Effects of non-uniform stiffness on the swimming performance of a passively-flexing, fish-like foil model

Effects of caudal fin stiffness on optimized forward swimming and turning maneuver in a robotic swimmer

In animal and robot swimmers of body and caudal fin (BCF) form, hydrodynamic thrust is mainly produced by their caudal fins, the stiffness of which has profound effects on both thrust and efficiency of swimming. Caudal fin stiffness also affects the motor control and resulting swimming gaits that correspond to optimal swimming performance; however, their relationship remains scarcely explored. Here using magnetic, modular, undulatory robots (μBots), we tested the effects of caudal fin stiffness on both forward swimming and turning maneuver. We developed six caudal fins with stiffness of more than three orders of difference. For aμBot equipped with each caudal fin (andμBot absent of caudal fin), we applied reinforcement learning in experiments to optimize the motor control for maximizing forward swimming speed or final heading change. The motor control ofμBot was generated by a central pattern generator for forward swimming or by a series of parameterized square waves for turning maneuver. In forward swimming, the variations in caudal fin stiffness gave rise to three modes of optimized motor frequencies and swimming gaits including no caudal fin (4.6 Hz), stiffness <10-4Pa m4(∼10.6 Hz) and stiffness >10-4Pa m4(∼8.4 Hz). Swimming speed, however, varied independently with the modes of swimming gaits, and reached maximal at stiffness of 0.23 × 10-4Pa m4, with theμBot without caudal fin achieving the lowest speed. In turning maneuver, caudal fin stiffness had considerable effects on the amplitudes of both initial head steering and subsequent recoil, as well as the final heading change. It had relatively minor effect on the turning motor program except for theμBots without caudal fin. Optimized forward swimming and turning maneuver shared an identical caudal fin stiffness and similar patterns of peduncle and caudal fin motion, suggesting simplicity in the form and function relationship inμBot swimming.

Locomotor effects of a fibrosis-based immune response in stickleback fish

The vertebrate immune system provides an impressively effective defense against parasites and pathogens. However, these benefits must be balanced against a range of costly side-effects including energy loss and risks of auto-immunity. These costs might include biomechanical impairment of movement, but little is known about the intersection between immunity and biomechanics. Here, we show that a fibrosis immune response to Schistocephalus solidus infection in freshwater threespine stickleback (Gasterosteus aculeatus) has collateral effects on their locomotion. Although fibrosis is effective at reducing infection, some populations of stickleback actively suppress this immune response, possibly because the costs of fibrosis outweigh the benefits. We quantified the locomotor effects of the fibrosis immune response in the absence of parasites to investigate whether there are incidental costs of fibrosis that could help explain why some fish forego this effective defense. To do this, we induced fibrosis in stickleback and then tested their C-start escape performance. Additionally, we measured the severity of fibrosis, body stiffness and body curvature during the escape response. We were able to estimate performance costs of fibrosis by including these variables as intermediates in a structural equation model. This model revealed that among control fish without fibrosis, there is a performance cost associated with increased body stiffness. However, fish with fibrosis did not experience this cost but rather displayed increased performance with higher fibrosis severity. This result demonstrates that the adaptive landscape of immune responses can be complex with the potential for wide-reaching and unexpected fitness consequences.

The morphological paradigm in robotics

In the paper, we are going to show how robotics is undergoing a shift in a bionic direction after a period of emphasis on artificial intelligence and increasing computational efficiency, which included isolation and extreme specialization. We assemble these new developments under the label of the morphological paradigm. The change in its paradigms and the development of alternatives to the principles that dominated robotics for a long time contains a more general epistemological significance. The role of body, material, environment, interaction and the paradigmatic status of biological and evolutionary systems for the principles of control are crucial here. Our focus will be on the introduction of the morphological paradigm in a new type of robotics and to contrast the interests behind this development with the interests shaping former models. The article aims to give a clear account of the changes in principles of orientation and control as well as concluding general observation in terms of historical epistemology, suggesting further political-epistemological analysis.

Locomotor effects of a fibrosis-based immune response in stickleback fish

The vertebrate immune system provides an impressively effective defense against parasites and pathogens. However, these benefits must be balanced against a range of costly side-effects including energy loss and risks of auto-immunity. These costs might include biomechanical impairment of movement, but little is known about the intersection between immunity and biomechanics. Here, we show that a fibrosis immune response in threespine stickleback (Gasterosteus aculeatus) has collateral effects on their locomotion. When freshwater stickleback are infected with the tapeworm parasite Schistocephalus solidus, they face an array of fitness consequences ranging from impaired body condition and fertility to an increased risk of mortality. To fight the infection, some stickleback will initiate a fibrosis immune response in which they produce excess collagenous tissue in their coelom. Although fibrosis is effective at reducing infection, some populations of stickleback actively suppress this immune response, possibly because the costs of fibrosis outweigh the benefits. Here we quantify the locomotor effects of the fibrosis immune response in the absence of parasites to investigate whether there are collateral costs of fibrosis that could help explain why some fish forego this effective defense. To do this, we induce fibrosis in stickleback and then test their C-start escape performance. Additionally, we measure the severity of fibrosis, body stiffness, and body curvature during the escape response. We were able to estimate performance costs of fibrosis by including these variables as intermediates in a structural equation model. This model reveals that among control fish without fibrosis, there is a performance cost associated with increased body stiffness. However, fish with fibrosis did not experience this cost but rather displayed increased performance with higher fibrosis severity. This result demonstrates that the adaptive landscape of immune responses can be complex with the potential for wide reaching and unexpected fitness consequences.

