Undulatory Gait Research Articles

Rigid flocks, undulatory gaits, and chiral foldamers in a chemically active polymer

Active matter systems—such as a collection of active colloidal particles—operate far from equilibrium with complex inter-particle interactions that govern their collective dynamics. Predicting the collective dynamics of such systems may aid the design of self-shaping structures comprised of active colloidal units with a prescribed dynamical function. Here, using simulations and theory, we study the collective dynamics of a chain consisting of active Brownian particles with internal interactions via trail-mediated chemicals, connected by harmonic springs in two dimensions to obtain design principles for active colloidal molecules. We show that two-dimensional confinement and chemo-repulsive interactions between the freely-jointed particles lead to an emergent rigidity of the chain in the steady-state dynamics. In the chemo-attractive regime, the chain collapses into crystals that abruptly halt their motion. Further, in a chain consisting of a binary mixture of monomers, we show that non-reciprocal chemical affinities between distinct species give rise to novel phenomena, such as chiral molecules with tunable dynamics, sustained undulatory gaits and reversal of the direction of motion. Our results suggest a novel interpretation of the role of trail-mediated interactions, in addition to providing active self-assembly principles arising due to non-reciprocal interactions.

Anisotropic swimming and reorientation of an undulatory microswimmer in liquid-crystalline polymers

Microorganisms can efficiently navigate in anisotropic complex fluids, but the precise swimming mechanisms remain largely unexplored. Their dynamics is determined by the interplay between multiple effects, including the fluid's orientation order, swimmer's undulatory gait and the finite length. Here, we extend the numerical study of the two-dimensional undulatory motions of a flexible swimmer in lyotropic liquid-crystalline polymers (LCPs) by Lin et al. (J. Fluid Mech., vol. 921, 2021, p. A25) to the scenario of arbitrary swimming directions with respect to the nematic director. The swimmer is modelled as a nearly inextensible yet flexible fibre with imposed travelling-wave-like actuation. We investigate the orientation-dependent swimming behaviours in nematic LCPs for an infinitely long sheet (i.e. Taylor's swimming sheet model) and finite-length swimmers. We demonstrate that the swimmer must be sufficiently stiff to produce undulatory deformations to gain net motions. Moreover, a motile finite-length swimmer can reorient itself to swim parallel with the nematic director, due to a net body torque arising from the asymmetric distribution of the polymer force along the body.

Lateral contact yields longitudinal cohesion in active undulatory systems.

Many animals and robots move using undulatory motion of their bodies. When the bodies are in close proximity undulatory motion can lead to novel collective behavior such as gait synchronization, spatial reconfiguration, and clustering. Here we study the role of contact interactions between model undulatory swimmers: three-link robots in experiment and multilink swimmers in simulation. The undulatory gait of each swimmer is generated through a time-dependent sinusoidal-like waveform which has a fixed phase offset, ϕ. By varying the phase relationship between neighboring swimmers we seek to study how contact forces and planar configurations are governed by the phase difference between neighboring swimmers. We find that undulatory actuation in close proximity drives neighboring swimmers into planar equilibrium configurations that depend on the actuation phase difference. We propose a model for stable planar configurations of nearest-neighbor undulatory swimmers which we call the gait compatibility condition, which is the set of planar and phase configurations in which no collisions occur. Robotic experiments with two, three, and four swimmers exhibit good agreement with the compatibility model. To study the contact forces and the time-averaged equilibrium between undulatory systems we perform simulations. To probe the interaction potential between undulatory swimmers we apply a small force to each swimmer longitudinally to separate them from the compatible configuration and we measure their steady-state displacement. These studies reveal that undulatory swimmers in close proximity exhibit attractive longitudinal interaction forces that drive the swimmers from incompatible to compatible configurations. This system of undulatory swimmers provides new insight into active-matter systems which move through body undulation. In addition to the importance of velocity and orientation coherence in active-matter swarms, we demonstrate that undulatory phase coherence is also important for generating stable, cohesive group configurations.

