**2.1 Electromechanical coupling model for piezoelectric stick-slip actuators**

The piezoelectric actuator part is an electromechanical coupling system. When a certain drive voltage is applied, the piezoelectric actuator generates a certain displacement and output force because of the inverse piezoelectric effect. The modeling of the piezoelectric actuator needs to reflect the relationship between the driving

#### **Figure 1.**

*Operation principle. (a) Driving principle of the stick-slip actuator. (b) Force analysis of the stick-slip actuator. (c) The actual object of the stick-slip actuator.*

voltage and the deformation and output force generated by the piezoelectric actuator at the same time, which are coupled with each other.

To quantitatively analyze a system, it is necessary to describe the dynamics of the system through a mathematical model. This enables more information about the system to be described, resulting in better system control. Adriaens et al. pointed out that if the piezoelectric positioning system is properly designed, a second-order approximate modeling approach can be used to represent the dynamics of the system very well [8]. Typically, it can be viewed as a simplified spring-damped mass second-order system with a friction bar. Its linear dynamics is represented as

$$m\ddot{\mathbf{x}} + c\dot{\mathbf{x}} + k\mathbf{x} = F\_p - F\_f \tag{1}$$

where *x* is the displacement of the slider, *c* and *k* denote the damping and stiffness of the piezoelectric drive stack, and *m* denotes the total mass of the piezoelectric stickslip actuator. *Fp* is the output force of the piezoelectric stack, and *F <sup>f</sup>* is the frictional reaction force between the slider and the friction rod.

There are both hysteresis and creep effects in the process of applying a voltage to the ends of the piezoelectric stack. When the nonlinear characteristics of the system are not considered, the piezoelectric actuator input voltage and output force can be expressed as

$$F\_p = K\_h u(t) \tag{2}$$

where *Kh* is the conversion ratio of input voltage to output force and *u t*ð Þ is the drive voltage of the upper piezoelectric driver.

### **2.2 Hysteresis model for piezoelectric stick-slip actuators**

In the ideal case, the output displacement of the piezoelectric stick-slip actuators is linearly related to the input control voltage curve. Due to the inherent characteristics of this material, there are certain nonlinear characteristics such as the hysteresis effect and creep phenomenon in the actual driving process. However, because the creep effect is so small that it is mostly ignored and the hysteresis effect is mainly considered in the existing literature. The hysteresis phenomenon refers to the non-coincidence between the boost displacement curve and the bulk displacement curve when the drive voltage is applied to the piezoelectric driver. The hysteresis model is used to represent the force generated by the piezoelectric driver with a new equation as

$$m\ddot{\mathbf{x}} + c\dot{\mathbf{x}} + k\mathbf{x} = H(t) - F\_f \tag{3}$$

Currently, they can be broadly classified into physical and phenomenological models based on the modeling principles. Physical models are based on the physical properties of materials, among which the Jiles-Atherton model and Ikuta-K model are more common [9, 10]. However, physical modeling is based on the physical properties of the material, so the model implementation is difficult and affects the generality of the physical hysteresis model. Phenomenological models are based on input-output relations of hysteresis systems and are described using similar mathematical models. There are three broad categories based on the modeling approach, operator hysteresis models, differential equation hysteresis models, and intelligent hysteresis models.

*A Review of Modeling and Control of Piezoelectric Stick-Slip Actuators DOI: http://dx.doi.org/10.5772/intechopen.103838*

#### *2.2.1 Operator hysteresis model*

The common operator hysteresis models are the Preisach model, Prandtl-Ishlinskii (PI) model, and Krasnosel'skii-Pokrovskii (KP) model.

The Preisach model was first used to describe the hysteresis phenomenon in ferromagnetic materials. It was then gradually extended to describe the hysteresis behavior of smart materials, such as piezoelectric ceramics and magnetically controlled shape memory alloys. It is one of the most frequently studied nonlinear models of hysteresis [11]. The model consists of an integral accumulation of relay operators in the Preisach plane. The relay operator is shown in **Figure 2a**. Its mathematical expression is given by

$$\mathcal{Y}(t) = \iint\_{P} \mu(a,\beta) \chi\_{a\beta}[u(t)] da d\beta \tag{4}$$

where *u t*ð Þ and *y t*ð Þ represent the input and output of the Preisach model, *μ α*ð Þ , *β* is the weight function of the relay operator corresponding to the Preisach plane, *γαβ* represents the output of the relay operator, *α* and *β* are the switching thresholds of the relay operator.

