elastic modulus is difficult to determine experimentally.[19] Lee et al.[73] introduced an equivalent beam and bimorph beam model for an IPMC actuator and determined Young's modulus and electromechanical coupling coefficient of the actuator using the rule of mixture, bimorph beam equations, and measured force–displacement data of a cantilevered IPMC. Ji et al.[74] noted that both the equivalent beam (Figure 6a) and the equivalent bimorph beam model are appropriate for analyzing the deformation of IPMC actuators with various thicknesses and polymer membranes. Previous research on beam models mainly focused on working in the air.[73,74] Considering that IPMCs can be activated in water,[7,75] some researchers have studied the model of IPMC as a biomimetic fish actuator. Brunetto et al.[10] proposed using the classical Euler–Bernoulli cantilever beam theory and hydrodynamic functions to describe the interplay between the actuator and the fluid, which has better estimation ability for underwater operation than the previously proposed IPMC actuator models. Chen et al.[17] combined Lighthill theory and mixed tail dynamics (Figure 6b) to obtain a computational model for the speed of IPMC propulsion robotic fishes. Wang et al.[76] applied the Lighthill theory on the elongated body to evaluate the thrust performance of robotic fish and obtained the beam dynamics of IPMC in fluid, thus obtaining the velocity model and thrust efficiency model of IPMC fishes. The velocity model in ref. [76] consists of finite terms, which is simpler than the velocity model in ref. [17], that includes infinite terms and approximates using the first three terms. More importantly, the article provides a thrust efficiency model. IPMC actuator exhibits significant nonlinear bending deformation through the complex coupling of mechanical, electrical, and chemical properties (Figure 6c).[77] A study utilized digital image correlation to derive the key relationship between the strain gradient and the excitation voltage,[77] as opposed to relying on assumptions and the elliptic integral method to develop an analysis model. Notably, the absolute node-coordinate formula (ANCF) has been shown to be advantageous in studying the dynamics of flexible multibody systems with high geometric nonlinearities due to the strong coupling between large global motion and large deformation of structures and the physical nonlinearities associated with soft materials.[78] Li et al.[79] observed that there is no report on the large deformation modeling theory based on ANCF in the research on IPMC-based underwater bionic robotic fishes. The team first explored the feasibility of ANCF in modeling IPMC-based fish tail dynamics, which contributes to the study of the large swing motion mechanism of biomimetic robotic fish.[79] The deformed 1D double-node beam element is shown in Figure 6d. IPMC is often used as a tail fin in robotic fish,[10] and Karthigan et al.[80] analyzed the dynamic characteristics of IPMC actuators as flapping fins.
The equivalent beam model equated the IPMC actuator to a cantilever beam and incorporated the material characteristics of the actuator into the model. Such models can predict actuator displacement,[10,17,73–80] develop controllers,[81–84] optimize actuator performance,[73–75,77] and develop IPMC-based devices.[17,78–80,83]
Multi-Segment ModelsAlthough the use of IPMC actuators in functional devices necessitates a nonlinear large deformation dynamic analysis of the system, these modeling strategies cannot account for the actuator's large bending deformation (geometric nonlinearity).[85] Given the large deformation of the IPMC actuator, Yim et al.[86] proposed an analytical framework for the segmental dynamic model of the IPMC. The RC model was utilized for the electromechanical characteristics of the IPMC, while the finite-element method (FEM) was devised for the mechanical portion of the IPMC. This model's proposal for the independent control of each segment is essential for developing seven complex motion robotics. Because its coordinate system does not account for each component's rigid body rotation, the model cannot accurately describe the large deflection of IPMCs.[87] Gutta et al.[87] improved the model[86] by modeling each segment of IPMCs as a dual-node beam finite element with a local coordinate system (Figure 7a), where the rigid body motion experienced by the coordinate system is equal to the rotation of its first node. Model[86] and model[87] are effective for large deformation prediction of IPMC actuator, but the derivation process of motion equation based on FEM is very complicated, and the centralized RC model is not accurate in predicting actuator performance.[85] In light of the aforementioned flaws, Amiri Moghadam et al.[85] proposed a dynamic model using the rigid finite-element (RFE) method. The RFE method, unlike the conventional FEM, discretized the flexible linkage into rigid elements, which aided in representing the inertial properties of the object. Similar to the principle of the RFE method, the pseudorigid body method analyzes the kinematics and dynamics characteristics of compliant mechanisms.[88] The pseudo rigid body method is applied to the IPMC actuator, which approximates the IPMC actuator as a compliant component and converts it into an appropriate equivalent rigid body mechanism (Figure 7b).[89] In order to predict the tip deformation of the IPMC actuator, the pseudorigid body modeling and the kinematics model of the Euler–Bernoulli method are used.[90] However, traditional four-bar crank rocker mechanisms made of rigid connecting rods can only produce one path through the rocker tip when the crank rotates once, the IPMC actuator can produce working volume with the rocker in a partially compliant four-bar mechanism.[91] Water content impacts the actuating performance of IPMCs, and the influence of dehydration on bending resistance and bending response can be evaluated using a variable parameter pseudorigid body model.[89] Other studies also adopted the thought of multisegment, decomposed an IPMC actuator into 20-link hyper-redundant serial manipulator, and solved the problem of IPMC's hyper-redundancy based on a new inverse kinematics method.[92] In order to propose a robust deterministic model that can control the performance of a reliable actuator, Chattaraj et al.[93] modeled IPMCs as multisegment chains connected to a rigid body (Figure 7c) and adopted an inverse kinematics solver based on circular coordinate descending to minimize the objective function in joint space through step-by-step iterative steps to solve the redundancy problem.
The modeling principle of multisegment body model is to treat IPMC as a multisegment rigid body and establish a dynamic analysis framework of multisegment rigid body. This kind of model can predict the displacement of the actuator[85–92] and optimize the performance of the actuator.[88,89] However, there is no literature to record its research on controller development and device development.
Physical-Based ModelsThe potential mechanism of IPMC actuator deformation is described by a physical-based model. These models characterize the mechanics and electrochemistry of IPMCs using constitutive equations and first principles.[94] In order to determine the response of IPMCs, only physical parameters derived from governing equations and constitutive equations and independent experiments are required.[94] The model based on microstructure analysis, Nernst–Planck (NP) equations, and thermodynamic theory makes up the majority of the currently available physical-based models. Considering the complexity and difficulty of physical model calculation, the simplification and calculation of multifield coupling models are also included in a single category. Table 2 summarizes the characteristics of physical-based models.
