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1.1 Introduction

This chapter is an introduction to the principal motivations of why using smart materials in the design and development of systems working at the micro and nano-scale, in particular for micro/nanopositioning applications. More focus will be given to piezoelectric materials as nine chapters of the book use such materials. The chapter is organized as follows. In Sect. 1.2, the principal difficulties encountered when using classical design to develop systems for micro/nano-scale are presented. Some smart materials are cited as possible alternative to the classical design (DC motors and articulations) and among the well-recognized smart materials for micro/nanopositioning, we find piezoelectric materials. Section 1.3 is therefore devoted to present the basic principle of piezoelectricity and Sect. 1.4 deals with their particular advantageous. In Sect. 1.5 we present some of characteristics (hysteresis, creep) that may limit the utilization of smart materials in general such that the readers could have an idea of the motivations of the modeling and control presented in the different chapters of the book.

1.2 Why Using Smart Materials in Micro/Nanopositioning

In positioning systems with classical dimensions (robots, manipulators, etc.), we often use several components to compose them. First, among the most used actuators, we find: DC motors, pneumatic actuators, magnetic actuators, etc. These actuators are themselves made up of several subcomponents such as stator, rotor, and movable part. In addition to the actuators, other mechanical components are utilized to transform the angular (resp. linear) motion into linear (resp. angular) motion to amplify the displacement stroke or to reduce the speed. In general, these mechanical amplifiers and transformers contain themselves several subcomponents such as passive articulations. At the macro-scale, the assembly of the different components is evident. Remind that many actuators and systems at this scale can perform theoretically infinite stroke, for instance the angle obtained with an electrical motor can be unlimited. Although the large stroke that can be obtained, the methods and the technologies used to design systems working at the macro- are not suitable for the design and the development of systems working at the micro-. The principal reasons of such incompatibility are as follows:

  • The sizes of systems used in macro are relatively bulky face to the available space when working at the micro-nano. For instance, the available space in an atomic force microscopy (AFM) is very reduced and could not welcome an actuator based on DC motor to position the sample of material to be scanned.

  • With “macro” systems, the consumed energy is huge and is not justified face to the small objects to be positioned. As an example, using a “macro” robot to precisely position a biological cell would require many electrical power and it would be more convenient to position the same object with lower power consumption systems such as smaller robot.

  • The mechanical clearances found in the articulations yield a limited resolution of positioning. This limited resolution is not often adapted to the resolution required in nanopositioning.

  • The fact that “macro” systems are based on the assembly of different components, they have minimal sizes that are irreducible to be convenient with the available space in micro. Remind that if we assumed the existence of similar but smaller components, the assembly itself would be very difficult at small scale and then still maintains the difficulty to fabricate assembled micro-components. In addition to that, some components are mobile relative to themselves. This yields some friction and therefore implies a loss of precision of the whole system.

The above reasons lead researchers and engineers to use new design of the systems devoted to micro/nanopositioning. The main idea is that, instead of using several assembled components (actuators and articulations), one employs smart materials, i.e. materials that react and that can generate motion when excited electrically, magnetically, thermally, etc. Indeed, it is possible to replace an actuator and related articulations by utilizing the same bulk of smart material which consequently removes many of the above cited limitations substantially. Figure 1.1a depicts a 2-degrees of freedom (2dof) classical manipulator based on two motors and bars, while Fig. 1.1b depicts its replacement with a cantilever structured smart material. Both systems can provide a displacement at their extremities. As we can see, the articulations are removed in the smart material-based actuator and the use and assembly of several components are bypassed. The resolution is highly increased as the friction and the mechanical clearances do not anymore exist. This resolution is only dependent on the minimal deformation that can perform the smart material, which should be theoretically infinite. Finally, the fabrication of smaller systems are possible since we can use the same bulk material to integrate different functions (actuation, mechanical amplification, etc.).

