Reflections on the potential of human power for transportation

Saturday, July 12, 2014

Graeme Obree's Beastie: The Lure of the Linear Pedal Drive

This post was inspired by Graeme Obree, two time world hour record holder on modified upright bikes and who, at the age of 48, rode his Beastie streamlined prone bicycle into the record books last September. Obree was attempting to break the flying 200m sprint for human-powered vehicles which stands near 83mph. At 56.6mph, Obree came up a bit short. Nonetheless, he did break the record for a streamlined vehicle using a prone rider position, and more significantly, in my opinion, he broke the record for a vehicle using a non-circular pedaling motion.

The two pictures below show a kinematic model of the Beastie's drive mechanism and an enlargement of the resulting pedal path. The path is an elongated ellipse whose major axis is tilted slightly downward front to back.

In the pictures above the cranks rotate in a clockwise direction and the pedals move in a counter-clockwise direction around the ellipse.

Obree bested the existing record for a linear-drive and prone-posture streamliner held by Richard Byrne on Steve Ball’s Dragonfly of 54.9mph. The Dragonfly used both arms and legs moving in straight paths for propulsion. The picture below is from Human Powered Vehicles by Abbott & Wilson.

For the purposes of the following discussion, I consider linear motion to include pedals moving in a straight  line, pedals moving in large arc over a small portion of a circle, and numerous curves generated by four-bar linkages having paths that are significantly longer (in the leg-extension direction) than they are wide. These coupler curves could be egg shaped, elliptical, figure-eight shaped among others. In the case of the Beastie, the pedal path is an elongated ellipse.
When someone of a technical bent takes a close look at the bicycle for the first time, they invariably comment that there has to be a more efficient means for the body to generate mechanical power than circular pedaling. Since a runner’s feet don’t go in circles, it makes no sense that feet going in circles on a bicycle are either natural or efficient. More often than not, the conclusion is that feet moving in a near-linear path would be a significant improvement.
The drive mechanisms that will be discussed here fall into two broad categories, oscillating treadles and constant-torque treadles

Oscillating treadles consist of an input link permanently connected to an output crank through an intermediate link. When the output crank moves continuously, the output link moves back and forth between its extreme positions or oscillates. For a constant crank speed, the speed of the input link varies over the cycle and often comes to a complete stop at the limits of travel. By its very nature, one important characteristic of the oscillating treadle is that one cycle of the input link results in only one rotation of the output crank. Some form of gearing is usually required between the crank and the wheel. These systems work best with fixed gearing so that the vehicle motion carries the pedals through their motion-dead spots.
 The crank slider (the core of every IC engine) is an oscillating treadle. The mechanism used in the Beastie was a offset-crank slider where the slider track is not lined up with the crank pivot. In addition, the pedal is located above the connecting rod. This produces a relatively horizontal-flattened ellipse located above the crank center, which accommodated the rider being located above the crank center.

Because friction associated with the slider can waste energy, a rocker link often replaces the slider for mechanism used for human power generation. The connecting rod moves through a short segment of a large arc instead of a straight path.

When used by Kirkpatrick McMillan in the mid 1800’s the crank-rocker mechanism was the first bicycle drive.

Oscar Egg, a world-hour record holder on upright bikes, used a crank rocker for a streamlined recumbent design. Notice that this is a fixed-gear system where the vehicle motion prevents the pedals from stopping at their dead spots.

And the prolific Gary Hale produces his Glider which employs a crank-rocker.

And the crank rocker is still used to propel most children’s kiddie cars.
Referring to the crank-rocker diagram again, if the pedals are located at point A, they will travel in a circle. If they are located at point B, they will travel in a large arc. If the pedals are attached to the connecting rod, at point C, they will travel through a hybrid of the circle and the arc, an elongated teardrop. These coupler curves have the advantage that the pedal continues to move at the ends of the stroke conserving some of the kinetic energy of the moving limbs. The downside of locating pedals on the connecting rod is the pedals are connected to the frame through two pivotal joints instead of one. This can result in more flexible connection (read sloppy) than a connection through only one joint.
Another oscillating treadle mechanism is a rocking slider. The slider is attached directly to the crank instead of through a connecting rod. To compensate for the transverse motion of the crank, the slider must rotate about its sliding point.
 The K drive, nicknamed because Miles Kingsbury used it on one of his streamliners, is derived from an elliptical trammel. The mechanism can produce a straight pedal path but as configured here it produced a long-thin ellipse. The pedal path produced by this configuration is very similar to that employed in the Beastie.

