Inspired by the relatively simple morphological blueprint provided by batoid fish such as stingrays and skates, we create a biohybrid system that enables an artificial animal, a tissue-engineered ray, * Correspondence to: K.K.P. 29 Oxford Street, Pierce Hall Cambridge, MA 02130. kkparker@seas.harvard.edu. Phone: 617-495-2850, 617-835-5920. Fax: 617-496-1793 Bioinspired design, as applied to robotics, aims at implementing naturally occurring features such as soft materials, morphologies, gaits, and control mechanisms in artificial settings to improve performance (1-4). For example, recent soft-robotics studies raised awareness on the importance of material properties (3, 4), shifting the focus from rigid elements to soft materials, while other investigations report successful mimicry of gaits or morphological features inspired by insects (5, 6), fish (7,8), snake (9), salamanders (10) and cheetahs (11). While recent advances have the promise of bridging the performance gap with animals, the current soft-robotic actuators based on, for instance, electroactive polymers, shape memory alloys or pressurized fluids, are yet to mature to the point of replicating the high-resolution complex movements of biological muscles (3, 4).In this context, biosensors and bioactuators (12) are intriguing alternatives, since they can intrinsically respond to a number of control inputs (such as electric fields and optical stimulation). Thanks to recent advances in genetic tools (13) and tissue engineering (12), these responses can be altered and tuned across a wide range of time and length scales. Some pioneering studies have exploited these technologies for self-propulsion, developing miniaturized walking machines (14-16), and flagellar (17) or jellyfish inspired (18) swimming devices. These biohybrid systems operate at high energy efficiency and harvest power from energy dense, locally available nutrients, although at present they require specialized environments (physiological solutions) that may limit their applicability. Moreover and most importantly, these biohybrid locomotors lack of the reflexive control (9, 19) necessary to enable adaptive maneuvering and thus of the ability to respond to spatiotemporally varying external stimuli.Here, we design, build and test a tissue-engineered analog of a batoid fish such as stingrays and skates. By combining soft materials and tissue engineering with optogenetics, we created an integrated sensory-motor system that allowed for coordinated undulating fin movement and phototactically controlled locomotion, that is guided via light stimuli. We drew from fish morphology, neuromuscular dynamics and gait control to implement a living, bio-hybrid system that leads to robust and reproducible locomotion and turning maneuvers. Batoid fish are ideal biological models in robotics (8) because their nearly planar bauplan is characterized by a broad dorsoventral disk, with a flattened body and extended pectoral fins, that enhances stability against roll (20). They swim with high energy efficien...

Inertial aquatic swimmers that use undulatory gaits range in length L from a few millimetres to 30 metres, across a wide array of biological taxa. Using elementary hydrodynamic arguments, we uncover a unifying mechanistic principle characterizing their locomotion by deriving a scaling relation that links swimming speed U to body kinematics (tail beat amplitude A and frequency ω) and fluid properties (kinematic viscosity ν). This principle can be simply couched as the power law Re ∼ Sw α , where Re = UL/ν 1 and Sw = ωAL/ν, with α = 4/3 for laminar flows, and α = 1 for turbulent flows. Existing data from over 1,000 measurements on fish, amphibians, larvae, reptiles, mammals and birds, as well as direct numerical simulations are consistent with our scaling. We interpret our results as the consequence of the convergence of aquatic gaits to the performance limits imposed by hydrodynamics. Aquatic locomotion entails a complex interplay between the body of the swimmer and the induced flow in the environment 1,2. It is driven by motor activity and controlled by sensory feedback in organisms ranging from bacteria to blue whales 3. Locomotion at low Reynolds numbers (Re = UL/ν 1) is governed by linear hydrodynamics and is consequently analytically tractable, whereas locomotion at high Reynolds numbers (Re 1) involves nonlinear inertial flows and is less well understood 4. Although this is the regime that most macroscopic creatures larger than a few millimetres inhabit, as shown in Fig. 1a, the variety of sizes, morphologies and gaits makes it difficult to construct a unifying framework across taxa. Thus, most studies have tried to take a more limited view by quantifying the problem of swimming in specific situations from experimental, theoretical and computational standpoints 5-7. Beginning more than fifty years ago, experimental studies 8-12 started to quantify the basic kinematic properties associated with swimming in fish, while providing grist for later theoretical models. Perhaps the earliest and still most comprehensive of these studies was performed by Bainbridge 8 , who correlated size and frequency f = ω/(2π) of several fish via the empirical linear relation U /L = (3/4)f − 1. However, he did not provide a mechanistic rationale based on fundamental physical principles. The work of Bainbridge served as an impetus for a variety of theoretical models of swimming. The initial focus was on investigating thrust production associated with body motion at high Reynolds numbers, wherein inertial effects dominates viscous forces 13,14. Later models also accounted for the elastic properties of the body and muscle activity 15-18. The recent advent of numerical methods coupled with the availability of fast, cheap computational resources has triggered a new generation of direct numerical simulations to accurately resolve the full three-dimensional problem 19-24. Although these provide detailed descriptions of the forces and flows during swimming, the large computational data sets associated with specific problems obscure the sea...

Soft slender structures are ubiquitous in natural and artificial systems, in active and passive settings and across scales, from polymers and flagella, to snakes and space tethers. In this paper, we demonstrate the use of a simple and practical numerical implementation based on the Cosserat rod model to simulate the dynamics of filaments that can bend, twist, stretch and shear while interacting with complex environments via muscular activity, surface contact, friction and hydrodynamics. We validate our simulations by solving a number of forward problems involving the mechanics of passive filaments and comparing them with known analytical results, and extend them to study instabilities in stretched and twisted filaments that form solenoidal and plectonemic structures. We then study active filaments such as snakes and other slender organisms by solving inverse problems to identify optimal gaits for limbless locomotion on solid surfaces and in bulk liquids.

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