Cyclic Shape Changes Research Articles

Optimal gaits for inertia-dominated swimmers with passive elastic joints.

Animals and some robots locomote by interacting with the environment through cyclic shape changes, or gaits. Many animals make significant use of passive dynamics with flexible tails or pendulum action to reduce the effort required to execute these gaits. Although geometric tools have been developed to study optimal passive gaits for swimmers in drag-dominated physics regimes, they have not yet been used to study larger-scale swimmers whose physics are dominated by inertial effects. In this paper, we leverage previous work in the geometric mechanics field to examine passive-elastic inertial swimmers and show that geometric mechanics can be used to rapidly determine many classes of optimal gaits for such systems. We also discuss how considering swimmer metabolic costs in addition to the mechanical costs of driving actuation is useful for discussing swimmer efficiency. In particular, we focus on two models of active-passive swimming inertial systems: the perfect-fluid three-link swimmer, and a swimmer with a passively flexible tail.

The Geometry of Optimal Gaits for Inertia-Dominated Kinematic Systems

Isolated mechanical systems—e.g., those floating in space, in free-fall, or on a frictionless surface—are able to achieve net rotation by cyclically changing their shape, even if they have no net angular momentum. Similarly, swimmers immersed in “perfect fluids” are able to use cyclic shape changes to both translate and rotate even if the swimmer-fluid system has no net linear or angular momentum. Finally, systems fully constrained by direct nonholonomic constraints (e.g., passive wheels) can push against these constraints to move through the world. Previous work has demonstrated that the displacement induced by these shape changes corresponds to the amount of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constraint curvature</i> that the gaits enclose. Properly assessing or optimizing the utility of a gait also requires considering the time or resources required to execute it: A gait that produces a small displacement per cycle, but that can be executed in a short time, may produce a faster average velocity than a gait that produces more displacement, but takes longer to complete a cycle at the same instantaneous effort. In this paper, we consider gaits under two instantaneous measures of effort. For each of these costs, we demonstrate that fixing the average instantaneous cost to a unit value allows us to transform the effort costs into time-to-execute costs for any given gait cycle. We then illustrate how the interaction between the constraint curvature and these costs leads to characteristic geometries for optimal cycles, in which the gait trajectories resemble elastic hoops distended from within by internal pressures.

Are the dynamic changes of the aortic root determinant for thrombosis or leaflet degeneration after transcatheter aortic valve replacement?

The role of the aortic root is to convert the accumulated elastic energy during systole into kinetic flow energy during diastole, in order to improve blood distribution in the coronary tree. Therefore, the sinuses of Valsalva of the aortic root are not predisposed to accept any bulky material, especially in case of uncrushed solid calcific agglomerates. This concept underlines the differences between surgical aortic valve replacement, in which decalcification is a main part of the procedure, and transcatheter aortic valve replacement (TAVR). Cyclic changes in shape and size of the aortic root influence blood flow in the Valsalva sinuses. Recent papers have been investigating the dynamic changes of the aortic root and whether those differences might be correlated with clinical effects, and this paper aims to summarize part of this flourishing literature. Post-TAVR aortic root remodeling, dynamic flow and TAVR complications might have a fluidodynamic background, and clinically observed side effects such as thrombosis or leaflet degeneration should be further investigated in basic researches. Also, aortic root changes could impact valve type and size selection, affecting the decision of over-sizing or under-sizing in order to prevent valve embolization or coronary ostia obstruction.

Gait modeling and optimization for the perturbed Stokes regime

Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that for swimming at the "Stokesian" (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection" model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime" where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer linear in shape change rate. We derive this model using results from singular perturbation theory, and the theory of noncompact normally hyperbolic invariant manifolds (NHIMs). Using the theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. This extends our previous work which assumed kinematic "connection" models. To compare the old and new algorithms, we analyze simulated swimmers over a range of inertia to damping ratios. Our new class of models performs well on the Stokesian regime, and over several orders of magnitude outside it into the perturbed Stokes regime, where it gives significantly improved prediction accuracy compared to previous work. In addition to algorithmic improvements, we thereby present a new class of models that is of independent interest. Their application to data-driven modeling improves our ability to study the optimality of animal gaits, and our ability to use hardware-in-the-loop optimization to produce gaits for robots.

