Mathematical Models of Locomotion: Legged Crawling, Snake-like Motility, and Flagellar Swimming

Three different models of motile systems are studied: a vibrating legged robot, a snake-like locomotor, and two kinds of agellar microswimmers. The vibrating robot crawls by modulating the friction with the substrate. This also leads to the ability to switch direction of motion by varying the vibration frequency. A detailed account of this phenomenon is given through a fully analytical treatment of the model. The analysis delivers formulas for the average velocity of the robot and for the frequency at which the direction switch takes place. A quantitative description of the mechanism for the friction modulation underlying the motility of the robot is also provided. Snake-like locomotion is studied through a system consisting of a planar, internally actuated, elastic rod. The rod is constrained to slide longitudinally without slipping laterally. This setting is inspired by undulatory locomotion of snakes, where frictional resistance is typically larger in the lateral direction than in the longitudinal one. The presence of constraints leads to non-standard boundary conditions, that lead to the possibility to close and solve uniquely the equations of motion. Explicit formulas are derived, which highlight the connection between observed trajectories, internal actuation, and forces exchanged with the environment. The two swimmer models (one actuated externally and the other internally) provide an example of propulsion at low Reynolds number resulting from the periodical beating of a passive elastic filament. Motions produced by generic periodic actuations are studied within the regime of small compliance of the filament. The analysis shows that variations in the velocity of beating can generate different swimming trajectories. Motion control through modulations of the actuation velocity is discusse

Modelling biological and bio-inspired swimming at microscopic scales: Recent results and perspectives

Abstract Some recent results on biological and bio-inspired swimming at microscopic scales are reviewed, and used to identify promising research directions for the future. We focus on broad conceptual principles such as looping in the space of shapes, loss of controllability of systems in which shape is only partially controlled, and steering by modulating the actuation rate. Moreover, we discuss propulsion mechanism that are most common for unicellular swimmers, such as flagellar and ciliary beating, and we examine amoeboid motion and flagellar propulsion in Euglena. The Helix Theorem, a universal law characterising orbits traced by ciliated and flagellated unicellular swimmers propelled by the periodic beating of cilia and flagella, is proved and discussed as a principle of self-assembly for helical structures

Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes

The flagellar swimming of euglenids, which are propelled by a single anterior flagellum, is characterized by a generalized helical motion. The 3D nature of this swimming motion, which lacks some of the symmetries enjoyed by more common model systems, and the complex flagellar beating shapes that power it make its quantitative description challenging. In this work, we provide a quantitative, 3D, highly resolved reconstruction of the swimming trajectories and flagellar shapes of specimens of Euglena gracilis. We achieved this task by using high-speed 2D image recordings taken with a conventional inverted microscope combined with a precise characterization of the helical motion of the cell body to lift the 2D data to 3D trajectories. The propulsion mechanism is discussed. Our results constitute a basis for future biophysical research on a relatively unexplored type of eukaryotic flagellar movement

Motion planning and motility maps for flagellar microswimmers

We study two microswimmers consisting of a spherical rigid head and a passive elastic tail. In the first one the tail is clamped to the head, and the system oscillates under the action of an external torque. In the second one, head and tail are connected by a joint allowing the angle between them to vary periodically, as a result of an oscillating internal torque. Previous studies on these models were restricted to sinusoidal actuations, showing that the swimmers can propel while moving on average along a straight line, in the direction given by the symmetry axis around which beating takes place. We extend these results to motions produced by generic (non-sinusoidal) periodic actuations within the regime of small compliance of the tail. We find that modulation in the velocity of actuation can provide a mechanism to select different directions of motion. With velocity-modulated inputs, the externally actuated swimmer can translate laterally with respect to the symmetry axis of beating, while the internally actuated one is able to move along curved trajectories. The governing equations are analysed with an asymptotic perturbation scheme, providing explicit formulas, whose results are expressed through motility maps. Asymptotic approximations are further validated by numerical simulations

Nonreciprocal oscillations of polyelectrolyte gel filaments subject to a steady and uniform electric field

Soft actuators typically require time-varying or spatially modulated control to be operationally effective. The scope of the present paper is to show, theoretically and experimentally, that a natural way to overcome this limitation is to exploit mechanical instabilities. We report experiments on active filaments of polyelectrolyte (PE) gels subject to a steady and uniform electric field. A large enough intensity of the field initiates the motion of the active filaments, leading to periodic oscillations. We develop a mathematical model based on morphoelasticity theory for PE gel filaments beating in a viscous fluid, and carry out the stability analysis of the governing equations to show the emergence of flutter and divergence instabilities for suitable values of the system’s parameters. We confirm the results of the stability analysis with numerical simulations for the nonlinear equations of motion to show that such instabilities may lead to periodic self-sustained oscillations, in agreement with experiments. The key mechanism that underlies such behaviour is the capability of the filament to undergo active shape changes depending on its local orientation relative to the external electric field, in striking similarity with gravitropism, the mechanism that drives shape changes in plants via differential growth induced by gravity. Interestingly, the resulting oscillations are nonreciprocal in nature, and hence able to generate thrust and directed flow at low Reynolds number. The exploitation of mechanical instabilities in soft actuators represents a new avenue for the advancement in engineering design in fields such as micro-robotics and micro-fluidics

The biomechanical role of extra-axonemal structures in shaping the flagellar beat of Euglena gracilis

We propose and discuss a model for flagellar mechanics in Euglena gracilis. We show that the peculiar non-planar shapes of its beating flagellum, dubbed 'spinning lasso', arise from the mechanical interactions between two of its inner components, namely, the axoneme and the paraflagellar rod. The spontaneous shape of the axoneme and the resting shape of the paraflagellar rod are incompatible. Thus, the complex non-planar configurations of the coupled system emerge as the energetically optimal compromise between the two antagonistic components. The model is able to reproduce the experimentally observed flagellar beats and the characteristic geometric signature of spinning lasso, namely, traveling waves of torsion with alternating sign along the length of the flagellum

Author response: The biomechanical role of extra-axonemal structures in shaping the flagellar beat of Euglena gracilis

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Morphable structures from unicellular organisms with active, shape-shifting envelopes: variations on a theme by Gauss

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We discuss some recent results on biological and bio-inspired morphing, and use them to identify promising research directions for the future. In particular, we consider issues related to morphing at microscopic scales inspired by unicellular organisms. We focus on broad conceptual principles and, in particular, on morphing approaches based on the use of Gauss’ theorema egregium (Gaussian morphing). We highlight some connections with biological cell envelopes containing filaments and motors, and discuss ideas for the implementation of Gaussian morphing in surfaces actuated by active shearing or stretching.Peer Reviewe

The Inversion of Motion of Bristle Bots: Analytical and Experimental Analysis

Bristle bots are vibration- driven robots actuated by the motion of an internal oscillating mass. Vibrations are translated into directed locomotion due to the alternating friction resistance between robots' bristles and the substrate during oscillations. Bristle bots are, in general, unidirectional locomotion systems. In this paper we demonstrate that motion direction of vertically vibrated bristle systems can be controlled by tuning the frequency of their oscillatory actuation. We report theoretical and experimental results obtained by studying an equivalent system, consisting of an inactive robot placed on a vertically vibrating substrate