Actuators, Volume 7, Issue 4 (December 2018) – 27 articles

In this paper, the author presents the adaptive control design and stability analysis of robotic manipulators based on two main approaches, i.e., Lyapunov stability theory and hyperstability theory. For the Lyapunov approach, the author presents the adaptive control of a 2-DOF (degrees of freedom) robotic manipulator. Furthermore, the adaptive control technique and Lyapunov theory are subsequently applied to the end-effector motion control and force control, as in most cases, one only considers the motion control (e.g., position control, trajectory tracking). To make the robot interact with humans or the environment, force control must be considered as well to achieve a safe working environment. For the hyperstability approach, a control system is developed through integrating a PID (proportional–integral–derivative) control system and a model reference adaptive control (MRAC) system, and also the convergent behavior and characteristics under the situation of the PID system, model reference adaptive control system, and PID+MRAC control system are compared. Full article

The work at hand focuses on an adaptive system aimed at improving the soundproof performance of car door seals at specific regimes (cruise), without interfering with the conventional opening and closing operations. The idea addresses the necessity of increasing seal effectiveness, jeopardized by aerodynamic actions that strengthen as the speed increases, generating a growing pressure difference between the internal and the external field in the direction of opening the door, and then deteriorating the acoustic insulation. An original expansion mechanism driven by a shape memory alloy (SMA) wire was integrated within the seal cavity to reduce that effect. The smart material was activated (heated) by using the Joule effect; its compactness contributed to the realization of a highly-integrable and modular system (expanding cells). In this paper, the system development process is described together with the verification and validation activity, aimed at proving the functionality of the realized device. Starting from industrial requirements, a suitable solution was identified by considering the basic phenomenon principle and the allowable design parameters. The envisaged system was designed and its executive digital mock-up (CAD, computer-aided design) was released. Prototyping and laboratory tests showed the reliability of the developed numerical models and validated the associated predictions. Finally, the system was integrated within the reference car. To demonstrate the insulation effect, the experimental campaign was carried out in an anechoic room, achieving significant results on the concept value. Full article

Flywheel Energy Storage Systems are interesting solutions for energy storage, featuring advantageous characteristics when compared to other technologies. This has motivated research effort focusing mainly on cost aspects, system reliability and energy density improvement. In this context, a novel shaftless outer-rotor layout is proposed. It features a semi-hard magnetic FeCrCo 48/5 rotor coupled with two bearingless hysteresis drives. The novelty lies in the use of the semi-hard magnetic material, lending the proposed layout advantageous features thanks to its elevated mechanical strength and magnetic properties that enable the use of bearingless hysteresis drives. The paper presents a study of the proposed layout and an assessment of its energetic features. It also focuses on the modeling of the radial magnetic suspension, where the electromagnets providing the levitating forces are modeled through a one-dimensional approach. The Jiles–Atherton model is used to describe the magnetic hysteresis of the rotor material. The proposed flywheel features a mass of 61.2 kg, a storage capability of 600 Wh at the maximum speed of 18,000 rpm and achieves an energy density of 9.8 Wh/kg. The performance of the magnetic suspension is demonstrated to be satisfactory and the influence of the hysteresis of the rotor material is highlighted. Full article

Compliant mechanisms based on flexure hinges are widely used in precision engineering applications. Among those are devices such as precision balances and mass comparators with achievable resolutions and uncertainties in the nano-newton range. The exact knowledge of the mechanical properties of notch hinges and their modeling is essential for the design and the goal-oriented adjustment of these devices. It is shown in this article that many analytical equations available in the literature for calculating the bending stiffness of thin semi-circular flexure hinges cause deviations of up to 12% compared to simulation results based on the three-dimensional finite element model for the considered parameter range. A close examination of the stress state within the loaded hinge reveals possible reasons for this deviation. The article explains this phenomenon in detail and shows the limitations of existing analytical models depending on specific geometric ratios. An accurate determination of the bending stiffness of semi-circular flexure hinges in a wide range of geometric parameters without the need for an elaborate finite element analysis is proposed in form of FEM-based correction factors for analytical equations referring to Euler-Bernoulli’s beam theory. Full article

