Austin Mini Maker Faire (and draft of Merfluke paper).

I answered the "Call for presenters" of the Austin Mini Maker Faire with a proposal about my Merfluke.  This places upon me a responsibility to get some performance numbers pronto!  Luckily, Spring has sprung here in Texas, so I should be able to get in the water soon.

I am also working on a technical paper which I hope to submit to the Open Hardware Journal.

In the spirit of Public Invention, here it is in its current state of unreadiness---typos, mistakes, awkward sentences, and muddle-headed thinking all stewed together. However, a public inventor must "Tell the truth, tell the whole truth, and tell it right now" as Buckminster Fuller said---even if that means you get to see the sausage being made.

I especially apologize for the incomplete technical references---I'm relying on some technical research about dolphins that I read years ago and would like to add to the references, but I suppose it will wait---don't be offended if you think your work should be cited---better yet, contact me and tell me so.

The Merfluke: A Machine for Human Thunniform Swimming

By Robert L. Read

-- Art by Ymki van den Berg

Thunniform Swimming

Tuna and whales, the fastest creatures in the ocean, swim by driving a relatively thin plane back and forth through the water at angles which they can precisely control so that their tail or fluke describes a sinusoid. This is called thunniform swimming and is mechanically quite different than what a trout-shaped fish or a human being wearing swim fins or a monofin can do, which is called carangiform swimming. Carangiform swimming is a total body wave of amplitude increasing from the head to the tail, which is broadened and flexible. Often human swimmers wearing swim fins use them independently, and don't really use a total body wave; however, competitive swimmers and monofin swimmers have solidly established the “total body wave” approach as the most efficient swimming style. Competitive swimmers use the underwater dolphin kick as much as the rules of their sport allow without the benefit of a broadened tail. Monofin users performing the same motion are the fastest and most efficient human swimmers at present. The goal of my investigation is to build a machine that allows a human being to swim using thunniform motion. There is a scant but real possibility that such a machine would allow human being to swim better in the future than the monofin does today.

Dolphins have flukes and tuna have tails. Dolphins have a ball joint between their spine and fluke that allows them to angle it up and down. Tuna and tuna-like fish, such as mackerel, have a peduncle, which serves the same purpose of allowing them to angle the tail independently of other body motion, and to do so forcefully. Dolphins and fish are symmetric in the direction in which they oscillate.

Although it remains an interesting area of research, I would like to make some educated guesses or supported assertions about how a dolphin swims, even if these points are not completely proven:

1. Dolphins move their bodies relatively little as the drive their flukes up and down.
2. At cruise, the fluke stays within the wake, beam, or shadow of the body.
3. If we call the wake of the dolphin a “beam”, the fluke stays withing the beam and furthermore is parallel to the direction of motion when it is at the extreme edges of the beam, and is at its greatest angle to the direction of motion when it is in the center of the beam.
4. Dolphins can probably sense the edge of their own “shadow” in the water as slight difference in water turbulence as their flukes reach the extreme end of the beam.
5. The split shape of the fluke improves its vortex shaping ability. (This is also true of perch and catfish, which use a different motion to similar shed counter-rotating vortices.)
6. From videos, it appears that the fluke of a dolphin bends relatively little during normal motion.
7. Dolphins can oscillate their flukes at 10 Hz at their top speed. Human being run at 3 Hz. Champion monofinners swim at 1.8 Hz. It seems unlikely that we can comfortably operate any higher than that.

Human Anatomy

To support human thunniform swimming, we can construct a machine, called a Merfluke, that allows us to angle a fluke up or down by rotating our ankles. The fluke can be driven up or down through the water with the muscles of the upper legs and the torso. I have developed seven distinct prototypes of the Merfluke over the last five years. The latest model works, in that it allows a human being (me) to swim in a thunniform fashion for a few hundred meters. Its performance is at present much less effectively than flexible swim fins, however.