The other drive mechanism that will be discussed is the constant-torque treadle.

Unlike the oscillating treadle, the ratio of input lever speed to output shaft speed is constant (and as a result, so is the torque). There is a one-way clutch located in the cable drum which allows the input lever to return to the beginning of stroke without reversing the motion of the output shaft. Additional stops must be inserted to limit the input link travel. The ratio of input link motion to output shaft motion can be adjusted by changing the position that the cable attaches to the input lever. This is one big advantage of the CTT; it can incorporate a very simple means to achieve multiple gearing. Some form of return device, usually a spring, must be used to reverse the input link motion at the end of travel.

The CTT is the mechanism most often reinvented by those who would improve the design of the bicycle propulsion mechanism. It is also has been the most prevalent drive system after the rotary crank. It was used on the American Star pre-safety bicycle in the late 1800’s. Paul De Vivie, aka Velocio, the father of the derailleur, experimented with CTTs in the early 1900's. Below is one of his own designs. Depending on where you placed you foot on the treadles, you could vary the effective gear ratio of the drive. A cable connected from one treadle to the other insured that the non-driving treadle moved up while the driving treadle moved down.

 You can still find numerous prototypes today. Steve Ball’s Dragonfly used a modified version of the constant-torque treadle, as did the Pedicar.

There are several reasons why a person designing a human-powered vehicle (HPV) would use a pseudo-linear pedaling motion.

1.       There is interference between the pedals and the steered wheel with circular pedaling motion.

W. D. Lydiard used a rocking-slider mechanism to reduce pedal-steered wheel interference on his entry for D. G. Wilson’s 1968 Human-powered-vehicle design competition, the Bicar. The picture is from the first edition of Bicycling Science by Witt & Wilson.

I experimented with a crank-rocker mechanism in an attempt to reduce pedal-steered wheel interference in my EcoVia commuter trike design.

I employed a two sided pedal in this design. The outboard side of the pedal holds the rocker link that supports the pedal. In this location it is spaced wide enough to clear the turning wheel. The inboard side of the pedal holds the connecting rod which is located above the wheel and includes a bend to clear the wheel. This is my interpretation of D.G. Wilson’s crank-rocker concept sketch for a recumbent bicycle.

2.       The foot and knee moving through a pseudo-linear motion take up less volume than circular pedaling.
A classic use of linear motion for this reason is in the Pressodyne streamliner form the late 1970’s. The article is from the Spring 1980 issue of Human Power.

Here are a few highlights relevant to the current discussion. The pedal motion was truly linear using rollers to support the pedal arms. The stilts-version of the Pressodyne used cables that connected the pedals to one-way clutches. This was a constant-torque treadle approach but no efficient means of limiting pedal travel was provided and the pedals crashed into the stops. The three-wheeled version used a crank-slider approach which was much smoother. The smoothness was also due to the fact that there was also no freewheel in the system (fixed gear). So, when the vehicle moved the pedals moved and there was no issue with dead spots in the motion.
The shape of the Pressodyne was not far of the mark for the optimal streamliner shape. Notice the similarity with probably the epitome of streamliner design, the Varna Tempest. The tempest required a bigger nose to house the circular pedaling but had a smaller canopy.

And reducing swept volume of the leg and foot is undoubted the reason Obree employed a teardrop-pedal path in his Beastie.
3.       The linear drive is simpler that a pair of cranks, a chain and two sprockets.
The Mergamobile was a pretty simple approach as was the 1921 J-Rad. One used the different pedal locations to obtain three different gear ratios. Both design use constant-torque treadles.