Manipulating micro objects has become an important task in several applications. Actuation is a crucial aspect of micromanipulation because there are physical restrictions which affect actuators’ performances at the micro or nano scale. One way of getting rid of these limitations is the use of an appropriate mechanical structure which enhances the elasticity of the material or provides mechanical advantage. This Special Issue of Actuators, which is dedicated to micromanipulation, offers a contribution to the development of some promising methods to actuate a microsystem for micromanipulation. Full article

Looking at new trends in global policies, electric vehicles (EVs) are expected to increasingly replace gasoline vehicles in the near future. For current electric vehicles, the motor current driving system and the braking control system are two independent issues with separate design. If a self-induced back-EMF voltage from the motor is a short circuit, then short-circuiting the motor will result in braking. The higher the speed of the motor, the stronger the braking effect. However, the effect is deficient quickly once the motor speed drops quickly. Traditional kinetic brake (i.e., in the short circuit is replaced by a resistor) and dynamic brake (the short circuit brake is replaced by a capacitor) rely on the back EMF alone to generate braking toque. The braking torque generated is usually not enough to effectively stop a rotating motor in a short period of time. In this research task, an integrated driving and braking control system is considered for EVs with an active regenerative braking control system where back electromagnetic field (EMF), controlled by the pulse-width modulation (PWM) technique, is used to charge a pump capacitor. The capacitor is used as an extra energy source cascaded with the battery as a charge pump. This is used to boost braking torque to stop the rotating motor in an efficient way while braking. Experiments are conducted to verify the proposed design. Compared to the traditional kinetic brake and dynamic brake, the proposed active regenerative control system shows better braking performance in terms of stopping time and stopping distance. Full article

A force generator module (FGM) based on magnetorheological fluid (MRF) was developed to provide force-feedback information for applications in tele-robotic bone biopsy procedures. The FGM is capable of rapidly re-producing a wide range of forces that are common in bone biopsy applications. As a result of the nonlinear nature of MRF, developing robust controllers for these mechanisms can be challenging. In this paper, we present a case study motivated by robotic bone biopsy. We use a non-linear Hammerstein-Wiener (H-W) estimator to address this challenge. The case is presented through three studies. First, an experiment to develop design constraints is presented and describes biopsy force measurements for various animal tissues. Required output forces were found to range between <1 N and <50 N. A second study outlines the design of the FGM and presents the experimental characterization of the hysteretic behavior of the MRF. This data is then used as estimators and validators to develop the nonlinear Hammerstein-Wiener (H-W) model of the MRF. Validation experiments found that the H-W model is capable of predicting the behavior of the MRF device with 95% accuracy and can eliminate hysteresis in a closed-loop control system. The third study demonstrates the FGM used in a 1-DOF haptic controller in a simulated robotic bone-biopsy. The H-W control tracked the input signal while compensating for magnetic hysteresis to achieve optimal performance. In conclusion, the MRF-based device can be used in surgical robotic operations that require a high range of force measurements. Full article

In this paper, two different torque control approaches for PMA-driven (PMA = Pneumatic muscle actuator) revolute joints are presented and tested. In previous work controllers for PMA-driven robots are typically customized for the use on a specific robotic system. In contrast, the proposed controllers define a general control interface for every robot that is actuated by PMA-driven joints. It will be shown that controlling the torque of a PMA-driven joint enables the use of standard robotic motion control frameworks, because the torque represents the natural input of the robotic equation of motion. Therefore, both proposed torque control approaches are interconnecting PMAs and their challenging characteristics on the one hand and “conventional” motion control strategies for robots on the other hand. After a detailed discussion of two different torque control approaches, we show that a torque controller handles all characteristics and dynamics of a PMA-driven joint internally, which implies that only its bandwidth and its static torque characteristic must be taken into account for the design of the outer motion control loop. This feature simplifies the integration of PMA-driven joints in robotic systems enormously, as will be demonstrated by a design of a cascade-structured, flatness-based motion controller for an exemplary robot with one degree of freedom. Full article

In production processes, many adjustment tasks have to be carried out manually. In order to automate these activities, there is a need for cost and space efficient actuators that can provide comparatively high forces. This paper presents a novel actuator concept based on the phase change material paraffin wax. Furthermore, a numerical modelling strategy is introduced enabling the prediction of actuator properties. The model considers paraffin wax as a deformable body. The temperature-dependent volume expansion data of the paraffin wax is obtained experimentally to allow for a realistic description of the thermal-mechanical properties. The simulation is verified, using experimental data from actuators with varying paraffin wax volumes. With a maximum deviation of 6%, the simulations show a good agreement with the experiments. Full article