Herein lies a problem: humans are not symmnetric front-to-back. We have feet that point forward and cannot be made to point backward. Further, we can jump. Jumping exerts a tremendous momentary force on the ground. We cannot pull up with the tops of our feet with nearly the same force that we can push down with the bottoms of our feet. This is a significant problem for monofin swimming. It tends to separate the use of a floppy fin into a power stroke and recovery stroke. The recovery stroke provides very little power, but is needed to position the fin for the power stroke. This problem can perhaps be abated by skillful employment of the whole-body wave, but cannot be eliminated.

Wave Theory and Strouhal Numbers

It is clear from watching videos of dolphins that when accelerating that begin with a very large amplitude, slow frequency wave and as they move faster their fluke makes smaller, higher frequency motions. This is relatively complicated to describe mathematically, but if we imagine a streamlined body “cruising” efficiently at a constant speed, we should be able to describe it with basic mathematics. This will be more accurate for a body which has very little drag.

Definitions:

Wavelength (λ) is the distance between the same points in a cycle, such as the point at which the fluke is highest, measured in meters. Frequency (f) is the number of oscillations per second, measured in Hz. Speed (v) is measured in meters/second. The basic equation relating these three equations is:

(0) v = λ*f

Of particular interest to us is the Strouhal number[1], because it is observed that swimming and flying animals tend to cruise in a limited range for Strouhal numbers.[2]

For our purpose the Strouhal number can be defined as the dimensionless ratio of the amplitude of the fluke (its up and down travel distance, or throw, R) to the wavelength (the distance traveled in one oscillation.)

(1) 1/S = λ/R

It has been observed that inverse of the Strouhal number tends to be between 3 and 7. If we guess an inverse Strouhal number to target for the human athlete, we can combine this with (0). I made an educated guess that I wanted to swim at 2m/sec with a frequency of 1 Hz. Champion monofin swimmers swim at frequency of 1.83 Hz, but that seems quite ambitious to me. Plugging these values into (0), we can compute a wavelength of 2 meters. Rewriting (1) to compute throw: 

 

(2) R = λ/(1/S)

To achieve an inverse Strouhal number of 3, we need a throw of 2/3rds of a meter, we need a throw R of 26.25 inches. To achieve an inverse Strouhal number of 7, we need a throw of 2/7ths of a meter, or 11.25 inches. I therefore chose the length of the Merfluke extension beyond the feet to be 18 inches, which is a guess that in a normal motion for myself the throw will be 18 inches. As is typical of this kind of work, it is difficult to be precise because we are not just building a machine---we are creating a prosthesis that extends the human machine.

If we oversimplify the swimming motion and describe it as a simple lever oscillating back and forth around a pivot point located in the hips, then the throw can also be computed by equation (3).

3. R = 2 * (hip-to-fluke distance) * sin (half angle of oscillation).

I am 5'10”, and have a 30” inseam. My waist is approximately one meter above the soles of my feet. Since the distance (measured along the direction of motion) from the ankle rotation point to the fluke rotation point is 15” and the fluke extends about 3” beyond the rotation point, we find:

4. 0.5 = 2 * 1.0 * sin (half anle of oscillation)
5. half angle = arcsin(1/4^th) ~= 15 degrees,
   
   which seems like an achievable degrees of oscillation. However, the pivot point might be closer to the knees than the waist. Since the swimming motion remains to some extent a total-body motion, if only because the torso must move to balance the motion of the legs, maximal angle of deflection cannot be theoretically predicted. So we have a theoretic hope that a human being of my size can cruise with the Merfluke 7 at a reasonable inverse Strouhal number. However, the combination of the human athlete and the Merfluke extension is so complicated that any theoretical result must be experimentally verified.
   Additionally, one desires a throw small enough that the Merfluke does not break the surface of the water when swimming with a snorkel.
   
   
   The diagrams in the paper are all drawn as if the dolphin or human athlete is swimming with an inverse Strouhal number of 3.

Design of the Machine

 

The basic parts of the machine are:

1. the frame and the leg braces,
2. the pedals,
3. the fluke, and
4. the connecting rod.

