Patent ID: 12240283

DETAILED DESCRIPTION

This disclosure describes devices that use the preferential buckling of curved beams which, by a passive reduction of effective area in recovery stroke, generates positive net thrust and moment. The devices utilize a design concept featuring an under-actuated compliant fin system that uses slender curved beams and their ability to buckle preferentially in one direction under symmetric motor inputs to produce net thrust and moments. The locomotion system is assessed using dynamic modeling and simulation, as well as experimental evaluation of a swimming robot that utilizes the proposed fin design to enhance maneuverability by switching between rowing and flapping gaits across different swimming scenarios. The air drag exerted on a wing utilizing curved beam buckling was also analyzed.

Buckling is a condition in which small geometric perturbations lead to drastic reductions in load-carrying capacity in structural systems.FIG.1Adepicts introduction of curvature into a beam. A flat, slender, compliant beam102shows little resistance towards bending. However, by using a template104to apply force to the beam102, the resulting curved beam106resists bending in the direction opposing its camber (known as opposite sense bending) more than when the curved beam is bent in the direction of the beam's camber (equal sense bending). The influence of curvature results in different buckling limits in equal and opposite sense bending as well. Furthermore, this phenomenon is also controllable by considering the effective length of the curved beam, as shown in plots provided inFIG.1Bfor a short curved beam110and a long curved beam112. When a curved beam with a shorter effective length is subjected to a load, both the stiffness and buckling occur more symmetrically, and beam stiffness is larger, when compared to a curved beam with a longer effective length subjected to a load.

FIGS.2A-2Ddepict different methods to achieve net thrust in locomotion.FIG.2Adepicts a mechanism in which the beam does not buckle. The fin (or wing)202is coupled to a motorized joint204by a rigid beam206. Nonzero net thrust or movement is produced by a faster power stroke in a first direction (e.g., downward) and a slower recovery stroke in a second direction that can be opposite the first direction (e.g., upward).FIG.2Bdepicts the use of a multi-actuation system to reconfigure the beam for recovery. In this mechanism two motorized joints204and208are used. This approach introduces the trajectory hysteresis needed for locomotion but can suffer from higher complexity as well as a heavier and less efficient system.FIG.2Cdepicts a soft robotic approach that uses a flexible joint210alongside rigid joint limit212to produce positive net thrust and moment. The deformation of flexible joints during recovery repositions the fin to reduce drag, while rigid joint limits prevent bending during the power stroke, keeping the fin system better-positioned to push against the surrounding fluid. This approach, while effective at reducing control complexity, is not actively reconfigurable.

Referring toFIG.2D, this disclosure describes passive rowing using curved beams214that preferentially buckle within a rowing or flapping cycle. This fin system design produces net thrust and moments through symmetric sinusoidal actuation of a single actuator, resulting in a simple and energy efficient approach. Moreover, the characteristics of slender curved beams provide an opportunity to tune the system's dynamic behavior by altering its stiffness as shown inFIG.1B. By changing the effective length of a curved beam, the buckling can be inhibited unidirectionally or bi-directionally, enabling a switch between a rowing gait with a net forward thrust (when actuated as a pair) and a flapping gait, generating lateral thrust. When a pair of such fins are used together, a number of other swimming modalities may be observed as well. This tunability, which is made possible through internal reconfiguration of the curved beam, splits the use of actuators according to their purpose—power and reconfiguration, permitting the use of machine learning approaches to find optimal gaits for different swimming modalities in a decoupled fashion, where tuning actuators are first determined, with a subsequent, independent optimization of the power actuator signal.

FIGS.3A-3Dillustrate the function of a fin (or wing) mechanism.FIG.3Ashows a load/displacement curve for a fin system302shown inFIG.3B. A fin304is coupled to an electric servo motor306by a compliant, curved beam308that buckles at two different locations along the positive310and negative312portion of its load/displacement curve, corresponding to opposite and equal sense bending directions.FIG.3Ashows that controlling the amount of force exerted on the end of the curved beam in positive and negative directions can avoid buckling in both directions, permit buckling in one direction, or buckle the curved beam in both directions. The curved beam is bistable. When actuated in a fluid such as air or water, dynamics of powered, symmetric flapping can result in general cyclic flapping patterns, as depicted inFIGS.3B-3E. A first regime is typified by slow flapping below the buckling limit in either direction, where drag and inertial forces remain low, shown inFIG.3D. In this case, the curved beam308acts like a simple bending beam. Referring toFIG.3E, little asymmetric behavior is observed in its flapping path or in the average thrust generated over a cycle. In a second regime, depicted inFIG.3B, the flapping velocity is sufficient to buckle the curved beam308in the equal-sense bending direction but not in the opposite direction. This results in the curved beam308undergoing large deflections about the buckling point during roughly half of its flapping cycle, which allows the larger surface area of the fin304to travel nearly parallel with the direction of motion, rather than perpendicular. This different angle of attack can result in reduced drag forces on the wing304during the recovery phase of the stroke. As the cycle reenters the power phase of the stroke, drag allows the wing to open back up in the other direction and remain perpendicular to the direction of motion. This difference in overall drag experienced by the wing in power and recovery phases generates nonzero average work over a single flapping cycle, depicted inFIG.3C, even with a symmetric input from the motor306, In a third regime, the curved beam buckles in both directions due at least in part to high torques exerted by the motor306that increase the drag and inertial forces experienced at the tip of the curved beam past the buckling limit in both directions.