Based on the previous work, Li et al. proposed that a multilayer neural network can be used to approximate the Preisach model. It can use any algorithm trained for neural networks to identify the model. The model is more flexible to adapt to different working conditions than the conventional model [12]. Later, Li et al. proposed a transformation operator for neural networks that can transform the multi-valued mapping of Lagrange into a one-to-one mapping. By adjusting its weights, the neural network model is made applicable to different operating conditions. The drawback that the Preisach model cannot be updated online is solved [13].

Both the PI model and the KP model evolved from the Preisach model. The PI model has a single threshold and two continuum hysteresis operators, the reciprocal inverse play operator and the stop operator [14], as shown in **Figure 2b** and **2c**. Therefore, the PI model can be derived from the PI inverse model by the stop operator, which can be easily used to design feedforward control compensators by the inverse model. The KP model uses a modified and improved Play operator with its corresponding density function superimposed for hysteresis modeling. While the traditional play operator has only one threshold that determines its width and

**Figure 2.** *Operators. (a) Relay operator. (b) Play operator. (c) Stop operator.*

symmetry, the KP operator has two different thresholds, enabling it to describe more complex hysteresis nonlinear behavior [15].

### *2.2.2 Differential equation hysteresis model*

The common differential equation hysteresis models include the Duhem hysteresis model, the Bouc-Wen hysteresis model, and the Backlash-like hysteresis model. P. Duhem et al. proposed the Duhem model, which is a differential equation. The model was later improved by Coleman and Hodgdon and applied to describe the hysteresis behavior of piezoelectric ceramics [16]. Its common mathematical expression is given by

$$
\dot{\mathbf{x}} = a\_D |\dot{u}| [f(u) - \varkappa] + \mathbf{g}(u)\dot{u} \tag{5}
$$

where *x* and *u* represent the output displacement and input voltage of the Duhem model, and *α<sup>D</sup>* represents the model parameters of the Duhem model, which is a positive constant. The functions *f u*ð Þ and *g u*ð Þ determine the shape and performance of the input-output hysteresis curve of the Duhem model.

Although the Duhem model is also applied to describe the piezoelectric ceramic hysteresis problem, its application in engineering is greatly limited due to the difficulty of solving the model inverse model. Su et al. proposed a simplified dynamic hysteresis Backlash-like model based on the Duhem model [17]. This model has fewer parameters compared to the Duhem model and is a first-order differential equation with the mathematical expression

$$
\dot{\mathfrak{x}} = \alpha\_B |\dot{u}| [\mathfrak{c}u - \mathfrak{x}] + \beta\_B \dot{u} \tag{6}
$$

where *x* and *u* represent the output displacement and input voltage of the Backlash-like model, *αB*, *c* and *β<sup>B</sup>* are constants.

The Bouc-Wen model was originally proposed as a differential equation by Bouc [18] and was later refined by Wen [19] to form the current Bouc-Wen model. The classical Bouc-Wen model can describe a large class of hysteresis phenomena and has a concise expression [20], which is given as follows

$$\begin{cases} \boldsymbol{\chi}(t) = k\boldsymbol{\nu}(t) - \boldsymbol{h}(t) \\ \dot{\boldsymbol{h}}(t) = a\boldsymbol{\dot{\nu}}(t) - \beta |\boldsymbol{\dot{\nu}}(t)| |\boldsymbol{h}(t)|^{n-1} \boldsymbol{h}(t) - \boldsymbol{\chi}\boldsymbol{\dot{\nu}}(t) |\boldsymbol{h}(t)|^{n} \end{cases} \tag{7}$$

where *k* denotes the scale factor of the system input to the output of the hysteresis part, *α*, *β*, *γ* , and *n* denote the parameters of the hysteresis part of the model. The Bouc-Wen output *y t*ð Þ consists of a proportional linear part *kv t*ð Þ and a hysteresis nonlinear part *h t*ð Þ.

The Bouc-Wen model is simple in form and has few identification parameters, which is convenient for controller design. However, it cannot completely describe the hysteresis characteristics of piezoelectric ceramics, its accuracy is low and it is only applicable to single frequency signals. The Bouc-Wen model is difficult to accurately describe the hysteresis phenomenon under the effect of frequency signals [21]. Therefore, the application of this model in practical engineering is greatly limited.

### *2.2.3 Intelligent hysteresis model*

In addition to the above hysteresis models, there are some other classes of hysteresis models used in hysteresis modeling of smart materials, such as neural network

*A Review of Modeling and Control of Piezoelectric Stick-Slip Actuators DOI: http://dx.doi.org/10.5772/intechopen.103838*

models polynomial models, and other nonlinear models. Gan et al. proposed a polynomial model for the hysteretic nonlinearity of piezoelectric actuators. Experimental results show that the proposed model has higher modeling accuracy than the conventional PI model [22]. Cheng et al. proposed a method for nonlinear model prediction. First, a multilayer neuron network is used to identify the nonlinear autoregressive sliding average model of piezoelectric ceramics. Then, the tracking control problem is transformed into an optimization problem for model prediction. Finally, the Levenberg-Marquardt method is used to solve the numerical solution of the nonlinear minimization [23].

