Large-amplitude Elongated-body Theory Research Articles

Nonlinear analysis of compliant robotic fish locomotion

Compliant robotic fish can achieve a better swimming performance than rigid-bodied robotic fish. Therefore, this article investigates the swimming behavior of the compliant robotic fish based on a new swimming model that combines the large-amplitude elongated-body theory with decoupled natural orthogonal complement matrices. The simulation reveals that the multi-order resonances are generated in tail-beat amplitude, forward speed, stride length, and transport efficiency when the compliant robotic fish is driven at the corresponding frequency. Moreover, the resonant effects demonstrate the nonlinear behaviors as the driving torque increases. A control strategy for resonance utilization is presented to improve the performance capabilities. The potential influence factors for resonant effects are also discussed, showing that the tail-generated hydrodynamic force significantly impacts the resonant effect. These nonlinear characteristics can provide important guidelines for the motion control of the compliant robotic fish.

Model Predictive Control-Based Path-Following for Tail-Actuated Robotic Fish

There has been an increasing interest in the use of autonomous underwater robots to monitor freshwater and marine environments. In particular, robots that propel and maneuver themselves like fish, often known as robotic fish, have emerged as mobile sensing platforms for aquatic environments. Highly nonlinear and often under-actuated dynamics of robotic fish present significant challenges in control of these robots. In this work, we propose a nonlinear model predictive control (NMPC) approach to path-following of a tail-actuated robotic fish that accommodates the nonlinear dynamics and actuation constraints while minimizing the control effort. Considering the cyclic nature of tail actuation, the control design is based on an averaged dynamic model, where the hydrodynamic force generated by tail beating is captured using Lighthill's large-amplitude elongated-body theory. A computationally efficient approach is developed to identify the model parameters based on the measured swimming and turning data for the robot. With the tail beat frequency fixed, the bias and amplitude of the tail oscillation are treated as physical variables to be manipulated, which are related to the control inputs via a nonlinear map. A control projection method is introduced to accommodate the sector-shaped constraints of the control inputs while minimizing the optimization complexity in solving the NMPC problem. Both simulation and experimental results support the efficacy of the proposed approach. In particular, the advantages of the control projection method are shown via comparison with alternative approaches.

Analytic Modeling for Precise Speed Tracking of Multilink Robotic Fish

In this paper, a novel analytical-based kinematic and dynamic modeling and control approach is developed for reference speed tracking of a multijoint robotic fish. The kinematic model is improved by the introduction of input signal for controlling body undulation amplitude without distorting the model structure. Next, the dynamic model is developed in the discrete-time domain primarily because fish swimming pattern generates oscillating continuous thrust where the effect of thrust over one cycle of undulation can be calculated through averaging. Hence, the dynamics can be discretized according to the undulation rate. The novelty in developing the discrete-time dynamical model is three fold. First, the thrust gradient is derived with assistance of Lighthill's large amplitude elongated body theory. More importantly, affine-in-control dynamical model form, essential for designing a real-time model-based control algorithm, is derived by inclusion of the updated kinematic model. Second, the thrust delay is studied via pulse input response of the motion system, and incorporated in the dynamical model. Third, drag force coefficient calculation is also proposed by comparing the thrust model and empirically recorded steady-state speeds of the robotic fish. Finally, discrete-time model-based sliding-mode control (SMC) is designed for speed tracking. Experimental results verify that SMC based on the proposed dynamical model, can significantly improve speed tracking performance of a multilink robotic fish.

Self-similar kinematics among efficient slender swimmers

We present an analysis of efficient undulatory propulsion for slender animals swimming at high Reynolds number. Using Lighthill’s large-amplitude elongated-body theory, we show that optimally efficient swimming kinematics can be characterized through a single dimensionless variable$\unicode[STIX]{x1D713}$. This variable,$\unicode[STIX]{x1D713}$, is defined by a simple function of a swimming animal’s body wave properties. Physically,$\unicode[STIX]{x1D713}$characterizes how the velocity of an animal’s tail varies throughout its swimming stroke. Lighthill’s model predicts that swimming efficiency is near optimal in the range$0.3<\unicode[STIX]{x1D713}<1.0$and peaks at$\unicode[STIX]{x1D713}=0.87$. At this point, the average magnitude of the tail velocity is minimized and swimming kinematics are tuned such that the thrust coefficient is as close to constant as possible throughout the swimming stroke. We use a compiled dataset of over 250 unique measurements to show that species across a wide range of size and shape fall within the optimal region.