Undulatory gait planning method of multi-legged robot with passive-spine

In this paper, we propose a new gait planning method for a multi-legged robot which has only 1 degree-of-freedom in each leg and has a passive body joint between two body segments. We firstly introduce the Finite State Machine (FSM) to the undulatory gait planning method of the 2n-legged robot. Then, the undulatory gait sequence for straight line motion is achieved by undulations motion. The idea that legged locomotion is achievable by less actuation of 2n-legged robot as well as the gait planning methods are verified finally by simulations and experiments.

Collective Synchronization of Undulatory Movement through Contact

Physical contact among simple robots with undulatory gaits leads to rich collective dynamics including synchronization, thus showing how some populations in nature sync their movements when in close proximity.

Neuromechanical Mechanisms of Gait Adaptation in C. elegans: Relative Roles of Neural and Mechanical Coupling

Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode \textit{C. elegans} is an ideal model organism for studying locomotion in an integrated neuromechanical setting because its neural circuit has a well-characterized modular structure and its undulatory forward swimming gait adapts to the surrounding fluid with a shorter wavelength in higher viscosity environments. This adaptive behavior emerges from the neural modules interacting through a combination of mechanical forces, neuronal coupling, and sensory feedback mechanisms. However, the relative contributions of these coupling modes to gait adaptation are not understood. The model consists of repeated neuromechanical modules that are coupled through the mechanics of the body, short-range proprioception, and gap-junctions. The model captures the experimentally observed gait adaptation over a wide range of mechanical parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces. The modularity of the model allows the use of the theory of weakly coupled oscillators to identify the relative roles of body mechanics, gap-junctional coupling, and proprioceptive coupling in coordinating the undulatory gait. The analysis shows that the wavelength of body undulations is set by the relative strengths of these three coupling forms. In a low-viscosity fluid environment, the competition between gap-junctions and proprioception produces a long wavelength undulation, which is only achieved in the model with sufficiently strong gap-junctional coupling.The experimentally observed decrease in wavelength in response to increasing fluid viscosity is the result of an increase in the relative strength of mechanical coupling, which promotes a short wavelength.

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.

Exploiting natural dynamics for gait generation in undulatory locomotion

ABSTRACTRobotic vehicles inspired by animal locomotion operate via periodic body movements. The pattern of body oscillation (gait) can be mimicked from animals, but understanding the principles underlying gait generation would allow for broad, flexible applications beyond nature's design. We hypothesise that travelling-wave oscillations observed in undulatory locomotion can be characterised as a natural oscillation of the locomotion dynamics and propose a formal definition of the natural gait for locomotion systems. We identify the essential dynamics and define the mode shape of natural oscillation by the free response of an idealised system. We then use body-environment resonance to define the amplitude and frequency. Explicit formulas for the natural gait are derived to provide insight into the mechanisms underlying undulatory locomotion. Examples of a swimming leech and a fictitious swimmer illustrate how undulatory gaits similar to those observed can be produced as the natural gait and modulated to achieve different swim speeds.

Robust Neuro-Optimal Control of Underactuated Snake Robots With Experience Replay

In this paper, the problem of path following for underactuated snake robots is investigated by using approximate dynamic programming and neural networks (NNs). The lateral undulatory gait of a snake robot is stabilized in a virtual holonomic constraint manifold through a partial feedback linearizing control law. Based on a dynamic compensator and Line-of-Sight guidance law, the path-following problem is transformed to a regulation problem of a nonlinear system with uncertainties. Subsequently, it is solved by an infinite horizon optimal control scheme using a single critic NN. A novel fluctuating learning algorithm is derived to approximate the associated cost function online and relax the initial stabilizing control requirement. The approximate optimal control input is derived by solving a modified Hamilton-Jacobi-Bellman equation. The conventional persistence of excitation condition is relaxed by using experience replay technique. The proposed control scheme ensures that all states of the snake robot are uniformly ultimate bounded which is analyzed by using the Lyapunov approach, and the tracking error asymptotically converges to a residual set. Simulation results are presented to verify the effectiveness of the proposed method.