Table 2 The properties of physical-based models
| Model type | Publication, Year, Reference | Mass transport | Diffusion or Convection | Back-Relaxation | Actuation mechanism | Model Characteristics |
| Models based on microstructure analysis | Micromechanics models | |||||
| Nemat-Nasser et al., 2000,[95] | Ion | Diffusion and Convection | × | Electrostatic stress |
|
|
| Nemat-Nasser, 2002,[96] | Water | Convection | × | Electrostatic stress |
|
|
| Nemat-Nasser et al., 2006,[99] | Ion and water | Diffusion | × | Osmotic, electrostatic, and elastic force |
|
|
|
||||||
| Davidson et al., 2008,[97] | Ion and water | Diffusion | √ | Electrostatic interactions |
|
|
| Peng et al., 2013,[98] | Ion and water | Diffusion | × | Electrostatic interactions |
|
|
| Frictional models | ||||||
| Tadokoro et al., 2000,[101] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Tadokoro et al., 2002,[100] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Gong et al., 2009,[102] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Gong et al., 2011,[103] | Ion and water | Diffusion | × | Swelling stress |
|
|
| He et al., 2011,[104] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Models based on NP equation | Research on internal mass transport and actuation characteristic | |||||
| Zhu et al., 2011,[22] | Ion and water | Diffusion and Convection |
|
|||
|
||||||
| Zhu et al., 2013,[105] | Ion and water | Diffusion and Convection | √ | Swelling stress |
|
|
|
||||||
| Schicker et al., 2013,[106] | Ion and water | Diffusion and Convection | √ | Osmotic and electrostatic forces |
|
|
| Zhu et al., 2013,[104] | Ion and water | Diffusion and Convection | √ | Swelling stress |
|
|
| Research on the structural characteristics of IPMC | ||||||
| Pugal et al., 2011,[111] | Ion | Diffusion | × | Swelling stress |
|
|
| Chang et al., 2014,[114] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Shen et al., 2015,[112] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Liu et al., 2018,[116] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Chang et al., 2019,[121] | Ion and water | Diffusion | × | Swelling stress |
|
|
|
||||||
| Wu et al., 2021,[14] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Annabestani et al., 2021,[119] | Ion and water | Diffusion and convection | × | Swelling stress |
|
|
|
||||||
| Rao et al., 2022,[122] | Ion and water | Diffusion | × | Swelling stress |
|
|
| Models based on thermodynamic theory | TIP models | |||||
| De Gennes et al., 2000,[124] | Ion and Water | Diffusion and convection |
|
|||
| Paquette et al., 2003,[125] | Ion and Water | Diffusion and convection | × | electro-phoretic pressure |
|
|
| Tixier et al., 2013,[128] | Ion and Water | Diffusion | × | Swelling stress |
|
|
| Tixier et al., 2015,[128] | Ion and Water | Diffusion | × | Swelling stress |
|
|
| TCC models | ||||||
| Wallmersperger et al., 2009,[131] | Ion and Water | Diffusion | × | Volume displacement and electrostatic forces |
|
|
| Nardinocchi et al., 2011,[132] | Ion and Water | Diffusion | × | Volmetric distortion field |
|
|
| Cha and Porfiri, 2014,[24] | Ion | Diffusion | √ | Osmotic and electrostatic forces |
|
|
| Boldini et al., 2020,[8] | Ion | Diffusion | √ | Osmotic pressure and Maxwell stress |
|
|
|
||||||
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| Olsen and Kim, 2021,[23] | Ion and Water | Diffusion | × | Osmaotic pressure and electrostic effects |
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| Zhang and Porfiri, 2022,[139] | Ion | Diffusion | √ | Osmotic pressure and Maxwell stress |
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In 2000, Nemat–Nasser and Li[95] published the first complete microscale physical model of IPMCs. The core of Nemat–Nasser's model is the formation of ion clusters by ion groups in Nafion (Figure 8a) and the redistribution of charges within IPMCs after the voltage is applied. Nemat–Nasser believed that electrostatic force was the main cause of electroinduced deformation of IPMC actuators. Afterward, Nemat–Nasser[96] considered the influence of IPMC electrodes, modified the model,[95] and focused on its micromechanics. The revised model further considered the impact of water migration caused by the redistribution of positive ions. Other studies[97,98] are also modeling based on the model.[96] They also believe that the deformation of IPMCs under voltage is dominated by electrostatic force. Davidson et al.[97] simulated the actuation of ionic liquid IPMCs. In order to describe the IPMC's actuation in air, the actuation model was changed to take into account the finite volume of the moving cations.[97] The model has good reliability and can guide the model development of novel actuators composed of novel polymers and solvents. Peng et al.[98] proposed an improved model based on Nemat–Nasser's model to explain the initial nonlinear response of electric actuation to fixed-end deformation. Nemat–Nasser et al.[99] established a model considering the effect of cations on IPMCs actuation. The model can be used to optimize and predict the electromechanical performance of IPMCs.
The white-box model of IPMC actuators established by Tadokoro et al.[100] based on the physicochemical assumption of motion principle is called the frictional model. The frictional model describes the transport process by establishing the force balance equation of specific components in the IPMC mass transport process, where friction is the key to introducing mutual coupling between each component.[22] Tadokoro et al.[101] pointed out that mobile cations quickly migrated through the actuator under the action of the electric field, thus dragging water molecules. The sodium ion is subject to electric field force, diffusion force (sodium ion and water), and viscous resistance (Figure 8b), and a model is established according to the force balance equation. Changes in volume due to water content, the electrostatic force due to the departure of the sulfonic group, and momentum conservation effects produced internal stresses, but the scale of the electrostatic force is smaller than the expansion stress due to the migration of ions and water. Later, Tadokoro et al.[100] improved the frictional model in the following aspects: the lateral strain caused by sodium ion migration and sulfonyl concentration was introduced into the model.
However, Water diffusion resistance was not considered in another study.[100,101] Gong et al.[102] used Tadokoro's model to derive a multifield finite-element method and put forward a numerical solution. Later, Gong et al.[103] developed a finite-element model coupled with an RC circuit based on Tadokoro's model, which could effectively establish the relationship between voltage and current in the membrane. Based on Tadokoro's model,[101] He et al.[104] quantitatively explored the influence of Nafion membrane thickness on the electromechanical performance of IPMCs from the theoretical model level.