Modelling of an actuated elastic swimmer

We study the force production dynamics of undulating elastic plates as a model for fish-like inertial swimmers. Using a beam model coupled with Lighthill’s large-amplitude elongated-body theory, we explore different localised actuations at one extremity of the plate (heaving, pitching and a combination of both) in order to quantify the reactive and resistive contributions to the thrust. The latter has the form of a quadratic drag in large Reynolds number swimmers and has recently been pointed out as a crucial element in the thrust force balance. We validate the output of a weakly nonlinear solution to the fluid–structure model using thrust force measurements from an experiment with flexible plates subjected to the three different actuation types. The model is subsequently used in a self-propelled configuration – with a skin friction model that balances thrust to produce a constant cruising speed – to map the reactive versus resistive thrust production in a parameter space defined by the aspect ratio and the actuation frequency. We show that this balance is modified as the frequency of excitation changes and the response of the elastic plate shifts between different resonant modes, the pure heaving case being the most sensitive to the modal response with drastic changes in the reactive/resistive contribution ratio along the frequency axis. We analyse also the role of the phase lag between the heaving and pitching components in the case of combined actuation, showing in particular a non-trivial effect on the propulsive efficiency.

Design and dynamic modeling of electrorheological fluid-based variable-stiffness fin for robotic fish

Fish actively control their stiffness in different swimming conditions. Inspired by such an adaptive behavior, in this paper we study the design, prototyping, and dynamic modeling of compact, tunable-stiffness fins for robotic fish, where electrorheological (ER) fluid serves as the enabling element. A multi-layer composite fin with an ER fluid core is prototyped and utilized to investigate the influence of electrical field on its performance. Hamilton's principle is used to derive the dynamic equations of motion of the flexible fin, and Lighthill's large-amplitude elongated-body theory is adopted to estimate the hydrodynamic force when the fin undergoes base-actuated rotation. The dynamic equations are then discretized using the finite element method, to obtain an approximate numerical solution. Experiments are conducted on the prototyped flexible ER fluid-filled beam for parameter identification and validation of the proposed model, and for examining the effectiveness of electrically controlled stiffness tuning. In particular, it is found that the natural frequency is increased by almost 40% when the applied electric field changes from 0 to .

Synchronized flutter of two slender flags

The interactions and synchronization of two parallel and slender flags in a uniform axial flow are studied in the present paper by generalizing Lighthill’s elongated body theory (EBT) and Lighthill’s large-amplitude elongated body theory (LAEBT) to account for the hydrodynamic coupling between flags. The proposed method consists of two successive steps, namely the reconstruction of the flow created by a flapping flag within the LAEBT framework and the computation of the fluid force generated by this non-uniform flow on the second flag. In the limit of slender flags in close proximity, we show that the effect of the wakes has little influence on the long-time coupled dynamics and can be neglected in the modelling. This provides a simplified framework extending LAEBT to the coupled dynamics of two flags. Using this simplified model, both linear and large-amplitude results are reported to explore the selection of the flapping regime as well as the dynamical properties of two side-by-side slender flags. Hydrodynamic coupling of the two flags is observed to destabilize the flags for most parameters, and to induce a long-term synchronization of the flags, either in-phase or out-of-phase.

Averaging Tail-Actuated Robotic Fish Dynamics Through Force and Moment Scaling

Averaging of robotic fish dynamics is of interest for the purposes of path planning and controller design due to the rhythmic movement of the robot. For a faithful dynamic model of robotic fish, however, classical averaging or geometric averaging typically cannot produce an average model that is accurate and in the meantime amenable to analysis or control design. In this paper, a novel averaging approach is proposed for tail-actuated robotic fish dynamics, in which the tail-generated hydrodynamic force is modeled with the Lighthill's large-amplitude elongated-body theory. The approach consists of scaling the force and moment terms and, then, conducting classical averaging. Numerical investigation reveals that the scaling function for the force term is a constant independent of tail-beat patterns, while the scaling function for the moment term depends linearly on the tail-beat bias. Extensive simulation and experimental results, comparing the predictions from the original and average models, are presented to support the effectiveness of the averaging approach. Existence and local stability of the equilibria for the average model are further analyzed. These equilibria are subsequently used to obtain an analytical solution for steady turning parameters, such as turning period and turning radius, without running simulation of the original or average dynamic models.