Experimental analysis of the sweepback angle effect on the thrust generation of a robotic penguin wing

Penguins have evolved excellent swimming skills as diving birds, benefiting from their agile wings. This paper experimentally analyzes the effects of the wing sweepback angle on thrust generation using a robotic penguin wing. A developed wing mechanism that can realize penguin-like flapping and feathering motion was used for actuating five alternative wing models, with different sweepback angles ranging from 0° to 50°. Force measurements under a steady water flow were conducted for both fixed and flapping states for all wing models. The results showed that small sweepback angles of 30° or less in the fixed state caused a steep lift curve and a moderate sweepback angle of 30° produced the largest lift-to-drag ratio. In the flapping state, the smaller sweepback wings generated a larger net thrust for the same wing motion, whereas the larger-sweepback wings produced more thrust under the same Strouhal number. The findings also revealed that larger sweepback wings more easily achieve the maximum net thrust in terms of less angle-of-attack control. In contrast, the hydrodynamic efficiency was not greatly affected by the sweepback. Regardless of the sweepback, the trend of the efficiency increasing with increasing flow speed indicates that the penguin wings can be more suitable for high-speed locomotion for higher hydrodynamic efficiency.

How Non-Uniform Stiffness Affects the Propulsion Performance of a Biomimetic Robotic Fish

Live fish in nature exhibit various stiffness characteristics. The anguilliform swimmer, like eels, has a relatively flexible body, while the thunniform swimmer, like the swordfishes, has a much stiffer body. Correspondingly, in the design of biomimetic robotic fish, how to balance the non-uniform stiffness to achieve better propulsion performance is an essential question needed to be answered. In this paper, we conduct an experimental study on this question. First, a customized experimental platform is built, which eases the adjustment of the non-uniform stiffness ratio, the stiffness of the flexible part, the flapping frequency, and the flapping amplitude. Second, extensive experiments are carried out, finding that to maximize the propulsion performance of the biomimetic robotic fish, the non-uniform stiffness ratio is required to adapt to different locomotor parameters. Specifically, the non-uniform stiffness ratio needs to be reduced when the robotic fish works at low frequency, and it needs to be increased when the robotic fish works at high frequency. Finally, detailed discussions are given to further analyze the experimental results. Overall, this study can shed light on the design of a non-uniform biomimetic robotic fish, which helps to increase its propulsion performance.

Internal vertebral morphology of bony fishes matches the mechanical demands of different environments

Fishes have repeatedly evolved characteristic body shapes depending on how close they live to the substrate. Pelagic fishes live in open water and typically have narrow, streamlined body shapes; benthic and demersal fishes live close to the substrate; and demersal fishes often have deeper bodies. These shape differences are often associated with behavioral differences: pelagic fishes swim nearly constantly, demersal fishes tend to maneuver near the substrate, and benthic fishes often lie in wait on the substrate. We hypothesized that these morphological and behavioral differences would be reflected in the mechanical properties of the body, and specifically in vertebral column stiffness, because it is an attachment point for the locomotor musculature and a central axis for body bending. The vertebrae of bony fishes are composed of two cones connected by a foramen, which is filled by the notochord. Since the notochord is more flexible than bony vertebral centra, we predicted that pelagic fishes would have narrower foramina or shallower cones, leading to less notochordal material and a stiffer vertebral column which might support continuous swimming. In contrast, we predicted that benthic and demersal fishes would have more notochordal material, making the vertebral column more flexible for diverse behaviors in these species. We therefore examined vertebral morphology in 79 species using micro-computed tomography scans. Six vertebral features were measured including notochordal foramen diameter, centrum body length, and the cone angles and diameters for the anterior and posterior vertebral cones, along with body fineness. Using phylogenetic generalized least squares analyses, we found that benthic and pelagic species differed significantly, with larger foramina, shorter centra, and larger cones in benthic species. Thus, morphological differences in the internal shape of the vertebrae of fishes are consistent with a stiffer vertebral column in pelagic fishes and with a more flexible vertebral column in benthic species.

Tail-stiffness optimization for a flexible robotic fish

Undulation regulation in a robotic fish propelled by a passive flexible tail is more similar to that of a natural fish than with a rigid tail, owing to the smooth curvature of the flexible tail. Moreover, it has been observed that fish change the stiffness of their bodies to adapt to various swimming states. Inspired by this, a stiffness optimization scheme is explored for a novel elastic tail, which can improve the performance of the robotic fish. Spring steels are used as passive flexible joints of the fishtail; these can be easily expanded into multi-joint structures and the joint stiffness can be altered by changing the joint size. In this study, the Lagrangian dynamic method is employed to establish a dynamic model of the robotic fish in which passive flexible joints are simplified by a pseudo-rigid-body model. In addition, the hydrodynamics of the head and tail are analyzed using the simplified Morison equation and quasi-steady wing theory, respectively. Furthermore, to determine unknown hydrodynamic parameters in the dynamic model, a parameter identification method is applied. The results show that the identified simulation speeds fit the experimental speeds well within a wide range of stiffness values. Finally, to improve performance, the influence of joint stiffness and frequency on swimming speed is investigated based on the identified dynamic model. At each frequency, the optimal joint stiffness distribution is one that reduces the stiffness from the front to the rear. At the maximum driving frequency of 2.5 Hz, the optimal swimming speed is 0.3 body lengths per second, higher than that when rigid joints are used.

Tailoring the bending pattern of non-uniformly flexible pitching hydrofoils enhances propulsive efficiency

We present new measurements of non-uniformly flexible pitching foils fabricated with a rigid leading section joined to a flexible trailing section. This construction enables us to vary the bending pattern and resonance condition of the foils independently. A novel effective flexibility, defined as the ratio of added mass forces to elastic forces, is proposed and shown to provide a scaling for the natural frequencies of the fluid-structural system. Foils with very flexible trailing sections ofEI< 1.81 × 10^-5N m²do not show a detectable resonance and are classified as 'non-resonating' as opposed to 'resonating' foils. Moreover, the non-resonating foils exhibit a novel bending pattern where the foil has a discontinuous hinge-like deflection instead of the smooth beam-like deflection of the resonating foils. Performance measurements reveal that both resonating and non-resonating foils can achieve high propulsive efficiencies of around 50% or more. It is discovered that non-uniformly flexible foils outperform their rigid and uniformly flexible counterparts, and that there is an optimal flexion ratio from 0.4 ⩽λ⩽ 0.7 that maximizes the efficiency. Furthermore, this optimal range coincides with the flexion ratios observed in nature. Performance is also compared under the same dimensionless flexural rigidity,R*, which highlights that at the same flexion ratio more flexible foils achieve higher peak efficiencies. Overall, to achieve high propulsive efficiency non-uniformly flexible hydrofoils should (1) oscillate above their first natural frequency, (2) have a flexion ratio in the range of 0.4 ⩽λ⩽ 0.7 and (3) have a small dimensionless rigidity at their optimal flexion ratio. Scaling laws for rigid pitching foils are found to be valid for non-uniformly flexible foils as long as the measured amplitude response is used and the deflection angle of the trailing sectionβ is < 45°. This work provides guidance for the development of high-performance underwater vehicles using simple purely pitching bio-inspired propulsive drives.

