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One of the oldest yet most common modalities of locomotion known among limbless animals is undulatory, also recognized for its stability compared to legged locomotion. Multiple forms of active mechanisms, e.g. active gait control, and passive mechanisms, e.g. body morphology and material properties, have adapted to different environments. The current research explores the passive role of body stiffness and internal losses in meeting terrain requirements. Furthermore, it addresses the influence of the environment on the resultant gait and how the interplay between various environments and body properties can lead to different speeds. We modelled undulatory locomotion in a dry friction environment where frictional anisotropy determines propulsion. We found that the body stiffness, the moment of inertia, the dry frictional coefficient ratio between normal and tangential frictional constants, and the internal damping of the body play an essential role in optimizing speed and animal adaptability to external conditions. Furthermore, we demonstrate that various known gaits like swimming, crawling and polychaete-like locomotion are achieved as a result of the interaction between body and environment parameters. Moreover, we validated the model by retrieving a corn snake's speed using data from the literature. This study demonstrates that the dependence between morphology, body material properties and environment can be exploited to design long-segmented robots to perform in specialized situations.

Sprawling posture animals with their bendable spine, such as salamanders, and geckos, can perform agile and versatile locomotion including walking, swimming, and climbing. Therefore, several roboticists have used them as templates for robot designs to investigate and generate efficient locomotion. Typically, walking and/or swimming abilities are realized by salamander-inspired robots with a bendable body, whereas climbing ability is achieved on gecko-inspired robots with an over-simplified fixed body. In this study, we propose optimal bendable body design with three degrees of freedom (DOFs). Its implementation on a sprawling posture robot is inspired by geckos for climbing enhancement. The robot leg and body movements are coordinated and driven by central pattern generator (CPG)-based neural control. As a consequence, the robot can climb using a combination of trot gait and lateral undulation of the bendable body with a C-shaped standing wave. Through the real robot experiments on a 3D force measuring platform, we demonstrate that, due to the dynamics of the bendable body movement, the robot can gain higher medio–lateral ( $F_x$ ) ground reaction forces (GRFs) at its front legs as well as anterior–posterior ( $F_y$ ) GRFs at its hind legs to increase the bending angular momentum ( $L_A{}_M$ ). This results in 52% and 54% reduced energy consumptions during climbing on steeper inclined solid and soft surfaces, respectively, compared to climbing with a fixed body. To this end, the study provides a basis for developing sprawling posture robots with a bendable body and neural control for energy-efficient inclined surface climbing with a possible extension towards agile and versatile locomotion, such as sprawling posture animals.

Locomotion is an essential behaviour for the survival of all animals. The neural circuitry underlying locomotion is therefore highly robust to a wide variety of perturbations, including injury and abrupt changes in the environment. In the short term, fault tolerance in neural networks allows locomotion to persist immediately after mild to moderate injury. In the longer term, in many invertebrates and vertebrates, neural reorganization including anatomical regeneration can restore locomotion after severe perturbations that initially caused paralysis. Despite decades of research, very little is known about the mechanisms underlying locomotor resilience at the level of the underlying neural circuits and coordination of central pattern generators (CPGs). Undulatory locomotion is an ideal behaviour for exploring principles of circuit organization, neural control and resilience of locomotion, offering a number of unique advantages including experimental accessibility and modelling tractability. In comparing three well-characterized undulatory swimmers, lampreys, larval zebrafish and Caenorhabditis elegans, we find similarities in the manifestation of locomotor resilience. To advance our understanding, we propose a comparative approach, integrating experimental and modelling studies, that will allow the field to begin identifying shared and distinct solutions for overcoming perturbations to persist in orchestrating this essential behaviour.

