1 General Facts About Actuator Design

Before a final selection of actuators is made, the appropriate kinematics and the control-engineering structure, according to the previous chapters, should have been fixed. However, in order to handle these questions in a reasonable way, some basic understanding about actuators is mandatory. Especially the available energy densities, forces and displacements should be estimated correctly for the intended haptic application. This section provides some suggestions and guidelines to help and preselect appropriate actuators according to typical requirements.

1.1 Overview of Actuator Principles

There are a certain number of approaches to transform an arbitrary energy source into mechanical energy. Each of these approaches is one actuation principle. The best known and most frequently used principles are:  

Electrodynamic principle:

 A force, so called Lorentz-force, acting upon a conductor conducting current.

Electromagnetic principle:

 A force, acting upon a magnetic circuit to minimize the enclosed energy.

Piezoelectric principle:

 A force, created by the atomic structure of a crystal due to applied voltage.

Capacitive principle:

 A force, resulting from charges trying to minimize the energy in a capacitor.

Magnetorheological principle:

 Viscosity change within a fluid resulting from particles trying to minimize the energy contained within a magnetic circuit.

Electrochemical principle:

 Displacement of or pressure within a closed system, whereby a substance emits or bounds a gas and consequently changes its volume due to the application of electrical energy.

Thermal principle:

 Change of length of a material due to controlled temperature changes, making use of the material’s coefficient of thermal expansion.

Shape-memory alloy:

 Sudden change of an object’s shape switching between two crystal structures when exposed to relatively small temperature changes.

Each of these principles is used in different embodiments. They mainly differ in the exact effective direction of e.g. a force vectorFootnote 1 or a building principle.Footnote 2 As a consequence a wide-spread terminology exists for naming actuators. The major terms are given here:  

Electric motor:

 The most general term of all. It may describe any electromechanic transformer. However, in most cases it refers to an actuator rotating continuously whose currents are commutated (mechanically or electronically), or which is equipped with a multiphase alternating current unit. Typically, it is a synchronous motor, a drive with a magnetic rotor moving synchronously to the rotating electromagnetic field. In a more general understanding the term includes hysteresis motors and squirrel-cage rotors/induction-machines, too. But the latter has not yet reached any relevance for haptic systems.

EC-motor:

 Specific embodiment of the synchronous motor and very common to haptic applications. Motor according to the electromagnetic or electrodynamic principle with an electronic control unit for the rotating field (electronic-commutated, electronically commutated).

DC-motor:

 In a general sense any motor operated with a DC-voltage. However in a more specific form a motor with a stationary stator-field and a alternating rotor-field. Used among haptic applications because of its cheapness and simplicity. This is an actuator according to the electromagnetic or electrodynamic principle with a mechanical control unit for the rotating field using switching contacts (mechanically commutated).

Resonance actuator:

 Generic term for a whole class of actuators with very different actuation principles. The term describes an actuator containing one component which is driven in its mechanical resonance mode (or nearby its resonance mode). Typically, parts of this component make an elliptic oscillation driving a second component in small steps via a frictional coupling. As a result of the high frequency, the movement of the second component seems uninterrupted. The term is most frequently applied to piezoelectric actuators.

Ultrasonic actuator:

 Resonance-actuator performing steps at a frequency within ultrasonic ranges (>15 kHz). These actuators are built almost always according to the piezoelectric principle.

Voice-coil-actuator:

 Translational drive according to the electrodynamic principle. Mainly consisting of a conductor wrapped around a cylinder. The cylinder itself is placed in a magnetic circuit, resulting in a Lorentz-force when a current is applied to the conductor. There are two major embodiments, either with a “moving coil”, another variant with a “moving magnet”.

Shaker/Exciter:

 A specific form of a voice-coil-actuator with an elastic suspension between the coil and the magnetic circuit. When current is applied to the coil an equilibrium condition between the suspension’s spring and the Lorentz-force is achieved at a specific dynamic displacement. With an alternating current a wide range of frequencies can be addressed. Actuators according to this structure are frequently used for fast and dynamic movements of masses. Whereas the origins can be found in vibration testing (this is where its name comes from), this actuator-type is meanwhile regularly used to create surface-haptics on touch-surfaces or vibrational-feedback in mobile-devices.

Plunger-type magnet:

 Actuator according to the electromagnetic principle. A rod made of ferromagnetic material is pulled into a magnetic circuit equipped with a coil. These actuators have very nonlinear force-displacement characteristics, but can create high forces with comparably little power required.

Stepper-motor:

 Generic term for all actuation principles moving forward step by step. In contrast to the resonance-drives no component of a stepper-motor is driven in resonance mode. Their step-frequency is below any resonance of the system and may vary. These actuators may even be used in a “microstep mode”, interpolating movement between so called “full-steps”, which are original to their mechanical design. The term is most frequently used for rotatory drives. Especially for those working according to the reluctance-principle or another electromagnetic actuation principle.

Pneumatic and Hydraulic:

 These actuation principles do not have a direct electric input value. They transform pressure and volume-flow into displacement and force. The media for pressure transmission is air in case of pneumatics and a fluid, typically oil, in case of hydraulics. Usually, the pressure itself is generated via actuators, e.g. electrical actuators attached to a compressor, and controlled via electrical valves.

Bending-actuator:

 Actuator with an active layer, frequently made of a piezoelectric material—attached to a passive mechanical substrate. By actuating the active layer, mechanical tensions between this layer and the substrate build up, resulting in a bending movement of the whole actuator.

Piezoelectric stack:

 A larger number of piezoelectric layers mechanically connected in series. Small displacements of each layer sum up to a large usable displacement of the whole actuator.

Piezoelectric motor/drive:

 Generic term for all drives according to the piezoelectric principle. It frequently refers to drives moving a rotor or translator with a frictional coupling. However, this movement does not need to happen in resonance mode.

Capacitive actuator:

 Actuator according to the capacitive principle and frequently used in microtechnology. Usually equipped with a comb-like structure of electrode-pairs, generating forces in millinewton range with micrometers of displacement.

Shape-memory wire:

 Wire on the basis of shape-memory alloys capable to shorten in the range of percents (\(\approx \) 8% of its total length) when changing its temperature (e.g. by controlling a current flowing through the wire. The current heats up the wire according to its thermal loss at the wire’s electrical resistance).

Surface-wave actuators:

 Generic term for a group of actuators generating high-frequency waves in mechanical structures or exciting the resonance-modes of structures. This actuator is frequently based on piezoelectric principles and is used for the generation of haptic textures.

Each of the above actuation principles can be found in tactile and/or kinaesthetic systems. To simplify the decision process for a new design, all actuators can be grouped into classes. Most of the physical working principles can be grouped either into “self-locking” or “free-wheeling” systems.Footnote 3 These groups are identical to:

  • Positional-sources (x) respectively angular-sources (\(\alpha \)) with high mechanical impedance

  • Force-sources (F) respectively torque-sources (M) with low mechanical impedance

According to the basic structures of haptic systems (Chap. 6) it is likely that both classes are used within different haptic systems. The correlation between basic structures of haptic systems and actuators is depicted in Table 9.1. This table shows a tendency towards typical applications. However actuators can be “impedance-matched” to a certain application. This happens by adding mechanical elements (springs, dampers) in series to the actuator. By this it is possible to use any actuator for any basic structure of haptic systems, trading in advantages for disadvantages which may be justified by the specific application and its requirements.

Table 9.1 Typical application areas for actuator principles in haptic systems

1.2 Actuator Selection Aid Based on Its Dynamics

Different actuator designs according to the same physical principle still cover wide performance ranges regarding their achievable displacements or forces. Based on the author’s experience, these properties are put into relation to the dynamical ranges relevant for haptic applications. In Fig. 9.1 the most important actuation principles are visualized in squares scaled according to their achievable displacements (a)Footnote 4 and typical output forces and torques (b). The area covered by a specific version of an actuator is typically smaller than the area shown here. The diagram should be read in such a way that e.g. for haptic applications, . electromagnetic linear actuators exist, providing a displacement up to \(5\,\)mm at \({\approx }\,50\,\)Hz. These designs are not necessarily the same actuators which are able to provide \({\approx }200\,\)N, as with electromagnetic systems the effectively available force increases with smaller maximum displacement (Sect. 9.4). The diagrams in Fig. 9.1 visualize the bandwidths of realization-possibilities according to a certain actuation principle and document the preferred dynamic range of their application. Using the diagrams, we have to keep in mind that the borders are fluent and have to be regarded in the context of the application and the actuator’s individual design.

Fig. 9.1
figure 1

Order of the actuator principles according to the achievable displacement (a) and forces resp. torques (b) in dependency from their dynamics. Further information can be found in [1]

1.3 Gears

In general machine engineering the usage of gears is a matter of choice for adapting actuators to their load and vice versa. Gears are available in many variants. A simple lever can be a gear; complex kinematics according to Chap. 8 are a strongly nonlinear gear. For haptic applications specialized gear designs are discussed for specific actuation principles in the corresponding chapters. However, there is one general aspect of the application of gears with relevance to the design of haptic systems which has to be discussed independently: the scaling of impedances.

Fig. 9.2
figure 2

Simple gear design with wheels (a), a lever (b) and a cable, rope or belt (c)

There is no principal objection to the use of gears for the design of haptic systems. Each gear (Fig. 9.2) may it be rotatory/rotatory (gearwheel or frictional wheel), translatory/translatory (lever with small displacements), rotatory/translatory (rope/cable/capstan) has a transmission  “tr”. This transmission ratio scales forces and torques neglecting loss due to friction according to

$$\begin{aligned} \frac{F_\text {out}}{F_\text {in}}=tr=\frac{l_2}{l_1}\,, \end{aligned}$$
(9.1)
$$\begin{aligned} \frac{M_\text {out}}{M_\text {in}}=tr=\frac{r_2}{r_1}\,, \end{aligned}$$
(9.2)
$$\begin{aligned} \frac{F_\text {out}}{M_\text {in}}=tr=\frac{1}{2\pi \,r_1}\,, \end{aligned}$$
(9.3)

and displacements respectively angles according to

$$\begin{aligned} \frac{x_\text {in}}{x_\text {out}}=tr=\frac{l_2}{l_1}\,, \end{aligned}$$
(9.4)
$$\begin{aligned} \frac{\alpha _\text {in}}{\alpha _\text {out}}=tr=\frac{r_2}{r_1}\,, \end{aligned}$$
(9.5)
$$\begin{aligned} \frac{\alpha _\text {in}}{x_\text {out}}=tr=\frac{1}{2\pi \,r_1}\,. \end{aligned}$$
(9.6)

The velocities and angular velocities scale to the differential of above equations. Assuming the impedance of the actuator \(\underline{Z}_\text {transl}=\frac{\underline{F}}{\underline{v}}\) resp. \(\underline{Z}_\text {rot}=\frac{\underline{M}}{\underline{\alpha '}}\), one consequence of the load-condition of a driven impedance \(\underline{Z}_\text {out}\) from the perspective of the motor is:

$$\begin{aligned} \underline{Z}_\text {transl}=\frac{\underline{F}_\text {in}}{\underline{v}_\text {in}}=\frac{\underline{F}_\text {out}}{\underline{v}_\text {out}}\,\frac{1}{tr^2}=\underline{Z}_\text {transl out}\,\frac{1}{tr^2} \end{aligned}$$
(9.7)
$$\begin{aligned} \underline{Z}_\text {rot}=\frac{\underline{M}_\text {in}}{\underline{\alpha '}}=\frac{\underline{M}_\text {out}}{\underline{\alpha '}}\frac{1}{tr^2}=\underline{Z}_\text {rot out}\,\frac{1}{tr^2} \end{aligned}$$
(9.8)

The transmission-ratio tr is quadratic for the calculation of impedances. From the perspective of an actuator, the driven impedance of a system is getting small with a gear showing transmission-ratios larger than one. This is favourable for the actuating system (and the original reason for the application of gears). For haptic applications, especially for impedance controlled ones, the opposite case has to be considered. In an idle situation and with a transmission ratio larger than oneFootnote 5 the perceived mechanical impedance of a system \(\underline{Z}_\text {out}\) increases to the power of two with the transmission-ratio. Another aspect makes this fact even more critical, as the increase of output force changes only in a linear way with the transmission ratio, whereas e.g. a motor’s unwanted moment of inertia is felt to increase quadratically. This effect is obvious to anyone who has ever tried to rotate a gear-motor with a high transmission ratio (e.g. tr > 100) at its output. The inertia and internal frictions upscaled by the gear are identical to a self-locking of the actuator.

 As a consequence, the usage of gears with force-controlled haptic systems makes sense only for transmission ratios of 1–20 (with emphasis on the lower transmission ratios between 3–6). For higher transmission ratios, designs according to Fig. 9.2c and Eq. (9.6) based on ropes or belts proved valid. They are used in many commercial systems, as with the aid of the definition \(tr=\frac{1}{2\pi \,r_1}\) and the included factor \(2\pi \) a comparably high transmission ratio can be achieved easily. In combination with rotatory actuators (typically EC-drives) with low internal impedances this design shows very impressive dynamic properties. Figure 9.3 shows an example for the application of such a gear to drive a delta mechanism [2].

Fig. 9.3
figure 3

Rope-based gear as widely used in haptic interfaces. The driven structure is connected to a lever, on which the driving rope is running. The driving rope is wound around the driving shaft of the motor. The number of revolutions around the shaft is determined by the amount of torque to be transmitted via the gear, threads are used to minimize friction and wearout between individual turns of the driving rope

Recently, a new type of gear came into view of several researchers [3]. The Twisted-String-Actuator (TSA) is based on a relatively small motor with large rotation speed that twists a string or a set of strings. Because of the twisting, the strings contract and provide pulling forces in the range of several ten newtons that can be transferred via bowden cables. Applications especially include exoskeletons as for example presented in [4] and other devices that are weight-sensitive.

Some advice may be given here out of practical experience: wheel-based gears are applicable for haptic systems but tend to generate unsteady and waving output torques due to their toothing. By a careful mechanical design, this unsteadiness can be reduced. The mechanical backlash should be minimized (which is typically accompanied by an increase in friction) for example by material combinations with at least one soft material. At least one gear should have spur/straight gearing, whereas the other one can keep involute gearing.

2 Electrodynamic Actuators

Electrodynamic actuators are the most frequently used type of drives for haptic applications. This popularity is a result of the direct proportion between their output value (force or torque) from their input values (the electrical current). In case of kinaesthetic applications they are typically used as open-loop controlled force sources. In tactile applications these very dynamic actuators are frequently used as oscillators or exciters to move a mass and by the inertia and system-reaction create a buzz-feeling. They can be found equally as much in form of rotary or translational actuators. Depending on the design either the electrical coil or the magnet is the moving component of the actuator, whereas the other part is fixed to the device. This section gives a short introduction to the mathematical basics of electrodynamic systems. Afterward some design variants are discussed in more details. The final subsection deals with the drive electronics necessary to control electrodynamic systems.

2.1 The Electrodynamic Effect and Its Influencing Variables

Electrodynamic actuators are based on the Lorentz-force

$$\begin{aligned} \textbf{F}_\text {Lorentz}=\textbf{i}\cdot l\times \textbf{B}, \end{aligned}$$
(9.9)

acting upon moving charges in a magnetic field. The Lorentz-Force is dependent on the current \(\textbf{i}\), the magnetic induction \(\textbf{B}\) such as the length of the conductor l, which is typically formed as a coil. This subsection deals with optimization of each parameter for the maximization of the generated output force \(F_\text {Lorentz}\). Any electrodynamic actuator is made of three components:

  • generator of the magnetic field (another coil, or most frequently a permanent magnet)

  • magnetic flux conductor (steel circuit, soft- or hard-magnetic core)

  • electrical conductor (frequently formed as a coil or a more complex winding)

After a shallow look at Eq. (9.9) a recommendation for the maximization of the output force could be to simply increase the current \(\textbf{i}\) in the conductor. However with a given and limited available space for the conductor’s length l (coil’s cross section), and a flux density \(\textbf{B}\) with an upper border (0.8 to 1.4 T), the effectiveness of this change has to be put into question. This can be shown with a simple calculation example.

Fig. 9.4
figure 4

Moving-coil actuator and corresponding functional elements

2.1.1 Efficiency Factor of Electrodynamic Actuators

For example a straight-forward design of an electrodynamic actuator similar to the AVN 20–10 (Fig. 9.4) is analyzed. It contains a wound coil with a permanent-magnet in a ferromagnetic core. The electrical power loss \(P_{\text {el}}\) of this electrodynamic system is generated mainly in a small moving coil with a pure ohmic resistance \(R_\text {coil}=3.5\,\varOmega \) and a nominal current \(i=0.78\,\)A:

$$\begin{aligned} P_{\text {el}}=R_\text {coil}\,i^2\,=3.5\,\varOmega \cdot 0.78\,\text {A}^2=2.13\,\text {W}. \end{aligned}$$
(9.10)

With this electrical power loss, and at a flux density \(\textbf{B}=\) 1.2 T, and with an orthogonal  conductor orientation, and a conductor length within the air gap \(l=\) 1.58 m, the actuator itself generates the force

$$\begin{aligned} F_{\text {Lorentz}}=i\,l\,B=0.78\,\text {A}\cdot 1.58\,\text {m}\cdot 1.2\,\text {T}=1.48\,\text {N}\,. \end{aligned}$$
(9.11)

Assuming the system being driven in idle mode—working against the coil’s own mass of \(m=\) 8.8 g only—being accelerated from idleness, and performing a displacement of \(x=10\) mm, above electrical power \(P_\text {el}\) is needed for a period of

$$\begin{aligned} t=\sqrt{2\,\frac{x}{a}}=\sqrt{2\,\frac{x\,m}{F}}=0.011\,s \end{aligned}$$
(9.12)

seconds. The electrical energy loss sums up to

$$\begin{aligned} W_\text {el}=P_\text {el}\cdot t=23,4\,mJ. \end{aligned}$$
(9.13)

This gives an efficiency factor of \(\frac{W_\text {mech}}{W_\text {el}+W_\text {mech}}=38 \%\) for idle mode and continuous acceleration. And this is a valid working point leading to exciter-type actuators (Fig. 9.5) whose efficiency and primary application in mobile applications derive from a highly dynamic movement.

Fig. 9.5
figure 5

Actuator as an exciter, moving mass-type actuator with fixed coil, Grewus Exciter EXS241408WA

Assuming now that such an actuator shall generate a force of 1 N against a finger tip for a period of e.g. two seconds, an electrical power of \(W_\text {el}=2.13\,\text {W}\cdot 2 \text {s}=4.26\,\)J is needed. This would be identical to an efficiency factor well below 1%. And indeed the efficiency factor of electrodynamic actuators in haptic applications lies in the area of low percentages due to the common requirement to generate quite static forces without much movement. This simple calculation points to one major challenge with electrodynamic actuators: The electrical power lost due to heat transmission extends the mechanically generated power by far. Consequently during the design of electrodynamic actuators the thermal management of energy losses is key.

2.1.2 Minimization of the Power Loss

Typical designs of electrodynamic actuators either have a wound conductor which in itself is self-supportive, or which is wound on a coil-carrier (Fig. 9.6). The available space for the electrical coil within the homogenous magnetic flux is limited (\(A_{Coil}\)). The number of coil-turns \(N_\text {Conductor}\) is limited too within this area due to the cross-sectional surface a single turn needs \(A_{Conductor}\). This cross-sectional surface is always more than the actual cross-section of the conductor, as the winding will have gaps in between single turns (Eq. (9.15)). Additionally the actual conducting core with the cross sectional surface \(A_\text {Core}\) will be smaller than the cross-section of the conductor itself due to its isolation. Both parameters describing the geometrical losses in cross sections which are available within tables of technical handbooks [5] and are assumed as factors \(k\ge 1\) according to Eq. (9.14). The length l of the conductor can be easily calculated by multiplication with the numbers of turns and the mean circumference Circ of the coil (Eq. (9.16)).

Fig. 9.6
figure 6

Cross-section through a cylindrical electrodynamic actuator according to the moving coil principle

The choice of the conductor’s diameter influences the resistance of the coil via the conducting area \(A_{Core}\). The specific length-based resistance \(R_{\text {spezf}}\) of a conductor is given according to Eq. (9.17). Big conducting diameters with large cross-sections \(A_\text {Core}\) allow coils able to conduct high currents at low voltages but—due to the limited volume available—few windings. Small diameters limit the necessary currents at high voltages and carry more windings. By a careful choice of wire-diameter the winding can be adjusted as a load to the corresponding source to drain the maximum available power.

The power loss \(P_\text {Loss}\) (Eq. (9.18)) acceptable within a given winding is limited. This limit is defined by the generated heat being able to dissipate. As a rule of thumb a standard copper winding can carry (if able to dissipated heat to one side) \(4 \frac{A}{mm^2}\) continuously. In case of printed-circuit-boards (PCBs) the current-density for copper can be increased to even \(20{ - }40 \frac{A}{mm^2}\) due to the very good thermal coupling between copper and environment. The real technical solution is dependent on the time of continuous operation, the thermal capacity resulting from the volume of the actuator and the materials it consists of, and a potential cooling system. A calculation of heat transmission is specific to the technical solution and can not be solved in general within this book. But the dependency of Lorentz-force on power loss can be formulated:

$$\begin{aligned} A_\text {Conductor}&=k\cdot A_{\text {Core}} \end{aligned}$$
(9.14)
$$\begin{aligned} N_\text {Conductor}&=\frac{A_\text {Coil}}{A_\text {Conductor}} \end{aligned}$$
(9.15)
$$\begin{aligned} l_\text {Conductor}&=N_\text {Conductor} \cdot Circ \end{aligned}$$
(9.16)
$$\begin{aligned} R_{\text {spezf.}}&=\frac{l_\text {Conductor}\,\rho }{A_\text {Conductor}} \end{aligned}$$
(9.17)
$$\begin{aligned} P_\text {Loss}&=i^2\cdot R_\text {Coil} \end{aligned}$$
(9.18)

From Eq. (9.18) follows

$$\begin{aligned} i=\sqrt{\frac{P_\text {Loss}}{R_\text {Coil}}} \end{aligned}$$
(9.19)

With Eq. (9.17) there is

$$\begin{aligned} i=\sqrt{\frac{P_\text {Loss}\,A_\text {Core}}{\rho \,l_\text {Conductor}}} \end{aligned}$$
(9.20)

put into Eq. (9.9) (keeping the direction of current flow \(\textbf{e}_i\)) there is

$$\begin{aligned} F_\text {Lorenz}= \sqrt{\frac{P_\text {Loss}\,A_\text {Core}\,l_\text {Conductor}}{\rho }} \textbf{e}_i\times \textbf{B} \end{aligned}$$
(9.21)

by considering Eqs. (9.15)–(9.16) the result is

$$\begin{aligned} F_\text {Lorenz}= \sqrt{\frac{P_\text {Loss}\,A_\text {Coil}\,N\,\text {Circ}}{\rho \, k}} \textbf{e}_i\times \textbf{B} \end{aligned}$$
(9.22)

The Eqs. (9.15)–(9.18) put into Eq. (9.9) gives a precise picture of the influence values on the Lorentz-force (Eq. (9.22)). The level of Lorentz-force is given by the power loss \(P_\text {Loss}\) acceptable within the coil. If there is room for modifications to the geometrical design of the actuator, the cross-sectional area of the coil and the circumference of the winding should be maximized. Additional a choice of alternative materials (e.g. alloy instead of copper) may minimize the electrical resistance. Furthermore the filling-factor k should be reduced. One approach could be the use of wires with a rectangular cross section to avoid empty spaces between the single turns.