These our components form a 4-bar linkage. The entire linkage is driven up and down through the water over the throw R. As this happens, the swimmer rotates the ankles, thereby changing the configuration of the linkage. The purpose of the linkage is two-fold: to make the fluke move through twice the angular deflection of the ankle, and to center this deflection when the foot is held in a natural, neutral position.

Additionally, the center point of deflection is 3” lower than the rotation point of the pedals which helps to keep the fluke under water when the swimmer is swimming at the surface and breathing with a snorkel.

Drag Testing

Incontrovertibly minimizing the drag of the total human/Merfluke combination is critical to its performance.

A homemade drag test was creating using a fishing rod and reel, two pulleys, a carabiner, and scale. A heavy fishing rod and reel was strong with 80-lb monofilament line. The line was run to a pulley attached to one of the eyes, and then back to a pulley attached to scale. The other end of the scale was attached to the handle of the fishing rod. In a steady state, the scale reads twice the force in pounds that the line is being pulled away from the rod.

The remaing fishing line was strung through the remaining eyes and a carabiner tied onto the end, so that I could comfortably hold on to something in the water.

I then swam out to clear water and had my son drag me at a constant walking speed while my wife observed the average force read by the scale. The results were:

	Average drag force
Human and swimsuit only	8 pounds
Human and swimsuit and monofin	9 pounds
Human and swimsuit and Merfluke	12 pounds

These measurements are valid relative to each other, thought they have no value for actually determining the drag coefficient of the monofin or Merfluke. 

 

This unfortunately does not bode well for the Merfluke achieving the goal of besting a monofin, since even if it allows greater propulsion, it will have to overcome significantly greater drag.

However, the drag of the Merfluke can be great improved with better craft technique than I have so far been able to apply in the Merfluke 7.

Safety

Never swim alone. Never swim without someone who is capable of rescuing you.

The Merfluke 4 used tight neoprene foot pockets, similar to swim fins or a monofin. I believe this is less safe than the current design, because if the Merfluke were to snag on a cable or weed under the water, the device cannot obviously be dismounted quickly. People often feel anxious for this reason the first time they try a monofin. The Merfluke 7 improves on this design by allow the machine to be dismounted by simply spreading the legs apart. It is also easier to mount than the Merfluke 4 (though dismount remains easier than mounting in the current design.

I believe using a Merfluke has several dangers. First, there is the danger of simply panicing, inhaling water, and drowing. Secondly, the Merfluke has more opportunity to snag on stone, cable, tree, or line under the water than do smooth fin swims. A diver can then dismount the Merfluke, but then must reach the surface without it. Thirdly, it shares a risk with swim fins and monofins---that of going fast enough that you strike a solid object and knock yourself unconscious. Never swim alone.

Finally, it is unusual for large marine animals to act aggressively toward human swimmers, but there is a small chance that a human swimming with a Merfluke would be treated more aggressively than a human swimming with normal fins for reasons that would remain to us unfathomable.

Fabrication

(Please refer to the Bill of Materials that follows.)

Frame

The frame shape is relatively complex, but is easily cut with saw blades designed for cutting aluminum. Every aluminum cut should be rubbed down a bit with a file, which greatly reduces the chance of the edge slicing human skin.

The leg braces can be cut for bars of aluminum and bent with a metal brake, or, in a pinch, with a hammer and a vice. After using bolts on previous models, I now prefer to rivet the braces to the frame and to each other.

The soccer shin guards are mounted onto the braces with screws, because there is a good chance you will want to replace them later or reposition them. In general the EVA foam inside the shin guards is thick enough that a flat-headed machine screw will not protrude enough to scratch the legs. Shin guards typically come with velcro straps to strap the guards to the legs. These should be carefully cut away, as it would be unsafe to use the straps in the water and only add drag. On some shin guard models, the stitching that holds the EVA in place also holds the straps in place, so be careful to not damage the stitching of the EVA, as it is needed for padding and comfort.