In this article, extend the uses of a deformable structure of magnetic elastomer to develop a bio-inspired locomotion system for millimeter-scaled robots. As proposed in other researches, this material allows a possibility of the motor-less mechanism powered by magnetic field. The actuating mechanism of the robot mainly relies on the body deformation due to magnetic alignment. On the other hand, herein, the magnetic soft robot with the triangular head-tail morphology and sine-based magnetization utilizes a high degree of freedom provided by magnetic compliance for mobility in a form of lateral undulation. Under the oscillating magnetic field, dynamic torque primarily acts to the head, the tail passively waves, and the whole body propagates a series of lateral body-waves for self-propulsion in fluid, instead of pushing out surrounding fluid to make a swimming gait. We achieve the independent control of the laterally undulating robot in force-free swimming, and demonstrate its tunable body deformation to swim in various diameters of the fluid-filled tubes. A by-product of the undulation can transfer a force rate to swim through the fabric-media. By the increase of the actuating frequency, the undulating robot propels faster to retain a stable swimming in the flow. The versatilities of the robot with the proposed mechanism can be applied to serve the diverse purposes as a reliable and effective locomotion system with a minimal control. Particularly in biomedical applications, miniature soft robots with a potential mechanism for self-propulsion can contribute the great results to pursue minimally invasive treatments.

In this paper, we investigate the hydrodynamics of swimmers with three caudal fins: a round one corresponding to snakehead fish ( Channidae), an indented one corresponding to saithe ( Pollachius virens), and a lunate one corresponding to tuna ( Thunnus thynnus). A direct numerical simulation (DNS) approach with a self-propelled fish model was adopted. The simulation results show that the caudal fin transitions from a pushing/suction combined propulsive mechanism to a suction-dominated propulsive mechanism with increasing aspect ratio ( AR). Interestingly, different from a previous finding that suction-based propulsion leads to high efficiency in animal swimming, this study shows that the utilization of suction-based propulsion by a high- AR caudal fin reduces swimming efficiency. Therefore, the suction-based propulsive mechanism does not necessarily lead to high efficiency, while other factors might play a role. Further analysis shows that the large lateral momentum transferred to the flow due to the high depth of the high- AR caudal fin leads to the lowest efficiency despite the most significant suction.

When flying snakes glide, they use aerial undulation. To determine if aerial undulation is a flight control strategy or a non-functional behavioural vestige of lateral undulation, we measured snake glides using high-speed motion capture and developed a new dynamical model of gliding. Reconstructions of the snake’s wing-body reveal that aerial undulation is composed of horizontal and vertical waves, whose phases differ by 90° and whose frequencies differ by a factor of two. Using these results, we developed a three-dimensional mathematical model of snake flight that incorporates aerodynamic and inertial effects. Although simulated glides without undulation attained some horizontal distance, they are biologically unrealistic because they failed due to roll and pitch instabilities. In contrast, the inclusion of undulation stabilized the rotational motion and markedly increased glide performance. This work demonstrates that aerial undulation in snakes serves a different function than known uses of undulation in other animals, and suggests a new template of control for dynamic flying robots. Observations of flying snakes inform the development of a dynamical model of gliding taking undulation into account. This work suggests that aerial undulation has a different function in snakes than in other animals.

Frictional anisotropy, as produced by the directionality of scales in snake skin, is necessary to propel snakes across flat, hard surfaces. This work illustrates the design, fabrication, and testing of a snake-inspired skin based on kirigami techniques that, when attached to a soft snake robot, improves the robot's locomotion capabilities when implementing a lateral undulation gait. Examination of snake scales in nature informed the shape and texture of the synthetic scales, which are activated through the buckling of kirigami lattices. Biological snakes have microornamentation on their scales, which is replicated by scoring ridges into the plastic skin. This microornamentation contributes to the lateral resistance necessary for lateral undulation. The skin's frictional properties were experimentally determined, as were their contributions to the locomotion of the robot across a flat, hard, textured surface. Contributions to locomotion from scale profile geometry, scale microornamentation, and scale angle of attack were identified. The range of longitudinal COF ratios was 1.0 to 3.0 and the range of lateral COF ratios was 0.9 to 3.3. The highest performing skin was the triangular scale profile with microornamentation, producing a velocity of 6 mm/s (0.03 BL/s) which is an increase of 335% over the robot with no skin when activated to maximum achievable curvature.