Fig. 1.1
figure 1

Replacement of a classical system (articulation with DC motors, see (a)) by utilizing a smart material (see (b))

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Among the smart materials that are commonly used to develop micro/nanopositioning systems, we find:

  • Shape memory alloys (SMA) and thermal shape memory alloys (TSMA) materials

  • Electropolymeric and magnetopolymeric materials

  • Magnetostrictive materials and fluid mechanics

  • Piezoelectric materials

To resume, the main benefit from smart materials is their ability to deform themselves under the electrical or magnetic excitation. However, there also exist other principles to deform materials that are not necessarily smart. They include:

  • The electrostatic principle which is based on the Coulomb force to deform the material

  • The magnetic principle, for instance by fixing a ferromagnetic small object at the tip of the cantilever of Fig. 1.1b which, under a magnetic field, is attracted/repulsed and then results a bending of this latter

  • The thermal principle based on a bilayered cantilever where each layer has different thermal coefficients

In these smart and active materials, piezoelectric one are among the well recognized thanks to several advantages that they can offer. In this book, nine chapters (Chaps. 2, 3, and 5–11) are devoted to piezoelectric materials used in nanopositioning systems and related applications. In order to give some preliminaries to the readers, the remaining sections will therefore be consecrated to these materials.

1.3 Basics on Piezoelectric Materials

Piezoelectricity is the ability of certain materials to create electrical charges in response to a mechanical stress. This phenomenon is called the direct piezoelectric effect. The piezoelectric effect is a reversible process and piezoelectric materials are able to generate internal mechanical strain when an electrical field is applied. A wide range of applications uses these two phenomena especially for sensors (pressure), actuators, energy harvesting or resonance applications (ultrasonic applications, filters, high sensitive mass sensors). We present here the principle at the microscopic and macroscopic scale before reminding the constitutive equations of a piezoelectric material. We end the section by classifying the different kinds of piezoelectric materials.

1.3.1 Microscopical Principle

Piezoelectric materials are composed of different kinds of atoms, having different electrical charges. At equilibrium (Fig. 1.2a), the electric charges of each compound are compensated, for example in quartz, oxygen atoms charged negatively sharing electrons with silicon atoms charged positively. However, under a mechanical stress, the material is deformed (Fig. 1.2b) and the electric equilibrium is broken. A polar moment appears in the solid which creates charges in the material. On the other hand, the separation of electrical charges, due to the application of electric potential on the material, induces a displacement of atoms which causes deformations in the solid.

Fig. 1.2
figure 2

Example of direct/converse piezoelectric effect on an elementary mesh of quartz. (a) The mesh at its equilibrium state (no mechanical force or electrical voltage is applied). (b) The mesh when a mechanical force or an electrical voltage is applied

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1.3.2 Macroscopical Principle and Equations

At macroscopical scale, the direct or inverse piezoelectric effect is visible by measuring, respectively, the electrical potential difference in function of the stress applied or the deformation generated by the application of an electrical potential difference. For an ideal piezoelectric material, these two relations are proportional and can be written as:

$$\displaystyle{ \left \{\begin{array}{l} S = dE\\ D = dT\\ \end{array} \right. }$$
(1.1)

where D is the electric displacement, T the stress, S the strain, E the electric field, and d the proportional coefficient, called the piezoelectric constant. If we add the relations that link stress with strain (Hooke’s law) and the electric displacement with the electric field which traduce, respectively, the mechanical and electrical behavior of a material, we obtain the piezoelectric constitutive equations:

$$\displaystyle{ \left \{\begin{array}{l} S = {s}^{E}T + dE \\ D = dT {+\varepsilon }^{T}E\\ \end{array} \right. }$$
(1.2)

where s is the stiffness and \(\varepsilon\) the permittivity for constant electric field and constant stress, respectively. These equations are generally combined with the equilibrium equations to determine the behavior of a device. The direct effect is usually used in sensor applications or energy harvesting and inverse effect for actuator applications (quasi-static case: expansion/compression or shear motions). Some devices combine both effects, especially for resonance applications using different modes of vibration: compression/expansion, flexion, thickness shear, or face shear.

A piezoelectric material can be viewed as an energy converter and its ability to convert electrical (resp. mechanical) energy into mechanical (resp. electrical) energy is given by the piezoelectric coupling factor:

$$\displaystyle\begin{array}{rcl}{ k}^{2}=\frac{\mbox{ mechanical (or electrical) energy converted to electrical (or mechanical) energy}} {\mbox{ input mechanical (or electrical) energy}} & &{}\end{array}$$
(1.3)

and can be expressed as:

$$\displaystyle{ {k}^{2} = \frac{U_{\mathrm{C}}^{2}} {U_{\mathrm{D}}U_{\mathrm{E}}} }$$
(1.4)

where U C is the coupling energy, U D the deformation energy, and U E the electrical energy.

