actuators Article On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment Kirill Poletkin 1,2   Citation: Poletkin, K. On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment. Actuators 2021, 10, 256. https://doi.org/10.3390/act10100256 Academic Editor: Jose Luis Sanchez-Rojas Received: 29 July 2021 Accepted: 24 September 2021 Published: 29 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Institute of Microstructure Technology—Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany; kirill.poletkin@kit.edu or k.poletkin@innopolis.ru 2 Institute of Robotics and Computer Vision, Innopolis University, 1 Universitetskaya Street, 420500 Innopolis City, Russia Abstract: This work presents the results of the experimental and theoretical study of the static pull-in of tilting actuation executed by a hybrid levitation micro-actuator (HLMA) based on the combination of inductive levitation and electrostatic actuation. A semi-analytical model to study such a pull-in phenomenon is developed, for the first time, as a result of using the qualitative technique based on the Lagrangian approach to analyze inductive contactless suspensions and a recent progress in the calculation of mutual inductance and force between two circular filaments. The obtained non-linear model, accounting for two degrees of freedom of the actuator, allows us to predict accurately the static pull-in displacement and voltage. The results of modeling were verified experimentally and agree well with measurements. Keywords: micro-actuators; micro-systems; micro-manipulators; levitation; mutual inductance; electrostatic pull-in; eddy current 1. Introduction Electromagnetic levitation micro-actuators employing remote ponderomotive forces, in order to act on a target environment or simply compensate a gravity force for holding stably a micro-object at the equilibrium without mechanical attachment, have already found wide applications and demonstrated a new generation of micro-sensors and -actuators with increased operational capabilities and overcoming the domination of friction over inertial forces at the micro-scale. There are number of techniques, which provide the implementation of electromagnetic levitation into a micro-actuator system and can be classified according to the materials used and the sources of the force fields in two major branches: electric levitation micro-actuators (ELMA) and magnetic levitation micro-actuators (MLMA). In particular, ELMA were suc- cessfully used as linear transporters [1] and in micro-inertial sensors [2,3]. MLAM can be further split into inductive (ILMA), diamagnetic (DLMA), superconducting micro-actuators and hybrid levitation micro-actuators (HLMA) [4], which have found applications in mi- crobearings [5–7], micromirrors [8,9], micro-gyroscopes [10,11], micro-accelerometers [12], bistable switches [13], nanoforce sensors [14], microtransporters [15], microaccelerators [16], micromotors [17–19] and resonators [20]. A wide spectrum of physical principles have been utilized and successfully imple- mented by using different techniques for microfabrication. However, recently developed 3D microcoil technology [21] together with the integration of a polymer magnetic compos- ite material for flux concentration, allows announcing inductive levitation micro-actuator systems—firstly, as systems with established micro-fabrication process in comparison to the other levitation actuator systems and, secondly, as high-performance systems. As a result of this progress, our group demonstrated the inductive levitation actuator system with the record lowest current consumption [7] around tens of mA. This permits us to avoid using standard bulky high-frequency current amplifiers for exciting the ILMA and to replace Actuators 2021, 10, 256. https://doi.org/10.3390/act10100256 https://www.mdpi.com/journal/actuators