The model based on microstructure analysis is a kind of physical-based model, which is mainly divided into two kinds: micromechanics model and frictional model. The two models include information about the electrode of the actuator, ions in the base membrane and water in the modeling range. The core of the micromechanics model is the ion cluster formed by the ion group in Nafion, and the core of the Frictional model is the analysis of the force on sodium ions and water. Both enable displacement prediction,[96–103] performance optimization,[97,99,101] and the development of IPMC-based devices. There is no relevant research on the development of IPMC devices, but in theory, it is possible to analyze the working state of devices using these two models.
Models Based on NP Equation Research on Internal Mass Transport and Actuation CharacteristicNemat–Nasser and Li[95] proposed the first microscale model of IPMCs. In fact, this model adopted a modified form of the NP equation, which was also the first time that the NP equation was used to establish the process model of ion distribution inside IPMCs. However, this model[95] only considered the convection effect of water, not the diffusion effect of water. Zhu et al.[22] proposed an improved IPMC transport model under the general NP theoretical framework. The model gave a complete water molecular flux equation, illustrating the relationship between the convective term and the elastic stress. The model took the convective effect rather than diffusion as the main transport mechanism and considered self-diffusion and electroosmotic resistance in the water flux equation. Zhu et al.[105] proposed a comprehensive multiphysical model to explain the complex deformation behavior of IPMCs based on their knowledge of the actuating mechanism. Different from the model,[22] the model[105] emphasized the influence of pressure on flux and water transport on deformation. It provided a convenient method for IPMC to deal with internal intrinsic stress and external mechanical load (Figure 9a). The model, such as in other studies ,[22,95,105] could not describe the back-relaxation phenomenon in the IPMC actuating process. Schicker et al.[106] used the cluster model developed by Nemat–Nasser to establish the IPMC multiphysical model based on Poison-NP (PNP) framework. Although Schicker's model correctly modeled the phenomenon of back-relaxation, it did not suggest what factors affect back-relaxation in IPMCs. Based on Zhu's model,[105] another study[104] described the back-relaxation phenomenon under varying water content, and when combined with the multiphysical model, the effects of various intrinsic stresses on the deformation were determined. In addition, the relationship between the osmotic pressure, the total electrostatic stress, and the back-relaxation phenomenon is provided, which adequately explains the back-relaxation phenomenon of IPMCs from a water content standpoint.
In a number of earlier theoretical models, IPMC electrodes were not considered[95] or were only regarded as optimal flat conductors.[22,104–106] According to the actual structural characteristics of IPMCs,[107] it is obviously unacceptable not to consider the electrodes on both sides of IPMCs. Relevant studies have shown that the electrode shape of IPMCs would affect the electrical characteristics of the actuator,[108] and the electrical characteristics of the actuator would affect the mechanical characteristics of its output.[109,110] Existing studies have shown that the electrode plays a very important role in the modeling and calculation of IPMC actuators, which is of great significance for improving the accuracy of calculation.[111,112] Pugal et al.[111] used the Ramo–Shockley theorem to couple the current in the polymer to the current in the IPMC continuous electrodes. This approach gives an idea of how to add other currents, which will allow the model to be used in higher applied voltage calculations. Shen et al.[112] proposed a model of the IPMC actuator based on the Poisson-NP equations of ion transport and charge dynamics in polymer membrane, combined with a physical model of surface resistance changed due to the deformation of the IPMC electrodes. According to other studies,[108–113] it is obvious that idealized electrodes are unscientific, and accurate characterization of electrode surface features is particularly important. Chang et al.[114] proposed a continuous regular curve to represent the IPMC electrode by analogy with the reality of the IPMC electrode. Bar-Cohen et al.[115] adopted the Koch curve to represent the electrode shape of IPMC. Despite the initial attempted to represent the characteristics of the electrode interface,[114,115] the influence of the rough interface is still based on idealized profiles and fixed parameters, and the real shape of the electrode interface is far from being revealed, so it is urgent to propose a model that can effectively represent the main characteristics of the real interface shape. Liu et al.[116] proposed a method to build its model in mass transport simulation by the Weirstrass–Mandelbrot (W–M) function (Figure 9b) and extracted actual interface curves from IPMC samples as reference and standard of W–M function. The W–M equation is more accurate relative to actual than the plate and Koch geometry. Although some achievements have been made in the study of rough interfaces, further studies are needed, such as contour extraction, selection of model parameters, influence of manufacturing processes, and interface formation mechanisms, to reveal the material transport mechanism within the IPMC. Yang et al.[117] established a 3D IPMC electromechanical model based on the W–M function.
The first 3D model of IPMCs describing internal physical processes was proposed by Ounaies et al.[118] Stalbaum et al.[13] proposed a tubular IPMC and introduced the mathematical modeling, manufacturing, and experimental performance of the tubular IPMC. The simulation using the model provided was in good agreement with the experimental results of simple harmonic input motion. Wu et al.[14] designed a spiral IPMC, built a multiphysics model, and simulated the model using Comsol Multiphysics. The simulation data of the model is in good agreement with the data measured by the prepared spiral IPMC sample. In addition, the above 3D models all believed that the ion migration of IPMC in the 3D model was oriented. Annabestani et al.[119] believed that ions in IPMC migrate in all directions and need to be solved in 3D space and put forward a fully analytical 3D physical ion transport model.
In general, the output force of IPMCs can be improved by simply stacking Nafion membranes,[12] but the thickening of the membrane will lose the flexibility of the actuator and reduce its range of motion, which is also illustrated by the theoretical model and experiment of He et al.[104]. A bending structure can improve the flexibility of the structure and also maintain its own mechanical force.[120] At the same time, with the development of the relevant electrochemicalmechanical coupling model theory, Chang et al.[121] proposed an IPMC with a crenellated structure, which uses the PNP model and different shape parameters to explore the crenellate ratio and mechanical output. After model simulation, it is found that the newly proposed crenellated structure can significantly improve the mechanical properties of IPMCs compared with the traditional structure, and the model can be used to assist the development of IPMCs. Inspired by the research of Chang et al.[121] Rao et al.[122] proposed an IPMC with a simple preparation process and a serrated electrode interface. Model simulations and experiments have shown that the new structure of IPMCs has high-performance electroactive displacement.