The question for the maximum current itself is only relevant in combination with the voltage available and in the context of adjusting the electrical load to a specific electric source. In this case for \(i_\text {Source}\) and \(u_\text {Source}\) the corresponding coil-resistance has to be chosen according to Eq. (9.23).

$$\begin{aligned} P_\text {Source}&=u_\text {Source}\cdot i_\text {Source}= i_\text {Source}^2 \cdot R_\text {Coil} \nonumber \\ R_\text {Coil}&=\frac{P_\text {Source}}{i_\text {Source}^2} \end{aligned}$$
(9.23)

Surprisingly from the perspective of a realistic design an increase in current is not necessarily the preferred option to increase the Lorentz-force according to Eq. (9.22). The possibility to optimize \(P_\text {Loss}\) by adding cooling, or to analyze the temporal pattern of on- and off-times is much more relevant. Additionally the flux-density \(\textbf{B}\) has a—compared to all other influence factors—quadratically higher influence on the maximum force.

2.1.3 Maximization of the Magnetic Flux-Density

For the optimization of electrodynamic actuators a maximization of the flux density \(\textbf{B}\) is necessary within the area where the conducting coils are located. This place is usually called air-gap and resembles an interruption of the otherwise closed ferromagnetic core conducting the magnetic flux. The magnetic flux density is influenced by

  • the choice of ferromagnetic material for the magnetic core,

  • the field winding/exciter winding of the static magnetic field, and

  • the geometrical design of the magnetic core.

In the context of this book some basic design criteria for magnetic circuits are given. For an advanced discussion and optimization process source [6] is recommended.

Basics for the Calculation of Magnetic Circuits

Calculating magnetic circuits show several parallels to the calculation of electrical networks. As shown in Table 9.2 several analogies between electrical and magnetic variables can be defined.  

Table 9.2 Analogies between electric and magnetic values

The direct analogy to the magnetic flux \(\phi \) is the electrical current I. Please note that this is an aid for thinking and not a mathematical reality, although it is very common. The actual direct analogy for the current I would be a time dependent magnetic flux \(\frac{d\phi }{dt}\), which is usually not defined with an own variable name. The great exception with this model is the magnetomotive force \(\varTheta \), which resembles the sum of all magnetic voltages V identical to a rotation within an electrical network. Or another way to say it: It is the source of potential difference in a magnetic network. Nevertheless it is treated differently, as many applications require generating a magnetomotive force \(\varTheta \) to be generated by a certain number of winding-turns N and a current I, often referred to as ampere turns. The coupling between field- and flux-variables is given by the permittivity \(\varepsilon \) in case of electrical values and by the permeability \(\mu \) in case of magnetic values. It is obvious that the field-constants \(\varepsilon _0\) differs from \(\mu _0\) by the factor \(10^{6}\). This is the main reason for the electromagnetic effect being the preferred physical realization of actuators in macroscopic systems.Footnote 6

However above dependencies although valid consider linearity. The electrical permittivity can be regarded as quite constant (Sect. 9.5) even for complex actuator designs, and can be approximated as linear around an operating point. The permeability \(\mu _r\) of typical flux-conducting materials however shows a strong nonlinear relationship, the materials are reaching saturation. The level of magnetic flux has to be limited to prevent saturation-effects in the design of magnetic core.

Magnetic Circuits

For the maximization of the magnetic flux density it is necessary to either analyze the magnetic circuit mathematical-analytically and/or do a numerical simulation of it. For the simulation of magnetic fields common CAD and FEM products are available.Footnote 7 For classification of the mathematical problem three solution levels exist: stationary, quasi-stationary, and dynamic magnetic fields. With stationary magnetic fields there is no time dependent change of the magnetic circuit. A steady state of flux density is assumed. With quasi-stationary field the induction is being considered resulting from changes within the current generating the magnetic field or a linearized change within the geometry of the magnetic circuit (e.g. a movement of an anchor). Dynamic magnetic fields consider additional effects covering the dynamic properties of moving mechanical components up to the change of the geometry of the magnetic circuit and the air gaps during operation. By dealing with electrodynamic actuators the analysis of static magnetic circuits is sufficient for a first dimensioning. The relevant dynamic drawbacks for electrodynamic actuators are presented in Sect. 9.2.1.4.

There are two principle possibilities to generate the magnetic flux densities within the volume of a conducting coil:

  1. 1.

    Generation via winded conductors with another coil (exciter winding)

  2. 2.

    Generation via a permanent magnet

Both approaches show specific pros and cons: With a winded conductor the flux density \(B=\mu \,(N\,I-H_{Fe}\,l_{Fe})\) can be raised without any theoretical limit. In practical application the flux-conducting material will reach saturation (Fig. 9.7) actually limiting the achievable maximum flux density. Additionally the ohmic resistance of the winding will generate electrical power losses, which will have to be dissipated in addition to the losses resulting from the electrodynamic principle itself (Sect. 9.2.1.1). Abandoning any flux-conducting material and using exciter-windings with extremely low electrical resistance extraordinary high field-densities can be reached.Footnote 8 Till now, such a technological effort for haptic devices is not made yet.

Building a magnetic circuit with a permanent magnet, the practical border for the flux density is given by the remanence flux density \(B_r\) of the magnetic material. Such a magnet can be compared to a source providing a certain magnetic power. The flux density—being the relevant quality for electrodynamic actuators—is not independent from the magnetic load attached to the permanent magnet. Additionally the relevant properties of the magnetic material are temperature-dependent, and wrong use of specific magnet-materials may harm its magnetic properties.Footnote 9

Fig. 9.7
figure 7

© Springer Nature, all rights reserved

Saturation curve of typical magnetic materials [6]

Nevertheless modern permanent-magnetic materials made of “rare earths” are the preferred source to generate static magnetic fields for electrodynamic actuators. The following section gives some basics on the calculation for simple magnetic circuits. In extension to what is shown here a more precise analytical calculation is possible [6]. However it is recommended to use simulation tools early within the design process. Especially leakage fields are a great challenge for the design of magnetic circuits. And especially beginners should develop a feeling for the look of these fields with the aid of simulation tools. 

Direct Current Magnetic Field

Figure 9.8a shows a magnetic circuit out of iron with a cross section A and an air-gap with the length \(\xi _G\) (\(G=\) Gap). The magnetic circuit has a winding with N turns conducting a current I. The medium length of the magnetic circuit is \(l_\text {Fe}\). For calculation the circuit can be transformed into a magnetic equivalent network (Fig. 9.8b). According to the analogies defined in Table 9.2 the magnetic induction generates a magnetomotive force \(\varTheta \) as a differential value. In combination with two magnetic resistances of the iron circuit \(R_\text {mFe}\) and the air gap \(R_\text {mG}\) a magnetic flux \(\phi \) can be identified.

Fig. 9.8
figure 8

Magnetic field generation B via a current-conducting coil with N turns (a), and derived equivalent circuit representation (b)

For the calculation of the flux density B in the air gap, it is assumed that this magnetic flux \(\phi \) is identical to the flux within the iron part of the circuit. Leakage fields are disregarded in this example.Footnote 10

$$\begin{aligned} B=\frac{\phi }{A} \end{aligned}$$

The magnetic resistance of materials and surfaces are dependent on the geometry and can be found in special tables [6]. For the magnetic resistance of a cylinder of the length l and the diameter d a resistance according to Eq. (9.24) is given.

$$\begin{aligned} R_m=\frac{4\,l}{\mu \,\pi \,d^2} \end{aligned}$$
(9.24)

For the magnetic circuit the magnetic resistances \(R_{m\text {Fe}}\) and \(R_{m\text {G}}\) can be regarded as known or at least calculable. The magnetic flux is given by

$$\begin{aligned} \phi =\frac{\varTheta }{R_{m\text {Fe}}+R_{m\text {G}}}, \end{aligned}$$
(9.25)

and the flux density by

$$\begin{aligned} B=\frac{\varTheta }{(R_{m\text {Fe}}+R_{m\text {G}})\,A}. \end{aligned}$$
(9.26)

Using this procedure a clever approximation of the magnetic resistances of any complex network of magnetic circuits can be made. In this specific case of a simple horseshoe-formed magnet an alternative approach can be chosen. Assuming that the magnetic flux density in the air-gap is identical to the flux density in the iron (no leakage fields, see above) the flux-density B is given by:

$$\begin{aligned} B=\mu _0\mu _r\,H \end{aligned}$$
(9.27)

Assuming that \(\mu _r\) is given either as a factor or with a characteristic curve (like in Fig. 9.7) only the magnetomotive force \(\varTheta \) within the iron has to be calculated. With

$$\begin{aligned} \varTheta =H_\text {Fe}\,l_\text {Fe}+H_\text {G}\,\xi _G =\frac{B}{\mu _0\mu _r}\,l_\text {Fe}+\frac{B}{\mu _0}\,\xi _G \end{aligned}$$
(9.28)

the flux density

$$\begin{aligned} B=\varTheta \,\frac{1}{\frac{l_\text {Fe}}{\mu _0\mu _r}+\frac{\xi _G}{\mu _0}}, \end{aligned}$$
(9.29)

results and can be written down immediately. The generalized model of a coil in a magnetic circuit is that of an ideal magnetic voltage source.

Permanent Magnets Generating the Magnetic Field

As stated earlier the typical approach to generate the magnetic field within an electrodynamic actuator is by using a permanent magnet. Permanent magnets are not just some ideal flux- or field-sources. Therefore some basic understanding of magnet technology will be necessary.

As a simple approach a magnet is a source of energy which is proportional to the volume of the magnet. Magnets are being made out of different magnetic materials (Table 9.3) differing in the maximum achievable flux density (remanence flux density \(B_r\)), the maximum field-strengths (coercive field strength \(H_{c\,B}\) and \(H_{c\,J}\))), and their energy density \(BH_{max}\), such as the temperature coefficient. Additionally identical materials are differentiated according to being isotropic or anisotropic. With isotropic magnets its substance is made of homogeneous material which can be magnetized in one preferred direction. With anisotropic material a magnetic powder was mixed with a binding material (e.g. epoxy) and formed via a casting or injection-molding process. Latter approach enables almost unlimited freedom for the magnet’s geometry and a very large influence concerning the pole-distribution on the magnet. However anisotropic magnets are characterized by slightly worse characteristic values in energy density such as maximum field-strengths and flux densities.

Table 9.3 Magnetic properties of permanent-magnet materials [6] © Springer Nature, all rights reserved

Figure 9.9 shows the second quadrant of the B-H-characteristic curve (only this quadrant is relevant for an application of a magnet within an actuator) of different magnetic materials. The remanence flux density \(B_r\) equals the flux density with short-circuit pole shoes (a magnet being surrounded by ideal iron as magnetic circuit). When there is an air gap within the magnetic circuit (or even by the magnetic resistance of the real magnetic circuit material itself), a magnetic field strength H appears as a load. As a reaction an operation point is reached, which is shown here as an example on a curve of NdFeB for a flux-density of \(\approx \)200 kA/m. The actually available flux density at the poles is decreased accordingly. As electrodynamic actuators for haptic applications face high requirements according to their energy density, there are almost no alternatives to the usage of magnet materials based on rare earths (NdFeB, SmCo). This is very accommodating for the design of the magnetic circuit, as nonlinear effects near the coercive field strength such as with AlNiCo or Barium-ferrite are of no relevance.Footnote 11 Rare earth magnets allow an approximation of their B/H-curve with a linear equation, providing a very nice relationship for their magnetic resistance (Fig. 9.10c):

$$\begin{aligned} R_{\text {Mag}}=\frac{V}{\phi }=\frac{H_c\,l_{\text {Mag}}}{B_r\,A} \end{aligned}$$
(9.30)

Equation 9.30 and Fig. 9.10c reveal the actual mental model of a permanent magnet in a circuit: At their working-point, they can be considered linear non-ideal magnetic voltage source \(V=H_c\,l_{\text {Mag}}\) with an internal resistance \(R_{\text {Mag}}\).

Fig. 9.9
figure 9

© Springer Nature, all rights reserved

Demagnetization curves of different permanent-magnet materials [6]

With this knowledge the magnetic circuit of Fig. 9.10a and the corresponding equivalent circuit (Fig. 9.10b) can be calculated identical to an electrically excited magnetic circuit.

Fig. 9.10
figure 10

Magnetic field generation B via permanent magnets (a), derived equivalent circuit (b), and dimensions of the magnet (c)

The flux density within the iron is once again given by

$$\begin{aligned} B=\frac{\phi }{A} \end{aligned}$$
(9.31)

For the given magnetic circuit the resistances \(R_{m\text {Fe}}\) and \(R_{m\text {G}}\) are assumed as known or calculable. From Eq. (9.30) the magnetic resistance of the permanent magnet is known. The source within the equivalent circuit is defined by the coercive field strength and the length of the magnets \(H_c\,l_\text {Mag}\). These considerations result in

$$\begin{aligned} \phi =\frac{H_c\,l_\text {Mag}}{R_{m\text {Fe}}+R_{m\text {G}}+R_{\text {Mag}}}, \end{aligned}$$
(9.32)

and the flux density

$$\begin{aligned} B=\frac{H_c\,l_\text {Mag}}{(R_{m\text {Fe}}+R_{m\text {G}}+R_{\text {Mag}})\,A}. \end{aligned}$$
(9.33)

Slightly rearranged and \(R_{\text {Mag}}\) included gives

$$\begin{aligned} B=\frac{B_r\,H_c\,\frac{l_\text {Mag}}{A}}{(R_{m\text {Fe}}+R_{m\text {G}})\,B_R+H_c\,\frac{l_\text {Mag}}{A}}. \end{aligned}$$
(9.34)

Equation (9.34) states by the factor \(B_r\,H_c\,\frac{l_\text {Mag}}{A}\) that it is frequently very helpful for achieving a maximum flux density B in the air gap to increase the length of a magnet with at the same time minimized cross-sectional area of the magnetic circuit—which is of course limited by the working distance within the air gap and the saturation field strengths of the magnetic circuit.

2.1.4 Additional Effects in Electrodynamic Actuator

To do a complete characterization of an electrodynamic actuator there are at least three more effects, whose influences will be sketched within the following paragraphs.

Induction

For a complete description of an electrodynamic actuator the dynamic properties needs to be considered next to the geometrical design of its magnetic circuit and the mechanical design of its winding and the considerations concerning electrical power losses. For this analysis the electrodynamic actuator is regarded as a bipolar transformer (Fig. 9.11).

Fig. 9.11
figure 11

Electrical and mechanical equivalent circuit of an electrodynamic actuator as being a transformer

A current \(\underline{i}_0\) generates via the proportional constant \(B\,l\) a force \(\underline{F}_0\), which moves the mechanical loads attached to the actuator. The movement itself results in a velocity \(\underline{v}_0\) which is transformed via the induction law and the proportional constant to an induced voltage \(\underline{u}_1\). By measurement of \(\underline{u}_1\) and a current source the rotational velocity or the movement velocity v can be measured, with a voltage source the measurement of \(\underline{i}_0\) provides a force- or torque-proportional signal. This is the approach taken by the variant of admittance controlled devices as a control value (Sect. 6.7).

The induction itself is a measurable effect, but should not be overestimated. Typically electrodynamic actuators are used as direct drives at small rotational or translational velocities in haptic systems. Typical coupling factors with rotatory drives are—depending on the size of the actuators—in an area between 100 to 10 \(\frac{\text {revolutions}}{s\,V}\). At a rotational speed which is already fast for direct drives of 10 Hz, induced voltage amplitude \(|\underline{u}_1|\) of 0.1–1 V can be achieved. This is around 1–5% of the control voltage’s amplitude.

Electrical Time Constant  

Another aspect resulting from the model according to Fig. 9.11 is the electrical transfer characteristics. Typical inductances L of electrodynamic actuators lie in the area of 0.1–2 mH. The ohmic resistance of the windings is largely depending on the actual design, but as a rule of thumb values between 10–100 \(\varOmega \) can be assumed. The step-response of the electrical transfer system \(\frac{\underline{i}_0}{\underline{u}_0}\) shows a time-constant \(\tau =\frac{L}{R}=10{-}30\,\upmu \textrm{s}\) and lies within a frequency range \(\gg \)10 kHz, which is clearly above the relevant dynamic area of haptics.

Field Response 

A factor which can not so easily be neglected when using electrodynamic actuators for high forces is the feedback of the magnetic field generated by the electromagnetic winding on the static magnetic field. Taking the actuator from the example at the beginning (Fig. 9.4) positive currents generate a field of opposite direction to the field generated by the magnet. This influence can be considered by substitution of both field sources. Depending on the direction of current this field either enforces or weakens the static field. With awkward dimensioning this can result in a directional variance of the actuator properties. The problem is not the potential damage to the magnet, modern magnetic materials are sufficiently stable, but a variation of the magnetic flux density available within the air gap. An intended application of this effect within an actuator can be found in an example according to Fig. 9.52.

A deeper discussion about electrodynamic actuators based on concentrated elements can be found in [7].

2.2 Actual Actuator Design

As stated earlier electrodynamic actuators are composed of three basic components: coil/winding, magnetic circuit, and magnetic exciter. The following section describes a procedure for the design of electrodynamic actuators based on these basic components. As the common principle for excitation a permanent magnet is assumed.

2.2.1 Actuator Topology

The most fundamental question for the design of an electrodynamic actuator is its topology. Usually it is known whether the system shall perform rotary or translation movements. Afterward the components magnetic circuit, the location of magnets, pole-shoes and the coil itself can be varied systematically. A few quite common structures are shown in Fig. 9.13 for translational actuators, and in Fig. 9.12 for rotatory actuators. For the design of electrodynamic actuators in any case the question should be asked, whether the coil or the magnetic circuit shall move. By this variation apparently complex geometrical arrangements can be simplified drastically. Anyway it has to be considered that a moving magnet has more mass and can typically be moved less dynamically than a coil. On the other hand there is no contact- or commutating problem to be solved with non-moving windings.

Fig. 9.12
figure 12

Variants of electrodynamic actuators for translational movement with moving magnets (a), moving coils (b), as plunger-type (c), and as flat-coil (d)

Fig. 9.13
figure 13

Variants of electrodynamic actuators for rotatory movements with self-supportive winding (a), and with disc-winding

Moving Coils 

Electrodynamic actuators according to the principle of moving coils with a fixed magnetic circuit are named “moving coil” in the case of a linear movement and “iron-less rotor” in the case of a rotatory actuator. They always combine few moving masses and as a result high dynamics. The translatory version shows displacements of a few millimeters, and is used especially within audio applications as loudspeaker. Actuators according to the principle of “moving coils” have two disadvantages:

  • As the coil is moving, the electrical contact is subject to mechanical stresses. Especially with high displacements the contact has to be mechanically robust.

  • If there is the idea to design moving coils as pure force sources with large displacements, always only a small area of the conducting coil is within the air-gap and therefore contributes to the force generation. With large displacements moving coils show an even lower efficiency factor. This can be compensated by switching the active coil areas, which again results in the necessity to have more contacts.

A similar situation happens with rotatory systems. Based on the electrodynamic principle there are two types of windings applicable to rotatory servo-systems: the Faulhaber and the Maxon-winding of the manufacturers with identical names. These actuators are also known as “iron-less” motors. Both winding principles allow the manufacture of self-supportive coils. A diagonal placement of conductors and a baking process after winding generates a structure sufficiently stable for the centrifugal forces during operation. The baked coils are connected with the rotating axis via a disk. The complete rotor (Fig. 9.14) is build of these three components. By the very small inertia of the rotor such actuators show impressive dynamic properties. The geometrical design allows placing the tubular winding around a fixed, diametral-magnetized magnet. This enables another volume reduction compared to conventional actuators as its housing has to close the magnetic circuit only instead of providing additional space for magnets.

Fig. 9.14
figure 14

Picture courtesy of Dr. Fritz Faulhaber GmbH, Schöneich, Germany, used with permission

Design of an electrodynamic actuator with self-supportive winding according to the Faulhaber-principle.

Within the self-supportive winding there are areas of parallel lying conductors combined to poles.Footnote 12 With moving coils there is always the need for a specialized contactor, either via contact rings, or electronic commutation or via mechanical switching. Depending on the number of poles all coils are contacted at several points. In case of mechanical switching these contacts are placed on the axis of the rotor and connected via brushes with the fixed part of the actuator named “stator”. This design enables a continuous movement of the rotor, whereas a change of the current flow is made purely mechanically by the sliding of the brushes on the contact areas of the poles on the axis. This mechanical commutation is a switching procedure with an inductance placed in parallel.

As such an actuator can be connected directly to a direct current source, they are known as “DC-drives”. As stated within Sect. 9.1 the term “DC-drive” is not only limited to actuators according to the electrodynamic principle but is also frequently applied to actuators following the electromagnetic principle (Sect. 9.4).

Moving Magnet 

In case of translatory (Fig. 9.12a) systems actuators according to the principle of a moving magnet are designed to provide large displacements with compact windings. The moving part of the actuator is composed almost completely of magnetic material. The polarity direction of this material may vary in its exact orientation. Actuators according to this principle are able to provide large power, but are expensive due to the quantity of magnet material necessary. Additionally the moving magnet is heavy; the dynamics of the actuator is therefore smaller than in the case of a moving coil. Nevertheless some very successful designs exists. A special form-factor can be found with the TapticEngine (Fig. 9.15) specialized for a very slim design at a still comparably large accelerated mass. The translator followed a moving-magnet-design with poles facing each other forcing the magnetic flux to exit through the airgap with coils wound flat on a magnetic back iron. The whole translator is spring-balanced and can operate in a wide frequency range with a clear resonance defined by the spring k and the moving-mass m: \(f_r=\frac{1}{2\pi }\sqrt{\frac{k}{m}}\). A related design but more straightforward as built on rotational symmetry is shown in Fig. 9.16 called HapCoilOne manufactured and sold by the French company actronika. Due to the large moving mass, a combined damper and spring-element and some very reasonable coiling such system allows a wide-bandwidth at an excellent power-level.

Fig. 9.15
figure 15

TapticEngine as used in mobile devices of the company Apple. Flat electrodynamic actuator with moving magnet on a translator. Figure shows principle sketch (a), assembled unit (b) and disassembled unit with lower translator with spring and upper body forming the magnetic back iron visible (c)

Fig. 9.16
figure 16

©2022 actronika, used with permission

Exciter-concept HapCoilOne by the company actronika with moving-magnet design for high-performance haptic applications,

In case of a rotatory system a design with moving magnet is comparable to a design with moving coil. Figure 9.17 shows such a drive. The windings fixed to the stator are placed around a diametral magnetized magnet. It rotates on an axis, which frequently additionally moves the magnetic circuit too. Providing the right current feed to the coil the orientation of the rotor has to be measured. For this purpose sensors based on the Hall-effect or optical code wheels are used.

Fig. 9.17
figure 17

Pictures courtesy of Dr. Fritz Faulhaber GmbH, Schöneich, Germany, used with permission

Components of a EC-drive.

Electrodynamic actuators with moving magnet are known as EC-drives (electronic-commutated) . This term is not exclusive to electrodynamic actuators, as there are electronic-commutated electromagnetic drives too. EC-drives—whether they are electrodynamic or electromagnetic—combined with the corresponding driver electronics are frequently known as servo-drives. Typically a servo-drive is an actuator able to follow a predefined movement path. Servo-drives are rarely used for haptic devices. However the usage of EC-drives for haptic application is very frequent, but then they are equipped with specialized driver electronics.

2.2.2 Commutation in the Context of Haptic Systems

If continuous rotations are required, there is the need to switch the direction of the current flow. This process is called commutation. This necessary commutation of the current for rotating actuators has a big influence upon the quality of force- respective torque-output.

Mechanically Commutating Actuators

With mechanically commutating actuators the current flow is interrupted suddenly. Two effects of switching contacts appear: The voltage at the contact point increases, sparks may become visible—an effect which is called electrical brush sparking. Additionally the remaining current flow induces a current within the switched-off part of the winding which itself results in a measurable torque. Depending on the size of the motor, this torque can be felt when interacting with a haptic system and has to be considered in the design.

The current- and torque changes can be reduced by the inclusion of resistors and capacitors into the coil. However this results into high masses of the rotor and worse dynamic properties. Beside that a full compensation is impossible. Nevertheless mechanically commutating actuators are in use for inexpensive haptic systems. The geomagic Touch from geomagic and the Falcon from Novint use such actuators.