Maneuvering Control of Planar Snake Robots Using Virtual Holonomic Constraints

This paper investigates the problem of maneuvering control for planar snake robots. The control objective is to make the center of mass of the snake robot converge to a desired path and traverse the path with a desired velocity. The proposed feedback control strategy enforces virtual constraints encoding a lateral undulatory gait, parametrized by the states of dynamic compensators used to regulate the orientation and forward speed of the snake robot.

Scaling macroscopic aquatic locomotion

Inertial aquatic swimmers that use undulatory gaits range in length L from a few millimetres to 30 metres, across a wide array of biological taxa. Using elementary hydrodynamic arguments, we uncover a unifying mechanistic principle characterizing their locomotion by deriving a scaling relation that links swimming speed U to body kinematics (tail beat amplitude A and frequency ω) and fluid properties (kinematic viscosity ν). This principle can be simply couched as the power law Re ∼ Swα, where Re = UL/ν ≫ 1 and Sw = ω AL/ν, with α = 4/3 for laminar flows, and α = 1 for turbulent flows. Existing data from over 1,000 measurements on fish, amphibians, larvae, reptiles, mammals and birds, as well as direct numerical simulations are consistent with our scaling. We interpret our results as the consequence of the convergence of aquatic gaits to the performance limits imposed by hydrodynamics.

Improved Lighthill fish swimming model for bio-inspired robots: Modeling, computational aspects and experimental comparisons

The best known analytical model of swimming was originally developed by Lighthill and is known as the large amplitude elongated body theory (LAEBT). Recently, this theory has been improved and adapted to robotics through a series of studies ranging from hydrodynamic modeling to mobile multibody system dynamics. This article marks a further step towards the Lighthill theory. The LAEBT is applied to one of the best bio-inspired swimming robots yet built: the AmphiBot III, a modular anguilliform swimming robot. To that end, we apply a Newton–Euler modeling approach and focus our attention on the model of hydrodynamic forces. This model is numerically integrated in real time by using an extension of the Newton–Euler recursive forward dynamics algorithm for manipulators to a robot without a fixed base. Simulations and experiments are compared on undulatory gaits and turning maneuvers for a wide range of parameters. The discrepancies between modeling and reality do not exceed 16% for the swimming speed, while requiring only the one-time calibration of a few hydrodynamic parameters. Since the model can be numerically integrated in real time, it has significantly superior accuracy compared with computational speed ratio, and is, to the best of our knowledge, one of the most accurate models that can be used in real-time. It should provide an interesting tool for the design and control of swimming robots. The approach is presented in a self contained manner, with the concern to help the reader not familiar with fluid dynamics to get insight both into the physics of swimming and the mathematical tools that can help its modeling.

Influence of surface traction on soft robot undulation

A pneumatically-driven robot traverses surfaces with different traction by adopting an undulatory mode of locomotion. The robot is composed of soft elastomer (elastic modulus ~100 kPa), an inextensible but flexible neutral plane, and embedded pneumatic channels. In contrast to conventional robots and wheeled vehicles, the robot deforms elastically to make ground contact over a relatively large area, where interfacial tractions have a unique role in controlling both the speed and direction of locomotion. Here, we demonstrate that for the same undulatory gait, the robot will either move forward or backward depending on the ground composition. Building on mathematical principles of elasticity and friction, we introduce a theoretical model that identifies the tribological properties that determine the direction of locomotion. Though overlooked in the past, this tribology-controlled phenomenon represents a central feature of undulation on smooth, soft, and slippery surfaces. These insights provide a starting point for identifying locomotion strategies that allow soft robots, like their natural invertebrate counterparts, to navigate a broad range of surfaces and terrains.

