The thermal state of the SMA actuators affects the Robojelly's bell deformation  , . The laser sheet is pointed at the bell. Cross correlation was performed on the high speed images to obtain the corresponding velocity fields. The mean error of this technique was estimated between 0.
Vortex identification was done using a method described by Holden  which consists of two different identification methods and comparing the results of both. The first method is based on Sperner's lemma which is adapted for vortex identification in . For this method, a first pass is done on the velocity field to label each velocity vectors in one of three equally spaced direction ranges. The second pass looks at the surrounding neighbors of every grid point and determines if all three direction labels are present.
If so, the current point is identified as a vortex center. After identifying possible vortex locations with Sperner labeling, the method was used to calculate the swirling strength of each possible vortex  , . The swirling strength is taken as the imaginary eigenvalue of. This is based on the fact that local streamlines can be represented as a function of real and complex eigenvalues.
See  for more details and a visual representation of this method. The maximum swirling strength over the velocity fields was determined in a preliminary pass. Erroneous vortex locations varying from the general trajectory of the starting vortex were manually removed. Circulation of the starting vortex was computed using the line integral of the velocity field , over the identified contour delimiting the vortex area:.
Vortex circulation and area were filtered using a second order Butterworth low pass filter. For scaling purposes, the bell margin was tracked as a function of time. Starting at rest, the margin position was detected manually using ImageJ. The distance traveled between each point was then calculated. The distances were smoothed using a second order Butterworth lowpass filter. Digitized A. The flap can be seen most distinct during the middle of the contraction.
Curvature for selected profiles in the relaxed, contracted, relaxing and contracting configurations are shown in Fig. The contracting profile shows a different curvature pattern than the others with a negative curvature towards the bell margin. The point at which curvature of the exumbrella profile goes from positive to negative is a possible lower bound of the flap. This location is called the inflexion point, defined as the location where a curve goes from positive curvature to negative curvature or vice-versa. Starting from the apex, the flap upper bound is simply the bell margin but several methods were considered to determine the location of the flap lower bound.
The inflexion point location varies depending on the deformation magnitude. A stronger contraction will result in larger flap deformation and change the inflexion point location. Therefore, this must also be considered when determining the flap lower bound location. From this point onward, the lower bound location of the flap will be referred to as the flexion point.
A Aurelia aurita bell kinematics showing the exumbrella profiles over a full swimming cycle. The exumbrella profiles shown were selected arbitrarily for clarity. The margin trajectories are also shown for each side of the exumbrella profile. B Bell trajectories at selected points along the exumbrella arclength over a full swim cycle. The different percentages correspond to exumbrella arclengths from the apex to the margin. A Curvature as a function of exumbrella arclength from apex to margin for exumbrella profiles in different stages. The profiles shown cover the four main states of the bell kinematics: relaxed, contracting, contracted and relaxing.
The curvature is normalized by exumbrella arclength in the relaxed state. B Average normalized curvature as a function of exumbrella arclength from apex to margin. The curvature is averaged over a full cycle which includes the left and right profiles of the Aurelia aurita. The flexion point can be determined by finding the point associated with the minimum curvature or largest negative curvature over a full cycle.
The largest negative curvature occurs during contraction. However, this method does not take into account the variation in curvature over time. The average flexion point location during strong contraction can be determined. A strong contraction can be defined as the region where the largest minimum curvature falls within a specified threshold. This method leads to a somewhat arbitrary selection of profiles determined by the threshold values.
The chosen method calculates the time average curvature for individual arclength locations and the associated standard deviation during an entire swimming cycle. In addition, the curvature for both the left and right side of the exumbrella profiles were averaged. The resulting average curvature as a function of exumbrella arclength is shown in Fig. The flap location was selected as the location where the average curvature reaches a local minimum with value of and where the standard deviation starts to diverge with value of.
If the transition location between the active and passive bell section was known, this could be a more direct method of determining the flexion point location but as previously mentioned, that location is unknown for the A. The bell margin follows looping trajectory with outer path during contraction and inner path during relaxation as shown in Fig.