Peduncle

The peduncle is fundamentally two 6” disks of aluminum held apart by scrap aluminum. Pop rivets are used to rivet through the two discs and the scrap aluminum, thus forming a single secure peduncle. The peduncle is attached to the frame through a hole in its center using a 1/4” shoulder bolt. Using two nuts separated by a lock nut has proved a very reliable and convenient way to building this joint and the other three moving joints.

The peduncle has a second hole for connecting the conrod. Both the conrod and the frame fit inside the two discs of the fluke. The conrod transfers rotational force to the peduncle from the pedals.

Note that the peduncle has a slot cut into in into which the fluke is inserted, which allows for effective transfer or rotational force into the fluke.

Pedals

The pedals are the most difficult component to manufacture.

My approach has been to keep one large piece of aluminum perfectly flat. This plane will be in contact with the frame. Hopefully, and apparently from my experience, a thin layer of water will lubricate these two large planes, so in fact there is little friction between them. This only works, however, if the all bends are made away from the plane, and no rivets or screws are placed on plane in contact with the frame.

Although my design can be improved upon, the basic design is keep a large plane, bend two pieces away from the frame that can be used to rivet the foot-deck in place, and bend the bottom piece up to hold a triangular brace and to attach the pedal shroud.

Additionally, two small pocket cuts must be made to create tabs so that the top-of-foot-block can be mounted. These should be started with holes that are drilled. A band saw, saber saw, or hack saw can then be used. Generally speaking the tabs can be bent into place with a pair of pliers and some hand strength.

It is very important to leave a small piece of aluminum for the heal to brace against, as this presents the foot from pulling away from the pedal on the up-stroke.

After the pedals are riveted together, cage that generally has the correct shape and is strong enough to support the foot. 

 

The top-of-foot blocks were cut from a 4x4 in an attempt to be both comfortable and to improve the hydrodynamics of the foot. For a human being prone in the water, the feet actually represent a large area facing the direction of flow, so the idea is to streamline them by decreasing the angle of attack. More importantly, that athlete must have something to pull against with the top of the foot to recover during the upstroke and to have control of the fluke angle.

After carefully attempting to cut a plane matching the shape of the top of my foot and rounding the blocks where the toes begin and on the back side, I coated with three coats of polyurethane coating. This delayed or prevented them absorbing water.

It is then absolutely essentially that some kind of padding be applied to the blocks where the foot contacts them. The human foot is very tough on the bottom and very delicate on the top. I found that a half-inch exercise mat cut into the appropriate shape and contact-cemented on is effective.

Although the Merfluke will function with pedals made only of aluminum, in the current design it is important to cover them and fill them with a buoyant material for two reasons. First, shadow or wake of the foot is hypothesized to be one of the main sources of drag. We want to produce a hydrodynamic shape when the pedal has a foot in it and is moving through the water at 2m/s. The best shape for this is complicated by that fact that it changes position during the swimming stroke. I therefore estimated that a generally rounded shape would be best.

I had a seamstress sew the pedal shrouds in generally rounded shape out of a material called marine vinyl. She did this by trial-and-error when I loaned her the pedals, so unfortunately I have no reproducible pattern for this complex shape. I attached the shrouds to the pedals by riveting them. I then filled them Tough Stuff ™ brand expanding foam, which cures into a solid mass which further strengthens the pedals and adheres the shrouds into place.

The overall pedal assembly is then bolted to the frame using a shoulder bolt.

Note that in the pedal design the conrod is attached quite far ahead of the toes. This is not obvious and took a great deal of trial-and-error to discover. I believe this helps to center the full range of motion of the foot around the neutral point of the fluke, and to allow the most balanced match between the rotational range of the foot in a prone position and the desired rotational range of the foot.

I am somewhat disappointed in the availability of free software for synthesizing 4-bar linkages, although there seems to be plenty that will analyze a designed linkage. I chose this design through trial and error with pencil and paper.