Biomechanical analysis of gait adaptation in the nematode Caenorhabditis elegans

To navigate different environments, an animal must be able to adapt its locomotory gait to its physical surroundings. The nematode Caenorhabditis elegans, between swimming in water and crawling on surfaces, adapts its locomotory gait to surroundings that impose approximately 10,000-fold differences in mechanical resistance. Here we investigate this feat by studying the undulatory movements of C. elegans in Newtonian fluids spanning nearly five orders of magnitude in viscosity. In these fluids, the worm undulatory gait varies continuously with changes in external load: As load increases, both wavelength and frequency of undulation decrease. We also quantify the internal viscoelastic properties of the worm's body and their role in locomotory dynamics. We incorporate muscle activity, internal load, and external load into a biomechanical model of locomotion and show that (i) muscle power is nearly constant across changes in locomotory gait, and (ii) the onset of gait adaptation occurs as external load becomes comparable to internal load. During the swimming gait, which is evoked by small external loads, muscle power is primarily devoted to bending the worm's elastic body. During the crawling gait, evoked by large external loads, comparable muscle power is used to drive the external load and the elastic body. Our results suggest that C. elegans locomotory gait continuously adapts to external mechanical load in order to maintain propulsive thrust.

Corrections to the theory and the optimal line in the swimming diagram of Taylor (1952).

The analysis of undulatory swimming gaits requires knowledge of the fluid forces acting on the animal body during swimming. In his classical 1952 paper, Taylor analysed this problem using a 'resistive-force' theory. The theory was used to characterize the undulatory gaits that result in the smallest energy dissipation to the fluid for a given swim velocity. The optimal gaits thus found were compared with data recorded from movies of a snake and a leech swimming. This report identifies and corrects a mathematical error in Taylor's paper, showing that his theory applies even better to animals of circular cross section.

LEARNING FROM GYMNOTIFORM SWIMMERS — DESIGN AND IMPLEMENTATION OF ROBOTIC KNIFEFISH NKF-II

Gymnotiform swimmer uses single undulating anal fin below the rigid body as the propulsor. This special morphological structure is generally thought to have advantages in reducing the friction drag in undulatory movements. The finned body form and the undulatory swimming gait give the inspiration to the design of underwater robot. We have developed a robotic fish (Nanyang KnifeFish, NKF-I), which learns from a Gymnotiform swimmer: Black Ghost Knifefish. A new version of biomimetic robotic fish (NKF-II) is presented in this paper. To mimic the actual flexible fin of real fish, an optimized fin-like propulsion mechanism is modelled with a series of connecting linkages for enhancing its propulsive performance. Workspace of the proposed mechanism is studied and optimized. Swimming motion function is also established for the planning of the undulatory gait movement. The manoeuvrability and buoyancy control has also been achieved by the integration of mechatronics design of a buoyancy module with the undulating fin. The design philosophy of modularity has also been incorporated into the design of the fish. Hence, the modularity of the prototypes not only provides versatility for the robot system and also facilitates in the maintenance and future development of fish with undulating fins/body, as new modules can be added to replace the existing modules easily. By reconfiguring the components of NKF-II, other version of robotic fish with undulating fins/body is derived. This evolution of the undulating fins is also discussed.

