Nonlinear dynamics and bifurcations of a planar undulating magnetic microswimmer.

Swimming microorganisms such as flagellated bacteria and sperm cells have fascinating locomotion capabilities. Inspired by their natural motion, there is an ongoing effort to develop artificial robotic nanoswimmers for potential in-body biomedical applications. A leading method for actuation of nanoswimmers is by applying a time-varying external magnetic field. Such systems have rich and nonlinear dynamics that call for simple fundamental models. A previous work studied forward motion of a simple two-link model with a passive elastic joint, assuming small-amplitude planar oscillations of the magnetic field about a constant direction. In this work, we found that there exists a faster, backward motion of the swimmer with very rich dynamics. By relaxing the small-amplitude assumption, we analyze the multiplicity of periodic solutions, as well as their bifurcations, symmetry breaking, and stability transitions. We have also found that the net displacement and/or mean swimming speed are maximized for optimal choices of various parameters. Asymptotic calculations are performed for the bifurcation condition and the swimmer's mean speed. The results may enable significantly improving the design aspects of magnetically actuated robotic microswimmers.

Fluid-Driven Traveling Waves in Soft Robots.

Many marine creatures, gastropods, and earthworms generate continuous traveling waves in their bodies for locomotion within marine environments, complex surfaces, and inside narrow gaps. In this work, we study theoretically and experimentally the use of embedded pneumatic networks as a mechanism to mimic nature and generate bidirectional traveling waves in soft robots. We apply long-wave approximation to theoretically calculate the required distribution of pneumatic network and inlet pressure oscillations needed to create desired moving wave patterns. We then fabricate soft robots with internal pneumatic network geometry based on these analytical results. The experimental results agree well with our model and demonstrate the propagation of moving waves in soft robots, along with locomotion capabilities. The presented results allow fabricating soft robots capable of continuous moving waves using the common approach of embedded pneumatic networks and requiring only two input controls.

Helical Klinotactic Locomotion of Two-Link Nanoswimmers with Dual-Function Drug-Loaded Soft Polysaccharide Hinges.

Inspired by the movement of bacteria and other microorganisms, researchers have developed artificial helical micro- and nanorobots that can perform corkscrew locomotion or helical path swimming under external energy actuation. In this paper, for the first time the locomotion of nonhelical multifunctional nanorobots that can swim in helical klinotactic trajectories, similarly to rod-shaped bacteria, under rotating magnetic fields is investigated. These nanorobots consist of a rigid ferromagnetic nickel head connected to a rhodium tail by a flexible hydrogel-based hollow hinge composed of chemically responsive chitosan and alginate multilayers. This design allows nanoswimmers switching between different dynamic behaviors-from in-plane tumbling to helical klinotactic swimming-by varying the rotating magnetic field frequency and strength. It also adds a rich spectrum of swimming capabilities that can be adjusted by varying the type of applied magnetic fields and/or frequencies. A theoretical model is developed to analyze the propulsion mechanisms and predict the swimming behavior at distinct rotating magnetic frequencies. The model shows good agreement with the experimental results. Additionally, the biomedical capabilities of the nanoswimmers as drug delivery platforms are demonstrated. Unlike previous designs constitute metallic segments, the proposed nanoswimmers can encapsulate drugs into their hollow hinge and successfully release them to cells.

Single-Input Control of Multiple Fluid-Driven Elastic Actuators via Interaction Between Bistability and Viscosity.

A leading concept in soft robotics actuation, as well as in microfluidics applications such as valves in lab-on-a-chip devices, is applying pressurized flow in cavities embedded within elastic bodies. Generating complex deformation patterns typically requires control of several inputs, which greatly complicates the system's operation. In this study, we present a novel method for single-input control of a serial chain of bistable elastic chambers connected by thin tubes. Controlling a single flow rate at the chain's inlet, we induce an irreversible sequence of transitions that can reach any desired state combination of all bistable elements. Mathematical formulation and analysis of the system's dynamics reveal that these transitions are enabled, thanks to bistability combined with pressure lag induced by viscous resistance. The results are demonstrated through numerical simulations combined with experiments for chains of up to five chambers, using water-diluted glycerol as the injected fluid. The proposed technique has a promising potential for development of sophisticated soft actuators with minimalistic control.

Leveraging Viscous Peeling to Create and Activate Soft Actuators and Microfluidic Devices.