This research explores a new realm of soft robotic materials where the stiffness magnitude, directionality, and spatial resolution may be precisely controlled. These materials mimic biological systems where localized muscle contractions and adjustment of tissue stiffness enables meticulous, intelligent movement. Here we propose the use of a low-melting-point (LMP) metal lattice structure as a rigid frame using localized heating to allow compliance about selectable axes along the lattice. The resulting shape of the lattice is modeled using product of exponentials kinematics to describe the serial chain of tunably compliant axes; this model is found to match the behavior of the physical test piece consisting of a Field’s metal (FM) lattice encased in silicone rubber. This concept could enable highly maneuverable robotic structures with significantly improved dexterity. Full article

Rings with Hirth couplings are primarily used for the accurate positioning of axial-symmetric components in the machine tool industry and, generally, in mechanical components. It is also possible to use Hirth rings as connection tools. Specific industries with special milling and grinding machines are able to manufacture both tailor made and standard Hirth rings available on stock. Unfortunately, no international standard (for instance ISO, DIN or AGMA) is available for the production and the design of such components. In the best-case scenario, it is possible to find simplified design formulae in the catalogue of the suppliers. The aim of this work is to provide some accurate formulae and computational methods for design to provide better awareness on the limitations and the potential of this type of connection. The work consists of five parts: (i) a review of the base calculation derived mainly from the catalogues of manufacturers; (ii) an improved calculation based on a new analytical method including the friction phenomenon; (iii) an experimentation run for validating the method; (iv) a case study applied to a machine tool; and, (v) a closed form formulation to determine an upper threshold for friction, thus ensuring the Hirth coupling regular performance. Full article

This paper presents both an experimental and a numerical study concerning the mechanical response of a silicon microgripper with bidirectional electrostatic actuation to externally applied excitations. The experimental set-up is composed of a probe station equipped with mobile probes that apply contact forces. This part of the investigation aims to test the device’s mechanical resistance, its mobility capability and possible internal contacts during the system deformation. The second part of the paper is dedicated to the study of the free undamped vibrations of the microsystem. Finite Element Analysis (FEA) is carried out to evaluate the system vibration modes. The analysis of the modes are useful to predict possible mechanical interference among floating and anchored fingers of the actuating comb drives. Full article

The plasma synthetic jet actuator (PSJA), also named as sparkjet actuator, is a special type of zero-net mass flux actuator, driven thermodynamically by pulsed arc/spark discharge. Compared to widely investigated mechanical synthetic jet actuators driven by vibrating diaphragms or oscillating pistons, PSJAs exhibit the unique capability of producing high-velocity (>300 m/s) pulsed jets at high frequency (>5 kHz), thus tailored for high-Reynolds-number high-speed flow control in aerospace engineering. This paper reviews the development of PSJA in the last 15 years, covering the major achievements in the actuator working physics (i.e., characterization in quiescent air) as well as flow control applications (i.e., interaction with external crossflow). Based on the extensive non-dimensional laws obtained in characterization studies, it becomes feasible to design an actuator under several performance constraints, based on first-principles. The peak jet velocity produced by this type of actuator scales approximately with the cubic root of the non-dimensional energy deposition, and the scaling factor is determined by the electro-mechanical efficiency of the actuator (O(0.1%–1%)). To boost the electro-mechanical efficiency, the energy losses in the gas heating phase and thermodynamic cycle process should be minimized by careful design of the discharge circuitry as well as the actuator geometry. Moreover, the limit working frequency of the actuator is set by the Helmholtz natural resonance frequency of the actuator cavity, which can be tuned by the cavity volume, exit orifice area and exit nozzle length. In contrast to the fruitful characterization studies, the application studies of PSJAs have progressed relatively slower, not only due to the inherent difficulties of performing advanced numerical simulations/measurements in high-Reynolds-number high-speed flow, but also related to the complexity of designing a reliable discharge circuit that can feed multiple actuators at high repetition rate. Notwithstanding these limitations, results from existing investigations are already sufficient to demonstrate the authority of plasma synthetic jets in shock wave boundary layer interaction control, jet noise mitigation and airfoil trailing-edge flow separation. Full article

