Net Locomotion Research Articles

Effect of internal damping on locomotion in frictional environments.

The gaits of undulating animals arise from a complex interaction of their central nervous system, muscle, connective tissue, bone, and environment. As a simplifying assumption, many previous studies have often assumed that sufficient internal force is available to produce observed kinematics, thus not focusing on quantifying the interconnection between muscle effort, body shape, and external reaction forces. This interplay, however, is critical to locomotion performance in crawling animals, especially when accompanied by body viscoelasticity. Moreover, in bioinspired robotic applications, the body's internal damping is indeed a parameter that the designer can tune. Still, the effect of internal damping is not well understood. This study explores how internal damping affects the locomotion performance of a crawler with a continuous, viscoelastic, nonlinear beam model. Crawler muscle actuation is modeled as a traveling wave of bending moment propagating posteriorly along the body. Consistent with the friction properties of the scales of snakes and limbless lizards, environmental forces are modeled using anisotropic Coulomb friction. It is found that by varying the crawler body's internal damping, the crawler's performance can be altered, and distinct gaits could be achieved, including changing the net locomotion direction from forward to back. We will discuss this forward and backward control and identify the optimal internal damping for peak crawling speed.

Helical Locomotion in Yield Stress Fluids.

We report three stages for locomotion of a helical swimmer in yield stress fluids. In the first stage, the swimmer must overcome the material's yield strain to generate rotational motion. However, exceeding the first threshold is not sufficient for locomotion. Only when the viscous forces are sufficiently strong to plastically deform the material to a finite distance away from the swimmer will net locomotion occur. Once locomotion is underway in the third stage, the yield stress retards swimming at small pitch angles. Conversely, at large pitch angles, yield stress dominates the flow by enhancing swimming speed. Flow visualizations reveal a highly localized flow near the swimmer in yield stress fluids.

Self-organized swimming with odd elasticity.

We theoretically investigate self-oscillating waves of an active material, which were recently introduced as a nonsymmetric part of the elastic moduli, termed odd elasticity. Using Purcell's three-link swimmer model, we reveal that an odd-elastic filament at low Reynolds number can swim in a self-organized manner and that the time-periodic dynamics are characterized by a stable limit cycle generated by elastohydrodynamic interactions. Also, we consider a noisy shape gait and derive a swimming formula for a general elastic material in the Stokes regime with its elasticity modulus being represented by a nonsymmetric matrix, demonstrating that the odd elasticity produces biased net locomotion from random noise.

Anisotropic magnetized tubular microrobots for bioinspired adaptive locomotion

Mimicry of motile microorganisms offers new concepts and strategies to design and manipulate microrobots. As a representative instance, Escherichia coli switches its motility between swimming and tumbling to adapt to surroundings. Inspired by this, we present a tubular microrobot behaving rolling and tumbling motion modes analogous to the motion of Escherichia coli, as a result of the anisotropic magnetized status with radially magnetized status at rolling and longitudinally magnetized one at tumbling. The motion behavior of microrobot as rolling or tumbling is determined by the frequency of rotating magnetic field. The threshold frequency for switching is closely related to the amplitude of magnetic field, viscosity of solution and size of microrobot. The net locomotion near substrate comes from the friction and pressure, and the motion switching is due to the magnetized statuses and discrepancy of drag force. The switch from rolling to tumbling endows the microrobot with adaptive locomotion to cross obstacles. Furthermore, microfluidic manipulation through rotational microrobot is achieved based on the local microvortex, which is generated from the hydrodynamic interaction between microrobot and substrate. This manipulation is applied in cargo delivery and release by altering surrounding microfluid via switchable motion modes. This work opens up new possibilities in improving moving ability and extending functions of microrobots through bioinspired motion behaviors.

Jet-driven viscous locomotion of confined thermoresponsive microgels

We consider the dynamics of micro-sized, asymmetrically coated thermoresponsive hydrogel ribbons (microgels) under periodic heating and cooling in the confined space between two planar surfaces. As the result of the temperature changes, the volume and, thus, the shape of the slender microgel change, which leads to repeated cycles of bending and elastic relaxation, and to net locomotion. Small devices designed for biomimetic locomotion need to exploit flows that are not symmetric in time (non-reciprocal) to escape the constraints of the scallop theorem and undergo net motion. Unlike other biological slender swimmers, the non-reciprocal bending of the gel centerline is not sufficient here to explain for the overall swimming motion. We show instead that the swimming of the gel results from the flux of water periodically emanating from (or entering) the gel itself due to its shrinking (or swelling). The associated flows induce viscous stresses that lead to a net propulsive force on the gel. We derive a theoretical model for this hypothesis of jet-driven propulsion, which leads to excellent agreement with our experiments.