Fluke

The fluke is a relatively simple plane that pushes the water. My current model is constructed of aluminum and polycarbonate riveted together. I'm not happy with my design put present it here anyway, although you may wish to improve on it.

My shoulders are about 24” across, so I made the fluke the same width. The fluke is braced with aluminum bars, which are necessary to strengthen it against the large forces that would bend it. However, in the future an internal frame or more hydrodynamic bracing might be better.

The lateral stiffness is provided by a plate of aluminum. The current fluke is cut into a “W” shape but I have no reason to believe that is a good shape. In general I think we should mimic a dolphin fluke until theory and experience allow us to design a better shape.

It is critical that the fluke be more flexible at the back end than at the front, so that when pushing water it can naturally conform to the shape of water flowing around it. This will minimize the turbulence produced in the water. Although a stiffer fluke may someday be used, a soft fluke is needed now to be forgiving of imperfection in the athletes motion. 

 

I conjecture that the split, dolphin-fluke shape allows a more coherent vortex to be shed when swimming. It is important to note that in a split design you have to drill out a circle of sufficient radius so that the bending forces are not concentrated in one place. Although the triangular braces prevent bending strain and polycarbonate is a wonderful tough and resilient material well suited to this purpose, it is not infinite strong---I cracked flukes on previous models.

Conrod

The conrod connects the pedal to the peduncle. It is the simplest component. Note however, that you can drill several holes in the conrod to allow for slight adjustments to the action of the linkage in the field.

The 4-Bar Linkage

When the conrod connects the peduncle to the pedals, the whole system forms a 4-bar linkage that allows the angle of the fluke relative to the frame to be controlled by pointing the toes (pushing against the pedals) or flexing the toes (pushing against the pedal blocks with the top of the feet.) The fluke should be in a neutral, gliding position when the foot and pedals are at the center of their range of motion, which seems to be about a 45 degree angle when the feet are relaxed in water. Extreme pointing of flexure should allow the fluke and peduncle to move through from the down-stop to the up-stop, about 150 degrees of motion.

The Swimming Experience

It is best to be in shallow water when you mount the Merfluke 7. I generally put on my mask and snorkel and carry it out to shallow water. Although you can in theory straddle the machine and simply push your legs together, in practice they catch on the edge of the shin guards, so I generally stab my feet down through the shin guards until the sole of my foot is flat against the pedal and my heel against the heel stop. Then I turn over (belly down, back to the sun) and push myself into deep enough water that the down-stroke of the Merfluke doesn't scrape the bottom. I place my arms straight out in the direction of travel, and try to squeeze my arms together in the “underwater dolphin-kick” pose used by swimmers after their turn.

The Merfluke is comfortable on the legs, and provides good leverage for a good hard push against the water. If you just wave it up and down with your feet relaxed, you will move forward. To gain velocity, however, I think to myself the sequence “point-kick-flex-pull”. This of course ideally should not be a strict sequence, but a coordinated movement, although it takes a while to learn and no doubt I will improve with practice. This sequence means “1. Point the toes--- 2. Dick Downward with the thighs---3. Flex the Toes toward the head--- 4. Pull Upward with the hams.”

If you do this vigorously (flexing the toes is the hardest part) you will go much faster, and begin to feel the water rushing past your face and making your snorkel thrum. With less vigor you can move steadily along. It is relatively easy to dive under the water and then come up, because if you are moving you have plenty of excess power for steering by simply directing your arms upward or downward.

Turning is harder, because at present the frame resists turning to the side. At present I just scull with my arms when I need to turn. I dream of the day that I have enough skill, depth of water, and visibility to perform an “Immelman turn”.