Dynamic Modeling of Robotic Fish With a Base-Actuated Flexible Tail

In this paper, we develop a new dynamic model for a robotic fish propelled by a flexible tail actuated at the base. The tail is modeled by multiple rigid segments connected in series through rotational springs and dampers, and the hydrodynamic force on each segment is evaluated using Lighthill's large-amplitude elongated-body theory. For comparison, we also construct a model using linear beam theory to capture the beam dynamics. To assess the accuracy of the models, we conducted experiments with a free-swimming robotic fish. The results show that the two models have almost identical predictions when the tail undergoes small deformation, but only the proposed multisegment model matches the experimental measurement closely for all tail motions, demonstrating its promise in the optimization and control of tail-actuated robotic fish.

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

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

A dynamic model for tail-actuated robotic fish with drag coefficient adaptation

In this paper we present a dynamic model for a tail-actuated robotic fish by merging rigid-body dynamics with Lighthill’s large-amplitude elongated-body theory. The model is validated with extensive experiments conducted on a robotic fish prototype. We investigate the role of incorporating the body motion in evaluating the tail-generated hydrodynamic force, and show that ignoring the body motion (as often done in the literature) results in significant overestimate of the thrust force and robot speed. By exploiting the strong correlation between the angle of attack and the tail-beat bias, a computationally efficient approach is further proposed to adapt the drag coefficients of the robotic fish, and its effectiveness is supported by experimental results.

Note on the swimming of an elongated body in a non-uniform flow

AbstractThis paper presents an extension of Lighthill’s large-amplitude elongated-body theory of fish locomotion which enables the effects of an external weakly non-uniform potential flow to be taken into account. To do so, the body is modelled as a Kirchhoff beam, made up of elliptical cross-sections whose size may vary along the body, undergoing prescribed deformations consisting of yaw and pitch bending. The fluid velocity potential is decomposed into two parts corresponding to the unperturbed potential flow, which is assumed to be known, and to the perturbation flow. The Laplace equation and the corresponding Neumann’s boundary conditions governing the perturbation velocity potential are expressed in terms of curvilinear coordinates which follow the body during its motion, thus allowing the boundary of the body to be considered as a fixed surface. Equations are simplified according to the slenderness of the body and the weakness of the non-uniformity of the unperturbed flow. These simplifications allow the pressure acting on the body to be determined analytically using the classical Bernoulli equation, which is then integrated over the body. The model is finally used to investigate the passive and the active swimming of a fish in a Kármán vortex street.

Biomechanics of swimming in the pufferfish Diodon holocanthus: propulsive momentum enhancement is an adaptation for thrust production in an undulatory median and paired-fin swimmer

A form of large-amplitude elongated-body theory appropriate for the analysis of undulatory fins attached to a rigid body of elliptical section suggests a benefit due to momentum enhancement relative to the fins on their own. This theoretical prediction is experimentally confirmed for the first time. Theoretical momentum enhancement factors for Diodon holocanthus (2.2 and 2.7 for the median and pectoral fins, respectively) compared well to inferred thrust values determined from particle-image velocimetry (PIV) wake measurements (2.2-2.4 and 2.7-2.9). Caudal fin mean theoretical thrust was not significantly different from measured (PIV) values (n = 24, P > 0.05), implying no momentum enhancement. Pectoral-fin thrust was half that of the median and caudal fins due to high fin-jet angles, low circulation and momentum. Average total fin thrust and fish drag were not significantly different (n = 24, P > 0.05). Vortex rings generated by the fins were elliptical, with size dependent on fin chord and stroke amplitude. Hydrodynamic advantages (thrust enhancement at no cost to hydrodynamic efficiency, reduction of side forces minimizing energy wasting yawing motions and body drag) are probably common among rigid-bodied organisms propelled by undulatory fins. A trade-off between momentum enhancement and the rate of momentum generation (thrust force) sets a practical limit to the former. For small fins whilst momentum enhancement is high, absolute thrust is low. In addition, previously suggested limitations on thrust enhancement set by reductions in propulsive force associated with progressive reductions in fin wavelength are found to be biologically unrealistic.