Magnetically Actuated Artificial Microswimmers as Mobile Microparticle Manipulators

AbstractMicroscale swimming robots have been envisaged for many biomedical applications such as targeted drug delivery, where the microrobot will be expected to navigate in a fluid environment while carrying a payload. We show that such a payload does not have to be physically bound to the swimmer, but may be instead manipulated by the microrobot through hydrodynamic interaction. We consider a magnetically actuated artificial microswimmer, whose locomotion induces a disturbance velocity field in the fluid, which moves a cargo particle in its vicinity. The problem investigated in this paper is therefore one of coupled locomotion-manipulation of two bodies in a fluid. The swimmer is actuated by a uniform, rotating magnetic field of constant strength leading to net motion in the direction perpendicular to the plane of rotation if the frequency associated with the periodic magnetic field is above a critical frequency. Below this critical frequency, the swimmer tumbles in place without net locomotion. Controlled motion of the particle and swimmer is achieved by switching the planes of rotation of the magnetic field and the frequency of the magnetic field above and below the critical frequency. The results of this paper show that microswimmers can be utilized as mobile manipulators of microparticles in a fluid.

Efficient sliding locomotion with isotropic friction.

Snakes' bodies are covered in scales that make it easier to slide in some directions than in others. This frictional anisotropy allows for sliding locomotion with an undulatory gait, one of the most common for snakes. Isotropic friction is a simpler situation (that arises with snake robots, for example) but is less understood. In this work we regularize a model for sliding locomotion to allow for static friction. We then propose a robust iterative numerical method to study the efficiency of a wide range of motions under isotropic Coulomb friction. We find that simple undulatory motions give little net locomotion in the isotropic regime. We compute general time-harmonic motions of three-link bodies and find three local optima for efficiency. The top two involve static friction to some extent. We then propose a class of smooth body motions that have similarities to concertina locomotion (including the involvement of static friction) and can achieve optimal efficiency for both isotropic and anisotropic friction.

A Novel Serpentine Gait Generation Method for Snakelike Robots Based on Geometry Mechanics

In this paper, a novel serpentine gait generation method is proposed based on geometry mechanics. From the view of differential geometry, the configuration of snake-like robots can be described by a fiber bundle, in which the net locomotion corresponds to the fiber space and the gait corresponds to the bases space. The serpentine gait can be generated by calculating the connection of the fiber bundle. Compared with the traditional gait generation method, such as the curve-based method and the central pattern generator-based method, the proposed method does not assume that the serpentine gait is a standard sinusoidal signal. Models have been developed based on geometric mechanics, which build the relation between the net locomotion and the serpentine gaits analytically. Based on these models, the serpentine gaits for snakelike robots with passive wheels can be generated by planning the quasi-velocity and the serpentine gaits for underwater snakelike robots can be generated by analyzing the integral of the height function. The serpentine gaits generated by the proposed method directly couple with the locomotion of the snakelike robot, which enables the desired locomotion to be realized efficiently. To validate the proposed method, both the snakelike robot with passive wheels and the underwater snakelike robot are taken to perform the experiments. Additionally the recursive Newton–Euler method with the model of hydrodynamic forces is also taken to test the underwater serpentine gaits. Experiments and simulation have validated our proposed method.

Passive swimming in viscous oscillatory flows.

Fluid-based locomotion at low Reynolds number is subject to the constraints of Purcell's scallop theorem: reciprocal shape kinematics identical under a time-reversal symmetry cannot cause locomotion. In particular, a single degree-of-freedom scallop undergoing opening and closing motions cannot swim. Most strategies for symmetry breaking and locomotion rely on direct control of the swimmer's shape kinematics. Less is known about indirect control via actuation of the fluid medium. To address how such indirect actuation strategies can lead to locomotion, we analyze a Λ-shaped model system analogous to Purcell's scallop but able to deform passively in oscillatory flows. Neutrally buoyant scallops undergo no net locomotion. We show that dense, elastic scallops can exhibit passive locomotion in zero-mean oscillatory flows. We examine the efficiency of swimming parallel to the background flow and analyze the stability of these motions. We observe transitions from stable to unstable swimming, including ordered transitions from fluttering to chaoticlike motions and tumbling. Our results demonstrate that flow oscillations can be used to passively actuate and control the motion of microswimmers, which may be relevant to applications such as surgical robots and cell sorting and manipulation in microfluidic devices.