Electrically-driven direct current (DC) motors are the core component of conventional robots thanks to the ease of computer control and high torque for their size. However, DC motors are often manually attached and soldered into robotic assemblies, and they are not flexible. For soft robotics, researchers have looked to new, compliant materials that are compatible with 3-D printing or other automated assembly methods. In this work we use a computer-controlled embroidery machine to create flat motor windings in flexible fabrics. We model their electromagnetic fields and present them as linear actuators that move a permanent magnet attached to a cable. The fabrication method puts some constraints on the coil design, which are discussed. However, the planar nature of the embroidered sheets enables the designer to use laminar fabrication methods, such as stacking or layering into parts, during 3-D printing. The soft motor windings produced static holding forces of up to 0.25 N and could lift a 0.3 g mass several cm using direct drive. A 3-D printed mechanical amplifier with two stages was able to quadruple the lifting mass, reducing the travel by a factor of 4. Machine embroidery-installed cables and motor coils could lead to “bolts and nuts free” fabrication of thin, electrically-driven cable actuators. Full article

Dielectric elastomers are in a special class of electro-active polymers known for generating expansion in plane and contraction in thickness under voltage application. This paper advances the understanding of a planar contractile dielectric elastomer actuator (cDEA) that is distinct from conventional multi-layer cDEAs but generates comparable contractile strains. Its structure has a rod-constrained rhombus-shaped electrode region, which undergoes simultaneous in-plane contraction and extension during actuation depending on the configuration of the rod-constraining. It is demonstrated that when the planar cDEA is driven by high voltages, off-plane deformation (i.e., wrinkling) in the direction of contraction causes the rod-constrained electrode region to lose tension and extend in the lateral direction, resulting in a significant increase in contraction strain. It also demonstrates that the contraction strain can be increased further by having biased bi-axial pre-strains. By incorporating both effects, the new cDEA generates a maximum contraction strain of 13%, twice that reported previously for planar cDEAs. A modified planar cDEA, having an additional rigid frame to maintain the pre-strain in the lateral direction to contraction was also developed to demonstrate contractile force actuation. Finally, a stability study shows that the planar cDEA has a primary failure mode of electrical breakdown close to the corners of the rod-constrained electrode region. Having inactive regions around the active cell is essential for generating contraction and eliminating buckling of the planar cDEA in the lateral direction. Full article

The mechanical characterization of biological samples is a fundamental issue in biology and related fields, such as tissue and cell mechanics, regenerative medicine and diagnosis of diseases. In this paper, a novel approach for the identification of the stiffness and damping coefficients of biosamples is introduced. According to the proposed method, a MEMS-based microgripper in operational condition is used as a measurement tool. The mechanical model describing the dynamics of the gripper-sample system considers the pseudo-rigid body model for the microgripper, and the Kelvin–Voigt constitutive law of viscoelasticity for the sample. Then, two algorithms based on recursive least square (RLS) methods are implemented for the estimation of the mechanical coefficients, that are the forgetting factor based RLS and the normalised gradient based RLS algorithms. Numerical simulations are performed to verify the effectiveness of the proposed approach. Results confirm the feasibility of the method that enables the ability to perform simultaneously two tasks: sample manipulation and parameters identification. Full article

Advancements in software engineering have enabled the robotics industry to transition from the use of giant industrial robots to more friendly humanoid robots. Soft robotics is one of the key elements needed to advance the transition process by providing a safer way for robots to interact with the environment. Electroactive polymers (EAPs) are one of the best candidate materials for the next generation of soft robotic actuators and artificial muscles. Lightweight dielectric elastomer actuators (DEAs) provide optimal properties such as high elasticity, rapid response rates, mechanical robustness and compliance. However, for DEAs to become widely used as artificial muscles or soft actuators, there are current limitations, such as high actuation voltage requirements, control of actuation direction, and scaling, that need to be addressed. The authors’ approach to overcome the drawbacks of conventional DEAs is inspired by the natural skeletal muscles. Instead of fabricating a large DEA device, smaller sub-units can be fabricated and bundled together to form larger actuators, similar to the way myofibrils form myocytes in skeletal muscles. The current study presents a novel fabrication approach, utilizing soft lithography and other microfabrication techniques, to allow fabrication of multilayer stacked DEA structures, composed of hundreds of micro-sized DEA units. Full article