Fig. 9.18
figure 18

Technologies for different commutation methods: sensor-less (a), block-commutation (b) and optical code-wheel (c)

Electronic Commutated Electrodynamic Actuators

Electronic commutated electrodynamic actuators differ from mechanically commutated actuators by the measurement technology used as a basis for switching currents. There are four typical designs for this technology:

  • In sensor-less designs (Fig. 9.18a) an induced voltage is measured within a coil. At zero-crossing point one pole is excited with a voltage after an interpolated 30 \({}^{\circ }\) phase delay dependent on the actual revolution speed of the rotor. In combination of measurement of the inductance followed by a switched voltage, a continuous rotation with batch-wise excitation is realized. This procedure can not be applied to low rotation speeds, as the induced voltage becomes too low and accordingly the switching point can hardly be interpolated. Additionally the concept of using one to two coils for torque generation results in a high torque variations at the output of up to 20%, making this approach not useful for haptic systems.

  • Block-commutating procedures (Fig. 9.18b) are based on the usage of simple hall-switches or field-plates for position detection of the rotor. Three sensors located at 120 \({}^{\circ }\) angular phase shift allow the detection of six different rotor positions. Reducing positioning information to six orientations per revolution makes this approach equally inappropriate for haptic applications, as the torque varies in a range of >15% for one revolution.

  • Sinus-commutating procedures with analogue hall-sensors are based on the measurement of the rotor position by at least two sensors. They are placed with an angle of either 120 \({}^{\circ }\) or 90 \({}^{\circ }\) at the front of the rotor. They provide voltages in an angular phase shift according to their geometrical position. By analyzing the polarity and the absolute height of the voltages absolute positioning information can be obtained and used for commutating the windings. If the phase lag between both sensor signals is identical to the phase lag between the poles of the winding a direct control of current-drivers can be performed without the need for a digitization or a specific calculation step.

  • Sinus-commutating with digital code-wheels (Fig. 9.18c) are based on the measurement of rotor position by the use of—usually optical—code discs. By reflective or transmissive measurement the rotor position is sampled with high resolution. This relative positioning information can be used for position measurement after an initial calibration. Depending on the code-wheels resolution a very smooth sinusoid commutation can be achieved with this method.

The sinus-commutating methods are the preferred solutions used for haptic applications due to the little torque variations and their applicability for slow revolution speeds typical to direct drives.

2.3 Actuator Electronics

Electrodynamic actuators require some specific electrical circuits. In the following section the general requirements on these electronics are formulated.

2.3.1 Driver Electronics

Driver electronics are electrical circuits transforming a signal of low power (several volts, some milli-ampere) into a voltage- or current level appropriate to drive an actuator. For electrodynamic actuators in haptic applications driver electronics have to provide a current in a dynamic range from static up to several kilohertz. This paragraph describes general concepts and approaches for such circuits.

Topology of Electric Sources

 

Driver electronics for actuators—independently from the actuation principle they are used for—are classified according to the flow of electrical energy (Fig. 9.19). There are four classes of driver electronics:

  • 1-quadrant controllers are capable of generating positive output currents and voltages. An actuator driven by them is able to move in one direction. These controller use only the first quadrant according to Fig. 9.16a.

  • Switched 1-quadrant controllers are capable of a direction change by the input of a logical signal. They are working within the 1st and 3rd quadrant according to Fig. 9.19a. The switching point is a nonlinear step in their characteristic curve.

  • Real 2-quadrant controllers are capable of providing a characteristic curve which is steady around the zero point. They function in the 1st and 3rd quadrant according to Fig. 9.19a, but are not capable to conduct currents and voltages with opposite directions.

  • 4-quadrant controllers function within all four quadrants of Fig. 9.19a. They are able to control currents and voltages in any combination of directions. Four-quadrant controllers allow energy recovery by induced currents to an energy storage, which is especially relevant for mobile applications.

For haptic application the switched 1-quadrant controller is frequently met, as many haptic systems do not have the necessity to control the device near the voltage- or current-zero point. However for systems with high dynamics and low impedance the 2-quadrant and the 4-quadrant controller are relevant, as the unsteadiness near the zero-point is perceivable with high quality applications.

Fig. 9.19
figure 19

Visualization of the four quadrants of an electric driver, formed by the directions of current and voltage

Pulse-Width-Modulation and H-Bridges

With the exception of some telemanipulators, the sources controlling the actuators are always digital processors. As actuators need an analogue voltage or current to generate forces and torques some transformer between digital signals and analogue control value is necessary. There are two typical realizations of these transformers:

  1. 1.

    Usage of a digital-analog converter (D/A-converter)

  2. 2.

    Usage of a \(\hookrightarrow \) Pulse-Width-Modulation (PWM) 

The use of D/A-converters as external components or integrated within a microcontroller is not covered further in this book, as it is, if necessary to use, extremely simple. It just requires some additional efforts in circuit layout. Latter results in it being not used much for the control of actuators.

With electrodynamic actuators the method of choice are driver electronics based on PWM (Fig. 9.20a). With the PWM a digital output of a controller is switched with a high frequency (>10 kHzFootnote 13). The period of the PWM is given by the frequency. The program controls the duty cycle between on- and off-times. Typically one byte is available to provide a resolution of 256 steps within this period. After filtering the PWM, either via an electrical low-pass or via the mechanical transfer-characteristics of an actuator, a smoothed output signal becomes available.

Fig. 9.20
figure 20

Principle of puls-width-modulation (PWM) at a digital \(\mu \)C-output (a), h-bride circuit principle (b), and extended h-bridge with PWM (S1) and current measurement at (\(\text {R}_\text {Sense}\)) (c)

Pulse-width-modulation is frequently used in the combination with H-bridges (Fig. 9.20b). The term H-bridge results from the H-like shape of the motor surrounded by four switches. The H-bridge provides two operation modes for two directions of movement and two operation modes for braking. If according to Fig. 9.20b the two switches S2 and S5 are on, the current I will flow through the motor in positive direction. If instead switches S3 and S4 are switched on, the current I will flow through the motor in negative direction. One additional digital signal acting upon the H-bridge will change the direction of movement of the motor. This is the typical procedure with switched 1-quadrant controllers. Additional switching-states are given by switching the groups S2 and S3 respectively S4 and S5. Both states results in short-circuit of the actuator and stops its movement. Other states like simultaneously switching S2 and S4 respectively S3 and S5 results in short-circuit of the supply voltage, typically destroying the integrated circuit of the driver.

To combine the H-bridge with a PWM either switch-groups S2 and S5 can be switched according to the timing of the PWM, or additional switches S1 (Fig. 9.20c) can be placed in series to the H-bridge modulating the supply voltage U. In a practical realization latter is the preferred design, as the timing of the switches S2 to S5 is very critical to prevent likely short circuits of the supply voltage. The effort to perform this timing between the switching is usually higher than the costs of another switch in series. The practical realization of H-bridges is done via field-effect transistors. The discrete design of H-bridges is possible, but not easy. Especially the timing between switching events, the prevention of short-circuits, and the protection of the electronics against induced currents is not trivial. There are numerous integrated circuits available at the market which already include appropriate protective circuitry and provide only a minimum of necessary control lines. The ICs L6205 (2A), L293 (2.8A) and VNH 35P30 (30A) are some examples common with test-bed developments. For EC drives there are specific ICs performing the timing for the field-effect transistors and reducing the number of necessary PWMs from the microcontroller. The IR213xx series switches three channels with one external half-bridge per channel built up from N-MOS transistors with a secure timing for the switching events.

The PWM described above with an H-bridge equals a controlled voltage source. For electrodynamic systems such a control is frequently sufficient to generate an acceptable haptic perception. Nevertheless for highly dynamic haptic systems a counter induction (Sect. 9.2.1.4) due to movement has to be expected, resulting in a variation of the current within the coils generating an uncontrolled change of the Lorentz-force. Additionally the power-loss within the coils (Sect. 9.2.1.1) may increase the actuator’s internal temperature resulting in a change of conductivity of the conductor’s material. The increasing resistance with increasing temperatures of the conductor results in less current flow at a constant voltage source. An electrodynamic actuator made of copper as conductive material would generate a reduced force when operated. With higher requirements on the quality of haptic output a controlled current should be considered. In case of a PWM a resistor with low resistance (\(\text {R}_\text {Sense}\) in Fig. 9.20c) has to be integrated, generating a current-proportional voltage \(\text {U}_\text {Sense}\), which itself can be measured with an A/D input of the controller. The control circuit is closed within the microcontroller. However the A/D transformation and the closing of the control circuit can be challenging for state of the art electronics with highly-dynamic systems with border frequencies of some kilohertz. Therefore analog circuits should be considered for closed-loop current controls too.

Haptic Driver ICs

Meanwhile for standard applications using excentric rotating mass (ERM) motors or linear resonant actuators (LRA) such as the engines shown in Fig. 9.15 or Fig. 9.16 integrated circuits with additional value exists. Texas Instruments (TI) for example offers the DRV2605 driver circuit with included PWM, controlled by an I2C protocol. It includes already some basic tactile patterns and by this offers a simple extension to any microcontroller to create basic patterns without loading the computing needs onto the main unit. And it even goes beyond that. For example with focus on industrial applications Maxim released the MAX11811, a driver combining resistive touchscreen measurement with haptic actuation. Almost all major manufacturers of integrated circuits meanwhile offer such drivers, which—for standard applications—makes it easy to create some level of haptic output especially for touchscreen-type of applications.

Analogue Current Sources 

Analogue current sources are—to make it simple—controlled resistors within the current path of the actuator. It should be noted that with the wide and easy access to PWMs this technology is not common anymore. However in terms of tactile performance, those sources are still a gold-standard as no high-frequency-component is involved into the signal generation. Their resistance is dynamically adjusted to provide the wished current flow. Identical to classical resistors analogue current sources transform the energy which is not used within the actuator into heat. Consequently in comparison to the switched H-bridges they are generating a lot of power loss. By the use of a discrete current control (Fig. 9.21a) analogue current sources for almost any output currents can be built by the choice of one to two field-effect-transistor (FET). For heat dissipation they are required to be attached to adequate cooling elements. There are only little requirements on the operational amplifiers themselves. They control the FET within its linear range proportional to the current-proportional-voltage generated at \(\text {R}_\text {Sense}\). Depending on the quadrant used within operational mode (1 or 3) either the N-MOS transistors or the P-MOS transistor is conductive. An alternative to such discrete designs is the use of power-amplifiers (e.g. LM675, Fig. 9.21b). It contains fewer components and is therefore less dangerous to make errors. Realized as non-inverting or inverting operational amplifier with a resistor for measurement \(\text {R}_\text {Sense}\), they can be regarded as a voltage controlled current source.

Fig. 9.21
figure 21

© Springer Nature, all rights reserved (a), and closed-loop current control with a power-operational-amplifier (b)

Discrete closed-loop current control [8]

2.3.2 Monitoring Temperature

Resulting from the low efficiency factor and the high dissipative energy from electrodynamic actuators it is useful to monitor the temperature nearby the coils. Instead of including a measuring resistor PT100 nearby the coil, another approach monitors the electrical resistance of the windings themselves. Depending on the material of the windings (e.g. cooper, Cu) the conductivity changes proportional to the coil’s temperature. With copper this factor is 0.39% per Kelvin temperature change. As any driver electronics either works with a known and controlled voltage or current, measurement of the other component immediately provides all information to calculate resistance and consequently the actual coil temperature.

2.4 Examples for Electrodynamic Actuators in Haptic Devices

Electrodynamic actuators are most frequently used as exciters for tactile systems also named linear-resonant-actuators (LRA) , or as force and torque sources within kinaesthetic systems. Especially EC-drives can be found in the products of Quanser, ForceDimension, Immersion, and SensAble/geomagic. Mechanically commutated electrodynamic actuators are used within less expensive devices, like the Phantom Omni or the Novint Falcon.

2.4.1 Cross-Coil System as Rotary Actuator

Beside self supportive coils electrodynamic actuators according the design of cross coils are one possibility to generate defined torques. Continental VDO developed a haptic rotary actuator device being a central control element for automotive applications (Fig. 9.22). It contains a diametral magnetized NdFeB-magnet. The magnet is surrounded by a magnetic circuit. The field-lines reach from the magnet to the magnetic circuit. The coils surround the magnet in an angular phase of 90 \({}^{\circ }\), and the electrodynamic active winding section lies in the air-gap between magnetic circuit and magnet. The rotary position control is made via two hall-sensors placed in a 90 \({}^{\circ }\) position. The actuator is able to generate a ripple-free torque of \(\approx \)25 mNm at a geometrical diameter of 50 mm, which is additionally increased by an attached gear to \(\approx \)100 mNm torque output.

Fig. 9.22
figure 22

Electrodynamic cross-coil system with moving magnet as haptic rotary actuator

2.4.2 Reconfigurable Keypad—HapKeys

Although the design shows similarities to Fig. 9.16, this design was built for kinaesthetic feedback. The electrodynamic linear actuators building the basis of this device are equipped with friction type bearings, and moving magnets with pole-shoes within cylindrically wound fixed coils as shown in Fig. 9.23. The coils have an inner diameter of 5.5 mm and an outer diameter of 8 mm. The magnetic circuit is decoupled from other nearby elements within the actuator-array. It is made of a tube with a wall thickness of 0.7 mm of a cobalt-iron alloy with very high saturation flux density. Each actuator is able to generate 1 N in continuous operation mode.

Fig. 9.23
figure 23

Electrodynamic linear actuator with moving magnet [9]

2.5 Conclusion About the Design of Electrodynamic Actuators

Electrodynamic actuators are the preferred actuators used for kinaesthetic impedance-controlled haptic devices due to their proportional correlation between the control value “current” and the output-values “force” or “torque”. The market of DC -and EC-drives offers a wide variety of solutions, making it possible to find a good compromise between haptic quality and price for many applications. Most suppliers of such components offer advice on how to dimension and select a specific model based on the mechanical, electrical and thermal properties as for example shown in [10].

If there are special requirements to be fulfilled, the design, development, and start of operation of special electrodynamic actuator variants are quite easy. The challenges by thermal and magnetic design are manageable, as long as some basic considerations are not forgotten. The examples of special haptic systems seen in the preceding section prove this impressively. Just driver electronics applicable to haptic systems and its requirements are still an exceptional component within the catalogs of manufactures from automation-technology. They must either be paid expensively or be built by oneself. Therefore commercial manufacturers of haptic devices, e.g. Quanser, offer their haptic-applicable driver electronics independent from the own systems for sale.

For the design of low-impedance haptic systems currently no real alternative to electrodynamic systems exists. Other actuation principles which are discussed within this book need a closed-loop control to overcome their inner friction and nonlinear force/torque-transmission. This always requires some kind of measurement technology such as additional sensors or the measurement of inner actuator states. The efforts connected with this are still a big advantage for electrodynamic actuators, which is gained by a low efficiency factor and as a consequence the relatively low energy density per actuator-volume.  

3 Piezoelectric Actuators

Next to the very frequently found electrodynamic actuators, the past few years piezoelectric actuators were used for a number of device designs. Especially their dynamic properties in resonance mode allow an application for haptics, which is very different from the common positioning application they are used for. As variable impedance a wide spectrum of stiffnesses can be realized. The following chapter gives the calculation basics for the design of piezoelectric actuators. It describes the design variants and their application in haptic systems. Beside specific designs for tactile and kinaesthetic devices approaches for the control of the actuators and tools for their dimensioning are presented.

3.1 The Piezoelectric Effect

The piezoelectric effect was discovered by Jacques and Pierre Curie first. The term is derived from the Greek word “piedein—piezo” = “to press” [11].

Figure 9.24 shows a scheme of a quartz crystal (chemical: SiO2). With force acting upon the crystal mechanical displacements of the charge-centers can be observed within the structure, resulting in microscopic dipoles within its elementary cells. All microscopic dipoles sum up to a macroscopic measurable voltage. This effect is called “reciprocal piezoelectric effect”. It can be reversed to the “direct piezoelectric effect”. If a voltage is applied on a piezoelectric material a mechanical deformation happens along the crystal’s orientation, which is proportional to the field strength in the material [12]. 

Fig. 9.24
figure 24

Crystal structure of quarz in initial state and under pressure

Piezoelectric materials are anisotropic—direction dependent—in their properties. Consequently the effect depends on the direction of the electrical field applied, and on the angle between the direction of the intended movement and the plane of polarization. For the description of these anisotropic properties the directions are labeled with indices. The index is defined by a Cartesian space with the axes being numbered with 1, 2 and 3. The plane of polarization of the piezoelectric material is typically orientated on direction 3. The shear at the axes is labeled with indices 4, 5 and 6.

Among all possible combinations, there are three major effects (Fig. 9.25), commonly used for piezoelectric applications: longitudinal- , transversal- and shear-effect.

Fig. 9.25
figure 25

Effects during applied voltage: longitudinal effect (left), transversal effect (center), shear effect (right)

The longitudinal effect acts in the same direction as the applied field and the corresponding field strength \(E_{3}\). As a consequence the resulting mechanical tensions \(T_{3}\) and strains \(S_{3}\) lie within plane 3 too.  With the transversal effect mechanical actions show normal to the electrical field. As a result from a voltage \(U_{3}\) with the electrical field strength \(E_{3}\) the mechanical tensions \(T_{1}\) and strains \(S_{1}\) appear.    The shear-effect happens with the electrical voltage U applied along plane 1 orthogonal to the polarization plane. The resulting mechanical tensions appears tangential to the polarization—in the direction of shear—and are labeled with the directional index 5.  

3.1.1 Basic Piezoelectric Equations

 The piezoelectric effect can be described most easily by state equations:

$$\begin{aligned} \ P = e \cdot T \end{aligned}$$
(9.35)

and

$$\begin{aligned} S = d \cdot E \end{aligned}$$
(9.36)
figure a

The piezoelectric coefficients are 

  • the piezoelectric coefficient of tension (also: coefficient of force) e (reaction of the mechanical tension on the electrical field)

    $$\begin{aligned} e_{ij,k} = \frac{\partial T_{ij}}{\partial E_{k}}\partial \end{aligned}$$
    (9.37)
  • and the piezoelectric coefficient of strain (also: coefficient of charge) d (reaction of the strain on the electrical field)

    $$\begin{aligned} d_{ij,k} = \frac{\partial \varepsilon _{ij}}{\partial E_{k}}\partial \end{aligned}$$
    (9.38)

The correlation of both piezoelectric coefficients is defined by the elastic constants \(C_{ijlm}\)

$$\begin{aligned} e_{ij,k} = \sum _{lm} \left( C_{ijlm} \cdot d_{lm,k}\right) \end{aligned}$$
(9.39)

      

Usually the tensors shown in the equation above are noted as matrix In this format, matrices result of six components identical to the defined axes. The matrix shown below describes the concatenation of the dielectrical displacement D, the mechanical strain S, the mechanical tension T, and the electrical field strength E.

This matrix can be simplified for the specific cases of a longitudinal and a transversal actuator. For a longitudinal actuator with electrical contact in direction 3 the following equations are the result.

  \(T_1\) \(T_2\) \(T_3\) \(T_4\) \(T _5\) \(T_6\) \(E_1\) \(E_1\) \(E_3\)
\(D_1\) 0 0 0 0 \(d_{15}\) 0 \(\varepsilon _{11}\) 0 0
\(D_2\) 0 0 0 \(d_{15}\) 0 0 0 \(\varepsilon _{11}\) 0
\(D_3\) \(d_{31}\) \(d_{31}\) \(d_{33}\) 0 0 0 0 0 \(\varepsilon _{11}\)
\(S_1\) \(s_{11}\) \(s_{12}\) \(s_{13}\) 0 0 0 0 0 \(d_{31}\)
\(S_2\) \(s_{12}\) \(s_{11}\) \(s_{13}\) 0 0 0 0 0 \(d_{31}\)
\(S_3\) \(s_{13}\) \(s_{13}\) \(s_{33}\) 0 0 0 0 0 \(d_{33}\)
\(S_4\) 0 0 0 \(s_{44}\) 0 0 0 \(d_{15}\) 0
\(S_5\) 0 0 0 0 \(s_{44}\) 0 \(d_{15}\) 0 0
\(S_6\) 0 0 0 0 0 \(2(s_{11}-s_{12})\) 0 0 0
$$\begin{aligned} D_{3}=\varepsilon _{33}^{T}E_{3}+d_{31}T_{1} \end{aligned}$$
(9.40)
$$\begin{aligned} S_{3}=d_{31}E_{3}+s_{11}^{E}T_{1}. \end{aligned}$$
(9.41)

Accordingly for a transversal actuator the correlation  

$$\begin{aligned} D_{3}=\varepsilon _{33}^{T}E_{3}+d_{33}T_{3} \end{aligned}$$
(9.42)
$$\begin{aligned} S_{3}=d_{33}E_{3}+s_{33}^{E}T_{3} \end{aligned}$$
(9.43)
figure b

becomes valid.

Therefore the calculation of piezoelectric coefficients simplifies into some handy equations: The charge constant d can be calculated for the electrical short-circuit— which is \(E=0\)— to

$$\begin{aligned} d_{E=0}=\frac{D}{T} \end{aligned}$$
(9.44)

and for the mechanical idle situation—which is \(T=0\)—to

$$\begin{aligned} d_{T=0}=\frac{S}{E}. \end{aligned}$$
(9.45)

The piezoelectric tension constant is defined as

$$\begin{aligned} g=\frac{d}{\varepsilon ^{T}}. \end{aligned}$$
(9.46)

The coupling factor k is given by Eq. (9.47). It is a quantity for the energy transformation and consequently for the strength of the piezoelectric effect. It is used for comparison among different piezoelectric materials. However note that it is not identical to the efficiency factor, as it does not include any energy  losses.

$$\begin{aligned} k=\frac{converted\,\,energy}{absorbed\,\,energy}. \end{aligned}$$
(9.47)

A complete description of the piezoelectric effect, a continuative mathematical discussion, and values for piezoelectric constants can be found in literature, such as [7, 13, 14].

3.1.2 Piezoelectric Materials

 Till 1944 the piezoelectric effect was observed with monocrystals only. These were quartz, turmalin, lithiumniobat, potassium- and ammonium-hydrogen-phosphat (KDP, ADP), and potassium sodium tartrate [12]. With all these materials the direction of the spontaneous polarization is given by the direction of the crystal lattice [11]. The most frequently used material was quartz.

The development of polarization methods made it possible to retrospectively polarize ceramics by the application of a constant exterior electrical field in 1946. By this approach “piezoelectric ceramics” (also “piezoceramics”) were invented. By this development of polycrystalline materials with piezoelectric properties the whole group of piezoelectric materials got an increased attention and technical significance. Today the most frequently used materials are e.g. barium titanate (MaTiO3) or lead zirconate titanate (PZT) [12]. C 82 is a piezoelectric ceramic suitable for actuator design due to its high k-factor. However as all piezoelectric ceramic materials it shows reduced long term stability compared to quartz. Additionally it has a pyroelectric effect which is a charge increase due to temperature changes of the material [7]. Since the 1960s the semi-crystalline synthetic material polyvinylidene fluoride (PVDF) is known. Compared to the materials mentioned before, PVDF excels by its high elasticity and reduced thickness (6–9 \(\upmu \)m).  

Table 9.4 shows different piezoelectric materials with their specific values.

Table 9.4 Selection of piezoelectric materials with characteristic values [7]

Looking at these values PZT is most suitable for actuator design due to its high coupling factor with large piezoelectric charge modulus and still a high Curie temperature . The Curie temperature represents the temperature at which the piezoelectric properties from the corresponding material are lost permanently. The value of the curie temperature depends on the material  (Table 9.4).

3.2 Designs and Properties of Piezoelectric Actuators

Actuators using the piezoelectric effect are members of the group of solid actuators (also: solid-state actuators). The transformation from electrical into mechanical energy happens without any moving parts, resulting in a very fast reaction time and high dynamics compared to other actuation principles. Additionally piezoelectric actuators have a high durability. The thickness changes are smaller—compared to other actuation principles. Although the generated forces are much higher.