The flexion point follows an inner path during contraction and outer path during relaxation, as shown in Fig. Looping in the bell margin trajectory is of significant importance because the range of motion, velocity and acceleration are highest in the flap region. This can be visualized with a half cross-section of the bell rotating about a pivot point located at the apex. The moment arm amplifies the magnitude of the hydrodynamic forces in that region. The outer path during contraction is favorable to produce thrust.
The length of the moment arm during the outer path will affect circulation of the starting vortex as shown in Section 3. The inner trajectory during relaxation is favorable to reduce drag on the vehicle and resistance to the passive relaxation. The flap has previously been modeled as a body passively pivoting at the bell periphery  , . Wilson and Eldredge  have modeled a two-dimensional solid body structure which loosely resembles a jellyfish.
The body consisted of solid elliptical bodies connected through hinges. They explored the effects of actively and passively controlling the different bodies with prescribed or passive motion at the hinges. They found that passively controlling the marginal bodies increased performance. McHenry and Jed  modeled the A. Thrust calculation with the jetting model based on  overestimated the measured A.
The paddle model approximated the flap as a flat plate oriented perpendicular to the flow and calculated thrust produced by the flap only. Thrust produced by paddling significantly underestimated the measured speed by an order of magnitude. Though the jetting model gave a better representation of the A.
The beginning of the contraction starts with an inner path and transitions about half way into an outer path. The same is true for the relaxation phase. The constant cross-section flap does show a negative curvature during contraction as seen with the A. The deformation profile of the tapered flap in Fig. The increased flap stiffness prevented higher curvature changes during relaxation.
Adding a curved profile caused the flap to already have a curvature during relaxation. Contraction of the curved and tapered flap shows an extension of the flap due to water resistance in Fig. The margin trajectory shows an outer path during contraction and an inner path during relaxation as achieved by the A. Margin trajectories are shown with arrowheads indicating time progression.
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Profiles were down sampled in this figure for clarity. These results show that it is indeed possible to replicate the important characteristics of the A. In terms of bio-inspired vehicle design, limiting the number of actuators reduces the complexity of the design dramatically and ultimately can lead to more efficient vehicles. This is based on the fact that similar kinematics are recreated using a passive flap. The Robojelly design assumes that the flexion point is where the transition from active to passive occurs and this greatly simplifies the architecture.
The role of circular muscles in the flap could be justified during a turning maneuver where the bell needs to achieve an asymmetric configuration and requires certain parts of the flap to stay stiff. A visual inspection of the exumbrella profiles during contraction Fig. This is mainly due to an increase in local stiffness due to the manufacturing constraints at the actuator tip.
The actuator tip has a rigid attachment for the SMAs as seen in Fig. As a result, the inflexion point occurs farther along the exumbrella arclength for a given contraction profile. Despite the differences in flap kinematics, a significant improvement in swimming performance was observed with the addition of a flap as shown in Fig. Superimposed exumbrella profile with flexion point location during contracting for the A Aurelia aurita and B Robojelly. The Robojelly inflexion point in B is for the single contraction profile shown.
The initial sinking state of the robot is shown along with the swimming state. The robot was not actuated during the initial sinking state. Figures 4 and 9 a show that the Robojelly underperformed the A. In addition to total displacement differences, the trajectories also differ. A Robojelly margin trajectories for different flap configurations. The trajectories are for a Robojelly normalized at its apex position 0, 0.
Arrows indicate margin direction. B Bell margin displacement as a funciton of time during actuation for the different flap lengths. Displacments are shown from the beginning of actuation till the end of motion in the negative x-direction. Flap displacement is calculated as the arclength traveled by the flap tip and is shown as a function of time for all flap configurations in Fig. The longer peak time with increasing flap length can be attributed to the the flap structural dynamics and SMA performance.
The longer the flap, the more bending it undergoes. As a result, a larger lag occurs at the margin relative to the actuator tip flexion point. The actuation difference between these two flap configurations is mainly related to actuator performance. SMA actuation performance can degrade because of overheating, shakedown, stretching and slipping at the connections. The bio flap was tested first which means the actuators were in their best condition.