Thrust Improvement of a Biomimetic Robotic Fish by Using a Deformable Caudal Fin

In nature, live fish has various deformable fins which are capable to promote the swimming speed, efficiency, stability, and thrust generation. However, this feature is rarely possessed by current man-made biomimetic robotic fishes. In this paper, a novel deformable caudal fin platform is proposed to improve thrust generation of biomimetic robotic fish. First, the design of the deformable caudal fin is given, which includes a servo motor, a gear-based transmission mechanism, fin bones, and silica membrane. Second, an improved Central Pattern Generator (CPG) model was developed to coordinately control the flapping of the tail and the deformation of the caudal fin. More specifically, three deformation patterns, i.e., conventional nondeformable mode, sinusoidal-based mode, instant mode, of the caudal fin are investigated. Third, extensive experiments are conducted to explore the effects of deformation of the caudal fin on the thrust generation of the biomimetic robotic fish. It was found that the instant mode of the caudal fin has the largest thrust, which sees a 27.5% improvement compared to the conventional nondeformable mode, followed by the sinusoidal-based mode, which also sees an 18.2% improvement. This work provides a novel way to design and control the deformation of the caudal fin, which sheds light on the development of high-performance biomimetic robotic fish.

Effect of Cross-Joints Fin on the Thrust Performance of Bionic Pectoral Fins

Cownose rays have a high forward propulsive performance due to their unique oscillating fin structure (named the cross-bracing structure), which differs from undulatory fish. The cross-bracing structure obtained through anatomy, on the other hand, is extremely complex. Hence, researchers used simple structures to model the biological structure to target the individual factors that affect cownose ray cruising performance. This paper simplified the cross-bracing fin structure to a cross-joints fin (CJF) structure with 18 designs. CJFs had five different joint widths (2 mm, 3 mm, 4 mm, 5 mm, and 6 mm) in both spanwise and chordwise directions, and these had two fin thicknesses (1.5 mm, 2.5 mm). The joint widths of CJF are related to the stiffness of the spanwise and chordwise fins (Fin stiffness increases with joint width). The experiments were conducted in a still water tank (1.5 m × 0.8 m × 0.8 m) with three stroke amplitudes (30°, 50°, 70°) and three flapping frequencies (0.4 Hz, 0.6 Hz, 0.8 Hz) for each fin, making up 162 distinct sets of data. The experimental results showed the following: (1) at low wingtip Reynolds numbers, the high stiffness of the CJF causes a significant reduction in thrust. In particular, high stiffness results in a low thrust averaged from all motion parameters; (2) at high wing tip Reynolds numbers, the effect of changing spanwise stiffness on thrust is more significant than the effect of changing chordwise stiffness. This paper compares the effects of spanwise and chordwise stiffness on thrust performance, indicating that the magnitude of spanwise stiffness should be considered when designing the bionic oscillating pectoral fin structure.

Fluid-structure interaction of bio-inspired flexible slender structures: a review of selected topics

Flexible slender structures are ubiquitous in biological systems and engineering applications. Fluid-structure interaction (FSI) plays a key role in the dynamics of such structures immersed in fluids. Here, we survey recent studies on highly simplified bio-inspired models (either mathematical or mechanical) that aim to revealthe flow physics associated with FSI. Various models from different sources of biological inspiration are included, namely flexible flapping foil inspired by fish and insects, deformable membrane inspired by jellyfish and cephalopods, beating filaments inspired by flagella and cilia of microorganisms, and flexible wall-mounted filaments inspired by terrestrial and aquatic plants. Suggestions on directions for future research are also provided.

A fish-like soft-robotic model generates a diversity of swimming patterns

Fish display a versatile array of swimming patterns, and frequently demonstrate the ability to switch between these patterns altering kinematics as necessary. Many hard and soft robotic systems have sought to understand a variety of aspects pertaining to undulatory swimming, but most have been built to focus solely on a subset of those swimming patterns. We have expanded upon a previous soft robotic model, the pneufish, so that it can now simulate a variety of swimming patterns, much like a real fish. We explore the performance space available for this longer soft robotic model, which we call the quad-pneufish, with particular attention to the effects on lateral forces and z-torques produced during locomotion. We show that the quad-pneufish is capable of achieving a variety of midline patterns - including more realistic, fish-like patterns - and introducing a slight amount of co-activation between the left and right sides maintains forward thrust while decreasing lateral forces, indicating an increase in swimming efficiency. Robotic systems that are capable of producing an array of swimming movement patterns hold promise as experimental platforms for studying the diversity of fish locomotor patterns.

Tunable stiffness in fish robotics: mechanisms and advantages

One of the emerging themes of fish-inspired robotics is flexibility. Adding flexibility to the body, joints, or fins of fish-inspired robots can significantly improve thrust and/or efficiency during locomotion. However, the optimal stiffness depends on variables such as swimming speed, so there is no one 'best' stiffness that maximizes efficiency in all conditions. Fish are thought to solve this problem by using muscular activity to tune their body and fin stiffness in real-time. Inspired by fish, some recent robots sport polymer actuators, adjustable leaf springs, or artificial tendons that tune stiffness mechanically. Models and water channel tests are providing a theoretical framework for stiffness-tuning strategies that devices can implement. The strategies can be thought of as analogous to car transmissions, which allow users to improve efficiency by tuning gear ratio with driving speed. We provide an overview of the latest discoveries about (1) the propulsive benefits of flexibility, particularlytunableflexibility, and (2) the mechanisms and strategies that fish and fish-inspired robots use to tune stiffness while swimming.