Biodegradable ionic polymer metallic composite (IPMC) electroactive polymers (EAPs) were fabricated using poly(ethylene oxide) (PEO) with various concentrations of lithium perchlorate. Nanocrystalline cellulose (NCC) rods created from a sulfuric acid hydrolysis process were added at various concentrations to increase the EAPs’ elastic modulus and improve their electromechanical properties. The electromechanical actuation was studied. PEONCC composites were created from combining a 35-mg/mL aqueous NCC suspension with an aqueous, PEO solution at varying vol.%. Due to an imparted space charge from the hydrolysis process, composites with an added 1.5 vol.% of NCC suspension exhibited an electromechanical tip displacement, strain, and elastic modulus that was 40.7%, 33.4% and 20.1% higher, respectively, than those for PEO IPMCs without NCC. This performance represented an increase of 300% in the energy density of these samples. However, the electromechanical response decreased when the NCC content was high. NCC without the space charge were also tested to verify the analysis. Additionally, the development of new relationships for modeling and evaluating the time-dependent instantaneous tip angular velocity and acceleration was discussed and applied to these IPMCs. Full article

Because they can achieve a high degree of molecular chain alignment in comparison with their bulk counterparts, the mechanical and thermal properties of polymer nanofibers are of great interest. However, due to their nanometer-scale size, it is difficult to manipulate, grip, and test these fibers. Here, we demonstrate simple repeatable methods to transfer as-drawn fibers to micrometer-scale test platforms where their properties can be directly measured. Issues encountered and methods to minimize measurement artifacts are also discussed. Full article

The evolution of microelectronic technologies is giving constant impulse to advanced micro-scaled systems which perform complex operations. In fact, the actual micro and nano Electro-Mechanical Systems (MEMS/NEMS) easily integrate information-gathering and decision-making electronics together with all sorts of sensors and actuators. Mechanical manipulation can be obtained through microactuators, taking advantage of magnetostrictive, thermal, piezoelectric or electrostatic forces. Electrostatic actuation, more precisely the comb-drive approach, is often employed due to its high versatility and low power consumption. Moreover, the device design and fabrication process flow can be simplified by compliant mechanisms, avoiding complex elements and unorthodox materials. A nano-scaled rotary comb drive is herein introduced and obtained using NEMS technology, with an innovative design which takes advantages of the compliant mechanism characteristics. A theoretical and numerical study is also introduced to inspect the electro-mechanical behavior of the device and to describe a new technological procedure for its fabrication. Full article

The hysteresis motor technology combined with the magnetic suspension makes bearingless hysteresis drives very appealing for high- and ultra-high-speed applications. Such systems exploit the magnetic behavior of the rotor material to achieve mechanical torque, but the hysteresis can significantly influence the magnetic suspension performance. The literature so far has focused mainly on the motor investigation. On the bearing side, the design and the performance assessment have been carried out by neglecting the hysteresis phenomenon of the rotor material. In those cases, the hysteresis of the rotor material is negligible and hence it slightly affects the force generation. In a wider perspective, this paper intends to investigate the force capability of electromagnetic actuators based on materials of large magnetic hysteresis behavior. To this purpose, the proposed numerical model, based on the finite element method, accounts for the magnetic hysteresis. The experimental results confirm the validity of the modeling approach, thus providing a useful tool for the design as well as the investigation of such systems. Full article

This paper investigates the basic deflection properties of polydimethylsiloxane (PDMS) membrane as an actuator component in a microfluidic device. Polydimethylsiloxane membrane is a widely used structure in various applications in microfluidics. Most of the applications using PDMS membrane as actuators are pumps, valves, microlenses, and cell stimulators. In these applications, PDMS membranes are deflected to function by applied pressure. However, based on our literature survey, correlations between thickness, applied air pressure, and the deflection properties of replaceable PDMS membrane have not been theoretically and experimentally investigated yet. In this paper, we first conducted a simulation to analyze the relationship between deflection of the replaceable PDMS membrane and applied pressure. Then we verified the deflection of the PDMS membrane in different experimental conditions. Finally, we demonstrated that the PDMS membrane functioned as a valve actuator in a cell-capturing device as one application. We expect this study would work as an important reference for research investigations that use PDMS membrane as an actuator. Full article

