Study of flexible fin and compliant joint stiffness on propulsive performance: theory and experiments

A Fast Online Elastic-Spine-Based Stiffness Adjusting Mechanism for Fishlike Swimming

Fish tunes fishtail stiffness by coordinating its tendons, muscles, and other tissues to improve swimming performance. For robotic fish, achieving a fast and online fishlike stiffness adjustment over a large-scale range is of great significance for performance improvement. This article proposes an elastic-spine-based variable stiffness robotic fish, which adopts spring steel to emulate the fish spine, and its stiffness is adjusted by tuning the effective length of the elastic spine. The stiffness can be switched in the maximum adjustable range within 0.26 s. To optimize the motion performance of robotic fish by adjusting fishtail stiffness, a Kane-based dynamic model is proposed, based on which the stiffness adjustment strategy for multistage swimming is constructed. Simulations and experiments are conducted, including performance measurements and analyses in terms of swimming speed, thrust, and so on, and online stiffness adjustment-based multistage swimming, which verifies the feasibility of the proposed variable stiffness robotic fish. The maximum speed and lowest cost of transport for robotic fish are 0.43 m/s (equivalent to 0.81 BL/s) and 7.14 J/(kg·m), respectively.

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.

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.

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.

Hydrodynamic stress maps on the surface of a flexible fin-like foil

We determine the time dependence of pressure and shear stress distributions on the surface of a pitching and deforming hydrofoil from measurements of the three dimensional flow field. Period-averaged stress maps are obtained both in the presence and absence of steady flow around the foil. The velocity vector field is determined via volumetric three-component particle tracking velocimetry and subsequently inserted into the Navier-Stokes equation to calculate the total hydrodynamic stress tensor. In addition, we also present a careful error analysis of such measurements, showing that local evaluations of stress distributions are possible. The consistency of the force time-dependence is verified using a control volume analysis. The flapping foil used in the experiments is designed to allow comparison with a small trapezoidal fish fin, in terms of the scaling laws that govern the oscillatory flow regime. As a complementary approach, unsteady Euler-Bernoulli beam theory is employed to derive instantaneous transversal force distributions on the flexible hydrofoil from its deflection and the results are compared to the spatial distributions of hydrodynamic stresses obtained from the fluid velocity field.

Comprehensive analysis of efficient swimming using articulated legs fringed with flexible appendages inspired by a water beetle

Drag-based swimming is usually accompanied with the shape change of rowing appendages to generate asymmetric force during the power stroke and recovery stroke. To implement this in an aquatic robot, one may actively control the surface area of its legs during the swimming. However, a small sized robot with a limited number of actuators should adjust the surface area of legs in passive manner. For this reason, we proposed a novel articulated leg with flexible appendages inspired by a water beetle. These leg structures were designed to implement an efficient recovery stroke with less resistive force during the recovery stroke, while its surface area was increased again if suitable relaxation time was applied to perform improved power stroke. To identify an optimal leg design, 36 different types were fabricated by changing the passive joint thickness, appendage materials, length, and morphology. Several correlations and dominant parameters were identified, and it was shown that the swimming leg with fixed joint and appendage stiffness cannot always generate the largest torque in all the swimming frequency. Also, a two-dimensional dynamic model was proposed based on an underactuated manipulator, and the model validation was proceeded by comparing with two selected leg designs. In addition, a 5.5 cm long robot with one pair of legs was built to further investigate their locomotory performance. By varying the beating frequency and relaxation time, thorough analysis was addressed in terms of the position, velocity, non-dimensional traveled distance, Strouhal number, and quasi-propulsive efficiency. Here, some important relationships between dimensionless numbers were established. Furthermore, it was found that introducing a relaxation phase between the power stroke and recovery stroke can increase the traveled distance per stroke with slight expense of propulsive efficiency.

Propulsive performance of an under-actuated robotic ribbon fin

Many aquatic animals propelled by elongated undulatory fins can perform complex maneuvers and swim with high efficiency at low speeds. In this propulsion, one or multiple waves travel along an elastic fin composed of flexible rays. In this study, we explore the potential benefits or disadvantages of passive fin motion based on the coupling of fluid-structure interactions and elasto-mechanical responses of the undulatory fin. The motivation is to understand how an under-actuated undulating fin can modify its active and passive fin motion to effectively control the hydrodynamic force and propulsive efficiency. We study the kinematics and propulsive performance of an under-actuated ribbon fin using a robotic device. During two experimental sets for fully-actuated fin and under-actuated fin respectively, we measured fin kinematics, surge forces and power consumption. Our results show that under-actuated fin can generate smaller thrust but consume less power comparing to a fully-actuated counterpart. The thrust generated by an under-actuated fin scales similarly to a fully-actuated fin-linear with the enclosed area and quadratic with the relative velocity. Power consumption scales with cube of lateral tangential velocity. Furthermore, we find that the under-actuated fin can keep the same propulsive efficiency as the fully-actuated fin at low relative velocities. This finding has profound implications to both natural swimmers and underwater vehicles using undulating fin-based propulsion, as it suggests that they can potentially exploit passive fin motion without decrementing propulsive efficiency. For underwater vehicles with undulatory fins, an under-actuated design can greatly simplify the mechanical design and control complexity of a versatile propulsion system.