Additionally, the added mass from the larger flaps prevented a fully relaxed position causing the actuation cycle to be smaller. An example vorticity field created by the Robojelly with bio flap is shown in Fig. Vorticity was calculated using:. This figure also shows the starting vortex and contour used for calculating circulation. Dimensional circulation as a function of time is shown in Fig. The constant cross-section flaps have a general increase in peak vortex strength as a function of flap length.
The time it takes for the vortex to reach peak circulation also increased with flap length. This is partly due to the deterioration of the actuator performance as stated in the previous section. To determine the underlying principles of flap length, varying parameters can be accounted for by making circulation and time non-dimensional. The Robojelly flap was scaled using two different methods. A Robojelly circulation as a function of time for different flap configurations. B Non-dimensional circulation as a function of time scaled by orifice diameter.
The first method approximates the Robojelly as an axi-symmetric body which ejects water from a varying orifice. This is similar to work on varying orifice pistons where circulation and time are scaled as a function of piston velocity and the orifice diameter is scaled as a function of time  , . Robojelly has a varying diameter but not a moving piston.
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Water displacement on the Robojelly is due to actuation of the bell or walls which contract or reduce the diameter of the Robojelly. The non-dimensional equations in  can be modified to take into account water displacement due to wall deformation:. The resulting non-dimensional circulation is shown in Fig. The results show a wide variability of the circulation peaks and no apparent scaling.
The second method for scaling circulation analyzes the Robojelly as a set of pitching panels distributed circularly about the bell apex. A comparative analysis of biomimetic robots showed that the presence of a passive flexible flap results in a significant increase in the swimming performance.
In this work we further investigate this concept by developing varying flap geometries and comparing their kinematics with A. It was shown that the animal flap kinematics can be replicated with high fidelity using a passive structure and a flap with curved and tapered geometry gave the most biomimetic performance. A method for identifying the flap location was established by utilizing the bell curvature and the variation of curvature as a function of time.
Flaps of constant cross-section and varying lengths were incorporated on the Robojelly to conduct a systematic study of the starting vortex circulation. The starting vortex circulation was scaled using a varying orifice model and a pitching panel model. The varying orifice model which has been traditionally considered as the better representation of jellyfish propulsion did not appear to capture the scaling of the starting vortex.
Flexible Margin Kinematics and Vortex Formation of Aurelia aurita and Robojelly
In contrast, the pitching panel representation appeared to better scale the governing flow physics and revealed a strong dependence on the flap kinematics and geometry. The results suggest that an alternative description should be considered for rowing jellyfish propulsion, using a pitching panel method instead of the traditional varying orifice model. Finally, the results show the importance of incorporating the entire bell geometry as a function of time in modeling rowing jellyfish propulsion.
Vehicle self-sustainability consists of autonomous control, robustness and energy independence. Energy independence can be achieved by energy harvesting and increasing the vehicle efficiency. Several energy sources can be harvested in ocean waters such as wave, solar, thermal and chemical energy but more research is required to adequately exploit these resources in order to make any practical use for AUVs. Vehicle efficiency is therefore of paramount importance for reducing the amount of energy needed for sustaining the vehicle.
More specifically, propulsion efficiency is critical when a vehicle must cover long distances or constantly propel itself to maintain a certain location. The propulsion system has three main sections where reducing losses becomes important: the actuators, the motion translating mechanism and the hydrodynamics.
Biological systems have been able to significantly minimize the losses and achieve efficiencies higher than any engineered system. We therefore take inspiration from biology for answering the questions related to vehicle efficiency and for determining fundamental principles that can lead towards self-sustainable AUVs. Jellyfish can be separated into two categories based on their mode of propulsion, namely: jetting and rowing . Jetting is utilized by smaller species of prolate medusa generally not exceeding a few centimeters in diameter. This method of propulsion is known for higher velocities and accelerations .