Experimental Performance

Lessons Learned

• Never swim alone, with or without a Merfluke or fins.
• Progress comes at the intersection of theory and practice. Every time I get in the water I learn something that surprises me.
• The Merfluke 4 used cables and sheaves to articulate the fluke. This was very complicated and not as sturdy as the simpler 4-bar linkage.
• A small amount of positive buoyancy is valuable to keep the device from sinking out of sight in deep water and to keep the swimming position prone. The book Total Immersion [4] discusses the importance of prone swimming position and other aspects of human hydrodynamics.
• Perhaps not unlike designing a bicycle, the pure engineering aspects of this kind of project are constantly tempered by the fact that it is an extension of a human being that has difficult to measure strength, power, and agility.
• The soccer shin guards provide a very comfortable application of pressure to drive the fluke up and down. However, more hydrodynamic solutions may be available.
• A boogie board is a valuable safety device when using the Merfluke for the first time.
• Because the Merfluke 7 does not lock the feet into tightly fitting neoprene pockets, there is a constant danger of relatively minor bruising and blistering due to the feet rubbing against improperly padded parts of the pedals.
• Placing a handle on the Merfluke was a major design improvement which made the device much easier to work with.

Possible Improvements

• Video recording to allow analysis of the swimming motion.
• Improvements in hydrodynamics:
  • Streamlining or reduction of bolts and nuts projecting into swimming stream.
  • Shrouding the entire area from the leg braces to the pedals.
  • Reducing or eliminating the fluke brace
  • Shrouding or removing the frontal area of the leg braces.
  • Better shaping a of the pedals behind the foot.
  • A fluke design that does not have protruding rivets.
• The addition of a spring to ease the pulling motion motion of the ankle (at the expense of pushing) to create a more balanced use of the asymmetric musculature of the lower leg.

Motivation for Continued Research

• It's fun to extend the range of human-powered capability.
• The lessons learned here could allow us to build a powered vessel that might be far more efficient and maneuverable than screw-driven vessels.
• The same oscillatory, non-propeller form of locomotion might be very valuable in propelling airships.

Bill of Materials

Tools

• Rotary drill with 1/8” and 1/4” bits
• (Optional, but helps) power saw with blade capable of cutting aluminum
• Hack saw
• Hand-powered pop riveter
• Aluminum pop-rivets with 1/8” diameter and 1/8” grab
• Aluminum pop-rivets with 1/8” diameter and 1/4” grab
• Metal File
• (Optional) Grinder
• Metal brake capable of bending aluminum at right angles (bigger is better but I use a cheap portable model that clamps to a table)
• Sewing machine
• Sand paper
• Disposable brushes

Frame

• (1) 4' x 1' sheet of aluminum ~1/8^th inch thick
  
• (2) Pairs of soccer shin guards with no fabric and EVA-style foam
• (6) feet of 1” x 1/8” inch aluminum bar
• (4) shoulder-bolts in 1/4” diameter
• (8) 1/2” machine screws with washers and nuts

Peduncle/foil

• (1) 3/32” thick aluminum sheet, 2' long and 1' wide (for fluke frame)
• (2) 1/16” right-angle aluminum pieces 1” wide and thick (for fluke outer rails)
• (1) Sheet of polycarbonate in a thin grade (1mm), 2' feet long and 1' wide
• (2) 1/8” aluminum bars
• (2) 3/32” aluminum sheet, 1' x 1' (for peduncle circles)
• scrap aluminum for spacing inside peduncle
• aluminum pop -rivets and hand-powered riveter

Pedals

• (1) sheet aluminum, between 1/32” and 1/16” or less in 1' width, about 8', for pedals
• (1) Marine vinyl, more than a foot wide, several yards
• (1) Sprayable polyurethane foam
• (2) feet of pine 4”x4” (or 3.5” x 3.5”) lumber
• (1) Brush-on polyurethane coating
• (4) Self-tapping screws
• (1) Contact cement
• (1) square foot of heavy foam, such as from a 1/2” thick yoga/exercise mat

Suppliers

• Online metals
• Metal brake
• Stanley Pop-riveter

References

[1] http://style.org/strouhalflight/

[2] http://www.nature.com/nature/journal/v425/n6959/images/nature02000-f2.2.jpg

[4] Total Immersion. Terry Laughlin, John Delves. Fireside, 1996. ISBN-10: 068481885X