Material and geometric effects on propulsion of a fish tail

We investigate the effects of material flexibility and aspect ratio on the propulsion of flapping tails. The tail, which is assumed to deform in the bending direction only, is modeled using the Euler-Bernoulli beam theory. The hydrodynamic loads generated by the flapping motion are calculated using the three-dimensional unsteady vortex lattice method. The finite element method is used to solve the coupled time-dependent equations of motion using an implicit solver for time integration. The results show improvement in the thrust and propulsive efficiency over a specific range of non-dimensional flexibility defined by the ratio of the elastic forces to fluid pressure forces. Structural and flow characteristics associated with the improved performance are discussed. As for geometric effects, the performance depends on the excitation frequency. At low frequencies, the improvement is continuous with increasing the aspect ratio in a manner similar to that of rigid tails. At higher frequencies, the improvement is limited to a region defined by aspect ratios that are less than 0.5. The extent of the improvement depends on the flexibility.

Fin-fin interactions during locomotion in a simplified biomimetic fish model

Fish median fins are extremely diverse, but their function is not yet fully understood. Various biological studies on fish and engineering studies on flapping foils have revealed that there are hydrodynamic interactions between fins arranged in tandem and that these interactions can lead to improved performance by the posterior fin. This performance improvement is often driven by the augmentation of a leading-edge vortex on the trailing fin. Past experimental studies have necessarily simplified fish anatomy to enable more detailed engineering analyses, but such simplifications then do not capture the complexities of an undulating fish-like body with fins attached. We present a flexible fish-like robotic model that better represents the kinematics of swimming fishes while still being simple enough to examine a range of morphologies and motion patterns. We then create statistical models that predict the individual effects of each kinematic and morphological variable. Our results demonstrate that having fins arranged in tandem on an undulating body can lead to more steady production of thrust forces determined by the distance between the fins and their relative motion. We find that these same variables also affect swimming speed. Specifically, when swimming at high frequencies, self-propelled speed decreases by 12%-26% due to out of phase fin motion. Flow visualization reveals that variation within this range is caused in part by fin-fin flow interactions that affect leading edge vortices. Our results indicate that undulatory swimmers should optimize both the positioning and relative motion of their median fins in order to reduce force oscillations and improve overall performance while swimming.

Contributions to a Discussion of Spinosaurus aegyptiacus as a Capable Swimmer and Deep-Water Predator

The new findings on Spinosaurus’ swim tail strongly suggest that Spinosaurus was a specialized deep-water predator. However, the tail must be seen in the context of the propelled body. The comparison of the flow characteristics of Spinosaurus with geometrically similar animals and their swimming abilities under water must take their Reynolds numbers into account and provide a common context for the properties of Spinosaurus’ tail and dorsal sail. Head shape adaptations such as the head crest reduced hydrodynamic disturbance and facilitated stealthy advance, especially when hunting without visual contact, when Spinosaurus could have used its rostral integumentary mechanoreceptors for prey detection. The muscular neck permitted ‘pivot’ feeding, where the prey’s escape abilities were overcome by rapid dorsoventral head movement, facilitated by crest-mediated lower friction.

Foretelling the Flex-Vertebral Shape Predicts Behavior and Ecology of Fishes

We modeled swimming kinematics and body mechanics of several fish species of varying habitat and body shape based on measurements of internal vertebral morphology.

SYNOPSIS

One key evolutionary innovation that separates vertebrates from invertebrates is the notochord, a central element that provides the stiffness needed for powerful movements. Later, the notochord was further stiffened by the vertebrae, cartilaginous, and bony elements, surrounding the notochord. The ancestral notochord is retained in modern vertebrates as intervertebral material, but we know little about its mechanical interactions with surrounding vertebrae. In this study, the internal shape of the vertebrae-where this material is found-was quantified in 16 species of fishes with various body shapes, swimming modes, and habitats. We used micro-computed tomography to measure the internal shape. We then created and mechanically tested physical models of intervertebral joints. We also mechanically tested actual vertebrae of five species. Material testing shows that internal morphology of the centrum significantly affects bending and torsional stiffness. Finally, we performed swimming trials to gather kinematic data. Combining these data, we created a model that uses internal vertebral morphology to make predictions about swimming kinematics and mechanics. We used linear discriminant analysis (LDA) to assess the relationship between vertebral shape and our categorical traits. The analysis revealed that internal vertebral morphology is sufficient to predict habitat, body shape, and swimming mode in our fishes. This model can also be used to make predictions about swimming in fishes not easily studied in the laboratory, such as deep sea and extinct species, allowing the development of hypotheses about their natural behavior.

High-Efficiency Can Be Achieved for Non-Uniformly Flexible Pitching Hydrofoils via Tailored Collective Interactions

New experiments examine the interactions between a pair of three-dimensional (AR = 2) non-uniformly flexible pitching hydrofoils through force and efficiency measurements. It is discovered that the collective efficiency is improved when the follower foil has a nearly out-of-phase synchronization with the leader and is located directly downstream with an optimal streamwise spacing of X*=0.5. The collective efficiency is further improved when the follower operates with a nominal amplitude of motion that is 36% larger than the leader’s amplitude. A slight degradation in the collective efficiency was measured when the follower was slightly-staggered from the in-line arrangement where direct vortex impingement is expected. Operating at the optimal conditions, the measured collective efficiency and thrust are ηC=62% and CT,C=0.44, which are substantial improvements over the efficiency and thrust of ηC=29% and CT,C=0.16 of two fully-rigid foils in isolation. This demonstrates the promise of achieving high-efficiency with simple purely pitching mechanical systems and paves the way for the design of high-efficiency bio-inspired underwater vehicles.