The research fields of microfluidics and soft robotics both involve complex small-scale internal channel networks, embedded within a solid structure. This study examines leveraging viscous peeling as a mechanism to create and activate soft actuators and microchannel networks, including complex elements such as valves, without the need for fabrication of structures with micron-scale internal cavities. We consider configurations composed of an internal slender structure embedded within another elastic solid. Pressurized viscous fluid is introduced into the interface between the two solids, thus peeling the two elastic structures and creating internal cavities. Since the gap between the solids is determined by the externally applied pressure, the characteristic size of the fluid network may vary with time and be much smaller than the resolution of the fabrication method. This study presents a model for the highly nonlinear elastic-viscous dynamics governing the flow and deformation of such configurations. Fabrication and experimental demonstrations of micron-scale valves and channel networks created from millimeter scale structures are presented, as well as the transient dynamics of viscous peeling-based soft actuators. The experimental data are compared with the suggested model, showing very good agreement.

Programmable Locomotion Mechanisms of Nanowires with Semihard Magnetic Properties Near a Surface Boundary.

We report on the simplest magnetic nanowire-based surface walker that is able to change its propulsion mechanism near a surface boundary as a function of the applied rotating magnetic field frequency. The nanowires are made of CoPt alloy with semihard magnetic properties synthesized by means of template-assisted galvanostatic electrodeposition. The semihard magnetic behavior of the nanowires allows for programming their alignment with an applied magnetic field as they can retain their magnetization direction after premagnetizing them. By engineering the macroscopic magnetization, the nanowires' speed and locomotion mechanism are set to tumbling, precession, or rolling depending on the frequency of an applied rotating magnetic field. Also, we present a mathematical analysis that predicts the translational speed of the nanowire near the surface, showing a very good agreement with experimental results. Interestingly, the maximal speed is obtained at an optimal frequency (∼10 Hz), which is far below the theoretical step-out frequency (∼345 Hz). Finally, vortices are found by tracking polystyrene microbeads, trapped around the CoPt nanowire, when they are propelled by precession and rolling motion.

Nonlinear Parametric Excitation Effect Induces Stability Transitions in Swimming Direction of Flexible Superparamagnetic Microswimmers.

Microscopic artificial swimmers have recently become highly attractive due to their promising potential for biomedical microrobotic applications. Previous pioneering work has demonstrated the motion of a robotic microswimmer with a flexible chain of superparamagnetic beads, which is actuated by applying an oscillating external magnetic field. Interestingly, they have shown that the microswimmer's orientation undergoes a 90°-transition when the magnetic field's oscillation amplitude is increased above a critical value. This unexpected transition can cause severe problems in steering and manipulation of flexible magnetic microrobotic swimmers. Thus, theoretical understanding and analysis of the physical origins of this effect are of crucial importance. In this work, we investigate this transition both theoretically and experimentally by using numerical simulations and presenting a novel flexible microswimmer with an anisotropic superparamagnetic head. We prove that this effect depends on both frequency and amplitude of the oscillating magnetic field, and demonstrate existence of an optimal amplitude achieving maximal swimming speed. Asymptotic analysis of a minimal two-link model reveals that the changes in the swimmer's direction represent stability transitions, which are induced by a nonlinear parametric excitation.

Optimizing an undulating magnetic microswimmer for cargo towing.

One of the promising capabilities of magnetic microswimmers is towing a cargo, which can be used for targeted drug delivery or performing tissue biopsy. A key question is what should be the optimal size ratio between the cargo and the swimmer's flexible tail. This question is addressed here for the simplest theoretical model of a magnetic microswimmer undergoing planar undulations-a spherical load connected by a torsion spring to a rigid slender link. The swimmer's dynamic is formulated and leading-order expressions for its motion are obtained explicitly under small-amplitude approximation. Optimal combinations of magnetic actuation frequency, torsion stiffness, and tail length for maximizing displacement, average speed, or energetic efficiency are obtained. The theoretical results are compared with reported experiments in several types of cargo-towing magnetic microswimmers.

Undulatory Locomotion of Magnetic Multilink Nanoswimmers.