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.

Theoretical and experimental study on a compliant flipper-leg during terrestrial locomotion

An amphibious robot with straight compliant flipper-legs can conquer various amphibious environments. The robot can rotate its flipper-legs and utilize their large deflection to walk on rough terrain, and it can oscillate the straight flipper-legs to propel itself underwater. This paper focuses on the dynamics of the compliant straight flipper-legs during terrestrial locomotion by modeling its deformation dynamically with large deflection theory and simulating it to investigate the parameters of locomotion such as trajectory, velocity, and propulsion. To validate the theoretical model of dynamic locomotion, a single-leg experimental platform is used to explore the flipper-legs in motion with various structural and kinematic parameters. Furthermore, a robotic platform mounting with four compliant flipper-legs is also developed and used to experiment with locomotion. The trajectories of the rotating axle of the compliant flipper-leg during locomotion were approximately coincidental in simulation and in experiments. The speed of locomotion and cost of transport during locomotion were explored and analyzed. The performance of different types of compliant flipper-legs during locomotion shows that varying the degrees of stiffness will have a significant effect on their locomotion. The dynamic model and analysis of the compliant flipper-leg for terrestrial locomotion facilitates the ability of amphibious robots to conquer complex environments.

Bio-inspired flexible joints with passive feathering for robotic fish pectoral fins

In this paper a novel flexible joint is proposed for robotic fish pectoral fins, which enables a swimming behavior emulating the fin motions of many aquatic animals. In particular, the pectoral fin operates primarily in the rowing mode, while undergoing passive feathering during the recovery stroke to reduce hydrodynamic drag on the fin. The latter enables effective locomotion even with symmetric base actuation during power and recovery strokes. A dynamic model is developed to facilitate the understanding and design of the joint, where blade element theory is used to calculate the hydrodynamic forces on the pectoral fins, and the joint is modeled as a paired torsion spring and damper. Experimental results on a robotic fish prototype are presented to illustrate the effectiveness of the joint mechanism, validate the proposed model, and indicate the utility of the proposed model for the optimal design of joint depth and stiffness in achieving the trade-off between swimming speed and mechanical efficiency.

The Effects of Fluidic Loading on Underwater Contact Sensing with Robotic Fins and Beams

As robots become more involved in underwater operations, understanding underwater contact sensing with compliant systems is fundamental to engineering useful haptic interfaces and vehicles. Despite knowledge of contact sensation in air, little is known about contact sensing underwater and the impact of fluid on both the robotic probe and the target object. The objective of this work is to understand the effects of fluidic loading, fin webbing, and target object geometry on strain sensation within compliant robotic fins and beams during obstacle contact. General descriptions of obstacle contact were sought for strain measurements in fins and beams. Multiple phases of contact were characterized where the robot, fluid, and object interact to affect sensory signals. Unlike in air, the underwater structure-fluid-structure interaction (SFSI) caused changes to strain in each phase of contact. The addition of webbing to beams created a mechanical coupling between adjacent beams, which changed contact strains. Complex obstacle geometries tended to make contact less apparent and caused stretch in fins. This work demonstrates several effects of fluidic loading on strain sensing with compliant robotic beams and fins as they contact obstacles in air and underwater, and provides guidance for future work in underwater active sensing with compliant manipulators.

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

Simple mechanical models emulating fish have been used recently to enable targeted study of individual factors contributing to swimming locomotion without the confounding complexity of the whole fish body. Yet, unlike these uniform models, the fish body is notable for its non-uniform material properties. In particular, flexural stiffness decreases along the fish's anterior-posterior axis. To identify the role of non-uniform bending stiffness during fish-like propulsion, we studied four foil model configurations made by adhering layers of plastic sheets to produce discrete regions of high (5.5 × 10(-5) Nm(2)) and low (1.9 × 10(-5) Nm(2)) flexural stiffness of biologically-relevant magnitudes. This resulted in two uniform control foils and two foils with anterior regions of high stiffness and posterior regions of low stiffness. With a mechanical flapping foil controller, we measured forces and torques in three directions and quantified swimming performance under both heaving (no pitch) and constant 0° angle of attack programs. Foils self-propelled at Reynolds number 21 000-115 000 and Strouhal number ∼0.20-0.25, values characteristic of fish locomotion. Although previous models have emphasized uniform distributions and heaving motions, the combination of non-uniform stiffness distributions and 0° angle of attack pitching program was better able to reproduce the kinematics of freely-swimming fish. This combination was likewise crucial in maximizing swimming performance and resulted in high self-propelled speeds at low costs of transport and large thrust coefficients at relatively high efficiency. Because these metrics were not all maximized together, selection of the 'best' stiffness distribution will depend on actuation constraints and performance goals. These improved models enable more detailed, accurate analyses of fish-like swimming.