1.3.3 Piezoelectric Materials

Two materials are widely used in devices: the PZT (lead zirconate titanate, Pb(Z x Ti1 − x )O3) which are generally used for actuators, and quartz crystals used for resonators. But various materials could be adapted depending on the applications.

Crystals, such as quartz, present generally a high stability, especially face to temperature variation, but they have a low piezoelectric module and high acoustic impedance. Monocrystals based on langasite (langasite, langatate, langanite) or lithium (lithium niobate or lithium tantalite) show higher piezoelectric coefficients although they are still lower than those of ceramics. Then, their costs of production and their fragility limit their applications. New monocrystals such as PMN-PT or PZN-PT, with the same composition as ceramics, offer high coupling properties and are under numerous research investigations nowadays, in the aim to substitute PZT ceramics in sensors or actuators applications.

Ceramics, in opposition to crystals, have a lower stability but a higher piezoelectric coefficient and a low cost of production. Ceramics are generally made of PZT with various concentration of titanium and it is possible to add some dopants to modify the material properties and make easier electrical exchanges. Two kinds of PZT are developed: soft and hard PZT. Hard PZT are doped with acceptor atoms which reduce loss in material but decrease piezoelectric constant. Soft PZT are doped with donator atoms which confer better piezoelectric constants but increase losses in material due to internal friction. Finally other piezoelectric materials are also found in small systems applications such as small sensors and small actuators. They include GaAs, AlN, ZnO, and piezoelectric polymers such as PVDF.

1.4 Gains Obtained with Piezoelectric Materials

Piezoelectric materials are commercially available from a number of companies, such as Noliac, Physik Instrumente (PI), and NEC/Tokin. Acting as an actuator, the piezoelectric material converts the electrical energy into motion. They are electromechanical devices for generating movements in the micrometer range. Piezoelectric actuators possess some attractive properties, such as compact size, high resolution, high bandwidth, ease of fabrication of small systems, avoidance of mechanical plays/clearances, and no electromagnetic interference. They can be operated over billions of cycles without wear or deterioration. Their bandwidth is very high, which is only limited by the inertia of the object being moved and the output capability of the electronic driver. In addition, virtually no power is consumed to maintain a piezoelectric actuator in an energized state. Moreover, they enable the capability of self-sensing [1]. Taking PI actuators as example, specific advantages of the piezoelectric actuators are enumerated as follows [2]:

  • Ultrahigh resolution: A piezoelectric actuator can produce extremely fine position changes down to the subnanometer range. The smallest changes in operating voltage are converted into smooth movements. Motion is not influenced by friction effect.

  • High bandwidth: Piezoelectric actuators offer the highest bandwidth available. Microsecond time characteristics can be easily obtained.

  • Large force generation: Piezoelectric actuators can generate a force of from several kN to several tons within a typical range of tens micrometer.

  • No magnetic fields: Piezoelectric actuators are especially well suited for applications where magnetic fields cannot be tolerated.

  • Low power consumption: The piezoelectric material absorbs electrical energy during movement only. Static operation, even holding heavy loads, does not consume power.

  • No wear and tear: A piezoelectric actuator has neither gears nor rotating shafts. Its displacement is based on pure solid-state effects and exhibits no wear and tear.

  • Vacuum and clean-room compatible: Piezoelectric actuators employ ceramic elements that do not need any lubricants and exhibit no wear or abrasion. This makes them clean-room compatible and ideally suited for ultra-high-vacuum applications.

  • Operation at cryogenic temperatures: The piezoelectric effect is based on electric fields, and it functions down to almost 0 K although at reduced specifications.

In view of the obtained gains mentioned above, piezoelectric actuators have been applied extensively in engineering applications. The most popular types of piezoelectric actuators are piezoelectric stacks, bending actuators, and shear actuators. Piezoelectric stack actuators are constructed by stacking multiple layers of piezoelectric materials together. They are usually adopted in micro-/nanopositioning stages [3, 4], auto focusing of cell phone camera, vibration sources, vibration controls, mirror/prism positioning, AFM [5], etc. Alternatively, piezoelectric bender actuators consist of one to several layers of piezoelectric materials. With one end fixed, the free-end of the bender delivers motion once powered. Such kind of cantilever-based actuators have been widely used in the scenarios of smart microgrippers [6], valves, active vibration damping, AFM [7], energy harvesting [8], etc. Besides, piezoelectric shear actuators present electrodes on top and bottom surfaces. They have been widely employed in the applications of active vibration control [9], structural health monitoring [10], microscopy, switches, etc.