FIGS.4A-4Ddepict different locomotion mechanisms used for a swimming mechanical system (e.g., a robot) in a fluid (e.g., water).FIG.4Adepicts a flapping mechanism for a single fin402, in which motion of the fin402is generally perpendicular to the direction of travel. The fin402is coupled to a curved beam404. The curved beam404is coupled to a motor406that is configured to impart a flapping motion to the curved beam404. The input from the motor406is symmetric. The curved beam404is configured to buckle at two different points along the positive and negative portions of its load/displacement curve, corresponding to opposite and equal sense bending directions, as shown inFIG.3A. For the flapping mechanism, the load is maintained below the buckling limit of the curved beam404in the positive and negative portions of the load/displacement curve, and the curved beam404does not buckle. The dashed arrows indicate the direction of travel for the end of the fin402, with movement in a first direction (e.g., downward or clockwise, as depicted by arrows p) defined as the power phase and movement in a second direction (e.g., upward or counterclockwise, as depicted by arrows r), defined as the recovery phase. The dashed lines indicate the position of the end of the fin402throughout the flapping cycle. For the flapping mechanism, the trajectory of the end of the fin402is the same for the power phase and the recovery phase.FIG.4Bdepicts a rowing mechanism and is characterized by reciprocating motions directed generally parallel with the direction of travel. During the recovery phase of the fin402, the load on the curved beam404is greater than the equal sense critical buckling limit, and the curved beam404buckles, reducing drag on the fin. Buckling of the curved beam404occurs only in the equal sense bending direction. The larger surface of the curved beam404travels substantially parallel with the direction of motion of the mechanical system. During the power phase, the load on the curved beam404is below the critical opposite sense buckling limit, and the fin402maintains full extension and experiences higher drag forces. The drag forces on the curved beam404in the power phase exceed drag forces on the curved beam404during the recovery phase, and this difference in drag forces generates nonzero average work over a rowing cycle.FIG.4Cdepicts lateral swimming using one fin flapping. The dashed arrows depict the position of the end of the fin throughout the recovery and power phases. The wide arrow depicts the direction of motion of the robot.FIG.4Ddepicts forward swimming using two fins408and410employing a rowing mechanism. The dashed lines depict the position of the end of the fins for the recovery (up) and power (down) phases, and the wide arrow shows the direction of motion of the robot.FIG.4Edepicts an undulating mechanism produced using two fins412and414driven by motor416in an asynchronous, symmetric flapping gait. The wide arrow depicts the direction of motion of the robot.FIG.4Fdepicts a turning mechanism produced by driving only one fin at its optimal gait.

A servomechanism can be coupled to the curved beam404and the motor406. The motor406can be configured to activate the servomechanism, and the servomechanism can be configured to impart the flapping motion to the curved beam404. The servomechanism can be configured to impart the flapping motion by controlling an amount of force exerted on an end of the curved beam404.

EXAMPLES

FIG.5shows the nonlinear behavior of two specimens with the same curvature (180°) and width (25.4 mm), and with effective lengths of 31.75 mm and 3.6 mm. For each specimen, the curved beam is coupled at one end to a fixed plate while a known force is applied to the other end. A force sensor mounted to the output of a linear actuator pushes on the curved beam by a small, 3D printed contact point. The linear actuator moves back and forth through a 50 mm range in 10 μm increments while forces and torques are logged. Since the curved beams are modeled as a flexible hinge, the sampled data is displayed by their equivalent torque and deformation angles. These data also show that by reducing the beam length, the critical buckling limit in equal sense bending increases from 0.1 Nm to 0.65 Nm. The limit for opposite sense buckling, however, increases from 0.76 Nm to 1.02 Nm. Thus, for fluid-dynamic loads between 0.1 Nm and 0.65 Nm, altering the effective length changes the buckling condition from unidirectional to bi-directional, resulting in the gait switching from rowing to flapping.

Dynamic modeling. The dynamics of the system are demonstrated by modeling the contribution of the wing's drag, the curved beam's stiffness (in the long configuration), and the inertial effects of each body. The model for the robot, depicted inFIG.6, uses two fins602and604coupled to the main robot's body606at a distance of do. Each fin is represented by rigid links608,610,612and614(d1, m1and d2, m2) with point masses616,618,620, and622located at their centers of mass, coupled by pin joints624and626and torsional springs628and630, with stiffness coefficient K, coupled in parallel. The nonlinear stiffness of the spring is represented by three linear regimes. The slopes of each of these regimes have been adjusted to best fit experimental data collected from the specimen in its long effective length configuration. Using the flat plate model, the forces on a fin due to a fluid are estimated by using equations can be derived from:
Fw=ρu2Asin α,  (1)
where ρ, u, A, and α are the density of fluid, the relative velocity of the plate, the area of the plate, and the angle-of-attack of the wing, respectively. α is 0 when parallel to the flow and 90° when perpendicular (in two dimensions). This force is perpendicular to the wing and acts as the fluid's dynamic load on the distal end of curved beam.