Micro- and nanorobots operating in low Reynolds number fluid environments require specialized swimming strategies for efficient locomotion. Prior research has focused on designs mimicking the rotary corkscrew motion of bacterial flagella or the planar beating motion of eukaryotic flagella. These biologically inspired designs are typically of uniform construction along their flagellar axis. This work demonstrates for the first time planar undulations of composite multilink nanowire-based chains (diameter 200 nm) induced by a planar-oscillating magnetic field. Those chains comprise an elastic eukaryote-like polypyrrole tail and rigid magnetic nickel links connected by flexible polymer bilayer hinges. The multilink design exhibits a high swimming efficiency. Furthermore, the manufacturing process enables tuning the geometrical and material properties to specific applications.

Simple model of a planar undulating magnetic microswimmer.

One of the most efficient actuation methods of robotic microswimmers for biomedical applications is by applying time-varying external magnetic fields. In order to improve the design of the swimmer and optimize its performance, one needs to develop simple theoretical models that enable explicit analysis of the swimmer's dynamics. This paper studies the dynamics of a simple microswimmer model with two magnetized links connected by an elastic joint, which undergoes planar undulations induced by an oscillating magnetic field. The nonlinear dynamics of the microswimmer is formulated by assuming Stokes flow and using resistive force theory to calculate the viscous drag forces. Key effects that enable the swimmer to overcome the scallop theorem and generate net propulsion are identified, including violation of front-back symmetry. Assuming small oscillation amplitude, approximate solution is derived by using perturbation expansion, and leading-order expressions for the swimmer's displacement per cycle X and average speed V are obtained. Optimal actuation frequencies that maximize X or V are found for given swimmer's parameters. An ultimate optimal choice of swimmer's parameters and actuation frequency is found, for which the average swimming speed V attains a global maximum. Finally, the theoretical predictions of optimal performance values are validated by comparison to reported experimental results of magnetic microswimmers.

Asymmetry and stability of shape kinematics in microswimmers' motion.

Many swimming microorganisms governed by low Reynolds number hydrodynamics utilize flagellar undulations for self-propulsion. Most of the existing theoretical models assume that the shape kinematics is directly controlled, while in reality, eukaryotes actuate internal bending moments along their flagellum. Under this control, the shape is dynamically evolving and periodic gaits may become unstable. This Letter addresses the problem by revisiting Purcell's three-link swimmer model where joint torques are controlled, and the geometric symmetries underlying the dynamics of the swimmer are analyzed. It is found that one has to break the front-back symmetry of the swimmer's structure and/or actuation profile in order to induce stable shape kinematics. The results may explain why most of the flagellated eukaryotes swim with their head forward.

Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall.

This paper studies a simple two-dimensional model of a swimmer at low-Reynolds-number near a no-slip wall by utilizing methods of complex analysis. The swimmer is propelled by purely tangential surface deformations and is modeled by moving point singularities. The nonlinear dynamics of the swimmer is formulated explicitly, and its motion near the wall is fully characterized. The results show qualitative agreement with predictions of three-dimensional models and with motion experiments on a robotic swimmer. The success and simplicity of the model suggest that it will provide a simple way to study the dynamics of low-Reynolds-number swimmers in more complicated geometries.

Dynamics and stability of Purcell's three-link microswimmer near a wall.

The motion of swimming microorganisms is strongly influenced by the presence of boundaries. Attraction of bacteria and sperm cells to surfaces is a well-known phenomenon which has been observed in experiments and confirmed by numerical simulations. This effect is studied in this work from a viewpoint of dynamical systems theory by analyzing a swimmer model which is a variant of the classical Purcell's three-link swimmer near an infinite plane wall. The underlying geometric structure of the swimmer's dynamics and its relation to stability are elucidated. It is found that a swimmer which breaks its fore-aft symmetry has a preferred swimming direction in which its motion is passively stable and converges to a fixed separation distance from the wall.

Dynamics and stability of a class of low Reynolds number swimmers near a wall.

We study the dynamic stability of low Reynolds number swimming near a plane wall from a control-theoretic viewpoint. We consider a special class of swimmers having a constant shape, focus on steady motion parallel to the wall, and derive conditions under which it is passively stable without sensing or feedback. We study the geometric structure of the swimming equation and highlight the relation between stability and reversing symmetry of the dynamical system. Finally, our numerical simulations reveal the existence of stable periodic motion. The results have implications for design of miniature robotic swimmers, as well as for explaining the attraction of micro-organisms to surfaces.