Three-dimensionally printed biological machines powered by skeletal muscle.

Combining biological components, such as cells and tissues, with soft robotics can enable the fabrication of biological machines with the ability to sense, process signals, and produce force. An intuitive demonstration of a biological machine is one that can produce motion in response to controllable external signaling. Whereas cardiac cell-driven biological actuators have been demonstrated, the requirements of these machines to respond to stimuli and exhibit controlled movement merit the use of skeletal muscle, the primary generator of actuation in animals, as a contractile power source. Here, we report the development of 3D printed hydrogel "bio-bots" with an asymmetric physical design and powered by the actuation of an engineered mammalian skeletal muscle strip to result in net locomotion of the bio-bot. Geometric design and material properties of the hydrogel bio-bots were optimized using stereolithographic 3D printing, and the effect of collagen I and fibrin extracellular matrix proteins and insulin-like growth factor 1 on the force production of engineered skeletal muscle was characterized. Electrical stimulation triggered contraction of cells in the muscle strip and net locomotion of the bio-bot with a maximum velocity of ∼ 156 μm s(-1), which is over 1.5 body lengths per min. Modeling and simulation were used to understand both the effect of different design parameters on the bio-bot and the mechanism of motion. This demonstration advances the goal of realizing forward-engineered integrated cellular machines and systems, which can have a myriad array of applications in drug screening, programmable tissue engineering, drug delivery, and biomimetic machine design.

Stokesian jellyfish: viscous locomotion of bilayer vesicles

Motivated by recent advances in vesicle engineering, we consider theoretically the locomotion of shape-changing bilayer vesicles at low Reynolds number. By modulating their volume and membrane composition, the vesicles can be made to change shape quasi-statically in thermal equilibrium. When the control parameters are tuned appropriately to yield periodic shape changes which are not time-reversible, the result is a net swimming motion over one cycle of shape deformation. For two classical vesicle models (spontaneous curvature and bilayer coupling), we determine numerically the sequence of vesicle shapes through an enthalpy minimization, as well as the fluid-body interactions by solving a boundary integral formulation of the Stokes equations. For both models, net locomotion can be obtained either by continuously modulating fore-aft asymmetric vesicle shapes, or by crossing a continuous shape-transition region and alternating between fore-aft asymmetric and fore-aft symmetric shapes. The obtained hydrodynamic efficiencies are similar to that of other low Reynolds number biological swimmers, and suggest that shape-changing vesicles might provide an alternative to flagella-based synthetic microswimmers.

Swimming due to transverse shape deformations

Balance laws are derived for the swimming of a deformable body due to prescribed shape changes and the effect of the wake vorticity. The underlying balances of momenta, though classical in nature, provide a unifying framework for the swimming of three-dimensional and planar bodies and they hold even in the presence of viscosity. The derived equations are consistent with Lighthill's reactive force theory for the swimming of slender bodies and, when neglecting vorticity, reduce to the model developed in Kanso et al. (J. Nonlinear Sci., vol. 15, 2005, p. 255) for swimming in potential flow. The locomotion of a deformable body is examined through two sets of examples: the first set studies the effect of cyclic shape deformations, both flapping and undulatory, on the locomotion in potential flow while the second examines the effect of the wake vorticity on the net locomotion. In the latter, the vortex wake is modelled using pairs of point vortices shed periodically from the tail of the deformable body.

Locomotion of Articulated Bodies in a Perfect Fluid

This paper is concerned with modeling the dynamics of N articulated solid bodies submerged in an ideal fluid. The model is used to analyze the locomotion of aquatic animals due to the coupling between their shape changes and the fluid dynamics in their environment. The equations of motion are obtained by making use of a two-stage reduction process which leads to significant mathematical and computational simplifications. The first reduction exploits particle relabeling symmetry: that is, the symmetry associated with the conservation of circulation for ideal, incompressible fluids. As a result, the equations of motion for the submerged solid bodies can be formulated without explicitly incorporating the fluid variables. This reduction by the fluid variables is a key difference with earlier methods, and it is appropriate since one is mainly interested in the location of the bodies, not the fluid particles. The second reduction is associated with the invariance of the dynamics under superimposed rigid motions. This invariance corresponds to the conservation of total momentum of the solid-fluid system. Due to this symmetry, the net locomotion of the solid system is realized as the sum of geometric and dynamic phases over the shape space consisting of allowable relative motions, or deformations, of the solids. In particular, reconstruction equations that govern the net locomotion at zero momentum, that is, the geometric phases, are obtained. As an illustrative example, a planar three-link mechanism is shown to propel and steer itself at zero momentum by periodically changing its shape. Two solutions are presented: one corresponds to a hydrodynamically decoupled mechanism and one is based on accurately computing the added inertias using a boundary element method. The hydrodynamically decoupled model produces smaller net motion than the more accurate model, indicating that it is important to consider the hydrodynamic interaction of the links.