Geometrically optimal gaits: a data-driven approach

The study of optimal motion of animals or robots often involves seeking optimality over a space of cyclic shape changes, or gaits, specified using a large number of parameters. We show a data-driven method for computing the gradient of a cost functional with respect to a large number of gait parameters by employing geometric properties of the dynamics to efficiently construct a local model of the system, and then using this model to rapidly compute the gradients. Our modeling step specifically applies to systems governed by connection-like models from geometric mechanics, which encompass a number of high-friction regimes. We demonstrate using our method for optimizing gaits under noisy, experiment-like conditions by simulating planar multi-segment serpent-like swimmers in a low Reynolds number (viscous friction) environment. Our optimization results recover known results for 3-segment swimmers with a 66 dimensional gait parameterization, and extend to optimizing the motion of a 9 segment swimmer with a 264 dimensional gait space, using only 30 simulation trials of 30 gait cycles each. The data-driven geometric gait optimization approach we present is designed to operate on noisy, stochastically perturbed dynamics—as noisy and variable as experimental data—and efficiently optimize a large number of parameters. We believe this approach has the potential to significantly advance our ability to optimize robot gaits with hardware in the loop and to study the optimality of animal gaits with respect to hypothesized cost functions.

Load response of shape-changing microswimmers scales with their swimming efficiency.

External forces acting on a microswimmer can feed back on its self-propulsion mechanism. We discuss this load response for a generic microswimmer that swims by cyclic shape changes. We show that the change in cycle frequency is proportional to the Lighthill efficiency of self-propulsion. As a specific example, we consider Najafi's three-sphere swimmer. The force-velocity relation of a microswimmer implies a correction for a formal superposition principle for active and passive motion.

Getting in shape and swimming: the role of cortical forces and membrane heterogeneity in eukaryotic cells.

Recent research has shown that motile cells can adapt their mode of propulsion to the mechanical properties of the environment in which they find themselves-crawling in some environments while swimming in others. The latter can involve movement by blebbing or other cyclic shape changes, and both highly-simplified and more realistic models of these modes have been studied previously. Herein we study swimming that is driven by membrane tension gradients that arise from flows in the actin cortex underlying the membrane, and does not involve imposed cyclic shape changes. Such gradients can lead to a number of different characteristic cell shapes, and our first objective is to understand how different distributions of membrane tension influence the shape of cells in an inviscid quiescent fluid. We then analyze the effects of spatial variation in other membrane properties, and how they interact with tension gradients to determine the shape. We also study the effect of fluid-cell interactions and show how tension leads to cell movement, how the balance between tension gradients and a variable bending modulus determine the shape and direction of movement, and how the efficiency of movement depends on the properties of the fluid and the distribution of tension and bending modulus in the membrane.

Shape change/memory actuators based on shape memory materials

Major techniques currently available to implement typical shape memory materials (such as shape memory alloys and polymers) for three basic types of shape switching actuations—one-time shape memory actuation, cyclic shape memory actuation and cyclic shape change actuation—were explored in detail. Typical actuators corresponding to these three types of actuations are systematically discussed. Possible combination of different types of shape memory materials/shape change materials and/or different stimuli for actuators with novel functions, which are not easily achievable, in particular at small scale, using conventional approaches, is presented to reveal the great potential of shape memory material based actuators in engineering applications. We provide a road map to guide engineers in the process of evaluation and selection of the right type of mechanism to meet the requirement(s) of a particular application.

Chemotactic Amoeboid-Like Shape Change of a Vesicle under a pH Gradient

Abstract A vesicle that exhibited the extrusion and contraction of a pseudopod-like structure under a pH gradient was studied. The vesicle is composed of oleate and oleic acid, and transforms from a double-spherical shape to a disk-like shape after numerous cyclic shape changes, including the reversal and rotation of the double-spherical vesicle. A pseudopod-like structure is extruded from the vesicle toward a pH gradient, created by NaOH diffusion, and is then contracted. This reversible motion is repeated many times. Notably, NaCl diffusion produces a monotonic and irreversible extrusion. The difference between the two systems could be explained by the cation permeability across the membrane. A mathematical model that accounted for this characteristic reproduced the experimental results semi-quantitatively. Vesicles exhibiting amoeboid-like behaviors, which may be useful in the design of amphiphilic molecular assemblies with biomimetic characteristics, have rarely been reported and are poorly understood. The present study may provide significant insight into the design of such biomimetic vesicles.

Enhanced Controllability of Low Reynolds Number Swimmers in the Presence of a Wall

Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers which is the regime of interest for micro-organisms and micro-robots. We focus on self-propelled stokesian robots composed of assemblies of balls and we prove that the presence of a wall has an effect on their motility. To rest on what has been done in Alouges et al. (Discrete Contin. Dyn. Syst., Ser. B 18(5):1189---1215, 2013) for such systems swimming on R 3, we demonstrate that a controllable swimmer remains controllable in a half space whereas the reachable set of a non fully controllable one is increased by the presence of a wall.