Using Eq. 1, the velocity of the plate (u) can be used to control the amount of drag force exerted on it, which, in conjunction with the load limits determined by the mechanics of the curved beam, determines whether and under what conditions buckling occurs. The flat plate model can best describe the fluid dynamics of a system when the Reynolds number is low and the system is in the laminar regime. Due at least in part to the simplicity of the flat plate model, it is used in the simulation to reduce computation time and keep the optimization process tractable. The simulation is performed with a Python-based dynamics package called Pynamics. This library derives the Equation of Motion (EOM) using Kane's method, which is then integrated to determine the system's state over time. The performance of the model is evaluated by comparing the moments generated by one fin against data collected experimentally. By defining two forces coupling the robot to the ground (kG and bG inFIG.6), the forces and moments exerted on the environment about the rotational axis when one fin is actuated can be measured in simulation.

When a sinusoidal input torque is applied to the base joint of a fin, the dynamic model demonstrates that the wing system transitions between a non-buckling flapping regime to a one-sided buckling regime when the input frequency increases. From the modeling data, the wing system transitions from the non-buckling regime to one-sided buckling at around 0.3 Hz, where the maximum positive torque increases with frequency in the power stroke, but the maximum negative torque in the recovery section remains low. Data in Table 1 show an acceptable correlation between the generated torques in this test and the values estimated by the dynamic model. Based on this performance, robot swimming is simulated by removing the forces holding the robot's main body (kG=bG=0). A drag force acting on the main body is also considered.

TABLE 1Torques (in Nm) generated in simulation and experimentSimulationExperimentFrequencyτminτmaxτminτmaxBuckling0.1−0.040.04−0.040.04No0.2−0.140.14−0.140.15No0.3−0.230.3−0.130.32One side0.4−0.240.46−0.120.44One side

Design optimization. Using the dynamic model introduced above, the design that maximizes forward swimming speed for symmetric rowing gaits can be found. In the optimization, the lengths of the fin's links and the distance between the robot's drive motors632and634are considered (d1, d2, d3and d0inFIG.6). The mass of each link is based on measurements of physical prototypes. Actuator inputs are optimized simultaneously. The torques at motorized joints track desired angular trajectories (θ1and θ2inFIG.6) by kGand bGas mentioned previously. Input signals are supplied as a pair of sinusoidal functions,
θ1=β1+α1sin(2πf1t)
θ2=β1+α2sin(2πf2t+φ)  (2)
where θiis actuator i's angle, and βi, αi, fi, and φi, are the sinusoidal signals' angular offset, amplitude, frequency, and phase shift, respectively. In order to have synchronized rowing gaits for the purposes of forward rowing, these parameters are set to α1=−α2, β1=−β2, f1=f2, and φ=0. Based on the design and input gaits parameters introduced above, there are at least seven parameters affecting the robot's swimming speed. A numerical optimization approach using an evolution strategy has been selected for finding the optimal parameters. While the parameter space can be searched for lower-dimensional problems, CMA-ES can be used as a way to find ideal parameters within a seven-dimensional space.

CMA-ES is an evolution strategy that uses stochastic methods to numerically solve nonlinear and non-convex optimization problems. Using an evolution strategy like CMA-ES in practical experiments can have advantages compared to other meta-heuristic and search-based algorithms. CMA-ES is a suitable example of an optimization tool in robotics due at least in part to its short evaluation time compared to other strategies, which has practical benefits including increasing the service life of motors, bearings, and gears that can become worn or damaged during training.

In the optimization process, the cost function can be defined as the negative of the swimming range that robot achieves in 10 seconds. The following assumptions and constraints are used to simplify the optimization process:

Assumptions: (i) Water drag is applied to the main body606and fins602and604(FBand FWinFIG.6), but not to the links. (ii) Drag is applied to the center of each geometry. (iii) Fins and main body have rectangular cross-sections with 80 and 50 mm widths, respectively. (iv) The robot body's mass, mostly driven by the mass of servos and electronics, is assumed to be constant.
Constraints: (i) Variables remain within the ranges defined by Table 2. (ii) The total length of the robot is under 560 mm (to fit the water tank). (iii) Actuation speed and power remain within the servo's nominal speed and power range. (iv) Loads on the curved beam remain below opposite sense critical load. (v) Design and gait parameters do not collide during actuation.

TABLE 2Parameter range in design and input optimizationParameterRangeParameterRanged0(mm)40-160d1(mm)30-160d2(mm)30-160d3(mm)30-160α (deg)0-90β (deg)−90-90f (Hz)0.1-1.2ϕ (deg)0-359

A penalty function is defined to exclude nonfeasible solutions, in which a large positive value proportional to the number of violated constraints is returned. The penalty function gradually restricts the large search space to converge within the feasible solution space of the problem. For feasible solutions, the dynamic simulation runs and the cost function returned. The results converged after 25 iterations, revealing that designs with a smaller distance between the fins (d0) as well as smaller second link length (d2) are preferential for maximizing swimming speed. The optimal design parameters (in mm) are d0=40, d1=112.1, d2=30.2, and d3=114.2.

FIG.7Ashows an embodiment of a swimming robot700using the above design parameters. Two Hitec D646WP waterproof servo motors702and704actuate the input joints. A slider mechanism including a pulley706and compression springs710and712control the position of sliders714and716with curved slots. The curved beams718and720are coupled to the fins722and724. The lengths of the rigid parts are calculated assuming that the curved beam bends at the midpoint. This assumption is made based on the observation of the curved beam bending underwater. The rigid links are 3D printed from Onyx. The fin is cut from 0.76 mm fiberglass sheets.

