Investigating The Energy Needed to Stay in Balance
Updated 4-28-05

Speculations on the unicycle by George Winters

One of the very fun things about unicycling, for me, is remembering how impossible it seemed when I first tried riding my homemade unicycle.  How is it that one can learn to move on such a completely slippery gizmo?

My next big surprise, after I actually started succeeding at riding the unicycle, was how tiring it was.

My initial investigation of the energy required to ride a unicycle suggests that there is Missing Energy.  This article attempts to resolve some of that disparity.  While looking at the mechanical process of riding the unicycle, I also see a hint that the ability to ride a unicycle builds on the often underestimated complexity of walking.

In the process of looking at the energy needed to unicycle I also propose that the challenge of riding a unicycle can help to balance the body and stretch the mind.  As I try to analyze this  complex mode of moving, it leads me appreciate a much wider array of
Proportion and Symetry
Click on image to find out more about unisk8tr and photos by John Childs*
systems in balance.

A friend asked me why I would want to ride a unicycle.  My flippant reply was, "Because it is as much fun as skiing, but I don't have to drive for 3 hours and spend $50 on a lift ticket."  After I said that, it slowly occurred to me: That's a fairly accurate assessment.

What is the energy demand to ride a unicycle?  
The skiing analogy seems apt to me (a novice at unicycling) because I have noticed that as I extended my riding to cover distances of a block or more, my thighs felt a similar stress to that which I have experienced when telemark skiing.

In his entertaining and informative book about his long distance unicycle adventure, One Wheel-Many Spokes, Lars Clausen notes that  a bicycle is one of the most energy efficient means of transportation.  A unicycle is not.  

Why not?

I did not find literature about testing the energy expenditure of a person riding a unicycle.

(I did read of a test where a pronghorn antelope was hooked up to a machine that measured the animal's oxygen consumption while it was running on a treadmill at 35 miles per hour.  Wow!  Imagine the challenge of doing that experiment!)

I am beginning to wonder if anyone has tested the biometrics of the energy requirement of unicycling.  Would it be possible to ride a unicycle on a treadmill while hooked up to a device so that a person's oxygen consumption and energy output could be measured?  Are there more general implications that could be learned from such an investigation?  

Maybe the answer has wider relevance than just having fun.

I can find lots of information on the energy demand to ride a bike, walk, run, or cross country ski.  What about the unicycle?

Unicycle vs Bike:  

An Initial Measure of the Energy Difference

I have recently read three accounts of endurance rides on unicycles, people who attempted to test the limit of distance that can be covered in an hour or a day.  It occurred to me that long distance efforts such as these will generate parallel tests of human energy output that allow comparisons across a range of similar activities.  The energy needed to run or walk, ski or bike for extended time periods places demands on similar muscle groups, will produce similar tests of the human "engine," and should have the same limiting factors related to the amount of energy that can be stored and processed in the human "gas tank."

A friend has suggested that my proposition would be best presented if I could chose performance efforts from comparable population samples.  If I had a large range of performance samples in each activity and I chose, say, the 90th percentile effort in each activity, it would be realistic to say that each of these human "engine" efforts produced the same amount of energy for some long period, an hour or a day.  But, I don't have access to that extent of data, especially in the activity of riding a unicycle where the population sample is relatively very small.

I am choosing a very subjective limited comparison while trying to keep in mind a larger relationship and primarily using limited data to show that a realistic comparison can be made without trying to claim a high level of precision.  I am trying to be accurate, while not necessarily precise.

The Basic Plan:

The Comparison Subjects: Each of these people either have, or are applying for Guinness recognition.   The performances they have established represent evolving and limited data.  Certainly they are very good athletes with training and experience, but they probably do not represent the extreme upper limit of the human engine capability.
The green line in the graph on the right represents the energy demand for the unicyclist evaluated with a Bike Energy Calculation as if it were a bike of appropriate weight and shape going the same average speed of the three endurance trails referenced.  The "George's Max." line represents the energy output I have managed (as a younger and slimmer self) riding a lightweight bike for each of those time periods.  I am not a stellar athlete.  The comparison with my bicycle energy output shows that as a bare minimum each of the individuals above must have been putting out at least 100 Kilo calories more per hour than the green line estimate to accomplish their feats of similar duration.  