SynopsisAnimals move in diverse ways, as indicated in part by the wide variety of gaits and modes that have been described for vertebrate locomotion. Much variation in the gaits of limbed animals is associated with changing speed, whereas different modes of snake locomotion are often associated with moving on different surfaces. For several decades different types of snake locomotion have been categorized as one of four major modes: rectilinear, lateral undulation, sidewinding, and concertina. Recent empirical work shows that the scheme of four modes of snake locomotion is overly conservative. For example, during aquatic lateral undulation, the timing between muscle activity and lateral bending changes along the length of the snake, which is unlike terrestrial lateral undulation. The motor pattern used to prevent sagging while bridging gaps also suggests that arboreal lateral undulation on narrow surfaces or with a few discrete points of support has a different motor pattern than terrestrial lateral undulation when the entire length of the snake is supported. In all types of concertina locomotion, the distance from the head to the tail changes substantially as snakes alternately flex and then extend different portions of their body. However, snakes climbing cylinders with concertina exert forces medially to attain a purchase on the branch, whereas tunnels require pushing laterally to form an anchoring region. Furthermore, different motor patterns are used for these two types of concertina movement. Some snakes climb vertical cylinders with helical wrapping completely around the cylinder, whereas all other forms of concertina bend regions of the body alternately to the left and right. Current data support rectilinear locomotion and sidewinding as being distinct modes, whereas lateral undulation and concertina are best used for defining categories of gaits with some unifying similarities. Partly as a result of different motor patterns, I propose recognizing five and four distinct types of lateral undulation and concertina, respectively, resulting in a total of 11 distinct gaits previously recognized as only four.

Animals modulate sensory processing in concert with motor actions. Parallel copies of motor signals, called corollary discharge (CD), prepare the nervous system to process the mixture of externally and self-generated (reafferent) feedback that arises during locomotion. Commonly, CD in the peripheral nervous system cancels reafference to protect sensors and the central nervous system from being fatigued and overwhelmed by self-generated feedback. However, cancellation also limits the feedback that contributes to an animal’s awareness of its body position and motion within the environment, the sense of proprioception. We propose that, rather than cancellation, CD to the fish lateral line organ restructures reafference to maximize proprioceptive information content. Fishes’ undulatory body motions induce reafferent feedback that can encode the body’s instantaneous configuration with respect to fluid flows. We combined experimental and computational analyses of swimming biomechanics and hair cell physiology to develop a neuromechanical model of how fish can track peak body curvature, a key signature of axial undulatory locomotion. Without CD, this computation would be challenged by sensory adaptation, typified by decaying sensitivity and phase distortions with respect to an input stimulus. We find that CD interacts synergistically with sensor polarization to sharpen sensitivity along sensors’ preferred axes. The sharpening of sensitivity regulates spiking to a narrow interval coinciding with peak reafferent stimulation, which prevents adaptation and homogenizes the otherwise variable sensor output. Our integrative model reveals a vital role of CD for ensuring precise proprioceptive feedback during undulatory locomotion, which we term external proprioception., Animals modulate sensory processing in concert with motor actions. A study of the corollary discharge in zebrafish reveals that it modulates the sensitivity of the lateral line during swimming to prevent sensor adaptation and maintain the high-quality feedback necessary for kinematic control.

A significant component of the repetitive dynamics during locomotion in vertebrates is generated within the spinal cord. The legged locomotion of mammals is most likely controled by a hierarchical, multi-layer spinal network structure, while the axial circuitry generating the undulatory swimming motion of animals like lamprey is thought to have only a single layer in each segment. Recent experiments have suggested a hybrid network structure in zebrafish larvae in which two types of excitatory interneurons (V2a-I and V2a-II) both make first-order connections to the brain and last-order connections to the motor pool. These neurons are connected by electrical and chemical synapses across segments. Through computational modeling and an asymptotic perturbation approach we show that this interleaved interaction between the two neuron populations allows the spinal network to quickly establish the correct activation sequence of the segments when starting from random initial conditions and to reduce the dependence of the intersegmental phase difference (ISPD) of the oscillations on the swimming frequency. The latter reduces the frequency dependence of the waveform of the swimming motion. In the model the reduced frequency dependence is largely due to the different impact of chemical and electrical synapses on the ISPD and to the significant spike-frequency adaptation that has been observed experimentally in V2a-II neurons, but not in V2a-I neurons. Our model makes experimentally testable predictions and points to a benefit of the hybrid structure for undulatory locomotion that may not be relevant for legged locomotion.