3.2.1 Basic Piezoelectric Actuator Designs

Depending on the application different designs may be used. One may require a large displacement; another one may require self-locking or high stiffnesses. The most frequently used actuator types are bending actuators and actuator staples. A schematic sketch of each design is given in Fig. 9.26a and c.

Fig. 9.26
figure 26

Further Information can be found in [11]

Piezoelectric transducers separated by longitudinal and transversal effect: longitudinal effect: a stack, b stack with lever transformation, change of length: \(x=d_{33}\cdot U_{B}\) transversal effect: c bending actuator, d cone, e band f bending disk, change of length: \(x=-d_{31}\cdot U_{B}\).

Stacked actuator are based on the longitudinal piezoelectric effect. For this purpose several ceramic layers of opposite polarity are stapled above each other. In between each layer contact electrodes are located for electrical control. A staple is able to generate high static forces up to several 10 kN. The achievable displacement of \(200\,\upmu \text {m}\) is low compared to other piezoelectric designs. By the use of levers Fig. 9.26b the displacement can be significantly increased (Fig. 9.26b). Voltages of several 100 V are necessary to drive a piezoelectric actuation staple.

Bending actuators are based on the transversal piezoelectric effect. Designed according to the so called bimorph principle—with two active layers—they are used in applications requiring large displacements. The transversal effect is characterized by comparably low controlling voltages [11, 12]. These electrical properties and the large displacements can be achieved by very thin ceramic layers in the direction of the electrical fields, and an appropriate geometrical design. Other geometrical designs using the transversal effect are tubular actuators, film actuators, or bending discs Fig. 9.26d–f. Due to their geometry they equal staple actuators in their mechanical and electrical characteristics. The achievable displacements of \(50\,\upmu \text {m}\) are comparably low, whereas the achievable forces excel bending actuators at several orders of magnitude.

The use of the shear effect is uncommon in actuator design. This is somewhat surprising as it shows charge modulus and coupling factor which is twice as much as the transversal effect. Additionally it is possible to increase the elongation \(x_{0}\) in idle mode (displacement without any load) by the optimization of the length to thickness (l/h) ratio. However the clamping force \(F_{k}\) of the actuator is not influenced by these parameters.

Table 9.5 summarizes the properties of different geometrical designs. Typical displacements, actuator forces and control voltages are shown.

Table 9.5 Properties of typical piezoelectric actuator designs based on [11]

3.2.2 Selection of Special Designs for Piezoelectric Actuators

Beside the standard designs shown above several variations of other geometrical designs exist. In this section, examples of an ultrasonic drive with resonator, oscillatory/traveling waves actuators and piezoelectric stepper motors are discussed. Ultrasonic actuators are differentiated according to resonators with bar like geometry and rotatory ring geometry.

Ultrasonic Actuators with Circular Resonators

As mentioned before beside actuators providing standing waves another group of actuators based on traveling waves exists. The traveling wave actuators known best are circular in their design. The first actuator according to this principle has been built in 1973 by Sashida [15]. Traveling waves actuators count to the group of ultrasonic actuators as their control frequencies typically lie in between 20 kHz up to 100 kHz.

This section is reduced to the presentation of ring-shaped traveling wave actuators with a bending wave. Other design variants for linear traveling wave actuators can be found in the corresponding literature [16,17,18]. 

Figure 9.27 shows an actuator’s stator made of piezoelectric elements. They have an alternating polarization all around the ring. The stator itself carries notches actually enabling the formation of the rotating traveling wave.

Fig. 9.27
figure 27

Piezoelectric traveling wave motor: left: stator disk with piezoelectric elements. right: schematic view of the functionality of a ring-shaped piezoelectric traveling wave motor [11]

Each point on the surface of the stator performs a local elliptic movement (trajectory). This movement is sketched schematically in Fig. 9.27 These individual elliptic movements overlay to a continuous wave on the stator. With a frictional coupling this movement is transferred on the rotor, resulting in a rotation. The contact between stator exists with the same number of contact points anytime during operation.

The movement equation of the traveling wave actuator is given by

$$\begin{aligned} u(x,t)=A \cos (kx-\omega t) \end{aligned}$$
(9.48)

By reshaping it the following form results  in

$$\begin{aligned} u(x,t)=A (\cos (kx))(\cos (wt)) + A (cos(kx-\pi /2))(cos(kx+\pi /2)) \end{aligned}$$
(9.49)

The second term of Eq. (9.49) includes important information for the control of traveling wave actuators. A traveling wave can be generated by two standing waves being spatially and timely different. Within typical realization the spatially difference of \(x_0 = \lambda / 4\) unis chosen, and a time phase lag of \(\Phi _0 = \pi / 2\). The usage of two standing waves is the only practical possibility for generating a traveling wave. The direction of rotor-movement can be switched by changing the phase lag from \(+ \pi / 2 \) to \(- \pi / 2 \) [15, 19,20,21].

Figure 9.28 shows the practical realization of a traveling wave motor. Big advantages of a traveling wave motor are the high torques possible to achieve at low rotational speeds. It has a low overall size and is of little weight. This enables a very thin design as shown in Fig. 9.28. In passive mode the traveling mode motor has a high locking torque of several Nm.

Other advantages are given by the good control capabilities, the high dynamics, and robustness against electromagnetic noise such as the silent movement [22]. Typical applications of traveling wave actuators are autofocus-functions in cameras.

Fig. 9.28
figure 28

All examples by Shinsei Corporation, Tokyo, JP, used with permission

Realization of a traveling wave motor. a cross-section model with functional parts, b motor model USR30 with a maximum speed of 300 rpm and a nominal torque of 0.05 Nm with driver D6030, c model USR60 with attached rotary encoder, maximum speed of 150 rpm and a nominal torque of 0.05 Nm.

Piezoelectric Stepper Motors  

Another interesting design can be found with the actuator PI Nexline. It combines the longitudinal effect with the piezoelectric shear effect, resulting in a piezoelectric stepper motor.

Fig. 9.29
figure 29

Piezoelectric stepper motor using the shear effect and the longitudinal effect [23]

The principle design is sketched in Fig. 9.29. The movement of the motor is similar to the inchworm-principle. Drive- and release-phases of the piezoelectric elements produce a linear movement of the driven rod. The piezoelectric longitudinal elements generate the clamping force in z-direction, the shear elements rotated by 90 \({}^{\circ }\) a translational movement in y-direction is possible too.

The advantage of this design is given by the high positioning resolution. Over the whole displacement of 20 mm a resolution of 0.5 nm can be achieved. The stepping frequency is given by—dependent on the control—up to 100 Hz and enables, depending on its maximum step-width, velocities of up to 1 mm/s. The step-width can be chosen continuously between 5 nm and 8 \(\upmu \)m. The intended position can be achieved either closed-loop or open-loop controlled. For the closed-loop control a linear encoder has to be added to the motor. In open-loop control, the resolution can be increased to 0.03 nm in a high resolution dithering mode.

The actuator can generate push- and pull-forces of 400 N maximum. The self-locking reaches up to 600 N. The typical driving voltage is 250 V. The specifications given above are based on the actuator N-215.00 Nexline\(^{\circledR }\) of the company Physik Instrumente (PI) GmbH & Co. KG, Karlsruhe, Germany [23]. Beside the impressive forces and positioning resolutions which can be achieved, these actuators have a high durability compared to other designs of piezoelectric actuators, as no friction happens between moving parts and stator.

Olsson et al. presented a haptic glove to display stiffness properties based on such an actuator as shown in Fig. 9.30 [24].

Fig. 9.30
figure 30

Figure taken from [24] © Springer Nature, all rights reserved

Hand exoskeleton for the display of stiffness parameters. Forces exerted by the user are recorded with thin force sensors and the actuator position is tracked magnetically.

3.3 Design of Piezoelectric Actuators for Haptic Systems

 Within the preceding section the basic designs of piezoelectric actuators have been discussed and special variants were shown. This section transfers this knowledge about the design of piezoelectric actuators focusing on haptic applications now.

Fig. 9.31
figure 31

Procedure of designing piezoelectric actuators

First of all the principle approach for designing is sketched. Hints are given about which designs are appropriate for which applications. Afterward three tools for practical engineering are shown: description via electromechanic networks, analytical formulations and finite-element simulations.

3.4 Procedure for the Design of Piezoelectric Actuators

Figure 9.31 gives the general procedure for the design of piezoelectric actuators.

The choice of a general design based on those shown in the prior section is largely dependent on the intended application. For further orientation Fig. 9.32 shows a decision tree for classifying the own application.

Fig. 9.32
figure 32

Decision tree for the selection of a type of piezoelectric actuator

The following paragraph describes the appropriate designs for specific application classes according to this scheme. The list has to be regarded as a point for orientation, but it does not claim to be complete. The creativity of an engineer will be able to find and realize other and innovative solutions beside those mentioned here. Nevertheless especially for the design of tactile devices some basic advice can be given based on the classification in Fig. 9.32:

1:

Braille displays have to act against the finger’s force. At typical resolutions this requires forces in the area of mN at displacement of around 100 \(\upmu \)m. The requirements on the dynamics are in the lower range of several Hertz. The smallest resolution of a pixel has to be in the area of 1 \(\times \) 1 mm\(^2\) and is defined by the lowest resolution at the finger’s tip. Looking at the force-amplitude diagram of Fig. 9.51 bending-actuators fit well to these requirements.

2:

In comparison to Braille-displays there are vibrotactile displays, which need higher frequencies and smaller displacements and forces to present a static shape to the user. With the diagram of Fig. 9.33 especially bending disc or staple actuators would be appropriate to fulfill these requirements, although there are overpowered concerning the achievable forces.

3, 4:

Such displays are subject to current research and are not yet applied in broad range. Their design is typically based on bending actuators, as shear the skin requires less force than forces acting normal to the skin by generating a comparable perception.

5–8:

In contrast to tactile displays the actuator selection for kinaesthetic systems is more dependent on the actual application. Forces, displacement and degrees of freedom influenced by the kinematics alter in wide ranges. Additionally the actuator’s volume may be a criterion for selection. Figure 9.33 gives an overview about piezoelectric actuation principles and has to be interpreted according to the specific kinaesthetic problem at hand. Generally speaking ultrasonic piezoelectric actuators are usually the matter of choice for kinaesthetic devices, although they have to be combined with a closed-loop admittance control.

Figure 9.33 gives an overview about piezoelectric actuation principles and has to be interpreted according to the specific kinaesthetic problem at hand. Generally speaking ultrasonic piezoelectric actuators are usually the matter of choice for kinaesthetic devices, although they have to be combined with a closed-loop admittance control.

Additional reference for actuator selection can be found in Sect. 9.3.2 are suitable for haptic applications, but still need some care in their usage due to high voltages applied and their sensitivity on mechanical damage. This effort is often rewarded by piezoelectric actuation principles, which can be combined to completely new actuators. And the only thing required is some creativity of the engineer.

Fig. 9.33
figure 33

Force-amplitude-diagram for the classification of the piezoelectric actuating types

After choosing the general actuator the design process follows next. For this purpose three different methods are available, which are presented in the following and discussed in their pros and cons. In addition some hints on further references are given.

3.4.1 Methods and Tools for the Design Process

There are three different engineering tools for the design of piezoelectric actuators:

  • Description via the aid of electromechanical concentrated networks

  • Analytical descriptions

  • Finite elements simulations

Description via the Aid of Electromechanical Concentrated Networks

The piezoelectric basic equations from Sect. 9.3.1.1 are the basis for the formulation of the electromechanic equivalent circuit of a piezoelectric converter.

Fig. 9.34
figure 34

© Springer Nature, all rights reserved

Piezoelectric actuator as a electromechanical schematic diagram a gyratory and b transformatory combination [7]

The piezoelectric actuator can be visualized as an electromechanical circuit. Figure 9.34 shows the converter with a gyratory coupling (a), alternatively a transformatory coupling (b) is possible too. For the gyratory coupling the Eqs. (9.50)–(9.53) summarize the correlations for the calculation of the values for the concentrated elements. They are derived from the constants \(e, c, \varepsilon \) such as the actuator’s dimensions l and A [7].

$$\begin{aligned} C_{b}= & {} \varepsilon \cdot \frac{A}{l} = (\varepsilon -d^2 \cdot c)\frac{A}{l} \qquad \,\, \text {with} \quad v=0 \end{aligned}$$
(9.50)
$$\begin{aligned} n_{K}= & {} \frac{1}{C} \cdot \frac{l}{A} = s \cdot \frac{A}{l} \qquad \qquad \quad \text {with}\quad U=0 \end{aligned}$$
(9.51)
$$\begin{aligned} Y= & {} \frac{1}{e} \cdot \frac{l}{A} = \frac{s}{d} \cdot \frac{l}{A} \end{aligned}$$
(9.52)
$$\begin{aligned} k^2= & {} \frac{e^2}{\varepsilon \cdot c} = \frac{d^2}{\varepsilon \cdot c} \end{aligned}$$
(9.53)

With the piezoelectric force constants

$$\begin{aligned} e = d \cdot c = \frac{d}{s} \end{aligned}$$
(9.54)

Which makes for the transformatory coupling

$$\begin{aligned} X = \frac{1}{\omega C_{b} \cdot Y } \qquad \text {and} \qquad n_{C} = Y^{2} \cdot C_{b} \end{aligned}$$
(9.55)

Figure 9.35 shows the sketch of a element \(\varDelta x \) taken out of a piezoelectric bimorph bending actuator (dimensions \(\varDelta l\) \(\times \) \(\varDelta h\) \(\times \) \(\varDelta b\)) as a electromechanical equivalent circuit.

Fig. 9.35
figure 35

© Springer Nature, all rights reserved

Piezoelectric bimorph bending element in a electromechanical schematic view in quasistatic state [7]

   It is:

$$\begin{aligned} C_{b}= & {} 4 \varepsilon _{33}^T (1-k_{L}^2) \frac{b \cdot \varDelta x}{h} \\ \varDelta n_{RK}\approx & {} 12 \, s_{11}^E \, \frac{(\varDelta x)^3}{b \cdot h^3} \\ \frac{1}{Y}= & {} \frac{1}{2} \, \frac{d_{31}}{s_{11}^E} \, \frac{b \cdot h}{\varDelta x} \end{aligned}$$

The piezoelectric lossless converter couples the electrical with the mechanical rotatory coordinates first, which are torque M and angular velocity \( \varOmega \). To calculate the Force F and the velocity v an additional transformatory coupling between rotatory and translatory mechanical network has to be introduced. As a result the complete description of a sub-element \(\varDelta x\) of a bimorph is given in a ten-pole equivalent circuit.

Analytical Calculations

A first approach for the design of piezoelectric actuators is given by the application of analytical equations. The advantage of analytical equations lies in the descriptive visualization of physical interdependencies. The influence of different parameters on a target value can be derived directly from the equations. This enables a high flexibility in the variation of e.g. dimensions and material properties. Additionally the processing power for the solution of equations can—compared to simulations—be neglected.

A disadvantage of analytical solution results from the fact, that they can only be applied to simple and frequently only symmetrical geometrical designs. Although already limited to such designs even simple geometries may result in very complex mathematical descriptions requiring a large theoretical background for their solution.

The following collection gives relevant literature for the familiarization with specific analytical challenges being faced with during the design of piezoelectric actuators:

  • Very compelling and complete works on the design of piezoelectric actuators are [15, 25,26,27].

  • The theory of the piezoelectric effect and piezoelectric elements are discussed in [13, 14, 28].

  • The mathematical description of traveling wave actuators can be found in [29, 30].

  • The contact behavior between stator and rotor with piezoelectric multilayer bending-actuators is analyzed in [31,32,33,34].

  • In [12] the static and dynamic behavior of multilayer beam bending actuators is described.

  • The description of the mechanical oscillations for resonance shapes is discussed in [35,36,37,38,39,40] elaborately.

Finite Element Simulation

The application of both approaches given before is limited to some limited geometrical designs. In reality complex geometrical structures are much more frequent which can not be solved with analytical solutions or mechanical networks. Such structures can be analyzed according to the method of finite element simulation (FEM).

For the design of piezoelectric actuators the use of coupled systems is relevant. One example of a FEM-simulation for a piezoelectric traveling wave motor is shown in Fig. 9.36.

Fig. 9.36
figure 36

Example FEM simulation of the oscillation shape of the stator of a piezoelectric travelling wave motor (view highly exaggerated)

3.5 Piezoelectric Actuators in Haptic Systems

Piezoelectric actuators are among the most frequently used actuation principles in haptic systems. The designs shown before can be optimized and adapted for a reasonable number of applications. One of the most important reasons for their usage is their effectiveness at a very small required space, which is identical to a high power density. To classify the realized haptic systems a division into tactile and kinaesthetic is done for the following paragraphs.

3.5.1 Piezoelectric Actuators for Tactile Systems

For the design of any tactile systems the application area is of major importance. The bandwidth ranges from macroscopic table top devices, which may be used for embossed printings in Braille being placed below a PC-keyboard, up to highly integrated systems, which may be used for mobile applications. Especially for latter use, the requirements on volume, reliable and silent operation, but also on low weight and energy consumption are enormous. The following examples are structured into two subgroups. Each of them addresses one of two directions of the penetration of the skin: lateral and normal.

Tactile Displays with Normal Stimulation

Braille Devices

A Braille character is encoded by a dot pattern formed by embossed points on a flat surface. By touching this pattern made of eight dots (two columns with four rows of dots each) combinations of up to 256 characters can be addressed. Since the seventies reading tablets for visually handicapped people are developed which are capable to present these characters with a 2 \(\times \) 4 matrix of pins. The most important technical requirements are a maximum stroke of \(0.1{ - }1\, \text {mm}\) and a counter force of \(200\, \text {mN}\). Already early in this development electromagnetic drives have been replaced by piezoelectric bimorph bending actuators. These actuators enable a thinner design, are more silent during operation and faster. At typical operating voltages of \(\pm 100{-}200\,\text {V}\) and a nominal current of \(300 \,\text {mA}\) they additionally need less energy than the electromagnetic actuators used before. Figure 9.37 shows the typical design of a Braille character driven by a piezoelectric bimorph actuator. A disadvantage of this system is the high price, as for 40 characters with eight elements each all together 320 bending actuators are needed. Additionally they require still a large volume as the bending elements have to show a length of several centimeters to provide the required displacements. This group of tactile devices is part of the shape-building devices. The statically deflected pins enable the user to detect the displayed symbol. 

Fig. 9.37
figure 37

Schematic setup of a Braille row with piezoelectric bending actuators

In the “HyperBraille”-project a two dimensional graphics-enabled display for blind computer users based on piezoelectric bending actuators is realized. The pin matrix of the portable tablet display consists of 60 rows with 120 pins each to present objects such as text blocks, tables, menus, geometric drawings and other elements of a graphical user interface. The array is an assembly of modules that integrate 10 pins, spaced at intervals of 2.5 mm as shown in Fig. 9.38. The benders raise the pins above the plate by 0.7 mm [41].

Fig. 9.38
figure 38

Pictures courtesy of metec AG, Stuttgart, Germany, used with permission

HyperBraille display: Whole device a [42], and single actuator module b [43].

Vibrotactile Devices 

With vibrotactile devices the user does not detect the displacement of the skin’s surface but the skin itself is put into oscillations. At smaller amplitude the sensation is similar to the static elongation. The general design of vibrotactile displays equals an extension of the Braille-character to an N \(\times \) N matrix which is actuated dynamically. The tactile image generated is not perceived by the penetration depth but by the amplitude of the oscillation [44]. Another impact factor is the oscillation frequency, as the tactile perception depends extremely on the frequency. With the knowledge of these inter-dependencies optimized tactile displays can be built generating a very well perceivable stimulation of the receptors. Important for displays according to this approach is a large surface, as movements performed by the own finger disturb the perception of the patterns.

The Texture Explorer presented in [45] is designed as a vibrating 2 \(\times \) 5 pin array. It is used for the research on the perception of tactile stimulation, such as the overlay of tactile stimulation with force-feedback. The surfaces displayed change according to their geometry and their roughness within the technical limits of the device. The contact-pins have a size of \(0,5\,\times \,0,5\,\text {mm}^2\) with a point-to-point distance of \(3\,\text {mm}\). Each pin is actuated separately by a bimorph bending actuator at a driving voltage of \(100 \,\text {V}\) and a frequency of \(250 \,\text {Hz}\). The maximum displacement of these pins with respect to surface level is \(22 \,\upmu \text {m}\) and can be resolved down to \(1\,\upmu \text {m}\) resolution.

An even more elaborate system is based on 100 individually controlled pins [46]. It can be actuated dynamically in a frequency range of \(20 {-} 400\,\text {Hz}\). Figure 9.39 shows a schematic sketch. 20 piezoelectric bimorph bending actuators (PZT-5H, Morgan Matrinic, Inc.) in five different layers one above the other are located in a circuit around the stimulation area. Each bending actuator carries one stimulation pin, which is placed \(1\,\text {mm}\) above the surface in idle state. The pins have a diameter of \(0.6\,\text {mm}\) and are located equally spaced at \(1\,\text {mm}\) distance. At a maximum voltage of \(\pm \,85\,\text {V}\) a displacement of \(\pm \, 50\,\upmu \text {m}\) is achieved. A circuit of equally high passive pins is located around the touch area to mark the borders of the active display.

Fig. 9.39
figure 39

© AIP Publishing, all rights reserved

Schematic setup of the 100-pin-array [46]

Another even more compelling system can be found in [47] (Fig. 9.40). The very compact 5 \(\times \) 6 array is able to provide static and dynamic frequencies of up to \({\approx }500\,\text {Hz}\). Once again, piezoelectric bending actuators are used to achieve a displacement of \(700\,\upmu \text {m}\). However the locking-force is quite low with a maximum of \(60\,\text {mN}\).

Fig. 9.40
figure 40

© Springer Nature, all rights reserved

Schematic setup of the 5 \(\times \) 6-pin-array [47]

Ubi-Pen

The Ubi-Penis one of the highest integrated tactile systems. Inside of a pen both components, a spinning disc motor and a tactile display, are assembled. The design of the tactile display is based on the “TULA35” ultrasonic linear drive (Piezoelectric Technology Co., Seoul, South Korea). The schematic sketch of the design is given in Fig. 9.41. The actuator is made of a driving component, a rod and the moving part. The two piezoelectric ceramic discs are set to oscillation resulting into the rod oscillating up- and downwards. The resulting movement is elliptical. To move upwards, the following procedure is applied: in the faster downward movement the inertial force is excelled by the speed of the operation and the element remains in the upper position. Whereas in the upwards movement the speed is controlled slow enough to carry the moving part up by the frictional coupling between moving element and central rod. The actuator discs have a diameter of \(4\, \text {mm}\) and a height of \(0.5\,\text {mm}\). The rod has a length of \(15\, \text {mm}\) and a diameter of \(1\,\text {mm}\). It can be used as contact-pin directly. The actuator’s blocking force is larger than \(200\,\text {mN}\) and at a control frequency of \(45\,\text {kHz}\) velocities of \(20 \,\text {mm/s}\) can be reached.

Fig. 9.41
figure 41

© Springer Nature, all rights reserved

Schematic setup of the ultrasonic motor “TULA35” [48]

Figure 9.42 shows the design of a 3 \(\times \) 3 pin array. Especially the very small size of the design is remarkable: all outer dimensions have a length of \(12\, \text {mm}\). The pins are distributed in a matrix of \(3\, \text {mm}\). On the area of \(1.44 \,\text {cm}^2\) nine separate drives are located. To achieve such a high actuator density the lengths of the rods have to be different, allowing the moving parts to be placed directly next to each other. If this unit is placed at the upper border of the display all pins move in- respectively out-of the plane. The weight of the whole unit is \(2.5\,\text {g}\). When the maximum displacement of \(1\,\text {mm}\) is used, a bandwidth of up to \(20\,\text {Hz}\) can be achieved.

Fig. 9.42
figure 42

© Springer Nature, all rights reserved

Tactile 3 \(\times \) 3-pin-array [47]

The integration in a pen including another vibratory motor at its tip is shown in Fig. 9.43. This additional drive is used to simulate the contact of the pen with a surface. The whole pen weights \(15\,\text {g}\).