There are some other models, for example, Zhang et al. proposed a proposed Rayleigh model to describe the hysteresis characteristics of the piezoelectric drive system. The parameters of the rate-dependent Rayleigh model were obtained and validated based on the functional and experimental data [24]. Li et al. proposed a simplified interval type 2 (IT2) fuzzy system for hysteresis modeling of piezoelectric drives. In the experiments, gradient resolution and inverse resolution are used to identify the IT2 fuzzy hysteresis model [25]. Although these models are not as widely applied as the three major classes of models, they can often achieve good results in some cases when dealing with the hysteresis characteristics in some specific situations.

#### **2.3 Selection of friction model**

The output of the drive system is ultimately transferred to the slider in the form of friction, so the choice of friction model will directly affect the accuracy of the stickslip drive platform model. At present, with the in-depth research of many international scholars on friction models, a variety of friction models have been established, which can be broadly divided into two categories, static friction models and dynamic friction models. Static friction models describe the friction force as a function of relative velocity. The dynamic friction model describes the friction force as a function of relative velocity and displacement. In contrast to static friction, which only considers the case where the relative velocity is not zero, the dynamic friction model uses differential equations to refer to the case where the relative velocity speed is zero. Therefore, in terms of accuracy, the dynamic friction model is more comprehensive and realistic than the static model. However, in addition to the accuracy of the friction model, it is also necessary to consider the complexity of the model, not all cases need to use the dynamic friction model.

#### *2.3.1 Static friction model*

The most widely used models in static friction modeling can be broadly classified into a series of coulomb and the stribeck model. Leonardo da Vinci took the lead in discovering that friction is related to the mass of an object and constructed a model. The model considers that the frictional force is proportional to the mass of the object and opposite to the direction of motion. Later this model was improved and called the coulomb model with the expression for friction

$$F\_f = F\_c \operatorname{sgn} \left( v \right) \tag{8}$$

where *F <sup>f</sup>* is the friction force, *Fc* is the Coulomb friction force, and sgn ð Þ*v* is the sign function.

In some studies, classical friction models have been used to represent the friction between the slider and the friction bar. The four static friction models commonly used in the early days are shown in **Figure 3** [26]. However, the slider step is only a few tens of nanometers to a few microns. The friction at this point is determined by the pre-slip displacement, which is the motion of the object before it is about to slide formally. When the friction surface is rough, the Coulomb friction model cannot accurately predict the friction force at pre-slip. Experiments have shown that in the pre-slip domain, the friction force depends on the micro-displacement between the two contacting surfaces [27]. However, the Coulomb friction model does not accurately predict this effect and will result in a relatively large error in the system. Therefore, a more accurate friction model is needed to describe the friction between the drive block and the terminal output.

These models need to exhibit some important static and dynamic properties of friction, such as the stribeck effect, Coulomb friction, stick friction, and pre-slip displacement. Li et al. proposed the stribeck model, which was the first model to describe dynamic and static frictional transition processes [28]. Its mathematical expression is as follows

$$F\_f = \left(F\_\varepsilon + (F\_\varepsilon - F\_\varepsilon)e^{-|\nu/v\_\imath|\xi\_\imath}\right) \text{sgn}\left(v\right) + bv \tag{9}$$

where *Fs* is the maximum static friction, *Fc* is the Coulomb friction force, *b* is the coefficient of stick friction, *vs* is the stribeck effect velocity value, and *ς<sup>s</sup>* is the empirical constant.

It not only reflects the linear relationship between dynamic friction and velocity but also expresses the change of friction during the transition between dynamic and static friction. It lays the groundwork for future research and the establishment of a dynamic friction model.

## *2.3.2 Dynamic friction models*

Dahl et al. proposed the Dahl model, which is a dynamic friction model [29]. It describes for the first time the pre-slip displacement in a friction model, represented by the partial differential equation

$$\frac{dF\_f}{d\mathbf{x}} = \sigma \left(\mathbf{1} - \frac{F\_f}{F\_c} \operatorname{sgn}(v)\right)^a \tag{10}$$

#### **Figure 3.**

*Four classical static friction models. (a) Classical Coulomb friction model. (b) Coulomb and stick friction models. (c) Static friction, coulomb, and stick friction models. (d) Stribeck model.*

*A Review of Modeling and Control of Piezoelectric Stick-Slip Actuators DOI: http://dx.doi.org/10.5772/intechopen.103838*

where *x* is the shape variable and *σ* is the stiffness coefficient. *α* determines the shape of the curve.