Effect of body stiffness distribution on larval fish-like efficient undulatory swimming

Energy-efficient propulsion is a critical design target for robotic swimmers. Although previous studies have pointed out the importance of nonuniform body bending stiffness distribution (k) in improving the undulatory swimming efficiency of adult fish-like robots in the inertial flow regime, whether such an elastic mechanism is beneficial in the intermediate flow regime remains elusive. Hence, we develop a class of untethered soft milliswimmers consisting of a magnetic composite head and a passive elastic body with different k These robots realize larval zebrafish-like undulatory swimming at the same scale. Investigations reveal that uniform k and high swimming frequency (60 to 100 Hz) are favorable to improve their efficiency. A shape memory polymer-based milliswimmer with tunable k on the fly confirms such findings. Such acquired knowledge can guide the design of energy-efficient leading edge-driven soft undulatory milliswimmers for future environmental and biomedical applications in the same flow regime.

The effect of variable stiffness of tuna-like fish body and fin on swimming performance

The work in this paper focuses on the examination of the effect of variable stiffness distributions on the kinematics and propulsion performance of a tuna-like swimmer. This is performed with the use of a recently developed fully coupled fluid-structure interaction solver. The two different scenarios considered in the present study are the stiffness varied along the fish body and the caudal fin, respectively. Our results show that it is feasible to replicate the similar kinematics and propulsive capability to that of the real fish via purely passive structural deformations. In addition, propulsion performance improvement is mainly dependent on the better orientation of the force near the posterior part of swimmers towards the thrust direction. Specifically, when a variable body stiffness scenario is considered, the bionic body stiffness profile results in better performance in most cases studied herein compared with a uniform stiffness commonly investigated in previous studies. Given the second scenario, where the stiffness is varied only in the spanwise direction of the tail, similar tail kinematics to that of the live scombrid fish only occurs in association with the heterocercal flexural rigidity profile. The resulting asymmetric tail conformation also yields performance improvement at intermediate stiffness in comparison to the cupping and uniform stiffness.

The role of suction thrust in the metachronal paddles of swimming invertebrates

An abundance of swimming animals have converged upon a common swimming strategy using multiple propulsors coordinated as metachronal waves. The shared kinematics suggest that even morphologically and systematically diverse animals use similar fluid dynamic relationships to generate swimming thrust. We quantified the kinematics and hydrodynamics of a diverse group of small swimming animals who use multiple propulsors, e.g. limbs or ctenes, which move with antiplectic metachronal waves to generate thrust. Here we show that even at these relatively small scales the bending movements of limbs and ctenes conform to the patterns observed for much larger swimming animals. We show that, like other swimming animals, the propulsors of these metachronal swimmers rely on generating negative pressure along their surfaces to generate forward thrust (i.e., suction thrust). Relying on negative pressure, as opposed to high pushing pressure, facilitates metachronal waves and enables these swimmers to exploit readily produced hydrodynamic structures. Understanding the role of negative pressure fields in metachronal swimmers may provide clues about the hydrodynamic traits shared by swimming and flying animals.

Enhanced flight performance in non-uniformly flexible wings

The flexibility of biological propulsors such as wings and fins is believed to contribute to the higher performance of flying and swimming animals compared with their engineered peers. Flexibility seems to follow a universal design rule that induces bending patterns at about one-third from the distal tip of the propulsor’s span. However, the aerodynamic mechanisms that shaped this convergent design and the potential improvement in performance are not well understood. Here, we analyse the effect of heterogeneous flexibility on the flight performance (range and descent angle) of passively tumbling wings. Using experiments, numerical simulations, and scaling analysis, we demonstrate that spanwise tip flexibility that follows this empirical rule leads to improved flight performance. Improvement in flight range seems to be related to flutter-induced drag reduction. This mechanism is independent of the wing’s auto-rotation and represents a more general trait of wings with non-uniform tip flexibility.

The effect of sensory feedback topology on the entrainment of a neural oscillator with a compliant foil for swimming systems

The sensorimotor system of fish endows them with remarkable swimming performance that is unmatched by current underwater robotic vehicles. To close the gap between the capabilities of fish and the capabilities of underwater vehicles engineers are investigating how fish swim. In particular, engineers are exploring the sensorimotor systems of fish that control the motion of fins. It is generally accepted that specialized neural circuits (known as central pattern generators) within the sensorimotor system produce the periodic drive signal that is used to control the motion of fins. An important aspect of these circuits is that their output signal can be modified by sensory feedback. Specifically, the way in which sensory feedback signals are applied to a CPG (i.e. the sensory feedback topology) affects the CPG's entrainment characteristics. This has been shown in simulation but has not been investigated in a robot interacting in the real-world. Furthermore, CPG-based control has only limitedly been applied to fish like robots and many questions remain as to how it should be applied to these types of systems. In this work we examine the effect of sensory feedback topology on the entrainment characteristics of a CPG-based neural oscillator driving three different foils swimming in flow. Additionally, we investigate how sensory feedback should be acquired from a foil and applied to a neural oscillator to promote beneficial swimming characteristics.

Fish-like aquatic propulsion studied using a pneumatically-actuated soft-robotic model

Fish locomotion is characterized by waves of muscle electrical activity that proceed from head to tail, and result in an undulatory pattern of body bending that generates thrust during locomotion. Isolating the effects of parameters like body stiffness, co-activation between the right and left sides of the body, and frequency on thrust generation has proven to be difficult in live fishes. We use a pneumatically-actuated fish-like model to investigate how these parameters affect locomotor force generation. We measure thrust as well as side forces and torques generated during propulsion. Using a statistical linear model we examine the effects of input parameter combinations on thrust generation. We show that both stiffness and frequency substantially affect swimming kinematics, and that there are complex interactive effects of these two parameters on thrust. The stiffer the backbone the more impact that increasing frequency has on thrust production. For stiffer models, increasing frequency resulted in higher values for both thrust and lateral forces. Large side forces reduce swimming efficiency but this effect could be mitigated by decreasing undulatory wavelength and allowing appropriate phasing of left and right body movements to reduce amplitudes of side force.