The slider mechanism alters the effective length of the curved beam based on the position of the sliders714and716relative to the curved beams718and720. In the configuration depicted inFIG.7A, the curved beams718and720are exposed and the curved beams are in a neutral buckling position. The sliders' neutral configuration is set by the compression springs710and712as the beam's natural length, which allows one-sided buckling—and thus rowing—at lower forces. To activate flapping, a single actuator coupled to pulley706pulls both fins' sliders714and716forward over the curved beams718and720by a tendon and pulley system, inhibiting buckling when the beams are loaded. The curved beam's stiffness in this configuration is higher and more symmetric (in the equal and opposite sense), as seen in the dashed line inFIG.5. By using the slider mechanism, the characteristics of the buckling wing can be actively reconfigured during use.FIGS.7B and7Cshow the position of the slider714for a long and short effective beam length, respectively.

Experimental Gait Optimization. Optimal gaits for various swimming maneuvers have been determined using the robot shown inFIG.7A, which was based on the optimal results obtained from simulations. To evaluate the performance of each gait, a test apparatus was implemented to sample the thrust and moment generated by the fins, or the swimming distance and speed. A linear carriage running on a rail is installed on the top of a 4×2×2 ft3tank (length×width×height). The robot's position was measured using an OptiTrack motion tracking system. An ATI Mini-40 force/torque sensor was coupled to the carriage; its distal connector coupled to the robot by an aluminum extension arm that held the robot underwater. An optimization process was then performed using CMA-ES to find the optimal parameters of each desired gait by determining optimal parameters for control rule.

Rowing Gaits: In this optimization process, the robot swam 5 seconds with the curved beam in its long configuration shown inFIG.7B. The distance traveled in that time was then measured. A servo and pulley resets the carriage to its initial position at the end of each trial. In order to protect the curved beams from damage due to gaits that exceed a safe operating range, a joint limit was temporarily coupled during the optimization process, and optimal gaits were then re-tested once they are determined to be safe. The forces generated during recovery stroke were limited to −1 N, while the power stroke achieved 7.5 N thrust at its peak.

Flapping Gaits: By reducing the effective length of the curved beam718using the slider configuration shown inFIG.7C, its stiffness increases in both the opposite and equal-sense directions, changing the curved beam's behavior to be both stiffer and more symmetric. Using this phenomenon, lateral thrusts can be generated by flapping one limb (α2=β2=f2=0) with a sinusoidal input while maintaining a neutral offset in the other (β1=0). A new set of optimal gait parameters was obtained by searching through the resulting two-dimensional space of gait parameters, but the lateral thrust found by this approach was unable to overcome rail stiction. As such, a thrust-based metric was adopted. The result shows that a flapping gait with relatively large input amplitude (α=87°) and low frequency (f=0.3 Hz) is optimal.

By commanding both limbs to perform asynchronous, symmetric flapping gaits (φ≠0°), the robot swimming mode changes to undulation as depicted inFIG.4E, similar to snakes, eels, and Purcell's three-link swimmer. The three-dimensional space of input parameters related to undulation was searched to obtain the highest lateral thrust undulating gait that exceeds the lateral single-limb maximum swimming speed found above. Stiction was addressed by a free-swimming prototype that was constructed by mounting the swimming mechanism to a floating platform that ensures the fins stay underwater, while keeping power electronics above the water. Using the optimal gaits obtained from the experimental search, the untethered robot's performance was evaluated.

Using the optimal rowing gait, the robot achieves a forward swimming speed of about 0.32 m/s. The swimming distance per rowing cycle is around 0.6 m. The robot is also able to turn, as depicted inFIG.4F, when only one limb is commanded by the same optimal gait. The turning speed is 25.7 deg/s. When flapping, the untethered robot achieves a lateral swimming speed of 0.17 m/s when only one limb is actuated. When undulating as depicted inFIG.4E, the robot achieves a swimming speed of 0.16 m/s. It was observed that, while the thrust data is better for an undulating strategy, the swimming speed of the untethered robot was lower than others, which is attributed to the higher drag of the floating platform when rotating.

Analysis of Curved Beam in Anisotropic Buckling Wings. Components of anisotropic buckling wings are described, focusing on modeling and characterization of curved beams embedded in these wings. To split the problem between aerodynamic and buckling domains, a wing system800was modeled as shown inFIG.8A. The model is based on a long thin beam coupled to a circular flat plate. This wing system800is coupled to a joint802that can be powered by a motor or transmission. The role of the circular plate804is to produce thrust and drag and apply resultant forces and torques through the curved beam808to the body of a mobile robot. The curved beam808is curved along its length by two curved sliders810and812. A symmetric propulsion allows the impact of buckling and deflection on thrust and force production as a function of wing configuration throughout its gait cycle.FIG.8Bdepicts a cross-section of the curved beam808with dimensional parameters.FIG.8Cdepicts a double beam design.