Fig. 9.43
figure 43

© Springer Nature, all rights reserved

Prototype of the “Ubi-Pen” [48]

The Ubi-Pen provides surface structures such as roughness and barriers or other extreme bumpy surfaces. To realize this, vibrations of the pins are superimposed with the vibratory motor. If the pen is in contact with a touch-sensitive surface (touch-panel), the shown graphical image may be displayed in its gray scale values by the pins of the tactile display. The system has been used for a number of tests for recognition of information displayed in different tactile modalities [49]. The results are very good with a mean recognition rate of \(80\%\) with untrained users.

Structural Impact Detection with Vibro-Haptic Interfaces

In this work, a new sensing model for structural impact detection by using vibro-haptic interfaces is developed. This haptic interface tries to obtain the touch sense of humans to provide information of structural impacts. This interface is designed as an arm-wearable one and provides a haptic stimulation to the human. This system helps humans to ‘feel’ structural responses and determine the structural conditions. The hardware and software parts are formed to achieve the vibro-haptic-based impact detection system. Some piezoelectric sensor arrays are used to measure the wing acoustic data. By processing the measured acoustic data, haptic signals are then created. Human arms will receive the haptic signals wirelessly, and the impact location, intensity, and possibility of subsequent damage initiation are identified with the vibro-haptic interface. The motors are installed at least 4 cm apart; thus, humans can detect the haptic vibration location (Fig. 9.44). The motors are operated as on and off states; thus, their frequency and amplitude cannot be directly modulated. Therefore, pulse width modulation (PWM) is used. The pulse length sends an intensity feeling such that the haptic signals amplitude and frequency could be simulated.

Fig. 9.44
figure 44

© IOP Publishing, all rights reserved

General view of the whole system [50].

Tactile Displays with Lateral Stimulation

Discrete Stimulation

The concept of discrete stimulation is based on an excitation lateral to the skin’s surface (“laterotactile display”) [51]. Figure 9.45a shows the schematic sketch of a one-dimensional array of actuators. An activation of a piezoelectric element results in its elongation and a deformation of the passive contact comb (crown). If the skin of a touching finger is stretched by this movement a contact point is perceived. Part (b) of Fig. 9.45 shows a 2-dimensional display on the right. With the extension from 1D to 2D array it is necessary to consider the more complex movement patterns resulting from it. A deeper analysis of the capabilities of such a system and the application for the exploration of larger areas can be found in [52, 53], proving the applicability of a laterotactile display as a virtual 6-point Braille device and to render surface properties. Further tests prove the generated tactile impression very realistic, it can hardly be distinguished from a moving pin below the finger surface.

Fig. 9.45
figure 45

© Springer Nature, all rights reserved, c integration of an optimized version of the 2D display mounted on a Pantograph haptic device to simulate texture on large surfaces [53]

a 1D array of the “laterotactile display” [51] b 2D \(\text {STReSS}^2\) display [52]

Continuous Stimulation

The transfer from the discrete points to a piezoelectric traveling wave drive is shown in [54]. The touching finger faces a closed and continuous surface. Due to this design the tactile display itself becomes less sensitive in its performance to movements of the finger. With the contact surface beyond the skin being excited as a standing wave, the user perceives a surface texture. With relative movement between wave and finger even a roughness can be perceived. By modifying the shape of the traveling wave a simulation of a touch-force perceivable by the moving finger can be achieved. Figure 9.46 shows the schematic sketch of the contact between finger and traveling wave, such as the corresponding movement direction.

Fig. 9.46
figure 46

© Springer Nature, all rights reserved

Contact area of finger and travelling wave [54]

In a test bed application [54] the stator of a traveling actuator USR60 from Shinsei has been used. This actuator provides a typical exploration speed by its tangential wave speed of \(15\,\text {cm/s}\) and forces up to \({\approx } 2\,\text {N}\). This system enables to generate continuous and braking impressions by the change of wave shapes. An additional modulation of the ultrasonic signals with a low frequency periodical signal generates the sensation of surface-roughness. Actual research is performed on the design of linear ultrasonic traveling wave displays.

Ultrasonic Touchscreens

Ultrasonic actuators received a lot of attention in the past years. They promise the opportunity that on hard surfaces (e.g. glas) tactile sensations can be generated without the need for complex suspensions or moving components. Hudin et al. [55] showed that with a reverse analysis of the mechanoacoustics of a touchscreen localized vertical exaggeration can be generated efficiently. Multiple demonstrators clearly showed that this approach is very impressive and promising. This lead to two directions for implementing ultrasonic touchscreen solutions:

  • Event-based (click-feeling) based on reverse analysis

  • Texture-based (textures) based on friction modulation

In case of texture-generation the key-challenge is given by overcoming the variability of users and conditions of use. Measuring the finger-movement is a basic feature. One further element to improve the reproducibility of perceived textures is by measuring the interaction forces and controlling the textures intensity [56]. However recently also the limitations of such systems were explored, as despite all success the textures do not feel natural. Bernard et al. compared the richness of texture sensation artificially generated with real textures [57] and identified, that the stimulated frequencies scale linearly with natural textures, what they can not do with ultrasonic textures. Furthermore, ultrasonic touchscreens are competing with capacitive systems based on electroadhesion [58] being discussed in Sect. 9.5. Both concepts rely on the same perceptual dynamics as fundamentally analyzed by Wiertlweksi et al. [59].

3.5.2 Piezoelectric Actuators for Kinaesthetic Systems

Piezoelectric actuators used in kinaesthetic systems are usually part of active systems (in a control-engineering sense). The user interacts with forces and torques generated by the actuator. A classic example is a rotational knob which is actuated by a traveling wave motor. With passive systems the actuator is used as a switching element, which is able to consume power from an actuator or user in either time-discrete or continuous operation. Examples are breaks and clutches.

Active Kinaesthetic Systems

Piezoelectric traveling wave actuators show a high torque to mass ratio if they are compared to other electrical actuation principles. They are predestined for the use in applications with a high torque at small rotational velocity, as they do not need an additional bearing or other transmission ratios. Kinaesthetic systems require exactly these properties. A very simple design can be used to build a haptic knob: a rotationally mounted plate with a handle attached for the user is pressed on the stator of a traveling wave piezoelectric motor. A schematic sketch of the critical part is shown in Fig. 9.27. Due to the properties specific to the traveling wave motor the rotational speed of the rotor can be adjusted easily by increasing the wave’s amplitude w at the stator. As this actuation principle is based on a mechanical resonance mode, it is actuated and controlled with frequencies nearby its resonance. Coincidentally, this is the most challenging part of the design, as the piezoelectric components show a very nonlinear behavior at the mechanical stator-rotor interface. Hence, the procedures for its control and its electronics have a large influence on its performance.

In such systems the torque [60] is highly dependent on the actual rotational speed and the amplitude. By monitoring the speed and controlling the phase and the wave amplitude, the system can be closed-loop controlled to a linear torque-displacement characteristic. In this mode a maximum torque of \({\approx } 120\,\text {mNm}\) can be achieved with this example. A deeper discussion of the phase control for a piezoelectric traveling wave motor according to this example is given in [61]. 

A specialized design of such a device is used in neurological research for the application with magneto-resonance tomography [62]. To close-loop control the admittance of the device the torque has to be measured. In the specific application near a large magnetic field glass fibers are used, measuring the reflection at a bending polymer body. Preventing disturbance from the device on the switching gradients of MRI and vice versa the device has been designed from specific non-conductive materials. It was based on a traveling wave motor in a special MR-compatible version of the “URS60” from Shinsei.

Hybrid Systems

Another class of kinaesthetic systems are so called hybrid systems. If there is the need of generating a wide bandwidth of forces and elongations with a single device, there is no actuator fulfilling all requirements alone. Due to this reason several hybrid systems are designed with two (or more) components complementing each other. A typical example is the combination of a dynamic drive with a brake, latter is used to provide large blocking torques. As seen in the above paragraph the closed-loop control of a traveling wave motor’s impedance is a challenging task. A simple approach to avoid this problem is the combination of a traveling wave actuator with a clutch. The main difference between a traveling wave actuator and other types of actuators is given by its property to rotate with a load-independent velocity. Providing a certain torque this system is accompanied with a clutch. Other designs add a differential gear or a break. Such a system is presented in [63]. If the system experiences a mechanical load, the operation of the break is sufficient to increase friction or even block the system completely. Consequently, the system provides the whole dynamic range of the traveling motor in active operation, whereas passive operation is improved significantly by the break. Due to the simple mechanical design and the reduction of energy consumption such systems are suitable for mobile applications, as well.

Passive Kinaesthetic Systems

Objects can be levitated by standing acoustic waves. To achieve this, an ultrasonic source has to provide a standing wave first. Within the pressure nodes of the wave a potential builds up, attracting a body placed nearby the node. The size of the object is important for the effect to take place, as with a large size the influence on the next node may be too high. A system based on this principle is described in [64]. It shows a design of an exoskeleton in form of a glove with external mechanical guides and joints. The joints are made of piezoelectric clutches. In their original state both discs are pressed together by an external spring generating a holding torque. With the vibrator being actuated the levitation mode is achieved between rotor and stator creating a gap h. This reduces the friction drastically allowing to turn both disks almost free from each other.

3.5.3 Summary

Tactile systems are distinguished according to their direction of movement. With a normal movement into the direction of the skin additional differences are made between passive systems simulating a more or less static surface by their pins, and active systems—so called vibrotactile systems—providing information by a dynamic excitement of the skin’s surface. The user integrates this information into a static perception. The advantages of this approach are given by the reduced requirements on the necessary force and displacements, as the dynamical perception of oscillations is higher than the perception of static slow-changing movements. When the display is not fixed to the finger however its fast movements will be a problem. With a static display in a fixed frame the user is able to repeatedly touch the display, increasing the dynamics of the haptic impression by own movements. With a dynamic display this interaction does not work as well anymore as periods of oscillations from the vibrating elements are lost.

Another alternative are tactile systems with a lateral movement of the skin. With an appropriate control the human can be “fooled” to feel punctual deformations analog to an impression of a normal penetration. Systems with a closed surface are very comfortable variants of such displays, but their dynamic control is demanding for finger positions moving across larger surfaces. Typically today’s solutions show smaller contact areas than with other variants, as the actuator elements can not be placed as close together as necessary.

Kinaesthetic (force-feedback) systems can be distinguished in active and passive systems according to a control-engineering sense. Active systems are able to generate counter forces and supportive forces. The spectrum of movements is only limited by the degrees-of-freedom achieved by the mechanical design. A stable control for active system tends to become very elaborate due to required measurement technology and complex control algorithm of sufficient speed. As with all active systems a danger remains: an error of its functionality of the system may harm the user. This danger increases for piezoelectric actuators, as the available forces and torques are high. Passive systems with breaks and clutches enable the user to feel the resistance against their own movement as reactive forces. These designs are simpler to build and less dangerous by definition of passivity. General disadvantages of passive systems can be found in their high reaction times, the change of their mechanical properties in long-time applications and their comparably large volume. Hybrid systems combining both variants—usually including another actuation principle—may enlarge the application area of piezoelectric actuators. Although the mechanical design increases in volume and size, the requirements on control may become less and large holding forces and torques can be achieved with low power consumption. From a standpoint of haptic quality they are one of the best actuator solutions for rotating knobs with variable torque/angle characteristics available today.

4 Electromagnetic Actuators

Electromagnetic actuators are the most frequently used actuator type within general automation-industry. Due to their simple manufacture and assembly they are the matter of choice. Additionally they do not necessarily need a permanent magnet, and their robustness against exterior influences is very high. They are used within coffee-machines, water pumps and for redirecting paper flow within office printers. But nevertheless their applicability for haptic devices, especially kinaesthetic devices, is limited. Their main fields of application are tactile systems. This can be reasoned by several special characteristics of the electromagnetic principle. Within this chapter the theoretical basics for electromagnetic actuators are given. Technical realizations are explained with examples, whereas first the general topology and later the specific designs are shown. The chapter closes with some examples of haptic applications of electromagnetic actuators.

4.1 Magnetic Energy

  The source responsible for the movement of a magnetic drive is the magnetic energy. It is stored within the flux-conducting components of the drive. These components are given by the magnetic core (compare Sect. 9.2.1.3) and the air gap, such as all leakage fields—which are neglected for the following analysis. It is known from Table 9.2, that stored magnetic energy is given by the products of fluxes and magnetic voltages in each element of the magnetic circuit:

$$\begin{aligned} W_\text {mag}=\sum \limits _{n}H_n\,l_n\,\cdot \,B_n\,A_n \end{aligned}$$
(9.56)

As every other system does, the magnetic circuit tries to minimize its inner energy.Footnote 14 Concentrating on electromagnetic actuators, the minimization of energy almost always refers to the reduction of an air gap’s magnetic resistance \(R_\text {mG}\). For this purpose two effects may be used, which can be found within electrostatic for electrical fields too (Sect. 9.5):

  • Electromagnetic longitudinal effect (Fig. 9.47a) (also: reluctance effect)

  • Electromagnetic transversal effect (Fig. 9.47b)

Fig. 9.47
figure 47

Electromagnetic transversal- (a) and longitudinal-effect (b)

The forces respectively torques generated with the individual effects are the derivations of the energy according to the corresponding direction,

$$\begin{aligned} \textbf{F}_\xi =\frac{dW_\text {mag}}{d\mathbf {\xi }}, \end{aligned}$$
(9.57)

being equal to a force in the direction of the change of the air gap

$$\begin{aligned} \textbf{F}_\xi =-\frac{1}{2}\phi ^2\frac{d\,R_\text {mG}}{d\mathbf {\xi }} . \end{aligned}$$
(9.58)
Fig. 9.48
figure 48

Electromagnetic transversal effect in the air-gap (a) and with a qualitative force plot (b)

Example: Transversal Effect

The magnetic resistance of an arbitrary homogenous element of length l between two boundary surfaces (Fig. 9.48a) with the surface A is calculated as

$$\begin{aligned} R_\text {m}=\frac{l}{\mu \,A}. \end{aligned}$$
(9.59)

This gives the stored energy \(W_\text {mag}\) within the magnetic resistance:

$$\begin{aligned} W_\text {mag}=(B\,A)^2\,\frac{l}{\mu \,A}. \end{aligned}$$
(9.60)

The flux density B is dependent on the length of the material. Assuming that the magnetic core contains one material only the magnetomotive force \(\varTheta \) is calculated as

$$\begin{aligned} \varTheta =\frac{B}{\mu \,l}=NI, \end{aligned}$$
(9.61)

which gives

$$\begin{aligned} B=NI\,\frac{\mu }{l}. \end{aligned}$$
(9.62)

This equation used to replace the flux density in Eq. (9.60), several variables canceled, finally results in the magnetic energy

$$\begin{aligned} W_\text {mag}=(NI)^2\,A\mu \,\frac{1}{l}. \end{aligned}$$
(9.63)

With the assumption about the magnetic energy concentrating within the air gap—which is identical to the assumption that the magnetic core does not have any relevant magnetic resistance—the approximation of the force for the transversal effect in the direction of l can be formulated

$$\begin{aligned} F_l=-\frac{1}{2}(NI)^2\,A\mu \,\frac{1}{l^2}. \end{aligned}$$
(9.64)

The force shows an anti-proportional quadratic coherence (Fig. 9.48b) to the distance l. The closer the poles become, the higher the force attracting the poles increases.

 

Example: Longitudinal Effect, Reluctance Effect

The same calculation can be repeated for the longitudinal effect. Assuming that the surfaces A from Eq. (9.63) is rectangular and its edges’ lengths are given by a and b, further assuming that a flux-conducting material is inserted into direction a, the forces in longitudinal direction can be calculated as

$$\begin{aligned} F_a=\frac{1}{2}(NI)^2\,b\mu \,\frac{1}{l}, \end{aligned}$$
(9.65)

and in direction b as

$$\begin{aligned} F_b=\frac{1}{2}(NI)^2\,a\mu \,\frac{1}{l}. \end{aligned}$$
(9.66)

The reluctance effect is—in contrast to the transversal effect—linear (Fig. 9.49b). The force is dependent on the length of the mobbing material’s edge only. Consequently, the stored energy within the magnetic circuit is necessary for the design of an electromagnetic actuator. Above examples have the quality of rough estimations. They are sufficient to evaluate the applicability of an actuation principle—no more, no less. The magnetic networks sufficient for a complete dimensioning should contain effects with magnetic leakage fields and the core’s magnetic resistance. Therefore it is necessary to further deal with the design of magnetic circuits and their calculation.

Fig. 9.49
figure 49

Electromagnetic longitudinal effect in the air gap (a) and as qualitative force plot (b)

4.2 Design of Magnetic Circuits

 The basic interdependencies for the design of magnetic circuits have already been discussed within Sect. 9.2.1.3 in the context of electrodynamic actuators. Taken from the approach of longitudinal- and transversal effect, several basic shapes (Fig. 9.50) can be derived applicable to electromagnetic actuators. In contrast to electrodynamic actuators the geometrical design of the air gap within electromagnetic actuators is freer. There is no need to guide an electrical conductor within the air gap anymore. Beside the designs shown in Fig. 9.50 there are numerous other geometrical variants too. For example all shapes can be transferred into a rotational-symmetrical design around one axis. Additional windings and even permanent magnets can be added. There are just two limits to their design:

  • A sufficient cross section of the flux conducting elements has to be guaranteed, to prevent the material from getting into saturation.

  • A sufficient distance between flux-conducting elements has to be kept to prevent a magnetic leakage-field via the air.

Fig. 9.50
figure 50

Basic shapes of electromagnetic actuators

4.2.1 Cross Section Surface Area—Rough Estimation

The calculation of the cross section surface area for dimensioning the magnetic core is simple. A common, easily available and within precision engineering and prototype-design gladly used material is steel ST37. The B/H-characteristic curve with its saturation is given in Fig. 9.9. For this example we choose a reasonable flux density of 1.2 T. This equals a field intensity of \(H\approx \)1000 A/m. Within the air gap a flux density of 1 T should be achieved. The magnetic flux within the air gap is given as

$$\begin{aligned} \phi =A_{\text {G}}\,B_{\text {G}}. \end{aligned}$$
(9.67)

As the magnetic flux is conducted completely via the magnetic core—neglecting leakage fields and other side bypasses—the relation

$$\begin{aligned} A_{\text {Iron}}\,B_{\text {Iron}}=A_{\text {G}}\,B_{\text {G}}, \end{aligned}$$
(9.68)

is given, and consequently with the concrete values from above:

$$\begin{aligned} \frac{A_{\text {Iron}}}{A_{\text {G}}}=\frac{B_{\text {G}}}{B_{\text {Iron}}}=0{,}833. \end{aligned}$$
(9.69)

At its tightest point the magnetic core may have 83% of the cross section of the air gap. Whereas more surface of the cross-section results in lower field intensities, which should be aimed at if is geometrically possible. Please note that \(A_{\text {G}}\le A_{\text {Iron}}\) is with almost all technical realization, as the boundary surface of the magnetic core is always one pole of the air gap.

4.2.2 Magnetic Energy in the Magnetic Core and Air Gap

 Within the preceding examples the assumption was made that the energy stored within the magnetic core is clearly less than the energy within the air gap. This assumption should be checked now for validity. Calculating the magnetic resistance of an arbitrary element

$$\begin{aligned} R_\text {m}=\frac{l}{\mu \,A}, \end{aligned}$$
(9.70)

the relation of two elements with identical length and cross-section scales via the magnetic resistance with the permeability \(\mu \):

$$\begin{aligned} \frac{R_\text {m1}}{R_\text {m2}}=\frac{\mu _2}{\mu _1} \end{aligned}$$
(9.71)

  The permeability \(\mu _r=\frac{B}{H\,\mu _0}\) is given by the relation between flux density versus field strength relatively to the magnetic constant. It is nonlinear (Fig. 9.51) for all typical flux-conducting materials within the flux-density areas relevant for actuator design in between 0.5 and 2 T. It is identical to the inverse gradient of the curves given in Fig. 9.9. The maximum permeability values are given in tables frequently, but refer to field strengths within the material only. They range from 6,000 for pure iron over 10,000 for nickel alloys up to 150,000 for special soft-magnetic materials.

Mechanical processing of flux-conducting materials and the resulting thermal changes within its microstructure will result in a considerable degradation of its magnetic properties. This change can be restored by an annealing process.

Generally speaking however even outside an optimum value for flux density the stored energy within typical materials is always several orders of magnitudes below the energy within the air gap. This legitimates to neglect this energy portion for the rough estimations in actuator design, but does show too that there is potential within the optimization of electromagnetic actuators. This potential can be used by the application of FEM-software, which is typically available as module for CAD software.Footnote 15

Fig. 9.51
figure 51

Qualitative change of the permeability for common flux-conducting materials

4.2.3 Permanent Magnets in Electromagnetic Actuators

Permanent magnets do not differ significantly in their properties from coils conducting a DC current. They generate a polarized field, which—in combination with another field—provide attraction or repulsion. For calculating combined magnetic circuits a first approach can be taken by substituting the sources within the magnetic equivalent circuit (neglecting saturation effects). The calculation is analog to the methods read in the chapter about electrodynamic actuators (Sect. 9.2.1.3). A permanent magnet within a circuit either allows

  • the realization of a state held without any current applied

  • or switching between two states with just one winding powered

A good example for a currentless held state [6] shows the calculation of a polarized magnetic clamp (Fig. 9.52). With non-active winding the flux is guided through the upper anchor and held securely. With active coil the magnetic flux via the upper anchor is compensated. The magnetic bypass above the coil prevents the permanent magnet from being depolarized by a counter field beyond the kink in the B/H-curve.

Fig. 9.52
figure 52

Permanent magnet in the magnetic circuit in shape (a), field-lines with inactive coil (b) and field-lines with active coil (c), releasing the anchor

4.3 Examples for Electromagnetic Actuators

Electromagnetic actuators are available in many variants. The following section gives typical designs for each principle and shows the corresponding commercial products. The knowledge about the designs will help to understand the freedom in the design of electromagnetic circuits more broadly.

4.3.1 Claw-Pole Stepper Motor

The electromagnetic claw-pole stepper motor (Fig. 9.53) is one of the most frequently used rotatory actuation principle. These actuators are made of two stamped metal sheets (1,2) with the poles—the so called claws—bended by 90 \({}^{\circ }\) to the interior of the motor. The metal sheets are the magnetic core for conducting the flux of one coil each (3). The permanent-magnet rotor (4) with a pole subdivision equalizing the claw pattern orientates to the claws in currentless state. In stepper mode the coils are powered subsequently, resulting in a combined attraction and repulsion of the rotor. The control of the coils’ currents may happen either by simply switching them or by a microstep mode with different current levels being interpolated between discrete steps. Latter generates stable states for the rotor not only in the positions of the claws but also in between.

Claw-pole stepper motors are available with varying number of poles, different numbers of phases and varying loads to be driven. As a result of the permanent-magnet they show a large holding torque with respect to their size. The frequency of single steps may reach up to 1 kHz for fast movements. By counting the steps the position of the rotor can be detected. Step-losses—the fact that no mechanical step happens after a control signal—are not very likely with a carefully designed power chain. Claw-pole stepper motors are the working horses of the electrical automation technology.

Fig. 9.53
figure 53

Figure based on [22] © Springer Nature, all rights reserved

Two-phase stepper motor made of stamped metal sheets and with a permanent magnet rotor in a 3D-sketch (a), cross-section (b), and with details of the claws-poles (c).

4.3.2 Reluctance Drives

The rotatory reluctance drives (Fig. 9.54) are based on the electromagnetic longitudinal effect. By clever switching of the windings (2) it is possible to keep the rotor (3) in a continuous movement with just minimal torque ripples. To make this possible the rotor has to have fewer poles than the stator. The rotor’s pole-angle \(\beta _r\) is larger than the pole-angle \(\beta _s\). Reluctance drives can be used as stepper motors by the integration of permanent magnets too. Generally speaking it excels by high robustness of the components and a high efficiency factor with—for electromagnetic drives—comparably little torque ripples.