Lateral undulation is the most widespread mode of terrestrial vertebrate limbless locomotion, in which posteriorly propagating horizontal waves press against environmental asperities (e.g. grass, rocks) and generate propulsive reaction forces. We hypothesized that snakes can generate propulsion using a similar mechanism of posteriorly propagating vertical waves pressing against suitably oriented environmental asperities. Using an array of horizontally oriented cylinders, one of which was equipped with force sensors, and a motion capture system, we found snakes generated substantial propulsive force and propulsive impulse with minimal contribution from lateral undulation. Additional tests showed that snakes could propel themselves via vertical undulations from a single suitable contact point, and this mechanism was replicated in a robotic model. Vertical undulations can provide snakes with a valuable locomotor tool for taking advantage of vertical asperities in a variety of habitats, potentially in combination with lateral undulation, to fully exploit the 3D structure of the habitat., Summary: Snakes are capable of generating propulsion via vertical undulations, which allows them to exploit their environment in 3D and allows more effective use of previously overlooked surfaces in cluttered habitats.

Inspired by the morphology characteristics of segmented worms and the combination properties of serial–parallel structures, a novel serial–parallel hybrid worm-like robot with nine degrees of freedom (DOFs) is proposed. This robot is constructed of two symmetrically arranged 3-RPS parallel mechanisms (PMs) with single-DOF expandable platforms. According to the retrograde peristaltic wave that the locomotion mechanism of segmented worms and the kinematics of the robot, four basic gaits of three undulation modes are developed, including start-up adjustment gait and straight-going gait of rectilinear mode, turning gait of lateral undulation mode, and stair-climbing gait of vertical undulation mode. Based on which, the locomotion performance of the robot on four typical terrains including the stair, the slope, the gap, and the narrow passage is discussed. Then the corresponding locomotion controller is designed. Finally, a prototype is manufactured, the experiment results show that the robot can move in an arbitrary given direction for any given distance, and the practical locomotion performance coincides to the theoretical analysis.

SummaryThe unique frictional properties conferred by snake ventral scales inspired the engineering and fabrication of surrogate mechanisms for a robotic snake. These artificial, biologically inspired scales produce anisotropic body-ground forcing patterns with various locomotion surfaces. The benefits they confer to robotic snake-like locomotion were evaluated in experimental trials employing rectilinear, lateral undulation, and sidewinding gaits over several distinct surface types: carpet, inhomogeneous concrete and homogeneous concrete. Enhanced locomotive performance, with respect to net displacement and heading stability, was consistently measured in scenarios that utilized the engineered scales, over equivalent scenarios where the anisotropic effects of scales were absent.

Robotic 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 und...

Sensory feedback during movement entails sensing a mix of externally- and self-generated stimuli (respectively, exafference and reafference). In many peripheral sensory systems, a parallel copy of the motor command, a corollary discharge, is thought to eliminate sensory feedback during behaviors. However, reafference has important roles in motor control, because it provides real-time feedback on the animal’s motions through the environment. In this case, the corollary discharge must be calibrated to enable feedback while avoiding negative consequences like sensor fatigue. The undulatory motions of fishes’ bodies generate induced flows that are sensed by the lateral line sensory organ, and prior work has shown these reafferent signals contribute to the regulation of swimming kinematics. Corollary discharge to the lateral line reduces the gain for reafference, but cannot eliminate it altogether. We develop a data-driven model integrating swimming biomechanics, hair cell physiology, and corollary discharge to understand how sensory modulation is calibrated during locomotion in larval zebrafish. In the absence of corollary discharge, lateral line afferent units exhibit the highly heterogeneous habituation rates characteristic of hair cell systems, typified by decaying sensitivity and phase distortions with respect to an input stimulus. Activation of the corollary discharge prevents habituation, reduces response heterogeneity, and regulates response phases in a narrow interval around the time of the peak stimulus. This suggests a synergistic interaction between the corollary discharge and the polarization of lateral line sensors, which sharpens sensitivity along their preferred axes. Our integrative model reveals a vital role of corollary discharge for ensuring precise feedback, including proprioception, during undulatory locomotion.