The calculation of effort to propel a unicycle in a long distance effort will require at least the addition of 100 food calories per hour above what is accounted for by the simple dynamics of moving the machine and system over level ground through the resistant air.

I have made an effort to find a comparable performance level in each of three other activities, running, walking, and biking to show energy outputs that would represent people who would be similar to the three individuals noted above.  My choice of comparisons is very sketchy, and is primarily listed just to show that there is an obvious disparity between the simple bicycle comparison and the real world energy demand.

The 24 hour energy requirements tend to converge because it is reasonable that a 24 hour event will tax the total energy storage limit of the individual.  The graph comparison shows that using the bike energy calculation for the unicyclist greatly under estimates the energy requirement.  So, if one uses the speed of a unicycle to calculate energy demand, the value attained by calculations of wind resistance, friction, and efficiency may miss  by half the real energy demand.  

Keeping an unstable dynamic system in balance can double the energy demand!

Limits to propelling a Unicycle at speed:

What is using up the extra energy?  
Modeling the System As An Inverted Pendulum:

Schematic of ballancing motion In the simplest analogy, the unicycle is like a long uniform rod balanced on one end.  The center of mass will pivot about the balance point moving in an arc, similar to an inverted pendulum.  As the center of mass tips away from the balance point, the arc of motion lowers the mass relative to the ground.  Each time the system tips, the energy needed to move back to the balance point will be equal to the energy needed to lift the mass of the system back up to the highest point.  This proposition applies the principle of  the conservation of energy.  The kinetic energy due to the center of mass changing height will equal the kinetic energy of the tipping rod which can be analyzed as a rotating mass.  
 Mgh = 1/2 I w2
M is the mass, g is the acceleration of gravity, h is the changed height, I is the inertial constant of the rotating body, and w is the angular speed of rotation.

Imagine the rider is in a hallway and regularly pushes off the adjacent wall to return to balance.  The energy represented here is the energy needed to continually push back off the walls.  In actual riding this energy is developed by steering, subtle changes of speed, and moving the body and arms.  Ultimately, the energy has to come from using the legs to push against the ground through the leverage action of the pedals to the wheel.  The energy required is the same as the push off the wall.  The direction of motion out of balance will be changing in  360 degrees of the horizon, but each individual return to balance can be analyzed as a rotation in two dimensions.

The question then becomes two fold:
  1. How far has the center of mass pivoted on the arc?
  2. How frequent is the need to apply energy to return to the balance point?
These two questions interact in the dynamic motion.  
The relationship between frequency and arc angle is not linear because the effect of gravity, g, is an accelerating contribution, and the speed of motion in the arc angle, w, is a squared factor.  So if the changing balance angle can be minimized by frequent corrective action, the energy output will be minimized.  This fact tends to show that the skill of the rider in correcting the balance change is very critical to total energy efficiency.  Energy not needed for balancing can then be available for propulsion in the desired direction.

Balancing Energy Requirement 1st Case
angle of tilt
angular velocity
watts of energy needed to return to balance
Kilo calories per hour
0.04 0.48
0.06 0.58
0.08 0.66
As the representational long rod tips the time it takes the tipping mass to reach each wider angle is shortened as the angular speed increases.  The force of gravity has a larger pull on the mass relative to the vertical support of the unicycle.  At the balance point all of the weight is vertical over the pivot point and there is no inherent motion to tip.  As the mass leans away from the balance point, it moves faster and faster.

The table at right shows kilo calorie energy requirements if various tipping motions were repeated regularly in the natural time it takes for gravity to move the mass to each of the listed tilt angles.  The angles are shown in radians, and 0.1 radian is approximately 6 degrees.  This list indicates that small angular displacement for balance does not require a lot of energy.  As I noted above, I am guessing that a minimum of 100 kilo calories per hour is used in the balancing activity.  This graph is only showing an energy requirement of at most 22 kilo calories per hour.  The angle size and frequency of balancing action could be measured with stop action photography.

Balancing Energy Requirement 2nd Case with overbalance velocity
angle of tilt
angular velocity
h meters
watts of energy needed to return to balance
Kilo calories per hour

The second table shows an alternative likely case in which with each balancing motion, a small excess energy causes the next cycle of moving out of balance to begin with an extra initial velocity.  Even a small initial velocity will significantly change the energy demand because the time available to reach each wider angle is shortened, and the increased angular velocity creates more energy that must be countered to prevent tipping.  This second table shows that at even small angles of moving balance, a very small over balance effort can add a large percentage increase in energy demand.
Timing is Everything
Learning to Ride the Unicycle Depends on Having Learned to Walk.