There are many IPMC actuator models based on the PNP framework. According to the focus of model research, it can be divided into research on internal mass transport and actuation characteristic and research on the structural characteristics of IPMC. It can predict IPMC displacement,[14,104,105,111,112,114–119,122] optimize IPMC performance,[114,121,122] and develop IPMC-based devices.[14,18,118,123]
Models Based on Thermodynamic Theory Thermodynamics of Irreversible Process (TIP) ModelsAccording to the thermodynamics of irreversible process (TIP), the process of mass transfer can be described by driving force and mass flux. Although the process is in a nonequilibrium state, it can be approximated by local equilibrium.[22] The generalized force and flux can be approximated as linear by the Onsager equation. De Gennes et al.[124] proposed the TIP model of ion and water molecule transport within IPMCs for the actuating process of IPMCs (Figure 10a). However, De Gennes et al.'s model did not include the parameter actuator model of IPMC itself. In order to predict the influence of surface particles on the performance of IPMC, Paquette et al.[125] incorporated the capacitance and resistance model into the TIP model. The model can predict the mechanical characteristics of the actuator and study the effect of the IPMC granular electrode. The model of ref. [125] cannot study the large nonlinear deformation of the IPMC actuator. Liu et al.[126] used the standard Onsager formula based on TIP to describe the actuation phenomenon and predict the deformation of the IPMC actuator. The model was used to predict the large nonlinear deformation behavior of the IPMC actuator with high accuracy. It is worth noting that the models[125,126] cannot describe the back-relaxation phenomenon of IPMCs. Sun et al.[127] proposed a method combining the TIP model (static model) and the dynamic model. The static model accurately predicted the static characteristics of the actuator according to the Onsager equation. The dynamic model revealed the back-relaxation characteristics of the actuator. Other studies have modeled the IPMC actuating process from a new perspective based on TIP models. Tixier et al.[128] proposed a detailed approach to this polymer material using the concept of a nonequilibrium thermodynamic process and then modeled the material through the coexistence of two phases (solid and liquid). The mass conservation equation, charge conservation equation, momentum conservation equation, and energy balance equation on the macroscopic scale of materials were derived by using the average technique. Then, based on the conservation law of electroactive polymers established in ref. [128], the team used the same method to derive the law of entropy equilibrium and thermodynamic relations.[129] The entropy generation was deduced, the generalized force and flux were identified, and the constitutive equation of the complete material was written. Based on the published TIP models,[127,128] the bending process of the IPMC actuator was numerically modeled. The model can also study the influence of actuator shape on tip displacement and blocking force.
While the underlying physical mechanism behind the actuation response of IPMCs has been reasonably well understood and agreed upon for some time, the continuum-mechanical modeling of the coupled chemoelectromechanical response of IPMCs in a thermodynamically consistent fashion has been a more recent effort.[130] This kind of model leverages continuum thermodynamics models that are derivable from potential functions. Within the continuum mechanics and thermodynamics framework, charge redistribution is typically governed by the so-called PNP system of partial differential equations (PDEs).[94] In PNP systems, the counterions redistribution is driven by diffusion and electromigration due to the electric field associated with the voltage difference applied to the two ends of the electrode. In this context, the voltage is obtained from the Poisson equation produced by Gauss’ law, relating the voltage and net charge in the membrane.[94] Wallmersperger et al.[131] used a thermodynamic formula developed by applying the Legendre transform to piezoelectric materials to describe the driving mechanism of IPMCs. The electrochemical transport model could calculate the charge density and the electric flux of the applied electric field distribution in space and time. Based on the laws of thermodynamics, a mechanical model describing the internal strain of materials was established. This model could predict the displacement behavior of the actuator well. In the context of 3D linear elasticity theory, Nardinocchi et al.[132] modeled the complex responses of IPMCs to electrical stimuli, including chemically induced deformation fields, which are thermodynamically consistent with the derivation of the final PDEs for multiple physical problems. According to the first principle, the redistribution of ions within the double layer will generate Maxwell stress, whose intensity is controlled by the voltage gradient at both ends of the double layer.[94] However, some studies did not consider the influence of Maxwell stress on the IPMC actuating process,[131,132] but the influence of Maxwell stress on the IPMC actuating process cannot be ignored.[133] Some studies considered the influence of Maxwell stress on the IPMC actuating process,[134,135] but these studies did not model the complex back-relaxation phenomenon over long time scales.
Based on the polyelectrolyte formula proposed by Hong et al.[136] Cha and Porfiri[24] proposed a modeling framework for studying the macrostatic deformation and electrochemistry of IPMCs based on the novel Helmholtz free energy density, which described the mixing of counterions and polymers and the electric polarization in IPMCs. On the basis of the Cha and Porfiri model framework,[24] Porfiri[137] proposed that Maxwell forces, which are frequently disregarded in IPMC models, play a prominent role in explaining back-relaxation. The IPMC actuation is due to the nonlinear interaction between osmotic and electrostatic phenomena, and the relationship between the applied voltage and the degree of back-relaxation is established. Porfiri et al.[25] investigated the role of steric effects and composite layer on back-relaxation based on the previously proposed mechanical and electrochemical consistent framework. To gain insight into the role of physical and geometric parameters, the resulting nonlinear PDEs are solved semianalytically using the method of matched asymptotic expansions, for the initial transient and the steady state.[25] A related study shows that the bending moment generated by Maxwell stress increases with the dielectric constant since the thickness of the electric double layers where Maxwell stress is relevant is larger.[133] The performance and size of 3D-printed IPMC actuators are comparable to that of traditional IPMCs, and 3D-printed IPMCs have been developed as crawling robots and manipulators.[72,138] With new advancements in additive manufacturing of IPMCs, it becomes of paramount importance to possess predictive modeling tools that could inform the selection of material constituents and the design of complex shapes toward specific objectives.[8] The study of IPMC multiaxial deformations (Figure 10b) by Boldini et al.[8] made modeling tools take an important step in this direction. In order to realize the application of IPMC actuators in robotics, medical equipment, and microsystem technology, Zhang and Porfiri[139] applied the thermodynamic consistent continuum model[24] proposed by Cha and Porfiri to study the actuation of arch IPMC (Figure 10c). An analytical framework for arch IPMC actuation is proposed, which quantitatively assesses the effect of bending on actuation, which may inform the future design of 3D-printed IPMCs. In addition, Olsen and Kim[23] developed a new actuator hyperelastic porous media framework for IPMCs using the principles of continuum thermodynamics and multiphasic materials. This model encompasses almost all the physical formulas in the literature.