Research on the Hydrodynamic Forces of the Torsional Wave Propulsion by Extended Large-Amplitude Elongated-Body Theory

Torsional wave propulsion is different from transversal wave propulsion. Based on the characteristic feature of fin ray’s motion, we divide the dorsal fin along the radial direction into a series of long narrow bands. For each band, the analysis method is the same as the method computing hydrodynamic force of transversal wave propulsion. Then we give an integral calculation over fin ray’ length. We analyze the influence of the swing amplitude, frequency and length of fin as well as wave numbers and ratio of wave speed to swimming speed on hydrodynamic force. The calculated results show that: 1. the average thrust on the dorsal fin is directly proportional to the square of swing amplitude and frequency of fin ray. 2. The average thrust on the dorsal fin is directly proportional to the biquadrate of ray’s length. 3. As the wave numbers increase, the average thrust begins with a little increase and then turns to decrease. When the wave number roughly equals to 1.5, it reaches to a maximum. 4. If the fin frequency is fixed, the average thrust is of maximum when the ratio of wave speed to swimming speed roughly equals to 1.4.

Three-dimensional extension of Lighthill's large-amplitude elongated-body theory of fish locomotion

The goal of this paper is to derive expressions for the pressure forces and moments acting on an elongated body swimming in a quiescent fluid. The body is modelled as an inextensible and unshearable (Kirchhoff) beam, whose cross-sections are elliptic, undergoing prescribed deformations, consisting of yaw and pitch bending. The surrounding fluid is assumed to be inviscid, and irrotational everywhere, except in a thin vortical wake. The Laplace equation and the corresponding Neumann boundary conditions are first written in terms of the body coordinates of a beam treating the body as a fixed surface. They are then simplified according to the slenderness of the body and its kinematics. Because the equations are linear, the velocity potential is sought as a sum of two terms which are linked respectively to the axial movements of the beam and to its lateral movements. The lateral component of the velocity potential is decomposed further into two sub-components, in order to exhibit explicitly the role of the two-dimensional potential flow produced by the lateral motion of the cross-section, and the role played by the curvature effects of the beam on the cross-sectional flow. The pressure, which is given by Bernoulli's equation, is integrated along the body surface, and the expressions for the resultant and the moment are derived analytically. Thereafter, the validity of the force and moment obtained analytically is checked by comparisons with Navier–Stokes simulations (using Reynolds-averaged Navier–Stokes equations), and relatively good agreements are observed.

Poincaré–Cosserat Equations for the Lighthill Three-dimensional Large Amplitude Elongated Body Theory: Application to Robotics

In this article, we describe a dynamic model of the three-dimensional eel swimming. This model is analytical and suited to the online control of eel-like robots. The proposed solution is based on the Large Amplitude Elongated Body Theory of Lighthill and a framework recently presented in Boyer et al. (IEEE Trans. Robot. 22:763–775, 2006) for the dynamic modeling of hyper-redundant robots. This framework was named “macro-continuous” since, at this macroscopic scale, the robot (or the animal) is considered as a Cosserat beam internally (and continuously) actuated. This article introduces new results in two directions. Firstly, it extends the Lighthill theory to the case of a self-propelled body swimming in three dimensions, while including a model of the internal control torque. Secondly, this generalization of the Lighthill model is achieved due to a new set of equations, which are also derived in this article. These equations generalize the Poincare equations of a Cosserat beam to an open system containing a fluid stratified around the slender beam.

Research on swimming by undulatory long dorsal fin propulsion

The kinematics of steady forward swimming of Gymnarchus niloticus is described. The geometric features of the body and locomotive characteristic and parameters of the flexible dorsal fin are discussed. On the basis of observation and experimental data, a simplified kinematic model on locomotion of the undulatory long dorsal fin propulsion is promoted. The hydromechanical performances of the undulatory long dorsal fin propeller of G. niloticus are estimated with the large-amplitude elongated-body theory. The hydromechanical efficiency of the undulatory long dorsal fin system ranged from 81.664% to 86.420% over a speed range of 0.728–0.985 length · s−1. It is suggested that the undulatory long dorsal fin propulsion is an adaptation to swimming with high hydromechanical efficiency.

RESEARCH ON SWIMMING BY UNDULATORY LONG DORSAL FIN PROPULSION

The kinematics of steady forward swimming of Gymnarchus niloticus are described. The geometric features of the body and locomotive characteristic and parameters of the flexible dorsal fin are discussed. A simplified kinematic model on locomotion of the undulatory long dorsal fin propulsion is based on the the observation and experimental data. The hy-dromechanical performances of the undulatory long dorsal fin propeller of Gymnarchus niloticus are estimated applying the large-amplitude elongated-body theory. The hydromechanical efficiency of the undulatory long dorsal fin system ranged from 82.284% to 87.455% over a speed rang of 0.728-0.985 lengthes· s-1. It is suggested that the undulatory long dorsal fin propulsion is an adaptation to swimming with high hydromechanical efficiency.