Fig. 9.54
figure 54

Figure based on [22] © Springer Nature, all rights reserved

Switched reluctance drive with pole- and coil-layout (a), in cross section (b), and with flux-lines of the magnetic excitation (c).

4.3.3 Electromagnetic Brakes

Electromagnetic brakes (Fig. 9.55) are based on the transversal effect. They make use of the high force increase at electromagnetic attraction to generate friction on a spinning disk (1). For this purpose usually rotational-symmetrically flux conducting magnetic cores (2) are combined with embedded coils (3). The frontal area of the magnetic core and braking disk (1) itself is coated with a special layer to reduce abrasion and influence positively the reproducibility of the generated torque. The current/torque characteristic of electromagnetic brakes is strongly nonlinear. At the one hand this is the result of the quadratic proportion between force and current of the electromagnetic transversal effect, but on the other hand this is also a result of its friction pairing. Nevertheless they are used in haptic devices for the simulation of ”hard contacts” and stop positions. A broad application for haptic devices is nevertheless not visible. This is likely a result of their limits in reproducibility of the generated torque, the resulting complex control of the current and the fact, that they can only be used as a break (passive) and not for active actuation.

Fig. 9.55
figure 55

Electromagnetic brake in cross section (a) and as technical realization for an airplane model, by Flight-Depot.com OHG, used with permission (b)

4.3.4 Plunger-Type Magnet

Electromagnetic plunger-type magnets (Fig. 9.56) are frequently based on the electromagnetic transversal effect. Their main uses are switching and control applications, requiring the actuation in specific states. With a magnetic core frequently made of bended iron steel sheets (2) a coil-induced (3) flux is guided to a central anchor, which itself is attracted by a yoke (4). The geometry of the yoke influences significantly the characteristic curve of the plunger-type magnet. By varying its geometry a linearization of the force-position curve is possible within certain limits. Even strongly nonlinear pulling-force characteristics can be achieved by such a modification. Plunger-type magnets are available with additionally magnets and with more coils. In these more complex designs they provide mono- and bi-stable switching properties. By variation of the wires diameter and the number of turns they can be adapted easily to any power level.

Fig. 9.56
figure 56

Plunger type magnet a with altered force-position curve (4), and realization as pulling-anchor b with metal-sheet-made magnetic circuit (2)

4.4 Magnetic Actuators in Haptic Devices

For haptic applications electromagnetic actuators are mainly used within tactile devices. Nevertheless admittance controlled devices can be found providing impressive output forces of high quality even by the use of stepper-motors. Beside the commercial systems such as HapticMaster of Moog FCS (Sect. 6.7) especially an idea of Lawrence attracted attention within the past few years.

4.4.1 Spring-Tendon Actuator

In [65] Lawrence describes an inexpensive actuator for kinaesthetic haptic systems based on an electromagnetic stepper motor coupled via a tendon to a pen and with a spring mechanically connected in parallel. Analog to other haptic devices the pen is the interface to the user. Between pen and tendon and spring there is a bending body with DMS as a force sensor. To additionally minimize the torque ripples of the stepper drive resulting from the latching of the poles, a high resolution external encoder has been attached to the motor, and a torque/angle curve was measured. A mathematical spline fit of this curve was used for the actuator’s control to compensate the torque oscillations. Beside this compensation the closed-loop control of the actuator via the force sensor near the pen includes a compensation of frictional effects too. The result of all these efforts is a force source, providing a force transmission with little noise and high stiffness up to 75 kN/m with movements of limited dynamics.

4.4.2 Electromagnetic Pin Array

The usage of electromagnetic actuators for the control of single pins in array design is very frequent. The earliest usages for haptic applications go back to printer-heads of dot matrix printers being used in the 80th an earl 90th of the last century. Modern designs are a lot more specific to haptics and make use of manufacturing technologies available from microtechnology. In [66] an actuator array is shown made of coils with 430 windings each and a 0.4 mm wide iron core. Above of it a magnet is embedded in a polymer-layer, being attracted by the flux induced into the core. With such an actuator and a diameter of 2 mm a maximum force of up to 100 mN is possible. A position measure is realized via inductive feedback. Further realizations of tactile arrays based on electromagnetic actuators and different manufacturing techniques can be found in the work of Streque et al. [67].

4.4.3 Electromagnetic Plunger-Type Magnet for the Tactile Transmission of Speech Information

One fundamental motivation for the design of haptic devices is the partially substitution of lost senses. Especially methods to communicate information from the sense of sight or hearing by the aid of tactile devices have some tradition. Blume designed and tested an electromagnetic plunger-type magnet according to the reluctance effect 1986 at the University of Technology, Darmstadt. Such actuators were attached to the forearm and stimulated up to eight points by mechanical oscillations encoded from speech signals. The actuator (Fig. 9.57) was made of two symmetrical plunger-type magnets (on the horizontal axis) acting upon a flux conducting element integrated into the plunger. The whole anchor takes a position symmetrically within the actuator due to the integrated permanent magnet. In this symmetrical position both magnetic circuits conduct a magnetic flux resulting in identical reluctance forces. In active mode either the flux in the upper or the lower magnetic circuit is amplified depending on the direction of current flow. The reluctance forces on the amplified side pull the anchor in a current proportional position working against the magnetic pretension from the permanent magnets, the mechanic pretension from the springs and the load of the actuator. The plunger is displaced in the direction of the weakened magnetic field. At a diameter of 20 mm this actuator covers a dynamic range of 500 Hz at a efficiency factor of 50%. The forces lie in the range of \(\approx \)4 N per ampere.

Fig. 9.57
figure 57

Electromagnetic actuator according to the reluctance principle in a “counteractive plunger type” design with permanent magnet: cross-section (a) and design incl. driver-electronics (b) [68]

4.5 Conclusion About the Design of Magnetic Actuators

Electromagnetic actuators are—identical to electrodynamic systems—mainly force sources. In rotary drives especially the reluctance effect is used to generate a continuous movement. With linear drives mainly plunger-type magnets are used based on the nonlinear transversal effect. Whereas there are exception to both showing some surprising properties (Sect. 9.4.4.3). The translational systems are usually used to actuate as either bistable switches between two discrete states or monostable against a spring (plunger-type, break, and valve). There are applications within haptics based on either of both effects. Whereas reluctance based actuators can be found equally often within kinaesthetic applications as drives and in an admittance controlled application in tactile systems as vibrating motor, switching actuators are almost exclusively found in tactile devices with single pins or pin arrays. In contrast to the highly dynamic electrodynamic drives, electromagnetic actuators excel in less dynamic applications with higher requirements on torque and self-holding. During switching in between two states however the acceleration and deceleration at the mechanical stop are a highly dynamic but almost uncontrollable action. The dynamic design of switching actions were not subject to this chapter, but are usually based on modeling a nonlinear force source of the electromagnet and assuming the moving parts as concentrated elements of masses, springs and dampers. Due to their relatively high masses within the moving parts, the hard to control nonlinearities of fluxes and forces, and the low efficiency factor of the transversal effect in many designs, electromagnetic actuators occupy niches within haptic applications only. However in those niches there is no way around their usage. If an appropriate area has been found they excel by an extremely high efficiency factor for the specific design and a big robustness against exterior influences.  

5 Electrostatic Actuators

Electrostatic transformers are part of the group of electric transformers, such as the piezoelectric actuators, too. Electric transformers show a direct coupling between the electrical value and the mechanical value. This is contrary to electrodynamic and electromagnetic actuators, which show an intermediate transformation into magnetic values as part of the actuation process. In principle the transformation may be used in both directions. Hence, all these actuators can be used as sensors, as well.

Electrostatic field actuators are utilized due to their simple design and low power consumption. As a result from the technical progress of micro-engineering, the advantages of integrated designs are fully utilized. Especially for miniaturized systems the electrostatic field actuators gain increased importance compared to all other actuator principles. This is even more surprising as their energy density is significantly lower in macroscopic designs. But during miniaturization the low efficiency factor and the resulting power loss and heat produced become limiting factors for magnetic actuators [69].

An important subgroup of electrostatic field actuators is given by solid state actuators, with an elastomeric dielectric. It has a high breakdown field strength compared to air, builds the substrate of the electrodes and can simultaneously provide an isolating housing, too.

Beside the classic field actuators mentioned above, electro-rheological fluids are part of electrostatic actuators, as well. With these actuators an electric field of an arbitrary external source results in a change of the physical (rheological) properties of the fluid.

5.1 Definition of the Electric Field

The following paragraphs define the electric field and relevant variables for the design of electrostatic actuators.

5.1.1 Force on Charge

The magnitude of a force F acting on two charges \(Q_1\) and \(Q_2\) with a distance r is given by Coulomb’s-law (Eq. (9.72)).

$$\begin{aligned} F = \frac{1}{4\pi \varepsilon _{0}} \frac{Q_{1}Q_{2}}{r^{2}} \end{aligned}$$
(9.72)

5.1.2 Electric Field

The electric field E describes the space where these forces are present. The field strength is defined as the relation of the force \(\textbf{F}\) acting on the charge in the field and the charge’s magnitude Q.

$$\begin{aligned} \textbf{E} = \frac{\textbf{F}}{Q} \end{aligned}$$
(9.73)

The charges cause the electric field; the forces on the charges within an electric field are the effect. Cause and effect are proportional. With the electric constant \(\varepsilon _{0} = 8{,}854 \cdot 10^{-12}\,\text {{C}/{Vm}}\) within vacuum and air Eq. (9.74) results:

$$\begin{aligned} \textbf{D} = \varepsilon _0 \textbf{E} \end{aligned}$$
(9.74)

The electric displacement field \(\textbf{D}\) describes the ratio of the bound charges and the area of the charges. The direction is given by the electric field pointing from positive to negative charges. If the electric field is filled with an insulating material (dielectric), the electric displacement field is bound partly due to the polarizing of the dielectric. Accordingly, the field strength drops from \(E_0\) to E (with still the same electric displacement field). Consequently, the ratio of the weakened field depends on the maximum polarization of the dielectric and is called “permittivity” \(\varepsilon _r = E_0/E\).

5.1.3 Capacity

The electrical capacity is defined as the ratio of charge Q on each conductor to the voltage U between them. A capacitor with two parallel-plates charged contrary with a surface of the plates A and a fixed distance d shows a capacity C depending on the dielectric:

$$\begin{aligned} C = \frac{Q}{U}=\varepsilon _0\varepsilon _r \frac{A}{d} \end{aligned}$$
(9.75)

5.1.4 Energy Storage

Work must be done by an external influence to move charges between the conductors in a capacitor. When the external influence is removed, the charge separation persists and energy is stored in the electric field. If charge is later allowed to return to its equilibrium position, the energy is released. The work done in establishing the electric field, and hence the amount of energy stored, is given by Eq. (9.76) and for the parallel-plate capacitor by the use of Eq. (9.75) according to Eq. (9.77).

$$\begin{aligned} W_{el} = \frac{1}{2} CU^2 =\frac{1}{2}\frac{Q^2}{C} \end{aligned}$$
(9.76)
$$\begin{aligned} W_{el} = \frac{1}{2} \varepsilon _0\varepsilon _r \frac{A}{d} U^2 \end{aligned}$$
(9.77)

This stored electric energy can be used to perform mechanical work according to Eq. (9.78).

$$\begin{aligned} W_{mech}=Fx \end{aligned}$$
(9.78)

5.2 Designs of Capacitive Actuators with Air-Gap

A preferred setup of electrostatic actuators is given by parallel-plate capacitors with air-gap. In these designs one electrode is fixed to the frame, while the other one is attached to an elastic structure, enabling the almost free movement in the intended direction (DoF). All other directions are designed stiff enough to prevent a significant displacement of this electrode. To perform physical work (displacement of the plate) the energy of the electric field according to Eq. (9.77) is used. Considering the principle design of these actuators two basic variants can be distinguished: the displacement may result in a change of the distance d, or the overlapping area A. Both variants are subject of discussion in the following paragraphs.

5.2.1 Movement Along Electric Field

Looking at the parallel-plate capacitor from Fig. 9.58, the capacity \(C_L\) can be calculated with

$$\begin{aligned} C_{L}=\varepsilon _0 \cdot \frac{A}{d} \end{aligned}$$
(9.79)
Fig. 9.58
figure 58

Parallel-plate capacitor with air-gap

As shown before the stored energy \(W_{el}\) can be calculated for an applied voltage U:

$$\begin{aligned} W_{el}=\frac{1}{2}CU^{2}=\frac{1}{2} \varepsilon _{0} \frac{A}{d} U^{2} \end{aligned}$$
(9.80)

The force between both plates in z-direction can be derived by the principle of virtual displacement:

$$\begin{aligned} {F_{z,el}}=\frac{\partial W}{\partial z}=\frac{1}{2} U^2 \frac{\partial C}{\partial z} \end{aligned}$$
(9.81)
$$\begin{aligned} \textbf{F}_{z,el}=-\varepsilon _0 \frac{A}{2d^2} U^2 \textbf{e}_z \end{aligned}$$
(9.82)

The inhomogeneities of the electric field at the borders of the plates are neglected for this calculation, which is an acceptable approximation for the given geometrical relations of a large plate surface A and a comparably small plate distance d. A spring pulls the moving electrode into its idle position. Consequently, the actuator has to work against this spring. The schematic sketch of this actuator is shown in Fig. 9.59. The plate distance d is limited by the thickness of the insulation layer \(d_I\). Analyzing the balance of forces according to Eq. (9.83) the interdependency of displacement z and electrical voltage U can be calculated:

$$\begin{aligned} F_z(z)=F_{spring}(z) + F_{z,el}(U,z)=0 \end{aligned}$$
(9.83)
$$\begin{aligned} -k \cdot z - \frac{1}{2} \varepsilon _0 A \frac{U^2}{(d+z)^2}=0 \end{aligned}$$
(9.84)
$$\begin{aligned} U^2 = -2 \frac{k}{\varepsilon _0A}(d+z)^2 \cdot z \end{aligned}$$
(9.85)
Fig. 9.59
figure 59

Schematic setup of an actuator with variable air-gap

Analyzing the electrical voltage U in dependancy of the displacement z, a maximum can be identified:

$$\begin{aligned} \frac{\text {d}U^2}{\text {d}z}=-2\frac{k}{\varepsilon _0A} (d^2 + 4dz +3z^2) = 0 \end{aligned}$$
(9.86)
$$ z^2+\frac{4}{3}dz+\frac{1}{3}d^2=0 $$
$$\begin{aligned} z_1=-\frac{1}{3}d; \,\, z_2=-d \end{aligned}$$
(9.87)

To use the actuator in a stable state, the force of the retaining spring has to be larger than the attracting forces of the charged plates. This condition is fulfilled for distances z

$$ 0>z>-\frac{1}{3}d $$

Smaller distances cause attracting forces larger than the retaining force and the moving plate is strongly pulled onto the fixed plate (“pull-in” effect). As this would immediately result in an electric short cut, typical designs include an insulating layer on at least one plate. Equations (9.85) and (9.87) are used to calculate the operating voltage for the pull-in:

$$\begin{aligned} U_{pull-in}=\sqrt{\frac{8}{27}\frac{k}{\varepsilon _0A}d^3} \end{aligned}$$
(9.88)

The retention force to keep this state is much less than the actual force at the point of pull-in. It should be noted that force increases quadratically with decreasing distance. A boundary value analysis for \(d\rightarrow 0\) provides the force \(F\rightarrow \infty \). Consequently, the insulation layer fulfills the purpose of a force limitation, too.

5.2.2 Moving Wedge Actuator

A special design of air-gap actuators with varying plate distance is given by the moving wedge actuator. To increase displacement, a bended flexible counter-electrode is placed on a base electrode with a non-conductive layer. The distance between the electrodes increases wedge-like from the fixation to its free end. The resulting electrical field is higher inside the area where the flexible electrode is closest to the counter-electrode and it decreases with increasing air-gap. When designing the stiffness of the flexible electrode it has to be guaranteed, that it is able to roll along with the tightest wedge on the isolation. Figure 9.60 shows the underlying principle in idle-state and during operation [70].

Fig. 9.60
figure 60

Schematic view of a moving wedge actuator

5.2.3 Movement Perpendicular to Electric Field

The major difference compared to the prior design is given by the fact, that the plates are moving in parallel to each other. The plate distance d is kept constant, whereas the overlapping area varies. Analog to Eq. (9.80) the forces for the displacement can be calculated in both direction of the plane:

$$\begin{aligned} {F}_{x}=\frac{\partial W}{\partial x}=\frac{1}{2} U^2 \frac{\partial C}{\partial x} \end{aligned}$$
(9.89)
$$\begin{aligned} \textbf{F}_{x}=\frac{1}{2}\varepsilon _0 \frac{b}{d} U^2 \textbf{e}_x \end{aligned}$$
(9.90)
$$\begin{aligned} F_{y}=\frac{\partial W}{\partial y}=\frac{1}{2} U^2 \frac{\partial C}{\partial y} \end{aligned}$$
(9.91)
$$\begin{aligned} \textbf{F}_{y}=\frac{1}{2}\varepsilon _0 \frac{a}{d} U^2 \textbf{e}_y \end{aligned}$$
(9.92)

The forces are independent on the overlapping length only. As a consequence they are constant for each actuator position. Figure 9.61 shows the moving electrode being attached to a retaining spring.

Fig. 9.61
figure 61

Electrostatic actuator with variable overlapping area

If an electrical voltage is applied on the capacitor, the surface A increases along the border a. Hence, the spring is deflected and generates a counter force \(\textbf{F}_F\) according to

$$\begin{aligned} \textbf{F}_F=-kx\textbf{E}_x \end{aligned}$$
(9.93)

The equilibrium of forces acting upon the electrode is given by

$$\begin{aligned} F_x(x)=F_F(x)+F_{x,el}(U) \end{aligned}$$
(9.94)

From idle position (\(F_x(x)=0\)) the displacement of the electrode in x-direction is given by:

$$\begin{aligned} x=\frac{1}{2}\varepsilon _0 U^2 \frac{b}{d}\frac{1}{k} \end{aligned}$$
(9.95)

Typically, this design is realized in a comb-like structure, with one comb of electrodes being engaged in a counter electrode comb. This equals an electrical parallel circuit of n capacitors, which is identical to a parallel circuit of force sources complementing each other. Figure 9.62 shows such a design. The area of the overlapping electrodes is given by a in x-direction and b in y-direction. With the plate distance d the capacity according to Eq. (9.96) can be calculated.

$$\begin{aligned} C_{Q}=\varepsilon _0 \cdot \frac{a b}{d} \cdot n \end{aligned}$$
(9.96)
Fig. 9.62
figure 62

Actuator with comb-electrodes and variable overlapping area

By differentiating the energy according to the movement direction the electromotive force can be calculated:

$$\begin{aligned} F_x= \frac{\partial W}{\partial x}= \frac{1}{2} U^2 \frac{\partial C}{\partial x} =\frac{1}{2} U^2 \varepsilon _0 \frac{b}{d} \cdot n \end{aligned}$$
(9.97)

5.2.4 Summary and Examples

For all actuators shown, the electrostatic force acts indirect against the user and is transmitted by a moveable counter-electrode. A much simpler design of tactile displays makes use of the user’s skin as a counter electrode, which experiences the whole electrostatic field force. Accordingly, tactile electrostatic applications can be distinguished in direct and indirect principles.

Direct Field Force

The simplest design combines one electrode, respectively a structured electrode array, and an isolating layer. A schematic sketch is given in Fig. 9.63. The user and his finger resemble the counter electrode. With the attractive force between the conductive skin and the electrodes, a locally distributed increase of friction can be achieved. It hinders a relative movement and can be perceived by the user. Such systems can be easily realized and excellently miniaturized. Their biggest disadvantage is their sensitivity on humidity on the surface, which is brought onto the electrodes during any use in form of sweat. This leads either to a blocking of the electrical field by the conductive watery layer above the isolation or it contributes by capillary effects to the field forming and the resulting perceived impression [71]. Nevertheless, the compactness of such solutions and the possibility to combine it with existing touchscreen-concepts lead to a relevant industrial and research-interest into it. Especially the group of Colgate did some very fundamental studies on comparing ultrasonic and electroadhesive vibrotactile effects and their perceptional basis in comparision to the physical domains involved [72].

Fig. 9.63
figure 63

Electrostatic stimulator with human finger as counter electrode [73], own visualization

Indirect Field Force

With these systems the field force is used to move an interacting surface. The finger of the user interacts with these surfaces (sliders) and experiences their movements as a perceivable stimulation. A realization with a comb of actuators moving orthogonal to the field direction is given in Fig. 9.64. The structural height is \(300\,\upmu \text {m}\) providing \(1\,\text {mN}\) at operating voltages of up to 100 V. The same design with an actuator made of parallel electrodes can achieve displacements of \(60\,\upmu \text {m}\). The comb-electrodes shown here displace \(100\,\upmu \text {m}\).

Fig. 9.64
figure 64

Electrostatic comb-actuator for tangential stimulation [74], own visualization

Electrostatic Break

Despite the area of electrostatic actuators is dominated by texture simulations using direct field forces and for kinaesthetic systems by the application of electroactive polymers (EAP) Sect. 9.5.4, there are still surprising solutions using this concept. Such a solution was shown by Hinchet et al. with the system DextrES for virtual reality application, where between two conducting elements an electrostatic force can be applied increasing friction and therefore creating a strong sensation of contact for finger-manipulations in virtual space. With the system forces \({>}20\) N can be realized, although they require voltages in the range of 1.5 kV. The dynamics of the system clearly proved beneficial in the study conducted to explore a VR-scenario [75].

5.3 Active Skin

One of the significant trends in robotic research is how to employ haptic feel in the human-and-machine interaction. Virtual reality, neurophysiology, and biomedical engineering require the haptic interface as their system primary function. In this work, a new haptic interface, active skin, by using a tactile sensor and a tactile stimulator is designed. By synchronizing the sensor and the stimulator, it generates a wide variety of haptic feel in response to the touch Fig. 9.65.

Fig. 9.65
figure 65

© (2010) Society of Photo-Optical Instrumentation Engineers (SPIE), all rights reserved

Configuration of the layers and working principle of the active skin [76]

Integration makes this active skin of tactile stimulator and tactile sensor into a single haptic unit. A layer of the tactile sensor is located on the tactile stimulator top. In this design, a dielectric elastomer layer is sandwiched between two conductive electrode layers. Two protective layers are located on the top and bottom of the active skin to save the system from damages. Figure 9.65 shows the layers’ structure.

The interaction part of the active skin is shown in Fig. 9.65 The tactile and capacitance sensors are responsible for detecting contact with an external object, as well as the contact position. The corresponding tactile stimulator is actuated according to the detected force. For representing different touch feelings, each cell of the active skin can be controlled independently. Therefore, as shown in Fig. 9.65, the position and magnitude of the force and touch feeling will be found by using different sensors.

Summary

Electrostatic drives with air gap achieve force in the range of mN to N. As the actuators are driven by fields, the compromise between plate distance and electrical operation voltage has to be validated for each individual application. The breakdown field strength of air (approx. 3 V\(/\upmu \)m) is the upper, limiting factor. The actuators’ displacement is limited to several \(\mu \)m. At the same time the operating voltages reach several hundred volts. Due to the very low achievable displacement the application of electrostatic actuators is limited to tactile stimulation only.

For the concrete actuator design it is recommended to deal with the modeling of such actuators, e.g. based on concentrated network parameters (see Sect. 4.3.2). This allows the analysis of the complete electromechanical system starting from the applicable mechanical load situation to the electrical control with a single methodological approach.

5.4 Dielectric Elastomer Actuators

As with many other areas, new synthetic materials replace classic materials such as metals in actuator design. Thanks to the enormous progress in material development their mechanical properties can be adjusted to a large spectrum of possible applications. Other big advantages are given by the very cheap material costs. Additionally, almost any geometrical shape can be manufactured with relatively small efforts.