1.5 Some Problems Encountered When Using Smart Materials

1.5.1 Background

Smart actuators invariably exhibit hysteresis, which is a path-dependent memory effect where the output relies not only on the current state but also on the past output history [11]. The presence of the hysteresis in smart actuators, such as piezoceramic, magnetostrictive, and SMA actuators has been widely associated with various performance limitations [12]. These include the oscillations in the responses of the open-as well as closed-loop systems, and poor tracking performance and potential instabilities in the closed-loop system [13]. Smart actuators have also shown strong creep effects in the output displacement during slow and fast operations [14, 15]. A number of studies are calling the creep effects at high excitation frequencies as rate-dependent hysteresis [15, 16]. These creep effects yield significant loss in precision when positioning is required over extended periods of time and high oscillations at high excitation frequencies [17].

Considerable continuing efforts are thus being made to seek methods for effective compensation of hysteresis and creep effects in order to enhance the tracking performance of smart actuators, particularly for closed-loop micro-positioning systems. The characterization and modeling of the hysteresis and creep properties of smart actuators, however, is vital for designing efficient compensation algorithms. Considering that the hysteresis properties of such actuators are strongly dependent upon the type of materials, magnitude of input and the rate of input in a highly nonlinear manner, the characterizations as well as modeling of the phenomenon pose considerable challenges. For instance, piezoceramic actuators generally exhibits symmetric convex minor and major hysteresis loops [18], while magnetostrictive and SMA actuators yield highly asymmetric concave hysteresis loops [12, 19], which further depend upon the rate of the applied input. Smart actuators also exhibit output saturation, which further contributes to the modeling challenge. Figure 1.3 shows measured hysteresis and creep nonlinearities in smart actuators.

Fig. 1.3
figure 3

(a) Measured displacement of a piezo micropositioning actuator under sinusoidal input voltage, (b) measured displacement of a piezocantilever when a step input reference is applied, (c) measured hysteresis loops of a piezo micropositioning actuator when a sinusoidal input voltage is applied at different excitation frequencies, and (d) measured hysteresis loops of a magnetostrictive actuator when a sinusoidal input current is applied at different excitation frequencies

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1.5.2 Hysteresis Models

A number of hysteresis models have been proposed in the literature for characterizing the hysteresis properties of smart actuators [13]. These models could be broadly classified into phenomenological models and physics-based models [11, 20]. Different dynamic models have been proposed to model the creep effects in smart actuators. These models could be classified into linear creep models and nonlinear creep models. Linear creep models characterize the creep effects using series connection between springs and dampers, while nonlinear creep models apply nonlinear equations.

The physics-based models are generally derived on the basis of a physical measure, such as energy, displacement, or stress–strain relationship. These hysteresis models generally require comprehensive knowledge of the physical phenomenon for the hysteretic system. Alternatively, the phenomenological models describe the hysteresis properties without attention to the physical properties of the hysteretic system [13]. Many of these models were initially proposed for specific physical systems and were later generalized for applications to other systems. The primary goal of these models is to accurately predict the hysteresis in order to study the hysteresis effects and to facilitate the design of controllers for compensating the hysteresis effects.

The most widely cited models based on the input and output behaviors include: the operator-based hysteresis models such as Preisach model [11] and Prandtl–Ishlinskii model [20] and differential equation-based hysteresis models such as Duhem model and Bouc–Wen model [21]. These models generally constitute a nonlinear differential equation relating the output to the magnitude and direction of the input. Unlike the differential equation-based model, the operator-based models are considered to be better suited for the design of control algorithms for compensating hysteresis effects due to their invertibility. Such models have been widely applied for modeling hysteresis nonlinearities in smart actuators, and are briefly described below.

1.5.2.1 The Preisach Model

The Preisach model has been most widely applied for characterizing the hysteresis properties of smart actuators, see, for example, [13, 18]. The Preisach model can be presented analytically as [11]:

$$\displaystyle{ \Gamma [v](t) =\iint \limits _{\alpha \geq \beta }\,p(\alpha,\beta )\gamma _{\alpha \beta }[v](t)\mathrm{d}\alpha \mathrm{d}\beta }$$
(1.5)

where γ αβ [v](t) is the output of the relay operator, α and β are thresholds, and p(α, β) is a positive integrable density function identified from the measured data for a particular smart actuator. The Preisach model is rate-independent hysteresis model.