Speed and Time-Energy Budget for Locomotion in Golden-Mantled Ground Squirrels

To determine the impact of locomotion on the total energy budget of a small mammal, we measured the frequency distribution of speeds and the time budget for locomotion in a population of Spermophilus saturatus, golden—mantled ground squirrels. Using these data together with laboratory—measured rates of energy expenditure for the walking and running gaits, we computed energy costs of locomotion in the field and compared these to field measurements of total daily energy expenditure. The animals moved in two distinct modes: walking, with mean speed 0.21 m/s (all values: <0.7 m/s), and running, with mean speed 3.63 m/s (range: 2—6 m/s). Of total daily time above ground mean: 7.37 h), 26.9% (°2 h) was spent walking and 3.6% (δ1 h) was running. The average daily movement distance (DMD) of 5 km/d consisted of 1.50 km walking and 3.53 km running. Walking required more time and net energy than running, yet walking resulted in less than half the distance covered by running. The net added energy cost of all moving around was 28.75 kJ/d. This net locomotion cost represents a 29% increase above basal metabolic rate (BMR) and comprises 13% of total daily energy expenditure (DEE). Our observation that this "ecological cost of transport" (ECT), being 13% of DEE, is so substanital suggests that energetic efficiencies should play an important role in determining speeds of locomotion. Our measurements suggest that the squirrels run when "commuting" around the home range rather than walking because it conserves both time and energy. We also suggest that the squirrels run at their maximum aerobic speed (3.6 m/s) rather than at their less physiologically demanding minimum speed for the running gait (2 m/s) to reduce exposure to predation. Our determination of ECT for golden—mantled ground squirrels is 15 times as great as an allometric prediction of ECT, due largely to a DMD that is 8 times as great as predicted. Daily movement distance should vary among species in relation to behavior. In S. saturatus the great magnitude of DMD is apparently a reflection of the dietary dependence of this species on hypogeous fungus. This food is distributed in patches and has low digestible energy and nitrogen content, necessitating that a large volume of the food be obtained, over a large area.

The organization of honey bee embryonic cells: I. Microtubules and amoeboid activity

In the development of the honey bee embryo, two lateral sheets of ectoderm overgrow a mid-ventral mesoderm plate, eventually meeting and fusing in the ventral midline. Cells from embryos at this stage (stage 6), isolated either singly or as multicellular monolayers, exhibit active pseudopod formation which, in the monolayers, is confined to the free edges of the marginal cells; this pseudopodal activity can sometimes be correlated with net locomotion in a direction normal to the free edge of the monolayer. It is therefore proposed that amoeboid movement by individual cells can account for the movement of lateral ectoderm in the intact embryo. Stage 6 embryonic cells also exhibit typical saltatory movements of cytoplasmic organelles. The following relationships have been observed: (1) Saltatory movements without pseudopod formation; (2) active pseudopod formation with minimal saltatory movements; (3) neither type of movement; (4) both movements together. In the latter case, when pseudopod formation has an obvious polarity (e.g., in the marginal cells of a monolayer), the saltatory movements may show a similar polarity, but they occur at a much more rapid rate. The evidence suggests that pseudopod formation and movement of cytoplasmic granules involve separate mechanisms in these cells. Whole-mount electron microscopy of stage 6 cells reveals that the cytoplasmic organelles are connected to one another and to the nuclear envelope by an extensive system of fine fibers. These fibers vary around 185 Å in diameter; they run in straight courses for distances of several microns, and in general, they conform with the organization of “microtubules” (recently described in other types of cell after thin-sectioning). Saltatory movement of cytoplasmic organelles in honey bee embryonic cells would appear to be due to longitudinal contraction of the extranuclear fibers or “microtubules.” On the basis of observations in this report, a possible mechanism of amoeboid movement is proposed, which depends on assembly of microtubules at the anterior end of the cell, accompanied by disassembly at the posterior end. In this scheme, net pseudopodal locomotion would depend on assembly of microtubules, while movement of cytoplasmic granules would depend on contraction of microtubules.