The small structures that decorate biological surfaces can significantly affect behavior, yet the diversity of animal–environment interactions essential for survival makes ascribing functions to structures challenging. Microscopic skin textures may be particularly important for snakes and other limbless locomotors, where substrate interactions are mediated solely through body contact. While previous studies have characterized ventral surface features of some snake species, the functional consequences of these textures are not fully understood. Here, we perform a comparative study, combining atomic force microscopy measurements with mathematical modeling to generate predictions that link microscopic textures to locomotor performance. We discover an evolutionary convergence in the ventral skin structures of a few sidewinding specialist vipers that inhabit sandy deserts—an isotropic texture that is distinct from the head-to-tail-oriented, micrometer-sized spikes observed on a phylogenetically broad sampling of nonsidewinding vipers and other snakes from diverse habitats and wide geographic range. A mathematical model that relates structural directionality to frictional anisotropy reveals that isotropy enhances movement during sidewinding, whereas anisotropy improves movement during slithering via lateral undulation of the body. Our results highlight how an integrated approach can provide quantitative predictions for structure–function relationships and insights into behavioral and evolutionary adaptations in biological systems.

Earthworms (Lumbricus terrestris) are characterized by soft, highly flexible and extensible bodies, and are capable of locomoting in most terrestrial environments. Previous studies of earthworm movement have focused on the use of retrograde peristaltic gaits in which controlled contraction of longitudinal and circular muscles results in waves of shortening/thickening and thinning/lengthening of the hydrostatic skeleton. These waves can propel the animal across ground as well as into soil. However, worms can also benefit from axial body bends during locomotion. Such lateral undulation dynamics can aid locomotor function via hooking/anchoring (to provide propulsion), modify travel orientation (to avoid obstacles and generate turns) and even generate snake-like undulatory locomotion in environments where peristaltic locomotion results in poor performance. To the best of our knowledge, the important aspects of locomotion associated with the lateral undulation of an earthworm body are yet to be systematically investigated. In this study, we observed that within confined environments, the worm uses lateral undulation to anchor its body to the walls of their burrows and tip (nose) bending to search the environment. This relatively simple locomotion strategy drastically improved the performance of our soft bodied robophysical model of the earthworm both in a confined (in an acrylic tube) and above-ground heterogeneous environment (rigid pegs), where the peristaltic gait often fails. In summary, lateral undulation facilitates the mobility of earthworm locomotion in diverse environments and can play an important role in the creation of low cost soft robotic devices capable of traversing a variety of environments.

Earthworms (Lumbricus terrestris) are characterized by soft, highly flexible and extensible bodies, and are capable of locomoting in most terrestrial environments. Previous studies of earthworm movement focused on the use of retrograde peristaltic gaits in which controlled contraction of longitudinal and circular muscles results in waves of shortening/thickening and thinning/lengthening of the hydrostatic skeleton. These waves can propel the animal across ground as well as into soil. However, worms benefit from axial body bends during locomotion. Such lateral bending and buckling dynamics can aid locomotor function via hooking/anchoring (to provide propulsion), modify travel orientation (to avoid obstacles and generate turns) and even generate snake-like undulatory locomotion in environments where peristaltic locomotion results in poor performance. To the best of our knowledge, lateral bending and buckling of an earthworm’s body has not yet been systematically investigated. In this study, we observed that within confined environments, worms use lateral bending and buckling to anchor their body to the walls of their burrows and tip (anterior end) bending to search the environment. This locomotion strategy improved the performance of our soft-bodied robophysical model of the earthworm both in a confined (in an acrylic tube) and above-ground heterogeneous environment (rigid pegs), where present peristaltic robots are relatively limited in terradynamic capabilities. In summary, lateral bending and buckling facilitates the mobility of earthworm locomotion in diverse terrain and can play an important role in the creation of low cost soft robotic devices capable of traversing a variety of environments.