Buckling under the assumption of end-loading conditions consisting of point loads and moments can be modeled from aerodynamic forces in the distal portion of the wing. The wing model depicted inFIG.8Acomprises several sections: a rigid plastic section814of length l0, a section816of length (x1−l0), a curved beam808of length (l=x2−x1), and a third section818of length (l1−x2), which is coupled to a circular plate804of diameter d. The curved beam808has a thickness t. The two curved plastic sliders810and812are located at x1and x2. These sliders induce a camber to the beam, which can be represented as a radius of curvature r. The sliders810and812and curved beam808of the wing are made of a single sheet of t-mm-thick polyester, whereas the rigid plastic section814containing the joint802is a sufficiently rigid 3D printed plastic. For the purposes of design and analysis, it is assumed that the position of sliders810and812is such that the curved beam808is the weakest and buckles first in the presence of flapping forces. Though camber of the wing may be observed along the beam, the circular plate804does not exhibit significant curvature due to increased material stiffness.

Theoretical Model for Curved Beam Buckling. Two different formulations are typically used to describe the buckling phenomenon of curved beams, namely the buckling of spherical shells and the behavior of folded tape-springs.

The buckling of spherical shells can be described as follows. In opposite sense bending, prestressed, curved material first passes through a flattened state via moments exerted on the shell's edge (Mxand My). Stress (σy) is the direct result the of curvature change in the y-direction, whereas (σx) is caused by Poisson's ratio. Considering that the material remains in its elastic range during this deformation, the stress distribution through the thickness stays linear and stress distribution can be determined.

This model finds critical buckling stress as a function of curvatures of the two stable phases, i.e., initial longitude curvature and final phase curvature. In this analysis, the system has no second stable phase. As a result, the value for final phase curvature is unknown and the value for critical buckling moment cannot be obtained based on this system of equations.

FIG.9Ashows the moment-curvature relationships for a tape spring subject to equal and opposite end moments. End moments can be obtained by integrating of moments about the transverse axis for the whole cross section of the tape spring by considering the beam as a slightly distorted axi-symmetric cylindrical shell. In this formulation

M=∫-s2s2(Ml-Nl⁢w)⁢dy=sD×(kl+vr-v⁡(1r+vkl)⁢F1+1kl⁢(1r+vkl)2⁢F2)(3)
where Mland Nlare the bending moment per unit length and the axial force per unit, respectively. w represents out-of-plan deflection, the y-axis corresponds to the longitudinal direction, and klis longitudinal curvature. The variables s and D are the width of the tape spring and bending stiffness, respectively and can be determined by the following equations:

s=2⁢r2⁢sin⁡(θ2)(4)D=Et312⁢(1-v2)(5)
where E, v, and t are Young's modulus, Poisson's ratio, and tape spring thickness, respectively. r and θ are the initial transverse radius and curvature angle of tape spring, respectively. F1and F2are calculated as follows:

F1=2λ⁢cosh⁢λ-cos⁢λsinh⁢λ+sin⁢λ(6)F2=F14-sinh⁢λ⁢sin⁢λ(sinh⁢λ+sin⁢λ)2(7)λ=3⁢(1-v2)⁢s4tkl(8)
The critical buckling moment (m+max), can be calculated by finding the maximum end moment in Eq. 3. The “steady moments” M+* and M−* can be calculated by considering that the curved region is approximately cylindrical
M+max=(1+v)Dθ(9)
M+max=−(1+v)Dθ(10)
This formulation is limited to the linear regime of the material's stress/strain curve.

To evaluate the theoretical model and provide better understanding of the curved beam, two specimens of a steel measuring tape and a curved polyester beam were considered. Both specimens have the same length (l). The polyester specimen is precurved so as to have the same radius of curvature (r) as the steel specimen. For each specimen, the curved beam is coupled at one end to a fixed plate, while a known force is applied to the other end. A force sensor mounted to the output of a linear actuator pushes on the beam via a small, 3-D printed contact point. The linear actuator moves back and forth through a 50 mm range in 10 μm increments; applied forces are sampled at each step.

FIG.9Bdepicts the result for both specimens in two cases of equal and opposite-sense bending. Both polyester and steel specimens exhibited the buckling behavior predicted in the theoretical model. This may be seen in the sudden drop in the resultant moment at high deflection. The buckling moments in opposite sense bending (M−max) are much higher than the equal-sense buckling moment (M+max) for both specimens. However, there are some notable differences between the theoretical model and experimental data. In both specimens, the deflection of the specimens inFIG.9Bdoes not follow the same path after buckling when forces are removed. This difference is more noticeable in the steel specimen compared to the polyester specimen. In the steel specimen, the values for M+* and M+maxare different, but in the polyester specimen, they have the same value. The sudden change in the experimental torque/displacement data may to be due to out-of plane deformation, pusher slip, and friction. Moreover, in the case of opposite-sense bending, the path during loading and unloading of the polyester specimen is closer to earlier theoretical model predictions than the steel specimen. This is attributed to plastic deformation that was observed in the steel specimen. While the theoretical model assumes that the buckling beam does not leave the elastic region, this analysis shows otherwise. This can be due to the fact that, like the drag force on the wing, the pusher produces a combination of force and moment on the edge of the curved beam instead of a pure moment. This force-moment combination produces a nonuniform stress distribution on the shell and, in some cases, deforms the plate after buckling in ways not predicted the earlier models. This deformation results in permanent damage to the beams if the moment exceeds (M−max). As a result, a safe region must be defined for the moment produced by the wing to ensure that the beam never undergoes opposite-sense buckling.