However, the model does not capture the variation of the static friction phase and also does not explain the stribeck phenomenon. It is far from an adequate description of the stick-slip-driven friction interface. However, due to its simple and accurate expressiveness, it provides a solid foundation for the subsequent dynamic modeling.

Based on the study of the Dahl model, the French scholar Canudas proposed the LuGre model [30]. Its principle formula is

$$\begin{cases} F\_f = \sigma\_0 z + \sigma\_1 \frac{dz}{dt} + \sigma\_2 v \\ \frac{dz}{dt} = v - \frac{z}{g(v)} |v| \\ g(v) = \frac{1}{\sigma\_0} \left[ F\_c + (F\_s - F\_c) e^{-\left(v/v\_t\right)^2} \right] \end{cases} \tag{11}$$

where *α*<sup>0</sup> denotes the stiffness coefficient of the elastic bristle, *α*<sup>1</sup> denotes the system damping coefficient, *α*<sup>2</sup> is the coefficient of stick friction, *v* is the relative velocity of the object surface, *z* is the mean deformation of the friction surface, and *g v*ð Þ is the described stribeck effect.

The LuGre model introduced the stribeck effect in addition to combining the idea of pre-slip displacement in the Dahl model. The idea of the bristle effect was designed to address the changing situation of the static friction phase. Swevers et al. proposed a new structure of the dynamic friction model [31]. The non-local memory hysteresis function and the modeling of arbitrary transition curves were added on the basis of the LuGre model. This allows the model to accurately describe the experimentally obtained friction characteristics, stribeck friction during slip, hysteresis behavior during slip, and stick-slip behavior. Since the structure of the obtained model is flexible, it can be further extended and generalized.

#### **2.4 Comprehensive model of piezoelectric stick-slip actuators**

By combining the above models and considering other influencing factors, a comprehensive dynamics model considering the electrical model of the piezoelectric stickslip actuator, the hysteresis effect, the linear dynamics performance, and the frictional characteristics of the system can be obtained, as shown in **Figure 4**.

In the stick-slip drive system, the output force of the piezoelectric stack is first obtained by the change of voltage at both ends of the piezoelectric element. Then, the electromechanical conversion model of the drive transmission system composed of the piezoelectric stack and the flexible transmission mechanism is transformed into the displacement and force output of the transmission system. Finally, the displacement is transferred to the slider by the kinematic friction conversion, and the final

**Figure 4.** *Flowchart of a comprehensive model of a piezoelectric stick-slip actuator*.

displacement output is obtained. The mathematical equation of its integrated model is as follows

$$\begin{cases} m\ddot{\mathbf{x}} + c\dot{\mathbf{x}} + k\mathbf{x} = H(t) - F\_f \\ m\ddot{\mathbf{x}}\_t = F\_f \\ \mathbf{x}\_t = \mathbf{x} - \mathbf{x}\_t \end{cases} \tag{12}$$

where *H t*ð Þ can be given by the previous hysteresis model [i.e., (4)–(7)]; *F <sup>f</sup>* can be given by the friction model [i.e., (8)–(11)]; *xs* is the backward displacement of the slider under the action of dynamic friction; and *xe* is the forward displacement of the slider.

Wang et al. developed a kinetic friction model of the actuator and investigated the effect of the input drive voltage on the viscous slip motion of the actuator through simulation [32]. Nguyen et al. used the method of dimensionality reduction to describe the frictional contact behavior of the stick-slip microactuator. The model accurately predicts the frictional contact behavior of the actuator on different geometric scales without using any empirical parameters [33]. Piezoelectric stick-slip actuators also have more complex dynamic characteristics. The ability to simulate a wide range of dynamic characteristics is the direction of increasing research. Shao et al. found that the contact behavior of the piezoelectric feed element produced an inconsistent displacement response. So the Hunt-Crossley kinetic model, LuGre model, distributed parameter method combined with Bouc-Wen hysteresis model were used to model the viscous slip actuator. This model can effectively model the step inconsistency in the front-to-back direction of the actuator [34]. Wang et al. proposed a stick-slip piezoelectric actuator dynamics model considering the overall system deformation. The model introduced stiffness coefficients and damping coefficients for the whole system and successfully simulated three single-step characteristics, namely, backward motion, smooth motion, and a sudden jump for the first time [35]. Due to a large number of parameters in the dynamic model, the accurate identification of each parameter will be quite difficult. Therefore, more accurate identification of simulation parameters needs further research, which may be our future work.