Rowing is found in larger jellyfish reaching up to 2 m in diameter which are more oblate in geometry. This is a more efficient mode of propulsion . Aurelia aurita fall in the category of rowers. The swimming mechanism for most jellyfish consists of circular muscles located in the subumbrella which collapses the flexible bell upon contraction. The collapse of the bell causes a volume change in the subumbrella which leads to the expulsion of water and thrust production. The relaxation phase is driven by the elastic energy stored in the bell structure during contraction  — .
For rowers, a stopping vortex is formed under the bell during relaxation. This is followed by a starting vortex formed during contraction. The starting and stopping vortices interact with each other to increase thrust . Recently, a robotic jellyfish Robojelly was developed mimicking the morphology and propulsion mechanism of the A.
The oscillating mode of propulsion utilized by jellyfish is not hydrodynamically efficient with a Froude efficiency ranging from 0. The cost of transport COT metric can be used to quantify the overall efficiency of a vehicle during transportation and is defined as:. The COT of jellyfish was found to be one of the lowest  , . The relatively low hydrodynamic efficiency is offset by the efficient metabolism and an elegant mechanical system. The design of a robotic jellyfish should attempt to incorporate all the relevant features that affect propulsion.
Using Robojelly, controlled experiments can be conducted to understand the fundamental principles influencing the thrust production and overall energy efficiency. Physical parameters of the Robojelly can be varied systematically that would otherwise disrupt the physiological functions of the animal and therefore prevent any conclusive analysis. In this study, we provide the fundamental understanding of the section of the bell towards the tip that was found to play a significant role in the propulsive performance of the Robojelly .
This section is referred to as a flap or flexible margin. The effect of the flexible margin on the bell kinematics and starting vortex are the focus of this study. When designing the Robojelly, the morphology of an A. The bell kinematics was then analyzed in order to determine where to position the actuators. It was noticed that the bell margin did not follow the same deformation pattern as the rest of the bell.
Figure 1 shows an A. Aurelia aurita half bell profile in the A relaxed, B contracting and C contracted position. These images of A. This test was conducted in ocean waters and the dye is seen as bright green in the images. The exumbrella profile of the video in Fig. Every two to four images were processed during the swimming cycle with higher temporal resolution during contraction.
Edge detection was done by importing the images in a computer aided design CAD software Inventor, Autodesk and manually selecting 10 to 15 points on each profile. It is critical to include margin points as accurately as possible in this process. A spline function was used to interpolate the points for a total of points per profile. These pixel locations of the exumbrella were converted to polar coordinates and filtered using second order lowpass Butterworth filter.
The points were then converted to Cartesian coordinates. The bell apex was selected as the middle point of the spline. The exumbrella points were then zeroed about the apex and normalized by a half exumbrella arclength in the fully relaxed position. Further information on the experimental procedure for tracking the deformation and image processing can be found in . Properly defining the flap location is critical for analyzing its function. The flap can be defined as the region of the subumbrella that deforms passively during contraction.
Chapman  demonstrated that subumbrellar muscles are located all the way to the bell margin which includes the flap region. It is unknown if the muscles in the flap region are actuated during contraction and if so, to what extent. Curvature is used to quantify the deformation along the arclength of the jellyfish bell. Curvature of the discrete exumbrella profiles were quantified using the definition of a circumcircle.
A circumcircle is a circle passing through three vertices A, B and C of a triangle with sides of length a, b and c. The diameter d of a circumcircle is:. The curvature equation for a discrete curve in Cartesian coordinates becomes:. Before investigating the flap's role on hydrodynamics, it is essential to replicate similar kinematics as the A. Initial observations made from bell kinematics lead to the hypothesis that it can be recreated using a passive flexible margin since the flap seems to deform with flow resistance, see Fig.
This implies that no actuators would be required in the flap region. A bio-inspired shape memory alloy composite BISMAC actuator capable of providing high curvature change was utilized to test this hypothesis  , . The BISMAC actuator was optimized by varying the stiffness along its length to recreate a deformation profile similar to that of the A. The three flap designs used for testing the hypothesis are shown in Fig. The first flap configuration tested had a rectangular cross-section of 0. The second flap configuration had a tapered cross-section and length of 3 cm as shown in Fig.