Finite Elemental Analysis (FEA) Study on Curved Beam Buckling. In order to customize the buckling behavior of curved beams, various design parameters depicted inFIGS.8A and8Bcan be adjusted, including the radius of curvature (r), beam width (rθ), beam length (l), and other material properties. Finite Elemental Analysis (FEA) may be used to predict the relationship between these parameters and the desired buckling conditions. Unlike the analytical formulation, which is based upon uniform geometry and specific loading assumptions, FEA methods permit use of a wider range of geometries with more nuanced loading combinations.

The behavior of a slender curved beam was analyzed by varying the curvature (θ), length (l), and width (rθ) of the beam as primary design parameters. The change in buckling factor of safety was monitored in linear, eigenvalue-based approach. To simplify the analysis model half the beam was modeled and a symmetric constraint was applied for the other half; a curvature-based mesh setting was used with a maximum element size of 0.4 and 0.02 mm tolerance. The proximal edge of the beam was fixed while a load is applied to the distal end. The load was a combination on nominal force and moment (1 N and 1 Nm).

First, it is demonstrated how adjusting the camber (or longitudinal curvature) of a beam can be used to alter the beam's stiffness and critical load to produce asymmetric flapping cycles and nonzero thrust. The curvature, θ, is defined inFIG.8Bso that θ=0 corresponds to a flat plate and θ=180 produces a half-cylinder. Using the results of an FEA calculation the deflection of a curved beam (of dimensions l=25.4 mm by rθ=25.4 mm by t=1 mm) loaded in equal and opposite sense differs noticeably.

The evolution of the differences in critical load for equal and opposite-sense bending was analyzed when the curvature of a beam is varied between 30° and 180°. The width (rθ) and length (l) of the undeformed half-beam was set to 25 mm, and the resulting critical loads were obtained when loads are applied in the equal and opposite orientation using a linear eigenvalue-based analysis. While exceeding the opposite-sense buckling limit leads to plastic deformation, exceeding the equal-sense buckling force reduces drag in the up-stroke portion of the swimming gait and increases the average thrust produced in swimming gaits without leading to beam failure.

To further analyze the relationship of beam width on buckling point, the curvature (θ) and length (l) of the undeformed beam were fixed at 180° and 25.4 mm, respectively, whereas the width of the beam is varied from 6.4 to 76.2 mm. The beam's, radius of curvature (r), volume, and mass change as a function of width. The result show the factor of safety corresponding to both equal and opposite-sense buckling increases as the width of the beam grows. The results also show that the difference in magnitude between equal and opposite-sense buckling limits grows with width.

In order to better understand how beam length (l) impacts buckling, the length of the beam was varied from 6.4 to 76.2 mm while keeping the curvature (θ) and width (rθ) of the undeformed half-beam fixed at 180° and 25.4 mm, respectively. The beam's volume and mass change as a function of length (l) while the radius of curvature (r) was held constant. Loading conditions were varied as a function of l in this since the loading conditions on the buckling portion of the system are defined by the moment and force combination generated by the forces exerted at the distal end of the beam.

The result of this analysis showed that the buckling limit decreases for both equal and opposite-sense buckling as the length grows. However, the difference between the magnitude of positive and negative buckling limits initially grows and then stays somewhat constant for l>25.4 mm.

Based on these results, a curved beam with θ=180° was selected for the rest of the analysis. The beam length (l), width (rθ), and thickness (t) remain free design variables that can be tuned in order to maximize the effects of one-sided buckling for use in conjunction with the drag and inertial forces acting on the fin across fluids of different viscosity.

Dynamic Modeling of Buckling Wing Propulsion. The dynamic behavior of a wing system was modeled by considering dynamic elements such as wing drag, curved beam stiffness, and rigid body dynamics. In this analysis, a model wing system shown inFIG.10Ais coupled to the ground at the base of the input joint1002, and the moments exerted on the environment about the rotational axis are recorded similar to the described experimental setup. The system is represented by two rigid links1004and1006with point masses1008and1010located at their centers of mass, coupled by a pin joint1012and torsional spring1014, with stiffness coefficient of K, coupled in parallel. The nonlinear stiffness of the spring is represented by three linear regimes; the slopes of each of these regimes have been adjusted to best fit experimental data collected from the prototype. The length (d1and d2) and mass (m1and m2) of each link match the measured values of the in-water prototype.

Using a flat plate model, the forces on a wing due to a fluid are estimated by the equations derived from:
FwD=ρutAsin2α  (10)
FwL=ρu2Acos α sin α  (11)
where ρ, u, A, and α are the density of fluid, the relative velocity of the plate, the area of the plate, and the angle-of-attack of the wing, respectively. FwDand FwL, correspond to the drag and lift elements of the aerodynamics forces on the plate. This model estimates the total force on a flat plate as
Fw=ρutAsin α  (12)
where α is 0 when parallel to the flow and 90° when perpendicular (in 2-D). This force is perpendicular to the wing and acts as the aerodynamic load on the curved beam.

Using Eq. 12, the velocity of the plate (u) can be used to control the amount of drag force exerted on it, which, in conjunction with the load limits determined by the mechanics of the curved buckling beam, determines whether and under what conditions buckling occurs.