Polymers are called “active polymers” if they are able to change their shape and form under the influence of external parameters. The causes for theses changes may be manifold: electric and magnetic fields, light and even ph-value. Being used within actuators, their resulting mechanical properties like elasticity, applicable force and deformation at simultaneously high toughness and robustness are quite comparable to biological muscles [77].

To classify the large variety of “active polymers” they are usually distinguished according to their physical working principle. A classification into “non-conductive polymers”, activated e.g. by light, pH-value or temperature, and “electrical polymers”, activated by an arbitrary electrical source. Latter are called “electroactive polymers” (EAP) and are further distinguished into “ionic” and “electronic” EAPs. Generally speaking electronic EAP are operated at preferably high field strengths near the breakdown field strength. Depending on the layer thickness of the dielectrics \( 1 {-} 20\,\text {kV}\) are typical operation voltages. Consequently, very high energy densities at low reaction times (in the area of milliseconds) can be achieved. In contrast, ionic EAP are operated at obviously lower voltages of \(1{ - }5\,\text {V}\). However, an electrolyte is necessary for transportation of the ions. It is frequently provided by a liquid solution. Such actuators are typically realized as bending bars, achieving large deformations at their tip with long reaction times (several seconds).

All EAP technologies are subject of actual research and fundamental development. However, two actuator types are already used in robotics: “Ionic polymer metal composite” (IPMC) and “dielectric elastomer actuators” (DEA). A summary and description of all EAP-types is offered by Kim [78]. Their functionality will be further discussed in the following paragraphs as they affiliate to the group of electrostatic actuators. A comparison between characteristic values of dielectric elastomer actuators and the muscles of the human is shown in Table 9.6. By the use of an elastomer actuator with large expansion additional mechanical components such as gears or bearings are needless. Additionally, the use of these materials may be combined to complex designs similar to and inspired by nature. One application is e.g. the locomotion of insects and fishes within bionic research [79].

Table 9.6 Comparison of human muscle and DEA according to Pei [80]

5.4.1 Dielectric Elastomer Actuators—Electrostatic Solid State Actuators

The design of dielectric elastomer actuators is identical to the design of a parallel-plate capacitor, but with an elastic dielectric (a polymer respectively elastomer) sandwiched by compliant electrodes. Hence, it is a solid state actuator. The schematic design of a dielectric elastomer actuator is visualized in Fig. 9.66, left. In an uncharged condition the capacity and the energy stored is identical to an air-gap actuator (Eqs. (9.75) and (9.76)). A change of this condition happens by the application of a voltage U and is visualized in Fig. 9.66, on the right: The charged capacitor contains more charges (\(Q+\varDelta Q\)), the electrode area increases (\(A+\varDelta A\)), whereas the distance (\(z-\varDelta z\)) simultaneously decreases. The change of energy after an infinitesimal change \(\text {d}Q\), \(\text {d}A\) and \(\text {d}z\) is calculated in Eq. (9.98):

Fig. 9.66
figure 66

DEA in initial state (left) and charged state (right)

$$\begin{aligned} \text {d}W = \left( \frac{Q}{C}\right) \text {d}Q + \left( \frac{1}{2} \frac{Q^2}{C} \frac{1}{z} \right) \text {d}z -\left( \frac{1}{2} \frac{Q^2}{C} \frac{1}{A} \right) \text {d}A \end{aligned}$$
(9.98)
$$\begin{aligned} \text {d}W = U \text {d}Q + W\left[ \left( \frac{1}{z}\right) \text {d}z - \left( \frac{1}{A}\right) \text {d}A\right] \end{aligned}$$
(9.99)

The internal energy change equals the change of the electrical energy by the voltage source and the mechanical energy used. Latter depends on the geometry (parallel (dz) and normal (dA) to the field’s direction). In comparison to the air-gap actuator in Sect. 9.5.2 an overlay of decreasing distance and increasing electrodes’ area occurs. This is causes by a material property which is common to all elastomers and to almost all polymers: the aspect of volume constancy. A body compressed in one direction will extend in the remaining two other dimensions if it is incompressible. This gives a direct relation between distance change and the change of electrodes’ area. As a consequence Eq. (9.100) results

$$\begin{aligned} A\,\text {d}z = -z \text {d}A \end{aligned}$$
(9.100)

simplifying Eq. (9.99) to

$$\begin{aligned} \text {d}W = U \text {d}Q + 2W\left( \frac{1}{z}\right) \text {d}z \end{aligned}$$
(9.101)

The resulting attractive force of the electrodes can be derived from this electrical energy. With respect to the electrode surface A the electrostatic pressure \(p_{el}\) at \(\text {d}Q=0\) is given according to Eq. (9.102)

$$\begin{aligned} p_{el} = \frac{1}{A} \frac{\text {d}W}{\text {d}z} = 2W \frac{1}{Az} \end{aligned}$$
(9.102)

and by the application of Eq. (9.76)

$$\begin{aligned} p_{el} = 2 \left( \frac{1}{2}\varepsilon _0 \varepsilon _r Az \frac{U^2}{z^2}\right) \frac{1}{Az} = \varepsilon _0 \varepsilon _r E^2 \end{aligned}$$
(9.103)

Comparing this result with Eq. (9.81) as a reference for a pressure of an air-gap actuator with variable plate distance, dielectric elastomer actuators are capable of generating a pressure twice as high with otherwise identical parameters [81].

Additional reasons for the obviously increased performance of the dielectric elastomer actuators are based on their material, too. The relative permittivity is given by \(\varepsilon _r >1\), depending on the material \(\varepsilon _r \simeq 3{ - }10\). By chemical processing and implementation of fillers the relative permittivity may be increased. However, it has to be noticed that other parameters (such as the breakdown field strength and the E-modulus) may become worse, possibly the positive effect of the increased \(\varepsilon _r\) gets lost. Especially the breakdown field strength is one of the most limiting factors. With many materials an increase in breakdown field strength could be observed after planar prestrain. In these cases breakdown field strengths of \(100{ - }400\,\text {V}/\upmu \text {m}\) are typical [82].

The pull-in effect does not happen at \(z=1/3 \cdot z_0\) (air-gap actuators), but at much higher deflections. With some materials mechanical prestrain of the actuator allows to displace the pull-in further, reaching the breakdown field strength before. The reason for this surprising property is the volume constant dielectric layer showing viscoelastic properties. It complies with a return spring with strong nonlinear force-displacement characteristics for large extensions. Its working point is displaced along the stress-strain-curve of the material as an effect of the mechanical prestrain.

For the application in dielectric elastomer actuators many materials may be used. The material properties cover an extreme wide spectrum ranging from gel-like polymers up to relatively rigid thermoplastics. Generally speaking, every dielectric material has to have a high breakdown field strength and elasticity beside a high relative permittivity. Silicone provides highest deformation-velocities and a high temperature resistance. Acrylics have high breakdown field strength and achieve higher energy densities. The following list is a selection of the dielectric materials most frequently used today:

  • silicone

    • HS 3 (Dow Corning)

    • CF 19-2186 (Nusil)

    • Elastosil P7670 (Wacker)

    • Elastosil RT625 (Wacker)

  • acrylics

    • VHB 4910 (3M)

The most frequently used materials for the elastic electrodes are graphite powder, conductive carbon, and carbon grease.

5.5 Designs of Dielectric Elastomer Actuators

As mentioned before, dielectric elastomer actuators achieve high deformations (compression in field direction) of \(10{ - }30\%\). To keep voltages within reasonable ranges, layer thicknesses of \(10 {-} 100\,\upmu \text {m}\) are used depending on the breakdown field strength. The resulting absolute displacement in field direction is too low to be useful. Consequently, there are several concepts to increase it. Two principle movement directions are distinguished for this purpose: the longitudinal effect in parallel to the field (thickness change), and the transversal effect orthogonal to field (surface area change). The importance of this discrimination lies in the volume constancy of the material: uni-axial pressure load equals a two-axial tension load in the remaining spatial directions. Hence, two transversal tensions within the surface result in a surface change. For materials fulfilling the concept of “volume constancy”, Eq. (9.104) is valid, providing the following properties for the longitudinal compression \(S_z\) and the transversal elongation \(S_x\):

  • with a longitudinal compression of \(62\%\) both extensions are identical;

  • for smaller values of the longitudinal compression the resulting transversal extension is smaller;

  • for a longitudinal compression \({>}62\%\) the transversal extensions increases much faster than the longitudinal compression.

$$\begin{aligned} S_x = \frac{1}{\sqrt{1-S_z}}-1 \end{aligned}$$
(9.104)

The extension of the surface area \(S_A\) depends on the longitudinal compression \(S_z\) according to Eq. (9.105):

$$\begin{aligned} S_A = \frac{dA}{A}=\frac{S_z}{1-S_z} \end{aligned}$$
(9.105)

The increase of the area with uni-axial compression is always larger than the change of thickness. Actuators built according to this principle are the most effective ones. Figure 9.67 shows three typical designs. A roll-actuator (left) being built as full or tubular cylinder can achieve length changes of more than \(10\%\). Kornbluh [83] describes an acrylic roll-actuator achieving a maximum force of \(29\,\text {N}\) with an own weight of no more than \(2,6\,\text {g}\) at a extension of \(35\,\text {mm}\). The manufacture of electrodes with a large area is very simple. On the other hand the rolling of the actuators with simultaneous pre-strain (up to \(500\%\)) can be very challenging. With a stack-actuator (middle) very thin dielectric layers down to 5 \(\upmu \)m with minimized operational voltages can be achieved, depending mainly on the manufacturing technique [84]. As the longitudinal effect is used extension is limited to approximately 10%. However, due to their design and fabrication process actuator arrays at high density can be built [85] and offer typically lifetimes of more than 100 million cycles depending on their electrical interconnection [86]. The most simple and most effective designs are based on a restrained foil, whose complete surface change is transformed into an up-arching (diaphragm-actuator, right) [87]. If this actuator experiences a higher external load, such as from a finger, an additional force source, e.g. a pressure, has to be provided to support the actuators own tension.

Fig. 9.67
figure 67

Typical designs of dielectric elastomer actuators: roll-actuator (left), stack-actuator (center) and diaphragm-actuator (right)

5.5.1 Summary and Examples

As with air-gap actuators, a dielectric solid state actuator’s major limit is given by the breakdown field strength of the dielectric. However, in contrast to the air-gap actuators, a carefully chosen design can easily avoid any pull-in effect. Consequently, these actuators show a larger workspace; and with the high number of different design variants a wide variety of applications can be realized, just depending on the requirements on displacement, maximum force and actuator density.

Tactile Displays

The simplest application of a tactile display is a Braille-device. Such devices are meant to display Braille-letters in patterns of small, embossed dots. In standard Braille six dots are being used, in computer compatible Euro-Braille eight dots. These points are arranged in a 2x3 or 2x4 matrix, respectively (Fig. 9.68). In a display-device 40 to 80 characters are displayed simultaneously. In state-of-the-art designs each dot is actuated with one piezoelectric-bending actuator (Sect. 9.3.5). This technical effort is reason for the high price of these devices. As a consequence there are several functional samples existing, which prove the applicability of less expensive drives with simplified mechanical designs, but still sufficient performance. Each of the three variants for dielectric elastomer actuators has already been used for this application.

Figure 9.68 shows the schematic sketch of roll-actuators formed to an actuator column [88]. Each roll-actuator moves one pin, which itself is pushed up above the base plate after applying a voltage. The elastomer film is coiled around a spring of \(60\,\text {mm}\) length with a diameter of \(1.37\,\text {mm}\). With an electric field of \(100\, \text {V}/\upmu \text {m}\) applied, the pre-tensioned spring can achieve a displacement of \(1\,\)mm at a force of \(750\,\)mN. The underlying force source is the spring with spring-constant \(225\,\)N/m pre-tensioned by a passive film. The maximum necessary displacement of \(500\, \upmu \text {m}\) is achieved at field strengths of \(60\,\text {V}/\upmu \text {m}\).

Fig. 9.68
figure 68

© Elsevier, all rights reserved

Presenting a Braille sign with roll-actuators, left: geometry, right: schematic setup of a Braille row [88]

The application of stack actuators according to Jungmann [85] is schematically sketched in Fig. 9.70, left. The biggest advantage of this variant is given by the extremely high actuator density combined with a simple manufacturing process. Additionally, the closed silicone elements are flexible enough to be mounted on an almost arbitrary formed surface. The surface—in itself being made of silicone—shows an adequate roughness and temperature conductivity. It is perceived as “convenient” by many users. With a field strength of \(30\,\text {V}/\upmu \text {m}\) a stack made of 100 dielectric layers displacements of \(500\,\upmu \text {m}\) can be achieved. The load of a reading finger on the soft substrate generates a typical contact pressure of \(4\,\text {kPa}\) resulting in a displacement of \(25\,\upmu \text {m}\). This extension is considerably less than the perception threshold of \(10\%\) of the maximum displacement. For control of the array it has to be noted, that the actuators are displaced in a negative logic. With applied voltage the individual pin is pulled downwards.

A remote control providing tactile feedback based on the same type of stack actuators is presented in [89]. The mobile user interface consists of five independent actuating elements. Beside the presentation of tactile feedback the stack transducers offer to acquire user’s input using the transducers intrinsic sensor functionality. The actuators are driven with a voltage of up to 1100 V generated out of a primary litium-ion battery cell. A free form touchpad providing tactile feedback to the human palm is described in [90]. Four actuators are integrated in a PC-mouse to enhance user experience and substitute visual feedback during navigation tasks (Fig. 9.69). The stacks consist of 40 dielectric layers each 40 \(\upmu \)m in thickness and are supplied by a maximum voltage of 1kV in a frequency range from 1.5 to 1 kHz. The mouse contains all the required driving electronics and can be customized by a software configuration tool.

Fig. 9.69
figure 69

© (2014) Society of Photo-Optical Instrumentation Engineers (SPIE), all rights reserved

Tactile feedback enhanced PC-mouse: CAD model (a), and demonstration device (b) [90]

The design of a Braille display with diaphragm-actuators according to Heydt [91] demonstrates the distinct properties of this variant. The increase of elastomer surface results in a notable displacement of a pin from the device’s contact area. However, a mechanical prestrain is necessary to provide a force. This can be either generated by a spring or air pressure below the actuator. Figure 9.70 on the right gives a schematic sketch for a single point being pretensioned by a spring with a diameter of \(1.6\,\text {mm}\). At an operating voltage of \(5.68\,\text {kV}\) the actuator displaces in idle mode \(450\,\upmu \text {m}\).

Fig. 9.70
figure 70

© 2003 John Wiley & Sons Inc, all rights reserved

left: Actuator row with stack-actuators [85], used with permission; right: Use of diaphrag-Actuators [91],

Carpi combines the principle of the diaphragm actuators with fluid-based hydrostatic transmission [92]. The result is a wearable tactile display intended for providing feedback during electronic navigation in virtual environments. The actuators are based on an incompressible fluid that hydrostatically couples a dielectric elastomer membrane to a passive membrane interfaced to the user’s finger. The transmission of actuation from the active membrane to the finger, without any direct contact allows suitable electrical safety. However, the actuator is driven with comparatively high voltages up to 4 kV.

5.6 Electro-Rheological Fluids

Fluids being influenced in their rheological properties (especially the viscosity) by electrical field varying in direction and strength are called \(\hookrightarrow \) Electro-Rheological Fluid (ERF). Consequently ERF are classified as non-Newton fluids, as they have a variable viscosity at constant temperature. The electro-rheological effect has been observed for the first time in 1947 at a suspension of cornstarch and oil by Willis Winslow.

Electro-rheological fluids include dipoles made of polarized particles, which are dispersed in a conductive suspension. These particles are aligned in an applied electrical field. An interaction between particles and free charge carriers happens. Chain-like microstructures are built between the electrodes [93,94,95] in this process. However, it seems as if this is not the only effect responsible for the viscosity change, as even when the microstructures [96] were destroyed a significant viscosity increase remained. The exact analysis of the mechanism responsible for this effect is subject of actual research.

The viscosity of the fluid changes depending on the strength of the applied electrical field. With an electric field of \(1{-} 10\,\text {kV/mm}\) the viscosity may change up to a factor of 1000 compared to the field-free state. This enormous change equals a viscosity difference between water and honey. A big advantage of this method can be found in the dynamics of the viscosity change. It is reversible and can be switched within one millisecond. Therefore, electro-rheological fluids are suitable for dynamic applications, too.

If large field strengths are assumed, the ERF can be modeled as Bingham-fluid. It has a threshold for linear flow characteristics: starting at a minimum tension \(\tau _{F,d}\) (flow threshold) the fluid actually starts to flow. The fluid starts flowing right below this threshold. The shear forces \(\tau \) are calculated according to Eq. (9.106):

$$\begin{aligned} \tau = \mu \dot{\gamma } + \tau _{F,d} \end{aligned}$$
(9.106)

With \(\mu \) being the dynamic viscosity, \(\dot{\gamma }\) the shear rate, and \(\tau _{F,d}\) is the dynamic flow limit. Latter changes quadratically with the electrical field strength (Eq. (9.107)). The proportional factor \(C_d\) is a constant provided with the material’s specifications.

$$\begin{aligned} \tau _{F,d} = C_d E^2 \end{aligned}$$
(9.107)

For complex calculations modelling the fluid’s transition to and from the state of flow, the model is extended to a nonlinear system according to Eq. (9.108) (for \(n=1\) this equals Eq. (9.106))

$$\begin{aligned} \tau = \tau _{F,d} + k \dot{\gamma ^n} \end{aligned}$$
(9.108)

This general form describes the shear force for visco-plastic fluids with flow-limit according to Vitrani [97]. For an analysis of idle state with shear rate \(\dot{\gamma }=0\) the static flow-limit \(\tau _{F,s}\) with \(\tau _{F,s} > \tau _{F,d}\) is introduced. When exceeding the static flow limit, the idle fluid is deformed. With the specific material constants \(C_s\) and \(E_{ref}\) Eq. (9.109) can be formulated:

$$\begin{aligned} \tau _{F,s} = C_s (E-E_{ref}) \end{aligned}$$
(9.109)

The materials used for the particles are frequently metal oxides, silicon an hydride, poly urethane and polymers with metallic ions. The diameter of particles is \(1{ - }100\,\upmu \textrm{m}\), their proportion on the fluid’s volume is 30–50%. As carrier medium typically oils (such as silicone oil) or specially treated hydrocarbon are used. To additionally improve the viscosity change, nanoscale particles are added in the electro-rheological fluids, too (“giant electro-rheological effect” [98, 99]). In [100, 101] further mathematical modelling is presented for the dynamic flow behavior of ER-fluids.

The central property of ERF—to reversibly change the viscosity—is used for force-feedback devices, haptic displays, and artificial muscles and joints. As the change in viscosity is mainly a change in counter-forces but not in shape or direct-forces, ERF actuators are counted to the group of “passive actuators”. For the characterization of their performance, the ratio between stimulated and idle state is used. They are built in three principle design variants [102] as described in the following sections.

5.6.1 Shear Mode

The ER-fluid is located in between two parallel plates, one fixed and one moving relatively to the fixed one. The only constrain is given by a fixed inter-plate distance d. If a force F is applied on the upper plate, it is displaced by a value x at a certain velocity v. For the configuration shown in Fig. 9.71 the mechanical control ratio \(\lambda \) can be calculated according to Eq. (9.112) from the ratio of dissipative forces (field-dependent flow-stresses, Eq. 9.115) and the field independent viscosity-term (Eq. (9.110)) [103]. \(\eta \) gives the basis viscosity of the ER-Fluid (in idle state) and \(\tau _y\) the low-stress depending on the electrostatic field.

Fig. 9.71
figure 71

Using ERF to vary the shear force

$$\begin{aligned} F_{\eta }&=\frac{\eta v a b}{d} \end{aligned}$$
(9.110)
$$\begin{aligned} F_{\tau }&=\tau _y a b \end{aligned}$$
(9.111)
$$\begin{aligned} \lambda&=\frac{F_{\tau }}{F_{\eta }}=\frac{\tau _y d}{\eta v} \end{aligned}$$
(9.112)

5.6.2 Flow Mode

The schematic sketch of this configuration is shown in Fig. 9.72. Both fixed plates form a channel, with the fluid flowing through it due to an external pressure difference p and a volume-flow \(\dot{V}\). With an electric field E applied between the plates the pressure loss increases along the channel and the volume flow is reduced. Analog to the prior design a field independent viscosity based pressure loss \(p_{\eta }\) and a field dependent pressure loss \(p_{\tau }\) can be calculated [103]:

$$\begin{aligned} p_{\eta }&=\frac{12\eta \dot{V} a}{d^3 b} \end{aligned}$$
(9.113)
$$\begin{aligned} p_{\tau }&=\frac{c\tau _y a}{d} \end{aligned}$$
(9.114)

The mechanical control ratio equals

$$\begin{aligned} \lambda =\frac{p_{\tau }}{p_{\eta }}=\frac{c \tau _y d^2 b}{12 \eta \dot{V}} \end{aligned}$$
(9.115)
Fig. 9.72
figure 72

Varying the flow channel’s resistivity with ERF-actuators

At an adequate dimensioning of the fluid, the flow-resistance can be increased by the electrical field to such a degree, that the fluid stops completely when exceeding a specific voltage. This makes the channel a valve without any moving mechanical components.

Fig. 9.73
figure 73

Varying the acoustic impedance with ERF-actuators under external forces

5.6.3 Squeeze Mode

A design to generate pressure is schematically sketched in Fig. 9.73. In contrast to the variants shown before, the distance between both plates is subject to change now. If a force acts on the upper plate, it moves downwards. This results in the fluid being pressed outside. A plate distance \(d_0\) is assumed at the beginning, and a relative movement of v of the plate moving downwards. The velocity dependent viscosity force \(F_\eta \) and the field dependent tension term \(F_\tau \) [104] are calculated according to:

$$\begin{aligned} F_{\eta }=\frac{3\pi \eta v r^4}{2(d_0-z)} \end{aligned}$$
(9.116)
$$\begin{aligned} F_{\tau }=\frac{4\pi \tau _y r^3}{3(d_0-z)} \end{aligned}$$
(9.117)

which gives the mechanical control ratio:

$$\begin{aligned} \lambda =\frac{8 \tau _y 3(d_0-z)^2}{9 \eta v r} \end{aligned}$$
(9.118)

With pressure (force on the upper plate) the fluid is pressed out of the gap. In this configuration the force-displacement-characteristics is strongly influenced by the electrical field strength. An analysis of the dynamic behavior of such an actuator is described in [105].

5.6.4 Designing ERF-Actuators

The maximum force \(F_\tau \) and the necessary mechanical power \(P_{mech}\) are the input values for the design of ERF-actuators from the perspective of an application engineer. Equations (9.110) and (9.118) can be combined to calculate the necessary volume for providing a certain power with all three actuator configurations.

$$\begin{aligned} V = k \frac{\eta }{\tau ^2_y}\lambda P_{mech} \end{aligned}$$
(9.119)

Consequently, the volume is defined by the mechanical control ratio, the fluid-specific values \(\eta \) and \(\tau _y\), such as a constant k dependent on the actual configuration. The electrical energy \(W_{el}\) necessary to generate the electrostatic field of the actuator (volume-dependent) is calculated according to Eq. (9.120).

$$\begin{aligned} W_{el} = V(\frac{1}{2}\varepsilon _0 \varepsilon _r E^2) \end{aligned}$$
(9.120)

5.6.5 Comparison to Magneto-Rheological Fluids

\(\hookrightarrow \) Magneto-Rheological Fluid (MRF) are very similar to electro-rheological fluids. However, the physical properties of the fluids are influenced by magnetic fields. All calculations which are shown before are applicable to MRF, too. Looking at the volume necessary for an actuator according to Eq. (9.119), considering the viscosities of electro-rheological and magneto-rheological fluids being comparable, a volume ratio proportional to the reciprocal ratio of the fluid-tensions’ square according to Eq. (9.121) results:

$$\begin{aligned} \frac{V_{ERF}}{V_{MRF}}= \frac{\tau ^2_{MRF}}{\tau ^2_{ERF}} \end{aligned}$$
(9.121)

In a rough but good approximation the flow-stress of a magneto-rheological fluid is one magnitude larger than of an ERF, resulting in a smaller (approximately factor 100) volume of a MRF actuator compared to the ERF. However, a comparison between both fluids going beyond the pure volume analysis for similar output power is hard: for an ERF high voltages at relatively small currents are required. The main power leakage is lost by leakage-currents through the medium (ERF) itself. With MRF-actuators smaller electrical voltages at very high currents become necessary to generate an adequate magnetic field. The energy for a MRF-actuator is calculated according to Eq. (9.122) with the magnetic flux density B and the magnetic field strength H.

$$\begin{aligned} W_{el,MRF} = V_{MRF}(\frac{1}{2}BH) \end{aligned}$$
(9.122)

The ratio between the energies for both fluids is calculated according to Eq. (9.123)

$$\begin{aligned} \frac{W_{el,ERF}}{W_{el,MRF}}= \frac{V_{ERF}}{V_{MRF}} \frac{\varepsilon _0 \varepsilon _r E^2}{BH} \end{aligned}$$
(9.123)

With typical values for all parameters the necessary electrical energy for actuator control is comparable for both fluids. A good overview on the design of actuators for both types of fluids is given in [106].