The effects of acute temperature change on swimming performance in bluegill sunfishLepomis macrochirus

Many fish change gait within their aerobically supported range of swimming speeds. The effects of acute temperature change on this type of locomotor behavior are poorly understood. Bluegill sunfish swim in the labriform mode at low speeds and switch to undulatory swimming as their swimming speed increases. Maximum aerobic swimming speed (U(max)), labriform-undulatory gait transition speed (U(trans)) and the relationships between fin beat frequency and speed were measured at 14, 18, 22, 26 and 30 degrees C in bluegill acclimated to 22 degrees C. At temperatures below the acclimation temperature (T(a)), U(max), U(trans) and the caudal and pectoral fin beat frequencies at these speeds were reduced relative to the acclimation level. At temperatures above T(a) there was no change in these variables relative to the acclimation level. Supplementation of oxygen levels at 30 degrees C had no effect on swimming performance. The mechanical power output of the abductor superficialis, a pectoral fin abductor muscle, was measured in vitro at the same temperatures used for the swimming experiments. At and below T(a), maximal power output was produced at a cycle frequency approximately matching the in vivo pectoral fin beat frequency. At temperatures above T(a) muscle power output and cycle frequency could be increased above the in vivo levels at U(trans). Our data suggest that the factors triggering the labriform-undulatory gait transition change with temperature. Muscle mechanical performance limited labriform swimming speed at T(a) and below, but other mechanical or energetic factors limited labriform swimming speed at temperatures above T(a).

Polychaete-Like Undulatory Robotic Locomotion in Unstructured Substrates

A biological paradigm of versatile locomotion and effective motion control is provided by the polychaete annelid worms, whose motion adapts to a large variety of unstructured environmental conditions (sand, mud, sediment, water, etc.), and could thus be of interest to replicate by robotic analogs. Their locomotion is characterized by the combination of a unique form of tail-to-head body undulations (opposite to snakes and eels), with the rowing-like action of numerous lateral appendages distributed along their long segmented body. Focusing on the former aspect of polychaete locomotion, computational models of crawling and swimming by such tail-to-head body undulations have been developed in this paper. These are based on the Lagrangian dynamics of the system and on resistive models of its interaction with the environment, and are used for simulation studies demonstrating the generation of undulatory gaits. Several biomimetic robotic prototypes have been developed, whose undulatory actuation achieves propulsion on sand and other granular unstructured environments. Extensive experimental studies demonstrate the feasibility of robot propulsion by tail-to-head body undulations in such environments, as well as the agreement of its qualitative and quantitative characteristics to the predictions of the corresponding computational models.

Eurypterid trackways from the Table Mountain Group (Ordovician) of South Africa

Eurypterid trackways in paralic facies of the Graafwater and Peninsula Formations, lower Table Mountain Group (Ordovician) of the Western Cape Province, South Africa, are reviewed. The ichnotaxonomy of trackways preserved on the ‘Brandenburg slab’ from the Graafwater area is reassessed, with the recognition of two distinct forms: (1) Palmichnium capensis (Anderson, 1975), consisting of a series of four tracks, many accessory imprints and (usually) an intermittent bilobed medial impression; and (2) Petalichnus brandenburgensis isp. nov., a much narrower trackway with nine to ten tracks per series and no medial impression. Palmichnium capensis is interpreted as the locomotory trace of an onychopterellan eurypterid (Chelicerata: Eurypterida). Petalichnus brandenbergensis is provisionally ascribed to the Trilobita. The likely walking techniques employed by onychopterellan eurypterids are assessed from the revised ichnological evidence, comparisons with extant taxa and modelling procedures; subaqueous locomotion was achieved by a series of short hops across the substrate and on land these animals were slow-moving and cumbersome, employing an ungainly, undulatory gait.

Contribution of gular pumping to lung ventilation in monitor lizards.

A controversial hypothesis has proposed that lizards are subject to a speed-dependent axial constraint that prevents effective lung ventilation during moderate- and high-speed locomotion. This hypothesis has been challenged by results demonstrating that monitor lizards (genus Varanus) experience no axial constraint. Evidence presented here shows that, during locomotion, varanids use a positive pressure gular pump to assist lung ventilation. Disabling the gular pump reveals that the axial constraint is present in varanids but it is masked by gular pumping under normal conditions. These findings support the prediction that the axial constraint may be found in other tetrapods that breathe by costal aspiration and locomote with a lateral undulatory gait.