4.       The linear drive is more efficient than circular pedaling motion.

The constant-torque treadle is the most popular design proposed for improving the efficiency of bicycle propulsion
The most publicized use of a constant-torque treadle was in the 1973 Pedicar.

Trevor Harris, a race car designer and designer of the iconoclastic Can Am Shadow produced the Harris Vertical in the mid 1970’s.

The Alenax Trans-bar bicycle was commercially produced in the 1980’s.

The Alenax was an almost a direct copy of the Svea manufactured in Sweden in the late 1890’s. Paul de Vivie, Velocio, the pioneer cyclo touriste, supposedly experimented with the Svea in his quest to find the perfect touring bicycle.
Notice both the Harris and the Alenax have adjustable cable-attach positions on their pedal levers for variable gearing and both have synchronization mechanisms to move the pedals in opposition to each other. The Harris uses a rocker linkage and the Alenax uses a cable loop.
When discussing the reinvention of the constant-torque treadle, I can’t help but hear Santayana’s quote “Those that cannot remember the past are condemned to repeat it”. In this case “know the past” is more appropriate. The following optimistic declaration that the bicycle has been greatly improved is a fun foil to discuss the shortcomings of constant-torque treadles

It also may be useful for the reader to review the section on Power in the following post.

Let us begin with why the constant-torque treadle appears to be more efficient than the rotary crank. Assume the rider exerts a force of F in a straight line with each leg. With a rotary crank the torque that is transferred to the wheel is F*sin(theta) where theta is the crank angle. The average torque over a cycle is 2F/Pi or .64F. So from the start, from a torque standpoint the CTT is 57% more efficient.

Unfortunately there are two factors that prevent this increase in torque from being converted to an increase in power
Our linear-motion bicycle salesman states that his drive develops full power from the beginning. That is not true. At the beginning of the pedal stroke, the foot is stopped but the output shaft is moving at full speed. It takes a portion of the pedal stroke for the pedals to catch up to the output shaft and during this catch-up phase no force is being produce and, as a result, no power is produced. Both the Dragonfly and the Harris Vertical incorporated cams to gear up the pedal stroke in the beginning to allow the pedal speed to more quickly match the speed of the output shaft. However the cam must be designed for a specific gear ratio. So on the Harris Vertical, the cam will only be effective for around one gear selection.

Another problem is the pedaling speeds that can be sustained with linear drives are significantly lower than those that can be sustained with a rotary crank. This is due to the kinetic-energy fluctuations of the moving limbs. With linear motion the foot stops at the pedal extremes and the kinetic energy drops to zero. With circular cranks, speeds of 300rpm have been achieved because the kinetic energy is relatively constant.  Since power moves the bike and since power is the product of torque times angular velocity, lower pedal speeds result in lower power levels.

The rider is very diplomatic when asked for his impressions riding the linear bicycle. He says it is much better than the first prototype but he doesn’t say it is better than a regular bicycle.

The linear bicycle riders comments that his leg muscles have gotten bigger riding the linear bicycle is an indication that things have become less efficient as opposed to more efficient. When Paul Dudley White, President Eisenhower’s personal physician and bike advocate, rode the Pedicar, he also noticed that it put more strain on the thigh muscles.

One factor that is inconsequential for light-weight vehicles like bicycles but becomes a problem for heavier commuter vehicles is that with the CTT drive, the vehicle cannot be rolled backward. The one-way clutches lock up going backward causing the input levers to jam against the motion stops. This is the reason the Pedicar incorporated a reverse gear at the cost of a significant increase in complexity of the transmission.
There appears to be a means of accelerating the foot at the beginning of the pedal stroke that remains effective throughout an adjustable gear range. When cams were used above, they were inserted in series with the drive cable. I advocate using springs in parallel with the drive cable.