Fluid-structure interaction modeling on a 3D ray-strengthened caudal fin

In this paper, we present a numerical model capable of solving the fluid-structure interaction problems involved in the dynamics of skeleton-reinforced fish fins. In this model, the fluid dynamics is simulated by solving the Navier-Stokes equations using a finite-volume method based on an overset, multi-block structured grid system. The bony rays embedded in the fin are modeled as nonlinear Euler-Bernoulli beams. To demonstrate the capability of this model, we numerically investigate the effect of various ray stiffness distributions on the deformation and propulsion performance of a 3D caudal fin. Our numerical results show that with specific ray stiffness distributions, certain caudal fin deformation patterns observed in real fish (e.g. the cupping deformation) can be reproduced through passive structural deformations. Among the four different stiffness distributions (uniform, cupping, W-shape and heterocercal) considered here, we find that the cupping distribution requires the least power expenditure. The uniform distribution, on the other hand, performs the best in terms of thrust generation and efficiency. The uniform stiffness distribution, per se, also leads to 'cupping' deformation patterns with relatively smaller phase differences between various rays. The present model paves the way for future work on dynamics of skeleton-reinforced membranes.

Stable formations of self-propelled fish-like swimmers induced by hydrodynamic interactions

Fish schools are fascinating examples of macro-scale systems with collective behaviours. According to conventional wisdom, the establishment and maintenance of fish schools probably need very elaborate active control mechanisms. Sir James Lighthill posited that the orderly formations in fish schools may be an emergent feature of the system as a result of passive hydrodynamic interactions. Here, numerical simulations are performed to test Lighthill's conjecture by studying the self-propelled locomotion of two, three and four fish-like swimmers. We report the emergent stable formations for a variety of configurations and examine the energy efficiency of each formation. The result of this work suggests that the presence of passive hydrodynamic interactions can significantly mitigate the control challenges in schooling. Moreover, our finding regarding energy efficiency also challenges the widespread idea in the fluid mechanics community that the diamond-shaped array is the most optimized pattern.

Intermittent locomotion of a fish-like swimmer driven by passive elastic mechanism

The intermittent locomotion performance of a fish-like elastic swimmer is studied numerically in this paper. The actuation is imposed only at the head and the locomotion is indirectly driven by passive elastic mechanism. For intermittent swimming, certain time durations of passive coasting are interspersed between two half-periods of active bursting. To facilitate the comparison of energy efficiencies in continuous and intermittent swimming at the same cruising speed, we consider both intermittent swimming at various duty cycles and also continuous swimming at reduced actuation frequencies. The result indicates that the intermittent style is more economical than the continuous style only when the cruising Reynolds number is sufficiently large and the duty cycle is moderate. We also explore the passive tail-beating pattern and wake structure for intermittent swimming. It is found that the kinematics of the tail contains a preparatory burst phase which lies in between the active bursting and the passive coasting phases. Three vortex streets are found in the wake structures behind the intermittent swimmers. The two oblique streets consist of strong vortex dipoles and the horizontal street is made up of weak vortices. The results of this study can provide some insight into the burst-and-coast swimming of fish and also inform the design of efficient bio-mimetic under-water vehicles.

Robotics-inspired biology

For centuries, designers and engineers have looked to biology for inspiration. Biologically inspired robots are just one example of the application of knowledge of the natural world to engineering problems. However, recent work by biologists and interdisciplinary teams have flipped this approach, using robots and physical models to set the course for experiments on biological systems and to generate new hypotheses for biological research. We call this approach robotics-inspired biology; it involves performing experiments on robotic systems aimed at the discovery of new biological phenomena or generation of new hypotheses about how organisms function that can then be tested on living organisms. This new and exciting direction has emerged from the extensive use of physical models by biologists and is already making significant advances in the areas of biomechanics, locomotion, neuromechanics and sensorimotor control. Here, we provide an introduction and overview of robotics-inspired biology, describe two case studies and suggest several directions for the future of this exciting new research area.

Dynamics and locomotion of flexible foils in a frictional environment

Over the past few decades, oscillating flexible foils have been used to study the physics of organismal propulsion in different fluid environments. Here, we extend this work to a study of flexible foils in a frictional environment. When the foil is oscillated by heaving at one end but is not free to locomote, the dynamics change from periodic to non-periodic and chaotic as the heaving amplitude increases or the bending rigidity decreases. For friction coefficients lying in a certain range, the transition passes through a sequence of N-periodic and asymmetric states before reaching chaotic dynamics. Resonant peaks are damped and shifted by friction and large heaving amplitudes, leading to bistable states. When the foil is free to locomote, the horizontal motion smoothes the resonant behaviours. For moderate frictional coefficients, steady but slow locomotion is obtained. For large transverse friction and small tangential friction corresponding to wheeled snake robots, faster locomotion is obtained. Travelling wave motions arise spontaneously, and move with horizontal speeds that scale as transverse friction coefficient to the power 1/4 and input power that scales as the transverse friction coefficient to the power 5/12. These scalings are consistent with a boundary layer form of the solutions near the foil's leading edge.

A pressure-based force and torque prediction technique for the study of fish-like swimming

Many outstanding questions about the evolution and function of fish morphology are linked to swimming dynamics, and a detailed knowledge of time-varying forces and torques along the animal’s body is a key component in answering many of these questions. Yet, quantifying these forces and torques experimentally represents a major challenge that to date prevents a full understanding of fish-like swimming. Here, we develop a method for obtaining these force and torque data non-invasively using standard 2D digital particle image velocimetry in conjunction with a pressure field algorithm. We use a mechanical flapping foil apparatus to model fish-like swimming and measure forces and torques directly with a load cell, and compare these measured values to those estimated simultaneously using our pressure-based approach. We demonstrate that, when out-of-plane flows are relatively small compared to the planar flow, and when pressure effects sufficiently dominate shear effects, this technique is able to accurately reproduce the shape, magnitude, and timing of locomotor forces and torques experienced by a fish-like swimmer. We conclude by exploring of the limits of this approach and its feasibility in the study of freely-swimming fishes.