Synthetic jet actuators (SJA) are emerging in various engineering applications, from flow separation and noise control in aviation to thermal management of electronics. A SJA oscillates a flexible membrane inside a cavity connected to a nozzle producing vortices. A complex interaction between the cavity pressure field and the driving electronics can make it difficult to predict performance. A reduced-order model (ROM) has been developed to predict the performance of SJAs. This paper applies this model to a canonical configuration with applications in flow control and electronics cooling, consisting of a single SJA with a rectangular orifice, emanating perpendicular to the surface. The practical implementation of the ROM to estimate the relationship between cavity pressure and jet velocity, jet velocity and diaphragm deflection and applied driving voltage is explained in detail. Unsteady Reynolds-averaged Navier Stokes computational fluid dynamics (CFD) simulations are used to assess the reliability of the reduced-order model. The CFD model itself has been validated with experimental measurements. The effect of orifice aspect ratio on the ROM parameters has been discussed. Findings indicate that the ROM is capable of predicting the SJA performance for a wide range of operating conditions (in terms of frequency and amplitude). Full article

Axial magnetic bearing actuators often lack the bandwidth necessary to achieve the desired closed loop performance due to their nonlaminated construction. Since bandwidth can be directly related to actuator material and geometric properties, an opportunity exists to improve closed loop performance through the optimization of these properties. This prospect is exploited herein, both to demonstrate the improvements that can be obtained and to illustrate the relationship between various parameters and dynamic performance. For the latter, Pareto-optimal curves are generated exploring the influence that disk outer radius, peak force, axial gap, and magnetic permeability have upon actuator bandwidth. Full article

This paper presents the design and testing of a novel flexure-based compliant compound constant-force mechanism (CCFM). One uniqueness of the proposed mechanism lies in that it achieves both constant-force input and constant-force output, which is enabled by integrating two types of sub-mechanisms termed active and passive constant-force structures, respectively. Unlike conventional structures, the active constant-force structure allows the reduction on input force requirement and thus the enlargement of motion stroke provided that the maximum stress of the material is within allowable value. While the passive one offers a safe environmental interaction during the contact process. Analytical model of the proposed CCFM is derived which is verified by simulation study with finite element analysis (FEA). A prototype mechanism is fabricated by a 3D printer to demonstrate the performance of the proposed CCFM design. Experimental results reveal the effectiveness of the reported CCFM. Full article

The number of studies on microgrippers has increased consistently in the past decade, among them the numeric simulations and material characterization are quite common, while the metrological issues related to their performance testing are not well investigated yet. To add some contribution in this field, an image analysis-based method for microgrippers displacement measurement and testing is proposed here: images of a microgripper prototype supplied with different voltages are acquired by an optical system (i.e., a 3D optical profilometer) and processed through in-house software. With the aim to assess the quality of the results a systematic approach is proposed for determining and quantifying the main error sources and applied to the uncertainty estimation in angular displacement measurements of the microgripper comb-drives. A preliminary uncertainty evaluation of the in-house software is provided by a Monte Carlo Simulation and its contribution added to that of the other error sources, giving an estimation of the relative uncertainty up to 3.6% at 95% confidence level for voltages from 10 V to 28 V. Moreover, the measurements on the prototype device highlighted a stable behavior in the voltage range from 0 V to 28 V with a maximum rotation of 1.3° at 28 V, which is lower than in previous studies, likely due to differences in system configuration, model, and material. Anyway, the proposed approach is suitable also for different optical systems (i.e., trinocular microscopes). Full article

Estimations of magnetic flux linkages, either between the stationary windings of the stator for the direct torque control (DTC), or between the stationary windings and the rotor for the sensorless field-oriented control (FOC), are based on integration of corresponding voltages. Integration of voltages with offsets that come from improperly calibrated measurements as well as from transient states generally produces unwanted drifts in the resulting magnetic flux linkages, which when used within any type of control of sensorless electrical drives results in instability. This paper addresses that problem and proposes a simple self-contained solution based on orthogonal properties of waveforms of input voltages and resulting magnetic flux linkages in the frame of reference fixed to the geometry of the stator. The proposed solution requires only two periodic orthogonal input waveforms with a distinct common fundamental harmonic, which as such is independent of the type and parameters of the used machine. The idea of the proposed solution is presented analytically, its stability is proven by means of the quadratic Lyapunov theory, and its functionality is demonstrated by standalone simulations and experiments within the sensorless FOC of a permanent magnet synchronous machine (PMSM). Full article