Assume a synchronizing linkage is used to connect the pedals and move them in opposition.  Springs are located so each pedal compresses the spring as the pedal is pushed forward. The springs exert no force at the beginning of the stroke and exert maximum force at the ends of the stroke. Assume the force at the end of the stroke is 2*F.

The combination of the synchronizer mechanism and the springs results in the force vs. pedal position shown in the first graph. With no external load, the zero-force position for the pedals will be at midstroke. From the beginning of the stroke to midstroke, the springs act to move the pedal forward, accelerating the foot. From midstroke to end of stroke, the springs resist forward motion and add to the force required to propel the vehicle. If the average force required to drive the vehicle is equal to F, then each pedal sees a force vs. displacement curve shown in the second graph. The pedal encounters an increasing force from the beginning to the end of the stroke. Since the leg can exert more force as it extends, this matches the pedal force to the legs ability to generate force.

One approach to determining the spring rate for theses springs is to select them so a resonant condition occurs with the moving leg mass. Let us say the moving leg mass for each leg is ½ the mass of the thigh plus the mass of the shin and the foot. From anthropometric data, that comes to about 13.5% of body weight. With a 170lb. rider that gives a moving mass of 23lb. Let the resonance be at 75rpm or 1.25Hz. That requires a spring rate of approx. 4lb./in. Assume a pedal stroke of 180mm or 7in., then F is 28lb.

28lb at 75rpm and a 14” stroke corresponds to 110W. So at a power level of 110W, the pedal force is zero at the beginning of the stroke and 56lb. at the end of the stroke.  The spring rate could be increased so negative to low forces are encountered at the beginning of the stroke for power levels higher than 110W.
After thinking so much about the Pedicar, I couldn’t resist speculating about a drive-system redesign that would address its shortcomings. I also assumed it would be a banking-three wheeler with front steering. As a result the pedal levers are two-piece with the support link outboard of a two-sided pedal and the input link inboard of the pedal. ( See my crank-rocker design for the EcoVia, above.)

 I have included a synchronizer linkage and accelerator springs to smooth out the pedaling. Instead of the Pedicar’s five speeds covering a range of 6:1, I use a shifting quadrant on the input link that can be rotated over an 8:1 ratio, but a range divided into 21 steps. The 8:1 ratio using 16t freewheels as the one-way clutches required a gear-up mechanism. I incorporated a forward and reverse gear set into that mechanism. Recall that, since the one-way clutches prevent the vehicle from being pushed backward, some means of disengaging the drive or having a reverse gear is necessary to move the vehicle backwards
When I stood back and looked at the design, I realized that although it addressed the Pedicar’s design deficiencies, it is probably no-less complicated than the Pedicar’s drive, and no lighter in weight. Since the cost of all-weather human-powered commuter vehicles seems to be the greatest factor preventing their popularity, this would not be a good design approach. An ultra-wide range cassette with a single chainring is cheaper, lighter weight and allows the vehicle to be pushed backwards.

So the next iteration of the EcoVia will pass on the constant-velocity treadle.

There is one circumstance where the constant-torque treadle performs significantly better than conventional circular pedaling is when climbing the very steep hills typically encountered in mountain biking. Outstanding hill climbing performance is mentioned in regard to the American Star of the late 1890s and the Pedicar.
A more detailed explanation of hill-climbing problems associated with conventional circular pedaling can be found in the Kinetic Energy and Cyclic Energy Storage section of the Why Hill Climbing is Hard post.

From the standpoint of power generation efficiency (mechanical power out/oxygen in) producing power in pulses interspaced with rest periods is better that producing power continuously. The extra energy produced during the pulses is used up during the rest periods and this energy is stored in changes in the kinetic energy of the vehicle. If the vehicle speed drops below a certain level, the power cannot remain pulsatile and the rider must produce power around the complete pedal cycle instead of the usual pulses produced from 1 to 5 o’clock in the pedal cycle.

The constant-torque treadle is cadence limited but this is not a problem because the cadences associated with steep hill climbing are low. The dead spots in the pedal cycle are only momentary with the CTT and torque is produced for almost all of the cycle while the foot only moves through its normal force generating range. Adding acceleration springs just improves the performance assisting foot motion at the beginning of the pedal stroke.