The tapering introduces a varying stiffness through the span of the flap. The third configuration had a taper and curvature which is more representative of the natural A. Reflective points distributed evenly along the side of the actuator and flap were tracked using image processing in Matlab. The flap can be added to the vehicle separately allowing the analysis of different configurations.
Figure 2 d and e shows the Robojelly with and without flap. It should be noted that the Robojelly bell is segmented as opposed to uniform as is the case of A. The segmentation was made to alleviate bell folding. The flap added to Robojelly in Fig. Further information on the silicone's mechanical properties can be found in  and a thorough description of the Robojelly design can be found in .
Both robot configurations were tested for their swimming performance. The vehicles were installed in a water tank with static water and were allowed to swim upwards. The robots were both actuated using a resistance feedback controller described in . The controller sends a peak current of 1. The vehicles were initially in a state of sinking to demonstrate that any thrust produced was from the propulsion mechanism and not the positive buoyancy. A thrust analysis described in  was used to quantify and compare the swimming performance of the Robojelly.
Thrust was measured using position over time to quantify the vehicle momentum and buoyancy state. The hydrodynamic forces were approximated using a jetting model and empirical models were used for drag and added mass  , . The constant cross-section flap had a thickness of 0. For the Robojelly with bell diameter of The Robojelly was first tested with the bio flap.
The Robojelly was then tested for each flap length by cutting out sections until no flap was left. The experimental setup consisted of a continuous laser sheet of nm wavelength and 2W power LaVision Inc. The robot was actuated for 19 cycles before a full cycle was recorded for analysis.
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Preliminary actuation was necessary for the shape memory alloy SMA actuators to reach a thermal steady state. The thermal state of the SMA actuators affects the Robojelly's bell deformation  , . The laser sheet is pointed at the bell. Cross correlation was performed on the high speed images to obtain the corresponding velocity fields. The mean error of this technique was estimated between 0. Vortex identification was done using a method described by Holden  which consists of two different identification methods and comparing the results of both.
The first method is based on Sperner's lemma which is adapted for vortex identification in . For this method, a first pass is done on the velocity field to label each velocity vectors in one of three equally spaced direction ranges. The second pass looks at the surrounding neighbors of every grid point and determines if all three direction labels are present. If so, the current point is identified as a vortex center. After identifying possible vortex locations with Sperner labeling, the method was used to calculate the swirling strength of each possible vortex  , .
The swirling strength is taken as the imaginary eigenvalue of. This is based on the fact that local streamlines can be represented as a function of real and complex eigenvalues. See  for more details and a visual representation of this method. The maximum swirling strength over the velocity fields was determined in a preliminary pass. Erroneous vortex locations varying from the general trajectory of the starting vortex were manually removed. Circulation of the starting vortex was computed using the line integral of the velocity field , over the identified contour delimiting the vortex area:.
Vortex circulation and area were filtered using a second order Butterworth low pass filter. For scaling purposes, the bell margin was tracked as a function of time. Starting at rest, the margin position was detected manually using ImageJ. The distance traveled between each point was then calculated. The distances were smoothed using a second order Butterworth lowpass filter. Digitized A. The flap can be seen most distinct during the middle of the contraction. Curvature for selected profiles in the relaxed, contracted, relaxing and contracting configurations are shown in Fig.
The contracting profile shows a different curvature pattern than the others with a negative curvature towards the bell margin. The point at which curvature of the exumbrella profile goes from positive to negative is a possible lower bound of the flap. This location is called the inflexion point, defined as the location where a curve goes from positive curvature to negative curvature or vice-versa. Starting from the apex, the flap upper bound is simply the bell margin but several methods were considered to determine the location of the flap lower bound.
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Pages: — Keywords: Guidance law,three-dimensional missile-target interception,finite-time stability,nonlinear disturbance observer Abstract. This is an electronic open access journal where the editorial board aims to keep an high publication standard. This number is focused on Plasmas for aeronautis. Plasmas for Aeronautics D. Full Article. Poggie, T. Mc Laughlin, S. Leonov Full Article. Knight Full Article. Elias Full Article.