The flat plate model describes the fluid dynamics of a system when the Reynolds number is low and the system is in the laminar regime. The Reynolds number of a flapping wing in fluid is formulated as follows

Re=u_⁢c_v(13)
where ū,cand v are the mean translational velocity of the wing tip, the wing mean chord, and the kinematic viscosity of the fluid, respectively. For the given flapping system, ū=2ΦfR, where Φ and f are flapping peak-to-peak angular amplitude and frequency and R is moment arm to the center of pressure of the wing. For this wing flapping in water, the Reynolds number varies from 1800 to 7200 when the flapping frequencies varies from 0.1 to 0.4 Hz, indicating that the flow regime changes from laminar to turbulent at higher flapping frequencies.

The flat plate model was analyzed using a computational fluid dynamic (CFD) analysis on the system wing. In this analysis, measurements were made of the average lift and drag exerted on the wing by uniform water flow with different flowrates as the angle-of-attack varies from 0 to 180°. The results for the flow of 0.1 m/s versus the flat plate model estimation indicate a high correlation between the flat plate model and CFD results for the latter speed for which the system is in laminar regime. At the maximum studied flapping frequency, the mean translational velocity of the wing reaches 0.41 m/s for which, in the worst case, the maximum error between flat plate model and CFD results is less than 15%.

When a sinusoidal torque input is applied to the base joint, the dynamic model demonstrates that the wing system transitions between a nonbuckling flapping regime to a one-sided buckling regime, as shown inFIG.3A, when the flapping frequency is increased. The wing system transitions from the nonbuckling regime to one-sided buckling at around 0.3 Hz. While the maximum positive torque increases with frequency in the power stroke, the torque in the recovery section remains low. The amount of work performed on the environment also grows with the emergence of buckling. The motion of the modeled wing through a full flapping cycle at 0.4 Hz is similar to the in-water flapping behavior.

Experimental Validation. The following results verify the effect of curvature on buckling force for a curved beam, as well as to demonstrate its potential for creating thrust and motion. Two case studies are considered (air and water) to validate the proposed methodology in order to underscore the generality of this concept, using the design principles from the previous section as a design guide.

Example 1

Wing Flapping in Air. In this example, the air drag exerted on a wing utilizing curved beam buckling is experimentally measured. The test apparatus is shown inFIG.10B. The flapping wing1020is coupled to a dc servo motor1022by a 3D printed mount, permitting rapid swapping of different wing designs. Control sliders1024and1026are configured adjacent to the curved beam buckling region1028. Forces and torques generated by flapping are measured with a six-axis ATI Mini-40 force/torque sensor1030mounted to the motor1022and ground and interfaced to a processor1032configured for data collection and motor control. The servo's position input signal is a triangular wave with a fixed amplitude of 66°; the frequency is varied in order to change the aerodynamic interactions experienced by the wing.

Variable Length (One Beam). Two different cases of symmetric flapping are used to demonstrate the effect of anisotropic buckling. In the first case, the sliders1024and1026are brought closer together; this shortens the exposed beam length (l) and prevents buckling in both directions of flapping and results in similar angle of attack and drag in both recovery phase upstroke and power phase downstroke. In the second case, the sliders1024and1026are arranged so that the gap between them is large enough to permit buckling in the equal-sense direction to occur during sinusoidal flapping. This longer buckling region allows the curved beam to buckle under drag forces in equal-sense bending, but is not sufficient to induce buckling in the opposite sense.

Plots of the moment generated by the wing during symmetric flapping as a function of the wing's angle and speed show that the shape of the nonbuckling curved beam's work loop is qualitatively symmetric (about torque τ=0). This indicates that the average work—the area of the work loop in the positive τ domain minus the area of the work loop in the negative τ domain—over several flapping cycles provided by a nonbuckling beam is near zero. In contrast, the buckling beam shows an asymmetric path (about torque τ=0), capable of producing nonzero work in the forward direction. The changes in power and work plots indicate the effectiveness of anisotropic buckling during symmetric flapping in generating nonzero thrust, power, and work.

The results demonstrate that the curved beam produces work in symmetric flapping when it is permitted to buckle. The average torque generated over one flapping cycle increases from 0.009 to 0.165 Nm in the presence of unidirectional buckling, as provided in Table 3. Though the wing-beam system is not optimized for energy efficiency, the mechanical energy efficiency increases from 1.86% to 29.5%. This is calculated by evaluating the ratio of useful work done over the total work done across a full flapping cycle.

TABLE 3Torque and Work Generated During Flapping in AirAverageFrequencyTorqueWorkMechanicalExperiment(Hz)(Nm)(J)EfficiencyBucklingVariable2.28−0.012−0.0091.86%NoLength2.280.1310.16529.50%One side1 Beam1.38−0.005−0.0052.98%No2.060.1490.15426.73%One side2.280.1310.16529.50%One side2 Beams2.060.0040.0132.29%No2.280.0310.0192.56%One side2.480.0770.09510.30%One side

Variable Frequency (One Beam). The effect of drag on buckling was tested by increasing the frequency of the triangular input signal for the same curved beam. The torque generated via a symmetric flapping gait with respect to time, servo angle, and angular velocity was measured for the three flapping rates of 1.38, 2.06, and 2.28 Hz. The torque generated by each successive increase in flapping speed increases the magnitude of torques experienced in the positive direction without similar magnitude increases in the negative direction. This results in work performed on the environment. At 1.38 Hz, the beam experiences no buckling; however, the faster two cases (2.06 and 2.28 Hz) result in one-sided buckling. The average torque, amount of work done on the environment, and mechanical efficiency are listed in Table 3. The data reveal that the buckling duration of a full flapping cycle increases from 25% to 42% in one-sided buckling cases between 2.06 and 2.28 Hz. Although the hysteretic gaits obtained here by anisotropic buckling during flapping resembles gaits generated by other techniques such as the split cycle method, the effect in this case is a result of designed system dynamics rather than asymmetric motor inputs.