Long-axis twisting during locomotion of elongate fishes

Fish live in a complex world and must actively adapt their swimming behavior to a range of environments. Most studies of swimming kinematics focus on two-dimensional properties related to the bending wave that passes from head to tail. However, fish also twist their bodies three dimensionally around their longitudinal axis as the bending wave passes down the body. We measured and characterized this movement, which we call 'wobble', in six species of elongate fishes (Anoplarchus insignis, Xiphister mucosus, Lumpenus sagitta, Pholis laeta, Apodichthys flavidus and Ronquilus jordani) from three different habitats (intertidal, nearshore and subtidal) using custom video analysis software. Wobble and bending are synchronized, with a phase shift between the wobble wave and bending wave. We found that species from the same habitats swim in similar ways, even if they are more closely related to species from different habitats. In nearshore species, the tail wobbles the most but, in subtidal and intertidal species, the head wobbles more than or the same as the tail. We also wanted to understand the relationship between wobble and the passive mechanics of the fish bodies. Therefore, we measured torsional stiffness and modulus along the body and found that modulus increases from head to tail in all six species. As wobble does not correlate with the passive properties of the body, it may play a different role in swimming behavior of fishes from different habitats.

How the bending kinematics of swimming lampreys build negative pressure fields for suction thrust

Swimming animals commonly bend their bodies to generate thrust. For undulating animals such as eels and lampreys, their bodies bend in the form of waves that travel from head to tail. These kinematics accelerate the flow of adjacent fluids, which alters the pressure field in a manner that generates thrust. We used a comparative approach to evaluate the cause-and-effect relationships in this process by quantifying the hydrodynamic effects of body kinematics at the body-fluid interface of the lamprey, Petromyzon marinus, during steady-state swimming. We compared the kinematics and hydrodynamics of healthy control lampreys to lampreys whose spinal cord had been transected mid-body, resulting in passive kinematics along the posterior half of their body. Using high-speed particle image velocimetry (PIV) and a method for quantifying pressure fields, we detail how the active bending kinematics of the control lampreys were crucial for setting up strong negative pressure fields (relative to ambient fields) that generated high-thrust regions at the bends as they traveled all along the body. The passive kinematics of the transected lamprey were only able to generate significant thrust at the tail, relying on positive pressure fields. These different pressure and thrust scenarios are due to differences in how active versus passive body waves generated and controlled vorticity. This demonstrates why it is more effective for undulating lampreys to pull, rather than push, themselves through the fluid.

Optimal specific wavelength for maximum thrust production in undulatory propulsion

What wavelengths do undulatory swimmers use during propulsion? In this work we find that a wide range of body/caudal fin (BCF) swimmers, from larval zebrafish and herring to fully-grown eels, use specific wavelength (ratio of wavelength to tail amplitude of undulation) values that fall within a relatively narrow range. The possible emergence of this constraint is interrogated using numerical simulations of fluid-structure interaction. Based on these, it was found that there is an optimal specific wavelength (OSW) that maximizes the swimming speed and thrust generated by an undulatory swimmer. The observed values of specific wavelength for BCF animals are relatively close to this OSW. The mechanisms underlying the maximum propulsive thrust for BCF swimmers are quantified and are found to be consistent with the mechanisms hypothesized in prior work. The adherence to an optimal value of specific wavelength in most natural hydrodynamic propulsors gives rise to empirical design criteria for man-made propulsors.

Maximizing propulsive thrust of a driven filament at low Reynolds number via variable flexibility

At low Reynolds numbers the locomotive capability of a body can be dramatically hindered by the absence of inertia. In this work, we show how propulsive performance in this regime can be significantly enhanced by employing spatially varying flexibility. As a prototypical example, we consider the propulsive thrust generated by a filament periodically driven at one end. The rigid case leads to zero propulsion, as so constrained by Purcell's scallop theorem, while for uniform filaments there exists a bending stiffness maximizing the propulsive force at a given frequency; here we demonstrate explicitly how considerable further improvement can be achieved by simply varying the stiffness along the filament. The optimal flexibility distribution is strongly configuration-dependent: while increasing the flexibility towards the tail-end enhances the propulsion of a clamped filament, for a hinged filament decreasing the flexibility towards the tail-end is instead favorable. The results reveal new design principles for maximizing propulsion at low Reynolds numbers, potentially useful for developing synthetic micro-swimmers requiring large propulsive force for various biomedical applications.

Effects of stiffness distribution and spanwise deformation on the dynamics of a ray-supported caudal fin

Using a fluid-structure interaction model, we study the effect of ray stiffness distribution on the deformation and performance of a caudal fin. By prescribing a simple swaying motion, our results show that through passive structural deformation alone it is possible to reproduce some complicated fin movements (e.g. the cup and 'W'-shape deformations) observed in real fish. Moreover, it has been numerically shown that, compared with a fin with uniform ray stiffness, at the same (average) ray stiffness fins with nonuniform stiffness distribution may achieve further performance enhancement, e.g. increase in propulsion efficiency and decrease in lateral force generation. This is attributed to spanwise deformations caused by phase differences between different rays.

Performance variation due to stiffness in a tuna-inspired flexible foil model

Tuna are fast, economical swimmers in part due to their stiff, high aspect ratio caudal fins and streamlined bodies. Previous studies using passive caudal fin models have suggested that while high aspect ratio tail shapes such as a tuna's generally perform well, tail performance cannot be determined from shape alone. In this study, we analyzed the swimming performance of tuna-tail-shaped hydrofoils of a wide range of stiffnesses, heave amplitudes, and frequencies to determine how stiffness and kinematics affect multiple swimming performance parameters for a single foil shape. We then compared the foil models' kinematics with published data from a live swimming tuna to determine how well the hydrofoil models could mimic fish kinematics. Foil kinematics over a wide range of motion programs generally showed a minimum lateral displacement at the narrowest part of the foil, and, immediately anterior to that, a local area of large lateral body displacement. These two kinematic patterns may enhance thrust in foils of intermediate stiffness. Stiffness and kinematics exhibited subtle interacting effects on hydrodynamic efficiency, with no one stiffness maximizing both thrust and efficiency. Foils of intermediate stiffnesses typically had the greatest coefficients of thrust at the highest heave amplitudes and frequencies. The comparison of foil kinematics with tuna kinematics showed that tuna motion is better approximated by a zero angle of attack foil motion program than by programs that do not incorporate pitch. These results indicate that open questions in biomechanics may be well served by foil models, given appropriate choice of model characteristics and control programs. Accurate replication of biological movements will require refinement of motion control programs and physical models, including the creation of models of variable stiffness.