Come to think of it, a few of the restored Alenax Trans-bar bikes were sporting mountain-bike tires.
If Graeme Obree had propelled his streamliner with convention circular pedaling, he probably would have gone faster, but he would only have a prone-rider record. I believe the linear-drive speed record is technically more interesting.



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  2. After studying treadle & linear bicycle mechanisms for a quarter century, this is the finest article on the subject I've had the pleasure of reading. Hopefully, this is just the beginning of the discussion and not the end.
    I should hope this will expand into the analysis of sinusoidal and linear "motion," and everything in between... and the analysis of sinusoidal and linear "function," and everything in between. The piston and crank with a cross slide is an example of the piston in linear motion, but still stuck in the sinusoidal function due to the rod and crankshaft. Graeme Obree's Beastie elliptical pedal motion is “in between” linear and circular, leaning more towards the linear, but it is still converted into a sinusoidal function working through a crank. 
    Since my prosthetic knee barely hinges ninety degrees, I'm no longer able to ride crank bicycles, but I can ride faster, farther and easier on my Alenax bikes. It is equivalent to pedaling from the 2 oclock to 5 oclock position, the "sweet spot," and not the whole 12 to 6 o'clock sine function. It has good power delivery the full stroke, which is a pleasant feeling. There's no denying, there's more area under the curve of a square wave than a sine wave, ergo, more work output for given thrust.
    For a decade or so, I had the pleasure of switching from my Alenaxes, mountain bikes and my Columbia 10 speed, so the different performances could be compared. The Alenax is more efficient, for the most part, but it has a number of failings- good design but poor execution. It may fall under the category of constant torque, but the short treadles are are still handcuffed to the sine wave. The lower 2 speeds drop down more into the 3 to 6 o'clock range and can be frustrating. The Alenax is nearly a clone of the Svea, Northfleet, Levocyclette and the Harris bicycles. The Harris MK 2 has been the most inspirational for my concept of a continuously variable speed long-treadle drive for an upright bike. I still study the 25 year old gray printout off the USPTO microfilm. 
    I disagree with the argument the reciprocating linear motion is inferior to a constant circular motion, and in fact, it is probably the other way around. Using the Alenax/Svea treadle vertical motion for example, it is obvious pedal movement stops at the top and bottom stroke, BUT, the all important “vertical component” of the crank pedals likewise reach zero speed. The crank will accelerate slower in the vertical direction than the treadles, because it is locked into sine function. The momentum of the leg will actually be less for pure vertical movement since it doesn't have the parasitic horizontal momentum of the lower leg. The idea there is a great deal of wasted horizontal motion with the crank is NOT an advantage. Treadles also offer the advantage of electing a shorter stroke for a break from long leg movements. 
    I've long planned to get into production with a long-treadle wedgie with continuously variable speeds, but have since elected to fabricate a continuously variable speed linear drive recumbent (no crank, derailers, forks or steering column) to test out the concepts with superior performance over the upright. Someday!

    JIM.... another left-handed cyclist

    (this reply was previously deleted in order to edit it)

    1. Thanks for your comment, and all those older designs names. I'll research all of that. I've added another comment below that you might like to read!
      Maybe we could get in touch to develop something.

  3. I'm happy and sad at the same time for having found this article. Happy because its very informative, and sad because, again, I've invented something that (in a big portion) already exists.