Bumps in the Road

a bumb in the road A bicycle is an incredibly efficient transport device over relatively smooth and level roads.  The size of the wheels, long wheelbase, and stiff frame structure allow the bike to carry its moving momentum quite successfully over small obstacles.  When a small bump is struck, the geometry of the bike transfers most of that energy into a vertical lift that is not counter productive to the forward momentum.  The light weight bicycle will effectively rock over the obstruction without much change in the rider's larger independent momentum.

The unicycle cannot absorb and re-direct this energy so effectively.  In fact, the obstruction in the road is almost certain to transfer a large amount of the force into the angular velocity as listed by the factor w in the tables above.  For a unicyclist traveling at 16 mph, a change of speed of 1 mph imparted to the angular velocity w would introduce an instantaneous demand for balancing effort that is equal to 100% of the energy that is being applied to forward motion.  This energy demand is not the same as the constant rhythmic motion proposed in the tables because the repetition over times depends on how often the bumps are encountered.  In addition to energy loss to bumps, similar losses will occur due to disturbances such as a change of wind, change of turning angle, and other dissipating forces.  These disturbances will potentially add large increases to the angular velocity that must be countered by applying energy to remain in balance.

Successfully applying energy to forward motion while on a unicycle is a constant challenge of sustaining momentum.

I see this as a close analogy to cross country skate skiing, where successful propulsion is a constant challenge of controlling balance and momentum.  Every action that takes away balance has a doubly counterproductive result by losing forward momentum and demanding that energy be used to regain balance rather than being applied to forward speed.

Notice in the drawing that the center of mass, the x, tends to be over the crank axle.  This is a logical expectation, since one's weight is what allows the application of force to the pedals.

How is energy applied to maintain balance?


torque causes precession Torque is force applied in a rotational reference.   A person standing is applying force on the ground due to the attraction of gravity acting between the person and the earth.  If the person stands on a lever such as a pedal and crank arm, the force is then described as torque.  As force or torque is applied over changing position, work is done.  As noted above, the bicycle frame geometry combined with efficient chain and gear connections is very good at using the pedal force to create the ability to do work at the wheel, propelling the mass of the bike and rider along the road.  The same torque at the pedals of the unicycle will result in a slight twist of the bike and wheel.  In the photo at the top of the page one can notice that Unisk8r has changed the geometry of the unicycle to add a handle structure.  This probably helps the rider control some of the torque at the pedal and more effectively direct the force into forward motion.  The long shape of a unicycle seat is a help in counteracting the torque of pedaling as well as being necessary to apply torque to steering.  My first attempt at learning to unicycle was on a home made unicycle.  I started with a normal touring bike seat.  With my fist effort on my homemade cycle, I realized that a long narrow seat was needed so that one's body can keep the unicycle from twisting (or make it twist as needed).

Unbalanced torque on a spinning wheel will cause precession, or an added rotation around a pivot point.  In an example above I suggested holding a spinning bike wheel to experience the effect of the conservation of angular momentum when the spinning wheel is turned.  In a similar way, if you hold the spinning bike wheel with one had supporting just one end of the axle, there is an added motion created where the spinning wheel wants to rotate horizontally (precession) about the support point at the end of the axle.  This precession is due to the torque caused by gravity that works at a right angle to the angular momentum of the spinning wheel.

On a unicycle there will be a counter productive torque applied to the spinning wheel by the leverage of the pedal and crank mechanism.  A bicycle frame with the stiff gear transmission structure and the added benefit of toe clips allows this torque to be applied very efficiently to forward motion.  On a unicycle some strategy of adding structure to the frame might diminish the otherwise counter productive effects of torque.  Toe clips might seem to make a logical structural improvement in the application of this torque, but as Lars Clausen describes in his book One Wheel-Many Spokes, Lars' attempt to use toe clips had other major drawbacks.  It is possible that in the specific case of riding a unicycle over a level track for long distance, having a heavier wheel would create more balancing torque from the spin of the wheel and allow the rider to put more pedaling effort into effective torque.  The increased angular momentum of a heavier wheel could act like a stiffer frame on a racing bike.  The extra effort needed to move the wheel up to speed would be offset by the increased ability to apply force to the pedal to maintain speed.

On the unicycle, torque at the pedal from the legs pushing is going to cause a rhythmic rotation of the wheel as a steering action.  This is a disrupting and energy consuming balance problem.  Using shorter crank arms and spinning faster might mitigate some of this torque.    The unicycle rider is probably constantly applying some extra force to both pedals simultaneously to counteract some of the unbalanced or uncontrolled torque of the driving leg.

Where The Rubber Meets the Road vs The Liquid AirGraph of Force Needed

Torque (which is force) has another important relation to the force needed to move the unicycle.  The tire on the road is where all the effort has to work to finally spin the energy into motion.

Air resistance is a major force that resists the rider as speed increases.  The power needed to push through the air increases with the square of the speed.  The rider on the unicycle is not very aerodynamic compared with a fast bike rider.  

The graph shows the difference in force that must be developed by the unicycle rider and a bike rider to move at a range of likely speeds.  The rider must apply force at the pedal, but the pedal has a reduced leverage ratio with the wheel pushing on the road.   The actual force applied at the pedal will be higher depending on the size of the wheel, the length of the crank, and the gear ratio.  A lightweight bicycle in high gear often will have an effective gear of about 104 inches.  This means that in high gear the ratio of pedal motion is the same as if the rider were pedaling a 104 inch diameter wheel directly.  The unicycle riders I have noted here were all using a 36 inch wheel.  Ken Looi and Lars Clausen write of changing the length of the crank to change their effective gear, and in general they use a crank that is shorter than the standard bicycle crank.  

It turns out that when I plot the actual force needed on the pedal for speeds of 10 to 15 mph, the choice of crank lengths used by Lars and Ken will create leverage ratios such that the force at the pedal closely matches the standard bicycle.  At 15 mph the torque needed at the pedal is about 22 pounds.  Note that this is a continuous force.  Neither the bike rider or the unicycle rider can apply continuous force.  Torque will be lost due to counter productive leverages and flex in the system.  It is also not possible to constantly apply the force perpendicular to the moving crank.  Toe clips can help, but in any event the rotating crank is putting some of the pedal force into the vertical push on the ground and some of the pedal force into tipping or turning the wheel.  These limits imply that the rider is trying to push with episodic forces in the range of 30 to 40 pounds in order to produce the 22 pound force needed to travel at 15 mph.  

The unicycle rider may approach a limit as to what force is mechanically possible, given the counterproductive losses noted above.  At 20 mph the continuous force at the pedal will need to be well over 30 pounds and momentary real torque at the pedal may be double that at perhaps 60 pounds.  

How the tire can create the needed push,
Good News/Bad News

The ability of the tire to push forward depends on friction between the rubber tire and the road.  The weight on the rear wheel of a bicycle will be some percentage (more than half) of the total weight of the bike and rider.  The ability of the tire to push against the road with the torque needed for forward motion is limited by the weight on the wheel and the coefficient of friction between the tire and the road.  The coefficient of friction between rubber and concrete is about 1.  Therefore the tire can provide almost as much forward push as the weight on the wheel.  The good news is that the limits of push that the tire can provide for forward propulsion is not reached at any realistic speed.  This is one factor that gives the advantage to the unicycle.  Since all of the weight of the system is on one wheel, the possible forward push is higher than is the case for the bike.  The bad news is that because of the unfavorable aerodynamics and lack of gears, the unicycle rider will need to put out more torque at the pedal than the rider's body weight at somewhere around 50 mph.

Spinning Your Wheel(s)

  As noted above, rotating mass is a primary factor in the energy demand of a bicycle or unicycle.  That is why racing bikes have expensive light weight wheels and tires.  Every time some resistance dissipates the rotating speed of any part in the system, it will take energy to increase the rotation speed of those parts in addition to the energy needed to return the entire mass to the desired speed.  Reducing the weight of rotating objects is paramount.  The mass and the radius of the rotating wheel are factors in this equation of energy. (See  A list of basic physics equations I have used.)  In the wheels, this same moment of inertia that will absorb energy when the rotating velocity is increased is also the moment of inertia that helps provide stabilizing balance.  One can ask two useful questions:
  1. Is there an optimal wheel diameter?
  2. How does one wheel vs two wheels play in this energy and balance process?
Wheel on a ramp In Question One, the radius, mass, and geometry of the wheel do affect the energy it takes to rotate the wheel.  Consider a simple example when a wheel is allowed to roll down a ramp.  The potential energy of the wheel at the top of the ramp (due to the mass of the wheel) will be converted into the rotational energy of the wheel and forward motion at the bottom of the ramp.  The potential energy is based on mass and the height of the ramp.  The rotational energy is proportional to mass and the radius squared.  To make a long story short, it turns out that if the proportional geometry of the wheel is kept constant, it will not matter how large the radius is.  All wheels will have the same velocity along the ground at the bottom of the ramp.   The velocity over the ground is the radius multiplied by the angular velocity.  If the radius is increased, the angular velocity will proportionally decrease and the velocity over the ground remains the same.   At the end of the ramp the speed will only depend on the change in height h.  This is the theoretical answer: Radius does not mater.  

With a real wheel spinning along the road, diameter will mater.  A larger wheel will roll over small bumps more easily (see above).  The structural features of making the wheel do come into play as the size of the wheel changes.  There is a practical limit of wheel sizes that will work with the human dimensions.

With Question Two, I have made rough calculations for the rotating moment of inertia for various

racing bike wheel set
mountain bike wheel set
36" wheel
24" wheel
angular momentum at 4 mph in kg * m2/sec
KE at 4 mph in joules
angular momentum at 15 mph in kg * m2/sec
KE at 15 mph in joules
wheels.  I have estimated the weight of the Coker type wheel by extrapolating on the dimensions of a light mountain bike wheel.    The table shows some comparisons of stabilizing momentum and energy requirements to move the wheels to 4 mph and 15 mph.  This table indicates numerically what one would intuitively guess.  At slow speed, a small unicycle wheel does not give much stability from its rotational momentum.  The 36" unicycle wheel does potentially provide some stabilizing balance.  The kinetic energy needed to rotate the 36" wheel from rest to 15 mph falls between the values for a racing bike and a mountain bike.  (Since writing this, I have had a short note from Lars Clausen  suggesting that I have underestimated the weight of the 36" wheel, but his note also reinforces my suggestion above, that extra weight in the wheel is helpful, giving much more stability.)

How to make the Unicycle Less Fun

The stabilizing effect of two wheels on a bike frame will have an additional benefit due to the Hypothetical racing unicycle length of the wheel base.  This is like adding stability to a table with a wider spread of legs.  The frame of the bike can use the extended leverage of the two spinning wheels to add another resistance to some of the adverse effects of torque at the pedals.  If one wants to make a more stable unicycle that more effectively transmits the human engine into forward motion, I propose a Hypothetical Unicycle as shown at right.  Widely spaced spinning wheels could add balance and torsional stiffness.  Small wheels closely spaced and spun at high rotational velocity could also ad stability.   Counter balances could be built into the drive wheel to minimize vibration from the unbalanced weight of cranks and pedals and perhaps even the moving legs.  (This would not work if the wheel has a gear ratio drive system.)

There is probably some structure that could improve the balance and usable energy of the unicycle, but that brings me back to my larger speculation hinted at above.  Riding the unicycle is partly fun because it doesn't stay in balance.  

A large part of the fun, and value, of skiing or biking or running or unicycling is the way these activities  demand an absolute connection to the dynamic pull of the world around us, being open to the subtle powers and the fine adjustment of the universe, and having a wide open view of space and environment.  The process of interacting with that dynamic balance within the environment gives lots of opportunity to appreciate the quality of the road and the need for assistance along the way.  Every change of light, wetting rain, and adjustment in the slope of the world tips ones attention into the here and now.  

On a unicycle, a passing glance or an unexpected "Hello" can alter one's fine tuned leverage with the world.  The balance is so precise, even changing one's mind can cause a fall.  

It seems to me that having a keen appreciation for the demand of balance and the necessity of putting effort into maintaining balance is a fine metaphor for the world around us.  Machines such as bicycles and unicycles have a very aesthetic place within that metaphor.

For further investigation:

Koyaanisqatsi: A Hopi word meaning life out of balance.  There is a fascinating artistic movie with this name that plays with the rhythm and pattern of modern life, check it out.
One Wheel-Many Spokes, the book noted above by Lars Clausen.
Physics for Scientists and Engineers, by Raymond A. Serway.  Second Edition.  The college physics text I used to find equations and methods.
Bike Energy Calculation

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