The Preisach model is completely characterized by two properties [11]: wiping-out and congruent minor-loop properties. The wiping out property means that the output is affected only by the current input and the history of the output, while the effect of all other inputs is wiped out. The congruent minor-loop property requires that all equivalent minor loops be similar. Two minor loops are said to be equivalent if they are generated under monotonically varying inputs of identical amplitudes. Different forms of the classical Preisach model have thus evolved to model hysteresis in various materials and smart actuators [11].

1.5.2.2 The Prandtl–Ishlinskii Model

The Prandtl–Ishlinskii model is constructed using the play hysteresis operator. Unlike the relay operators in the Preisach model, the play operator is continuous hysteresis operator characterized by the input v and the threshold r. A detailed discussion about these operators can be found in [20]. The play operator has been described by the motion of a piston within a cylinder of length 2r [20]. Analytically the output of the Prandtl–Ishlinskii model is expressed as [20]:

$$\displaystyle{ \Psi [v](t) =\int \limits _{ 0}^{R}\,p(r)F_{ r}[v](t)\mathrm{d}r, }$$
(1.6)

where F r [v](t) is the output of the play operator and p(r) a positive integrable density function identified from the measured data for a particular smart actuator. The Prandtl–Ishlinskii model is a rate-independent hysteresis model, attributed to the time independent play operator that the model employs.

The Prandtl–Ishlinskii model has been applied to characterize hysteresis effects of different smart actuators. The model, however, is limited to symmetric hysteresis loops, such as those observed in many piezoceramic actuators, which is attributed to the play operator. The model, thus, cannot be applied for predicting asymmetric input–output hysteresis, which is invariably observed in SMA and magnetostrictive actuators. Furthermore, unbounded nature of the play operator does not permit the Prandtl–Ishlinskii model applications for saturation property, which is widely observed in SMA actuators. Different developments have been carried out in a number of studies to enhance the performance of the Prandtl–Ishlinskii model. These studies include the generalized Prandtl–Ishlinskii model [13], the classical Prandtl–Ishlinskii model [22], the modified Prandtl–Ishlinskii model [23], the rate-dependent Prandtl–Ishlinskii model [16].

1.5.3 Hysteresis Compensation

The hysteresis in smart actuators has been associated with oscillations and poor tracking performance of the closed-loop system [24]. Consequently, considerable efforts have been made towards the design of controllers for compensation of hysteresis. A vast number of controllers have been proposed to reduce the error due to hysteresis effects. The proposed control algorithms could be classified into two broad categories, namely non-inverse-based control methods and inverse-based control methods.

1.5.3.1 Model-Based Control Methods

Compensation of hysteresis nonlinearities have been carried out in many studies without considering the inverse of the hysteresis models. Model-based hysteresis compensation methods employ the phenomenological hysteresis models to construct controllers to compensate for the actuator hysteresis. A number of control methods have been proposed to compensate for smart actuators such as adaptive control [25, 26], energy-based control methods [12], and sliding model control systems [27], which employ the hysteresis model of the actuator for constructing the controller.

1.5.3.2 Inverse-Based Control Methods

Control algorithms based on inverse compensators have been suggested to be more effective in compensating the hysteresis and creep effects [28–30]. The inverse model-based hysteresis compensation methods generally employ a cascade of a hysteresis model and its inverse together with a controller to compensate for the hysteresis effects. These methods, however, necessitate the formulation of the hysteresis model inverse, which is often a challenging task.

Some reported hysteresis models have thus been employed for deriving the inverse hysteresis models to serve as a compensator for the hysteresis effects, particularly these based on the Preisach model and Prandtl–Ishlinskii models [13]. The Preisach model is not analytically invertible; numerical methods are thus employed to obtain approximate inversions of the model. The effectiveness of the approximate inversions in conjunction with different controllers in hysteresis compensation has been demonstrated in a number of studies, see, for example, [31].

1.6 Conclusion

This chapter presented the main motivations of using smart materials as core components in systems working at the micro/nanoscale, in particular systems for micro/nanopositioning. Piezoelectric materials are considered as one of the well recognized among the existing smart materials in these applications. Indeed nine chapters of the book are devoted to the use of these materials. Hence, this chapter provided a remind of the basics of piezoelectricity and of their main advantages. Finally, the chapter presented some of the main behaviors (hysteresis and creep) that limit the performances of smart materials and in particular of piezoelectric materials. Most of the chapters in the book will treat these behaviors.