    I have to disagree on efficiency opinions:
    1) Cyclists trained for years on circular cycling said linear is about the same or inferior. But for a unbiased comparison one would have to train with both bikes for a whole year and test later. Or test with nin-cyclists athletes, like a runner who hasnt cycled for two years.
    2) Leg muscle growth indicates inefficiency: new movement will use new muscle groups. A cyclist will probably notice if a non-cycling muscle starts to pop out. But he probably never realised how strong his glutes and quads already are! And calves, if you use your cleats close to the tip of the foot.
    3)Lower cadence on linear bike: but consider also neuro-motor training. Ive been circular cycling since I was 4! I can spin 200rpm downhill on my fixie, and I can probably train to do the same linearly. Also I've rowed in the past, that probably will help the day I get my linear bike built.
    4)If not using a fixed gear system, each side would waste energy in the begining of each stroke as the freewheel engages again, because the wheel is spinning fast and the leg has stopped: I like the spring idea! But maybe you don't need it to start compressing in the middle of the stroke, but more towards the extreme. Also there are non linear springs (conical springs) that have even more compression force the more they get compressed. So you can play with that to fine tune the most efficient spring setting. But I have to question if this engagement delay is really a big problem, because theres also no resistance force. Circular cycling also has those dead spots where you are not really pushing any power, just moving the crank along until it gets to 1 o clock so that you can push it hard again. How many people really "pedal in circles" all the time? I only do it sprinting or hammering uphill on a big gear. You can also build a linear path that is much longer than anyone will actually push. So the feet will just stop and move back because your leg isn't longer and you dont want to hit your knee on your chin.

    Finally, I said that i have, in a big part, reinvented linear drive. But I have something new, even after reading this article. I won't get in detail here for now as I have some more resarch to do - you never know if your idea doesnt already exists.

    But I can add what features I belive my system can achieve, if it works as intended, even considering the problems above:
    -CVT transmission. Probably automatic and electronic will actually work better and easier, but i have to hire an electronic engeneer to help me with that... $$$. A mechanical CVT for my idea actually seems to be more complicated to build.
    -Ridiculously wide range: above 600%, maybe as much as 900%, without any efficiency change throughout the range (eg: NuVinci CVT hub wastes energy on the top range. Rohloff or nexus only has no waste on 1:1 ratio).
    -Power modulation throughout the stroke: the beggining will be lighter to push, as the bent knee has less power. The bottom will be harder to push, as the extended leg is much stronger. I still have to research what is the proportion if this strengh difference for a kneew angle function. This is similar to that added spring idea, but adressed in a very different way. Still my design might benefit from a smaller spring added just for a little help in the extreme of the movement ti shift direction of feet, or avoid harsh impact.

    That's it for now. But if you want to get in touch,

  4. I forgot to mention my design should be much lighter than all of those swinging arms, and more compact. Probably lighter than crank+derrailleur systems. And lower maintence. Sounds like magic, no? I believe in it, looking to make it become true.

  5. I have a question for you all since you’ve obviously studied this much more than me. Was recently pedaling and was very annoyed by the dead spots at top and bottom of my stroke and the backlash I got consistently when I was not actively producing power in the crank cycle. It occurred to me that a flywheel linked to the crank would help carry through those dead spots and get us closer to the feeling of direct drive in place of freewheeling hub. Have people tried this? I imagine it would take extra energy to excite the freewheel, but would it pay off in helping to create a more consistent cadence and less strain on joints etc? Excellent article by the way!

    1. Hi Nathan, sorry for not getting back to you sooner. Flywheels have been used on bicycles in the past and they can smooth out the pedaling dead spots. The two negatives are the weight of the flywheel and the flywheels resistance to changing pedaling cadence. A fixed gear gives a similar effect but they prevent changing gear ratios. You and experiment with changing your seat height to try and smooth out your pedaling. A lower seat height with more knee bend allows greater pedaling cadences which the more constant kinetic energy of the leg masses allows. Keep in mind that smooth pedaling has produced cadences of 6 revolutions per sec!

  6. I saw a paper on this, hold on a sec....

    There we go. It's by the University of Liverpool, Department of Mechanical Engineering (1979). The conclusion was that a Constant Torque Pedal system was 33% less efficient then the more common rotary action and that most of that loss was due to overcoming the legs own inertia and that of the long levers.

    1. Oh, a bit got cut somehow. I was going to say that their testing was not very through though, testing only one type/format of CTT. And from appearances, a fairly inefficient design with very long treadles.