The control laws presented in Chaps. 2, 3 and 4 were aimed at achieving improved tracking performance utilizing least control effort for the planar snake robot model. This particular form of locomotion for a snake robot is inspired by the commonest mode of snake motion called lateral undulation. Other modes of generating propagating force for a snake robot, in channel or around a cylinder for example, have not been extensively studied. This chapter is targeted to establish and obtain a mathematical model of a snake robot inside a constrained space like a pipe or a channel. The model has been employed to design control laws for the robot’s efficient motion through the pipe. To address uncertainties that may occur during such a motion, a flatness-based adaptive robust control approach presented in Chap. 5 has been employed as well.

We propose the use of bio-inspired robotics equipped with soft sensor technologies to gain a better understanding of the mechanics and control of animal movement. Soft robotic systems can be used to generate new hypotheses and uncover fundamental principles underlying animal locomotion and sensory capabilities, which could subsequently be validated using living organisms. Physical models increasingly include lateral body movements, notably back and tail bending, which are necessary for horizontal plane undulation in model systems ranging from fish to amphibians and reptiles. We present a comparative study of the use of physical modeling in conjunction with soft robotics and integrated soft and hyperelastic sensors to monitor local pressures, enabling local feedback control, and discuss issues related to understanding the mechanics and control of undulatory locomotion. A parallel approach combining live animal data with biorobotic physical modeling promises to be beneficial for gaining a better understanding of systems in motion.

Synopsis Lateral undulation and trunk flexibility offer performance benefits to maneuverability, stability, and stride length (via speed and distance traveled). These benefits make them key characteristics of the locomotion of tetrapods with sprawling posture, with the exception of turtles. Despite their bony carapace preventing lateral undulations, turtles are able to improve their locomotor performance by increasing stride length via greater limb protraction. The goal of this study was to quantify the effect of reduced lateral flexibility in a generalized sprawling tetrapod, the tiger salamander (Ambystoma tigrinum). We had two potential predictions: (1) either salamanders completely compensate by changing their limb kinematics, or (2) their performance (i.e., speed) will suffer due to the reduced lateral flexibility. This reduction was performed by artificially limiting trunk flexibility by attaching a 2-piece shell around the body between the pectoral and pelvic girdles. Adult tiger salamanders (n = 3; SVL = 9–14.5 cm) walked on a 1-m trackway under three different conditions: unrestricted, flexible shell (Tygon tubing), and rigid shell (PVC tubing). Trials were filmed in a single, dorsal view, and kinematics of entire midline and specific body regions (head, trunk, tail), as well as the fore and hind limbs, were calculated. Tygon individuals had significantly higher curvature than both PVC and unrestricted individuals for the body, but this trend was primarily driven by changes in tail movements. PVC individuals had significantly lower curvature in the trunk region compared with unrestricted individuals or Tygon; however, there was no difference between unrestricted and Tygon individuals suggesting the shells performed as expected. PVC and Tygon individuals had significantly higher curvature in the tails compared with unrestricted individuals. There were no significant differences for any limb kinematic variables among treatments including average, minimum, and maximum angles. Thus, salamanders respond to decreased lateral movement in their trunk by increasing movements in their tail, without changes in limb kinematics. These results suggest that tail undulations may be a more critical component to sprawling-postured tetrapod locomotion than previously recognized.

The objective of this study was to imitate undulatory motion, which is a commonly observed swimming mechanism of rays, using a soft morphing actuator. To achieve the undulatory motion, an artificial muscle built with shape memory alloy (SMA) based soft actuators was exploited to control the shape changing behavior of a soft fin membrane. Artificial undulating fins were divided into two categories according to the method of generating the wave motion: single and multiple actuator-driven fins. For empirical research on the transformation and propulsion behavior of each fin type, the design and construction of bound propulsors were undertaken to mimic the structural and behavioral aspects of animals. To visualize the effect of undulatory motion on the swimming efficiency test of the fin beat frequency, a simplified soft undulating fin with a rectangular propulsor was constructed and tested. Additionally, dynamic modeling of the fin tip in wave-traveling was conducted for comparison and optimization. To optimize the thrust and propulsion efficiency of robot speed, the effects of the wave amplitude control and actuator sequence on the fin behavior were examined. Untethered robots was constructed according to the experimental results of the propulsors. Both exhibited exceptional swimming efficiency and maneuverability. The multiple actuator-driven ray robot exhibited a maximum swimming speed of 0.25 body lengths per second which is almost similar swimming speed with previously reported robot. The developed robot achieved directional swimming (forward and backward) and turning (including rotation). Underwater exploration in an artificial environment was performed using the robot.