Models based on thermodynamic theory are the focus of recent research. This model can also predict the displacement of IPMC,[23,124,130–132,139] optimize the performance of IPMC,[25,125,133,137,139] and develop IPMC-based devices.[140]
Applications Microelectromechanical Systems (MEMSs)A series of IPMC-based microelectromechanical devices, including optical focusing devices, micropumps, and microfluidic devices, have been developed. Kim et al.[141] proposed the application of the hook actuator of IPMCs in the autofocus (AF) lens actuator, which can achieve focusing by adjusting the loading voltage (Figure 11a). An annular IPMC was used as a micromirror actuator to adjust the tilt of the lens (Figure 11b).[142] The chip IPMC serves as a soft engine stretching Fresnel zone plate to tune the focus of the ultrasound beam, and its main structure and working principle are shown in Figure 11c.[143] Microvalves in microfluidic applications are also important applications of IPMCs. The IPMC bends to actuate the PDMS to connect the inlet and outlet chambers so that fluid can pass through the microvalve (Figure 11d).[144] Annabestani et al.[145] used the IPMC as a micromixer for microfluidics by exploiting the periodic oscillation characteristics of the chip IPMC at AC voltage (Figure 11e). The microfluid pump is also an important active component of the microfluid system. According to the working principle, the pump can be divided into the peristaltic pump and the closed pump. The microflow pump (peristaltic pump) designed by Sideris et al.[15] can achieve a pumping rate of 669 pL s−1 and a pumping rate of 9.18 nL s−1. Nam et al.[146] designed a closed pump to separate the IPMC from the pumping fluid, and the diaphragm was designed to deform up to 0.4 mm. Switches such as p–i–n diodes, varactors, and microelectromechanical systems (MEMS) can be employed to construct a reconfigurable antenna by altering the electric length or changing the loading reactance of the antenna.[147] These methods are more traditional and have the disadvantages of complexity and high-power consumption. Chang et al.[147] integrated IPMC actuators into radio frequency identification tag antennas for the first time to realize frequency reallocation. The IPMC swings up and down to achieve a wide frequency tuning range within one degree of freedom, and the three views and bias circuits of its system are shown in Figure 11f. Cheong et al.[148] and Chang et al.[18] both proposed wirelessly powered IPMC operating devices based on external radio frequency (RF) magnetic field operation. Using the developed axially actuating IPMCs, Wu et al.[14] developed a three-degree of freedom microparallel platform (Figure 11g), which can realize complex coupled motion of pitch, roll, and yaw.
Natural organisms are a significant source of inspiration for scientific studies. Li et al.[149] designed the flytrap gripper based on the biological flytrap. The opening and closing of this device are actuated by IPMCs, and its structure is shown in Figure 12a. Other plants that can achieve opening and closing movements, such as tulips and forsythia, are also imitated objects.[150] Feng et al.[151] proposed a miniature hand-shaped gripper whose instantaneous maximum output force is 5mN. A micromechanical catcher for debris capture is proposed, whose actuation device is coated with SU-8 mesh and polydimethylsiloxane film (Figure 12f),[152] which can operate in the air for a long time. Similarly, He et al.[153] prepared IPMC grippers with high durability in air through chemical modification. Compared with the rigid mechanism, the mechanism composed of IPMCs has high compliance. Jain et al.[154] used a flexible four-bar mechanism composed of IPMC to insert the pin into the hole (Figure 12) and could control the motion path of the mechanism. The team proposed a selective compliant assembly robot arm using an IPMC-based microgripper for peg-in-hole assembly (Figure 12c).[155] A more flexible and compliant two-link flexible manipulator is used in robot assembly (Figure 12e).[16] Soft fingers and joints have great potential in robot assembly flexible link manipulators. To improve the sensitivity of the gripper, Chen et al.[156] designed a two-finger gripper consisting of an IPMC actuator and an ultrasensitive force sensor (Figure 12d). Cheong et al.[157] designed a wirelessly powered IPMC microgripper. When the device is exposed to a magnetic field and the frequency matches, the gripper will be activated. Its design and working principle are shown in Figure 12g. Some studies have mentioned tuning fork grippers and arched auxiliary grippers.[158,159]
In addition to MEMSs and soft grippers, bionic robotic systems using IPMC actuators have been explored. Yamakita et al.[160] developed a snake moving robot using an IPMC actuator, which serves as both the connecting rod and the actuator for smooth propulsion (Figure 13a). In order to develop a low-power robot, Vahabi et al.[161] proposed a quadruped pipe robot and a biped pipe robot (Figure 13b), which are microrobots capable of moving at a speed of about 1 mm s in a narrow, confined area. Stasik et al.[162] developed an IPMC self-actuation roller device with a mass block in the middle of the rim, which is attached to the external wheel by four IPMCs (Figure 13c). The periodic contraction of the IPMC can realize the rotation of the wheel robot. Similarly, Firouzed et al.[163] proposed a circular robot composed of six IPMCs connected for mobility (Figure 13d). The underwater robot is a research hotspot, and underwater microrobot working in narrow space has broad application prospects.[164] Guo et al.[164] developed a new type of underwater microrobot (Figure 13e). Ten IPMC actuators are used as bionic legs or fingers, and four actuators are used as legs to achieve walking, rotating, and floating movements. The other six actuators are used as fingers to grip small objects. Shi et al.[165] developed a mother–child robot system based on IPMC actuators. The claw of the subrobot is in the head, six legs play the role of support, forward, and rotation, and the tail wing actuates forward. The team developed a mobile bionic flytrap inspired by the Venus flytrap (Figure 13f).[166] The device can maintain a maximum payload of 36 mN at 7 V voltage. Kodaira et al.[167] proposed a monolithic thin film robot based on an IPMC multilayer casting process that does not require mechanical or electrical assembly and could swim (Figure 13g). Wang et al.[168] combined dielectric elastomer (DE) with IPMC to develop a jellyfish robot with 2D operation capability (Figure 13h). When the DE receives a sinusoidal signal of 5 kV with a frequency of 1.4 Hz, the average speed of the robot is about 4.8 mm s−1, and the maximum average course angle change can reach 3.02°. Chen et al.[17] applied IPMC to the tail fin of the bionic robotic fish (Figure 13i). In fact, the design of the propulsion device of the underwater vehicle is also very important, because the chip IPMC can only achieve bending motion, not more complex twisting motion. Therefore, research on propulsion devices for underwater systems is ongoing, and Kim et al.[169] could realize the possibility of a patterned electrode IPMC for torsional action as a fin. In addition, Chen et al.[170] combined multiple IPMCs on the same passive film to produce bionic fins, which could also achieve torsion and other complex deformation capabilities (Figure 13j). Most IPMCs used in the above applications are 2D, and 3D printing can produce 3D IPMCs. Carrico et al.[83] used fuse 3D printing to create 3D IPMC structures, fabricating IPMC robotic hands (Figure 13k) and a caterpillar-inspired 3D-printed IPMC crawling robot (Figure 13l).[171]
As actuators utilized in the development of medical devices, IPMC can be developed into active interventional catheters, auxiliary rehabilitation devices, and scalpel drive devices by virtue of its own superb characteristics. By changing the shape of traditional chip IPMCs and casting Nafion into solid or hollow cylinders, two types of tubular IPMCs can be prepared.[172,173] By embedding the IPMC into a flexible tube formed of organosilicon (Figure 14a), this IPMC-based catheter can avoid the environmental impact of the IPMC.[174] He et al.[175] proposed a square-shaped rod-shaped IPMC with multidegree of freedom movement, which can be successfully inserted into the organ model (Figure 14b). The IPMC catheter devices with the same morphology and embedded optical fiber are used in biomedical applications (such as microendoscopic eye surgery) (Figure 14c).[45] The IPMC can be developed as an assisted rehabilitation device. Feng et al.[176] assembled a multistage IPMC on a polyacrylamide hydrogel/polyp-xylene tube to develop a medical device for assisting peristalsis (Figure 14d). Fu et al.[84] proposed an active robotic surgical instrument for force control based on IPMC actuation and integrated sensing (Figure 14e). The device has a rotary joint connected with a scalpel, which is actuated by an IPMC, a strain gauge embedded as a feedback unit, and controlled by a PI controller, capable of maintaining a cutting force of 1 gf.[84] At present, the need for single-cell manipulation is increasing rapidly. McDaid et al.[177] described a 2-degree of freedom IPMC-actuated micromechanical device for cell processing (Figure 14), which is much safer than conventional devices. Wearable and implantable biomedical devices are also the focus of research. Chang et al.[18] designed a wireless drug delivery device based on IPMC (its structure and principle are shown in Figure 14g). Wireless drug delivery only releases drugs in local areas, reducing the use of drugs and avoiding the side effects caused by traditional implantable devices. For eyelid dysfunction caused by facial palsy, Hosseini et al.[123] developed an eyelid motor function-assist device (Figure 14h), which was used to close and open the eyelid.