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 1. Biological background, and analysis by elongated-body theory

Elongated-body theory has been fruitfully applied over twenty years to the biofluiddynamic analysis of modes of locomotion of elongated fishes by means of body flexure, with special emphasis on the anguilliform mode using undulatory body movements, and on the carangiform mode where oscillatory movements of only a fish's posterior end(including the caudal fin) exhibit phase lag of posterior movements behind anterior movements just as in an undulation yet not nearly as much as a whole wavelength is apparent at any one time. The extension of elongated-body theory to analyse the locomotion of elongated fishes with elongated median fins (dorsal and/or anal) in modes where the body (together with any caudal fin) remains rigid, being propelled forwards by undulations or oscillations of those median fins, has long been recognized as desirable but is here presented for the first time.In many large groups of fishes, evolutionary adaptation to limited environments (such as coral reefs) favoured a development of defensive ‘armour’ at the expense of speed, to such an extent that bodies became essentially inflexible, with locomotion achieved by fin movements alone. In one principal group of such fishes, however (the sub-order Balistoidei including the trigger-fishes), a later evolutionary development restored a capacity for relatively high-speed movement even though the body remained essentially rigid. The balistiform mode of locomotion, with propulsion achieved by synchronized movements of the dorsal and anal fins, exists in two alternative forms, with either undulatory or oscillatory movements of these median fins, that are analogous to the anguilliform and carangiform modes of body flexure, respectively.Analysis in this paper throws light on the puzzling question of why trigger-fishes are able to move so fast notwithstanding the modest extent of their fin movements. A form of the large-amplitude elongated-body theory, specially adapted to balistiform locomotion, allows a direct comparison of thrust and efficiency for different modes of propulsion. The conclusions in brief are that thrust is dominated by the mean rate of shedding of backward momentum at the posterior end of the fish's propulsive apparatus and that, for movements of median fins attached to a deep, essentially rigid body, this momentum is increased (above the momentum expected for the same movements of the fins ‘on their own’) by a momentum enhancement factor β of around 3 or a little more. Yet there is no such enhancement of the rate of shedding of ‘unproductive’ energy into the wake; accordingly, overall efficiency is improved. Also, especially for the undulatory mode of balistiform locomotion, sideforces are minimized so that the fish body avoids sideslip and yaw; accordingly, the body drag which fm thrust must overcome is reduced by another large factor.Alongside discussion of the Balistoidei, this paper reviews and analyses balistiform locomotion as observed in several other groups, including groups of flexible-bodied fishes that regularly use this mode as a low-energy-cost alternative to locomotion by means of body flexure. Finally, we similarly analyse gymnotiform locomotion, in which the body is again held rigid, being propelled by undulations in just a single (ventral) fin, and compare and contrast different interpretations of its advantages.

New Kinematic Data From High Speed Cine Film Recordings of Swimming Cod (Gadus Morhua)

1. We described a method to determine the values of the parameters of the body wave: speed v, wavelength λ b , wave period T b and of the sinusoidal track in space of different points of the body of a swimming fish: speed u, wavelength λ s , wave period T s and amplitude A. 2. These characteristic wave parameters were measured from high speed cine recordings of straight forward swimming behaviour of one 0.42 m Cod. 3. The instantaneous propulsive force of the tail was determined using LIGHTHILL'S (1971) large amplitude elongated body theory. 4. The same theory predicts and data in this paper affirm, that, to increase propulsive force per unit virtual mass, a fish should increase the lateral speed of the tail tip and should keep the angle between tail blade and direction of movement not far from values between -10° and + 10°. 5. Results show that swimming at higher speeds as well as acceleration is always attended with an increase of v as a result of a decrease of the period of the waves of the body. The body wavelength λ b does not change with speed. The wavelength of the tracks in space, λ s , slightly decreases with an increase of speed. 6. Relationships between some of the characteristic features of the cycles of swimming movements are discussed. With the aid of a simple model, limits of swimming abilities of the fish studied could be determined. 7. We came to the conclusion that propeller efficiency during steady swimming at cruising speeds decreases with increasing speed.