Snake-like robots move flexibly in complex environments due to their multiple degrees of freedom and various gaits. However, their existing 3-D models are not accurate enough, and most gaits are applicable to special environments only. This work investigates a 3-D model and designs hybrid 3-D gaits. In the proposed 3-D model, a robot is considered as a continuous beam system. Its normal reaction forces are computed based on the mechanics of materials. To improve the applicability of such robots to different terrains or tasks, this work designs hybrid 3-D gaits by mixing basic gaits in different parts of their bodies. Performances of hybrid gaits are analyzed based on extensive simulations. These gaits are compared with traditional gaits including lateral undulation, rectilinear, and sidewinding ones. Results of simulations and physical experiments are presented to demonstrate the performances of the proposed model and hybrid gaits of snake-like robots.

Some creatures, such as eels, snakes and slender fish use body undulations to move the fluid around them and create propulsion; similarly, travelling waves in structures can be used as an alternative propulsion system. Based on this bio-inspired mechanism, the application of travelling waves is widespread in engineering, e.g. motors, pump systems and transport devices. However, generating high amplitude travelling waves is not as easy as generating standing waves, since travelling waves are generally observed away from resonance. Also, the reflection of the waves at the boundaries can interact destructively further reducing their amplitude. This paper investigates the characteristics of the experimental rig built for emulating the propulsion mechanism used in undulatory locomotion, introducing the conceptual design, and investigating the features that promote travelling waves. Finally, an analysis of the influence of individual components in the design is carried out.

Undulatory locomotion is a common and powerful strategy used in nature at different biological scales by a broad range of living organisms, from flagellated bacteria to prehistoric snakes, which have overcome the complexity of living in ?flowable? media. By taking inspiration from this evolution-induced strategy, we aim at modelling the locomotion in a granular and viscous environment with the objective to provide more insights for designing robots for soil-like media exploration. Moreover, in contrast to common types of movement, the granular locomotion is still not well understood and is an open and challenging field. We approached this phenomenon with several tools: (i.) numerically, via coupling the Finite Element Method (FEM) with the Discrete Element Method (DEM) using ABAQUS; (ii.) analytically, by employing the Lagrangian formalism to derive the equations of motion of a discrete and continuous system subject to non-conservative forces, and (iii.) experimentally, by creating an ad-hoc set up in order to observe the migration of microfibres used for the treatment of spinal cord injuries. The computational attempts to model the motion in a granular medium involved the simulation of the dynamics of an elastic beam (FEM) surrounded by rigid spherical particles (DEM). A propulsion mechanism was introduced by sinusoidally forcing the beam?s tip normally to the longitudinal axis, while the performance of the locomotion was evaluated by means of a parametric study. Depending on the parameters of the external excitation, after a transient phase, the slender body reached a steady-state with a constant translational velocity. In order to gain physical insights, we studied a simplified version of the previous continuous beam by introducing a discrete multi-bar system. The dynamics of the latter was analytically derived, by taking into account the forces exchanged between the locomotor and the environment, according to the Resistive Force Theory. By numerically solving the equations of motion and evaluating the input energy and dissipations, we were able to define the efficiency and thus provide an effective tool to optimise the locomotion. It is worth mentioning that the two approaches, despite the different physical hypothesis, show a qualitatively and quantitatively good accordance. The numerical and analytical models previously analysed have shown promising results for the interpretation of "ad-hoc" experiments that demonstrate the migration of a microfibre embedded in a spinal cord-like matrix. This migration needs to be avoided, once the regenerative microfibre is implanted in the lesioned spinal cord, for the sake of the patients health.