This article reviews the modeling methods and the applications of IPMC actuators. They are divided into four major categories, nonphysical identification models, partial-physical models, physical-based models, and applications. Their key techniques and characteristics are summarized and discussed. The characteristics of different types of models are shown in Table 3.
Table 3 IPMC actuator model comparison
| Type of model | Nonphysical models | Partial-physical models | Physical-based models |
| Comprehensiveness | Low to medium | Medium | High |
| Accuracy | Low to medium | Medium | High |
| Computational complexity | Medium to high | Medium | Very high |
| Configuration effort | Low to medium | Low to medium | High |
| Time | Less time-consuming as prior IPMC knowledge is not required | Simple and easily understand, so medium time-consuming | Solving control equations consumes a lot of time |
| Explanatory actuating mechanism | Non | Low to medium | High |
| Merits | Do not rely on the IPMC model, eliminating the tedious process of physical modeling. Can quickly establish the relationship between input voltage and output displacement. | Simple structure from the natural sciences of electricity, mechanics, and physics. The model parameters are identified by the experiment | The mathematical model established by the knowledge of electrochemical theory and mechanics theory can better reflect the characteristics of the IPMC actuating process |
| Demerits | The estimation accuracy depends heavily on the number of samples. When the sample size is small, and the numerical error rate is high, the model will be over-fitted and under-fitted | It cannot reflect the internal characteristics of the IPMC well | The complex physical mechanisms of interplay within IPMC are not fully understood |
| Applications | Prediction, Control | Prediction, Control, Optimization performance, Development devices | Prediction, Optimization performance, Development devices |
As a modeling method that only depends on the electric field (the same IPMC actuator outputs different displacements at different voltages), the time series mathematical relationship between the input voltage and output displacement is established. From the process of establishing this kind of mathematical relationship, it is a process of identifying different displacements under different voltages. According to the method of establishing mathematical relations, the models dependent on the electric field can be divided into the classical identification model and the intelligent identification model. The common feature of classical identification models is that they are built with nonintelligent models. The difference is that they use different modeling methods. The black-box identification model is widely used in the category of classical identification model. However, the classical identification model cannot describe the bending behavior of IPMC actuators well when facing inherent problems such as nonlinear and hysteresis. The common characteristic of the intelligent identification model is the model built using an intelligent model. The difference lies in using different models or combining different models to build the model. Compared with the classical model, the intelligent model can represent the nonlinear actuation of IPMCs with higher precision. In the category of intelligent identification models, the models can be divided into autoregressive models and nonautoregressive models according to different data sources. The precision of the autoregressive model is higher than that of the nonautoregressive model, but in some cases where it is not possible to monitor the displacement data of IPMC actuators in real time, the nonautoregressive model shows great advantages. The identification of displacement under large deformation is proposed, which is inseparable from the progress of displacement data measurement. In the nonphysical identification model, the time series displacement under different voltages can be well recognized and predicted and is widely used in control. However, this model cannot reveal the internal physical characteristics of IPMC actuators.
Partial-Physical ModelsPartial-physical models are usually related to electrical and mechanical theories, which are developed for the actuation characteristics of IPMC actuators. The modeling process usually uses the knowledge of natural sciences such as electricity, mechanics, and physics, and the parameters of the model are identified by experiments. This part mainly describes the establishment process of some physical models of IPMCs but does not give too much overview of the parameter identification process of the models. The parameter identification content of the model is placed in the section nonphysical identification model. This kind of model can be divided into four categories: the equivalent circuit model is equivalent to IPMC actuators as a circuit and combined with the mechanical model; the equivalent beam model is obtained by equivalent IPMC actuators to the cantilever beam combined with the electromechanical coupling model; the multisegment body model is equivalent to the IPMC actuator as a multisegment rigid body, which is obtained by combining the voltage-dependent position relation or the voltage-dependent mechanical relation. Some physical models have not fully revealed the internal laws of the actuator, but it is easy to get the analytical solution of the model and the transfer function of the system. The equivalent circuit model is widely used in the construction of other models and control. The equivalent beam model is widely used in the modeling of bionic fish. Multisegment body model provides some new ideas to solve the motion modeling of the continuum.