5.6.6 Summary and Examples

Electro-rheological fluids are also called partly-active actuators, as they are not transforming the electrical values into a direct movement, but change their properties due to the electrical energy provided. The change of their properties covers a wide range. Naturally, their application in haptics ranges from small tactile displays to larger haptic systems.

Tactile Systems

The first application of ERF as tactile sensor in an artificial robot hand has been made in 1989 by Kenaley [107]. Starting from this work several ideas developed to use ERF in tactile arrays for improving systems for virtual reality applications. Several tactile displays, among them a 5 \(\times \) 5 Matrix from Taylor [108] and another one from Böse [109], were built. Figure 9.74 shows the schematic design of such a tactile element. A piston is pressed in an ERF filled chamber by the user. Varying counter forces are generated depending on the actuation state of the ERF. Elastic foam is connected to the piston as a spring to move it back to its resting position. With an electric field of \(3\,\text {V/}\upmu \text {m}\) a force of \(3.3\,\text {N}\) can be achieved at a displacement of \(30\,\text {mm}\). Switching the electrical voltages is realized by light emitting diodes and corresponding receivers (GaAs-elements) on the backplane.

Fig. 9.74
figure 74

© Elsevier, all rights reserved

Schematic setup of a tactile actuator based on ER fluids [109]

Haptic Operating Controls

Another obvious application for ERF in haptic systems is their use as a “variable brake”. This is supported by the fact that typical applications beside haptic systems are variable brakes and bearings (e.g. adaptive dampers). There are several designs with a rotary knob mobbing a spinning disk within an ERF or MRF generating varying counter torques as shown in [110]. In this case, the measurement of the rotary angle is solved by a potentiometer. In dependency on the rotary angle the intended counter force respectively torque is generated. The user can perceive a “latching” of the rotary knob with a mature system. The latching depth itself can be varied in a wide range. By the varying friction hard stops can be simulated, too, such as sticking and of course free rotation.

An extension of the one-dimensional system is presented in [111]. Two systems based on ERF are coupled to a joystick with two DoF. A counter force can be generated in each movement direction of the joystick. As ERF are able to generate higher torques with less energy required compared to a normal electrical drive, they are especially suitable to mobile applications like in cars.

Senkal et al. presented a combination of an MRF brake and pneumatic actuators for a 2D joystick as shown in Fig. 9.75. This hybrid concept uses pneumatic actuators because of the high energy density and the MR brake to increase the fidelity of rigid objects [112]. Further realizations of MRF operating controls can be found in [113].

Fig. 9.75
figure 75

© Elsevier, all rights reserved

Haptic joystick based on pneumatic actuators and a MRF-brake as presented in [112]

Force-Feedback Glove

A force-feedback glove was designed as a component for a simulator of surgeries [114]. Surgical interventions shall be trained by the aid of haptic feedback. The system MEMICO (“Remote Mechanical Mirroring using Controlled stiffness and Actuators”) shall enable a surgeon to perform the treatment with a robot in telemanipulation, whereas the haptic perception is retained. ERF actuators are used for both ends: on the side of the end-effector, and for the haptic feedback to the user. The adjustable elasticity is based on the same principle as with tactile systems. For generating forces a force source is necessary. A new ECFS actuator (“Electronic Controlled Force and Stiffness”) is used for this application. The schematic design is shown in Fig. 9.76. It is an actuator according to the inchworm-principle, wherein both brakes are realized by the ER-fluid surrounding it. The driving component for the forward- and backward movement is realized by two electromagnets. Both actuators are assembled within a haptic exoskeleton. They are mounted on the rim of a glove to conserve the mobility of the hand. With the actuators in between all finger-joints arbitrary forces and varying elasticities can be simulated independently. The ECFS actuators are operated at voltages of \(2\,\text {kV}\) and generate a force up to \(50\,\text {N}\).

Fig. 9.76
figure 76

©  (2001) Society of Photo-Optical Instrumentation Engineers (SPIE), all rights reserved

Schematic setup of an ERF-Inchworm motor [114]

6 Special Designs of Haptic Actuators

The actuation principles discussed so far are the most common approaches to the actuation of haptic devices. Besides these principles, there are numerous research projects, singular assemblies, and special classes of devices. The knowledge of these designs is an enrichment for any engineer, yet it is impossible to completely cover the variety of all haptic designs in a single book. This section, nevertheless, intends to give a cross section of alternative, quaint and unconventional systems for generating kinaesthetic and tactile impressions. This cross-section is based on the authors’ subjective observations and knowledge and does not claim to be exhaustive. The discussed systems have been selected, as examples suited best to cover one special class of systems and actuators, each. They are neither the first systems of their kind, nor necessarily the best ones. They are thought to be crystallization points of further research, if specific requirements have to be chosen for special solutions. The systems shown here are meant to be an inspiration and an encouragement not to discard creative engineering approaches to the generation of haptic impressions too early during the design process.

6.1 Haptic-Kinaesthetic Devices

Haptic-kinaesthetic devices of this category excel primarily due to their extraordinary kinematics and not to very special actuation principles. Nevertheless, every engineer may be encouraged to be aware of the examples of this device class and let this knowledge influence his/her own work.

6.1.1 Rope-Based Systems

With rope-based systems, actuators and the point of interaction are connected with ropes, i.e. mechanical elements that can only convey pulling forces. They are especially suited for lightweight systems with large working spaces, as for example simulation of assembly tasks, rehabilitation and training. von Zitzewitz describes the use of rope-based systems for sport simulation and training (tennis and rowing) as well as an experimental environment to investigate vestibular stimulation in sleeping subjects [115].

Another system, the Spidar (Fig. 9.77) is based on the work of Prof. Sato and has frequently been used in research projects [116, 117] as well as in commercial systems. It is composed of an interaction handle—usually a ball—held by eight strings. Each string is operated by an actuator, which is frequently (but not obligatorily) mounted in the corners of a rectangular volume. The drives are able to generate pulling forces on the strings, enabling the generation of forces and torques in six DoFs on the handle. Typically the actuators used are based on electrodynamic electronic—commutated units. The Spidar-system can be scaled to almost any size, ranging from table-top devices to room-wide installations. It convinces by the small number of mechanical components and the very small friction. As strings are able to provide pull forces only, it is worth noting that just two additional actuators are sufficient to compensate this disadvantage.

Fig. 9.77
figure 77

© Haption, used with permission

a Desktop-version of the Spidar with ball-like interaction handle by Shoichi Hasegawa, used with permission, b room-size version INCA 6D with 3D visualization environment by

6.1.2 Screw-and-Cable-System

In several projects relating to medical rehabilitation and master slave teleoperation the CEA-LIST Interactive Robotics Unit use their Screw-and-Cable-System (SCS) shown in a first prototype in 2001 [118,119,120,121]. In this prototype 6 Screw-Cable actuators are used to motorize a master arm in a teleoperation system enabling high-fidelity force feedback. In the meantime the master arm is commercialized by Haption S.A. under the name Virtuose 6D 4040 [122]. The patented SCS basic principle can be seen in Fig. 9.78.

Fig. 9.78
figure 78

Figure courtesy of CEA LIST

SCS basic principles: a driving unit b and particular patented mounting [118].

A rotative joint is driven by a standard push-pull cable. On one side, the cable is driven by a ball-screw which translates directly in its nut (the screw is locked in rotation thanks to rollers moving into slots). The nut is rotating in a fixed bearing and is driven by the motor thanks to a belt transmission [121].

Using SCS allows significantly mass and volume reduced driving units for joint torque control. The low friction threshold and high backdrivability enable true linear torque control without a sensor, avoiding drift and calibration procedure. Low inertia of the structure lead to a high transparency. In the upper limb exoskeleton ABLE 4D the SCS is embedded in the moving parts of the arm resulting in reduced cable length and simplified routing (Fig. 9.79). The two SCS integrated in the arm module perform alike artificial electrical muscles. Further information about the design of such systems can be found in [123].

Fig. 9.79
figure 79

Figure courtesy of CEA LIST

ABLE arm module: a optimized architecture to be integrated, b structure with its 2 integrated actuators [118].

6.1.3 Magnetorheological Fluids as Three-Dimensional Haptic Display

The wish to generate an artificial haptic impression in a volume for free interaction is one of the major motivations for many developments. The rheological systems shown in Sect. 9.5 provide one option to generate such an effect. For several years the team of Bicchi has been working on the generation of spatially resolved areas of differing viscosity in a volume (Fig. 9.80) to generate force-feedback on an exploring hand. Lately, the results were summarized in [124]. The optimization of such actuators is largely dependent on the control of the rheological fluid [125]. The psycho-physical experiments performed until today show that the identification of simple geometrical structures can be achieved on the basis of a 4x4 pattern inside the rheological volume.

Fig. 9.80
figure 80

© Springer Nature, all rights reserved

Magnetorheological actuation principle for full-hand interaction based on a 4x4 pattern [124]

6.1.4 Self-induction and Eddy Currents as Damping

An active haptic device is designed to generate forces resp. torques in any direction. By the concept of “active” actuation the whole spectrum of mechanical interaction objects (e.g. masses, springs, dampers, other force sources like muscles, and moving objects) is covered. Nevertheless, only a slight portion of haptic interaction actually is “active”. This has the side effect (of control engineering approaches) that active systems have continuously to be monitored for passivity. An alternative approach to the design of haptic actuators is given by choosing technical solutions able to dissipate mechanical energy. A frictional brake would be such a device, but its properties are strongly nonlinear and hard to control. Alternatives are therefore highly interesting. The team of Colgate showed in [126] how to increase the impedance of an electronic—commutated electrodynamic actuator, whereby two windings were bypassed by a variable resistor. The mutual induction possible by this bypass damped the motor significantly. In [127] the team of Hayward went even further by implementing an eddy current break into a pantograph-kinematics (Fig. 9.81). This break is a pure damping element with almost linear properties. By this method a controlled dynamic damping up to 250 Hz was achieved.

Fig. 9.81
figure 81

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Principle of eddy currents damping a rotating disc (a) and realization as a haptic device (b) by [127]

6.1.5 Serial Coupled Actuators

 Serial coupled actuators include an additional mechanical coupling element between actuator and the driven element of the system. In the majority of cases this is a elastic element that was originally inserted to ease force control of actuators interacting with stiff environments [128]. These so called serial-elastic actuators allow the replacement of direct force control by the position control of both sides of the series elasticity and are used in applications like rehabilitation or man-machine interaction [129]. An example is shown in Fig. 9.82. For haptic applications, this configuration is especially interesting for the display of null-forces and -torques, i.e. free space movements.

Fig. 9.82
figure 82

Example for the realization of a serial elastic actuator for use in an active knee orthesis [129]. The bevel gear is needed for better integration of the actuator near the knee of the wearer. Picture courtesy of Roman Müller, Institute of Electromechanical Design, Technische Universität Darmstadt, used with permission

Another application of serial coupled actuators was introduced by Mohand-Ousaid et al. To increase dynamics, lower the impact of inertia and increase transparency, a serial arrangement of two actuators connected with an viscous coupler based on eddy currents was presented in [130] and is shown in Fig. 9.83. By using two motors, the range of displayable forces/torques can be extended, because of the low inertia of the smaller motor higher dynamics can be archived. The viscous clutch couples slip velocity to transmitted torque that is used in the control of the device. With this approach, inertia is decoupled from the delivered torque effectively.

Fig. 9.83
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©  SAGE Publications, all rights reserved

Setup with two serial actuators coupled with an eddy-current clutch as presented in [130]

6.1.6 MagLev—Butterfly Haptics

In the 1990s the team of Hollis developed a haptic device [131] based on the electrodynamic actuation principle (Fig. 9.84). Since recently, the device has been sold commercially by Butterfly Haptics. It is applied, e.g., to ongoing research projects on psychophysical analysis of texture perception. Six flat coils are mounted in a hemisphere with a magnetic circuit each. The combination of Lorentz-forces of all coils allows an actuation of the hemisphere in three translational and three rotational directions. Via three optical sensors—each of them measuring one translation and one rotation—the total movement of the sphere is acquired. Besides the actuation within its space, the control additionally includes compensation of gravity with the aid of all six actuators. This function realizes a bearing of the hemisphere with Lorentz-forces, only. The air-gap of the coils allows a translation of 25 mm and a rotation of ±8 \({}^{\circ }\) in each direction. Resolutions of 2 \(\upmu \)m (1 \(\sigma \)) and stiffnesses of up to 50 N mm\(^{-1}\) can be reached. As a consequence of the small mass of the hemisphere, the electrodynamic actuator principle as a drive and the abandonment of mechanical bearings, forces of a bandwidth of 1 kHz can be generated.

Fig. 9.84
figure 84

Images courtesy of Butterfly Haptics, LLC, Pittsburgh, PA, USA, used with permission

MagLev device a inner structure, b use of two devices in bimanual interaction.

6.2 Haptic-Tactile Devices

Haptic-tactile devices of this category are intelligent combinations of well-known actuator principles of haptic systems with either high position resolutions or extraordinary, dynamic properties.

6.2.1 Pneumatic

Due to their working principle pneumatic systems are a smart way to realize flexible high resolution tactile displays. But these systems suffer acoustic compliance, low dynamics and the requirement of pressurized air. A one-piece pneumatically-actuated tactile 5 \(\times \) 5 matrix molded from silicone rubber is described in [132, 133]. The spacing is 2.5 mm with 1 mm diameter tactile elements. Instead of actuated pins an array of pressurized chambers without chamber leakage and seal friction is used (Fig. 9.85). 25 solenoid 3-way valves are used to control the pressure in each chamber resulting in a working frequency of 5 Hz. Instead of closed pressurized chambers the direct contact between the fingertip and the compressed air is used for tactile stimulation in [134]. The interface to the skin consists of channels each 2 mm in diameter. A similar display is shown in [135]. Using negative air pressure the tactile stimulus is generated by suction through 19 channels 2.5 mm in diameter with 5 mm intervals.

Fig. 9.85
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Pneumatic actuated tactile display: sketch (a) bidigital teletaction, and realization (b) [133], used with permission

6.2.2 Thermo-Pneumatic

A classic problem of tactile pin-arrays is given by the high density of stimulator points to be achieved. The space below each pin for control and reconfiguration of the pin’s position is notoriously finite. Consequently, a large number of different designs has been tested until today. In [136] a thermo-pneumatic system is introduced (Fig. 9.86) based on tubes filled with a fluid (methyl-chloride) with a low boiling point. The system allows a reconfiguration of the pins within 2 s. however, it has high power requirements, although the individual elements are very cheap.

Fig. 9.86
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Thermo-pneumatic actuation principle in a schematic sketch (a), and as actual realization (b) [136]

6.2.3 Shape Memory Materials

Materials with shape memory property are able to remember their initial shape after deformation. When the material is heated up its internal structure starts to change and the materials forms back in its pre-deformed state. Due to the material intrinsic actuating effect high resolution tactile displays are achievable. The low driving frequencies caused by thermal inertia and needed heating and/or cooling systems are the drawback of this technology.

Shape Memory Alloys

In [137] a pin-array with 64 elements is realized covering an area of 20 \(\times \) 40 \(\text {mm}^2\). The display consists of 8 modules each containing eight dots (Fig. 9.87). Each element comprise a 120 mm length NiTi SMA wire pre-tensioned by a spring. When an electrical current flows through the wire it heats up and starts to shorten. The result is a contraction of up to 5 mm. Driving frequencies up to a few Hz using a fan to cool the SMA wires down can be reached.

Fig. 9.87
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©  The Institution of Engineering and Technology, all rights reserved

Sketch of tactile display using SMA: Top view (a), and side view of one module (b) [137]

Bistable Electroactive Polymers

Bistable Electroactive Polymers (BSEP) combine the large-strain actuation of dielectric elastomers with shape memory properties. The BSEP provide bistable deformation in a rigid structure. These polymers have a glass transition temperature \(T_g\) slightly above ambient temperature. Heated above \(T_g\) it is possible to actuate the materials like a conventional dielectric elastomer.

Using a chemically crosslinked poly(tert-butyl acrylate) (PTBA) as BSEP a tactile display is presented in [138]. The display contains a layer of PTBA diaphragm actuators and an incorporated heater element array. Figure 9.88 shows the fabricated refreshable Braille display device with size of a smartphone screen.

Fig. 9.88
figure 88

© (2012) Society of Photo-Optical Instrumentation Engineers (SPIE), all rights reserved

Bistable BSEP Braille display: actuator array (a) [138], and zoom of Braille dots in “OFF” and “ON” state (b) [139]

6.2.4 Texture Actuators

Besides an application in Braille-related tasks, the design of tactile displays is relevant for texture perception, too. Instead of vibrotactile stimulation on a user’s finger the modification of the friction between a sliding finger and a touch screen surface is a promising new direction in touch screen haptics. This kind of displays are mostly based on two basic technologies. In displays based on the electrovibration effect a periodic electrical signal is injected into a conductive electrode coated with a thin dielectric layer. The result is an alternating electrostatic force that periodically attracts and releases the finger, producing friction-like rubbery sensations as mentioned in Sect. 9.5.2.4.

In friction displays based on the squeeze film effect a thin cushion of air under the touching finger is created by a layer which is placed on top of the screen and is vibrated at an ultrasonic frequency. The modulation of the frequency and intensity of vibrations allows to put the finger touching the surface in different degrees of levitation, thus actually affecting the frictional coefficient between the surface and the sliding finger [140].

In 2007, Winfield impressively demonstrated a simple tactile texture display called TPaD based on the squeeze film effect. The actuating element is a piezoelectric bending disk driven in resonance mode [141]. By the aid of an optical tracking right above the disc, and with a corresponding modulation of the control signal, perceivable textures with spatial resolutions were generated. The 25 mm diameter piezoelectric disk is bonded to a 25 mm diameter glass disk and supported by an annular mount (Fig. 9.89).

Fig. 9.89
figure 89

Figure based on [93]

Schematic view of the T-PaD with convex (a) and concave (b) oscillation states: Piezoelectric disc underneath a glass substrate modulates the friction between finger and display based on the position of the finger and the model of the simulated texture.

A similar display with an increased surface area of 57 \(\times \) 76 \(\text {mm}^2\) is shown in [86]. The vibrations are created by piezoelectric actuators bonded along one side of a glass plate placed on top of an LCD screen.

6.2.5 Flexural Waves

A transparent display providing localized tactile stimuli is presented in [142]. The working principle is based on the concept of computational time reversal and is able to stimulate one or several regions, and hence several fingers, independently. According to the wave propagation equation the direct solution of a given propagation problem is a diverging wave front and a converging wave front for the time reversed one if the initial condition is an impulse force. Consequently it is possible to generate peaks of deflection localized in space and in time using constructive interference caused by a multiple of stimulating actuators. Of course the quality of the focusing process is increasing with the number of transducers and has to be optimized. Depending on the requirements the noise occurring at the passive areas might be reduced below the tactile perception threshold.

Relating to their patented bending wave technology the company Redux Laboratories offers electromagnetic transducers for medium and large applications as well as piezo exciters for small form factor applications such as mobile devices (Fig. 9.90).

Fig. 9.90
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Redux transducers for bending wave technology: Moving coil exciters (a), and multi-layered piezos

6.2.6 Volume-Ultrasonic Actuator

Iwamoto built tactile displays which are made of piezoelectric actuators and are actuated in the ultrasonic frequency range. They use sound pressure as a force transmitter. The underlying principle is given by generating a displacement of the skin and a corresponding haptic perception by focused sound pressure. Whereas in the first realization an ultrasonic array had been used to generate tactile dots in a fluid [143], later developments used the air for energy transmission [144]. The pressures generated by the designs (Fig. 9.91) provide a weak tactile impression, only. But especially the air-based principle works without any mechanical contact and could therefore become relevant for completely new operation concepts combined with gesture recognition.

Fig. 9.91
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© Springer Nature, all rights reserved

Tactile display based on ultrasonic sound pressure as array of senders for a transmission in the air [144]

The concept developed further, especially as applications in public areas were getting into focus where operation of machines is preferred without direct contact to surfaces touched by others. Scientifically the group around Shinoda explores the limites in size and sound pressure of such devices [145]. On a commercial level the concept was taken and extended by ultraleap offering a wide variety of products and solutions into multiple industries. A key-challenge for successful application lies in the fact that ultrasonic interfaces require unique and new tactile effects for operation [146], but offers at the same time a holographic-like volumetric experience [147].

6.2.7 Ungrounded Haptic Displays

In case of the interaction with large virtual worlds it is frequently necessary to design devices which are worn on the body, i.e. that do not exhibit a fixed ground connection. An interesting solution has been shown in [148], generating a tactile sensation with belts at the palm and at each finger. The underlying principle is based on two actuators for each belt, generating a shear force to the skin when being operated in the same direction, and a normal force when being operated in the opposite direction. This enables to provide tactile effects when grasping or touching objects in a virtual world, but without the corresponding kinaesthetic effects.

Other realizations of ungrounded devices include the use of gyro effects and the variation of angular momentum and designs incorporating non-linear perception properties of the human user as presented in [149]. The device shown in Fig. 9.92 is based on the display of periodical steep inertial forces generated by a spring-mass system and an electrodynamic voice coil actuator.

Fig. 9.92
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Ungrounded haptic display to convey pulling impressions as shown in [149]

6.2.8 Electro-Tactile

As haptic receptors can be stimulated electrically, it is not far-fetched to design haptic devices able to provide low currents to the tactile sense organs. The design of such devices can be traced back to the 1970s. One realization is presented in [150] (Fig. 9.93). Electro-tactile displays do work—no doubt—however they have the disadvantage also to stimulate noci-receptors for pain sensation beside the mechano-receptors. Additionally, the electrical conductivity between display and skin is subject to major variations. These variations are inter-person differences due to variations in skin-thickness, but they are also a time-dependent result of electro-chemical processes between sweat and electrodes. The achievable tactile patterns and the abilities to distinguish tactile patterns are subject to current research.

Fig. 9.93
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© Springer Nature, all rights reserved

Electro-tactile display worn on the forehead: Electrodes (a), and edge recognition and signal conditioning principle (b) [150]

Recommended Background Reading

  •  [12] Ballas, R.: Piezoelectric multilayer beam bending actuators: static and dynamic behavior and aspects of sensor integration. Springer, 2007.

    Dynamic modeling of piezoelectric bimorph actuators based on concentrated and distributed network elements

  •  [22] Kallenbach, E.; Stölting, H.-D. & Amrhein, W. (Eds.): Handbook of Fractional-Horsepower Drives, Springer, Heidelberg, 2008.

    Overview about the most common fractional horsepower actuators and transmissions with focus on the design parameters.

  •  [1] Huber, J.; Fleck, N. & Ashby, M.: The Selection of Mechanical Actuators Based on Performance Indices In: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 1997, 453, 2185-2205.

    Analysis of several performance indices for different actuation principles.