Vol.:(0123456789) 1 3 International Journal of Intelligent Robotics and Applications https://doi.org/10.1007/s41315-018-0075-5 REGULAR PAPER Pairwise controllability and motion primitives for micro‑rotors in a bounded Stokes fow Jake Buzhardt 1  · Vitaliy Fedonyuk 1  · Phanindra Tallapragada 1 Received: 31 August 2018 / Accepted: 1 November 2018 © Springer Nature Singapore Pte Ltd. 2018 Abstract Micro-robots that can propel themselves in a low Reynolds number fuid fow by converting their rotational motion into translation have begun attracting much attention due to their ease of fabrication. The dynamics and controllability of the motion of such microswimmers are investigated in this paper. The microswimmers under consideration here are spinning spheres (or rotors) whose dynamics are approximated by rotlets, a singularity solution of the Stokes equations. While sin- gularities of Stokes fows are commonly used as theoretical models for microswimmers and micro-robots, rotlet models of microswimmers have received less attention. While a rotlet alone cannot generate translation, a pair of rotlets can interact and execute net motion. Taking the control inputs to be the strengths of the micro rotors, the positions of a pair of rotors are not controllable in an unbounded planar fuid domain. However, in a bounded domain, which is often the case of practical interest, we show that the positions of the micro rotors are controllable. This is enabled by the interaction of the rotors with the boundaries of the domain. We show how control inputs can be constructed based on combinations of Lie brackets to move the rotors from one point to another in the domain. Another contribution of this paper is the creation of a framework for path planning and control of the motion of Stokes singularities that model the dynamics of microswimmers. This can be extended to microswimmers with other shapes moving in confned fuid domains with complex boundaries. Keywords Micro-robot · Controllability · Motion planning 1 Introduction The ability to precisely maneuver a micro-robot or a collec- tion of small robots in a small scale fuid environment has enormous implications for areas such as targeted drug deliv- ery Nelson et al. (2010); Zhang et al. (2012); Peyer et al. (2013); Gao et al. (2012), particle and cell manipulation Zhang et al. (2012); Peyer et al. (2013); Gao et al. (2012); Petit et al. (2012), cell identifcation and diagnostics Ding et al. (2016), and the fabrication of novel materials with tunable properties Snezhko and Aranson (2011). In recent years, magnetic microswimmers actuated by means of a time-varying external magnetic feld have attracted much attention. This is due to the feasibility of fabrication of such robots and the numerous potential applications for remotely controlled locomotion Dreyfus et al. (2005); Zhang et al. (2009); Ghosh et al. (2013); Cheang et al. (2014); Meshkati and Fu (2014); Chen et al. (2017). The fabrication of a magnetic swimmer with a special- ized geometry is necessary because of the reversibility of fow at low Reynolds numbers Purcell (1977). In general, the actuation of a magnetic micro-robot could be in the form of a force or a torque, but a net force that directly leads to a translation of the robot requires large gradient of the mag- netic feld. On the other hand, a torque can be efciently exerted on the magnetic body via a time-varying magnetic feld. Such a torque causes the body to rotate. For bodies possessing the necessary asymmetries, the interaction of the spinning body with the viscous fuid produces a propulsive force Happel and Brenner (1983); Kim and Karrila (2005). Inspired by the fagellar propulsion of micro-organisms, microrobot geometries with helicoidal symmetry are com- monly used to achieve this sort of locomotion Peyer et al. (2013); Petit et al. (2012). More recently, spherical micro- beads have been used to construct simpler chiral geometries possessing two perpendicular planes of symmetry, but not three Cheang et al. (2014); Meshkati and Fu (2014). These * Phanindra Tallapragada ptallap@clemson.edu 1 Department of Mechanical Engineering, Clemson University, Clemson, SC, USA