Physical-Based ModelsIt can be seen from the mechanism of IPMC actuators accurately describing the mass transport process inside polymer membrane is the key to establishing the physical model of the actuator and understanding the response of IPMC electrical stimulation. The essence of the actuating mechanism is that the transport of water and ions within the polymer causes the internal stress of the IPMC actuator. At its core is the NP equation. The model based on microstructure analysis includes the study of the micromechanical model and frictional model. The NP-based model includes some NP-based studies of the IPMC actuating mechanism, electrodes, and geometry. The models based on thermodynamic theory include the irreversible thermodynamic model based on the Onsager equation and the continuum thermodynamic model based on continuum mechanics and thermodynamic framework. Considering the complexity of the multifield coupling model, the related research on the simplification and calculation of the multiphysics model is introduced. Studies on the mechanism of back relaxation are also summarized, which are respectively explained from the perspective of water content and Maxwell stress. The physical-based model describing the physical mechanism of IPMC actuation can well explain the electrochemical and mechanical response of IPMC actuators under electrical stimulation, and it is a model that can clearly get the internal law. However, the calculation process of this kind of model is complicated, which is not conducive to obtaining the transfer function of the system. Therefore, this model is mostly used to study the actuating mechanism and performance optimization of actuators, but it is not paid attention to in the field of control because of its complexity. The simplified part of the physical-based model is the research to apply the physical-based model to the engineering field. The research has achieved good results and also provides the idea for the engineering application of the physical-based model.
ApplicationsIPMC actuators have been widely developed for MEMSs, soft grippers, bionic robots, and biomedical devices due to their superior characteristics. However, some performance defects of IPMCs and their own characteristics result in the unsatisfactory use effect of IPMC actuator devices. In order to better understand and use IPMC actuators, researchers have carried out a variety of mathematical modeling on IPMCs to explain their actuating process. These mathematical models establish mathematical relationships between key features in the IPMC actuation process and quantify the actuation process of IPMCs. More influencing factors are considered in the model, and the influence of these factors on the IPMC actuating process can be explored by modifying related factor parameters, thus achieving the purpose of performance optimization. IPMCs with more complex shapes can be manufactured by 3D printing compared with traditional preparation methods. However, it is obvious that no matter whether the IPMC is prepared by traditional methods or the IPMC manufactured by 3D printing, its mechanics, displacement, and internal mass transport characteristics need to be studied through physical-based models. A better understanding of the actuating process can lay the foundation for further use, optimization, and manufacturing. In the development of devices based on IPMC actuators, no matter the nonphysical identification model, the partial-physical model or the physical-based model can be used to simulate the motion process of IPMC actuators so as to further simulate the working process of IPMC-based devices. In addition, the accurate operation of IPMC-based devices cannot be separated from the precise control of IPMC actuators, and most of the IPMC control is dependent on the model. Most of the models used for control are linear empirical models. This model is simple and easy to calculate, but it cannot accurately describe the nonlinear behavior of IPMCs. More accurate physics-based models require more computing power, which is impractical to use in control applications. In addition to inversion control, almost all other methods require feedback. The control system developed using the IPMC's self-sensing model does not require complex sensing feedback. The feedforward method can also be used in the absence of sensors. The traditional feedforward method cannot compensate for the uncertainty of modeling. The control method based on feedforward learning can change this status quo. The IPMC self-sensing model or the feedforward learning control method can eliminate the complicated sensor system of IPMC-based devices and achieve more precise control.
Future TrendsWhether it is a nonphysical identification model, a partial-physical model, and a physical-based model, all these models are designed to represent the relationship between the input voltage and output displacement, and the most important evaluation index to evaluate this relationship is accuracy (precision, error). On the premise of ensuring accuracy (error, precision), there are two other important evaluation indicators: applicability (actuator model can be applied to control, performance optimization, explanation of actuator mechanism, or device-assisted development) and simplicity and speed of calculation. At the same time, the important role of the model in the development of IPMC-based devices is also one of the focuses of this article. Its future direction will be given at the end. The relationship between the development of future IPMC-based devices and the model is shown in Figure 15.
As shown in Figure 15, the block diagram on the left shows the characteristics of the model, as shown in Section 6.2.1, 6.2.2, and 6.2.3, respectively. The circular graph on the right represents the model-based development of IPMC-based devices, Section 6.2.4 of this article. The gray arrows from the block diagram on the left to the circle diagram on the right indicate the influence of the superiority of future models on the model-based development of IPMC-based devices. First, these superior characteristics directly affect the IPMCs modeling process, resulting in better-performing models, as shown in the inner circle on the right. Second, the blue arrows represent the multiple functions of the IPMC model. These functions include predicting IPMC displacement, optimizing IPMC performance, controlling development, and guiding equipment development (see the middle circle of the circle chart). Finally, the yellow arrows indicate the model's functions contribute to developing IPMC-based prototypes. The outermost ring of the circular pattern represents typical IPMC-based applications, including MEMSs, soft grippers, bionic robots, and biomedical devices.
Comprehensiveness and AccuracyAs mentioned earlier, IPMC actuator models are mostly small deformation models, and the sensing characteristics of laser displacement sensors are an essential reason for this phenomenon. A CCD camera can measure the large deformation tip displacement of the IPMC without deviation error. However, the speed of manual calculation is slow. In the future, a computer vision system can be proposed to automatically measure the IPMC tip displacement, which can significantly improve the measurement efficiency and accurately map the IPMC's actuating behavior. However, for systems that do not easily obtain displacement information (such as biomedical devices and robotics), it is necessary to use nonautoregressive models. Still, the accuracy of nonautoregressive models is low. The hybrid of distributed equivalent circuit model and ML model can greatly improve the accuracy of the nonautoregressive model. It is a future trend to introduce a hybrid model combining multiple models according to the characteristics of problems. More influencing factors are introduced into the model, such as the aging problem of IPMCs, so that the whole service cycle of IPMC actuators can be predicted. Therefore, the future model can be established using different models to map different deformation stages and life stages of IPMCs, choosing an appropriate model mix according to model characteristics to solve a problem or integrating multiple models to improve model accuracy. In the future, the identification of model parameters should have the characteristics of real time, and the data obtained from the computer vision measurement system should be input into the model for automatic identification of model parameters. It is foreseeable that more factors that cannot be quantified at this stage will be incorporated into the model.
Ubiquitousness of IPMC Model Application ScenariosAt present, the model is mostly used in control, displacement prediction, performance improvement, actuation mechanism research, and auxiliary development of IPMC-based devices, but the model has strong specificity. The nonphysical identification model and partial-physical model are mainly used for control and displacement prediction. The partial-physical model can be used to improve the performance of the actuator. Physical-based models contain more physical information and are used for displacement prediction, performance improvement, and actuation mechanism research. The application scope of the model is relatively fixed, and the barriers between the models are obvious. Improving the applicability of the model is an important development direction in the future. Therefore, it is necessary to propose a model for multiple application scenarios. One direction is to simplify the physical-based model to produce a simple model that retains more physical information and can be used in a variety of applications. Another direction of development is to propose a universal model that can be used in all application scenarios. When the model is applied to different scenarios, the switch between different application scenarios can be realized by adding or reducing internal modules.