Transient oscillation of shape and membrane conductivity changes by field pulse-induced electroporation in nano-sized phospholipid vesicles

The results of electrooptical and conductometrical measurements on unilamellar lipid vesicles (of mean radius a = 90 nm), filled with 0.2 M NaCl solution, suspended in 0.33 M sucrose solution of 0.2 mM NaCl, and exposed to a stepwise decaying electric field (time constant τE = 154 μs) in the range 10 ≤ E0 (kV cm(-1)) ≤ 90, are analyzed in terms of cyclic changes in vesicle shape and vesicle membrane conductivity. The two peaks in the dichroitic turbidity relaxations reflect two cycles of rapid membrane electroporation and slower resealing of long-lived electropores. The field-induced changes reflect structural transitions between closed (C) and porated (P) membrane states, qualified by pores of type P1 and of type P2, respectively. The transient change in the membrane conductivity and the transient shape oscillation are based on changes in the pore density of the (larger) P2-pores along a hysteresis cycle. The P2-pore formation leads to transient net ion flows across the P2-pores and to transient changes in the membrane field. The kinetic data are numerically processed in terms of coupled structural relaxation modes. Using the torus-hole pore model, the mean inner pore radii are estimated to be r1 = 0.38 (±0.05) nm and r2 = 1.7 (±0.1) nm, respectively. The observation of a transient oscillation of membrane electroporation and of shape changes in a longer lasting external field pulse is suggestive of potential resonance enhancement, for instance, of electro-uptake by, and of electro-release of biogenic molecules from, biological cells in trains of long-lasting low-intensity voltage pulses.

Geometric motion planning: The local connection, Stokes’ theorem, and the importance of coordinate choice

The locomotion of articulated mechanical systems is often complex and unintuitive, even when considered with the aid of reduction principles from geometric mechanics. In this paper, we present two tools for gaining insights into the underlying principles of locomotion: connection vector fields and connection height functions . Connection vector fields illustrate the geometric structure of the relationship between internal shape changes and the system body velocities they produce. Connection height functions measure the curvature of their respective vector fields and capture the net displacement over any cyclic shape change, or gait , allowing for the intuitive selection of gaits to produce desired displacements. Height function approaches have been previously attempted, but such techniques have been severely limited by their basis in a rotating body frame, and have only been useful for calculating planar rotations and infinitesimal translations. We circumvent this limitation by introducing a notion of optimal coordinates defining a body frame that rotates very little in response to shape changes, while still meeting the requirements of the geometric mechanics theory on which the vector fields and height functions are based. In these optimal coordinates, the height functions provide close approximations of the net displacement resulting from a broad selection of possible gaits.

Effective broaching in mills of different design

Helical broaching mills used in the production of large-diameter pipe with variable dimensions of the blank (sleeve) are considered. The influence of the type of mill on the cyclic shape change is discussed, taking account of the supply angle. Recommendations are made regarding the readjustment of the mills and optimal broaching conditions for different purposes.

The Role of the Scar/WAVE Complex in the Mechanics of Cell Migration

Cell motility is integral to a wide spectrum of biological phenomena. It requires the spatiotemporal coordination of underlying biochemical processes, resulting in cyclic shape changes associated with mechanical events (the motility cycle). A major driving force of cell migration is the dendritic polymerization of actin at the leading edge, regulated through the nucleation activity of the Arp2/3 complex, activated by the Scar/WAVE complex. Our aim is to understand the effect of the different components of the Scar/WAVE complex in the mechanics and in particular the motility cycle of migrating cells. For this purpose, we acquired time-lapse recordings of cell shape and traction forces of Dictyostelium cells migrating on deformable substrates. We compared results for wild-type cells and cells lacking the Scar/WAVE complex proteins PIR121 (Sra-1/CYFIP/GEX-2) (pirA-) and SCAR (scrA-). We find that mutantcells move slower than wild-type, while maintaining the overall characteristics of the mechanical interaction with the substrate, attaching at front and back and contracting inwards. Although the distribution of applied forces is unchanged, their magnitude is lower than in wild-type for scrA- cells and higher for pirA- cells. This correlates with the F-actin content of the different cell lines corroborating a role for F-actin in determining the level of the traction stresses. In pirA- cells regularity of the motility cycle (quasiperiodic repetition of shape changes and strain energy deposited) seems to be reduced compared to wild-type. This suggests that proper regulation of the Scar/WAVE complex and its role in F-actin turnover is essential for amoeboid motility.