Optimal chordwise stiffness profiles of self-propelled flapping fins

The versatility of fish to adapt to different swimming requirements is attributed to their complex muscular system. Fish modulate their fin stiffness and shape for maximized performance. In this paper, optimal chordwise stiffness profiles that maximize the propulsive performance have been predicted using theoretical studies. An experimental setup has been fabricated to measure the stiffness profiles of real fish caudal fins. Chordwise varying stiffness robotic fins fabricated using carbon fiber reinforced composites (CFRC) have been tested in the water tunnel to evaluate their performance over constant stiffness fins. It is observed that the varying stiffness fins produce larger thrusts and efficiencies compared to constant stiffness fins for all the operating conditions considered in this work. A comparison of the digital image correlation (DIC) measured deformations of the fins showed that the better performance of varying stiffness fins is due to their larger curvatures and trailing edge amplitudes. These theoretical and experimental studies provide a greater understanding of the role of stiffness in fish fins for locomotion.

Development of a bio-inspired transformable robotic fin

Fish swim by oscillating their pectoral fins forwards and backwards in a cyclic motion such that their geometric parameters and aspect ratios change according to how fast or slow a fish wants to swim; these complex motions result in a complicated hydrodynamic response. This paper focuses on the dynamic change in the shape of a fin to improve the underwater propulsion of bio-inspired mechanism. To do this, a novel transformable robotic fin has been developed to investigate how this change in shape affects the hydrodynamic forces acting on the fin. This robotic fin has a multi-link frame and a flexible surface skin where changes in shape are activated by a purpose designed multi-link mechanism driven by a transformation motor. A drag platform has been designed to study the performance of this variable robotic fin. Numerous experiments were carried out to determine how various controlling modes affect the thrust capability of this fin. The kinematic parameters associated with this robotic fin include the oscillating frequency and amplitude, and the drag velocity. The fin has four modes to control the cyclic motion; these were also investigated in combination with the variable kinematic parameters. The results will help us understand the locomotion performance of this transformable robotic fin. Note that different controlling modes influence the propulsive performance of this robotic fin, which means its propulsive performance can be optimized in a changing environment by adapting its shape. This study facilitates the development of bio-inspired unmanned underwater vehicles with a very high swimming performance.

Self-propelled swimming simulations of bio-inspired smart structures

This paper presents self-propelled swimming simulations of a foldable structure, whose folded configuration is a box. For self-locomotion through water the structure unfolds and undulates. To guide the design of the structure and understand how it should undulate to achieve either highest speed or maximize efficiency during locomotion, several kinematic parameters were systematically varied in the simulations: the wave type (standing wave versus traveling wave), the smoothness of undulations (smooth undulations versus undulations of rigid links), the mode of undulations (carangiform: mackerel-like versus anguilliform: eel-like undulations), and the maximum amplitude of undulations. We show that the swimmers with standing wave are slow and inefficient because they are not able to produce thrust using the added-mass mechanism. Among the tested types of undulation at low Reynolds number (Re) regime of [Formula: see text] (Strouhal number of about 1.0), structures that employ carangiform undulations can swim faster, whereas anguilliform swimmers are more economic, i.e., using less power they can swim a longer distance. Another finding of our simulations is that structures which are made of rigid links are typically less efficient (lower propulsive and power efficiencies and also lower swimming speed) compared with smoothly undulating ones because a higher added-mass force is generated by smooth undulations. The wake of all the swimmers bifurcated at the low Re regime because of the higher lateral relative to the axial velocity (high Strouhal number) that advects the vortices laterally creating a double row of vortices in the wake. In addition, we show that the wake cannot be used to predict the performance of the swimmers because the net force in each cycle is zero for self-propelled bodies and the pressure term is not negligible compared to the other terms.

Mechanisms of anguilliform locomotion in fishes studied using simple three-dimensional physical models

Physical models enable researchers to systematically examine complex and dynamic mechanisms of underwater locomotion in ways that would be challenging with freely swimming animals. Previous research on undulatory locomotion, for example, has used rectangular flexible panels that are effectively two-dimensional as proxies for the propulsive surfaces of swimming fishes, but these bear little resemblance to the bodies of elongate eel-like swimming animals. In this paper we use a polyurethane rod (round cross-section) and bar (square cross-section) to represent the body of a swimming Pacific hagfish (Eptatretus stoutii). We actuated the rod and bar in both heave and pitch using a mechanical controller to generate a propulsive wave at frequencies between 0.5 and 2.5 Hz. We present data on (1) how kinematic swimming patterns change with driving frequency in these elongate fish-like models, (2) the thrust-generating capability of these simple models, (3) how forces and work done during propulsion compare between cross-sectional shapes, (4) the wake flow patterns in these swimming models using particle image velocimetry. We also contrast kinematic and hydrodynamic patterns produced by bar and rod models to comparable new experimental data on kinematics and wake flow patterns from freely swimming hagfish. Increasing the driving frequency of bar and rod models reduced trailing edge amplitude and wavelength, and above 2 Hz a nodal point appeared in the kinematic wave. Above 1 Hz, both the rod and bar generated net thrust, with the work per cycle reaching a minimum at 1.5 Hz, and the bar always requiring more work per cycle than the rod. Wake flow patterns generated by the swimming rod and bar included clearly visible lateral jets, but not the caudolaterally directed flows seen in the wakes from freely swimming hagfish.