Simplification and Computational Rapidity of IPMC ModelsPhysical-based models consist of nonlinear PDEs, and their analysis and simulation processes are often complex and time-consuming. However, most of the current models do not pay attention to model simplification and improving the solving speed. There are two ways to solve this problem: one is to simplify the nonlinear PDE model and second is to optimize the numerical solution method. For the first solution direction, the future development direction can be: 1) deleting the items that have little influence on the model; 2) linearizing the nonlinear model; 3) applying a new dimensionality reduction mapping method; and 4) using relevant parameters for a rough approximation. For the second problem, the promising development directions can be as follows: 1) Using a more reasonable FEM and mesh division method; 2) A numerical solution method more suitable for solving the multiphysical coupling problem of IPMC actuator and a multiphysical simulator independent of commercial simulation software; 3) The semianalytical solution by combining the analytical solution and the numerical solution when the calculation amount of the numerical solution is large; and 4) adopting more reasonable computer hardware and software.
Model-Based Development of IPMC-Based DevicesThe model and the application of IPMC are currently relatively independent. Nevertheless, given that the IPMC model is a potent tool, independent development is obsolete and cannot leverage the model's strength in the IPMC-based device development process. Existing IPMC-based devices rely primarily on IPMC actuators with simple geometries and a single mode of actuation, which cannot meet the varied requirements and application circumstances. The performance of the existing IPMC actuator cannot satisfy the actual use requirements. In the future, models that account for a greater number of actuating factors will be used to investigate the enhancement of the performance of IPMC actuators. The optimization process does not involve manually entered parameter changes but instead provides the developer with a visual single/multifactor performance impact curve. Using 3D printing technology, IPMC actuators with more intricate shapes can be produced. For mechanical properties of complex-shaped IPMC outputs, the IPMC model can be used for preliminary analysis. This preliminary analysis considers the impact of material, shape, and other variables on the output performance of IPMCs with complex shapes. For the manufacturing process, it is anticipated that the ability to control IPMC physical parameters will continue to improve. This is because the uncontrollability of the IPMC manufacturing process physical parameters has a direct impact on the actuator's performance. Additionally, the aging rate of IPMC is distinct. This poses a significant modeling and control challenge for IPMC actuators. Future models and model-based controls of IPMC should leverage the self-sensing capabilities of IPMC and the potent tool ML. Complex shape IPMCs can thus be modeled and controlled without the need for complex sensor systems. In order to account for the influence of a complex external environment, the simulation of the operating parameters of IPMC-based devices (such as the speed of bionic robots) at the present juncture is conducted under ideal environmental conditions. In the future, the simulation process should be a multifield model that considers the external environment's complexity, which can provide more useful information for the IPMC-based device development process.
SummaryIn this article, the modeling methods and applications of existing IPMC actuators are investigated. The modeling methods of IPMC actuators are divided into the nonphysical identification model, the partial-physical model, and the physical-based model. This classification method is divided according to the degree to which the model reveals the physics of IPMC actuators. The nonphysical model is mainly developed through the input voltage and output displacement of the actuator. The partial-physical model is modeled according to different analogical objects, and some parameters are derived from experiments. The physical-based model is developed to describe the mechanics and electrochemistry of IPMCs based on first principles and constitutive equations without any parameters obtained from the experiment. Due to the development of theory in different stages or the modeling based on different analog characteristics, the model can be divided into several subcategories. Model improvement is the adoption of new modeling methods or the incorporation of more parameters into the model in order to better explain and represent the actuating process.
IPMC actuators have great potential in the fields of wearable devices, medical devices, and bionic robots. Researchers need to better understand the actuating process of IPMCs and apply IPMCs, which is the starting point of model development. The application of IPMC, that is, the development of devices based on IPMC actuators, is the ultimate goal of model development and improvement. Manufacturing, modification, and control are the major research directions of IPMC. It is worth noting that the development of these three directions cannot be separated from the promotion of the IPMC model. The quantification of the parameters of manufactured IPMCs enables researchers to better understand the effect of changing manufacturing parameters on the performance of IPMC actuators. The modification process can be simulated by the model to predict the IPMC actuating result so as to reduce the number of experiments and time. The control of IPMC is mostly model-based, and the control based on the self-sensing model can achieve high precision control without the use of sensors. Therefore, the development of relevant mathematical models can promote the improvement and application of IPMC. Better understanding and use of IPMC cannot do without the development and improvement of models. The characteristics of IPMC-based devices (such as short service life) and the characteristics of the model (such as the non-physical identification model contains few physical characteristics and the calculation difficulty of the physical-based model) these deficiencies will be the focus of future IPMC model research. In this article, the current situation of IPMC actuator modeling and application is reviewed, and the future research direction is determined. This will benefit the comprehensiveness and accuracy, ubiquitousness, simplification, and computational rapidity of IPMC models. Moreover, the most important review of the model is of great significance to the development of IPMC-based devices.
AcknowledgementsThis work was supported by the Natural Science Foundation of Heilongjiang Province (grant no. LH2021E081); the Fundamental Research Foundation for Universities of Heilongjiang Province (grant no. LGYC2018JQ016); and the Research Launch Foundation for Heilongjiang Province Postdoctoral.
Conflict of InterestThe authors declare no conflict of interest.
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Abstract
Compared to conventional actuators, the soft ionic polymer–metal composite (IPMC) actuator has significant advantages in specific applications, and the mathematical model of IPMC actuators is essential to comprehending and applying IPMCs. Due to the inherent characteristics of IPMCs and the impact of the manufacturing and measurement processes, it is challenging to developa reliable model. This article provides a comprehensive overview of the developments in IPMC actuator modeling. In particular, three types of models are examined and contrasted: the nonphysical identification model, the partial-physical model, and the physical-based model. In order to comprehend the current state of numerous IPMC actuator models, the characteristics, evolution, and functions of each type of model are discussed. Afterward, the evolution of the IPMC actuators’ applications is discussed. Finally, promising research directions for IPMC actuator models are identified that can more effectively facilitate the development of IPMC-based devices.
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Details
; Lin, Chuan 1 ; Xu, Shuainan 1 ; Yu, Yanxin 1 ; Yao, Liang 1 ; Huang, Zhiyuan 2 1 Key Laboratory of Advanced Manufacturing and Intelligent Technology, Ministry of Education, Harbin University of Science and Technology, Harbin, Heilongjiang, P. R. China
2 State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, Heilongjiang, P. R. China