Crack growth of 10M Ni–Mn–Ga material in cyclic mechanical loading

The 10M martensitic Ni–Mn–Ga single crystal materials are usually applied in the magneto-mechanical actuators. Therefore, it is important to know the possible effect of the long-term cyclic shape changes on their structure and behavior. This can be evaluated with the mechanical fatigue testing. In the present study, the single crystal 10M Ni–Mn–Ga samples of different compositions were applied to strain-controlled uniaxial mechanical cycling in the multivariant state at ambient temperature. The experiments revealed distinctive changes of the twin variant structure, especially in the mobile twin area, density of twin boundaries, and in the tendency for fatigue crack growth. Characterization of the crack surface showed that the cracks in the microscale grow in a step-wise manner on specific crystallographic planes, i.e, twin boundary planes, but that the macroscopic crack does not occur only along crystallographic directions.

In vivo imaging of the cyclic changes in cross‐sectional shape of the ventricular segment of pulsating embryonic chick hearts at stages 14 to 17: A contribution to the understanding of the ontogenesis of cardiac pumping function

The cardiac cycle-related deformations of tubular embryonic hearts were traditionally described as concentric narrowing and widening of a tube of circular cross-section. Using optical coherence tomography (OCT), we have recently shown that, during the cardiac cycle, only the myocardial tube undergoes concentric narrowing and widening while the endocardial tube undergoes eccentric narrowing and widening, having an elliptic cross-section at end-diastole and a slit-shaped cross-section at end-systole. Due to technical limitations, these analyses were confined to early stages of ventricular development (chick embryos, stages 10-13). Using a modified OCT-system, we now document, for the first time, the cyclic changes in cross-sectional shape of beating embryonic ventricles at stages 14 to 17. We show that during these stages (1) a large area of diminished cardiac jelly appears at the outer curvature of the ventricular region associated with formation of endocardial pouches; (2) the ventricular endocardial lumen acquires a bell-shaped cross-section at end-diastole and becomes compressed like a fireplace bellows during systole; (3) the contracting portions of the embryonic ventricles display stretching along its baso-apical axis at end-systole. The functional significance of our data is discussed with respect to early cardiac pumping function.

Swimming in an inviscid fluid

We present a set of equations governing the motion of a body due to prescribed shape changes in an inviscid, planar fluid with nonzero vorticity. The derived equations, when neglecting vorticity, reduce to the model developed in Kanso et al. (J Nonlinear Sci 15:255–289, 2005) for swimming in potential flow, and are also consistent with the models developed in Borisov et al. (J Math Phys 48:1–9, 2007), Kanso and Oskouei (J Fluid Mech 800:77–94, 2008), Shasikanth et al. (Phys Fluids 14(3):1214–1227, 2002) for a rigid body interacting dynamically with point vortices. The effects of cyclic shape changes and the presence of vorticity on the locomotion of a submerged body are discussed through examples.

Optimal Strokes for Low Reynolds Number Swimmers: An Example

Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers. This is the regime of interest for micro-organisms and micro- or nano-robots. We focus in this paper on a simple yet representative example: the three-sphere swimmer of Najafi and Golestanian (Phys. Rev. E, 69, 062901–062904, 2004). For this system, we show how to cast the problem of swimming in the language of control theory, prove global controllability (which implies that the three-sphere swimmer can indeed swim), and propose a numerical algorithm to compute optimal strokes (which turn out to be suitably defined sub-Riemannian geodesics).

A shape memory-based multifunctional structural actuator panel

A multifunctional panel concept is presented in which a lightweight, structurally efficient sandwich panel is able to undergo a reversible change in shape upon application of a localized thermal stimulus while supporting significant load. The shape change is effected by shape memory alloy face sheet elements, which exploit a “one-way” shape memory effect only. Unlike other related designs, no external or bias forces are required to complete the full cycle of shape change. The reversible shape change is accomplished by a core design which forces an inactive face sheet to martensitically deform in tension when the opposing face sheet undergoes a length change. By alternately heating one face sheet and then the other, the sandwich panel is able to perform fully reversible cyclic shape changes. The performance of the sandwich panel, in terms of required thermal power, actuation frequency, peak load bearing capacity, stiffness, and weight, can be optimized by proper selection of face sheet material and its thickness, the overall core thickness, core member thickness and length, and the design of the joint connecting core members and face sheet.

Swimming in Newtonian space–time: Orbital changes by cyclic changes in body shape

I investigate whether, in the context of Newtonian physics, a satellite can change its orbit by cyclic shape changes without any external forces other than gravity. This question has been discussed in the context of general relativity, and Wisdom has estimated that a swimmer near the surface of the Earth using meter-sized deformations can achieve a displacement of ∼10−23 m. I show that much larger displacements, ∼10−7 m, can occur for a Newtonian swimmer. Such a swimmer can change its orbit by rotating about its center of mass and trading chemical, solar, or other energy for orbital energy, without the need for any propellant.