Variable Frequency (Two Beams). To address the nonnegligible torsional effects visible in the wing during flapping, the system was stiffened in torsion by coupling two beams—40 mm apart from each other, in parallel—to the wing, as depicted inFIG.8C. This reduced the noticeable effects of torsion on long thin beams and produced slightly different torques throughout flapping cycles at different speeds. The tests were conducted at 2.06 Hz, 2.28 Hz and 2.48 Hz. The results, provided in Table 3, show similar trends and behavior with the previous one-beam case, but because the system is stiffer (due to two beams in parallel), it requires higher velocities (and higher drag) to initiate buckling. This can be seen in the of 2.06 Hz case, which experiences no buckling in contrast to the single-beam trial. The data sampled in the two-beam case are smoother, with less high-frequency noise; this can be attributed both to a reduction in torsional effects as well as the impact of the altered stiffness on the resonant frequencies of the system.

Example 2

Flapping in Water. Flapping was also demonstrated in water, using a remote control (RC) servo to produce symmetric flapping while measuring the torques produced by the fluidic interactions. Test were performed on a single flapping cycle of a wing with a precurved buckling beam. The recovery stroke and a power stroke of sinusoidal control signal was observed. Hysteresis was visible between these strokes, indicating that the dynamic interactions between inertia, drag, and buckling play a role in deforming the beam anisotropically.

A sinusoidal input signal with constant amplitude and variable frequency was used to analyze the impact of flapping speed on buckling and torque. The torque generated for 0.1, 0.2, 0.3, and 0.4 Hz frequencies over several cycles was measured. The results demonstrate the effect of anisotropic beam buckling. The maximum positive torque increased from 0.05 to 0.43 Nm between 0.1 and 0.4 Hz while the negative torque generated during a flapping cycle was limited across all experiments to no less than −0.12 Nm. Table 4 shows the comparison between the generated torques in this experiment and values estimated by the dynamic model. The results of the two-beam design shown inFIG.8Cshow that increased mechanism stiffness increases the torque that can be supported by the beam in recovery, undesirable from the perspective of gait efficiency. This design can require optimization against other design parameters to simultaneously reduce the effect of torsion and increase efficiency.

TABLE 4Generated Torques in Simulation and ExperimentalSimulationExperimentτminτmaxτminτmaxFrequency(Nm)(Nm)(Nm)(Nm)Buckling0.1−0.040.04−0.040.04No0.2−0.140.14−0.140.15No0.3−0.230.3−0.130.32One side0.4−0.240.46−0.120.44One side

Using these results, a water-based robotic platform has been developed that leverages buckling during flapping. The robot uses curved beams coupled to two rigid fins made from 0.76 mm fiberglass sheet. The buckling portions of the links are made from a laminated composite of fabric, adhesive and 0.18 mm-thick polyester, which is used to reinforce the material during buckling.

Based on the properties of the curved beam, if the combination of force and moment experienced at the fin is between the equal and opposite-sense buckling values discussed earlier, the curved beam will buckle unidirectionally, resulting in a different angle of attack, which impacts the lift and drag forces acting on the fin by the fluid. As a result, drag on the robot will be different in power stroke and recovery stroke, creating a thrust differential over a gait cycle, which makes the robot swim forward. The magnitude of forces and moments caused by fin propulsion can be adjusted by controlling the amplitude and speed of the servo movements, size of the fin, length of the beam (l), and radius of curvature (r). The left and right fin servos follow a sinusoidal control signal of the form
y1=Aisin(2πfit+ai)+bi(14)
where Airepresents an adjustable amplitude, firepresents the frequency, airepresents a phase offset, and birepresents an amplitude offset from the neutral point, which is nominally set to bi=0 throughout these trials. This symmetric motion about the transverse and bilaterally symmetric robot guarantees that any forward locomotion can be attributed to the changes in drag caused by the buckling curved beam coupled to the fin. The forward thrust generated by symmetrical flapping of the two wings was measured for 0.1, 0.2, 0.3, and 0.4 Hz frequencies.

In water trials, the swimming robot was able to swim with an average speed of 0.1 m/s when y0=y1. The robot was able to rotate by using only one limb at a time. A nonbuckling fin acts more like a fish caudal fin and causes the robot to move laterally; because of buckling, the fin produces nonzero average torque, resulting in the robot turning.

Particular embodiments of the subject matter have been described. Other embodiments, alterations, and permutations of the described embodiments are within the scope of the following claims as will be apparent to those skilled in the art. While operations are depicted in the drawings or claims in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed (some operations may be considered optional), to achieve desirable results.

Accordingly, the previously described example embodiments do not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure.